<%BANNER%>

Modeling and Control of MEMS Micromirror Arrays with Nonlinearities and Parametric Uncertainties

Permanent Link: http://ufdc.ufl.edu/UFE0021233/00001

Material Information

Title: Modeling and Control of MEMS Micromirror Arrays with Nonlinearities and Parametric Uncertainties
Physical Description: 1 online resource (218 p.)
Language: english
Creator: Bronson, Jessica R
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: adaptive, control, feedback, mems, micromirrors, nonlinear, optics, parametric, uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Micromirror arrays have resulted in some of the most successful and versatile microelectromechanical system (MEMS) devices for applications including optical switches, scanning and imaging, and adaptive optics. Many of these devices consist of large arrays of micromirrors, and it is desirable to ensure accurate positioning capabilities for each mirror in the array despite the presence of nonlinearities or parametric uncertainties from the fabrication process. This research develops analytical models in the electrostatic and mechanical domains to study the effect of fabrication tolerances and uncertainties, electrostatic pull-in, and hysteresis on the performance of micromirror arrays, and presents solutions to improve device performance. To achieve these goals, extensive modeling of the electrostatic micromirror arrays is presented. As with many MEMS devices that operate in multiple physical domains, the modeling considers both electrical and mechanical characteristics. The electrical model consists of determining the electrostatic torque produced when an actuation voltage is applied. The mechanical model considers the opposing torque provided by the supporting torsion spring. These models are also used to evaluate the sensitivity of the micromirrors to parametric uncertainties from the fabrication process by considering the effect of each fabrication tolerance individually and also their combined effects using Monte Carlo simulations. Additional characterization of the system dynamics is presented through modal analysis in which the results for the full 6 degree-of-freedom (DOF) device are compared to the 1 DOF model assumptions. The devices are characterized by measuring the micromirror rotation as a function of the actuation voltage using an optical profiler to determine static performance, as well as measuring the electrostatic pull-in and hysteresis behavior. The measurements, taken for multiple mirrors across three different arrays, validate the results from analytical models, and demonstrate the need to compensate for differences in performance. Results from the modeling and characterization are used to develop passive and active control techniques to ensure accurate position tracking across an array of devices in the presence of model uncertainties. A passive design method is presented called a progressive-linkage that is intended to eliminate the occurrence of electrostatic pull-in and hysteresis. Also, classical and optimal feedback control techniques are utilized to further delineate the impact of the parametric uncertainties on the system performance. As these mirrors are arranged in an array, the performance of individual mirrors is examined, and then this control is extended to the problem of controlling an array. This array control problem is explored by considering different types of feedback error metrics and the sensors that may be used to provide the feedback signal for this system. The impact of the work presented in this dissertation is an increased understanding of the complexities of designing and operating arrays of electrostatic micromirrors for high-precision applications. The modeling methods developed may be extended in future work to include design optimization to decrease the effects of parametric uncertainty on the micromirror performance, as well as developing systems that can easily incorporate feedback mechanisms for implementation of the closed-loop control algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jessica R Bronson.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Wiens, Gloria J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021233:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021233/00001

Material Information

Title: Modeling and Control of MEMS Micromirror Arrays with Nonlinearities and Parametric Uncertainties
Physical Description: 1 online resource (218 p.)
Language: english
Creator: Bronson, Jessica R
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: adaptive, control, feedback, mems, micromirrors, nonlinear, optics, parametric, uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Micromirror arrays have resulted in some of the most successful and versatile microelectromechanical system (MEMS) devices for applications including optical switches, scanning and imaging, and adaptive optics. Many of these devices consist of large arrays of micromirrors, and it is desirable to ensure accurate positioning capabilities for each mirror in the array despite the presence of nonlinearities or parametric uncertainties from the fabrication process. This research develops analytical models in the electrostatic and mechanical domains to study the effect of fabrication tolerances and uncertainties, electrostatic pull-in, and hysteresis on the performance of micromirror arrays, and presents solutions to improve device performance. To achieve these goals, extensive modeling of the electrostatic micromirror arrays is presented. As with many MEMS devices that operate in multiple physical domains, the modeling considers both electrical and mechanical characteristics. The electrical model consists of determining the electrostatic torque produced when an actuation voltage is applied. The mechanical model considers the opposing torque provided by the supporting torsion spring. These models are also used to evaluate the sensitivity of the micromirrors to parametric uncertainties from the fabrication process by considering the effect of each fabrication tolerance individually and also their combined effects using Monte Carlo simulations. Additional characterization of the system dynamics is presented through modal analysis in which the results for the full 6 degree-of-freedom (DOF) device are compared to the 1 DOF model assumptions. The devices are characterized by measuring the micromirror rotation as a function of the actuation voltage using an optical profiler to determine static performance, as well as measuring the electrostatic pull-in and hysteresis behavior. The measurements, taken for multiple mirrors across three different arrays, validate the results from analytical models, and demonstrate the need to compensate for differences in performance. Results from the modeling and characterization are used to develop passive and active control techniques to ensure accurate position tracking across an array of devices in the presence of model uncertainties. A passive design method is presented called a progressive-linkage that is intended to eliminate the occurrence of electrostatic pull-in and hysteresis. Also, classical and optimal feedback control techniques are utilized to further delineate the impact of the parametric uncertainties on the system performance. As these mirrors are arranged in an array, the performance of individual mirrors is examined, and then this control is extended to the problem of controlling an array. This array control problem is explored by considering different types of feedback error metrics and the sensors that may be used to provide the feedback signal for this system. The impact of the work presented in this dissertation is an increased understanding of the complexities of designing and operating arrays of electrostatic micromirrors for high-precision applications. The modeling methods developed may be extended in future work to include design optimization to decrease the effects of parametric uncertainty on the micromirror performance, as well as developing systems that can easily incorporate feedback mechanisms for implementation of the closed-loop control algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jessica R Bronson.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Wiens, Gloria J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021233:00001


This item has the following downloads:


Full Text





MODELING AND CONTROL OF MEMS MICROMIRROR ARRAYS WITH
NONLINEARITIES AND PARAMETRIC UNCERTAINTIES























By

JESSICA RAE BRONSON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007





































O 2007 Jessica Rae Bronson


































To my parents









ACKNOWLEDGMENTS

I thank first of all, my supervisory committee chair, Gloria Wiens, for the opportunities to

attend the University of Florida and to conduct this work. She is also responsible for providing

me with the exceptional experience of working closely with Sandia National Laboratories

through the summer internship program. I also thank James Allen for his guidance as my mentor

at Sandia National Laboratories and for shaping the scope of this work. I thank all my

committee members, Louis Cattafesta, Norman Fitz-Coy, and Toshikazu Nishida for their time

and consideration.

Additionally I thank my family, especially my parents, for their loving support in all of my

endeavors. I also thank my many friends and classmates in the Space, Automation, and

Manufacturing Mechanisms (SAMM) Laboratory and in the Department of Mechanical and

Aerospace Engineering for their friendship and camaraderie. In particular, I thank Adam

Watkins for the many roles he plays as friend, colleague, mentor and partner.












TABLE OF CONTENTS
Page

ACKNOWLEDGMENT S .............. ...............4.....


LI ST OF T ABLE S ................. ...............8................


LI ST OF FIGURE S .............. ...............9.....


AB S TRAC T ............._. .......... ..............._ 17...


CHAPTER


1 INTRODUCTION ................. ...............19.......... ......


1.1 M otivation ................. ................. 19..............
1.2 Research Objectives ................. ...............20................

2 LITERATURE REVIEW ................. ...............23................


2.1 Microelectromechanical Systems .............. ...............23....
2.2 Micromirrors and Applications ................. ...............24................
2.3 Electrostatic Actuation and Instability ................. ....__. ...._.._ ...........2
2.3.1 Modeling, Pull-in and Hysteresis............... ...............2
2.3.2 Design Techniques to Eliminate Pull-in .............. .............. ...........3
2.3.3 Capacitive and Charge Control Techniques to Eliminate Pull-in ........._.._........32
2.3.4 Closed-loop Voltage Control to Eliminate Pull-in ........._.._.... .....................33
2.4 Feedback Control Techniques Applied to MEMS ................ .......... ................3 4
2.4.1 Linear Control .............. ...... ...............35.
2.4.2 Adaptive and Robust Control ................. ...............37........... ...
2.4.3 Nonlinear Control .............. ...............39....
2.5 Sensing Methods for Feedback ................. ...............41........... ...
2.6 Summary Remarks ................. ...............43........... ....

3 MICROMIRROR MODELING AND STATIC PERFORMANCE ................. ................. 45


3.1 Description of the SUMMiT V Microfabrication Process ............__ .........__ ......45
3.2 Micromirror Actuator Description ....._ .....___ .........__ ............4
3.3 Electrostatic Actuation and Instability ........................_. ......... ...........5
3.3.1 Parallel Plate Electrostatics ................. ...............50................
3.3.2 Parallel Plate Torsion Actuator ................. ...............56........... ...
3.4 Model for Vertical Comb Drive Actuator ................. ...............60........... ..
3.4.1 Mechanical Model ................. ...............61........... ....
3.4.2 Electrostatic M odel .............. ...............63....
3.4.3 Electromechanical Model .............. ...............66....
3.4.4 Linear Approximation............... .............7
3.4.5 Bifurcation Analysis .............. ...............73....
3.5 Chapter Summary............... ...............78











4 UNCERTAINTY ANALYSIS AND EXPERIMENTAL CHARACTERIZATION ............81


4.1 Parametric Uncertainty and Sensitivity Analysis............... ...............81
4.1.1 Effects of Individual Parameter Variations ................. ......... ................84
4. 1.2 M onte Carlo Simulations .............. ...............91....
4.2 Experimental Characterization............... ...........10
4.2.1 Equipment Description .............. ...............100....
4.2.2 Static Results for Single Micromirrors .............. ...............103....
4.2.3 Static Results for Micromirror Arrays .............. ...............105....
4.3 Chapter Summary ................. ...............111................

5 DYNAMIC MODEL AND HYSTERESIS STUDY ............_...... .__ ............._..113


5.1 Dynamic Model and Resonant Frequency Determination............. ..__.........__ ....113
5.1.1 M odal Analysis ............ _...... ._ ...............114...
5.1.2 Dynamic Characterization ............ .....___ ...............119.
5.2 Open-Loop Step Response ............... .... ...___ ... ...............122.
5.2.1 Effects of Parametric Uncertainty on Step Response ............__................123
5.2.2 Effects of Pull-in and Hysteresis on Open-Loop Response.............._._. .........123
5.2.3 Continuous Characterization of Micromirror Arrays ............ ................126
5.3 Hysteresis Case Study: Progressive-Linkage........... ...............12
5.3.1 Progressive-Linkage Design ........._._... ... .....__. .....__............12
5.3.2 Open-Loop Response Using a Progressive-Linkage ................... ...............135
5.3.3 Parametric Sensitivity of the Progressive-Linkage............_._. .........._._. ...13 5
5.3.4 Progressive-Linkage Prototype ........._._._ ...._. ...............139..
5.5 Chapter Summary............... ...............144

6 CONTROL DESIGN AND SIMULATION ....__ ......_____ .......___ ............4


6.1 PID Control ............ ... ... ...............147..
6.1.1 PID Control Theory .............. ...............147....
6. 1.2 PID Results .............. ... .......... ..........14
6.1.3 PID Controller Re sponse to Hy steresi s ...._.__._ ..... ... .__. ......_._......14
6.2 LQR Control ................. ...............154..
6.2.1 LQR Control Theory ........._.___..... .___ ...............154...
6.2.2 State Estimation .............. ...............159....
6.2.3 LQR Results .........._.... .... ...._.__......_. ............16
6.2.4 LQR Controller Response to Hysteresis .............. ...............163....
6.3 Modeling the Micromirror Array ................. ......... ...............164 ....
6.3.1 Modeling the Array of Mirrors ................. ...............165..............
6.3.2 Sensor Types ................ ...............167................
6.4 Modeling the Sensor Response ................ ...............170........... ...
6.4.1 PSD Response ................. ...............174................
6.4.2 CCD Response .................. .......... ...............179......
6.4.3 Summary of Sensor Analysis ................. ...............181........... ...
6.5 Chapter Summary............... ...............18












7 CONCLUSIONS AND FUTURE WORK ................. ...............183........... ...


APPENDIX


A MODEL GEOMETRY ................. ...............188................


B MONTE CARLO SIMULATION INPUTS ....__ ......_____ .......___ ............8


C LASER DOPPLER VIBROMETER RESULTS ................. ...............200........... ...


LIST OF REFERENCES ................. ...............208................


BIOGRAPHICAL SKETCH .............. ...............218....










LIST OF TABLES


Table page

2-1 Summary of feedback control papers discussed in the literature review. ................... .......36

3-1 Mean and standard deviation of fabrication variations for layer thickness in the
SUMMiT V surface micromachining process. ............. ...............47.....

3-2 Mean and standard deviation of fabrication variations of line widths in SUMMiT V......47

3-3 Values output from finite element analysis of mechanical spring stiffness. ...................63

3-4 Comparison of polynomial fit for approximation of capacitance function. .................. .....66

3-5 Comparison of polynomial fit for approximation of capacitance function. .................. .....66

3-6 List of parameters used for this analysis ....___ ................ ................ ...........78

4-1 Mean and standard deviation of fabrication variations for layer thickness in the
SUMMiT V surface micromachining process. ............. ...............82.....

4-2 Mean and standard deviation of fabrication variations of line widths in SUMMiT V......82

4-3 Spring stiffness values for changing dimensional and material parameters. .....................85

4-4 Results from the Monte Carlo simulations for the capacitance values in terms of
mean, standard deviation, and the percent change from nominal............... .................9

4-5 Mean and standard deviation for pull-in angle and voltage from sets of mirrors on all
three arrays tested. ............. ...............108....

5-1 Modal analysis results for first 10 modes and their natural frequencies, and the
participation factors and ratios for each direction. ......___ .... ... .__ .............._..118

5-2 The first three natural frequencies determined from the LDV experiment. ....................122

5-3 Link length dimensions used for progressive-linkage design............._. .........._._....134

5-4 Joint dimensions used for progressive-linkage design............... ...............134

5-5 Uncertainties in the joint dimensions for a proposed progressive-linkage design. ..........138











LIST OF FIGURES


Figure page

2-1 The SEM images of MEMS devices created using SUMMiT V microfabrication
process............... ...............25

2-2 Images of micromirror arrays developed in industry ......___ ..... ... ._ ........_......26

2-3 Adaptive optics (AO) mirror used for wavefront correction. ............. ......................2

2-4 Use of an AO MEMS programmable diffraction grating for spectroscopy.......................27

3-1 Drawing of the SUMMiT V structural and sacrificial layers .........._.._.. ............. .......46

3-2 Area with nominal dimensions L and w with the dashed line indicating the actual
area due to error in the line width. ............. ...............47.....

3-3 Images of the micromirror array ........... _...... ._ ...............48.

3-4 Illustration of mirrors operating as an optical diffraction grating ................. ................ .48

3-5 Micrograph of an array of mirrors and schematic of mirror with hidden vertical comb
drive. ............. ...............49.....

3-6 Solid model of micromirror showing polysilicon layer names from SUMMiT V. ...........49

3-7 Schematic of a parallel plate electrostatic actuator modeled as a mass-spring-damper
system ............. ...............51.....

3-8 Static equilibrium relationship for the parallel plate electrostatic actuator. ......................54

3-9 Electrostatic force for different voltages and mechanical force showing pull-in for
the electrostatic parallel plate actuator. ...._ ........__... ....._.._...........5

3-10 Pull-in function for the parallel-plate electrostatic actuator. ............. .....................5

3-11 Static equilibrium relationships for the parallel plate actuator using different spring
constants ................. ...............56.._._._.......

3-12 Schematic of a torsion electrostatic actuator. ............. ...............57.....

3-13 Static equilibrium relationships for the torsion actuator ................. .....__. ..............59

3-14 Pull-in function for the torsion actuator. ................._.._.._ .......... ..........6

3-15 Drawing of the mechanical spring that supports the micromirrors and provides the
restoring force. ............. ...............61.....










3-16 Image of the mechanical spring that supports the micromirror indicating boundary
conditions and location for applying displacement loads for finite element analysis. ......62

3-17 Image from ANSYS of the deformed spring and the outline of the undeformed shape
after displacements are applied.. ............. ...............63.....

3-18 Solid model geometry of the unit cell used in the electrostatic FEA simulation............_...65

3-19 Capacitance calculation as a function of rotation angle, 6, calculated using 3D FEA
and varying orders of polynomial curve fit approximations ................. ............. .......66

3-20 Plot of the Pull-in function PI(6) for the micromirror with the vertical comb drive
actuator showing that pull-in occurs at 16.5 degrees. .................... ...............6

3-21 Electrostatic and Mechanical torque as a function of rotation angle, theta, and
voltage for different voltage values. ............. ...............68.....

3 -22 Torque as a function of rotation angle, theta, and voltage for different values of
mechanical spring constant. .............. ...............69....

3-23 Plot of static equilibrium behavior, showing pull-in and hysteresis, predicted from
the model............... ...............69.

3 -24 Static equilibrium relationships for the nonlinear plant model, and the linear plant
approximation. ............. ...............70.....

3-25 Static equilibrium relationships for the nonlinear plant model, and the small signal
model linearized about an operating point (60, Vo).................. .............................72

3-26 Illustration of piecewise linearization about multiple operating points.............._..._. .........73

3-27 Plot showing the roots of the function F(xl) occur where the function crosses zero.........76

3-28 Bifurcation diagram for a MEMS torsion mirror with electrostatic vertical comb
drive actuator. ............. ...............79.....

3-29 Bifurcation diagram showing the effects of different spring constants. ............................79

4-1 Fabrication tolerances can changes the thicknesses of the layers, resulting in changes
in the final geometry dimensions. .............. ...............82....

4-2 Fabrication tolerances can change the dimensions of a fabricated geometry, affecting
the final shape, volume, and mass. ............. ...............83.....

4-3 Nominal dimensions used to calculate the volume of the moving mass. ................... .......83

4-4 Capacitance functions for the electrostatic model with parametric changes in the
layer thickness of the structural polysilicon. ....._ .....___ ........___ ...........8










4-5 Capacitance functions for the electrostatic model with parametric changes in the
layer thickness of the Dimple3 backfill and the sacrificial oxide ................. ................ .88

4-6 Capacitance functions for the electrostatic model with parametric changes in the
linewidth error of the structural polysilicon layers ................. .............................89

4-7 Static displacement relationships for the micromirror model with parametric changes
in the layer thickness of the structural polysilicon ................. ............... ......... ...89

4-8 Static displacement relationships for the micromirror model with parametric changes
in the layer thickness of the Dimple3 backfill and the sacrificial oxide. ................... ........90

4-9 Static displacement relationships for the micromirror model with parametric changes
in the linewidth error of the structural polysilicon layers ................. ........... ...........90

4-10 Sensitivity of voltage with respect to changes in line width for each value of 0..............92

4-11 Sensitivity of voltage with respect to changes in layer thickness for each value of 0.......92

4-12 Gaussian distribution with a mean of 0 and standard deviation of 1................. ...............94

4-13 Histogram for mechanical stiffness when accounting for variations in thickness of
MMPoly1 and Young's modulus............... ...............95

4-14 Histogram for mechanical stiffness taking into account variations in thickness of
MMPolyli, Young' s modulus, and linewidth of MMPolyl1................. ............ .........95

4-15 Results from the capacitance simulation for 250 random variable sets that show the
effects of parametric uncertainty on the electrostatic model. ................ .....................96

4-16 Static displacement results of 250 Monte Carlo simulations with random Gaussian
distributed dimensional variations. ............. ...............98.....

4-17 Histogram of values from the Monte Carlo simulations for the layer thickness of
Sacox3 ................ ...............99.................

4-18 Static displacement curves from the Monte Carlo simulations that indicate the effect
of large variations in the Sacox3 layer thickness ................. ...............99..............

4-19 Diagram of an optical profiler measurement system ................ ......... ................10 1

4-20 Six mirrors from the micromirror array measured with the optical profiler system........102

4-21 Data records from the SureVision display that show the cross-section profile of the
tilt angle measurements............... .............10

4-22 Micrograph image of a single micromirror ................. ...............104........... ..











4-23 Experimental static results taken from individual micromirrors that are not in an
array. ............. ...............104....

4-24 Approximate locations of data collection on all three arrays. ............. ..................... 106

4-25 Experimental results from array 1, area A. .............. ...............106....

4-26 Experimental results from array 2, areas D and E. ............. ...............107....

4-27 Experimental results from array 3, areas A and D ................. ................ ......... .10

4-28 Nominal model with experimental data ................. ...............109..............

4-29 Model-predicted results from 100 simulations with parameters determined by
random Gaussian variations, shown with experimental data ................. ............... ....109

5-1 Open-loop nonlinear plant response to a step input of 7 degrees for different damping
ratios ................. ...............114................

5-2 Solid model created for modal analysis ................. ...............118........... ..

5-3 Time series data of the micromirror response to an acoustic impulse taken with a
laser doppler vibrometer. ............. ...............120....

5-4 Results from the LDV experiment showing resonant peaks ................. .....................121

5-5 Open-loop response to a step input of 7 degrees for the nonlinear plant dynamics and
variations in spring stiffness, km. ............. ...............124....

5-6 Open-loop nonlinear plant response to a step input of 7 degrees for 50 random
parameter variations ................. ...............124................

5-7 Open-loop responses to a sinusoidal input showing hysteresis .............. ....................12

5-8 Open-loop responses to a step command showing overshoot that result in pull-in.........126

5-9 Results from dynamic study showing pull-in and hysteresis ................. ............... .....127

5-10 Results showing the hysteretic behavior of the micromirrors .........___..... .............. ..128

5-11 Diagram of four-bar mechanism for progressive linkage analysis. ............. ..................130

5-12 Cantilever beam with cross-section w x t, and length L. ............. ....................13

5-13 Free body diagrams for each member of the linkage. ................. ...._._ ................132

5-14 Progressive-linkage behavior for different values of ro in Cpm........ ............... 132










5-15 Progressive linkage output for ro equal to 9 Cpm along with the electrostatic torque
curves and the linear restoring torque. ................ ...._.._ ...............133...

5-16 Static e-V relationship for micromirror with a progressive-linkage .............. ..............134

5-17 Bifurcation diagram for micromirror using a progressive-linkage to avoid pull-in
b ehavi or ........._._. ._......_.. ...............136....

5-18 Bifurcation diagram for the micromirrors using a progressive-linkage to avoid pull-in
behavior for different values of mechanical stiffness. ................. ...._._ ............... 136

5-19 Open-loop responses to a sinusoidal input for the device using a progressive-linkage. ..137

5-20 Open-loop response to a step input for device using a progressive-linkage. ................... 137

5-21 Results of parametric analysis for individual errors in j oint fabrication of the
progressive-linkage. .............. ...............140....

5-22 Fifty Monte Carlo simulation results for varying the joint fabrication parameters for
the progressive-linkage design............... ...............140

5-23 Schematic drawing of the prototype progressive linkage spring ................. ..................141

5-24 Micrograph of the prototype micromirror with a progressive linkage spring. ................141

5-25 Experimental data collected for the prototype of the micromirror with the
progressive-linkage ................. ...............143................

5-26 Results from FEA of the prototype progressive-linkage design for linear and
nonlinear deflection analysis shows the prototype progressive-linkage fails to
produce the desired stiffness profile. ............. ...............143....

6-1 Basic block diagram with unity feedback ................. ...............146........... ..

6-2 Block diagram with PID controller ................. ...............148..............

6-3 Step responses for PID controller ................. ......... ...............150 ...

6-4 Closed-loop PID response to different step inputs when the spring constant is varied
by +10% .............. ...............150....

6-5 Closed-loop PID response to a step input of 7 degrees for 50 random sets of
parameteric variations ..........._.... ...............151.._.__.......

6-6 Closed-loop PID response to a commanded position in the unstable region. ..................1 53

6-7 Closed-loop step responses for PID controller for a system using a progressive-
linkage. ........... ..... .. ...............153..











6-8 General block diagram for LQR controller problem .............. ...............154....

6-9 Block diagram of LQR control with an internal model for tracking a step command. ...158

6-10 Block diagram of LQR controller using a state-estimator for a plant without an
integrator. .............. ...............159....

6-11 Step responses for LQR controller. ........._._ ....__. ...............161

6-12 Closed-loop LQR response to a step input of 7 degrees for 50 random parameter
variations. ........._ ...... .. ...............162...

6-13 Closed-loop step responses for LQR controller for a system using a progressive-
linkage. ........._ ...... .. ...............164...

6-14 Schematic of modeling an array of mirrors as a SIMO system ................ ................. .165

6-15 Schematic drawing of an array of 5 mirrors. ............. .....................166

6-16 Illustration of the measured center of gravity (CG) on a 1-D PSD when there are
errors in the spacing and linearity of the micromirrors ................. ........................168

6-17 Illustration of the measured center of gravity (CG) on a 2-D PSD when there are
errors in the spacing and linearity of the micromirrors ................. ........................168

6-18 Illustration of the measured errors of the reflected light from two micromirror arrays
onto a CCD. ............. ...............169....

6-19 Schematic of the beam steering experiment with only one micromirror. ................... .....171

6-20 Geometry used to determine the angle of incidence and reflection. .............. .... ........._..172

6-21 Calculating the reflected ray of light. .......___......... ___ .........___ ....___......173

6-22 The intersection of the line from BO to R and the plane C occurs at point P. ................. 174

6-23 Schematic of 5 micromirrors in an array reflecting light onto a PSD sensor ..................1 75

6-24 Calibration of the PSD for ideal case of five micromirrors. ................ .....................176

6-25 Open-loop results to a step response for an array of 5 micromirrors with model
uncertainty ................. ...............176................

6-26 Open-loop response for system with one broken mirror and 4 ideal mirrors, measured
by a PSD .............. ...............177....

6-27 Incorporating feedback control into array model ................. ..............................17

6-28 Controlled PID step response using PSD sensor ................. ...............178.............











6-29 Controlled LQR step response using PSD sensor ................. ............... ......... ...17

6-30 Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a
PID controller and a PSD sensor. ............. ...............179....

6-31 Schematic of 5 micromirrors in an array reflecting light onto a CCD sensor where
each separate location of the light can be measured ................. ...........................180

6-32. Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a PID
controller and a CCD sensor ................. ...............180........... ...

A-1 Geometry dimensions in Cpm for creating electrostatic model. ............. ....................188

B-1 Histogram of values for the thickness of layer MMPoly0......._____ ...... .....__.........189

B-2 Histogram of values for the thickness of layer MMPolyl1......____ ...... ....__..........190

B-3 Histogram of values for the thickness of layer MMPoly2. ....._____ ...... .....__.........190

B-4 Histogram of values for the thickness of layer MMPoly3 ........._._. .... ...._._...........191

B-5 Histogram of values for the thickness of layer MMPoly4 ................_ ............ ........191

B-6 Histogram of values for the thickness of Dimple3 backfill. ................ .....................192

B-7 Histogram of values for the thickness of layer Sacoxl ................_ ......... ..............192

B-8 Histogram of values for the thickness of layer Sacox2. ................ ....___ .............193

B-9 Histogram of values for the thickness of layer Sacox3 ................. .........................193

B-10 Histogram of values for the thickness of layer Sacox4 ................. ................. ...._194

B-11 Histogram of values for the linewidth variation of layer MMPoly2. ............._ .............194

B-12 Histogram of values for the linewidth variation of layer MMPoly3. .........._... .............195

B-13 Histogram of values for the linewidth variation of layer MMPoly4. ............. .... ...........195

B-14 Histogram of the mass values calculated from the parametric variation data. ..............196

B-15 Histogram of values from the Monte Carlo simulations for the linewidth error of
M M Poly2 ................ ...............196................

B-16 Static displacement curves from the Monte Carlo simulations that indicate the effect
of large variations in the linewidth error of MMPoly2. ....._____ ... ......_ ..............197

B-17 Histogram of values from the Monte Carlo simulations for the layer thickness of
M M Poly l ................ ...............197......... ......











B-18 Static displacement curves from the Monte Carlo simulations that indicate the effect
of large variations in the thickness of MMPolyl1................. ...............198........... .

B-19 Histogram of values from the Monte Carlo simulations for the layer thickness of
Sacox4 ................. ...............198................

B-20 Static displacement curves from the Monte Carlo simulations that indicate the effect
of large variations in the thickness of Sacox4. ............. ...............199....

C-1 Magnitude results of FFT for device 1, trial 1. ............. ...............200....

C-2 Magnitude results of FFT for device 1, trial 2. ................ ...............200...........

C-3 Magnitude results of FFT for device 1, trial 3 ................ ...............201...........

C-4 Magnitude results of FFT for device 1, trial 4. ................ ...............201...........

C-5 Magnitude results of FFT for device 1, trial 5 ................ ...............202...........

C-6 Magnitude results of FFT for device 2, trial 1 ................ ...............202............

C-7 Magnitude results of FFT for device 2, trial 2. ................ ...............203............

C-8 Magnitude results of FFT for device 2, trial 3 ................ ...............203............

C-9 Magnitude results of FFT for device 2, trial 4. ................ ...............204............

C-10 Magnitude results of FFT for device 2, trial 5 ................ ...............204............

C-11 Magnitude results of FFT for device 3, trial 1 ................ ...............205............

C-12 Magnitude results of FFT for device 3, trial 2. ................ ...............205............

C-13 Magnitude results of FFT for device 3, trial 3 ................ ...............206............

C-14 Magnitude results of FFT for device 3, trial 4. ................ ...............206............

C-15 Magnitude results of FFT for device 3, trial 5 ................ ...............207............









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MODELING AND CONTROL OF MEMS MICROMIRROR ARRAYS WITH
NONLINEARITIES AND PARAMETRIC UNCERTAINTIES

By

Jessica Rae Bronson

December 2007

Chair: Gloria J. Wiens
Major: Mechanical Engineering

Micromirror arrays have resulted in some of the most successful and versatile

microelectromechanical system (MEMS) devices for applications including optical switches,

scanning and imaging, and adaptive optics. Many of these devices consist of large arrays of

micromirrors, and it is desirable to ensure accurate positioning capabilities for each mirror in the

array despite the presence of nonlinearities or parametric uncertainties from the fabrication

process. This research develops analytical models in the electrostatic and mechanical domains to

study the effect of fabrication tolerances and uncertainties, electrostatic pull-in, and hysteresis on

the performance of micromirror arrays, and presents solutions to improve device performance.

To achieve these goals, extensive modeling of the electrostatic micromirror arrays is

presented. As with many MEMS devices that operate in multiple physical domains, the

modeling considers both electrical and mechanical characteristics. The electrical model consists

of determining the electrostatic torque produced when an actuation voltage is applied. The

mechanical model considers the opposing torque provided by the supporting torsion spring.

These models are also used to evaluate the sensitivity of the micromirrors to parametric

uncertainties from the fabrication process by considering the effect of each fabrication tolerance

individually and also their combined effects using Monte Carlo simulations. Additional










characterization of the system dynamics is presented through modal analysis in which the results

for the full 6 degree-of-freedom (DOF) device are compared to the 1 DOF model assumptions.

The devices are characterized by measuring the micromirror rotation as a function of the

actuation voltage using an optical profiler to determine static performance, as well as measuring

the electrostatic pull-in and hysteresis behavior. The measurements, taken for multiple mirrors

across three different arrays, validate the results from analytical models, and demonstrate the

need to compensate for differences in performance.

Results from the modeling and characterization are used to develop passive and active

control techniques to ensure accurate position tracking across an array of devices in the presence

of model uncertainties. A passive design method is presented called a progressive-linkage that is

intended to eliminate the occurrence of electrostatic pull-in and hysteresis. Also, classical and

optimal feedback control techniques are utilized to further delineate the impact of the parametric

uncertainties on the system performance. As these mirrors are arranged in an array, the

performance of individual mirrors is examined, and then this control is extended to the problem

of controlling an array. This array control problem is explored by considering different types of

feedback error metrics and the sensors that may be used to provide the feedback signal for this

sy stem.

The impact of the work presented in this dissertation is an increased understanding of the

complexities of designing and operating arrays of electrostatic micromirrors for high-precision

applications. The modeling methods developed may be extended in future work to include

design optimization to decrease the effects of parametric uncertainty on the micromirror

performance, as well as developing systems that can easily incorporate feedback mechanisms for

implementation of the closed-loop control algorithms.









CHAPTER 1
INTTRODUCTION

1.1 Motivation

Micromirror arrays have resulted in some of the most successful and versatile

microelectromechanical system (MEMS) devices for applications including optical switches for

telecommunications, scanning and imaging for proj section displays, diffraction gratings for

spectroscopy, and adaptive optics for wave front correction. Many of these devices consist of

large arrays of micromirrors. As such, it is desirable to ensure accurate positioning capabilities

for each mirror in the array despite the presence of outside disturbances or variations from the

fabrication process. The errors due to the fabrication process can be attributed in part to small

deviations in dimensional or material properties. It is the effects of these errors that can have

significant impact on the performance of the final product. As such, it is important to evaluate

the sensitivity of the micromirror design to determine the potential limitations on the device

performance .

The diminutive scale of MEMS devices makes electrostatic actuation a popular and

effective means of driving micromirrors. One limiting factor to most electrostatic actuators is

the electrostatic pull-in instability that occurs when the electrostatic force overcomes the

mechanical restoring force. When pull-in occurs, the device can no longer maintain an

equilibrium position and will move to its fully actuated position, limiting the full scanning range

available. Another phenomenon associated with this instability is that once the mirror has

pulled-in, the voltage required to maintain the fully actuated position is lower than the pull-in

voltage. The mirror will not return from this position until the actuating voltage has been

reduced below a certain threshold. In order to understand these phenomena, theoretical models

may be developed for the electrostatic and mechanical domains. These models can then be









utilized to evaluate the effects of fabrication errors and determine the performance limitations of

the micromirrors. These issues can be mitigated through the successful application of design

methods as well as through feedback control.

Currently, state-of-the-art micromirror arrays rely on open-loop actuation that may limit

the device to on/off digital operation or require extensive calibration for analog performance [1],

[2]. Many of today's emerging technologies, however, require true analog positioning

capabilities. Therefore, in order to guarantee precision and accuracy of the mirror position for

analog operation, closed-loop feedback control techniques are considered essential. Feedback

control has long been used in many macro-scale systems, yet limited work has been done to

apply these techniques to MEMS systems. An additional need arises in the use of very large

arrays of micromirror devices. While control of one mirror may be a straightforward task, it

becomes much more difficult to extend that control to a very large system. The micromirror

arrays in this research are constrained such that the micromirrors are not individually

controllable, creating a unique control application to a single-input/multiple-output system

(SIMO). This also gives rise to the question of obtaining an appropriate feedback signal for a

system of arrays. The types of sensing used to gather the feedback information as well as how

this information is used are critical issues.

1.2 Research Objectives

The obj ective of this research is to develop analytical models to study the effect of

fabrication tolerances and uncertainties, electrostatic pull-in and hysteresis on the performance of

micromirror arrays that are used in adaptive optics applications requiring precise and accurate

positioning. The modeling techniques allow for analysis of the system in both the electrostatic

and the mechanical domains using a combination of analytical models and finite element

analysis (FEA). These theoretical results are compared to experimental characterization data.









The models are also used to determine the potential effects of parametric uncertainties in the

fabrication process, and to estimate the sensitivity of the micromirror design to these

uncertainties. This information is valuable in determining the possible limits on performance

that can be achieved through only open-loop actuation methods. These models also characterize

the effects of electrostatic instability and the resulting hysteresis. The modeling is extended from

the initial quasi-static approximation to include the dynamic behavior of the system. After

thorough analysis and characterization of the system behavior, several solutions are presented to

improve the performance and positioning accuracy of the micromirror devices. These solutions,

including passive and active controls, are developed to ensure that the device performance will

be robust in the presence of system nonlinearities and parametric uncertainties. A passive design

solution, called a progressive-linkage, is presented that will eliminate the effects of electrostatic

pull-in and hysteresis, thus extending the stable range of motion for the micromirrors. The

theoretical design approach is presented along with discussion and analysis regarding the

sensitivity of the linkage to fabrication errors, as well as an initial prototype attempt. Active

control solutions, including classical and optimal control design, are presented as an

investigation of feedback control methodologies for use on micromirrors that can be used to

achieve high precision positioning. The sources of nonlinearities and parametric uncertainties

previously identified and quantified during the modeling and characterization of the devices now

forms an error basis for examining the robust performance of the control algorithms.

In Chapter 2, an overview of previous work from the literature on micromirror arrays, their

applications, and control of MEMS devices is presented for identifying the underlying issues

impeding further development and implementation. This led to the motivation of the work

presented in the remaining chapters of this dissertation including analytical modeling and the










study of effects of fabrication tolerances and uncertainties, dynamic performance and passive

control. Chapter 3 provides the static modeling for the micromirrors and Chapter 4 analyzes the

sensitivity of the devices with respect to fabrication tolerances, comparing the model results to

experimental characterization data. Chapter 5 discusses the dynamic system, including

characterizing the resonant modes and studying the effect of electrostatic hysteresis. The

progressive-linkage design is presented as a solution to the problems caused by electrostatic pull-

in. Upon identifying the model uncertainties and behavioral characteristics of these

micromirrors, an investigation of closed-loop control methods is conducted in Chapter 6 to

further delineate the impact of the parametric uncertainties on system performance. The

controllers are compared and evaluated in simulation to determine their effectiveness for position

control in the presence of model uncertainties for a single mirror and a spectrum of uncertainties

across the array. In order to evaluate the array performance, the method of sensing the position

of the micromirrors is critical, and several sensor types are considered. Finally, the conclusions

of this study are given in Chapter 7, along with directions for future work.










CHAPTER 2
LITERATURE REVIEW

In this chapter, a review of the literature concerning applications of micromirrors,

modeling of electrostatic devices, and the design of feedback controllers for MEMS devices and

related adaptive optics systems is presented. This review is intended to provide an overview of

the current state of research on electrostatic micromirrors and the control of MEMS devices so as

to identify underlying issues impeding further development and implementation. A brief

introduction to MEMS and microfabrication methods is presented, followed by a discussion of

applications for micromirror technology. Electrostatic actuation is used widely for MEMS

devices, and it is the actuation method used by the micromirror arrays presented in this

dissertation; therefore a discussion of electrostatic actuation and the pull-in instability is given,

including modeling methods and the different methods that are dedicated to addressing pull-in.

Control applications to MEMS is a relatively new area of research, therefore a thorough

discussion is included of control methods that have been applied to a variety of MEMS devices

with many different methods of actuation in addition to electrostatic. The chapter closes with

remarks summarizing the findings of this review and outlining the specific areas of research that

are currently lacking, and that will be addressed in the remainder of this document.

2.1 Microelectromechanical Systems

Microelectromechanical Systems (MEMS) refer to mechanical and electrical structures

used to create sensors and actuators with feature sizes ranging from 1 Cpm to 1 mm. MEMS have

found successful applications in many markets, most notably nozzles for inkj et printing,

accelerometers for automotive airbags, blood pressure sensors for health care, optical switches

and arrays for communications and projection displays. This remarkable technology is

continuing to expand and promises to bring revolutionary capabilities to nearly every industry.









MEMS are batch fabricated, typically making them inexpensive, using a microfabrication

process such as bulk or surface micromachining [3]. This technology is founded on fabrication

techniques first used for integrated circuits (IC) and utilizes the same lithographic patterning

techniques. Bulk micromachining techniques rely on selective etching to remove material from

the whole to form structures with wells and trenches [4]. Surface micromachining is considered

an additive technique that creates mechanisms by layering a structural layer, such as polysilicon,

with a sacrificial layer, such as silicon dioxide (oxide) [5]. Through a repeated series of

lithography, etching and deposition, freestanding structures are created.

As with any manufacturing or machining process, fabrication tolerances can give rise to

parametric uncertainties causing the dimensions of fabricated device to vary slightly from the

intended design. For microfabrication this is due to small over or under etching of layers as well

as variations in material properties, and misalignment between layers [7]-[12]. All these

variations can occur across the wafer as well as from batch to batch. Chapter 3 provides further

analysis on the influence of fabrication variations on device performance.

The process utilized to create the devices discussed in this research is Sandia's Ultra-

planar, Multi-level MEMS Technology (SUMMiT V), developed by Sandia National

Laboratories that utilizes five structural layers of polysilicon [6]. The specifics of the fabrication

process are discussed further in Chapter 3. Examples of structures that can be created using this

process seen in Figure 2-1 show scanning electron micrographs (SEMs) of a mechanical gear

hub and a cross-section of a pin-j oint that allows rotation. These are excellent examples of the

complex structures created from layering simple, 2-D geometry.

2.2 Micromirrors and Applications

Micromirrors are one of the most widely used and commercially viable applications for

MEMS technology. The small size of these devices makes them ideal for optical switching and





























Figure 2-1. The SEM images of MEMS devices created using SUMMiT V microfabrication
process. A) Micromachined gears. B) Micromachined gears. C) A cross-section view
of a pin-j oint that allows for gear rotation. (Courtesy of Sandia National Laboratories,
SUMMiT Technologies, www.mems.sandia.gov).

scanning operations at very high speeds. Both single mirrors and large arrays are used for

optical switches for communications [13]-[19], scanning and imaging for projection displays [2],

[20], diffraction gratings for optical spectroscopy [21]-[25], and beam steering for adaptive

optics [26]-[32] and free-space communication [33], [34].

An example of micromirrors that have been commercially successful is the Texas

Instruments' Digital Micromirror DeviceTM (DMD) that uses millions of torsional electrostatic

micromirrors to manipulate light. Applications for the DMD include projection displays,

televisions, laser printers, image processing, light modulation, and optical switching [2], [20].

The success of many of these applications relies on purely digital functioning that is not suitable

for more advanced applications that require analog operation, such as adaptive optics (AO).

Sandia National Laboratories developed electrostatic micromirror arrays to be used as

instrumentation for adaptive optics in space applications [35]. Images of Texas Instruments'

DMD and the Sandia micromirrors are shown in Figure 2-2.



























Figure 2-2. Images of micromirror arrays developed in industry. A) Texas Instruments' Digital
Micromirror Device (DMD) and B) Sandia National Laboratories' AO micromirror
array. (Courtesy of Texas Instruments, www.ti.com, and Sandia National
Laboratories, SUMMiT Technologies, www.mems. sandia.gov).

Adaptive optics (AO) refers to optical components such as mirrors or lenses that are able to

change shape or orientation in order to manipulate a light source. Adaptive arrays of large

mirrors (on the order of meters in diameter) have long been used in astronomy to correct for

atmospheric distortions in images from space [36] This same concept can be achieved with

MEMS micromirror arrays for use in wavefront corrections and spectroscopy. Figure 2-3 shows

a general schematic of how wavefront correction is achieved using adaptive optics. A distorted

wavefront is reflected onto an adaptive optics device which is deformed accordingly to eliminate

the distortions in the original wavefront. The newly corrected wavefront is split and sent to a

detector (e.g. camera) and to a sensor that measures the wavefront and sends this signal to a

control system that directs the motions of the deformable mirror. These kinds of systems

traditionally rely on expensive wavefront sensors to sense the wavefront and direct the mirror' s

actions. However there are many new applications that are utilizing MEMS micromirrors and

lenslet arrays to replace the traditional wavefront sensors. Horenstein et al. demonstrate

wavefront correction using the Texas Instruments' DMD [30]. Another example of










Detector


Corrected
Incoming i Wavefront
Lig ht



Distorted
Wrvavefront
F: Wavefront
SAO Mirror Sensor

Control System


Figure 2-3. Adaptive optics (AO) mirror used for wavefront correction.

micromirrors used for AO include Boston Micromachine's Deformable Mirrors (DM), which

have been used for image correction in telescopes, microscopes, and Optical Coherence

Tomography (OCT) [29], [31i], [32]. AO micromirrors are also being used for imaging of the

human retina [37], [38].

Another variation of AO uses arrays of micromirrors to create programmable diffraction

gratings for use in spectroscopy [21]-[25]. As shown in Figure 2-4, light sent through a sample,

such as a chemical, gas, or material, is diffracted into its spectrum by a fixed grating. This is


Light Source AO MEMS
~-c~I Detector









Fixed Grating

Figure 2-4. Use of an AO MEMS programmable diffraction grating for spectroscopy.









then sent to the MEMS diffraction grating that is set to filter light in specific regions of the

spectrum. The filtered light is sent back to the fixed grating and then collected by a detector.

The light measured in the detector can be used to determine the material composition of the

sample.

2.3 Electrostatic Actuation and Instability

The examples of micromirrors presented in Section 2.2 all use electrostatic actuation,

which is popular in MEMS as it is easy to implement using the silicon-based semiconductor

structural materials available in most MEMS fabrication processes. The theory of operation for

electrostatic actuation is presented in detail in Chapter 3, and is discussed here more generally to

give an understanding of the current modeling methods and the challenges with this type of

actuation, including nonlinear behavior and electrostatic instability.

2.3.1 Modeling, Pull-in and Hysteresis

The theory of electrostatic actuation relies on established relationships regarding the

energy generated in an electric field when a charge differential is applied to two bodies, such as

in a capacitor [3]. The energy in this electric field creates an attractive force between the two

plates, and this is the principle exploited for electrostatic actuation. The equations used to

describe the electrostatic forces are derived from the energy in the electric field between the

charged electrodes, and often assumptions are made in calculating the capacitance using

analytical expressions that neglect the fringe field effects. It is typical in MEMS devices, such as

parallel plate actuators or torsion micromirrors, for one set of the charged electrodes to be

stationary, and the other electrode to be supported by a flexible suspension or spring that allows

it to move. The spring suspension counteracts the attractive electrostatic force with an opposing

mechanical force that can constrain the degrees of freedom of the moving plate and ensure that

the two electrodes to not come into contact.










Many electrostatic actuators exhibit the well-documented phenomenon of electrostatic

pull-in. The electrostatic force is nonlinear as it is inversely proportional to the square of the

electrode gap. Pull-in, sometimes called snap-down, occurs when the electrostatic force

generated by the actuator exceeds the mechanical restoring force of the structure. The result is

that the device reaches an unstable position and subsequently is pulled down to the substrate at

its maximum displacement. The electrostatic instability has been studied extensively and the

pull-in characteristics can be modeled fairly accurately [3], [14], [39]-[52], [106]. Pull-in for a

parallel-plate actuator occurs at one-third of the separation gap, which greatly limits the actuator

stroke.

Another phenomenon associated with pull-in instability is that once the mirror has pulled-

in, the voltage required to maintain the pull-in position is lower than the pull-in voltage. The

mirror will not return from this position until the actuating voltage has been reduced below the

holding-voltage. The result of this holding effect is hysteresis. Electrostatic hysteresis behaves

differently from hysteresis that is common in piezoelectric or thermal actuators where continuous

motion is possible in both directions. Electrostatic systems experience a dead-band after pull-in

in which no actuation is even possible until the applied voltage drops below the holding

threshold. The effects of pull-in and hysteresis are a challenge in achieving stable, controllable

actuation over the maximum range of motion of an electrostatic micromirror.

The behavior of electrostatic actuators has been modeled throughout the literature using

analytical expressions for cases of simple electrode geometry, such as parallel plate actuation [3],

[24], [41]-[43], [49], [64], [106]. When the electrode geometry becomes more complex, such as

the case when the actuators use vertical comb drives, finite element analysis (FEA) can be used

to numerically calculate the properties of the electric field. Hah, et al. use a 2D Maxwell solver









and then integrate the results over the length of the mirror to predict the 3D electric field [14],

[46]. This method can be advantageous for computational efficiency, as a 2D FEA simulation

will likely take less time than a 3D model. There can be benefits to using a 3D FEA solution,

which is the modeling method that is employed in this dissertation. A full 3D electrostatic model

can allow easily for evaluation of the effects of complex electrode shapes, such as shapes that do

not have a constant cross-section along the length of the device. These non-constant cross-

sections could be designed on purpose to study the effects of changing electrode shape, or can be

the result of processing. Etching procedures in both bulk and surface micromachining can

inherently result in sloping sidewalls or uneven surfaces [3]-[5]. Therefore, 3D analysis may be

more computationally intensive, but it also allows for the study of more sophisticated

geometries. Regardless of the modeling method used, it is possible to describe the static behavior

of the actuators and the position and voltage at which pull-in and the release will occur.

Electrostatic instability is also an example of bifurcation behavior, and once an equation of

motion is determined for the device, the pull-in can be examined from stability theory [106].

Bifurcation analysis is demonstrated in Chapter 3 of this dissertation.

The modeling performed for electrostatic devices typically assumes that they are operated

below the resonant frequencies of the device. The pull-in phenomena is affected by resonance

and it has been shown that parallel plate actuators driven at their resonant frequency have a

greater range of motion compared to the one-third gap limitation for frequencies below

resonance [40]. Additional model assumptions that are commonly made are that the device

operates only in its intended degrees of freedom as prescribed by the operating conditions and

the mechanical suspension design. In cases with multiple degrees of freedom, such as 2DOF









mirrors, positioners, or gyroscopes that have coupling between the DOF, it is crucial to take this

into account during the model development [10], [1l].

Many of the first generation of micromirror devices, such as Texas Instruments' DMD, use

pull-in as an advantage that allows for open-loop, on/off binary actuation at reduced voltages [2],

[53]. While the actual pull-in voltage of the device may vary slightly from mirror to mirror due

to variations in dimension and material properties, reliable open-loop operation can still be

guaranteed by ensuring that the actuation voltage is sufficiently high enough to capture the pull-

in effects for all the mirrors despite these variations. The hysteresis phenomena can also be

beneficially exploited, since once a mirror is pulled-in it can be held there at a reduced voltage,

which decreases power consumption. While the electrostatic instability can be advantageous for

digital applications, it is an obstacle for the application of micromirrors with continuous, analog

actuation capabilities.

The issue of electrostatic pull-in has been thoroughly documented and there has been a

considerable amount of research conducted to find ways to avoid pull-in for electrostatic

micromirrors in order to move beyond binary positioning capabilities and achieve full, analog

positioning for applications such as scanning and adaptive optics. Attempted solutions to this

problem have included design techniques to alter the electrostatic or mechanical forces of the

device, capacitive and charge control techniques, and closed-loop feedback control. A review of

these methods is given in the following sections.

2.3.2 Design Techniques to Eliminate Pull-in

There are multiple design methods researchers have employed to address the problem of

electrostatic pull-in to achieve an extended range of travel for electrostatic actuators. Some have

employed geometrical design changes to achieve increased stability. These methods have

included tailoring the electrode geometry [54] or applying insulating layers of dielectric material










[55]. Changes in the electrode geometry are especially effective for torsional microactuators as

they do not have a constant electrostatic force generated over the surface of the actuator as it tilts.

Changes to device geometry are sometime limited by other design or fabrication constraints.

The use of nonlinear flexures has also been used to ensure that, as the electrostatic force

increases, the mechanical restoring force of the devices also increases to compensate. Burns and

Bright developed nonlinear flexures that utilize a series of linear flexural elements that are

designed to engage the device at predetermined deflections [56]. This effectively creates a

piecewise linear stiffness profile. A similar concept of creating nonlinear stiffness has been

explored by Bronson et al. in [57], [58] and will be discussed further in Section 5.3. The

leveraged-bending approach introduced by Hung and Senturia [24] uses the stress-stiffening of a

fixed-fixed beam to generate the nonlinear mechanical force needed to achieve controllable

positioning over the entire range of motion of a polychrometer programmable diffraction grating

[21]-[23]. The cost of using these techniques is a higher actuation voltage needed to achieve

large, stable deflections.

2.3.3 Capacitive and Charge Control Techniques to Eliminate Pull-in

The issue of controlling the electrostatic instability has been addressed by using capacitive

and charge control methods. Seeger and Crary [59] proposed a simple method that incorporates

a capacitor in series with the actuator to provide stabilizing negative feedback. This passively

controls the voltage across the actuator electrodes as the gap width changes. They showed

theoretically that this method can be used to stabilize across the entire gap. The tradeoff is that

higher voltages are required to stabilize the actuator using this method. This concept is extended

by Seeger and Boser using a switched-capacitor circuit to control charge across the actuator and

reduces the actuation voltage requirements [60]. Seeger and Crary neglected to take into account

nonlinear deformation of the elastic members of the actuator. Once these nonlinear deformation









terms are considered however, the method is found to only partially stabilize the system [61],

[62]. Other issues such as residual charge and parasitic capacitance addressed by Chan and

Dutton [61], [63] were shown to limit the actuator travel to less than full range. Chan and Dutton

also introduced a folded capacitor design that could be fabricated in the surface micromachining

MUMPS process as part of the device itself and showed that this series capacitor method can be

used to increase the stable range of electrostatic torsion actuators up to 60% of the initial gap

with the cost of using higher actuation voltages.

Other work has used similar charge control strategies that have resulted in reduced voltage

penalties and extended travel [64]-[66]. Current leakage has been shown to create drift of

steady-state positions and this can be overcome using discharge methods that resemble sigma-

delta operations, but the results can lead to 'ringing', or chatter about the steady-state position

[66]. A related method uses an inductor and capacitor in series and has been shown to increase

the stable range of travel at lower voltages, but this technique cannot be easily implemented with

MEMS technology due to a lack of inductors available in integrated circuits that meet the high

inductance requirements [67]. These methods show that charge control schemes can be utilized

to extend the range of travel and in some cases even improve the transient response as well [64].

In order to overcome the limitations imposed by parasitics, leakage, and residual charge more

involved methods must be employed using charging/discharging cycles, controlling clock

frequencies and complex circuit implementations.

2.3.4 Closed-loop Voltage Control to Eliminate Pull-in

There are cases where a closed-loop control technique has been used for attenuating and

stabilizing electrostatic instability. Voltage control methods have been explored to achieve

stabilization beyond the pull-in point [68], [69]. Chu, and Pister discuss the effect of introducing

a voltage control law into a system of electrostatically actuated parallel-plates and shows









theoretical stability at small gap distances [68]. Chen, et al., introduced a method for extending

the travel range of a torsional actuator by implementing voltage control to achieve desired

electrostatic torque profiles that can by-pass the pull-in point. This method was successful up to

approximately 80% of the initial gap [69].

2.4 Feedback Control Techniques Applied to MEMS

The previous discussion highlights several of the problems with current electrostatic

devices that have impeded the development of highly accurate and precise analog micromirror

arrays. These problems, including the limitations imposed by electrostatic instability as well as

the variable behaviors that result from fabrication uncertainties, have been addressed using

closed-loop control methods. Feedback control can help to increase the stable region of

operation for electrostatically actuated devices, provide accurate and precise positioning that is

robust with respect to variations in device fabrication, and also rej ect outside disturbances such

as vibrations and other noise sources. As seen in recent literature and summarized in Table 2-1,

controllers have been successful at both extending travel range of electrostatic actuators and for

improving tracking, disturbance rejection, transient response, system bandwidth and stability,

and reducing steady-state errors. Within the work that has been done to design and implement

feedback control systems on MEMS devices, a wide array of techniques and methods have been

employed, including linear-time-invariant (LTI) techniques such as proportional-integrator-

derivative (PID), robust, adaptive, and nonlinear control design. Some researchers address both

achieving actuation in the unstable range of motion and improved transient performance [15],

[70]-[74]. The control techniques presented in this review of the literature are not limited to

electrostatic micromirrors, but include a variety of devices and actuation methods to illustrate the

range of methods that have been employed for control of MEMS devices.









The controls literature reveals the many methods have been suggested as improvements to

facing the problems outlined above. Linear methods in some cases are insufficient, and more

advanced techniques have not been implemented due to the complexities required. An extended

review is given here of these controller methods and their applications for the benefit of the

reader. However, the work in this dissertation focuses mostly on modeling the behavior and

examining the effects of nonlinearity and uncertainties and the impact these have on control

implementation.

2.4.1 Linear Control

While all real systems will have nonlinearities, it is common engineering practice to treat

them as linear whenever possible. These assumptions and approximations, when acceptable,

greatly simplify analytical models as well as allow for the use of a wide range of linear control

methods. The use of classical, linear controller design such as PID, lead-lag, and state-variable

is adequate for these systems for which the systems operate in a small range of motion avoiding

nonlinear behavior [75]-[77], or in which the nonlinearities are small enough to be neglected

[78]-[80].

In the case of systems with large nonlinearities, such as those from electrostatics, it can be

a challenge to apply linear control design and ensure that a controller designed for the linear

system will be able to operate on the actual nonlinear plant. Despite the considerable

nonlinearities associated with electrostatic actuation, linearization of the plant model is often

done to allow for the use of linear-time-invariant (LTI) control methods. The nonlinear effects

of electrostatic actuation are perhaps most evident for parallel-plate actuator systems. Lu and

Fedder used a linearized plant model for a parallel-plate type actuator and designed a LTI

controller for both extended range of travel and position control [71]. The LTI controller was

designed and simulated on the linearized plant model and showed theoretically that very large











Table 2-1. Summary of feedback control papers discussed in the literature review.


Experimental/
Feedback Type


Author


Control Type


Control Objective

Increase Stability
Stability, Position
Tracking

Position Tracking

Position Control

Stability
Increase Stability,
Position Tracking

Position Control
Position Control,
Dynamic Response
(Settling Time),
Disturbance Rejection

Position Control

Increase Stability,
Position Tracking

Increase Stability,
Position Tracking

Position Control


System Type
Parallel-Plate
Electrostatic
Electromagnetic
MEMS Motor
Electrostatic Lateral
Comb Drive
Parallel-Plate
Electrostatic
Parallel-Plate
Electrostatic
Parallel-Plate
Electrostatic
Parallel-Plate
Electrostatic


Magnetic
Micromirror
Electrostatic Lateral
Comb Drive


[68] [Chu, Pister, 1994]

[84] [Lyshevski, 2001]
[Piyabongkarn, et al.,
[72] 2005]

[85] [Zhu, et al., 2006]
[Miathripala, et al.,
[74] 2003]


[73] [Sane, 2006]

[77] [Horsley, et al., 1999]


Nonlinear

Nonlinear

Nonlinear

Nonlinear

Nonlinear


No

No

Yes, Capacitive

No

No


Nonlinear
Classical (PD, Phase-
lead)


No
Yes, Capacitive, Laser
Doppler Vib. (LDV)


Yes, Position Sensing
Detector (PSD)

Yes, Capacitve


[78] [Pannu, et al., 2000] Classical (PID)

[75] [Cheung, et al., 1996] State-Feedback


[70], [Lu, Fedder, 2002,
[71] 2004]


Parallel-Plate
Electrostatic


Classical (P1


Yes, Capacitve


Electrostatic Torsion Yes, Current Meas. and
Micromirror (2DOF) PSD


[15]
[79],
[80]


[Chu, et al., 2005]
[Messenger, et al.,
2004, 2006]


State-Feedback
Classical (P, PI,Lead-
Lag)


Thermal actuator


Yes, Piezoresistive


Yes, unspecified


Yes, PSD


Yes, PSD


Yes, Capacitive, LDV


[Hernandez, et al.,
[82] 1999]


[27] [Kim, et al., 2004]

[Arancibia, et al.,
[26] 2004]


[76] [Liao, et al., 2005]
[10], [Park, Horowitz, 2001,
[ll] 2003]


[83] [Liaw, et al., 2006]
[33], [Gorman, et al., 2003,
[34] 2005]

[86] [Lee, et al., 2000]


Dual-Stage Disk
Robust (Mu-Synthesis) Position Tracking Drive
Piezoelectric Torsion
Adaptive control, Disturbance Rejection Mirror (2DOF, not
Robust (H-Infinity) (Wavefront Correction) MEMS)
Electromagnetic
Disturbance Rejection MEMS Torsion
Adaptive Control (Wavefront Correction) Mirror (2DOF)


Electrostatic Torsion
Micromirror


Adaptive Control

Adaptive Control

Sliding Mode Control
(SMC)

SMC

SMC


Position Control


Disturbance Rejection MEMS Gyroscope No


Piezoelectric
Acuators (PEA) (not
MEMS)

PEA (not MEMS)
Dual-Stage Disk
Drive
Electromagnetic
MEMS Torsion
Mirror (2DOF)
Electromagnetic
MEMS Torsion
Mirror


Position Tracking

Position Tracking

Position Tracking


Increase Stability

Dynamic response
(reduce rise time)


Yes, unspecified

No

No


Yes, PSD


No


[81] [Yazdi, et al., 2003] SMC


[87] [Chiou, et al., 2002] Fuzzy Logic









stable deflections could be achieved for this linearized plant. The LTI controller did not account

for the higher order nonlinear effects of the actuator, initial conditions or external disturbances,

and when the controller was implemented on the nonlinear plant, the maximum achievable stable

travel range was insufficient to reach the stated goal for stable range of motion. The LTI

controller was unable to satisfy both the stability conditions and disturbance rejection for large

deflections of the actuator, meaning that it could not attain the large deflections predicted for the

given controller design [71]. This illustrates the importance of considering robust operation of

the controller, especially when using a linearized plant model for a highly nonlinear system.

Linearized control is limited by the true nonlinearities of the system including the effects of

unmodeled dynamics, parameter uncertainties, disturbances, and stability, and it is most

appropriate for cases in which these effects are small. It is crucial to have an understanding of

the system behavior and its nonlinearities prior to the implementation of such control methods.

2.4.2 Adaptive and Robust Control

In utilizing closed-loop feedback control techniques for MEMS devices, robustness

becomes a commonly desired quality [26], [27], [70]-[72], [76], [81], [82]. Robustness is

important in MEMS control systems as there can be many uncertainties introduced through

variations in the device parameters, including geometry and material properties that arise from

the fabrication process, as well as nonlinearities in the dynamics and disturbances from noise or

other external influences. There are many ways to compensate for these uncertainties and

develop robustly stable systems.

An advantage of adaptive control over other methods, like PID, is that the controller can

compensate for uncertainties from fabrication, reject disturbance, and achieve desired tracking

obj ectives by continuously updating the controller parameters according to the actual system

performance [76]. When applying adaptive control it is very important to have an accurate










system model. The actual system output is compared to the estimated output predicted by the

model and this error is used to determine the controller gains during each step. If the predictive

plant model does not reflect the actual system behavior well, then large errors can lead to poor

performance and sometimes cause the system to go unstable [26], [27]. Calculating the

controller gains at each step in real-time can be difficult to implement, requires computationally

intensive algorithms and cannot be done compactly in an analog circuit.

Adaptive methods have been employed to account for parametric uncertainties within the

plant that arise from variations from the fabrication process. For actuators with performance that

is highly sensitive to fabrication variations, adaptive techniques may also be used for parameter

estimation. In the case of [72], the actuator dynamics of lateral electrostatic comb drives are

sensitive to fabrication errors arising from the alignment tolerances of bulk-micromachining.

Adaptive control has also been applied to MEMS gyroscopes, which are known to suffer from

parametric variations from the fabrication process that degrade the performance [10], [l l].

References [26], [27] demonstrate the use of adaptive control techniques for rejecting

disturbances that occur in adaptive optics applications when there is turbulence in the

atmosphere that affects the optical wave front. Kim et al. examined the control of piezoelectric

mirrors. These mirrors are not MEMS devices, however the control methods and application to

adaptive optics still warrants discussion. This work showed that using a combination of linear

time invariant (LTI) HI-infinity control and adaptive control resulted in good disturbance

rej section of band-limited noi se and the HI-infinity controller improved performance by

eliminating steady-state drift and reducing noise [26].

There are few examples of robust control design methods such as HI-infinity and mu-

synthesis that have been applied to MEMS systems. In addition to the use of HI-infinity control









demonstrated by Kim, et al. for a non-MEMS micromirror system [26], mu-synthesis controller

design was applied to a dual-stage actuator system for track-following in a hard-disk drive [82].

The controller design was successful in simulations, but no experimental work has been done so

far. The application of mu-synthesis to design robust controllers has not been specifically

applied to a strictly MEMS device. Difficulties in implementing these types of controllers arise

if the order of the controller is very high, in which case model order reduction can be used. In

summary, adaptive and robust control techniques appear promising at solving the issues of

controlling MEMS devices that are fabricated with parametric uncertainties, but only if the

system has very accurate models, and the sources of the uncertainties are clearly identified

within the model. In addition, these methods have largely only been evaluated in simulation thus

far because of implementation issues including high order controllers, lack of adequate sensing

methods, and difficulty in realizing the control in hardware.

2.4.3 Nonlinear Control

The instability problem posed by parallel-plate electrostatic systems has been a fertile area

for applications of nonlinear control techniques that incorporate Lyapunov stability analysis [68],

[72]-[74], [76], [85]. A general overview of Lyapunov stability analysis and how it applies to

nonlinear controller design or MEMS is given by [84]. It is clear that this method is

mathematically intensive and that proving global asymptotic stability of the Lyapunov function

is not a trivial matter. In the case of Maithrapala, et al., the researchers use a nonlinear state

feedback controller with a nonlinear observer to stabilize an electrostatic parallel plate actuator

in its unstable range and to improve the performance by reducing overshoot and decreasing

settling time [74]. The resulting control law is determined to have good performance at 80% of

the electrode gap in simulation; however it is only locally asymptotically stable. Several

researchers have developed controllers to extend the range of stability for parallel-plate









electrostatics, and have achieved excellent results based on numerical simulations [73], [85].

However, like other advanced control techniques discussed here, the resulting control laws are

not be easily amenable to implementation in analog circuitry and thus the results have not been

tested experimentally.

Additional control techniques that have been used include sliding mode control (SMC),

which can also be robust to plant variations, have good disturbance rej section and compact

implementation schemes. SMC is a digital, nonlinear control method generally good for systems

with nonlinearities and parametric uncertainties and tends to produce low order controllers. Lee

et al. used a discrete-time SMC for a dual-stage actuator for hard-disk drives to track a desired

traj ectory so as to avoid unwanted excitation of any resonant modes [86]. SMC was also applied

to the problem of electrostatic pull-in instability of two-axis torsion micromirrors [81]. SMC

operates through switching pulses that can result in chattering of the actuated device about the

steady state value, although attempts have been made to reduce this effect [83].

Although electrostatic systems are known to have hysteresis, there is little work examining

its effects on system performance and control. Piezoelectric actuators have significant hysteresis

in both traditional piezoelectric stack actuators and newer MEMS devices that utilize

piezoelectric materials. Liaw, et al. examines a traditional piezoelectric stack actuator, which is

in itself not a MEMS device but is used for micro and nano-scale manipulation [83]. A robust

sliding mode controller is developed that takes into account bounded parametric uncertainties

and hysteresis. The controller was implemented in an experimental system and found to have

good trajectory tracking with minimal tracking error and hysteretic behavior. Thermal actuators

also have hysteretic behavior, and Gorman et al. designed a robust controller for a thermally

actuated, microfabricated nanopositioner that uses a multi-loop control scheme based on SMC










[33]. This robust motion controller is shown in simulation to be able to track traj ectories and

rej ect disturbances to the system given a priori knowledge of the model uncertainty.

Chiou et al. [87] examine the use of fuzzy control for a micromirror that is actuated using

an array of electrodes that allow for a large number of positions using programmed digital

operation. The fuzzy controller showed improvement in the transient response over the open-

loop system in simulation, but issues concerning feedback signal and controller implementation

are not addressed.

In summary, nonlinear control techniques have been shown to be effective at addressing

the control of MEMS devices in theory, but like with adaptive and robust control, experimental

validation is thus far missing. It is clear from examining these various control methods that as

the techniques become more complex to account for robust performance and system

nonlinearities, the implementation issues also become more complicated. While many of the

papers in the literature discuss robustness of the control system, very few go into great depth of

defining the system uncertainties and determining the acceptable margins for the uncertainty.

Therefore it is not always clear if meaningful robustness is achieved for the system. Detailed

exploration of the uncertainties and the nonlinear behaviors is needed to further understand these

1SSUeS.

2.5 Sensing Methods for Feedback

In order to implement closed-loop control, a feedback signal is required. Optical beam

steering methods are considered in the scope of this research; however it is important to note

other sensing methods that may be used. There are multiple sensing mechanisms that have been

employed to produce feedback of position and rate for MEMS actuators. These include optical,

capacitive, and piezoresistive methods. One optical method that has been shown to produce a

good feedback signal is one in which the micromirrors steer a laser beam to a target










photosensitive diode (PSD) to track the position of the mirror [15], [26], [27], [78] [81]. Like

many optical methods, beam steering does not always offer the benefit of reducing the size of

physical implementation that can be achieved with capacitive or piezoresistive methods. Size

may be reduced in some cases by utilizing vertical cavity surface emitting lasers (VCSELs) as

the laser source, as was done in [13]. Other optical methods include using an atomic force

microscope (AFM) or laser Doppler vibrometer (LDV) [77]. Both of these methods have been

used and require special equipment that is only practical to use in a laboratory setup.

Capacitive sensing can be done by measuring changes in capacitance as the electrostatic

device moves. This method can produce very good signals, but does require additional circuitry

to use the signal [71], [72], [75]-[77], [96]. Depending on the complexity and fabrication

process abilities, this circuitry is able to be incorporated directly onto the chip as an analog signal

processor [71]. In some cases, estimators and observers must be employed to estimate and

extract the states of the system (position, velocity) from the sensor data. A Kalman filter, which

uses an observer and compares the actual response to the observer response, was used by

Cheung, et al. to estimate position and velocity based on the change in capacitance [75].

Piezoresistive sensing has already successfully been used in pressure sensors, shear sensors

[88]-[91] and acoustic sensor applications [92], [93]. It is relatively easy to implement in silicon

surface-micromachining processes by utilizing a Wheatstone bridge and does not require CMOS

to obtain a signal. The piezoresistive properties of silicon and polysilicon make it suitable for

feedback applications. Although polysilicon has a lower piezoresistive effect than single crystal

silicon, it has been used successfully as a sensing mechanism. Piezoresistive sensing created

within the SUMMiT fabrication process is demonstrated in [91] and [94]. Messenger, et al. has

successfully demonstrated the use of surface micromachined polysilicon to sense displacement









of a linear thermal actuator and then use that information to perform PID position control [79],

[80]. Drawbacks to piezoresistive sensing include a large area needed for the resistor elements

and drift due to temperature and time.

Noise is the limiting factor for any type of sensor. Microsensors are susceptible to

Brownian motion noise, 1/f noise, and thermal noise. Piezoresistive sensors have been shown in

the past to be most affected by 1/f noise [95]. Many researchers have experienced the limits of a

high signal-to-noise ratio and it can limit the bandwidth of the system [79], [80]. In some cases

the noisy sensor output can be filtered to achieve better response characteristics.

2.6 Summary Remarks

The results of this literature review reveal that there is still work that remains to be

completed toward the development of robust micromirror devices. The issue of electrostatic

pull-in and hysteresis has been addressed by making design modifications to the electrostatic

devices as well as with feedback control methods including LTI control, nonlinear control, and

sliding mode control. The literature has demonstrated cases in which electrostatic pull-in has

been successfully mitigated, but not entirely eliminated. A disadvantage to methods that

incorporate nonlinear mechanical springs into the system is that they require higher actuation

voltages. In this dissertation, electrostatic pull-in is addressed by introducing a novel design

technique called the progressive-linkage to create a nonlinear restoring force. This progressive-

linkage has the advantage of having a continuous spring force over other designs that use

discontinuous, piece-wise defined stiffness profies. While this approach still has the

disadvantage of higher actuation voltages, the benefits gained via this continuous passive control

approach of the nonlinearities in the system reduce the need for the complex control approaches

identified in the above literature review. This passive control approach should minimize the

degree of hysteresis resulting from the pull-in phenomenon, an issue that has largely been









unaddressed. This is an issue in which feedback control methods can also be applied to help

reduce the recovery time for hysteresis that occurs after pull-in. Bifurcation theory is used in this

dissertation as another method for capturing these nonlinear behaviors in the dynamic modeling.

It is also evident that there has been considerable study regarding parallel-plate

electrostatic actuators for which analytical relationships are known and are well defined from

physics. There has been less work done to model more intricate electrostatic configurations such

as those of vertical comb drives. Hah, et al. use 2D electrostatic models to determine the

electrostatic performance of vertical comb-drive actuated micromirrors [14]. While this

approach is intended to be more computationally efficient, it can limit the types and range of

electrode geometries that can be easily analyzed. In this dissertation, 3D FEA modeling is used

to determine the electrostatic characteristics of the micromirrors, and the FEA need only be done

one time for a given dimensional configuration, thus the computational costs remain low. In

addition, this work presents a detailed modeling approach to study the effects of fabrication

uncertainties along with characterization data for multiple devices that demonstrate variations in

actuator response.

Different control methods including PID, and LQR, are applied to the micromirror arrays

in this dissertation to compare the performance of each method and to further delineate the

impact of the parametric uncertainties on system performance. While a variety of controller

design methods have been utilized for MEMS devices, very few have considered optimal control

applications to electrostatic micromirrors. This dissertation also addresses a unique issue of how

to control an array of micromirrors that are not individually controllable. The micromirror arrays

examined here have single-input/multiple-output (SIMO) characteristics, providing an interesting

challenge to determining the appropriate sensors and error metrics to apply to feedback.









CHAPTER 3
MICROMIRROR MODELING AND STATIC PERFORMANCE

This chapter presents the micromirror array devices chosen for in depth study and

experimental validation. These devices are arrays of electrostatic micromirrors developed by

Sandia National Laboratories (SNL) for application to adaptive optics diffraction gratings like

those discussed in Section 2.2. A description of the SUMMiT V surface micromachining

process shows how these devices are made and gives some insight into sources of parametric

uncertainties that arise through the fabrication process. The static performance, described in

terms of the relationship of the actuation voltage applied and the resulting rotation angle of the

micromirror, is examined by developing models for the mechanical and electrostatic behaviors.

Electrostatic instability can also be predicted in terms of the pull-in angle, voltage, and

hysteresis. The static performance model is developed and presented here along with analysis of

the nonlinear behaviors of electrostatic instability and hysteresis.

3.1 Description of the SUMMiT V Microfabrication Process

The micromirror array is fabricated in the SUMMiT V surface micromachining process at

SNL. Figure 3-1 shows a diagram of the fabrication process from the SUMMiT V design

manual in which the five alternating polysilicon structural layers (mmpoly) and four silicon

dioxide sacrificial layers (sacox) are labeled along with their nominal thickness values [6]. (For

further information on surface micromachining fabrication, see [5].)

As with all manufacturing processes, there are machining tolerances in surface

micromachining that affect the final dimensions of the finished product. These tolerances can

result in slight deviations of the dimensions from the intended nominal values. Material

properties, such as Young's modulus and Poisson's ratio, are also variable and dependent on film

thickness and processing methods [12], [97]. The result can be that the fabricated devices will


























Substrate
6 inch wafer, <100>, n-type-


--0.2 Clm dimple4 backfill


o.4 pm dimple3 backfill
0.3 pm Scx
1.0 pm mmpolv1
2.0 lum sacox1


0.3 pm mmpolyO


I I o.5pm dimplel gap


Figure 3-1. Drawing of the SUMMiT V structural and sacrificial layers. (Courtesy of Sandia
National Laboratories, SUMMiT Technologies, www.mems. sandia.gov)

not behave as predicted, or that devices of the same design can behave differently from one

another. Dimensional variations can affect spring constants, resonant frequencies, and electrical

characteristics [7]-[9], [98]. Information on fabrication tolerances for the SUMMiT process is

available in the design manual [6], and those values relevant to this discussion are listed in

Tables 3-1 and 3-2. This information was gathered through diagnostic process testing as

described in [98]. Table 3-1 gives the mean and standard deviations of the thicknesses of the

layers of polysilicon and silicon dioxide. Table 3-2 gives values for variations in the dimensions

of the line widths of the device design. Figure 3-2 illustrates the effect of line width variation,

showing that for a desired area of dimension L by w, the actual fabricated area may be slightly

less, indicated by the dashed lines. Note that negative values indicate an inward bias resulting in

the actual size being smaller than drawn. The variability of the Young' s modulus, E, is not listed

in the SUMMiT design manual, however information published in the literature has found it to

be 164.3 GPa with a standard deviation of 3.2 GPa, which indicates a variation of 2% [97].









This information is useful for considering the effects of parametric uncertainties from the

fabrication process on the device performance. This subject will be considered more fully in

Chapter 4.

Table 3-1. Mean and standard deviation of fabrication variations for layer thickness in the
SUMMiT V surface micromachining process.
Layer Mean (Cpm) Std. Dev. (Cpm)
MMPOLYO 0.29 0.002
SACOX1 2.04 0.021
MMPOLY1 1.02 0.0023
SACOX2 0.3 0.0044
MMPOLY2 1.53 0.0034
SACOX3 1.84 0.54
DIMPLE3 Backfill 0.4 0.0053
MMPOLY3 2.36 0.0099
SACOX4 1.75 0.0045
MMPOLY4 2.29 0.0063

Table 3-2. Mean and standard deviation of fabrication variations of line widths in SUMMiT V.
Layer Mean (Cpm) Std. Dev. (pm )
MMPOLY2 -0.08 0.03
MMPOLY3 -0.07 0.05
MMPOLY4 -0.24 0.05





'L ~mean bias I




Figure 3-2. Area with nominal dimensions L and w with the dashed line indicating the actual
area due to error in the line width.

3.2 Micromirror Actuator Description

The micromirror arrays are shown packaged in a standard 24-pin dual in-line package

(DIP) in Figure 3-3. A magnified view of the surface of the array is also shown. The device

contains six groupings of micromirror arrays, and the particular grouping that is studied here is

indicated by a box drawn around it. This array contains 416 micromirrors arranged in 32 rows









and 13 columns. Each individual mirror is 20 x 156 Cpm2. These arrays were originally designed

at SNL to create a programmable diffraction grating for use in making spectral measurements.

Figure 3-4 illustrates the operation of the arrays as a diffraction grating in which the light source

striking normal to the surface of the mirrors when they are flat is reflected back on the same

path. When several mirrors are tilted, some light is reflected off at an angle. The result of this is

that the light is selectively diffracted.

The micromirrors are one-degree of freedom, actuated electrostatically and are shown

schematically in Figure 3-5. The electrostatic micromirror arrays have a ground plane and a















Figure 3-3. Images of the micromirror array. A) Packaged device. B) Micrograph of the surface
of the array.











A B Cr


Figure 3-4. Illustration of mirrors operating as an optical diffraction grating. A) When the
mirrors are not actuated (i.e. flat), the incident light is reflected straight back. B) For
mirrors that are actuated (i.e. tilted), the incident light is reflected off at an angle. C)
This results in a diffraction pattern of the light.









series of vertically offset comb fingers, all contained underneath a flat mirror surface. Having

the vertical comb drive beneath the mirror rather than just a parallel-plate capacitor attenuates the

electrostatic field and increases the stable range of motion of the device. This also allows for

large arrays with high fill factors, making them a good choice for analog scanning devices. A


Figure 3-5. Micrograph of an array of mirrors and schematic of mirror with hidden vertical comb
drive. A) The torsion spring. B) The full device. C) A 2-D cross-section view of a
unit cell (figure not to scale).


Mirror Surface
MM Poly 4


Moving Fingers
MMPoly 3


Figure 3-6. Solid model of micromirror showing polysilicon layer names from SUMMiT V.


tZ
M MMPoly 4 I

SMMPoly 2Mol X ~-
MMoyO MMPoly 1









voltage potential is applied across the fixed fingers and the moving fingers of the device creating

an electrostatic force. This force causes the mirror to rotate about an axis supported by the

hidden spring suspension, shown separately in Figure 3-5(a). Not shown in the drawing is a

design constraint that restricts the motion of the fixed-end of the mirror plate from moving a

large distance in the Z-direction. While some motion may occur, the assumption is made that

this device acts in one degree-of-freedom by rotating about the x-axis. Figure 3-6 shows a 3D

model identifying the fabrication layers used to create the micromirrors.

3.3 Electrostatic Actuation and Instability

Many electrostatic actuators exhibit the well-documented phenomenon of electrostatic

pull-in. The electrostatic force is nonlinear, as it is inversely proportional to the square of the

electrode gap. Pull-in, sometimes called snap-down, occurs when the electrostatic force

generated by the actuator exceeds the mechanical restoring force of the structure. The result is

that the device reaches an unstable position and subsequently is pulled down to the substrate at

its maximum displacement. Another phenomenon associated with pull-in instability is that once

the mirror has pulled-in, the voltage required to maintain the pull-in position is lower than the

pull-in voltage. The mirror will not return from this position until the actuating voltage has been

reduced below the holding-voltage. The result of this holding effect is hysteresis.

This section will examine the modeling of the electrostatic-mechanical system and the

instability phenomena. The case of parallel plate electrostatics is examined and used to derive

general relationships for modeling the system. This is extended to a torsion electrostatic actuator

to illustrate the complications that arise from adding complexity to the system geometry.

3.3.1 Parallel Plate Electrostatics

Consider a parallel plate capacitor, such as shown in Figure 3-7, in which the top plate is

supported by a spring, with spring constant km, and the bottom plate is fixed. Damping in the










system is represented by the damping coefficient, b. The plates are separated by a distance of xo,

and have an overlapping area of A.






LJm A










Figure 3-7. Schematic of a parallel plate electrostatic actuator modeled as a mass-spring-damper
sy stem.

The equation of motion for this mass-spring-damper system is derived by the balance of

the forces on the system from Newton's second law

CF = m (3-1)

where m is the mass of the moving plate. When the top plate is displaced in the positive x-

direction, shown in Figure 3-7, the motion is opposed by the force from the mechanical spring,

which is assumed to be linear, and follows Hooke's law. The mechanical spring constant is k;;.

Fm = -kmx (3 -2)

The damping force is assumed to be linearly proportional to the velocity by a factor of b, the

damping coefficient.

F, = -bx (3-3)

When a voltage potential is applied across the two plates, an electrostatic force is generated that

attracts the top plate to the bottom. The electrostatic force for a system operating in air is

derived from the energy, U, of an electric field, E, integrated over a volume, v.










U = So E 2d (3-4)


where so is the permittivity of free space, 8.854 x10-12 F/m. The electric field is given by


E =- (3-5)
coA

where Q is the electric charge. The charge, Q, can be written as

Q = CV (3-6)

where C is the capacitance, and Vis the voltage. The capacitance between two parallel plate

actuators is given in terms of the overlapping area of the plates, A, and the distance between the

two plates.

coA
C(x) = (3 -7)
(xo x)

Equation 3-4 can be rewritten as


Ulc= CV2 = (3-8)
2 2(xo x)

The electrostatic force is thus written as

dU 1 dC 2 0,AV2
F = V2 (3-9)
8(xo x) 2 8?(xo x) 2 (xo x)2

The force balance for the system yields the equation of motion.

Fe = mit+ b + kmx (3-10)

The static equilibrium for the system reduces to only the electrostatic force, and the mechanical

force.

1 dC 2 ,A V2
V2 kmx (3-11)
2 8(xo x) 2 (xo x)2









Equation 3-11 can be interpreted to show the relationship between the voltage and the

displacement, as plotted in Figure 3-8 for system of parallel plates with the area, A, equal to 100

x 100 Cpm2, an initial gap, xo, equal to 10 Cpm, and a mechanical spring constant of k, equal to 1

N/Cpm. From this, it is clear that there is a maximum voltage for the system, and that there can be

multiple solutions for the same applied voltage. This behavior is the result of the electrostatic

pull-in instability. It turns out that the solutions in the lower portion of Figure 3-8 are stable

solutions and the solutions in the upper portion are unstable. The maximum voltage value

corresponds to the actuation voltage at which pull-in occurs, and the maximum stable position

for parallel plate actuator occurs at one-third the gap between the electrodes.

To further explore the pull-in phenomena, the static relationship in Equation 3-11 can be

examined graphically, by plotting the electrostatic force and the mechanical force separately in

Figure 3-9. The electrostatic force is a function of both the displacement and the voltage. Static

equilibrium occurs where the electrostatic force lines and the mechanical force line cross each

other. As was shown in Figure 3-8, there are instances where the mechanical and electrostatic

lines intersect at more than one point. Because of the nonlinear behavior of the electrostatic

force, there is a point at which the electrostatic torque exceeds the ability of the mechanical

spring and equilibrium can no longer be maintained. This is referred to as electrostatic pull-in.

At the pull-in point, both the electrostatic and mechanical torques are equal in magnitude and

slope and thus only have one point of intersection between these forces on the graph [14], [46].

Stable static solutions occur before the pull-in point, while unstable solutions occur after. This

slope equality is written by taking the first derivative with respect to the displacement of

Equation 3-11.

1 d2C
y V = kM, (3-12)
2 8?(xo x)2









0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.'1


10 20 30 40 50
Voltage (V)


Figure 3-8. Static equilibrium relationship for the parallel plate electrostatic actuator.


\ e~ Pull-in at
Stable 1/~3 gap

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Displacemnent x/x~


0.8 0.9


Figure 3-9. Electrostatic force for different voltages and mechanical force showing pull-in for
the electrostatic parallel plate actuator.










Substituting Equation 3-12 into 3-11 and evaluating at the pull-in position results in the

following relationship that is only a function of the capacitance and the pull-in position, xpn.


dC 82C

8(xo x) 8.x (xo x)21 ~=o(-3


Assuming that the restoring springs are linearly deformed in the range of actuation, the

pull-in angle is independent of the spring stiffness, and depends only on the angle of rotation. A

pull-in function, PI(x), is defined to determine the pull-in angle, which occurs when PI(x) is

equal to zero.

dC 82C
PI(xo x) = ax (xo x) a~,-x"(3-14)


In turn, once the pull-in angle is determined, the pull-in voltage can be calculated by the

following expression,


VP 2mPI (3-15)
8r (xo x)a 'p



The pull-in function for the parallel plate electrostatic actuators is shown in Figure 3-10,

0.01

0.005 -Pull-in at 1/3 gap



a -0.005

-0.01

-0.015

-0.02
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Displacement x/xo


Figure 3-10. Pull-in function for the parallel-plate electrostatic actuator.










and verifies that pull-in does occur at 1/3 the gap between the plates. The pull-in voltage is

calculated from Equation 3-15 to be equal to 57.85 V.

To further investigate the effects of changing the spring constant on the pull-in, the static

equilibrium relationships are plotted for different values of the mechanical spring constant in

Figure 3-11. This shows that even for a different spring constant, the pull-in displacement

location remains at 1/3 the gap, while the pull-in voltage changes.


~ 1 0.8
20 *II~
/ o 0.7




0 0.1 0. .5 06 07 0 0.9 0 2 5 5 10 2 5
AL Noralze Diplceen xx Votg (v B
Figure" 3-11 Sttceulbimrltosip o h aallpaeatutruigdfeetsrn
contats A)o Th elcrsai and mehnclfocs )Thttcdipaeet
volag relationships.4

3.. PaalelPat Trion Actutor




an th dspacmaient m pay edescrbe inx tem fteanlfta oltation, 6.

Thgue gen.Sateral ibiu relationships for ecto tati pactation thatwee detatriveding Sifeint 3.3.1ma



al o be dervdfr this type of aneetrorstaion aale actuator intemso trqe insta of fores. The tsumof






the torques for the system describes the equation of motion for the system. The micromirror can

be considered as a one-degree-of-freedom mass-spring damper system of the form

J + b + kB = T,(8,Vy) (3-16)














VW


Figure 3-12. Schematic of a torsion electrostatic actuator.

where Jis the mass moment of inertia, b is the damping coefficient, km is the mechanical spring

constant, and Te is the electrostatic torque, which is represented by the following,

1 dC
Te V2y (3-17)
S2 80

where C is the capacitance, 8 is the angle of rotation about the X-axis, and Vis the voltage

potential. The mechanical system is governed by the stiffness of the support structure of the

mirrors. It is assumed that the spring suspension provides a linear mechanical restoring torque,

Tm, to the system that can be represented as,

Tm = kmB (3-18)

where km is the rotational spring constant. Static equilibrium occurs in the device when the

electrostatic torque is equal to the mechanical restoring torque. Therefore, the static device

behavior, which is the relationship of the actuation voltage, V, to the rotation angle, 8, is

determined by equating Equations 3-17 and 3-18.

1 BC\
V2 Iy = km9 (3-19)
2 80!

As was previously shown for parallel-plate electrostatics, there is a point at which the

electrostatic torque exceeds the ability of the mechanical spring and equilibrium can no longer be









maintained. This is referred to as electrostatic pull-in. At the pull-in point, both the electrostatic

and mechanical torques are equal in magnitude and slope [14], [46]. As was shown previously

for the parallel-plate actuator, electrostatic pull-in can be considered as the mechanical and

electrostatic torques being equal, as in Equation 3-19, and their first derivatives being equal.

V1 2Cd2 = k (3-20)


Combining Equations 3-19 and 3-20 and evaluating at the pull-in angle results in the following

relationship that is only a function of the capacitance and the pull-in angle, BpI.


-_,, )PI= (3-21)

Assuming that the restoring springs are linearly deformed in the range of actuation, the pull-in

angle is independent of the spring stiffness, and depends only on the angle of rotation. A pull-in

function, PI(0), is defined to determine the pull-in angle, which occurs when PI(0) is equal to

zero.


PI(0) =(c 6a (3-22)


In turn, once the pull-in angle is determined, the pull-in voltage can be calculated by the

following expression,

2k, PI
Vr = cal= (3-23)



Thus far, the only difference between modeling the torsion actuator and the parallel plate

actuator is that the parallel plate actuator has linear displacement, while the torsional has

rotational motion. Therefore, the equations for each system are very similar. The difference in










evaluating the torsion actuator becomes apparent however when the capacitance for the system is

calculated. Unlike the parallel-plate actuator, the torsion actuator does not have a constant gap

between the top and bottom electrodes when it moves. Considering the system drawn in Figure

3-12, the capacitance for a torsion actuator in terms of the angle of rotation about its axis is given

as



C(0) = I1- -n (3-24)
max max


where a is L3/L2, iS L2/L1 and Imax is Ho L1 [46]. From Equation 3-24 it is possible to calculate

the performance for a torsion actuator. The pull-in function for this system is


32 2
3O, B BB ,, -4 138 4 1 I(-5
PI(0) = oW max Smma x x max + 31n max(325
82 2I 2IP

max max 8,


As an example, consider a system with the following geometric variables: km = 1 N-Cpm,

Omax= 10o, Wm = 100 Cpm, a = 0.5, r = 0.5. The static equilibrium for this system can be

evaluated by examining the electrostatic and mechanical torques, as shown in Figure 3-13. The


0.4, 1
0.35~ /-1
0.3 E 0.8 -
5 Pull-in at 82%
,0.25r
p ie 0.6

0.2.



0 0.2 0.4 0.6 0.8 1 00
A Normalized Displacement elemax Voltage (v) B

Figure 3-13. Static equilibrium relationships for the torsion actuator. A) The electrostatic and
mechanical torques. B) The static rotation-voltage relationship.










displacement as a function of the voltage is also shown in Figure 3-13. The pull-in function from

Equation 3-25 is plotted in Figure 3-14. From these figures, it is found that the pull-in for this

system occurs at 82% of maximum rotation angle for the system. For the given spring constant,

the pull-in voltage is 6.91 V. As with the parallel plate actuator, the pull-in angle will remain the

same despite the mechanical spring constant, but the pull-in voltage will change. The pull-in

angle can change, however, if the system geometry is changed. This is different from the

parallel-plate actuator, which always pulls in at 1/3 the gap.

3x10 3







u- -1
Pull-in at
0.82




0 0.2 0.4 0.6 0.8 1
Normalized Displacement a/e


Figure 3-14. Pull-in function for the torsion actuator.

3.4 Model for Vertical Comb Drive Actuator

From the previous section it becomes clear that analytically describing the performance of

an electrostatic actuator becomes more difficult as the geometry of the electrodes becomes more

complex. In fact, it is very difficult to describe the capacitance for the micromirror devices that

operate via vertical comb drive actuators. Thus it becomes necessary to employ FEA to assist in

developing the system model. The device operates in both the mechanical and the electrical

domains. Therefore, the model is developed for the mechanical and the electrostatic functions

separately. The following analysis presents the model first in the mechanical domain and then









the electrostatic. The two models are then combined to determine theoretically the static

behavior of the mirrors, including pull-in and hysteresis.

3.4.1 Mechanical Model

The mechanical spring is shown in Figure 3-15 with the fabrication layers labeled. The

spring has two anchor points that connect to the ground layer (MMPoly0) and thin beams in the

MMPoly 1 layer to provide the restoring force. The layer MMPoly3, which is used to create the

moving comb fingers, has a dimple cut in the center of the spring mechanism that, when

actuated, comes into contact with the anchor (MMPoly2) and allows the mirror surface to pivot

about this point. An alternative depiction is shown in Figure 3-16 in which the spring is

considered as thin beams that are fixed to the substrate in two places. The length and cross-

sectional area of the MMPoly1 beams is given in Figure 3-16 to be 33 Cpm and 1x1 Cpm2,

respectively. The value for Young's modulus is 164.3 GPa, and Poisson's ratio is 0.22. To

determine the stiffness of this mechanical spring, a simple model is created in ANSYS Finite

Element Analysis (FEA) program using Beaml89 elements, which are capable of nonlinear large

deflection analysis [99]. The boundary conditions constrain all motion in six-degrees-of-

freedom at the two anchor points.



Mirror Surface 1'Dimple Cut


MMPoly 4
MMPoly 3
~ MMPolyZ
~MMPolyl


IM MM*oly 0
Anchor Points Beam


Figure 3-15. Drawing of the mechanical spring that supports the micromirrors and provides the
restoring force.










Displacement loads are applied in all six-degrees-of-freedom at the point indicated in

Figure 3-16 that corresponds to the pivot point created by the MMPoly3 dimple. The FEA

determines the forces and stresses in the beam elements after the displacement loads are applied.

The deformed shape of the structure is shown in Figure 3-17. Assuming Hooke's law for the

force applied to a linear spring, the spring stiffness in all six degrees-of-freedom can be

calculated. The linear spring assumption is verified by performing nonlinear FEA over the entire

range of motion of the spring displacement from zero to nineteen degrees. The results are listed

in Table 3-3, retaining 4 significant figures. The stiffness in X, Y, and Z refer to the stiffness of

the spring in each respective axis direction, and qX, qY, and qZ refer to the rotational stiffness

about the axes X, Y, and Z, respectively. It is clear that the spring is not very stiff in the Y and Z

directions. The torsional stiffness about the X axis, qX, is lower than those about the Y or Z

axes, meaning that the mirror is able to rotate about the X axis, while it is resistant to off-axis

rotations about the Y or Z. It is the value of qX equal to 612.4 pN-m/rad that is used for ks, in

Equation 3-17.







Apply
~- ~di a meant
Area cross-section
of beam: zI


w = 1pm Boundary
Conditions:
DOF





Figure 3-16. Image of the mechanical spring that supports the micromirror indicating boundary
conditions and location for applying displacement loads for finite element analysis.
























Boundary
Candltlons 0 In
all DOF





Figure 3-17. Image from ANSYS of the deformed spring and the outline of the undeformed
shape after displacements are applied. The displacement is amplified by a scale
factor of 4.

Table 3-3. Values output from Einite element analysis of mechanical spring stiffness.
Parameter Value
X stiffness 744.7 pN/m
Z stiffness 7.946 pN/m
Y stiffness 1.266 pN/m
qX stiffness 612.4 pN-m/rad
qZ stiffness 11360 pN-m/rad
qY stiffness 16310 pN-m/rad

3.4.2 Electrostatic Model

In order to compute the electrostatic torque values in Equation 3-16, it is necessary to Eind

an expression for the capacitance as a function of the rotation angle. For parallel-plate

electrostatics, this can be done quite easily as an analytical expression is known. Because of the

more complex electrode geometry created by the inclusion of the vertical comb drive, the

capacitance of the device cannot be as easily derived. To determine the charge created by the

electrostatic Hield, 3-D FEA is used to calculate the capacitance as a function of 8.









The symmetry of the device design makes it convenient to model only a small section of

the device, termed the unit cell. A cross-section of a unit cell made up of one-half of one moving

comb finger and one half the associated fixed comb finger and portions of the ground plane and

mirror surface is shown in Figure 3-5(c). The model of the geometry in Figure 3-18 is created in

ANSYS. The nominal dimensions used to create this model are given in Appendix A. For an

electrostatic analysis, the volume of the surrounding fluid, in this case ambient air, is created

around the device geometry, and it is this air volume that is meshed and analyzed to determine

the electrostatic field generated as the mirror and moving finger rotate about an axis parallel to

the X-axis in the figure. An arbitrary voltage differential, V, is applied as shown in the drawing.

The only relevant material properties needed in this analysis are the permittivity of free space, so,

which is 8.854 x10-12 F/m, and the relative permittivity of the dielectric medium, e, which in this

case for air, is equal to 1. The analysis calculates the total charge of the electric field, W, and

then calculates the capacitance for a given 8 position as


C =2 2 (3-25)


Using numerical values generated in the electrostatic FEA model, Equation 3-24 is applied to

calculate the capacitance at discrete points as the geometry of the mirror surface and moving

comb finger rotate through a range of motion from 0 to 19 degrees. A polynomial least-squares

fit of these capacitance values is used to find an analytical expression for the capacitance. The

capacitance as a function of a is approximated with an nth order polynomial curve fit.

C(0) = N(10" + P2B" n-+ -+ POB+ P, ) (3 -26)

where the coefficients of the polynomial are P,, (i = 1, 2,..., n, n 1), and N is the total number of

unit cells. The results of this analysis are plotted in Figure 3-19 along with a comparison of first,

second, third, and fourth order polynomial curve fit approximations of the data. The coefficients










for these curve fit approximations are listed in Table 3-4. Table 3-5 compares the quality of the

different order polynomial approximations compared to the FEA data points. One metric to

evaluate the fit quality for a curve fit is the norm of the residuals, normr. The smaller the value

of normr is, the better the approximation. Another standard metric is the sum of the square of

the residuals, r2, which is calculated from normr by

2 OTmT
r2 =1-- (3 -27)
(n 1)s 2

where n is the number of data points (FEA data), and s is the standard deviation of the curve fit

approximation from the data. A value of r2 equal to one indicates a perfect fit. It is clear that a

higher order polynomial does a slightly better j ob of capturing the nature of the capacitance data.

However, the first order linear curve fit can still be sufficient for analysis in the stable range of

motion. It will not be as accurate at predicting the pull-in behavior. The advantage of using the

first order fit is that its derivative which is used in Equation 3-16 is a constant, thus simplifying

the plant model to a linear approximation in V2. In Order to capture the nonlinear behaviors of

pull-in and hysteresis, the fourth order polynomial curve fit approximation is used in Section

3.4.4. The effects of different linear approximations in the model are discussed further in

Chapter 5.

Mirror Surface

(mmpoly4)
Moving Finger
(mmpoly3)

Fixed Finger and
Ground Planes Ground Plane
(electrical isolation) 7 Va
l (mmpoly2)
(mmpoly0)
(mmpo~ly1 )



Figure 3-18. Solid model geometry of the unit cell used in the electrostatic FEA simulation.




















2. -7 + FEA data
a #~ ----- 4th order
15~ ----- 3rd order
2nd order
I st order

0 5 10 15 20
-Theta (deg)


Figure 3-19. Capacitance calculation as a function of rotation angle, 6, calculated using 3D FEA
and varying orders of polynomial curve fit approximations.

Table 3-4. Comparison of polynomial fit for approximation of capacitance function
Order Pl P2 P3 P4 P5
4 0.023 120 -0.013678 0.004164 0.000109 0.000106
3 0.000848 0.001280 0.000299 0.000103 --
2 0.001680 0.000250 0.000104 -- --
1 0.000777 0.000078 -- -- --

Table 3-5. Comparison of polynomial fit for approximation of capacitance function
Order normr s n r2
4 1.1192E-05 0.000185 18 0.999785
3 1.1463E-05 0.000185 18 0.999775
2 3 .6691E-05 0.000185 18 0.997691
1 8.608E-05 0.0001854 18 0.987166

3.4.3 Electromechanical Model

Taking both the mechanical and electrostatic models into account, the static behavior of the

system can now be predicted using Equations 3-16 to 3-20. Equations 3-18 to 3-20 calculate the

electrostatic pull-in characteristics of the device. A plot of the pull-in function is shown in

Figure 3-20 where pull-in occurs when the function equals zero at 16.5 degrees. Using this value

in Equation 3-20, the pull-in voltage is 71.06 V.










x 103









-0.5 Pull-in
Angle



-1 .5-

0 2 4 6 0 10 12 14 16 10 20
Theta [degrees]


Figure 3-20. Plot of the Pull-in function PI(6) for the micromirror with the vertical comb drive
actuator showing that pull-in occurs at 16.5 degrees.

The static equilibrium behavior can also be evaluated from Equations 3-16, and 3-17,

respectively. When the mechanical and electrostatic torques are equal to each other, the system

is in static equilibrium. This can be shown graphically by plotting these values. Figure 3-21

shows the electrostatic torque as a function of rotation angle for different values of voltage

ranging from 10 volts to 80 volts. The straight line on the plot corresponds to the mechanical

restoring torque of the spring from Equation 3-17. At every point where the mechanical torque

and the electrostatic torque lines cross, they are in equilibrium indicating a stable position. There

is a point at which this line runs tangent to the electrostatic torque, and this indicates the

electrostatic pull-in point, which corresponds to the calculated values of 16.50, and 71.06 volts.

The electrostatic torque curve at the pull-in voltage, Vn, is also indicated in Figure 3-21. The

pull-in angle for a linear spring is determined completely by the electrostatic torque. For a

different value of the mechanical spring constant, km, the slope of the mechanical torque line









would be different, but it would still run tangent to the electrostatic torque at the same pull-in

angle. Only the value of the pull-in voltage would be affected. This is shown in Figure 3-22.

The pull-in instability is known to cause hysteresis in the device behavior, and this too can be

predicted using this modeling approach. After the device has pulled-in, it is possible to reduce

the voltage below the pull-in voltage without releasing the device. This is referred to the holding

voltage. Once the voltage has been reduced below this holding voltage threshold, the device will

release from its pulled-in position, but it will return to a position different from the pull-in

position. From this electromechanical analysis, it is determined that the holding voltage is 68.89

V. The static behavior of the device is shown in Figure 3-23, including the pull-in point and the

hysteresis loop. This type of curve will be referred to as a 8-V profile, and represents the static

calibration for the device.


250


200 ,
Pull-in/








f10 2 0 12 1 6 1
Thet [dges
Fiue3-1 lctotti n ecaialtru a uctino roaionage tean
votg fo iffrn otgaus









1000-


800-



400 -I



2 00 Y.


0 2 4 68 "10 "12 14 16 18
Theta [degrees]

Figure 3-22. Torque as a function of rotation angle, theta, and voltage for different values of
mechanical spring constant.


