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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 FitzCoy, 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, Pullin and Hysteresis............... ...............2 2.3.2 Design Techniques to Eliminate Pullin .............. .............. ...........3 2.3.3 Capacitive and Charge Control Techniques to Eliminate Pullin ........._.._........32 2.3.4 Closedloop Voltage Control to Eliminate Pullin ........._.._.... .....................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 OpenLoop Step Response ............... .... ...___ ... ...............122. 5.2.1 Effects of Parametric Uncertainty on Step Response ............__................123 5.2.2 Effects of Pullin and Hysteresis on OpenLoop Response.............._._. .........123 5.2.3 Continuous Characterization of Micromirror Arrays ............ ................126 5.3 Hysteresis Case Study: ProgressiveLinkage........... ...............12 5.3.1 ProgressiveLinkage Design ........._._... ... .....__. .....__............12 5.3.2 OpenLoop Response Using a ProgressiveLinkage ................... ...............135 5.3.3 Parametric Sensitivity of the ProgressiveLinkage............_._. .........._._. ...13 5 5.3.4 ProgressiveLinkage 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 21 Summary of feedback control papers discussed in the literature review. ................... .......36 31 Mean and standard deviation of fabrication variations for layer thickness in the SUMMiT V surface micromachining process. ............. ...............47..... 32 Mean and standard deviation of fabrication variations of line widths in SUMMiT V......47 33 Values output from finite element analysis of mechanical spring stiffness. ...................63 34 Comparison of polynomial fit for approximation of capacitance function. .................. .....66 35 Comparison of polynomial fit for approximation of capacitance function. .................. .....66 36 List of parameters used for this analysis ....___ ................ ................ ...........78 41 Mean and standard deviation of fabrication variations for layer thickness in the SUMMiT V surface micromachining process. ............. ...............82..... 42 Mean and standard deviation of fabrication variations of line widths in SUMMiT V......82 43 Spring stiffness values for changing dimensional and material parameters. .....................85 44 Results from the Monte Carlo simulations for the capacitance values in terms of mean, standard deviation, and the percent change from nominal............... .................9 45 Mean and standard deviation for pullin angle and voltage from sets of mirrors on all three arrays tested. ............. ...............108.... 51 Modal analysis results for first 10 modes and their natural frequencies, and the participation factors and ratios for each direction. ......___ .... ... .__ .............._..118 52 The first three natural frequencies determined from the LDV experiment. ....................122 53 Link length dimensions used for progressivelinkage design............._. .........._._....134 54 Joint dimensions used for progressivelinkage design............... ...............134 55 Uncertainties in the joint dimensions for a proposed progressivelinkage design. ..........138 LIST OF FIGURES Figure page 21 The SEM images of MEMS devices created using SUMMiT V microfabrication process............... ...............25 22 Images of micromirror arrays developed in industry ......___ ..... ... ._ ........_......26 23 Adaptive optics (AO) mirror used for wavefront correction. ............. ......................2 24 Use of an AO MEMS programmable diffraction grating for spectroscopy.......................27 31 Drawing of the SUMMiT V structural and sacrificial layers .........._.._.. ............. .......46 32 Area with nominal dimensions L and w with the dashed line indicating the actual area due to error in the line width. ............. ...............47..... 33 Images of the micromirror array ........... _...... ._ ...............48. 34 Illustration of mirrors operating as an optical diffraction grating ................. ................ .48 35 Micrograph of an array of mirrors and schematic of mirror with hidden vertical comb drive. ............. ...............49..... 36 Solid model of micromirror showing polysilicon layer names from SUMMiT V. ...........49 37 Schematic of a parallel plate electrostatic actuator modeled as a massspringdamper system ............. ...............51..... 38 Static equilibrium relationship for the parallel plate electrostatic actuator. ......................54 39 Electrostatic force for different voltages and mechanical force showing pullin for the electrostatic parallel plate actuator. ...._ ........__... ....._.._...........5 310 Pullin function for the parallelplate electrostatic actuator. ............. .....................5 311 Static equilibrium relationships for the parallel plate actuator using different spring constants ................. ...............56.._._._....... 312 Schematic of a torsion electrostatic actuator. ............. ...............57..... 313 Static equilibrium relationships for the torsion actuator ................. .....__. ..............59 314 Pullin function for the torsion actuator. ................._.._.._ .......... ..........6 315 Drawing of the mechanical spring that supports the micromirrors and provides the restoring force. ............. ...............61..... 316 Image of the mechanical spring that supports the micromirror indicating boundary conditions and location for applying displacement loads for finite element analysis. ......62 317 Image from ANSYS of the deformed spring and the outline of the undeformed shape after displacements are applied.. ............. ...............63..... 318 Solid model geometry of the unit cell used in the electrostatic FEA simulation............_...65 319 Capacitance calculation as a function of rotation angle, 6, calculated using 3D FEA and varying orders of polynomial curve fit approximations ................. ............. .......66 320 Plot of the Pullin function PI(6) for the micromirror with the vertical comb drive actuator showing that pullin occurs at 16.5 degrees. .................... ...............6 321 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.... 323 Plot of static equilibrium behavior, showing pullin and hysteresis, predicted from the model............... ...............69. 3 24 Static equilibrium relationships for the nonlinear plant model, and the linear plant approximation. ............. ...............70..... 325 Static equilibrium relationships for the nonlinear plant model, and the small signal model linearized about an operating point (60, Vo).................. .............................72 326 Illustration of piecewise linearization about multiple operating points.............._..._. .........73 327 Plot showing the roots of the function F(xl) occur where the function crosses zero.........76 328 Bifurcation diagram for a MEMS torsion mirror with electrostatic vertical comb drive actuator. ............. ...............79..... 329 Bifurcation diagram showing the effects of different spring constants. ............................79 41 Fabrication tolerances can changes the thicknesses of the layers, resulting in changes in the final geometry dimensions. .............. ...............82.... 42 Fabrication tolerances can change the dimensions of a fabricated geometry, affecting the final shape, volume, and mass. ............. ...............83..... 43 Nominal dimensions used to calculate the volume of the moving mass. ................... .......83 44 Capacitance functions for the electrostatic model with parametric changes in the layer thickness of the structural polysilicon. ....._ .....___ ........___ ...........8 45 Capacitance functions for the electrostatic model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide ................. ................ .88 46 Capacitance functions for the electrostatic model with parametric changes in the linewidth error of the structural polysilicon layers ................. .............................89 47 Static displacement relationships for the micromirror model with parametric changes in the layer thickness of the structural polysilicon ................. ............... ......... ...89 48 Static displacement relationships for the micromirror model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide. ................... ........90 49 Static displacement relationships for the micromirror model with parametric changes in the linewidth error of the structural polysilicon layers ................. ........... ...........90 410 Sensitivity of voltage with respect to changes in line width for each value of 0..............92 411 Sensitivity of voltage with respect to changes in layer thickness for each value of 0.......92 412 Gaussian distribution with a mean of 0 and standard deviation of 1................. ...............94 413 Histogram for mechanical stiffness when accounting for variations in thickness of MMPoly1 and Young's modulus............... ...............95 414 Histogram for mechanical stiffness taking into account variations in thickness of MMPolyli, Young' s modulus, and linewidth of MMPolyl1................. ............ .........95 415 Results from the capacitance simulation for 250 random variable sets that show the effects of parametric uncertainty on the electrostatic model. ................ .....................96 416 Static displacement results of 250 Monte Carlo simulations with random Gaussian distributed dimensional variations. ............. ...............98..... 417 Histogram of values from the Monte Carlo simulations for the layer thickness of Sacox3 ................ ...............99................. 418 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the Sacox3 layer thickness ................. ...............99.............. 419 Diagram of an optical profiler measurement system ................ ......... ................10 1 420 Six mirrors from the micromirror array measured with the optical profiler system........102 421 Data records from the SureVision display that show the crosssection profile of the tilt angle measurements............... .............10 422 Micrograph image of a single micromirror ................. ...............104........... .. 423 Experimental static results taken from individual micromirrors that are not in an array. ............. ...............104.... 424 Approximate locations of data collection on all three arrays. ............. ..................... 106 425 Experimental results from array 1, area A. .............. ...............106.... 426 Experimental results from array 2, areas D and E. ............. ...............107.... 427 Experimental results from array 3, areas A and D ................. ................ ......... .10 428 Nominal model with experimental data ................. ...............109.............. 429 Modelpredicted results from 100 simulations with parameters determined by random Gaussian variations, shown with experimental data ................. ............... ....109 51 Openloop nonlinear plant response to a step input of 7 degrees for different damping ratios ................. ...............114................ 52 Solid model created for modal analysis ................. ...............118........... .. 53 Time series data of the micromirror response to an acoustic impulse taken with a laser doppler vibrometer. ............. ...............120.... 54 Results from the LDV experiment showing resonant peaks ................. .....................121 55 Openloop response to a step input of 7 degrees for the nonlinear plant dynamics and variations in spring stiffness, km. ............. ...............124.... 56 Openloop nonlinear plant response to a step input of 7 degrees for 50 random parameter variations ................. ...............124................ 57 Openloop responses to a sinusoidal input showing hysteresis .............. ....................12 58 Openloop responses to a step command showing overshoot that result in pullin.........126 59 Results from dynamic study showing pullin and hysteresis ................. ............... .....127 510 Results showing the hysteretic behavior of the micromirrors .........___..... .............. ..128 511 Diagram of fourbar mechanism for progressive linkage analysis. ............. ..................130 512 Cantilever beam with crosssection w x t, and length L. ............. ....................13 513 Free body diagrams for each member of the linkage. ................. ...._._ ................132 514 Progressivelinkage behavior for different values of ro in Cpm........ ............... 132 515 Progressive linkage output for ro equal to 9 Cpm along with the electrostatic torque curves and the linear restoring torque. ................ ...._.._ ...............133... 516 Static eV relationship for micromirror with a progressivelinkage .............. ..............134 517 Bifurcation diagram for micromirror using a progressivelinkage to avoid pullin b ehavi or ........._._. ._......_.. ...............136.... 518 Bifurcation diagram for the micromirrors using a progressivelinkage to avoid pullin behavior for different values of mechanical stiffness. ................. ...._._ ............... 136 519 Openloop responses to a sinusoidal input for the device using a progressivelinkage. ..137 520 Openloop response to a step input for device using a progressivelinkage. ................... 137 521 Results of parametric analysis for individual errors in j oint fabrication of the progressivelinkage. .............. ...............140.... 522 Fifty Monte Carlo simulation results for varying the joint fabrication parameters for the progressivelinkage design............... ...............140 523 Schematic drawing of the prototype progressive linkage spring ................. ..................141 524 Micrograph of the prototype micromirror with a progressive linkage spring. ................141 525 Experimental data collected for the prototype of the micromirror with the progressivelinkage ................. ...............143................ 526 Results from FEA of the prototype progressivelinkage design for linear and nonlinear deflection analysis shows the prototype progressivelinkage fails to produce the desired stiffness profile. ............. ...............143.... 61 Basic block diagram with unity feedback ................. ...............146........... .. 62 Block diagram with PID controller ................. ...............148.............. 63 Step responses for PID controller ................. ......... ...............150 ... 64 Closedloop PID response to different step inputs when the spring constant is varied by +10% .............. ...............150.... 65 Closedloop PID response to a step input of 7 degrees for 50 random sets of parameteric variations ..........._.... ...............151.._.__....... 66 Closedloop PID response to a commanded position in the unstable region. ..................1 53 67 Closedloop step responses for PID controller for a system using a progressive linkage. ........... ..... .. ...............153.. 68 General block diagram for LQR controller problem .............. ...............154.... 69 Block diagram of LQR control with an internal model for tracking a step command. ...158 610 Block diagram of LQR controller using a stateestimator for a plant without an integrator. .............. ...............159.... 611 Step responses for LQR controller. ........._._ ....__. ...............161 612 Closedloop LQR response to a step input of 7 degrees for 50 random parameter variations. ........._ ...... .. ...............162... 613 Closedloop step responses for LQR controller for a system using a progressive linkage. ........._ ...... .. ...............164... 614 Schematic of modeling an array of mirrors as a SIMO system ................ ................. .165 615 Schematic drawing of an array of 5 mirrors. ............. .....................166 616 Illustration of the measured center of gravity (CG) on a 1D PSD when there are errors in the spacing and linearity of the micromirrors ................. ........................168 617 Illustration of the measured center of gravity (CG) on a 2D PSD when there are errors in the spacing and linearity of the micromirrors ................. ........................168 618 Illustration of the measured errors of the reflected light from two micromirror arrays onto a CCD. ............. ...............169.... 