20
18
16
14
i12


r8
6
4
2


20 40 60
Voltage (V)


Figure 3-23. Plot of static equilibrium behavior, showing pull-in and hysteresis, predicted from
the model.










3.4.4 Linear Approximation

Recall from the discussion in Section 3.4.2 of the electrostatic model development that the

capacitance function is approximated using a polynomial curve fit, and that different orders of

polynomial can be used. For this system, the nonlinear behavior of the electrostatic instability is

best captured using a higher order polynomial; however a first order function is still able to

approximate the system performance. Using a first order approximation makes the derivative

term of the capacitance a constant value, which greatly simplifies the dynamics and allows the

system to be modeled as linear. The effects of using a higher order curve fit versus the first order

are more apparent by looking at the static equilibrium relationship between the applied voltage,

V, and the rotation angle, 6. This is shown in Figure 3-24 for the fourth order fit, called the

nonlinear model, and the first order fit, called the linear capacitance approximation model. It is

clear that by using the lower order model approximation there is a difference between the



18
nonlinear model
16~ ---*linear capacitance approx. /
14t -






0-





0 20 40 60 80
Voltage (V)

Figure 3-24. Static equilibrium relationships for the nonlinear plant model, and the linear plant
approximation.









predicted static performances. To establish the effects of model uncertainty on micromirror

arrays, the linear model is used as a basis for designing controllers in Chapter 5. The linear

model is suitable to the design of the controller, but the resulting control law must still be able to

perform well on the nonlinear system.

For a system in which the capacitance cannot be adequately modeled as linear, such as the

case of parallel plate electrostatics, a higher order approximation is required. In this case, it is

possible to linearize the second order dynamic model in Equation 3-16 about an operating point

(60, Vo) using the Taylor series expansion (TSE) [36]. This can be considered as the small signal

model approximation about 30 and 3V Doing so yields the following linear system model,

J6O + b30 + k,3 = ke30 + C3V (3-28)

The linearization in Equation 3-28 includes a term that is dependent only on the rotation

angle that can be considered the electrostatic spring force, ke [20].


k, =~ d2C V (3-29)
dB2,


The nonlinear torque approximation is reduced to a constant.

dC
C, = Vo1 (3 3 0)
dB I

When linearizing a function about an operating point, it is desirable that the linear model

will provide an adequate estimate of the nonlinear function within a small range about that

operating point. For systems that are operating over a large range or have very nonlinear

characteristics, this linearization may not provide a satisfactory estimate of the nonlinear

function. To illustrate the effect of the small signal linearization, Figure 3-25 shows the static

equilibrium relationship between rotation angle and actuation voltage for the nonlinear system









model and for the small signal model linearized about the operating point (7 degrees, 54 volts).

The inset shows the small signal response for 6O, 6yV

It is clear in Figure 3-25 that this linear estimate of the nonlinear system does not capture

all of the static performance characteristics over the entire range of operation, but is adequate

enough for a portion of the range from 5 to 14 degrees. In order to cover the full range of

actuation, a piecewise linearization can be done at different operating points. This piecewise

linearization approach would represent the system response as shown in Figure 3-26.

The linearized models discussed above are important when considering control design

techniques that require a linear transfer function or state-space model for the design process. Of

the two linearization methods discussed, the first method of using a linear capacitance

approximation is used throughout this dissertation whenever the linear system model is required.

This method was chosen for its ease of use.


20
-nonlinear model
--*linearization about operating point
operating point
15




.C 5
1-
0 10 20 30





0 10 20 30 40 50 60 70 80
Voltage (V)

Figure 3-25. Static equilibrium relationships for the nonlinear plant model, and the small signal
model linearized about an operating point (60, Vo).























0" '"TO 20 30 40 50 60 70 80
Volta8ge (V)

Figure 3-26. Illustration of piecewise linearization about multiple operating points.

3.4.5 Bifurcation Analysis

Electrostatic instability is an example of bifurcation, and the stability of the system can be

examined by looking at the dynamics of the actuator and finding the fixed-point solutions [106],

[107]. One advantage of evaluating the bifurcation behavior of the device is that unlike the

methods used in Equations 3-14 and 3-15, the mechanical spring constant is not required to be

linear. This analysis will be used again in Chapter 5 to determine the effects of a nonlinear

spring constant on the electrostatic pull-in. Here, the spring constant is still assumed to be linear,

and the results may be compared to those obtained using Equations 3-14 and 3-15.

The state space model for the system is

x, =
x, = 0 (3-31)

.1 x,

x -fT,(x ) -- 'x


Recall that Te is a function of the capacitance expression from Equation 3-26. In order to capture

the nonlinear effects of the system, a fourth-order curve fit approximation is used. The fixed

points occur at xz = 0 and









Te (x, ) k, x, = 0 (3-32)

This can be expressed in full as


N(44Fx3 + 3Px2 12 +2Px, + P)V2 kx, = 0 (3-33)


Equation 3-33 is a cubic polynomial equation for which finding the roots has been the subj ect of

considerable study [110]. One solution is to write the polynomial as

Ax,3 + Bx12 C1 + D= A( 81(x, 8,)(x, -21 8e3) = 0 (3-34)

where 8,, 8,2 and 8,3 are the three roots, and the coefficients A, B, C and D are

A = 2NV24F (3 -3 5)


B = NV2P (3-36)


c 3 NV2P3 km (3 -3 7)


D= NV2P4 (3-38)


Further, define

9ABC 27A2D 2B3
q = (3-39)
54A3


u = 3A Bg2 2~ (3-40)


s =J4 (3 -4 1)

t =~r (3-42)

The roots of Equation 3-34 are


,= s+ t -- (3-43)
3A









1 B
8 (s +t)- -+-(s -t)i (3 -44)
2 3A 2

I B
=e (s +t) (s -t)i (3-45)
2 3A 2

The roots of Equation 3 -3 3 can be found to determine the static voltage-di splacement

relationship, as was done previously in Section 3.4.3. Solving this equation gives the Eixed

points as functions of the control parameter V. The roots of this expression can be examined

graphically by defining a function F(xy) as

1 d C
F(x,) = Te Tm = V- k,,x, (3 -46)
2 dx

In Figure 3-27, F(xy) is plotted for varying values of voltage, V. The roots of F(xy) correspond to

the zero crossings on the Eigure. Notice that there are three roots for each line of constant

voltage, and this corresponds to F(x,) being a third order polynomial. The roots that occur to the

left of zero degrees theta are solutions that are non-physical solutions and are thus ignored. The

solutions of function F(xy) that occur for positive values of theta can have either two roots, one

root, or zero roots. For a sufficiently small voltage, there are two roots. In this case the

electrostatic force is low enough that the linear spring force can balance it, creating a stationary

state. As the voltage is increased, the electrostatic force increases, eventually overwhelming the

linear spring force and all the steady-state solutions disappear. This is another description of the

pull-in instability caused by the disappearance of all physically possible steady-state solutions

[106].

Now that the steady state solutions of the system can be determined, it is the stability of those

solutions that must be determined. A Jacobian matrix is found by taking the Taylor series

expansion of Equation 3-3 1 and retaining only the first order terms [107].










400


300


200-







-'100
Direction of
increasing V
-2q0 -10 0 10 20 30 40
Theta (deg)

Figure 3-27. Plot showing the roots of the function F(xl) occur where the function crosses zero.


Df (x) 1 dT, (xl)C1 kJ, 1b (3-47)


where


1N(124Fx,2 + 6P x, + 2P,)V2 (3-48)
ax, 2

The Jacobian defined in Equation 3-47 relates the perturbation of the states from equilibrium as

= Df (Y) a" (3-49)
~Al x, 0 x

The stability is determined by evaluating the matrix in Equation 3-47 at the fixed points and

determining the eigenvalues. The fixed point solution is stable when the real part of the

eigenvalues is less than zero. The eigenvalues, 3t for j = 1, 2, are calculated for each fixed point

solution (i.e., roots 8,, 8,, and 8,,). This is expressed as










1 1 b j
Ai ---+- +4 "
S2 J 2 J \J dx, J
j =1, 2 (3-50)
i=1, 2, 3

Substituting Equation 3-48 into 3-50 gives the expression for the eigenvalue problem in terms of

the expression for the capacitance.


1 2 Jb 21 b1 k ,
i =---+- -+ 20+P 2)
(t 2J(t 2j28 J6, 2J 2P) J
j =1, 2 (3 -5 1)
i=1, 2, 3

To evaluate the eigenvalues and their stability, an expression for the damping in the system

must be defined. In a MEMS system such as this, the dominant source of damping comes from

the squeeze-film effect, in which air that is compressed between very small spaces begins to act

as a viscous fluid [3]. Squeeze-film damping is dependent on the device geometry, and

expressions are known for parallel plate actuators and for torsion plate actuators. As was the

case with the electrostatic model development, the complex geometry of the vertical comb drive

micromirrors makes determining the squeeze-film damping coefficient analytically difficult. For

the purpose of this discussion, an approximation is made to consider the squeeze-film damping

term for a torsional plate developed by Pan, et al [100].

rLw5
b = Kror (3-52)


where L is the length of the plate, w the width, g is the gap between the plates, and rl is the

absolute viscosity of the fluid. The term Kror is

48
rYot l r + (3-53)









.Table 3-6 lists the values of additional parameters for this analysis. This estimate for squeeze-

film damping is used here for simplicity.

The resulting bifurcation diagram in Figure 3-28 shows a saddle node bifurcation at 16.5

degrees and 71.06 V. This is in agreement with the pull-in results from Section 3.4.3. Figure 3-

29 shows the bifurcation diagram for different values of the mechanical spring constant, k;;, to

illustrate how changing the spring constant for a linear spring only affects the pull-in voltage.

Table 3-6. List of parameters used for this analysis.
Parameter Value
p, density of polysilicon 2331 kg/m3
YI, absolute viscosity of air 1.73e-5 N-s/m2
L, length of mirror 20 Cpm
w, width of mirror 100 Cpm
g, gap between plates 11.25 Cpm
N, number of unit cells 54

3.5 Chapter Summary

The electrostatic modeling in this chapter reveals the performance characteristics of a

micromirror based on the nominal design parameters of the device. The model is developed by

considering the mechanical spring element and the electrostatic actuation forces separately.

Doing so allows for greater understanding of the role of each energy domain in determining the

performance of the electromechanical device. It can also be useful in the design stages of an

electrostatic micromirror to see the effects of changing the design to have a different spring

stiffness or electrode shape. The electrostatic instability phenomenon is described in analytic

terms that can be used to predict the pull-in angle, pull-in voltage, and the hysteresis behaviors of

the device. The electrostatic behaviors are also examined through bifurcation analysis.

It is discussed in the description of the fabrication process in Section 3.1 that there are

certain errors that occur in the geometry and the material properties during fabrication.
















lu Bifurcation point
8 71.06 V, 16.50

S4


-2-
-4-

-10

0; 10 20 30 40 50 60 70 80 90 100
Voltage (V)


Figure 3-28. Bifurcation diagram for a MEMS torsion mirror with electrostatic vertical comb
drive actuator.


40 60
Control parameter V


Figure 3-29. Bifurcation diagram showing the effects of different spring constants.


*Unstable
- Stable










Information on these errors is available in the process design manual, and gives a MEMS

designer a reasonable expectation of the precision available from the micromachining process.

The next chapter will use the modeling methods developed here to examine the effects of

parametric uncertainties that come from the fabrication process, and what these errors in

dimensions and material properties can do to the performance of a microdevice.









CHAPTER 4
UNCERTAINTY ANALYSIS AND EXPERIMENTAL CHARACTERIZATION

Chapter 3 presented the description of the micromirrors and demonstrated the modeling

methods used to predict the static behavior of the devices. While one may assume that the

micromirrors were fabricated exactly to the nominal design specifications for dimension and

material properties, it is well established that surface micromachining processes have machining

tolerances that result in small parametric errors in the finished devices. The effects of these

fabrication variations in dimension and material property are examined utilizing the modeling

methods put forth in Chapter 3 for the electrostatic micromirrors. The effects of varying a single

parameter at a time are examined first to determine the sensitivity of the design to a given

parametric uncertainty. Then, combinations of uncertainties are evaluated using Monte Carlo

simulations. The results obtained from the models in Chapters 3 and 4 are then compared to

experimental characterization data that was obtained using an optical profiler.

4.1 Parametric Uncertainty and Sensitivity Analysis

Recall from the discussion in Section 3.1, that fabrication tolerances for surface

micromachining processes can result in final dimensions that differ from the intended design.

The SUMMiT V design manual gives values of dimensional tolerances in layer thickness and

linewidth error, shown first in Tables 3-1 and 3-2 respectively, and reprinted in this chapter for

convenience as Tables 4-1 and 4-2. These show that dimensions can vary by as much as eight

percent in layer thickness, and as much as twenty-nine percent for width dimensions on a feature

size of 2 microns [6]. The result can be that the fabricated devices will not behave as predicted,

or that devices of the same design can behave differently from one another. Dimensional

variations can affect spring constants, resonant frequencies, and electrical characteristics.










Table 4-1. Mean and standard deviation of fabrication variations for layer thickness in the
SUMMiT V surface micromachining process.
Layer Mean (Cpm) Std. Dev. (Cpm)
MMPOLYO 0.29 0.002
SACOX1 2.04 0.021
MMPOLY1 1.02 0.0023
SACOX2 0.3 0.0044
MMPOLY2 1.53 0.0034
SACOX3 1.84 0.54
DIMPLE3 Backfill 0.4 0.0053
MMPOLY3 2.36 0.0099
SACOX4 1.75 0.0045
MMPOLY4 2.29 0.0063

Table 4-2. Mean and standard deviation of fabrication variations of line widths in SUMMiT V.
Layer Mean (Cpm) Std. Dev. (pm )
MMPOLY2 -0.08 0.03
MMPOLY3 -0.07 0.05
MMPOLY4 -0.24 0.05

Changes in layer thickness result in differences in the vertical spacing of the final device

dimensions, as shown in Figure 4-1. The thickness of the structural polysilicon layers have an

obvious impact on the final device dimensions, however the thickness of the sacrificial oxide

layers plays an important role in determining the intermediate spacing of the structural layers.

The linewidth variations of the polysilicon layers also contribute to the final fabricated

dimensions of a given geometry being different from the nominal, designed values. Figure 4-2

shows that changes in any of the dimensions can result in a final geometry that is different from

the nominal design, which affects the size, shape, volume, and mass of the device.









Figure 4-1. Fabrication tolerances can changes the thicknesses of the layers, resulting in changes
in the final geometry dimensions.










L L+6L



li 1; o/ w+ dw

Nominal Geometry Actual Geometry

Figure 4-2. Fabrication tolerances can change the dimensions of a fabricated geometry, affecting
the final shape, volume, and mass.

The mass for the micromirror array devices can be estimated from the volume of the

moving components, which are the mirror surface and the moving comb fingers. The nominal

dimensions for these components are shown in Figure 4-3. Once the fabrication tolerances are

considered, it becomes clear that the mass of these parts will be affected by the changes in the

geometry. Calculating the volume and multiplying by the density of polysilicon (233 1

kg/m3), the nominal mass of these components is 2.34 x 10-"1 kg.

The mechanical spring constant is affected by changes to the geometry of the spring and

variations in the Young's Modulus. The electrostatic model is also affected by these changes.

The following sections will examine the effects of the dimensional tolerances on the

performance of the devices using the modeling methods developed in Chapter 3. First, the

156 pm

20 pm Mirror Surfac e MMPoly4iknss225~

152 pm



17 ~millllllllllllllll 11111 Dimple Cut Fngr MMPoly3
Thickness 1.65 pm




Figure 4-3. Nominal dimensions used to calculate the volume of the moving mass.









contributions of each individual parameter variation are considered to try to identify the effect of

any given parameter on the final device performance. Through sensitivity analysis, it can be

determined which key parameters have the most effect on the final device performance. Because

these variations can occur in any combination with each other, there are an exceedingly large

number of possibilities. Therefore, in order to understand the effects of these fabrication

variations on the device performance, Monte Carlo simulations are done to give an idea of the

combined effects of multiple parameter variations.

4.1.1 Effects of Individual Parameter Variations

To understand the effects of a single parameter variation on the system, the device

performance is determined using the modeling methods developed in Chapter 3 as only one

parameter is allowed to change at a time. There are fourteen parameter variations to be

considered, and they include ten variations in layer thickness listed in Table 4-1, three linewidth

variations listed in Table 4-2, and one material property variation for the Young's modulus of

polysilicon. A change in a single parameter can cause both the mechanical spring constant and

the electrostatic capacitance to change from the nominal model. First, the effects on the

mechanical model are examined, followed by the electrostatic.

The mechanical model described in Section 3.4.1 is a spring in which the stiffness is

determined by the dimensions of the beam members, as well as the material properties of

Young's modulus and Poisson's ratio. Recall that the main structural element of the spring is a

set of two thin beams constructed in the MMPoly1 layer, which was shown in Figure 3-7. The

length of this beam and the cross-sectional area are the most critical dimensions for

determination of the beam stiffness. Therefore, the dimension variation in the thickness of the

MMPoly 1 layer is considered, as well as uncertainty in the Young's modulus as calculated by

Jensen et al. to be 164.3 GPa +3.2 GPa [97]. Poisson's ratio is still assumed to be a constant at










0.22 as there is no available data to suggest that it varies. Table 4-3 shows the effects of

changing the MMPoly 1 thickness as well as the Young' s modulus on the value of the spring

constant.

While there is no data given in the SUMMiT V design manual [6] regarding line width

variations for MMPolyl, it is possible that this variation does occur. The layers MMPoly1 and

MMPoly2 are most often used together to create one thicker, laminate layer of polysilicon,

therefore, diagnostic data is only collected for MMPolyl/2 laminate [98]. As an additional

study, analysis is done here for cases in which line width variations for MMPoly 1 are considered

to be equal to those of MMPoly2, as 80 nm + 30 nm. This analysis is also included in the results

of Table 4-3.

The first entry in Table 4-3 is the nominal model value, and each subsequent value of the

mechanical spring constant, km, is compared to this value in terms of the percent change. When

only the thickness of layer MMPoly 1 and the Young' s modulus are considered, the spring

constant is found to vary between -1.95% to 5.66% from the nominal spring constant. By

Table 4-3. Spring stiffness values for changing dimensional and material parameters.
Layer Change in
Thickness Young's Linewidth Spring % change
MMPoly1 Modulus, E MMPoly1 Stiffness, Km from
Cpm GPa Cpm pN-m nominal
1.0000 164.30 0.00 612.35 0.00
1.0200 164.30 0.00 634.72 3.65
1.0223 164.30 0.00 637.33 4.08
1.0177 164.30 0.00 632.12 3.23
1.0000 167.50 0.00 624.28 1.95
1.0000 161.10 0.00 600.43 -1.95
1.0200 167.50 0.00 647.03 5.66
1.0200 161.10 0.00 622.36 1.63
1.0200 164.30 0.08 730.98 19.37
1.0200 164.30 -0.08 546.91 -10.69
1.0000 164.30 0.08 706.55 15.38
1.0000 164.30 -0.08 526.56 -14.01









considering the effects of variations in the linewidth of MMPolyl1, the resulting spring constants

are found to vary significantly from -14.01% to 19.37% from the nominal value. From this it is

clear that including the effects of linewidth variation can have a significant effect on the spring

constant. As stated previously, there is no available recorded data to indicate that linewidth

variations do occur in MMPoly 1. However, it is reasonable to assume linewidth variations do

exist for MMPoly 1 as these variations are present in all other layers. For the remaining analysis

in this section, linewidth variations in MMPoly 1 will be omitted from consideration and are only

included here to demonstrate that these errors can have a very large impact on structural

stiffness.

In the case of the mechanical spring constant, there are only a few parametric variations to

consider. As the capacitance for the device is dependent upon the geometric spacing of the

device components, the electrostatic model will be affected much more by any changes in layer

thickness or in linewidth. To see the effects of the individual parameters, electrostatic analysis

was done for each of the thirteen structural parameters in which each parameter was allowed in

turn to be increased by a value of its standard deviation as listed in Tables 4-1 and 4-2.

The results are shown in terms of the capacitance in Figures 4-4, 4-5, and 4-6. Figure 4-4

shows the capacitance function for changes in the thickness of the polysilicon structural layers,

MMPoly0, MMPolyl, MMPoly2, MMPoly3, and MMPoly4. The nominal capacitance function

is shown for a comparison using the nominal dimensions of the device. It is evident that making

changes individually to these parameters has little effect on the electrostatic model for the

device. Figure 4-5 shows the capacitance function for changes in the thickness of the Dimple3

backfill, and the sacrificial oxide layers Sacoxl, Sacox2, Sacox3, and Sacox4. In the case of

Sacox3, it is clear that this parameter alone plays a significant role in determining the









electrostatic characteristics of the micromirror. Sacox3 is the sacrificial layer that determines the

spacing between the fixed comb fingers in layer MMPoly2, and the moving comb fingers in

layer MMPoly3. Figure 4-6 shows the capacitance functions calculated for changing the area

dimensions of the device in the linewidths of layers MMPoly2, MMPoly3, and MMPoly4. The

capacitance curve does deviate some from the nominal model for these parametric variations,

particularly in MMPoly2.

This analysis is extended to see the combined electromechanical effect of the parametric

variations in terms of the static displacement curves. Figure 4-7, 4-8, and 4-9 show these results.

Figure 4-7 shows the 8-V curves for the micromirrors when the structural polysilicon layers are

each varied. The results here are similar to the results for the capacitance function in Figure 4-4,

in that changes in these parameters do not appear to have a significant affect on the device

performance. It is worth noting however that the layer thickness of MMPoly 1 does have a slight

effect on the altering the systems static behavior and this is because the layer MMPoly 1 plays a

significant role in determining the mechanical spring stiffness. Figure 4-8 demonstrates the

sensitivity of the micromirror to variations in the thickness of Sacox3, similar to that seen in the

capacitance function of Figure 4-5. Likewise, Figure 4-9 shows small deviations in the static

displacement curves when the linewidths of the polysilicon layers are changed. It is clear that

some parameters have a larger effect on the final static performance of the device, most

prominently is Sacox3. Parametric sensitivity analysis is another way to examine how sensitive

the modeled system is to variations in a given parameter. Sensitivity, S, can be defined as the

percent change in the output of the system divided by the percent change in the parameter of

interest, a. In this case, the output of the system can be considered as the voltage required to

achieve a desired position, 8. That is,












-MMPoly0
.......... MMPoly1
16 MMPoly2
MMPoly3l
MMPoly4
a, 14-
o ------ Nominal





10-



0 2 4 6 8 10 '12 14 16 '18
Theta [deg]

Figure 4-4. Capacitance functions for the electrostatic model with parametric changes in the
layer thickness of the structural polysilicon.


'18


16


14 -4


.9 12
t~ I I Dimple3
ca .......... Saclolx
0 10 -Sacox2

Sacox3
8- Sacox4


0 2 4 6 8 '10 '12 '14 "16 '18
Theta [deg]

Figure 4-5. Capacitance functions for the electrostatic model with parametric changes in the
layer thickness of the Dimple3 backfill and the sacrificial oxide.










'18

SMM Poly2 :
16C --------- MMPoly3
MM Poly4
t6 ------ Nominal



a,1 0-,






0 2 4 6 8 10 12 14 16 18
Theta [deg]

Figure 4-6. Capacitance functions for the electrostatic model with parametric changes in the
linewidth error of the structural polysilicon layers.


'18
MMFJPoly0
"16
......... MMPoly1
14 MMPoly2
MMIVPoly3
S12~ MMPoly4
-a 10 ----- Nominal









0 '10 20 30 40 50 60 70 80
Voltage (V)

Figure 4-7. Static displacement relationships for the micromirror model with parametric changes
in the layer thickness of the structural polysilicon.













16-

14-

12-
a 0
a -




4 -

2 -

0
-


-Dimple3
---------- Sacorx1
Sacox2
Sacox3
-Sacox4
------ Nominal


10 20 30 40
Voltage (1/


50 60 70 80


Figure 4-8. Static displacement relationships for the micromirror model with parametric changes
in the layer thickness of the Dimple3 backfill and the sacrificial oxide.


16t MMPoly2
.......... MuMPoly3
114-
MMPoly4
12C -'----- Nominal

a '1 0


I--
6-


'10 20 30 40
Voltage


50
(V)


60 70 80


Figure 4-9. Static displacement relationships for the micromirror model with parametric changes
in the linewidth error of the structural polysilicon layers.










S=~= (4-1



where S is the sensitivity with respect to parameter a, Va(0) is the voltage required to achieve a

position of 8 for a model with a variation in parameter a, ao is the nominal value of the

parameter, and Vno,,(0) is the voltage required to achieve a position of 8 for the nominal model.

Figure 4-10 displays the sensitivity of the system to changes in line widths. The same

analysis for variations in layer thickness is given in Figure 4-11. The four parameters with the

highest sensitivities are the thicknesses of layers MMPolyl, Sacoxl, Sacox3, and Sacox4.

Variations in the parameters Dimple3 backfill and Sacox2 have the lowest sensitivities; nearly

zero for the entire range of motion. This analysis reveals which geometric parameters in the

device design are expected to be the most sensitive to the changes in dimensions from fabrication

tolerances. This kind of analysis can also be very useful during the design stage of a new device

as it can be used in conjunction with optimal design methods to reduce the effects of parametric

uncertainty on the operation of the completed device. However examination of the individual

parametric effects will only reveal a partial understanding of the effects of the fabrication

tolerances on device performance, and it is therefore beneficial to consider the effects on the

system when multiple fabrication errors are present. This is done in the following section using

Monte Carlo simulations.

4.1.2 Monte Carlo Simulations

As in the previous section, there are fourteen different parameters of interest in this

analysis, and performing the model analysis for every possible combination of parametric

variation would be a very large and time-consuming task. Each of these parameters is assumed

to vary within a Gaussian distribution defined by the mean and standard deviation information















0.7 -f MMPoly3


0.6-


0.5-


S0.4-


0.3-


0.2 -

MMPoly4
0.1



O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Theta (deg)


Figure 4-10. Sensitivity of voltage with respect to changes in line width for each value of 8.


O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Theta (deg)


Figure 4-11i. Sensitivity of voltage with respect to changes in layer thickness for each value of 8.









from the fabrication data in Tables 4-1 and 4-2, and from the studies on material properties in

[97]. Monte Carlo simulations provide an effective method for examining the effects of these

parametric uncertainties by randomly choosing values from the Gaussian distribution and

running a large number of model simulations. In the analysis here, 250 simulations are done.

From these 250 sets of randomly generated Gaussian parameters, the effects of the fabrication

tolerances on the system performance can be determined. The histograms for each parametric

variable are shown in Appendix B to show the distribution of each variable generated in the

simulations. The histogram of the resulting mass that is calculated for each of the 250 sets of

variables is also shown in Appendix B.

It is possible that the fabrication tolerances could have some systematic correlations, such

as all of the polysilicon layers for a given fabrication run having thicker layers at the same time.

As there is no data to support this idea however, it can only be assumed that each parameter is

allowed to vary independently from the others. A Gaussian, or normal, distribution is given by

the following expression [109]


Gx (x) = e-a-Ix)'2 20 (4-2)


where Xis the mean value of the data set, o-is the standard deviation, and x is the data being

measured. The Gaussian distribution for a set of data with a mean of zero and a standard

deviation of one is plotted in Figure 4-12. This figure also illustrates that 95% of the values of x

fall between -1.96 and 1.96, which is considered the 95% confidence interval for this

distribution. This is also very close to falling between -20 and 2 o, which constitutes 95.45%.

Using the randomly generated variables, it is possible to analyze the impact of these

fabrication variations on the mechanical spring constant. First, this is done for the case in which

only the layer thickness of MMPoly1 and the Young's modulus are allowed to vary. The


















QI
S0.2

0.15

0.19 .9
-.695% Confidepce Interval 19
0.05

-40 -30 -20 1 0 10 20 30 40


I 95.45%
Figure 4-12. Gaussian distribution with a mean of 0 and standard deviation of 1.

resulting spring constants, km, had a mean of 634.21 pN-m and a standard deviation of 12. 17

pN-m. Using twice the standard deviation (120) to represent the 95% confidence interval (CI)

for the mechanical stiffness values, one can say that there is a 95% chance that the mechanical

stiffness will fall between the values of 609.87 pN-m and 658.55 pN-m. This corresponds to a

variation in the mechanical spring stiffness of 3.84% from the mean.

It was shown previously in Table 4-3 that if the linewidth of MMPoly 1 is allowed to vary by the

same level of uncertainty as the MMPoly2 linewidth, there is a significant impact on the spring

constant. Repeating that analysis here for the 250 Monte Carlo simulations but this time

mechanical spring constant results for each analysis respectively. The effect of the MMPoly 1

linewidth variation is included here to once again show the large effect this variable has on the

mechanical spring stiffness, however the MMPoly 1 linewidth variation will not be considered in

the remaining analyses.

The Monte Carlo simulations are conducted on the electrostatic model as well, using the

same set of 250 randomly generated variables that are used in the mechanical spring constant
















IllI


I 1, I
80 600


62U 64U


Spring Constant Km [pN-prn]

Figure 4-13. Histogram for mechanical stiffness when accounting for variations in thickness of
MMPoly1 and Young's modulus.


16
14-

r 10
O





00 4


400 500 600 700
Spring Constant Km [pN-gn]


Figure 4-14. Histogram for mechanical stiffness taking into account variations in thickness of
MMPolyl, Young's modulus, and linewidth of MMPoly1.


II |
680










analysis. These variables are kept consistent throughout this analysis to ensure the results will be

accurate when the electrostatic and mechanical simulation results are combined. As was seen in

the above analyses, the structural polysilicon layers affect the device dimensions, and the

sacrificial oxide layers affect the spacing of the geometry in the Z-direction. The capacitance

function is affected by both these changes in dimensions. Material properties do not play a role

in the electrostatic analysis. Figure 4-15 shows the results of the calculated capacitance

functions for 250 simulations using randomly chosen sets of variables. In order to classify the

capacitance simulation results in terms of the mean and standard deviation, it is necessary to look

at the capacitance values calculated at each value of theta and determine the mean and standard

deviation at each point. This is done in Table 4-4, and the 95% confidence interval values for

capacitance vary from 18.47% at zero degrees of rotation and 27.83% at eight degrees of

rotation.


25



20








S'5




0 2 4 6 8 10 12 14 16 18
Theta [deg]

Figure 4-15. Results from the capacitance simulation for 250 random variable sets that show the
effects of parametric uncertainty on the electrostatic model.










Table 4-4. Results from the Monte Carlo simulations for the capacitance values in terms of
mean, standard deviation, and the percent change from nominal.
Theta Mean Capacitance St. Dev. + % change
(deg) (fF) (fF) (95% CI)
0 0.0005 4.73E-05 18.47
1 0.0285 0.0028 19.65
2 0.0295 0.0031 20.81
3 0.0306 0.0034 22.16
4 0.0318 0.0037 23.58
5 0.0331 0.0041 24.88
6 0.0346 0.0045 26.06
7 0.0363 0.0049 26.92
8 0.0381 0.0052 27.59
9 0.0399 0.0056 27.81
10 0.0419 0.0058 27.83
11 0.0440 0.0061 27.61
12 0.0461 0.0063 27.11
13 0.0483 0.0065 27.10
14 0.0504 0.0065 25.88
15 0.0526 0.0066 25.26
16 0.0547 0.0068 24.72
17 0.0570 0.0072 25.31
18 0.0592 0.0078 26.30

Taking the results of the mechanical and electrostatic analyses together gives a picture of

the overall effect that parametric fabrication errors can have on the system performance in terms

of the 8-V profile. Figure 4-16 shows the results of doing this for the 250 simulations using the

randomly chosen variables. Given the large number of possible combinations of dimensions that

affect both the mechanical and electrostatic models, using 250 samples may not be enough to

give a complete statistical representation of all the numerous possible combinations; however it

is sufficient to show trends in the model predicted results. These results are compared to

experimental characterization data in Section 4.2.

It is evident from these results that the parametric uncertainty that arises from the

fabrication process alone can have a significant performance effect on the static displacement

behavior of the micromirrors. Because the variable sets used in these simulations are randomly

generated, it is difficult to obtain a sense for the role that each individual parameter, or even

combinations of parameters have on the overall performance of the micromirrors. Recall from













'12


16









0 20 40 60 80
Voltage (V)

Figure 4-16. Static displacement results of 250 Monte Carlo simulations with random Gaussian
distributed dimensional variations.

the sensitivity analysis in Section 3.5.1 that some variables had a significantly larger effect on

the system performance, most notable the layer thickness of Sacox3. To understand the impact

this particular variable had in the results from the Monte Carlo simulations, it is possible to try to

isolate the contribution from Sacox3 by first considering only the results that occur for large

deviation in Sacox3 thickness. Figure 4-17 shows the histogram of the Sacox3 values used in the

Monte Carlo simulations. The values in blue correspond to those that lie within the 95%

confidence interval. The values in red represent the other 5% of values that fall at the extreme

ends of the distribution. Figure 4-18 shows the simulation results for the 8-Vprofiles that are

colored to correspond to the values of Sacox3 thickness. The lines in blue are the results that

correspond to Sacox3 values within the 95% confidence interval, while the lines in red are the

results from the parametric variations that lie outside this interval. This gives a clear indication

that for extreme differences in the Sacox3 thickness, the resulting e-V proHile will also have the

most extreme behavior. This analysis was done for additional variables to try to determine a



















1 1.5 2 2.5 3 3.5 4
Thickness of Sacox3 [pm]


Figure 4-17. Histogram of values from the Monte Carlo simulations for the layer thickness of
Sacox3. Values in blue lie within the 95% confidence interval, and values in red lie
without.


5:


0 '10


20 30 40 50 60 70
Voltage (V)


Figure 4-18. Static displacement curves from the Monte Carlo simulations that indicate the
effect of large variations in the Sacox3 layer thickness. Curves in blue have Sacox3
values that lie within the 95% confidence interval, and lines in red have Sacox3
values that lie in the remaining 5% of the distribution.


HM


0 0.5










pattern of contributions; however the results for the other parameters did not show any detectable

correlations to the performance. This same analysis for the variables of linewidth in MMPoly2,

thickness of MMPolyli, and thickness of Sacox4 are included in Appendix B. Changes in each

of these variables show a cluster of profiles in the middle region of the randomly generated 8-V

profiles, which is the opposite of the impact of changes to Sacox3.

4.2 Experimental Characterization

This section presents experimental characterization and validation of the models developed

in the preceding sections. Static characterization measurements for the micromirror device were

taken using a WYKO NT 1100 Optical Profiler to determine the 8-V profiles for the mirrors

[101]. This measurement tool is able to make measurements of out-of-plane deflections as the

micromirrors are given different actuation signals. This information can be used to determine

how variable the 8-V profiles are for mirrors within the arrays, and from one array to another.

Measurements were taken with the system in static mode, in which the voltage is applied at

different values, returning to zero voltage between each deflection measurement. Static

measurement results are provided for the arrays of micromirrors described, as well as for a set of

single micromirrors that are not part of an array. These results are compared to the model

predictions, validating the results of the model in determining the static performance, and pull-in

behavior. The experimental results taken from different micromirrors across three different

arrays demonstrate significant differences in behavior among them. This further illustrates the

presence of parameter variations within a given array as well as between arrays of the same

device design.

4.2.1 Equipment Description

The WYKO NT1100 optical profiling system uses interferometric measurements to

determine the out-of-plane measurements of a surface. The working principle of the instrument










is shown in Figure 4-19. Light travels from the light source and is divided by a beam splitter.

One beam is sent to the reference mirror of the Mirau interferometer, and the other beam is

directed onto the measurement sample. The reflections of the two beams are recombined into

one beam, and because they have traveled different distances in their respective paths, they are

no longer in phase. Thus, the newly recombined beams form interference fringes which are

recorded by an optical detector array. The digital information from the detector is processed to

determine the surface measurement of the sample.


Detetor rray-i -Digitized Intensity



file,, Beamsplitter

Illuminatc -:::.-- III---- Translator

Microsco e
Light Source
Field Objective
Aperture Stop
Stop Mirau
Interferometer


Sample

Figure 4-19. Diagram of an optical profiler measurement system.

The optical profiling system is able to take measurements in static mode, in which the

MEMS device is not in motion when the measurement is taken, as well as in dynamic mode,

capturing the motion of the device under excitation. The surface measurements are recorded into

a database, specified by the user, and an example of a surface measurement taken for the

micromirror arrays is shown as a 3D image in Figure 4-20. This image shows six mirrors from

the array, four of which are tilted by an applied actuation voltage of 60 V. The two mirrors in

the center are left without any actuation, and this arrangement proves useful as these mirrors can

become a zero reference from which the other measurements are taken. While the data for the









tilted mirrors is recorded into a database, it is also possible to review each individual

measurement that has been taken. This is helpful to ensure that the data is recorded accurately,

and gives insight into how the angular tilt measurement of the mirrors is determined. The data

can be reviewed using WYKO SureVision software, which accompanies the optical profiling

system. This program allows the user to examine 3D images, such as that in Figure 4-20, as well

as look at cross-sections of the data. Figure 4-21 shows a cross section of the micromirror data














I I 20 pm



Figure 4-20. Six mirrors from the micromirror array measured with the optical profiler system.

in which the four tilted mirrors appear as diagonal lines. The tilt angle measurement is

determined from the displacement measurements in the vertical, out-of-plane, Z-direction, and

the horizontal, in-plane X-direction. Thus, the angle of tilt is found from the tangent relationship

of the X and Z measurements. Any measurement errors in X or Z will result in an error in the

angle measurement as well. This error will be discussed in more detail in the following sections.

Figure 4-21 also shows an example of a measurement in which the profiling system failed to

properly record the data. This illustrates the difficulties encountered in obtaining these

measurements, as the micromirrors are actuated to very large angular displacements that are

more difficult for the system to record. A poorly constructed data record such as the one shown














































Im


1 -1-


in Figure 4-21 is too sparse to be relied upon for a measurement and should be discarded.


Unfortunately, these incomplete and sometimes erroneous records are sometimes recorded into


the database files. For this reason, each of the data records has been individually examined and


verified to ensure the most accurate of measurement results.


N
to

I
3


X-direction (pLm)


h


c
o
~ -1-
u
111
L
U
r~
i-

"'-


B


3n


X-direction ( om)


Figure 4-21. Data records from the SureVision display that show the cross-section profile of the
tilt angle measurements. A) An example that clearly shows the cross-sectional
measurements. B) An example of a poorly recorded data file that cannot be used.


4.2.2 Static Results for Single Micromirrors


To validate the single micromirror models, a set of single micromirrors were fabricated and


analyzed. These mirrors, shown in a micrograph in Figure 4-22, were characterized in the static


mode of testing, in which voltages are reset to zero for each measurement, using a WYKO



103










NT 1100 optical profiler at Sandia National Laboratories. In Figure 4-22, the square bond pad on

the left is 100x100 Cpm2 and the micromirror on the right has dimensions 156x20 Cpm2. The

results from these single mirrors are shown in Figure 4-23. It is clear that the pull-in point for

this set of experimental data is similar to the data collected on the arrays, and the pull-in angle,

13.870, is at the lower range of the pull-in angles for the arrays of mirrors. The pull-in voltage is

71.5 V, similar to the values for the micromirror arrays and very close to the predicted value. At

the time this data was recorded, the calibration and resolution of the machine were not recorded;

therefore it is not possible to discuss the specific errors that are associated with this data.

However, the standard operation of the WYKO NT1 100 is supposed to be on the order of

nanometers .









100 plm


Figure 4-22. Micrograph image of a single micromirror.

20

16 Pull-in at
12




12





0 10 20 30 40 50 60 70 80 90 100
Voltage (V)


Figure 4-23. Experimental static results taken from individual micromirrors that are not in an
array.









4.2.3 Static Results for Micromirror Arrays

Experimental data on the performance of the micromirror arrays was acquired using a

WYKO NT1100 Optical Profiler located at the Veeco company offices in Chads Ford, PA. This

machine was calibrated to a National Institute of Standards and Technology (NIST) traceable

standard to be accurate to one-half of one percent (0.5%) of an 82 nm step. This corresponds to

height measurements accurate to 0.410 nm. As the tilt angle measurements are determined from

the inverse tangent of the Z over the X measurement, shown above in Figure 4-21, this amount

of error in the Z-direction corresponds to an error in the tilt angle measurement of +0.023 50

This amount of error is too small to even demonstrate on the plots of the data as error bars.

While the measurement equipment is believed to operate true to its calibration standards, there is

evidence from researchers in [1 12] that this optical profiling system may be subj ect to larger

errors. Measurements of the 8-V profile for micromirrors taken from different sections across

the array for three different miromirror arrays were taken. These results were obtained using the

static mode of measurement in which the voltage signal is reset to zero between each

measurement. The approximate locations of data collection for all three arrays are shown in

Figure 4-24 and these locations are labeled. These areas were chosen to try to gain an

understanding of any changes in the performance across the array.

Shown in Figures 4-25, 4-26, and 4-27, data from 5 different areas (consisting of four

mirrors actuated and two mirrors for reference) on the arrays from among the 3 arrays reveals

that there is considerable variation in the behaviors of the individual mirrors. Each array

consisted of 416 mirrors arranged in 32 rows and 13 columns. Data was collected from different

areas in the arrays in order to examine how the micro performance varies in different locations

within the array. Table 4-5 gives a summary of the pull-in angle and voltages for the data. The
























.1 mm .


Figure 4-24. Approximate locations of data collection on all three arrays.

average pull-in angles for arrays 1, 2, and 3 are 14.270, 13.540, and 15.890, respectively. While

these values do not agree exactly with the predicted pull-in value of 16.50 from the analytical

model, the lowest value is within 20 percent. Also, the values listed in Table 4-6 are averaged

values over multiple data sets. From Figures 4-25 through 4-27, it is evident that in many cases

the mirrors did experience pull-in very close to the predicted angle of 16.50. The pull-in voltages


20



16-

14 -

3i12 -i

10-


m Mirror 1
6 -1
A Mirror 2
4 -1 d Mirror 3
L x Mirror 4



0 20 40 60 80
Voltage (V)


Figure 4-25. Experimental results from array 1, area A.
















16


14

~3 12





6

4

2
-


m Mirror 1
First run
g ~ Mrror Mirror 3

4B 5 Mirror 4




20 40 60 80 "1


Voltage (V)


Figure 4-26. Experimental results from array 2, areas D and E.


aa

a n
iM


Area A


1-

6

4-


2-
0


r I


a


'A'rea D
40 60 80


Voltage (V)


Figure 4-27. Experimental results from array 3, areas A and D.


SMrr or 2
SMirror 2
SMirror 3


*, 1










Table 4-5. Mean and standard deviation for pull-in angle and voltage from sets of mirrors on all
three arrays tested.
6PI (Deg) VPI (V)
Array # Area Mean St. Dev. Mean St. Dev.
1 D 14.27 0.85 62.27 1.75
2 E 13.93 0.62 68.81 2.57
2 D 13.15 0.88 67. 17 2.31
3 A 15.89 0.53 64.4 0.62
3 D 15.88 0.49 83.53 1.17

for arrays 1, 2, and 3 are 62.27 V, 67.99 V, and 73.96 V, respectively. It should be noted that for

array 3, there is a large difference in the pull-in voltage observed at two different locations on the

array. Compared to the predicted pull-in voltage of 71 V, these values are within 12 percent.

Measurements on the mirrors in these experiments were often conducted such that tests

were performed repeatedly on the same set of micromirrors before changing the location of data

collection, or switching to a different array. It was observed during the experiments that after a

device had sat idle without actuation voltage applied, the devices behaved differently when

actuated for the first time, as opposed to subsequent measurements taken on the same mirrors

directly afterward. The likely reason for this is a charging effect that occurs after the first

actuation of the device after it has sat idle for some time. Figure 4-26 shows this occurred for

array 2 when multiple sets of data were taken.

Figure 4-28 shows the data from all three devices together along with the model predicted

behavior of the device using the nominal model geometry of the micromirror design presented in

Section 3.4.3. The nominal geometry refers to the dimensions of the micromirror based on the

original design, not considering any fabrication-induced variations. It is clear that the nominal

model falls close to the middle of the widely scattered experimental results. Section 4.1.2

presented the results of the electromechanical model for 250 randomly varied sets of dimensional

and material parameters. These modeled variations are compared to the experimental data in









Figure 4-29, and it is evident that the experimental values fall mostly within the bounds of the


modeled variation results.


20
18 -Model
"16C Array 1
14 Array 2
Array 3
S12 -






2- I *

0o 10 20 30 40 50
Voltage (V)

Figure 4-28. Nominal model with experimental data.


"10 20 30


40 50 60
Voltage (V)


70 80 90 100


Figure 4-29. Model-predicted results from 100 simulations with parameters determined by
random Gaussian variations, shown with experimental data.


Model `csr
Array 1
Array 2
Array 3I









The model results were calculated based on known fabrication tolerances, but this alone

does not entirely explain the variations in the device performances. Fabrication variations are

known to occur across the wafer as well as from one process batch to the next, but it is not

definitively known if large variations occur locally such that they can have significant effect on

the micromirrors within each array. The experimental data presented above showed variations

between results for different areas in array 3. This indicates the presence of fabrication

variations across the array. However within each area on the array, the group of mirrors

exhibited relatively small differences in their results until their individual pull-in voltages. In

addition, the differences in the pull-in voltage could indicate that the mechanical stiffness used to

calculate the modeled value is different from the actual stiffness values of the micromirrors.

While it is not completely known the causes of these differences in performance, it is apparent

from the data that considerable performance variation can occur.

The effects of fabrication variation on the performance are best illustrated by the case of

Sacox3, which was shown in Figure 4-8 to have a significant effect on the 6-V profie, causing it

to deviate outward to the right of the other curves. This same behavior is seen again in the

Monte Carlo simulation results of Figure 4-18 in which those cases with large variations of

Sacox3 outside the 95 percent confidence interval. Upon comparison of the of the Monte Carlo

simulation results and the experimental results in Figure 4-29, it is seen that the experimental

results do not exhibit behavior that is consistent with that of very large Sacox3 variations. This

suggests that in the fabrication of these particular micromirror arrays, a large variation of the

Sacox3 layer thickness did not occur. Plots included in Appendix B study the effects of large

variations in the linewidth of MMPoly2, and the layer thickness of MMPoly 1 and Sacox4.

These plots did not indicate a clear connection between the Monte Carlo simulation results and










the effects of these three fabrication errors; therefore it is not possible to make a conclusion from

the experimental results as to the presence or magnitude of fabrication errors in these three

variables. To do so properly would require diagnostic data regarding the exact layer thicknesses

and linewidth errors collected for a given array of micromirrors, and this data is not available

here.

4.3 Chapter Summary

This chapter continues the electromechanical device modeling for the micromirrors that

was developed first in Chapter 3, and expands the analysis to include the effects of fabrication

tolerances on the performance of the micromirrors. By looking at the individual contributions of

particular parameters, it is evident that the layer thickness of Sacox3 has the largest effect on the

static displacement behavior for the micromirrors. The other parameters appear through

sensitivity analysis to also play less distinct roles when considered individually, but when

multiple parametric uncertainties are considered, the overall effect of the fabrication variations is

evident. Monte Carlo simulations are conducted to examine the effects of parametric

uncertainties, and this reveals the full extent to which the precision of the micromachining

process can dictate performance.

The micromirror modeling is then compared to static experimental characterization data

that was collected using an optical profiler that is capable of making non-contact displacement

measurements. The results are reported for some individual micromirrors tested at Sandia

National Labs, and then additional results are given for the micromirror arrays tested on a

separate measurement system at Veeco, Inc. From these measurements, the static equilibrium

behaviors of the micromirrors is determined, as well as the pull-in angle, and pull-in voltage.

Taking measurements at different location on three different micromirror arrays begins to show

that there can be considerable variation in the performance. When these experimental results are










compared to the uncertainty modeling results, it reinforces the notion that this variation can be

the result of microfabrication errors. While the manufacturers of the optical profiling system do

claim a very high level of accuracy for measurements made using their equipment, recent studies

of the machine conducted by Mattson show that the measurements can be susceptible to larger

errors [112]. It is not known if the measurements taken for the micromirror devices are in fact

showing larger deviations in the data due to this kind of measurement error. This type of study

would be valuable for future work.









CHAPTER 5
DYNAMIC MODEL AND HYSTERESIS STUDY

Previous results only considered the static performance of the micromirrors after they

have reached a steady-state value. Here, the dynamics of the system are taken into consideration

in order to examine the effects of natural frequency and damping on the time response of the

system. Modal analysis and dynamic characterization are performed to determine the natural

frequencies of the micromirror and the mode shapes. It becomes clear that parametric

uncertainty in the micromirrors also affects the dynamic performance of these mirrors. Most

notably, the effects of the uncertainty on the behavior of the electrostatic instability may be seen.

In addition to modeling the pull-in and hysteresis behaviors of the open-loop system, a case

study is presented for a progressive-linkage that can be applied to alter the stiffness of the system

to avoid these undesirable behaviors.

5.1 Dynamic Model and Resonant Frequency Determination

It is convenient to rewrite the model dynamics in Equation 3-16 in terms of natural

frequency, an,, and the damping ratio, i.


m,, = (5 -1)


g = (5 -2)
2J,

Written in state-space form, the system is described as follows,

x, =
x, = 0 (5-3)

O+ 1 dC

1:)=[-2 S~i~,IIX2J ~Lc1dry











From the linearized dynamic model discussed in Section 3.4.4 using a first order polynomial


approximation for the capacitance function, the derivative of the capacitance is a constant.

Therefore, the natural frequency of the lumped-parameter model determined from Equation 5-1

is found to be approximately 188 k
As stated previously in Section 3.4.5, the squeeze-film damping coefficient is difficult to


predict analytically for this model, and based on values from similar devices in [71], the damping

ratio is assumed to be approximately 0.3. The damping ratio has a significant effect on the open-


loop performance of the system, as seen in Figure 5-1 for damping ratios ranging from 0.1 to 1.


1 d = 01
d=02
10~ d P= 03
d=04
:L d=05






d=08
2~ 1= 09
a= I

0 1 2 3 4 5
Time (sec) x 10


Figure 5-1. Open-loop nonlinear plant response to a step input of 7 degrees for different damping
ratios.

5.1.1 Modal Analysis

In addition to using the lumped parameter model to estimate the natural frequency of the

micromirror devices, modal analysis is done to determine the natural frequencies and the mode


shapes. The analysis is performed for an undamped system, and the equation of motion

expressed in matrix notation is


[M] {ii) + [K] {uf = {0) (5-4)









where M~ and K are the mass and stiffness matrices, respectively, and u is the displacement

vector. Free harmonic vibrations of the structure are of the form

{u) = {#}z cosmat (5-5)
where (), is 'the eigenvector representing 'the ith natural frequency, we~ is the ith natural frequency


(rad/s), and t is time. Substituting Equation 5-5 into 5-4 yields

(-m~ [M]+ [K]) { ), = {0) (5-6)

Ignoring the trivial solution to Equation 5-6, which is {#}z = {0) then the following expression

must be true.

I[K]-m? [M} = (5-7)

Equations 5-6 and 5-7 form the eigenvalue problem, and the solutions are the natural

frequencies#, and the eigenvectors { ),

The participation factor is related to the eigenvector, and it identifies the amount each

mode contributes to the total response in a particular direction [113]. A small participation factor

means that an excitation in that direction will not excite the mode in that direction. A large

participation factor indicates that the mode can be excited by motion in that direction. The

participation factor can be used to determine the direction of motion in each mode that dominates

the response. As defined in reference [99], the participation factor for the ith mode, y is given

by

y = (#} [M] {D) (5-8)

The vector D describes the excitation direction and is of the form

{D)= [Tl{e) (5-9)









where {e] are the six- possible- unit' vectors. {D} is furthe dsrie in--;- term of-- -Cthe individual

excitations, D ", for DOF j in direction a. The directions of excitation, a, can be either X, Y, Z,

or rotations about these axes, ROTX, ROTY, ROTZ.


(D) = D D(D(...] (5-10)

The matrix [T] is

1 0 0 0 (Z Zo) -(Y 0)
0 1 0 -(Z Zo ) 0 (X Xo)
0 0 1 (Y Yo) -(X Xo) 0
[T]= (5-11)
0 00 1 0 0
0 00 0 1 0
0 00 0 0 1

in which X, Y, and Z represent the global Cartesian coordinates, and Xo, Yo, and Zo are the

global Cartesian coordinates of a point about which the rotation are done.

Modal analysis is performed for the micromirrors using the ANSYS finite element analysis

software. The solid model of the structure is shown in Figure 5-2 and consists of the mechanical

spring, the mirror surface and the moving comb fingers. The fixed comb finger electrodes may

be ignored as they are not part of the moving structure. The structure is anchored to ground in all

degrees-of-freedom at the base of the mechanical springs. This solid model is meshed with

solid92 elements which have 3DOF at each node. The modal analysis is performed using the

Block Lanczos method which is appropriate for large symmetric eigenvalue problems [99]. The

results from the analysis give the first ten natural frequencies, as well as modal participation

factors, listed in Table 5-1. The ratio of each participation factor to the largest participation

factor value for a given direction is also listed in Table 5-1, in which a ratio of one indicates the

mode that contributes the most to the response in that direction.









The mass calculated from the modal analysis is 2.44 x10-"1 kg. The mass result that was

reported in Chapter 4 based on the volume of the moving geometry was 2.34 x10-"1 kg, which

matches the ANSYS calculated result within 4 percent. The difference in these values arises

from the inclusion of additional components in the ANSYS model that are not included in the

volume calculation done in previously. These additional components include the mechanical

spring and its supports.

The results of this analysis indicate that the first mode of vibration for the micromirror

structure occurs at 84.74 k
in Section 5.1 where it was assumed that the micromirror acts only in one degree-of-freedom,

rotating about the X-axis (ROTX). It is likewise assumed that the first natural frequency will

occur in this rotational direction and be given by Equation 5-1. The results from the modal

analysis for the first mode at 84.74 kHz do in fact show that the dominant direction of the

response at this frequency is in the ROTX direction. This is determined by comparing the values

of the participation factors for each direction for this mode and it validates the one-degree-of-

freedom assumption for the model in Equation 5-3. The largest participation factor is 6.5E-05

for the ROTX direction, and this is an order of magnitude larger than the next largest

participation factor which occurs in the Z-direction. While it is verified that the primary motion

for the first resonant frequency occurs in the ROTX direction, the modal analysis results reveal

that the resonant motion is more complex than one degree-of-freedom motion and in fact, the

first resonant frequency excites motion in both the X-axis (ROTX) and the Z-direction. The

motion that occurs in the Z-direction will affect the compliance of the system, which will result

in a different natural frequency than that predicted using Equation 5-1, which assumes one

degree-of-freedom motion about the X-axis only. The spring stiffness results presented in Table












L;


I

spring


A anchor B


Figure 5-2. Solid model created for modal analysis. A) View of the top and back. B) View of
the bottom showing the comb fingers.

Table 5-1. Modal analysis results for first 10 modes and their natural frequencies, and the
participation factors and ratios for each direction.


X-Direction
Participation
Factor
2.1183E-10
9.5920E-07
-1.3215E-07
8.3800E-07
8.0827E-10
4.6105E-08
-2.4312E-06
3.1366E-07
3.0772E-07
-2.9854E-07
ROTX-
Direction
Participation
Factor
6.4831E-05
-1.9988E-08
9.4879E-06
1.5072E-06
-6.3384E-06
1.3954E-07
2.7564E-08
5.5281E-09
4.2753E-07
4.5237E-08


Y-Direction
Participation
Factor
2.7197E-06
8.8566E-10
3.5837E-06
5.5914E-07
1.7035E-06
-2.2081E-08
-3.6331E-09
5.4173E-08
2.2398E-08
1.9532E-07
ROTY-
Direction
Participation
Factor
-8.7467E-08
-2.2171E-04
-9.8134E-07
7.3266E-06
-8.7374E-08
9.0970E-08
-3.4525E-06
-2.5788E-07
1.0185E-07
2.3618E-08


Z-Direction
Participation
Factor
3.5955E-06
-4.3162E-09
-3.0841E-06
-4.8219E-07
9.7970E-08
7.5056E-08
2.7242E-09
-3.3755E-09
1.1384E-08
-1.2589E-08
ROTZ-
Direction
Participation
Factor
4.3414E-08
-1.5238E-05
3.4382E-05
-2.2001E-04
-2.8728E-07
-1.8579E-08
-3.1098E-06
8.3555E-06
3.2656E-07
-1.8531E-06


Mode
1
2
3
4
5
6
7
8
9
10




Mode
1
2
3
4
5
6
7
8
9
10


Freq. (Hz)
84736.51
120372.52
162970.10
164493.10
391530.45
1208580.00
1310412.38
1610211.37
1696417.45
1853628.28




Freq. (Hz)
84736.51
120372.52
162970.10
164493.10
391530.45
1208580.00
1310412.38
1610211.37
1696417.45
1853628.28


Ratio
0.000087
0.394546
0.054358
0.344691
0.000332
0.018964
1.000000
0.129016
0.126574
0.127970




Ratio
1.000000
0.000308
0.146350
0.232480
0.977690
0.002152
0.000425
0.000085
0.006595
0.000698


Ratio
0.758918
0.000247
1.000000
0.156023
0.475336
0.006161
0.001014
0.015116
0.006250
0.054501




Ratio
0.000395
1.000000
0.004426
0.033047
0.000394
0.000410
0.015572
0.001163
0.000459
0.000107


Ratio
1.000000
0.001200
0.857751
0.134109
0.252526
0.020875
0.000758
0.000939
0.003166
0.003501




Ratio
0.000197
0.069261
0.156274
1.000000
0.001306
0.000084
0.014135
0.037978
0.001484
0.008423


3-3 previously show that the spring is very compliant in the Z-direction with a stiffness of 7.94


pN/m. This additional compliance will lower the overall spring constant for the mode and result

in a lower resonant frequency that when only the rotational motion is considered. The evidence


J


cornb fingers










of motion in additional degrees of freedom at resonance does not however invalidate the

assumption that the mirror will rotate about the X-axis for excitations that occur below the

resonant frequency. Furthermore, the electrostatic force that is applied to the micromirror is

always an attractive force, drawing the moving electrode down toward the fixed electrode. Thus,

if resonance is avoided, smooth rotational motion in one degree-of-freedom is still accomplished.

This does, however, show the limitations of the 1DOF model assumption, which limits the

analysis to only low frequency responses where resonant behavior may be avoided. Table 5-1

also includes the resonant frequencies and their participation factors for modes 2 through 10. It

is noticed that several of the modes have motion that acts in more than one direction.

5.1.2 Dynamic Characterization

In addition to the lumped parameter estimation and the modal analysis results to determine

the natural frequencies of the micromirrors, some experimental data was obtained using a Laser

Doppler Vibrometer (LDV), courtesy of the Integrated Microsystems Group at the University of

Florida. This device measures the velocity of a point on a device as it is excited over a range of

frequencies. The excitation signal can be a swept sine wave, or chirp signal, or it can also be

white noise, which will excite the device at all frequencies in the given range. Due to limited

signal generation capabilities and time constraints, the excitation signal chosen for this

experiment was an acoustic impulse, generated by firing a small cap-gun, which produces a loud

noise. This effectively generates a white noise signal that can excite the microdevice, and the

resulting velocity of the device is recorded by the LDV. Generating the pulse in this manner is

simple and does not require signal generation; however the acoustic impulse is not guaranteed to

be the same signal each time it is produced. This experiment was performed five times on each

of the three micromirror arrays. Figure 5-3 shows an example of the time response of the

micromirrors to the acoustic impulse taken for device 2, trial 1.











0.03

0.02-

., 0.01 -



-0.01-

-0.02-

-0.03
0 0.001 0.002 0.003 0.004 0.005 0.006
Time [s]


Figure 5-3. Time series data of the micromirror response to an acoustic impulse taken with a
laser doppler vibrometer. This is the response of device 2, trial 1.

The time series data can be examined in the frequency domain by a Fast Fourier Transform


(FFT) of the velocity of the micromirror surface. Dominant spikes in the FFT indicate a resonant

frequency for the device. Figure 5-4 shows examples of the FFT results for several of the tests.

It is clear from these results that there is considerable noise occurring in the measurements, the

source of which has not been identified. As such, it can make it more difficult to identify which


peaks are in fact resonant frequencies. The complete FFT results for each LDV measurement are

given in Appendix C. In each of the measurements, there appear consistently to be three results

that stand out. All of the measurements had a large resonant peak that occurred in the range of

40 k

signal was given, and is therefore considered to be result of noise in the environment. This noise

could be caused by another piece of laboratory equipment or system in the area, and

unfortunately the cause was never identified. It is assumed that this frequency is not in fact a

resonant behavior. The results are summarized for here in Table 5-2 for the two dominant


resonant frequencies of each test, excluding the lower frequency 40 k

appearance of resonance that occurs throughout the LDV measurements occurs in the range


























































































.Results from the LDV experiment showing resonant peaks.
Device 2, trial 1. C) Device 3, trial 4.


8.OE-04

7.OE-04

6.OE-04

5.OE-04

4.OE-04

3.OE-04

2.OE-04

1.OE-04

0.OE+OO


100 200 300 400

Frequency [kHz]


500


7.OE-04

6.OE-04

5.OE-04

4.OE-04

3.OE-04

2.OE-04

1.OE-04

0.OE+OO


O 100 200 300 400
Frequency [kHz]


500


3.5E-04


3.OE-04


2.5E-04


2.OE-04


1.5E-04

1.OE-04


5.OE-05


0.OE+OO


O 100 200 300

Frequency [kHz]


400 500





A) Device 1, trial 4. B)


C


Figure 5-4










of 80 k
results obtained from the FEA modal analysis in which the first natural frequency was found to

occur at 84 k
for devices 1 and 3, large responses occurring in the range of 180 k
k
Table 5-2. The first three natural frequencies determined from the LDV experiment. Results
from the linear model, using Equation 5-1, and the modal FEA are included for
comparison.
Frequency (k Device Trial 1st 2nd
1 81.41 186.88
2 81.41 187.81
1 3 81.71 187.03
4 82.66 186.88
5 81.56 187.19
1 82.19 140.78
2 85.63 139.53
2 3 85.31 136.56
4 92.02 137.03
5 85.31 136.10
1 80.91 182.34
2 90.31 183.91
3 3 81.41 180.63
4 83.13 183.28
5 83.28 183.28
Model
Eq. 5-1 182 -
Modal
84.74 120.37
Analysis

5.2 Open-Loop Step Response

The open-loop response of the system is determined by the actuation voltage signal that is

given to the micromirrors. For open-loop operation, it is necessary to determine a calibration

relationship between the desired angular position and the actuation voltage needed to achieve

such position. This relationship is often determined experimentally. If variations in the devices

due to fabrication tolerances or other system disturbances are present, then the calibration must










be performed for each separate micromirror device to ensure the correct calibration is obtained.

This approach of individually calibrating each micromirror device is not practical or efficient.

The effects of parametric uncertainty on the device performance using a given calibration are

examined for the step response. The effects of pull-in and hysteresis are also examined.

5.2.1 Effects of Parametric Uncertainty on Step Response

To illustrate the effects of parametric uncertainty on the system, the open-loop response of

the plant model is considered using different values of stiffness, ks,. Figure 5-5 shows the

response to a step input command of 7 degrees (0. 12 radians) for the nominal stiffness value, and

for variations of 10%. To further illustrate this concept, all of the parameters in the system

described in Equation 5-3 are subj ect to parametric variation, including the mass moment of

inertia, J, the damping, b, the spring stiffness, ks,, and the electrostatic torque, Te. If each of these

parameters is allowed to vary by 10% from the nominal value, there are a very large number of

possible plants to consider. It is assumed that calibration is performed on the device for the

nominal parameter values. Figure 5-6 shows the open-loop plant responses of the nonlinear

plant model to a step input of 7 degrees of the system model for 50 randomly generated sets of

parameters J, b, ks,, and Te that are allowed to vary by 10% of their nominal values. It is clear

that with the presence of uncertainties, a step input to the open-loop plant will result in steady-

state error in the response. In order to correct for this in open-loop operation, the system must be

carefully recalibrated for each device to ensure the proper response is achieved.

5.2.2 Effects of Pull-in and Hysteresis on Open-Loop Response

Electrostatic instability and hysteresis can also greatly affect the system response in open-

loop operation. Recall from the discussion in Section 3.3, that pull-in occurs when the

electrostatic force generated by the actuator exceeds the mechanical restoring force of the

structure, causing the mirror to be pulled down to the substrate at its maximum displacement.













8






e4


2


SCommand
-Nominal
---+10% k


-10% k
m


0 0.5 1 1.5
Time (sec) x 105
Figure 5-5. Open-loop response to a step input of 7 degrees for the nonlinear plant dynamics and
variations in spring stiffness, km.


12


10~


4/


0 0.5 1 1.5 2
Time (sec) x 10d
Figure 5-6. Open-loop nonlinear plant response to a step input of 7 degrees for 50 random
parameter variations.









The mirror will remain in this position until the actuating voltage has been reduced below the

holding-voltage, causing hysteresis. The effects of pull-in and hysteresis for the static response

are investigated in Chapters 3 and 4, but there are dynamic effects that can affect pull-in as well.

It is known that pull-in is affected by resonance, and it is therefore assumed that the

micromirrors operate at frequencies below resonance [40]. If the system is driven dynamicallyby

a voltage that is greater than the holding voltage and less than the pull-in voltage, it is still

possible for the inertial effects to cause the mirror to experience pull-in and remain pulled in

until the applied voltage is reduced below the holding voltage. In order to incorporate this effect

into the dynamic model, the system response is subj ected to a set of discontinuous, piecewise

defined behaviors. When the angle, 8, becomes greater than or equal to the pull-in angle, On, the

system response sets theta equal to the final pull-in position, eF. After pull-in has occurred, the

system response remains pulled-in until the voltage drops below the holding voltage, Va. The

system then returns to the released position, Is. This response is shown in Figure 5-7 for

sinusoidal commands of amplitudes of 14.90, 16.60, and 17.20. The corresponding voltage

command is also shown in the figure. Again, for commands beyond the pull-in angle of 16.50,

the response shows pull-in and remains in this state until the actuation voltage is reduced below

the holding voltage of 68 V.

In the case of a step command, overshoot in the system response becomes very critical

when driving the device to a position that is near the pull-in point. In the case of large overshoot

in the response, the device will pull-in and will not be released as the voltage command for a step

input is constant. Figure 5-8 shows the open-loop step response of the system for commands of

120, 140, and 170. It is expected that the command input of 170 will result in pull-in as it is

greater than the pull-in angle. However in this case, overshoot in the response for a step













80


50
10-

S40
30
5 20
******* Command
Result 10

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
A Time [s] Time [s] B


Figure 5-7. Open-loop responses to a sinusoidal input showing hysteresis. A) Results of angle of
rotation over time. B) Voltage signals that correspond to the command inputs.


20



15





******* Command
5C -I Result




0 0.5 1 1.5 2
Time [s] x 10~


Figure 5-8. Open-loop responses to a step command showing overshoot that result in pull-in.

command of 140 also results in pull-in of the response as the overshoot causes the device to

move beyond the pull-in point, and the actuation voltage applied is not less than the holding

voltage required to release it. This is another example of the effects of hysteresis on the response

of the system where the inertial effects plays a role, referred to as dynamic pull-in [1 11].


Dynamic pull-in can result in cases where the velocity of the actuator is high as it approaches the

pull-in point. This can be caused in the case of applying instantaneous actuation voltages, and it











can cause the actuator to pull-in at a lower voltage than the static pull-in voltage. This dynamic


effect is difficult to model, and is affected by the damping of the system. For zero damping in a


parallel plate system, the dynamic pull-in can occur at an 8% lower voltage than the static pull-in

voltage; however the presence of damping in the system decreases this effect.


5.2.3 Continuous Characterization of Micromirror Arrays

The optical profiler measurement system described in Section 4.2.1 used to collect static


performance data was also used to apply continuous voltage as a partial sine wave. The voltage

was increased and decreased without resetting to zero in between measurements, which allows


the effect of hysteresis to be studied. This is done by applying a voltage signal such as that


shown in Figure 5-9 with amplitudes ranging from 44 volts to 85 volts following a partial sine


wave, with measurements taken at every ten degrees of phase. The results for a set of four


mirrors from array 3 are shown as a function of phase in Figure 5-9, and as a function of voltage


in Figure 5-10. In this instance, only two of the micromirrors, 1 and 3, experienced pull-in and


hysteresis, while the other two, 2 and 4, did not.

90 20
-tMirror 1
85 18 Mirror 2
~Mirror3
80 *" 16Mirror 4

75 14
S70 -12



55 6
50 ..~ 4-
45 2 -
40 0
0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180
A Phase (Deg) Phase (Deg) B


Figure 5-9. Results from dynamic study showing pull-in and hysteresis. A) Actuation signal
applied for dynamic study. B) Results from applying the actuation signal.











20 20
18- 18-
16- 16-
14 -1 14-

9i12 -1
E10 0

/ -Mirror 2Up
6 -- Mirror 1 Up 6 /- -- Mirror 2 Down
4 -- Mirror 1 Down 4 -1 Mirror 4 Up
-- -.-~~~~ Mirror 3 Up irr4Dw
2 ~-, Mirror 3 Down2-
0 0
40 45 50 55 60 65 70 75 80 85 90 40 45 50 55 60 65 70 75 80 85 90
A Voltage (v Voltae M B


Figure 5-10. Results showing the hysteretic behavior of the micromirrors. A) Mirrors 1 and 3
show pull-in and hysteresis. B) Mirrors 2 and 4 do not have pull-in.

5.3 Hysteresis Case Study: Progressive-Linkage

As discussed in the literature review Section 2.3, there are ways that researchers have used


nonlinear flexure designs to mitigate electrostatic pull-in and hysteresis. One such nonlinear


flexure design is presented here, called a progressive-linkage [57], [58]. The design and function


of the linkage is presented and it is analyzed to show how it affects the electrostatic instability


and hysteresis in open-loop operation. The results presented are only theoretical and have not


been fully realized in fabrication.


5.3.1 Progressive-Linkage Design

Electrostatic instability occurs when the electrostatic force becomes too great for the


mechanical spring to handle. If the characteristics of the mechanical restoring force can be


altered such that this pull-in never occurs, then the micromirror device could operate


continuously over its full range of motion, from 0 to 19 degrees for the micromirror designs of


studied in this dissertation. This done at the cost of increased actuation voltages. The following


analysis proposes a new design for the spring that has a nonlinear restoring force such that the

stiffness characteristics increase significantly as the spring is rotated.









The analysis for this design is based on an equivalent four-bar model as depicted in Figure

5-11. The geometric relationships between the links are also shown in Figure 5-11. The

kinematics of the mechanism can be denoted by the following vector sum where the vectors

denote the position and orientation of each side of the mechanism shown in Figure 5-1 1.

rz = ro + rl '3 (5-12)

Since the four-bar mechanism is a one-degree-of-freedom device, the angles 82 and 83 can

be described as a function of 03 That is, the length and orientation of each side can be used to

determine the relationships of the angles 82 and 83. By using the y and : components of the

vectorrs, an expression for 82 is given as



,= tan (5-13)


In order to determine the angle Os, begin with the relationship

r,2 = rd2 + 32 2rzrd COsY3 (5-14)

This yields an expression for y,.


73 co (5-15)



An expression for ed is found from


Od = tan 'ir4 + r~ (5-16)


The angle 83 is given as

03 = Od + 3 (5-17)

To realize this design scheme in a surface micromachined device, the design will be

subject to the limits and constraints of the micromachining process. One of the challenges













1- .. ~.E.~-
i; .. --- ,--- '-r- '-'
~
.r
r
H
~ I
I',

... i
I '
: i
'" 'I-. ~-
II

[i1~7: II. 1


I'
z'


.... I.:i
i .

I' I
I' ,I
I '

,L ~
I I
,,

A~ Po


Figure 5-11i. Diagram of four-bar mechanism for progressive linkage analysis. A) The vectors
and geometry for kinematic analysis. B) The springs and angles for force and moment
analysis.

to realizing this mechanism in a surface micromachining process is to find suitable joint


configurations that will allow for the creation of a four-bar mechanism. For the sake of this


discussion it is assumed that this 2-D representation of the four-bar linkage is created using a


series of thin beams, kinematically spaced by r, (i = 0, 1,2,3), each j oint may be considered as a


beam in torsion that provides a restoring force to the system. Seen in Figure 5-12, a beam of


length L with a rectangular cross-section of dimensions w x t is used to model the stiffness at the


joints. The restoring torque on the member can be calculated by


T K ss,-0sz) (5-18)


E
is the shear modulus, (0,,, -0s,,0) is the change
2(1 + v)


for each joint i = 0, 1, 2, 3, where G


in the rotation at the j oint from its unloaded position (the free length configuration of the


torsional spring), and K, is given as


(5-19)


tw3 16 w w
K = 3.3 6- 1 2tw )
16 3 t 1t









when t > w. For the case of t < w, the expression is


K 3.36--t Ir~t 1 (5-20)













Figure 5-12. Cantilever beam with cross-section w x t, and length L.

The resulting static force and moment equations can be determined from the free body

diagrams in Figure 5-13.

F~lbarl = F, + F2 =0(-1

SFar2 = 2 F3 = 0 (5-22)

Fbar3r = -F3 + Fo; = 0 (5-23)

SnJomtl = -s T + TS2 + rxF2 =0 (5-24)


C yln't2 T2 T3-gZx F3 =0 (5-25)

CMio'nt3 0 3s,-T4rx Fo =0 (5-26)

The relationships above combine to determine the torque output for a progressive linkage

design. The dimension of the mechanism that is the easiest to change in the design is the

horizontal distance separating the anchor points of the device, referred to above as0 r. Figure 5-

14 shows the output of the progressive linkage for different values ofro. For a value of ro less

than 10 Cpm, the structure will become very stiff before the mirror reaches its maximum angle and










it will not be able to fully rotate. This is seen for values of ro equal to 4, 6, and 8 pm. As the


value of ro is increased, the structure becomes more compliant. Figure 5-15 shows plots of the



~Ijoint 2


-TS


F T,
joint 1 F~


Figure 5-13. Free body diagrams for each member of the linkage.


1000


400 8
0 ro=l

200 '0 =6
ro ro 1102

0 2 4 6 8 10 12 14 16 18 20
Theta [degrees]

Figure 5-14. Progressive-linkage behavior for different values of ro in pm.









behavior of the progressive linkage for r, equal to 9 Cpm overlaying the electrostatic torque

curves from Figure 3-21. Tables 5-3 and 5-4 give the link length and joint dimensions used for

this progressive-linkage design. The Young's Modulus is assumed to be 164.3 GPa and the

Poisson's ratio is 0.22. The linear restoring force from Figure 3-21 is also included for

comparison. The requirements for pull-in to occur are that the electrostatic and mechanical

torques be equal in magnitude and slope. The progressive linkage creates a stiffness profile that

eliminates the occurrence of the second condition such that the stiffness curve does not at any

point run tangent to the electrostatic torque curves and therefore does not exhibit pull-in

behavior. The static 6-V profile for a device using a progressive linkage is shown in Figure 5-

16. The cost of this extended actuation range is that larger voltages are required.


500



400 Tri Pro~gressive
-T Linear

300 m

20


1200 -'B

0 2 0 12 1 6 1
Tht [dgres
Fiur 51. roresvelikaeoupu frroeqalt 9pmaln wthth letrsttc oru
cuve ndte ina rsorngtrqe










Table 5-3. Link length dimensions used for progressive-linkage design.
Link Length (Clm)
ro 9.000
rl 8.625
r2 9.953
r3 4.375

Table 5-4. Joint dimensions used for progressive-linkage design.
Dimension (Clm)
Joint T w L
0 2.50 1.00 66
1 2.50 1.00 66
2 2.25 3.00 111
3 2.25 3.50 111


20



15













0 20 40 60 80 1 00 '120 '140
Voltage (V)

Figure 5-16. Static e-V relationship for micromirror with a progressive-linkage.

Recall from Section 3.4.5 that a bifurcation analysis may be used to examine the

electrostatic pull-in behavior for a system with a nonlinear spring constant. The equations for the

analysis now include the progressive spring constant that is a function of the rotation angle,

expressed as k;,(0). The expression for the fixed point solutions is now










1 b 1 b 12 8 1(,)k,
Ai ---+- +4
S2 J 2 JI J Sx, J
j =1, 2 (5-27)
i=1, 2, 3

Applying this analysis to the device using the progressive linkage yields the bifurcation

diagram shown in Figure 5-17. It is clear from this analysis that the device is able to reach

angles up to 18 degrees using higher voltages of up to 130 V. Figure 5-18 shows how the

bifurcation plot will change as the progressive stiffness profile increases or decreases by a factor

of 2.

5.3.2 Open-Loop Response Using a Progressive-Linkage

Because this device does not experience pull-in, it is assumed that there is no hysteresis in

the response. Therefore the system will be able to respond to actuation signals such as a sine

wave or step command without having pull-in. The system open-loop response to sinusoidal

inputs is shown in Figure 5-19. Figure 5-20 shows the step response to inputs of 120, 14.30,

17.10, and 180. Unlike the system with a linear spring force, this device is able to achieve

positions beyond the pull-in angle. As stated before, the actuation voltages for this device with a

progressive linkage will be higher than for the results using a linear spring in Figures 5-7and 5-8.

5.3.3 Parametric Sensitivity of the Progressive-Linkage

It was shown in Chapter 4 how the mechanical spring that consisted of only one set of

beams was sensitive to fabrication tolerances. It thus seems logical to assume that by adding

complexity to the spring design in terms of the progressive-linkage will add to the effects of this

sensitivity. The following discussion will examine the effects of fabrication tolerances on the

progressive-linkage design. The methods of analysis will follow that of Section 4. 1, in which

first the effects of changing only one parameter at a time are examined. Then, Monte Carlo



















20




-5

-'10
0 20 40 60 80 100 120
Voltage (V)


Figure 5-17. Bifurcation diagram for micromirror using a progressive-linkage to avoid pull-in
behavior.


30

25

20

~i15

'10





-5


0 25 50 75


100 125 150 175 200 225
Voltage (V)


Figure 5-18. Bifurcation diagram for the micromirrors using a progressive-linkage to avoid pull-
in behavior for different values of mechanical stiffness.


1Ro











-----Command
- Result


Time [s]


Time [s]


Figure 5-19. Open-loop responses to a sinusoidal input for the device using a progressive-
linkage. A) Results of angle of rotation over time. B) Voltage signals that correspond
to the command inputs.






15







rr 0



0 1 2 3 4 5
Time [s] x 105

Figure 5-20. Open-loop response to a step input for device using a progressive-linkage.

simulations are done to look at the effects of randomly varying all of the uncertain parameters.

The uncertain parameters are assumed to vary in a Gaussian distribution, identified by a mean

and standard deviation given from known fabrication tolerances.

Table 5-3 and 5-4 gave numbers for the progressive-linkage design variables that are used

to evaluate the four bar linkage model. The torsional spring constants, calculated by Equation 5-









1 1, depend on the values of w and t from Table 5-4, as well as the Young' s modulus, E. The

dimensions of the j points are subj ect to the fabrication tolerances of the surface micromachining

process. Assume for this given design, that the joints 0, 1, 2, and 3 are fabricated as beams in the

layers MMPolyl1, MMPolyl1, MMPoly4, and MMPoly3, respectively. This means that each joint

will be subj ect to the errors in layer thickness and linewidth that are defined from the fabrication

tolerances given for the manufacturing process. Table 5-5 lists this information, including the

nominal joint dimensions, t and w, the respective fabrication layer used to make each joint, and

the associated fabrication errors given in terms of mean and standard deviation. For example,

joints 0 and 1 are to be fabricated in layer MMPolyl1, making their dimensions prone to variation

in the thickness of MMPoly 1. All of the layers are subject to variation in Young' s Modulus,

previously stated to be 164.3 & 3.2 GPa. Other errors in the fabrication can occur that will affect

the design of the four-bar type linkage in terms of the link lengths, however these are neglected

here, and only the errors associated with the joint stiffness are being considered in this analysis.

Table 5-5. Uncertainties in the j oint dimensions for a proposed progressive-linkage design.
Dimension Uncertainty of Dimension
(Clm) Fabrication Mean & St. Dev.(Clm)
Joint T w Layer t (thickness) w (linewidth)
0 2.50 1.00 MMPoly 1 1.02 & 0.0023 --
1 2.50 1.00 MMPoly 1 1.02 & 0.0023 --
2 2.25 3.00 MMPoly4 2.29 & 0.0063 -0.07 & 0.05
3 2.25 3.50 MMPoly3 2.36 & 0.0099 -0.24 & 0.05

The sensitivity of the progressive-linkage design is examined when only one variable is

altered at a time. Figure 5-21 shows the results from this analysis in terms of the mechanical

torque as a function of rotation angle as each variable is changed by one standard deviation from

the mean. The two variables that have the greatest effect on the stiffness profile of the nonlinear

spring are the thickness of layer MMPoly3, and the linewidth of MMPoly3. In order to examine

the effects of changing multiple variables at the same time, Monte Carlo simulation is done in









the same fashion as in Section 4.1.2. Each variable is randomly varied according to a Gaussian

distribution defined by the mean and standard deviation of that variable. For this progressive-

linkage design, 50 simulations are performed, and the results in terms of the torque-theta profie

are shown in Figure 5-22. It is striking to see the very large effects of these very small

parametric perturbations, and from a qualitative point-of-view, it becomes evident that the

current proposed design will be very sensitive to the fabrication. In a case such as this, design

optimization is recommended to Eind a design for the linkage that is less sensitive to these errors.

This is suggested for future work to explore alternative j oint designs and variations of the

progressive linkage that will make it less prone to parametric uncertainties.

5.3.4 Progressive-Linkage Prototype

Despite the limitations of the design that are revealed through the parametric analysis in

Section 5.3.3, a prototype of the micromirror with the progressive-linkage has been developed.

This design, illustrated in Figure 5-23, was developed and fabricated in the SUMMiT V

micromachining process with j oint configurations consisting of a series of thin beams as

modeled in Section 5.3.1. Figure 5-24 shows a micrograph image of the progressive-linkage and

the micromirror device. Because of the planar fabrication requirements of surface

micromachining, the diagonal top member of the four-bar device, r,, can be acquired via a

kinematically equivalent L-shaped beam, shown in Figure 5-24. It should be noted that for an

array of micromirrors that required close spacing, this is perhaps not an ideal design

implementation as the linkage itself occupies a significant amount of space behind the

micromirror. A more compact implementation that could be located underneath the mirror or to

the side would be preferred. Due to time-constraints with the available fabrication run, in-depth

analysis of the device performance was not conducted before the Einal design was submitted for

























































Figure 5-22. Fifty Monte Carlo simulation results for varying the joint fabrication parameters for
the progressive-linkage design. The nominal spring value is shown for comparison.


- Nominal


....... E
thickness MMPoly1 .
-thickness MMPoly3 l
******* thickness MMRPoly4
----- linewidth MMPo3ly3
---** inewidth MMPoly4F /


600


500


400


300


200


100


2 4 68 '10 '12
Theta (deg)


14 1618


Figure 5-21. Results of parametric analysis for individual errors in j oint fabrication of the
progressive-linkage.


500


400


300-


200


100-


12 14 16 18


-1 00 L
0


2 4 6 8 10
Theta (deg)










fabrication. This is an unfortunate but sometimes common occurrence encountered by MEMS-

designers who may be restricted by the time-tables of foundry services and available project

funding. It also gives a good example of the consequences of incomplete a priori analysis.




Proglressive-linkage
Mirror Surface









L-slasx~ca-quivalent link C






Figure 5-23. Schematic drawing of the prototype progressive linkage spring. A) Progressive
linkage spring. B) Spring attached to the micromirror. C) Drawing of L-shaped
equivalent beam.


Figure 5-24. Micrograph of the prototype micromirror with a progressive linkage spring.

As previously stated, the proposed linkage design prototype was fabricated, and

subsequently tested using the WYKO NT 1100 optical profiler located at Sandia National

Laboratories in Albuquerque, NM. This is the same optical profiler discussed in Section 4.2.2.

The results of this static experimentation are shown in Figure 5-25 as the rotation angle, Bi, that

was measured for an applied actuation voltage. It is clear that the voltages required to actuate the









micromirror with the progressive-linkage are higher than for the micromirror without the

progressive linkage. It is not clear however if the progressive device was able to accomplish the

nonlinear spring behavior desired. After the device was rotated to approximately 14 degrees, all

subsequent measurements failed to record proper data files. This issue was first discussed in

Section 4.2, where for high angles of rotation, the measurement machine routinely had difficulty

taking measurements. Thus, it is inconclusive to state whether the pull-in point of the

micromirrors was in fact delayed by the spring design or not. It is suspected however that the

linkage did not perform its intended function, and the data beyond 14 degrees of rotation did not

record because the mirror had in fact pulled in.

In order to investigate the device performance to try to identify if the proposed

progressive-linkage design implemented is working properly, the structure of the progressive-

linkage has been examined using FEA. Just as with the previous mechanical spring analysis of

Section 3.4, the progressive-linkage is modeled in ANSYS using beaml89 elements [99]. As the

structure is displaced about the X-axis, is soon becomes clear from looking at the resulting

displacement of the linkage, that the design is not operating as the intended four-bar model, but

is instead deflecting in the positive Z-direction, out-of-plane. This Z-direction deflection

prevents the joints, which are fabricated as thin beams, from rotating as they are intended.

Figure 5-26 shows the results of the FEA analysis of the prototype design for both linear

deflection analysis, and nonlinear, large-deflection analysis. The nonlinear analysis begins to

deviate from the linear results for very large deflections, but does not produce the desired

nonlinear stiffness profile for the range of motion of the micromirror. It becomes evident from

this deflection, that the progressive-linkage in this current design implementation is not

providing the appropriate motion that is capable of providing the nonlinear stiffness profile to










affect the pull-in behavior of the device. This becomes a very good example of the importance

of performing careful analysis of a MEMS design prior to fabrication. The above theoretical

model presented for the progressive linkage is still valid. The challenge remains however to find

the appropriate design implementation that will carry out the four-bar linkage design principles.

This remains as future work.

20
18
16. No Data
14 1






4 *


0 50 100 150 200
Voltage (V)


Figure 5-25. Experimental data collected for the prototype of the micromirror with the
progressive-linkage

3000
-*- Nonlinear
2500-
--- Linear

3.2000-

S1 500-



500-


0 10 20 30 40 50
Theta (deg)

Figure 5-26. Results from FEA of the prototype progressive-linkage design for linear and
nonlinear deflection analysis shows the prototype progressive-linkage fails to produce
the desired stiffness profile.









5.5 Chapter Summary

The work presented in this chapter expanded upon the static modeling methods developed

in Chapters 3 and 4 to examine the dynamic characteristics of the system. In keeping with

previous modeling assumption, the lumped parameter model for the micromirrors is presented as

a one degree-of-freedom mass-spring-damper system. The damping characteristics are assumed

to have a low damping ratio based on the results from similar devices in the literature. The

natural frequency of the device is determined from the mass, which is estimated from the volume

of the moving micromirror, and the spring constant that was calculated and characterized in

Chapter 3. This determined the natural frequency of the micromirrors to be 188 k
Modal analysis performed using FEA on the structure determined the first natural

frequency to be lower, at 84 k
one degree-of-freedom. An examination of the participation factors for the response of the first

mode in each direction reveals that the primary direction of the response is in the rotational X-

direction (ROTX), which corresponds to the one-degree-of-freedom model assumption.

However, it is clear that motion in other directions, namely the Z-direction affects the

compliance of the system and the response, resulting in a lower than predicted first natural

frequency. This additional degree of freedom acting in the Z-axis direction significantly lowered

the effective spring constant for this mode, thus lowering the natural frequency. The modal

analysis results are verified by experimental measurements taken with a LDV to determine

resonant behavior for the devices. While the results from these experiments were affected by

noise, it is clear that resonant peaks do occur near the values predicted by the modal analysis

results.

It is clear that the first mode does respond primarily in the ROTX direction, and the

evidence of motion in additional degrees of freedom at resonance does not invalidate the









assumption that the mirror will rotate about the X-axis for excitations that occur below the

resonant frequency. The electrostatic force that is applied to the micromirror is always an

attractive force, drawing the moving electrode down toward the fixed electrode. Thus, if

resonance is avoided, smooth rotational motion in one degree-of-freedom is still accomplished.

This does, however, show the limitations of the 1DOF model assumption, which limits the

analysis to only low frequency responses where resonant behavior may be avoided.

Additionally, the hysteresis behavior for the micromirrors is examined, and it is found that

the theoretical model is able to predict not only the pull-in, as demonstrated previously, but also

the point at which the mirrors will release from pull-in as the actuation voltage is reduced.

Experimental results from the optical profiler validate these findings. Hysteresis causes a

deadband in the actuation capabilities that can be detrimental to the performance of the

micromirrors, and thus actuation within the hysteresis loop should be avoided. The effects of

pull-in and hysteresis also have the ability to negatively affect the dynamic response for

actuation signals that occur below the pull-in voltage. To alleviate the problems associated with

electrostatic instability, a novel solution is presented, called a progressive linkage.

The progressive linkage creates a nonlinear mechanical restoring force that increases as the

electrostatic force increases. It is shown through theoretical predictions that this method can be

effective at eliminating pull-in, with the cost of requiring increased actuation voltages.

Sensitivity analysis reveals however that this design is very sensitive to the fabrication

tolerances, and therefore should be optimized to ensure better performance. A prototype of the

progressive linkage design is presented along with some experimental data that unfortunately is

inconclusive. Further design development and analysis of the progressive linkage device is

considered as future work.









CHAPTER 6
CONTROL DESIGN AND SIMULATION

Now that a dynamic model has been developed for the micromirrors, controllers may be

designed for the system with the goal of ensuring steady-state performance regardless of changes

to the plant dynamics. As seen in recent literature and the work presented in Chapter 5, active

and passive control approaches have been successful at both extending travel range of

electrostatic actuators and for improving tracking, disturbance rej section, transient response,

system bandwidth and stability, and in reducing steady-state errors. For active control design

considerations, in this dissertation the linearized model of the system was used for determining

the controller gains before implementing them on the nonlinear plant models. The general form

of a feedback control system is shown in Figure 6-1 for unity feedback. In this chapter, PID and

LQR controllers are developed and implemented to further quantify the significance of model

uncertainties, pull-in and hysteresis. The PID and LQR control designs in Sections 6.1 and 6.2

only consider the performance of single micromirrors. The model and performance of the

micromirrors as an array is discussed in Section 6.3. Here, the unique issue of how to control an

array of micromirrors that are not individually controllable is explored. This section will

demonstrate a model of multiple mirrors as a single-input/multiple-output (SIMO) system and

will discuss the feedback signals available by considering two different kinds of optical sensors:

position detecting sensors (PSD) and charge-coupled devices (CCD). The performance of these

sensors is considered as well as the impact they will have on implementation of closed-loop

control system on the array of micromirrors.


r + Contllrole Plant .~- Y
Desired angle Actual angle



Figure 6-1. Basic block diagram with unity feedback.









6.1 PID Control

Proportional-Integral-Derivative (PID) control is perhaps the most widely used kind of

control scheme [102]. The appeal of PID control is that it applies to almost any system, even

those for which a system model is not known. There are many techniques that may be used to

define the control gains and to tune them for the best performance. It is popular because it is

easy to design and fairly intuitive to determine the control parameters for systems modeled with

second-order dynamics.

6.1.1 PID Control Theory

The general form of the transfer function for a PID controller is

1 + s 1+ Ts 61
Gc (s) = KP + K, KS=K i 61
s 71s
where

K,
T P (6-2)
'K
and

K,
T, =D(6 -3)
KP
The block diagram of this system is shown in Figure 6-2. The closed-loop transfer

function for this block diagram, with the plant modeled as a linear second order system is

C(s) Gc (s)GP (S)
(6-4)
R(s) 1 + Gc (s)GP (S)H(s)
Assuming unity feedback, that is H(s) = 1, and substituting Equation 6-1 into 6-4 gives an

expression for the closed-loop transfer function.

C(s) (s2KD + SKP +KI )
(6-5)
R(s) s3 + (2goi, + KD )S2 + (KP + (0 ,)S + KI


















Controller G,(s)


Sensor
H(s)

Figure 6-2. Block diagram with PID controller.

The proportional term is a gain that attenuates the magnitude of the system response. The

integral term seeks to eliminate steady-state error in the system. The derivative term, as seen in

the denominator of Equation 6-5, is associated with the damping term, and as KD inCreaSes, the

system damping will also increase affecting the rise time and settling time of the response. The

gain KD is also in the numerator, and can act as a high pass filter that will amplify high frequency

noise. Design methods, such as root locus, can be employed to help derive the proper control

gains for a particular desired performance [102].

This details in general, how a PID controller affects a linear second-order system. For the

micromirror array models presented in the previous chapters, a linearized version has been

developed using a linear first order approximation of the capacitance function in Section 3.4.4.

A set of PID control gains are chosen using trial and error to yield a linearized closed-loop

response characteristic of a an overdamped system with zero overshoot, and to drive the steady-

state error to zero. This controller is implemented on both a linear and nonlinear plant.

6.1.2 PID Results

A PI controller is implemented on the system in an effort to ensure zero steady-state error

despite the presence of model uncertainty. Using only a simple proportional controller (P-









controller) on the system is not sufficient to ensure zero steady-state error for different plant

variations, therefore an integral term is included. The controller gains are chosen as the

proportional gain, Kp = 100, and the integral gain, KI = 100,000. It was found that the derivative

controller term, KD is not needed. To compare the effects of the nonlinear terms in the

electrostatic model, the step response of the linearized plant model is compared to that of the

nonlinear plant model in Figure 6-3. Step responses are shown for both models for different step

values ranging from 2 degrees to 16 degrees, and the effects of the nonlinear terms begin to

appear as the transient response of the nonlinear plant is affected by the magnitude of the step

input. The closed-loop system has no overshoot, which is important in electrostatic systems that

experience pull-in. For a system application with strict transient performance requirements, this

set of gains however may not be sufficient at very low command angles.

As has been shown in the experimental characterization data for these micromirrors in

Chapters 4 and 5, an important control obj ective is to drive the response to have zero tracking

errors in the presence of plant uncertainties. Figure 6-4 shows the effects of model uncertainty

for the nonlinear plant response, including model variations of 10% variation in the spring

stiffness, ks,. Open-loop analysis in Chapter 5 presented the open-loop plant responses of the

system for 50 randomly generated sets of parameters m, b, ks,, and Te that are allowed to vary by

10% of their nominal values. The closed loop response of those same 50 plants is shown in

Figure 6-5 for parametric variations ranging from 10% to a very high value of 190%. It is clear

that even this simple PI controller drives all of the plants to zero steady-state error, achieving the

goal of position tracking.

6.1.3 PID Controller Response to Hysteresis

While it is preferable to avoid driving the micromirrors in the unstable range of motion, it

is possible that this could occur, especially as the pull-in point is known to vary for different











'16 16
14~ 14
12 -1 12

a 1 0 --~ -- 310 -










r1 -




8-





00
0 1 15 2 3 350 0.5 1 1.5 2 2.5 3 3.5
A ~ ~ ~ ~ Tm (ms)m) im ms


Figure 6-4. Clo rsedlopsso PID resonseto diferen ste Lin puats whden the spring consant is ared.




by... +10 10%
micomiror and for dyaic opertn codtos The dicsinnScio ..












systrem with alsdl PID cotrlerin pae.t Thfe resut isthat ifpt e the mroispn commndedtoant svr








unstable position, and subsequently pulls-in, the controller will see this position error and


























Time (ms)


Time (ms)
B


) 0.5 1 1.5 2
Time (ms)


2.5 3 3.5


Time (ms)
D


1.5 2
Time (ms)
F


Time (ms)
E


Figure 6-5. Closed-loop PID response to a step input of 7 degrees for 50 random sets of
parameteric variations. Parameters are allowed to vary by A) 10%, B) 20%, C) 30%,
D) 40%, E) 50%, and F) 90%.


S30% Uncertainty









seek to correct it. The controller will command the actuator with lower voltages until the

holding voltage is reached, and the mirror will release from pull-in. The position of the released

mirror will still not be the correct commanded position, which is an unstable position that cannot

be reached. So this cycle will repeat itself, as shown in Figure 6-6, resulting in a fast switching

behavior until the commanded position of the mirror returns to the stable range of motion. In

Figure 6-6, it is clear that this switching behavior would be undesirable for the system, and could

even result in damage to the micromirrors; however one benefit of the controller response is that

the effect of the hysteresis is mitigated by the controller, and the mirror position returns from

pull-in at an earlier time than the response without the controller. This control behavior

demonstrates potentially undesirable behavior that could result, and it is not suggested that this

PID control implemented for motion in the unstable range is ideal. The control algorithm can

easily be augmented to detect electrostatic pull-in conditions to keep the switching response from

occurring, and thus avoid potentially damaging the device, but still keep the added benefit of

reducing the effect of hysteresis. This discussion is also useful to show once again, the need for

eliminating this electrostatic instability in the response, which may be done with the progressive

linkage design proposed previously.

Section 5.3 presented the design of a progressive-linkage that can be utilized to eliminate

the effects of pull-in and hysteresis. It was demonstrated theoretically that this device can

provide actuation over an extended range of the mirror' s motion at higher actuation voltages.

Using this progressive linkage to eliminate pull-in however does not guarantee that the effects of

fabrication tolerances will not play a role in device performance. With the added complexity in

the design, parametric uncertainty in the dimensions of the linkage could contribute even more to

variations in the system performance; hence closed-loop control is still necessary. Figure 6-7











shows the closed-loop PID step responses for the micromirrors with the progressive linkage, and


they are in fact able to achieve stable rotation above the pull-in limit of 16.5 degrees. Also

shown are the PID step responses for the 50 random plant variations with +10% variation of


model parameters.


20 -i








10 -,
.CI Time (s)


Open-loop



0 0.5 1 1.5 2 2.5 3
Time (s)


Figure 6-6. Closed-loop PID response to a commanded position in the unstable region.


16
14
12


S1--

6
4
2


0 0.5 1 1.5 2 2.5 3 3.5 0O 0.5 1 1.5 2 2.5 3 3.5
A Time (ms) Time (ms) B


Figure 6-7. Closed-loop step responses for PID controller for a system using a progressive-
linkage. A) Step responses of different magnitudes. B) Step responses of 50 plants
with model uncertainties.










6.2 LQR Control

Linear quadratic regulator (LQR) control is an optimal control method that uses a linear

state-space model of the plant to design a stable controller that seeks to minimize the response of

the system states and the control actuation. LQR control design is concerned with minimizing a

cost function that balances the control effort with the system states according to defined weights.

This type of control requires full-state feedback and that the system is completely controllable.

In order to apply LQR control to the system of micromirrors in which only position information

is available, a state estimator must be employed for the velocity state.

6.2.1 LQR Control Theory

First, the LQR control problem will be considered for the regulator problem in which the

controller will seek to rej ect noise and disturbances, and drive all the states of the system to zero.

LQR control can also be used to track an input traj ectory, and this case will be considered

second. The LQR regulator problem, shown in the block diagram in Figure 6-8, assumes full-

state feedback. Cases without full-state feedback will require the use of an estimator and will be

discussed in section 6.2.1.

noise, ~ l n~
disturbance






Figure 6-8. General block diagram for LQR controller problem

The plant is modeled as a continuous time, linear system described by a set of state-space

equations

x = Ax + Bu

y = Cx + Du(6)









It is desired to find a controller that minimizes a cost function, J,


min J = (x'Qx + ru'Rudd (6-7)


where Q is a matrix that relates to tracking performance and R is a matrix related to control

actuation [102]. The values of Q and R are chosen to apply penalties to the states and actuator

commands. The Q and R matrices are either a positive-definite Hermitian or a real symmetric

matrix. A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. The

superscript next to a variable denotes its complex conjugate. The optimal controller, K, that

minimizes this cost function is

u = -Kx(t) (6-8)

Substituting Equation 6-8 into Equation 6-6 gives

x = Ax -BKx = (A -BK)x (6-9)

Thus, Equation 6-7 becomes


J = (x'Qx +x'KIRKx)dt = ~x' (Q + KIRK)xdt (6-10)

The following relationship sets a condition that restricts K to be finite.


x'(Q + K*RK)x = -(x'Px) (6-1 1)


where P is a positive-definite Hermitian or real symmetric matrix. Evaluating the right hand side

of Equation 6-11 and substituting in Equation 6-9 yields

-i'Px-x"'Pi= -' (A-K'+(BKC) PPA-K (6-12)

Equation 6-12 must hold true for any x, therefore

(A BK)*P + P(A BK) = -(Q + K*RK) (6-13)
IfA-BK is a stable matrix, there exists a positive-definite matrix P that satisfies Equation 6-13.
In order to determine this matrix P, evaluate the cost function J.










J = [x'(Q +K'RK)xdt = -xPxo (6-1 4)
Since A-BK is stable, all of the eigenvalues are assumed to have negative real components and
x(oo)4 0 Equation 6-14 becomes
J = x*(0)Px(0) (6-15)
Because R is defined as a positive-definite Hermitian matrix, it can be written in terms of a

nonsingular matrix, T.

R = T*T (6-16)

Substitute Equation 6-16 into 6-13 to get

(A' K*B')P +P(A -BK) +Qe+K*T*TK =
(6-17)
A'P +PA +[TK (T*)-'B'P]*[TK (T*)-'B'P] -PBR-'B'P +Q= 0

The minimization of the cost function J with respect to K requires the minimization of

x*[TK -(T*') 'BP]*[TK -(T*') 'BP]x (6-18)

with respect to K. This expression is nonnegative and the minimum occurs when it is zero, or

when

TK = (T*)-'B'P (6-19)

Hence, the optimal matrix K is found by

K = T '(T*') 'B*P= R 'B*P (6-20)

The matrix P in Equation 6-20 must satisfy the reduced-matrix Riccati equation

A'P +PA -PBR-'B'P+ Q = 0 (6-21)

Equation 6-21 must be solved for the matrix P, whose existence guarantees that the system is

stable. Once P is found, it is substituted back into Equation 6-20 to find the optimal gain matrix

K that is used in the control law of Equation 6-8.









The above development for the LQR controller considered the development of an optimal

controller for the case of driving all of the states in the system to zero. LQR can also be

designed for tracking a desired input traj ectory, r. Consider the traj ectory, described by

r = Fz
(6-22)
z =Hx

for some observable matrices F, and H. In this case, z represents the actual traj ectory of the

system as a function of the states, and this can include any noise in the sensor as well. An error

signal, e, is defined as the difference between the reference (desired) input, r, and the actual

traj ectory.

e =r -z (6-23)

For this problem, the cost function J can be defined in terms of the error signal.


J = l(e'Qee+ u'Reu~dt (6-24)

Equation 6-24 can be rewritten as follows


J = ((x'x + u'Ru + 2x~~'Nud (6-25)

with

Q = (HA FH)" ep (HA FH) (6-26)

R = B*HQeHB + Re (6-27)


N = (HA FH)QeHB (6-28)

Qe = I, Re = pl (6-29)

where p is a constant value describing the weighting function on the control effort.









The goal is to find the optimal controller that minimizes the cost function of Equation 6-

25, and this is determined from solving the following algebraic Riccati equation with an

additional term describing the error signal.

A*P +PA -(PB +N)R '(B*P +N* )Q = 0 (6-3 0)

The solution of Equation 6-30 results in the matrix P such that the controller is described as

K =[R-'B'P R-'N']=[Kfb K,.] (6-31)

where Kgb andKffare the feedback and feedfoward controller gains, respectively. The control

law is thus written as

u= -(Kybx+Kfse) (6-32)

The use of this LQR control law for tracking a reference command with zero steady-state error

requires that the system include an internal model of the reference command. In the case of a

step command, the system must include an integrator and be what is called a type-one system

[108]. If the system model does not already include an internal model, then it must be included

in the controller. LQR optimal control for a tracking control of a step input for a plant that does

not include an integrator has the block diagram shown in Figure 6-9. There is a feed-forward

gain, Ky, and a feed-back gain, Kfb, as in Equation 6-32. In this case, the error signal is the


r L ~ R +x = Ax + Bu


Figure 6-9. Block diagram of LQR control with an internal model for tracking a step command.










difference between the desired reference command, r, and the position state, xl. It is assumed

that full-state feedback is available for this system for the feedback loop. In cases where full-

state feedback is not available, state-estimation is required. This situation is discussed in Section

6.2.2.

6.2.2 State Estimation

The derivation of the control law for LQR control assumes that full-state feedback is

available for the controller. In many cases, full-state information is not available, and state-

estimation must be used. A block diagram representation for the control system using a state

estimator is given in Figure 6-10. L is the estimator gain matrix. State estimation estimates the

state variables of the system based on the measurements of the output and the control variables.















Figure 6-10. Block diagram of LQR controller using a state-estimator for a plant without an
integrator.

In the case shown in the block diagram, assume that there are two states, xl and x2, but only xl is

available, hence C = [1 0], and the output y is






Let a~ represent the vector of the estimated states. The control law of Equation 6-32 becomes

u= -(Kex+Kfse) (6-34)









The mathematical model for the estimator is similar to the plant model of Equation 6-6 with

additional terms included to estimate the error to compensate for inaccuracies in matrices A and

B. The estimation error is the difference between the measured output and the estimated outputs.

The mathematical model for the estimator is

i = A + Bu +L(y ) (6-3 5)

where the A, B, C, D are the matrices of the plant model from Equation 6-6, and 9 = Ci .

One method to design the estimator matrix gain, L, is to use Ackermann's formula for

pole placement for single-input systems. In this method, the gain L is calculated such that the

state feedback signal places the closed-loop poles of the estimator at desired closed-loop pole

locations, h. Ackermann's. formula is


L= [0 0--0 1 [BiABi--BA" B] 1(A) (6-36)

for an arbitrary integer n. The term #(A) is the characteristic polynomial of matrix A.


~(A) = A" + aA"' + a,A"2 +- + a,A +a,,I (6-37)

The coefficients a have a relationship with the roots of the polynomial, h, which are also the

closed-loop pole locations.

(s j)(s A,).--(s A,) = sn + azsl + +aes"2 + -+ an,,s + a, (6-3 8)

This approach of designing the state estimator depends on the proper placement of the

desired pole locations. The most frequently used approach is to choose pole locations from the

root locus such that they are far to the left of the dominant poles of the plant.

6.2.3 LQR Results

An LQR controller is designed along with a state-estimator for the micromirror system

using the linear plant model. The control design is done in Matlab using the 'lqr' command, and










the estimator is designed using the 'acker' command. The results are simulated on the linear

plant model and the nonlinear model.

The weights Q and R are chosen to be

00 0
Q=00 0
(6-39)
0 0 100000
R = 0.0001

The gain matrix from the LQR design produces a feedforward gain, Ky- = [-100000.00], and a

feedback gain vector, Kfb= [0.0585388701 0.0000000952]. The open-loop poles of the linear

plant model of the micromirrors are p = -307489.41 d 977754.00i. Therefore, the desired closed-

loop pole locations for the state-estimator are chosen to be 31= [-3 x 10 -3 x 10 ]. These closed-

loop poles for the estimator are chosen as they lie far to the left of the open-loop poles on the real

axis of the S-plane. Here, they are chosen to be repeated poles because the real part of the open-

loop poles are repeated, but it is not required that they be the same value.

The response of the linear plant model and the nonlinear plant are for step inputs of

different magnitudes are seen in Figure 6-11. The closed-loop system response for the LQR


14 .q

12-
12
10 - 10 -








model.



















4 -4
ic3 3
23 2


1 10% Uncertainty 1 20% Uncertainty

00 0.5 1 1.5 2 2.5 3 3.5 00 0.5 1 1.5 2 2.5 3 3.5
Time (ms) Time (ms)
A B



7~ 7 -





F3 3-

2~ 2-

1 30% Uncertainty 1 40% Uncertainty

0 0.5 1 1.5 2 2.5 3 3." .
Time (ms) Time (ms)
C D


7~--- ---- 7-


4~ -4




2~818 -1 2

S50% Uncertainty le. 90% Uncertainty

0 ""' 0'
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
Time (ms) Time (ms)
E F



Figure 6-12. Closed-loop LQR response to a step input of 7 degrees for 50 random parameter
variations.









controller is shown to be similar to that from the PID. The speed of the system response is

dictated by the choices of Q and R in the control design. The effects of parametric model

uncertainty are examined by testing the controller for the 50 plant models with variations from

10% on the model parameters up to 90% and the response to a step input is shown in Figure 6-

12. The results for the LQR appear to be consistent with the results for the PID controller.

6.2.4 LQR Controller Response to Hysteresis

For the PID controller, discussion is presented in Section 6.1.3 concerning the response of

the closed-loop system when the micromirror is commanded to an unstable position, and thus

experiences electrostatic pull-in and hysteresis. In that demonstration, the behavior of the

controller was found to result in an undesirable switching behavior that nevertheless did improve

the hysteretic response. For the same conditions operating with an LQR controller using state-

estimation, the controller would not be able to function in this unstable range of motion. Recall

from LQR control theory presented in Section 6.2.1 and 6.2.2 that the state-estimation requires

full controllability of the system, and this is not the case in the unstable range of motion. As a

result, for implementing an LQR controller on this system it is particularly beneficial to avoid the

electrostatic instability through the use of a progressive-linkage.

Based on the similarity between the responses from the PID and LQR controllers, it safe to

assume that the closed-loop LQR performance for the system with a progressive-linkage will be

very similar to that of Figure 6-7. Figure 6-13 shows the closed-loop LQR step responses for the

micromirrors with the progressive linkage, and they are able to achieve stable rotation above the

pull-in limit of 16.5 degrees. Also shown are the LQR step responses for the 50 random plant

variations with 10% variation of model parameters.












16 -1
14~ / 14
12~ 12
S10 -n -10


6 -6
4~ -4


0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
A Time (ms) Time (ms) B


Figure 6-13. Closed-loop step responses for LQR controller for a system using a progressive-
linkage. A) Step responses of different magnitudes. B) Step responses of 50 plants
with model uncertainties.

6.3 Modeling the Micromirror Array

The work thus far has focused on modeling and control of just a single micromirror from

an array of mirrors, assuming a single-input/single-output (SISO) system. In reality, these

micromirrors are part of an array that is single-input/multiple-output (SIMO) since there is only

one actuation voltage applied, but each individual mirror is capable of having a unique response.

This section will demonstrate a model of multiple mirrors as a SIMO system and will discuss the

feedback signals available by considering two different kinds of optical sensor: position

detecting sensors (PSD) and charge-coupled devices (CCD). PSDs measure the locations and

intensity of the incident light and output the position of the center of gravity (CG) of the total

light distribution. These devices are inexpensive and easy to use; however the positions of the

individual micromirrors are obscured. By contrast, a CCD sensor is able to output the locations

of the individual light sources; however they are much more expensive devices and require

considerably more computation and processing methods to utilize the sensor information.

The controllers developed in Section 6.1 and 6.2 are implemented on the array model to

determine their effectiveness at reducing the steady-state error of the system as a whole when









model uncertainties are present. Considering the system of micromirrors as they function in an

array is a critical step in expanding the application of feedback control from just one device, to

being able to control very large arrays that are required for many adaptive optics applications.

6.3.1 Modeling the Array of Mirrors

The preceding chapters have developed analytical models for individual micromirror

components that have a SISO structure. Extending this to a SIMO model that includes multiple

micromirror arrays is accomplished by simply adding multiple mirror models in parallel as the

plant of the system. Figure 6-14 shows this system architecture in which a single input is given

to the array of mirrors, and multiple outputs from that system are produced. These outputs are

the position states for each individual micromirror. Figure 6-15 shows schematically what this

system architecture looks like for a system that assumes 5 micromirrors in the array. This image

is not drawn to scale so that the individual mirrors and rays of light can be seen. While in reality

the array is much larger, using only 5 mirrors allows for a more tractable demonstration of array

performance in the simulation environment. It can be difficult to compare the results for a larger

numb er of mirrors.

Just as it was shown for the open-loop dynamic model in Chapter 5, if the model

parameters vary, the response of each mirror will vary for a given input signal. If all the mirrors


Mirror 1

Mirror 2 Sensor
Coman, cs Mirror 3 3 Geometry

determination
N of light
Mirror N reflection

Array

Figure 6-14. Schematic of modeling an array of mirrors as a SIMO system.


















2




y 0 Mirror Array

10 15



Figure 6-15. Schematic drawing of an array of 5 mirrors.

in the array have the same plant model, then they will all have the same response. However, if

the model parameters of each mirror in the array are allowed to independently take on values

subj ect to uncertainty in mass, stiffness, damping ratio, and capacitance, then the results are not

so well behaved. The challenge comes from determining one overall error metric that can be

used for the feedback controller such that the errors in the system can be decreased. Thus, the

goal becomes trying to decrease the total amount of error in the system, which means it is

possible for the individual errors in the mirror responses to still exist. While model uncertainty

can be controlled effectively for one mirror at a time, trying to implement control for this SIMO

array system is a more difficult problem.

One problem with controlling this array system comes from choosing the appropriate

measurement to use as a feedback signal. In the case that each mirror could be controlled

independently, then one approach is to treat it as multiple SISO systems in parallel and provide

one control signal for each micromirror and measure its individual performance. In that case, the

problem quickly becomes one of scale for determining the best way to accomplish this for a very










large array. The case for SIMO system does not have to deal with the issue of scaling multiple

control algorithms, but rather how to apply a single controller to a group of mirrors. While each

mirror can behave independently, there is still only one available control input to the system.

The type of sensor chosen to provide the measurement is critical in determining the overall

performance metric for the system, and the type of error signal used for the feedback control

system. To better understand this, several available sensor types are considered for determining

the impact each would have on detecting and interpreting the system performance. The sensors

considered here are position sensing detectors (PSD), and charge-coupled devices (CCD).

6.3.2 Sensor Types

When a light source is incident on the surface of a PSD, the sensor will output a current or

voltage signal that corresponds to the location of the center of the total distribution of the light

intensity on the sensor surface. This location can be considered as the center of gravity (CG) of

the total light on the sensor surface. PSDs can be one dimensional, which means that they are

able to detect the CG of the light in only one direction, or two dimensional, detecting the CG of

the light in two directions. Consider in Figure 6-16, the case of light from one array of mirrors

reflecting onto a 1-D PSD in which there are errors in the actual positions compared to the

desired positions. Errors in spacing between the spots of light can result in CG measurement that

is different from that desired. The control system seeing this error will try to correct such that

the CG error goes to zero, when in fact this can cause the actual deflections of the micromirrors

in the array to be different values from what is desired. Figure 6-16 also shows the 1-D PSD

with errors in linearity that could be caused by off axis rotations of the mirrors. For small

rotations, the same problem of error in the CG occurs. Since the 1-D array is only able to

measure the CG in terms of one direction (y-axis shown), the off-axis deflection cannot be










directly measured. There is also the case that for a very large error in spacing or linearity, some

of the light could be deflected off of the 1-D PSD entirely, also affecting the location of the CG.

For a single micromirror array being measured by a 2-D PSD, similar problems in

calculating the CG occur, except in this case it is possible to locate the CG in both the x and y-

axes. As seen in Figure 6-17, errors in spacing can shift the CG in the y-direction, but linearity




84 b,
-~ I b2 C) CG









Desired Errors In spacing Srnall errors in Large errors in
Ilnearlty linearlty

Figure 6-16. Illustration of the measured center of gravity (CG) on a 1-D PSD when there are
errors in the spacing and linearity of the micromirrors.




--o *~
aI cb
81 1ct
-' o b2t~









Desired Errors In spacing and
linearity

Figure 6-17. Illustration of the measured center of gravity (CG) on a 2-D PSD when there are
errors in the spacing and linearity of the micromirrors.



168









errors can also shift the CG in the x-direction. Despite the limitations of PSDs, they offer the

advantages of giving an analog signal with a very fast response. In addition, PSDs are typically

more affordable than CCDs.

To avoid the problems such as those described for using a PSD and to allow for the

simultaneous measurement of light from multiple micromirrors in an array, one may use a CCD.

The CCDs are an array of metal-oxide-semiconductor (MOS) diodes that are able to provide

digital information of the light intensity of each pixel in the CCD array. This information can be

interpreted using an image processing algorithm to determine the location of each separate spot

of light from the micromirror arrays. Then it is possible to obtain x and y-direction displacement

measurements for each spot and compare that with the desired positions. This is illustrated in

Figure 6-18 for light from two arrays in which the dark spot indicates the actual position of the






oDesired Poition e
Actual Position






Mirror Array 1 Mirror Array 2

Figure 6-18. Illustration of the measured errors of the reflected light from two micromirror arrays
onto a CCD.

reflected light, and the white spots indicate the desired positions. The error is drawn as a vector

from the actual to the desired positions. If each row of mirrors were given a separate actuation

voltage signal, it would be possible to control the position of each spot to reduce the individual

error signals. Because the mirror arrays discussed here have only one actuation signal available









for the entire array, it will only be possible to reduce the overall error signal by perhaps using a

sum of the squares of the displacement error vectors. An additional error metric could be to

consider statistical yields, such as trying to achieve desired performance goals for a certain

percentage of mirrors. The ability to reduce the error is further limited in that the mirrors have

only one axis of rotation; thus the x and y errors are not independent.

6.4 Modeling the Sensor Response

Including the sensor model into the simulation of the micromirror arrays involves taking

the geometry of the problem into account. Assuming that the locations of the light source, the

micromirrors, and the sensor are known, this becomes a calculation of the system geometry to

determine the location of the reflected light. Figure 6-19 shows a schematic of beam steering

with only one micromirror. The light source, each micromirror, and the sensor, are given a

coordinate frame such that they can be located and oriented in space with respect to a global

reference frame, E. Light that travels to the micromirror is defined by the vector r .,,

Following the laws of reflection, light reflecting off of a flat mirror will have an angle of

reflection that is equal to the angle of incidence, such as that shown in Figure 6-20. In general,

the angle, 'P, between two vectors, a and b as shown in Figure 6-20, can be calculated using the

dot product relationship.

a = a Ibcos'F (6-40)

In this case, the two vectors are the vector r ,,, and the unit normal vector of the mirror

surface, bk Therefore, the angle of incidence ~, is given as,



= o j,*b (6-41)









Knowing the angle of incidence, # now allows for the vector of the reflected light to be

calculated. It is possible to determine the distance between the unit normal vector of the mirror

and the light source as shown in Figure 6-21 by calculating two vectors 3, and 32 that are

perpendicular to each other at point N, and form a right triangle with the vector ronso as the

hypotenuse. The vector 32 is along the bk, unit vector. The magnitudes of these two vectors are

1 = inso in #(6-42)

2 0480~o~ s COs # (6-43)

A location for the reflected light, point R, can be found by reflecting vector 3, about the unit

normal vector bk at point N, resulting in a new vector, r6,, that reveals a location through which

the reflected ray passes. Now there is a known relationship for the reflected ray of light,

represented by the vector, rBO->R

Lighrt Source
Sensor-
Frame Aa jC rm



i kCC


li~i i
ii 'sm C


Fal e ^


Global Reference

Figure 6-19. Schematic of the beam steering experiment with only one micromirror.









Angle of A~ngle of
incidence reflection



A 0 BO 0 BO ->Cba



A B

Figure 6-20. Geometry used to determine the angle of incidence and reflection. A) The angle of
incidence is equal to the angle of reflection. B) For two vectors a and b, the angle
between them, 'F, can be determined from the dot product.

Next, the intersection of this ray of light with the plane of the sensor can be calculated.

Referring to Figure 6-22, let the sensor plane in Frame C be defined by three points, C1, C2, and

C3 which have global coordinates (Czx, Cy, Czz), where the sub script i may equal 1, 2, or 3. The

vector of the reflected ray of light, rBO->R, iS given by two points, the origin of the B frame, BO,

and the point R, which are known to have global coordinates (BOx, BOy, BOz) and (Rx, Ry, Rz),

respectively. The orientation of the B frame will represent the angle of rotation of the

micromirrors as they are actuated, and will rotate about the be unit vector. The intersection of

the vector with the plane, at point P, is found by simultaneously solving the following four

equations for the variables x, y, z, and t.


Cx Cy Czl

12x C2y C2z
C3x C3y C3z (-4

BOx +(Rx -BOx)t = x (6-45)

BOr + R,~y-BOy)t = y (6-46)

BOz + (R BOZ)t= z (6-47)









Solving for t yields,


111 1
Cx Cx 3x, BOx
C, Cy 3y, BO,
C, C, C, BO
1z 2z 3z=t (6-48)
1 11 0
Cix Cx 3x, BOx
C,y C2y C3y BO,
CI, Cz 3z BOz

This value for t calculated from Equation 6-36 may be substituted back into Equations 6-45, 6-46

and 6-47 to solve for the (x, y, z) global coordinates of the intersection point P. This process can

be repeated for multiple mirrors to determine the coordinates of their reflected light on the

sensor. Now that the reflected light can be located, it is possible to calculate the sensor

measurement for the system depending upon the type of sensor used.

Light Source
Frame A
k" Frame C


T
AO->B0


~BO->R


ei Frame B
Frame E Mro
Global Reference

Figure 6-21. Calculating the reflected ray of light.










Frame C
C,




BO -g R Cz

b. BrR,


Frame B



Figure 6-22. The intersection of the line from BO to R and the plane C occurs at point P.

6.4.1 PSD Response

A 1-D PSD is incorporated into the model for the system of arrays by calculating the

locations of the reflected light and determining the center of gravity of the light. Laser light is

known to have a Gaussian distribution of light intensity, with the light being more intense in the

center of the beam, and reducing toward the outside of the beam [105]. Therefore, for the model

of sensor performance, the light from the mirrors is weighted accordingly such that the light in

the center has a higher intensity, following a Gaussian distribution. Center of gravity may be

calculated as


CG; = [w, (6-49)


where W represents the total weighting of the light intensity, w, represents the weight of the light

intensity for one ray of light, and r, represents the position of the ray of light on the sensor for n

total rays of light.

Figure 6-23 shows a representation of a system of 5 micromirrors in which the light is

reflected onto a sensor. This figure is not drawn to scale so that the individual light rays are










more easily seen. The CG of the light is calculated for this ideal case as equal to the position of

the center mirror. A 2-D representation of the reflection on the sensor is also shown. The CG is

output as a voltage between 10 V, where a value of 0 V indicates the CG is at the center, +10 V

indicates the CG is at the top, and -10 V indicates that the CG is at the bottom of the sensor

array. In order to relate this sensor value of the CG back to a meaningful measurement in terms

of the angle of rotation of the mirrors, 8, the sensor output is determined first for an ideal set of

mirrors. This calibration then allows the sensed CG to be converted to an angle corresponding to

the angle of the micromirrors, 8. The calibration result for the case of 5 micromirrors is given in





Lih Source










15
Y -2 '0 '5 10
A X B

Figure 6-23. Schematic of 5 micromirrors in an array reflecting light onto a PSD sensor. A) The
CG of the measurement is calculated in the sensor plane. B) A 2-D view of the
reflected light on the sensor plane with the CG marked in the red star.

Figure 6-24. If any of the reflected light is directed off of the sensor, then this light is not

recorded and its contribution is neglected in calculating the CG. This can cause a shift in the

measured value of the CG.

In order to evaluate the effectiveness of this sensor at providing feedback signals to the

system, the sensor response is determined for the case of 5 micromirrors with randomly varied






























































i-- -- -


CG measure
.......... mirror 1
--mirror 2
mirror 3
mirror 4
-*- mirror 5


models. Figure 6-25 shows the resulting open-loop step response for each mirror and the overall


CG measurement. Also shown are the position errors for each mirror and the overall error. The


sensor outputs the location of the CG of all 5 mirror responses showing that on average, the 5


mirrors of the array have a steady-state error. In the case of a broken device or a mirror in the


Theta (cleg)


Figure 6-24. Calibration of the PSD for ideal case of five micromirrors.


16

14

'12


310


-- 6

4

2


O
U
V
L
O
t
W
c
o

o
a


0 1 2 3 4 5 -0 1 2 3 4 5
A Time (s) x10-6 Time (s) x1o' B



Figure 6-25. Open-loop results to a step response for an array of 5 micromirrors with model
uncertainty. A) Step response. B) Position error.


array with very deviant behavior, the CG calculation can be greatly affected. If one mirror is


broken and remains stationary, the CG calculation for the overall array will be affected. This is


shown in Figure 6-26 in which one mirror is broken and does not actuate while the other mirrors


121
CG measure
10~ ---------- mirror 1
-- --mirror 2
mi error 3
6 mirror 4
-*- mirror 5






-2

4- tt-











are considered to have the ideal model with no uncertainty. It is clear that the inclusion of model


uncertainty in the other mirrors would only add to the calculated error of the CG. This could


also be the case that occurs when some mirrors in the array experience pull-in at different times.


This illustrates a limitation of using a PSD for the sensing mechanism.


The model of the array of mirrors and the sensor can be incorporated into a control system


like that in Figure 6-27. To illustrate the effects of using a PSD as the sensor for this


14 12

12C t* 10 -~ CG measure
I I ------ mirror 1
10- II -I a mirror 2
ill 6 mirror 3
S8t 11 I mirror 4
m Lu 4~ -*- mirror 5
S6C -I -CG measure S
I-- ------- mirror 1 g 2
4- -mirror 2 0
mirror 3 7
21 mirror 4 -2 .
-* mirror 5
0 -4 5 512
A Time (s) x 10-5 Time (s) x 10 B


Figure 6-26. Open-loop response for system with one broken mirror and 4 ideal mirrors,
measured by a PSD. A) Step response. B) Position error.


Mirror 1-
Command, 8,c Mirror 2 Sno
SController Mirror 3 -Go er
based
determination
Mirror N -N of light
reflection
Array





Figure 6-27. Incorporating feedback control into array model.

micromirror array system, the closed loop response is determined. Figure 6-28 shows the


response for a PID controller, and likewise, Figure 6-29 shows the response using the LQR


controller. In both cases, the CG measurement is used as the feedback signal, and the controllers










thereby only see this average error measurement. The controllers are both able to reduce the

average error of the system, but this is really only accomplished by shifting the responses of the

5 mirrors. In this case, the two mirrors, 3 and 4, that had the least amount of error in the open-


O 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8
A Time (s) Time (s)


Figure 6-28. Controlled PID step response using PSD sensor. A) Step response. B) Position
error.


O 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.81
A Time (s) Time (s) B


Figure 6-29. Controlled LQR step response using PSD sensor. A) Step response. B) Position
error.

loop response are actually shifted so that in the controlled system, they have more error. Another

weakness is illustrated in closed-loop control in the case of a device in which one mirror in the

array is not functioning. Shown in Figure 6-30 is the closed-loop PID response of the system










with one mirror broken, and it is clear that in order to compensate for the malfunctioning mirror,

the system instead drives the other four mirrors to an incorrect position.


16 1 12
14 / j~------------------ 10
12 g CG measure
B mirror 1
i,10 6 -mirror 2
8 mirror 3
-r C G measure Lu4mro
6 0 ~mirror 5
~ 6 mirror 1 .~ 2
4- --mirror 2 g 0
mirror 3
2 mirror 4 -1 -2 L
-*-mirror 5
00 0.2 0.4 0.6 0.8 1 -0 0.2 0.4 0.6 0.8 1
A Time (s) Time (s) B


Figure 6-30. Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a
PID controller and a PSD sensor. A) Step response. B) Position error.

6.4.2 CCD Response

CCD sensors have an advantage over the PSDs in that they can measure and interpret the

response of each mirror in the array separately, such as in the Figure 6-3 1. The limitation of the

SIMO system still imposes a requirement that these sensor measurements be compiled into only

one metric. Using a CCD for this SIMO system has many of the same limitations of the PSD.

The error metric used to compare the actual micromirror position to the desired micromirror

position may be limited to represent some average of the errors of all of the mirrors. In this case,

the results for using a CCD are not an improvement over using a PSD. However, the ability of

the CCD to identify individual positions of the micromirrors does allow for some advantages.

For instance, in the case of a damaged or broken micromirror, the actuation for that one mirror

may remain at zero, or have a drastically deviant behavior compared to a mirror that is working

properly. Using a PSD sensor, the measurement from the damaged mirror will remain part of the

CG calculation, thus skewing the overall results. With the appropriate processing algorithm, the











data from the CCD can be used to identify any mirrors that are broken or have highly unusual

behavior and eliminate those mirrors from the control consideration. This is demonstrated


simplistically in Figure 6-32 in which the PID closed-loop system is able to identify the broken

micromirror that remains at zero degrees actuation, and thus eliminates that measurement from

the error metric. Additional error metrics may also be defined, such as identifying a yield for the


10~

8

6

N 4


Light Source


sensor


2\

YO


-2 '


Figure 6-31. Schematic of 5 micromirrors in an array reflecting light onto a CCD sensor where
each separate location of the light can be measured.


0.4 C


0.4 0.6


A Time (s) Time (s)


Figure 6-32. Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a
PID controller and a CCD sensor. A) Step response. B) Position error.










array such that a certain percentage of the mirrors are guaranteed to have minimal error, even if it

means that other mirrors will have larger errors. Weights can be assigned to the measurements

to determine those mirrors that have a higher priority in the error measurement.

6.4.3 Summary of Sensor Analysis

It is clear that the limitations in the controllability of the individual micromirrors inhibit

the ability of the controller to affect only some aggregate response for the system. The PSD

sensor, while fast, inexpensive, and easy to use, is not able to differentiate the responses of the

single mirrors, and is therefore most affected by deviations in the single mirrors responses. The

CCD sensor is a more expensive option, both in purchase cost and computational efficiency,

however it does allow for more flexible parsing of the error signal that can be used to concentrate

the control effort on a subset of selected micromirror responses. In the case of trying to control

more than one array of mirrors, or for a system of mirrors with SISO controllability, then the

CCD array would be the obvious choice of sensor because it can detect multiple locations of

light. The issue of sensor noise was not taken into account in this study, but this too will affect

the outcome of the control system. The level of noise will vary depending on the sensor chosen,

as well any noise from the environment such as vibrations. While noise levels for a given PSD

or CCD product vary by the make and model of the sensor, CCDs typically have lower noise.

6.5 Chapter Summary

The control algorithms explored in this chapter, including PID and LQR, are designed

based on the 1DOF model developed in previous chapters, and the closed-loop system is

analyzed in simulation to explore the effectiveness of these control schemes and examine unique

issues that may be encountered, such as the electrostatic instability phenomena. Other

implementation issues are addressed, including choosing the appropriate sensing elements with

which to detect the micromirror position for feedback. The different sensor types discussed are









all optical, that is they can measure the position of light reflected from the micromirrors, and

depending on the type of sensor chosen, they can operate in one or two degrees of freedom.

PSDs are only able to report the aggregated results for all light incident on the sensor surface,

while CCDs are able to report individual signals from different sources. Due to the actuation

limitations for the micromirror arrays in this study, it is concluded that a PSD sensor is adequate

for the system, but there are still advantages that can be obtained from the use of a CCD. The

next step in this work, which is included in the list of future work, is to develop an optical

testbed to implement the control algorithms presented here, and to determine their ability to

influence the precision and accuracy of the micromirror arrays. The optical testbed must also

consider the implementation issues of noise in the feedback look from the sensor and from the

environment. Addition studies concerning control design include examining the PID and LQR

controllers for response at higher frequencies and exploring further nonlinear dynamic behaviors

that result from the electrostatic instability.











CHAPTER 7
CONCLUSIONS AND FUTURE WORK

The work presented here is an effort to model and analyze the behavior of MEMS

micromirror arrays that have inconsistent behaviors caused by parametric uncertainties and

nonlinear effects from electrostatic actuation. The micromirror arrays are evaluated first by

extensive analytical modeling and experimental validation to determine their performance and

understand the effects of fabrication variations. Using tolerance information from the fabrication

process, it was shown that it is possible to model the effects of fabrication variations on the

performance of the mirrors and to determine the sensitivity of that performance with respect to a

particular parameter. These modeled results are compared to open-loop characterization data

obtained using an optical profiler. It is apparent that there exists varying behaviors for the

mirrors of the arrays in terms of the static voltage-displacement relationships and the

electrostatic pull-in and hysteresis that can affect the dynamic system response as well.

Electrostatic instability is addressed here through the introduction of a progressive-linkage that

provides a continuous, nonlinear restoring force to the device that allows it to theoretically

achieve stable actuation over the entire range of motion of the micromirror. Bifurcation theory

was used to further characterize the electrostatic behaviors and the effectiveness of the

progressive linkage to mitigate these behaviors.

To validate the dynamic modeling, modal analysis was performed using FEA on the

structure and validated experimentally using measurements obtained using a Laser Doppler

Vibrometer. An examination of the participation factors for the response of the first mode in

each direction reveals that the primary direction of the response is in the rotational X-direction

(ROTX), which corresponds to the one-degree-of-freedom model assumption. However, it is

clear that motion in other directions, namely the Z-direction (vertical) affects the compliance of









the system and the response, resulting in a lower than predicted first natural frequency. Because

the electrostatic force that is applied to the micromirror is always an attractive force, drawing the

moving electrode down toward the fixed substrate and if resonance is avoided, smooth rotational

motion in one degree-of-freedom is still accomplished. The presence of extra degrees of

freedom does, however, show the limitations of the 1DOF model assumption, which limits the

analysis to only low frequency responses where resonant behavior may be avoided.

To further evaluate the effects of uncertain system behavior, simple feedback controllers

are developed using a linear system model and then applied to the nonlinear model. This work

demonstrates the use of PID and LQR control, and tests these controllers on nonlinear plant

models with varying parameters. The results from both controller designs show that they are

able to provide stable actuations with no overshoot for a range of plant models. The cost of

applying these control methods comes in terms of the speed of the response. The open-loop

dynamics, while exhibiting some overshoot behavior in the transient response, operates on a very

fast time-scale, on the order of Cps. Closing the loop on the system slows the response time by

several orders of magnitude to ms; however, this is still a sufficiently fast response time for

many applications, and the added benefits of the controllers at eliminating overshoot and

correcting system response in the presence of model uncertainty are clearly worthwhile.

After modeling and developing controllers considering only one micromirror at a time, the

system is evaluated as an entire array of devices. The SIMO structure of the system puts

limitations on the ability to control each micromirror individually, and it is important to consider

the type of feedback information available and how it is utilized. Both PSD and CCD optical

sensors are considered and it is found that with both sensors, it is possible to correct for the

average errors of the system, while not guaranteeing that each micromirror in the array will in









itself attain perfect position tracking. Use of a CCD sensor does have advantages however that

can allow for more advances sensor processing allowing for selective control of the sensor data,

such as identifying outliers and ensuring their measurements are not retained in the feedback

signal. An optical testbed is developed in order to study the effectiveness of control

implementation on the actual micromirror arrays. Laser beam steering and a PSD sensor are

used for position feedback, and preliminary results illustrate the ability to implement feedback

control of these systems.

This research presented in this dissertation provides a validated theoretical model basis that

allows for the development of micromirrors for adaptive optics applications that are robust to

parametric uncertainties that commonly arise through microfabrication processes as well as to

disturbance rej section and plant nonlinearities. Future work includes exploration of dynamic

response of the system at higher frequencies, and development of optimally designed devices

that are less sensitive to the effects of variations in the fabrication process. In addition, the

passive (progressive linkage) and active controller development presented in this dissertation,

additional work is needed to be expanded to refine the designs with inclusion of design

optimization and expansion of the modeling techniques used. While many researchers develop

models of the system performance, very few use these analytical techniques to optimize the

device performance. The application of optimal design methods and closed-loop control

techniques will enable both cost reduction as the devices will no longer require extensive

calibration for open-loop performance, as well as improved performance and reliability. The

impact of this work is not limited to the application of micromirror or micro-optics design. The

design and optimization methods used in the creation of these new actuator designs will create a

general design framework that can be used in the development of many new MEMS devices.










This will aid researchers in all future design efforts and improve the design and development

process.

The PID and LQR controllers presented in Chapter 6 can be adapted and refined to meet

specific performance metrics defined by the application requirements. The gains proposed for

the controllers are quite high, and limitations in hardware capabilities may require these gains to

be lowered, and the stability of the system must always be maintained. Additional study is

required to determine the effects of noise and disturbances on the feedback loop, as well as how

this affects the stability of the system. The results of the modal analysis in Chapter 4 show that

the one-degree-of-freedom motion of the system is not valid during resonant behavior, therefore

it is recommended to avoid driving the system to resonance. However, it would be very

interesting to study the nonlinear dynamics of the system at higher frequencies to identify the

effects relating to resonance and to electrostatic pull-in.

In order to design a robust microsystem that can be deployed in a wide variety of

scenarios, the device should have on-chip sensing capabilities built in so that the actuation,

sensing, and control can be packaged into a complete system. The development of such sensing

and control strategies will contribute to the advancement of precision optical applications. The

incorporation of on-chip sensing mechanisms into the device will allow for compact realization

of complete microsystems. The method proposed in [91] for using piezoresistive methods within

SUMMiT VTM fabrication is novel and its success will open up new areas of device applications.

Several feedback mechanisms should be investigated, including piezoresistive, capacitive, and

optical sensing methods. There is also a need to integrate sensing mechanisms at the device

level to allow for the realization of complete, compact microsystems. Piezoresistive and










capacitive methods seem very promising in this area, however noise in the sensor output will

need to be carefully examined and minimized.

The development of an experimental test bed was also initiated at the University of Florida

as part of the research where further development is still needed before implementation and

validation of the presented closed-loop controllers can be realized. In doing so, this work will

provide a greater impact on the development of micromirrors for adaptive optics applications

that are robust to parametric uncertainties that commonly arise through microfabrication

processes as well as to disturbance rej section and plant nonlinearities.









APPENDIX A
MODEL GEOMETRY

The dimensions used for creating the electrostatic model for one unit cell of the device

geometry are shown in Figure A-1 by layer. All dimensions in Cpm are shown for layer MMPoly

0, and the subsequent layer dimensions are shown in relation to the MMPoly 0 ground plane.

The model is created by drawing these areas in the X-Y plane, and extruding the thickness in the

positive Z-axis.

MMPoly 0 (Ground Plane)






MMPoly 1 (Fixed Finger)






MMPoly 2 (Fixed Finger)





MMPoly 3 (Moving Finger)





MMPoly 4 (Mirror Surface)




-11.5 9 .


Figure A-1. Geometry dimensions in Cpm for creating electrostatic model.







APPENDIX B
MONTE CARLO SIMULATION INPUTS
This appendix shows the values used to perform the Monte Carlo simulations in Chapter
4. The values were determined from a random number generator in order to have a normal
distribution about a mean and standard deviation. Five-hundred sets of random values were
generated, and are shown as histograms here. Also shown is the histogram of the calculated
mass values.


16
14-
12-
10-


6
4


0.284 0.286


ulu


lmM~l
0.288 0.29 0. 292 0.294
Thickness of MMPoly0 [pm]


0.296 0.298


Figure B-1. Histogram of values for the thickness of layer MMPoly0.















































I .3Lo 1.00 1.000


14

12-

10


O




1. 1 1 1 1 .0 2 2 2 .2
Thcns fMrl1[m
Fiue -. itorm fvaus o tethcnesoflye Ml1


14

12

10-



6

4

2-

o I
1.515 1.52


I


lI


O
1.54


Thickness of MMPoly2 [pm]

Figure B-3. Histogram of values for the thickness of layer MMPoly2.














14


12


10-



O 8-
O





L4-


2-



2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39
Thickness of MMPoly3 [pm]



Figure B-4. Histogram of values for the thickness of layer MMPoly3.



14


12


10-



oO 8-
O





L4-


2-



2.27 2. 275 2.28 2. 285 2.29 2. 295 2.3 2. 305 2.31 2.315
Thickness of MMPoly4 [pm]



Figure B-5. Histogram of values for the thickness of layer MMPoly4.





2.1 2.12


14



12



10



8-



6-



4-



2-


I


.415 0.42


0.385 0.39 0.395 0.4 0.405 0.41 0
Thickness of Dimple3 backfill [pm]



Figure B-6. Histogram of values for the thickness of Dimple3 backfill.


14


12



10-



8-



6-



4-


1.98~ 2 Z.uZ Z.u4 Z.ub; Z.ua
Thickness of Sacox1 [pm]



Figure B-7. Histogram of values for the thickness of layer Sacoxl.






































0.29 0.295 0.3 0. 305 0.31 0.315 0.32
Thickness of Sacox2 [pm]


Figure B-8. Histogram of values for the thickness of layer Sacox2.


20

18

16

a, 14

S12






S6






0 0.5 1 1.5 2 2.5 3 3.5 4
Thickness of Sacox3 [pLm]


Figure B-9. Histogram of values for the thickness of layer Sacox3.














I I


Ll
2.5 3


0 0.5 1 1.5 2
Thickness of Sacox4 [pm]
Figure B-10. Histogram of values for the thickness of layer Sacox4.


I I


-0.16 -0.14 -0.12 -0.1 -0. 08 -0. 06 -0. 04 -0.02 0 0.
Linewidth variation for MMPoly2 [Cim]
Figure B-11i. Histogram of values for the linewidth variation of layer MMPoly2.















































UI


O 10
o

5
I. .


-0.25 -02 -0.15 -0.1 -0.05 0 0.05 0.1
Linewidth variation for MMPoly3 [pm]

Figure B-12. Histogram of values for the linewidth variation of layer MMPoly3.


a,
~10
o
o
o
o
x
o
~
a,
g! 5
LL


03 513 155 lou lo 170 175
Youngs Modulus E [GPa]

Figure B-13. Histogram of values for the linewidth variation of layer MMPoly4.


Irli


111~


I


O











LI


2.34 2.36
Mass [kg]


Figure B-14. Histogram of the mass values calculated from the parametric variation data.


0 0.02


Figure B-15. Histogram of values from the Monte Carlo simulations for the linewidth error of
MMPoly2. Values in blue lie within the 95% confidence interval, and values in red
lie without.


,O I III
2.28 2.3 2.32


m,5. 1
2.38 2.4 2.42
x 10M1


I .1 M1llm il
-1 16 -0 14 -0.12 -0.1 -0 OB -0 06 -0 04 -0.02
Linewricith Error of MMPoly2 [Crn]








I~


L


O 10 20 30 40
Voltage (V


60 60 70 BO


Figure B-16. Static displacement curves from the Monte Carlo simulations that indicate the
effect of large variations in the linewidth error of MMPoly2.


1.016 1.018 1.02 1.022
Thickness of MMPoly1 [Cpm]


1.024 1.026 1.028


Figure B-17. Histogram of values from the Monte Carlo simulations for the layer thickness of
MMPoly 1.


1.012 1.014





0 10 20 30 40 50 60 70 80
Voltage (V
Figure B-18. Static displacement curves from the Monte Carlo simulations that indicate the
effect of large variations in the thickness of MMPoly 1.


cll


I I


2.5 3


0.5 1 1.5 2
Thickness of Sacox4 [pLm]


Figure B-19. Histogram of values from the Monte Carlo simulations for the layer thickness of
Sacox4.






















12-

S10-






O 0 2 0 4 0 6 0 B
Votg (V


FiueB20 ttc ipaemn uve rmth ot Crosmuain ta nict h






Figue 2effeatct of larg me vritin inve the thikes Moft Saclox4. ltin ha nict












APPENDIX C
LASER DOPPLER VIBROMETER RESULTS


2.50E 04
44.06 kHz

81.41 kHz
2.00E-04-




S1.50E-04-




S1.00E-04 -I 1II 111 186.88 kHz



5.00E-05-




0.00E+00
0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-1. Magnitude of FFT results for device 1, trial 1.


4.50E-04

4.00E-04

3.50E-04

3.00E-04

2.50E-04

2.00E-04

1.50E-04

1.00E-04

5.00E-05


0.00E+00 U" l'' '', "~ ~''''" '1"" "~ "'ve mrwwu*cl
0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-2. Magnitude of FFT results for device 1, trial 2.


200















81.71 kHz


42.03 kHz









187.03 kHz










50 100 150 200 250 300 350 400 450 500
Frequency [kHz]


4.50E-04


4.00E-04


3.50E-04


3.00E-04


,E 2.50E-04

'B2.00E-04


1.50E-04


1.00E-04


5.00E-05


0.00E+00


Figure C-3. Magnitude of FFT results for device 1, trial 3.


8.00E 04

186.88 kHz
7.00E-04
35.78 kHz
6.00E-04


5.00E-0482.66 kHz

S4.00E-04


3.00E-04


2.00E-04


1.00E-04


0.00E I00
0 50 100 150 200 250 300 350 400 450 500

Frequency [kHz]



Figure C-4. Magnitude of FFT results for device 1, trial 4.













2.50E-04


2.00E-04-



51.25 kHz
S1.50E-04-



o 81.56 kHz
J 1.00E-04 I187.19 kHz
123.75 kHz



5.00E-05-




0.00E+00
0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-5. Magnitude of FFT results for device 1, trial 5.


7.00E-04


6.0-4-43.44 kHz

5.00E-04-


4.00E-04-

82.19 kHz

140.78 kHz
3.00E-04-


2.00E-04-


1.00E-04-


0.00E100 --
0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-6. Magnitude of FFT results for device 2, trial 1.


202













5.00E-04

4.50E-04 43.13 kHz

4.00E-04-

3.50E-04-

S3.00E-04-

S2.50E-04- 139.53 ktHz

S2.00E-04- -~ I 85.63 kHz

1.50E-04-






0.00E 100
0 50 100 150 200 250 300 350 400 450 500

Frequency [kHz]



Figure C-7. Magnitude of FFT results for device 2, trial 2.


3.00E-04

43.44 kHz

2.50E-04



2.00E-04
85.31 kHz

25.63
1.0E0 kHz 118.59 kHz
I ,I II I I 136.56 kHz

1.00E-04








0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-8. Magnitude of FFT results for device 2, trial 3.


203













2.00E 04

1.80E-0427.34 kHz
1.60E-04

1.40E-04


S1.20E-04 -( i43.59 kHz

S1.00E-04

3 8.00E-05 I 92.03 kHz
137.03 kHz
6.00E-05

4.00E-05|





0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-9. Magnitude of FFT results for device 2, trial 4.


4.50E-04
41.88 kHz
4.00E-04-

25.6
3.50E-04-kz


3.00E-04-
136.10 kHz
,E, 2.50E-04-


.o 2.00E-04-
8 1 111,1185.31kz
1.50E-04-


1.00E-04-


5.00E-05-


0.00E 100 -
0 50 100 150 200 250 300 350 400 450 500

Frequency [kHz]



Figure C-10. Magnitude of FFT results for device 2, trial 5.


204













2.50E-04


2.00E-04 'L.u5
kHz

50.16
kHz 105.31 kHz
~i1.50E-04-




S1.00E-04-




5.00E-05-




0.00E 100
0 50 100 150 200 250 300 350 400 450 500

Frequency [kHz]



Figure C-11i. Magnitude of FFT results for device 3, trial 1.


2.50E 04

46.88 kHz

2.00E-04

183.91 kHz
31.25
~'1.50E-04 kHz

P 116.56 kHz
o 90.31



1.00E-04 5







0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-12. Magnitude of FFT results for device 3, trial 2.


205













3.00E-04
81.41 kHz
42.34 kHz

2.50E-04-



2.00E-04-

I II, (112.97 kHz
132.66 kHz
S1.50E-04-I
8 1 I.,1II YII 1 180.63 kHz


1.00E-04- ,



5.00E-05-



0.00E100
0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-13. Magnitude of FFT results for device 3, trial 3.


3.50E-04


3.00E04 -183.28 kHz



2.50E-04 -1 48.13 kHz



i~2.00E-04-


S1.50E-04-


1.00E-04-


5.00E-05-


0.00E 100
0 50 100 150 200 250 300 350 400 450 500

Frequency [kHz]



Figure C-14. Magnitude of FFT results for device 3, trial 4.


206















1.80E-04-

1.60E-04-
183.28 kHz
1.40E-04 -I 57.34 kHz

S1.20E-04-
I ,183.28 kHz
S1.00E-04-

S8.00E-05- IIIdII 1 106.56 kHz

6.00E-05-



040000E 0 0


0 50 100 150 200 250 300 350 400 450 500
Frequency [kHz]



Figure C-15. Magnitude of FFT results for device 3, trial 5.


207










LIST OF REFERENCES


[1] O. Solgaard, F.S.A. Sandejas, and D.M. Bloom, "Deformable grating optical modulator,"
Optics Letters, vol. 17, no. 9, pp. 688-690, 1992.

[2] D. Dudley, W. Duncan, and J. Slaughter, "Emerging digital micromirror device (DMD)
applications," Proc. SPIE, vol. 4985, 2003.

[3] S.D. Senturia, M~icrosystem Design, Boston, MA: Kluwer Academic Publishers, 2001.

[4] G.T. Kovacs, N.I. Maluf, and K.E. Petersen, "Bulk micromachining of silicon," Proc.
IEEE, vol. 86, no. 8, pp. 1536-1551, 1998.

[5] J.M. Bustillo, R.T. Howe, and R.S. Muller, "Surface micromachining for
microelectromechanical systems," Proc. IEEE, vol. 86, no. 8, pp. 1552-1574, 1998.

[6] Sandia National Laboratories. (2005, Sept. 9). SUMMiT V Design Manual. Last Access:
Sept. 9, 2007. Available e-mail: memsinfo@sandia.gov Message: SUMMiT Design
Manual

[7] J.W. Wittwer, T. Gomm, and L.L. Howell, "Surface micromachined force gauges:
uncertainty and reliability," J. M~icromech. M~icroeng., vol. 12, pp.13-20, 2002.

[8] J. Wittwer, "Predicting the effects of dimensional and material property variations in micro
compliant mechanisms," M.S. Thesis, Dept. Mech. Eng., Brigham Young Univ., 2001.

[9] J.W. Wittwer, M. S. Baker, and L.L. Howell, "Robust design and model validation of
nonlinear compliant micromechanisms," J. M~icroelectromech. Syst., vol. 15, no. 1, 2006.

[10] S. Park, and R. Horowitz, "Adaptive control for z-axis MEMS gyroscopes," in Proc. Amer.
Controls Conf~, Arlington, VA, 2001, pp. 1233-1228.

[1l] S. Park, and R. Horowitz, "Adaptive control for the conventional mode of operation of
MEMS gyroscopes," J. M~icroelectromech. Syst., vol. 12, no. 1, pp. 101-108, 2003.

[12] W. N. Sharpe, Jr., B. Yuan, and R. Vaidyanathan, "Measurements of Young's modulus,
Poisson' s ratio, and tensile strength of polysilicon," in Proc. IEEE 10th Ann. Int. Workshop
M~icroelectromech. Syst., Nagoya, Japan, 1997, pp. 424-429.

[13] A. Tauntranont, V.M. Bright, J. Zhang, W. Zhang, J. Neff, and Y.C. Lee, "MEMS-
controllable microlens array for beam steering and precision alignment in optical
interconnect systems," in Proc. Solid-State Sensor and Actuator Workshop, Hilton Head
Island, SC, 2000, pp. 101-104.

[14] D. Hah, S.T. Huang, J.C. Tsai, H. Toshiyoshi, and M.C. Wu, "Low-voltage, large-scan
angle MEMS analog micromirror arrays with hidden vertical comb-drive actuators2," J.
M~icroelectromech. Syst., vol. 13, no. 2, pp. 279-289, 2004.


208











[15] P.B. Chu, I. Brener, C. Pu, S.S. Lee, J.I. Dadap, S. Park, K. Bergman, N.H. Bonadeo, T.
Chau, M. Chou, R.A. Doran, R. Gibson, R. Harel, J.J. Johnson, C.D. Lee, D.R. Peale, B.
Tang, D.T.K. Tong, M.J. Tsai, Q. Wu, W. Zhong, E.L. Goldstein, L.Y. Lin, J.A. Walker,
"Design and nonlinear servo control of MEMS mirrors and their performance in a large
port-count optical switch," J. M~icroelectromech. Syst., vol 14, no. 2, pp. 261-273, 2005.

[16] S. Huang, J-C. Tsai, D. Hah, H. Toshioshi, and M.C. Wu, "Open-loop operation of MEMS
WDM routers with analog micromirror array," in Proc. IEEE/LEOS Int. Conf: on Optical
M~EMS, Lugano, Switzerland, 2002, pp. 179-180.

[17] L. Zhou, M. Last, V. Milanovic, J.M. Kahn, and K.S.J. Sister, "Two-axis scanning mirror
for free-space optical communication between UAVs," in IEEE/LEOS OpticalM2\EM\S
2003, Hawaii, USA, 2003.

[18] Lucent Technologies. (Last Access: 2007, June 18), LambdaRouter Mirror. [Online].
Available: http ://www.alcatel-lucent. com/wps/PA1lA_10OB/images/Photos/Other
Photos/Lambda~mirrorLR.jpg

[19] B. Mi, D.A. Smith, H. Kahn, F.L. Merat, A.H. Heuer, and S.M. Phillips, "Static and
Electrically Actuated Shaped MEMS Mirrors," J. M~icroelectromech. Syst., vol. 14, no. 1,
pp. 29-36, 2005.

[20] L. J. Hornbeck. (2006, June 13). Digital Light ProcessingTM: A new MEMS-based display
technology. Texas Instruments. [Online]. Last Access: Sept. 11, 2007. Available:
http://www. dlp. com/downl oads/default. aspx?&ref=/downloads/white_papers/ 117 Digital
Light ProcessingMEMS_display technology .pdf

[21] M.A. Butler, E.R. Deutsch, S.D. Senturia, M.B. Sinclair, W.C. Sweatt, D.W. Younger, and
G.B. Hocker, "A MEMS-based programmable diffraction grating for optical holography in
the spectral domain," in IEDM~ Tech. Digest Int. Electron Devices M~eeting, Washington,
D.C., 2-5, 2001, pp. 41.1.1-41.1.4.

[22] E.R. Deutsch, J.P. Bardhan, S.D. Senturia, G.B. Hocker, D.W. Youngner, M.B. Sinclair,
and M.A. Butler, "A large-travel vertical planar actuator with improved stability," in 12th
Int. Conf: on Solid State Sensors, Actuators and Microsystems, Boston, MA, 2003, pp. 352-
355.

[23] G.B. Hocker, D. Younger, E. Deutch, A. Volipicelli, S. Senturia, M. Butler, M. Sinclair, T.
Plowman, and A.J. Ricco, "The polychrometer: a MEMS diffraction grating for synthetic
spectra, in Proc. Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC,
2000, pp. 89-91.

[24] E.S. Hung, and S.D. Senturia, "Extending the travel range of analog-tuned electrostatic
actuators," Journal M~icroelectromechanical Systems, vol. 8, no. 4, pp. 497-505, 1999.


209










[25] S.D. Senturia, "Diffractive MEMS: the polychrometer and related devices," in IEEE Conf:
OpticalM~EMS, Lugano, Switzerland, 2002.

[26] N.O.P. Arancibia, S. Gibson, and T.-C. Tsao, "Adaptive control of MEMS mirrors for
beam steering," in Proc. ASM~E Int. Mechanical Eng. Congress and Expo., Anaheim, CA,
2004.

[27] B.-S. Kim, S. Gibson, and T-C. Tsao, "Adaptive control of a tilt-mirror for laser beam
steering," Proc. American Control Conference, Boston, MA, 2004, pp. 3417-3421.

[28] N. G. Dagalakis, T. LeBrun, and J. Lippiatt, "Micro-mirror array control of optical tweezer
trapping beams," in Proc. 2ndlEEE Conf: Nanotechnology, Washington D.C., 2002. pp.
177-180.

[29] T.G. Bifano, J. Perreault, R.K. Mali, and M.N. Horenstein, "Micromechanical deformable
mirrors," IEEE J. ofSel. Topics Quantum Electron., vol. 5, no. 1, pp.83-89, 1999.

[30] M.N. Horenstein, S. Pappas, A. Fishov, and T.G. Bifano, "Electrostatic micromirrors for
subaperturing in an adaptive optics system," J. Electrostatics, vol.54, pp. 321-332, 2002.

[31] T. Weyrauch, M.A. Vorontsov, T.G. Bifano, J.A. Hammer, M. Cohen, and G.
Cauwenberghs, "Microscale adaptive optics: wave-front control with a CL-mirror array and a
VLSI stochastic gradient descent controller," Applied Optics, vol. 40, no. 24, pp. 4243-
4253, 2001.

[32] J.B. Stewart, T.G. Bifano, P. Bierden, S. Cornelissen, T. Cook, and B.M. Levine, "Design
and development of a 329-segment tip-tilt piston mirrorarray for space-based adaptive
optics," in Proc. SPIE, vol. 6113, 2006, pp. 181-189.

[33] J.J. Gorman, and N.G. Dagalakis, "Modeling and disturbance rejection control of a
nanopositioner with application of beam steering," in Proc. ASM~E Int. M~ech. Eng. Cong.,
Washington, D.C., 2003.

[34] J.J. Gorman, N.G. Dagalakis, and B.G. Boone, "Multi-loop control of a nanopositioning
mechanism for ultra-precision beam steering," in Proc. SPIE, vol. 5160, 2005, pp. 170-181.

[35] Sandia National Laboratories. (1999, Nov. 11). Sandia micromirrors may be part of next
generation space telescope. [Online]. Last Access: Sept. 11, 2007. Available:
http://www. sandi a.gov/medi a/NewsRel/NR 1999/space.htm

[36] J.R.P. Angel, "Ground-based imaging of extrasolar planets using adaptive optics," Nature,
vol. 368, pp. 203-207, 1994.

[37] A. Roorda, and D.R. Williams, "The arrangement of the three cone classes in the living
human eye," Nature, vol. 397, pp. 520-522,1999.


210










[38] R.D. Ferguson, D.X. Hammer, C.E. Bigelow, N.V. Iftimia, T.E. Ustun, S.A. Burns, A.E.
Elsner, and D.R. Williams, "Tracking adaptive optics scanning laser ophthalmoscope," in
Proc. SPIE, vol. 6138, 2006, pp. 232-240.

[39] R.K. Gupta, and S.D. Senturia, "Pull-in time dynamics as a measure of absolute pressure,"
in Proc. IEEE 10th AnnuallInt. Workshop on M~EMS, 1997, pp. 290-294.

[40] J.I. Seeger, and B.E. Boser, "Parallel-plate driven oscillations and resonant pull-in," in
Tech. Dig. Solid-State Sensor, Actuator, and2~icrosyst. Workshop, 2002, pp. 313-316.

[41] D. Bernstein, R. Guidotti, and J.A. Pelesko, "Analytical and numerical analysis of
electrostatically actuated MEMS devices," in Proc. Modeling Simulation M~icrosyst., 2000,
pp. 489-492.

[42] O. Degani, E. Socher, A. Lipson, T. Leitner, D.J. Setter, S. Kaldor, and Y. Nemirovsky,
"Pull-in study of an electrostatic torsion microactuator," J. M~icroelectromech. Syst., vol. 7,
no. 4, pp. 373-379, 1998.

[43] G. Flores, G.A. Mercado, and J.A. Pelesko, "Dynamics and touchdown in electrostatic
MEMS," in Proc. oflDETC/CIE 19th ASM~E Biennial Conf: Mech. Vibration and Noise,
Chicago, IL, 2003.

[44] O. Francais, and I. Dufour, "Dynamic simulation of an electrostatic micropump with pull-in
hysteresis phenomena," Sensors and Actuators A, vol. 70, pp. 56-60, 1998.

[45] M.-A. Gretillat, Y.-J. Yang, E.S. Hung, V. Rabinovich, G.K. Ananthasuresh, N.F. de Rooij,
and S.D. Senturia, "Nonlinear electromechanical behavior of an electrostatic microrelay,"
in Transd'ucers '97 IEEE Int. Conf: Solid-State Sensors and Actuators, Chicago, IL, 1997,
pp. 1141-1144.

[46] D. Hah, H. Toshiyoshi, and M.C. Wu, "Design of electrostatic actuators for MOEMS
applications," in Design, Test, Integration, and Packaging ofM\~EM\S/M\~OEMS~ 2002 Proc.
SPIE, vol. 4755.

[47] D. Hah, P.R. Patterson, H.D. Nguyen, H. Toshiyoshi, and M.C. Wu, "Theory and
experiments of angular vertical comb-drive actuators for scanning micromirrors," IEEE J
Sel. Topics Quantum Electron., vol. 10, no. 3, pp. 505-513, 2004.

[48] R. Legtenberg, J. Gilbert, S.D. Senturia, and M. Elwenspoek, "Electrostatic curved
electrode actuators," J. M~icroelectromech. Syst., vol. 6, no. 3, pp. 257-265, 1997.

[49] Y. Nemirovsky, and O. Bochobza-Degani, "A methodology and model for the pull-in
parameters of electrostatic actuators," J. M~icroelectromech. Syst., vol. 10, no. 4, pp. 601-
615, 2001.










[50] Z. Xiao, XT Wu, W. Peng, and K.R. Farmer, "An angle-based approach for rectangular
electrostatic torsion actuators," J. M~icroelectromech. Syst., vol. 10, no. 4, pp. 561-568,
2001.

[51] Z. Xiao, W. Peng, and K.R. Farmer, "Analytical behavior of rectangular electrostatic
torsion actuators with nonlinear spring bending," J. M~icroelectromech. Syst., vol. 12, no. 6,
pp. 929-936, 2003.

[52] X.M. Zhang, F.S. Chau, C. Quan, Y.L Lam, and A.Q. Liua, "A study of the static
characteristics of a torsional micromirror, Sensors and Actuators A, vol. 90, pp. 73-81,
2001.

[53] J. Zhang, Y.C. Lee, V.M. Bright, and J. Neff, "Digitally positioned micromirror for open-
loop controlled applications," in Proc. IEEEM~EMS, 2002, pp. 536-539.

[54] Z. Xiao, and K.R. Farmer, "Instability in micromachined electrostatic torsion actuators with
full travel range," in Transducers 2003 12th Int. Conf: Solid State Sensors, Actuators and
M~icrosyst., Boston, MA, 2003, pp. 1431-1434.

[55] X. Wu, R.A. Brown, S. Mathews, and K.R. Farmer, "Extending the travel range of
electrostatic micro-mirrors using insulator coated electrodes," in 2000 IEEE/LEOS Int.
Conf: OpticalM~EMS, 2000, pp. 151-152.

[56] D.M. Burns, and V.M. Bright, "Nonlinear flexures for stable deflection of an
electrostatically actuated micromirror," in Proc. SPIE, vol. 3226, 1997, pp. 125-136.

[57] J.R. Bronson, G.J. Wiens, and J.J. Allen, "Modeling and alleviating instability in a MEMS
vertical comb drive using a progressive linkage," in Proc. IDETC/CIE 2005 ASM~E Int.
Design Eng. Tech. Conf Long Beach, CA, 2005.

[58] J.J. Allen, J.R. Bronson, and G.J. Wiens, "Extended-range tiltable micromirror," US Patent
SD7962S106240 (pending). Technical Advance filed March 1, 2005. Application filed
July 28, 2006.

[59] J.I. Seeger, and S.B. Crary, "Stabilization of electrostatically actuated mechanical devices,"
in Transducers '97 IEEE Int. Conf: on Solid-State Sensors and Actuators, Chicago, IL,
1997, pp. 1133-1136.

[60] J.I. Seeger, and B.E. Boser, "Dynamics and control of parallel plate actuators beyond the
electrostatic instability," in Transducers '99 10th Int. Conf: on Solid-State Sensors and
Actuators, Sendai, Japan, 1999, pp. 474-477.

[61] E.K. Chan, and R. W. Dutton, "Electrostatic micromechanical actuator with extended range
of travel," J. M\~icroelectromechan. Syst., vol. 9, no. 3, pp. 321-328, 2000.


212










[62] J.A. Pelesko, and A.A. Triolo, "Nonlocal problems in MEMS device control," J. Eng
Math., vol. 41: pp. 345-366, 2001.

[63] E.K. Chan, and R. W. Dutton, "Effects of capacitors, resistors and residual charge on the
static and dynamic performance of electrostatically actuated devices," in Proc. SPIE, vol.
3680, 1999, pp. 120-30.

[64] J.I. Seeger, and B.E. Boser, "Charge control of parallel-plate, electrostatic actuators and the
tip-in instability," J. M~icroelectromech. Syst., vol. 12, no. 5, pp. 656-671, 2003.

[65] R.N. Guardia, R. Aigner, W. Nessler, M. Handtmann, L.M Castaner, "Control positioning
of torsional electrostatic actuators by current driving," in Thirdlnt. Euro. ConJ Advanced
Semiconductor Devices and2~icrosyst., Sinolenice Castle, Slovakia, 2000, pp. 91-94.

[66] R. Nadal-Guardia, A. Dehe, R. Aigner, and L.M. Castaner, "Current drive methods to
extend the range of travel of electrostatic microactuators beyond the voltage pull-in point, "
J. M~icroelectromech. Syst., vol. 11, pp. 255-263, 2002.

[67] J.M. Kyynaraninen, A. S. Oj a, and H. Seppa, "Increasing the dynamic range of a
micromechanical moving plate capacitor," Anal. Intergr. Circuits Signal Processing, vol.
29, no. 1-2, pp. 61-70, 2001.

[68] P.B. Chu, and K.S.J. Sister, "Analysis of closed-loop control of parallel-plate electrostatic
microgrippers," in Proc. Int. ConJ Robotics and Automation, San Diego, CA, 1994, pp.
820-825.

[69] J. Chen, W. Weingartner, A. Azarov, and R.C. Giles, "Tilt-angle stabilization of
electrostatically actuated micromechanical mirrors beyond the pull-in point," J.
M~icroelectromech. Syst., vol. 13, no. 6, pp. 988-997, 2004.

[70] M. S. Lu, and G. Fedder, "Closed-loop control of a parallel-plate microactuator beyond the
pull-in limit," in Solid-State Sensor Actuator and Microsyst. Workshop, Hilton Head, SC,
2002, pp. 255-258.

[71] M. S.-C. Lu, and G.K. Fedder, "Position control of parallel-plate microactuators for probe-
based data storage," J. M~icroelectromech. Syst., vol. 13, no. 5, pp. 759-769, 2004.

[72] D. Piyabongkarn, Y.Sun, R. Rajamani, A. Sezen, and B.J. Nelson, "Travel range extension
of a MEMS electrostatic microactuator," IEEE Trans. Control Syst. Tech., vol. 13, no. 1,
pp. 138-145, 2005.

[73] H.S. Sane, "Energy-based control for MEMS with one-sided actuation," in Proc. Amer.
Controls ConJ, Minneapolis, MN, 2006, pp. 1209-1214.


213










[74] D.H. S. Maithripala, J.M. Berg, and W.P. Dayawansa, "Nonlinear dynamic output feedback
stabilization of electrostatically actuated MEMS," in Proc. 42nd IEEE Conf: Decision and
Control, Maui, HI, 2003, pp. 61-66.

[75] P. Cheung, R. Horowitz, ad R. Howe, "Design, fabrication, position sensing and control of
an electrostatically-driven polysilicon microactuator," IEEE Trans. Magnetics, vol. 32, pp.
122-128, 1996.

[76] K.M. Liao, Y.C. Wang, C.H. Yeh, and R. Chen, "Closed-loop adaptive control for
electrostatically driven torsional micromirrors," J. M~icrolith., M~icrofab., Microsyst., vol. 4,
no. 4, 2005.

[77] D.A. Horsley, N. Wongkomet, N. Horowitz, and A.P. Pisano, "Precision positioning using
a microfabricated electrostatic actuator," IEEE Trans. Magnetics, vol. 35, no. 2, pp. 993-
999, 1999.

[78] S. Pannu, C. Chang, R.S. Muller, and A.P. Pisano, "Closed-loop feedback-control system
for improved tracking in magnetically actuated micromirrors," in IEEE/LEOS Opt. M\EM\S
2000, pp. 107-108.

[79] R.K. Messenger, T.W. McLain, L.L. Howell, "Feedback control of a thermomechanical
inplane microactuator using piezoresistive displacement sensing," in ASM~EInt.
Mechanical Eng. Congress Expo, Anaheim, CA, 2004.

[80] R.K. Messenger, T.W. McLain, L.L. Howell, "Piezoresistive feedback for improving
transient response of MEMS thermal actuators," in Proc. SPIE, vol. 6174, 2006.

[81] N. Yazdi, H. Sane, T.D. Kudrle, and C.H. Mastrangelo, "Robust sliding-mode control of
electrostatic torsional micromirrors beyond the pull-in limit," Transducers 2003 IEEE Int.
Conf: Solid-State Sensors and Actuators, 2003, pp. 1450-1453.

[82] D. Hernandez, S.-S. Park, R. Horowitz, A.K. Packard, "Dual-stage track-following servo
design for hard disk drives," in Proc. Amer. Control Conf 1999.

[83] H.C. Liaw, D. Oetomo, B. Shirinzadeh, and G. Alici, "Robust motion tracking control of
piezoelectric actuation systems," in Proc. IEEE Int. Conf: Robotics and Automation,
Orlando, FL, 2006, pp. 1414-1419.

[84] S.E. Lyshevski, "Nonlinear microelectromechanical systems (MEMS) analysis and design
via the lyapunov stability theory," in Proc. IEEE Conf: Decision and Control, vol. 5, 2001,
pp. 4681-4686.

[85] G. Zhu, J. Penet, and L. Saydy, "Robust control of an electrostatically actuated MEMS in
the presence of parasitics and parametric uncertainties," in Proc. American Controls Conf ,
Minneapolis, MN, 2006, pp. 1233-1238.


214










[86] S.H. Lee, S.E. Bak, and Y.H. Kin, "Design of a dual-stage actuator control system with
discrete-time sliding mode for hard disk drives," in Proc. IEEE Conf: Decision and'
Control, vol. 4, 2000, pp. 3120-3125.

[87] J.C. Chiou, Y.C. Lin, and S.D. Wu, "Closed-loop fuzzy control for torsional micromirror
with multiple electrostatic electrodes," in IEEE/LEOS Opt. M~EMS, 2002, pp. 85-86.

[88] D.V. Dao, T. Toriyama, J. Wells, and S. Sugiyama, "Silicon piezoresistive six-degree of
freedom force-moment micro sensor," Sensors and2aterials, vol. 15, no. 3, pp. 113-135,
2002.

[89] D.V. Dao, T. Toriyama, J. Wells, and S. Sugiyama, "Six-degree of freedom micro force-
moment sensor for applications in geophysics," in 15th Annual IEEE Conf: onM2EM~S, Las
Vegas, NV, pp. 312-315.

[90] Y. Li, M. Papila, T. Nishida, L. Cattafesta, and M. Sheplak, "Modeling and optimization of
a side-implanted piezoresistive shear stress sensor," in Proc. SPIE 13th AnnuallInt. Symp.
Smart Structures andMaterials, San Diego, CA, 2006.

[91] G. Roman, J. Bronson, G. Wiens, J. Jones, and J. Allen, "Design of a piezoresistive surface
micromachined three-axis force transducer for microassembly," in Proc. Int. M~ech. Eng.
Conf~, Orlando, FL, 2005.

[92] R. Schellin, and G. Hess, "Silicon subminiature microphone based on piezoresistive
polysilicon strain gauges," Sensors and'Actuators A, vol. 32, no. 1-3, pp. 555-559,1992.

[93] D.P. Arnold, T. Nishida, L.N. Cattafesta, and M. Sheplak, "A directional acoustic array
using silicon micromachined piezoresistive microphones," J. Acoust. Soc. Amer., vol. 113,
no. 1, pp. 289-298, 2003.

[94] H.L. Stalford, C. Apblett, S.S. Mani, W.K. Schubert, and M. Jenkins, "Sensitivity of
piezoresistive readout device for microfabricated acoustic spectrum analyzer," in Proc.
SPIE, vol. 5344, pp.36-43.

[95] R. Dieme, G. Bosman, M. Sheplak, and T. Nishida, "Source of excess noise in silicon
piezoresistive microphones," J. Acoust. Soc. Amer., vol. 119, pp. 2710-2720, 2006.

[96] H. Xie, and G.K. Fedder, "Vertical comb-finger capacitive actuation and sensing for
CMOS-MEMS," Sensors & Actuators A, vol. 95, pp. 212-221, 2002.

[97] B.D. Jensen, M.P de Boer, N.D. Masters, F. Bitsie, and D.A. LaVan, "Interferometry of
actuated microcantilevers to determine material properties and test structure nonidealities
in MEMS," J. M~icroelectromech. Syst., vol. 10, no. 3, 2001.


215










[98] S. Limary, H. Sterart, L. Irwin, J. McBrayer, J. Sniegowski, S. Montague, J. Smith, M. de
Boer, and J. Jakubczak, "Reproducability data on SUMMiT," in Proc. SPIE, vol. 3874,
1999, pp. 102-112.

[99] ANSYS, Inc. (2007, June 19). ANSYS Product Information and Documentation. [Online].
Last Access: Sept. 11, 2007. Available: http://www.ansys.com

[100] F. Pan, J. Kubby, E. Peeters, A.T. Tran, and S. Mukherjee, "Squeeze-film damping effect
on the dynamic response of a MEMS torsion mirror," in Tech. Proc. Int. Cong: Modeling
and Simulation of2~icrosystems, 1998, pp. 474-479.

[101] M. Zecchino, and E. Novak. (2001, June 19). MEMS in motion: a new method for
dynamic MEMS metrology. Veeco, Inc. website. [Online]. Last Access: Sept. 11, 2007.
Available: http://www.veeco.com/ pdfs.php/69

[102] K. Ogata, M~odern Control Engineering, Fourth Ed., Upper Saddle River, NJ: Prentice
Hall, 2002.

[103] K. Zhou, and J.C. Doyle, Essentials of Robust Control, Upper Saddle River, NJ: Prentice
Hall, 1998.

[ 104] G.E. Dullerud, and F. Paganini, A Course in Robust Control Theory: A Convex Approach,
New York, NY: Springer, 1999.

[105] A. Ashkin, "History of optical trapping and manipulation of small-neutral particle, atoms,
and molecules," IEEE J. Sel. Topics Quantum Electronics, vol. 6, no. 6, pp. 841-856, 2000.

[106] J.A. Pelesko, and D.H. Bernstein, M\~odelingM\~EM\~S andNEM~S, Boca Raton, FL: Chapman
and Hall/CRC, 2003.

[ 107] A.H. Nayfeh, and B. Balachandran, Applied Nonlinear Dynamics: Analytical,
Computational, and Experimental M~ethods, New York, NY: John Wiley and Sons, 1995.

[108] R.C. Dorf, and R.H. Bishop, M~odern Control Systems 9th ed., Upper Saddle River, NJ:
Prentice Hall, 2001.

[ 109] J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical
Measurements 2nd ed., Sausalito, CA: University Science Books, 1982.

[1 10] L. Guilbeau, "The history of the solution of the cubic equation," Mathemat~l,(iL \ News
Letters, vol. 5, no. 4, 1930.

[1 11] D. Elata, and H. Bamberger, "On the dynamic pull-in of electrostatic actuators with
multiple degrees of freedom and multiple voltage sources," J. M~icroelectromech. Syst., vol.
15, no. 1, 2006.


216










[112] L. Mattsson, "Experiences and challenges in multi material micro metrology," in Proc. 2nd
Annuallnt. Conf: M\icromanufa~cutring, Clemson, SC, 2007, pp. 318-322.

[1 13] W.A. Moussa, H. Ahmed, W. Badawy, and M. Moussa, "Investigating the reliability of the
electrostatic comb-drive actuators utilized in microfluidic and space systems using finite
element analysis," Canad'ian J. Electrical Computer Syst., vol. 27, no. 4, pp. 195-200,
2002.


217









BIOGRAPHICAL SKETCH

Jessica Bronson graduated with honors from the University of Missouri at Columbia with a

B.S. in mechanical engineering in December 2002. Ms. Bronson began her graduate studies in

January 2003 under Professor Gloria Wiens in the Space Automation and Manufacturing

Mechanisms Laboratory at the University of Florida in Gainesville. Shortly after beginning

graduate school, Ms. Bronson was awarded an internship at Sandia National Laboratories in

Albuquerque, New Mexico as a fellow through the Microsystems, Engineering, and Science

Applications (MESA) Institute at Sandia. In 2004, she was granted the Sandia National

Laboratories Campus Executive Fellowship that allowed her to continue to develop her research

program at the university, in addition to returning to New Mexico for internships at Sandia each

summer for the next three years. The focus of her Ph.D. research is to develop and implement

closed-loop control systems for Microelectromechanical Systems (MEMS) micromirrors. The

impact of this research is that it will increase accuracy, performance and repeatability leading to

advances in imaging and communications technology. Upon completion of her Ph.D., Ms.

Bronson hopes to continue her work in MEMS and control systems by obtaining a position at a

leading research laboratory.


218





PAGE 1

1 MODELING AND CONTROL OF ME MS MICROMIRROR ARRAYS WITH NONLINEARITIES AND PARAMETRIC UNCERTAINTIES By JESSICA RAE BRONSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

PAGE 2

2 2007 Jessica Rae Bronson

PAGE 3

3 To my parents

PAGE 4

4 ACKNOWLEDGMENTS I thank first of all, my supe rvisory committee chair, Gloria Wiens, for the opportunities to attend the University of Florida and to conduct this work. She is also responsible for providing me with the exceptional experience of worki ng closely with Sandia National Laboratories through the summer internship progra m. I also thank James Allen for his guidance as my mentor at Sandia National Laboratories and for shaping the scope of this work. I thank all my committee members, Louis Cattafesta, Norman F itz-Coy, and Toshikazu Nishida for their time and consideration. Additionally I thank my family, especially my pa rents, for their loving support in all of my endeavors. I also thank my many friends and classmates in the Space, Automation, and Manufacturing Mechanisms (SAMM) Laboratory a nd in the Department of Mechanical and Aerospace Engineering for their friendship and camaraderie. In particular, I thank Adam Watkins for the many roles he plays as friend, colleague, mentor and partner.

PAGE 5

5 TABLE OF CONTENTS Page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............17 CHAPTER 1 INTRODUCTION..................................................................................................................19 1.1 Motivation.....................................................................................................................19 1.2 Research Objectives......................................................................................................20 2 LITERATURE REVIEW.......................................................................................................23 2.1 Microelectromechanical Systems.................................................................................23 2.2 Micromirrors and Applications.....................................................................................24 2.3 Electrostatic Actuati on and Instability..........................................................................28 2.3.1 Modeling, Pull-in and Hysteresis......................................................................28 2.3.2 Design Techniques to Eliminate Pull-in...........................................................31 2.3.3 Capacitive and Charge Control Techniques to Eliminate Pull-in.....................32 2.3.4 Closed-loop Voltage Contro l to Eliminate Pull-in............................................33 2.4 Feedback Control Techniques Applied to MEMS........................................................34 2.4.1 Linear Control...................................................................................................35 2.4.2 Adaptive and Robust Control............................................................................37 2.4.3 Nonlinear Control.............................................................................................39 2.5 Sensing Methods for Feedback.....................................................................................41 2.6 Summary Remarks........................................................................................................43 3 MICROMIRROR MODELI NG AND STATIC PERFORMANCE......................................45 3.1 Description of the SUMMiT V Microfabrication Process............................................45 3.2 Micromirror Actuator Description................................................................................47 3.3 Electrostatic Actuati on and Instability..........................................................................50 3.3.1 Parallel Plate Electrostatics...............................................................................50 3.3.2 Parallel Plate Torsion Actuator.........................................................................56 3.4 Model for Vertical Comb Drive Actuator.....................................................................60 3.4.1 Mechanical Model.............................................................................................61 3.4.2 Electrostatic Model...........................................................................................63 3.4.3 Electromechanical Model.................................................................................66 3.4.4 Linear Approximation.......................................................................................70 3.4.5 Bifurcation Analysis.........................................................................................73 3.5 Chapter Summary..........................................................................................................78

PAGE 6

6 4 UNCERTAINTY ANALYSIS AND EXPE RIMENTAL CHARACTERIZATION............81 4.1 Parametric Uncertainty a nd Sensitivity Analysis..........................................................81 4.1.1 Effects of Individual Parameter Variations.......................................................84 4.1.2 Monte Carlo Simulations..................................................................................91 4.2 Experimental Characterization....................................................................................100 4.2.1 Equipment Description...................................................................................100 4.2.2 Static Results for Single Micromirrors...........................................................103 4.2.3 Static Results for Micromirror Arrays............................................................105 4.3 Chapter Summary........................................................................................................111 5 DYNAMIC MODEL AND HYSTERESIS STUDY...........................................................113 5.1 Dynamic Model and Resonant Frequency Determination..........................................113 5.1.1 Modal Analysis...............................................................................................114 5.1.2 Dynamic Characterization...............................................................................119 5.2 Open-Loop Step Response..........................................................................................122 5.2.1 Effects of Parametric Uncertainty on Step Response.....................................123 5.2.2 Effects of Pull-in and Hysteresis on Open-Loop Response............................123 5.2.3 Continuous Characterization of Micromirror Arrays......................................126 5.3 Hysteresis Case Study: Progressive-Linkage..............................................................128 5.3.1 Progressive-Linkage Design...........................................................................128 5.3.2 Open-Loop Response Using a Progressive-Linkage......................................135 5.3.3 Parametric Sensitivity of the Progressive-Linkage.........................................135 5.3.4 Progressive-Linkage Prototype...................................................................................139 5.5 Chapter Summary........................................................................................................144 6 CONTROL DESIGN AND SIMULATION........................................................................146 6.1 PID Control.................................................................................................................147 6.1.1 PID Control Theory........................................................................................147 6.1.2 PID Results.....................................................................................................148 6.1.3 PID Controller Response to Hysteresis...........................................................149 6.2 LQR Control...............................................................................................................154 6.2.1 LQR Control Theory.......................................................................................154 6.2.2 State Estimation..............................................................................................159 6.2.3 LQR Results....................................................................................................160 6.2.4 LQR Controller Response to Hysteresis.........................................................163 6.3 Modeling the Micromirror Array................................................................................164 6.3.1 Modeling the Array of Mirrors.......................................................................165 6.3.2 Sensor Types...................................................................................................167 6.4 Modeling the Sensor Response...................................................................................170 6.4.1 PSD Response.................................................................................................174 6.4.2 CCD Response................................................................................................179 6.4.3 Summary of Sensor Analysis..........................................................................181 6.5 Chapter Summary........................................................................................................181

PAGE 7

7 7 CONCLUSIONS AND FUTURE WORK...........................................................................183 APPENDIX A MODEL GEOMETRY.........................................................................................................188 B MONTE CARLO SIMULATION INPUTS.........................................................................189 C LASER DOPPLER VIBROMETER RESULTS..................................................................200 LIST OF REFERENCES.............................................................................................................208 BIOGRAPHICAL SKETCH.......................................................................................................218

PAGE 8

8 LIST OF TABLES Table page 2-1 Summary of feedback control papers discussed in the l iterature review...........................36 3-1 Mean and standard deviation of fabri cation variations for layer thickness in the SUMMiT V surface micromachining process...................................................................47 3-2 Mean and standard deviation of fabricat ion variations of line widths in SUMMiT V......47 3-3 Values output from finite element an alysis of mechanical spring stiffness.......................63 3-4 Comparison of polynomial fit for approximation of cap acitance function........................66 3-5 Comparison of polynomial fit for approximation of cap acitance function........................66 3-6 List of parameters used for this analysis............................................................................78 4-1 Mean and standard deviation of fabri cation variations for layer thickness in the SUMMiT V surface micromachining process...................................................................82 4-2 Mean and standard deviation of fabricat ion variations of line widths in SUMMiT V......82 4-3 Spring stiffness values for changing dimensional and material parameters......................85 4-4 Results from the Monte Carlo simulations for the capacitance values in terms of mean, standard deviation, and the percent change from nominal......................................97 4-5 Mean and standard deviation for pull-in a ngle and voltage from sets of mirrors on all three arrays tested............................................................................................................108 5-1 Modal analysis results for first 10 mode s and their natural frequencies, and the participation factors and ra tios for each direction............................................................118 5-2 The first three natural frequencies de termined from the LDV experiment.....................122 5-3 Link length dimensions used for progressive-linkage design..........................................134 5-4 Joint dimensions used fo r progressive-linkage design.....................................................134 5-5 Uncertainties in the joint dimensions for a proposed progressive-linkage design...........138

PAGE 9

9 LIST OF FIGURES Figure page 2-1 The SEM images of MEMS devices created using SUMMiT V microfabrication process........................................................................................................................ ........25 2-2 Images of micromirror arra ys developed in industry.........................................................26 2-3 Adaptive optics (AO) mirror used for wavefront correction.............................................27 2-4 Use of an AO MEMS programmabl e diffraction grating for spectroscopy.......................27 3-1 Drawing of the SUMMiT V structural and sacrificial layers............................................46 3-2 Area with nominal dimensions L and w w ith the dashed line indicating the actual area due to error in the line width......................................................................................47 3-3 Images of the micromirror array........................................................................................48 3-4 Illustration of mirrors operating as an optical diffraction grating......................................48 3-5 Micrograph of an array of mirrors and schematic of mirro r with hidden vertical comb drive.......................................................................................................................... .........49 3-6 Solid model of micromirror showi ng polysilicon layer names from SUMMiT V............49 3-7 Schematic of a parallel plate electrosta tic actuator modeled as a mass-spring-damper system......................................................................................................................... .......51 3-8 Static equilibrium relationship for th e parallel plate elect rostatic actuator.......................54 3-9 Electrostatic force for different voltage s and mechanical force showing pull-in for the electrostatic para llel plate actuator...............................................................................54 3-10 Pull-in function for the parall el-plate electrostatic actuator..............................................55 3-11 Static equilibrium relationships for the parallel plate actuator using different spring constants...................................................................................................................... .......56 3-12 Schematic of a torsion electrostatic actuator.....................................................................57 3-13 Static equilibrium relations hips for the torsion actuator....................................................59 3-14 Pull-in function for the torsion actuator.............................................................................60 3-15 Drawing of the mechanical spring that supports the micromirrors and provides the restoring force................................................................................................................ ....61

PAGE 10

10 3-16 Image of the mechanical spring that supports the micromirro r indicating boundary conditions and location for applying displacem ent loads for finite element analysis.......62 3-17 Image from ANSYS of the deformed spri ng and the outline of the undeformed shape after displacements are applied..........................................................................................63 3-18 Solid model geometry of the unit cell us ed in the electrostatic FEA simulation...............65 3-19 Capacitance calculation as a function of rotation angle, calculated using 3D FEA and varying orders of polynomial curve fit approximations..............................................66 3-20 Plot of the Pull-in function PI( ) for the micromirror with the vertical comb drive actuator showing that pull-in occurs at 16.5 degrees.........................................................67 3-21 Electrostatic and Mechani cal torque as a function of rotation angle, theta, and voltage for different voltage values...................................................................................68 3-22 Torque as a function of rotation angle, theta, and voltage for different values of mechanical spring constant................................................................................................69 3-23 Plot of static equilibrium behavior, s howing pull-in and hysteresis, predicted from the model...................................................................................................................... ......69 3-24 Static equilibrium relationships for the nonlinear plant model, and the linear plant approximation.................................................................................................................. ..70 3-25 Static equilibrium relationships for the nonlinear plant model, and the small signal model linearized about an operating point ( 0, V0)............................................................72 3-26 Illustration of piecewise linearizati on about multiple operating points.............................73 3-27 Plot showing the roots of the function F(x1) occur where the function crosses zero.........76 3-28 Bifurcation diagram for a MEMS torsion mirror with electrostatic vertical comb drive actuator................................................................................................................. ....79 3-29 Bifurcation diagram showing the e ffects of different spring constants.............................79 4-1 Fabrication tolerances can changes the thicknesses of the layers, resulting in changes in the final geometry dimensions.......................................................................................82 4-2 Fabrication tolerances can change the dime nsions of a fabricated geometry, affecting the final shape, volume, and mass.....................................................................................83 4-3 Nominal dimensions used to calculate the volume of the moving mass...........................83 4-4 Capacitance functions for the electrosta tic model with parametric changes in the layer thickness of the structural polysilicon.......................................................................88

PAGE 11

11 4-5 Capacitance functions for the electrosta tic model with parametric changes in the layer thickness of the Dimple3 b ackfill and the sacrificial oxide......................................88 4-6 Capacitance functions for the electrosta tic model with parametric changes in the linewidth error of the structural polysilicon layers............................................................89 4-7 Static displacement relationships for the micromirror model with parametric changes in the layer thickness of the structural polysilicon.............................................................89 4-8 Static displacement relationships for the micromirror model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide............................90 4-9 Static displacement relationships for the micromirror model with parametric changes in the linewidth error of the structural polysilicon layers..................................................90 4-10 Sensitivity of voltage with respect to change s in line width for each value of ...............92 4-11 Sensitivity of voltage with respect to changes in layer thickness for each value of .......92 4-12 Gaussian distribution with a mean of 0 and standard deviation of 1.................................94 4-13 Histogram for mechanical stiffness when accounting for variations in thickness of MMPoly1 and Youngs modulus.......................................................................................95 4-14 Histogram for mechanical stiffness taki ng into account varia tions in thickness of MMPoly1, Youngs modulus, and linewidth of MMPoly1...............................................95 4-15 Results from the capacitance simulation fo r 250 random variable sets that show the effects of parametric uncertain ty on the electrostatic model.............................................96 4-16 Static displacement results of 250 Mont e Carlo simulations with random Gaussian distributed dimensional variations.....................................................................................98 4-17 Histogram of values from the Monte Ca rlo simulations for the layer thickness of Sacox3......................................................................................................................... .......99 4-18 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the Sacox3 layer thickness...............................................................99 4-19 Diagram of an optical profiler measurement system.......................................................101 4-20 Six mirrors from the micromirror array m easured with the opti cal profiler system........102 4-21 Data records from the SureVision display that show the cross-section profile of the tilt angle measurements....................................................................................................103 4-22 Micrograph image of a single micromirror......................................................................104

PAGE 12

12 4-23 Experimental static result s taken from individual microm irrors that are not in an array.......................................................................................................................... .......104 4-24 Approximate locations of data collection on all three arrays..........................................106 4-25 Experimental results from array 1, area A.......................................................................106 4-26 Experimental results from array 2, areas D and E...........................................................107 4-27 Experimental results from array 3, areas A and D...........................................................107 4-28 Nominal model with experimental data...........................................................................109 4-29 Model-predicted results from 100 simu lations with parameters determined by random Gaussian variations, show n with experimental data...........................................109 5-1 Open-loop nonlinear plant response to a step input of 7 degrees for different damping ratios......................................................................................................................... ........114 5-2 Solid model created for modal analysis...........................................................................118 5-3 Time series data of the micromirror re sponse to an acoustic impulse taken with a laser doppler vibrometer..................................................................................................120 5-4 Results from the LDV experi ment showing resonant peaks............................................121 5-5 Open-loop response to a step input of 7 degrees for the nonlinear plant dynamics and variations in spring stiffness, km......................................................................................124 5-6 Open-loop nonlinear plant response to a step input of 7 de grees for 50 random parameter variations.........................................................................................................124 5-7 Open-loop responses to a sinus oidal input showing hysteresis.......................................126 5-8 Open-loop responses to a step command s howing overshoot that result in pull-in.........126 5-9 Results from dynamic study showing pull-in and hysteresis...........................................127 5-10 Results showing the hysteretic behavior of the micromirrors.........................................128 5-11 Diagram of four-bar mechanism for progressive linkage analysis..................................130 5-12 Cantilever beam with cross-section w x t and length L ..................................................131 5-13 Free body diagrams for each member of the linkage.......................................................132 5-14 Progressive-linkage behavi or for different values of ro in m.........................................132

PAGE 13

13 5-15 Progressive linkage output for ro equal to 9 m along with the electrostatic torque curves and the linear restoring torque..............................................................................133 5-16 Static -V relationship for micromirror with a progressive-linkage................................134 5-17 Bifurcation diagram for micromirror usi ng a progressive-linkage to avoid pull-in behavior....................................................................................................................... .....136 5-18 Bifurcation diagram for the micromirrors using a progressive-linka ge to avoid pull-in behavior for different values of mechanical stiffness......................................................136 5-19 Open-loop responses to a sinusoidal input for the device using a progressive-linkage...137 5-20 Open-loop response to a step input fo r device using a progressive-linkage....................137 5-21 Results of parametric analysis for indi vidual errors in joint fabrication of the progressive-linkage..........................................................................................................140 5-22 Fifty Monte Carlo simulati on results for varying the join t fabrication parameters for the progressive-linkage design.........................................................................................140 5-23 Schematic drawing of the prot otype progressive linkage spring.....................................141 5-24 Micrograph of the prototype micromirro r with a progressive linkage spring.................141 5-25 Experimental data collected for the prototype of the mi cromirror with the progressive-linkage..........................................................................................................143 5-26 Results from FEA of the prototype progressive-linkage design for linear and nonlinear deflection analysis shows the pr ototype progressive-linkage fails to produce the desired stiffness profile................................................................................143 6-1 Basic block diagram with unity feedback........................................................................146 6-2 Block diagram with PID controller..................................................................................148 6-3 Step responses for PID controller....................................................................................150 6-4 Closed-loop PID response to different step inputs when the spring constant is varied by %........................................................................................................................ ...150 6-5 Closed-loop PID response to a step input of 7 degrees for 50 random sets of parameteric variations......................................................................................................151 6-6 Closed-loop PID response to a co mmanded position in the unstable region...................153 6-7 Closed-loop step responses for PID c ontroller for a system using a progressivelinkage........................................................................................................................ ......153

PAGE 14

14 6-8 General block diagram fo r LQR controller problem.......................................................154 6-9 Block diagram of LQR control with an internal model for tracking a step command....158 6-10 Block diagram of LQR c ontroller using a state-estimat or for a plant without an integrator..................................................................................................................... .....159 6-11 Step responses for LQR controller...................................................................................161 6-12 Closed-loop LQR response to a step i nput of 7 degrees for 50 random parameter variations..................................................................................................................... .....162 6-13 Closed-loop step responses for LQR c ontroller for a system using a progressivelinkage........................................................................................................................ ......164 6-14 Schematic of modeling an array of mirrors as a SIMO system.......................................165 6-15 Schematic drawing of an array of 5 mirrors....................................................................166 6-16 Illustration of the measured center of gravity (CG) on a 1-D PSD when there are errors in the spacing and linea rity of the micromirrors....................................................168 6-17 Illustration of the measured center of gravity (CG) on a 2-D PSD when there are errors in the spacing and linea rity of the micromirrors....................................................168 6-18 Illustration of the measured errors of th e reflected light from two micromirror arrays onto a CCD..................................................................................................................... .169 6-19 Schematic of the beam steering experiment with only one micromirror.........................171 6-20 Geometry used to determine the angle of incidence and reflection.................................172 6-21 Calculating the refl ected ray of light...............................................................................173 6-22 The intersection of the line from B0 to R and the plane C occurs at point P..................174 6-23 Schematic of 5 micromirrors in an array reflecting light onto a PSD sensor..................175 6-24 Calibration of the PSD for idea l case of five micromirrors.............................................176 6-25 Open-loop results to a step response for an array of 5 microm irrors with model uncertainty.................................................................................................................... ....176 6-26 Open-loop response for system with one broken mirror and 4 ideal mirrors, measured by a PSD....................................................................................................................... ...177 6-27 Incorporating feedback control into array model.............................................................177 6-28 Controlled PID step response using PSD sensor.............................................................178

PAGE 15

15 6-29 Controlled LQR step response using PSD sensor............................................................178 6-30 Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a PSD sensor.....................................................................................179 6-31 Schematic of 5 micromirrors in an arra y reflecting light onto a CCD sensor where each separate location of the light can be measured........................................................180 6-32. Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a CCD sensor............................................................................................180 A-1 Geometry dimensions in m for creating electrostatic model.........................................188 B-1 Histogram of values for the thickness of layer MMPoly0...............................................189 B-2 Histogram of values for the thickness of layer MMPoly1...............................................190 B-3 Histogram of values for the thickness of layer MMPoly2...............................................190 B-4 Histogram of values for the thickness of layer MMPoly3...............................................191 B-5 Histogram of values for the thickness of layer MMPoly4...............................................191 B-6 Histogram of values for the thickness of Dimple3 backfill.............................................192 B-7 Histogram of values for the thickness of layer Sacox1....................................................192 B-8 Histogram of values for the thickness of layer Sacox2....................................................193 B-9 Histogram of values for the thickness of layer Sacox3....................................................193 B-10 Histogram of values for the thickness of layer Sacox4....................................................194 B-11 Histogram of values for the linew idth variation of layer MMPoly2...............................194 B-12 Histogram of values for the linew idth variation of layer MMPoly3...............................195 B-13 Histogram of values for the linew idth variation of layer MMPoly4...............................195 B-14 Histogram of the mass values calculated from the parametric variation data.................196 B-15 Histogram of values from the Monte Ca rlo simulations for the linewidth error of MMPoly2........................................................................................................................ .196 B-16 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the linewidth error of MMPoly2....................................................197 B-17 Histogram of values from the Monte Ca rlo simulations for the layer thickness of MMPoly1........................................................................................................................ .197

PAGE 16

16 B-18 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of MMPoly1.............................................................198 B-19 Histogram of values from the Monte Ca rlo simulations for the layer thickness of Sacox4......................................................................................................................... .....198 B-20 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of Sacox4.................................................................199 C-1 Magnitude results of FFT for device 1, trial 1.................................................................200 C-2 Magnitude results of FFT for device 1, trial 2.................................................................200 C-3 Magnitude results of FFT for device 1, trial 3.................................................................201 C-4 Magnitude results of FFT for device 1, trial 4.................................................................201 C-5 Magnitude results of FFT for device 1, trial 5.................................................................202 C-6 Magnitude results of FFT for device 2, trial 1.................................................................202 C-7 Magnitude results of FFT for device 2, trial 2.................................................................203 C-8 Magnitude results of FFT for device 2, trial 3.................................................................203 C-9 Magnitude results of FFT for device 2, trial 4.................................................................204 C-10 Magnitude results of FFT for device 2, trial 5.................................................................204 C-11 Magnitude results of FFT for device 3, trial 1.................................................................205 C-12 Magnitude results of FFT for device 3, trial 2.................................................................205 C-13 Magnitude results of FFT for device 3, trial 3.................................................................206 C-14 Magnitude results of FFT for device 3, trial 4.................................................................206 C-15 Magnitude results of FFT for device 3, trial 5.................................................................207

PAGE 17

17 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODELING AND CONTROL OF ME MS MICROMIRROR ARRAYS WITH NONLINEARITIES AND PARAMETRIC UNCERTAINTIES By Jessica Rae Bronson December 2007 Chair: Gloria J. Wiens Major: Mechanical Engineering Micromirror arrays have resulted in so me of the most successful and versatile microelectromechanical system (MEMS) devices for applications including optical switches, scanning and imaging, and adaptive optics. Many of these devices consis t of large arrays of micromirrors, and it is desirable to ensure accura te positioning capabilitie s for each mirror in the array despite the presence of non linearities or parametric uncertainties from the fabrication process. This research develops analytical mode ls in the electrostatic and mechanical domains to study the effect of fabrication to lerances and uncertainties, electr ostatic pull-in, and hysteresis on the performance of micromirror a rrays, and presents solutions to improve device performance. To achieve these goals, extensive modeling of the electrostatic micromirror arrays is presented. As with many MEMS devices that operate in multiple physical domains, the modeling considers both electrical and mechanical characteristics. The electrical model consists of determining the electrostatic torque produced when an actuation volta ge is applied. The mechanical model considers the opposing torq ue provided by the supporting torsion spring. These models are also used to evaluate the se nsitivity of the micromirrors to parametric uncertainties from the fabrication process by cons idering the effect of each fabrication tolerance individually and also their combined eff ects using Monte Carlo simulations. Additional

PAGE 18

18 characterization of the system dynamics is presen ted through modal analysis in which the results for the full 6 degree-of-freedom (DOF) device ar e compared to the 1 DOF model assumptions. The devices are characterized by measuring th e micromirror rotation as a function of the actuation voltage using an optical profiler to dete rmine static performance, as well as measuring the electrostatic pull-in and hysteresis behavior The measurements, taken for multiple mirrors across three different arrays, vali date the results from analytic al models, and demonstrate the need to compensate for differences in performance. Results from the modeling and characterizati on are used to devel op passive and active control techniques to ensure accu rate position tracking across an array of devices in the presence of model uncertainties. A passive design method is presented called a progressive-linkage that is intended to eliminate the occurren ce of electrostatic pu ll-in and hysteresis. Also, classical and optimal feedback control techniques are utilized to further delineate the impact of the parametric uncertainties on the system performance. As these mirrors are arranged in an array, the performance of individual mirrors is examined, a nd then this control is extended to the problem of controlling an array. This array control probl em is explored by considering different types of feedback error metrics and the sensors that may be used to provide the feedback signal for this system. The impact of the work presented in this di ssertation is an increas ed understanding of the complexities of designing and operating arrays of electrostatic micromirrors for high-precision applications. The modeling methods developed may be extended in future work to include design optimization to decrease the effects of parametric uncertainty on the micromirror performance, as well as developing systems that can easily incorporate feedback mechanisms for implementation of the closed -loop control algorithms.

PAGE 19

19 CHAPTER 1 INTRODUCTION 1.1 Motivation Micromirror arrays have resulted in so me of the most successful and versatile microelectromechanical system (MEMS) devices fo r applications including optical switches for telecommunications, scanning and imaging for pr ojection displays, diffraction gratings for spectroscopy, and adaptive optics for wave front correction. Many of these devices consist of large arrays of micromirrors. As such, it is de sirable to ensure accurate positioning capabilities for each mirror in the array despite the presence of outside disturbances or variations from the fabrication process. The errors due to the fabric ation process can be attrib uted in part to small deviations in dimensional or materi al properties. It is the effect s of these errors that can have significant impact on the performance of the final pr oduct. As such, it is important to evaluate the sensitivity of the micromirror design to determine the potential limitations on the device performance. The diminutive scale of MEMS devices make s electrostatic actuation a popular and effective means of driving microm irrors. One limiting factor to most electrostatic actuators is the electrostatic pull-in instability that occu rs when the electrostatic force overcomes the mechanical restoring force. When pull-in occurs, the device can no longer maintain an equilibrium position and will move to its fully actuated position, limiting the full scanning range available. Another phenomenon associated with this instability is th at once the mirror has pulled-in, the voltage required to maintain the fu lly actuated position is lower than the pull-in voltage. The mirror will not return from this position until the actuating voltage has been reduced below a certain threshold. In order to understand these phenomen a, theoretical models may be developed for the electrostatic and mech anical domains. These models can then be

PAGE 20

20 utilized to evaluate the effects of fabrication e rrors and determine the performance limitations of the micromirrors. These issues can be mitigat ed through the successful application of design methods as well as through feedback control. Currently, state-of-the-art micromirror arrays rely on open-loop actuation that may limit the device to on/off digital operation or require extensive calibration for analog performance [1], [2]. Many of todays emerging technologi es, however, require true analog positioning capabilities. Therefore, in order to guarant ee precision and accuracy of the mirror position for analog operation, closed-loop feedback control techniques are consid ered essential. Feedback control has long been used in many macro-scale systems, yet limited work has been done to apply these techniques to MEMS systems. An a dditional need arises in the use of very large arrays of micromirror devices. While control of one mirror may be a straightforward task, it becomes much more difficult to extend that cont rol to a very large system. The micromirror arrays in this research are constrained such that the micromirrors are not individually controllable, creating a unique control application to a single-input/multiple-output system (SIMO). This also gives rise to the question of obtaining an appropriate feedback signal for a system of arrays. The types of sensing used to gather the feedback information as well as how this information is used are critical issues. 1.2 Research Objectives The objective of this research is to develop analytical m odels to study the effect of fabrication tolerances and uncertainties, electros tatic pull-in and hysteresi s on the performance of micromirror arrays that are used in adaptive op tics applications requi ring precise and accurate positioning. The modeling techniques allow for anal ysis of the system in both the electrostatic and the mechanical domains using a combinati on of analytical models and finite element analysis (FEA). These theoretical results are co mpared to experimental characterization data.