619 Schematic of the beam steering experiment with only one micromirror. ................... .....171 620 Geometry used to determine the angle of incidence and reflection. .............. .... ........._..172 621 Calculating the reflected ray of light. .......___......... ___ .........___ ....___......173 622 The intersection of the line from BO to R and the plane C occurs at point P. ................. 174 623 Schematic of 5 micromirrors in an array reflecting light onto a PSD sensor ..................1 75 624 Calibration of the PSD for ideal case of five micromirrors. ................ .....................176 625 Openloop results to a step response for an array of 5 micromirrors with model uncertainty ................. ...............176................ 626 Openloop response for system with one broken mirror and 4 ideal mirrors, measured by a PSD .............. ...............177.... 627 Incorporating feedback control into array model ................. ..............................17 628 Controlled PID step response using PSD sensor ................. ...............178............. 629 Controlled LQR step response using PSD sensor ................. ............... ......... ...17 630 Closedloop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a PSD sensor. ............. ...............179.... 631 Schematic of 5 micromirrors in an array reflecting light onto a CCD sensor where each separate location of the light can be measured ................. ...........................180 632. Closedloop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a CCD sensor ................. ...............180........... ... A1 Geometry dimensions in Cpm for creating electrostatic model. ............. ....................188 B1 Histogram of values for the thickness of layer MMPoly0......._____ ...... .....__.........189 B2 Histogram of values for the thickness of layer MMPolyl1......____ ...... ....__..........190 B3 Histogram of values for the thickness of layer MMPoly2. ....._____ ...... .....__.........190 B4 Histogram of values for the thickness of layer MMPoly3 ........._._. .... ...._._...........191 B5 Histogram of values for the thickness of layer MMPoly4 ................_ ............ ........191 B6 Histogram of values for the thickness of Dimple3 backfill. ................ .....................192 B7 Histogram of values for the thickness of layer Sacoxl ................_ ......... ..............192 B8 Histogram of values for the thickness of layer Sacox2. ................ ....___ .............193 B9 Histogram of values for the thickness of layer Sacox3 ................. .........................193 B10 Histogram of values for the thickness of layer Sacox4 ................. ................. ...._194 B11 Histogram of values for the linewidth variation of layer MMPoly2. ............._ .............194 B12 Histogram of values for the linewidth variation of layer MMPoly3. .........._... .............195 B13 Histogram of values for the linewidth variation of layer MMPoly4. ............. .... ...........195 B14 Histogram of the mass values calculated from the parametric variation data. ..............196 B15 Histogram of values from the Monte Carlo simulations for the linewidth error of M M Poly2 ................ ...............196................ B16 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the linewidth error of MMPoly2. ....._____ ... ......_ ..............197 B17 Histogram of values from the Monte Carlo simulations for the layer thickness of M M Poly l ................ ...............197......... ...... B18 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of MMPolyl1................. ...............198........... . B19 Histogram of values from the Monte Carlo simulations for the layer thickness of Sacox4 ................. ...............198................ B20 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of Sacox4. ............. ...............199.... C1 Magnitude results of FFT for device 1, trial 1. ............. ...............200.... C2 Magnitude results of FFT for device 1, trial 2. ................ ...............200........... C3 Magnitude results of FFT for device 1, trial 3 ................ ...............201........... C4 Magnitude results of FFT for device 1, trial 4. ................ ...............201........... C5 Magnitude results of FFT for device 1, trial 5 ................ ...............202........... C6 Magnitude results of FFT for device 2, trial 1 ................ ...............202............ C7 Magnitude results of FFT for device 2, trial 2. ................ ...............203............ C8 Magnitude results of FFT for device 2, trial 3 ................ ...............203............ C9 Magnitude results of FFT for device 2, trial 4. ................ ...............204............ C10 Magnitude results of FFT for device 2, trial 5 ................ ...............204............ C11 Magnitude results of FFT for device 3, trial 1 ................ ...............205............ C12 Magnitude results of FFT for device 3, trial 2. ................ ...............205............ C13 Magnitude results of FFT for device 3, trial 3 ................ ...............206............ C14 Magnitude results of FFT for device 3, trial 4. ................ ...............206............ C15 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 pullin, 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 degreeoffreedom (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 pullin 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 progressivelinkage that is intended to eliminate the occurrence of electrostatic pullin 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 highprecision 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 closedloop 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 pullin instability that occurs when the electrostatic force overcomes the mechanical restoring force. When pullin 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 pulledin, the voltage required to maintain the fully actuated position is lower than the pullin 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, stateoftheart micromirror arrays rely on openloop 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, closedloop feedback control techniques are considered essential. Feedback control has long been used in many macroscale 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 singleinput/multipleoutput 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 pullin 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 openloop actuation methods. These models also characterize the effects of electrostatic instability and the resulting hysteresis. The modeling is extended from the initial quasistatic 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 progressivelinkage, is presented that will eliminate the effects of electrostatic pullin 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 progressivelinkage 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 closedloop 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 pullin instability is given, including modeling methods and the different methods that are dedicated to addressing pullin. 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, Multilevel 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 21 show scanning electron micrographs (SEMs) of a mechanical gear hub and a crosssection of a pinj oint that allows rotation. These are excellent examples of the complex structures created from layering simple, 2D 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 21. The SEM images of MEMS devices created using SUMMiT V microfabrication process. A) Micromachined gears. B) Micromachined gears. C) A crosssection view of a pinj 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 freespace 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 22. Figure 22. 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 23 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 23. 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 24, 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 24. 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 siliconbased 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, Pullin 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 welldocumented phenomenon of electrostatic pullin. The electrostatic force is nonlinear as it is inversely proportional to the square of the electrode gap. Pullin, sometimes called snapdown, 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 pullin characteristics can be modeled fairly accurately [3], [14], [39][52], [106]. Pullin for a parallelplate actuator occurs at onethird of the separation gap, which greatly limits the actuator stroke. Another phenomenon associated with pullin instability is that once the mirror has pulled in, the voltage required to maintain the pullin position is lower than the pullin voltage. The mirror will not return from this position until the actuating voltage has been reduced below the holdingvoltage. 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 deadband after pullin in which no actuation is even possible until the applied voltage drops below the holding threshold. The effects of pullin 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 crosssection along the length of the device. These nonconstant 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 pullin 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 pullin 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 pullin 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 onethird 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 pullin as an advantage that allows for openloop, on/off binary actuation at reduced voltages [2], [53]. While the actual pullin voltage of the device may vary slightly from mirror to mirror due to variations in dimension and material properties, reliable openloop 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 pulledin 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 pullin has been thoroughly documented and there has been a considerable amount of research conducted to find ways to avoid pullin 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 closedloop feedback control. A review of these methods is given in the following sections. 2.3.2 Design Techniques to Eliminate Pullin There are multiple design methods researchers have employed to address the problem of electrostatic pullin 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 leveragedbending approach introduced by Hung and Senturia [24] uses the stressstiffening of a fixedfixed 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 Pullin 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 switchedcapacitor 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 steadystate 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 steadystate 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 Closedloop Voltage Control to Eliminate Pullin There are cases where a closedloop control technique has been used for attenuating and stabilizing electrostatic instability. Voltage control methods have been explored to achieve stabilization beyond the pullin point [68], [69]. Chu, and Pister discuss the effect of introducing a voltage control law into a system of electrostatically actuated parallelplates 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 bypass the pullin 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 closedloop 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 21, 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 steadystate 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 lineartimeinvariant (LTI) techniques such as proportionalintegrator 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, leadlag, and statevariable 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 lineartimeinvariant (LTI) control methods. The nonlinear effects of electrostatic actuation are perhaps most evident for parallelplate actuator systems. Lu and Fedder used a linearized plant model for a parallelplate 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 21. 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 ParallelPlate Electrostatic Electromagnetic MEMS Motor Electrostatic Lateral Comb Drive ParallelPlate Electrostatic ParallelPlate Electrostatic ParallelPlate Electrostatic ParallelPlate 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] StateFeedback [70], [Lu, Fedder, 2002, [71] 2004] ParallelPlate 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] StateFeedback 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] DualStage Disk Robust (MuSynthesis) Position Tracking Drive Piezoelectric Torsion Adaptive control, Disturbance Rejection Mirror (2DOF, not Robust (HInfinity) (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) DualStage 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 closedloop 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 realtime 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 bulkmicromachining. 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) HIinfinity control and adaptive control resulted in good disturbance rej section of bandlimited noi se and the HIinfinity controller improved performance by eliminating steadystate drift and reducing noise [26]. There are few examples of robust control design methods such as HIinfinity and mu synthesis that have been applied to MEMS systems. In addition to the use of HIinfinity control demonstrated by Kim, et al. for a nonMEMS micromirror system [26], musynthesis controller design was applied to a dualstage actuator system for trackfollowing in a harddisk drive [82]. The controller design was successful in simulations, but no experimental work has been done so far. The application of musynthesis 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 parallelplate 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 parallelplate 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 discretetime SMC for a dualstage actuator for harddisk 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 pullin instability of twoaxis 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 nanoscale 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 multiloop 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 closedloop 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 surfacemicromachining 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 signaltonoise 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 pullin 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 pullin 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 pullin is addressed by introducing a novel design technique called the progressivelinkage to create a nonlinear restoring force. This progressive linkage has the advantage of having a continuous spring force over other designs that use discontinuous, piecewise 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 pullin 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 pullin. 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 parallelplate 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 combdrive 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 singleinput/multipleoutput (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 pullin 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 31 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>, ntype 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 31. 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 31 and 32. This information was gathered through diagnostic process testing as described in [98]. Table 31 gives the mean and standard deviations of the thicknesses of the layers of polysilicon and silicon dioxide. Table 32 gives values for variations in the dimensions of the line widths of the device design. Figure 32 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 31. 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 32. 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 32. 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 24pin dual inline package (DIP) in Figure 33. 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 34 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 onedegree of freedom, actuated electrostatically and are shown schematically in Figure 35. The electrostatic micromirror arrays have a ground plane and a Figure 33. Images of the micromirror array. A) Packaged device. B) Micrograph of the surface of the array. A B Cr Figure 34. 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 parallelplate 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 35. 