PAGE 21

21 The models are also used to determine the potential effects of parametric uncertainties in the fabrication process, and to estimate the sens itivity of the micromirror design to these uncertainties. This information is valuable in determining the possible limits on performance that can be achieved through only open-loop actua tion methods. These models also characterize the effects of electrostatic instability and the re sulting hysteresis. The modeling is extended from the initial quasi-static approximation to include the dynamic behavior of the system. After thorough analysis and characterizati on of the system behavior, several solutions are presented to improve the performance and positioning accuracy of the micromirror devices. These solutions, including passive and active controls, are develo ped to ensure that the device performance will be robust in the presence of sy stem nonlinearities and parametric uncertainties. A passive design solution, called a progressive-linkage is presented that will eliminat e the effects of electrostatic pull-in and hysteresis, thus extending the stab le range of motion for the micromirrors. The theoretical design approach is presented along with discussion and analysis regarding the sensitivity of the linkage to fabrication errors, as well as an initial prototype attempt. Active control solutions, including cl assical and optimal control design, are presented as an investigation of feedback cont rol methodologies for use on microm irrors that can be used to achieve high precision positioning. The sources of nonlinearities and para metric uncertainties previously identified and quantif ied during the modeling and charac terization of the devices now forms an error basis for examining the robus t performance of the control algorithms. In Chapter 2, an overview of previous work fr om the literature on mi cromirror arrays, their applications, and control of ME MS devices is presented for identifying the underlying issues impeding further development and implementation. This led to the motivation of the work presented in the remaining chapters of this di ssertation including anal ytical modeling and the

PAGE 22

22 study of effects of fabrication tolerances and uncertainties, dy namic performance and passive control. Chapter 3 provides the static modeling for the micromirrors and Chapter 4 analyzes the sensitivity of the devices with respect to fabrication tolerances, comparing the model results to experimental characterizati on data. Chapter 5 discusse s the dynamic system, including characterizing the resonant modes and studying th e effect of electrosta tic hysteresis. The progressive-linkage design is presented as a so lution to the problems cau sed by electrostatic pullin. Upon identifying the model uncertainties and behavioral characteristics of these micromirrors, an investigation of closed-loop control methods is conducted in Chapter 6 to further delineate the impact of the parametric uncertainties on system performance. The controllers are compared and evaluated in simula tion to determine their effectiveness for position control in the presence of model uncertainties fo r a single mirror and a spectrum of uncertainties across the array. In order to evaluate the arra y performance, the method of sensing the position of the micromirrors is critical, and several sensor types are considered. Finally, the conclusions of this study are given in Chapter 7, along with directions for future work.

PAGE 23

23 CHAPTER 2 LITERATURE REVIEW In this chapter, a review of the literature concerning a pplications of micromirrors, modeling of electrostatic devices and the design of feedback c ontrollers for MEMS devices and related adaptive optics systems is presented. This review is intended to pr ovide an overview of the current state of research on el ectrostatic micromirrors and the control of MEMS devices so as to identify underlying issues impeding furthe r development and implementation. A brief introduction to MEMS and micr ofabrication methods is presented, followed by a discussion of applications for micromirror technology. Electro static actuation is used widely for MEMS devices, and it is the actuation method used by the micromirror arrays presented in this dissertation; therefore a discussi on of electrostatic actuation and th e pull-in instability is given, including modeling methods and the different methods that are dedicated to addressing pull-in. Control applications to MEMS is a relatively new area of research, therefore a thorough discussion is included of control methods that have been applied to a variety of MEMS devices with many different methods of actuation in additi on to electrostatic. Th e chapter closes with remarks summarizing the findings of this review and outlining the specific areas of research that are currently lacking, and that will be addressed in th e remainder of this document. 2.1 Microelectromechanical Systems Microelectromechanical Systems (MEMS) refer to mechanical and electrical structures used to create sensors and actuators with feature sizes ranging from 1 m to 1 mm. MEMS have found successful applications in many markets, most notably nozzles for inkjet printing, accelerometers for automotive airbags, blood pressu re sensors for health care, optical switches and arrays for communications and projection displays. This remarkable technology is continuing to expand and promises to bring revolu tionary capabilities to nearly every industry.

PAGE 24

24 MEMS are batch fabricated, typically making them inexpensive, using a microfabrication process such as bulk or surface micromachining [3]. This technology is founded on fabrication techniques first used for integrated circuits (IC) and utilizes the same lithographic patterning techniques. Bulk micromachining techniques rely on selective etch ing to remove material from the whole to form structures with wells and tren ches [4]. Surface micromachining is considered an additive technique that creates mechanisms by layering a structural layer, such as polysilicon, with a sacrificial layer, such as silicon dioxi de (oxide) [5]. Through a repeated series of lithography, etching and deposition, frees tanding structures are created. As with any manufacturing or machining proce ss, fabrication toleran ces can give rise to parametric uncertainties causing the dimensions of fabricated device to vary slightly from the intended design. For microfabrication this is due to small over or under etching of layers as well as variations in material pr operties, and misalignment between layers [7]-[12]. All these variations can occur across the wa fer as well as from batch to batc h. Chapter 3 provides further analysis on the influence of fabricat ion variations on device performance. The process utilized to create the devices discussed in this research is Sandias Ultraplanar, Multi-level MEMS Technology (SU MMiT V), developed by Sandia National Laboratories that utilizes five stru ctural layers of polysilicon [6]. The specifics of the fabrication process are discussed further in Ch apter 3. Examples of structures that can be created using this process seen in Figure 2-1 show scanning electr on micrographs (SEMs) of a mechanical gear hub and a cross-section of a pin-jo int that allows rotation. Thes e are excellent examples of the complex structures created from layering simple, 2-D geometry. 2.2 Micromirrors and Applications Micromirrors are one of the most widely us ed and commercially viable applications for MEMS technology. The small size of these devices makes them ideal for optical switching and

PAGE 25

25 A B C Figure 2-1. The SEM images of MEMS device s created using SUMMi T V microfabrication process. A) Micromachined gears. B) Micr omachined gears. C) A cross-section view of a pin-joint that allows for gear rotati on. (Courtesy of Sandia National Laboratories, SUMMiT Technologies, www.mems.sandia.gov ). scanning operations at very high speeds. Both single mirrors and large arrays are used for optical switches for communications [13]-[19], scanning and imagi ng for projection displays [2], [20], diffraction gratings for optical spectroscopy [21]-[25], an d beam steering for adaptive optics [26]-[32] and free-space communication [33], [34]. An example of micromirrors that have b een commercially successful is the Texas Instruments Digital Micromirror Device (DMD) that uses millions of torsional electrostatic micromirrors to manipulate light. Applicati ons for the DMD include projection displays, televisions, laser printers, image processing, li ght modulation, and optical switching [2], [20]. The success of many of these applications relies on purely digital functioning that is not suitable for more advanced applications that require analog operation, such as adaptive optics (AO). Sandia National Laboratories developed electros tatic micromirror arrays to be used as instrumentation for adaptive optics in space appl ications [35]. Images of Texas Instruments DMD and the Sandia micromirrors are shown in Figure 2-2.

PAGE 26

26 A B Figure 2-2. Images of micromirror arrays develope d in industry. A) Texas Instruments Digital Micromirror Device (DMD) and B) Sandia National Laboratories AO micromirror array. (Courtesy of Texas Instruments, www.ti.com and Sandia National Laboratories, SUMMiT Technologies, www.mems.sandia.gov ). Adaptive optics (AO) refers to optical components such as mirrors or lenses that are able to change shape or orientation in order to manipul ate a light source. Adaptive arrays of large mirrors (on the order of meters in diameter) ha ve long been used in astronomy to correct for atmospheric distortions in images from space [36] This same concept can be achieved with MEMS micromirror arrays for us e in wavefront corrections a nd spectroscopy. Figure 2-3 shows a general schematic of how wavefront correction is achieved using adaptive optics. A distorted wavefront is reflected onto an adaptive optics device which is deformed accordingly to eliminate the distortions in the original wavefront. The ne wly corrected wavefront is split and sent to a detector (e.g. camera) and to a sensor that m easures the wavefront and sends this signal to a control system that directs th e motions of the deformable mirror. These kinds of systems traditionally rely on expensive wavefront sensors to sense the wavefront and direct the mirrors actions. However there are many new applications that are utilizing MEMS micromirrors and lenslet arrays to replace the traditional wavefr ont sensors. Horenstein et al. demonstrate wavefront correction using the Texas Instru ments DMD [30]. Another example of

PAGE 27

27 Figure 2-3. Adaptive optics (AO) mirro r used for wavefront correction. micromirrors used for AO include Boston Micromachines Deformable Mirrors (DM), which have been used for image co rrection in telescopes, micros copes, and Optical Coherence Tomography (OCT) [29], [31], [32]. AO micromirro rs are also being used for imaging of the human retina [37], [38]. Another variation of AO uses arrays of micr omirrors to create programmable diffraction gratings for use in spectroscopy [21]-[25]. As shown in Figure 24, light sent through a sample, such as a chemical, gas, or material, is diffracted into its spectrum by a fi xed grating. This is Figure 2-4. Use of an AO MEMS programma ble diffraction grating for spectroscopy.

PAGE 28

28 then sent to the MEMS diffraction grating that is set to filter light in specific regions of the spectrum. The filtered light is sent back to the fixed grating and then collected by a detector. The light measured in the detector can be used to determine the material composition of the sample. 2.3 Electrostatic Actuation and Instability The examples of micromirrors presented in Section 2.2 all use electrostatic actuation, which is popular in MEMS as it is easy to implement using the silicon-based semiconductor structural materials available in most MEMS fabr ication processes. The theory of operation for electrostatic actuation is presented in detail in Chapter 3, and is di scussed here more generally to give an understanding of the cu rrent modeling methods and the ch allenges with this type of actuation, including nonlinear behavior and electrostatic instability. 2.3.1 Modeling, Pull-in and Hysteresis The theory of electrostatic actuation relies on established relationships regarding the energy generated in an electric field when a charge differential is applied to two bodies, such as in a capacitor [3]. The energy in this electric field creates an attractiv e force between the two plates, and this is the principl e exploited for electrostatic act uation. The equations used to describe the electrostatic forces are derived fr om the energy in the el ectric field between the charged electrodes, and often assumptions ar e made in calculating the capacitance using analytical expressions that neglect the fringe field effects. It is typical in MEMS devices, such as parallel plate actuators or torsion micromirrors for one set of the ch arged electrodes to be stationary, and the other electrode to be supported by a flexible suspension or spring that allows it to move. The spring suspension counteracts th e attractive electrostatic force with an opposing mechanical force that can constrain the degrees of freedom of the moving plate and ensure that the two electrodes to no t come into contact.

PAGE 29

29 Many electrostatic actuators exhibit the we ll-documented phenomenon of electrostatic pull-in. The electrostatic force is nonlinear as it is inversely proportional to the square of the electrode gap. Pull-in, sometim es called snap-down, occurs when the electrostatic force generated by the actuator exceeds the mechanical re storing force of the structure. The result is that the device reaches an unstable position and s ubsequently is pulled down to the substrate at its maximum displacement. The electrostatic in stability has been studied extensively and the pull-in characteristics can be modeled fairly accura tely [3], [14], [39]-[52 ], [106]. Pull-in for a parallel-plate actuator occurs at one-third of the separation gap, which greatly limits the actuator stroke. Another phenomenon associated with pull-in in stability is that once the mirror has pulledin, the voltage required to maintain the pull-in position is lower than the pull-in voltage. The mirror will not return from this position until th e actuating voltage has been reduced below the holding-voltage. The result of this holding effect is hysteresis. Electrostatic hysteresis behaves differently from hysteresis that is common in pi ezoelectric or thermal act uators where continuous motion is possible in both directi ons. Electrostatic systems experience a dead-band after pull-in in which no actuation is even possible until the applied voltage drops below the holding threshold. The effects of pull-in and hysteresis ar e a challenge in achieving stable, controllable actuation over the maximum range of motion of an electrostatic micromirror. The behavior of electrostatic actuators ha s been modeled throughout the literature using analytical expressions for cases of simple electrod e geometry, such as parallel plate actuation [3], [24], [41]-[43], [49], [6 4], [106]. When the electrode geometry becomes more complex, such as the case when the actuators use vertical comb driv es, finite element analysis (FEA) can be used to numerically calculate the proper ties of the electric field. Hah, et al. use a 2D Maxwell solver

PAGE 30

30 and then integrate the results ove r the length of the mirror to pred ict the 3D electric field [14], [46]. This method can be advantageous for co mputational efficiency, as a 2D FEA simulation will likely take less time than a 3D model. There can be benefits to using a 3D FEA solution, which is the modeling method that is employed in th is dissertation. A full 3D electrostatic model can allow easily for evaluation of the effects of complex electrode shapes, such as shapes that do not have a constant cross-secti on along the length of the devi ce. These non-constant crosssections could be designed on pur pose to study the effects of cha nging electrode shape, or can be the result of processing. Etching procedures in both bulk and surface micromachining can inherently result in sloping sidewalls or uneven su rfaces [3]-[5]. Therefore, 3D analysis may be more computationally intensive, but it also allows for the study of more sophisticated geometries. Regardless of the modeling method used, it is possible to describe the static behavior of the actuators and the position and voltage at which pull-in and the release will occur. Electrostatic instability is also an example of bifurcation behavior, and once an equation of motion is determined for the device, the pull-in can be examined from st ability theory [106]. Bifurcation analysis is demonstrated in Chapter 3 of this dissertation. The modeling performed for electrostatic device s typically assumes th at they are operated below the resonant frequencies of the device. The pull-in phenomena is affected by resonance and it has been shown that parall el plate actuators driven at th eir resonant frequency have a greater range of motion compared to the one -third gap limitation for frequencies below resonance [40]. Additional model assumptions that are commonly made are that the device operates only in its intended degrees of freedom as prescribed by the operating conditions and the mechanical suspension design. In cases with multiple degrees of freedom, such as 2DOF

PAGE 31

31 mirrors, positioners, or gyroscopes that have coup ling between the DOF, it is crucial to take this into account during the model development [10], [11]. Many of the first generation of micromirror devi ces, such as Texas Instruments DMD, use pull-in as an advantage that allows for open-loo p, on/off binary actuation at reduced voltages [2], [53]. While the actual pull-in voltage of the device may vary slightly from mirror to mirror due to variations in dimension and material prope rties, reliable open-loop operation can still be guaranteed by ensuring that the ac tuation voltage is sufficiently high enough to capture the pullin effects for all the mirrors despite these vari ations. The hysteresis phenomena can also be beneficially exploited, since once a mirror is pulled-in it can be he ld there at a reduced voltage, which decreases power consumption. While the elect rostatic instability can be advantageous for digital applications, it is an obstacle for the app lication of micromirrors with continuous, analog actuation capabilities. The issue of electrostatic pu ll-in has been thoroughly docum ented and there has been a considerable amount of research conducted to find ways to avoid pull-in for electrostatic micromirrors in order to move beyond binary positioning capabilities and achieve full, analog positioning for applications such as scanning and adaptive optics. Attempted solutions to this problem have included design techniques to alter the electrostatic or mechanical forces of the device, capacitive and charge co ntrol techniques, and closed-loop f eedback control. A review of these methods is given in the following sections. 2.3.2 Design Techniques to Eliminate Pull-in There are multiple design methods researcher s have employed to address the problem of electrostatic pull-in to achieve an extended range of travel for elec trostatic actuators. Some have employed geometrical design changes to achieve increased stability. These methods have included tailoring the electrode geom etry [54] or applying insulating layers of dielectric material

PAGE 32

32 [55]. Changes in the electrode geometry are espe cially effective for torsional microactuators as they do not have a constant electrost atic force generated over the surface of the actuator as it tilts. Changes to device geometry are sometime limited by other design or fabri cation constraints. The use of nonlinear flexures has also been used to ensure that, as the electrostatic force increases, the mechanical restoring force of the devices also increases to compensate. Burns and Bright developed nonlinear flexures that utilize a series of linear flexural elements that are designed to engage the device at predetermined deflections [56] This effectively creates a piecewise linear stiffness profile A similar concept of creati ng nonlinear stiffness has been explored by Bronson et al. in [ 57], [58] and will be discussed further in Section 5.3. The leveraged-bending approach introduced by Hung and Se nturia [24] uses the stress-stiffening of a fixed-fixed beam to generate the nonlinear mech anical force needed to achieve controllable positioning over the entire range of motion of a polychrometer programmable diffraction grating [21]-[23]. The cost of using these techniques is a higher actua tion voltage needed to achieve large, stable deflections. 2.3.3 Capacitive and Charge Control Techniques to Eliminate Pull-in The issue of controlling the elect rostatic instability has been addressed by using capacitive and charge control methods. Seeg er and Crary [59] proposed a si mple method that incorporates a capacitor in series with the actuator to provide stabilizing negative feed back. This passively controls the voltage across the actuator electrod es as the gap width changes. They showed theoretically that th is method can be used to stabilize acros s the entire gap. The tradeoff is that higher voltages are required to stab ilize the actuator using this me thod. This concept is extended by Seeger and Boser using a switc hed-capacitor circuit to control charge across the actuator and reduces the actuation voltage require ments [60]. Seeger and Crary ne glected to take into account nonlinear deformation of the elas tic members of the actuator. Once these nonlinear deformation

PAGE 33

33 terms are considered however, the method is fo und to only partially stab ilize the system [61], [62]. Other issues such as residual charge and parasitic cap acitance addressed by Chan and Dutton [61], [63] were s hown to limit the actuator travel to less than full range. Chan and Dutton also introduced a folded capacitor design that coul d be fabricated in the surface micromachining MUMPS process as part of the device itself and showed that this series capacitor method can be used to increase the stable range of electrosta tic torsion actuators up to 60% of the initial gap with the cost of using hi gher actuation voltages. Other work has used similar charge control stra tegies that have result ed in reduced voltage penalties and extended travel [ 64]-[66]. Current leakage has be en shown to create drift of steady-state positions and this can be overcome using discharge methods that resemble sigmadelta operations, but the results can lead to r inging, or chatter about the steady-state position [66]. A related method uses an inductor and capacitor in series and has been shown to increase the stable range of travel at lo wer voltages, but this technique cannot be easily implemented with MEMS technology due to a lack of inductors available in integrated circuits that meet the high inductance requirements [67]. Thes e methods show that charge c ontrol schemes can be utilized to extend the range of travel and in some cases even improve the transient response as well [64]. In order to overcome the limitations imposed by parasitics, leakage, and residual charge more involved methods must be employed using char ging/discharging cycles, controlling clock frequencies and complex circuit implementations. 2.3.4 Closed-loop Voltage Control to Eliminate Pull-in There are cases where a closed -loop control technique has been used for attenuating and stabilizing electrostatic instabil ity. Voltage control methods ha ve been explored to achieve stabilization beyond the pull-in point [68], [69]. Chu, and Pister di scuss the effect of introducing a voltage control law into a system of electro statically actuated parallel-plates and shows

PAGE 34

34 theoretical stability at small ga p distances [68]. Chen, et al., introduced a method for extending the travel range of a torsional actuator by im plementing voltage control to achieve desired electrostatic torque profiles that can by-pass th e pull-in point. This method was successful up to approximately 80% of the initial gap [69]. 2.4 Feedback Control Techniques Applied to MEMS The previous discussion highlights several of the problems with current electrostatic devices that have impeded the development of highly accurate and precise analog micromirror arrays. These problems, includi ng the limitations imposed by electrostatic instability as well as the variable behaviors that re sult from fabrication uncertaintie s, have been addressed using closed-loop control methods. F eedback control can help to increase the stable region of operation for electrostatically act uated devices, provide accurate a nd precise positioning that is robust with respect to variations in device fabrication, and also re ject outside disturbances such as vibrations and other noise sources. As seen in recent literature and summarized in Table 2-1, controllers have been successful at both extending travel range of electrostatic actuators and for improving tracking, disturbance re jection, transient response, sy stem bandwidth and stability, and reducing steady-state errors. Within the work that ha s been done to design and implement feedback control systems on MEMS devices, a wide array of tec hniques and methods have been employed, including linear-time-inva riant (LTI) techniques such as proportional-integratorderivative (PID), robust, adaptiv e, and nonlinear control design. Some researchers address both achieving actuation in the unsta ble range of motion and improve d transient performance [15], [70]-[74]. The control technique s presented in this review of the literature are not limited to electrostatic micromirrors, but include a variety of devices and actuation methods to illustrate the range of methods that have been employed for control of MEMS devices.

PAGE 35

35 The controls literature reveals the many methods have been suggested as improvements to facing the problems outlined above. Linear methods in some cases are insufficient, and more advanced techniques have not been implemented due to the complexities required. An extended review is given here of these controller methods and their applications for the benefit of the reader. However, the work in this disserta tion focuses mostly on modeling the behavior and examining the effects of nonlinearity and uncerta inties and the impact these have on control implementation. 2.4.1 Linear Control While all real systems will have nonlinearities, it is common engineering practice to treat them as linear whenever possible. These a ssumptions and approximations, when acceptable, greatly simplify analytical models as well as allo w for the use of a wide range of linear control methods. The use of classical, li near controller design such as PI D, lead-lag, and state-variable is adequate for these systems for which the syst ems operate in a small range of motion avoiding nonlinear behavior [75]-[77], or in which the nonlinearities ar e small enough to be neglected [78]-[80]. In the case of systems with large nonlinearities, such as those from electrostatics, it can be a challenge to apply linear control design and en sure that a controller designed for the linear system will be able to operate on the actual nonlinear plant. Desp ite the considerable nonlinearities associated with electrostatic actuation, lineari zation of the plant model is often done to allow for the use of linear-time-invarian t (LTI) control methods. The nonlinear effects of electrostatic actuation are pe rhaps most evident for parallel-pl ate actuator systems. Lu and Fedder used a linearized plant model for a para llel-plate type actuat or and designed a LTI controller for both extended range of travel and position control [71]. The LTI controller was designed and simulated on the linearized plant m odel and showed theoretically that very large

PAGE 36

36 Table 2-1. Summary of feedback control pape rs discussed in the literature review. Ref. Author Control Type Control Objective System Type Experimental/ Feedback Type [68] [Chu, Pister, 1994] Nonlinear Increase Stability Parallel-Plate Electrostatic No [84] [Lyshevski, 2001] Nonlinear Stability, Position Tracking Electromagnetic MEMS Motor No [72] [Piyabongkarn, et al., 2005] Nonlinear Position Tracking Electrostatic Lateral Comb Drive Yes, Capacitive [85] [Zhu, et al., 2006] Nonlinear Position Control Parallel-Plate Electrostatic No [74] [Miathripala, et al., 2003] Nonlinear Stability Parallel-Plate Electrostatic No [73] [Sane, 2006] Nonlinear Increase Stability, Position Tracking Parallel-Plate Electrostatic No [77] [Horsley, et al., 1999] Classical (PD, Phaselead) Position Control Parallel-Plate Electrostatic Yes, Capacitive, Laser Doppler Vib. (LDV) [78] [Pannu, et al., 2000] Classical (PID) Position Control, Dynamic Response (Settling Time), Disturbance Rejection Magnetic Micromirror Yes, Position Sensing Detector (PSD) [75] [Cheung, et al., 1996] State-Feedback Position Control Electrostatic Lateral Comb Drive Yes, Capacitve [70], [71] [Lu, Fedder, 2002, 2004] Classical (P) Increase Stability, Position Tracking Parallel-Plate Electrostatic Yes, Capacitve [15] [Chu, et al., 2005] State-Feedback Increase Stability, Position Tracking Electrostatic Torsion Micromirror (2DOF) Yes, Current Meas. and PSD [79], [80] [Messenger, et al., 2004, 2006] Classical (P, PI,LeadLag) Position Control Thermal actuator Yes, Piezoresistive [82] [Hernandez, et al., 1999] Robust (Mu-Synthesis) Position Tracking Dual-Stage Disk Drive Yes, unspecified [27] [Kim, et al., 2004] Adaptive control, Robust (H-Infinity) Disturbance Rejection (Wavefront Correction) Piezoelectric Torsion Mirror (2DOF, not MEMS) Yes, PSD [26] [Arancibia, et al., 2004] Adaptive Control Disturbance Rejection (Wavefront Correction) Electromagnetic MEMS Torsion Mirror (2DOF) Yes, PSD [76] [Liao, et al., 2005] Adaptive Control Position Control Electrostatic Torsion Micromirror Yes, Capacitive, LDV [10], [11] [Park, Horowitz, 2001, 2003] Adaptive Control Disturbance Rejection MEMS Gyroscope No [83] [Liaw, et al., 2006] Sliding Mode Control (SMC) Position Tracking Piezoelectric Acuators (PEA) (not MEMS) Yes, unspecified [33], [34] [Gorman, et al., 2003, 2005] SMC Position Tracking PEA (not MEMS) No [86] [Lee, et al., 2000] SMC Position Tracking Dual-Stage Disk Drive No [81] [Yazdi, et al., 2003] SMC Increase Stability Electromagnetic MEMS Torsion Mirror (2DOF) Yes, PSD [87] [Chiou, et al., 2002] Fuzzy Logic Dynamic response (reduce rise time) Electromagnetic MEMS Torsion Mirror No

PAGE 37

37 stable deflections could be achieve d for this linearized plant. The LTI controller did not account for the higher order nonlinear eff ects of the actuator, in itial conditions or external disturbances, and when the controller was implemented on the non linear plant, the maximum achievable stable travel range was insufficient to reach the stat ed goal for stable range of motion. The LTI controller was unable to satisfy both the stability conditions and disturbance rejection for large deflections of the actuator, meaning that it could not attain the large deflections predicted for the given controller design [71]. This illustrates the importance of considering robust operation of the controller, especially when using a linea rized plant model for a highly nonlinear system. Linearized control is limited by the true nonlinearities of the system including the effects of unmodeled dynamics, parameter uncertainties, di sturbances, and stabili ty, and it is most appropriate for cases in which thes e effects are small. It is cr ucial to have an understanding of the system behavior and its nonlin earities prior to th e implementation of such control methods. 2.4.2 Adaptive and Robust Control In utilizing closed-loop feedback control techniques for MEMS devices, robustness becomes a commonly desired quali ty [26], [27], [70] -[72], [76], [81], [8 2]. Robustness is important in MEMS control systems as ther e can be many uncertain ties introduced through variations in the device paramete rs, including geometry and material properties that arise from the fabrication process, as well as nonlinearities in the dynamics and disturbances from noise or other external influences. Th ere are many ways to compensate for these uncertainties and develop robustly stable systems. An advantage of adaptive contro l over other methods, like PID, is that the controller can compensate for uncertainties from fabrication, re ject disturbance, and achieve desired tracking objectives by continuously updating the controlle r parameters according to the actual system performance [76]. When applyi ng adaptive control it is very im portant to have an accurate

PAGE 38

38 system model. The actual system output is compared to the estimated output predicted by the model and this error is used to determine the cont roller gains during each step. If the predictive plant model does not reflect the actual system beha vior well, then large errors can lead to poor performance and sometimes cause the system to go unstable [26], [ 27]. Calculating the controller gains at each step in real-time can be difficult to implement, requires computationally intensive algorithms and cannot be done compactly in an analog circuit. Adaptive methods have been employed to accoun t for parametric uncertainties within the plant that arise from variations from the fabrication process. For actuators with performance that is highly sensitive to fabrication variations, adaptive techniques may also be used for parameter estimation. In the case of [72], the actuator dyna mics of lateral electrostatic comb drives are sensitive to fabrication errors arising from the alignment tolera nces of bulk-micromachining. Adaptive control has also been applied to ME MS gyroscopes, which are known to suffer from parametric variations from the fabrication pr ocess that degrade the performance [10], [11]. References [26], [27] demons trate the use of adaptive control techniques for rejecting disturbances that occur in adaptive optics a pplications when there is turbulence in the atmosphere that affects the optical wave front. Kim et al. examined the control of piezoelectric mirrors. These mirrors are not MEMS devices, how ever the control methods and application to adaptive optics still warrants disc ussion. This work showed that using a combination of linear time invariant (LTI) H-infinity control and ad aptive control resulted in good disturbance rejection of band-limited noi se and the H-infinity cont roller improved performance by eliminating steady-state drift and reducing noise [26]. There are few examples of robust control de sign methods such as H-infinity and musynthesis that have been applied to MEMS systems. In addition to the use of H-infinity control

PAGE 39

39 demonstrated by Kim, et al. for a non-MEMS mi cromirror system [26], mu-synthesis controller design was applied to a dual-stage actuator system for track-following in a hard-disk drive [82]. The controller design was successf ul in simulations, but no experi mental work has been done so far. The application of mu-synthesis to desi gn robust controllers has not been specifically applied to a strictly MEMS devi ce. Difficulties in implementing these types of controllers arise if the order of the cont roller is very high, in which case m odel order reduction can be used. In summary, adaptive and robust control techniques a ppear promising at solving the issues of controlling MEMS devices that ar e fabricated with parametric uncertainties, but only if the system has very accurate models, and the source s of the uncertainties are clearly identified within the model. In addition, these methods have largely only b een evaluated in simulation thus far because of implementation issues including hi gh order controllers, lack of adequate sensing methods, and difficulty in realizi ng the control in hardware. 2.4.3 Nonlinear Control The instability problem posed by parallel-plate electrostatic sy stems has been a fertile area for applications of nonlinear control techniques th at incorporate Lyapunov stability analysis [68], [72]-[74], [76], [85]. A gene ral overview of Lyapunov stability analysis and how it applies to nonlinear controller design or MEMS is given by [84]. It is clear that this method is mathematically intensive and that proving globa l asymptotic stability of the Lyapunov function is not a trivial matter. In the case of Maithra pala, et al., the researchers use a nonlinear state feedback controller with a nonlinea r observer to stabilize an electr ostatic parallel plate actuator in its unstable range and to improve the pe rformance by reducing overshoot and decreasing settling time [74]. The resulting control law is determined to have good performance at 80% of the electrode gap in simulation; however it is only locally asymptotic ally stable. Several researchers have developed cont rollers to extend the range of stability for parallel-plate

PAGE 40

40 electrostatics, and have achieve d excellent results based on nume rical simulations [73], [85]. However, like other advanced control technique s discussed here, the resulting control laws are not be easily amenable to implementation in analog circuitry and thus the results have not been tested experimentally. Additional control techniques th at have been used include sliding mode control (SMC), which can also be robust to plant variations have good disturbance rejection and compact implementation schemes. SMC is a digital, nonlinear control method generally good for systems with nonlinearities and parametric uncertainties and tends to produce low order controllers. Lee et al. used a discrete-time SMC for a dual-stage ac tuator for hard-disk drives to track a desired trajectory so as to avoid unwante d excitation of any re sonant modes [86]. SMC was also applied to the problem of electrostatic pull-in instability of two-axis torsion micromirrors [81]. SMC operates through switching pulses that can result in chattering of the actuated device about the steady state value, although attempts have been made to reduce this effect [83]. Although electrostatic systems are known to have hysteresis, there is little work examining its effects on system performance and control. Piezoelectric actuators ha ve significant hysteresis in both traditional piezoelectric stack actua tors and newer MEMS devices that utilize piezoelectric materials. Liaw, et al. examines a traditional piezoel ectric stack actuator, which is in itself not a MEMS device but is used for mi cro and nano-scale manipulation [83]. A robust sliding mode controller is developed that ta kes into account bounded pa rametric uncertainties and hysteresis. The controller was implemented in an experimental system and found to have good trajectory tracking with minimal tracking erro r and hysteretic behavior. Thermal actuators also have hysteretic behavior, and Gorman et al. designed a robust controller for a thermally actuated, microfabricated nanopositioner that us es a multi-loop control scheme based on SMC

PAGE 41

41 [33]. This robust motion controlle r is shown in simulation to be able to track trajectories and reject disturbances to the system given a priori knowledge of the model uncertainty. Chiou et al. [87] examine the use of fuzzy cont rol for a micromirror that is actuated using an array of electrodes that allow for a large number of positions using programmed digital operation. The fuzzy controller showed improv ement in the transient response over the openloop system in simulation, but issues concerning feedback signal and controller implementation are not addressed. In summary, nonlinear control techniques have been shown to be effective at addressing the control of MEMS devices in theory, but like with adaptive and robust control, experimental validation is thus far missing. It is clear from examining these various control methods that as the techniques become more complex to account for robust performance and system nonlinearities, the implementation issues also become more complicated. While many of the papers in the literature discuss robustness of the control system, very few go into great depth of defining the system uncertainties and determinin g the acceptable margins for the uncertainty. Therefore it is not always clear if meaningful robustness is achi eved for the system. Detailed exploration of the uncertainties and the nonlinear behaviors is need ed to further understand these issues. 2.5 Sensing Methods for Feedback In order to implement closed-loop control, a feedback signal is required. Optical beam steering methods are considered in the scope of this research; however it is important to note other sensing methods that may be used. There are multiple sensing mechanisms that have been employed to produce feedback of position and rate for MEMS actuators. These include optical, capacitive, and piezoresistive methods. One opti cal method that has been shown to produce a good feedback signal is one in which the micr omirrors steer a laser beam to a target

PAGE 42

42 photosensitive diode (PSD) to track the position of the mirror [15], [26], [27] [78] [81]. Like many optical methods, beam steering does not always offer the benefit of reducing the size of physical implementation that can be achieved with capacitive or piezoresistive methods. Size may be reduced in some cases by utilizing vertical cavity surface emitting lasers (VCSELs) as the laser source, as was done in [13]. Other optical methods include using an atomic force microscope (AFM) or laser Doppler vibrometer (LDV) [77]. Both of these methods have been used and require special equipment that is only practical to use in a laboratory setup. Capacitive sensing can be done by measuring changes in capacitance as the electrostatic device moves. This method can produce very g ood signals, but does requir e additional circuitry to use the signal [71], [72], [7 5]-[77], [96]. Depending on th e complexity and fabrication process abilities, this circuitry is able to be inco rporated directly onto the chip as an analog signal processor [71]. In some cases, estimators and observers must be employed to estimate and extract the states of the system (position, velocity) from the sensor data. A Kalman filter, which uses an observer and compares the actual re sponse to the observer response, was used by Cheung, et al. to estimate position and veloci ty based on the change in capacitance [75]. Piezoresistive sensing has alrea dy successfully been used in pr essure sensors, shear sensors [88]-[91] and acoustic sensor applic ations [92], [93]. It is relativ ely easy to implement in silicon surface-micromachining processes by utilizing a Wheatstone bridge and does not require CMOS to obtain a signal. The piezoresistive properties of silicon and polysilicon make it suitable for feedback applications. Although pol ysilicon has a lower piezoresisti ve effect than single crystal silicon, it has been used successfully as a sens ing mechanism. Piezoresistive sensing created within the SUMMiT fabrication process is demonstr ated in [91] and [94]. Messenger, et al. has successfully demonstrated the use of surface mi cromachined polysilicon to sense displacement

PAGE 43

43 of a linear thermal actuator and then use that information to perform PID position control [79], [80]. Drawbacks to piezoresisti ve sensing include a large area needed for the resistor elements and drift due to temperature and time. Noise is the limiting factor fo r any type of sensor. Microsensors are susceptible to Brownian motion noise, 1/f noise, an d thermal noise. Piezoresistive sensors have been shown in the past to be most affected by 1/f noise [95]. Many researchers have experienced the limits of a high signal-to-noise ratio and it can limit the bandwid th of the system [79], [80]. In some cases the noisy sensor output can be filtered to achieve better response characteristics. 2.6 Summary Remarks The results of this literature review reveal that there is still work that remains to be completed toward the development of robust micr omirror devices. The i ssue of electrostatic pull-in and hysteresis has been addressed by ma king design modifications to the electrostatic devices as well as with feedback control methods including LTI control, nonlinear control, and sliding mode control. The literature has demons trated cases in which el ectrostatic pull-in has been successfully mitigated, but not entirely eliminated. A disadvantage to methods that incorporate nonlinear mechan ical springs into the system is that they require higher actuation voltages. In this dissertati on, electrostatic pull-in is addressed by introducing a novel design technique called the progre ssive-linkage to create a nonlinear re storing force. This progressivelinkage has the advantage of having a conti nuous spring force over other designs that use discontinuous, piece-wise defined stiffness prof iles. While this approach still has the disadvantage of higher actuation voltages, the bene fits gained via this continuous passive control approach of the nonlinearities in the system reduce the need for the complex control approaches identified in the above literat ure review. This passive control approach should minimize the degree of hysteresis resulting from the pull-in phenomenon, an issue th at has largely been

PAGE 44

44 unaddressed. This is an issue in which feedback control methods can also be applied to help reduce the recovery time for hysteresi s that occurs after pull-in. Bifu rcation theory is used in this dissertation as another method fo r capturing these nonlinear behavi ors in the dynamic modeling. It is also evident that there has been considerable study regarding parallel-plate electrostatic actuators for which analytical re lationships are known and are well defined from physics. There has been less work done to model more intricate el ectrostatic configurations such as those of vertical comb drives. Hah, et al use 2D electrostatic models to determine the electrostatic performance of ve rtical comb-drive actuated micromirrors [14]. While this approach is intended to be more computationally effi cient, it can limit the types and range of electrode geometries that can be easily analyzed. In this disse rtation, 3D FEA modeling is used to determine the electrostatic ch aracteristics of the micromirrors, and the FEA need only be done one time for a given dimensional configuration, t hus the computational costs remain low. In addition, this work presents a detailed modeli ng approach to study the effects of fabrication uncertainties along with characterization data for multiple devices that dem onstrate variations in actuator response. Different control methods including PID, and LQR, are applied to the micromirror arrays in this dissertation to compare the performan ce of each method and to further delineate the impact of the parametric uncertainties on system performance. While a variety of controller design methods have been utilized for MEMS devi ces, very few have considered optimal control applications to electrostatic micromirrors. This dissertation also addresses a unique issue of how to control an array of micromirrors that are not individually controllable. The micromirror arrays examined here have single-input/multiple-output (SIMO) characteristics, providing an interesting challenge to determining the appropriate sens ors and error metrics to apply to feedback.

PAGE 45

45 CHAPTER 3 MICROMIRROR MOD ELING AND STATIC PERFORMANCE This chapter presents the micromirror ar ray devices chosen for in depth study and experimental validation. These devices are arrays of electrost atic micromirrors developed by Sandia National Laboratories (SNL) for applicati on to adaptive optics diffraction gratings like those discussed in Section 2.2. A descrip tion of the SUMMiT V surface micromachining process shows how these devices are made and gi ves some insight into sources of parametric uncertainties that arise through th e fabrication process. The stat ic performance, described in terms of the relationship of the actuation voltage applied and the resulting rotation angle of the micromirror, is examined by developing models fo r the mechanical and electrostatic behaviors. Electrostatic instability can also be predicte d in terms of the pullin angle, voltage, and hysteresis. The static performa nce model is developed and presen ted here along with analysis of the nonlinear behaviors of electrostatic instab ility and hysteresis. 3.1 Description of the SUMMiT V Microfabrication Process The micromirror array is fabricated in th e SUMMiT V surface micromachining process at SNL. Figure 3-1 shows a diagram of the fa brication process from the SUMMiT V design manual in which the five altern ating polysilicon structural laye rs (mmpoly) and four silicon dioxide sacrificial layers (sacox) are labeled along with their nomi nal thickness values [6]. (For further information on surface micromachining fabrication, see [5].) As with all manufacturing processes, th ere are machining tolerances in surface micromachining that affect the final dimensions of the finished product. These tolerances can result in slight deviations of the dimensions from the intended nominal values. Material properties, such as Youngs modulus and Poissons ratio, are also variable and dependent on film thickness and processing methods [ 12], [97]. The result can be th at the fabricated devices will

PAGE 46

46 Figure 3-1. Drawing of the SUMMiT V structural and sacrificial layers. (Courtesy of Sandia National Laboratories, SUMMiT Technologies, www.mems.sandia.gov ) not behave as predicted, or that devices of th e same design can behave differently from one another. Dimensional variations can affect spring constants, res onant frequencies, and electrical characteristics [7]-[9], [98]. Information on fa brication tolerances for the SUMMiT process is available in the design manual [6], and those valu es relevant to this discussion are listed in Tables 3-1 and 3-2. This information was ga thered through diagnosti c process testing as described in [98]. Table 3-1 gi ves the mean and standard deviat ions of the thic knesses of the layers of polysilicon and silicon di oxide. Table 3-2 gives values fo r variations in the dimensions of the line widths of the device design. Figure 3-2 illustrates the effect of line width variation, showing that for a desired area of dimension L by w the actual fabricated area may be slightly less, indicated by the dashed lines. Note that negative values indicate an inward bias resulting in the actual size being smaller than draw n. The variability of the Youngs modulus, E is not listed in the SUMMiT design manual, however informa tion published in the lite rature has found it to be 164.3 GPa with a standard deviation of .2 GP a, which indicates a va riation of % [97].

PAGE 47

47 This information is useful for considering th e effects of parametric uncertainties from the fabrication process on the device performance. Th is subject will be considered more fully in Chapter 4. Table 3-1. Mean and standard deviation of fabr ication variations for layer thickness in the SUMMiT V surface micromachining process. Layer Mean ( m)Std. Dev. ( m) MMPOLY0 0.29 0.002 SACOX1 2.04 0.021 MMPOLY1 1.02 0.0023 SACOX2 0.3 0.0044 MMPOLY2 1.53 0.0034 SACOX3 1.84 0.54 DIMPLE3 Backfill0.4 0.0053 MMPOLY3 2.36 0.0099 SACOX4 1.75 0.0045 MMPOLY4 2.29 0.0063 Table 3-2. Mean and standard deviation of fabri cation variations of lin e widths in SUMMiT V. Layer Mean ( m) Std. Dev. ( m ) MMPOLY2 -0.08 0.03 MMPOLY3 -0.07 0.05 MMPOLY4 -0.24 0.05 Figure 3-2. Area with nominal dime nsions L and w with the dashed line indicating the actual area due to error in the line width. 3.2 Micromirror Actuator Description The micromirror arrays are shown packaged in a standard 24-pin dual in-line package (DIP) in Figure 3-3. A magnified view of the surface of the array is also shown. The device contains six groupings of micromi rror arrays, and the particular gr ouping that is studied here is indicated by a box drawn around it. This array contains 416 microm irrors arranged in 32 rows

PAGE 48

48 and 13 columns. Each individual mirror is 20 x 156 m2. These arrays were originally designed at SNL to create a programmable diffraction grat ing for use in making spectral measurements. Figure 3-4 illustrates the operation of the arrays as a diffraction gr ating in which the light source striking normal to the surface of the mirrors when they are flat is reflected back on the same path. When several mirrors are tilted, some light is reflected off at an angle. The result of this is that the light is selectively diffracted. The micromirrors are one-degree of freedom actuated electrostatically and are shown schematically in Figure 3-5. The electrostati c micromirror arrays have a ground plane and a A B Figure 3-3. Images of the micromirror array. A) Packaged device. B) Micrograph of the surface of the array. Figure 3-4. Illustration of mirrors operating as an optical diffr action grating. A) When the mirrors are not actuated (i.e. flat ), the incident light is refl ected straight back. B) For mirrors that are actuated (i.e. tilted), the incident light is re flected off at an angle. C) This results in a diffrac tion pattern of the light.

PAGE 49

49 series of vertically offset comb fingers, all contained underneath a flat mirror surface. Having the vertical comb drive beneath th e mirror rather than just a para llel-plate capacitor attenuates the electrostatic field and increases the stable range of motion of the device. This also allows for large arrays with high fill f actors, making them a good choice for analog scanning devices. A Figure 3-5. Micrograph of an array of mirrors and schematic of mi rror with hidden vertical comb drive. A) The torsion spring. B) The full de vice. C) A 2-D crosssection view of a unit cell (figure not to scale). Figure 3-6. Solid model of micromirror show ing polysilicon layer names from SUMMiT V.

PAGE 50

50 voltage potential is applied acro ss the fixed fingers and the moving fingers of the device creating an electrostatic force. This force causes the mirror to rotate about an axis supported by the hidden spring suspension, shown separately in Figure 3-5(a). Not shown in the drawing is a design constraint that restrict s the motion of the fixed-end of the mirror plate from moving a large distance in the Z-direction. While some motion may occur, the assumption is made that this device acts in one degree-of-freedom by ro tating about the x-axis. Figure 3-6 shows a 3D model identifying the fabrication layers used to create the micromirrors. 3.3 Electrostatic Actuation and Instability Many electrostatic actuators exhibit the we ll-documented phenomenon of electrostatic pull-in. The electrostatic force is nonlinear, as it is inversely proportional to the square of the electrode gap. Pull-in, sometim es called snap-down, occurs when the electrostatic force generated by the actuator exceeds the mechanical re storing force of the structure. The result is that the device reaches an unstable position and s ubsequently is pulled down to the substrate at its maximum displacement. Another phenomenon asso ciated with pull-in instability is that once the mirror has pulled-in, the voltage required to maintain the pull-in position is lower than the pull-in voltage. The mirror will not return from this position until the actuating voltage has been reduced below the holding-voltage. The result of this holding effect is hysteresis. This section will examine the modeling of the electrostatic-mechanical system and the instability phenomena. The case of parallel plat e electrostatics is examined and used to derive general relationships for modeling th e system. This is extended to a torsion electrostatic actuator to illustrate the complications that arise from adding complexity to the system geometry. 3.3.1 Parallel Plate Electrostatics Consider a parallel plate capacitor, such as shown in Figure 3-7, in which the top plate is supported by a spring, with spring constant km, and the bottom plate is fixed. Damping in the

PAGE 51

51 system is represented by the damping coefficient, b. The plates are separated by a distance of x0, and have an overlapping area of A Figure 3-7. Schematic of a parall el plate electrostatic actuator modeled as a mass-spring-damper system. The equation of motion for this mass-spring-damper system is derived by the balance of the forces on the system from Newtons second law x m F (3-1) where m is the mass of the moving plate. When th e top plate is displaced in the positive xdirection, shown in Figure 3-7, the motion is op posed by the force from the mechanical spring, which is assumed to be linear, and follows Hooke s law. The mechanical spring constant is km. mmFkx (3-2) The damping force is assumed to be linearly proportional to the velocity by a factor of b the damping coefficient. x b Fb (3-3) When a voltage potential is applie d across the two plates, an electr ostatic force is generated that attracts the top plate to the botto m. The electrostatic force for a system operating in air is derived from the energy, U of an electric field, E integrated over a volume, v

PAGE 52

52 2 02vUEdv (3-4) where 0 is the permittivity of free space, 8.854 x10-12 F/m. The electric field is given by A Q E0 (3-5) where Q is the electric charge. The charge, Q, can be written as CV Q (3-6) where C is the capacitance, and V is the voltage. The capacita nce between two parallel plate actuators is given in terms of the overlapping area of the plates, A, and the distance between the two plates. 0 0() () A Cx x x (3-7) Equation 3-4 can be rewritten as ) ( 2 2 10 2 0 2x x A V CV U (3-8) The electrostatic force is thus written as 2 0 2 0 2 0 0) ( 2 1 2 1 ) ( x x AV V x x C x x U Fe (3-9) The force balance for the system yields the equation of motion. emFmxbxkx (3-10) The static equilibrium for the system reduces to only the electrostatic force, and the mechanical force. 2 2 0 2 0011 2()2()mAV C Vkx xxxx (3-11)

PAGE 53

53 Equation 3-11 can be interprete d to show the relationship between the voltage and the displacement, as plotted in Figure 3-8 for system of parallel plates with the area, A equal to 100 x 100 m2, an initial gap, x0, equal to 10 m, and a mechanical spring constant of km equal to 1 N/ m. From this, it is clear that there is a maxi mum voltage for the system, and that there can be multiple solutions for the same applied voltage. This behavior is the result of the electrostatic pull-in instability. It turns out that the soluti ons in the lower portion of Figure 3-8 are stable solutions and the solutions in the upper porti on are unstable. The maximum voltage value corresponds to the actuation voltage at which pull-in occurs, and the maximum stable position for parallel plate actuator occurs at onethird the gap between the electrodes. To further explore the pull-in phenomena, the static relationship in Equation 3-11 can be examined graphically, by plotting the electrostatic force and the mechanical force separately in Figure 3-9. The electrostatic force is a function of both the displacement and the voltage. Static equilibrium occurs where the electrostatic for ce lines and the mechanical force line cross each other. As was shown in Figure 3-8, there are instances where th e mechanical and electrostatic lines intersect at more than one point. Because of the nonlinear behavior of the electrostatic force, there is a point at whic h the electrostatic torque exceeds the ability of the mechanical spring and equilibrium can no longer be maintained. This is referred to as electrostatic pull-in. At the pull-in point, both the electrostatic and mechanical torques are equal in magnitude and slope and thus only have one point of intersection between these fo rces on the graph [14], [46]. Stable static solutions occur before the pull-in poi nt, while unstable solutions occur after. This slope equality is written by ta king the first derivative with respect to the displacement of Equation 3-11. 2 2 2 01 2mC Vk xx (3-12)

PAGE 54

54 Figure 3-8. Static equilibriu m relationship for the parallel plate electrostatic actuator. Figure 3-9. Electrostatic force for different voltages and mechan ical force showing pull-in for the electrostatic para llel plate actuator. Stable Unstable

PAGE 55

55 Substituting Equation 3-12 into 3-11 and evalua ting at the pull-in position results in the following relationship that is only a function of the capacitance and the pull-in position, xPI. 2 2 000 ()()PIPIPI xxxxCC x xxxx (3-13) Assuming that the restoring springs are linear ly deformed in the range of actuation, the pull-in angle is independent of the spring stiffness, and depends only on the angle of rotation. A pull-in function, PI(x) is defined to determine the pull-in angle, which occurs when PI(x) is equal to zero. 2 00 2 00()() ()() CC PIxxxx x xxx (3-14) In turn, once the pull-in angle is determined, the pull-in voltage can be calculated by the following expression, 002 () P ImPI PI xxxkx V C xx (3-15) The pull-in function for the parallel plate electr ostatic actuators is shown in Figure 3-10, Figure 3-10. Pull-in function for the pa rallel-plate electrostatic actuator.

PAGE 56

56 and verifies that pull-in does o ccur at 1/3 the gap between the pl ates. The pull-in voltage is calculated from Equation 3-15 to be equal to 57.85 V. To further investigate the eff ects of changing the spring consta nt on the pull-in, the static equilibrium relationships are plotted for different values of the mechani cal spring constant in Figure 3-11. This shows that even for a diffe rent spring constant, th e pull-in displacement location remains at 1/3 the gap, wh ile the pull-in voltage changes. A B Figure 3-11. Static equilibrium relationships for the parallel plate actuato r using different spring constants. A) The electrostatic and mech anical forces. B) The static displacementvoltage relationships. 3.3.2 Parallel Plate Torsion Actuator Consider the case of an electrostatic parallel plate actuator that is supported by torsion springs, such as shown in Figure 3-12. This actuato r rotates about th e axis of the to rsion springs, and the displacement may be described in terms of the angle of that rotation, The general relationships for el ectrostatic actuation that were derived in Section 3.3.1 may also be derived for this type of torsion actuator in terms of torques instead of forces. The sum of the torques for the system describes the equation of motion for the system. The micromirror can be considered as a one-degree-of-freedom mass-spring damper system of the form (,)meJbkTV (3-16)

PAGE 57

57 Figure 3-12. Schematic of a to rsion electrostatic actuator. where J is the mass moment of inertia, b is the damping coefficient, km is the mechanical spring constant, and Te is the electrostatic torque, which is represented by the following, 21 2eC TV (3-17) where C is the capacitance, is the angle of rotati on about the X-axis, and V is the voltage potential. The mechanical system is governed by the stiffness of the support structure of the mirrors. It is assumed that the spring suspensi on provides a linear mechan ical restoring torque, Tm, to the system that can be represented as, m mk T (3-18) where km is the rotational spring constant. Static equilibrium occurs in the device when the electrostatic torque is equal to the mechanical restor ing torque. Therefor e, the static device behavior, which is the relati onship of the actuation voltage, V to the rotation angle, is determined by equating Equations 3-17 and 3-18. 21 2mC Vk (3-19) As was previously shown for parallel-plate electrostatics, there is a point at which the electrostatic torque excee ds the ability of the mechanical spring and equilibrium can no longer be

PAGE 58

58 maintained. This is referred to as electrostatic pull-in. At the pull-in point, both the electrostatic and mechanical torques are equal in magnitude a nd slope [14], [46]. As was shown previously for the parallel-plate actuator, electrostatic pull-in can be cons idered as the mechanical and electrostatic torques bein g equal, as in Equation 3-19, and th eir first derivatives being equal. 2 2 21 2mC Vk (3-20) Combining Equations 3-19 and 3-20 and evaluating at the pull-in angle results in the following relationship that is only a function of the capacitan ce and the pull-in angle, PI. 02 2 PI PIC CPI (3-21) Assuming that the restoring springs are linearly deformed in the range of actuation, the pull-in angle is independent of the spring stiffness, and depends only on the angle of rotation. A pull-in function, PI( ) is defined to determine the pull-in angle, which occurs when PI( ) is equal to zero. 2 2) ( C C PI (3-22) In turn, once the pull-in angle is determined, the pull-in voltage can be calculated by the following expression, PIC k VPI m PI 2 (3-23) Thus far, the only difference between modeling the torsion actuator and the parallel plate actuator is that the parallel plate actuator ha s linear displacement, while the torsional has rotational motion. Therefore, the equations for each system are very similar. The difference in

PAGE 59

59 evaluating the torsion actuator becomes apparent however when the capacitance for the system is calculated. Unlike the parallel-plate actuator, the torsion actuator does not have a constant gap between the top and bottom electrodes when it move s. Considering the system drawn in Figure 3-12, the capacitance for a torsion actuator in terms of the angle of rotation about its axis is given as 0 maxmax()ln1ln1mW C (3-24) where is L3/L2, is L2/L1 and max is H0/L1 [46]. From Equation 3-24 it is possible to calculate the performance for a torsion actuator. The pull-in function for this system is 22 maxmaxmaxmax 0 max 22 2 max maxmax3434 1 ()3ln 1 11mW PI (3-25) As an example, consider a system w ith the following geometric variables: km = 1 Nm, max = 10, Wm = 100 m, = 0.5, = 0.5. The static equilibrium for this system can be evaluated by examining the electrostatic and mechan ical torques, as shown in Figure 3-13. The A B Figure 3-13. Static equilibrium relationships for the torsion actu ator. A) The electrostatic and mechanical torques. B) The static rotation-voltage relationship.

PAGE 60

60 displacement as a function of the voltage is also shown in Figure 3-13. The pull-in function from Equation 3-25 is plotted in Figure 3-14. From thes e figures, it is found that the pull-in for this system occurs at 82% of maximum rotation angle for the system. For the given spring constant, the pull-in voltage is 6.91 V. As with the parall el plate actuator, the pull-in angle will remain the same despite the mechanical spring constant, but the pull-in voltage will change. The pull-in angle can change, however, if the system geometry is changed. This is different from the parallel-plate actuator, which alwa ys pulls in at 1/3 the gap. Figure 3-14. Pull-in functi on for the torsion actuator. 3.4 Model for Vertical Comb Drive Actuator From the previous section it becomes clear that analytically describing the performance of an electrostatic actuator becomes more difficult as the geometry of the electrodes becomes more complex. In fact, it is very difficult to descri be the capacitance for the micromirror devices that operate via vertical comb drive ac tuators. Thus it becomes necessa ry to employ FEA to assist in developing the system model. The device operate s in both the mechanical and the electrical domains. Therefore, the model is developed fo r the mechanical and the electrostatic functions separately. The following analysis presents the model first in the mechanical domain and then

PAGE 61

61 the electrostatic. The two mode ls are then combined to determine theoretically the static behavior of the mirrors, incl uding pull-in and hysteresis. 3.4.1 Mechanical Model The mechanical spring is shown in Figure 3-15 with the fabric ation layers labeled. The spring has two anchor points that connect to the ground layer (MMP oly0) and thin beams in the MMPoly1 layer to provide the restoring force. The layer MMPoly3, which is used to create the moving comb fingers, has a dimple cut in the center of the spring mechanism that, when actuated, comes into contact with the anchor (MMPoly2) and allows the mirror surface to pivot about this point. An alternative depiction is shown in Figure 3-16 in which the spring is considered as thin beams that are fixed to th e substrate in two places The length and crosssectional area of the MMPoly1 beam s is given in Figure 3-16 to be 33 m and 1x1 m2, respectively. The value for Y oungs modulus is 164.3 GPa, and Poissons ratio is 0.22. To determine the stiffness of this mechanical spri ng, a simple model is created in ANSYS Finite Element Analysis (FEA) program using Beam189 elements, which are capable of nonlinear large deflection analysis [99]. The boundary conditio ns constrain all motion in six-degrees-offreedom at the two anchor points. Figure 3-15. Drawing of the mechanical spring th at supports the micromirrors and provides the restoring force.

PAGE 62

62 Displacement loads are applied in all six-degr ees-of-freedom at th e point indicated in Figure 3-16 that corresponds to the pivot point created by the MMPoly3 dimple. The FEA determines the forces and stresses in the beam el ements after the displacement loads are applied. The deformed shape of the structure is shown in Figure 3-17. Assuming Hookes law for the force applied to a linear spring, the spring st iffness in all six degrees-of-freedom can be calculated. The linear spring assumption is veri fied by performing nonlinear FEA over the entire range of motion of the spring disp lacement from zero to nineteen degrees. The results are listed in Table 3-3, retaining 4 significan t figures. The stiffness in X, Y, and Z refer to the stiffness of the spring in each respective axis direction, and qX, qY and qZ refer to the rotational stiffness about the axes X, Y, and Z, respectively. It is cl ear that the spring is not very stiff in the Y and Z directions. The torsional stiffn ess about the X axis, qX, is lowe r than those about the Y or Z axes, meaning that the mirror is ab le to rotate about the X axis, wh ile it is resistant to off-axis rotations about the Y or Z. It is the value of qX equal to 612.4 pN-m/rad that is used for km in Equation 3-17. Figure 3-16. Image of the mechanical spring th at supports the micromi rror indicating boundary conditions and location for applying displacem ent loads for finite element analysis.

PAGE 63

63 Figure 3-17. Image from ANSYS of the deform ed spring and the outline of the undeformed shape after displacements are applied. The displacement is amplified by a scale factor of 4. Table 3-3. Values output from finite element analysis of mechanical spring stiffness. Parameter Value X stiffness 744.7 pN/m Z stiffness 7.946 pN/m Y stiffness 1.266 pN/m qX stiffness612.4 pN-m/rad qZ stiffness 11360 pN-m/rad qY stiffness16310 pN-m/rad 3.4.2 Electrostatic Model In order to compute the electros tatic torque values in Equation 3-16, it is necessary to find an expression for the capacitance as a function of the rotation angle. For parallel-plate electrostatics, this can be done quite easily as an analytical expression is known. Because of the more complex electrode geometry created by the inclusion of the vertical comb drive, the capacitance of the device cannot be as easily derived. To determine the charge created by the electrostatic field, 3-D FEA is used to calculate the capacitance as a function of

PAGE 64

64 The symmetry of the device design makes it c onvenient to model only a small section of the device, termed the unit cell. A cross-secti on of a unit cell made up of one-half of one moving comb finger and one half the associated fixed comb finger and portions of the ground plane and mirror surface is shown in Figure 3-5(c). The mode l of the geometry in Figure 3-18 is created in ANSYS. The nominal dimensions used to create this model are given in Appendix A. For an electrostatic analysis, the volume of the surrounding fluid, in this case ambient air, is created around the device geometry, and it is this air volume that is meshed and analyzed to determine the electrostatic field generated as the mirror and moving finger rotate about an axis parallel to the X-axis in the figure. An arbitrary voltage differential, V is applied as show n in the drawing. The only relevant material properties needed in this analysis are the permittivity of free space, 0, which is 8.854 x10-12 F/m, and the relative permitt ivity of the dielectric medium, which in this case for air, is equal to 1. The analysis calculates the tota l charge of the electric field, W and then calculates the capacitance for a given position as 22 W C V (3-25) Using numerical values generated in the electrostatic FEA model, Equation 3-24 is applied to calculate the capacitance at discrete points as the geometry of the mirror surface and moving comb finger rotate through a ra nge of motion from 0 to 19 degr ees. A polynomial least-squares fit of these capacitance values is used to find an analytical expression for the capacitance. The capacitance as a function of is approximated with an nth order polynomial curve fit. 1 121()()nn nnCNPPPP (3-26) where the coefficients of the polynomial are Pi, (i = 1, 2,, n, n+1) and N is the total number of unit cells. The results of this analysis are plot ted in Figure 3-19 along with a comparison of first, second, third, and fourth order polynomial curve f it approximations of the data. The coefficients

PAGE 65

65 for these curve fit approximations are listed in Table 3-4. Table 3-5 compares the quality of the different order polynomial approxim ations compared to the FEA data points. One metric to evaluate the fit quality for a curve fit is the norm of the residuals, normr The smaller the value of normr is, the better the approximation. Another stan dard metric is the sum of the square of the residuals, r2, which is calculated from normr by 2 2 21 (1) normr r ns (3-27) where n is the number of data points (FEA data), and s is the standard devi ation of the curve fit approximation from the data. A value of r2 equal to one indicates a perfect fit. It is clear that a higher order polynomial does a slightly better job of capturing the na ture of the capacitance data. However, the first order linear curv e fit can still be sufficient for an alysis in the stable range of motion. It will not be as accurate at predicting th e pull-in behavior. The advantage of using the first order fit is that its derivative which is us ed in Equation 3-16 is a constant, thus simplifying the plant model to a linear approximation in V2. In order to capture th e nonlinear behaviors of pull-in and hysteresis, the fourth order polynomia l curve fit approximation is used in Section 3.4.4. The effects of different linear approxi mations in the model are discussed further in Chapter 5. Figure 3-18. Solid model geometry of the unit ce ll used in the electros tatic FEA simulation.

PAGE 66

66 Figure 3-19. Capacitance calculation as a function of rotation angle, calculated using 3D FEA and varying orders of polynomi al curve fit approximations. Table 3-4. Comparison of polynomial fit for approxima tion of capacitance function Order P1 P2 P3 P4 P5 4 0.023120 -0.013678 0.0041640.000109 0.000106 3 0.000848 0.001280 0.0002990.000103 -2 0.001680 0.000250 0.000104--1 0.000777 0.000078 ---Table 3-5. Comparison of polynomial fit for approxima tion of capacitance function Order normr s n r2 4 1.1192E-05 0.000185 18 0.999785 3 1.1463E-05 0.000185 18 0.999775 2 3.6691E-05 0.000185 18 0.997691 1 8.608E-05 0.0001854 18 0.987166 3.4.3 Electromechanical Model Taking both the mechanical and electrostatic mode ls into account, the static behavior of the system can now be predicted using Equations 3-16 to 3-20. Equations 3-18 to 3-20 calculate the electrostatic pull-in characteristics of the device. A plot of the pull-in function is shown in Figure 3-20 where pull-in occurs when the function equals zero at 16.5 degrees. Using this value in Equation 3-20, the pull-in voltage is 71.06 V.

PAGE 67

67 Figure 3-20. Plot of th e Pull-in function PI( ) for the micromirror with the vertical comb drive actuator showing that pull-in occurs at 16.5 degrees. The static equilibrium behavior can also be evaluated from Equations 3-16, and 3-17, respectively. When the mechanical and electrosta tic torques are equal to each other, the system is in static equilibrium. This can be shown graphically by plotting these values. Figure 3-21 shows the electrostatic torque as a function of rotation angle for different values of voltage ranging from 10 volts to 80 volts. The straight line on the plot corresponds to the mechanical restoring torque of the spring from Equation 3-17. At every point where the mechanical torque and the electrostatic torque lines cross, they are in equilibrium indicating a stable position. There is a point at which this line runs tangent to the electrostatic torque, and this indicates the electrostatic pull-in point, whic h corresponds to the calculated values of 16.5, and 71.06 volts. The electrostatic torque curve at the pull-in voltage, VPI, is also indicated in Figure 3-21. The pull-in angle for a linear spring is determined completely by the electro static torque. For a different value of the mechanical spring constant, km, the slope of the mechanical torque line

PAGE 68

68 would be different, but it would st ill run tangent to the electrostat ic torque at the same pull-in angle. Only the value of the pull-in voltage wo uld be affected. This is shown in Figure 3-22. The pull-in instability is known to cause hysteresi s in the device behavior, and this too can be predicted using this modeling approach. After the device has pulled-in, it is possible to reduce the voltage below the pull-in voltage without releasing the device. This is referred to the holding voltage. Once the voltage has been reduced below this holding voltage thre shold, the device will release from its pulled-in position, but it will re turn to a position different from the pull-in position. From this electromechanical analysis, it is determined that the holding voltage is 68.89 V. The static behavior of the device is shown in Figure 3-23, including the pull-in point and the hysteresis loop. This type of curve will be referred to as a V profile, and represents the static calibration for the device. Figure 3-21. Electrostatic and M echanical torque as a function of rotation angle, theta, and voltage for different voltage values.

PAGE 69

69 Figure 3-22. Torque as a function of rotation angl e, theta, and voltage for different values of mechanical spring constant. Figure 3-23. Plot of static equi librium behavior, showing pull-in and hysteresis, predicted from the model.

PAGE 70

70 3.4.4 Linear Approximation Recall from the discussion in S ection 3.4.2 of the electrostatic model development that the capacitance function is approximate d using a polynomial curve fit, and that different orders of polynomial can be used. For this system, the nonlinea r behavior of the electrostatic instability is best captured using a higher order polynomial; how ever a first order func tion is still able to approximate the system performance. Using a first order approximation makes the derivative term of the capacitance a constant value, whic h greatly simplifies the dynamics and allows the system to be modeled as linear. The effects of using a higher orde r curve fit versus the first order are more apparent by looking at the static equilibrium relations hip between the applied voltage, V, and the rotation angle, This is shown in Figure 3-24 fo r the fourth order fit, called the nonlinear model, and the first order fit, called the linear capacitance approximation model. It is clear that by using the lower order model approximation ther e is a difference between the Figure 3-24. Static equilibrium relationships for the nonlinear plant model, and the linear plant approximation.

PAGE 71

71 predicted static performances. To establish the effects of model unc ertainty on micromirror arrays, the linear model is used as a basis for designing controllers in Ch apter 5. The linear model is suitable to the design of the controller, but the resulting c ontrol law must still be able to perform well on the nonlinear system. For a system in which the capacitance cannot be adequately modeled as linear, such as the case of parallel plate electrostatic s, a higher order approximation is required. In this case, it is possible to linearize the second order dynamic model in Equation 3-16 about an operating point ( 0, V0) using the Taylor series expa nsion (TSE) [36]. This can be considered as the small signal model approximation about andV Doing so yields the following linear system model, meJbkkCV (3-28) The linearization in Equation 3-28 includes a te rm that is dependent only on the rotation angle that can be considered the electrostatic spring force, ke [20]. 02 2 1 0 2 2 edC kV d (3-29) The nonlinear torque approximation is reduced to a constant. 00 TdC CV d (3-30) When linearizing a function about an operating po int, it is desirable that the linear model will provide an adequate estimate of the nonlin ear function within a small range about that operating point. For systems that are operati ng over a large range or have very nonlinear characteristics, this linearization may not pr ovide a satisfactory estimate of the nonlinear function. To illustrate the effect of the small si gnal linearization, Figure 3-25 shows the static equilibrium relationship between rotation angle and actuation vo ltage for the nonlinear system

PAGE 72

72 model and for the small signal mode l linearized about the operating point (7 degrees, 54 volts). The inset shows the small signal response for V. It is clear in Figure 3-25 that this linear esti mate of the nonlinear system does not capture all of the static performance char acteristics over the entire range of operation, but is adequate enough for a portion of the range from 5 to 14 de grees. In order to cover the full range of actuation, a piecewise lin earization can be done at different operating points. This piecewise linearization approach would represent the system response as s hown in Figure 3-26. The linearized models discussed above are important when considering control design techniques that require a linear transfer function or state-space model for the design process. Of the two linearization methods discussed, th e first method of using a linear capacitance approximation is used throughout this dissertation whenever the linear system model is required. This method was chosen for its ease of use. 0 10 20 30 40 50 60 70 80 0 5 10 15 20 Voltage (V)Theta (deg) nonlinear model linearization about operating point operating point Figure 3-25. Static equilibrium relationships for the nonlinear plant model, and the small signal model linearized about an operating point ( 0, V0). 0 10 20 30 0 5 10 15 20 V (V) (deg)

PAGE 73

73 Figure 3-26. Illustration of piecewise lin earization about multiple operating points. 3.4.5 Bifurcation Analysis Electrostatic instability is an example of bifur cation, and the stability of the system can be examined by looking at the dynamics of the actuato r and finding the fixed-point solutions [106], [107]. One advantage of evaluating the bifurcati on behavior of the device is that unlike the methods used in Equations 3-14 and 3-15, the mech anical spring constant is not required to be linear. This analysis will be used again in Chapter 5 to determine th e effects of a nonlinear spring constant on the electrostatic pull-in. Here, th e spring constant is still assumed to be linear, and the results may be compared to thos e obtained using Equations 3-14 and 3-15. The state space model for the system is 1 2 2 1 2 1211 ()m ex x x x k b x Txxx JJJ (3-31) Recall that Te is a function of the capacitance expression from Equation 3-26. In order to capture the nonlinear effects of the syst em, a fourth-order curve fit approximation is used. The fixed points occur at x2 = 0 and

PAGE 74

74 0 ) (1 1 x k x Tm e (3-32) This can be expressed in full as 322 112131411 (432)0 2mNPxPxPxPVkx (3-33) Equation 3-33 is a cubic polynomial equation for whic h finding the roots has been the subject of considerable study [110]. One soluti on is to write the polynomial as 32 111111213()()()0eeeAxBxCxDAxxx (3-34) where 123 and eee are the three roots, and the coefficients A, B, C and D are 2 12 A NVP (3-35) 2 23 2 B NVP (3-36) 2 3mCNVPk (3-37) 2 41 2DNVP (3-38) Further, define 23 39272 54ABCADB q A (3-39) 3 2 2 23 9 ACB uq A (3-40) 3squ (3-41) 3tqu (3-42) The roots of Equation 3-34 are 13e B st A (3-43)

PAGE 75

75 213 ()() 232eB ststi A (3-44) 313 ()() 232eB ststi A (3-45) The roots of Equation 3-33 can be found to determine the static voltage-displacement relationship, as was done previously in Section 3.4.3. Solving this equation gives the fixed points as functions of the control parameter V. The roots of this expression can be examined graphically by defining a function F(x1) as 2 11 11 () 2emmC FxTTVkx x (3-46) In Figure 3-27, F(x1) is plotted for varying values of voltage, V. The roots of F(x1) correspond to the zero crossings on the figure. Notice that there are three roots for each line of constant voltage, and this corresponds to F(x1) being a third order polynomial. The roots that occur to the left of zero degrees theta are so lutions that are non-physical solu tions and are thus ignored. The solutions of function F(x1) that occur for positive values of theta can have either two roots, one root, or zero roots. For a sufficiently small vo ltage, there are two roots. In this case the electrostatic force is low enough th at the linear spring force can ba lance it, creating a stationary state. As the voltage is increased, the electros tatic force increases, eventually overwhelming the linear spring force and all the steady-state solutions disappear. This is another description of the pull-in instability caused by the disappearance of all physically possible steady-state solutions [106]. Now that the steady state solutions of the system can be determined, it is the stability of those solutions that must be determined. A Jacobian matrix is found by taking the Taylor series expansion of Equation 3-31 and retaini ng only the first order terms [107].

PAGE 76

76 Figure 3-27. Plot showing th e roots of the function F(x1) occur where the function crosses zero. J b J k x x T J x Dfm e 1 1) ( 1 1 0 ) ( (3-47) where 22 1 11213 1() 1 (1262) 2eTx NPxPxPV x (3-48) The Jacobian defined in Equation 3-47 relates the perturbation of the states from equilibrium as 11 0 22eix x x Dfx x x (3-49) The stability is determined by evaluating the ma trix in Equation 3-47 at the fixed points and determining the eigenvalues. The fixed point so lution is stable when the real part of the eigenvalues is less than zero. The eigenvalues, j for j = 1, 2, are calculated for each fixed point solution (i.e., roots 123 and eee ). This is expressed as

PAGE 77

77 2 1() 111 4 22 1,2 1,2,3eeim jTk bb JJJxJ j i (3-50) Substituting Equation 3-48 into 3-50 gives the expr ession for the eigenvalue problem in terms of the expression for the capacitance. 2 22 123111 41262 222 1,2 1,2,3m jeieik bb NPPPV JJJJ j i (3-51) To evaluate the eigenvalues and their stabilit y, an expression for the damping in the system must be defined. In a MEMS system such as this, the dominant source of damping comes from the squeeze-film effect, in whic h air that is compressed between very small spaces begins to act as a viscous fluid [3]. Squeeze-film damp ing is dependent on the device geometry, and expressions are known for parallel plate actuators and for torsion plate actuators. As was the case with the electrostatic model development, th e complex geometry of the vertical comb drive micromirrors makes determining the squeeze-film da mping coefficient analytically difficult. For the purpose of this discussion, an approximation is made to consider the squeeze-film damping term for a torsional plate developed by Pan, et al [100]. 5 3 rot L w bK g (3-52) where L is the length of the plate, w the width, g is the gap between the plates, and is the absolute viscosity of the fluid. The term Krot is 4 482 6 L w rotK (3-53)

PAGE 78

78 Table 3-6 lists the values of additional parameters for this analysis. This estimate for squeezefilm damping is used here for simplicity. The resulting bifurcation diagram in Figure 328 shows a saddle node bifurcation at 16.5 degrees and 71.06 V. This is in agreement with the pull-in results from Section 3.4.3. Figure 329 shows the bifurcation diagram for different values of the mechanical spring constant, km, to illustrate how changing the spring constant for a linear spring only affects the pull-in voltage. Table 3-6. List of parameters used for this analysis. Parameter Value density of polysilicon 2331 kg/m3 absolute viscosity of air 1.73e-5 N-s/m2 L length of mirror 20 m w width of mirror 100 m g gap between plates 11.25 m N number of unit cells 54 3.5 Chapter Summary The electrostatic modeling in this chapter reveals the performance characteristics of a micromirror based on the nominal design parameters of the device. The model is developed by considering the mechanical spring element and th e electrostatic actuation forces separately. Doing so allows for greater understanding of the role of each energy domain in determining the performance of the electromechanical device. It can also be usef ul in the design stages of an electrostatic micromirror to see the effects of changing the design to have a different spring stiffness or electrode shape. The electrostati c instability phenomenon is described in analytic terms that can be used to predict the pull-in angl e, pull-in voltage, and the hysteresis behaviors of the device. The electrostatic behaviors are also examined th rough bifurcation analysis. It is discussed in the descri ption of the fabrication proce ss in Section 3.1 that there are certain errors that occur in the geometry and the material properties during fabrication.

PAGE 79

79 Figure 3-28. Bifurcation diagram for a MEMS tors ion mirror with electrostatic vertical comb drive actuator. Figure 3-29. Bifurcation diagram showing th e effects of different spring constants. Bifurcation point 71.06 V, 16.5

PAGE 80

80 Information on these errors is available in the process design manua l, and gives a MEMS designer a reasonable expectation of the precision available from the micromachining process. The next chapter will use the modeling methods developed here to examine the effects of parametric uncertainties that come from the fa brication process, and what these errors in dimensions and material properties can do to the performance of a microdevice.

PAGE 81

81 CHAPTER 4 UNCERTAINTY ANALYSIS AND EXPERIMENTAL CHARACTERIZATION Chapter 3 presented the descri ption of the micromirrors and demonstrated the modeling methods used to predict the static behavior of the devices. While one may assume that the micromirrors were fabricated exactly to the nominal design specifications for dimension and material properties, it is well established that surface micromachining processes have machining tolerances that result in small parametric errors in the finished devices. The effects of these fabrication variations in dimension and materi al property are examined utilizing the modeling methods put forth in Chapter 3 for the electrostatic micromirrors. The eff ects of varying a single parameter at a time are examined first to determine the sensitivity of the design to a given parametric uncertainty. Then, combinations of uncertainties are evaluated using Monte Carlo simulations. The results obtained from the models in Chapters 3 and 4 are then compared to experimental characterization data that was obtained using an optical profiler. 4.1 Parametric Uncertainty and Sensitivity Analysis Recall from the discussion in Section 3.1, that fabrication tolerances for surface micromachining processes can result in final dime nsions that differ from the intended design. The SUMMiT V design manual gives values of dime nsional tolerances in layer thickness and linewidth error, shown first in Tables 3-1 and 3-2 respectively, a nd reprinted in this chapter for convenience as Tables 4-1 and 4-2. These show th at dimensions can vary by as much as eight percent in layer thickness, and as much as twen ty-nine percent for width dimensions on a feature size of 2 microns [6]. The result can be that th e fabricated devices will not behave as predicted, or that devices of the same design can behave differently from one another. Dimensional variations can affect spring cons tants, resonant frequencies, and electrical characteristics.

PAGE 82

82 Table 4-1. Mean and standard deviation of fabr ication variations for layer thickness in the SUMMiT V surface micromachining process. Layer Mean ( m)Std. Dev. ( m) MMPOLY0 0.29 0.002 SACOX1 2.04 0.021 MMPOLY1 1.02 0.0023 SACOX2 0.3 0.0044 MMPOLY2 1.53 0.0034 SACOX3 1.84 0.54 DIMPLE3 Backfill0.4 0.0053 MMPOLY3 2.36 0.0099 SACOX4 1.75 0.0045 MMPOLY4 2.29 0.0063 Table 4-2. Mean and standard deviation of fabri cation variations of lin e widths in SUMMiT V. Layer Mean ( m) Std. Dev. ( m ) MMPOLY2 -0.08 0.03 MMPOLY3 -0.07 0.05 MMPOLY4 -0.24 0.05 Changes in layer thickness result in differences in the vertical spaci ng of the final device dimensions, as shown in Figure 41. The thickness of the structur al polysilicon layers have an obvious impact on the final device dimensions, ho wever the thickness of the sacrificial oxide layers plays an important role in determining th e intermediate spacing of the structural layers. The linewidth variations of th e polysilicon layers also contribute to the final fabricated dimensions of a given geometry being different from the nominal, designed values. Figure 4-2 shows that changes in any of the dimensions can re sult in a final geometry that is different from the nominal design, which affects the size, shape, volume, and mass of the device. Figure 4-1. Fabrication tolerances can changes the thickn esses of the layers, resulting in changes in the final geometry dimensions.

PAGE 83

83 Figure 4-2. Fabrication tolerances can change the dimensions of a fabricated geometry, affecting the final shape, volume, and mass. The mass for the micromirror array devices can be estimated from the volume of the moving components, which are th e mirror surface and the moving comb fingers. The nominal dimensions for these components are shown in Fi gure 4-3. Once the fabrication tolerances are considered, it becomes clear that the mass of th ese parts will be affected by the changes in the geometry. Calculating the volume and multiplying by the density of polysilicon (2331 kg/m3), the nominal mass of these components is 2.34 x 10-11 kg. The mechanical spring constant is affected by changes to the geomet ry of the spring and variations in the Youngs Modulus. The electrostatic model is also affected by these changes. The following sections will examine the eff ects of the dimensional tolerances on the performance of the devices using the modeling methods developed in Chapter 3. First, the Figure 4-3. Nominal dimensions used to calculate the volume of the moving mass.

PAGE 84

84 contributions of each individual parameter variation are considered to try to identify the effect of any given parameter on the final device performan ce. Through sensitivity analysis, it can be determined which key parameters have the most effect on the final device performance. Because these variations can occur in any combination w ith each other, there are an exceedingly large number of possibilities. Theref ore, in order to understand th e effects of these fabrication variations on the device performance, Monte Carl o simulations are done to give an idea of the combined effects of multiple parameter variations. 4.1.1 Effects of Individual Parameter Variations To understand the effects of a single para meter variation on the system, the device performance is determined using the modeling methods developed in Chapter 3 as only one parameter is allowed to change at a time. Th ere are fourteen parame ter variations to be considered, and they include ten variations in layer thickness listed in Table 4-1, three linewidth variations listed in Table 4-2, and one materi al property variation fo r the Youngs modulus of polysilicon. A change in a single parameter can cause both the mechanical spring constant and the electrostatic capacita nce to change from the nominal model. First, the effects on the mechanical model are examined, followed by the electrostatic. The mechanical model described in Section 3.4.1 is a spring in which the stiffness is determined by the dimensions of the beam memb ers, as well as the material properties of Youngs modulus and Poissons ratio. Recall that th e main structural element of the spring is a set of two thin beams constructed in the MMPol y1 layer, which was shown in Figure 3-7. The length of this beam and the cross-sectional area are the most critical dimensions for determination of the beam stiffness. Therefore, the dimension variation in the thickness of the MMPoly1 layer is considered, as well as uncer tainty in the Youngs m odulus as calculated by Jensen et al. to be 164.3 GPa .2 GP a [97]. Poissons ratio is sti ll assumed to be a constant at

PAGE 85

85 0.22 as there is no available data to suggest that it varies. Table 4-3 shows the effects of changing the MMPoly1 thickness as well as the Youngs modulus on the value of the spring constant. While there is no data given in the SUMMiT V design manual [6] regarding line width variations for MMPoly1, it is po ssible that this variation does occur. The layers MMPoly1 and MMPoly2 are most often used together to create one thicker, laminate layer of polysilicon, therefore, diagnostic data is only collected fo r MMPoly1/2 laminate [98]. As an additional study, analysis is done here for cases in which li ne width variations for MMPoly1 are considered to be equal to those of MMPoly2, as 80 nm 30 nm. This analysis is also included in the results of Table 4-3. The first entry in Table 4-3 is the nominal model value, and each subsequent value of the mechanical spring constant, km, is compared to this value in terms of the percent change. When only the thickness of layer MMPoly1 and the Youngs modulus are considered, the spring constant is found to vary between -1.95% to 5.66% from the nominal spring constant. By Table 4-3. Spring stiffness values for cha nging dimensional and material parameters. Layer Thickness MMPoly1 Young's Modulus, E Change in Linewidth MMPoly1 Spring Stiffness, Km m GPa m pN-m % change from nominal 1.0000 164.30 0.00 612.35 0.00 1.0200 164.30 0.00 634.72 3.65 1.0223 164.30 0.00 637.33 4.08 1.0177 164.30 0.00 632.12 3.23 1.0000 167.50 0.00 624.28 1.95 1.0000 161.10 0.00 600.43 -1.95 1.0200 167.50 0.00 647.03 5.66 1.0200 161.10 0.00 622.36 1.63 1.0200 164.30 0.08 730.98 19.37 1.0200 164.30 -0.08 546.91 -10.69 1.0000 164.30 0.08 706.55 15.38 1.0000 164.30 -0.08 526.56 -14.01

PAGE 86

86 considering the effects of variations in the li newidth of MMPoly1, the resulting spring constants are found to vary significantly fr om -14.01% to 19.37% from the nominal value. From this it is clear that including the effects of linewidth variation can have a significant effect on the spring constant. As stated previously, there is no ava ilable recorded data to indicate that linewidth variations do occur in MMPoly1. However, it is reasonable to assume linewidth variations do exist for MMPoly1 as these variations are present in all other layers. Fo r the remaining analysis in this section, linewidth variations in MMPoly1 will be omitted from consideration and are only included here to demonstrate that these errors can have a very large impact on structural stiffness. In the case of the mechanical spring constant, there are only a few parametric variations to consider. As the capacitance for the device is dependent upon the geom etric spacing of the device components, the electrostatic model will be affected much more by any changes in layer thickness or in linewidth. To see the effects of the individual parameters, electrostatic analysis was done for each of the thirteen structural para meters in which each parameter was allowed in turn to be increased by a value of its standard deviation as listed in Tables 4-1 and 4-2. The results are shown in terms of the capacita nce in Figures 4-4, 45, and 4-6. Figure 4-4 shows the capacitance function fo r changes in the thickness of th e polysilicon structural layers, MMPoly0, MMPoly1, MMPoly2, MMPoly3, and MMPoly 4. The nominal capacitance function is shown for a comparison using the nominal dimensi ons of the device. It is evident that making changes individually to these parameters has l ittle effect on the electrostatic model for the device. Figure 4-5 shows the capacitance functi on for changes in the thickness of the Dimple3 backfill, and the sacrificial oxide layers Sacox1, Sacox2, Saco x3, and Sacox4. In the case of Sacox3, it is clear that this pa rameter alone plays a significa nt role in determining the

PAGE 87

87 electrostatic characteristics of th e micromirror. Sacox3 is the sacr ificial layer that determines the spacing between the fixed comb fingers in la yer MMPoly2, and the moving comb fingers in layer MMPoly3. Figure 4-6 shows the capacitanc e functions calculated for changing the area dimensions of the device in the linewidth s of layers MMPoly2, MMPoly3, and MMPoly4. The capacitance curve does deviate some from the nominal model for these parametric variations, particularly in MMPoly2. This analysis is extended to see the combined electromechanical effect of the parametric variations in terms of the static displacement cu rves. Figure 4-7, 4-8, and 4-9 show these results. Figure 4-7 shows the -V curves for the micromirrors when the structural polysilicon layers are each varied. The results here are similar to th e results for the capacitanc e function in Figure 4-4, in that changes in these parameters do not a ppear to have a significant affect on the device performance. It is worth noting however that the layer thickness of MMP oly1 does have a slight effect on the altering the systems static behavior and this is because the layer MMPoly1 plays a significant role in determining the mechanical spring stiffness. Figur e 4-8 demonstrates the sensitivity of the micromirror to variations in th e thickness of Sacox3, simila r to that seen in the capacitance function of Figure 4-5. Likewise, Figure 4-9 shows sma ll deviations in the static displacement curves when the linewidths of the po lysilicon layers are cha nged. It is clear that some parameters have a larger effect on the final static performan ce of the device, most prominently is Sacox3. Parametric sensitivity an alysis is another way to examine how sensitive the modeled system is to variations in a given parameter. Sensitivity, S can be defined as the percent change in the output of the system divided by the percen t change in the parameter of interest, a In this case, the output of the system ca n be considered as the voltage required to achieve a desired position, That is,

PAGE 88

88 Figure 4-4. Capacitance functions for the electr ostatic model with parametric changes in the layer thickness of the structural polysilicon. Figure 4-5. Capacitance functions for the electr ostatic model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide.

PAGE 89

89 Figure 4-6. Capacitance functions for the electr ostatic model with parametric changes in the linewidth error of the structural polysilicon layers. Figure 4-7. Static displacement re lationships for the micromirror model with parametric changes in the layer thickness of th e structural polysilicon.

PAGE 90

90 Figure 4-8. Static displacement re lationships for the micromirror model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide. Figure 4-9. Static displacement re lationships for the micromirror model with parametric changes in the linewidth error of the structural polysilicon layers.

PAGE 91

91 00() () ()()1 1a nomnomV V VV aa aaS (4-1) where S is the sensitivity with respect to parameter a Va( ) is the voltage required to achieve a position of for a model with a variation in parameter a a0 is the nominal value of the parameter, and Vnom( ) is the voltage required to achieve a position of for the nominal model. Figure 4-10 displays the sensitiv ity of the system to changes in line widths. The same analysis for variations in layer thickness is given in Figure 4-11. The four parameters with the highest sensitivities are the thicknesses of layers MMPoly1, Sacox1, Sacox3, and Sacox4. Variations in the parameters Dimple3 backfill a nd Sacox2 have the lowest sensitivities; nearly zero for the entire range of motion. This analysis reveals which geometric parameters in the device design are expected to be the most sensitive to the changes in dimensions from fabrication tolerances. This kind of analysis can also be very useful durin g the design stage of a new device as it can be used in conjunction with optimal de sign methods to reduce the effects of parametric uncertainty on the operation of the completed de vice. However examination of the individual parametric effects will only re veal a partial understanding of th e effects of the fabrication tolerances on device performance, and it is theref ore beneficial to consider the effects on the system when multiple fabrication errors are present. This is done in the following section using Monte Carlo simulations. 4.1.2 Monte Carlo Simulations As in the previous section, there are fourteen different parameters of interest in this analysis, and performing the m odel analysis for every possible combination of parametric variation would be a very large and time-consuming task. Each of these parameters is assumed to vary within a Gaussian dist ribution defined by the mean and standard deviation information

PAGE 92

92 Figure 4-10 Sensitivity of voltage with respect to changes in line width for each value of Figure 4-11. Sensitivity of voltage with respect to changes in layer thickness for each value of

PAGE 93

93 from the fabrication data in Tables 4-1 and 42, and from the studies on material properties in [97]. Monte Carlo simulations provide an eff ective method for examining the effects of these parametric uncertainties by randomly choosing values from the Gaussian distribution and running a large number of model simulations. In the analysis here, 250 simulations are done. From these 250 sets of randomly generated Gaussi an parameters, the eff ects of the fabrication tolerances on the system performance can be de termined. The histograms for each parametric variable are shown in Appendix B to show the distribution of each variable generated in the simulations. The histogram of th e resulting mass that is calculat ed for each of the 250 sets of variables is also shown in Appendix B. It is possible that the fabrication tolerances could have some systematic correlations, such as all of the polysilicon layers for a given fabric ation run having thicker layers at the same time. As there is no data to support this idea however, it can only be assumed that each parameter is allowed to vary independently from the others. A Gaussian, or normal, distribution is given by the following expression [109] 22()/2 ,1 () 2xX XGxe (4-2) where X is the mean value of the data set, is the standard deviation, and x is the data being measured. The Gaussian distribution for a set of data with a mean of zero and a standard deviation of one is plotted in Fi gure 4-12. This figure also illustra tes that 95% of the values of x fall between -1.96 and 1.96, which is consider ed the 95% confidence interval for this distribution. This is also ve ry close to falling between -2 and 2 which constitutes 95.45%. Using the randomly generated variables, it is possible to analyze the impact of these fabrication variations on the mech anical spring constant. First, th is is done for the case in which only the layer thickness of MMPoly1 and the Y oungs modulus are allowed to vary. The

PAGE 94

94 Figure 4-12. Gaussian distribution with a mean of 0 and standard deviation of 1. resulting spring constants, km, had a mean of 634.21 pN-m and a standard deviation of .17 pN-m. Using twice the standard deviation ( ) to represent the 95% confidence interval (CI) for the mechanical stiffness valu es, one can say that there is a 95% chance that the mechanical stiffness will fall between th e values of 609.87 pN-m and 658.55 pN -m. This corresponds to a variation in the mechanical spring st iffness of .84% from the mean. It was shown previously in Tabl e 4-3 that if the linewidth of MMPoly1 is allowed to vary by the same level of uncertainty as the MMPoly2 linewid th, there is a significant impact on the spring constant. Repeating that analysis here for the 250 Monte Carlo simulations but this time mechanical spring constant results for each an alysis respectively. The effect of the MMPoly1 linewidth variation is included here to once again show the large effect th is variable has on the mechanical spring stiffness, however the MMPoly1 linewidth variation will not be considered in the remaining analyses. The Monte Carlo simulations are conducted on th e electrostatic model as well, using the same set of 250 randomly generated variables that are used in th e mechanical spring constant

PAGE 95

95 Figure 4-13. Histogram for mechanical stiffness when accounting for variations in thickness of MMPoly1 and Youngs modulus. Figure 4-14. Histogram for mechanical stiffness taking into account vari ations in thickness of MMPoly1, Youngs modulus, and linewidth of MMPoly1.

PAGE 96

96 analysis. These variables are kept consistent throughout th is analysis to ensure the results will be accurate when the electrostatic and mechanical si mulation results are combined. As was seen in the above analyses, the structur al polysilicon layers affect the device dimensions, and the sacrificial oxide layers affect the spacing of the geometry in the Z-direction. The capacitance function is affected by both these changes in dime nsions. Material properties do not play a role in the electrostatic analysis. Figure 4-15 s hows the results of the calculated capacitance functions for 250 simulations using randomly chosen se ts of variables. In order to classify the capacitance simulation results in te rms of the mean and standard de viation, it is nece ssary to look at the capacitance values calculate d at each value of theta and de termine the mean and standard deviation at each point. This is done in Table 4-4, and the 95% confidence interval values for capacitance vary from 18.47% at zero degrees of rotation and .83% at eight degrees of rotation. Figure 4-15. Results from the capacitance simulati on for 250 random variable sets that show the effects of parametric uncertain ty on the electrostatic model.

PAGE 97

97 Table 4-4. Results from the Monte Carlo simula tions for the capacitance values in terms of mean, standard deviation, and the percent change from nominal. Theta (deg) Mean Capacitance (fF) St. Dev. (fF) % change (95% CI) 0 0.0005 4.73E-05 18.47 1 0.0285 0.0028 19.65 2 0.0295 0.0031 20.81 3 0.0306 0.0034 22.16 4 0.0318 0.0037 23.58 5 0.0331 0.0041 24.88 6 0.0346 0.0045 26.06 7 0.0363 0.0049 26.92 8 0.0381 0.0052 27.59 9 0.0399 0.0056 27.81 10 0.0419 0.0058 27.83 11 0.0440 0.0061 27.61 12 0.0461 0.0063 27.11 13 0.0483 0.0065 27.10 14 0.0504 0.0065 25.88 15 0.0526 0.0066 25.26 16 0.0547 0.0068 24.72 17 0.0570 0.0072 25.31 18 0.0592 0.0078 26.30 Taking the results of the mechanical and electr ostatic analyses togeth er gives a picture of the overall effect that parametric fabrication erro rs can have on the system performance in terms of the -V profile. Figure 4-16 shows th e results of doing this for the 250 simulations using the randomly chosen variables. Given the large numbe r of possible combinations of dimensions that affect both the mechanical and electrostatic m odels, using 250 samples may not be enough to give a complete statistical representation of all the numerous possible combinations; however it is sufficient to show trends in the model pred icted results. These results are compared to experimental characterizati on data in Section 4.2. It is evident from these results that the parametric uncertainty that arises from the fabrication process alone can have a significant performance effect on th e static displacement behavior of the micromirrors. Because the variab le sets used in these simulations are randomly generated, it is difficult to obtain a sense for the role that each individual parameter, or even combinations of parameters have on the overall performance of the micromirrors. Recall from

PAGE 98

98 Figure 4-16. Static displacement results of 250 Monte Carlo simulations with random Gaussian distributed dimensional variations. the sensitivity analysis in Section 3.5.1 that some variables had a significantly larger effect on the system performance, most notable the laye r thickness of Sacox3. To understand the impact this particular variable had in the results from the Monte Carlo simulations, it is possible to try to isolate the contribution from S acox3 by first considering only the results that occur for large deviation in Sacox3 thickness. Figure 4-17 shows the histogram of the Sac ox3 values used in the Monte Carlo simulations. The va lues in blue correspond to th ose that lie within the 95% confidence interval. The values in red represent the other 5% of values that fall at the extreme ends of the distribution. Figure 4-18 shows the simulation results for the -V profiles that are colored to correspond to the values of Sacox3 thickness. The lines in blue are the results that correspond to Sacox3 values within the 95% confid ence interval, while the lines in red are the results from the parametric variations that lie ou tside this interval. This gives a clear indication that for extreme differences in the Sacox3 thickness, the resulting -V profile will also have the most extreme behavior. This analysis was done for additional variables to try to determine a

PAGE 99

99 Figure 4-17. Histogram of values from the Mont e Carlo simulations for the layer thickness of Sacox3. Values in blue lie within the 95% confidence interval, a nd values in red lie without. Figure 4-18. Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the Sacox3 la yer thickness. Curves in blue have Sacox3 values that lie within the 95% confiden ce interval, and lines in red have Sacox3 values that lie in the rema ining 5% of the distribution.

PAGE 100

100 pattern of contributions; however the results for the other para meters did not show any detectable correlations to the performance. This same an alysis for the variables of linewidth in MMPoly2, thickness of MMPoly1, and thickness of Sacox4 are included in Appendix B. Changes in each of these variables show a cluster of profiles in the middle region of the randomly generated -V profiles, which is the opposite of the impact of changes to Sacox3. 4.2 Experimental Characterization This section presents experimental characteri zation and validation of the models developed in the preceding sections. Static characterizatio n measurements for the micromirror device were taken using a WYKO NT1100 Optical Profiler to determine the -V profiles for the mirrors [101]. This measurement tool is able to make measurements of out-of-plane deflections as the micromirrors are given different actuation signals. This information can be used to determine how variable the -V profiles are for mirrors within the arra ys, and from one array to another. Measurements were taken with the system in st atic mode, in which the voltage is applied at different values, returning to zero voltage between each deflection measurement. Static measurement results are provided fo r the arrays of micromirrors de scribed, as well as for a set of single micromirrors that are not part of an a rray. These results are compared to the model predictions, validating the results of the model in determining the static performance, and pull-in behavior. The experimental re sults taken from different micromirrors across three different arrays demonstrate significant diffe rences in behavior among them. This further illustrates the presence of parameter variations within a given array as well as between arrays of the same device design. 4.2.1 Equipment Description The WYKO NT1100 optical profiling system us es interferometric measurements to determine the out-of-plane measurements of a surf ace. The working principle of the instrument

PAGE 101

101 is shown in Figure 4-19. Light travels from the light source and is divided by a b eam splitter. One beam is sent to the reference mirror of th e Mirau interferometer, and the other beam is directed onto the measurement sample. The refl ections of the two beams are recombined into one beam, and because they have traveled different distances in their respective paths, they are no longer in phase. Thus, the newly recombined beams form interfer ence fringes which are recorded by an optical detector ar ray. The digital information from the detector is processed to determine the surface measurement of the sample. Figure 4-19. Diagram of an optical profiler measurement system. The optical profiling system is able to take measurements in static mode, in which the MEMS device is not in motion when the measur ement is taken, as well as in dynamic mode, capturing the motion of the device under excitation. The surface meas urements are recorded into a database, specified by the user, and an ex ample of a surface measurement taken for the micromirror arrays is shown as a 3D image in Figure 4-20. This image shows six mirrors from the array, four of which are tilted by an applied actuation voltage of 60 V. The two mirrors in the center are left without any ac tuation, and this arrangement prove s useful as these mirrors can become a zero reference from which the other me asurements are taken. While the data for the

PAGE 102

102 tilted mirrors is recorded into a database, it is also possible to re view each individual measurement that has been taken. This is helpful to ensure that the data is recorded accurately, and gives insight into how the a ngular tilt measurement of the mirrors is determined. The data can be reviewed using WYKO SureVision software, which accompanies the optical profiling system. This program allows the user to examine 3D images, such as that in Figure 4-20, as well as look at cross-sections of the data. Figure 4-21 shows a cross s ection of the micromirror data Figure 4-20. Six mirrors from the micromirror arra y measured with the optical profiler system. in which the four tilted mirrors appear as di agonal lines. The tilt angle measurement is determined from the displacement measurements in the vertical, out-of-plane, Z-direction, and the horizontal, in-plane X-directi on. Thus, the angle of tilt is f ound from the tangent relationship of the X and Z measurements. Any measurement e rrors in X or Z will resu lt in an error in the angle measurement as well. This error will be disc ussed in more detail in the following sections. Figure 4-21 also shows an example of a measur ement in which the profiling system failed to properly record the data. This illustrates th e difficulties encountered in obtaining these measurements, as the micromirrors are actuated to very large angular displacements that are more difficult for the system to record. A poorly constructed data record such as the one shown

PAGE 103

103 in Figure 4-21 is too sparse to be relied upon for a measurement and should be discarded. Unfortunately, these incomplete and sometimes erroneous records are sometimes recorded into the database files. For this reason, each of the data records has been individually examined and verified to ensure the most accurate of measurement results. Figure 4-21. Data records from the SureVision disp lay that show the crosssection profile of the tilt angle measurements. A) An example that clearly shows the cross-sectional measurements. B) An example of a poorly recorded data file that cannot be used. 4.2.2 Static Results for Single Micromirrors To validate the single micromirror models, a set of single micromirrors were fabricated and analyzed. These mirrors, shown in a micrograph in Figure 4-22, were characterized in the static mode of testing, in which volta ges are reset to zero for each measurement, using a WYKO

PAGE 104

104 NT1100 optical profiler at Sandia National Laboratories. In Fi gure 4-22, the square bond pad on the left is 100x100 m2 and the micromirror on the right has dimensions 156x20 m2. The results from these single mirrors are shown in Figur e 4-23. It is clear that the pull-in point for this set of experimental data is similar to the data collected on the arrays, and the pull-in angle, 13.87, is at the lower range of the pull-in angles for the arrays of mirrors. The pull-in voltage is 71.5 V, similar to the values for the micromirror arra ys and very close to th e predicted value. At the time this data was recorded, the calibration and resolution of the machine were not recorded; therefore it is not possible to disc uss the specific errors that ar e associated with this data. However, the standard operation of the WY KO NT1100 is supposed to be on the order of nanometers. Figure 4-22. Micrograph image of a single micromirror. Figure 4-23. Experimental static results taken fr om individual micromirrors that are not in an array.

PAGE 105

105 4.2.3 Static Results for Micromirror Arrays Experimental data on the performance of the micromirror arrays was acquired using a WYKO NT1100 Optical Profiler located at the Veeco company offices in Chads Ford, PA. This machine was calibrated to a National Institute of Standards and Technology (NIST) traceable standard to be accurate to one-half of one percen t (0.5%) of an 82 nm step. This corresponds to height measurements accurate to 0.410 nm. As th e tilt angle measurements are determined from the inverse tangent of the Z ove r the X measurement, shown above in Figure 4-21, this amount of error in the Z-direct ion corresponds to an er ror in the tilt angle m easurement of .0235. This amount of error is too small to even demons trate on the plots of the data as error bars. While the measurement equipment is believed to operate true to its calibration standards, there is evidence from researchers in [112] that this opti cal profiling system may be subject to larger errors. Measurements of the -V profile for micromirrors taken from different sections across the array for three different miromirror arrays we re taken. These results were obtained using the static mode of measurement in which the voltage signal is rese t to zero between each measurement. The approximate locations of data collection for all thr ee arrays are shown in Figure 4-24 and these locations are labeled. Th ese areas were chosen to try to gain an understanding of any changes in th e performance across the array. Shown in Figures 4-25, 4-26, and 4-27, data from 5 different areas (consisting of four mirrors actuated and two mirrors for reference) on the arrays from among the 3 arrays reveals that there is considerable variation in the be haviors of the individual mirrors. Each array consisted of 416 mirrors arranged in 32 rows and 13 columns. Data was collected from different areas in the arrays in order to examine how the micro performance varies in different locations within the array. Table 4-5 give s a summary of the pull-in angle and voltages for the data. The

PAGE 106

106 Figure 4-24. Approximate locations of data collection on all three arrays. average pull-in angles for arrays 1, 2, and 3 are 14.27, 13.54, and 15.89, respectively. While these values do not agree exactly with the predic ted pull-in value of 16.5 from the analytical model, the lowest value is within 20 percent. Also, the values listed in Table 4-6 are averaged values over multiple data sets. From Figures 425 through 4-27, it is evident that in many cases the mirrors did experience pull-in very close to th e predicted angle of 16.5. The pull-in voltages Figure 4-25. Experimental re sults from array 1, area A.

PAGE 107

107 Figure 4-26. Experimental results from array 2, areas D and E. Figure 4-27. Experimental result s from array 3, areas A and D. Area A Area D First run

PAGE 108

108 Table 4-5. Mean and standard deviation for pull-in angle and voltage from sets of mirrors on all three arrays tested. PI (Deg) VPI (V) Array # Area Mean St. Dev. Mean St. Dev. 1 D 14.27 0.85 62.27 1.75 2 E 13.93 0.62 68.81 2.57 2 D 13.15 0.88 67.17 2.31 3 A 15.89 0.53 64.4 0.62 3 D 15.88 0.49 83.53 1.17 for arrays 1, 2, and 3 are 62.27 V, 67.99 V, and 73.96 V, respectively. It should be noted that for array 3, there is a large difference in the pull-in voltage observed at two different locations on the array. Compared to the predicte d pull-in voltage of 71 V, these values are within 12 percent. Measurements on the mirrors in these experi ments were often conducted such that tests were performed repeatedly on the same set of micromirrors before changing the location of data collection, or switching to a different array. It was observed during the ex periments that after a device had sat idle without actuation voltage a pplied, the devices behaved differently when actuated for the first time, as opposed to subsequent measurements taken on the same mirrors directly afterward. The likely reason for this is a charging effect that occurs after the first actuation of the device after it has sat idle for some time. Figure 4-26 shows this occurred for array 2 when multiple sets of data were taken. Figure 4-28 shows the data from all three devi ces together along with the model predicted behavior of the device using the nominal model ge ometry of the micromirror design presented in Section 3.4.3. The nominal geometry refers to the dimensions of the micromirror based on the original design, not considering a ny fabrication-induced variations. It is clear th at the nominal model falls close to the middle of the widely scattered experi mental results. Section 4.1.2 presented the results of the electromechanical m odel for 250 randomly varied sets of dimensional and material parameters. These modeled variatio ns are compared to the experimental data in

PAGE 109

109 Figure 4-29, and it is evident that the experiment al values fall mostly within the bounds of the modeled variation results. Figure 4-28. Nominal model with experimental data. Figure 4-29. Model-predicted re sults from 100 simulations with parameters determined by random Gaussian variations, s hown with experimental data.

PAGE 110

110 The model results were calculated based on kno wn fabrication tolerances, but this alone does not entirely explain the variations in the device performances. Fabrication variations are known to occur across the wafer as well as from one process batch to th e next, but it is not definitively known if large variatio ns occur locally such that they can have significant effect on the micromirrors within each array. The experime ntal data presented above showed variations between results for different areas in array 3. This indicates the presence of fabrication variations across the array. However within each area on th e array, the group of mirrors exhibited relatively small differences in their resu lts until their individual pull-in voltages. In addition, the differences in the pullin voltage could indicate that th e mechanical stiffness used to calculate the modeled value is different from th e actual stiffness values of the micromirrors. While it is not completely known the causes of thes e differences in performance, it is apparent from the data that considerable pe rformance variation can occur. The effects of fabrication variation on the pe rformance are best illu strated by the case of Sacox3, which was shown in Figure 4-8 to have a significant effect on the V profile, causing it to deviate outward to the right of the other curves. This same behavior is seen again in the Monte Carlo simulation results of Figure 4-18 in which those cases with large variations of Sacox3 outside the 95 percent confidence interval Upon comparison of the of the Monte Carlo simulation results and the experimental results in Figure 4-29, it is seen that the experimental results do not exhibit behavior that is consistent with that of ve ry large Sacox3 variations. This suggests that in the fabrication of these particul ar micromirror arrays, a large variation of the Sacox3 layer thickness did not occur. Plots incl uded in Appendix B study the effects of large variations in the linewidt h of MMPoly2, and the layer th ickness of MMPoly1 and Sacox4. These plots did not indicate a clear connection between the Monte Carlo simulation results and

PAGE 111

111 the effects of these three fabrication errors; ther efore it is not possible to make a conclusion from the experimental results as to the presence or magnitude of fabrication errors in these three variables. To do so properly would require diag nostic data regarding the exact layer thicknesses and linewidth errors collected for a given array of micromirrors, and this data is not available here. 4.3 Chapter Summary This chapter continues the electromechanical device modeling for the micromirrors that was developed first in Chapter 3, and expands the analysis to include the effects of fabrication tolerances on the performance of the micromirrors. By looking at the individual co ntributions of particular parameters, it is evid ent that the layer thic kness of Sacox3 has the largest effect on the static displacement behavior for the micromi rrors. The other parameters appear through sensitivity analysis to also play less distinct roles when considered individually, but when multiple parametric uncertainties are considered, the overall effect of the fabrication variations is evident. Monte Carlo simulations are conduc ted to examine the effects of parametric uncertainties, and this reveals the full extent to which the precision of the micromachining process can dictate performance. The micromirror modeling is then compared to static experimental characterization data that was collected using an op tical profiler that is capable of making non-contact displacement measurements. The results are reported for some individual micromirro rs tested at Sandia National Labs, and then additional results are given for the micromirror arrays tested on a separate measurement system at Veeco, Inc. From these measurements, the static equilibrium behaviors of the micromirrors is determined, as well as the pull-in angle, and pull-in voltage. Taking measurements at different location on three different micromirror arrays begins to show that there can be considerable va riation in the performance. When these experimental results are

PAGE 112

112 compared to the uncertainty modeling results, it reinforces the notion that this variation can be the result of microfabrication errors. While the manufacturers of the optical profiling system do claim a very high level of accuracy for measuremen ts made using their equipment, recent studies of the machine conducted by Mattson show that th e measurements can be susceptible to larger errors [112]. It is no t known if the measurements taken for the micromirror devices are in fact showing larger deviations in the data due to this kind of measurement error. This type of study would be valuable for future work.

PAGE 113

113 CHAPTER 5 DYNAMIC MODEL AND HYSTERESIS STUDY Previous results only considered the static performance of the micr omirrors after they have reached a steady-state value. Here, the dynamics of the syst em are taken into consideration in order to examine the effects of natural fr equency and damping on the time response of the system. Modal analysis and dynamic characteri zation are performed to determine the natural frequencies of the micromirror and the mode shapes. It becomes clear that parametric uncertainty in the micromirrors also affects the dynamic performance of these mirrors. Most notably, the effects of the uncertain ty on the behavior of the electros tatic instability may be seen. In addition to modeling the pull-in and hystere sis behaviors of the open-loop system, a case study is presented for a progressive-l inkage that can be applied to al ter the stiffness of the system to avoid these undesira ble behaviors. 5.1 Dynamic Model and Resonant Frequency Determination It is convenient to rewrite the model dyna mics in Equation 3-16 in terms of natural frequency, n, and the damping ratio, m nk J (5-1) 2mb kJ (5-2) Written in state-space form, the system is described as follows, 1 2 11 2 22 10 01 1 2 2nnx x xx dC V xx Jdx (5-3)

PAGE 114

114 From the linearized dynamic model discussed in Section 3.4.4 using a first order polynomial approximation for the capacitance function, the de rivative of the capacita nce is a constant. Therefore, the natural frequency of the lumped-parameter model determined from Equation 5-1 is found to be approximately 188 kHz. As stated previously in Section 3.4.5, the sque eze-film damping coefficient is difficult to predict analytically for this model, and based on values from similar devices in [71], the damping ratio is assumed to be approximately 0.3. The da mping ratio has a signifi cant effect on the openloop performance of the system, as seen in Fi gure 5-1 for damping ratios ranging from 0.1 to 1. 0 1 2 3 4 5 x 1 0 -5 0 2 4 6 8 10 12 Time (sec)Theta (deg) d = 0.1 d = 0.2 d = 0.3 d = 0.4 d = 0.5 d = 0.6 d = 0.7 d = 0.8 d = 0.9 d = 1 Figure 5-1. Open-loop nonlinear plan t response to a step input of 7 degrees for different damping ratios. 5.1.1 Modal Analysis In addition to using the lumped parameter mode l to estimate the natural frequency of the micromirror devices, modal analysis is done to determine the natural fr equencies and the mode shapes. The analysis is performed for an undamped system, and the equation of motion expressed in matrix notation is 0MuKu (5-4)

PAGE 115

115 where M and K are the mass and stiffness matrices, respectively, and u is the displacement vector. Free harmonic vibrations of the structure are of the form cosi iut (5-5) where i is the eigenvector representing the ith natural frequency, i is the ith natural frequency (rad/s), and t is time. Substituting Equation 5-5 into 5-4 yields 20i iMK (5-6) Ignoring the trivial solution to Equation 5-6, which is 0i, then the following expression must be true. 20iKM (5-7) Equations 5-6 and 5-7 form the eigenvalue problem, and the solutions are the natural frequencies,i and the eigenvectors i The participation factor is related to the eigenvector, and it identifies the amount each mode contributes to the to tal response in a particul ar direction [113]. A sm all participation factor means that an excitation in that direction will no t excite the mode in that direction. A large participation factor indicates that the mode can be excited by motion in that direction. The participation factor can be used to determine the direction of motion in each mode that dominates the response. As defined in reference [99], the participation factor for the ith mode,i is given by T i i M D (5-8) The vector D describes the excitation direction and is of the form DTe (5-9)

PAGE 116

116 where {e} are the six possible unit vectors. {D} is further described in terms of the individual excitations,a iD for DOF j in direction a The directions of excitation, a can be either X, Y, Z, or rotations about these ax es, ROTX, ROTY, ROTZ. 123 T aaaDDDD (5-10) The matrix [T] is 00 00 001000()() 010()0() 001()()0 000100 000010 000001 Z ZYY ZZXX YYXX T (5-11) in which X, Y, and Z represent the global Cartesian coordinates, and X0, Y0, and Z0 are the global Cartesian coordinates of a po int about which the rotation are done. Modal analysis is performed for the micromirro rs using the ANSYS finite element analysis software. The solid model of the structure is sh own in Figure 5-2 and consists of the mechanical spring, the mirror surface and the moving comb fi ngers. The fixed comb finger electrodes may be ignored as they are not part of the moving structure. The stru cture is anchored to ground in all degrees-of-freedom at the base of the mechanical springs. This solid model is meshed with solid92 elements which have 3DOF at each node. The modal analysis is performed using the Block Lanczos method which is appropriate for la rge symmetric eigenvalue problems [99]. The results from the analysis give the first ten natural frequencies, as well as modal participation factors, listed in Table 5-1. The ratio of each participation factor to the largest participation factor value for a given direction is also listed in Table 5-1, in which a ra tio of one indicates the mode that contributes the most to the response in that direction.

PAGE 117

117 The mass calculated from the modal analysis is 2.44 x10-11 kg. The mass result that was reported in Chapter 4 based on the volume of the moving geometry was 2.34 x10-11 kg, which matches the ANSYS calculated result within 4 percent. The difference in these values arises from the inclusion of additional components in the ANSYS model that are not included in the volume calculation done in previously. These additional components include the mechanical spring and its supports. The results of this analysis indicate that th e first mode of vibrat ion for the micromirror structure occurs at 84.74 kHz. This is consid erably lower than the value of 188 kHz calculated in Section 5.1 where it was assumed that the mi cromirror acts only in one degree-of-freedom, rotating about the X-axis (ROTX). It is likewise assumed that the first natural frequency will occur in this rotational directi on and be given by Equation 5-1. The results from the modal analysis for the first mode at 84.74 kHz do in fact show that the dominant direction of the response at this frequency is in the ROTX direction. This is determined by comparing the values of the participation factors for each direction fo r this mode and it validates the one-degree-offreedom assumption for the model in Equation 5-3. The largest participation factor is 6.5E-05 for the ROTX direction, and this is an orde r of magnitude larger than the next largest participation factor which occurs in the Z-directi on. While it is verified that the primary motion for the first resonant frequency occurs in the RO TX direction, the modal analysis results reveal that the resonant motion is more complex than one degree-of-freedom motion and in fact, the first resonant frequency excites motion in both the X-axis (ROTX) and the Z-direction. The motion that occurs in the Z-dire ction will affect the compliance of the system, which will result in a different natural frequency than that predicted using Equation 5-1, which assumes one degree-of-freedom motion about the X-axis only. Th e spring stiffness results presented in Table

PAGE 118

118 Figure 5-2. Solid model created for modal analysis. A) View of the top and back. B) View of the bottom showing the comb fingers. Table 5-1. Modal analysis results for first 10 modes and their natural frequencies, and the participation factors and ra tios for each direction. X-Direction Y-Direction Z-Direction Mode Freq. (Hz) Participation Factor Ratio Participation Factor Ratio Participation Factor Ratio 1 84736.51 2.1183E-10 0.000087 2.7197E-06 0.758918 3.5955E-06 1.000000 2 120372.52 9.5920E-07 0.394546 8.8566E-10 0.000247 -4.3162E-09 0.001200 3 162970.10 -1.3215E-07 0.054358 3.5837E-06 1.000000 -3.0841E-06 0.857751 4 164493.10 8.3800E-07 0.344691 5.5914E-07 0.156023 -4.8219E-07 0.134109 5 391530.45 8.0827E-10 0.000332 1.7035E-06 0.475336 9.7970E-08 0.252526 6 1208580.00 4.6105E-08 0.018964 -2.2081E-08 0.006161 7.5056E-08 0.020875 7 1310412.38 -2.4312E-06 1.000000 -3.6331E-09 0.001014 2.7242E-09 0.000758 8 1610211.37 3.1366E-07 0.129016 5.4173E-08 0.015116 -3.3755E-09 0.000939 9 1696417.45 3.0772E-07 0.126574 2.2398E-08 0.006250 1.1384E-08 0.003166 10 1853628.28 -2.9854E-07 0.127970 1.9532E-07 0.054501 -1.2589E-08 0.003501 ROTXDirection ROTYDirection ROTZDirection Mode Freq. (Hz) Participation Factor Ratio Participation Factor Ratio Participation Factor Ratio 1 84736.51 6.4831E-05 1.000000 -8.7467E-08 0.000395 4.3414E-08 0.000197 2 120372.52 -1.9988E-08 0.000308 -2.2171E-04 1.000000 -1.5238E-05 0.069261 3 162970.10 9.4879E-06 0.146350 -9.8134E-07 0.004426 3.4382E-05 0.156274 4 164493.10 1.5072E-06 0.232480 7.3266E-06 0.033047 -2.2001E-04 1.000000 5 391530.45 -6.3384E-06 0.977690 -8.7374E-08 0.000394 -2.8728E-07 0.001306 6 1208580.00 1.3954E-07 0.002152 9.0970E-08 0.000410 -1.8579E-08 0.000084 7 1310412.38 2.7564E-08 0.000425 -3.4525E-06 0.015572 -3.1098E-06 0.014135 8 1610211.37 5.5281E-09 0.000085 -2.5788E-07 0.001163 8.3555E-06 0.037978 9 1696417.45 4.2753E-07 0.006595 1.0185E-07 0.000459 3.2656E-07 0.001484 10 1853628.28 4.5237E-08 0.000698 2.3618E-08 0.000107 -1.8531E-06 0.008423 3-3 previously show that the spring is very comp liant in the Z-direction with a stiffness of 7.94 pN/m. This additional compliance will lower the overall spring constant for the mode and result in a lower resonant frequency that when only th e rotational motion is considered. The evidence

PAGE 119

119 of motion in additional degrees of freedom at resonance does not however invalidate the assumption that the mirror will rotate about th e X-axis for excitations that occur below the resonant frequency. Furthermore, the electrostatic force that is applied to the micromirror is always an attractive force, drawing the moving el ectrode down toward the fixed electrode. Thus, if resonance is avoided, smooth rotational motion in one degree-of-freedom is still accomplished. This does, however, show the limitations of the 1DOF model assumption, which limits the analysis to only low frequency responses where resonant behavior may be avoided. Table 5-1 also includes the resonant frequencies and their participation factors for modes 2 through 10. It is noticed that several of the modes have motion that acts in more than one direction. 5.1.2 Dynamic Characterization In addition to the lumped parameter estimation and the modal analysis results to determine the natural frequencies of the micromirrors, some experimental data was obtained using a Laser Doppler Vibrometer (LDV), courtesy of the Integr ated Microsystems Group at the University of Florida. This device measures the velocity of a point on a device as it is excited over a range of frequencies. The excitation signa l can be a swept sine wave, or chirp signal, or it can also be white noise, which will excite the device at all fr equencies in the given range. Due to limited signal generation capabilities and time constrai nts, the excitation signal chosen for this experiment was an acoustic impulse, generated by firing a small cap-gun, which produces a loud noise. This effectively generates a white noise signal that can excite the microdevice, and the resulting velocity of the device is recorded by th e LDV. Generating the pulse in this manner is simple and does not require signal generation; ho wever the acoustic impulse is not guaranteed to be the same signal each time it is produced. This experiment was performed five times on each of the three micromirror arrays. Figure 5-3 shows an example of the time response of the micromirrors to the acoustic impulse taken for device 2, trial 1.

PAGE 120

120 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 00.0010.0020.0030.0040.0050.006 Time [s]Velocity [m/s] Figure 5-3. Time series data of the micromi rror response to an acousti c impulse taken with a laser doppler vibrometer. This is the response of device 2, trial 1. The time series data can be examined in th e frequency domain by a Fast Fourier Transform (FFT) of the velocity of the micr omirror surface. Dominant spikes in the FFT indicate a resonant frequency for the device. Figure 5-4 shows examples of the FFT results for several of the tests. It is clear from these results that there is cons iderable noise occurring in the measurements, the source of which has not been identified. As such it can make it more difficult to identify which peaks are in fact resonant frequencies. The co mplete FFT results for each LDV measurement are given in Appendix C. In each of the measurements there appear consistently to be three results that stand out. All of the measurements had a la rge resonant peak that occurred in the range of 40 kHz. A spike occurs in this frequency ra nge for LDV measurements in which no impulse signal was given, and is therefore considered to be result of noise in the environment. This noise could be caused by another piece of laboratory equipment or system in the area, and unfortunately the cause was never identified. It is assumed that this frequency is not in fact a resonant behavior. The results are summarized for here in Ta ble 5-2 for the two dominant resonant frequencies of each test, excluding the lower frequency 40 kHz range results. The next appearance of resonance that occurs throughou t the LDV measurements occurs in the range

PAGE 121

121 A 0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04 6.0E-04 7.0E-04 8.0E-04 0100200300400500 Frequency [kHz]Velocity [m /s]186.88 kHz 82.66 kHz 35.78 kHz B 0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04 6.0E-04 7.0E-04 0100200300400500 Frequency [kHz]Velocity [m /s]43.44 kHz 82.19 kHz 140.78 kHz C 0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04 2.5E-04 3.0E-04 3.5E-04 0100200300400500 Frequency [kHz]Velocity [m /s]48.13 kHz 183.28 kHz Figure 5-4. Results from the LDV experiment showing resonant peak s. A) Device 1, trial 4. B) Device 2, trial 1. C) Device 3, trial 4. 83.13 kHz

PAGE 122

122 of 80 kHz and it is assumed that this is the first resonant mode. In this case, this validates the results obtained from the FEA modal analysis in which the first natural frequency was found to occur at 84 kHz. Higher freq uency resonances occur in each of the measurements, showing that for devices 1 and 3, large response s occurring in the range of 180 kHz, and for device 2, near 140 kHz. These higher order responses are also cons istent with the modal analysis results. Table 5-2. The first three natural frequencies determined from the LDV experiment. Results from the linear model, using Equation 5-1, and the modal FEA are included for comparison. Frequency (kHz) DeviceTrial1st 2nd 1 81.41 186.88 2 81.41 187.81 3 81.71 187.03 4 82.66 186.88 1 5 81.56 187.19 1 82.19 140.78 2 85.63 139.53 3 85.31 136.56 4 92.02 137.03 2 5 85.31 136.10 1 80.91 182.34 2 90.31 183.91 3 81.41 180.63 4 83.13 183.28 3 5 83.28 183.28 Model Eq. 5-1 188.12 -Modal Analysis 84.74 120.37 5.2 Open-Loop Step Response The open-loop response of the system is determined by the actuation voltage signal that is given to the micromirrors. For open-loop operat ion, it is necessary to determine a calibration relationship between the desired angular position and the actuation voltage needed to achieve such position. This relationship is often determin ed experimentally. If va riations in the devices due to fabrication tolerances or other system di sturbances are present, then the calibration must

PAGE 123

123 be performed for each separate micromirror device to ensure the correct calibration is obtained. This approach of individually calibrating each micr omirror device is not practical or efficient. The effects of parametric uncertainty on the device performance using a given calibration are examined for the step response. The effects of pull-in and hysteresis are also examined. 5.2.1 Effects of Parametric Uncertainty on Step Response To illustrate the effects of parametric uncerta inty on the system, the open-loop response of the plant model is considered using different values of stiffness, km. Figure 5-5 shows the response to a step input command of 7 degrees (0 .12 radians) for the nominal stiffness value, and for variations of %. To further illustrate this concept, all of the parameters in the system described in Equation 5-3 are subject to parame tric variation, including the mass moment of inertia, J the damping, b the spring stiffness, km, and the electrostatic torque, Te. If each of these parameters is allowed to vary by % from the nominal value, there are a very large number of possible plants to consider. It is assumed th at calibration is performed on the device for the nominal parameter values. Figure 5-6 shows the open-loop plant responses of the nonlinear plant model to a step input of 7 degrees of th e system model for 50 randomly generated sets of parameters J b km, and Te that are allowed to vary by % of their nominal values. It is clear that with the presence of uncertainties, a step input to the open-loop pl ant will result in steadystate error in the response. In or der to correct for this in open-l oop operation, the system must be carefully recalibrated for each device to en sure the proper response is achieved. 5.2.2 Effects of Pull-in and Hyster esis on Open-Loop Response Electrostatic instability and hysteresis can also greatly affect the system response in openloop operation. Recall from the discussion in Section 3.3, that pull-in occurs when the electrostatic force generated by the actuator exceeds the mech anical restoring force of the structure, causing the mirror to be pulled down to the substrate at its maximum displacement.

PAGE 124

124 0 0.5 1 1.5 x 10-5 0 2 4 6 8 Time (sec)Theta (deg) Command Nominal +10% km -10% km Figure 5-5. Open-loop response to a step input of 7 degrees for the nonlin ear plant dynamics and variations in spring stiffness, km. 0 0.5 1 1.5 2 x 1 0 -5 0 2 4 6 8 10 12 Time (sec)Theta (deg) Figure 5-6. Open-loop nonlinear plant response to a step input of 7 degrees for 50 random parameter variations.

PAGE 125

125 The mirror will remain in this position until th e actuating voltage has been reduced below the holding-voltage, causing hysteresis. The effects of pull-in and hysteresis for the static response are investigated in Chapters 3 and 4, but there ar e dynamic effects that can affect pull-in as well. It is known that pull-in is affected by re sonance, and it is therefore assumed that the micromirrors operate at frequencies below resonan ce [40]. If the system is driven dynamicallyby a voltage that is greater than the holding volt age and less than the pullin voltage, it is still possible for the inertial effects to cause the mi rror to experience pull-in and remain pulled in until the applied voltage is reduced below the holding voltage. In order to incorporate this effect into the dynamic model, the system response is subjected to a set of discontinuous, piecewise defined behaviors. When the angle, becomes greater than or equal to the pull-in angle, PI, the system response sets theta equal to the final pull-in position, F. After pull-in has occurred, the system response remains pulled-in until the voltage drops below the holding voltage, VH. The system then returns to the released position, R. This response is shown in Figure 5-7 for sinusoidal commands of amplitudes of 14.9, 16.6, and 17.2. The corresponding voltage command is also shown in the figure. Again, for commands beyond the pull-in angle of 16.5, the response shows pull-in and remains in this st ate until the actuation voltage is reduced below the holding voltage of 68 V. In the case of a step command, overshoot in the system response becomes very critical when driving the device to a position that is near the pull-in point. In the case of large overshoot in the response, the device will pu ll-in and will not be released as the voltage command for a step input is constant. Figu re 5-8 shows the open-loop step respon se of the system for commands of 12, 14, and 17. It is expected that the co mmand input of 17 will result in pull-in as it is greater than the pull-in angle. However in this case, overshoot in the response for a step

PAGE 126

126 A B Figure 5-7. Open-loop responses to a sinusoidal input showing hystere sis. A) Results of angle of rotation over time. B) Voltage signal s that correspond to the command inputs. Figure 5-8. Open-loop responses to a step co mmand showing overshoot that result in pull-in. command of 14 also results in pull-in of the response as the overshoot causes the device to move beyond the pull-in point, and the actuation voltage applied is not less than the holding voltage required to release it. This is another exampl e of the effects of hysteresis on the response of the system where the inertial effects plays a role, referred to as dynamic pull-in [111]. Dynamic pull-in can result in cases where the velocity of the actuator is high as it approaches the pull-in point. This can be caused in the case of applying instantaneous ac tuation voltages, and it

PAGE 127

127 can cause the actuator to pull-in at a lower voltage than th e static pull-in vo ltage. This dynamic effect is difficult to model, and is affected by the damping of the system. For zero damping in a parallel plate system, the dynamic pull-in can occur at an 8% lower voltage than the static pull-in voltage; however the presence of damping in the system decreases this effect. 5.2.3 Continuous Characterizati on of Micromirror Arrays The optical profiler measurement system describe d in Section 4.2.1 used to collect static performance data was also used to apply continuo us voltage as a partial sine wave. The voltage was increased and decreased without resetting to zero in between measurements, which allows the effect of hysteresis to be studied. This is done by applying a voltage signal such as that shown in Figure 5-9 with amplitudes ranging from 44 volts to 85 volts following a partial sine wave, with measurements taken at every ten degrees of phase. The results for a set of four mirrors from array 3 are shown as a function of pha se in Figure 5-9, and as a function of voltage in Figure 5-10. In this instance, only two of the micromirrors, 1 and 3, experienced pull-in and hysteresis, while the other two, 2 and 4, did not. A B Figure 5-9. Results from dynamic study showing pull-in and hysteresis. A) Actuation signal applied for dynamic study. B) Results from applying the actuation signal.

PAGE 128

128 A B Figure 5-10. Results showing the hysteretic behavior of the micromirrors. A) Mirrors 1 and 3 show pull-in and hysteresis. B) Mirrors 2 and 4 do not have pull-in. 5.3 Hysteresis Case Study: Progressive-Linkage As discussed in the literature re view Section 2.3, there are ways that researchers have used nonlinear flexure designs to mitigate electrostati c pull-in and hysteresis. One such nonlinear flexure design is presented here, called a progres sive-linkage [57], [58]. The design and function of the linkage is presented and it is analyzed to show how it affects the electrostatic instability and hysteresis in open-loop operation. The re sults presented are only th eoretical and have not been fully realized in fabrication. 5.3.1 Progressive-Linkage Design Electrostatic instability occurs when the el ectrostatic force becomes too great for the mechanical spring to handle. If the characteri stics of the mechanical restoring force can be altered such that this pull-in never occurs then the micromirror device could operate continuously over its full range of motion, from 0 to 19 degrees for the micromirror designs of studied in this dissertation. This done at the cost of increased actuati on voltages. The following analysis proposes a new design for the spring that has a nonlinear restoring force such that the stiffness characteristics increase signifi cantly as the spring is rotated.

PAGE 129

129 The analysis for this design is based on an eq uivalent four-bar model as depicted in Figure 5-11. The geometric relationships between th e links are also shown in Figure 5-11. The kinematics of the mechanism can be denoted by the following vector sum where the vectors denote the position and orientation of each side of the mechanism shown in Figure 5-11. 3 1 0 2r r r r (5-12) Since the four-bar mechanism is a one-degree-of-freedom device, the angles 2 and 3 can be described as a function of 1. That is, the length and orientation of each side can be used to determine the relationships of the angles 2 and 3. By using the y and z components of the vector 2r, an expression for 2 is given as y zr r2 2 1 2tan. (5-13) In order to determine the angle 3, begin with the relationship 3 3 2 3 2 2 2cos 2d dr r r r r (5-14) This yields an expression for3 3 2 2 2 3 2 1 32 cosr r r r rd d (5-15) An expression for d is found from y y z z dr r r r1 0 1 0 1tan (5-16) The angle 3 is given as 3 3 d. (5-17) To realize this design scheme in a surface micromachined device, the design will be subject to the limits and constraints of the micromachining process. One of the challenges

PAGE 130

130 A B Figure 5-11. Diagram of four-bar mechanism for progressive linkage analysis. A) The vectors and geometry for kinematic analysis. B) Th e springs and angles for force and moment analysis. to realizing this mechanism in a surface micr omachining process is to find suitable joint configurations that will allow for the creation of a four-bar mechanism. For the sake of this discussion it is assumed that this 2-D representa tion of the four-bar linka ge is created using a series of thin beams, kinematically spaced by ri ( i = 0,1,2,3), each joint may be considered as a beam in torsion that provides a restoring force to the system. Seen in Figure 5-12, a beam of length L with a rectangular cross-section of dimensions w x t is used to model the stiffness at the joints. The restoring torque on the member can be calculated by L G K Ti s i s i i s) (0 (5-18) for each joint i = 0, 1, 2, 3, where ) 1 ( 2 E G is the shear modulus, ,,0 sisi is the change in the rotation at the joint from its unloaded position (the free length configuration of the torsional spring), and Ki is given as 34 416 3.361 16312itwww K tt (5-19)

PAGE 131

131 when t > w For the case of t < w the expression is 34 416 3.361 16312iwttt K ww (5-20) Figure 5-12. Cantilever beam with cross-section w x t and length L The resulting static force and moment equa tions can be determined from the free body diagrams in Figure 5-13. 02 1 1F F Fbar (5-21) 03 2 2F F Fbar (5-22) 00 3 3F F Fbar (5-23) int112120joSeSMTTTrF (5-24) int223230joSSMTTrF (5-25) int303300joSSMTTrF (5-26) The relationships above combine to determine the torque output for a progressive linkage design. The dimension of the mechanism that is the easiest to change in the design is the horizontal distance separating the anchor po ints of the device, referred to above as 0r Figure 514 shows the output of the progressive linkage for different values of0r For a value of 0r less than 10 m, the structure will become very stiff befo re the mirror reaches its maximum angle and

PAGE 132

132 it will not be able to fully rotate. This is seen for values of 0r equal to 4, 6, and 8 m. As the value of 0r is increased, the structure becomes more co mpliant. Figure 5-15 shows plots of the Figure 5-13. Free body diagrams for each member of the linkage. 0 2 4 6 8 10 12 14 16 18 20 0 200 400 600 800 1000 Theta [degrees]Torque [pN-m] r0 = 8 r0 = 6 r0 = 4 r0 = 10 r0 =12 Figure 5-14. Progressive-linkage be havior for different values of ro in m.

PAGE 133

133 behavior of the progressive linkage for 0r equal to 9 m overlaying the electrostatic torque curves from Figure 3-21. Tables 5-3 and 5-4 give the link length and joint dimensions used for this progressive-linkage design. The Youngs Modulus is assumed to be 164.3 GPa and the Poissons ratio is 0.22. The linear restoring force from Figure 3-21 is also included for comparison. The requirements for pull-in to occur are that the electrostatic and mechanical torques be equal in magnitude and slope. The prog ressive linkage creates a stiffness profile that eliminates the occurrence of the second condition such that the stiffness curve does not at any point run tangent to the electrostatic torque curves and therefore doe s not exhibit pull-in behavior. The static -V profile for a device using a progre ssive linkage is shown in Figure 516. The cost of this extended actuation ra nge is that larger voltages are required. Figure 5-15. Progressive linkage output for ro equal to 9 m along with the elec trostatic torque curves and the linear restoring torque.

PAGE 134

134 Table 5-3. Link length dimensions used for progressive-linkage design. Link Length ( m) r0 9.000 r1 8.625 r2 9.953 r3 4.375 Table 5-4. Joint dimensions used for progressive-linkage design. Dimension ( m) Joint T w L 0 2.50 1.00 66 1 2.50 1.00 66 2 2.25 3.00 111 3 2.25 3.50 111 Figure 5-16. Static -V relationship for micromirror with a progressive-linkage. Recall from Section 3.4.5 that a bifurcation analysis may be used to examine the electrostatic pull-in behavior for a system with a nonlinear spring constant. The equations for the analysis now include the progressive spring cons tant that is a function of the rotation angle, expressed as km( ). The expression for the fixed point solutions is now

PAGE 135

135 2 1()() 111 4 22 1,2 1,2,3eeimei jTk bb JJJxJ j i (5-27) Applying this analysis to the device using the progressive linkage yields the bifurcation diagram shown in Figure 5-17. It is clear from this analysis that the device is able to reach angles up to 18 degrees usin g higher voltages of up to 130 V. Figure 5-18 shows how the bifurcation plot will change as the progressive s tiffness profile increases or decreases by a factor of 2. 5.3.2 Open-Loop Response Using a Progressive-Linkage Because this device does not experience pull-in, it is assumed that th ere is no hysteresis in the response. Therefore the system will be able to respond to actuation signals such as a sine wave or step command without having pull-in. The system open-loop response to sinusoidal inputs is shown in Figure 5-19. Figure 5-20 sh ows the step response to inputs of 12, 14.3, 17.1, and 18. Unlike the system with a linear spring force, this device is able to achieve positions beyond the pull-in angle. As stated be fore, the actuation voltages for this device with a progressive linkage will be higher than for the results using a linear spring in Figures 5-7and 5-8. 5.3.3 Parametric Sensitivity of the Progressive-Linkage It was shown in Chapter 4 how the mechanical spring that consisted of only one set of beams was sensitive to fabrication tolerances. It thus seems logical to assume that by adding complexity to the spring design in terms of the pr ogressive-linkage will add to the effects of this sensitivity. The following discussion will exam ine the effects of fabrication tolerances on the progressive-linkage design. The methods of analys is will follow that of Section 4.1, in which first the effects of changing only one paramete r at a time are examined. Then, Monte Carlo

PAGE 136

136 Figure 5-17. Bifurcation diagram for micromirror using a progressive-linkage to avoid pull-in behavior. Figure 5-18. Bifurcation diagram for the micromi rrors using a progressive-linkage to avoid pullin behavior for different valu es of mechanical stiffness.

PAGE 137

137 A B Figure 5-19. Open-loop responses to a sinusoid al input for the device using a progressivelinkage. A) Results of angle of rotation over time. B) Voltage si gnals that correspond to the command inputs. Figure 5-20. Open-loop response to a step i nput for device using a progressive-linkage. simulations are done to look at th e effects of randomly varying all of the uncertain parameters. The uncertain parameters are assumed to vary in a Gaussian distribution, identified by a mean and standard deviation given from known fabrication tolerances. Table 5-3 and 5-4 gave numbers for the progres sive-linkage design variables that are used to evaluate the four bar linkage model. The to rsional spring constants, calculated by Equation 5-

PAGE 138

138 11, depend on the values of w and t from Table 5-4, as well as the Youngs modulus, E. The dimensions of the joints are subject to the fa brication tolerances of the surface micromachining process. Assume for this given design, that the join ts 0, 1, 2, and 3 are fabricated as beams in the layers MMPoly1, MMPoly1, MMPoly4, and MMPoly3, respectively. This means that each joint will be subject to the errors in layer thickness a nd linewidth that are defined from the fabrication tolerances given for the manufacturing process. Table 5-5 lists this information, including the nominal joint dimensions, t and w the respective fabrication layer used to make each joint, and the associated fabrication errors given in terms of mean and standard deviation. For example, joints 0 and 1 are to be fabricated in layer MM Poly1, making their dimensions prone to variation in the thickness of MMPoly1. All of the layers are subject to variation in Youngs Modulus, previously stated to be 164.3 3.2 GPa. Other errors in the fabrication can occur that will affect the design of the four-bar type linkage in terms of the link lengths, however these are neglected here, and only the errors associated with the join t stiffness are being considered in this analysis. Table 5-5. Uncertainties in the joint dimensi ons for a proposed progressive-linkage design. Dimension ( m) Uncertainty of Dimension Mean St. Dev.( m) Joint T w Fabrication Layer t (thickness) w (linewidth) 0 2.501.00 MMPoly1 1.02 0.0023-1 2.501.00 MMPoly1 1.02 0.0023-2 2.253.00 MMPoly4 2.29 0.0063-0.07 0.05 3 2.253.50 MMPoly3 2.36 0.0099-0.24 0.05 The sensitivity of the progressive-linkage desi gn is examined when only one variable is altered at a time. Figure 5-21 shows the results from this analysis in terms of the mechanical torque as a function of rotation angle as each vari able is changed by one standard deviation from the mean. The two variables that have the greates t effect on the stiffness profile of the nonlinear spring are the thickness of layer MMPoly3, and th e linewidth of MMPoly3. In order to examine the effects of changing multiple variables at th e same time, Monte Carlo simulation is done in

PAGE 139

139 the same fashion as in Section 4.1.2. Each va riable is randomly varied according to a Gaussian distribution defined by the mean and standard de viation of that variable. For this progressivelinkage design, 50 simulations are performed, and the results in terms of the torque-theta profile are shown in Figure 5-22. It is striking to see the very la rge effects of these very small parametric perturbations, and from a qualitat ive point-of-view, it becomes evident that the current proposed design will be ve ry sensitive to the fabrication. In a case such as this, design optimization is recommended to find a design for th e linkage that is less sensitive to these errors. This is suggested for future work to explore alternative joint designs and variations of the progressive linkage that will make it le ss prone to parametric uncertainties. 5.3.4 Progressive-Linkage Prototype Despite the limitations of the design that ar e revealed through the parametric analysis in Section 5.3.3, a prototype of th e micromirror with the progressive -linkage has been developed. This design, illustrated in Figure 5-23, wa s developed and fabricated in the SUMMiT V micromachining process with joint configurations consisting of a series of thin beams as modeled in Section 5.3.1. Figure 5-24 shows a micrograph image of the progressive-linkage and the micromirror device. Because of the pl anar fabrication requirements of surface micromachining, the diagonal top member of the four-bar device, 2r, can be acquired via a kinematically equivalent L-shaped beam, shown in Figure 5-24. It should be noted that for an array of micromirrors that required close sp acing, this is perhaps not an ideal design implementation as the linkage itself occupi es a significant amount of space behind the micromirror. A more compact implementation that could be located underneath the mirror or to the side would be preferred. Due to time-constra ints with the available fabrication run, in-depth analysis of the device performance was not conducted before the final design was submitted for

PAGE 140

140 Figure 5-21. Results of parametric analysis for individual errors in joint fabrication of the progressive-linkage. Figure 5-22. Fifty Monte Carlo simulation results for varying the joint fabrication parameters for the progressive-linkage design. The nomina l spring value is shown for comparison.

PAGE 141

141 fabrication. This is an unfortunate but so metimes common occurrence encountered by MEMSdesigners who may be restricted by the time-tab les of foundry services and available project funding. It also gives a good example of the co nsequences of incomplete a priori analysis. Figure 5-23. Schematic drawing of the prototype progressive linkage spri ng. A) Progressive linkage spring. B) Spring attached to th e micromirror. C) Drawing of L-shaped equivalent beam. Figure 5-24. Micrograph of the prototype micr omirror with a progre ssive linkage spring. As previously stated, the proposed link age design prototype was fabricated, and subsequently tested using the WYKO NT1100 op tical profiler located at Sandia National Laboratories in Albuquerque, NM. This is the same optical profiler discussed in Section 4.2.2. The results of this static experimentation ar e shown in Figure 5-25 as the rotation angle, 1, that was measured for an applied actuation voltage. It is clear that the voltages required to actuate the

PAGE 142

142 micromirror with the progressive-linkage are higher than for the micromirror without the progressive linkage. It is not clear however if the progressive device was able to accomplish the nonlinear spring behavior desired. After the de vice was rotated to approximately 14 degrees, all subsequent measurements failed to record proper data files. This issue was first discussed in Section 4.2, where for high angles of rotation, the measurement machine routinely had difficulty taking measurements. Thus, it is inconclusive to state whether the pull-in point of the micromirrors was in fact delayed by the spring desi gn or not. It is suspected however that the linkage did not perform its inte nded function, and the data beyond 14 degrees of rotation did not record because the mirror had in fact pulled in. In order to investigate the device performan ce to try to identify if the proposed progressive-linkage design implemented is worki ng properly, the structure of the progressivelinkage has been examined using FEA. Just as with the previous mechanical spring analysis of Section 3.4, the progressive-linkage is modeled in ANSYS using beam189 elements [99]. As the structure is displaced about the X-axis, is so on becomes clear from looking at the resulting displacement of the linkage, that the design is no t operating as the intende d four-bar model, but is instead deflecting in the positive Z-directi on, out-of-plane. This Z-direction deflection prevents the joints, which are fabricated as thin beams, from rotating as they are intended. Figure 5-26 shows the results of the FEA analysis of the pr ototype design for both linear deflection analysis, and nonlinear, large-deflectio n analysis. The nonlinear analysis begins to deviate from the linear results for very large deflections, but does not produce the desired nonlinear stiffness profile for the range of motion of the micromirror. It becomes evident from this deflection, that the progressive-linkage in this current design implementation is not providing the appropriate motion that is capable of providing the nonlinear stiffness profile to

PAGE 143

143 affect the pull-in behavior of the device. Th is becomes a very good example of the importance of performing careful analysis of a MEMS design prior to fabric ation. The above theoretical model presented for the progressive linkage is st ill valid. The challenge remains however to find the appropriate design implementation that will ca rry out the four-bar linkage design principles. This remains as future work. Figure 5-25. Experimental data collected fo r the prototype of the micromirror with the progressive-linkage Figure 5-26. Results from FE A of the prototype progressiv e-linkage design for linear and nonlinear deflection analysis shows the prototype progressi ve-linkage fails to produce the desired stiffness profile.

PAGE 144

144 5.5 Chapter Summary The work presented in this chapter expanded upon the static modeling methods developed in Chapters 3 and 4 to examine the dynamic char acteristics of the system. In keeping with previous modeling assumption, the lumped parameter model for the micromirrors is presented as a one degree-of-freedom mass-spring-damper syst em. The damping characteristics are assumed to have a low damping ratio based on the result s from similar devices in the literature. The natural frequency of the device is determined fr om the mass, which is estimated from the volume of the moving micromirror, and the spring cons tant that was calculated and characterized in Chapter 3. This determined the natural frequency of the mi cromirrors to be 188 kHz. Modal analysis performed using FEA on the structure determined the first natural frequency to be lower, at 84 kHz, and it was fo und that this frequency responded in more than one degree-of-freedom. An examination of the part icipation factors for the response of the first mode in each direction reveals th at the primary direction of the response is in the rotational Xdirection (ROTX), which corresponds to the one-degree-of-freedom model assumption. However, it is clear that motion in other di rections, namely the Z-direction affects the compliance of the system and the response, resu lting in a lower than pr edicted first natural frequency. This additional degr ee of freedom acting in the Z-axis direction significantly lowered the effective spring constant for this mode, t hus lowering the natural frequency. The modal analysis results are verified by experimental measurements taken with a LDV to determine resonant behavior for the devices. While the results from these experiments were affected by noise, it is clear that resonant peaks do occur near the values predicted by the modal analysis results. It is clear that the first mode does respon d primarily in the ROTX direction, and the evidence of motion in additional degrees of freedom at resonance does not invalidate the

PAGE 145

145 assumption that the mirror will rotate about th e X-axis for excitations that occur below the resonant frequency. The electrostatic force that is applied to the micromirror is always an attractive force, drawing the moving electrode down toward the fixed electrode. Thus, if resonance is avoided, smooth rotational motion in one degree-of-freedom is still accomplished. This does, however, show the limitations of the 1DOF model assumption, which limits the analysis to only low frequency responses where resonant behavior may be avoided. Additionally, the hysteresis behavior for the mi cromirrors is examined, and it is found that the theoretical model is able to predict not only the pull-in, as demonstrat ed previously, but also the point at which the mirrors will release from pull-in as the actuation voltage is reduced. Experimental results from the optical profiler validate these findings. Hysteresis causes a deadband in the actuation capabi lities that can be detrimental to the performance of the micromirrors, and thus actuation within the hyste resis loop should be avoi ded. The effects of pull-in and hysteresis also have the ability to negatively affect the dynamic response for actuation signals that occur below the pull-in voltage. To alleviate the problems associated with electrostatic instability, a novel solution is presented, called a progressive linkage. The progressive linkage creates a nonlinear mech anical restoring force that increases as the electrostatic force increases. It is shown through theo retical predictions that this method can be effective at eliminating pull-in, with the cost of requiring increased actuation voltages. Sensitivity analysis reveals however that this design is very sensitive to the fabrication tolerances, and therefore should be optimized to ensure better performan ce. A prototype of the progressive linkage design is pres ented along with some experiment al data that unfortunately is inconclusive. Further design development and an alysis of the progress ive linkage device is considered as future work.

PAGE 146

146 CHAPTER 6 CONTROL DESIGN AND SIMULATION Now that a dynamic model has been develope d for the micromirrors, controllers may be designed for the system with the goal of ensuri ng steady-state performanc e regardless of changes to the plant dynamics. As seen in recent literat ure and the work presented in Chapter 5, active and passive control approaches have been su ccessful at both extending travel range of electrostatic actuators and for improving tracking, disturbance rejection, transient response, system bandwidth and stability, and in reducing steady-state errors. For active control design considerations, in this dissertation the linearized model of the system was used for determining the controller gains before implementing them on the nonlinear plant models. The general form of a feedback control system is shown in Figure 61 for unity feedback. In this chapter, PID and LQR controllers are developed and implemented to further quantify the significance of model uncertainties, pull-in and hysteresis. The PID an d LQR control designs in Sections 6.1 and 6.2 only consider the performance of single micromirrors. The model and performance of the micromirrors as an array is discussed in Section 6.3. Here, the unique issue of how to control an array of micromirrors that are not individually controllable is explored. This section will demonstrate a model of multiple mirrors as a single-input/multiple-output (SIMO) system and will discuss the feedback signals available by considering two different kinds of optical sensors: position detecting sensors (PSD) and charge-coupled devices (CCD). The performance of these sensors is considered as well as the impact th ey will have on implementation of closed-loop control system on the array of micromirrors. Figure 6-1. Basic block diagram with unity feedback.

PAGE 147

147 6.1 PID Control Proportional-Integral-Derivative (PID) control is perhaps the most widely used kind of control scheme [102]. The appeal of PID control is that it applies to almost any system, even those for which a system model is not known. Th ere are many techniques that may be used to define the control gains and to tune them for the best performance. It is popular because it is easy to design and fairly intuitive to determine the control parameters for systems modeled with second-order dynamics. 6.1.1 PID Control Theory The general form of the transfer function for a PID controller is 11 ()1CPIDPd iGsKKKsKTs sTs (6-1) where P i IK T K (6-2) and D d P K T K (6-3) The block diagram of this system is shown in Figure 6-2. The closed-loop transfer function for this block diagram, with the plan t modeled as a linear second order system is ()() () ()1()()()CP CPGsGs Cs R sGsGsHs (6-4) Assuming unity feedback, that is H(s) = 1, and substituting Equation 6-1 into 6-4 gives an expression for the closed-loop transfer function. 2 322() () ()(2)()DPI nDPnIsKsKK Cs R ssKsKsK (6-5)

PAGE 148

148 Figure 6-2. Block diagra m with PID controller. The proportional term is a gain that attenuates the magnitude of the system response. The integral term seeks to eliminate steady-state error in the system. The derivative term, as seen in the denominator of Equation 6-5, is associated with the damping term, and as KD increases, the system damping will also increase affecting the rise time and settling time of the response. The gain KD is also in the numerator, and can act as a high pass filter that will amplify high frequency noise. Design methods, such as root locus, can be employed to help derive the proper control gains for a particular desired performance [102]. This details in general, how a PID controller affects a linear secondorder system. For the micromirror array models presented in the prev ious chapters, a lineari zed version has been developed using a linear first order approximation of the capacitance functi on in Section 3.4.4. A set of PID control gains are chosen using tr ial and error to yield a linearized closed-loop response characteristic of a an overdamped system with zero overshoot, and to drive the steadystate error to zero. This controller is implem ented on both a linear and nonlinear plant. 6.1.2 PID Results A PI controller is implemented on the system in an effort to ensure zero steady-state error despite the presence of model uncertainty. Usin g only a simple proportional controller (P-

PAGE 149

149 controller) on the system is not sufficient to en sure zero steady-state error for different plant variations, therefore an integral term is incl uded. The controller gains are chosen as the proportional gain, KP = 100, and the integral gain, KI = 100,000. It was found that the derivative controller term, KD is not needed. To compare the e ffects of the nonlinear terms in the electrostatic model, the step response of the lin earized plant model is co mpared to that of the nonlinear plant model in Figure 6-3. Step respon ses are shown for both models for different step values ranging from 2 degrees to 16 degrees, an d the effects of the nonlinear terms begin to appear as the transient response of the nonlinear plant is affected by the magnitude of the step input. The closed-loop system has no overshoot, wh ich is important in electrostatic systems that experience pull-in. For a system application with strict transient performance requirements, this set of gains however may not be suffici ent at very low command angles. As has been shown in the experimental char acterization data for these micromirrors in Chapters 4 and 5, an important control objectiv e is to drive the respons e to have zero tracking errors in the presence of plant uncertainties. Figure 6-4 shows the effects of model uncertainty for the nonlinear plant response, including mode l variations of % variation in the spring stiffness, km. Open-loop analysis in Chapter 5 presen ted the open-loop plan t responses of the system for 50 randomly generated sets of parameters m b km, and Te that are allowed to vary by % of their nominal values. The closed loop response of those same 50 plants is shown in Figure 6-5 for parametric variations ranging from % to a very hi gh value of %. It is clear that even this simple PI controller drives all of the plants to zero steadystate error, achieving the goal of position tracking. 6.1.3 PID Controller Response to Hysteresis While it is preferable to avoid driving the micromirrors in the unstable range of motion, it is possible that this could occur, especially as the pu ll-in point is known to vary for different

PAGE 150

150 A B Figure 6-3. Step responses for PI D controller. A) Linear plant model. B) Nonlinear plant model. Figure 6-4. Closed-loop PID response to different step inputs when the spring constant is varied by %. micromirrors and for dynamic operating conditions. The discussion in Section 5.2.2 demonstrated the open-loop hysteretic behavi or of the micro devi ces when subject to electrostatic pull-in. Here, this same scenario is considered for the case of the closed-loop system with a PID controller in place. The resu lt is that if the mirror is commanded to an unstable position, and subsequently pulls-in, th e controller will see this position error and

PAGE 151

151 Figure 6-5. Closed-loop PID resp onse to a step input of 7 degrees for 50 random sets of parameteric variations. Parameters are allo wed to vary by A) 10%, B) 20%, C) 30%, D) 40%, E) 50%, and F) 90%. 10% Uncertainty 30% Uncertainty 20% Uncertainty 40% Uncertainty 50% Uncertainty 90% Uncertainty A B D C E F

PAGE 152

152 seek to correct it. The controller will comma nd the actuator with lower voltages until the holding voltage is reached, and the mirror will rel ease from pull-in. The position of the released mirror will still not be the correct commanded position, which is an unstable position that cannot be reached. So this cycle will repeat itself, as shown in Figure 6-6, resulting in a fast switching behavior until the commanded positi on of the mirror returns to the stable range of motion. In Figure 6-6, it is clear that this switching behavi or would be undesirable for the system, and could even result in damage to the micromirrors; however one benefit of the contro ller response is that the effect of the hysteresis is mitigated by the controller, and the mirror position returns from pull-in at an earlier time than the response w ithout the controller. This control behavior demonstrates potentially undesirable behavior that could result, and it is not suggested that this PID control implemented for motion in the unstabl e range is ideal. The control algorithm can easily be augmented to detect electrostatic pull-in conditions to keep th e switching response from occurring, and thus avoid potentially damaging the device, but still keep the added benefit of reducing the effect of hysteresis. This discussion is also useful to show once again, the need for eliminating this electrostatic inst ability in the response, which ma y be done with the progressive linkage design proposed previously. Section 5.3 presented the design of a progress ive-linkage that can be utilized to eliminate the effects of pull-in and hysteresis. It was demonstrated theoretically that this device can provide actuation over an extended range of the mirrors motion at higher actuation voltages. Using this progressive linkage to eliminate pull-in however does not guarantee that the effects of fabrication tolerances will not play a role in devi ce performance. With the added complexity in the design, parametric uncertainty in the dimensions of the linkage could contribute even more to variations in the system performance; hence closed-loop contro l is still necessary. Figure 6-7

PAGE 153

153 shows the closed-loop PID step responses for the micromirrors with the progressive linkage, and they are in fact able to achieve stable rotatio n above the pull-in limit of 16.5 degrees. Also shown are the PID step responses for the 50 rand om plant variations with % variation of model parameters. Figure 6-6. Closed-loop PID response to a commanded position in the unstable region. A B Figure 6-7. Closed-loop step responses for PI D controller for a system using a progressivelinkage. A) Step responses of different ma gnitudes. B) Step responses of 50 plants with model uncertainties.

PAGE 154

154 6.2 LQR Control Linear quadratic regulator (LQR ) control is an optimal control method that uses a linear state-space model of the plant to design a stable controller that seeks to minimize the response of the system states and the control actuation. LQ R control design is concerned with minimizing a cost function that balances the control effort with the system states according to defined weights. This type of control requires full-state feedback and that the system is completely controllable. In order to apply LQR control to the system of micromirrors in which only position information is available, a state estimator must be employed for the velocity state. 6.2.1 LQR Control Theory First, the LQR control problem will be consid ered for the regulator problem in which the controller will seek to reject noise and disturbances, and drive all the states of the system to zero. LQR control can also be used to track an input trajectory, and this case will be considered second. The LQR regulator problem, shown in th e block diagram in Figure 6-8, assumes fullstate feedback. Cases without full-state feedback will require the use of an estimator and will be discussed in section 6.2.1. Figure 6-8. General block diagra m for LQR controller problem The plant is modeled as a continuous time, lin ear system described by a set of state-space equations x AxBu y CxDu (6-6)

PAGE 155

155 It is desired to find a controller that minimizes a cost function, J 0min()KJxQxuRudt (6-7) where Q is a matrix that relates to tracking performance and R is a matrix related to control actuation [102]. The values of Q and R are chosen to apply penaltie s to the states and actuator commands. The Q and R matrices are either a positive-definite Hermitian or a real symmetric matrix. A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. The superscript next to a variable denotes it s complex conjugate. The optimal controller, K, that minimizes this cost function is ()uKxt (6-8) Substituting Equation 6-8 into Equation 6-6 gives () x AxBKxABKx (6-9) Thus, Equation 6-7 becomes 00()() JxQxxKRKxdtxQKRKxdt (6-10) The following relationship sets a condition that restricts K to be finite. ()()d x QKRKxxPx dt (6-11) where P is a positive-definite Hermitian or real symm etric matrix. Evaluating the right hand side of Equation 6-11 and substituting in Equation 6-9 yields x PxxPxxABKPPABKx (6-12) Equation 6-12 must hold true for any x, therefore ()()()ABKPPABKQKRK (6-13) If A-BK is a stable matrix, there exists a positive-definite matrix P that satisfies Equation 6-13. In order to determine this matrix P, evaluate the cost function J.

PAGE 156

156 0 0() JxQKRKxdtxPx (6-14) Since A-BK is stable, all of the eigenvalues are assu med to have negative real components and 0 x Equation 6-14 becomes (0)(0)JxPx (6-15) Because R is defined as a positive-definite Hermitian matrix, it can be written in terms of a nonsingular matrix, T. R TT (6-16) Substitute Equation 6-16 into 6-13 to get 111()() [()][()]0 AKBPPABKQKTTK APPATKTBPTKTBPPBRBPQ (6-17) The minimization of the cost function J with respect to K requires the minimization of 11[()][()] x TKTBPTKTBPx (6-18) with respect to K. This expression is nonnegative and the minimum occurs when it is zero, or when 1()TKTBP (6-19) Hence, the optimal matrix K is found by 111()KTTBPRBP (6-20) The matrix P in Equation 6-20 must satisfy the reduced-matrix Riccati equation 10APPAPBRBPQ (6-21) Equation 6-21 must be solved for the matrix P, whose existence guarantees that the system is stable. Once P is found, it is substituted back into E quation 6-20 to find the optimal gain matrix K that is used in the control law of Equation 6-8.

PAGE 157

157 The above development for the LQR controller considered the development of an optimal controller for the case of driving all of the stat es in the system to zero. LQR can also be designed for tracking a de sired input trajectory, r. Consider the trajectory, described by rFz zHx (6-22) for some observable matrices F, and H. In this case, z represents the actual trajectory of the system as a function of the states, and this can include any noise in the sensor as well. An error signal, e, is defined as the difference betw een the reference (desired) input, r, and the actual trajectory. erz (6-23) For this problem, the cost function J can be defined in term s of the error signal. 0()eeJeQeuRudt (6-24) Equation 6-24 can be rewritten as follows 0(2) JxQxuRuxNudt (6-25) with ()()eQHAFHQHAFH (6-26) ee R BHQHBR (6-27) ()eNHAFHQHB (6-28) ,eeQIRI (6-29) where is a constant value describing the weighting function on the control effort.

PAGE 158

158 The goal is to find the optimal controller that minimizes the cost function of Equation 625, and this is determined from solving th e following algebraic Riccati equation with an additional term describing the error signal. 1()()0APPAPBNRBPNQ (6-30) The solution of Equation 630 results in the matrix P such that the controller is described as 11[][] f bffKRBPRNKK (6-31) where Kfb and Kff are the feedback and feedfoward controll er gains, respectively. The control law is thus written as ()fbffuKxKe (6-32) The use of this LQR control law for tracking a reference command with zero steady-state error requires that the system include an internal mode l of the reference command. In the case of a step command, the system must include an integr ator and be what is ca lled a type-one system [108]. If the system model does no t already include an internal m odel, then it must be included in the controller. LQR optimal control for a trac king control of a step input for a plant that does not include an integrator has th e block diagram shown in Figure 69. There is a feed-forward gain, Kff, and a feed-back gain, Kfb, as in Equation 6-32. In this case, the error signal is the Figure 6-9. Block diagram of LQ R control with an internal model for tracking a step command. Internal model for a step command

PAGE 159

159 difference between the desired reference command, r, and the position state, x1. It is assumed that full-state feedback is available for this sy stem for the feedback loop. In cases where fullstate feedback is not ava ilable, state-estimation is required. Th is situation is discussed in Section 6.2.2. 6.2.2 State Estimation The derivation of the control law for LQR c ontrol assumes that full-state feedback is available for the controller. In many cases, full -state information is not available, and stateestimation must be used. A block diagram repr esentation for the control system using a state estimator is given in Figure 6-10. L is the estimator gain matrix. State estimation estimates the state variables of the system based on the measurem ents of the output and the control variables. Figure 6-10. Block diagram of LQ R controller using a state-estim ator for a plant without an integrator. In the case shown in the block diagram, assume that there are two states, x1 and x2, but only x1 is available, hence C = [1 0], and the output y is 1 210 x yCx x (6-33) Let x represent the vector of the estimated stat es. The control law of Equation 6-32 becomes ()fbffuKxKe (6-34)

PAGE 160

160 The mathematical model for the estimator is similar to the plant model of Equation 6-6 with additional terms included to estimate the error to compensate for inaccuracies in matrices A and B. The estimation error is the difference between the measured output and the estimated outputs. The mathematical model for the estimator is () x AxBuLyy (6-35) where the A, B, C, D are the matrices of the plant model from Equation 6-6, and y Cx One method to design the estimator matrix gain, L, is to use Ackermanns formula for pole placement for single -input systems. In this method, the gain L is calculated such that the state feedback signal places th e closed-loop poles of the estimat or at desired closed-loop pole locations, Ackermanns formula is 1 10001()nLBABABA (6-36) for an arbitrary integer n. The term A is the characteristic polynomial of matrix A. 12 121()nnn nnAAAAAI (6-37) The coefficients have a relationship with th e roots of the polynomial, which are also the closed-loop pole locations. 12 12121()()()nnn nnnsssssss (6-38) This approach of designing the state estim ator depends on the proper placement of the desired pole locations. The most frequently used approach is to choose pole locations from the root locus such that they are far to the le ft of the dominant poles of the plant. 6.2.3 LQR Results An LQR controller is designe d along with a state-estimator for the micromirror system using the linear plant model. The control design is done in Matlab using the lqr command, and

PAGE 161

161 the estimator is designed using the acker comm and. The results are simulated on the linear plant model and the nonlinear model. The weights Q and R are chosen to be 000 000 00100000 0.0001 Q R (6-39) The gain matrix from the LQR design produces a feedforward gain, Kff = [-100000.00], and a feedback gain vector, Kfb = [0.0585388701 0.0000000952]. The open-loop poles of the linear plant model of the micromirrors are p = -307489.41 977754.00 i Therefore, the desired closedloop pole locations for the state-estimator are chosen to be = [-3 x 108, -3 x 108]. These closedloop poles for the estimator are chosen as they lie far to the left of the open-loop poles on the real axis of the S-plane. Here, they are chosen to be repeated poles because the real part of the openloop poles are repeated, but it is not required that th ey be the same value. The response of the linear plant model and th e nonlinear plant are for step inputs of different magnitudes are seen in Figure 6-11. The closed-loop system response for the LQR A B Figure 6-11. Step responses for LQR controller. A) Linear plan t model. B) Nonlinear plant model.

PAGE 162

162 Figure 6-12. Closed-loop LQR response to a step input of 7 degrees for 50 random parameter variations. 10% Uncertainty 30% Uncertainty 20% Uncertainty 40% Uncertainty 50% Uncertainty 90% Uncertainty A B D C E F

PAGE 163

163 controller is shown to be simila r to that from the PID. The speed of the system response is dictated by the choices of Q and R in the control design. The effects of parametric model uncertainty are examined by testing the controller for the 50 plant models with variations from % on the model parameters up to % and the response to a step input is shown in Figure 612. The results for the LQR appear to be consistent with the results for the PID controller. 6.2.4 LQR Controller Response to Hysteresis For the PID controller, discussi on is presented in Section 6.1.3 concerning the response of the closed-loop system when the micromirror is commanded to an unsta ble position, and thus experiences electrostatic pull-in and hysteresis. In that dem onstration, the behavior of the controller was found to result in an undesirable switching behavior that nevertheless did improve the hysteretic response. For the same conditions operating with an LQR controller using stateestimation, the controller would not be able to f unction in this unstable range of motion. Recall from LQR control theory presented in Secti on 6.2.1 and 6.2.2 that the st ate-estimation requires full controllability of the system, and this is not the case in the unstable range of motion. As a result, for implementing an LQR controller on this sy stem it is particularly beneficial to avoid the electrostatic instability through the use of a progressive-linkage. Based on the similarity between the responses from the PID and LQR cont rollers, it safe to assume that the closed-loop LQR performance for the system with a progressive-linkage will be very similar to that of Figure 6-7. Figure 6-13 shows the closed-loop LQR step responses for the micromirrors with the progressive linkage, and they are able to achieve st able rotation above the pull-in limit of 16.5 degrees. Also shown are the LQR step responses for the 50 random plant variations with % variation of model parameters.

PAGE 164

164 A B Figure 6-13. Closed-loop step responses for LQR controller for a system using a progressivelinkage. A) Step responses of different magnitudes. B) Step responses of 50 plants with model uncertainties. 6.3 Modeling the Micromirror Array The work thus far has focused on modeling and control of just a single micromirror from an array of mirrors, assuming a single-input/singl e-output (SISO) system. In reality, these micromirrors are part of an array that is single -input/multiple-output (SIMO) since there is only one actuation voltage app lied, but each individual mirror is ca pable of having a unique response. This section will demonstrate a model of multiple mirrors as a SIMO system and will discuss the feedback signals available by considering two different kinds of op tical sensor: position detecting sensors (PSD) and ch arge-coupled devices (CCD). PSDs measure the locations and intensity of the incident light a nd output the position of the center of gravit y (CG) of the total light distribution. These devices are inexpensive and easy to use; however the positions of the individual micromirrors are obscure d. By contrast, a CCD sensor is able to output the locations of the individual light sources; however they are much more expensive devices and require considerably more computation and processing methods to utili ze the sensor information. The controllers developed in Section 6.1 and 6.2 are implemented on the array model to determine their effectiveness at reducing the stea dy-state error of the system as a whole when

PAGE 165

165 model uncertainties are present. Considering the sy stem of micromirrors as they function in an array is a critical step in expanding the applica tion of feedback control from just one device, to being able to control very large arrays that are required for many adaptive optics applications. 6.3.1 Modeling the Array of Mirrors The preceding chapters have developed anal ytical models for individual micromirror components that have a SISO structure. Extendi ng this to a SIMO model that includes multiple micromirror arrays is accomplished by simply adding multiple mirror models in parallel as the plant of the system. Figure 6-14 shows this syst em architecture in which a single input is given to the array of mirrors, and multiple outputs from that system are produced. These outputs are the position states for each indi vidual micromirror. Figure 6-15 shows schematically what this system architecture looks like for a system that a ssumes 5 micromirrors in the array. This image is not drawn to scale so that the individual mirrors and rays of light can be seen. While in reality the array is much larger, using only 5 mirrors allows for a more tractable demonstration of array performance in the simulation environment. It can be difficult to compare the results for a larger number of mirrors. Just as it was shown for the open-loop dynamic model in Chapter 5, if the model parameters vary, the response of eac h mirror will vary for a given i nput signal. If all the mirrors Figure 6-14. Schematic of modeling an ar ray of mirrors as a SIMO system.

PAGE 166

166 Figure 6-15. Schematic drawi ng of an array of 5 mirrors. in the array have the same plant model, then they will all have the same response. However, if the model parameters of each mirror in the array are allowed to independently take on values subject to uncertainty in mass, stiffness, dampi ng ratio, and capacitance, then the results are not so well behaved. The challenge comes from dete rmining one overall error metric that can be used for the feedback controller such that the er rors in the system can be decreased. Thus, the goal becomes trying to decrease the total amount of error in the system, which means it is possible for the individual errors in the mirror re sponses to still exist. While model uncertainty can be controlled effectively for one mirror at a time, trying to implement control for this SIMO array system is a more difficult problem. One problem with controlling this array sy stem comes from choosing the appropriate measurement to use as a feedback signal. In the case that each mirror could be controlled independently, then one approach is to treat it as multiple SISO systems in parallel and provide one control signal for each micromirror and measure its individual performance. In that case, the problem quickly becomes one of scale for determin ing the best way to accomplish this for a very

PAGE 167

167 large array. The case for SIMO system does not ha ve to deal with the issue of scaling multiple control algorithms, but rather how to apply a sing le controller to a group of mirrors. While each mirror can behave independently, there is still only one available control input to the system. The type of sensor chosen to provide the meas urement is critical in determining the overall performance metric for the system, and the type of error signal used fo r the feedback control system. To better understand this, several available sensor types are considered for determining the impact each would have on detecting and in terpreting the system performance. The sensors considered here are position sensing detector s (PSD), and charge-c oupled devices (CCD). 6.3.2 Sensor Types When a light source is incident on the surface of a PSD, the sensor will output a current or voltage signal that corresponds to the location of the cen ter of the total distribution of the light intensity on the sensor surface. This location can be considered as the cent er of gravity (CG) of the total light on the sensor surface. PSDs can be one dimensional, which means that they are able to detect the CG of the light in only one direction, or two dimensiona l, detecting the CG of the light in two directions. Consider in Figur e 6-16, the case of light fr om one array of mirrors reflecting onto a 1-D PSD in which there are erro rs in the actual positions compared to the desired positions. Errors in spac ing between the spots of light can result in CG measurement that is different from that desired. The control system seeing this error will tr y to correct such that the CG error goes to zero, when in fact this can cause the actual deflectio ns of the micromirrors in the array to be different values from what is desired. Figure 6-16 also shows the 1-D PSD with errors in linearity that could be caused by off axis rotations of the mirrors. For small rotations, the same problem of e rror in the CG occurs. Since the 1-D array is only able to measure the CG in terms of one direction (y-a xis shown), the off-axis deflection cannot be

PAGE 168

168 directly measured. There is also the case that fo r a very large error in spacing or linearity, some of the light could be deflected off of the 1-D PSD entirely, also affecting the location of the CG. For a single micromirror array being meas ured by a 2-D PSD, similar problems in calculating the CG occur, except in this case it is possible to locate the CG in both the x and yaxes. As seen in Figure 6-17, e rrors in spacing can shift the CG in the y-direc tion, but linearity Figure 6-16. Illustration of the measured center of gravity (CG) on a 1-D PSD when there are errors in the spacing and lineari ty of the micromirrors. Figure 6-17. Illustration of the measured center of gravity (CG) on a 2-D PSD when there are errors in the spacing and lineari ty of the micromirrors.

PAGE 169

169 errors can also shift the CG in the x-direction. Despite the lim itations of PSDs, they offer the advantages of giving an analog signal with a very fast respons e. In addition, PSDs are typically more affordable than CCDs. To avoid the problems such as those descri bed for using a PSD and to allow for the simultaneous measurement of light from multiple micromirrors in an array, one may use a CCD. The CCDs are an array of metal-oxide-semiconduc tor (MOS) diodes that are able to provide digital information of the light intensity of each pixel in the CCD array. This information can be interpreted using an image processing algorithm to determine the location of each separate spot of light from the micromirror arrays. Then it is possible to obtain x and y-direction displacement measurements for each spot and compare that with the desired positions. This is illustrated in Figure 6-18 for light from two arrays in which the dark spot indicates the actual position of the Figure 6-18. Illustration of the measured errors of the reflected light from two micromirror arrays onto a CCD. reflected light, and the white spot s indicate the desired po sitions. The error is drawn as a vector from the actual to the desired pos itions. If each row of mirrors were given a separate actuation voltage signal, it would be possible to control the position of each spot to reduce the individual error signals. Because the mirror arrays discusse d here have only one actuation signal available

PAGE 170

170 for the entire array, it will only be possible to reduce the overall error si gnal by perhaps using a sum of the squares of the displ acement error vectors. An additional error metric could be to consider statistical yields, su ch as trying to achieve desired performance goals for a certain percentage of mirrors. The ability to reduce the error is further limited in that the mirrors have only one axis of rotation; thus the x and y errors ar e not independent. 6.4 Modeling the Sensor Response Including the sensor model in to the simulation of the micromirror arrays involves taking the geometry of the problem into account. Assumi ng that the locations of the light source, the micromirrors, and the sensor are known, this be comes a calculation of the system geometry to determine the location of the reflected light. Figure 6-19 shows a schematic of beam steering with only one micromirror. The light source, each micromirror, and the sensor, are given a coordinate frame such that they can be locate d and oriented in space with respect to a global reference frame, E. Light that travels to the micromirror is defined by the vector 0 0B Ar. Following the laws of reflection, light reflecting off of a flat mirror will have an angle of reflection that is equal to the angle of incidence, such as th at shown in Figure 6-20. In general, the angle, between two vectors, a and bas shown in Figure 6-20, can be calculated using the dot product relationship. cosabab (6-40) In this case, the two vectors are the vector 0 0B Arand the unit normal vector of the mirror surface, kb Therefore, the angle of incidence, is given as, 1 00 00 cos ABk ABkrb rb (6-41)

PAGE 171

171 Knowing the angle of incidence, now allows for the vector of the reflected light to be calculated. It is possible to determine the distan ce between the unit normal vector of the mirror and the light source as shown in Fi gure 6-21 by calculating two vectors 1 and 2 that are perpendicular to each other at point N, and form a righ t triangle with the vector 0 0B Ar as the hypotenuse. The vector 2 is along the kb unit vector. The magnitude s of these two vectors are sin0 0 1 B Ar (6-42) cos0 0 2 B Ar (6-43) A location for the reflected light, point R, can be found by reflecting vector 1 about the unit normal vector kb at point N, resulting in a new vector, 1r, that reveals a lo cation through which the reflected ray passes. Now there is a know n relationship for the reflected ray of light, represented by the vector, R Br 0. Figure 6-19. Schematic of the beam steeri ng experiment with only one micromirror.

PAGE 172

172 A B Figure 6-20. Geometry used to determine the a ngle of incidence and reflection. A) The angle of incidence is equal to the angle of reflection. B) For two vectors a and b, the angle between them, can be determined from the dot product. Next, the intersection of this ra y of light with the plane of the sensor can be calculated. Referring to Figure 6-22, let the sensor plan e in Frame C be defined by three points, C1, C2, and C3 which have global coordinates (Cix, Ciy, Ciz), where the subscript i may equal 1, 2, or 3. The vector of the reflected ray of light, R Br 0, is given by two points, the origin of the B frame, B0, and the point R, which are known to have global coordinates (B0x, B0y, B0z) and (Rx, Ry, Rz), respectively. The orientation of the B frame will represent the angle of rotation of the micromirrors as they are actuated, and will rotate about the ib unit vector. The intersection of the vector with the plane, at point P, is found by simultaneously solving the following four equations for the variables x, y, z, and t. 0 1 1 1 13 3 3 2 2 2 1 1 1z y x z y x z y xC C C C C C C C C z y x (6-44) 00xxx B RBtx (6-45) 0-0yyy B RBty (6-46) 00zzz B RBtz (6-47)

PAGE 173

173 Solving for t yields, t B C C C B C C C B C C C B C C C B C C C B C C Cz z z z y y y y x x x x z z z z y y y y x x x x 0 0 0 0 1 1 1 0 0 0 1 1 1 13 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 (6-48) This value for t calculated from Equation 6-36 may be s ubstituted back into Equations 6-45, 6-46 and 6-47 to solve for the (x, y, z) global coordinates of the inters ection point P. This process can be repeated for multiple mirrors to determine th e coordinates of their reflected light on the sensor. Now that the reflected light can be located, it is pos sible to calculate the sensor measurement for the system dependi ng upon the type of sensor used. Figure 6-21. Calculating the reflected ray of light.

PAGE 174

174 Figure 6-22. The intersection of the line from B0 to R and the plane C occurs at point P. 6.4.1 PSD Response A 1-D PSD is incorporated into the model for the system of arrays by calculating the locations of the reflected light and determining th e center of gravity of the light. Laser light is known to have a Gaussian distribution of light inte nsity, with the light bei ng more intense in the center of the beam, and reducing toward the outside of the beam [105]. Th erefore, for the model of sensor performance, the light from the mirrors is weighted accordingly such that the light in the center has a higher intensity, following a Gaussian distribution. Center of gravity may be calculated as n i i ir w W CG11 (6-49) where W represents the total weighting of the light intensity, wi represents the weight of the light intensity for one ray of light, and ri represents the position of the ray of light on the sensor for n total rays of light. Figure 6-23 shows a representation of a system of 5 micromirrors in which the light is reflected onto a sensor. This figure is not drawn to scale so that the i ndividual light rays are

PAGE 175

175 more easily seen. The CG of the light is calculat ed for this ideal case as equal to the position of the center mirror. A 2-D represen tation of the reflection on the sens or is also shown. The CG is output as a voltage between V, where a value of 0 V indicates the CG is at the center, +10 V indicates the CG is at the top, and V indicates that the CG is at the bottom of the sensor array. In order to relate this sensor value of the CG back to a meaningful measurement in terms of the angle of rotation of the mirrors, the sensor output is determin ed first for an ideal set of mirrors. This calibration then allows the sensed CG to be converted to an angle corresponding to the angle of the micromirrors, The calibration result for the case of 5 micromirrors is given in A B Figure 6-23. Schematic of 5 micr omirrors in an array reflecting light onto a PSD sensor. A) The CG of the measurement is calculated in th e sensor plane. B) A 2-D view of the reflected light on the sensor plane wi th the CG marked in the red star. Figure 6-24. If any of the reflec ted light is directed off of the sensor, then this light is not recorded and its contribution is neglected in cal culating the CG. This can cause a shift in the measured value of the CG. In order to evaluate the effec tiveness of this sensor at prov iding feedback signals to the system, the sensor response is determined for the case of 5 micromirrors with randomly varied CG

PAGE 176

176 models. Figure 6-25 shows the resulting open-lo op step response for each mirror and the overall CG measurement. Also shown are the position errors for each mirror and the overall error. The sensor outputs the locatio n of the CG of all 5 mirror respons es showing that on average, the 5 mirrors of the array have a steadystate error. In the case of a br oken device or a mirror in the Figure 6-24. Calibration of the PSD fo r ideal case of five micromirrors. A B Figure 6-25. Open-loop results to a step response for an array of 5 micr omirrors with model uncertainty. A) Step response. B) Position error. array with very deviant behavior the CG calculation can be greatly affected. If one mirror is broken and remains stationary, the CG calculation fo r the overall array will be affected. This is shown in Figure 6-26 in which one mirror is brok en and does not actuate while the other mirrors

PAGE 177

177 are considered to have the ideal model with no uncertainty. It is clear that the inclusion of model uncertainty in the other mirrors would only add to the calculated error of the CG. This could also be the case that occurs when some mirrors in the array experience pull-in at different times. This illustrates a limitation of using a PSD for the sensing mechanism. The model of the array of mirrors and the sensor can be incorporated into a control system like that in Figure 6-27. To illustrate the eff ects of using a PSD as the sensor for this A B Figure 6-26. Open-loop response for system w ith one broken mirror and 4 ideal mirrors, measured by a PSD. A) Step response. B) Position error. Figure 6-27. Incorporating feedb ack control into array model. micromirror array system, the closed loop resp onse is determined. Figure 6-28 shows the response for a PID controller, and likewise, Figure 6-29 shows the re sponse using the LQR controller. In both cases, the CG measurement is used as the fee dback signal, and the controllers

PAGE 178

178 thereby only see this average error measurement. The controllers are both able to reduce the average error of the system, but this is really only accomplished by shifting the responses of the 5 mirrors. In this case, the two mirrors, 3 and 4, that had the least amount of error in the openA B Figure 6-28. Controlled PID step response usi ng PSD sensor. A) Step response. B) Position error. A B Figure 6-29. Controlled LQR step response usin g PSD sensor. A) Step response. B) Position error. loop response are actually shifted so that in the c ontrolled system, they have more error. Another weakness is illustrated in closed-loop control in the case of a device in which one mirror in the array is not functioning. Shown in Figure 6-30 is the closed-loop PID response of the system

PAGE 179

179 with one mirror broken, and it is cl ear that in order to compensate for the malfunctioning mirror, the system instead drives the other four mirrors to an incorrect position. A B Figure 6-30. Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a PSD sensor. A) Step response. B) Position error. 6.4.2 CCD Response CCD sensors have an advantage over the PSDs in that they can measure and interpret the response of each mirror in the arra y separately, such as in the Fi gure 6-31. The limitation of the SIMO system still imposes a requirement that these sensor measurements be compiled into only one metric. Using a CCD for this SIMO system has many of the same limitations of the PSD. The error metric used to compare the actual micromirror position to the desired micromirror position may be limited to represent so me average of the errors of all of the mirrors. In this case, the results for using a CCD are not an improveme nt over using a PSD. Ho wever, the ability of the CCD to identify individual positions of the micromirrors does allow for some advantages. For instance, in the case of a damaged or broke n micromirror, the actuation for that one mirror may remain at zero, or have a drastically devian t behavior compared to a mirror that is working properly. Using a PSD sensor, the measurement from the damaged mirror will remain part of the CG calculation, thus skewing the ov erall results. With the appropriate processing algorithm, the

PAGE 180

180 data from the CCD can be used to identify a ny mirrors that are broke n or have highly unusual behavior and eliminate those mirrors from the control consideration. This is demonstrated simplistically in Figure 6-32 in which the PID cl osed-loop system is able to identify the broken micromirror that remains at zero degrees actua tion, and thus eliminates that measurement from the error metric. Additional error metrics may also be defined, such as identifying a yield for the Figure 6-31. Schematic of 5 microm irrors in an array reflecting light onto a CCD sensor where each separate location of the light can be measured. A B Figure 6-32. Closed-loop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a CCD sensor. A) Step response. B) Position error.

PAGE 181

181 array such that a certain percentage of the mirrors are guaranteed to have minimal error, even if it means that other mirrors will have larger errors Weights can be assigned to the measurements to determine those mirrors that have a hi gher priority in the error measurement. 6.4.3 Summary of Sensor Analysis It is clear that the limitations in the controll ability of the individual micromirrors inhibit the ability of the controller to affect only so me aggregate response for the system. The PSD sensor, while fast, inexpensive, and easy to use, is not able to differentia te the responses of the single mirrors, and is therefore most affected by de viations in the single mirrors responses. The CCD sensor is a more expensive option, both in purchase cost and computational efficiency, however it does allow for more flexible parsing of the error signal that can be used to concentrate the control effort on a subset of selected micromi rror responses. In the ca se of trying to control more than one array of mirrors, or for a system of mirrors with SISO co ntrollability, then the CCD array would be the obvious choice of sensor because it can detect multiple locations of light. The issue of sensor noise was not taken in to account in this study, but this too will affect the outcome of the control system. The level of noise will vary depending on the sensor chosen, as well any noise from the environment such as vibrations. While noise levels for a given PSD or CCD product vary by the make and model of the sensor, CCDs typically have lower noise. 6.5 Chapter Summary The control algorithms explored in this chapter, includi ng PID and LQR, are designed based on the 1DOF model developed in previ ous chapters, and the closed-loop system is analyzed in simulation to explore the effectiven ess of these control schemes and examine unique issues that may be encountered, such as th e electrostatic instability phenomena. Other implementation issues are addressed, including choosing the appropriate sensing elements with which to detect the micromirror position for feedback. The differe nt sensor types discussed are

PAGE 182

182 all optical, that is they can measure the positi on of light reflected from the micromirrors, and depending on the type of sensor chosen, they can operate in one or two degrees of freedom. PSDs are only able to report the aggregated results for all light incident on the sensor surface, while CCDs are able to report indi vidual signals from different s ources. Due to the actuation limitations for the micromirror arrays in this stud y, it is concluded that a PSD sensor is adequate for the system, but there are stil l advantages that can be obtained from the use of a CCD. The next step in this work, which is included in th e list of future work, is to develop an optical testbed to implement the control algorithms presented here, and to determine their ability to influence the precision and accuracy of the micromirror arrays. The optical testbed must also consider the implementation issues of noise in the feedback look from the sensor and from the environment. Addition studies concerning control design incl ude examining the PID and LQR controllers for response at highe r frequencies and exploring fu rther nonlinear dynamic behaviors that result from the electrostatic instability.

PAGE 183

183 CHAPTER 7 CONCLUSIONS AND FUTURE WORK The work presented here is an effort to model and analyze the behavior of MEMS micromirror arrays that have inconsistent behaviors caused by parametric uncertainties and nonlinear effects from electrostatic actuation. The micromirror arrays are evaluated first by extensive analytical modeling a nd experimental validation to de termine their performance and understand the effects of fabricati on variations. Using tolerance in formation from the fabrication process, it was shown that it is possible to model the effects of fabrication variations on the performance of the mirrors and to determine the se nsitivity of that perfor mance with respect to a particular parameter. These modeled results are compared to open-l oop characterization data obtained using an optical profiler. It is appare nt that there exists va rying behaviors for the mirrors of the arrays in terms of the stat ic voltage-displacement relationships and the electrostatic pull-in and hyster esis that can affect the dynamic system response as well. Electrostatic instability is addr essed here through th e introduction of a prog ressive-linkage that provides a continuous, nonlinear re storing force to the device that allows it to theoretically achieve stable actuation over the entire range of motion of the mi cromirror. Bifurcation theory was used to further characterize the electros tatic behaviors and the effectiveness of the progressive linkage to mitigate these behaviors. To validate the dynamic modeling, modal analysis was performed using FEA on the structure and validated experimentally using measurements obtained using a Laser Doppler Vibrometer. An examination of the participati on factors for the response of the first mode in each direction reveals that the primary direction of the response is in the rotational X-direction (ROTX), which corresponds to the one-degree-of -freedom model assumption. However, it is clear that motion in other direct ions, namely the Z-direction (ver tical) affects the compliance of

PAGE 184

184 the system and the response, resulting in a lower than predicted first natural frequency. Because the electrostatic force that is applied to the micromirror is al ways an attractive force, drawing the moving electrode down toward the fixed substrate and if resonance is av oided, smooth rotational motion in one degree-of-freedom is still accompli shed. The presence of extra degrees of freedom does, however, show the limitations of the 1DOF model assumption, which limits the analysis to only low frequency responses where resonant behavior may be avoided. To further evaluate the effects of uncertain system behavior, simple feedback controllers are developed using a linear syst em model and then applied to th e nonlinear model. This work demonstrates the use of PID and LQR control, and tests these contro llers on nonlinear plant models with varying parameters. The results fr om both controller designs show that they are able to provide stable actuations with no oversho ot for a range of plant models. The cost of applying these control methods comes in terms of the speed of the response. The open-loop dynamics, while exhibiting some overshoot behavior in the transient respons e, operates on a very fast time-scale, on the order of s. Closing the loop on the sy stem slows the response time by several orders of magnitude to ms; however, this is still a suffi ciently fast response time for many applications, and the added benefits of the controllers at el iminating overshoot and correcting system response in the presence of model uncertain ty are clearly worthwhile. After modeling and developing co ntrollers considering only one micromirror at a time, the system is evaluated as an entire array of de vices. The SIMO structure of the system puts limitations on the ability to control each micromi rror individually, and it is important to consider the type of feedback information available and how it is utilized. Both PSD and CCD optical sensors are considered and it is found that with both sensors, it is possible to correct for the average errors of the system, while not guarantee ing that each micromirror in the array will in

PAGE 185

185 itself attain perfect position track ing. Use of a CCD sensor does have advantages however that can allow for more advances sensor processing a llowing for selective control of the sensor data, such as identifying outliers and ensuring their measurements are not retained in the feedback signal. An optical testbed is developed in order to study the eff ectiveness of control implementation on the actual micromirror arrays Laser beam steering and a PSD sensor are used for position feedback, and preliminary results illustrate the ability to implement feedback control of these systems. This research presented in this dissertation pr ovides a validated theoretical model basis that allows for the development of micromirrors for adaptive optics applicatio ns that are robust to parametric uncertainties that commonly arise through microfabrication processes as well as to disturbance rejection and plant nonlinearities. Future work includes exploration of dynamic response of the system at higher frequencies, and development of optimally designed devices that are less sensitive to the eff ects of variations in the fabrica tion process. In addition, the passive (progressive linkage) a nd active controller development presented in this dissertation, additional work is needed to be expanded to refine the designs with inclusion of design optimization and expansion of the modeling tech niques used. While many researchers develop models of the system performance, very few us e these analytical techniques to optimize the device performance. The appl ication of optimal design me thods and closed-loop control techniques will enable both cost reduction as the devices will no longer require extensive calibration for open-loop performance, as well as improved performance and reliability. The impact of this work is not limited to the appli cation of micromirror or micro-optics design. The design and optimization methods used in the creati on of these new actuator designs will create a general design framework that can be used in the development of many new MEMS devices.

PAGE 186

186 This will aid researchers in all future design efforts and improve the design and development process. The PID and LQR controllers presented in Chap ter 6 can be adapted and refined to meet specific performance metrics defined by the appli cation requirements. The gains proposed for the controllers are quite high, and limitations in hardware capabiliti es may require these gains to be lowered, and the stability of the system must always be maintained. Additional study is required to determine the effects of noise and di sturbances on the feedback loop, as well as how this affects the stability of the system. The results of the modal analysis in Chapter 4 show that the one-degree-of-freedom motion of the system is not valid duri ng resonant behavior, therefore it is recommended to avoid driving the system to resonance. However, it would be very interesting to study the nonlinear dynamics of the system at higher frequencies to identify the effects relating to resonance and to electrostatic pull-in. In order to design a robust microsystem th at can be deployed in a wide variety of scenarios, the device should have on-chip sens ing capabilities built in so that the actuation, sensing, and control can be packaged into a comp lete system. The development of such sensing and control strategies will contribute to the ad vancement of precision optic al applications. The incorporation of on-chip sensing mechanisms in to the device will allow for compact realization of complete microsystems. The method proposed in [91] for using piezoresistive methods within SUMMiT V fabrication is novel and its success w ill open up new areas of device applications. Several feedback mechanisms should be investig ated, including piezoresis tive, capacitive, and optical sensing methods. There is also a need to integrate sensing mechanisms at the device level to allow for the realization of complete, compact microsystems. Piezoresistive and

PAGE 187

187 capacitive methods seem very promising in this area, however noise in the sensor output will need to be carefully examined and minimized. The development of an experimental test bed wa s also initiated at the University of Florida as part of the research where further development is still n eeded before implementation and validation of the presented closed-loop controllers can be realized. In doing so, this work will provide a greater impact on the development of micromirrors for adaptive optics applications that are robust to parametric uncertaintie s that commonly arise through microfabrication processes as well as to disturbanc e rejection and pl ant nonlinearities.

PAGE 188

188 APPENDIX A MODEL GEOMETRY The dimensions used for creating the electr ostatic model for one unit cell of the device geometry are shown in Figure A-1 by layer. All dimensions in m are shown for layer MMPoly 0, and the subsequent layer dimensions are show n in relation to the MMPoly 0 ground plane. The model is created by drawing these areas in th e X-Y plane, and extruding the thickness in the positive Z-axis. Figure A-1. Geometry dimensions in m for creating electrostatic model. (Ground Plane) (Fixed Finger) (Fixed Finger) (Moving Finger) (Mirror Surface)

PAGE 189

189 APPENDIX B MONTE CARLO SIMULATION INPUTS This appendix shows the values used to pe rform the Monte Carlo simulations in Chapter 4. The values were determined from a random number generator in order to have a normal distribution about a mean and standard deviati on. Five-hundred sets of random values were generated, and are shown as histograms here. Also shown is the histogram of the calculated mass values. 0.284 0.286 0.288 0.29 0.292 0.294 0.296 0.298 0 2 4 6 8 10 12 14 16 Thickness of MMPoly0 [m]Frequency of Occurance Figure B-1. Histogram of values fo r the thickness of layer MMPoly0.

PAGE 190

190 1.012 1.014 1.016 1.018 1.02 1.022 1.024 1.026 1.028 0 2 4 6 8 10 12 14 Thickness of MMPoly1 [m]Frequency of Occurance Figure B-2. Histogram of values fo r the thickness of layer MMPoly1. 1.515 1.52 1.525 1.53 1.535 1.54 0 2 4 6 8 10 12 14 Thickness of MMPoly2 [m]Frequency of Occurance Figure B-3. Histogram of values fo r the thickness of layer MMPoly2.

PAGE 191

191 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 0 2 4 6 8 10 12 14 Thickness of MMPoly3 [m]Frequency of Occurance Figure B-4. Histogram of values fo r the thickness of layer MMPoly3. 2.27 2.275 2.28 2.285 2.29 2.295 2.3 2.305 2.31 2.315 0 2 4 6 8 10 12 14 Thickness of MMPoly4 [m]Frequency of Occurance Figure B-5. Histogram of values fo r the thickness of layer MMPoly4.

PAGE 192

192 0.385 0.39 0.395 0.4 0.405 0.41 0.415 0.42 0 2 4 6 8 10 12 14 Thickness of Dimple3 backfill [m]Frequency of Occurance Figure B-6. Histogram of values for the thickness of Dimple3 backfill. 1.98 2 2.02 2.04 2.06 2.08 2.1 2.12 0 2 4 6 8 10 12 14 Thickness of Sacox1 [m]Frequency of Occurance Figure B-7. Histogram of values for the thickness of layer Sacox1.

PAGE 193

193 0.29 0.295 0.3 0.305 0.31 0.315 0.32 0 2 4 6 8 10 12 14 Thickness of Sacox2 [m]Frequency of Occurance Figure B-8. Histogram of values for the thickness of layer Sacox2. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 12 14 16 18 20 Thickness of Sacox3 [m]Frequency of Occurance Figure B-9. Histogram of values for the thickness of layer Sacox3.

PAGE 194

194 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 Thickness of Sacox4 [m]Frequency of Occurance Figure B-10. Histogram of values for the thickness of layer Sacox4. -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0 2 4 6 8 10 12 Linewidth variation for MMPoly2 [m]Frequency of Occurance Figure B-11. Histogram of values for th e linewidth variation of layer MMPoly2.

PAGE 195

195 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0 5 10 15 Linewidth variation for MMPoly3 [m]Frequency of Occurance Figure B-12. Histogram of values for th e linewidth variation of layer MMPoly3. 150 155 160 165 170 175 0 5 10 15 Youngs Modulus E [GPa]Frequency of Occurance Figure B-13. Histogram of values for th e linewidth variation of layer MMPoly4.

PAGE 196

196 2.28 2.3 2.32 2.34 2.36 2.38 2.4 2.42 x 10-11 0 2 4 6 8 10 12 14 Mass [kg]Frequency of Occurance Figure B-14. Histogram of the mass values calc ulated from the parametric variation data. Figure B-15. Histogram of values from the Mont e Carlo simulations for the linewidth error of MMPoly2. Values in blue lie within the 95% confidence interval, and values in red lie without.

PAGE 197

197 Figure B-16. Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the linewidth erro r of MMPoly2. Figure B-17. Histogram of values from the Mont e Carlo simulations for the layer thickness of MMPoly1.

PAGE 198

198 Figure B-18. Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of MMPoly1. Figure B-19. Histogram of values from the Mont e Carlo simulations for the layer thickness of Sacox4.

PAGE 199

199 Figure B-20. Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of Sacox4.

PAGE 200

200 APPENDIX C LASER DOPPLER VIBROMETER RESULTS 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]44.06 kHz 81.41 kHz 186.88 kHz Figure C-1. Magnitude of FFT results for device 1, trial 1. 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04 4.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]47.03 kHz 81.41 kHz 187.81 kHz Figure C-2. Magnitude of FFT results for device 1, trial 2.

PAGE 201

201 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04 4.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]42.03 kHz 81.71 kHz 187.03 kHz Figure C-3. Magnitude of FFT results for device 1, trial 3. 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-04 7.00E-04 8.00E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]186.88 kHz 82.66 kHz 35.78 kHz Figure C-4. Magnitude of FFT results for device 1, trial 4.

PAGE 202

202 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]187.19 kHz 43.75 kHz 51.25 kHz 81.56 kHz 123.75 kHz Figure C-5. Magnitude of FFT results for device 1, trial 5. 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-04 7.00E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]43.44 kHz 82.19 kHz 140.78 kHz Figure C-6. Magnitude of FFT results for device 2, trial 1.

PAGE 203

203 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04 4.50E-04 5.00E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]43.13 kHz 139.53 kHz 85.63 kHz Figure C-7. Magnitude of FFT results for device 2, trial 2. 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 050100150200250300350400450500Frequency [kHz]Velocity [m/s]43.44 kHz 85.31 kHz 118.59 kHz 25.63 kHz 136.56 kHz Figure C-8. Magnitude of FFT results for device 2, trial 3.

PAGE 204

204 0.00E+00 2.00E-05 4.00E-05 6.00E-05 8.00E-05 1.00E-04 1.20E-04 1.40E-04 1.60E-04 1.80E-04 2.00E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]27.34 kHz 43.59 kHz 92.03 kHz 137.03 kHz Figure C-9. Magnitude of FFT results for device 2, trial 4. 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04 4.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]25.63 kHz 41.88 kHz 136.10 kHz 85.31 kHz Figure C-10. Magnitude of FFT results for device 2, trial 5.

PAGE 205

205 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]42.03 kHz 105.31 kHz 80.94 kHz 50.16 kHz 182.34 kHz Figure C-11. Magnitude of FFT results for device 3, trial 1. 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]31.25 kHz 116.56 kHz 46.88 kHz 90.31 kHz 183.91 kHz Figure C-12. Magnitude of FFT results for device 3, trial 2.

PAGE 206

206 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]112.97 kHz 81.41 kHz 42.34 kHz 132.66 kHz 180.63 kHz Figure C-13. Magnitude of FFT results for device 3, trial 3. 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]48.13 kHz 183.28 kHz Figure C-14. Magnitude of FFT results for device 3, trial 4.

PAGE 207

207 0.00E+00 2.00E-05 4.00E-05 6.00E-05 8.00E-05 1.00E-04 1.20E-04 1.40E-04 1.60E-04 1.80E-04 2.00E-04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]83.28 kHz 57.34 kHz 43.44 kHz 106.56 kHz 183.28 kHz Figure C-15. Magnitude of FFT results for device 3, trial 5.

PAGE 208

208 LIST OF REFERENCES [1] O. Solgaard, F.S.A. Sandejas, and D.M. Bloom, "Deformable grating optical modulator," Optics Letters, vol. 17, no. 9, pp. 688-690, 1992. [2] D. Dudley, W. Duncan, and J. Slaughter, "Emerging digital micromirror device (DMD) applications," Proc. SPIE, vol. 4985, 2003. [3] S.D. Senturia, Microsystem Design, Boston, MA: Kluwer Academic Publishers, 2001. [4] G.T. Kovacs, N.I. Maluf, and K.E. Petersen, Bulk micromachining of silicon, Proc. IEEE, vol. 86, no. 8, pp. 1536-1551, 1998. [5] J.M. Bustillo, R.T. Howe, and R.S. Muller, Surface micromachining for microelectromechanical systems, Proc. IEEE, vol. 86, no. 8, pp. 1552-1574, 1998. [6] Sandia National Laboratories. (2005, Sept. 9). SUMMiT V Design Manual. Last Access: Sept. 9, 2007. Available e-mail: memsinfo @sandia.gov Message: SUMMiT Design Manual [7] J.W. Wittwer, T. Gomm, and L.L. Ho well, Surface micromachined force gauges: uncertainty and reliability, J. Micromech. Microeng., vol. 12, pp.13-20, 2002. [8] J. Wittwer, Predicting the effects of dime nsional and material property variations in micro compliant mechanisms, M.S. Thesis, Dept. Mech. Eng., Brigham Young Univ., 2001. [9] J.W. Wittwer, M.S. Baker, and L.L. Howell, Robust design and model validation of nonlinear compliant micromechanisms, J. Microelectromech. Syst., vol. 15, no. 1, 2006. [10] S. Park, and R. Horowitz, Adaptiv e control for z-axis MEMS gyroscopes, in Proc. Amer. Controls Conf., Arlington, VA, 2001, pp. 1233-1228. [11] S. Park, and R. Horowitz, Adaptive co ntrol for the conventiona l mode of operation of MEMS gyroscopes, J. Microelectromech. Syst., vol. 12, no. 1, pp. 101-108, 2003. [12] W. N. Sharpe, Jr., B. Yuan, and R. Vaidyanathan, Measurements of Youngs modulus, Poissons ratio, and tensile st rength of polysilicon, in Proc. IEEE 10th Ann. Int. Workshop Microelectromech. Syst., Nagoya, Japan, 1997, pp. 424-429. [13] A. Tauntranont, V.M. Bright, J. Zha ng, W. Zhang, J. Neff, and Y.C. Lee, "MEMScontrollable microlens array for beam st eering and precision alignment in optical interconnect systems," in Proc. Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, 2000, pp. 101-104. [14] D. Hah, S.T. Huang, J.C. Tsai, H. Tosh iyoshi, and M.C. Wu, "L ow-voltage, large-scan angle MEMS analog micromirror arrays with hidden vertical comb-drive actuators2," J. Microelectromech. Syst., vol. 13, no. 2, pp. 279-289, 2004.

PAGE 209

209 [15] P.B. Chu, I. Brener, C. Pu, S.S. Lee, J. I. Dadap, S. Park, K. Bergman, N.H. Bonadeo, T. Chau, M. Chou, R.A. Doran, R. Gibson, R. Harel, J.J. Johnson, C.D. Lee, D.R. Peale, B. Tang, D.T.K. Tong, M.J. Tsai, Q. Wu, W. Zhong E.L. Goldstein, L.Y. Lin, J.A. Walker, "Design and nonlinear servo cont rol of MEMS mirrors and th eir performance in a large port-count optical switch," J. Microelectromech. Syst., vol 14, no. 2, pp. 261-273, 2005. [16] S. Huang, J-C. Tsai, D. Hah, H. Toshiosh i, and M.C. Wu, "Open-loop operation of MEMS WDM routers with analog micromirror array," in Proc. IEEE/LEOS Int. Conf. on Optical MEMS, Lugano, Switzerland, 2002, pp. 179-180. [17] L. Zhou, M. Last, V. Milanovic, J.M. Ka hn, and K.S.J. Pister, "Two-axis scanning mirror for free-space optical comm unication between UAVs," in IEEE/LEOS Optical MEMS 2003, Hawaii, USA, 2003. [18] Lucent Technologies. (L ast Access: 2007, June 18), Lam bdaRouter Mirror. [Online]. Available: http://www.alcatel-lucent.com/wps /PA_1_A_1OB/images/Photos/Other_ Photos/Lambda_mirror_LR.jpg [19] B. Mi, D.A. Smith, H. Kahn, F.L. Merat, A.H. Heuer, and S.M. Phillips, Static and Electrically Actuated Shaped MEMS Mirrors, J. Microelectromech. Syst., vol. 14, no. 1, pp. 29-36, 2005. [20] L. J. Hornbeck. (2006, June 13). Digital Light Processing: A new MEMS-based display technology. Texas Instruments. [Online]. Last Access: Sept. 11, 2007. Available: http://www.dlp.com/downloads/default.aspx?& ref=/downloads/white_papers/117_Digital_ Light_Processing_MEMS _display_technology.pdf [21] M.A. Butler, E.R. Deutsch, S.D. Senturia, M.B. Sinclair, W.C. Sweatt, D.W. Younger, and G.B. Hocker, "A MEMS-based programmable di ffraction grating for optical holography in the spectral domain," in IEDM Tech. Digest Int. Electron Devices Meeting, Washington, D.C., 2-5, 2001, pp. 41.1.1-41.1.4. [22] E.R. Deutsch, J.P. Bardhan, S.D. Sentur ia, G.B. Hocker, D.W. Youngner, M.B. Sinclair, and M.A. Butler, "A large-travel vertical planar actuator with im proved stability," in 12th Int. Conf. on Solid State Sens ors, Actuators and Microsystems, Boston, MA, 2003, pp. 352355. [23] G.B. Hocker, D. Younger, E. Deutch, A. Volip icelli, S. Senturia, M. Butler, M. Sinclair, T. Plowman, and A.J. Ricco, "The polychrometer: a MEMS diffraction grating for synthetic spectra," in Proc. Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, 2000, pp. 89-91. [24] E.S. Hung, and S.D. Senturia, "Extending the travel range of an alog-tuned electrostatic actuators," Journal Microelectro mechanical Systems, vol. 8, no. 4, pp. 497-505, 1999.