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 2D crosssection view of a unit cell (figure not to scale). Mirror Surface MM Poly 4 Moving Fingers MMPoly 3 Figure 36. 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 35(a). Not shown in the drawing is a design constraint that restricts the motion of the fixedend of the mirror plate from moving a large distance in the Zdirection. While some motion may occur, the assumption is made that this device acts in one degreeoffreedom by rotating about the xaxis. Figure 36 shows a 3D model identifying the fabrication layers used to create the micromirrors. 3.3 Electrostatic Actuation and Instability Many electrostatic actuators exhibit the welldocumented phenomenon of electrostatic pullin. The electrostatic force is nonlinear, as it is inversely proportional to the square of the electrode gap. Pullin, sometimes called snapdown, 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 pullin instability is that once the mirror has pulledin, the voltage required to maintain the pullin position is lower than the pullin voltage. The mirror will not return from this position until the actuating voltage has been reduced below the holdingvoltage. The result of this holding effect is hysteresis. This section will examine the modeling of the electrostaticmechanical 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 37, 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 37. Schematic of a parallel plate electrostatic actuator modeled as a massspringdamper sy stem. The equation of motion for this massspringdamper system is derived by the balance of the forces on the system from Newton's second law CF = m (31) where m is the mass of the moving plate. When the top plate is displaced in the positive x direction, shown in Figure 37, 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 (33) 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 (34) where so is the permittivity of free space, 8.854 x1012 F/m. The electric field is given by E = (35) coA where Q is the electric charge. The charge, Q, can be written as Q = CV (36) 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 34 can be rewritten as Ulc= CV2 = (38) 2 2(xo x) The electrostatic force is thus written as dU 1 dC 2 0,AV2 F = V2 (39) 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 (310) The static equilibrium for the system reduces to only the electrostatic force, and the mechanical force. 1 dC 2 ,A V2 V2 kmx (311) 2 8(xo x) 2 (xo x)2 Equation 311 can be interpreted to show the relationship between the voltage and the displacement, as plotted in Figure 38 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 pullin instability. It turns out that the solutions in the lower portion of Figure 38 are stable solutions and the solutions in the upper portion are unstable. The maximum voltage value corresponds to the actuation voltage at which pullin occurs, and the maximum stable position for parallel plate actuator occurs at onethird the gap between the electrodes. To further explore the pullin phenomena, the static relationship in Equation 311 can be examined graphically, by plotting the electrostatic force and the mechanical force separately in Figure 39. 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 38, 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 pullin. At the pullin 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 pullin point, while unstable solutions occur after. This slope equality is written by taking the first derivative with respect to the displacement of Equation 311. 1 d2C y V = kM, (312) 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 38. Static equilibrium relationship for the parallel plate electrostatic actuator. \ e~ Pullin 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 39. Electrostatic force for different voltages and mechanical force showing pullin for the electrostatic parallel plate actuator. Substituting Equation 312 into 311 and evaluating at the pullin position results in the following relationship that is only a function of the capacitance and the pullin 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 pullin angle is independent of the spring stiffness, and depends only on the angle of rotation. A pullin function, PI(x), is defined to determine the pullin angle, which occurs when PI(x) is equal to zero. dC 82C PI(xo x) = ax (xo x) a~,x"(314) In turn, once the pullin angle is determined, the pullin voltage can be calculated by the following expression, VP 2mPI (315) 8r (xo x)a 'p The pullin function for the parallel plate electrostatic actuators is shown in Figure 310, 0.01 0.005 Pullin 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 310. Pullin function for the parallelplate electrostatic actuator. and verifies that pullin does occur at 1/3 the gap between the plates. The pullin voltage is calculated from Equation 315 to be equal to 57.85 V. To further investigate the effects of changing the spring constant on the pullin, the static equilibrium relationships are plotted for different values of the mechanical spring constant in Figure 311. This shows that even for a different spring constant, the pullin displacement location remains at 1/3 the gap, while the pullin 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" 311 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 onedegreeoffreedom massspring damper system of the form J + b + kB = T,(8,Vy) (316) VW Figure 312. 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 (317) S2 80 where C is the capacitance, 8 is the angle of rotation about the Xaxis, 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 (318) 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 317 and 318. 1 BC\ V2 Iy = km9 (319) 2 80! As was previously shown for parallelplate 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 pullin. At the pullin point, both the electrostatic and mechanical torques are equal in magnitude and slope [14], [46]. As was shown previously for the parallelplate actuator, electrostatic pullin can be considered as the mechanical and electrostatic torques being equal, as in Equation 319, and their first derivatives being equal. V1 2Cd2 = k (320) Combining Equations 319 and 320 and evaluating at the pullin angle results in the following relationship that is only a function of the capacitance and the pullin angle, BpI. _,, )PI= (321) Assuming that the restoring springs are linearly deformed in the range of actuation, the pullin angle is independent of the spring stiffness, and depends only on the angle of rotation. A pullin function, PI(0), is defined to determine the pullin angle, which occurs when PI(0) is equal to zero. PI(0) =(c 6a (322) In turn, once the pullin angle is determined, the pullin voltage can be calculated by the following expression, 2k, PI Vr = cal= (323) 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 parallelplate 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 312, the capacitance for a torsion actuator in terms of the angle of rotation about its axis is given as C(0) = I1 n (324) max max where a is L3/L2, iS L2/L1 and Imax is Ho L1 [46]. From Equation 324 it is possible to calculate the performance for a torsion actuator. The pullin 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 NCpm, 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 313. The 0.4, 1 0.35~ /1 0.3 E 0.8  5 Pullin 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 313. Static equilibrium relationships for the torsion actuator. A) The electrostatic and mechanical torques. B) The static rotationvoltage relationship. displacement as a function of the voltage is also shown in Figure 313. The pullin function from Equation 325 is plotted in Figure 314. From these figures, it is found that the pullin for this system occurs at 82% of maximum rotation angle for the system. For the given spring constant, the pullin voltage is 6.91 V. As with the parallel plate actuator, the pullin angle will remain the same despite the mechanical spring constant, but the pullin voltage will change. The pullin angle can change, however, if the system geometry is changed. This is different from the parallelplate actuator, which always pulls in at 1/3 the gap. 3x10 3 u 1 Pullin at 0.82 0 0.2 0.4 0.6 0.8 1 Normalized Displacement a/e Figure 314. Pullin 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 pullin and hysteresis. 3.4.1 Mechanical Model The mechanical spring is shown in Figure 315 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 316 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 316 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 sixdegreesof 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 315. Drawing of the mechanical spring that supports the micromirrors and provides the restoring force. Displacement loads are applied in all sixdegreesoffreedom at the point indicated in Figure 316 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 317. Assuming Hooke's law for the force applied to a linear spring, the spring stiffness in all six degreesoffreedom 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 33, 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 offaxis rotations about the Y or Z. It is the value of qX equal to 612.4 pNm/rad that is used for ks, in Equation 317. Apply ~ ~di a meant Area crosssection of beam: zI w = 1pm Boundary Conditions: DOF Figure 316. 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 317. 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 33. 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 pNm/rad qZ stiffness 11360 pNm/rad qY stiffness 16310 pNm/rad 3.4.2 Electrostatic Model In order to compute the electrostatic torque values in Equation 316, it is necessary to Eind an expression for the capacitance as a function of the rotation angle. For parallelplate 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, 3D 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 crosssection of a unit cell made up of onehalf 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 35(c). The model of the geometry in Figure 318 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 Xaxis 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 x1012 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 (325) Using numerical values generated in the electrostatic FEA model, Equation 324 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 leastsquares 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 319 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 34. Table 35 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 pullin behavior. The advantage of using the first order fit is that its derivative which is used in Equation 316 is a constant, thus simplifying the plant model to a linear approximation in V2. In Order to capture the nonlinear behaviors of pullin 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 318. 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 319. Capacitance calculation as a function of rotation angle, 6, calculated using 3D FEA and varying orders of polynomial curve fit approximations. Table 34. 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 35. Comparison of polynomial fit for approximation of capacitance function Order normr s n r2 4 1.1192E05 0.000185 18 0.999785 3 1.1463E05 0.000185 18 0.999775 2 3 .6691E05 0.000185 18 0.997691 1 8.608E05 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 316 to 320. Equations 318 to 320 calculate the electrostatic pullin characteristics of the device. A plot of the pullin function is shown in Figure 320 where pullin occurs when the function equals zero at 16.5 degrees. Using this value in Equation 320, the pullin voltage is 71.06 V. x 103 0.5 Pullin Angle 1 .5 0 2 4 6 0 10 12 14 16 10 20 Theta [degrees] Figure 320. Plot of the Pullin function PI(6) for the micromirror with the vertical comb drive actuator showing that pullin occurs at 16.5 degrees. The static equilibrium behavior can also be evaluated from Equations 316, and 317, 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 321 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 317. 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 pullin point, which corresponds to the calculated values of 16.50, and 71.06 volts. The electrostatic torque curve at the pullin voltage, Vn, is also indicated in Figure 321. The pullin 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 pullin angle. Only the value of the pullin voltage would be affected. This is shown in Figure 322. The pullin instability is known to cause hysteresis in the device behavior, and this too can be predicted using this modeling approach. After the device has pulledin, it is possible to reduce the voltage below the pullin 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 pulledin position, but it will return to a position different from the pullin 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 323, including the pullin point and the hysteresis loop. This type of curve will be referred to as a 8V profile, and represents the static calibration for the device. 250 200 , Pullin/ f10 2 0 12 1 6 1 Thet [dges Fiue31 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 322. 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 323. Plot of static equilibrium behavior, showing pullin 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 324 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 324. 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 316 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 (328) The linearization in Equation 328 includes a term that is dependent only on the rotation angle that can be considered the electrostatic spring force, ke [20]. k, =~ d2C V (329) 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 325 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 325 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 326. The linearized models discussed above are important when considering control design techniques that require a linear transfer function or statespace 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 325. 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 326. 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 fixedpoint solutions [106], [107]. One advantage of evaluating the bifurcation behavior of the device is that unlike the methods used in Equations 314 and 315, 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 pullin. Here, the spring constant is still assumed to be linear, and the results may be compared to those obtained using Equations 314 and 315. The state space model for the system is x, = x, = 0 (331) .1 x, x fT,(x )  'x Recall that Te is a function of the capacitance expression from Equation 326. In order to capture the nonlinear effects of the system, a fourthorder curve fit approximation is used. The fixed points occur at xz = 0 and Te (x, ) k, x, = 0 (332) This can be expressed in full as N(44Fx3 + 3Px2 12 +2Px, + P)V2 kx, = 0 (333) Equation 333 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 (334) 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 (336) c 3 NV2P3 km (3 3 7) D= NV2P4 (338) Further, define 9ABC 27A2D 2B3 q = (339) 54A3 u = 3A Bg2 2~ (340) s =J4 (3 4 1) t =~r (342) The roots of Equation 334 are ,= s+ t  (343) 3A 1 B 8 (s +t) +(s t)i (3 44) 2 3A 2 I B =e (s +t) (s t)i (345) 2 3A 2 The roots of Equation 3 3 3 can be found to determine the static voltagedi 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 327, 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 nonphysical 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 steadystate solutions disappear. This is another description of the pullin instability caused by the disappearance of all physically possible steadystate 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 33 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 327. Plot showing the roots of the function F(xl) occur where the function crosses zero. Df (x) 1 dT, (xl)C1 kJ, 1b (347) where 1N(124Fx,2 + 6P x, + 2P,)V2 (348) ax, 2 The Jacobian defined in Equation 347 relates the perturbation of the states from equilibrium as = Df (Y) a" (349) ~Al x, 0 x The stability is determined by evaluating the matrix in Equation 347 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 (350) i=1, 2, 3 Substituting Equation 348 into 350 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 squeezefilm effect, in which air that is compressed between very small spaces begins to act as a viscous fluid [3]. Squeezefilm 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 squeezefilm damping coefficient analytically difficult. For the purpose of this discussion, an approximation is made to consider the squeezefilm damping term for a torsional plate developed by Pan, et al [100]. rLw5 b = Kror (352) 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 + (353) .Table 36 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 328 shows a saddle node bifurcation at 16.5 degrees and 71.06 V. This is in agreement with the pullin 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 pullin voltage. Table 36. List of parameters used for this analysis. Parameter Value p, density of polysilicon 2331 kg/m3 YI, absolute viscosity of air 1.73e5 Ns/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 pullin angle, pullin 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 328. Bifurcation diagram for a MEMS torsion mirror with electrostatic vertical comb drive actuator. 