PAGE 210

210 [25] S.D. Senturia, "Diffractive MEMS: th e polychrometer and related devices," in IEEE Conf. Optical MEMS, Lugano, Switzerland, 2002. [26] N.O.P. Arancibia, S. Gibson, and T.-C. Tsao, "Adaptive control of MEMS mirrors for beam steering," in Proc. ASME Int. Mechanical Eng. Congress and Expo., Anaheim, CA, 2004. [27] B.-S. Kim, S. Gibson, and T-C. Tsao, "A daptive control of a til t-mirror for laser beam steering," Proc. American Control Conference, Boston, MA, 2004, pp. 3417-3421. [28] N. G. Dagalakis, T. LeBrun, and J. Lippia tt, "Micro-mirror array c ontrol of optical tweezer trapping beams," in Proc. 2nd IEEE Conf. Nanotechnology, Washington D.C., 2002. pp. 177-180. [29] T.G. Bifano, J. Perreault, R.K. Mali, and M.N. Horenstein, Micromechanical deformable mirrors, IEEE J. of Sel. Topics Quantum Electron., vol. 5, no. 1, pp.83-89, 1999. [30] M.N. Horenstein, S. Pappas, A. Fishov, an d T.G. Bifano, Electrostatic micromirrors for subaperturing in an ad aptive optics system, J. Electrostatics, vol.54, pp. 321-332, 2002. [31] T. Weyrauch, M.A. Vorontsov, T.G. Bifano, J.A. Hammer, M. Cohen, and G. Cauwenberghs, Microscale adaptive op tics: wave-front control with a -mirror array and a VLSI stochastic gradient descent controller, Applied Optics, vol. 40, no. 24, pp. 42434253, 2001. [32] J.B. Stewart, T.G. Bifano, P. Bierden, S. Cornelissen, T. Cook, and B.M. Levine, "Design and development of a 329-segment tip-tilt piston mirrorarray for space-based adaptive optics," in Proc. SPIE, vol. 6113, 2006, pp. 181-189. [33] J.J. Gorman, and N.G. Dagalakis, Mode ling and disturbance rejection control of a nanopositioner with application of beam steering, in Proc. ASME Int. Mech. Eng. Cong., Washington, D.C., 2003. [34] J.J. Gorman, N.G. Daga lakis, and B.G. Boone, Mul ti-loop control of a nanopositioning mechanism for ultra-precis ion beam steering, in Proc. SPIE, vol. 5160, 2005, pp. 170-181. [35] Sandia National Laboratorie s. (1999, Nov. 11). Sandia micromi rrors may be part of next generation space telescope. [Online]. Last Access: Sept. 11, 2007. Available: http://www.sandia.gov/media/NewsRel/NR1999/space.htm [36] J.R.P. Angel, Ground-based imaging of extrasolar planets using adaptive optics, Nature, vol. 368, pp. 203-207, 1994. [37] A. Roorda, and D.R. Williams, The arrang ement of the three cone classes in the living human eye, Nature, vol. 397, pp. 520-522,1999.

PAGE 211

211 [38] R.D. Ferguson, D.X. Hammer, C.E. Bigelow, N.V. Iftimia, T.E. Ustun, S.A. Burns, A.E. Elsner, and D.R. Williams, "Tracking adaptive optics scanning laser ophthalmoscope," in Proc. SPIE, vol. 6138, 2006, pp. 232-240. [39] R.K. Gupta, and S.D. Senturia, "Pull-in ti me dynamics as a measure of absolute pressure," in Proc. IEEE 10th Annual Int. Workshop on MEMS, 1997, pp. 290-294. [40] J.I. Seeger, and B.E. Bose r, "Parallel-plate driven oscillat ions and resonant pull-in," in Tech. Dig. Solid-State Sensor, Actuator, and Microsyst. Workshop, 2002, pp. 313-316. [41] D. Bernstein, R. Guidotti, and J.A. Pelesko, "Analytical and numerical analysis of electrostatically actuat ed MEMS devices," in Proc. Modeling Simulation Microsyst., 2000, pp. 489-492. [42] O. Degani, E. Socher, A. Lipson, T. Leitn er, D.J. Setter, S. Kaldor, and Y. Nemirovsky, "Pull-in study of an electrost atic torsion microactuator," J. Microelectromech. Syst., vol. 7, no. 4, pp. 373-379, 1998. [43] G. Flores, G.A. Mercado, and J.A. Pe lesko, "Dynamics and touchdown in electrostatic MEMS," in Proc. of IDETC/CIE 19th ASME Bienni al Conf. Mech. Vibration and Noise, Chicago, IL, 2003. [44] O. Francais, and I. Dufour, "Dynamic simula tion of an electrostatic micropump with pull-in hysteresis phenomena," Sensors and Actuators A, vol. 70, pp. 56-60, 1998. [45] M.-A. Gretillat, Y.-J. Yang, E.S. Hung, V. Rabinovich, G.K. Ananthasuresh, N.F. de Rooij, and S.D. Senturia, "Nonlinear electromechanical behavior of an electr ostatic microrelay," in Transducers '97 IEEE Int. Conf. Solid-State Sensors and Actuators, Chicago, IL, 1997, pp. 1141-1144. [46] D. Hah, H. Toshiyoshi, and M.C. Wu, "Design of elect rostatic actuators for MOEMS applications," in Design, Test, Integration, and Packaging of MEMS/MOEMS 2002 Proc. SPIE, vol. 4755. [47] D. Hah, P.R. Patterson, H.D. Nguyen, H. Toshiyoshi, and M.C. Wu, "Theory and experiments of angular vertical comb-dri ve actuators for scanning micromirrors," IEEE J. Sel. Topics Quantum Electron., vol. 10, no. 3, pp. 505-513, 2004. [48] R. Legtenberg, J. Gilbert, S.D. Sent uria, and M. Elwenspoek, "Electrostatic curved electrode actuators," J. Microelectromech. Syst., vol. 6, no. 3, pp. 257-265, 1997. [49] Y. Nemirovsky, and O. Bochobza-Dega ni, "A methodology and model for the pull-in parameters of electro static actuators," J. Microelectromech. Syst., vol. 10, no. 4, pp. 601615, 2001.

PAGE 212

212 [50] Z. Xiao, XT Wu, W. Pe ng, and K.R. Farmer, "An angle-ba sed approach for rectangular electrostatic torsion actuators," J. Microelectromech. Syst., vol. 10, no. 4, pp. 561-568, 2001. [51] Z. Xiao, W. Peng, and K.R. Farmer, "Ana lytical behavior of r ectangular electrostatic torsion actuators with nonlinear spring bending," J. Microelectromech. Syst., vol. 12, no. 6, pp. 929-936, 2003. [52] X.M. Zhang, F.S. Chau, C. Quan, Y.L La m, and A.Q. Liua, "A study of the static characteristics of a torsional micromirror," Sensors and Actuators A, vol. 90, pp. 73-81, 2001. [53] J. Zhang, Y.C. Lee, V.M. Bright, and J. Neff, "Digitally positioned micromirror for openloop controlled applications," in Proc. IEEE MEMS, 2002, pp. 536-539. [54] Z. Xiao, and K.R. Farmer, "Instability in micromachined electrostatic torsion actuators with full travel range," in Transducers 2003 12th Int. Conf. So lid State Sensors, Actuators and Microsyst., Boston, MA, 2003, pp. 1431-1434. [55] X. Wu, R.A. Brown, S. Mathews, and K.R. Farmer, "Extending the travel range of electrostatic micro-mirrors using insulator coated electrodes," in 2000 IEEE/LEOS Int. Conf. Optical MEMS, 2000, pp. 151-152. [56] D.M. Burns, and V.M. Bright, "Nonlin ear flexures for stable deflection of an electrostatically actuated micromirror," in Proc. SPIE, vol. 3226, 1997, pp. 125-136. [57] J.R. Bronson, G.J. Wiens, and J.J. Allen, Modeling and alleviating instability in a MEMS vertical comb drive using a progressive linkage, in Proc. IDETC/CIE 2005 ASME Int. Design Eng. Tech. Conf., Long Beach, CA, 2005. [58] J.J. Allen, J.R. Bronson, and G.J. Wiens, Extended-range tiltable micromirror, US Patent SD7962S106240 (pending). Technical Advance f iled March 1, 2005. Application filed July 28, 2006. [59] J.I. Seeger, and S.B. Crary, "Stabilization of electrostatically actua ted mechanical devices, in Transducers '97 IEEE Int. Conf. on Solid-State Sensors and Actuators, Chicago, IL, 1997, pp. 1133-1136. [60] J.I. Seeger, and B.E. Boser, "Dynamics and control of parallel plate actuators beyond the electrostatic instability," in Transducers '99 10th Int. Conf on Solid-State Sensors and Actuators, Sendai, Japan, 1999, pp. 474-477. [61] E.K. Chan, and R. W. Dutton, "Electrosta tic micromechanical actuator with extended range of travel," J. Microelectromechan. Syst., vol. 9, no. 3, pp. 321-328, 2000.

PAGE 213

213 [62] J.A. Pelesko, and A.A. Triolo, "N onlocal problems in MEMS device control," J. Eng. Math., vol. 41: pp. 345-366, 2001. [63] E.K. Chan, and R. W. Du tton, "Effects of capacitors, resist ors and residual charge on the static and dynamic performance of el ectrostatically actuated devices," in Proc. SPIE, vol. 3680, 1999, pp. 120-30. [64] J.I. Seeger, and B.E. Boser, "Charge control of pa rallel-plate, electrostatic actuators and the tip-in instability," J. Microelectromech. Syst., vol. 12, no. 5, pp. 656-671, 2003. [65] R.N. Guardia, R. Aigner, W. Nessler M. Handtmann, L.M Castaner, "Control positioning of torsional electros tatic actuators by current driving," in Third Int. Euro. Conf. Advanced Semiconductor Devices and Microsyst., Sinolenice Castle, Slovakia, 2000, pp. 91-94. [66] R. Nadal-Guardia, A. Dehe, R. Aigner, and L.M. Castaner, "Current drive methods to extend the range of travel of electrostatic microactuators be yond the voltage pull-in point," J. Microelectromech. Syst., vol. 11, pp. 255-263, 2002. [67] J.M. Kyynaraninen, A.S. Oja, and H. Seppa, "Increasing th e dynamic range of a micromechanical moving plate capacitor," Anal. Intergr. Circuits Signal Processing, vol. 29, no. 1-2, pp. 61-70, 2001. [68] P.B. Chu, and K.S.J. Pist er, "Analysis of closed-loop contro l of parallel-plat e electrostatic microgrippers," in Proc. Int. Conf. Robotics and Automation, San Diego, CA, 1994, pp. 820-825. [69] J. Chen, W. Weingartne r, A. Azarov, and R.C. Giles, "Tilt-angle stabilization of electrostatically actuated micromechan ical mirrors beyond the pull-in point," J. Microelectromech. Syst., vol. 13, no. 6, pp. 988-997, 2004. [70] M.S. Lu, and G. Fedder, "Closed-loop c ontrol of a paralle l-plate microactuator beyond the pull-in limit," in Solid-State Sensor Actuator and Microsyst. Workshop, Hilton Head, SC, 2002, pp. 255-258. [71] M.S.-C. Lu, and G.K. Fedder, "Position c ontrol of parallel-plate mi croactuators for probebased data storage," J. Microelectromech. Syst., vol. 13, no. 5, pp. 759-769, 2004. [72] D. Piyabongkarn, Y.Sun, R. Rajamani, A. Sezen, and B.J. Nelson, "Travel range extension of a MEMS electrostatic microactuator," IEEE Trans. Control Syst. Tech., vol. 13, no. 1, pp. 138-145, 2005. [73] H.S. Sane, Energy-based control for MEMS with one-sided actuation, in Proc. Amer. Controls Conf., Minneapolis, MN, 2006, pp. 1209-1214.

PAGE 214

214 [74] D.H.S. Maithripa la, J.M. Berg, and W.P. Dayawansa, Nonlinear dynamic output feedback stabilization of electrostatically actuated MEMS, in Proc. 42nd IEEE Conf. Decision and Control, Maui, HI, 2003, pp. 61-66. [75] P. Cheung, R. Horowitz, ad R. Howe, "Des ign, fabrication, position sensing and control of an electrostatically-driven polysilicon microactuator," IEEE Trans. Magnetics, vol. 32, pp. 122-128, 1996. [76] K.M. Liao, Y.C. Wang, C.H. Yeh, and R. Chen, "Closed-loop adaptive control for electrostatically driven torsional micromirrors," J. Microlith., Microfab., Microsyst., vol. 4, no. 4, 2005. [77] D.A. Horsley, N. Wongkomet, N. Horo witz, and A.P. Pisano, "Precision positioning using a microfabricated electrostatic actuator," IEEE Trans. Magnetics, vol. 35, no. 2, pp. 993999, 1999. [78] S. Pannu, C. Chang, R.S. Muller, and A. P. Pisano, "Closed-loop f eedback-control system for improved tracking in magnetical ly actuated micromirrors," in IEEE/LEOS Opt. MEMS 2000, pp. 107-108. [79] R.K. Messenger, T.W. McLain, L.L. Howe ll, "Feedback control of a thermomechanical inplane microactuator using piezore sistive displacement sensing," in ASME Int. Mechanical Eng. Congress Expo, Anaheim, CA, 2004. [80] R.K. Messenger, T.W. McLain, L.L. Ho well, "Piezoresistive feedback for improving transient response of MEMS thermal actuators," in Proc. SPIE, vol. 6174, 2006. [81] N. Yazdi, H. Sane, T.D. Kudrle, and C. H. Mastrangelo, "Robust sl iding-mode control of electrostatic torsional micromi rrors beyond the pull-in limit," Transducers 2003 IEEE Int. Conf. Solid-State Sensors and Actuators, 2003, pp. 1450-1453. [82] D. Hernandez, S.-S. Park, R. Horowitz A.K. Packard, "Dual-stage track-following servo design for hard disk drives," in Proc. Amer. Control Conf., 1999. [83] H.C. Liaw, D. Oetomo, B. Shirinzadeh, and G. Alici, Robust mo tion tracking control of piezoelectric actuation systems, in Proc. IEEE Int. Conf. Robotics and Automation, Orlando, FL, 2006, pp. 1414-1419. [84] S.E. Lyshevski, "Nonlin ear microelectromechanical systems (MEMS) analysis and design via the lyapunov stability theory," in Proc. IEEE Conf. Decision and Control, vol. 5, 2001, pp. 4681-4686. [85] G. Zhu, J. Penet, and L. Saydy, Robust co ntrol of an electrostati cally actuated MEMS in the presence of parasitics a nd parametric uncertainties, in Proc. American Controls Conf., Minneapolis, MN, 2006, pp. 1233-1238.

PAGE 215

215 [86] S.H. Lee, S.E. Bak, and Y.H. Kin, "Desi gn of a dual-stage actuator control system with discrete-time sliding mode fo r hard disk drives," in Proc. IEEE Conf. Decision and Control, vol. 4, 2000, pp. 3120-3125. [87] J.C. Chiou, Y.C. Lin, and S.D. Wu, "Close d-loop fuzzy control for torsional micromirror with multiple electrostatic electrodes," in IEEE/LEOS Opt. MEMS, 2002, pp. 85-86. [88] D.V. Dao, T. Toriyama, J. Wells, and S. Sugiyama, Silicon piezoresistive six-degree of freedom force-moment micro sensor, Sensors and Materials, vol. 15, no. 3, pp. 113-135, 2002. [89] D.V. Dao, T. Toriyama, J. Wells, and S. Sugiyama, Six-degree of freedom micro forcemoment sensor for applications in geophysics, in 15th Annual IEEE Conf. on MEMS, Las Vegas, NV, pp. 312-315. [90] Y. Li, M. Papila, T. Nishida, L. Cattafe sta, and M. Sheplak, "Modeling and optimization of a side-implanted piezoresistive shear stress sensor," in Proc. SPIE 13th Annual Int. Symp. Smart Structures and Materials, San Diego, CA, 2006. [91] G. Roman, J. Bronson, G. Wiens, J. Jones, and J. Allen, Design of a piezoresistive surface micromachined three-axis force transducer for microassembly, in Proc. Int. Mech. Eng. Conf., Orlando, FL, 2005. [92] R. Schellin, and G. Hess, "Silicon su bminiature microphone based on piezoresistive polysilicon strain gauges," Sensors and Actuators A, vol. 32, no. 1-3, pp. 555-559,1992. [93] D.P. Arnold, T. Nishida, L.N. Cattafesta, and M. Sheplak, "A di rectional acoustic array using silicon micromachined piezoresistive microphones," J. Acoust. Soc. Amer., vol. 113, no. 1, pp. 289-298, 2003. [94] H.L. Stalford, C. Apblett, S.S. Mani, W.K. Schubert, and M. Jenkins, Sensitivity of piezoresistive readout device for microfabri cated acoustic spectrum analyzer, in Proc. SPIE, vol. 5344, pp.36-43. [95] R. Dieme, G. Bosman, M. Sheplak, and T. Nishida, "Source of excess noise in silicon piezoresistive microphones," J. Acoust. Soc. Amer., vol. 119, pp. 2710-2720, 2006. [96] H. Xie, and G.K. Fedder, "Vertical comb-finger capacitive actuation and sensing for CMOS-MEMS," Sensors & Actuators A, vol. 95, pp. 212-221, 2002. [97] B.D. Jensen, M.P de Boer, N.D. Masters, F. Bitsie, and D.A. La Van, Interferometry of actuated microcantilevers to determine materi al properties and test structure nonidealities in MEMS, J. Microelectromech. Syst., vol. 10, no. 3, 2001.

PAGE 216

216 [98] S. Limary, H. Sterart, L. Irwin, J. McBray er, J. Sniegowski, S. Montague, J. Smith, M. de Boer, and J. Jakubczak, Reproducability data on SUMMiT, in Proc. SPIE, vol. 3874, 1999, pp. 102-112. [99] ANSYS, Inc. (2007, June 19). ANSYS Pro duct Information and Documentation. [Online]. Last Access: Sept. 11, 2007. Available: http://www.ansys.com [100] F. Pan, J. Kubby, E. Peeters, A.T. Tran, and S. Mukherjee, Squeeze-film damping effect on the dynamic response of a MEMS torsion mirror, in Tech. Proc. Int. Conf. Modeling and Simulation of Microsystems, 1998, pp. 474-479. [101] M. Zecchino, and E. Novak. (2001, June 19). MEMS in motion: a new method for dynamic MEMS metrology. Veeco, Inc. website. [Online]. Last Access: Sept. 11, 2007. Available: http://www.veeco.com/ pdfs.php/69 [102] K. Ogata, Modern Control Engineering, Fourth Ed., Upper Saddle River, NJ: Prentice Hall, 2002. [103] K. Zhou, and J.C. Doyle, Essentials of Robust Control, Upper Saddle River, NJ: Prentice Hall, 1998. [104] G.E. Dullerud, and F. Paganini, A Course in Robust Contro l Theory: A Convex Approach, New York, NY: Springer, 1999. [105] A. Ashkin, History of optical trapping and manipulation of small-neutral particle, atoms, and molecules, IEEE J. Sel. Topics Quantum Electronics, vol. 6, no. 6, pp. 841-856, 2000. [106] J.A. Pelesko, and D.H. Bernstein, Modeling MEMS and NEMS, Boca Raton, FL: Chapman and Hall/CRC, 2003. [107] A.H. Nayfeh, and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Ex perimental Methods, New York, NY: John Wiley and Sons, 1995. [108] R.C. Dorf, and R.H. Bishop, Modern Control Systems 9th ed., Upper Saddle River, NJ: Prentice Hall, 2001. [109] J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements 2nd ed., Sausalito, CA: University Science Books, 1982. [110] L. Guilbeau, The history of the solution of the cubic equation, Mathematics News Letters, vol. 5, no. 4, 1930. [111] D. Elata, and H. Bamberger, On the dyna mic pull-in of electrost atic actuators with multiple degrees of freedom and multiple voltage sources, J. Microelectromech. Syst., vol. 15, no. 1, 2006.

PAGE 217

217 [112] L. Mattsson, Experiences and chal lenges in multi material micro metrology, in Proc. 2nd Annual Int. Conf. Micromanufacutring, Clemson, SC, 2007, pp. 318-322. [113] W.A. Moussa, H. Ahmed, W. Badawy, and M. Moussa, Investigating the reliability of the electrostatic comb-drive actuato rs utilized in microfluidic and space systems using finite element analysis, Canadian J. Electrical Computer Syst., vol. 27, no. 4, pp. 195-200, 2002.

PAGE 218

218 BIOGRAPHICAL SKETCH Jessica Bronson graduated with honors from the University of Missouri at Columbia with a B.S. in mechanical engineering in December 2002. Ms. Bronson began her graduate studies in January 2003 under Professor Gloria Wiens in the Space Automation and Manufacturing Mechanisms Laboratory at the University of Flor ida in Gainesville. Shortly after beginning graduate school, Ms. Bronson was awarded an in ternship at Sandia Nati onal Laboratories in Albuquerque, New Mexico as a fellow through the Microsystems, Engineering, and Science Applications (MESA) Institute at Sandia. In 2004, she was granted the Sandia National Laboratories Campus Executive Fellowship that allo wed her to continue to develop her research program at the university, in addi tion to returning to New Mexico for internships at Sandia each summer for the next three years. The focus of her Ph.D. research is to develop and implement closed-loop control systems for Microelectromech anical Systems (MEMS) micromirrors. The impact of this research is that it will increas e accuracy, performance and repeatability leading to advances in imaging and communications tec hnology. Upon completion of her Ph.D., Ms. Bronson hopes to continue her work in MEMS a nd control systems by obtaining a position at a leading research laboratory.


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101211_AAAAAK INGEST_TIME 2010-12-11T09:19:44Z PACKAGE UFE0021233_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 36848 DFID F20101211_AAAIOI ORIGIN DEPOSITOR PATH bronson_j_Page_067.pro GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
cb79b01744f51a7acd1dcac47da193a8
SHA-1
6a1e2d5953f6acc62e6bb2aa9b957adacfc1e752
42266 F20101211_AAAINT bronson_j_Page_046.pro
590fda50ca2db662fd4734b5d8ab7f8a
b93ae46a9846faaecae246281893547611b2acfd
588 F20101211_AAAHLG bronson_j_Page_196.txt
1d52f9e956d4558d570840712269d6d9
a4636d424dd808f1f0ded4136bcff16d2e5c1381
7000 F20101211_AAAHKR bronson_j_Page_050thm.jpg
3e67bef0efd7e624a155604072c40ca9
be6b34fec8984186db3bdea46ef71c75da882e1b
30994 F20101211_AAAIOJ bronson_j_Page_068.pro
f20a6dd0e09d42cda3ad9d962b44786f
ded17aee3d88119055ace525c12ba19956cb506d
35031 F20101211_AAAINU bronson_j_Page_047.pro
1eb30f9f7c44f0136fbbd702f1cb5b26
1940036fe110fa3c55cd34ba5ca6f26e2d01a606
118158 F20101211_AAAHLH bronson_j_Page_044.jp2
608a53e87295c7d050087674ef816c5f
2d7f57d9364e23b6b110a7eda8fd0accb79beeb9
24034 F20101211_AAAHKS bronson_j_Page_143.pro
282c59ddfeabc6e0a2400252d86cb565
3948851228f0eefb4967009193d6f19022255d9f
10270 F20101211_AAAIOK bronson_j_Page_069.pro
1abed365980f6e2d6685e83d95e8a674
e252cd8cb2d6246cd6d1d6d4e7d9c77aa758e6f6
21413 F20101211_AAAINV bronson_j_Page_049.pro
50776b4dd5d9e2eba1f7489f1b660132
3c0fc7c674525cd34d37360bd9dc5810313d43f1
25271604 F20101211_AAAHLI bronson_j_Page_049.tif
fba05b62842d1a941e00b5cf6316ea6c
1075b3213032355a5d3640c0604a1b7331e10ca0
108118 F20101211_AAAHKT bronson_j_Page_019.jp2
68454f1428fb727f4e1017199431f8d6
310f5895c460fe88a7272a64d2dcc6c0440a955b
56475 F20101211_AAAINW bronson_j_Page_050.pro
6e15dc148ca4739ae9f2649f920e3035
2f22bd510e7efadae74d1a5bd70ca7d652060ae4
16905 F20101211_AAAHKU bronson_j_Page_168.QC.jpg
3ab90b089b6f59f106bd59bf5ad5f403
f09b76a587b3ac8b5a80b0f61cc0c1d425ff0d5c
15402 F20101211_AAAIPA bronson_j_Page_089.pro
e03b8fb3ab59e332bcc8f6e66c55c6bc
05b085a4822fa869b70dd332ee385b29973dad6c
34371 F20101211_AAAIOL bronson_j_Page_070.pro
76e7a2e9701dee1feb47b215f83bcd9f
762a8bebddb7c889d1aa4c3638728995acac234d
29796 F20101211_AAAINX bronson_j_Page_051.pro
7cf1ed1c17d3925b57a268a1e51bff85
d2190187a5649f78d7cc383c4b18d3eb35d98031
46320 F20101211_AAAHLJ bronson_j_Page_131.jpg
43ab5ee887e9f55e85a434ac1b4fa8ef
a201adc66beb760168450968b4f906c7a85c7690
1725 F20101211_AAAHKV bronson_j_Page_061.txt
03ece845a4f286ce397a7c277a1d290f
3ff2ac75a9ec9f8eb25eaa132f2fe19d4e39256d
16821 F20101211_AAAIPB bronson_j_Page_090.pro
fafbbcbe8f51b3fc6dc721f3ff164e40
85e2b25c24ef1b2848fbfb868a70e92b4eb2b30f
44822 F20101211_AAAIOM bronson_j_Page_071.pro
301acd6e75b43a40d03c33752d128d5e
f061c7e1722e885234f01ea4426787739fde882d
24382 F20101211_AAAINY bronson_j_Page_052.pro
fc386fc76d764f1f573754ea049aef80
c293fd757a1e7a3363f112e5a9b6cace0956b9e1
77830 F20101211_AAAHLK bronson_j_Page_040.jpg
23fcda1d504c5cfc525ed4ade93dfa8b
05920988bca63df3a323418cb37b5b5288d48357
F20101211_AAAHKW bronson_j_Page_076.tif
c3b12323dd9d6f27b2344a4f6433127f
ff1d0ddfb59eb84784e3d250ae816075b20dfe28
49275 F20101211_AAAIPC bronson_j_Page_091.pro
a3f6fe6a0c6b363b2ea96b46fffd5545
145ae5c995cef98e40c6780ea04aefdb4ce4bf69
18450 F20101211_AAAION bronson_j_Page_074.pro
da16545647780c3e2c32d9259254dea9
73f934e2c53f8ac4338f7ed7e12e9e5ee991fd6b
11613 F20101211_AAAINZ bronson_j_Page_054.pro
f0134163390371958a8d732dece04428
90a83dd0a74ff013d9dd4656230c23392ed64233
19753 F20101211_AAAHMA bronson_j_Page_154.QC.jpg
9fd6e07e104b0bf61c13de93a0072b74
db98efc2bf05896ef8cd87e3fd4530e208413f35
12332 F20101211_AAAHLL bronson_j_Page_189.QC.jpg
eea8d306d23f75c8566069dc0736fe77
a30d9f4fd53159871a8737fe3c26a61aa31b8f94
868937 F20101211_AAAHKX bronson_j_Page_067.jp2
84773957f2fac0c5ddb1099439227ea8
9f48ab7d40521f798f4d66a576b00692275b5bf2
10960 F20101211_AAAIPD bronson_j_Page_092.pro
20f879dba6eed95cbd80d21691e2540c
4031daa8dd851511daeab4ab5b7e2c958484e32b
21078 F20101211_AAAIOO bronson_j_Page_076.pro
e08705270db6d588391562bd479869dc
a248948ca8207f78881464ea773af2bb25b76b70
111891 F20101211_AAAHMB bronson_j_Page_108.jp2
18f02008420b713424958ce0568c3f65
7487561b1832e48025e349230327094a06ba5a77
27059 F20101211_AAAHLM bronson_j_Page_123.QC.jpg
3d46380b96939b6baa1c6fc84dd5d397
c60829b2b78b1b74aad17632bbc8475ab78874ea
17956 F20101211_AAAHKY bronson_j_Page_104.QC.jpg
9292865515e32e6724b868b10a23dcc5
c19bca05040ba3cf09dcbda5b9e7be6865346410
52454 F20101211_AAAIPE bronson_j_Page_093.pro
dfe94131544e290cf91f5aed636a3f5e
c5b271dae31f58408a690a013c951939bed1c464
36639 F20101211_AAAIOP bronson_j_Page_077.pro
080b49845e2c57bb7da6f06ab52db374
989daa0e47105614c67c2e640e983eeffeb49377
20978 F20101211_AAAHMC bronson_j_Page_173.pro
ad067485244237b134e415b0acc8c907
5bdfd77e4a8e51223fab4e19ac434653167c6ee3
57544 F20101211_AAAHLN bronson_j_Page_172.jpg
fe93c3549afea2f10dae526bdfa1b6e7
b8a5f01effce5244dcd1aa41423f9ad2d9833dc0
1053954 F20101211_AAAHKZ bronson_j_Page_020.tif
acbd49f6e229704444f5d72f018ca036
6c2b3292ad3ba4b596c89b6300a5215753af214b
38001 F20101211_AAAIPF bronson_j_Page_094.pro
cc33fc50ee4fab6e427e30b97b0f1f26
a7d3f58c22d764bbbd43ec4aa7e74d26ff5c8bd2
48910 F20101211_AAAIOQ bronson_j_Page_078.pro
99908ba6d3e848f8de2b57c7964388d5
3390c637957ed5fb99207dac1578256618a28bd4
25469 F20101211_AAAHMD bronson_j_Page_164.QC.jpg
665e7535b365135651693f93804d84f8
48cbd2d007c8fd3b48463c08b033657028912059
1375 F20101211_AAAHLO bronson_j_Page_051.txt
5fe683b9f97ebcffa8346f0a8a89deaf
b8066985c46f01fe144dbdd8d93a0f5af278b67a
11653 F20101211_AAAIPG bronson_j_Page_095.pro
3ad5fa7a1102efbbf4268dbc73230a33
5eb4cca7462d1ceb1dd93716687afe463beb31c6
9716 F20101211_AAAIOR bronson_j_Page_079.pro
0e88774c5297a14583e2e8a5217f9d58
46886b2f23b6428628eae39a8d5542d19f878234
34107 F20101211_AAAHME bronson_j_Page_202.jpg
7063117ce0d4313b42d70831143ef6bc
006d75d98a5e3e3fe4c37b95c33965ef59c234f5
70954 F20101211_AAAHLP bronson_j_Page_135.jpg
221137fd1ba21f5b776e8cdb0e23f8c9
02ed2129685999ae5ac0041f2d1b87ab8cadbc97
17100 F20101211_AAAIPH bronson_j_Page_099.pro
294137f576162ddb9de2481c1d592d69
056f629b31c0753aeabfad24f970278f03569634
11730 F20101211_AAAIOS bronson_j_Page_080.pro
8d904feb22d2bf8394913da633744037
9214ccaeb2065513a6cb81ffb2d11cd069da9e92
3474 F20101211_AAAHMF bronson_j_Page_005.txt
b13184335e8e45100ae98af3b68055c8
9b001df615a1ed60bd011354475a841c317741df
21435 F20101211_AAAHLQ bronson_j_Page_170.QC.jpg
4bad2963e50ecd2d4ae0b33a03fc0b86
f6a6efbada820690bdc9d9a105f3a47424700470
53343 F20101211_AAAIPI bronson_j_Page_100.pro
a7789da56c79017b39b726f34eec5f65
784edbf1ee205a4efc3e02b766d4a07315478991
52138 F20101211_AAAIOT bronson_j_Page_081.pro
b0365d988875cab0053facf903c5cfe2
0973ade505f1d5decba919481885ac36dcc9b52a
25871 F20101211_AAAHMG bronson_j_Page_048.QC.jpg
4871d8a54f00fe36d4e9d03cd6d591d4
6d1e1fa5a1599fd57af038da6134c0ddb54fd933
641757 F20101211_AAAHLR bronson_j_Page_088.jp2
2284c0412a713be14e79cdf6374493a3
9abad60e8f1b4a1c9948fccf43dc0cd6b2d4c098
40576 F20101211_AAAIPJ bronson_j_Page_102.pro
ea11170051c27da11e6a0fbd435ab293
dd492d1925a7ba8d45d5b4e9a485e3fed896a986
37981 F20101211_AAAIOU bronson_j_Page_082.pro
bb90c1f03d941477c4fcd62f08e75f28
661211c097e8d31ac1bccc073862c0a31306648e
19084 F20101211_AAAHMH bronson_j_Page_073.QC.jpg
e1915bd6046b4b68bed4c30760a27943
d31d096d55e9f1286de9e1385cdf5b1359bdbf67
69585 F20101211_AAAHLS bronson_j_Page_156.jp2
a8ca89617cf45add397e5be65cc2b571
2f981eb51a46a9c130c9ab023a2e30cbdb11611a
28342 F20101211_AAAIPK bronson_j_Page_103.pro
6aeb8a81ed3faa0cb1d2e4b3f77381a1
6c803f68f192345f362178a079bd6357a5934eac
34399 F20101211_AAAIOV bronson_j_Page_083.pro
c60d4e43d103c4b621c44b4702e5365b
0517ebbc045720c55029f06d04542b63849d9c86
22529 F20101211_AAAHMI bronson_j_Page_127.QC.jpg
e151c97c660a3ba9a226c82450d5d776
749c63ac0c8ec05c1764e7b9c610879a6ef21561
1219 F20101211_AAAHLT bronson_j_Page_106.txt
ae5a62ad0291b1240dcb74a316da1002
f57aca2acbaef996d6a6855b452cbc0babd8427f
54798 F20101211_AAAIPL bronson_j_Page_105.pro
ac46fc5f10e37e1cf1255ce631e1418c
b2c3110b4e259e0980044899a955adddd8d9241e
56133 F20101211_AAAIOW bronson_j_Page_084.pro
9fcb7df01e2326afae9e041ec303b455
217eabccd7055d47e87a5c3d09e8769bb19c519c
3935 F20101211_AAAHMJ bronson_j_Page_193thm.jpg
6418f2e803002e675c94bb4583b0c06b
2c572076e749e9807172fefdec44a38efa2b08ec
7210 F20101211_AAAHLU bronson_j_Page_179thm.jpg
29e3688dd10d8b765e777c8e662a3fd4
41bc3b2d485896b347a1797d98b5c9f115867e4f
44490 F20101211_AAAIQA bronson_j_Page_120.pro
a1df28dc6df7f72645caaa1659d298f9
9486aab836a159f36a190014d64a8f821ee0fe5a
45939 F20101211_AAAIOX bronson_j_Page_085.pro
967b31c3431facf79af00d4b6b6aaa1d
9e4b59e7fbd25688b4298304d0b6e3f48692edcc
26013 F20101211_AAAHLV bronson_j_Page_119.QC.jpg
dceebf568815c29ef6cbc474244b9f2f
c4e36a6c2d954f509b1c071112a9192f1ee73b6f
13229 F20101211_AAAIQB bronson_j_Page_121.pro
84c3247b75cf949b0d9e42fb1f78a88f
db96035876e123366f94e4892826bf9657250227
19678 F20101211_AAAIPM bronson_j_Page_106.pro
5fd3bc5deb113dfa28691f6a77015ac5
11644a29ac0bea763692a074a7075113494f51c3
55426 F20101211_AAAIOY bronson_j_Page_086.pro
9f6a38203f086ed259b83664bcb07760
0e39557f10286c66b5bed655109e2ef2cb04c8e4
76696 F20101211_AAAHMK bronson_j_Page_087.jpg
f61bb789c038f05739c58fd3091c6083
1e21519e25af028f898193ad6643956c8c81400a
2809 F20101211_AAAHLW bronson_j_Page_209.txt
9fbd363a8e64cbdf7461bda6a2bbfabb
dd5f203e04d84ba9ed74b7f2250b9f302dae197c
40298 F20101211_AAAIQC bronson_j_Page_122.pro
98e4f02b176aef32bae86b74f3e86572
2c7f4c7816a66d52a0349e66a7746b60b5fa088f
10699 F20101211_AAAIPN bronson_j_Page_107.pro
334862d4434b4e2d56a344016ae7ebee
d260c107f61d155179ccc582b2f8a6febd10166d
16214 F20101211_AAAIOZ bronson_j_Page_088.pro
15f1b4746477ac2ff84f4bbec7a66ce7
f8ea416a3c763f3e606ebfec2f88150f77e71259
4040 F20101211_AAAHML bronson_j_Page_089thm.jpg
b26df17668a3f2dcd67e07ad20d918ed
8c116957ea8d44a77736822032403283dc870f7a
F20101211_AAAHLX bronson_j_Page_195.tif
ca44e8376c12587cb935b187ddcff8bd
90d036cd74ae7ad22691ab9764a67657352c5208
71674 F20101211_AAAHNA bronson_j_Page_019.jpg
9402314fcd786ee6fd629c34c475206e
e34f82df8e4c465d874ecf04f4524bfb755c3036
58638 F20101211_AAAIQD bronson_j_Page_123.pro
338cee261a3f4a4a107f44ebdcc14c0e
61d2f076be8f04a38af82604717a12086855eea0
52570 F20101211_AAAIPO bronson_j_Page_108.pro
8113e23e1620910b1630bb7dc33bf4ac
82c0dde0b7ab4723429c0dcdf7acf3332fc00bb1
1051986 F20101211_AAAHMM bronson_j_Page_133.jp2
9e2b353bc576871e7c7d65ae97603886
3b2f8a66933f297583415743a4863398a92c04b1
66732 F20101211_AAAHLY bronson_j_Page_150.jpg
3780d96dda4ccf0646ec5d88d1e26654
1b0b2d327ab800f6bb2fa764370ee49ea479ac9b
1051971 F20101211_AAAHNB bronson_j_Page_164.jp2
e32ff56a2574a9a9228075139eb6ae4b
6df159035ca71d1e13c3de8ab8074d109d066e24
56585 F20101211_AAAIQE bronson_j_Page_125.pro
b41f144b289694d050819cdf30da61de
7ee0edbf1957aeb3ead51410eb7c2b7031e1b7ec
14413 F20101211_AAAIPP bronson_j_Page_109.pro
1c91d259c453328ae63805c1b68b68f1
da77aab83d44e0c73c0740b472238e23bec007c2
1043387 F20101211_AAAHMN bronson_j_Page_120.jp2
5ca7c94d2089207f938bced4c9647293
390adcf35fa298bc386535e295e81a87dadf774a
30062 F20101211_AAAHLZ bronson_j_Page_073.pro
6193cebbfb10e1c579b6617b7d250260
1ed4f22556b3bbd4c14d636294b3fe25b3d915f9
85926 F20101211_AAAHNC bronson_j_Page_009.jpg
271cd1c1069c44f72717903e6240c900
1badb304127cccf3640f818dce56c6285c14cb8f
28654 F20101211_AAAIQF bronson_j_Page_126.pro
9908eb7e43e28d7b2c67c7dc88b6cc3f
06027fec72b0b54a9d437ea05078fde1f20e2670
57043 F20101211_AAAIPQ bronson_j_Page_110.pro
c936c063cf90befb5a98398ab2ab4a8c
5841962f27b808caa7d49d7477a08198f0478024
1495 F20101211_AAAHMO bronson_j_Page_143.txt
b0ff7a7950768e516ffb70f00bce67a3
ea9458d456e4fd6270f20ceb857c5243c0fbddcb
119602 F20101211_AAAHND bronson_j_Page_181.jp2
15d02e76ff3e7b61ae0bfbd19e2fa5b1
4bbd16825a2e8dcada21220cda32fb2d862063b1
48599 F20101211_AAAIQG bronson_j_Page_128.pro
206fb46bb25e2bb6b9c2760b4890ea67
fd298dcb89484e2236296942e2c55cfc8e9f4e8a
54351 F20101211_AAAIPR bronson_j_Page_111.pro
ff3b4f0bd1bc6840bf37f3f96d3ca730
64f38ee9170fac2ab8f149f46a442c034449d716
F20101211_AAAHMP bronson_j_Page_165.tif
9ecf309c7c7862831319268272752f7d
dbd1279685e26eb9ce0be24c2eaa1635e59c4617
28865 F20101211_AAAHNE bronson_j_Page_011.QC.jpg
316a27e5586b57f0cd077671d5a626e6
534cc57095ef92ee009b25ec14a428a4e81a5bd8
32061 F20101211_AAAIQH bronson_j_Page_129.pro
3fb57bb8612ce2148620335d633dc0a3
f611a631c8bf54a7f4d399e607cb514b0caa5611
16235 F20101211_AAAIPS bronson_j_Page_112.pro
a8519e403ee367ec9a3148d39883a8f4
61fe564ea202e31d43dbb03482bc463fb77113e2
42606 F20101211_AAAHMQ bronson_j_Page_026.pro
75d08abdd42ac8501c9ccc5a69aaca88
dca7f5fc0ff933f29113338f88478fa2ffc66df7
54201 F20101211_AAAHNF bronson_j_Page_144.pro
7cbcbba4f33aa23ac702174405d086b1
fa34e012881f0cfd4f1b8b67145ba6da21190090
35475 F20101211_AAAIQI bronson_j_Page_130.pro
fbb38f9c8332c1a1707ea50292c4d809
25e38f4b27baad93c504d6c69e52446addbaf025
33244 F20101211_AAAIPT bronson_j_Page_113.pro
6c26e8c55062b251526d9431748d0541
c0ca451ecf9b2516cccd40ed175f1955419fdbac
523148 F20101211_AAAHMR bronson_j_Page_095.jp2
47ebc2da74e1f7326db84ad5452b5267
73f6c905a1002be911b097e69af3e375cfe43cb3
28267 F20101211_AAAHNG bronson_j_Page_063.pro
daf729e000b1a631845c93a67522f62a
308057db3ef77f36c295c8fbb19e3f3d10fe3ee1
26213 F20101211_AAAIQJ bronson_j_Page_131.pro
23b5e5c2e16651bd7261a876c15679a2
ed93172cf3361a15f3b92790f73d050f6c9bde5d
33342 F20101211_AAAIPU bronson_j_Page_114.pro
3490eabc6bb0eedaac208fe216bf0391
91304c6427464948bc06f40459d1a8477b9aa42f
5297 F20101211_AAAHMS bronson_j_Page_151thm.jpg
e88293c79efd8efdfb7e6a1f05c3b7ab
133b109c51b7f2dded416fd3f9f5fc9a9f881d1c
23182 F20101211_AAAHNH bronson_j_Page_065.QC.jpg
550b68e479a0c39647df0a594da9de7a
56fa97cd219ebaf314e9e7e35964338f4cc0c3a4
12916 F20101211_AAAIQK bronson_j_Page_132.pro
464ebb00bd7e239287d5a0b9a709f7db
f956488bfe2e24b9161494df78777ffb552d0251
35372 F20101211_AAAIPV bronson_j_Page_115.pro
c8f245ba3cc2c35bd203dbfbf009621c
6bf82da34b1dc1f2c35539ea261aa2d719676fd2
1707 F20101211_AAAHMT bronson_j_Page_172.txt
45a23bb2f0e29542d02b57001fdcb975
dd3dcf334141fffed88e2b53d2a792ef73d4aa3e
6827 F20101211_AAAHNI bronson_j_Page_117thm.jpg
dd1575c027b360629bda7026e6abdaa3
11f1231dd41c8ef5c360ddaab48eefe68ba1440e
29706 F20101211_AAAIQL bronson_j_Page_133.pro
38be8c6aabc61c314bcc615079c3415a
fbfaa84eeace9f5a4464bfffbf513d2f4b8897d3
43081 F20101211_AAAIPW bronson_j_Page_116.pro
8f4b4e7e31136a8459f4684a6c9386c2
874a5f9c92af616befe680c107b95955f8c13e54
53177 F20101211_AAAHMU bronson_j_Page_115.jpg
39c4a6a52fa700ac381d7176c7fb364f
1ade95dc5b018019f66b3c3cdb1edfa8aef5f6fd
49125 F20101211_AAAHNJ bronson_j_Page_097.pro
0746cd2bdf65dc3a344dec116d3a3dbb
24f4b2cebf470f71a5ab9a7178db519bf3bfded3
37791 F20101211_AAAIRA bronson_j_Page_154.pro
79e327445148c47ee24080b227638f1a
017913c3c92d0913c072e4bd256aa66118cb5c25
20893 F20101211_AAAIQM bronson_j_Page_134.pro
3b4671cb7790530fa926ee64e599a1b8
a42c6ecf031b30604c67a39482660ca2b5617e0e
55301 F20101211_AAAIPX bronson_j_Page_117.pro
b50825a36e191fa60cc63272940f3992
2d11e1b2053381737ae23bb8e5a89d7287cb769b
43705 F20101211_AAAHMV bronson_j_Page_088.jpg
47bdcc0f66987122f27c0a8b6cb1da7d
e096274b21a38511ba63fead6d8d011d412566e5
44291 F20101211_AAAHNK bronson_j_Page_165.pro
2e4c370a833ac53d1e239dc7a3f6c587
2e5fa16e818a2a0d04a23c9729142a577236803a
31259 F20101211_AAAIRB bronson_j_Page_156.pro
38aae593c0fc5b16736922174dd3b919
abf5545eb40f42c7cf036e8b61e1c75b39c3f309
66118 F20101211_AAAIPY bronson_j_Page_118.pro
846c63dda7bbb5861b9f594fa6601f30
4a94732b48aa7437d0d881d62209e6be884752d7
114982 F20101211_AAAHMW bronson_j_Page_045.jp2
158535e3bb8179924845c774617aeed0
e2b32bc4da86bd8610d61221293b8238d89bb489
27123 F20101211_AAAIRC bronson_j_Page_157.pro
31c37649550283fe6bae32340e6fcbd5
5869f078be6846ce42eaf56540905bfb557de39a
48247 F20101211_AAAIQN bronson_j_Page_135.pro
9614f85d35b877e9866145cff486b9af
b5ed15c4169d3d77b156b1c4cf498a74d9cd7bb5
57531 F20101211_AAAIPZ bronson_j_Page_119.pro
b07c840320929f5fc8a6894ddddb0a87
7acd399840dd086bc97b5e9b02adda1ecb90198e
6083 F20101211_AAAHOA bronson_j_Page_083thm.jpg
49d605d1d22579bdc6f745bf81a73e16
76cd14cbbfac8e776cfc7b4e9d9d6df73942180b
26219 F20101211_AAAHNL bronson_j_Page_152.QC.jpg
793cffa933aba7efdfadc67eebda4478
e8e73761eb8e0da513dba0bb5697bc095304ec1b
55479 F20101211_AAAHMX bronson_j_Page_218.jpg
39a7515cdf1468d6fd434573dfd469ed
bddbadfbb6911d8976cb2161e6505c680f1d5498
30973 F20101211_AAAIRD bronson_j_Page_159.pro
f93ee6ea404ad0eec54d1b09cfadf76a
c62bcd3715edad306c32ca37e595e3ba31955db4
11022 F20101211_AAAIQO bronson_j_Page_136.pro
360436388ed685150296a8ecc5e0912c
8863f76e5e6a0335c37ef7f48d84cc2fe924b2a7
116467 F20101211_AAAHOB bronson_j_Page_087.jp2
1881f34bcf63a6953578f080d1948861
6aa2bdf70d64076d6c28d85ca26a536876758f7a
481480 F20101211_AAAHNM bronson_j_Page_196.jp2
01a42c9c170548b5422533371b80fd31
1f56362611e88bdef4fda313b08e1dccf0b33534
34802 F20101211_AAAHMY bronson_j_Page_056.pro
53534afedb8f0972bf9090868681871d
37956da309ddaa8141c7034a0e2faf8e70787933
43242 F20101211_AAAIRE bronson_j_Page_160.pro
ab96a2ca5d5f314450632e316c86608e
cbecab96594d7c9778561ad9a6a9af4dfe12f970
22382 F20101211_AAAIQP bronson_j_Page_137.pro
3af75f6ed3157a1873a04103a327c900
b0f290d5546a862bbc56e3a9c5a11e8e10bd0390
651997 F20101211_AAAHOC bronson_j_Page_089.jp2
2c6c74b285ff152435eac661ea4990d1
a7fc74e9a9806bdb49b638067ab51fe3be9555cc
81149 F20101211_AAAHNN bronson_j_Page_155.jp2
0328458c7bd62ffff5dfd07810e37340
9c424848ba0d2e7849494201fb1fc4cbf7b719d3
19258 F20101211_AAAHMZ bronson_j_Page_122.QC.jpg
2f753bf8bf789425b693b2086b897741
841f1aa520750b61b716324844be38a92d00be17
29078 F20101211_AAAIRF bronson_j_Page_161.pro
79748cd154d6e53ceaa9a9ada3e0e35e
37d58c202c2362122140a706aac8df95e9b3b54e
59080 F20101211_AAAIQQ bronson_j_Page_138.pro
56468defa588466b4b2f604e89aa88d9
08614b73eee128c976aae36e37dedd1f56de8c50
1225 F20101211_AAAHOD bronson_j_Page_076.txt
2ef132c25f99c53ba1f1b0b8011f98f5
4af22d92bb6f43eaa78c1d62ecee4b5b8bafd6f8
9972 F20101211_AAAHNO bronson_j_Page_187.QC.jpg
66b544310db7c421f03a2e05e34fb4f9
7d52db0a42f5562b2cd2b7b4a194e4b3c95bd15f
45057 F20101211_AAAIRG bronson_j_Page_164.pro
3591f7e13a5856fe8e98ecb8039d8525
a09a8c4ad37f592a9326c8dfbb3f9c19dd5d0856
16868 F20101211_AAAIQR bronson_j_Page_140.pro
3273be7c0ca9e46ea021a19f339fbc09
39fe3f9e94ec9405225ef82b572919f565f50bca
40563 F20101211_AAAHOE bronson_j_Page_062.pro
d8e17fa8dc50563c6c42888e9a92ddae
dc4b435a7879460a85a9136eeed13db73f860d3a
3687 F20101211_AAAHNP bronson_j_Page_202thm.jpg
890fe5f4a73bc5652be077ef0710d688
b742ffdc0c78f5879bd5e0c9dc482d880272fd38
36427 F20101211_AAAIRH bronson_j_Page_166.pro
34947c3b657c253b8007d21cd28c3b6a
0eb7e45036af6c42df336f335c73d66995a43048
29274 F20101211_AAAIQS bronson_j_Page_141.pro
71cd4c89d5aa56e235b246f8c1883eac
629db31480c4a3162e94940206a48800daadd187
15212 F20101211_AAAHOF bronson_j_Page_204.pro
647ea534624fb79fc6a7aa35160ab15c
943d47711b1870946dafdc29921d57181c019c60
7028 F20101211_AAAHNQ bronson_j_Page_045thm.jpg
f56fbbe354d0e893a435eb99933eb57c
c28b94da44313ea6cc3dcd765f6159bd323fe48b
56108 F20101211_AAAIRI bronson_j_Page_167.pro
49ea4bae511df093f9f8559a2e9ae123
a6c9e7b939951b79255710b6c2915b089b1ff212
56097 F20101211_AAAIQT bronson_j_Page_142.pro
48316696169373aaef191bced2e5c2bb
66a1db578a36f8947e3599fa7bc3c270b658f003
55870 F20101211_AAAHOG bronson_j_Page_051.jpg
471c524eae29a1e8d8701ea094410aa4
1f10962989cd1d4d6868b014f48bc29366dd56e6
1485 F20101211_AAAHNR bronson_j_Page_096.txt
00ef793fe3a82cf962511d75c6c1f42b
c46f94a727bf659fd762e32c2b640be869b8a593
25674 F20101211_AAAIRJ bronson_j_Page_168.pro
5e876025f0258c5d156eb9d3c9c52003
aa19d4a07dc76e6f9135723f1182109006a96bf0
54733 F20101211_AAAIQU bronson_j_Page_145.pro
629447cf96b28f4b11029819a7a64d6e
aea47eb3209bd6713419823b99b5d0298e0354c1
7272 F20101211_AAAHOH bronson_j_Page_015thm.jpg
ba481917f4259d1ea25e982375ff4f82
f7abc53f711f0f3944dec7ea5d94883c43ea9c16
54356 F20101211_AAAHNS bronson_j_Page_113.jpg
fdc3f1b0ed52b6c5261ea23e56c54428
7eb380753634dc5fbbdcb845aed2a372a1861f64
41581 F20101211_AAAIRK bronson_j_Page_169.pro
ded7c7b1e4749a50927b2ae3ce4fc2c1
a91ba6b80150db71c19fdf74778764c311e06824
31043 F20101211_AAAIQV bronson_j_Page_147.pro
bc11b24a7411f690a913a4c92cad586a
8a81eb68eecdba625cd5d553b040f23a48d8fa84
51352 F20101211_AAAHOI bronson_j_Page_186.pro
b55e991b7244c88497cd6f5aa1d1b0ff
af5ce721d38b1fa878ebc15c56eaf9d1305621c9
F20101211_AAAHNT bronson_j_Page_178.tif
6dda7000f9f1dd13cc58b6f3d5f71bf5
623ec889d2f1c1e9ce28f34f2f31eb818668e798
44054 F20101211_AAAIRL bronson_j_Page_170.pro
5cf5e2fbb94468399d63015ac5fbe97c
c50c0e4f3d106f0c93c1c42acdb14133b8f9299f
39107 F20101211_AAAIQW bronson_j_Page_148.pro
1044083bd62966c451ba8ee9dd6aac59
f5fca44edb38193385ade34f552034fe98374893
35054 F20101211_AAAHOJ bronson_j_Page_072.pro
b11bb5eaea766b49afa55086e038a4c2
7c8615d2e1fbfe3e73324106a1c31c7672db3e29
36842 F20101211_AAAHNU bronson_j_Page_155.pro
78443298f699f664e3e0026d1f514edd
2101d4aca078b5bf858b548cd0cf439c76f95811
15814 F20101211_AAAISA bronson_j_Page_189.pro
8e3e4500e7c7c8fa42d7b5b74fcf8534
ed000bff5184da9cb341b78b7e102c2897814f78
27613 F20101211_AAAIRM bronson_j_Page_171.pro
38ed7231ff253ea5306c567e2ec3bb5e
0579ca01a83559fd2dcab4d34018bf310ea1b3d8
15351 F20101211_AAAIQX bronson_j_Page_150.pro
4a2560645a860e6f7a57172f86730b60
a621830c41015a92c6d231cb28f183cd8afe7ff0
118672 F20101211_AAAHOK bronson_j_Page_185.jp2
7d7be976d316654102929e8ddfa7725d
1a6e7abcd8486fcfd1ee473c4cfb0f6f95d3efeb
57044 F20101211_AAAHNV bronson_j_Page_149.pro
dcfe58a03c0c9db4b49ab1fb22848906
b0e69b59b727bd6c519e149e78750c0d23bc9473
9365 F20101211_AAAISB bronson_j_Page_191.pro
1aab0b8956661f45d0edfde56471c5c4
2a392d37da4a757959aa3eed900c1f80d7ff8e3a
33955 F20101211_AAAIRN bronson_j_Page_172.pro
7129f122ed2a21b1e4daece871391991
6b34c2d8b1bf6ec326acc769d21d697834c60035
9497 F20101211_AAAIQY bronson_j_Page_151.pro
2e6e97bfc32819093043a9ff538cdf09
7d1e5639cc15446939df268fc8427d9a69d34aa9
43047 F20101211_AAAHOL bronson_j_Page_101.pro
b62e18ef00b456bbb4b571b85b481268
1fcd1f9daeb2419a169fec83459bc5ea4e12d3b7
44269 F20101211_AAAHNW bronson_j_Page_127.pro
b8c6594e6edf05a5b70d56978e3af5bd
d64d2a9cce301af5c30b0fd305dc9163c5113dc4
9713 F20101211_AAAISC bronson_j_Page_192.pro
d0a55884ef62bc3d422ee9c2baa25900
6411907744acdc60130cc630fb5daf71d32020a4
57469 F20101211_AAAIQZ bronson_j_Page_152.pro
b5d43d37df22502166c40e313c4127c0
874c01d4eca326773e7cdda52c301045a3e9bed3
579 F20101211_AAAHNX bronson_j_Page_192.txt
b26253ed35991b2320f1263431aecf8a
d8384dc6847caddfe8af983e6ba489697d135b1b
54897 F20101211_AAAHPA bronson_j_Page_139.pro
ab3adcc12a4d961fe924ffde49744077
15c6add9e0c4ecb1d500c9dc6719819dcdcbc8c2
8615 F20101211_AAAISD bronson_j_Page_194.pro
b0f58c74bc1be71c8bf3015ead633c03
ea56e27a439b4022ecdc57f71ba4d6fa79933e98
31699 F20101211_AAAIRO bronson_j_Page_174.pro
1bef8034e514d8300cbd8d917dda759d
a90cb4d8a28359d7d225b00c96c6fd91e56f1429
4639 F20101211_AAAHOM bronson_j_Page_197thm.jpg
613b104818666fa83f273fd6e7067328
aa8354da0748167c9c094efded08cc1d8a043b5e
116986 F20101211_AAAHNY bronson_j_Page_139.jp2
c8c079b8a00af68ed004d396d952e8b5
06c7d9b397d7807702593fec4f7e28b1c223e009
1239 F20101211_AAAHPB bronson_j_Page_182.txt
69a5fe394eb37a45d8c93cde3b41f169
bbaac3698e19d1876cc5e2159a366c6d5f39e316
10983 F20101211_AAAISE bronson_j_Page_195.pro
abeb8c2fc9f6a5f90e4636dbf1c10694
d01bdbaa43ee27670be5f90f4a895e9649d7d3f1
39317 F20101211_AAAIRP bronson_j_Page_175.pro
0d3eb0841feea075de56e2d892ea19b9
b8e0c6f94c4dfbc6e279618a4508647df4bc73fc
25244 F20101211_AAAHON bronson_j_Page_207.jpg
9b26ab4042e0721a234dca086ac3c927
7cbd8d58d1906b54a31973fad96b9fc9ea2fcae6
8266 F20101211_AAAHNZ bronson_j_Page_190.pro
810295ae39bfec114f3527303032f8f6
e482b399ac38b10dd814f03e3f45c1f4cc182c4b
F20101211_AAAHPC bronson_j_Page_005.tif
5b6010d991879aab53f99ae37ff66489
823a3d01f217158c5e9f2ee8d820da20ac833b7d
13587 F20101211_AAAISF bronson_j_Page_196.pro
a9a58e37f172264a8c3b103355f89d55
91cafc0c979a3031130f94799782c1892ecbc8af
34581 F20101211_AAAIRQ bronson_j_Page_176.pro
8f741aa20c331aaca624530bc43e26ff
a3aff2b5cfecd5499291f531a023623be4cb9efc
24998 F20101211_AAAHOO bronson_j_Page_009.QC.jpg
2c4694f44690910e2ce133c0a4d56b02
3730c52de0c089081f4117c61101e3c5557a80dc
4636 F20101211_AAAHPD bronson_j_Page_194thm.jpg
888d21c1f6cdb9148cc70525224ed5a4
611b6d4e7778de5f5cf41fe2d42e097daefb21fa
11016 F20101211_AAAISG bronson_j_Page_197.pro
0bfec6300b41e3fb68c202c413354d72
6d6aa39b59a7994c92fd3f4dc1eb997a819a8ae8
24390 F20101211_AAAIRR bronson_j_Page_178.pro
710c18ef98b881d77bf0aca88bc0a678
008a284407cbd42488fa2254f40d526231b2f54e
44043 F20101211_AAAHOP bronson_j_Page_004.jpg
5645da9028a934f7dad8fa8ed6225bab
26c1cff1c96d9682d1ac8c5c81132cca9f3ece72
87267 F20101211_AAAHPE bronson_j_Page_213.jpg
64218aaa94246854f1c464be6d26ad02
78e17f254485ad8722a8e6b4c6a0ac97970e3fb3
9772 F20101211_AAAISH bronson_j_Page_198.pro
9a8356eb7914bb151e6489c3b91991dd
8a433d8d78e4d206d7ca6ea753eedf5d2475f643
48254 F20101211_AAAIRS bronson_j_Page_179.pro
1ee3292167c702b9090467e76f4a680a
0862f0518ffa734a62e8c386a936a0637927f479
6782 F20101211_AAAHOQ bronson_j_Page_183thm.jpg
eb4906feb06a776097dbef88cb6a3158
36cf3f98a1bb2b202ee5ac7175b64b72b2e1ae62
1051984 F20101211_AAAHPF bronson_j_Page_012.jp2
00900ac1d176d14ce01171e043ef5493
9756698831233d1881b8b89a4808a8c8a74aed54
5221 F20101211_AAAISI bronson_j_Page_199.pro
20d02a7e3fcfd1cf9aec07b30be38dc9
eea8ab2bc6fc54cef00019390e4f1bd726604e55
24138 F20101211_AAAIRT bronson_j_Page_180.pro
a9a1882cb869ea9aa1fafd61c3fc6e9e
f6a587ec9c2f7b0b6e81845ce957d548792c468e
3698 F20101211_AAAHOR bronson_j_Page_204thm.jpg
f7f21c06c2e150a33f31d2d14ddeded9
1058e8203191ada904542d498aa10da403e2c112
32035 F20101211_AAAHPG bronson_j_Page_048.pro
077aaedcbd1c2e17ff7018007df067c7
a06402f5c9d9db652ce677d936c86e8d068364e4
14735 F20101211_AAAISJ bronson_j_Page_200.pro
87be6ebf58c883f75bedd84f46f76e5f
7d72b5848c7a310f962390bb0905c96eb70fd45a
56450 F20101211_AAAIRU bronson_j_Page_181.pro
8711036e7964a44b0e1e246fd3a44bfc
d17839c4fb167b08cdf9c1fa7eaa1ea6e1ee95f9
24205 F20101211_AAAHOS bronson_j_Page_025.QC.jpg
0c0f59416d0d86eb314e81197daf2b6d
9aff9b5670d82f63770e7a3fa5e64f27989646e7
847162 F20101211_AAAHPH bronson_j_Page_047.jp2
2b787e276b8160704a37be0da63f7c23
8a4bb6d0e3de2a87da7575c3865458a6df535d48
12835 F20101211_AAAISK bronson_j_Page_201.pro
c9a83566f6f0a972952225f86e5672a1
78445af3101cc90a0700f9dd19c4d1913a7c0ff4
31299 F20101211_AAAIRV bronson_j_Page_182.pro
fdb9c4bbadfc9123064f2af1e4e50e2f
8617efe9b1d642647bfb493e991956cb8fef378c
104103 F20101211_AAAHOT bronson_j_Page_011.jpg
c4d9748336c3d604f8d3f1a02b603dc8
663ed6250c35dbe663a96d5f281eef166ec1b148
67049 F20101211_AAAHPI bronson_j_Page_097.jpg
c961c5962ef5bf60473d92456101ed10
b0fd866c4159af14642483615c201e3d649c79dd
13354 F20101211_AAAISL bronson_j_Page_202.pro
2691f94aec71b6bd2feb84dedd526b0a
a6c3f765429b4af03fd66b01a81854f4a00b5dab
54267 F20101211_AAAIRW bronson_j_Page_183.pro
82a5b0a1868aa553938d15b69c507984
1db696cae86ee046627bbd29eaa5bc180bd502fb
16688 F20101211_AAAHOU bronson_j_Page_188.pro
2301a1edfba0cb10f1f6ef5c138b90e6
8b5c09443fcde82eaabc288ab97ab7b07231df47
17786 F20101211_AAAHPJ bronson_j_Page_153.QC.jpg
7cef678571d1d6f9dc69f90cde8acdba
8e2cff58b6ce330b579e8ae937e2b99e6f29ad7d
80 F20101211_AAAITA bronson_j_Page_003.txt
dc77786b5c1717680485a503fcd825e0
a72f5bf28ffda001d0ca8e46418fa440af4a57c9
13203 F20101211_AAAISM bronson_j_Page_203.pro
d1db9510030533255873645d006a8249
ed26b13a2fe580c6f1d76e5d1ac7714c22397136
56909 F20101211_AAAIRX bronson_j_Page_184.pro
c143cb6aa6477afaeb87d29959edd763
a33371fb5794f6044cafa93e7d939084998da394
48874 F20101211_AAAHOV bronson_j_Page_017.pro
5ba138826f6f02c212b9bf8822cf0c6e
0590f70fa75a174f533ddbc042e4fadbfdcfae61
F20101211_AAAHPK bronson_j_Page_170.tif
747f06b4a754c0c5532f7fa186915566
bbf30bf76be7bd870e5a1a4308e1b4b236c439f9
1106 F20101211_AAAITB bronson_j_Page_004.txt
fc905c78de434e0c5c2c557d389fdcda
ae92e378a9de474419426586cbe9c55451f39d2a
11771 F20101211_AAAISN bronson_j_Page_205.pro
36b307cb80031aa104b5681411a72faf
45daae5823df54e4e8bfa07a24f2174797b36580
55127 F20101211_AAAIRY bronson_j_Page_185.pro
d4172fad696775404790316dcc862cd3
26c9e7ad63c858c13e4d87a066fb7cab941da348
122477 F20101211_AAAHOW bronson_j_Page_042.jp2
ad7078fc794279bbd9c572b4fddf3f14
2b88817cc1989cc3d00f6bd329bc6de4bd0fa84b
2247 F20101211_AAAHPL bronson_j_Page_149.txt
7477511df35419b737c86a3cf68c6e4f
40ad92f9efb043fac9b8ce6fff8c27c2c30e6f0e
3862 F20101211_AAAITC bronson_j_Page_006.txt
5baad7e88b54252bba1c6036a9b9c793
1a04a81b230dc4cd93a3534bde24c5026269d6c7
13581 F20101211_AAAISO bronson_j_Page_206.pro
39659b7a6ed106c7add8ba5e3b47dccb
8cc90deffc8c801b0a1e927e6923580797d33b3e
17912 F20101211_AAAIRZ bronson_j_Page_187.pro
faf27f3eaaa302976e1cedff89fd7fbf
161cc99a29b5a5a2c1afe965e7c71504444b3da6
F20101211_AAAHOX bronson_j_Page_093.tif
0e29b0fbf9bee664e88fddf34173ad59
4a19f7c403d02a9f3bbad1fc98c4c88c9fb7faaa
4336 F20101211_AAAHQA bronson_j_Page_191thm.jpg
ff5f21988c71fdd3bb953e2197f9ee4a
3607374fd83bd28ea78011db75b0a097994c907e
118507 F20101211_AAAHPM bronson_j_Page_149.jp2
2d90e60c08d24073a18fede01e261b16
55ae2a1acf4a47b2a1cf08923536337815f8e486
569 F20101211_AAAITD bronson_j_Page_007.txt
977b8b26f8c6e063790db578c10bfafc
fe09358aec0da0cb7e868d5ea8425ba25e933682
23221 F20101211_AAAHOY bronson_j_Page_078.QC.jpg
4d445420257bfb187dedabd0d1c471f5
be19134bbfcbc6a02d2b99b66afea336e87f83f9
38060 F20101211_AAAHQB bronson_j_Page_098.pro
522b03f6c87488772f44e0d9f9bc89fd
d3a80d0ce7b67c3cf555a5cfc5c3cc9713150606
2248 F20101211_AAAITE bronson_j_Page_008.txt
d7b7852202a564e3d438630e93d02f0f
5fc7cddb415f716400e28fc1364c34f30a41163d
7395 F20101211_AAAISP bronson_j_Page_207.pro
7602661f7ac2b3a62e1ecc7019e780cf
f5d1e42a6c152cdb3d33c100994dcdb065fb7988
2798 F20101211_AAAHOZ bronson_j_Page_012.txt
4999a4f0200b193afac0c54dc0a7d4f0
bcd8794433822b33e3481b47cf49647c27a884df
6764 F20101211_AAAHQC bronson_j_Page_105thm.jpg
065b1fe5e56e869689db8c39f33e797a
0cb845bde5892ca16cb5864b1a56555041fc22d7
29746 F20101211_AAAHPN bronson_j_Page_104.pro
0dc47cdfaccd1f957894f2d1da8bec4a
92b10054fbe4d8bc98f89bc4c24277922ae09606
2624 F20101211_AAAITF bronson_j_Page_009.txt
156affa02781fb9fab4249d3d9df666f
5042eb5bb0b4fbc32e5d2363749cfc5fd8bea666
65231 F20101211_AAAISQ bronson_j_Page_208.pro
18e0929ca93a2f09412d717337e99812
c730e1b7b21f87f2476f30ea9ca87beff3b7e900
54557 F20101211_AAAHQD bronson_j_Page_053.pro
0086d86dba830f239d9c4dd2d5d1e132
d5da299f901e91df38d4f74f5d8b4f4726e65a66
2612 F20101211_AAAHPO bronson_j_Page_214.txt
4674367f955a3a31192963b7a0cb94b5
85f16bbb43d3f50a5c0e315f6724c04bf57ae488
2973 F20101211_AAAITG bronson_j_Page_011.txt
d0c744b9e9204d8d1e751a90ca2ea784
6f70225af1e02b1992d2773a1713a4b8e0e5d610
69481 F20101211_AAAISR bronson_j_Page_209.pro
ae3583bf1c5be55145d479d21fed0b78
97976cb862b9d0e21eee2f02f36d9c7c2872701e
20505 F20101211_AAAHQE bronson_j_Page_153.pro
48c8abb63910cfe239983a6865654035
a5b32a5027f435c995af724300a0752af7d96a39
3026 F20101211_AAAHPP bronson_j_Page_010.txt
1e15666d374bb31492f30c4f43431dd5
3fa14848f30755979122b33c8f04b340712a0ea2
2936 F20101211_AAAITH bronson_j_Page_013.txt
55668fbdc86f2beb5d4ad93d1e697b75
b92164d08122ad3c926b9576031a2c51480fa052
64150 F20101211_AAAISS bronson_j_Page_210.pro
e5f8c517b05b593894d767b8d21a41dd
0811f6ff5554d22df490a34872e4d80e09b19efd
F20101211_AAAHQF bronson_j_Page_161.tif
1823a43349d8567aa195647b82edaacb
c5c12ba382cedcf142004e28807ee499c9392ca8
33788 F20101211_AAAHPQ bronson_j_Page_096.pro
aba03ee020e82a7fbddf5afb5db66af6
4ec18ee7c07c06ad5f3768e90e9867eb314bac20
2932 F20101211_AAAITI bronson_j_Page_014.txt
27832f5a075aad297c00d728c184a77d
2e64c6607232332bd2cea86dad39a48857fb4bfd
64115 F20101211_AAAIST bronson_j_Page_211.pro
43fbaf541211b1cef0821864f8cd8bd1
5746f7297c7ebba01a5eeb57ef2771891408459e
91703 F20101211_AAAHQG bronson_j_Page_160.jp2
bceb028ec06e94f928ed367495998cdc
d5e218fd6df94fd7e8742b87fff21f412dd50911
42411 F20101211_AAAHPR bronson_j_Page_090.jpg
9bbbba3802b3f6a4042ff574ab40699c
1c77dfc6d6544a42b01750e09548fadd0ce42c03
2892 F20101211_AAAITJ bronson_j_Page_015.txt
e2be6150fed473b5abf7df76955d1dc1
9f5e8187ea1282b9486838375a6d2e5aceaf209c
63553 F20101211_AAAISU bronson_j_Page_212.pro
5abc83813980c58441049aa6549ff9da
f740a26b58d2d07a8c8dd5937a1c9da5f616aa28
22530 F20101211_AAAHQH bronson_j_Page_017.QC.jpg
0a020bf22cc5b6cc3412ee9a1a04ef73
a1675cd86497a17009ae53ecc061218b607e59d9
25912 F20101211_AAAHPS bronson_j_Page_181.QC.jpg
4a39a732e80c7c9d78d2dc8b68059b54
c1308767a0fae5230de6f733bf59552423b60488
2175 F20101211_AAAITK bronson_j_Page_016.txt
ca830ae780f737d8dd235bfad8f97b36
ac6e2b6d2443f2a971ac0fd6c33d970f4fccd3c3
63207 F20101211_AAAISV bronson_j_Page_213.pro
61265d0db6a8bee23d3f7ff91f9e66e8
a5dde4818fb49772c87bf791df9a6853687d4f5e
62674 F20101211_AAAHQI bronson_j_Page_154.jpg
8ce32feb5e3dd132aea77d0445446183
71c3d841d8a74c0006c3feaf26b07db3fe87c11f
842488 F20101211_AAAHPT bronson_j_Page_106.jp2
9dd76691505dd45add471834170e604a
3aa81d00d89a1deff97d8ae04d4f555606b4652d
2116 F20101211_AAAITL bronson_j_Page_017.txt
fc39292ac3cf747f1e6abcabbbc813e2
f3851e4c5238e2c6b7897d1f81d45fe62fab1e76
58448 F20101211_AAAISW bronson_j_Page_216.pro
80bcefe722aa731e14e7f18cea4b2c12
d0aa691a64061eead67a285036b806973f7a4f35
11212 F20101211_AAAHQJ bronson_j_Page_201.QC.jpg
64bfc123f81d52973c3791ccb20700a3
dce20296b1d323bea9375c3a5f950c72b866f4c8
1051981 F20101211_AAAHPU bronson_j_Page_011.jp2
d77f485e771eb80b6b08dda7e6a43c4c
24cafb3039e6e2c258cc7aa027ff84848ebed7a3
3275 F20101211_AAAIUA bronson_j_Page_036.txt
52da3dc4dd7e240a0aa13cf8fd4d5e59
88d465cf713d470ddfb4460a75c9669daee83b0d
2093 F20101211_AAAITM bronson_j_Page_018.txt
9c658e8bfdc12e7b55b521f9d1c98562
517525bfafdbff2bea30814484c96e7de5c6fed6
12145 F20101211_AAAISX bronson_j_Page_217.pro
6aeaa44f9464b7ae2493c590650bb779
55395d827930b7bd7c396bae6c400908c260cada
6883 F20101211_AAAHQK bronson_j_Page_214thm.jpg
a676c6e6ba65715dd9c0d9b7b49623b1
1570d96dbe19106979e2d18422ee2d47854b461b
2083 F20101211_AAAHPV bronson_j_Page_101.txt
dcd4d18c909f1d2c523bd31458c05ea4
0a5b8a6ecddbbee46b4041445f09c6f1dcc3c7a4
2174 F20101211_AAAIUB bronson_j_Page_038.txt
261f777374c071f9a8b2366228bd1d1e
0c6537d781e82b62448d7994945df5296e2714f0
2123 F20101211_AAAITN bronson_j_Page_019.txt
b46d28a85b7738bfa526a12a961f8f40
9599a31b378056b106cd3b5ee1179da680631d11
34887 F20101211_AAAISY bronson_j_Page_218.pro
daed5c0ffdea80ee72acc45f20eeba38
92059de24923bc6a6a80a3043ccc41ab819b3e40
1354 F20101211_AAAHQL bronson_j_Page_171.txt
03aacfe248fffdb975e1d23e45962a0d
07eafe4158f93c5e1c3af93a6dfbd3686b79fe11
117921 F20101211_AAAHPW bronson_j_Page_034.jp2
ab90d15c881b64cfc9df3ba460d40a23
bddb9652a26e8f50b2c9556880a72748a4075965
2252 F20101211_AAAIUC bronson_j_Page_039.txt
48dd83cd01d6a5c2a97f601265777694
def61d76652317be6174831dd7cac51099ccb372
2198 F20101211_AAAITO bronson_j_Page_020.txt
dcd98eaca4f20f38e83d235a01869610
0e3c8f1bbe3cf45cab8d4206f5927635b7c829ff
99 F20101211_AAAISZ bronson_j_Page_002.txt
05852cb344709fe84968d2bedf746665
62a9355c655b8d5bb0e850a0b2d8229a366bf26b
251495 F20101211_AAAHRA UFE0021233_00001.mets FULL
74df10460a623dbba922aeed47485941
876a27aa45f3365b7e4aa83ec2457daddbe538be
2369 F20101211_AAAHQM bronson_j_Page_080thm.jpg
30d320d0061c4ec091770621aca9e636
5b6384e151a0ee6a762a79ea283a05d6f7823edb
559 F20101211_AAAHPX bronson_j_Page_190.txt
e61c7096f9c9780c0ab9599ab021d11c
d431073ed14e00e511ac84ee9f99f3e626ca0821
2176 F20101211_AAAIUD bronson_j_Page_040.txt
546298195163ed749ab6a68dd6b22718
40ded655a9b088c32e78cb72f2ce961b898136be
2191 F20101211_AAAITP bronson_j_Page_023.txt
eb9754fa0f5f76b773a9a0f9f325fa34
c654aba8eb1b995997ed95c78c4dd655c0a2a37d
6833 F20101211_AAAHQN bronson_j_Page_028thm.jpg
4471bee682e12e63d8a70d3d2c00524f
233b38c80fe70d9b483d01afdc3bbd630d608068
48532 F20101211_AAAHPY bronson_j_Page_069.jpg
df13ac8437b3e503ec4f2921167ac660
bd6d972284811c4d6a7bdd3bf594d51849924162
2251 F20101211_AAAIUE bronson_j_Page_042.txt
1dca36950dc3cc351a541cb8b3d2319f
969d332f7d9707876988fc46dad155f7b9aff295
13662 F20101211_AAAHPZ bronson_j_Page_197.QC.jpg
1957b4990f96d356686248485e75a640
64104476dc7b013458864bb50e83b119aa512a89
2241 F20101211_AAAIUF bronson_j_Page_043.txt
eed75676e07281bda65de83a4f8730a6
b8287c76c5f9aaba85bfcb204e9cb7b103270c62
1558 F20101211_AAAITQ bronson_j_Page_025.txt
e70466344e3d790145a76682e53d2354
117fbca2d82c7d801141fbe6167935bf8e8f6de3
24066 F20101211_AAAHRD bronson_j_Page_001.jpg
19456290eb1644d944bf73c327665f9c
db541114dcb6b53f5b027f5ad06bb0bd856b3976
18273 F20101211_AAAHQO bronson_j_Page_027.QC.jpg
e88a429cab513eee7132d7d6ac61e441
37534bd7ab5ae2c5477615cbd525a1f51e9391b6
6014 F20101211_AAAJAA bronson_j_Page_005thm.jpg
4ef4e7d5a08f6cb75edac8845a3071dd
e0179d8cb1a4e6b7d72cba4ac73a922418174b21
2180 F20101211_AAAIUG bronson_j_Page_044.txt
30d289c50a05e1f77b3cab731db23be2
f57267564a170f34c90453f5c8fe60e0488f2ea8
1361 F20101211_AAAITR bronson_j_Page_027.txt
77217a9a4d5c43f1b4895c1764bacb14
0114a22cea0e1bdbcfdd616b8f15df4eea7cb408
9456 F20101211_AAAHRE bronson_j_Page_003.jpg
cd9a5639845c1001cbcfb5b58657147f
f8f8fc0276a881f1fc9a9a8f9adb394661801b19
10443 F20101211_AAAHQP bronson_j_Page_124.pro
45a549cf2463c731e285a87d606cd62b
0c23767dfd05c705a71733d4f7ddd9546176e780
27012 F20101211_AAAJAB bronson_j_Page_006.QC.jpg
6a1e6d6feb8bf033c53750b63fc4a6cc
824ee23e5a42093315b5d282600fc44e0f801f0c
2157 F20101211_AAAIUH bronson_j_Page_045.txt
0485c06ca1c19fcb2c1e6fc76e4f1c5f
0ef07d6385b84180af55e93ea3b2248a5730d1d4
2129 F20101211_AAAITS bronson_j_Page_028.txt
efcf3b420282280b4cc69ec8ba4e4355
57b78a4b406ce3e0abfb5cde8d0dbe4acd50ff7a
90536 F20101211_AAAHRF bronson_j_Page_005.jpg
ab9b0c5459a65ea80f40efa925ec80ac
8a494e9ac035b725635d0731073e88df3c49d07f
1357 F20101211_AAAHQQ bronson_j_Page_022.txt
05315b47f5f11faa41d7f82912d04661
8d666989fcdd5d8282ead769683f44c667a695ed
6560 F20101211_AAAJAC bronson_j_Page_006thm.jpg
42c80cee23a76689aedf8493935d05c1
3c9d0d480cb5a09718921060e92eb20916ce1729
1749 F20101211_AAAIUI bronson_j_Page_047.txt
8763bb1c463f7f2dc7880ca7c5758a9e
903a6e5f52cb890961af0219656a6a6de09a6d5d
2193 F20101211_AAAITT bronson_j_Page_029.txt
f396d8dd39de0501d69fa1f4e380f4bf
98b802bf00511840e3a4d825005d97df38718a7f
107613 F20101211_AAAHRG bronson_j_Page_006.jpg
4b84100901e0d0d5bea8a607e5bca12c
2052828e83bef074ab09185600917fe58017676a
65898 F20101211_AAAHQR bronson_j_Page_083.jpg
7ca21aa61831c1d7a0337896484cd862
9c2d46c36d1fa0f7f5e5266e1416775367d357b7
7809 F20101211_AAAJAD bronson_j_Page_007.QC.jpg
e9e158b706a2926fecc6bb4e12b94572
3c76ed98a28603eab0d94ee3471ee6856a456380
896 F20101211_AAAIUJ bronson_j_Page_049.txt
f35668e1a7695be4192fa58ad55996a3
dabc9dfde927bfc56ba2534661c46a6fe613066b
2117 F20101211_AAAITU bronson_j_Page_030.txt
520bac5fd712aab7ff3adeb8ea50d6b3
5b63ba906078a2fb8784abcad6a0dad4ea36e39c
25289 F20101211_AAAHRH bronson_j_Page_007.jpg
6703f54ccc04e563d088c98a7fe6e596
e19df1a16c8784a4b757822c123a6ff00bbc31ba
78075 F20101211_AAAHQS bronson_j_Page_105.jpg
abd0662d2f812b44111d7d57b66fc773
4de09aa03d9d289c0226502ae15e48c3263841cf
2457 F20101211_AAAJAE bronson_j_Page_007thm.jpg
de3b022c55130755ef30523ff1dcf290
14e6ac68731e2661e984070b33f0c03ce0dc18cb
F20101211_AAAIUK bronson_j_Page_050.txt
c4306a522f6d397747e354740cbc8b52
82a3918c302d74e3bf027da29e51f84bec2124fe
2173 F20101211_AAAITV bronson_j_Page_031.txt
3bea9cda67d530a3e07d413bf6d87813
9ffb2e36ffd7aeb1309dd5e04b0abbe2c05978c6
83086 F20101211_AAAHRI bronson_j_Page_008.jpg
61439034d272d8fd27e84bbb4b36c717
4ec424a50f3e88d258f1989bba259355ac76701c
3208 F20101211_AAAHQT bronson_j_Page_199thm.jpg
a6c075e9a324d760f6f8c2006814d921
d298384baec3cdaf26f48a952e99b9e824f71dc0
23294 F20101211_AAAJAF bronson_j_Page_008.QC.jpg
879b2c4054f69117c9fe66384ee49753
1c6f1c027dc7f3615011319697855c906d4122c7
1622 F20101211_AAAIUL bronson_j_Page_052.txt
194f96cf610641104f73d6396d2aa88a
50c46b8c4b7517c147d9d008a3fecde4f39e9631
2288 F20101211_AAAITW bronson_j_Page_032.txt
f9bd664c5549c6ed9c9bd2c183e5a6c9
e7e388696e8cb79fa7b1fc84a33faa871e5f90b0
105643 F20101211_AAAHRJ bronson_j_Page_010.jpg
fdd836b34458a2eef4632c2acaa99b6a
2f63cd4b2f589c551a78b7ea91e0135d0faeff90
75244 F20101211_AAAHQU bronson_j_Page_108.jpg
90c307a07400142ed9ab55b74b7a4495
fc9242a5c7945a087b9885f497c238bcad19f580
6108 F20101211_AAAJAG bronson_j_Page_008thm.jpg
170b5f98bbb1949c853343066835b5f2
fc2fe39baed25211d541b92ba7018d848d6661a9
1999 F20101211_AAAIVA bronson_j_Page_071.txt
74d116291e881dc549f4b4bc17da42ed
5d5a0ff33d9350690030301f7476befc21c6cb53
2294 F20101211_AAAIUM bronson_j_Page_053.txt
8f26832b3a264c9b5feba096b6a78b93
67af1d81a10118a74e3eae87d516b94061cdce23
2237 F20101211_AAAITX bronson_j_Page_033.txt
41fa807d858bc580cca069466cee3246
35b9efa616d49b1d163f4a43e1a869627451660d
89921 F20101211_AAAHRK bronson_j_Page_012.jpg
b906313dd9e02a0f8db1774f08cbd6bd
666a9a8f83008d4e8ad29f2f9b4d621f8ed4fd6f
867 F20101211_AAAHQV bronson_j_Page_140.txt
4bfc355fc9b36529164f36edff26afa0
e29021a7d052642963600541930f146cfb084438
6647 F20101211_AAAJAH bronson_j_Page_009thm.jpg
86dce76730b0fa189f6a845d9001f077
a284cba2dbd0dfccdf613fe505166f0c058bebcb
1642 F20101211_AAAIVB bronson_j_Page_072.txt
99fd2f4a07a05af63575e6711828fd90
4eaddb4539d85fbe110274a48696b91eebe44b87
538 F20101211_AAAIUN bronson_j_Page_054.txt
cad5df2254873b4d62fb999ef6fe8fd7
2609610393bda3cef12c17f6814794a35b3d3e2c
2167 F20101211_AAAITY bronson_j_Page_034.txt
b9e1798828c227fa953ec0d15621c3a0
2ffe63593457685a1f75d7dc5caa3742ec09cb1c
95781 F20101211_AAAHRL bronson_j_Page_013.jpg
44396b34e41b440f94d851cb658ce70c
db1dd1a3c9f919191ecd933c11a82a86295c8d8c
20443 F20101211_AAAHQW bronson_j_Page_016.QC.jpg
12e8cb5bf24f84ca6073ac6593334794
f72fb3390cafd1a38cda028f3e6ceb385d99994e
29642 F20101211_AAAJAI bronson_j_Page_010.QC.jpg
7148bceb8d1f9d73e09f5768ec36c73c
32b38d2bbf882146e0ebc05128a602ee38055c89
1453 F20101211_AAAIVC bronson_j_Page_073.txt
626a85b7911601b70e06cefe2b017ae3
7cd718c77d2161dd3a794e7e8b697c30b2000d6a
1599 F20101211_AAAIUO bronson_j_Page_055.txt
b75b48f761983a1b4aa3ea35c9f1a67d
b4935f02889dd5ae747d1fed83192daa5e7505d7
2114 F20101211_AAAITZ bronson_j_Page_035.txt
28e925b50cce2259eee455e0657a782a
18b4c0df310daa61d494c27488335038970132b7
93731 F20101211_AAAHRM bronson_j_Page_014.jpg
87d5c2e2c67b9cd8ff891ec26e27cfe7
cf7b099c2aee083e9d8b5a5e9e416edb380b4e08
78729 F20101211_AAAHQX bronson_j_Page_110.jpg
3422518c49ae5d321d6a7209397ba5f3
4c7748a1dcd5bf13132dfe89cb6acfc6524a70aa
77125 F20101211_AAAHSA bronson_j_Page_029.jpg
fb990ae5a39a66fce9e7c20d28e9ce43
bf17ad17e70d6ffa472cb083e66750fee33285da
7399 F20101211_AAAJAJ bronson_j_Page_010thm.jpg
78cefa6414aaa8c065d99ec4f32b0053
77837fff4b2c8fd4404d4b9a70c40fba2d7e7f78
1334 F20101211_AAAIVD bronson_j_Page_074.txt
4d586bbca16d26a1c75c68509a2324c1
e4c12d49558087ac5f2b71e65a32262017bcc36a
1588 F20101211_AAAIUP bronson_j_Page_056.txt
1e30043704b6144d48253d479a6e46f8
1e475f8dd6b69d7ff964bcc542e793d3aed02446
94777 F20101211_AAAHRN bronson_j_Page_015.jpg
00dfa32711dcaef4055c3a1eacf386a5
dea0e6316860ae1bc047b3b05c7b9ed60b840707
567741 F20101211_AAAHQY bronson_j_Page_203.jp2
2b8a7712d8214d9153c08397d587267d
3a56c74d59472ffc18aa8c95bcbb7049c211999f
76894 F20101211_AAAHSB bronson_j_Page_030.jpg
aa1600685740aae05b312278ec529f4b
355d23a34e260db416274d593990adc039117054
7201 F20101211_AAAJAK bronson_j_Page_011thm.jpg
9faa2672637e8b4891303f9f90d9c6da
d66f42b6c693262a3072083a7fea6364974c203c
F20101211_AAAIVE bronson_j_Page_075.txt
96e368b4083d8e7c9a2848db24226704
476772f22e9ad901fc0fd20804d0e4f0bd04513b
1565 F20101211_AAAIUQ bronson_j_Page_057.txt
f7fa7a0ad63b089c0737d176e5a561d2
a31f375a8b95e69012ac6d2eec1d58c560299b4c
69713 F20101211_AAAHRO bronson_j_Page_016.jpg
5d18868a799894294756d3a5302e2fe1
04e0a462a78da6ff1ea9b87d6deb8fcebed7ccb9
5726 F20101211_AAAHQZ bronson_j_Page_176thm.jpg
6d0c9b970a3da9b704bc459eb6d1e390
347ac3f4858da2face0c42ce6cf41d17bef7ae49
77892 F20101211_AAAHSC bronson_j_Page_031.jpg
ae9b1c20e1907037c7f78f84bf73e287
c9b339ed0d89545c43064a369f39ef5bc38a6eaf
25809 F20101211_AAAJAL bronson_j_Page_012.QC.jpg
58d0258dcfc9b81fa05416bebde36208
d366f338d500c77f7e0928f661335e42dd651ace
2013 F20101211_AAAIVF bronson_j_Page_077.txt
f930af5e8c1197b76954c12f44f2ce71
7e8ee44c0b9d058c26605ad77e0d1b40993fca24
81173 F20101211_AAAHSD bronson_j_Page_032.jpg
0f2004bf524d5124af5b5aadb4138a4a
e01babdb0a0b98eda0087a846d04adeb19cb8696
26201 F20101211_AAAJBA bronson_j_Page_021.QC.jpg
796d9d8a903c70c9569c9e92da02b870
3e6ebfb64dee8b810bf04abfc015cddb0bbe97e8
6834 F20101211_AAAJAM bronson_j_Page_012thm.jpg
c01da80aa53d591ad1ac1af8864cc039
9018cde682b48f9a984152d5c0462484dfd92b36
2080 F20101211_AAAIVG bronson_j_Page_078.txt
4ac3f09cd1be72fe16f4d78cd410f0ca
7542cb02e302b26d7de9f4ebed2aa99d5a940e2f
1744 F20101211_AAAIUR bronson_j_Page_058.txt
311b0e41af6a5c7ffa2c1f6d159173ae
77c46b385e28524ac40a49f68e24d2c2d55f148f
73273 F20101211_AAAHRP bronson_j_Page_017.jpg
58ac5016d98e9ce273002bae6dbf9e0e
3c51160d253606e7199dea3a1235ddf171f5c2df
76960 F20101211_AAAHSE bronson_j_Page_034.jpg
7215a840fe1f1a69347bbcd10b909fb7
c36fb38b7d3d2f4a9ee687cc1c706e907587b09c
6937 F20101211_AAAJBB bronson_j_Page_021thm.jpg
e023ef831d05e4f958d7411134c732ca
ee8fc29fe8af89a5c377056059338d15922f6265
27187 F20101211_AAAJAN bronson_j_Page_013.QC.jpg
46b5db42a6860427488d0da505ca07d5
89f9966d080eaa22c7592eb3336b358ed25774b6
585 F20101211_AAAIVH bronson_j_Page_079.txt
762e226ac0736899123efd6a9f0c8ffa
8e33630df9ff1529e44c8e9260cc89a319272973
1829 F20101211_AAAIUS bronson_j_Page_059.txt
b257795111c7f03ff61062a345f9c6fc
88b7fa53daa45bdeeaa0d34cd13b416f85a95dd8
75247 F20101211_AAAHRQ bronson_j_Page_018.jpg
c636347e0b2afce70b0423677626f65f
6e8b53f278a4709c03c4924b53d43ef3b89d1123
74470 F20101211_AAAHSF bronson_j_Page_035.jpg
06ff47f42738c6fd7f37a5b03d0d3c17
c32d244f94c7c32a25cc2294e442cb8a8d84859a
16977 F20101211_AAAJBC bronson_j_Page_022.QC.jpg
0ad85901c702d684d0b2e8c46ff3f895
2ebbf9b511e1df7b42e1cbcd32cd060d4ca729e6
7353 F20101211_AAAJAO bronson_j_Page_013thm.jpg
7af402dd44ecbe2c3459668209e8b987
e83dbab190bd01891041c5d9a537bb76c525bff7
469 F20101211_AAAIVI bronson_j_Page_080.txt
13bede6d8ce36897a69d5141e475757b
9cf93d5f068b6659d0c146bfaf883c1693f9f937
1813 F20101211_AAAIUT bronson_j_Page_060.txt
5bb6bd2764409df919b3bfe3b8d99406
bd23e6159efd8439d73d13fd805570680c519d79
76916 F20101211_AAAHRR bronson_j_Page_020.jpg
df76708f7bac80a9b30fcd1658f55188
04532b76d9b7ea82bbb38a28e0b4515a3e20bde3
73395 F20101211_AAAHSG bronson_j_Page_036.jpg
a12492cfff425d9c9a38f10948b96f52
3730682c0b6885d2e2bce16efe759c82b65b8c71
4794 F20101211_AAAJBD bronson_j_Page_022thm.jpg
341adf7126d3815e5a551af3ffd800ba
3282e3cf0ba88fdb75c277725ea3e5a25a5810c9
26977 F20101211_AAAJAP bronson_j_Page_014.QC.jpg
8d12d65122e18bec4bd03e4a433e1583
68b5588b29178b4146b0aaa8788b68f462512c19
2118 F20101211_AAAIVJ bronson_j_Page_081.txt
e032811fb07a03a6ebf4894b6cb710c0
b4381a834664a0a94a842e1db9de2f8c4dd6b9f6
1904 F20101211_AAAIUU bronson_j_Page_062.txt
32355493ac105da967f4ba840f8f505d
d6768bd0e59c2af187ac2415542813039d9cc8e2
80388 F20101211_AAAHRS bronson_j_Page_021.jpg
700350f0d70575091c8f56689a2f5da3
5463e2476561e6b11b0a76ff9e46c6beb314a73e
79442 F20101211_AAAHSH bronson_j_Page_037.jpg
56465e2cddfad7e5ffc37b1893046569
6927c3a16afc4a6709a6ea9ae83a93694f317994
24407 F20101211_AAAJBE bronson_j_Page_023.QC.jpg
c3507e42a117846e030bd13084a02bb6
c2542657133e7f48021b8470b68d4268e8b4547f
7055 F20101211_AAAJAQ bronson_j_Page_014thm.jpg
ea181bfb753508047264cc3e42d1cdd2
1e87d6b8a235ec328b9500e3a0eeec8de42bd3ca
1850 F20101211_AAAIVK bronson_j_Page_082.txt
4a5b7db4ce1e8bcf6a44be8ff33d8205
e5900122c9105e49c5541f18be66718c0bc8c9bd
1476 F20101211_AAAIUV bronson_j_Page_063.txt
1ba7860180540917e8d78f2e91710545
d3d2fc46bf5d66952c5920110bc0566975910622
77622 F20101211_AAAHSI bronson_j_Page_038.jpg
b7f0f360a95e27a1bd9939ee61ee9b26
ef5f5ac8bf4fcfab2579f002efd8204163bcb131
51468 F20101211_AAAHRT bronson_j_Page_022.jpg
9f1de6d047a856f4e88bfe571e9824fc
6190aa29a3043d3d365918153cb5b496cfb582b0
6769 F20101211_AAAJBF bronson_j_Page_023thm.jpg
7936ac273a2c51752fa94307154a6474
42b30e9854f93b004a22139f493abfd16defff01
27145 F20101211_AAAJAR bronson_j_Page_015.QC.jpg
d004ba37d6d06ab989a6ac81a324a5f7
b30dbd85a77bb1c27705ce7954d6e3c8d65dc91d
1645 F20101211_AAAIVL bronson_j_Page_083.txt
d54131759d83fbf1d883a8cb1addd69c
09859c8580042aed8025c41da71ef8c7d6dd686d
2229 F20101211_AAAIUW bronson_j_Page_064.txt
b5d5118a61a91d7bfaf10fd28f78e78a
288e6937d01c4a4d2edae821b243ed801bd9ac67
80522 F20101211_AAAHSJ bronson_j_Page_039.jpg
45e3d4d29a328a4d0467621bc775cb81
7e31ce75dfd1d8ddee0620ebdb3cd5717f52e8a7
77122 F20101211_AAAHRU bronson_j_Page_023.jpg
74027518c00995aa451caf62cff2953c
3498ffa64fdedb30a017b5b2d55897aa5d28bd80
26335 F20101211_AAAJBG bronson_j_Page_024.QC.jpg
5432d8c9bf5b4f1c1e4c7d131e10081d
5777bf87a78f09f12e181d461e159dc0ea87904b
5451 F20101211_AAAJAS bronson_j_Page_016thm.jpg
93def054a353fbfce80bc884b8832d0f
cfeaebdc177dffe8e1643853b321423151e14a7a
2133 F20101211_AAAIWA bronson_j_Page_100.txt
babad5bd3bd2d5a5200219140a124d9a
9557bd3e274a133b37a4b599403a309af2c99220
2077 F20101211_AAAIVM bronson_j_Page_085.txt
fda20e932bf0ecda833184adc054d90a
173b4691994b49cb795f639e87a9ab5a7db575e5
2276 F20101211_AAAIUX bronson_j_Page_065.txt
4d3a7913dc7610b96bb034f6977e4538
6338fa5d3c6d5a74e3fce02087bbd4264d88f3a8
74533 F20101211_AAAHSK bronson_j_Page_041.jpg
4fb2e23a0a3f932edbb753a23436d57a
1ab3bab790b057613f321e0ae3668d5ce8a32578
80436 F20101211_AAAHRV bronson_j_Page_024.jpg
8e51f3afecd2cb2e32570f703ba185a0
072f0cea3d5ee5192201e02063c818e03604e0af
7074 F20101211_AAAJBH bronson_j_Page_024thm.jpg
f63e2ac022f8c29adf6c8449ab6dd214
c7948d51c816a9d2a4de788c6c65115cdc10c4fa
6363 F20101211_AAAJAT bronson_j_Page_017thm.jpg
d22646438d0a9b45d235cef1ee955c13
cd3f97de60cc44bad441210e5516e54397b26553
1625 F20101211_AAAIWB bronson_j_Page_102.txt
eaa81714dc3e3dd07daddadc5848f1f6
5165aa4a81ab657fe5c5d29683ed50c06c648aeb
F20101211_AAAIVN bronson_j_Page_086.txt
bb80616ef516b233ce7798ff0b6825a5
06c4f2f72b95ebd5f5863375dfe02ce48e8a4ca1
1638 F20101211_AAAIUY bronson_j_Page_067.txt
7b12dc36dae7c33e87b328f3d5cb409a
c935798418a66f49c7ed6c10358ed7acca19debe
80400 F20101211_AAAHSL bronson_j_Page_042.jpg
95ce4aa848f0e10d73971a96d26c2099
ea631eece783adc216f7f3d7513bd5834be5f14b
79536 F20101211_AAAHRW bronson_j_Page_025.jpg
b156ee025824f142ca3d11e47af9d784
6dbdab1a064a6fb8de95ad81e8e4b5d7be5e8ca6
6603 F20101211_AAAJBI bronson_j_Page_025thm.jpg
e352b488aa595e47cc530699f2d0b45c
b66996b9007c20558d5e89f33a352be8d342f621
25050 F20101211_AAAJAU bronson_j_Page_018.QC.jpg
48787a5b1e26b5361ad69d0cd10dadfb
85f1707eca4d397bba7e6df497ee1e3768eb46a4
1327 F20101211_AAAIWC bronson_j_Page_103.txt
2d637f56f631952cd6590167cd4523d7
8b90d5d9d61e3d3109c716db7ef347c8ccc31d90
2141 F20101211_AAAIVO bronson_j_Page_087.txt
39ba7e17d4d55f83b0e1cc60f9ff3078
baa6369496f25ef49db68b570204671410459cc8
590 F20101211_AAAIUZ bronson_j_Page_069.txt
ff3dccaddc56128098fb08662445407f
02682b1d43ea95068f1cdd66170938ff3ee09435
74428 F20101211_AAAHTA bronson_j_Page_061.jpg
0af730dc2880d3719e418d0b1829e64f
6bcc17bd363ad2076d369c5cf36fd6f6ce7dfffc
77371 F20101211_AAAHSM bronson_j_Page_044.jpg
d8d0ec902fbb16236b079c639c2f9ea1
de36e306fd663fc3c6fa4dbb572c03c7fd663011
91008 F20101211_AAAHRX bronson_j_Page_026.jpg
0dd15da02a69becdd56a59834c1f5e79
b44e8d2cd62fd5345304916fa7023d4b81bc7375
27272 F20101211_AAAJBJ bronson_j_Page_026.QC.jpg
00ac72f9a4c65f23ea6de951f63a46c0
509c93a74701b0e9805fd7c9e15eec63bfe85f7e
6778 F20101211_AAAJAV bronson_j_Page_018thm.jpg
adcbaee91a147909ee5bf4516026b7ed
7a5d8c21e67812f9538c0a18df30c0f6f628f192
2154 F20101211_AAAIWD bronson_j_Page_105.txt
c83c109f033fca74ada1a1644f9d8fcf
d103b2c224a186a78e6a773f4957ba418a3da175
1224 F20101211_AAAIVP bronson_j_Page_088.txt
5d045ac48267556419b4522569bb4a9e
0a03615a254a32e15dd19e56e5023432a439b63d
71055 F20101211_AAAHTB bronson_j_Page_062.jpg
09cc2e72471ecb35daa17b68a207e46d
ed7c79bbde6f14ffb2fcbc02dcc4246fe7fdb978
77153 F20101211_AAAHSN bronson_j_Page_045.jpg
28710e6336f89028bae53570fdae5f95
a2426742a130cc88d138fe6cc3171d9e9adb8fe3
60764 F20101211_AAAHRY bronson_j_Page_027.jpg
0f1a4b627b5892412d98a819d622afe2
b0b52457d8e60ff85e4134f10560eafb7fef3b1b
7475 F20101211_AAAJBK bronson_j_Page_026thm.jpg
91758ca899983c1cc00bbe6807ffbed0
89d51911fb814f28720f99e3007a12093e26121f
23370 F20101211_AAAJAW bronson_j_Page_019.QC.jpg
2dc9f1f776d4620f08bd99e82ea1a561
e37280ddbf0cd4b0303702486082ea48b5dd1db0
695 F20101211_AAAIWE bronson_j_Page_107.txt
e7ebcbc20f845eb083aa1692b573ae14
c820dea5793854c24dad086704e7faa7bffc5499
1069 F20101211_AAAIVQ bronson_j_Page_089.txt
50c4762b44f8963711368e6e77e0a592
b7021d59fd868c22ccfa6f1324e3759b89dc99db
56270 F20101211_AAAHTC bronson_j_Page_063.jpg
f53002a83586218922a46becf8593dba
13c92f218aae8459014d116de6626e875627a35f
75464 F20101211_AAAHSO bronson_j_Page_046.jpg
9a8f232adccba68a55867f9903568349
3a2912335192d470bdfcc4f580486f3c7ac28200
73540 F20101211_AAAHRZ bronson_j_Page_028.jpg
7d3c4b08ba141ad14b053ef7f13f6369
e12d5b64b1bfb6e31ee0210c996d7486052f716b
5310 F20101211_AAAJBL bronson_j_Page_027thm.jpg
b2fded8b4455b85ffe75ebebecaa6833
7aa73d080a4f40cbdf3ac1a5ed620d187fcbca6f
6664 F20101211_AAAJAX bronson_j_Page_019thm.jpg
5abfd9ef0c6c6d5dab3f2ee783ca3ef1
5b31599557e1caf0c014f2274d247e37b92b732c
F20101211_AAAIWF bronson_j_Page_108.txt
496cd8db60ade342aff2a00f550792a8
0f0ba532e062078e9abbe6ef66506938b56c5d9f
834 F20101211_AAAIVR bronson_j_Page_090.txt
c1b602d8d6100d98a945dc992a8ae926
c667a4d0aed5ce856db69eb49fff584fdd5c80b3
76540 F20101211_AAAHTD bronson_j_Page_064.jpg
d329b3e33450f2b79447f5b932dc9145
4406d08fd4a6d283cfa9e46854de56a654b66e0b
63825 F20101211_AAAHSP bronson_j_Page_047.jpg
9f4b5184f8d126631aaea63b3e92f67b
6066d2e7242ed9b448225a1cf90d760b01a61a81
F20101211_AAAJCA bronson_j_Page_036thm.jpg
607c58f0deb8960177737c4a80ddf524
045e3cb3ec951cd3091ec01234963ce594cfe4e0
25078 F20101211_AAAJBM bronson_j_Page_028.QC.jpg
3d2b5e69ef0ef112c63ed277da116223
e607d6b7eae77983f89afdb0b0c464037fd150af
25494 F20101211_AAAJAY bronson_j_Page_020.QC.jpg
f327b02aee666491a8fde5e4f0d8e6e7
7b4c006d6fecdcb65aee19158a7ff6611a5fcfef
788 F20101211_AAAIWG bronson_j_Page_109.txt
711043093374e11ed43f1bbda0a94643
9c8b9bf356a1cd143b426b6bec9bf53323ee1d1a
71862 F20101211_AAAHTE bronson_j_Page_065.jpg
95d7e55095b3764b5cebb133d1533e3b
0b7e0ef6a2f3fcafbca173a15efb217473278be0
26615 F20101211_AAAJCB bronson_j_Page_037.QC.jpg
4dd9c2822e2664c1ccaa40067e57df99
578ae20e6ba7a0f112aefea11fec3fc49879f381
25558 F20101211_AAAJBN bronson_j_Page_029.QC.jpg
aed3febf0b47d6462f6c4f74d7a03caf
37618dd4c83ebcefea7b681277fa98373bd60e16
6876 F20101211_AAAJAZ bronson_j_Page_020thm.jpg
4db5ded1d4ddf0730804b7d3e9cdab8d
e32696444f063d092c913b89f1ed4c45c4852f04
2238 F20101211_AAAIWH bronson_j_Page_110.txt
febbdec63d454ab04c8c089aba9f11ce
b684a0829770aef9c1d430cfebd9d3be1af8977f
1983 F20101211_AAAIVS bronson_j_Page_091.txt
9ea5d71e7516103e8f66524a43f0508a
bb8d4bc98432e6ee30b6c198a9f50d6d16fded53
62096 F20101211_AAAHTF bronson_j_Page_066.jpg
a17a8a7c1946c5f63d4e49f8994a5211
70a81bd84139ecfd0542586db0328a542569b233
86341 F20101211_AAAHSQ bronson_j_Page_048.jpg
b45919f3fcc0450cca352a997d557f70
9338a9cb31783d704b97bc3bfec7e6c746c27e07
25501 F20101211_AAAJCC bronson_j_Page_038.QC.jpg
b059306f773956fbaf212962038c71f3
e606718fea4df2098899f0004de35f429a88de8e
6836 F20101211_AAAJBO bronson_j_Page_029thm.jpg
e46605a67cc2c3f2aa9d70a9caf0457a
2e9ae09af5b1a896526df777887653e8a2811555
2166 F20101211_AAAIWI bronson_j_Page_111.txt
ca80bcc73ac7464f3e40e9f05c7d06a7
65df3c8530d4117d13eeb2d3f455208ba2a8e9c6
727 F20101211_AAAIVT bronson_j_Page_092.txt
e0ccb28875d59d05641a2e82be941bba
afb1b8cb7a30c0a3aae4da144206c42752e83a05
66176 F20101211_AAAHTG bronson_j_Page_067.jpg
64afc7a23dbe1bdfbe3d681e4d22744a
1627febfbd5d9314e55662f8d9e77a907d586a2a
68280 F20101211_AAAHSR bronson_j_Page_049.jpg
e801b5be9bebf993e88dafefeb663d03
8be75d12d19333f8aa70570ac2bb483722c5f19a
6865 F20101211_AAAJCD bronson_j_Page_038thm.jpg
7381c5f1260b13d6d798557581803994
a9bafecbb87156086cb032efc3cfab1106825f53
24715 F20101211_AAAJBP bronson_j_Page_030.QC.jpg
0a5b6a9e8609d06aa13f8c50a7314682
2b5969fa56b97a65b2bdd4bd7c71e1cbf1a795b3
647 F20101211_AAAIWJ bronson_j_Page_112.txt
04db24af71ef6e1b575945714cc9472b
16c988ed752328462ade5090d158843e1676da10
2099 F20101211_AAAIVU bronson_j_Page_093.txt
9f0138fb61857ef52663e5bc7b8943e0
863b78512d610f0eb3b575308c6c15bd862d07cf
72628 F20101211_AAAHTH bronson_j_Page_068.jpg
ac143f0347ecb224df316d0c7df1a41e
df85ef158f697298d091c28e4516ff974cb75c1f
37363 F20101211_AAAHSS bronson_j_Page_052.jpg
efa244988ae6057376de0445e861506c
b212b605036621b3dc5111c744ad3cf96ee9ceb0
26935 F20101211_AAAJCE bronson_j_Page_039.QC.jpg
9fccf64528e58ab21d592219c2a52ea6
5101ffd713e821290872eb14a570b0ff71188a91
6849 F20101211_AAAJBQ bronson_j_Page_030thm.jpg
22f0915e0f21c684cdf73aa6272ece7b
81ff86bcd683d6d4f716394b0924a69a5091f48b
1668 F20101211_AAAIWK bronson_j_Page_113.txt
e541288a5152cf4bcb64cedaf5c77b9d
91e8037dfd886d491751d9e005ea159343db8eae
1754 F20101211_AAAIVV bronson_j_Page_094.txt
3462c6b3dbbee29f404bc9b54e0d1e66
e1ab4b01af2c97ce38587899c9abffc84c2a15e2
62530 F20101211_AAAHTI bronson_j_Page_070.jpg
cd9c40b41bd8473a33455ba73018c8ef
617d54bee238b879f5a43789c51a1be4cf2468dc
75459 F20101211_AAAHST bronson_j_Page_053.jpg
7ceda531d453e17031f2dfbb4cdb5b6f
527722ba5a5e989f910506e2d7f9eca8f766ff9b
7110 F20101211_AAAJCF bronson_j_Page_039thm.jpg
5880e66c1bb96de1d8ab66dd1149a085
3e080bfbd28fbe8db20e5d43629ca1886e305b22
25574 F20101211_AAAJBR bronson_j_Page_031.QC.jpg
f8610aa7afcbad13172c9cfb81a7c011
dfff23b722c778285929e9671f15e8b0992f5edd
1719 F20101211_AAAIWL bronson_j_Page_114.txt
55b77822fb359092f61dc88c86d084e3
e66ba33f2d660d6097d0650385f84066a68b7159
546 F20101211_AAAIVW bronson_j_Page_095.txt
2ca0c4bb4488971a0571061960419cc7
c94abedfff92b137f34ac8dbc87f4c7b72299fa1
64257 F20101211_AAAHTJ bronson_j_Page_071.jpg
ff495764163ef2507d1a6b301681a100
b79db251a02922a0a46aa1cba1982cb722246eb8
54795 F20101211_AAAHSU bronson_j_Page_054.jpg
9d0f14825ebf5863dd1c4d956a25c445
216c4af745587b66f2f44276f873884c257fc87d
25950 F20101211_AAAJCG bronson_j_Page_040.QC.jpg
3852904b66910f7ce3dcdf2e6b2b1260
5c571da210d5a9aa128c33bdb10226f4d788b173
6837 F20101211_AAAJBS bronson_j_Page_031thm.jpg
8339f0972e35b9fefe322a2a05765d63
188bee0f3459ae4e9e8e2ae7383edc40385a5516
1992 F20101211_AAAIXA bronson_j_Page_130.txt
014feb3a57f2ed682d9cf57d6d6ce9be
3443fde8c105336b25e2995c48d0101a7c59375f
1601 F20101211_AAAIWM bronson_j_Page_115.txt
bcb5e2eff78a8a91507f2bc22b5dca6e
3b98ec5e1a1a699245701b52f4ce7b95d5c71118
2284 F20101211_AAAIVX bronson_j_Page_097.txt
d1d9da469d67804c641e0d84780f394e
f58977bf65376d3d1c1483fab4ae855b27c70c30
63771 F20101211_AAAHTK bronson_j_Page_072.jpg
93f226850db14389634cf0563ed30f8b
f7da8b17d6c9a9db3174560b1eef299994cb1427
50556 F20101211_AAAHSV bronson_j_Page_055.jpg
f8f45fc70fb5ac50ab6fc69de168faa7
7edf5e364cdb8c983ed9d006a6339a53145a2b3a
6896 F20101211_AAAJCH bronson_j_Page_040thm.jpg
8e96a487ec22a1193836212ee5f27f7c
93cc71d108d3ceedd3664bf1004eacb80ed73049
7248 F20101211_AAAJBT bronson_j_Page_032thm.jpg
bf3b8161578aca6338be77dcee132515
ca0a21192d9dca207849b5967e374cb28a3367c4
F20101211_AAAIXB bronson_j_Page_131.txt
e5f9d866a4d816d6b4c95b0b5aa3b5a0
a5891d9ebb29be50d0c71fb46c017f93cbc5e1d6
1907 F20101211_AAAIWN bronson_j_Page_116.txt
9b087a7a4577844e23e18fd55de724e3
773a5622c266244f784f35c70b3174852c09d6d6
1627 F20101211_AAAIVY bronson_j_Page_098.txt
80bad6271290f971fbcf2fcf29e35818
6504df100a9ad6af5678d24e38452579be37dad3
65257 F20101211_AAAHTL bronson_j_Page_075.jpg
da244425d8ff4d8d38feffa67c8f40cf
6711c9d1a9156b2fefd2b119fc6904a30e68963a
86583 F20101211_AAAHSW bronson_j_Page_056.jpg
ad60bb7044e8c1eb39c83476a508da8c
4db1adf177b1ee6dde8a0a44d0c6db59bbd05729
25062 F20101211_AAAJCI bronson_j_Page_041.QC.jpg
f6d4bee6661f1f35f135f00acd1a5779
e49110591e6135864aac1a80cd8ef21823814e57
26619 F20101211_AAAJBU bronson_j_Page_033.QC.jpg
c35eb3b23c24e33b0b39758032a22d1a
8c5cf4ab626288aa1fce7e22abdaac3fe4c32741
864 F20101211_AAAIXC bronson_j_Page_132.txt
7c745bc9ea58285e18baa4919fe5fdb1
05301fc7477df5324f82783dc1459d58485c6825
2171 F20101211_AAAIWO bronson_j_Page_117.txt
6d217ee97edb08a76b5ba02862375684
e54b4dfb689a0febc731052235cafb07619968f3
733 F20101211_AAAIVZ bronson_j_Page_099.txt
7545c5fbda6f96d09ed4eef9d8d11289
363524ea323064662ca88cfff4ffef08b6640d3a
53245 F20101211_AAAHTM bronson_j_Page_076.jpg
ca8524933ec6d494a6bf7ca0b537344f
5288cea5eb1df5da9e255031534826b15bd0fc29
57313 F20101211_AAAHSX bronson_j_Page_057.jpg
3faf1d9428576e4f875372f41a714dc3
284e30390a4a42269f7f9dd301aceda8b729b9ae
71804 F20101211_AAAHUA bronson_j_Page_096.jpg
72288771ec95d098ed343cc87464b98f
314e3842a0e9e925d55a9a101bc4cdbe72ea13bb
6815 F20101211_AAAJCJ bronson_j_Page_041thm.jpg
248592c164429b97826cee4525f5aa16
962aab2a4fd1f925ec817b5028984c2c2d8cecec
7001 F20101211_AAAJBV bronson_j_Page_033thm.jpg
896cb6bd2ec431025666061ef41bf67c
9d649355e6093299ab0d0c5042011c92e9aeb00c
1374 F20101211_AAAIXD bronson_j_Page_133.txt
434fdecaa95bb76a0ab5df550e44b95f
de11eed02be4aff012fe0639ff851ab2190f7c5b
2815 F20101211_AAAIWP bronson_j_Page_118.txt
7fe020cd1c0e1a17576f1fa04e9c53ec
117b828024d0f212a3fe85cb1b64dcc1734849cf
54389 F20101211_AAAHTN bronson_j_Page_077.jpg
e62324530a9431cf394920354fb01a18
aaca5a22f8ac07b296c777c22595af4037f2e0e7
56813 F20101211_AAAHSY bronson_j_Page_058.jpg
f586c19727fd231073b1f5f0063bb30f
553c9986f02c01b476e36d1900bc588734254c43
54397 F20101211_AAAHUB bronson_j_Page_099.jpg
483f6e1248aaa2f8c1e290dcd09a95ec
de9d5ccf7707ba137b9cb564028e88c743bd5f0f
26612 F20101211_AAAJCK bronson_j_Page_042.QC.jpg
8e24ab6d6abcd183552dba4d797d1e60
af1f79b6acf9a359a86c1ee511f43178180fe1e3
25847 F20101211_AAAJBW bronson_j_Page_034.QC.jpg
ce3f3e4127d0548ad91f778a168d119d
6e31eb9493abf83695191f2f7b7d20e3990eeef0
1316 F20101211_AAAIXE bronson_j_Page_134.txt
fa5c33cda7b168f364ed1a551448a7be
c356382a85a3bd14d9904ad2ef3bace455b31948
F20101211_AAAIWQ bronson_j_Page_119.txt
9cc43e1c4f2bc3e8ae4b49c5266e034e
0aa769e9cebbb76f25db17ed261098e735cec874
70490 F20101211_AAAHTO bronson_j_Page_078.jpg
bde6eb7b9aa256225b0dd1194382fc53
aa18a3e77c899e3992c9d2c767c97c28fcaa862b
69790 F20101211_AAAHSZ bronson_j_Page_059.jpg
588e3b4b2cf6588b860176b5546a4a41
0a92d2effb5d7a2c6dff959355d5f481a838f0fc
75280 F20101211_AAAHUC bronson_j_Page_100.jpg
24bd9bbf9f9e46e56eabc1aa5ad93fbc
c8b8de0d145a5b38803a69d8715fcd4d7b244e86
7026 F20101211_AAAJCL bronson_j_Page_042thm.jpg
1dfc71bc0b88c3daef807fa9d1a6b02c
b790a62628ea9f1839f4c643b2fe67a9024db846
6974 F20101211_AAAJBX bronson_j_Page_034thm.jpg
8d0e48a54d41a0b35e173655cb04c9a7
f28e32f74a8434210e4d042ac43bd2f7d861d066
2073 F20101211_AAAIXF bronson_j_Page_135.txt
f1c23afa9bc8b6aaee9ddf805ac22c5b
ece9964ef714524b6ad461153b90d8e7c78c26e6
1978 F20101211_AAAIWR bronson_j_Page_120.txt
a81c043da6a5c36d611cdaa6c4034f4d
95c970e81fa14c9d9b7bea67474ee7d35b067b73
40849 F20101211_AAAHTP bronson_j_Page_079.jpg
60e89de6311bd9c1ad32048a85575a57
f8edb88eaac7fc9baa0c3799f29a0acf7a940fac
70587 F20101211_AAAHUD bronson_j_Page_101.jpg
576adae59e4629d6465c635052e0e6cc
fabc6debd93875991bafc9ed258564709eef3a8c
12992 F20101211_AAAJDA bronson_j_Page_052.QC.jpg
dbb49baa4afe231eab304d6ecce0c0bc
dbeb05cf3a11f11bc7297a1d9705b1a8273fb565
F20101211_AAAJCM bronson_j_Page_043.QC.jpg
ff258cf30b77d252fbb1605195da0c78
246874b34c5a8940d57225bacf00f099eeed0cd7
6808 F20101211_AAAJBY bronson_j_Page_035thm.jpg
12e0928b0de541ab451903dd559f0d7a
39d3870d561491645ceeec6ce0301a4656968b80
617 F20101211_AAAIXG bronson_j_Page_136.txt
da59b4472ca59a8f441d39801d85d105
d3b8fff9e57d5d1827635e19646ad356556d7cf4
840 F20101211_AAAIWS bronson_j_Page_121.txt
f83b2941e44528c6ff2b4da3acb600b6
c27fa4cf704a329e53f4df5cb0337e8187cc42b1
75596 F20101211_AAAHTQ bronson_j_Page_081.jpg
85eddebe4d03ed4ad74c7fbc161ef430
4e96e4c9b04188d31628f3fcadb1641949358e97
78050 F20101211_AAAHUE bronson_j_Page_102.jpg
d0d8cd4cd12d9fd3a8f37ae0b04e0506
50541da6cf016a52f791a0b14751e6829cdd3466
4190 F20101211_AAAJDB bronson_j_Page_052thm.jpg
1574538d0f6fc23edc5cf3f04cbcac8e
c8e7c0df8b6ec00c2308d494e7ada96a31add52c
7172 F20101211_AAAJCN bronson_j_Page_043thm.jpg
b5b8a402fa6b8b50a5f6e5130fcb15b4
f4960d122565c959a508b9ac5cb37c4d4da0b672
22405 F20101211_AAAJBZ bronson_j_Page_036.QC.jpg
a2bbf6be9f053073f854304cc46b6f04
91c4f1995351a20a8ebb4eb1799889f283e3d2ed
1042 F20101211_AAAIXH bronson_j_Page_137.txt
581dd762d37432b597bdbead70710d40
492592c864cbf7fcfaf51b6bc34761073bb58470
64430 F20101211_AAAHUF bronson_j_Page_103.jpg
1e713c78f0b820de8c527ad55cf46d02
f8fe7b74f819312fabf2a6b1ed453546dcbbe392
6826 F20101211_AAAJDC bronson_j_Page_053thm.jpg
72686eefa975c1ae59ab2caaa2127a12
c542fd56dd321c3c1c61f7699a80fbbd5b8243e4
25355 F20101211_AAAJCO bronson_j_Page_044.QC.jpg
4dfb4eccc8d1dfa24f0d6ac7f2de3ef2
4b4180defad22ea7d1f7e55b9e26011c5f134140
2435 F20101211_AAAIXI bronson_j_Page_138.txt
c4a38d7153ea7db1b33ce7d10bda2144
e32100e5163a0bb66e5e9623a2e815ba88cbdad5
2398 F20101211_AAAIWT bronson_j_Page_122.txt
7ac798106385cf8bbf8c63198c9d4957
fddda6dbf70c06de5b9135392b8268a3eb95128e
73668 F20101211_AAAHTR bronson_j_Page_082.jpg
063111feeeff9fe0e22a2b5df2b720cc
75a4b97d97b1673b689798a645012b3bce2d1299
923219 F20101211_AAAIAA bronson_j_Page_072.jp2
29fcbf329d0d117623ae357758484d40
2f94624abaf48d3585e18848c462e87cbdeeeccc
56009 F20101211_AAAHUG bronson_j_Page_104.jpg
813ed7f384ce9642487acddbe00e1bc7
227fec88b4808426f30b19c61b1cc70f2fd4d6e9
17635 F20101211_AAAJDD bronson_j_Page_054.QC.jpg
264fa8e1d45a78b3e49094c6529f2bad
a93e45c46f8508843af03b11f64e2b59370cb0dc
6992 F20101211_AAAJCP bronson_j_Page_044thm.jpg
f973d1c1226c5c2f7c5bdd523d28f349
dc9d0b197830eaf7a00e343616de973a78183acc
F20101211_AAAIXJ bronson_j_Page_139.txt
39988a48739c1b217c1046cdfbb46019
6fe8cb221294d553d38e06785ae6aeaff2faed1e
2306 F20101211_AAAIWU bronson_j_Page_123.txt
568dd0dd9246600c610d10c27c67fc93
9244dac89a1784044f4128b0cbbbf9e7ab72df67
74731 F20101211_AAAHTS bronson_j_Page_085.jpg
30c98b3033898989bd428c76f58c9248
19fbee552b2cac69c731445214b610354a4c468e
774401 F20101211_AAAIAB bronson_j_Page_073.jp2
6823f66d0e6bc45d846fcd8dd2e1164a
ad0afb3254da93f3c1e6f3cc8cf3f0652b255f60
48226 F20101211_AAAHUH bronson_j_Page_106.jpg
94bb36cff0377a9a59d62ba91de178d7
ad35b01ec69f5cf44d4dc7cb85e0d03873882a97
5470 F20101211_AAAJDE bronson_j_Page_054thm.jpg
c82b0253ec6fec52cce44626b292bf7f
3132eeb749d61add27bb2b055ec75802f2bf775d
25202 F20101211_AAAJCQ bronson_j_Page_045.QC.jpg
54e9f412ad22d94f9789b4dc5100b91e
858eb820d3ee8f7de33102f754b0fada682de5d1
1371 F20101211_AAAIXK bronson_j_Page_141.txt
ed46bf1a7387d9e2418155fe2696a962
3470c975dbd9e4d3af564c8bcc9061c3410c2c33
F20101211_AAAIWV bronson_j_Page_124.txt
01e4b789530af487ec83cf894a2b79d9
6f9eb955c20a93469bd834f53526ca3120c5ec75
77042 F20101211_AAAHTT bronson_j_Page_086.jpg
1d5bac69731d4736236c8cabb9c14876
f8be51bac2546f788732b9d0319e984f15d8b6f5
96078 F20101211_AAAIAC bronson_j_Page_075.jp2
14831a025413fb656434a2b5ad9ad8d7
e4c24b1f9a61d7a18958d1ce48ec93f2a37e6513
35362 F20101211_AAAHUI bronson_j_Page_107.jpg
83d3da9965c194b5953cd7e5a163cba3
bbd2c45c0413eae744a380aa2f34a0fa7e7a0eb0
16350 F20101211_AAAJDF bronson_j_Page_055.QC.jpg
177636edbf3ec9a493cda17a943f1a61
f68b7b263f895f6b3fb832c54e781bc12f3eaa80
6873 F20101211_AAAJCR bronson_j_Page_046thm.jpg
3eaaaa1bb18a0ca8c2794567fc5c3eae
8733af7830d7a3c116ecbe2d66233ce4df374a6e
2196 F20101211_AAAIXL bronson_j_Page_142.txt
6a805d6a1a5264073a7d96e73ec200db
f793c7d1aebf27dc25c210a5bf2a55656c1449f8
2214 F20101211_AAAIWW bronson_j_Page_125.txt
9922d21f6c9a0acfb6752050f42a0f03
3d871708fceeb1adb93583ab418797f7fbddec08
39885 F20101211_AAAHTU bronson_j_Page_089.jpg
b6fc7c0cb2d1cc7f6123f3e26de37f1d
1156c8535c637a7664f61dc04c690911b5968847
776921 F20101211_AAAIAD bronson_j_Page_076.jp2
97d8b4107cf67db67162aa1ae25072e3
ad48bed2f9c553a68f04620b292da1a72cc96c8f
55094 F20101211_AAAHUJ bronson_j_Page_109.jpg
21da12a3ea308ff9945393e9380bcf46
41d33ae15b1ec9d1f66c4b46c49b23145929cab5
4962 F20101211_AAAJDG bronson_j_Page_055thm.jpg
1f3eb5bab923bfccafaf47f129d5310f
eb3b16a147546e779d5901fdae5d491e0053aff8
19813 F20101211_AAAJCS bronson_j_Page_047.QC.jpg
1c94aaab8c6b6ba6acc5b43d5374b4c6
cf67346597018c24b08b610b39fa6f011074176e
F20101211_AAAIYA bronson_j_Page_162.txt
33dfe80c0fd6f52ecb774bd82d075122
f25874367c8eb06cda715f6ad67e2aa822118cb2
2172 F20101211_AAAIXM bronson_j_Page_144.txt
6a629f16a6aa276abc1e2a0c8b849432
d35097fff4f3f0726c3b7cf1150d350d06a1a176
1623 F20101211_AAAIWX bronson_j_Page_126.txt
09882f14fcc719da918c5637a3f68fb1
63124bed3a7c85ee296a15b793439f3d285d7e3a
71455 F20101211_AAAHTV bronson_j_Page_091.jpg
132883af9c6daa501c92159f85a73fcb
106ce410f27f1b442b03bdfac5de2b25cf4470ba
78425 F20101211_AAAIAE bronson_j_Page_077.jp2
1976a75203b8cd5360194407a7700c68
bc933dd0e98269422a27145279736bb836a1b919
75727 F20101211_AAAHUK bronson_j_Page_111.jpg
922d4d23fc84798155f99d14ef5a6edd
e3d4d781e8862fdb8d190c2d16a2ecb43f6a2966
26567 F20101211_AAAJDH bronson_j_Page_056.QC.jpg
a1923371ce44ceeb4fd6cbafb599db8c
b43f6c4fd0d6bb0dcd6e9b83dbefd99d2fd18a75
5830 F20101211_AAAJCT bronson_j_Page_047thm.jpg
6db8e2494f99a14efb993ec6312e0822
346def9d7d32d8016676c44c68cfb07f2a8bc5dc
2057 F20101211_AAAIYB bronson_j_Page_163.txt
22b097886552cf5c9362e6061bc7fd47
717e8ce129a5106de70d5b113213bc9783eff480
2105 F20101211_AAAIXN bronson_j_Page_146.txt
0e0242c7ad4261aec6ba604802e276ab
0608de4ed2c5490a186367cd7f9ce98b217fb159
2072 F20101211_AAAIWY bronson_j_Page_127.txt
7e3826605925ccae732015c5bdb86e66
a2ae3d13391c3fdcd86c5a0e8af18f86354c1e71
51353 F20101211_AAAHTW bronson_j_Page_092.jpg
dfffb3e6babd3744d1ae7d3e25ba5fae
998fdad1ba276cc7822fe2b00f0f3436e46b4fde
104630 F20101211_AAAIAF bronson_j_Page_078.jp2
d3dd00fd6c21b0df15df2b98878003b3
937e0e051852868987f30287c46a0e621563929e
28445 F20101211_AAAHUL bronson_j_Page_112.jpg
26295b596ba68a6e476aae318519701b
b4c0231fb19dd495a85ae410ab68fe1b7476f6ea
6972 F20101211_AAAJDI bronson_j_Page_056thm.jpg
d90a31d0407bb60adc50b8771c509b64
6bc69d0e837ff5a3dc78e4f2222d57a1bd618a8d
7135 F20101211_AAAJCU bronson_j_Page_048thm.jpg
b18115ab31a0aeeae6ddbfd3e86130c3
0fd229866eef21f391f0a7449a9b516943d14b37
1882 F20101211_AAAIYC bronson_j_Page_164.txt
55c023ffc086d78d368d0b77240505c2
d09411696f49ceda9bd394290a05d907bc37a942
1965 F20101211_AAAIXO bronson_j_Page_147.txt
5b373fe46103949d2bc21dbc34774a3f
66d747d67ad393b259a4ba6ebb414927cb9fd321
1571 F20101211_AAAIWZ bronson_j_Page_129.txt
5c43e932bd23cbbb8f562edf9c4690fe
2db64bf348f17312a81c2c6d30634fdfdd39f45a
74360 F20101211_AAAHTX bronson_j_Page_093.jpg
01298726505700dd4753942e3c6e16b2
7ec95ea71672905d26744eea2ad4a318c0366357
411329 F20101211_AAAIAG bronson_j_Page_079.jp2
6b3c7e0ccc34d9fda9f53989a780a08d
5d2caeadd7cb1a372f55c22a6e9889d7215f4500
66923 F20101211_AAAHVA bronson_j_Page_130.jpg
4e1a913508ad1c2fd2340c98b345c55d
500e3953a3c7ff8f2ac95334e74052073314d3fa
59508 F20101211_AAAHUM bronson_j_Page_114.jpg
6b4449055d68f916efe1fd115d120595
0b53d8ea03a86f7b0ff720b2d7a8e78ade349dcb
18896 F20101211_AAAJDJ bronson_j_Page_057.QC.jpg
f31f8c071b26e06d1d8c944e2f592a70
fa206c0dbb64e4823d5999b988f36edc4046f612
21989 F20101211_AAAJCV bronson_j_Page_049.QC.jpg
cec9f0ce3e002a1f68f8907362ce558e
8f2b5145bc31b7ab6faa812405320ca0a03ffbf9
1944 F20101211_AAAIYD bronson_j_Page_165.txt
2c1651323646b6c6f22e7e4ad0bc3575
6182aba928ae96243f69fc0f527b54b62444c1ef
1702 F20101211_AAAIXP bronson_j_Page_148.txt
0bdb2dd596960cc19417614bab7f17bb
655e2afd2e1f9bd0bcb79c87f2c2a160ba5da509
66616 F20101211_AAAHTY bronson_j_Page_094.jpg
9c2bbf3ab359e51b1693f8345a8b377e
9d6555e3172b472859ee14658f3f52b69dab1d6f
29238 F20101211_AAAIAH bronson_j_Page_080.jp2
ccad366ecbee8f7379ab00917272e186
73ce3826b4f858faa2d54f25220ea6c31d1056e2
35006 F20101211_AAAHVB bronson_j_Page_132.jpg
f1143c4ffe56acc1a454de3d95cf11c2
0d53515388b35ad24ebdc999dedc27bcd46866ab
63347 F20101211_AAAHUN bronson_j_Page_116.jpg
00569bef557fad594262d430ea747b6f
c01be6e61f133b4215e093c1b8642955a0c3a196
5737 F20101211_AAAJDK bronson_j_Page_057thm.jpg
3213a7ba62067a492aebb953f8605a4b
c8ff15a00cb82325132f427c40ceeaf7dfd763f5
6866 F20101211_AAAJCW bronson_j_Page_049thm.jpg
73a512863b1f0b8f5e3ef4b8d433fa4d
ce488f20185291a4772603c32385451e4ba5f3fe
1559 F20101211_AAAIYE bronson_j_Page_166.txt
dce8ecaf3eac18c21e14e29d0437060e
77b71357ed383376348cc8eea8b2b743c421222f
794 F20101211_AAAIXQ bronson_j_Page_150.txt
7df8bb6475aaebd8d5cac362df45be3f
fb1c832fae516a9261188297957154efe086a16e
39847 F20101211_AAAHTZ bronson_j_Page_095.jpg
ff57044a8694bd6ea1dd6e8b793a3828
55e474bd4fd7a856e87a417cc8fc727f7677eeed
985947 F20101211_AAAIAI bronson_j_Page_082.jp2
7aed062d1f2df1cd284f0b48b311bb97
de5f1bf5d529065728fc1159e93f6367f28bec31
75697 F20101211_AAAHVC bronson_j_Page_133.jpg
be0dd092ca00f89b26fbac871430fbc4
487314a475241fee2cae9cebff8f2a39df89758a
77630 F20101211_AAAHUO bronson_j_Page_117.jpg
277b53c450edeaf77e6628aadb0394f1
4d4ca6dad13ad83ce45b03d06628fdf060497fc3
18023 F20101211_AAAJDL bronson_j_Page_058.QC.jpg
e3a4872b3a27799facddfcf190fc8883
ec97a06ba669e8479c18e3202e4e65d8b1485794
26200 F20101211_AAAJCX bronson_j_Page_050.QC.jpg
1899b1a4b0ea9859aaadb30d478719ab
5977bc2ede1b1ecdbda4fa3e35237a054703b46f
2200 F20101211_AAAIYF bronson_j_Page_167.txt
29afefba7c9f94cc8bcbcc48ed70113d
baaa74f3cf2b3081e13cad92be83e0400dd414f7
453 F20101211_AAAIXR bronson_j_Page_151.txt
ec085f3ca85bed01bcc392967a644500
ee6db63af8d7d25e138667d0eaac246beb554832
876660 F20101211_AAAIAJ bronson_j_Page_083.jp2
b4fa2768bf96c57a24100ce03a8e1b08
b4f094ef4e1ab6affa8357e03069ecb19f526f8f
42837 F20101211_AAAHVD bronson_j_Page_134.jpg
f9d939c407372c44f993fdd83b872313
dbee2949fcfc4a01fbe72e4d22cda833bdfe6977
100420 F20101211_AAAHUP bronson_j_Page_118.jpg
7675b979c9786402ac5eec2967eff378
5ed1c76384b4bd5efaad4161ef2ebdb77295aec9
21756 F20101211_AAAJEA bronson_j_Page_067.QC.jpg
0d421d66072facd05b4eaff824cd70a4
2ab9eadaabf23568858309769f07ee59a424761e
5473 F20101211_AAAJDM bronson_j_Page_058thm.jpg
97ca59469ff4acf864a262b12e4284c3
a781a5594542c4befdc3111321b8aeb727ca2bc5
18230 F20101211_AAAJCY bronson_j_Page_051.QC.jpg
567e3d18f11be6503eb4449cc4732fb1
fb7fffa59c27d956b0f13dba5c591837451657b3
1734 F20101211_AAAIYG bronson_j_Page_169.txt
1fd154d613be443ef4c5beca9cf5f2f7
8041b514a50c2a05255ba382dbeba1d9ca930102
F20101211_AAAIXS bronson_j_Page_152.txt
365865c10dec2b3f74a1d005c2c08c34
5c251476e0234aeddc8c3cc372830755a206a3ee
118169 F20101211_AAAIAK bronson_j_Page_084.jp2
e6e53cc755663536c367fdebcbc49153
15332cad88743deff2dc3c6402aef94757ac4133
46977 F20101211_AAAHVE bronson_j_Page_136.jpg
f6168b73cdd4255fed0196e84a80ed7d
46577a15e4ca9dd758103bc42c9c81c80ef91d4d
80150 F20101211_AAAHUQ bronson_j_Page_119.jpg
7afe9a4c6e44fac49c826e720d7fab75
51d2519ac97af78f0b38a95a8b1759e9663ad361
6629 F20101211_AAAJEB bronson_j_Page_068thm.jpg
c957565708b6f11910f3992bf4c6c02a
24b011da68cb22d670a526c1dfdd3c8fbbf4307a
21499 F20101211_AAAJDN bronson_j_Page_059.QC.jpg
3b77358c02aedd74053dcac84220b37e
0cb123b02e28ac4404d3ddf79b34044211e74aad
5786 F20101211_AAAJCZ bronson_j_Page_051thm.jpg
65e03481ce2e4f1f4abccd2b1ecbf7ab
f00c42e608f324dfd722a06928466aa36749caaf
1873 F20101211_AAAIYH bronson_j_Page_170.txt
afda756a54cdc10655fdabd835ad69cb
da3b578173153cc2dc54b771b309911183c4a144
1704 F20101211_AAAIXT bronson_j_Page_154.txt
0b06e26c325ac1489f7ce30a7606d642
56270dd0550884613cf1263976fb7557dc1c5c0e
116922 F20101211_AAAIAL bronson_j_Page_086.jp2
0009759187959cd349001e9c2f4e9c45
26296e23661bfeb5353c300cc3a6835eeeca3a5d
61339 F20101211_AAAHVF bronson_j_Page_137.jpg
5c412414a7089bd1d4721c8b11627d58
46d401855d2415fc6ba7ba020aba896a85f518b0
72394 F20101211_AAAHUR bronson_j_Page_120.jpg
d9e9b84788dd210912fdcb5f13a2f076
2b2f8c8b93df4735b06acd4b7dde1cf9a58dd08b
15423 F20101211_AAAJEC bronson_j_Page_069.QC.jpg
cf3718424719ac66780e19a2d47aff8b
e5f943394919c37b0d84020f5ae9c4c23c4adac8
6574 F20101211_AAAJDO bronson_j_Page_059thm.jpg
c88753545838c41e3e06f34bf503623d
dce80e0d2228d33bad1666c9a88841b29ba8a9b8
1185 F20101211_AAAIYI bronson_j_Page_173.txt
6afbd08db90cbd20257a5098397bb5fc
1c7f1a871c23b60653df5b28076d29cfeb7bf79a
107404 F20101211_AAAIAM bronson_j_Page_091.jp2
df297f9e3d8ee1819b9cb31b99c9bf9f
d7c0a2179aaad0845e0a6fc5ff1f717371588d90
82913 F20101211_AAAHVG bronson_j_Page_138.jpg
4036abe38f22ec425fc79bdaf3109d38
c08afa98143ecb9705f52e7d322a747ccc5130b7
546520 F20101211_AAAIBA bronson_j_Page_107.jp2
4c5d0fe61f19972b3947ae1c57b171d9
82f071de0c743aafd1b2034dc6c1f1061c073027
4750 F20101211_AAAJED bronson_j_Page_069thm.jpg
6fdf1a155917a30e1dd5e7051c59b631
5abed1451271b7a65e14737e8e6d44421c4c8007
22237 F20101211_AAAJDP bronson_j_Page_060.QC.jpg
a675be4e9ae6021832d990cd20090b11
455f90cda8c46a8b5e2e82ea824180626f018567
1524 F20101211_AAAIYJ bronson_j_Page_174.txt
6fbf3af9bab541f64779457643b8acf7
526c065c5e4fe6bb520fd365f72d62a1cd8083bf
1716 F20101211_AAAIXU bronson_j_Page_155.txt
ad4e57ab93674a9c54500cee6e338bf1
5390fd60316b5aacde0a20d56b487ad0d6cfc28b
911998 F20101211_AAAIAN bronson_j_Page_092.jp2
5dce4793bd92f6b0eeb76cdd6d20926b
4ffbc045e309641654c6303831f08622f27f3c67
76742 F20101211_AAAHVH bronson_j_Page_139.jpg
81d01cc78e1c6c2ea5217be74ec478ed
c14b75ae7ec1419d9d90d3d72fbc944123011bb7
48762 F20101211_AAAHUS bronson_j_Page_121.jpg
d6cdb974a1887d5aefc45db593f1dc31
fe627f61e3adaa5f1308cf15aa246b52992e8ec8
1051881 F20101211_AAAIBB bronson_j_Page_109.jp2
dce8f0bc16a7537da37a46d86716d4cc
84cb654360fd626617cc1df9d57a9fa29d90162a
19844 F20101211_AAAJEE bronson_j_Page_070.QC.jpg
8fccbd58356b3940ea9be736dc150a6e
f23fac9ee66739fd2223236ea80d5014a6d319a9
6020 F20101211_AAAJDQ bronson_j_Page_060thm.jpg
bb1ca9343fbaa225590000afcff1d6bb
7e639eecffd60be65c23b0e95e0a3687bb1aa8c1
F20101211_AAAIYK bronson_j_Page_175.txt
541eb9174b07a9ce521c093906ae04e2
bfa84772836ceb2d5730ce119cb7757c5a23eb16
1631 F20101211_AAAIXV bronson_j_Page_156.txt
9b4262937bf9f16f05342225bdd0afec
6ff3a20615d3f47523b49e6ebfcce41d382e9a66
111472 F20101211_AAAIAO bronson_j_Page_093.jp2
07432a47247611fbf01e484324f235b4
c29aba6540ae6eba10d2324e5587c13b72a5589f
67250 F20101211_AAAHVI bronson_j_Page_140.jpg
7e6db0c67c96c2d2ebf96fa8992e4c74
b4a425674a6832c12a865e82291ad8f1ffdee58f
61420 F20101211_AAAHUT bronson_j_Page_122.jpg
951db8866390ef4b988888ea81513e68
b9994d7f4e6029e42cf3b86b97f9cfde068b5099
119299 F20101211_AAAIBC bronson_j_Page_110.jp2
98796fc057f7e2845ee067f2580820ef
ac5690550fd195dceeb057a719221ae483703331
20632 F20101211_AAAJEF bronson_j_Page_071.QC.jpg
15fd40931d2d33dfa90c165538aac2d8
563a5d7155a256b4ac2a1b4629ab9a5e70da7fcf
23967 F20101211_AAAJDR bronson_j_Page_061.QC.jpg
885e6f20251980bb52a245511cbf94ce
11a865c6873786201ccb99ce7aa60110621bcfa7
1815 F20101211_AAAIYL bronson_j_Page_176.txt
6ca5c88ad06ac6248e8694d0b1b24793
5292bd39b6a243aeecde70084c58107d315cd89c
1515 F20101211_AAAIXW bronson_j_Page_157.txt
971be9ae1b8ef638544d09084186a17b
56a5d6cfd946b1606406a95f2bedd495994a2b4a
919430 F20101211_AAAIAP bronson_j_Page_094.jp2
62bae1f03e01ff42ac1d248eb230fc4d
b46590e03f796e7187adebd2026375e5d5fb537c
70337 F20101211_AAAHVJ bronson_j_Page_141.jpg
76078141d4cf65e7cf235602979a4236
3f8603a92795ad7954ef60d2acf035c80c17e287
80410 F20101211_AAAHUU bronson_j_Page_123.jpg
6eb2e64abf9150a8f05c82b7c522d6f6
dacb81811ea9edf95e0bad5d954f09a1e891b04b
115390 F20101211_AAAIBD bronson_j_Page_111.jp2
180a205281f76b208bd3e7eb7580032f
528750ee8bd120a4f226c4d26a5f2876a5a1c56f
20180 F20101211_AAAJEG bronson_j_Page_072.QC.jpg
4ca2d167c71b7c558bb7543fdae0cb73
fa7dd9ed7cc94b16deacd7d49c52d5a182ac21a0
7029 F20101211_AAAJDS bronson_j_Page_061thm.jpg
64aaf5976050a126794e8f2678a8954b
013352b3a4d6222beb38dba7261d7444b33368f0
449 F20101211_AAAIZA bronson_j_Page_194.txt
3d7392528292f04c4c663c712f16b0f5
d12664031c42d619d48587878b6d4c2df9839eb7
2387 F20101211_AAAIYM bronson_j_Page_177.txt
0ba19acfd8c5f83419becf3299efe12e
1c5c37fea329c39b3ca5bdc2e25c50dd2b7a83b7
1545 F20101211_AAAIXX bronson_j_Page_158.txt
655bb37f7432480b7a92aaf36ab393e8
6270a5776b2ba8c611da71bf39f07cf8821947d9
F20101211_AAAIAQ bronson_j_Page_096.jp2
daea8b203eea2651ef2b9efe8155868f
3d65e681ee1548de33706b249d978032ee984ce9
78573 F20101211_AAAHVK bronson_j_Page_142.jpg
701e31c33565d9a40a1e3e3f1500b7a5
15de32d03a25d090e6a4a39e97e44066884875b2
49184 F20101211_AAAHUV bronson_j_Page_124.jpg
35cfc20952f64a57a759149e5cc14b92
40f8acc92bdc5fe061f41919d14ae04799b30ada
76852 F20101211_AAAIBE bronson_j_Page_113.jp2
c1df21800df291f7b94ff3b6b530da1b
22a54975bb032b5011f64273b276e6a4aafcfc2d
5898 F20101211_AAAJEH bronson_j_Page_072thm.jpg
bede27ea4b6694ee0bb5d456d0824905
c695fc239e503958b3bed054913fb39a98357349
6093 F20101211_AAAJDT bronson_j_Page_062thm.jpg
4fa2a1cc9b0b7f2baa24d9e06f227ca6
826c6811960fd45ae156f37d63da4f8de2dbf3d2
659 F20101211_AAAIZB bronson_j_Page_195.txt
b9ba75193ee1a8350573a188c1c72a08
57c5882e05086598bd6af7a5ff01dbe0bf8401bc
988 F20101211_AAAIYN bronson_j_Page_178.txt
b054e701c534550ff039ce3a6d61e219
64e2b6694593f4aa3fbf098adf5d62703a0b1f3e
1318 F20101211_AAAIXY bronson_j_Page_159.txt
658e55238b19848ef4c11afcddd1d469
b27994906a9cf5c50f67a3544d409d6d3ff51da1
102591 F20101211_AAAIAR bronson_j_Page_097.jp2
09bbd7f7df46c622bfb0e7f64154a3e4
b47c21f5fc0940869fddb894989a4eb4a96f87cc
47808 F20101211_AAAHVL bronson_j_Page_143.jpg
44737369716accf46db9395bb1d98674
f54b021996e816b2d870d809bac32559218920bb
77631 F20101211_AAAHUW bronson_j_Page_125.jpg
b96819e1827768760e5a494d83c39041
420f68cced8f0a0f47173a9285df32a518bf26e2
884431 F20101211_AAAIBF bronson_j_Page_114.jp2
f97363a96319f6f02791edebc5712c85
12cc2d5675e1d2c43ea900f50b7852a7c2d61085
5413 F20101211_AAAJEI bronson_j_Page_073thm.jpg
0e668af422d2d160f99a7123cb61e3b5
cfb29ef805a2ec5508f5a88078cc36495add9b9e
4931 F20101211_AAAJDU bronson_j_Page_063thm.jpg
57eacf27b6a52fde63c6a25fb18d7113
9b0d782d5095bb1c348d83b82489479f72b14bea
486 F20101211_AAAIZC bronson_j_Page_197.txt
f84cea90ec924f24f0b480d1c0ff2e4d
c36a888c9d68773d51b2adcb7d3d3cf376684e51
2155 F20101211_AAAIYO bronson_j_Page_179.txt
aea1be02b508485a99c3936f5c2b29c3
53de7f554beb5bc493d8fff3bf077de9e6cb96cc
1533 F20101211_AAAIXZ bronson_j_Page_161.txt
5db1f529dfe64616bdc3679add2f7cae
2e91f90bffe4bc7029842f40562a42b4c72ea34e
61738 F20101211_AAAHWA bronson_j_Page_160.jpg
c69664ff2c154ad1a635b37be8cfb4fa
14d00e92a98d2cfffeb03bfc2456c75ec3a7c640
1051966 F20101211_AAAIAS bronson_j_Page_098.jp2
4551c1667b63c82183997be4e4efe1ec
5ad2595fc6f21722478ebc33a2d6c0b90e9984b7
77613 F20101211_AAAHVM bronson_j_Page_144.jpg
6438a1013fc55d064bb5628dc28a98d0
6e83a585bb189275eaca23173e7db51c0836ceb7
72635 F20101211_AAAHUX bronson_j_Page_127.jpg
e8bfa894c48e7981e7a43714be5d73e0
8942b7bdedfa64f08f071863961b527ae0339434
78571 F20101211_AAAIBG bronson_j_Page_115.jp2
7be72235dd83efafb9c4bfabdd8ff459
37099a762061a7e444b22a1dab0f6ac4d4f0f8a2
10561 F20101211_AAAJEJ bronson_j_Page_074.QC.jpg
a2c18c2cde5c80f89955fc8d56300391
5993312a1da69c4c6cce5d42a4c29652992eba42
25284 F20101211_AAAJDV bronson_j_Page_064.QC.jpg
e9355bdbf0f73c5ef60f81816927f78a
2b6e8093e047c2966b82496a1517df38c8154318
465 F20101211_AAAIZD bronson_j_Page_198.txt
98cb67f32dc9c9e2b0c638653b52954b
aa367fe3da19609c6e2ae780eb860e55e1204637
1003 F20101211_AAAIYP bronson_j_Page_180.txt
50f72bf0b0a8ede9133183bb0d501c77
96fc332ae9296da2bc9e04293c65f12361ce8101
65842 F20101211_AAAHWB bronson_j_Page_161.jpg
6bbc9b696620deb46284cf8350ad9f73
d54984a7cfe22393cd9892370f7a39627952e04e
1045390 F20101211_AAAIAT bronson_j_Page_099.jp2
9d312ba9b43dbc4c16fe6b5c0e78895c
091c49bedf79c16a02709cd21cd683c07c8b411d
76808 F20101211_AAAHVN bronson_j_Page_145.jpg
6034da42af955fbc21587c64a33be5b5
b51620b94c4f509ec04a038f45b5e894437e4181
74185 F20101211_AAAHUY bronson_j_Page_128.jpg
7813aef262f2c43f9701c632f23c154b
7a820fa13344f2ce27b28ec0f671116b7aa61724
90876 F20101211_AAAIBH bronson_j_Page_116.jp2
6cedafe430881a097ef410fad71528ab
77b4e5d5986bccb2960df79d54a0b9ef50565a1a
3792 F20101211_AAAJEK bronson_j_Page_074thm.jpg
7517cdc51c4343fc4c7268603a687245
37569408202d201e69ef894800669df8397cad92
6908 F20101211_AAAJDW bronson_j_Page_064thm.jpg
19abfa15a2b4e9bc8ee83b2bd6964a09
a9025e190924a8672cc29f1946cb572784026853
325 F20101211_AAAIZE bronson_j_Page_199.txt
4816bfe5019efe63aa0264f23b11b561
cd871b680204eb899a2144230d998c5875c6538e
F20101211_AAAIYQ bronson_j_Page_181.txt
684d6f429d5f88265d2712d66734fccc
639b4975b71e285c6be1cdbe3011fc5ef121eb85
51799 F20101211_AAAHWC bronson_j_Page_162.jpg
457b72f2f7b2b643154884d8205de8d1
470b11f6e70f8d55159c6f60f81847944e7759ce
114557 F20101211_AAAIAU bronson_j_Page_100.jp2
0ae386cf88648752acfb7724755a1f59
434d5b23c71cb4b78c404f98937a8583881a7c83
82496 F20101211_AAAHVO bronson_j_Page_146.jpg
cc2ae18610c582d35143a93d7f684e9a
8a2f35f1f7793d164f81e24c69d889139e887b56
48865 F20101211_AAAHUZ bronson_j_Page_129.jpg
8dfc25a4025ec09edfa5ec6c03bbb772
d5b29318d5489c8274aa941c75b022e0f9db9b5f
116015 F20101211_AAAIBI bronson_j_Page_117.jp2
827b9099304a0fd49bc880ed43d85dfc
0a74772719b19f0d614daedfe7552990cb72b747
26230 F20101211_AAAJFA bronson_j_Page_084.QC.jpg
56dfcb9f3c1ae1cec2b40b7d5582237d
44478df4ccc2ff81b0df95d33c33e1b0343053e7
21450 F20101211_AAAJEL bronson_j_Page_075.QC.jpg
8d24c3bb7024cac5eea7303c4ec21a9c
a2b1c297245c0d28ed1f28514499b5fd7116756d
6395 F20101211_AAAJDX bronson_j_Page_065thm.jpg
8734fa4126f48ad84e038e2fde9eaace
7083d3913b1979a9cfd1be39be8b5387286833d9
949 F20101211_AAAIZF bronson_j_Page_200.txt
726ff282386ff8c1ff7e470b91564d99
d0d97de6e32094052c77ddc36912b96798365f16
2195 F20101211_AAAIYR bronson_j_Page_183.txt
6a03e3f2353bd6ef35ea7ae5889a4377
fbc0944eab4554c03543daed82794fb0852026d3
73425 F20101211_AAAHWD bronson_j_Page_163.jpg
3e148b58274f804ac0c174c38ad0a9a8
0c67a5cebc91cc1513010831374d7112dd185f08
1051945 F20101211_AAAIAV bronson_j_Page_101.jp2
00670ec5ff4901e1bd68891d94b1e15a
7f175c38ba504822cec545ac77a1868f16b64d28
48078 F20101211_AAAHVP bronson_j_Page_147.jpg
59143843e9a9167e8d276fd8534f3f7d
0e3727652553259a5d9e119bacbba88493632c85
1051983 F20101211_AAAIBJ bronson_j_Page_118.jp2
7aa053246d64a4390d6ca8a9f563233b
9c8ef03448b37f82aac9b07679d9326c49285809
6015 F20101211_AAAJEM bronson_j_Page_075thm.jpg
ab94169375039687f54a78a6615a8c72
04272c83ca224f89b4fb01b7b851e65dbdfcd165
18603 F20101211_AAAJDY bronson_j_Page_066.QC.jpg
135050e8ae2efce97177fc5dce879b6c
546902fdabe705aac77ca0dbba4dafc917364510
1037 F20101211_AAAIZG bronson_j_Page_201.txt
b4d00b28d80cf7148e01acd269ee2632
23180760fea856953e84b5b502267fc5d04c8c79
2232 F20101211_AAAIYS bronson_j_Page_184.txt
ca36c5779cf0461735dbfa085521b66a
c5bf27f0623f05d49cdf1358000d78c75b079e30
80172 F20101211_AAAHWE bronson_j_Page_164.jpg
d18ac9990ed6261d8d206923da2ef990
0ace32af5157e91fbca393950e7ceb2530b41159
F20101211_AAAIAW bronson_j_Page_102.jp2
b0f35d67c7814076dfb21f90933f84c2
015106dca085a4a3211961ef13c12897aba7c0c5
67900 F20101211_AAAHVQ bronson_j_Page_148.jpg
5a1736a6deca544cd249224cbd46c2a4
cf95ec4d59e094b4b5b79eebd2b3258f46d37d9a
121466 F20101211_AAAIBK bronson_j_Page_119.jp2
a170085ab10982069dbfe4721ef6c829
543f353b062d049a8a508a27390f4f53c2cf7636
7020 F20101211_AAAJFB bronson_j_Page_084thm.jpg
c00c0393115b679db57c257e64a6ced4
c876e3eb0bfc2dd40f49438b9f3e17412cbe0ff7
17037 F20101211_AAAJEN bronson_j_Page_076.QC.jpg
5413f98a0ae23d7df031877dcb6477f5
5ca5040e599c66e5ac270e3d808e872cb1c38884
5405 F20101211_AAAJDZ bronson_j_Page_066thm.jpg
f47264fad796c86f596794baf216318b
fb864d6996387887778f599b311dc9142e08aa12
1091 F20101211_AAAIZH bronson_j_Page_202.txt
3012bbf332e16ea7e2987c9c28197160
0c074ebfde169cee7a19f1bb36639243cb79847e
2158 F20101211_AAAIYT bronson_j_Page_185.txt
635534a10a4f8c6d9a8ce526665bbaf5
9474aa904baafc256e79f8fc38acb55256b4a639
68437 F20101211_AAAHWF bronson_j_Page_166.jpg
74d97184cbb707c50c4cb81d2788d6ca
ed773e3d0e257dc27938352b5426dd33587b0328
910011 F20101211_AAAIAX bronson_j_Page_103.jp2
4cb84582384e32979580af36633580ec
50b4f3f26ad3944f846b6c89a3b36258820d36f2
79718 F20101211_AAAHVR bronson_j_Page_149.jpg
4b1e680c99bb2bc58aba76b943c17c7c
a85894bc63077e868ffec09120605774e9720229
723556 F20101211_AAAIBL bronson_j_Page_121.jp2
9505493d49c75fe278cc252fcd9d844f
d3e5d4c575e181031c00e3e0dcfd62fcd543e4da
22884 F20101211_AAAJFC bronson_j_Page_085.QC.jpg
c429b80bde6842fb8f00f319fb49e5fc
97904d66140fc7fd6ca0240585f5e2bcf2e28bab
5390 F20101211_AAAJEO bronson_j_Page_076thm.jpg
8f6370ad042fda6041b16bfc3de1ec2c
fd7651a70977ec0e7533f15ccbadcbdca9f2264d
1040 F20101211_AAAIZI bronson_j_Page_203.txt
fb8e2458d377c29fdb0913c0dbdcc30f
34eaf9ae072c8189476273e7b78fb087a3de0ed1
2026 F20101211_AAAIYU bronson_j_Page_186.txt
f749f2a56284e21c5f76094d7d4ed43d
bef92160b0f6f2447cf57fda6f411aed57a72820
78555 F20101211_AAAHWG bronson_j_Page_167.jpg
ba7d45b2900ab4f8cf58052cfdb10037
d028b1dda6865a958cfb92258909dd4a09e49bb1
788406 F20101211_AAAIAY bronson_j_Page_104.jp2
3eab2c93b7ea1b7b0be021db6b8ff49b
048f4943f7dcc4fbbf34992a5c004069c5e3867f
52494 F20101211_AAAHVS bronson_j_Page_151.jpg
341afc6e16b2977e8d78d05debe458be
6c32e58b5f66e34dd3efb9d154520fb87a1dcd4d
122885 F20101211_AAAICA bronson_j_Page_138.jp2
dfc76c20b152dd33e1b72210c1e35014
7157523d9365b7642fe1f657808e0715bb0669e5
86064 F20101211_AAAIBM bronson_j_Page_122.jp2
6a7b9240cf1c18df7deca4a839f2d012
875f8dd62f4d1ce77890aa72db2ea049f9484692
6306 F20101211_AAAJFD bronson_j_Page_085thm.jpg
ffd33fb794fe7ae7bb5a423dddf801b3
96329d3233c229105e319a17ff45ab387c0c8f70
18192 F20101211_AAAJEP bronson_j_Page_077.QC.jpg
a004a29c218c72b206916ffb9c876c3b
1f3b587294def4680bd44556f9bda09f69cebe01
1111 F20101211_AAAIZJ bronson_j_Page_206.txt
999d3ec567f6b61317854de0a37020a5
d2db30b4bfce113cac0cf0e277205ab94d456f10
50698 F20101211_AAAHWH bronson_j_Page_168.jpg
f3800c24419de7bb69407edf2a4691df
9a5202fecbbf55ea2d9f737525b621c913a43004
116432 F20101211_AAAIAZ bronson_j_Page_105.jp2
93a96386f27fcd0277c07d8c3037ef24
2115f719815f148c1099483b4a06f6c0efac1572
1051977 F20101211_AAAICB bronson_j_Page_140.jp2
dc337072894b4a999953aa05f6c649d7
943eb8509efae8cbc9a761b730bd04e99a5c6351
121826 F20101211_AAAIBN bronson_j_Page_123.jp2
c9af7cf9b0d230798c4c81565d1420f1
7498ef69b646a7d214aa4dc8ab338ce5cb4aad90
25814 F20101211_AAAJFE bronson_j_Page_086.QC.jpg
e8c17157ba413ab929edf2a7a49aa4bb
1ba88b504f68d57cc93e38c702ccb7868a42a05e
5371 F20101211_AAAJEQ bronson_j_Page_077thm.jpg
21f987c2604a2cd5b44f25945363f615
56b6c1bcf7bf941ff63d37d568e481da90415aaf
587 F20101211_AAAIZK bronson_j_Page_207.txt
ff3f84bd832fa10ccf9e1eadc559a9ba
2667e9c52c0cff8750079815a713832ec4b785b6
720 F20101211_AAAIYV bronson_j_Page_187.txt
85c47fd76477c93bd961d7a1ef010a13
296cf7e8623595e0a312752db6461589b3841a75
70832 F20101211_AAAHWI bronson_j_Page_169.jpg
3a8292a87f0a02cdd8ebdb926f0f855d
e29890f979767d073b1139659b8fa82af3aee59a
79309 F20101211_AAAHVT bronson_j_Page_152.jpg
5fecd4d47c8413b2551201c83672082d
8f80a634986a3dd552a197eed249272bc62950c4
1039780 F20101211_AAAICC bronson_j_Page_141.jp2
0bc780525c17ae921b3146320f932008
09eea3b1956f9eb59e3010e9f5fce883b4fa4462
1051750 F20101211_AAAIBO bronson_j_Page_124.jp2
f566c8475a488ebcf204b5c17a35fd1e
1531f162268f0b7aad2a3337456b550c37f6c75e
7067 F20101211_AAAJFF bronson_j_Page_086thm.jpg
fb4a7e05458e218e058ddea5fa267126
3da675e48f5ee0f56abcbc2c3a02a4e51b79f7c4
F20101211_AAAJER bronson_j_Page_078thm.jpg
0a977070be491c037ceb438dda22a18e
9614a3feca4d678f58c845687e937da30c903228
2610 F20101211_AAAIZL bronson_j_Page_208.txt
cc13856e5569ffb09385092ded146f8e
91e8a2d6f2f207b7cccc2494902b5813e6ba18c9
816 F20101211_AAAIYW bronson_j_Page_188.txt
9a48153ea69bf33c2ec8aa3c71690ea9
c83bae43c1cb537ae41d38e03f4bb901f8d16148
63333 F20101211_AAAHWJ bronson_j_Page_170.jpg
c9275dc36aa2cb8973a93b3661e53655
f3a9c0aa465a772100725b8ecaad08b736f71bed
56142 F20101211_AAAHVU bronson_j_Page_153.jpg
8d3584f4a9a599862e37bc88d90eb9ee
a51774e291753115d93ccd0639aa64c88834121c
118807 F20101211_AAAICD bronson_j_Page_142.jp2
ea111a0fc37950f4604f013f3df61b31
43da91d8fae1400f7cb369a2b0d87d123d687028
978708 F20101211_AAAIBP bronson_j_Page_126.jp2
f38982bb197b2de6073fd4a8dd84e26f
932b0d998697a464b82626e2cafa206ea765aaba