40 60 Control parameter V Figure 329. 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 31 and 32 respectively, and reprinted in this chapter for convenience as Tables 41 and 42. These show that dimensions can vary by as much as eight percent in layer thickness, and as much as twentynine 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 41. 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 42. 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 41. 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 42 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 41. 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 42. 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 43. 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 43. 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 41, three linewidth variations listed in Table 42, 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 37. The length of this beam and the crosssectional 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 43 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 43. The first entry in Table 43 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 43. 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 pNm 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 41 and 42. The results are shown in terms of the capacitance in Figures 44, 45, and 46. Figure 44 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 45 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 46 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 47, 48, and 49 show these results. Figure 47 shows the 8V 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 44, 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 48 demonstrates the sensitivity of the micromirror to variations in the thickness of Sacox3, similar to that seen in the capacitance function of Figure 45. Likewise, Figure 49 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 44. 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 45. 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 46. 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 47. 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 48. 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 49. Static displacement relationships for the micromirror model with parametric changes in the linewidth error of the structural polysilicon layers. S=~= (41 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 410 displays the sensitivity of the system to changes in line widths. The same analysis for variations in layer thickness is given in Figure 411. 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 timeconsuming 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 410. 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 411i. Sensitivity of voltage with respect to changes in layer thickness for each value of 8. from the fabrication data in Tables 41 and 42, 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) = eaIx)'2 20 (42) where Xis the mean value of the data set, ois 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 412. 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 412. Gaussian distribution with a mean of 0 and standard deviation of 1. resulting spring constants, km, had a mean of 634.21 pNm and a standard deviation of 12. 17 pNm. 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 pNm and 658.55 pNm. This corresponds to a variation in the mechanical spring stiffness of 3.84% from the mean. It was shown previously in Table 43 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 [pNprn] Figure 413. 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 [pNgn] Figure 414. 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 Zdirection. The capacitance function is affected by both these changes in dimensions. Material properties do not play a role in the electrostatic analysis. Figure 415 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 44, 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 415. Results from the capacitance simulation for 250 random variable sets that show the effects of parametric uncertainty on the electrostatic model. Table 44. 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 4.73E05 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 8V profile. Figure 416 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 416. 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 417 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 418 shows the simulation results for the 8Vprofiles 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 eV 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 417. 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 418. 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 8V 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 8V profiles for the mirrors [101]. This measurement tool is able to make measurements of outofplane deflections as the micromirrors are given different actuation signals. This information can be used to determine how variable the 8V 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 pullin 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 outofplane measurements of a surface. The working principle of the instrument is shown in Figure 419. 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 rrayi Digitized Intensity file,, Beamsplitter Illuminatc :::. III Translator Microsco e Light Source Field Objective Aperture Stop Stop Mirau Interferometer Sample Figure 419. 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 420. 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 420, as well as look at crosssections of the data. Figure 421 shows a cross section of the micromirror data I I 20 pm Figure 420. 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, outofplane, Zdirection, and the horizontal, inplane Xdirection. 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 421 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 421 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 Xdirection (pLm) h c o ~ 1 u 111 L U r~ i "' B 3n Xdirection ( om) Figure 421. Data records from the SureVision display that show the crosssection profile of the tilt angle measurements. A) An example that clearly shows the crosssectional 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 422, 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 422, 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 423. It is clear that the pullin point for this set of experimental data is similar to the data collected on the arrays, and the pullin angle, 13.870, is at the lower range of the pullin angles for the arrays of mirrors. The pullin 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 422. Micrograph image of a single micromirror. 20 16 Pullin at 12 12 0 10 20 30 40 50 60 70 80 90 100 Voltage (V) Figure 423. 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 onehalf 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 421, this amount of error in the Zdirection 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 8V 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 424 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 425, 426, and 427, 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 45 gives a summary of the pullin angle and voltages for the data. The .1 mm . Figure 424. Approximate locations of data collection on all three arrays. average pullin 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 pullin value of 16.50 from the analytical model, the lowest value is within 20 percent. Also, the values listed in Table 46 are averaged values over multiple data sets. From Figures 425 through 427, it is evident that in many cases the mirrors did experience pullin very close to the predicted angle of 16.50. The pullin 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 425. 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 426. 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 427. Experimental results from array 3, areas A and D. SMrr or 2 SMirror 2 SMirror 3 *, 1 Table 45. Mean and standard deviation for pullin 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 pullin voltage observed at two different locations on the array. Compared to the predicted pullin 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 426 shows this occurred for array 2 when multiple sets of data were taken. Figure 428 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 fabricationinduced 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 429, 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 428. Nominal model with experimental data. "10 20 30 40 50 60 Voltage (V) 70 80 90 100 Figure 429. Modelpredicted 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 pullin voltages. In addition, the differences in the pullin 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 48 to have a significant effect on the 6V 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 418 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 429, 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 noncontact 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 pullin angle, and pullin 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 steadystate 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 pullin and hysteresis behaviors of the openloop system, a case study is presented for a progressivelinkage 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 316 in terms of natural frequency, an,, and the damping ratio, i. m,, = (5 1) g = (5 2) 2J, Written in statespace form, the system is described as follows, x, = x, = 0 (53) 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 lumpedparameter model determined from Equation 51 is found to be approximately 188 k As stated previously in Section 3.4.5, the squeezefilm 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 51 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 51. Openloop 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) (54) 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 (55) where (), is 'the eigenvector representing 'the ith natural frequency, we~ is the ith natural frequency (rad/s), and t is time. Substituting Equation 55 into 54 yields (m~ [M]+ [K]) { ), = {0) (56) Ignoring the trivial solution to Equation 56, which is {#}z = {0) then the following expression must be true. I[K]m? [M} = (57) Equations 56 and 57 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) (58) The vector D describes the excitation direction and is of the form {D)= [Tl{e) (59) 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(...] (510) 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]= (511) 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 52 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 degreesoffreedom 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 51. The ratio of each participation factor to the largest participation factor value for a given direction is also listed in Table 51, 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 degreeoffreedom, rotating about the Xaxis (ROTX). It is likewise assumed that the first natural frequency will occur in this rotational direction and be given by Equation 51. 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 onedegreeof freedom assumption for the model in Equation 53. The largest participation factor is 6.5E05 for the ROTX direction, and this is an order of magnitude larger than the next largest participation factor which occurs in the Zdirection. 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 degreeoffreedom motion and in fact, the first resonant frequency excites motion in both the Xaxis (ROTX) and the Zdirection. The motion that occurs in the Zdirection will affect the compliance of the system, which will result in a different natural frequency than that predicted using Equation 51, which assumes one degreeoffreedom motion about the Xaxis only. The spring stiffness results presented in Table L; I spring A anchor B Figure 52. Solid model created for modal analysis. A) View of the top and back. B) View of the bottom showing the comb fingers. Table 51. Modal analysis results for first 10 modes and their natural frequencies, and the participation factors and ratios for each direction. XDirection Participation Factor 2.1183E10 9.5920E07 1.3215E07 8.3800E07 8.0827E10 4.6105E08 2.4312E06 3.1366E07 3.0772E07 2.9854E07 ROTX Direction Participation Factor 6.4831E05 1.9988E08 9.4879E06 1.5072E06 6.3384E06 1.3954E07 2.7564E08 5.5281E09 4.2753E07 4.5237E08 YDirection Participation Factor 2.7197E06 8.8566E10 3.5837E06 5.5914E07 1.7035E06 2.2081E08 3.6331E09 5.4173E08 2.2398E08 1.9532E07 ROTY Direction Participation Factor 8.7467E08 2.2171E04 9.8134E07 7.3266E06 8.7374E08 9.0970E08 3.4525E06 2.5788E07 1.0185E07 2.3618E08 ZDirection Participation Factor 3.5955E06 4.3162E09 3.0841E06 4.8219E07 9.7970E08 7.5056E08 2.7242E09 3.3755E09 1.1384E08 1.2589E08 ROTZ Direction Participation Factor 4.3414E08 1.5238E05 3.4382E05 2.2001E04 2.8728E07 1.8579E08 3.1098E06 8.3555E06 3.2656E07 1.8531E06 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 33 previously show that the spring is very compliant in the Zdirection 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 Xaxis 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 degreeoffreedom 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 51 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 capgun, 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 53 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 53. 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 54 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 52 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.OE04 7.OE04 6.OE04 5.OE04 4.OE04 3.OE04 2.OE04 1.OE04 0.OE+OO 100 200 300 400 Frequency [kHz] 500 7.OE04 6.OE04 5.OE04 4.OE04 3.OE04 2.OE04 1.OE04 0.OE+OO O 100 200 300 400 Frequency [kHz] 500 3.5E04 3.OE04 2.5E04 2.OE04 1.5E04 1.OE04 5.OE05 0.OE+OO O 100 200 300 Frequency [kHz] 400 500 A) Device 1, trial 4. B) C Figure 54 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 52. The first three natural frequencies determined from the LDV experiment. Results from the linear model, using Equation 51, and the modal FEA are included for comparison. Frequency (k 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. 51 182  Modal 84.74 120.37 Analysis 5.2 OpenLoop Step Response The openloop response of the system is determined by the actuation voltage signal that is given to the micromirrors. For openloop 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 pullin 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 openloop response of the plant model is considered using different values of stiffness, ks,. Figure 55 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 53 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 56 shows the openloop 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 openloop plant will result in steady state error in the response. In order to correct for this in openloop operation, the system must be carefully recalibrated for each device to ensure the proper response is achieved. 