25027 F20101211_AAAJFG bronson_j_Page_087.QC.jpg
cc70cdf5c5750ca0a38211e13eda6aba
42094ee1180d8b01d7b15a42a32f94664bb7964b
13503 F20101211_AAAJES bronson_j_Page_079.QC.jpg
b0ad4365b588d3d851d939f4e47508df
d576bb7aaf1b16ab6eb79644f7c326ecf2ffe8fc
2588 F20101211_AAAIZM bronson_j_Page_211.txt
56d47b5ffaad8d0d8ca07dff3d96a926
fd173da3f09dfb6f24c30f93103c19310c7d4676
705 F20101211_AAAIYX bronson_j_Page_189.txt
08724338ca65f08ec4e71a80975b27eb
16a5403a17971a9c148670a0965acef190aa1ff2
56098 F20101211_AAAHWK bronson_j_Page_171.jpg
3dc9713ad3d93c3faae50486b0b54959
db523f9e03484327644fbccde0e4d16698a51a93
54222 F20101211_AAAHVV bronson_j_Page_155.jpg
bedb7ebe0f58a7d6ade6db0e5a9d95ba
eb0e77fb2c0f48a2e4736db67d749cc8c7953442
115930 F20101211_AAAICE bronson_j_Page_144.jp2
1d53740481d426314bdbabf80e41af8d
aaf0da8f15fa135394f39124a895d61b4d90235a
1020101 F20101211_AAAIBQ bronson_j_Page_127.jp2
3712e4f82f69bf74012aaf7ce912cdc1
f435ab8bea320640c95665737d7c2c8a95d84ca3
6813 F20101211_AAAJFH bronson_j_Page_087thm.jpg
a9aad40278d65f916b90a5eb944ddc7b
27236ecd79fe12428ba4b3743d34feaa2ce9ba5b
4501 F20101211_AAAJET bronson_j_Page_079thm.jpg
b45a6218fe82c649c53b007f3dc75811
f3c8a21dbca6af17d64be7ef92da84442fe3c064
2567 F20101211_AAAIZN bronson_j_Page_212.txt
3b45e4a50872913e8acfcc045e38adf6
5efa7a2422097c4f1ef05a22ddd7a0f60907a88f
804 F20101211_AAAIYY bronson_j_Page_191.txt
2758e64a1e4648b64675a8d9f77ad8e8
ae0c21a4e2d8764cadb8c0bca3c5d46d82f88b21
57288 F20101211_AAAHWL bronson_j_Page_174.jpg
f9ad7eaad4313731fcdc108c6ce691c7
ac3c6ca611c3f213d967d557ece23d1bd0772570
48657 F20101211_AAAHVW bronson_j_Page_156.jpg
bb2275ab06cf059e319425b7ce5ce984
12862184cdb9ac09b291003cbdf46c2118b49935
115690 F20101211_AAAICF bronson_j_Page_145.jp2
60b9932742b2f5c09140e745a458e681
1a427bdcd70db03c45e464310a8061ceab6523c1
F20101211_AAAIBR bronson_j_Page_128.jp2
0528fddbcce2ce1b5b2151292e7eb82e
d5a3150fc1278a39fc8d40e9f38135e7320fffc8
14291 F20101211_AAAJFI bronson_j_Page_088.QC.jpg
c3e12f82c5815cd742d6d1ef9a961ff2
60d6fb0e5844fb3647147727ad1768698b515ec7
7617 F20101211_AAAJEU bronson_j_Page_080.QC.jpg
be1aaa66da5ae1bcc8175b3d1f11d59a
8073f7f351f2e905e6a21a8396d9950ee3311f03
2362 F20101211_AAAIZO bronson_j_Page_216.txt
826b28a5892372a8f8298f0491f400cc
f99d9b1eef0ebcdab87e8e438cf7c4f7b5909106
626 F20101211_AAAIYZ bronson_j_Page_193.txt
21ea6d90a203747462e8a042c584332f
76e3329b7dd56c316c00a75b595667fdc788d272
74076 F20101211_AAAHWM bronson_j_Page_175.jpg
5767b761439b9336251c2894725225ab
e5111d0ba5d63ef3e49867971a10cfd3574b6a5f
42362 F20101211_AAAHVX bronson_j_Page_157.jpg
6b329a7b9126f0609890404d8fd8d8b8
3609f5d385675a2de7986bf74e4c000141da2288
1051947 F20101211_AAAICG bronson_j_Page_146.jp2
797cad71515527b8f53b79c952fffe5c
eb8c12623258496bb97abe9940ec4a0a3874c05b
33048 F20101211_AAAHXA bronson_j_Page_190.jpg
d9e4c410013b2224b37295a9bc6fb46b
b9a390a18b0b3e72f9e627e00ecd64c3f49e4ab0
71451 F20101211_AAAIBS bronson_j_Page_129.jp2
f02b6179b1eba5798730417ed29f11e8
62cbb344ec48252116b5884f049e8228632cdb60
12721 F20101211_AAAJFJ bronson_j_Page_089.QC.jpg
d4a840b58f1087182136cc05ea5161ab
4c7dbfda356c23f3ab9e405377ed76527babb07d
24741 F20101211_AAAJEV bronson_j_Page_081.QC.jpg
b81508bf7dde666b46c416ab7e9d0007
29b5ffa16de255208f757c259385afa84dca14f7
503 F20101211_AAAIZP bronson_j_Page_217.txt
028ed9bb35554f3d8dfdbdfa9272cf1c
6782c2d705e9d761e7dbc52224a716aff28ef09b
68993 F20101211_AAAHWN bronson_j_Page_177.jpg
67180b6f9025b8b2b4f9f7dc87f22b69
5fa71fb50b66ad44041f0657cf6a6ecbca68db4e
64711 F20101211_AAAHVY bronson_j_Page_158.jpg
c56851d5d837ca1dfe93898954c2ffb7
4d51a174ac9205fbe4d596b8a60bcd902fe0a3f3
67752 F20101211_AAAICH bronson_j_Page_147.jp2
04b9a74208ffd904b79c4972fa36d78d
5b7987ef303939f33dda15d7f4b5c6495456da62
33308 F20101211_AAAHXB bronson_j_Page_191.jpg
324d34548bd6c3a953f8b6e9d2e301f9
31d484a30c23c1860f2beedc69507537634260a8
866537 F20101211_AAAIBT bronson_j_Page_130.jp2
c7d4e63aaee4f40d0ff1ae3609f8b1e2
0267f47ad07c7dc53aa8b440301b48a274d388a2
12804 F20101211_AAAJFK bronson_j_Page_090.QC.jpg
7a17722b9518513794a78a112ec625a2
3f2d8963326f1fefeb67850122b0c3c41af967a5
6731 F20101211_AAAJEW bronson_j_Page_081thm.jpg
8b95b3d39065bca96f354f82d7c006c1
071629117a3c12ad76f05fc1fb1f5488da0716e1
1421 F20101211_AAAIZQ bronson_j_Page_218.txt
5606f769ddee5e0bed7ea46e4f2d0845
6f21537066b34d597d909886a6842ad4007723a9
66668 F20101211_AAAHWO bronson_j_Page_178.jpg
92037ca396b506fd1222c27ed66b1cfe
fdae0eb0cf607886bbd47de9e159a4f62df63d57
62095 F20101211_AAAHVZ bronson_j_Page_159.jpg
5303707ea16d2461bc3caffe48891814
d061afad8bba50a86863545ff776a0ff7d46186c
909170 F20101211_AAAICI bronson_j_Page_148.jp2
6b929707637a17dce17a599a4f4512d3
21631adfee41e33300f01bd3bcdeb009f9e3cc2f
35896 F20101211_AAAHXC bronson_j_Page_192.jpg
3352f06d2a9171774bbcb2f1e8a5ee0b
72e45608cb14767282a526c637f9710f570f1628
595516 F20101211_AAAIBU bronson_j_Page_131.jp2
3d8877c9e0a850ab58213b6f715ddbb1
c7b5a236867498a53c26ae053ade8c5689e37fce
6374 F20101211_AAAJGA bronson_j_Page_098thm.jpg
80fe7362216ee39608f3388186b20ab4
29b61688e7e08a6fc7e7e6662888fc939815140b
4451 F20101211_AAAJFL bronson_j_Page_090thm.jpg
a00e4d5ec7d5e55655fa8714e32c6a41
a0b4e75baf8b7e5faa3344b9be9ea585ee5c2cac
22296 F20101211_AAAJEX bronson_j_Page_082.QC.jpg
688ec7e1dbfd67674a72291dc1a8fb7b
6fd59fbe01e19aea7a952be9c01d498849733ccd
2314 F20101211_AAAIZR bronson_j_Page_001thm.jpg
2d90c8657075d0636ce6931272255a83
b2f06f52694105e2ce0b8bf2151447550550e0df
78719 F20101211_AAAHWP bronson_j_Page_179.jpg
a17b14a1f817f2dd72cfd2a0d7f8b82f
53acf585c3208e94aacba18d25a008b1abd34b54
1051970 F20101211_AAAICJ bronson_j_Page_150.jp2
c2d32a626285f4b2205966d1e60e18c5
a8650246cee749ae5e4af0648ef25ce230e3d201
32694 F20101211_AAAHXD bronson_j_Page_193.jpg
a111b9dbad604abf5a153b4bb1c02c13
a1e2cc6026866a65e368fd40bdae29a87e21a88e
414173 F20101211_AAAIBV bronson_j_Page_132.jp2
d175aa1d5b7b2ffd0f803c0e5343cfe0
7c9e3d7b5ce98c99c9b3715244fae7a2b9f82789
15499 F20101211_AAAJGB bronson_j_Page_099.QC.jpg
e17396e704439f4a9627c5e3e7a366f1
5efdb877d2c2fe1b420bbd6cf9644bf669331ad0
23414 F20101211_AAAJFM bronson_j_Page_091.QC.jpg
72768d8476d3af5e02aa7e9ef9ff0ce2
0aa6e4312bec2e49e5cf9e58e276fabb1e4153a7
6549 F20101211_AAAJEY bronson_j_Page_082thm.jpg
dee777d5d2cd319d954ee27f0b0281a4
795ba9d41eaba36c23468d56d6f5b9bf44dbfe06
325909 F20101211_AAAIZS UFE0021233_00001.xml
9da62dc7293ffa20374b64e74d2d89d3
52534bd0417a3a1d111727a44692cb4e88e58dfb
62301 F20101211_AAAHWQ bronson_j_Page_180.jpg
39b06c8cf233201326dd83666c5f2a42
1b016e7239a635319c97e2410c97fa04836e3a71
1008961 F20101211_AAAICK bronson_j_Page_151.jp2
4b85b54bacab0b72852bf3e1cd95e0be
e8a0eaae1a0976f032953952f8529ad92b01df15
36852 F20101211_AAAHXE bronson_j_Page_194.jpg
d9e653f30145d5c67396f6fcd4edf030
b84393fbcc3a9fa8ef11f06e2388ecc05cd04e86
557324 F20101211_AAAIBW bronson_j_Page_134.jp2
ab553896b8f16aca413cbbdb1114f853
490a7da15835de358807b8446988e11a7c13d812
6212 F20101211_AAAJFN bronson_j_Page_091thm.jpg
5e6ce06928b37eadc179f23ad4ad789d
731e2ac3b835b71f682f377fd34a9fe054952e65
20950 F20101211_AAAJEZ bronson_j_Page_083.QC.jpg
ec021405b84fcb44f1c5f6bd2f8afc53
becf9167b8e3e11f1d32a5865744df357e283388
7484 F20101211_AAAIZT bronson_j_Page_001.QC.jpg
c57f3c4f01a1fa6173f53b0d47ac2414
f02d03a238bd9d326586adfa6c3c96bb63134ccc
119542 F20101211_AAAICL bronson_j_Page_152.jp2
f0f438498ba5b37eb98754eb3bcbd643
b245c791a5dd272989293c3b7eba352fda303998
33394 F20101211_AAAHXF bronson_j_Page_195.jpg
d9783e1eb74dc6ae9be0529638a02ec8
689d71c9fb604b34674cb8ecf991d61c0c3b7715
103400 F20101211_AAAIBX bronson_j_Page_135.jp2
64034736fd21171a4ac9539efe992706
93f80d447e584832f683b933e78aba366a970457
78643 F20101211_AAAHWR bronson_j_Page_181.jpg
d6680f626b4f9a8510c989208e823a37
300132aab8649bdcf79e49b3e5e34d2997132b3c
4895 F20101211_AAAJGC bronson_j_Page_099thm.jpg
50bd4cc8d83b273c52eabaa7c8f0da2c
87c9a7fca5ad434416e47e781d77f83cb50f6ab1
16599 F20101211_AAAJFO bronson_j_Page_092.QC.jpg
1e2daf9858ee595da756f156fa575a55
6279ceb0cc09c181252897bed91e68338efacdfb
3100 F20101211_AAAIZU bronson_j_Page_002.QC.jpg
fa8bf3cd23ad00772877e962d1fe5811
3b8ef92e525f74d3067183dbe52e2e3bf050b928
689068 F20101211_AAAIDA bronson_j_Page_171.jp2
6cd0e60b8769809376ba36f3e89306c5
bf5d45ed76842f06dd3a724b62dae9bb936e46ef
941314 F20101211_AAAICM bronson_j_Page_153.jp2
9aad9cd42382a37d4fb5d1577cd49005
fa08766b66422cef35e1f780191d4526cc273bef
40081 F20101211_AAAHXG bronson_j_Page_196.jpg
b5b628b123f30902ec81b6d6d89b6da7
24f44c47bfc0583d82332ffca7112624b9a3d498
623885 F20101211_AAAIBY bronson_j_Page_136.jp2
f8087e8f14d7fb72341350316a65984b
8fe85f1df92b116c70b01b20ef36c8e4b63f4480
47798 F20101211_AAAHWS bronson_j_Page_182.jpg
7bceef858db7c773a750a8db5face3a4
9746fbead0f178b619c1db2500abf89f6c00d411
25268 F20101211_AAAJGD bronson_j_Page_100.QC.jpg
e7c239c080d911cc4cd7b3a09c91bc97
144fb0a12f8eeb95443759a0191e4010be43bb69
5507 F20101211_AAAJFP bronson_j_Page_092thm.jpg
94156777291560e6c37d247135c10c09
b7a4d04b54a8316923ae009ccbd68d08a3d2b3ca
1337 F20101211_AAAIZV bronson_j_Page_002thm.jpg
4f3b8561fda4d3200323d6498b7760a7
9ed2f9e6f9eec011414bb27220a38920c45c57b0
575342 F20101211_AAAIDB bronson_j_Page_173.jp2
94a35204afd3046c7afa0970d89293e1
13a67efcc1c4116181d8f0b3bcfde3ef0aa57943
867715 F20101211_AAAICN bronson_j_Page_154.jp2
19187a6796cbbff6a6d9f09338441395
0330ebb10597e53b514d1e31962d53fcc801b3f8
43336 F20101211_AAAHXH bronson_j_Page_197.jpg
1ea618a209e81e9767bdf5129486069c
1729af22c463429ab684006a320c5706df60f4b3
1011923 F20101211_AAAIBZ bronson_j_Page_137.jp2
0b954b6bc1f15d9102b9a0878f73786b
9d9edbcdd2e24959afa8716e3a487aad6675a860
77750 F20101211_AAAHWT bronson_j_Page_183.jpg
fc9f9a62c3462a9ddaa5629bc8db4cc1
2463dd8675cfa22437dfd7083ea0c1f93a50c4b9
6987 F20101211_AAAJGE bronson_j_Page_100thm.jpg
a11b6cea3a7f1710f528053f2f173333
fbe7f99209c388ffd09fae5ceef5aefd41239652
24353 F20101211_AAAJFQ bronson_j_Page_093.QC.jpg
da69705d473fd1ef51a2e15d69063c4b
685d4fc8456d73a356eb37c636b536bbaa88e69a
746744 F20101211_AAAIDC bronson_j_Page_174.jp2
255a28b9f019b4aefbd4fbed7e8c7786
bd2436e5aa462471bf9b7bae523e7bc558055268
61783 F20101211_AAAICO bronson_j_Page_157.jp2
82005b066942e6e4546c633995fb142c
99fbbee6123b4eb815d1e32ece7b951177f0c055
45844 F20101211_AAAHXI bronson_j_Page_198.jpg
382d3dabe35b5f53c94bfc1a925682bd
889d933313daffab6cfdac9ec4001e40776a760a
23174 F20101211_AAAJGF bronson_j_Page_101.QC.jpg
261d27973eedefd71763411d3be2cb14
1808e1c700ef7b0c9ab3ee260e2800d247685ad6
6787 F20101211_AAAJFR bronson_j_Page_093thm.jpg
b8f20110008f1c20a540fe962cc1ffbe
0cc3306429a5e1093c5250b4060c86dd5c840432
2961 F20101211_AAAIZW bronson_j_Page_003.QC.jpg
cf59d9e840d1d892f2a50da303c22a35
4449e519dc8f07d9a411af552c3041507d2488f8
1051976 F20101211_AAAIDD bronson_j_Page_175.jp2
f53ddb21bcb9300cdaf7f3987abc52f7
8fc597b323e9b1b29ae0246ac5ef1b325a17a2d7
877271 F20101211_AAAICP bronson_j_Page_158.jp2
909458a1c26cddc0cedeb50b544c7361
ac24b736c15e42a9a8f4884d8c2ea8067c6cb1c6
30300 F20101211_AAAHXJ bronson_j_Page_199.jpg
8143fd55737995fb19d88876d6012b7e
2fa7c112fcf18fd62d0aa742caea70692e05feb6
79034 F20101211_AAAHWU bronson_j_Page_184.jpg
28045542d56c52eaa02f6a47b6d01cbe
d6aa797fadbac7000acca68412e2bfb35b88ab09
6495 F20101211_AAAJGG bronson_j_Page_101thm.jpg
bab1116ba80cf3b65af4206c51905259
a1f3b2cda323f3483b91a70f0a5dba72a990f507
22274 F20101211_AAAJFS bronson_j_Page_094.QC.jpg
658f72ef7a81f173f4cf5052ce90b3b5
084c7cb4a30b933c90fe1d286e936cbc64ff8f07
F20101211_AAAIZX bronson_j_Page_004.QC.jpg
dc7d464a3584d6244c5400f1ad16c383
5f39af63928645d1e2dcad9ae259956415043e07
981726 F20101211_AAAIDE bronson_j_Page_176.jp2
e96ff59809d2ab40c216e66595d3747a
24ab16fa5541e3faf03474c07b4d9d7e8460a044
844622 F20101211_AAAICQ bronson_j_Page_159.jp2
ed31ba39c441fcbdd91152bf361c190e
ba355c150dc80f65a9e388e867ad8b084a492623
40389 F20101211_AAAHXK bronson_j_Page_200.jpg
919f9f8e58589a89ad9beb64233f8b02
fed009d74d91ba3f4d726450f9a56d33dfe8bf27
77747 F20101211_AAAHWV bronson_j_Page_185.jpg
6721ed9554854b0752d7b482df00879f
34095024b5f274180c8118e45a70ea9e7a217d8c
6919 F20101211_AAAJGH bronson_j_Page_102thm.jpg
5174d80e13e66486087800352a71de2e
7cac3aab1277f546768cf129614bbd4907844be5
6146 F20101211_AAAJFT bronson_j_Page_094thm.jpg
7e3b7fe9f7706bfd39efae625f896493
89eb03e954b0f199e291c5b0cd299149d0d414fa
4145 F20101211_AAAIZY bronson_j_Page_004thm.jpg
723e665c0c22096c3a8d99b0dcf91081
adebe07ae9bc21b165c410949f2d3102d9781303
1046545 F20101211_AAAIDF bronson_j_Page_177.jp2
2c817b7d537876ee3cabb82a0f03516a
c2f71381c03e5774deedf2931f0f2a57656abfff
989101 F20101211_AAAICR bronson_j_Page_161.jp2
01285cc25afb8845417c9fa1360f6355
be7e5d5a5d798777d174fdb3070b52591ee456e3
35536 F20101211_AAAHXL bronson_j_Page_201.jpg
2250d0f61970322eef0187be8bc2894e
6fd9119be2b4fc6c1ca6f9ed93494832f40f9cc7
73455 F20101211_AAAHWW bronson_j_Page_186.jpg
4e44039f82f80ede33e5b23d9710d9f9
d64a05c8ebdbadf85a23823ef1402205871adbbc
19886 F20101211_AAAJGI bronson_j_Page_103.QC.jpg
2e2e85c19a138f96fbb19ed0596daba9
a1ccf3b83e885bbabbec527a0d28e6f8831b2f65
13374 F20101211_AAAJFU bronson_j_Page_095.QC.jpg
4c9c550f090a297b8afd9813e0c354b6
781eabd9ec186c6a76f82bf83b2ae54e89412399
23570 F20101211_AAAIZZ bronson_j_Page_005.QC.jpg
03ed31ed1fdae962a14776eeb6724a47
6e77697cc4de247f0ec137ed19bd2fceb8c893a4
1051953 F20101211_AAAIDG bronson_j_Page_178.jp2
cce1265b005d7228c4ce6d4613e05f23
d7742156e2c37c6e0e01211d8e900e3c77a49af9
5643 F20101211_AAAHYA bronson_j_Page_002.jp2
7827059903c4572b1bb6e9fe8ef08a09
cd6cfcf65b4a0bf01b724e05b9d538bab7222aa3
993593 F20101211_AAAICS bronson_j_Page_162.jp2
b38a2b8007058fd38d955a189da1358a
b59a50b4410c66c0f61b8fe2a4530edaa301c1ee
37026 F20101211_AAAHXM bronson_j_Page_203.jpg
d318f86757186b78bcfeae1bdbf92d0c
6f7a444eaa614fe567c081fe51f1eb1708f4b27a
30481 F20101211_AAAHWX bronson_j_Page_187.jpg
671a9f6c069dc276865179679f2db6b7
2765abbc2151fd7058a7ceb091df371e9b9d7a05
5754 F20101211_AAAJGJ bronson_j_Page_103thm.jpg
1632b3679dfdddaebf4c9771de5e9cc3
80591c3fb97fe6202735271ef0a8c49b8f20c1bc
21716 F20101211_AAAJFV bronson_j_Page_096.QC.jpg
45cdddf1face9ed96010b29ab3138d01
6e5a3d9bb86578639cdd835b419b6a8a382c23c3
F20101211_AAAIDH bronson_j_Page_179.jp2
373271d1947634d1c829af85550effaf
2380a66221113bd5833c00186b65f9521532bd6e
61938 F20101211_AAAHYB bronson_j_Page_004.jp2
c93b17dde93b2d2d1027264c377da6a6
442e504713a9e610c628f9f10b87b2dcb706e350
110676 F20101211_AAAICT bronson_j_Page_163.jp2
b8841ed3e59b473393a8b76ca600e28b
d4ea51f8f7340e1b902d09f30dd9e553e83dce70
36286 F20101211_AAAHXN bronson_j_Page_204.jpg
dfcff53a8de63fa8b76f4cfe65f06675
7175f924398f8506472fdb0b1cc0049a8327d3e5
45912 F20101211_AAAHWY bronson_j_Page_188.jpg
69a8cb8311c7a4240d65c1efc6d06337
a05681caeb9c2352176dfd72efc50c7752ed8f8f
5344 F20101211_AAAJGK bronson_j_Page_104thm.jpg
5bcea16be98ebae839c8a962aa15030b
9062979abe07411e26e7b772007cf69820fa4d99
6231 F20101211_AAAJFW bronson_j_Page_096thm.jpg
2fc7108b8ad81effa3e2b7988d3fee55
fef984deb288322234e6fbe2067f3addea4dbfdc
998310 F20101211_AAAIDI bronson_j_Page_180.jp2
0420b5514cbc12b0279524cbe2fc49a4
6e5e3c31f45ce6f774641dbe48f120df10979388
F20101211_AAAHYC bronson_j_Page_005.jp2
c047994f4d7e1b1815ee1e8ecdaaf28c
6990a103ab671af79be9acef9c55c157ce56b674
1051975 F20101211_AAAICU bronson_j_Page_165.jp2
2ebdb2c570458a5344f70741479b4734
d3084df1e9db35517f5d36044f6f9808fea7770e
39025 F20101211_AAAHXO bronson_j_Page_205.jpg
327f5bea0d6713e1e1b99684b778b6c6
7c3031e8cd2ef68f2266ea2731d6ca21f822badb
38417 F20101211_AAAHWZ bronson_j_Page_189.jpg
b12ab1b9663d204ae8995518633b4bdc
ea64934ad5aff046a96998daec3a64eec14950b4
18137 F20101211_AAAJHA bronson_j_Page_113.QC.jpg
a0f31d2e3a70a3c557b72e520a175048
ba2ac6c4e61406c0495090be87b42d88ddf6a4fb
25827 F20101211_AAAJGL bronson_j_Page_105.QC.jpg
89a19f8288488a9eb1e2857d94e234a4
074393fb2dd2a4ea01a6d42e07703697c2315196
20959 F20101211_AAAJFX bronson_j_Page_097.QC.jpg
6290992dd0fe848e27c51826f04a09b0
fa7a84017d8daef331497f331f416824904fe3fb
67998 F20101211_AAAIDJ bronson_j_Page_182.jp2
b54bb349345ac3d1776f335cc5040de0
5ba9f097fac5e10e7766c3a9dbe8138d9abd7035
516159 F20101211_AAAHYD bronson_j_Page_007.jp2
6f373588e60776d071ccf575cb5f1eb7
8e7762b9573a2d6c54e881add943de28e44556f1
1006817 F20101211_AAAICV bronson_j_Page_166.jp2
3fa259c38de16371cad0795c63540b27
96055d7590b240c7155380680326f68440a919ef
39415 F20101211_AAAHXP bronson_j_Page_206.jpg
0714c6d7ba2e66aa77fe34ffcae95f62
081ba5a4d7d41774a333b2b833f5e0000b8148d0
5177 F20101211_AAAJHB bronson_j_Page_113thm.jpg
59d7b24806b615f40d93072595682454
07f6be80417aa4e070d29a35d80b4546ed274568
15209 F20101211_AAAJGM bronson_j_Page_106.QC.jpg
05e3b8bee17d86062132ab285a7a9022
cc1067dd6fc76b34fdf121b9ee680173804e978e
6042 F20101211_AAAJFY bronson_j_Page_097thm.jpg
3f5d6d6deecb724d2e6f88b15b678873
7e0aeebc7b2891a9423fa8a568912f55f73cbafb
116213 F20101211_AAAIDK bronson_j_Page_183.jp2
97ad4b5e594c84df0ff68ddec239e568
967b974103ce7fb74ff798388c95ded446770f42
F20101211_AAAHYE bronson_j_Page_008.jp2
df03228ce6bf0659e4cb4e85b8a17432
fe1c4159740d7efb1620bb25fbd5392a61ca3d44
117422 F20101211_AAAICW bronson_j_Page_167.jp2
bb2a60b41a1f5f8f9ec03e4e84f966fd
b154c8f63541e4acf7063bf6076bf3dda6c32b87
93035 F20101211_AAAHXQ bronson_j_Page_208.jpg
66e2128b947d4d9b5902a78ddd2544f0
8f22e9836f5ee55f46e4745d111a298d34a79b17
18684 F20101211_AAAJHC bronson_j_Page_114.QC.jpg
bf443a7b0dbf081f474695d697caca2b
594fe1d910c70b7fb8c6967439a83b36b82a8733
5021 F20101211_AAAJGN bronson_j_Page_106thm.jpg
4abbb2171d3f8b67cf5d13348f585740
a92f53759005a83c07bd41c23373fbdffed68f27
21929 F20101211_AAAJFZ bronson_j_Page_098.QC.jpg
8c34f71a38d3d74cd4ad02020b70212b
c201c7e5b51c8fa02302514c92bf897f4b2c1f6f
120415 F20101211_AAAIDL bronson_j_Page_184.jp2
ac454c837a02c5632e35c6bf69f10a62
c283bdd1ebd4223d6615c6498bfdd1d5a6ca2030
1051980 F20101211_AAAHYF bronson_j_Page_013.jp2
b189e438237b729d6cbd88b5c7785c5d
5b21bb9ccd94d360e2e7fa9aad279d72d6404f4d
694611 F20101211_AAAICX bronson_j_Page_168.jp2
01303a04d1fbe4bdf4e524ea799aadc8
da5a5aeb35581c795bb5046f071c98aaa0efb0ba
105150 F20101211_AAAHXR bronson_j_Page_209.jpg
eee90cf51b949e581fc80b01402e3318
4418229d32ff432c268470c26bfda1ef9d6270ae
543399 F20101211_AAAIEA bronson_j_Page_202.jp2
4dbf0116620f9ede17e15703719cb643
293cd308c00b778b294ae4cab3d43cfa988fd9ee
11693 F20101211_AAAJGO bronson_j_Page_107.QC.jpg
3b7971c76c35ac5c9a58739f7ac594d1
9c017d1c785f66e89c76f184ddce549248696809
110708 F20101211_AAAIDM bronson_j_Page_186.jp2
0462eeb30c2182dda348352727afff43
295cbd6500b4823b15cb36578c1925603d0535b5
1051982 F20101211_AAAHYG bronson_j_Page_014.jp2
840474344b3593a8977d820a5aa9b340
e6b85ba1559fb127c3d801aa0e15083d23a37644
950351 F20101211_AAAICY bronson_j_Page_169.jp2
a6707f543989d90bb819636cee412a08
57b0e713143d128e20057deb4a8213a178c52d36
98250 F20101211_AAAHXS bronson_j_Page_210.jpg
c7e651fe420ef3a065475e0acfc140fc
e7401c3b752781736a376d186d318df73792e978
5570 F20101211_AAAJHD bronson_j_Page_114thm.jpg
217bc5728bfd73c0d43e25cc55a51193
8013cdb68bf0c9fdf1e8f010f3764e327b64515c
3868 F20101211_AAAJGP bronson_j_Page_107thm.jpg
7faf2c5bd5bc7ee5d08e6ab7ecbe8541
ab59be69c54837430617e5f99f8175f1da364091
41464 F20101211_AAAIDN bronson_j_Page_187.jp2
689952af0cd0bbe4839b32782c4fdee5
7ae174e07101c5dbfbd2aa792d120da6a8f54f73
1051979 F20101211_AAAHYH bronson_j_Page_015.jp2
9265c06be06343aef086fa3317d46aa0
ff16297a2620b1ecf67a81aa15486a6490be58bf
95260 F20101211_AAAICZ bronson_j_Page_170.jp2
a75fb2e84a79acbff911b22747951e8e
b2e76c099cef17b79f15e621020bb87ba369de8e
88563 F20101211_AAAHXT bronson_j_Page_211.jpg
25d61fd39cac9f6d5ba7998a23e25ee8
c213c20e4260f5e2027446ddd9cdb66496daafd2
620598 F20101211_AAAIEB bronson_j_Page_205.jp2
ffb1281b829db804ca5bf90a1f2acc6b
6ba2a195d3f71e472acafccd540944bfc7998d93
18480 F20101211_AAAJHE bronson_j_Page_115.QC.jpg
5cee62db127d29dcf2aebb8bc116b812
9bde3f9fc68dd3f50b6628343c1066056c780b25
24613 F20101211_AAAJGQ bronson_j_Page_108.QC.jpg
4c963714d14714dac745a0bf5481ed65
5df8f6f153457b465b80537c17cd74a3c18fcec9
F20101211_AAAHYI bronson_j_Page_016.jp2
a0803409b71288a91ee32766f2d55588
394f97e267cdfc2c83238720ac5c62cf4bd41516
85895 F20101211_AAAHXU bronson_j_Page_212.jpg
27803bfc0df62cb2232f10b09f672980
40b2f2452e6a9a69c435ac9d0f73fbd8acab4382
627728 F20101211_AAAIEC bronson_j_Page_206.jp2
96e2301a2a304be10038ff778a83082c
48144f633563fcb1514e1e20f55a51ffe1d3ac6b
525565 F20101211_AAAIDO bronson_j_Page_188.jp2
22c4d2c887d6a898474f4ca22256961e
a68cdd408d6b9fd600af9ee407b73ff2f1a31b65
5736 F20101211_AAAJHF bronson_j_Page_115thm.jpg
505b09409859448335c42d83abf494bb
7ba019974c0c08cff76f0254cc7fac2caa6c9326
6546 F20101211_AAAJGR bronson_j_Page_108thm.jpg
6e1269b2041f34c80dc7c30d87e1dd4a
73734fc9a4a14d493e0081415c2c14d830dd78dd
107448 F20101211_AAAHYJ bronson_j_Page_017.jp2
ca657651940af54a908265e4ffb9fc19
6060a3015e6661671e44174883c57582b83ba106
339985 F20101211_AAAIED bronson_j_Page_207.jp2
4a59bc1d3a60e5707fba38ebad211f40
c53e68f9eae3072fd27c406e09ac5bf30c873ac4
462598 F20101211_AAAIDP bronson_j_Page_189.jp2
e026365aa60b9dbc82f997e43aba10cb
bc2c2db6bf396bb6c12a7730669ca1b13dbbe5eb
21770 F20101211_AAAJHG bronson_j_Page_116.QC.jpg
258763ac728ff07cdf69947e7583dc3c
9bf47560144bb976d80b176f4f20b9ce9d2e62a5
17691 F20101211_AAAJGS bronson_j_Page_109.QC.jpg
876d0dc244ce3ab0c445f9c4f907deb5
eb0d6ad5c135fc91ba4b58b3ddddaf1ec214dea3
114544 F20101211_AAAHYK bronson_j_Page_018.jp2
31e54abfffd1dee67c2319bcd2189742
d85fc8981ecb0ffb8c488b7e3044797efd56e6a8
86830 F20101211_AAAHXV bronson_j_Page_214.jpg
496c7a4f84b18cf8b0b39c936f8f8644
8ddb2da2018f7747aa054cf7871f3324ce08300e
135625 F20101211_AAAIEE bronson_j_Page_208.jp2
c40eaacc6a6b184cbfa8b0b39f2a486e
0e30a9d74aa5ec0797cbc1c47c6703157c0114de
387663 F20101211_AAAIDQ bronson_j_Page_190.jp2
45e7c928263f76c4d6cf18b237086233
408d5f5e920f993b4f8cf56b12f192891bf7c8a7
5988 F20101211_AAAJHH bronson_j_Page_116thm.jpg
06f2ef8be422f7e3f9206ea7ae3a7fdd
3a94cb425f0d5e8e67f44da103582c4543fc7953
5083 F20101211_AAAJGT bronson_j_Page_109thm.jpg
bba594cd7857eb9debff679f68dea7a4
d60d3ba5787d8562c8eb19029218901dd0fda8cc
117531 F20101211_AAAHYL bronson_j_Page_020.jp2
295a8e20772c7987375f86d218720465
fe1d27518ac4345d5e293a4b59365f256e783fe3
88607 F20101211_AAAHXW bronson_j_Page_215.jpg
edd958c5a55dab018f15819fe41a1f8c
ab8c6dfb7b171e6368459e060e6ddb1810de7a4b
1051956 F20101211_AAAIEF bronson_j_Page_209.jp2
78f6c2586e5a712dd09f77f9d53a80ba
253b05770b56e03b4fd8745dde5ba5a7212f8453
394609 F20101211_AAAIDR bronson_j_Page_191.jp2
7440c886565666d5f3a42187c486ff31
1f50f1ff911324b21d6c1cbcf5daffac269ba007
25443 F20101211_AAAJHI bronson_j_Page_117.QC.jpg
6923e565c6fae12305f5512a64c4bfc9
6e55870568ace414b1757b203f016145a0558de6
26231 F20101211_AAAJGU bronson_j_Page_110.QC.jpg
3c3501797a1e2fa49f63876b2fe26055
e9741ffe1221843825ab2cff01b4fa8d34c1ca81
121639 F20101211_AAAHYM bronson_j_Page_021.jp2
335dc1349f28721f6db1873876a620db
6e2b551080a0470dffa1717787136d548e104db0
80992 F20101211_AAAHXX bronson_j_Page_216.jpg
fcca8ba7257e6feb05764c421d5c5978
144b50bb2a44b711cc6c17d90fba4d49c3eeea57
1051961 F20101211_AAAIEG bronson_j_Page_210.jp2
6a5facf92602d838d9d5e9f354d7ebe0
400216372c4bc58d59af298afbb52c1a638b96c2
118871 F20101211_AAAHZA bronson_j_Page_040.jp2
e77d587b9224f1d20969b79f02a23c95
babb6e53af52ff434e5482a35295fa8ab0d7453a
396133 F20101211_AAAIDS bronson_j_Page_192.jp2
765c364729564a5a0ae4199c87d36215
a460e58a581ab73471b9cb7dc4c8e83811d0966d
28441 F20101211_AAAJHJ bronson_j_Page_118.QC.jpg
7511bd06a84d34ad83cf55c092344bd0
f06182c01ff3e6972a35ae805ab05a145838a1e1
6856 F20101211_AAAJGV bronson_j_Page_110thm.jpg
c635ffb08c2119f3fa5ce363744e5135
c6e4554a20e8a72b80f56c540ad3d31c329dd88f
114620 F20101211_AAAHYN bronson_j_Page_023.jp2
f576b1c53b59ba6784db561d0a4f8e80
209d71a38b86dbfc7f50ded646368e7116b2d5c5
22839 F20101211_AAAHXY bronson_j_Page_217.jpg
631b0610fb6b2ae4123914a2b4da3ae8
e990e6b233a60e14a3dc6a21ff672093fa3cd8d3
133289 F20101211_AAAIEH bronson_j_Page_211.jp2
ab6d72954b432ba4531a6f2767336840
6c79aa8111b16fc9a7b9034a91ff79ef32d83680
114071 F20101211_AAAHZB bronson_j_Page_041.jp2
8b023201e816de0c64ee594ea47089fb
83a0a73948b16cb9e50effb973943c2fe1b0801e
362839 F20101211_AAAIDT bronson_j_Page_193.jp2
5ace424c4dc577587ae65a14ef3d774a
e8c549771717c7994746ec4a39b631b8be091171
7531 F20101211_AAAJHK bronson_j_Page_118thm.jpg
a9ac9a31a2e282fdb1377b26845665cc
f0c5ba4b10ad040e03475f408710f0faacdf7251
25006 F20101211_AAAJGW bronson_j_Page_111.QC.jpg
a14fd4061258cb00f417dfdf92697ee4
d60c20525a60c6d3d9f6db9d01fe65be6382691f
120046 F20101211_AAAHYO bronson_j_Page_024.jp2
29b9eeb3a5e7040ceccac5786479d5bd
4b02f48638faba32a23cc5d363b4bfa329bb1ee9
26800 F20101211_AAAHXZ bronson_j_Page_001.jp2
395374e801af44c4b14d8af3e48042da
a0fa0a80d6936b267ee3e5b7e1beab0e99ecc5d5
132344 F20101211_AAAIEI bronson_j_Page_212.jp2
5e08de07d9eb1b3e8d3ca75250528e52
d4d5531be673c4083af0514e320e71d8e1e0170d
118854 F20101211_AAAHZC bronson_j_Page_043.jp2
c2ff96b63800f32c4e8240ceddcc8f01
a9dbbf7d1ee7620e2ed6e58f6f0125f160c71452
415830 F20101211_AAAIDU bronson_j_Page_194.jp2
4412f1f56d6329baa00a3b30e628da8f
3588343661b45f11ab8a2e78d2cffc20e5801dbd
6392 F20101211_AAAJIA bronson_j_Page_128thm.jpg
216bd354f3b4d19cda4985246a8f46eb
4dad8ccd93917f54d9810bd74c3d0fbab26c4713
7195 F20101211_AAAJHL bronson_j_Page_119thm.jpg
8165156d9991667ecf58ac46776851c8
159b3d9b164b9f9442f4135eb6fbfc0b7ce7811f
6962 F20101211_AAAJGX bronson_j_Page_111thm.jpg
9956272be9e318a2a9f34f06f6ac1732
8a40181857082ca46d3c3187e8862130e4ca45e0
1051938 F20101211_AAAHYP bronson_j_Page_025.jp2
1482e84adb86909256c7f52b1b40cb51
e3ada95486944d8bcae634b7423fd5e16be3e183
130742 F20101211_AAAIEJ bronson_j_Page_213.jp2
928d7a89194d2313999e39077bb06a8a
db8f59f8fa48dcf67842e7a28db53a3b9a1353d3
1051906 F20101211_AAAHZD bronson_j_Page_046.jp2
dab8f96c2adaf5583e92d65367ca7bbf
b381ba5f3e905108511ef1e1f4d93686a9800a62
364005 F20101211_AAAIDV bronson_j_Page_195.jp2
6b685d982b4b36d62e5dc24f748b9411
48cafa7aad4c20534e62b071a693195205325740
15920 F20101211_AAAJIB bronson_j_Page_129.QC.jpg
11a4f33d35a521dbf8cd2cce2aae8585
561affb0b881b40ad8e899c2adb6c2d58331659a
23074 F20101211_AAAJHM bronson_j_Page_120.QC.jpg
42cadd7e4fd2a0ec54033dae6c0e03d0
41f0cd6828732f0c7b01b20502c26e0e262fe873
9676 F20101211_AAAJGY bronson_j_Page_112.QC.jpg
35c258dba560a95cdcd913c92901398c
e68fdb2886be845651f5212e016c47480c564661
F20101211_AAAHYQ bronson_j_Page_026.jp2
525886b95eeb6654ccfbe239bf8f5b13
7e15c1cf23880f309a98ba7b6e708a67df117c57
136104 F20101211_AAAIEK bronson_j_Page_214.jp2
86f635c240dd1bf03b746ab897f8b7ba
982764670941fbe9235cac0bf202fc420d198882
F20101211_AAAHZE bronson_j_Page_048.jp2
6b9f158b93e70a4e3d57e3a92a7ee2a2
11557855a4939a0a132676179661e3720c914be5
909030 F20101211_AAAIDW bronson_j_Page_197.jp2
7dc0e31cc207b6e804a9629859fa5b25
867f910150837f1d28621924ded5b8e58280032a
5010 F20101211_AAAJIC bronson_j_Page_129thm.jpg
a02388c264133c037601d3242297faf7
2f1adc7d47f4fe772e3d0f5acaedc3be79c62f63
6544 F20101211_AAAJHN bronson_j_Page_120thm.jpg
5672e74d145216531ab49ae42fc3693f
ae42be24bdb6ab53a1a5a54366f24ae49dc41be3
2911 F20101211_AAAJGZ bronson_j_Page_112thm.jpg
2348957a93140f4cace19301bdad95b8
e2d2af1c588983d6807c3689c7f90c99649199db
1051985 F20101211_AAAHYR bronson_j_Page_027.jp2
060dff2bb6b14b9451970d7cbd90a7d6
e7482c8f323cefc70bc1672ed2c377129d60a992
F20101211_AAAIFA bronson_j_Page_015.tif
e36537018d8bfc9683999726551c456b
e20c57e797166b617efef44ea23ce7db942af63a
132167 F20101211_AAAIEL bronson_j_Page_215.jp2
b11cc6c535006fd82d23959bda21d5e8
adc65d5e485de3c3e5c922fc38f6e6adb3342171
F20101211_AAAHZF bronson_j_Page_049.jp2
276ea9c94e79ffa6597f1181d1cb3156
bbd5c4769983659ca72c66683cd9bc58884360d1
950623 F20101211_AAAIDX bronson_j_Page_198.jp2
e3d6c34f300dcbee5dcbad1ba401739c
6e99f7ea9e7104e7c70dc68e9b698424afdc3edc
20362 F20101211_AAAJID bronson_j_Page_130.QC.jpg
e695abcc34f4415fd4d5d6bee8caf432
8699c08b582cf939ec1e6001720932d4576fe587
15743 F20101211_AAAJHO bronson_j_Page_121.QC.jpg
ad1ba2dfbf51b9da70613ab8a04d7678
2c6b29d920b28dd64fc6d278f799106a4fc54653
113655 F20101211_AAAHYS bronson_j_Page_028.jp2
c9c53fe295214815cfd669ca2a3b81a7
58870a16e5ba9de7d26b4acbe5115036426aa0fa
F20101211_AAAIFB bronson_j_Page_016.tif
a02df7baa04c89c51d829145033b0cbd
0b45b8126436727a83de49d8f12edeecd2383f85
125342 F20101211_AAAIEM bronson_j_Page_216.jp2
ab1cd37a72127eea3c71b3ebb13203a6
271fdefc86a1ff3dee696659624da6c6ab2d2ec3
118728 F20101211_AAAHZG bronson_j_Page_050.jp2
40de5b05a1cc4bb73212cae79245ad00
0929b8ef6d90a305ca6eab0839e362cd7157c0bf
735487 F20101211_AAAIDY bronson_j_Page_199.jp2
d7ed05ed60036c5476bc56291654fdd6
788ed9a23e9acd4aaca148218e0464b113dcb748
4761 F20101211_AAAJHP bronson_j_Page_121thm.jpg
cdecdea4277c0d01a83d9d2fa905a80a
32319989c525e9c270f397ca4e220589a309d038
116058 F20101211_AAAHYT bronson_j_Page_029.jp2
70f8bb9eb0628875994cad99c246e5e0
32c51d3bbc063012904f86d44f79f5da07059fc8
28743 F20101211_AAAIEN bronson_j_Page_217.jp2
6bb0d85b94258ccd649e8c9f6ce68eba
ed25ca78eb077778588d3c8595a9f1766318b475
723022 F20101211_AAAHZH bronson_j_Page_051.jp2
1b86905df2345b822f4cb2b076e1abc6
ad6890180c27df54e5920eeca9d448d20af1f105
632069 F20101211_AAAIDZ bronson_j_Page_200.jp2
918e0d00cff0edef1b0e3858395322d9
7b5fca3b2f05be915a3c8281bde997301334e315
6026 F20101211_AAAJIE bronson_j_Page_130thm.jpg
d00004e71cf8083d7a3e3410fef064be
3dd752723de06aa4cf56296a6111257bb1c72711
5360 F20101211_AAAJHQ bronson_j_Page_122thm.jpg
968e6c27eada9c92d282a67eeb709d66
dd465d5110e6f1c7d9e0c2f9fb2ca85a9ee295b8
114924 F20101211_AAAHYU bronson_j_Page_030.jp2
aa7a46eaa165be89c6bdb95dd8695d31
8d178d4d81b51f91b7f944fdf7faed7740bd0459
F20101211_AAAIFC bronson_j_Page_017.tif
d13427bce8255fc60acdb9692827e121
6d816cdca9eb88ebe293fddbecc152e7d4307a45
80114 F20101211_AAAIEO bronson_j_Page_218.jp2
1419e9cf23a6b9b13fbf58a463acba73
bfcebdf4a644655f2a921b29247581097682fb57
54166 F20101211_AAAHZI bronson_j_Page_052.jp2
db79fe52370d8e2caeaf0fca6d66a52a
3fe2eb705cc1c9a29d30d4b8bdc47b88a64a9520
14251 F20101211_AAAJIF bronson_j_Page_131.QC.jpg
a6fb22558c4627469a9fc2d75aff691e
34c5f05c1cd81992f4dc51e8d07825f50dc58176
7273 F20101211_AAAJHR bronson_j_Page_123thm.jpg
6b724b3b069d08d30b7dda0a67d2c8ac
2c414feeb4ad0f63bebfbf01fa1ed6ef371feba0
117104 F20101211_AAAHYV bronson_j_Page_031.jp2
2c91acfb6b4a785b385febdd246e658d
e852176773e50545b5caba5af7b58ff2148a1a73
F20101211_AAAIFD bronson_j_Page_018.tif
9d87eee1c832dce178a90390532f3bfa
84e9dc15611754e65493aa9b4cad124135b15d5f
F20101211_AAAIEP bronson_j_Page_001.tif
62123d242759d4c6c7ab2f9a86e00fa6
e753befeb50eb6f426a3b7ea3cf36572a6a14991
111942 F20101211_AAAHZJ bronson_j_Page_053.jp2
187f450b1da9d7d9c630e567e9ef697d
e2b0e7c0234cc50db6cf03536d064f65ef522b24
4802 F20101211_AAAJIG bronson_j_Page_131thm.jpg
3437db041f31cbd28d6669aadc19738d
074750d55c8dc425f2c131b950be94be72400c75
14951 F20101211_AAAJHS bronson_j_Page_124.QC.jpg
80f3743e5960b5502b67579389be3944
e91ada6333315dc3aac2aabf455af5e8373eccb7
F20101211_AAAIFE bronson_j_Page_019.tif
f43495a13938828325d2bdba9d770b0c
52b651599cd1274a0afd651f8a6085424bff12f0
F20101211_AAAIEQ bronson_j_Page_002.tif
10aef5e92a2555b9dc0f7e19c37cfdf0
1b752e9ef652bf314bb07799b1b7564867663848
846642 F20101211_AAAHZK bronson_j_Page_054.jp2
0ae05c5b562aee25b63b2674500ea009
fbdf230c3be78ad44ce9e6d0c369414322d65148
12248 F20101211_AAAJIH bronson_j_Page_132.QC.jpg
d537370234564f426fbfafed619fc0c4
e80203db6b6826f43634c67f575b81ddf9cd474c
4747 F20101211_AAAJHT bronson_j_Page_124thm.jpg
951bc13767a5a5ef43bdcd8ddeaa0809
bb7f1dccdb81afcf8cb4e76b29a02bb89b4e86c3
118591 F20101211_AAAHYW bronson_j_Page_036.jp2
eb2c79797e726d60b0fcc93ad117d006
3f8abcdf7330fa6c6d1c7776d89eb9d39568c20b
F20101211_AAAIFF bronson_j_Page_021.tif
0846222addb2292153f1d537852b08a6
8e7210cf2319b2da96690773c8c94d0459abc57b
F20101211_AAAIER bronson_j_Page_003.tif
b09265a319cb660e8da153cd7da3da6f
9ee9c4e0d8c26a1771b1fa223eae5c6650f2972b
685410 F20101211_AAAHZL bronson_j_Page_055.jp2
f64d1f6d46761a27eafa7cdc4fc95879
c4e75cb6ce3a3422fa814e31d604acc88ab70915
3944 F20101211_AAAJII bronson_j_Page_132thm.jpg
ce4b947e1038a215fc71ee41a0ba12f1
02dba74ea5e14967e93746fd7b344dadf97824a4
25927 F20101211_AAAJHU bronson_j_Page_125.QC.jpg
86c8476225f7380fd5adf5933e5cae2d
ccbe1d64e5b9908624556d3d5fbea51a5fa3d3e8
119918 F20101211_AAAHYX bronson_j_Page_037.jp2
23d1782e82026a60b1144ef5b46586e8
570c0820f40cc822918f7978fc4046df4ff9a987
F20101211_AAAIFG bronson_j_Page_023.tif
579b4833b93b94fefad32dadaa168c58
b90e27c2794faf822781d76192d8bc296406c783
F20101211_AAAIES bronson_j_Page_004.tif
940275cb84a487c36513bb98c1fb561b
99217bd7059ef6d9461d17f8afc3539a1efe5ec2
F20101211_AAAHZM bronson_j_Page_056.jp2
1ead7899794f84935b2524fcdeca33b7
dec724b56e22c55d0fd6a432280e2e056ad97284
23946 F20101211_AAAJIJ bronson_j_Page_133.QC.jpg
317497ed498f385259a6612650dd3b3e
70b48bcea161bfc43c25bce99dfb6d9b3e4672eb
7045 F20101211_AAAJHV bronson_j_Page_125thm.jpg
beb0a5901a8628a44461c3a0ad5381f4
beed6abe7fb4b3b6291cc0432096fe943dc10ee7
116932 F20101211_AAAHYY bronson_j_Page_038.jp2
85117633132cdf862f3ca7c3f7da4bd9
257fe544b637e95c8a991238c55abb405d7c137f
F20101211_AAAIFH bronson_j_Page_024.tif
ccf8ef16912785f85c128bce05df8978
876315091e69c33a9497d649cc2ce3dc1ede5d73
F20101211_AAAIET bronson_j_Page_006.tif
7752ffa314971ea5e25877382af29dc3
3be9265a73c12c5e1eb5075a12f0ca8b54b61d0a
757060 F20101211_AAAHZN bronson_j_Page_057.jp2
6f4d4a092f43b9bf20522569379ec096
fe1271f33af2c06359f0b49f592d193e36ff9217
6488 F20101211_AAAJIK bronson_j_Page_133thm.jpg
7467a9587c41ca05ac9940598d82091a
5aec07e317225e3e045d39946c765eb5bedf2c96
19589 F20101211_AAAJHW bronson_j_Page_126.QC.jpg
02b74ee14529fdbd01f5143f342525ce
12282145585b17df5faf7d6527ff57eb6299d7cb
121012 F20101211_AAAHYZ bronson_j_Page_039.jp2
7725cd3743f3e6a780a54b0a3958cee0
bbf17dbeacc2cf970ee5c7501f93f59e1cde02cf
F20101211_AAAIFI bronson_j_Page_025.tif
8fae6389a26d87abbf1c430a01dd4046
bc8807ae51a81839d95ccae0d6476468935ffccd
F20101211_AAAIEU bronson_j_Page_007.tif
a1468101099987eb86579e528eb6fe8f
16dd48d912db7f1831ac1b52da42b9b39dd95598
82485 F20101211_AAAHZO bronson_j_Page_058.jp2
63ef3aebfd91aa0812ecc97094453bcf
147e8b2043f70e29c989b863c4d84584c0c0c1e9
4880 F20101211_AAAJJA bronson_j_Page_143thm.jpg
e56c6ca24947e93d626479eff14462fb
e1ea1fac3ac8fdc117e7f268c18d1cec1c13f960
13876 F20101211_AAAJIL bronson_j_Page_134.QC.jpg
b8a7da5e696e9e62e190c6ff1c35f9a5
7cd6ce3e785c8b683916d879dbb04da264d8a358
5714 F20101211_AAAJHX bronson_j_Page_126thm.jpg
d913063e67b7143608cb92753d2f56bc
18f55acdb6d13bcf25365a98ef1dff1ede947e36
F20101211_AAAIFJ bronson_j_Page_026.tif
589e594719550442edf0edc43253484c
362de034d3ed2a0d803b8b8c6b63e539c03cbd17
F20101211_AAAIEV bronson_j_Page_008.tif
711cfdfad6ccce65dcdf3d3638295d3f
6cf9650107d9ef968d08ef6bd38a124d25d3be8a
1024698 F20101211_AAAHZP bronson_j_Page_059.jp2
2a5499132acf555f3c6ebf28fac05ce6
2587a40c59d487e54de1ef309538ea119ca120e2
24926 F20101211_AAAJJB bronson_j_Page_144.QC.jpg
6c0034f0e416d134c2647491a48dd151
dcf033c59d26932c59c1b2463a561e18ba452d65
4077 F20101211_AAAJIM bronson_j_Page_134thm.jpg
3c8f8892003da31f0437c179786c031e
3c1f48838a9741637038e8c843d408aaf9538944
6314 F20101211_AAAJHY bronson_j_Page_127thm.jpg
f7fe3b430fc0a6f645cca55cf57ae14e
e651676f50b70676b2a49bedc9092be34c4871c3
F20101211_AAAIFK bronson_j_Page_027.tif
edcd0233f9bfd92ff8f8e9ce5adedef5
8cac70673855f18ea30528827a71c3c89bede659
F20101211_AAAIEW bronson_j_Page_009.tif
74714447465349a40484c50570637059
93ae34be1be70c28175ab8ac5302c3351ed4c227
946409 F20101211_AAAHZQ bronson_j_Page_060.jp2
96c2c164bb43b6a4c22694900dc4ef6a
967c23780c000bbfdf94a642b59543bdcd10ef4f
F20101211_AAAJJC bronson_j_Page_144thm.jpg
4d6d3d7187509834f26199adc053e790
313add4d30fd38101f0025a12a96cd4917f8cf7c
23200 F20101211_AAAJIN bronson_j_Page_135.QC.jpg
d282bc8db6e74909593f401f5d0e464f
31e31ad977e9096e923b16ef9a2216b9ec19e050
23332 F20101211_AAAJHZ bronson_j_Page_128.QC.jpg
e5e02ec903503e5963fa6ef6a2973fb8
b4df42f58cf38b39afd341712c2d3c7efd618b90
F20101211_AAAIFL bronson_j_Page_028.tif
23b959dd1c2b7da65d25b6d518f8f54e
e5b31ffb063f1f389b46e6b121a08be7cd08d8b5
F20101211_AAAIEX bronson_j_Page_010.tif
c2b82c4299495c1c1a2d27798b7067a6
a78108886a3ffa7529472e4436e1904f29a40d26
994769 F20101211_AAAHZR bronson_j_Page_061.jp2
7f4c90dc8dba2abdaea4896be39e0f7d
0b5f7891977a878bbdfffeaddbb1097cccc5abb8
F20101211_AAAIGA bronson_j_Page_044.tif
9f193b1834a1190d6dd2a1908f7b4b35
1004d3e341fc768f7c032a5158911d665833f765
24965 F20101211_AAAJJD bronson_j_Page_145.QC.jpg
8c39a0e86c748547a5e7e2d693a6d4bf
feda8cb4e2c1e930617eae16a9886b5b1fe7b201
6298 F20101211_AAAJIO bronson_j_Page_135thm.jpg
d6a117842d47c00b0ac66e0f082e67e6
0063baa5490244fb18da218e0e4dcc72368992b7
F20101211_AAAIFM bronson_j_Page_029.tif
6d02cd3c6b850644fac268d7dd1fc5b8
095dc613518fc49e1cb4b51227c1901696578ddd
F20101211_AAAIEY bronson_j_Page_011.tif
59e0f6f53b7423ab0a6afd2dfcbcbf10
24c5d36436bc56b3c49500631ab24563c6cd14fa
948568 F20101211_AAAHZS bronson_j_Page_062.jp2
b98c3debd15c69f6daab72be1d347861
62065a3c771bfbf4fd4f2db2d97c5530a123000c
F20101211_AAAIGB bronson_j_Page_045.tif
98cf37fb23ccd6e9af12bb117db7543f
972a61ca5f0766fdd288fc83235ab9c52c2d31c9
6748 F20101211_AAAJJE bronson_j_Page_145thm.jpg
7984d6dc4b1ed49e7b7fa9167b1d7685
93e7c671c95b2f63093dd11c62e8510d12737616
5112 F20101211_AAAJIP bronson_j_Page_136thm.jpg
5521281efccd4d81d9ee037482353157
5263fc22bf7d0f419f811fe8913175fcf4615f4b
F20101211_AAAIFN bronson_j_Page_030.tif
b31a6544034aff4c86d63115eb7fd140
a4cf9434585420aeca0b2201024b0e9f928bc3ce
F20101211_AAAIEZ bronson_j_Page_013.tif
1349af3eae4573a5524927f82db17c9c
e827466fb4a249fcefcce1388aa91045f39c9024
876227 F20101211_AAAHZT bronson_j_Page_063.jp2
c60b1b90beaf31955096c39dddd40504
4b5f1960782dbf8892577f4f235657050c418c31
F20101211_AAAIGC bronson_j_Page_046.tif
9e5ea82f205d10ef402e7e19214f4917
00edabf73b3888c40eb2199f39c630574c6c0932
18790 F20101211_AAAJIQ bronson_j_Page_137.QC.jpg
e5c1ce9ac798057a1c2ed793e82e691c
4bf5df6d492f863675680457d0c9f6e22b93d383
F20101211_AAAIFO bronson_j_Page_031.tif
91e1e4d7fd0a70d6df162df08eaeee0f
6c023745d17462511d7af1c18b21aa84cc3b64af
113664 F20101211_AAAHZU bronson_j_Page_064.jp2
3ecb2f1e5a25a6180efb6aef01c64eb1
405d5c5f324ed2b86381ae593db469f47624b181
25929 F20101211_AAAJJF bronson_j_Page_146.QC.jpg
0b95118d8cdfda39908aa5e9c093b0a9
27cc6d5658c8f3f5fd5089b010f40e40d0aaad10
26709 F20101211_AAAJIR bronson_j_Page_138.QC.jpg
c60866864a3ce338b113ef284b65d819
55a4b749c8a778b9d5553b81f043dd6892ffbcde
F20101211_AAAIFP bronson_j_Page_032.tif
d7e6b01658e9db3b4d9cca553c2fe10b
00659a97c8e0ac655b094b2585a6a5ca19dabe98
F20101211_AAAHZV bronson_j_Page_065.jp2
3ffb1fc1b13e4bb4bd27d53f215887a3
8bcbbc300f7a683bd33b46c036b06ca6a26b9ef1
F20101211_AAAIGD bronson_j_Page_047.tif
d893485b8127d7f09db5950dcb575759
c73988b096c44a7ee7a9098ee36e749efebf5cfc
7125 F20101211_AAAJJG bronson_j_Page_146thm.jpg
e9c9fa2e9acead69ecd42e7d34aa7c29
31a5b35e58995a853c78a8ac956e6e032c4c3f06
F20101211_AAAJIS bronson_j_Page_139thm.jpg
d17a16417caf49d4c994a3e279d25b6b
f4e25202fb67a2948a733cebaa6b8c3c7817b84b
F20101211_AAAIFQ bronson_j_Page_033.tif
67a72fa595814d0a1df95e5cf3d4a929
faa59531696a569d6b61fef70829d422f5072379
909749 F20101211_AAAHZW bronson_j_Page_066.jp2
96b2eeb91fce6eeea9db48573ba919ef
2461e74f6ddd14c49d9517b6a2673d63d3fa69d5
F20101211_AAAIGE bronson_j_Page_048.tif
17df84c8e5b5446cb804603a4d8ce231
92bbe42304a5e394ac50bf688ecbf4e3bdb067bd
4725 F20101211_AAAJJH bronson_j_Page_147thm.jpg
4b4c578e76858e2e0f2ccd8425613d23
7f3d70f61672338946d65cfdf18239c0ed5c2770
20337 F20101211_AAAJIT bronson_j_Page_140.QC.jpg
17c01b987e6436129e6313a1d0394d25
7e5f709f758cfc898e78d0b05455638a3535e69e
F20101211_AAAIFR bronson_j_Page_034.tif
178fd99921d6202e929b410eb5410b0e
095b6d491c5a42305fc65f308040254e61a6b6b7
F20101211_AAAIGF bronson_j_Page_050.tif
ba25cc60a2758e71447571b0836d7a51
11dc15bd5d6f60155595ddd17bb697599c738ec4
21139 F20101211_AAAJJI bronson_j_Page_148.QC.jpg
d9565ca4592862c56d6cec2d71653627
5947448b18e7257a52feef9495cf17959d514934
5887 F20101211_AAAJIU bronson_j_Page_140thm.jpg
e946e955800a598e71f059b87295fe92
d7b578da4f456dd33235aeaf43428b7a246f91f0
F20101211_AAAIFS bronson_j_Page_035.tif
eaa2ca13640f16508cff01ff2320165e
9986594d957b09bdb4dd879b7104cd306d65efbd
1051964 F20101211_AAAHZX bronson_j_Page_068.jp2
2cd40df4fea86b76ef587d86aa9559d8
bd345284a38f988c9fce1a0d1ba8c3ea448942e1
F20101211_AAAIGG bronson_j_Page_051.tif
2c3dc2dc548c5163722dd539d5682521
733d096d3e898d943494b54986ccb194d232edef
5985 F20101211_AAAJJJ bronson_j_Page_148thm.jpg
408135fd0a129d828cbc25bf9009aeed
df791b630421d041a2e2f648f2fb94289402a7d6
22912 F20101211_AAAJIV bronson_j_Page_141.QC.jpg
db2a80fea6d73982f625e95f01847475
02cb2a090e487dbf8aa0d5c9703581c0cd0809b9
F20101211_AAAIFT bronson_j_Page_036.tif
8ce7712d7d00b03c4ccbf5d7674018b3
794ad5a3293278b55b3ead6f657ce0d879b7e5d0
764661 F20101211_AAAHZY bronson_j_Page_069.jp2
8e2213de65190afd9d323c1d3ac9ae9c
565236d838c4c3b6830e3d9cd5dac4322f804ff9
F20101211_AAAIGH bronson_j_Page_052.tif
4cdb83f9a71e34e9639bae911861424a
6274f49bfe5730e1c7089bc94083d9291dec8cf8
26214 F20101211_AAAJJK bronson_j_Page_149.QC.jpg
d22f0d28ad91cd71e149b5fe3d6c06e5
abb9e1609d95807f44894bafed8ec32b8671d77b
6406 F20101211_AAAJIW bronson_j_Page_141thm.jpg
9861cb1e7c8d0a1c26f4ed905c766e73
b642cf3f3597d9fe55975c2399cb3ab2a5ca465b
F20101211_AAAIFU bronson_j_Page_037.tif
6ff299d29d582c3ae096928105cf7b83
ae5cb6e8642ce29b7362649aa1633dd462255d67
96187 F20101211_AAAHZZ bronson_j_Page_071.jp2
f4458faa463251b4346e604dce24a70a
b2e6fd4de1d2d25a6ccea14e56c077124ca948a9
F20101211_AAAIGI bronson_j_Page_053.tif
11f11938b493c24e7d4edca64b07f13d
586bd241b9e0ff2c82fd0db374a56dff681c1ca4
20304 F20101211_AAAJKA bronson_j_Page_159.QC.jpg
f0cbc85d7e6958b49300d5c5aa6bfa1b
8d40a8f0f087efaea1c4408bb9fe165b5d3f1815
7177 F20101211_AAAJJL bronson_j_Page_149thm.jpg
73521717e70d39a137c7d05052cad864
66b864fdfb0e671811c2af32fbb287ef0fc4c204
25905 F20101211_AAAJIX bronson_j_Page_142.QC.jpg
fddd815d5ad668bb78980f50e0134ef6
81a8ee442df33caa821cb95c17d583a243756287
F20101211_AAAIFV bronson_j_Page_039.tif
4bf942775c3017cba6e6c500fb4bde3f
2f5c3584ad774600ab306c0ecd0b5c23b0863808
F20101211_AAAIGJ bronson_j_Page_054.tif
7383d26753b0a4394d2f33a7347abd9d
9f80663b48305035d112cfc865a308cca41ab992
5912 F20101211_AAAJKB bronson_j_Page_159thm.jpg
8ac1533bcb82a220e782ed4247c030e3
8527933a2c3b1de0954d9c6e8cc5a8498ab67af6
20691 F20101211_AAAJJM bronson_j_Page_150.QC.jpg
eabd7802b2788fee4466366204a7e6f3
5bc0908aea2a70c2d200acfb514b9e7466689f3d
7047 F20101211_AAAJIY bronson_j_Page_142thm.jpg
7ba87712ffc52bf11ce8d7824aa0f58a
6958a0fa5a95a870fbbb0a7d3b0b51ba3a1e9a9a
F20101211_AAAIFW bronson_j_Page_040.tif
ae1952a7adcc9b6c567cd5d7741780b2
5f077eebcafc5cc1945243917c8ed4c417ba4e1c
F20101211_AAAIGK bronson_j_Page_055.tif
4c3700e37f62e4901a4bb8b1cadb23fd
b70c538c31f8d1bda2cececcfcf83c70b07304ed
20909 F20101211_AAAJKC bronson_j_Page_160.QC.jpg
6edff24dc2e52780078498461bb88314
6667afdceef5c2c6e7a2d0f5b4c788be858732ad
5895 F20101211_AAAJJN bronson_j_Page_150thm.jpg
ba3a175ce0c416f083785b707794e814
03363440ac81b38288ecf48066e20105dcc35cf3
15344 F20101211_AAAJIZ bronson_j_Page_143.QC.jpg
5a34f0c388a8c37de093f0c87897b184
ca47ad074da2d4b08fabe30cc808f04006b570b8
F20101211_AAAIFX bronson_j_Page_041.tif
6e877fe67ceb4c0edaab62d1605a69be
10cbcc9571bda018a6283403a3bc8ef4c01a20ee
F20101211_AAAIHA bronson_j_Page_072.tif
9d91f56b54e944d31b22e3a6860b2824
e9d644f5266d13284978ab59a4bc007478c0fec5
F20101211_AAAIGL bronson_j_Page_056.tif
ba85077afa43abdbc30a7195c6749efa
9fbc5ad6bdd2d89805bd23fb31e1ac5dc9645ccd
6284 F20101211_AAAJKD bronson_j_Page_160thm.jpg
b44fb98109ed602c4d77ef0d0911c33e
222e4f2d763e93c305e6e8613fcad1159e273df6
16410 F20101211_AAAJJO bronson_j_Page_151.QC.jpg
e5792b099f0590990401012d43f2971f
e20ed90d7ed1e917b914c44f82859f34ff9e35a5
F20101211_AAAIFY bronson_j_Page_042.tif
fb93b672a99728da17d8bf39af265d76
9ef39a801967aef6c97e37b875142133740dd874
F20101211_AAAIHB bronson_j_Page_073.tif
c72fabcd3b97eedc2f6ddad8958b99ef
6accc56003bed57909d57c4437f81234247983cc
F20101211_AAAIGM bronson_j_Page_058.tif
880428f960de92fe59b283b2d3058b98
9e19aeae3566052e70767e9274c5e64387b99817
22362 F20101211_AAAJKE bronson_j_Page_161.QC.jpg
63f922c69458bc2deade3ec76dba6910
7e506f7af2abcf43734916780915278d28adf21d
7147 F20101211_AAAJJP bronson_j_Page_152thm.jpg
5e3f4cdf932cf2993efae9fe9a008b38
9faf5152afc0c0794c5d8a84efe2121d635beb7f
F20101211_AAAIFZ bronson_j_Page_043.tif
f4b89cce872c790c8d5f6228ccbc46b2
cd6c1ec696f96fcc3c52af3013c1c5d7237d39a2
F20101211_AAAIHC bronson_j_Page_074.tif
fe423cda07eb93763d51b357516e00b4
79fe5e6c8ce48e22aaa9110c921ee45d17612e31
F20101211_AAAIGN bronson_j_Page_059.tif
08acc89820bc82ae6d1628a0d83b16d5
1f6a87895eac82f970ef0a7d7b6fc9654d0ea95e
6095 F20101211_AAAJKF bronson_j_Page_161thm.jpg
c59c02635f881441f258887d3397e4ea
d7c568248b18a2fe9b548979abf3581f165b05f5
5651 F20101211_AAAJJQ bronson_j_Page_153thm.jpg
8682f7e27961da3f36358bbf42528717
ffcf72fb354b045c32dad3aaa3dd63e095d9527e
F20101211_AAAIHD bronson_j_Page_077.tif
84377a82053fdc5b90fa3d20664ecd8c
5764c0804b76bdc4793657562eab15257d56d536
F20101211_AAAIGO bronson_j_Page_060.tif
ad75c4f532a2857d30df3976cc9090e4
65a5549b23f436c276632d106a34efd4183d1057
5867 F20101211_AAAJJR bronson_j_Page_154thm.jpg
9a3a1c16641a5fbe5037d7f7cde09e22
28c238fe8f149dc57700bd00eaeaf2659362d3c6
F20101211_AAAIGP bronson_j_Page_061.tif
254b0fb6acbb9d9563bf4ddacb92f3cc
772f243152dcad9c82d9867d1e9878db96792604
5196 F20101211_AAAJKG bronson_j_Page_162thm.jpg
ad70b271aa43b6249bbd204abef06b33
a0086404fdf82450939f6cf0eb1b3b2ef3347c48
17039 F20101211_AAAJJS bronson_j_Page_155.QC.jpg
615e190aa40e0836bd9b354f56bbe181
f0c3cc9761eb6e028ddb76c82c5380d227875ca5
F20101211_AAAIHE bronson_j_Page_078.tif
50535afb79a666b761b75f9ae9325f93
4409643995e509747431be23bd7b183c7d28e2da
F20101211_AAAIGQ bronson_j_Page_062.tif
fca31b3743427a98d49efd0cc39f3ae0
2bef7ae881f13e3234af006f9c5a73abcd1f442e
24359 F20101211_AAAJKH bronson_j_Page_163.QC.jpg
2189fb8601e9fed2e429a23d8cfdc7d8
fa7e7333cb0bf767367c7c4ae9c1aa013433b32c
5565 F20101211_AAAJJT bronson_j_Page_155thm.jpg
7762e19ceee72b791a72a684ca392b56
3d503919705d9e25598d9bfd068eb316a60c8f30
F20101211_AAAIHF bronson_j_Page_079.tif
c0aad0b294fd1994312bc2e35ee5daae
90a58f4e5ab1c19196fa69b4a836a8d202edadea
F20101211_AAAIGR bronson_j_Page_063.tif
d70c5affb1c1356a4283a7218e2b6963
57d47dc72b80f321a6d73021bcecfc8105e61ad6
6982 F20101211_AAAJKI bronson_j_Page_164thm.jpg
9071771d1e964ce4605ae99d15e99fcd
84148091d7885ac672b8a6a3e738a4aabae71fb8
16573 F20101211_AAAJJU bronson_j_Page_156.QC.jpg
a260aa527fea2058d81a8ec49393ca1b
3fd2735b1f8cf22243e1d6d91278c6353bdbe341
F20101211_AAAIHG bronson_j_Page_080.tif
c183fa0759e701321e7cc7b7f4cdde39
63d8d2433b07762cdf795a73ff0e38fafd4f93d0
F20101211_AAAIGS bronson_j_Page_064.tif
806efa00315463b98481a91d04c02a09
97da634441941e4efd40dc453c8f6213922308cb
24509 F20101211_AAAJKJ bronson_j_Page_165.QC.jpg
32a1c65df3f6e731c52690c1927f9278
a27382e5a2690ee83e5b5256f8920fcfe17a458f
4824 F20101211_AAAJJV bronson_j_Page_156thm.jpg
7146d55d098e82f8506a71fb998de4dc
ec9ca82f7472d7a775f5d770cf2fde55b6581a71
F20101211_AAAIHH bronson_j_Page_081.tif
531f75f1e488751514bb5fe65c2bccca
051b8f09cfb182e64505fff4e79c6297ba40681e
F20101211_AAAIGT bronson_j_Page_065.tif
db449daae26d7c9e18ed1a5e387a6e22
17b4ce559ae4ecb750e1d2f5f1eb87bcc61f85ba
7011 F20101211_AAAJKK bronson_j_Page_165thm.jpg
ca8d532cdd3e44105b4ba6dc609283c9
25d44884103c15b46cf0efb6f2c9961e2d5e2165
14842 F20101211_AAAJJW bronson_j_Page_157.QC.jpg
622237f4b061b824068e7b4ffb4f2435
849cc6c9bee1df8adc882c6f83513c6127ee3cdf
F20101211_AAAIHI bronson_j_Page_082.tif
36f11e66c8de3f7254de486d599ef2c8
242756099e4cc752232d379361dbaadee85a05d0
F20101211_AAAIGU bronson_j_Page_066.tif
678f99f5af286f43baa439d76d329ad0
9a62b73872b07458ff62860fe9c68c3c4cb26985
5486 F20101211_AAAJLA bronson_j_Page_174thm.jpg
2db25805723c4086981024f37b96fb0f
e936f3adb6c40b69c35806396cfe5fb838fde798
21085 F20101211_AAAJKL bronson_j_Page_166.QC.jpg
77c4f89fe0fc6ad42bc684b72f8e20e6
b8c1e94b08f70c62cf50d6738039f2f6cf4d1a0b
4573 F20101211_AAAJJX bronson_j_Page_157thm.jpg
e8e9aba3032d1cdcc102435c0e0a3253
9224d89f75f6d6c1da3d4f3c466c2324ce1fb55a
F20101211_AAAIHJ bronson_j_Page_083.tif
47ea568cfab3d8361e9d8e45f9378020
76b7ab716ffc978f7eca8fe4c623bb2a3059b0b8
F20101211_AAAIGV bronson_j_Page_067.tif
5a84c93eefb4273c497e4990ae853d7b
11a2ebc61352496daa90290bd3a82454983b8922
21864 F20101211_AAAJLB bronson_j_Page_175.QC.jpg
98c1d78ccf7cd52704304fc4a0045b75
733ddbaec1e5c46a623103d49023735614dc9b99
6155 F20101211_AAAJKM bronson_j_Page_166thm.jpg
f53190d3cc368be30bf102258744a70e
d17eb5a3f9fb09de2ada925abe506df6ccaf788f
21074 F20101211_AAAJJY bronson_j_Page_158.QC.jpg
4653116b40ab63cd1df13e5e7228d801
c4ead4111a20c5c5baad8fc320c724781fd20147
F20101211_AAAIHK bronson_j_Page_084.tif
b3cfb4fc8de867f2f8b089ac265e1a00
dd6e75ea5c9d879f6a98dd419875a3c631ff0572
F20101211_AAAIGW bronson_j_Page_068.tif
f6dd366955cbb8e58f88898704012f51
2af85e2faf61a999be1b696901fafc9fe32249ec
6283 F20101211_AAAJLC bronson_j_Page_175thm.jpg
372d18a61a19f46f75187bee8bc46f4a
4d522beda4705fbdf6918639f1ec497b8c2ec056
26182 F20101211_AAAJKN bronson_j_Page_167.QC.jpg
75482fe08230b0ebf35a194e08baaee9
46785c8d0e0db326f2aa45dc12b73d1c4757f3c9
6105 F20101211_AAAJJZ bronson_j_Page_158thm.jpg
e952ce73d7b67a4cde28a2ee1a2a1e58
c7764809a3ab343f43dee0d4495385d8e37f0485
F20101211_AAAIIA bronson_j_Page_101.tif
74cf191023b4584a7bc9a607493b2b0a
ac1587e0fff50d3821fa69f90ca603c04ee70134
8423998 F20101211_AAAIHL bronson_j_Page_085.tif
9a64a8f3f0bbc86d56eab2b23b047b18
8bfeb9cdeae217e7e9dbed48a87ae9d13f90342f
F20101211_AAAIGX bronson_j_Page_069.tif
ef74540395411e88d58ccaea6ebf8358
0a0f55962f83cce13452e22f42c0de9aaa7936e2
19439 F20101211_AAAJLD bronson_j_Page_176.QC.jpg
43389748d599e60188d1bec46b6389ca
f4b7a92f895e9b2ea38739212f59ef933c5d8bd9
7018 F20101211_AAAJKO bronson_j_Page_167thm.jpg
88c142fd877c42b97b771255553efec4
80845391db05c59832564109723422e97d830c02
F20101211_AAAIIB bronson_j_Page_102.tif
4fcd789b3732ac3f919e14293c0616c2
699ca5fb8572f7e5afca7c010b5b7f517a5590de
F20101211_AAAIHM bronson_j_Page_086.tif
f1d2d2316be5f61f24943ba636b81006
e3c01ec0f5911bc4d21c2f6ebf23005bc7f98b90
F20101211_AAAIGY bronson_j_Page_070.tif
540c68e023c34c57f763d81387c14681
d115276742938661a79625962ed75ccbaca14082
21861 F20101211_AAAJLE bronson_j_Page_177.QC.jpg
08ebf437d221df643f43627d0c0615ee
18894dd88137bfcb79cf293cd196207f830f97ba
5457 F20101211_AAAJKP bronson_j_Page_168thm.jpg
c68a1d148b19307fee605c0e5fb25550
fe1885a06a9f7c0ed3ee49291b813c637eb7d3e9
F20101211_AAAIIC bronson_j_Page_103.tif
c36b30868883a56b43de769e04b79e6e
bc04466ae98f0bb231e3cba22652414228b1a9ef
F20101211_AAAIHN bronson_j_Page_087.tif
91caf84c6f062f68e56f1c2b04bec1e4
8aed1bc8fe00e691bdea7b6e2f67e7e1cf0f200b
F20101211_AAAIGZ bronson_j_Page_071.tif
fa9aca8f36f7e76a1f99685d044b5d04
1dfea57fc5e357b671db8fc641389e8bac8f4abd
6428 F20101211_AAAJLF bronson_j_Page_177thm.jpg
d38f622d11f07d9bb589b42d5a643dc9
00a91da6604d6c23094932991746cfdebf8e2f9b
22288 F20101211_AAAJKQ bronson_j_Page_169.QC.jpg
14e4d18782639180cbfea1b1e1d64234
22742110bbdeee5aa997627e27dfb3ea9e21f2c4
F20101211_AAAIID bronson_j_Page_104.tif
b9542acefb88d8e7d74064a3591410e7
bdc4d8dd987f1a24a50d808f6f9756f7124dc6f0
F20101211_AAAIHO bronson_j_Page_088.tif
b76bfb2472a8c049f12dd7c60451abc9
a126bd3693d15848fc0207ce6293a5afad7a145d
21371 F20101211_AAAJLG bronson_j_Page_178.QC.jpg
b5f3cf5d415e2394b3e14244f71952ef
5ac12ecce1d704f8a432a66772b4b3834a961321
6474 F20101211_AAAJKR bronson_j_Page_169thm.jpg
7a4dc65be1dff89e2fcae5175a7c6b97
23671bf0c3839315ebdb095844e619dc7510b270
F20101211_AAAIIE bronson_j_Page_105.tif
626f515b6f7a19cb3962aaac7ac7bee5
c21fc595fcc81d92c9a2efbd880837099d31c1d1
F20101211_AAAIHP bronson_j_Page_089.tif
bedf2ba9b9b616b737bb27f1b184727d
c65b45ef166d1a969fab849003b7b9966903f04e
6189 F20101211_AAAJKS bronson_j_Page_170thm.jpg
635469af89d87af1f5e0e3b7948b4274
861d0e3f733baf5bc7073093dea64808b14cf09b
F20101211_AAAIHQ bronson_j_Page_090.tif
afaff7a5330d76a4229767d9938aabba
504c942a06f7055ddcdbd48b0419495212f720bc
6251 F20101211_AAAJLH bronson_j_Page_178thm.jpg
b9ecfa07aaf24667b48f7eb7237a0563
46a88f2a1b03f9fee0f18abe9363f6b6fcb8c26e
19145 F20101211_AAAJKT bronson_j_Page_171.QC.jpg
7f3a680356a3f9410dffca1349908299
07d55a3b1abe9d04345da23b5cf3fcb3b29f0178
F20101211_AAAIIF bronson_j_Page_106.tif
f613b8ea01e5ab05dc9c15efc604c6fd
bf6be72280c10b588a836b7e90a432a1c230a365
F20101211_AAAIHR bronson_j_Page_091.tif
22ae8313e27678d0914d6425fabce386
db41323a0a2edc532efd273bcf762a0121950f6d
25772 F20101211_AAAJLI bronson_j_Page_179.QC.jpg
301b0c8ad0e1132c22d46298dd817090
0883fc44befdb9824d6d64aa2ee260ee9486aa9b
5627 F20101211_AAAJKU bronson_j_Page_171thm.jpg
1b9add0ee8c6d23f6d6d0c068205a54e
e189cbeb6d6a2722bbc0c96058e02bd97e10b06a
F20101211_AAAIIG bronson_j_Page_108.tif
c316a81e0a7c8dbed6e70ebc0fdc759a
156d22e670abfc0357161a8b2a4447a1c85e24fe
F20101211_AAAIHS bronson_j_Page_092.tif
f3d885b940555ca94f6df6fc4cf9b712
e9b5b37f52f9a6d080908f0c550d29c4c785d70f
19313 F20101211_AAAJLJ bronson_j_Page_180.QC.jpg
1fa0f5efbee051322e320a661bb01261
d9443c8cf7ea204a55fceff4570d9895c1f1cf2e
18561 F20101211_AAAJKV bronson_j_Page_172.QC.jpg
3302d714fa50fa7f767a78be1892f607
59daaef58006fdd5fdd4a441952fc5465667c86b
F20101211_AAAIIH bronson_j_Page_109.tif
0bbf3b9dd3d68e57c349d255ad1c3760
ad724663b0b242a4987efec8f09fd22ba64f004a
F20101211_AAAIHT bronson_j_Page_094.tif
df160eea17a26b71d79d58ea8f7bd33b
7d5908ab5f803b91d83c82eaf41ba8971d9ab0b5
5697 F20101211_AAAJLK bronson_j_Page_180thm.jpg
6c22180f2cea66e1021ceb20ce621123
8c9bbb1411f71f6191670728a6f6ee311267b50b
5394 F20101211_AAAJKW bronson_j_Page_172thm.jpg
be951376076d1302cc72e95a26a52dac
716517f23884fe032c9ce98c89c879b9fd5c9bc3
F20101211_AAAIII bronson_j_Page_111.tif
39f41b66f30ad84489e8474b0a595ec9
415183ee02e2b3b156aae94ebb6242c87f7d5b6c
F20101211_AAAIHU bronson_j_Page_095.tif
70224a385362f4feb52e76dd1565cc84
ed9b63e4462902007e9ba8c7c8198be9b69cffdc
4278 F20101211_AAAJMA bronson_j_Page_190thm.jpg
ea21a36a5aafc27e3859ae2b6aebbc0c
89032613023c42c8e1f5779028d790e9a687243b
7111 F20101211_AAAJLL bronson_j_Page_181thm.jpg
ea1d4b9a2990a9a22536e778992247ac
08dfb68eab32d9f17e490fb7464f4c58aa05c650
16081 F20101211_AAAJKX bronson_j_Page_173.QC.jpg
c1f6ce7341b343b30c8895d8bc198175
2906b11ec46497f39515736e8574070366117d2b
F20101211_AAAIIJ bronson_j_Page_112.tif
c5fe4f63b99e8897916c025bba8ba733
32832f0d138edba07df1e9d322a378c260ce911f
F20101211_AAAIHV bronson_j_Page_096.tif
52f0e785ea697d828460b8afc6b85de7
f03b7b5fd8e7f990d2591e8de4ae48e13cbc2584
11185 F20101211_AAAJMB bronson_j_Page_191.QC.jpg
b722c23b00dadda415acd13c0f8d6a54
a753f347a0bb196725631d75dcd8a1375f90bdd5
15902 F20101211_AAAJLM bronson_j_Page_182.QC.jpg
779f3172a2a05d511901c9782eb0645e
72573f953f7dee8a7b1b49ffd475a1bf21db974c
4815 F20101211_AAAJKY bronson_j_Page_173thm.jpg
aabb42fed430652441213f78bac71b99
6a4c08554d4c4334ccdc40a4e3e0a42fab1e6812
F20101211_AAAIIK bronson_j_Page_113.tif
ed4caf8ad289a3b028fe7d646873c1d4
02c76031f08c9f5bfa528889d282388e25a07789
F20101211_AAAIHW bronson_j_Page_097.tif
f0dd0c8e177abeafc71453006ffe2de8
4731e4945f1224a5ace2831e1a6afcc78714c619
11807 F20101211_AAAJMC bronson_j_Page_192.QC.jpg
72eb757dba91483b7cfff825163ea47d
672bbe0e797d60c4d0c0eade109ddb48bc0fac51
4396 F20101211_AAAJLN bronson_j_Page_182thm.jpg
3b7084e68db5de677a76f6afb87085dc
5537ae4f60770acadc3c354f2b96a5493b9dab9b
18713 F20101211_AAAJKZ bronson_j_Page_174.QC.jpg
1511d38c5e0e79c87b493ae5fce0b4b7
833f992b6a50e62782e304ae34cf2446dc1423d0
F20101211_AAAIHX bronson_j_Page_098.tif
2be179951cf760f06721c005c8c7a12f
0fdeb1fe20d63abec873a4b29615d48ce905e078
F20101211_AAAIJA bronson_j_Page_130.tif
1f0c66549ca94d7e9a90be6e1c769527
ca5fd5d92a65efe41031431208014414e64e85e2
F20101211_AAAIIL bronson_j_Page_114.tif
17a5b6a2cb12881fda6e39d7f975ae2d
776d30aa82b557ede2e4c78c7595e3966f678f18
4365 F20101211_AAAJMD bronson_j_Page_192thm.jpg
82cb213c40cb69c0fea123b7d38cd621
591a613436e92eabb6a0e05b268d005976e38501
25039 F20101211_AAAJLO bronson_j_Page_183.QC.jpg
e01217886af6362d610b60fce2fe371f
56c0574205c9e9f181d96756076cfeee27007196
F20101211_AAAIHY bronson_j_Page_099.tif
7c67af8d1cd19f943736d0c7ba673be4
c51e616572864f2fd324dcf8c85c784419b0ad59
F20101211_AAAIJB bronson_j_Page_131.tif
7912c4f53603e22c0f5c74f3606c7353
71d1582a9040759c8277018ac77705497def0524
F20101211_AAAIIM bronson_j_Page_115.tif
fb5fe2b73a54754ac98de625289c3c1a
4ed18d96e378528b46f5ec92bd2387410c6eb76c
11412 F20101211_AAAJME bronson_j_Page_193.QC.jpg
471090f309a7f905f38757d688a48d6a
e4346b3eecc2cdf0ddc4ed5f0db650816b9d62c5
26106 F20101211_AAAJLP bronson_j_Page_184.QC.jpg
a09d1890b3f296b4065956cc232be3f4
9cc74cc0e1522b69c2bc94d106b55d00daf26b26
F20101211_AAAIHZ bronson_j_Page_100.tif
fabe0ce8e672c1279c0785fa2068cdc7
fa58130d6534d760c5488c0347997eddb3314075
F20101211_AAAIJC bronson_j_Page_132.tif
c3d11b557cfbcd1fd21c7bbe128154af
d6ea35c5494523c3e9033e64b0271a4c7c0470c4
F20101211_AAAIIN bronson_j_Page_116.tif
056cffcb74f878b8025b737838e9cf47
ac0cf734fb5091ca5b73ad64f9bda0986b8319c4
12424 F20101211_AAAJMF bronson_j_Page_194.QC.jpg
bdeb2db4cea09833d819d42e892b40be
3d569326c404cfdf2ce57d820e429af888f77b69
7021 F20101211_AAAJLQ bronson_j_Page_184thm.jpg
3db3ba25d9cccc61763ce69c83aee4d2
ac12188d5892a4879393e1d5d454ded926dd5f59
F20101211_AAAIJD bronson_j_Page_133.tif
33ee9a8f2b77b43aecd17132b291d83d
e344e24da97ba30ac1a796d78560a8925a66b54a
F20101211_AAAIIO bronson_j_Page_117.tif
54c5ae79dd7b8451c7caf4f00fd3d140
5ab5f50722ff798d0822f8cc7331c13d9dc973cc
4157 F20101211_AAAJMG bronson_j_Page_195thm.jpg
813c183654fbea1bc831639cd2d79a21
7e9d6260c700b6ad8f397f8932eb9eae7e46fccd
25735 F20101211_AAAJLR bronson_j_Page_185.QC.jpg
be61845416291336fb54e909e6c868b6
ee59eadc2e3d88921631f1267ac58510a4246108
F20101211_AAAIJE bronson_j_Page_134.tif
3aa24a70db17c5d6c0af999c4a42b097
2635af657bb6f53479dc13ef4820961b5b7d5a58
F20101211_AAAIIP bronson_j_Page_118.tif
6413975eda1a835fb6aadba747594556
44f061caec82866aacc7d70fe03df6f5048a1a3f
12541 F20101211_AAAJMH bronson_j_Page_196.QC.jpg
f23b9c25833e37255fce7ef064ef6cfd
101bcc7fcf830bbc1ca9e01d309c59e26ba94035
6716 F20101211_AAAJLS bronson_j_Page_185thm.jpg
d57566c126bcf4398621fb4a45fcc4b8
55ed0685c55abf78a499427780e8bc5d9a2af34b
F20101211_AAAIJF bronson_j_Page_135.tif
e063dc4ef436549e5bc7d71da6017e48
0194f87f43c9a7a842888729cefae3f16e51a9f8
F20101211_AAAIIQ bronson_j_Page_119.tif
253b62da7f9d2d2d12bff51f3f097e08
5c714aa68ee6731f625bd9e20564e05a78fb68dc
23543 F20101211_AAAJLT bronson_j_Page_186.QC.jpg
7441bf35eb3b44f49aefbfc0363b27ee
8ac5b096bc4724945753eeae9a6c34e548321b19
F20101211_AAAIIR bronson_j_Page_120.tif
91597a2a4ab755d1f9be9155f625d99b
a27d4e710532219a30a52c1b8e50b027c744a9ab
4454 F20101211_AAAJMI bronson_j_Page_196thm.jpg
6a0e1c9b9a2746de8d7bcbe81f287c55
c4d8fc79bfee8e83c049f4234bdd79a36e75555d
6516 F20101211_AAAJLU bronson_j_Page_186thm.jpg
0079f33812c834d7f069e51a88948633
093f06fe16ad2361df8f17af763558547f27e087
F20101211_AAAIJG bronson_j_Page_136.tif
eea3ad686c5a58602b963c0a72b99aca
f44e19ba6d51b7e6c9225a65ea642f11c323cd57
F20101211_AAAIIS bronson_j_Page_121.tif
e2241094e2bc6b5c0492b8c0e4ab22bb
08a39d4d27b7945776abe3bb6b7a9a4c28dfc6ef
13843 F20101211_AAAJMJ bronson_j_Page_198.QC.jpg
01fbd7c5b7d5fcdeafa05bf721645432
d284c7812277013a37c7910531355b3ba2811566
3060 F20101211_AAAJLV bronson_j_Page_187thm.jpg
3863155b2b8ae42e7ff127d3836de12b
55dcc19cc42f7e4b83354e39d291632031fcdd9d
F20101211_AAAIJH bronson_j_Page_137.tif
417a824b041f5b9331684b87642f5651
6944c2bb565c992e9c468692c10140260499503d
F20101211_AAAIIT bronson_j_Page_122.tif
12be50a0eed5c3ecd18e89f98dadb53a
d3d931a0aea4082af5fed6b1b59a1366b8491828
4514 F20101211_AAAJMK bronson_j_Page_198thm.jpg
f99aa1937738dfb6c6475ffd8b61d9e4
87a0c4bd80efc6839b7482b259145a8668b32056
15652 F20101211_AAAJLW bronson_j_Page_188.QC.jpg
ac18ed6d8ef7970f1cd1762f9fa72cfc
05cea634ff4e1ab4f94a75702d437c1a10eb2e20
F20101211_AAAIJI bronson_j_Page_138.tif
f80d560305b64723735e653242bcd6bc
61025a5ec7395ee454d19d5287cb2b20721ccec7
F20101211_AAAIIU bronson_j_Page_123.tif
199e4b1adbbdba5bc7e6cf1338e1e79a
4242f6133a34b64bf068c28854c1da48475980dd
7383 F20101211_AAAJNA bronson_j_Page_209thm.jpg
d18e43d6ddcef75cc16df6e147b79d69
1b153e4a2f317983d600616a06ba93a0529859b1
9000 F20101211_AAAJML bronson_j_Page_199.QC.jpg
bcbbe9c199099f7b89d02189551b6a27
5634ee8df7d197826c3b0a77804846bbd09bab18
4874 F20101211_AAAJLX bronson_j_Page_188thm.jpg
1681db4b9f090e23e9d0211c658aa29d
623bc8629f0fa580311daf34cbc0dda1c012911e
F20101211_AAAIJJ bronson_j_Page_139.tif
1772cfd7df5daf906d5e045927ef87fa
45a6d77c297e3b823137b4384761a3caea19bf88
F20101211_AAAIIV bronson_j_Page_124.tif
92e127b95557e00b3c69f450fe24717e
5098649042b0053fc3b1b777038dc3028c1a5e45
27737 F20101211_AAAJNB bronson_j_Page_210.QC.jpg
5e92ede76763fb06e01002f013511098
2d2e64408b019b2efdba38ab344322f62d073786
12428 F20101211_AAAJMM bronson_j_Page_200.QC.jpg
3cfdada2a1d794d67bac5d2540c8d47b
0855459a3c954a65129fbad277065ce55df4a9a3
4097 F20101211_AAAJLY bronson_j_Page_189thm.jpg
4490f32ac8b801bbf1403cf7cb1b2447
f1b5033fcddcc17a03cdb3703f90fb85a4ce814b
F20101211_AAAIJK bronson_j_Page_140.tif
626805ac70485d71d50d69d30ef242b0
29aa966b288452d77383c46c521b129b98a11d65
F20101211_AAAIIW bronson_j_Page_126.tif
e81b0bec57257e33f17a90f40874be8a
ad1c141cbdecbba27f66a949c693fd166084f3cd
7423 F20101211_AAAJNC bronson_j_Page_210thm.jpg
7b950bfdbf20857add374c2ba10206fd
e86c346ea0b04356cfdf5569a64d32759c086dfa
4067 F20101211_AAAJMN bronson_j_Page_200thm.jpg
c0b550ed493dd1953e86455d9d17d32e
32ed814acff86d9b0a603fc02314a2b21740bb25
11194 F20101211_AAAJLZ bronson_j_Page_190.QC.jpg
08dd340f03b0646089d4a1d0722076d5
b268920f0194c1b4e58fa9cbb52979f879cfcb38
F20101211_AAAIKA bronson_j_Page_156.tif
198b622fff63d1f69a35c07506fdd00a
70517b8b7e7c4248beceb69a6c203790401496be
F20101211_AAAIJL bronson_j_Page_141.tif
8160649240db0402a8fb90ec4031756a
bd444b99d92003f105609d02bc54b619855428f5
F20101211_AAAIIX bronson_j_Page_127.tif
d7fb6ff72584aa7c49278d5501af1a9c
69a0990e7189d403bd9c17299845a28db08f1e4d
25615 F20101211_AAAJND bronson_j_Page_211.QC.jpg
ad013441cc9b6d2e8232489baedff48f
ed27fac06c59f0828d5aacd37b2a8786cd7c311e
3649 F20101211_AAAJMO bronson_j_Page_201thm.jpg
a555bf5baa1fe7e9f3a006afb749de96
c8ab9d27657c9d16b64e356763cc15b7615ea62e
F20101211_AAAIKB bronson_j_Page_157.tif
5a14563c5629da5991a6d0297a0ae9d9
2e3c5a2c5890e29cecd86e27ad8be3ac35d717de
F20101211_AAAIJM bronson_j_Page_142.tif
52658376fa274d35d99a59aec6ccc414
a3b7c1f7da8e3ab89501092672c40ac0dfd5e464
F20101211_AAAIIY bronson_j_Page_128.tif
bb5623b8998c6abd69a2e8198e69a963
5d7b5b3283d82b8f8e530126b1fcd1e82ab1a661
7065 F20101211_AAAJNE bronson_j_Page_211thm.jpg
9f5a967397e55974cc8d050cd6b979a5
74cb5f3d39bdde86dd0bc64c48b8a19d098d3b9c
11205 F20101211_AAAJMP bronson_j_Page_202.QC.jpg
5650d6fd628717717e53904a8e5bf1e7
3a2e5cbe7a1660f01d267dc4b7c607ba49b19187
35609 F20101211_AAAHHA bronson_j_Page_158.pro
5db950e611710175197ba641b3fce5ff
6a70a30af833912f5e11158337d372342db56d59
F20101211_AAAIKC bronson_j_Page_159.tif
be88ebf48e7ebb9682656ef14a32361a
35bd13036d71170747e3b48b40a19d41a171ed99
F20101211_AAAIJN bronson_j_Page_143.tif
a927414c8b84a448f884b3a78b166e0f
644879012f9957fd68274c073a010806b9870d92
F20101211_AAAIIZ bronson_j_Page_129.tif
0b4c9d08f77b7bef60667587a3f4315a
d20322f7f4b033c7c01fbc3f0f6a33ded0e7645d
25103 F20101211_AAAJNF bronson_j_Page_212.QC.jpg
d8e5ec4e15edbf8071e4321808acd6d5
777546225e51b829bdc2ecd36ad244cde91ec7ac
3723 F20101211_AAAJMQ bronson_j_Page_203thm.jpg
6fb525221315e453ec264cfd9e70e1ba
d99cbbb0d2dd3e1b02745bccd62caa7088b3dba0
F20101211_AAAHHB bronson_j_Page_145.txt
8e7afce91c9065dc4eeec4d2cab412c4
e0f261a5a6ad30c47bbb2c180d57ca12e66e7da3
F20101211_AAAIKD bronson_j_Page_160.tif
66896dd4ff0a9e610117e8507017cfbf
47f7f3a54ada8b6431bfe54180e63b48643214d8
F20101211_AAAIJO bronson_j_Page_144.tif
928d774c83eecf85e0abaaafe25d0fba
d7fc3ccf5b84b031f46c5a4155b51cccfe034a6c
6963 F20101211_AAAJNG bronson_j_Page_212thm.jpg
31f33b6ccb2807591d28f2ca34bb85c1
7e354ef7e67bbf6a365b3d11298d6267b6c29caf
11519 F20101211_AAAJMR bronson_j_Page_204.QC.jpg
5870ee92fa5fa5f708569ae26c04b236
11da4554690e2ae6a3435680c8359c4800fa0775
56793 F20101211_AAAHHC bronson_j_Page_073.jpg
4887975119c5a4f4b6405031c93ae6da
d6b703384543cd74678289b7dfe8c4f72f3f40a9
F20101211_AAAIKE bronson_j_Page_162.tif
12f0fb5b81a8776d262bcf690bed68ac
d98f6293513943bd425c36873ef0e0c52515ca8e
F20101211_AAAIJP bronson_j_Page_145.tif
72c83e38984f075deaaa1b26e13b825e
916ae923a2c65111cd17215680a5c44cced68d68
25048 F20101211_AAAJNH bronson_j_Page_213.QC.jpg
57c3d11f4e2c49f4067e1ce4f9020ee2
ac428088614dc8d42c6ce4d088be3f70c00f3af4
F20101211_AAAJMS bronson_j_Page_205.QC.jpg
e7ec141080fc528a7cd6de4dc844c172
bf04b5a8d3d23b43d389160dc23b5becac74efb7
762554 F20101211_AAAHHD bronson_j_Page_172.jp2
c8b39dade8c241d0f916ea3065f68fc3
8199074738dc617d6a1e7529cb3118218a3b8b23
F20101211_AAAIKF bronson_j_Page_163.tif
570ef8ab2bac6a91a60f8c437bbf1453
c55ddc251e448dba9921b2eeca1858e3362b671d
F20101211_AAAIJQ bronson_j_Page_146.tif
8276db457daf9f6f39031f94fb51e600
466fe384bf88ca0eac8e4860c6e1fce8a4858cc7
7121 F20101211_AAAJNI bronson_j_Page_213thm.jpg
b8c25992fd8a664bbf8b4353d1c22925
638f72f41dfec010b7cd8943d29b36bb05c53835
4084 F20101211_AAAJMT bronson_j_Page_205thm.jpg
51db0234686d5c261399ceb7a4502571
9679d85641a468b1dff71d51598d55dca3cbb424
1858 F20101211_AAAHHE bronson_j_Page_046.txt
2e98eafa5dfccd031508acd2b3efbdca
dec00b9296884bc7053a70297cecba5a979652ad
F20101211_AAAIKG bronson_j_Page_164.tif
c6ee1325c7372288d1f5dbc1707a4c2e
3331afa4b9df01a8a5636a3b7c2abbafc70a8be1
F20101211_AAAIJR bronson_j_Page_147.tif
2e4081ebe92c49a933f2fbc44e243020
355699e9fba15436ec3fcd67f16ce45e3b5a2978
12043 F20101211_AAAJMU bronson_j_Page_206.QC.jpg
0a83091cc70bae39480a7f6e177ea7bc
07b2bb006545c1cedb2ae7d4068c9b411a941f4d
F20101211_AAAIJS bronson_j_Page_148.tif
4ed99e795ca516ab49e6364b0cf32e41
910e52a09c4d8feaed7b726362b267b7018c41ca
25520 F20101211_AAAJNJ bronson_j_Page_214.QC.jpg
0fcbb490e3bae522559f5597d44e02ee
efe1c9aa7f80850968b2c418dc41c58da1e4361a
7911 F20101211_AAAJMV bronson_j_Page_207.QC.jpg
5612271dfa9a187d149edcd7840aea60
6a81fc1beeb2e0d9bf8875192b78b6fa4ff2baa6
78327 F20101211_AAAHHF bronson_j_Page_043.jpg
88e0fe4a7040cfb6abe86f5cd9dd1f38
46c8dbba9d4449caa50823ab7597a03b05a18cc9
F20101211_AAAIKH bronson_j_Page_166.tif
9d240630e971650c74fb9a468f16d485
1588ff63388fac49be9b788aebf5913425950a24
F20101211_AAAIJT bronson_j_Page_149.tif
837f69a813a5f90c2b693b0225b6bb90
31be4bcd9c2336b5a9006dbefdaf953a3653c618
25379 F20101211_AAAJNK bronson_j_Page_215.QC.jpg
9d051b6f18015acafe4e8676af5306ab
7c73bce9f31df31e100ad71819c69f1f9b44b517
2715 F20101211_AAAJMW bronson_j_Page_207thm.jpg
f84c5ad721bad9a17ff7cdd43fde2fbf
a8eb8253499533e0f344a4afa533939239d03bb9
1449 F20101211_AAAHHG bronson_j_Page_068.txt
42da71ff35f2f444e454c19960dd9af0
2412e587d29854943bea0e8289b44e8eb2788f75
F20101211_AAAIKI bronson_j_Page_167.tif
4a7f993cf1d7e614723355617e513c33
715b0d7edf96c57d2be8b8c6e1899b398806d95d
F20101211_AAAIJU bronson_j_Page_150.tif
32f70774c269c5db04ec1614da761cec
e2a2440d25036ad963a7b3fac0d0ea3aa54e28bc
6749 F20101211_AAAJNL bronson_j_Page_215thm.jpg
44c8e32c5aa700c99bf44370effa6f95
993a4bc54e9ae0c258f67d36f6d222a95602c2e7
27148 F20101211_AAAJMX bronson_j_Page_208.QC.jpg
a80c2638b38c1d204ef044e7226a2f40
53ee243df0fd1e4875b813b71e797231e5ebe9c0
1305 F20101211_AAAHHH bronson_j_Page_003thm.jpg
961b42826092febaa32e72b85c7ea937
9283fad0e72ebae0c57e28a58a26249d645bee52
F20101211_AAAIKJ bronson_j_Page_169.tif
65fe2b694cee532abf315e9e05ce13af
8634802a40a8e9bdfd16843c1c646aa916cd4253
F20101211_AAAIJV bronson_j_Page_151.tif
63029b3472bdf0757453a1451dd92d55
5f7fe3170e0893d849d932fd38566326e33ce143
24655 F20101211_AAAJNM bronson_j_Page_216.QC.jpg
a927493ccc481fb3c34367a97a980695
aa80249f7f5e4bd0e5d0e69af591fe5b04d8bb92
7150 F20101211_AAAJMY bronson_j_Page_208thm.jpg
301698bb09c9444825b03887d93e40c9
71fcb005fce52625e85c1070a05aef8745401000
4426 F20101211_AAAHHI bronson_j_Page_003.jp2
26c9cd99843bf6b7b84387cca28d9533
9ab6e3a2c3698d1ad084d4debeef342bb1852d0c
F20101211_AAAIKK bronson_j_Page_171.tif
06212e4c32bd8436a6d5cfed9059df5f
343be61a1fcd8fa482dd617561ce74d658453e13
F20101211_AAAIJW bronson_j_Page_152.tif
464cd6b8b94a4152f1be58c1f1d5b300
ebcb8e715e68381d0a8f4da59c798e672e309b4a
6783 F20101211_AAAJNN bronson_j_Page_216thm.jpg
159e44872988538cbf5ed9d4943aa4aa
e00afbd5839fcf863fb10724f5c15828e972d3ee
28934 F20101211_AAAJMZ bronson_j_Page_209.QC.jpg
8a027fd77a3c018b642d1f0b2a081085
296c0194721a8c88ab83b870d28b4c685c0ead29
1036948 F20101211_AAAHHJ bronson_j_Page_085.jp2
b38032b4b0dfcf9f5ed80941bd3e4355
e68a14adddf22d496f0ff7c733f3961c0897f018
F20101211_AAAIKL bronson_j_Page_172.tif
82a726fa768f95b3e863559c46cb5f9a
259513444e55cf54bd1d0b88d54f31bb86a38805
16231 F20101211_AAAHGV bronson_j_Page_136.QC.jpg
e36a6e7584c51a082cb05b932db9081b
bbe58fa795e91a1d6216d621b373737b53219c43
F20101211_AAAIJX bronson_j_Page_153.tif
fea95e0e18da15ff525b1b96ce9da40c
0cc6a80b51232df5383f4f35af51caec374ff71d
F20101211_AAAILA bronson_j_Page_189.tif
a908765a3ce79327bd695d094a14fb3d
6f0c330af7adcaf975b593ac6154da7725f2b64c
6988 F20101211_AAAJNO bronson_j_Page_217.QC.jpg
b3a484a97fbea29a04020bf2a6dc1abc
ae8a1a82f9dd2fea1429f2cb58a8e2425140c45c
F20101211_AAAILB bronson_j_Page_190.tif
a3acc0043873ae4a3f37b690c42f78ab
be81c8eaa54c2a2454430b457254305dbeeadf7e
F20101211_AAAHHK bronson_j_Page_057.tif
6774231835a9cac581a39da8291b089c
3d623f77296f6b70d10f5c30661180c6238a7b3c
F20101211_AAAIKM bronson_j_Page_174.tif
e18f02fa064957b3a1b217060b8ea771
8e183c6a8765d295771f319f5751b4ddfd9f3084
637259 F20101211_AAAHGW bronson_j_Page_143.jp2
758a73e06464509587be8d465d510f3a
b888372749a7863c0f2d98162045d53b607bed41
F20101211_AAAIJY bronson_j_Page_154.tif
2cf5abce4199362fbbfc1a73b8d559f9
71fd26ad53be6718ba23c9af102612d106541c88
2209 F20101211_AAAJNP bronson_j_Page_217thm.jpg
369466c504f4e733e2b00cfb19368110
4bfe7803b40b0c1a717a5b06fc77f8e8bafefaa5
F20101211_AAAILC bronson_j_Page_191.tif
223349859fb2501a3af0b0f403ed7e84
b54ac564ff810a04460a069fdbe0e2aa61b68553
64669 F20101211_AAAHHL bronson_j_Page_214.pro
29ba8e30a317e08a142618bdb1d115db
223084bb295b30e61166b33766f6d4c6af7c9af5
F20101211_AAAIKN bronson_j_Page_175.tif
b66542b3ec20aaa2bd793e201d0e346d
c6c165942f4ceaa5fc012b86746a6a0721e60afb
121748 F20101211_AAAHGX bronson_j_Page_033.jp2
56dd5c65fc81fa8a4b9faa95f01ccdf6
e05b38b5e31dcabbd1dc5c3c70bfe30712566257
F20101211_AAAIJZ bronson_j_Page_155.tif
3029b377b22b3257353411bd00ddad69
78030ef463682bfab451f9c6902cc9f5c14b9803
16945 F20101211_AAAHIA bronson_j_Page_063.QC.jpg
5dd704628c69586169704670fb77aa7f
fff086ffc181e7d351914f1f90d86bf25a1564d5
F20101211_AAAJNQ bronson_j_Page_218.QC.jpg
5c843476bf4f51e88baeef364577d616
c130fe390f6f439bd8a5f10a50237ab209e30112
F20101211_AAAILD bronson_j_Page_192.tif
a0943da9c962cdab040867a4b7f0b46f
1a759de87ba54129ac6fd360550cfbf50fefd412
116838 F20101211_AAAHHM bronson_j_Page_125.jp2
210e6279a60647f3a56d81660e1af392
77f74b26b34fb60c9d8dd8a31bf90ee520856c68
F20101211_AAAIKO bronson_j_Page_176.tif
a894c4ce14fb902230b36774bd23c8e1
1c58c5970f22f7d2313ec4f1585eef82b0eca9dc
23035 F20101211_AAAHGY bronson_j_Page_068.QC.jpg
cbfbf588504e0379d265b55da08843c6
1d70c761f17a6ef97bb9a257280c4b1a57fe8e65
16145 F20101211_AAAHIB bronson_j_Page_147.QC.jpg
1a8495a6c8d9972eb9a7efe005835509
f405ed8bfb9fbd824271a3b6d91e3b66c91a65b3
5110 F20101211_AAAJNR bronson_j_Page_218thm.jpg
bda8cd6ce55898afc7710f77ec1cd0c7
6987c3b1b4630a4166f40add05ce6002d059de3a
F20101211_AAAILE bronson_j_Page_193.tif
ff75134db26b7a1e8068c1faec2f7ae7
b459cdeeab7632a5e83733e300c1e91faff20ca6
F20101211_AAAHHN bronson_j_Page_009.jp2
dbef162d90f983edeb988595dde9b170
b88ddb7492a9c205b70c3474b72f8b19a4be985a
F20101211_AAAIKP bronson_j_Page_177.tif
6cfbc976035991154652d8d5fb9f6203
d510b8eb1c87a16d9e7761f27a1809016d5dc03e
63558 F20101211_AAAHGZ bronson_j_Page_215.pro
3fed88cc05811d3b3c2a962f5582d219
cb2a0732489259c72855b85f6a0a102900124f84
1388 F20101211_AAAHIC bronson_j_Page_048.txt
458407534ed46a764c778eedaa7c633c
cbf8056476fb4b09a740113ab033744f081ac387
F20101211_AAAILF bronson_j_Page_194.tif
75ba584f13b419520cd5ba9f9bf57c3e
dddae233655c95bf6039b59d53c1360a5ac43202
52098 F20101211_AAAHHO bronson_j_Page_163.pro
7ef0e6d01affd93b2338d879ad58fafe
c8d563297ed8b7faa9757d205a1f235c42ee34be
F20101211_AAAIKQ bronson_j_Page_179.tif
589726be42cb70bdf48ee9b4e7b80417
b3661615544142e222d2d314bdbac0ae7f71ced0
24875 F20101211_AAAHID bronson_j_Page_053.QC.jpg
e077e7222ccff6d581f3643bde3cd8ec
84b41f0e9782d372e4e2aef123d6115e422c312a
F20101211_AAAILG bronson_j_Page_196.tif
321b0a39eba8423b5c51ffd7d3bca6f0
7cc36ce748b0c4357ff1997c89ec645698b56f2c
1760 F20101211_AAAHHP bronson_j_Page_066.txt
70a9bc0641db571600534382ab49ba45
cd879bedc847b46c5495a4dfaad188070012b614
F20101211_AAAIKR bronson_j_Page_180.tif
e9baf1e67c942572f07cfcaab41cc296
ce4dbdc14c4b8252505a4ab13682798ac7cf488e
78045 F20101211_AAAHIE bronson_j_Page_084.jpg
ace7de6e3466a013099d8c49202362bb
81d17965ae846cfa8b1268078c210ef8d5c7f318
F20101211_AAAILH bronson_j_Page_197.tif
26eff8b8baef41e6ee86e8554ec6c018
924e463624fdd865414e14ed604baf2b1fbf6d4e
77960 F20101211_AAAHHQ bronson_j_Page_050.jpg
471bf5d4313e77e10da54d382504502c
146de9a0cbba81e10c4952b492909a5a1d223d2a
F20101211_AAAIKS bronson_j_Page_181.tif
2126aa9dd6fe6829c3670ce9c0b0c089
7f44c22a7179512dc6c7559b52bea78a0611410e
6800 F20101211_AAAHIF bronson_j_Page_163thm.jpg
1c948091741a96e228a38461f2681eed
7b8dcb72a7cbc1e2f4bfc65f04d8ccbd685c9c0e
25241 F20101211_AAAHHR bronson_j_Page_102.QC.jpg
70ba2067fa8400153c1e80829591e462
efff521626fb27025bd5a750bf31ab9b29cddbc4
F20101211_AAAIKT bronson_j_Page_182.tif
b8af91857c2c2cb1e81c5cb4fee1c31c
652fa733e2a99cf7bfbf95a11233984a024fb7ee
F20101211_AAAILI bronson_j_Page_198.tif
c6db1f4198fea2fc882ea0b4b41dc397
3e621971ef5f3406968dca2d9c2b55fd0c3f891a
60406 F20101211_AAAHHS bronson_j_Page_176.jpg
719a78c9ead329da80c99c05b0188f59
8f535d82cca28dc51d82d33f3ff48560a2963199
F20101211_AAAIKU bronson_j_Page_183.tif
1330081397bda5b307096935296666bb
37880b1404ce6ae0cbc3b0257d164664057622a1
6116 F20101211_AAAHIG bronson_j_Page_067thm.jpg
b2cb6781a8034d31f904ecb82c54a7bb
bd7899354ea72a0dc66b45153c756320ebf16482
F20101211_AAAILJ bronson_j_Page_199.tif
52f8d5890e343f060208ca9b47660580
9dd93b8168d37807d646ed9d215aea1c123b2a87
4748 F20101211_AAAHHT bronson_j_Page_095thm.jpg
60f0d192b63307c4cbd2e52913acd9da
c45c9eb67da30a1107a0b8bfe299589e435bc4b6
F20101211_AAAIKV bronson_j_Page_184.tif
16b1431d963ca48d09724cb9231b77de
9593185ef2100f0e451f85f499ab28c3d83fc2de
1285 F20101211_AAAHIH bronson_j_Page_204.txt
41b8c4b97f8e661acc4574b800021d8a
601b5c4e661a2770fb9a906066cbc4b7b686f75d
F20101211_AAAILK bronson_j_Page_200.tif
c98431be7d41f3d700ee3efbff355285
3cec2893c0f8dcd1245c17601d02d4ef4d41430c
113568 F20101211_AAAHHU bronson_j_Page_035.jp2
efc36f7fdf0f875e65cb1dfd6eda0cfb
0e8b98bb9f459b10d944047d714dd5f8ceb02878
F20101211_AAAIKW bronson_j_Page_185.tif
c6ed831ce43a8bf982c9b35e5fdd1c7a
62591d2e8211994f993f707a5a82d8bbf3ec7987
1859 F20101211_AAAHII bronson_j_Page_160.txt
8b4af84361ff35af4673fedc4683fc9a
c1ac673698123747746b9294939f216ffd0b7bab
F20101211_AAAIMA bronson_j_Page_216.tif
5b2517b3ecdbccd41c7c72e8e1c15fd9
ebafb239191e59b199743073aa06e33fde9bf95d
F20101211_AAAILL bronson_j_Page_201.tif
aeb7d213ad718bc30199fd0a032c5d91
3cf7d26d2bee4b8a73376c052b0bfdcb93883e43
F20101211_AAAHHV bronson_j_Page_075.tif
f9b1a0ab127c40efb7a9686c0b9a08c9
805c5b1f6cc7d07f525d689be48560de36e0066f
F20101211_AAAIKX bronson_j_Page_186.tif
0e882ad57430f08a9f411937c82d6e40
b61ec4f5f3cba711763c14a8322b5e4c96ed28e8
37513 F20101211_AAAHIJ bronson_j_Page_112.jp2
eda1aa89649df57bd2483a6ef8520c10
536485f2b3916b32098be7c482bcc0336f567b9a
F20101211_AAAIMB bronson_j_Page_217.tif
acac5d13128a7de0247735885cd0edcc
2ab8eb0f0efe4bdc90bbc5ca7f272740ded183c3
F20101211_AAAILM bronson_j_Page_202.tif
471492ba789781ffea8e93f3c055b61d
c5bd23e7e88544a770efdf5b5259ccd96fbc6779
906075 F20101211_AAAHHW bronson_j_Page_070.jp2
320358e025d26db6361298c50bd67ca5
b94a14df013ed3db5d214cbebbb58025990e2246
F20101211_AAAIKY bronson_j_Page_187.tif
eb1d437934e001ea5da48059873bc0f4
7d6d863f398957da9306bf4bb29cc6230f3045f7
1439 F20101211_AAAHIK bronson_j_Page_104.txt
2ea9bb5043561ab7c1fb0f26ac685dce
f17e0c2b1a9f949fa353f07526e03021167d4619
F20101211_AAAIMC bronson_j_Page_218.tif
5ee4b832f6b47a0172e0d342688ec434
46b68a1167ad47260f765ea7311d3f0d80cb0f43
F20101211_AAAILN bronson_j_Page_203.tif
501802a6c59d6c523cf3d3a678a13818
0674ef383b0db4c4d0e047d3b600eaa09b95d5e1
F20101211_AAAHHX bronson_j_Page_084.txt
56606506a78ee651ffdb3ef78f6df687
3d9b08768996dcea1d34fa191f786710037e3e56
F20101211_AAAIKZ bronson_j_Page_188.tif
559b10dcc3b251b1748b414567f48699
5db59d152e936a092e97fb14f19479c9a97724d6
67305 F20101211_AAAHJA bronson_j_Page_060.jpg
28d8db811e63c5ce4999f21077358061
12643cf1e533c6446e6df574454059bb3d653c7b
76196 F20101211_AAAHIL bronson_j_Page_165.jpg
2478635a2180f4f7208c88fe613d7896
a0f4d9d6bccd53d5ea5a76d9ec6c8e51699473d1
8737 F20101211_AAAIMD bronson_j_Page_001.pro
45057d847b776a6346dde37a1ba1647e
d7ba70e595714e3e0b988b1a4dc6228844a0e3be
F20101211_AAAILO bronson_j_Page_204.tif
4e429e5bf548a93fdebfda2f4df7baee
27785123debf91a6c82a9a68e8bc1141f1bb55c1
7146 F20101211_AAAHHY bronson_j_Page_138thm.jpg
289ff0ce85d56f7f2b016494dbba1b18
fa41a9a47beb70f3c3618bdbb6b7af4d6ca08afb
24560 F20101211_AAAHJB bronson_j_Page_046.QC.jpg
9014235627407307b18806822d03b876
e792acbeec975e76b6cf9a17ccb82bf0daeca358
4142 F20101211_AAAHIM bronson_j_Page_206thm.jpg
8f57c839778a555c1dca8231fe1ce3b0
11f9265104b824a7cee58fef0d456cb27557ab53
961 F20101211_AAAIME bronson_j_Page_002.pro
1cc16742103200c0fc92ac2c93317fbb
1b90a45bd6e1e55fdbe56e6011e097dc1969d125
F20101211_AAAILP bronson_j_Page_205.tif
fd0a8dccd07e75525f1cdd8ef1e224db
bbd7e824d2d2bf93375781906101b2c0451a0465
4234 F20101211_AAAHJC bronson_j_Page_088thm.jpg
f835b963f20644b02481ff9d2e1bc63a
acd3feea9abd732668f8f8146a1290802b7dbd10
575440 F20101211_AAAHIN bronson_j_Page_204.jp2
4f9f43606e74b37f5af9bdbac6c4d59e
5323922b35fe114d13dfa74c65907e49739340ba
6953 F20101211_AAAHHZ bronson_j_Page_037thm.jpg
ff2e9eefbafd3d58f4cb1957e14a4ecd
682020930d90c26c3d459342e996efdf2b1012f0
623 F20101211_AAAIMF bronson_j_Page_003.pro
774c9c8efa368d97e40fe6f61b429d18
c77ba8bba50485787574de8958001907f2d17e16
F20101211_AAAILQ bronson_j_Page_206.tif
78156c3079b08d262a454a95a322da45
2ccfafb160cc11dfc1d5a30eab0eb721108225bb
1520 F20101211_AAAHJD bronson_j_Page_168.txt
c7677280c50a2bec589cb1b67b28a1cd
790257a33e4afd60ac2f86cb676f926900583596
2244 F20101211_AAAHIO bronson_j_Page_021.txt
dbe577b0d0b8e4eec47e3a4c6063cafe
20fa354e33d99498afa96142399dd423ac602701
26750 F20101211_AAAIMG bronson_j_Page_004.pro
cb772d20c49ca090faff808c5e8e074d
60ccecbd6e274da8d451e9604d483d13e967851c
F20101211_AAAILR bronson_j_Page_207.tif
da5021cac2f2f50398de13378e914b7b
ebab515e45f0ac9741b57f26b8ef5a7c6e22acb3
44559 F20101211_AAAHJE bronson_j_Page_074.jp2
8b771a78d3d194cae095382b7b93e910
fa42697e0c697753f7d9c72912b6820429b51443
F20101211_AAAHIP bronson_j_Page_215.txt
9d02f06e6b3d896599f44f7638e2ad3a
6710abeb920d1512d351f59215dd991da24dc52e
78857 F20101211_AAAIMH bronson_j_Page_005.pro
5e356e1f80d3e97ca4ee2a40b212af07
1c5a824b9de621fabc9f242c5a795a1625edb4d8
F20101211_AAAILS bronson_j_Page_208.tif
aab350f4a9b0024322bc55e24098d1ca
d07df6a106a26fb8d201878ada62ac1786830c66
F20101211_AAAHJF bronson_j_Page_010.jp2
c14b8099f6f2f1adefc1e113bf2cc7a8
45e5a888bb7c6ef9cc03a5f3603b40a22fce6571
F20101211_AAAHIQ bronson_j_Page_070.txt
91a5688e020efa4a53edd8298b653593
f38565f8ecfc8a3f58b34d8a3ba08ee00ceb7e97
90253 F20101211_AAAIMI bronson_j_Page_006.pro
2e99617e722cfb10f964b94a96ae8cf3
e98851a6dbf530c23e2517e6b854d9d2b22897f0
F20101211_AAAILT bronson_j_Page_209.tif
261375ee0a4c5fc2ed454663c082f6ae
f9b27ee71ece1e98928e7be5b5010ac33b9f8927
12615 F20101211_AAAHJG bronson_j_Page_203.QC.jpg
246b6292ab15a0d7015960b1e1786137
1f36b784493b5a65c80a0073e23bb87b5a0a2680
F20101211_AAAHIR bronson_j_Page_006.jp2
031bb624c026c7891f17e5734dca7b59
fb5ec489d55e5a019e7320fbe56281b355359185
F20101211_AAAILU bronson_j_Page_210.tif
5feac22aa881292401c4131e0c2eaa6d
9755a1bfd96538d462418f28dc76f45f24f3c7a8
F20101211_AAAHIS bronson_j_Page_110.tif
f21fcd80435da0c01b24491cb9b9b354
ac7d2929d211c92a2c4c2f891e4efeee3130769b
13411 F20101211_AAAIMJ bronson_j_Page_007.pro
0f8c7eb6a6505372656f2742fcdc7c43
5d9f285de673baaa85ccd9d11d6a5f9dc92fe9aa
F20101211_AAAILV bronson_j_Page_211.tif
f8ec09d1a6b5e0bfc12762beed316341
2a07567845683430e654fb35ca06730bdab97b59
124843 F20101211_AAAHJH bronson_j_Page_032.jp2
b00289eda30be0109e7f8525c66ada93
86a696763b8c974e27d5489763bae2dc36b097cd
21728 F20101211_AAAHIT bronson_j_Page_062.QC.jpg
dc9af09af90db08b8348431fd068c436
6a8396d645c850793182d227739584c33330164a
55457 F20101211_AAAIMK bronson_j_Page_008.pro
6c11b9ce97f96c908135b1b003badcb3
f718c2e20752301ae8a8d4c7befc6e0335bbb287
F20101211_AAAILW bronson_j_Page_212.tif
121a2742bd9b069ef05b3bfdac5855fe
5a811d497a47d5e5d584fda00e6b0d7d3297213b
1721 F20101211_AAAHJI bronson_j_Page_026.txt
c4a67bf6f7036eb8cbcc9ffbf7216fbb
7e3ff3845a34577ed613cfbcac36b82fa87f87de
112166 F20101211_AAAHIU bronson_j_Page_081.jp2
805f5a6e64745e9fdb15b0ba84dba0f0
4cb0fa8827c36b3c879a6fba5f331897a07b0951
24216 F20101211_AAAINA bronson_j_Page_027.pro
a59a5071758088b02e86c3534d0b968c
eb1a94f213c1f708126bd69d59acbe7e3bb8bbb3
63271 F20101211_AAAIML bronson_j_Page_009.pro
fe7e7a87a06dcf4f9e919620c13d96a0
d64f3e2155d03b3b5838dd24a0b83aed3e16bdfe
F20101211_AAAILX bronson_j_Page_213.tif
a817e4c773345f8789841db17b20b060
ba8dbb8590767c344287b2a74edac31280a95370
F20101211_AAAHJJ bronson_j_Page_158.tif
98c673f1430389c51962e65e58e45050
671ebc75931fe08e2ae20bb679bce21d19a8ab7f
16488 F20101211_AAAHIV bronson_j_Page_162.pro
0079f3f103bc230c7bdb07df3a1d60e9
39525a6f81a2dd99c1be6d1544c68dce13d1fc5b
53579 F20101211_AAAINB bronson_j_Page_028.pro
1716f75641ce788a761bb5ad57afc057
2b524e9d54a74fa6665d3a561bacce76a74e38cb
73158 F20101211_AAAIMM bronson_j_Page_010.pro
0b74928801817b0c8fe5a37c75aeee4d
13641fa271d91c42a9cda63b290dc2f6fe762157
F20101211_AAAILY bronson_j_Page_214.tif
842bb77596802eb524a029b2acef9c16
1ab4d59263a7b3b06a04d5d902710122ce39544c
2591 F20101211_AAAHJK bronson_j_Page_210.txt
3e49f936c1fa602735263d2a213fa011
111f8fed879b22c9911f56b6038d24ecf9d3c17a
F20101211_AAAHIW bronson_j_Page_022.tif
e02ef767518628319b90d3e3a36c8b2d
33fd7d9f1d1ae7d9b013c845939d3f1b08c39478
55699 F20101211_AAAINC bronson_j_Page_029.pro
35219e11530c6daabb0cc3f1e9cc9a7c
7207be436c46d5a5877dca720cca883fd72538d1
73265 F20101211_AAAIMN bronson_j_Page_011.pro
5a05370d2aca4db8d3bd615ae5a004fc
f44d8407959c8254f9f99dc13687a45b3d8b196e
F20101211_AAAILZ bronson_j_Page_215.tif
3738e8913790d9b18c89f77e97be12f8
983f039abd3d384473419dc54df88bf0e9c35dc9
48383 F20101211_AAAHJL bronson_j_Page_173.jpg
43029e7285db67a487966c7e4f5e2c65
2e066b6b4672284ba72b913b5ee6b1c6a554fccc
75585 F20101211_AAAHIX bronson_j_Page_022.jp2
4451119b77011e297e7ce9f8be725ed0
eb5dcd7abf8e4c5bc493523c56e16af929a64aec
F20101211_AAAHKA bronson_j_Page_125.tif
5f5b5bfa92a47bb393ed638995792382
757c8ea3e6e776a96fee0c8f84bd8122895209fc
53922 F20101211_AAAIND bronson_j_Page_030.pro
0593aef4ab07a64fe1213fe32b9958ae
1a61bc9a678cde981407cc7f9a617c16ff617b2b
67920 F20101211_AAAIMO bronson_j_Page_012.pro
25f168eff5ae3ddce594837a3d30278e
04fa52941658e7544317cbd0a559b27a66e12cc4
1031 F20101211_AAAHJM bronson_j_Page_153.txt
d46eef76bce39cb1e02688f2e6845527
cc7c62d287be9c813ad02970892cfbc1231f551a
46074 F20101211_AAAHIY bronson_j_Page_075.pro
7b6029cd5023548e5b7bcdabefba574a
6498a41854dc92bcecd46d42c6cc75555a335075
535641 F20101211_AAAHKB bronson_j_Page_201.jp2
d2bfb5e2ccb161472fd6eeba667b7b86
24bd2ed486583500141a71e8447838bc5c7c08e1
55240 F20101211_AAAINE bronson_j_Page_031.pro
ef8ec6201541e7c72a8b66e58802f604
855ce3d07d0c3f2071177e4c08607dd840fb6d7b
70895 F20101211_AAAIMP bronson_j_Page_013.pro
362a325e333adca71fca1642ce269560
eb3bf23230a1570c709d7778c85558598c241e71
F20101211_AAAHJN bronson_j_Page_012.tif
d982fb7926b6dd3c5563d9df2e2c7a8e
d89446b3632dd56dabd3d8aa0b0d98b979316912
1035 F20101211_AAAHIZ bronson_j_Page_205.txt
1a79cb13b902cf8a4d1769b8a8d19b0e
a615f949949edb9ded566caed7a2c84b5f389133
F20101211_AAAHKC bronson_j_Page_014.tif
48232d07b676ac53a1710b103fa403c9
44f54c6bd739a688d44700801e47918af59ee90f
58403 F20101211_AAAINF bronson_j_Page_032.pro
9238a16432f59abe06943ea68b33cb03
4e450d214748d705aeaa6c175674f4b5f9fc4f3c
70827 F20101211_AAAIMQ bronson_j_Page_014.pro
eef31534e0d8df347bd51c73b0accea4
5339b77e6c9a19db3317d4d8a7d0dcafd927acdb
24730 F20101211_AAAHJO bronson_j_Page_139.QC.jpg
4f55b44b475e8a633fd87c497a96556b
041adfb471cc641224aae24a18fd47bbd5359d0e
17030 F20101211_AAAHKD bronson_j_Page_162.QC.jpg
da43c452e7d651c386468c1a464a7568
2cb5427ea03e9290b4e4b8facd705d32d8db6774
57075 F20101211_AAAING bronson_j_Page_033.pro
8cdf21127482bb1f86757e7f53145464
a0d8c6f71a9f5dda771d7393b48a4699c617fd07
53611 F20101211_AAAIMR bronson_j_Page_016.pro
47a5070995f77cf9b35ae64fd5907fb8
ef8f2e87ca963d63d370fe7a0e7c96f56080984d
5671 F20101211_AAAHJP bronson_j_Page_137thm.jpg
7ecd46b728387e101d57d97b51e0edb6
686ec9758506aa67284f364270d878badbfb73e0
10083 F20101211_AAAHKE bronson_j_Page_002.jpg
3e7afc84b961d7345a5d226049e465cf
63a8e5fbeed199d9e3e39ba74ed7f6578bc7c0e9
54784 F20101211_AAAINH bronson_j_Page_034.pro
3862ff005879db73d460fe4aaa4c3d5f
b4cb7aebd3f5b859164e9b042274c075c684ab7d
53128 F20101211_AAAIMS bronson_j_Page_018.pro
f336549f2c885808db06aceb6e4fb277
606ffa9f92e6221167b396b14790110360e268ab
32973 F20101211_AAAHJQ bronson_j_Page_074.jpg
ef989b713585612df71a77ce2ca4b717
640da3bb712e005659e8f3ac7feb3db94ed0a996
2125 F20101211_AAAHKF bronson_j_Page_041.txt
eaedb99449891064f88132de593eb97a
f67ae19a2ed939ff399cf524b6e082b950711431
53821 F20101211_AAAINI bronson_j_Page_035.pro
a0445708c53a0fa85a584da3f6df3275
7959f75cfc608e9c2ec2f2ac01c9483aacfcae95
51114 F20101211_AAAIMT bronson_j_Page_019.pro
f581000527396b93c4606f927740026b
c18479c9723b110d5b1b68dc9fabf218b1490935
2205 F20101211_AAAHJR bronson_j_Page_128.txt
95fd99c2bb44c26cfed7cb8718714789
8d972438ea56a19d322517384711db341e35f0d7
25195 F20101211_AAAHKG bronson_j_Page_035.QC.jpg
33e6a95e30e5adaba9e40fc74ccce698
76bcc2da2565f310869e4804edbe5afa6f8e90e6
72800 F20101211_AAAINJ bronson_j_Page_036.pro
50a022d813f7af05c7f69b566c104722
e5b6b162fc8a1e1b5bc5d7821e9204e7a4dc7be3
55255 F20101211_AAAIMU bronson_j_Page_020.pro
8d3a04aebe421bb048952ac87cfe5947
40eda01dba4645afbac9816d102b50562b3580c4
80682 F20101211_AAAHJS bronson_j_Page_033.jpg
0327f05c1930029ab9b158449dc7d5d7
987f21c620b5cb3e2522695d7b29a30726a8fe17
F20101211_AAAHKH bronson_j_Page_032.QC.jpg
3ffdca10154c507247fa94cb27517279
853c892fb03bdbdcc1bc0d22a20d3d8c2332d7ac
57356 F20101211_AAAIMV bronson_j_Page_021.pro
beb5f4383b4099b0dc71fa56e8429a0c
51ec7d4a38627b288f095160e74aabe64b9a94ec
70815 F20101211_AAAHJT bronson_j_Page_098.jpg
985c152a233f228127ea836086ba142f
9f6955c534168da42c90a239a799a334dec01ca1
56917 F20101211_AAAINK bronson_j_Page_037.pro
ff12b2d58e9e1502693281f48158cc4b
0c0d6411f50aea34f1822e6591975282491ead30
34254 F20101211_AAAIMW bronson_j_Page_022.pro
ca1d30813c9e41cf0567b73a8dad4cc9
4b365906970eaea3b749586ab0835b5678578cf4
2233 F20101211_AAAHJU bronson_j_Page_037.txt
519c72f9d3e6ada84a53dc51322db42a
2455d020dbf29a7d8868af195da048313b55a676
50273 F20101211_AAAHKI bronson_j_Page_146.pro
a314064c8b05f92fee311bc2be755067
df40a4029177fd5bb6bb7bf4ae7ce0fe89d7c512
27606 F20101211_AAAIOA bronson_j_Page_055.pro
4deade602dff2fd7de6b214b32461edb
8ee82a651b630e895b1d7b7d51fc16a89f65576d
55234 F20101211_AAAINL bronson_j_Page_038.pro
c1e8087c5d6c05cadab564598d998fe5
9e915d1b4af450a9c13e9d6586064e9b45197a32
53332 F20101211_AAAIMX bronson_j_Page_023.pro
d6d9ea45ccadac7a1f42c56ffe5a473d
9f07057d24cde32c5fbf28f5a368a6a2e29a148c
F20101211_AAAHJV bronson_j_Page_070thm.jpg
78d5d6645a552ef622f91e66168b1a0d
cf1916d602f3694348fff215d8370c47bf5c908b
664212 F20101211_AAAHKJ bronson_j_Page_090.jp2
86222327493af6fbd8d3b2ee32eae0fb
03dc9b0b198a9ac3ebb525c351f119ff2754ea63
31700 F20101211_AAAIOB bronson_j_Page_057.pro
03f099b815616a902966e505294d4baf
9dfd2d42986f685411e05a694b7267e8ea445ddf
57634 F20101211_AAAINM bronson_j_Page_039.pro
bfcb6ad86e858aba4dfd5b91cb85161f
43fe11744bd0743625948eeb497dbbab9b313ae3
56043 F20101211_AAAIMY bronson_j_Page_024.pro
eb91bc64abf5daf00a6230d2f297f226
600fe58094c883d1e4536af5330744d0131e1898
71970 F20101211_AAAHJW bronson_j_Page_015.pro
c1410c34e1d6008beb1467311de2beb7
c32b205e202cedfc0133049a88cf836f42112451
487 F20101211_AAAHKK bronson_j_Page_001.txt
a3fcd5a7c45f0ca695af2f05bbeee218
9d4e45a47a74ef46c97fdde968859d0c5bc71092
37230 F20101211_AAAIOC bronson_j_Page_058.pro
64ada3e46d3bc40df44a27b777caa1b2
25f72da06c13f0bad5c0e43b373333ac141a39cc
55435 F20101211_AAAINN bronson_j_Page_040.pro
5eabe9699fae830f009f5faa4008a9a5
fe6aaa9e014e82708bd061cfbbcdcb7e27193c0c
38279 F20101211_AAAIMZ bronson_j_Page_025.pro
ab69b83a5ce2ab332b330555c9e35193
30b538d299a32f99018f848ec27e266f02394538
6263 F20101211_AAAHJX bronson_j_Page_071thm.jpg
23ec429e23300dd23b870a16855e9a1e
85ee242ff71f56e1f43f5bb90018debf4bc1d32e
F20101211_AAAHLA bronson_j_Page_168.tif
05b1e1135711f756fbf40ddce0d838bc
045776ed3e2ec88755b32d1f95dd495e8a1fda42
22999 F20101211_AAAHKL bronson_j_Page_080.jpg
8994d487c638a7e1967078200e6bf93a
abe1f17322459b634518a02423916714fdad1ea8
36466 F20101211_AAAIOD bronson_j_Page_059.pro
1d862f4a48b901911a5288dcf0e15dcc
032421a2cc8ef2378e6becf5a237036458f09365
53277 F20101211_AAAINO bronson_j_Page_041.pro
48f800b6320491cb8d8e367cbdeac0bc
d8999ead3424850f70dd66d6befe43efd419a846
3969740 F20101211_AAAHJY bronson_j.pdf
263d3b867ddf385ff96fdc5c8baf6500
842c0743dec868e27b5cc2d40e6c71df58e83da0
F20101211_AAAHLB bronson_j_Page_107.tif
eb01de0aee25ae052cd4d3990db026b9
539960ae536ff38125b1dc5770899a052cae855b
11176 F20101211_AAAHKM bronson_j_Page_195.QC.jpg
cd81e69bdb848caebf6f1acdfbe4f0d6
01d50b375d514abe5f3109fdd6f61d23c9b0563f
39678 F20101211_AAAIOE bronson_j_Page_060.pro
24dd5775aec198e87c3e0452ea92cad0
a8c1c6eefab565da8027e71380e6a3e9e9f641bc
57343 F20101211_AAAINP bronson_j_Page_042.pro
2485e3336154a59379398b7eb0ca0fa8
705228da31ac12ee11ca2dfcd4e4d6632b9d08aa
42167 F20101211_AAAHJZ bronson_j_Page_177.pro
09ee8ea2158e3deb0c819afebc6dbebe
353d51adf3398f5c8d95d0259ac606628e7abbe6
2558 F20101211_AAAHLC bronson_j_Page_213.txt
c309e2c2fd60dab0bead2fcf8ae02d87
f1e67981d96beedbc82b97df6b4ff7098249aa55
54585 F20101211_AAAHKN bronson_j_Page_087.pro
67709487a2ce0181b47d0d0809daf695
1e7c4d0efd2403650ecd5dc5e7f6d789bcbd041c
54693 F20101211_AAAIOF bronson_j_Page_064.pro
6eb6a91bb06afea0fb010d54894dcd23
18b006fa7aae29491d3421572e8f4cd86437cb6e
56345 F20101211_AAAINQ bronson_j_Page_043.pro
1ed3ac9bbaa4706277b26f718df1cf7a
0cf93dee4d95f94842ee497e1a9083440a6e33d9
2234 F20101211_AAAHLD bronson_j_Page_024.txt
e7cb8c4defdd041830e9552c63f9ab58
de2357f0b98065358c225c95a0a59973282394a9
7584 F20101211_AAAHKO bronson_j_Page_193.pro
244ced6929f5e34ca4da2ed325516409
23fadb62bc406a1647a5ef836543f6a5c82d8aef
43693 F20101211_AAAIOG bronson_j_Page_065.pro
f7f257a586eb6152fd6a601f19d4632b
eefc3980fdfca754dda028338fc20afc00358bd0
55566 F20101211_AAAINR bronson_j_Page_044.pro
a5d636c27f18bf414d06b91b69f73be4
9324ba39d1c865d684d97593211fe97da0a48420
40147 F20101211_AAAHLE bronson_j_Page_061.pro
60e2647d4b025032b2941d7f13b59bd9
e0741afa8c5f41c9c7a6024cdaa14c4dc18b8915
F20101211_AAAHKP bronson_j_Page_173.tif
239b62edb8567dd7aac08f457a4f0414
5db4b0972f4727695a1f7a319b8d647fc7b90bc2
33389 F20101211_AAAIOH bronson_j_Page_066.pro
d581a2883e28f609e575a3ce55385318
3eb401f7a8d4bd7e4b32e36519c6d0d184e1db1f
53068 F20101211_AAAINS bronson_j_Page_045.pro
1f92df02cb4b12f25ba48eb6a3587b36
5fdbdfc090637b2c265e932954b361b05c3564ea
F20101211_AAAHLF bronson_j_Page_038.tif
9d8bb78b24dbe88318df89fad208986d
b44bee15ca39ab6ace85c8f3eceddf0e083b123b
62207 F20101211_AAAHKQ bronson_j_Page_126.jpg
6becdb10d8704620dbeb3b8a5ec817c8
4775d00c0c73d4b272b4310c549859847d1bfe8e