5.2.2 Effects of Pullin and Hysteresis on OpenLoop 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 pullin 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 55. Openloop 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 56. Openloop 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 holdingvoltage, causing hysteresis. The effects of pullin and hysteresis for the static response are investigated in Chapters 3 and 4, but there are dynamic effects that can affect pullin as well. It is known that pullin 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 pullin voltage, it is still possible for the inertial effects to cause the mirror to experience pullin 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 pullin angle, On, the system response sets theta equal to the final pullin position, eF. After pullin has occurred, the system response remains pulledin until the voltage drops below the holding voltage, Va. The system then returns to the released position, Is. This response is shown in Figure 57 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 pullin angle of 16.50, the response shows pullin 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 pullin point. In the case of large overshoot in the response, the device will pullin and will not be released as the voltage command for a step input is constant. Figure 58 shows the openloop step response of the system for commands of 120, 140, and 170. It is expected that the command input of 170 will result in pullin as it is greater than the pullin 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 57. Openloop 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 58. Openloop responses to a step command showing overshoot that result in pullin. command of 140 also results in pullin of the response as the overshoot causes the device to move beyond the pullin 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 pullin [1 11]. Dynamic pullin can result in cases where the velocity of the actuator is high as it approaches the pullin point. This can be caused in the case of applying instantaneous actuation voltages, and it can cause the actuator to pullin at a lower voltage than the static pullin 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 pullin can occur at an 8% lower voltage than the static pullin 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 59 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 59, and as a function of voltage in Figure 510. In this instance, only two of the micromirrors, 1 and 3, experienced pullin 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 59. Results from dynamic study showing pullin 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 510. Results showing the hysteretic behavior of the micromirrors. A) Mirrors 1 and 3 show pullin and hysteresis. B) Mirrors 2 and 4 do not have pullin. 5.3 Hysteresis Case Study: ProgressiveLinkage As discussed in the literature review Section 2.3, there are ways that researchers have used nonlinear flexure designs to mitigate electrostatic pullin and hysteresis. One such nonlinear flexure design is presented here, called a progressivelinkage [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 openloop operation. The results presented are only theoretical and have not been fully realized in fabrication. 5.3.1 ProgressiveLinkage 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 pullin 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 fourbar model as depicted in Figure 511. The geometric relationships between the links are also shown in Figure 511. 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 51 1. rz = ro + rl '3 (512) Since the fourbar mechanism is a onedegreeoffreedom 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 (513) In order to determine the angle Os, begin with the relationship r,2 = rd2 + 32 2rzrd COsY3 (514) This yields an expression for y,. 73 co (515) An expression for ed is found from Od = tan 'ir4 + r~ (516) The angle 83 is given as 03 = Od + 3 (517) 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 511i. Diagram of fourbar 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 fourbar mechanism. For the sake of this discussion it is assumed that this 2D representation of the fourbar 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 512, a beam of length L with a rectangular crosssection 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) (518) 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 (519) 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 3.36t Ir~t 1 (520) Figure 512. Cantilever beam with crosssection w x t, and length L. The resulting static force and moment equations can be determined from the free body diagrams in Figure 513. F~lbarl = F, + F2 =0(1 SFar2 = 2 F3 = 0 (522) Fbar3r = F3 + Fo; = 0 (523) SnJomtl = s T + TS2 + rxF2 =0 (524) C yln't2 T2 T3gZx F3 =0 (525) CMio'nt3 0 3s,T4rx Fo =0 (526) 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 515 shows plots of the ~Ijoint 2 TS F T, joint 1 F~ Figure 513. 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 514. Progressivelinkage 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 321. Tables 53 and 54 give the link length and joint dimensions used for this progressivelinkage 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 321 is also included for comparison. The requirements for pullin 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 pullin behavior. The static 6V 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 53. Link length dimensions used for progressivelinkage design. Link Length (Clm) ro 9.000 rl 8.625 r2 9.953 r3 4.375 Table 54. Joint dimensions used for progressivelinkage 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 516. Static eV relationship for micromirror with a progressivelinkage. Recall from Section 3.4.5 that a bifurcation analysis may be used to examine the electrostatic pullin 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 (527) i=1, 2, 3 Applying this analysis to the device using the progressive linkage yields the bifurcation diagram shown in Figure 517. 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 518 shows how the bifurcation plot will change as the progressive stiffness profile increases or decreases by a factor of 2. 5.3.2 OpenLoop Response Using a ProgressiveLinkage Because this device does not experience pullin, 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 pullin. The system openloop response to sinusoidal inputs is shown in Figure 519. Figure 520 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 pullin 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 57and 58. 5.3.3 Parametric Sensitivity of the ProgressiveLinkage 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 progressivelinkage will add to the effects of this sensitivity. The following discussion will examine the effects of fabrication tolerances on the progressivelinkage 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 517. Bifurcation diagram for micromirror using a progressivelinkage to avoid pullin behavior. 30 25 20 ~i15 '10 5 0 25 50 75 100 125 150 175 200 225 Voltage (V) Figure 518. Bifurcation diagram for the micromirrors using a progressivelinkage to avoid pull in behavior for different values of mechanical stiffness. 1Ro Command  Result Time [s] Time [s] Figure 519. Openloop 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 520. Openloop response to a step input for device using a progressivelinkage. 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 53 and 54 gave numbers for the progressivelinkage 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 54, 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 55 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 fourbar 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 55. Uncertainties in the j oint dimensions for a proposed progressivelinkage 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 progressivelinkage design is examined when only one variable is altered at a time. Figure 521 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 torquetheta profie are shown in Figure 522. It is striking to see the very large effects of these very small parametric perturbations, and from a qualitative pointofview, 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 ProgressiveLinkage 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 progressivelinkage has been developed. This design, illustrated in Figure 523, 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 524 shows a micrograph image of the progressivelinkage and the micromirror device. Because of the planar fabrication requirements of surface micromachining, the diagonal top member of the fourbar device, r,, can be acquired via a kinematically equivalent Lshaped beam, shown in Figure 524. 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 timeconstraints with the available fabrication run, indepth analysis of the device performance was not conducted before the Einal design was submitted for Figure 522. Fifty Monte Carlo simulation results for varying the joint fabrication parameters for the progressivelinkage 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 521. Results of parametric analysis for individual errors in j oint fabrication of the progressivelinkage. 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 timetables of foundry services and available project funding. It also gives a good example of the consequences of incomplete a priori analysis. Proglressivelinkage Mirror Surface Lslasx~caquivalent link C Figure 523. Schematic drawing of the prototype progressive linkage spring. A) Progressive linkage spring. B) Spring attached to the micromirror. C) Drawing of Lshaped equivalent beam. Figure 524. 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 525 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 progressivelinkage 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 pullin 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 progressivelinkage 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 progressivelinkage is modeled in ANSYS using beaml89 elements [99]. As the structure is displaced about the Xaxis, is soon becomes clear from looking at the resulting displacement of the linkage, that the design is not operating as the intended fourbar model, but is instead deflecting in the positive Zdirection, outofplane. This Zdirection deflection prevents the joints, which are fabricated as thin beams, from rotating as they are intended. Figure 526 shows the results of the FEA analysis of the prototype design for both linear deflection analysis, and nonlinear, largedeflection 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 progressivelinkage in this current design implementation is not providing the appropriate motion that is capable of providing the nonlinear stiffness profile to affect the pullin 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 fourbar linkage design principles. This remains as future work. 20 18 16. No Data 14 1 4 * 0 50 100 150 200 Voltage (V) Figure 525. Experimental data collected for the prototype of the micromirror with the progressivelinkage 3000 * Nonlinear 2500  Linear 3.2000 S1 500 500 0 10 20 30 40 50 Theta (deg) Figure 526. Results from FEA of the prototype progressivelinkage design for linear and nonlinear deflection analysis shows the prototype progressivelinkage 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 degreeoffreedom massspringdamper 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 degreeoffreedom. 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 onedegreeoffreedom model assumption. However, it is clear that motion in other directions, namely the Zdirection 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 Zaxis 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 Xaxis 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 degreeoffreedom 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 pullin, as demonstrated previously, but also the point at which the mirrors will release from pullin 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 pullin and hysteresis also have the ability to negatively affect the dynamic response for actuation signals that occur below the pullin 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 pullin, 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 steadystate 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 steadystate 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, pullin 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 singleinput/multipleoutput (SIMO) system and will discuss the feedback signals available by considering two different kinds of optical sensors: position detecting sensors (PSD) and chargecoupled devices (CCD). The performance of these sensors is considered as well as the impact they will have on implementation of closedloop control system on the array of micromirrors. r + Contllrole Plant .~ Y Desired angle Actual angle Figure 61. Basic block diagram with unity feedback. 6.1 PID Control ProportionalIntegralDerivative (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 secondorder 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 (62) 'K and K, T, =D(6 3) KP The block diagram of this system is shown in Figure 62. The closedloop transfer function for this block diagram, with the plant modeled as a linear second order system is C(s) Gc (s)GP (S) (64) R(s) 1 + Gc (s)GP (S)H(s) Assuming unity feedback, that is H(s) = 1, and substituting Equation 61 into 64 gives an expression for the closedloop transfer function. C(s) (s2KD + SKP +KI ) (65) R(s) s3 + (2goi, + KD )S2 + (KP + (0 ,)S + KI Controller G,(s) Sensor H(s) Figure 62. 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 steadystate error in the system. The derivative term, as seen in the denominator of Equation 65, 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 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 closedloop 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 steadystate error despite the presence of model uncertainty. Using only a simple proportional controller (P controller) on the system is not sufficient to ensure zero steadystate 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 63. 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 closedloop system has no overshoot, which is important in electrostatic systems that experience pullin. 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 64 shows the effects of model uncertainty for the nonlinear plant response, including model variations of 10% variation in the spring stiffness, ks,. Openloop analysis in Chapter 5 presented the openloop 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 65 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 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 pullin 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 64. 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 pullsin, 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 65. Closedloop 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 pullin. 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 66, resulting in a fast switching behavior until the commanded position of the mirror returns to the stable range of motion. In Figure 66, 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 pullin 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 pullin 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 progressivelinkage that can be utilized to eliminate the effects of pullin 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 pullin 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 closedloop control is still necessary. Figure 67 shows the closedloop PID step responses for the micromirrors with the progressive linkage, and they are in fact able to achieve stable rotation above the pullin 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) Openloop 0 0.5 1 1.5 2 2.5 3 Time (s) Figure 66. Closedloop 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 67. Closedloop 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 statespace 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 fullstate 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 68, assumes full state feedback. Cases without fullstate feedback will require the use of an estimator and will be discussed in section 6.2.1. noise, ~ l n~ disturbance Figure 68. General block diagram for LQR controller problem The plant is modeled as a continuous time, linear system described by a set of statespace 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 (67) 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 positivedefinite 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) (68) Substituting Equation 68 into Equation 66 gives x = Ax BKx = (A BK)x (69) Thus, Equation 67 becomes J = (x'Qx +x'KIRKx)dt = ~x' (Q + KIRK)xdt (610) The following relationship sets a condition that restricts K to be finite. x'(Q + K*RK)x = (x'Px) (61 1) where P is a positivedefinite Hermitian or real symmetric matrix. Evaluating the right hand side of Equation 611 and substituting in Equation 69 yields i'Pxx"'Pi= ' (AK'+(BKC) PPAK (612) Equation 612 must hold true for any x, therefore (A BK)*P + P(A BK) = (Q + K*RK) (613) IfABK is a stable matrix, there exists a positivedefinite matrix P that satisfies Equation 613. In order to determine this matrix P, evaluate the cost function J. J = [x'(Q +K'RK)xdt = xPxo (61 4) Since ABK is stable, all of the eigenvalues are assumed to have negative real components and x(oo)4 0 Equation 614 becomes J = x*(0)Px(0) (615) Because R is defined as a positivedefinite Hermitian matrix, it can be written in terms of a nonsingular matrix, T. R = T*T (616) Substitute Equation 616 into 613 to get (A' K*B')P +P(A BK) +Qe+K*T*TK = (617) 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 (618) with respect to K. This expression is nonnegative and the minimum occurs when it is zero, or when TK = (T*)'B'P (619) Hence, the optimal matrix K is found by K = T '(T*') 'B*P= R 'B*P (620) The matrix P in Equation 620 must satisfy the reducedmatrix Riccati equation A'P +PA PBR'B'P+ Q = 0 (621) Equation 621 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 620 to find the optimal gain matrix K that is used in the control law of Equation 68. 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 (622) 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 (623) For this problem, the cost function J can be defined in terms of the error signal. J = l(e'Qee+ u'Reu~dt (624) Equation 624 can be rewritten as follows J = ((x'x + u'Ru + 2x~~'Nud (625) with Q = (HA FH)" ep (HA FH) (626) R = B*HQeHB + Re (627) N = (HA FH)QeHB (628) Qe = I, Re = pl (629) 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 (63 0) The solution of Equation 630 results in the matrix P such that the controller is described as K =[R'B'P R'N']=[Kfb K,.] (631) where Kgb andKffare the feedback and feedfoward controller gains, respectively. The control law is thus written as u= (Kybx+Kfse) (632) The use of this LQR control law for tracking a reference command with zero steadystate 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 typeone 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 69. There is a feedforward gain, Ky, and a feedback gain, Kfb, as in Equation 632. In this case, the error signal is the r L ~ R +x = Ax + Bu Figure 69. 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 fullstate feedback is available for this system for the feedback loop. In cases where full state feedback is not available, stateestimation 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 fullstate feedback is available for the controller. In many cases, fullstate 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 610. 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 610. Block diagram of LQR controller using a stateestimator 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 632 becomes u= (Kex+Kfse) (634) The mathematical model for the estimator is similar to the plant model of Equation 66 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 ) (63 5) where the A, B, C, D are the matrices of the plant model from Equation 66, and 9 = Ci . One method to design the estimator matrix gain, L, is to use Ackermann's formula for pole placement for singleinput systems. In this method, the gain L is calculated such that the state feedback signal places the closedloop poles of the estimator at desired closedloop pole locations, h. Ackermann's. formula is L= [0 00 1 [BiABiBA" B] 1(A) (636) 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 (637) The coefficients a have a relationship with the roots of the polynomial, h, which are also the closedloop pole locations. (s j)(s A,).(s A,) = sn + azsl + +aes"2 + + an,,s + a, (63 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 stateestimator 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 (639) 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 openloop 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 stateestimator 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 openloop poles on the real axis of the Splane. 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 611. The closedloop 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 612. Closedloop 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 closedloop system when the micromirror is commanded to an unstable position, and thus experiences electrostatic pullin 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 stateestimation 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 progressivelinkage. Based on the similarity between the responses from the PID and LQR controllers, it safe to assume that the closedloop LQR performance for the system with a progressivelinkage will be very similar to that of Figure 67. Figure 613 shows the closedloop LQR step responses for the micromirrors with the progressive linkage, and they are able to achieve stable rotation above the pullin 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 613. Closedloop 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 singleinput/singleoutput (SISO) system. In reality, these micromirrors are part of an array that is singleinput/multipleoutput (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 chargecoupled 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 steadystate 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 614 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 615 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 openloop 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 614. Schematic of modeling an array of mirrors as a SIMO system. 2 y 0 Mirror Array 10 15 Figure 615. 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 chargecoupled 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 616, the case of light from one array of mirrors reflecting onto a 1D 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 616 also shows the 1D 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 1D array is only able to measure the CG in terms of one direction (yaxis shown), the offaxis 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 1D PSD entirely, also affecting the location of the CG. For a single micromirror array being measured by a 2D 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 617, errors in spacing can shift the CG in the ydirection, but linearity 84 b, ~ I b2 C) CG Desired Errors In spacing Srnall errors in Large errors in Ilnearlty linearlty Figure 616. Illustration of the measured center of gravity (CG) on a 1D 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 617. Illustration of the measured center of gravity (CG) on a 2D PSD when there are errors in the spacing and linearity of the micromirrors. 168 errors can also shift the CG in the xdirection. 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 metaloxidesemiconductor (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 ydirection displacement measurements for each spot and compare that with the desired positions. This is illustrated in Figure 618 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 618. 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 619 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 620. In general, the angle, 'P, between two vectors, a and b as shown in Figure 620, can be calculated using the dot product relationship. a = a Ibcos'F (640) 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 (641) 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 621 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 #(642) 2 0480~o~ s COs # (643) 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 619. 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 620. 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 622, 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 (645) BOr + R,~yBOy)t = y (646) BOz + (R BOZ)t= z (647) Solving for t yields, 111 1 Cx Cx 3x, BOx C, Cy 3y, BO, C, C, C, BO 1z 2z 3z=t (648) 1 11 0 Cix Cx 3x, BOx C,y C2y C3y BO, CI, Cz 3z BOz This value for t calculated from Equation 636 may be substituted back into Equations 645, 646 and 647 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 621. Calculating the reflected ray of light. Frame C C, BO g R Cz b. BrR, Frame B Figure 622. The intersection of the line from BO to R and the plane C occurs at point P. 6.4.1 PSD Response A 1D 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, (649) 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 623 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 2D 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 623. 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 2D view of the reflected light on the sensor plane with the CG marked in the red star. Figure 624. 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 625 shows the resulting openloop 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 steadystate error. In the case of a broken device or a mirror in the Theta (cleg) Figure 624. 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) x106 Time (s) x1o' B Figure 625. Openloop 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 626 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 pullin 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 627. 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 105 Time (s) x 10 B Figure 626. Openloop 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 627. Incorporating feedback control into array model. micromirror array system, the closed loop response is determined. Figure 628 shows the response for a PID controller, and likewise, Figure 629 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 628. 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 629. 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 closedloop control in the case of a device in which one mirror in the array is not functioning. Shown in Figure 630 is the closedloop 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 630. Closedloop 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 63 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 632 in which the PID closedloop 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 631. 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 632. Closedloop 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 closedloop 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 openloop 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 voltagedisplacement relationships and the electrostatic pullin and hysteresis that can affect the dynamic system response as well. Electrostatic instability is addressed here through the introduction of a progressivelinkage 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 Xdirection (ROTX), which corresponds to the onedegreeoffreedom model assumption. However, it is clear that motion in other directions, namely the Zdirection (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 degreeoffreedom 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 openloop dynamics, while exhibiting some overshoot behavior in the transient response, operates on a very fast timescale, 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 closedloop control techniques will enable both cost reduction as the devices will no longer require extensive calibration for openloop performance, as well as improved performance and reliability. The impact of this work is not limited to the application of micromirror or microoptics 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 onedegreeoffreedom 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 pullin. In order to design a robust microsystem that can be deployed in a wide variety of scenarios, the device should have onchip 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 onchip 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 closedloop 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 A1 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 XY plane, and extruding the thickness in the positive Zaxis. MMPoly 0 (Ground Plane) MMPoly 1 (Fixed Finger) MMPoly 2 (Fixed Finger) MMPoly 3 (Moving Finger) MMPoly 4 (Mirror Surface) 11.5 9 . Figure A1. 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. Fivehundred 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 B1. 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 B3. 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 B4. 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 B5. 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 B6. 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 B7. 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 B8. 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 B9. 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 B10. 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 B11i. 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 B12. 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 B13. Histogram of values for the linewidth variation of layer MMPoly4. Irli 111~ I O LI 2.34 2.36 Mass [kg] Figure B14. Histogram of the mass values calculated from the parametric variation data. 0 0.02 Figure B15. 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 B16. 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 B17. 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 B18. 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 B19. 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 2.00E04 S1.50E04 S1.00E04 I 1II 111 186.88 kHz 5.00E05 0.00E+00 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C1. Magnitude of FFT results for device 1, trial 1. 4.50E04 4.00E04 3.50E04 3.00E04 2.50E04 2.00E04 1.50E04 1.00E04 5.00E05 0.00E+00 U" l'' '', "~ ~''''" '1"" "~ "'ve mrwwu*cl 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C2. 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] 4.50E04 4.00E04 3.50E04 3.00E04 ,E 2.50E04 'B2.00E04 1.50E04 1.00E04 5.00E05 0.00E+00 Figure C3. Magnitude of FFT results for device 1, trial 3. 8.00E 04 186.88 kHz 7.00E04 35.78 kHz 6.00E04 5.00E0482.66 kHz S4.00E04 3.00E04 2.00E04 1.00E04 0.00E I00 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C4. Magnitude of FFT results for device 1, trial 4. 2.50E04 2.00E04 51.25 kHz S1.50E04 o 81.56 kHz J 1.00E04 I187.19 kHz 123.75 kHz 5.00E05 0.00E+00 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C5. Magnitude of FFT results for device 1, trial 5. 7.00E04 6.0443.44 kHz 5.00E04 4.00E04 82.19 kHz 140.78 kHz 3.00E04 2.00E04 1.00E04 0.00E100  0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C6. Magnitude of FFT results for device 2, trial 1. 202 5.00E04 4.50E04 43.13 kHz 4.00E04 3.50E04 S3.00E04 S2.50E04 139.53 ktHz S2.00E04 ~ I 85.63 kHz 1.50E04 0.00E 100 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C7. Magnitude of FFT results for device 2, trial 2. 3.00E04 43.44 kHz 2.50E04 2.00E04 85.31 kHz 25.63 1.0E0 kHz 118.59 kHz I ,I II I I 136.56 kHz 1.00E04 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C8. Magnitude of FFT results for device 2, trial 3. 203 2.00E 04 1.80E0427.34 kHz 1.60E04 1.40E04 S1.20E04 ( i43.59 kHz S1.00E04 3 8.00E05 I 92.03 kHz 137.03 kHz 6.00E05 4.00E05 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C9. Magnitude of FFT results for device 2, trial 4. 4.50E04 41.88 kHz 4.00E04 25.6 3.50E04kz 3.00E04 136.10 kHz ,E, 2.50E04 .o 2.00E04 8 1 111,1185.31kz 1.50E04 1.00E04 5.00E05 0.00E 100  0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C10. Magnitude of FFT results for device 2, trial 5. 204 2.50E04 2.00E04 'L.u5 kHz 50.16 kHz 105.31 kHz ~i1.50E04 S1.00E04 5.00E05 0.00E 100 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C11i. Magnitude of FFT results for device 3, trial 1. 2.50E 04 46.88 kHz 2.00E04 183.91 kHz 31.25 ~'1.50E04 kHz P 116.56 kHz o 90.31 1.00E04 5 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C12. Magnitude of FFT results for device 3, trial 2. 205 3.00E04 81.41 kHz 42.34 kHz 2.50E04 2.00E04 I II, (112.97 kHz 132.66 kHz S1.50E04I 8 1 I.,1II YII 1 180.63 kHz 1.00E04 , 5.00E05 0.00E100 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C13. Magnitude of FFT results for device 3, trial 3. 3.50E04 3.00E04 183.28 kHz 2.50E04 1 48.13 kHz i~2.00E04 S1.50E04 1.00E04 5.00E05 0.00E 100 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C14. Magnitude of FFT results for device 3, trial 4. 206 1.80E04 1.60E04 183.28 kHz 1.40E04 I 57.34 kHz S1.20E04 I ,183.28 kHz S1.00E04 S8.00E05 IIIdII 1 106.56 kHz 6.00E05 040000E 0 0 0 50 100 150 200 250 300 350 400 450 500 Frequency [kHz] Figure C15. 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Bamberger, "On the dynamic pullin 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. 318322. [1 13] W.A. Moussa, H. Ahmed, W. Badawy, and M. Moussa, "Investigating the reliability of the electrostatic combdrive actuators utilized in microfluidic and space systems using finite element analysis," Canad'ian J. Electrical Computer Syst., vol. 27, no. 4, pp. 195200, 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 closedloop 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 itzCoy, 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, Pullin and Hysteresis......................................................................28 2.3.2 Design Techniques to Eliminate Pullin...........................................................31 2.3.3 Capacitive and Charge Control Techniques to Eliminate Pullin.....................32 2.3.4 Closedloop Voltage Contro l to Eliminate Pullin............................................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 OpenLoop Step Response..........................................................................................122 5.2.1 Effects of Parametric Uncertainty on Step Response.....................................123 5.2.2 Effects of Pullin and Hysteresis on OpenLoop Response............................123 5.2.3 Continuous Characterization of Micromirror Arrays......................................126 5.3 Hysteresis Case Study: ProgressiveLinkage..............................................................128 5.3.1 ProgressiveLinkage Design...........................................................................128 5.3.2 OpenLoop Response Using a ProgressiveLinkage......................................135 5.3.3 Parametric Sensitivity of the ProgressiveLinkage.........................................135 5.3.4 ProgressiveLinkage 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 21 Summary of feedback control papers discussed in the l iterature review...........................36 31 Mean and standard deviation of fabri cation variations for layer thickness in the SUMMiT V surface micromachining process...................................................................47 32 Mean and standard deviation of fabricat ion variations of line widths in SUMMiT V......47 33 Values output from finite element an alysis of mechanical spring stiffness.......................63 34 Comparison of polynomial fit for approximation of cap acitance function........................66 35 Comparison of polynomial fit for approximation of cap acitance function........................66 36 List of parameters used for this analysis............................................................................78 41 Mean and standard deviation of fabri cation variations for layer thickness in the SUMMiT V surface micromachining process...................................................................82 42 Mean and standard deviation of fabricat ion variations of line widths in SUMMiT V......82 43 Spring stiffness values for changing dimensional and material parameters......................85 44 Results from the Monte Carlo simulations for the capacitance values in terms of mean, standard deviation, and the percent change from nominal......................................97 45 Mean and standard deviation for pullin a ngle and voltage from sets of mirrors on all three arrays tested............................................................................................................108 51 Modal analysis results for first 10 mode s and their natural frequencies, and the participation factors and ra tios for each direction............................................................118 52 The first three natural frequencies de termined from the LDV experiment.....................122 53 Link length dimensions used for progressivelinkage design..........................................134 54 Joint dimensions used fo r progressivelinkage design.....................................................134 55 Uncertainties in the joint dimensions for a proposed progressivelinkage design...........138 PAGE 9 9 LIST OF FIGURES Figure page 21 The SEM images of MEMS devices created using SUMMiT V microfabrication process........................................................................................................................ ........25 22 Images of micromirror arra ys developed in industry.........................................................26 23 Adaptive optics (AO) mirror used for wavefront correction.............................................27 24 Use of an AO MEMS programmabl e diffraction grating for spectroscopy.......................27 31 Drawing of the SUMMiT V structural and sacrificial layers............................................46 32 Area with nominal dimensions L and w w ith the dashed line indicating the actual area due to error in the line width......................................................................................47 33 Images of the micromirror array........................................................................................48 34 Illustration of mirrors operating as an optical diffraction grating......................................48 35 Micrograph of an array of mirrors and schematic of mirro r with hidden vertical comb drive.......................................................................................................................... .........49 36 Solid model of micromirror showi ng polysilicon layer names from SUMMiT V............49 37 Schematic of a parallel plate electrosta tic actuator modeled as a massspringdamper system......................................................................................................................... .......51 38 Static equilibrium relationship for th e parallel plate elect rostatic actuator.......................54 39 Electrostatic force for different voltage s and mechanical force showing pullin for the electrostatic para llel plate actuator...............................................................................54 310 Pullin function for the parall elplate electrostatic actuator..............................................55 311 Static equilibrium relationships for the parallel plate actuator using different spring constants...................................................................................................................... .......56 312 Schematic of a torsion electrostatic actuator.....................................................................57 313 Static equilibrium relations hips for the torsion actuator....................................................59 314 Pullin function for the torsion actuator.............................................................................60 315 Drawing of the mechanical spring that supports the micromirrors and provides the restoring force................................................................................................................ ....61 PAGE 10 10 316 Image of the mechanical spring that supports the micromirro r indicating boundary conditions and location for applying displacem ent loads for finite element analysis.......62 317 Image from ANSYS of the deformed spri ng and the outline of the undeformed shape after displacements are applied..........................................................................................63 318 Solid model geometry of the unit cell us ed in the electrostatic FEA simulation...............65 319 Capacitance calculation as a function of rotation angle, calculated using 3D FEA and varying orders of polynomial curve fit approximations..............................................66 320 Plot of the Pullin function PI( ) for the micromirror with the vertical comb drive actuator showing that pullin occurs at 16.5 degrees.........................................................67 321 Electrostatic and Mechani cal torque as a function of rotation angle, theta, and voltage for different voltage values...................................................................................68 322 Torque as a function of rotation angle, theta, and voltage for different values of mechanical spring constant................................................................................................69 323 Plot of static equilibrium behavior, s howing pullin and hysteresis, predicted from the model...................................................................................................................... ......69 324 Static equilibrium relationships for the nonlinear plant model, and the linear plant approximation.................................................................................................................. ..70 325 Static equilibrium relationships for the nonlinear plant model, and the small signal model linearized about an operating point ( 0, V0)............................................................72 326 Illustration of piecewise linearizati on about multiple operating points.............................73 327 Plot showing the roots of the function F(x1) occur where the function crosses zero.........76 328 Bifurcation diagram for a MEMS torsion mirror with electrostatic vertical comb drive actuator................................................................................................................. ....79 329 Bifurcation diagram showing the e ffects of different spring constants.............................79 41 Fabrication tolerances can changes the thicknesses of the layers, resulting in changes in the final geometry dimensions.......................................................................................82 42 Fabrication tolerances can change the dime nsions of a fabricated geometry, affecting the final shape, volume, and mass.....................................................................................83 43 Nominal dimensions used to calculate the volume of the moving mass...........................83 44 Capacitance functions for the electrosta tic model with parametric changes in the layer thickness of the structural polysilicon.......................................................................88 PAGE 11 11 45 Capacitance functions for the electrosta tic model with parametric changes in the layer thickness of the Dimple3 b ackfill and the sacrificial oxide......................................88 46 Capacitance functions for the electrosta tic model with parametric changes in the linewidth error of the structural polysilicon layers............................................................89 47 Static displacement relationships for the micromirror model with parametric changes in the layer thickness of the structural polysilicon.............................................................89 48 Static displacement relationships for the micromirror model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide............................90 49 Static displacement relationships for the micromirror model with parametric changes in the linewidth error of the structural polysilicon layers..................................................90 410 Sensitivity of voltage with respect to change s in line width for each value of ...............92 411 Sensitivity of voltage with respect to changes in layer thickness for each value of .......92 412 Gaussian distribution with a mean of 0 and standard deviation of 1.................................94 413 Histogram for mechanical stiffness when accounting for variations in thickness of MMPoly1 and Youngs modulus.......................................................................................95 414 Histogram for mechanical stiffness taki ng into account varia tions in thickness of MMPoly1, Youngs modulus, and linewidth of MMPoly1...............................................95 415 Results from the capacitance simulation fo r 250 random variable sets that show the effects of parametric uncertain ty on the electrostatic model.............................................96 416 Static displacement results of 250 Mont e Carlo simulations with random Gaussian distributed dimensional variations.....................................................................................98 417 Histogram of values from the Monte Ca rlo simulations for the layer thickness of Sacox3......................................................................................................................... .......99 418 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the Sacox3 layer thickness...............................................................99 419 Diagram of an optical profiler measurement system.......................................................101 420 Six mirrors from the micromirror array m easured with the opti cal profiler system........102 421 Data records from the SureVision display that show the crosssection profile of the tilt angle measurements....................................................................................................103 422 Micrograph image of a single micromirror......................................................................104 PAGE 12 12 423 Experimental static result s taken from individual microm irrors that are not in an array.......................................................................................................................... .......104 424 Approximate locations of data collection on all three arrays..........................................106 425 Experimental results from array 1, area A.......................................................................106 426 Experimental results from array 2, areas D and E...........................................................107 427 Experimental results from array 3, areas A and D...........................................................107 428 Nominal model with experimental data...........................................................................109 429 Modelpredicted results from 100 simu lations with parameters determined by random Gaussian variations, show n with experimental data...........................................109 51 Openloop nonlinear plant response to a step input of 7 degrees for different damping ratios......................................................................................................................... ........114 52 Solid model created for modal analysis...........................................................................118 53 Time series data of the micromirror re sponse to an acoustic impulse taken with a laser doppler vibrometer..................................................................................................120 54 Results from the LDV experi ment showing resonant peaks............................................121 55 Openloop response to a step input of 7 degrees for the nonlinear plant dynamics and variations in spring stiffness, km......................................................................................124 56 Openloop nonlinear plant response to a step input of 7 de grees for 50 random parameter variations.........................................................................................................124 57 Openloop responses to a sinus oidal input showing hysteresis.......................................126 58 Openloop responses to a step command s howing overshoot that result in pullin.........126 59 Results from dynamic study showing pullin and hysteresis...........................................127 510 Results showing the hysteretic behavior of the micromirrors.........................................128 511 Diagram of fourbar mechanism for progressive linkage analysis..................................130 512 Cantilever beam with crosssection w x t and length L ..................................................131 513 Free body diagrams for each member of the linkage.......................................................132 514 Progressivelinkage behavi or for different values of ro in m.........................................132 PAGE 13 13 515 Progressive linkage output for ro equal to 9 m along with the electrostatic torque curves and the linear restoring torque..............................................................................133 516 Static V relationship for micromirror with a progressivelinkage................................134 517 Bifurcation diagram for micromirror usi ng a progressivelinkage to avoid pullin behavior....................................................................................................................... .....136 518 Bifurcation diagram for the micromirrors using a progressivelinka ge to avoid pullin behavior for different values of mechanical stiffness......................................................136 519 Openloop responses to a sinusoidal input for the device using a progressivelinkage...137 520 Openloop response to a step input fo r device using a progressivelinkage....................137 521 Results of parametric analysis for indi vidual errors in joint fabrication of the progressivelinkage..........................................................................................................140 522 Fifty Monte Carlo simulati on results for varying the join t fabrication parameters for the progressivelinkage design.........................................................................................140 523 Schematic drawing of the prot otype progressive linkage spring.....................................141 524 Micrograph of the prototype micromirro r with a progressive linkage spring.................141 525 Experimental data collected for the prototype of the mi cromirror with the progressivelinkage..........................................................................................................143 526 Results from FEA of the prototype progressivelinkage design for linear and nonlinear deflection analysis shows the pr ototype progressivelinkage fails to produce the desired stiffness profile................................................................................143 61 Basic block diagram with unity feedback........................................................................146 62 Block diagram with PID controller..................................................................................148 63 Step responses for PID controller....................................................................................150 64 Closedloop PID response to different step inputs when the spring constant is varied by %........................................................................................................................ ...150 65 Closedloop PID response to a step input of 7 degrees for 50 random sets of parameteric variations......................................................................................................151 66 Closedloop PID response to a co mmanded position in the unstable region...................153 67 Closedloop step responses for PID c ontroller for a system using a progressivelinkage........................................................................................................................ ......153 PAGE 14 14 68 General block diagram fo r LQR controller problem.......................................................154 69 Block diagram of LQR control with an internal model for tracking a step command....158 610 Block diagram of LQR c ontroller using a stateestimat or for a plant without an integrator..................................................................................................................... .....159 611 Step responses for LQR controller...................................................................................161 612 Closedloop LQR response to a step i nput of 7 degrees for 50 random parameter variations..................................................................................................................... .....162 613 Closedloop step responses for LQR c ontroller for a system using a progressivelinkage........................................................................................................................ ......164 614 Schematic of modeling an array of mirrors as a SIMO system.......................................165 615 Schematic drawing of an array of 5 mirrors....................................................................166 616 Illustration of the measured center of gravity (CG) on a 1D PSD when there are errors in the spacing and linea rity of the micromirrors....................................................168 617 Illustration of the measured center of gravity (CG) on a 2D PSD when there are errors in the spacing and linea rity of the micromirrors....................................................168 618 Illustration of the measured errors of th e reflected light from two micromirror arrays onto a CCD..................................................................................................................... .169 619 Schematic of the beam steering experiment with only one micromirror.........................171 620 Geometry used to determine the angle of incidence and reflection.................................172 621 Calculating the refl ected ray of light...............................................................................173 622 The intersection of the line from B0 to R and the plane C occurs at point P..................174 623 Schematic of 5 micromirrors in an array reflecting light onto a PSD sensor..................175 624 Calibration of the PSD for idea l case of five micromirrors.............................................176 625 Openloop results to a step response for an array of 5 microm irrors with model uncertainty.................................................................................................................... ....176 626 Openloop response for system with one broken mirror and 4 ideal mirrors, measured by a PSD....................................................................................................................... ...177 627 Incorporating feedback control into array model.............................................................177 628 Controlled PID step response using PSD sensor.............................................................178 PAGE 15 15 629 Controlled LQR step response using PSD sensor............................................................178 630 Closedloop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a PSD sensor.....................................................................................179 631 Schematic of 5 micromirrors in an arra y reflecting light onto a CCD sensor where each separate location of the light can be measured........................................................180 632. Closedloop response for system with one broken mirror and 4 ideal mirrors, for a PID controller and a CCD sensor............................................................................................180 A1 Geometry dimensions in m for creating electrostatic model.........................................188 B1 Histogram of values for the thickness of layer MMPoly0...............................................189 B2 Histogram of values for the thickness of layer MMPoly1...............................................190 B3 Histogram of values for the thickness of layer MMPoly2...............................................190 B4 Histogram of values for the thickness of layer MMPoly3...............................................191 B5 Histogram of values for the thickness of layer MMPoly4...............................................191 B6 Histogram of values for the thickness of Dimple3 backfill.............................................192 B7 Histogram of values for the thickness of layer Sacox1....................................................192 B8 Histogram of values for the thickness of layer Sacox2....................................................193 B9 Histogram of values for the thickness of layer Sacox3....................................................193 B10 Histogram of values for the thickness of layer Sacox4....................................................194 B11 Histogram of values for the linew idth variation of layer MMPoly2...............................194 B12 Histogram of values for the linew idth variation of layer MMPoly3...............................195 B13 Histogram of values for the linew idth variation of layer MMPoly4...............................195 B14 Histogram of the mass values calculated from the parametric variation data.................196 B15 Histogram of values from the Monte Ca rlo simulations for the linewidth error of MMPoly2........................................................................................................................ .196 B16 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the linewidth error of MMPoly2....................................................197 B17 Histogram of values from the Monte Ca rlo simulations for the layer thickness of MMPoly1........................................................................................................................ .197 PAGE 16 16 B18 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of MMPoly1.............................................................198 B19 Histogram of values from the Monte Ca rlo simulations for the layer thickness of Sacox4......................................................................................................................... .....198 B20 Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of Sacox4.................................................................199 C1 Magnitude results of FFT for device 1, trial 1.................................................................200 C2 Magnitude results of FFT for device 1, trial 2.................................................................200 C3 Magnitude results of FFT for device 1, trial 3.................................................................201 C4 Magnitude results of FFT for device 1, trial 4.................................................................201 C5 Magnitude results of FFT for device 1, trial 5.................................................................202 C6 Magnitude results of FFT for device 2, trial 1.................................................................202 C7 Magnitude results of FFT for device 2, trial 2.................................................................203 C8 Magnitude results of FFT for device 2, trial 3.................................................................203 C9 Magnitude results of FFT for device 2, trial 4.................................................................204 C10 Magnitude results of FFT for device 2, trial 5.................................................................204 C11 Magnitude results of FFT for device 3, trial 1.................................................................205 C12 Magnitude results of FFT for device 3, trial 2.................................................................205 C13 Magnitude results of FFT for device 3, trial 3.................................................................206 C14 Magnitude results of FFT for device 3, trial 4.................................................................206 C15 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 pullin, 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 degreeoffreedom (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 pullin 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 progressivelinkage that is intended to eliminate the occurren ce of electrostatic pu llin 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 highprecision 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 pullin instability that occu rs when the electrostatic force overcomes the mechanical restoring force. When pullin 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 pulledin, the voltage required to maintain the fu lly actuated position is lower than the pullin 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, stateoftheart micromirror arrays rely on openloop 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, closedloop feedback control techniques are consid ered essential. Feedback control has long been used in many macroscale 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 singleinput/multipleoutput 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 pullin 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 openloop actua tion methods. These models also characterize the effects of electrostatic instability and the re sulting hysteresis. The modeling is extended from the initial quasistatic 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 progressivelinkage is presented that will eliminat e the effects of electrostatic pullin 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 progressivelinkage 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 closedloop 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 pullin instability is given, including modeling methods and the different methods that are dedicated to addressing pullin. 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, Multilevel 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 21 show scanning electr on micrographs (SEMs) of a mechanical gear hub and a crosssection of a pinjo int that allows rotation. Thes e are excellent examples of the complex structures created from layering simple, 2D 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 21. The SEM images of MEMS device s created using SUMMi T V microfabrication process. A) Micromachined gears. B) Micr omachined gears. C) A crosssection view of a pinjoint 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 freespace 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 22. PAGE 26 26 A B Figure 22. 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 23 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 23. 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 24. 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 siliconbased 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, Pullin 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 lldocumented phenomenon of electrostatic pullin. The electrostatic force is nonlinear as it is inversely proportional to the square of the electrode gap. Pullin, sometim es called snapdown, 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 pullin characteristics can be modeled fairly accura tely [3], [14], [39][52 ], [106]. Pullin for a parallelplate actuator occurs at onethird of the separation gap, which greatly limits the actuator stroke. Another phenomenon associated with pullin in stability is that once the mirror has pulledin, the voltage required to maintain the pullin position is lower than the pullin voltage. The mirror will not return from this position until th e actuating voltage has been reduced below the holdingvoltage. 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 deadband after pullin in which no actuation is even possible until the applied voltage drops below the holding threshold. The effects of pullin 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 crosssecti on along the length of the devi ce. These nonconstant 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 pullin 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 pullin 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 pullin 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 pullin as an advantage that allows for openloo p, on/off binary actuation at reduced voltages [2], [53]. While the actual pullin voltage of the device may vary slightly from mirror to mirror due to variations in dimension and material prope rties, reliable openloop 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 pulledin 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 llin has been thoroughly docum ented and there has been a considerable amount of research conducted to find ways to avoid pullin 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 closedloop f eedback control. A review of these methods is given in the following sections. 2.3.2 Design Techniques to Eliminate Pullin There are multiple design methods researcher s have employed to address the problem of electrostatic pullin 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 leveragedbending approach introduced by Hung and Se nturia [24] uses the stressstiffening of a fixedfixed 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 Pullin 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 hedcapacitor 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 steadystate 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 steadystate 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 Closedloop Voltage Control to Eliminate Pullin 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 pullin point [68], [69]. Chu, and Pister di scuss the effect of introducing a voltage control law into a system of electro statically actuated parallelplates 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 bypass th e pullin 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 closedloop 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 21, 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 steadystate 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 lineartimeinva riant (LTI) techniques such as proportionalintegratorderivative (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, leadlag, and statevariable 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 lineartimeinvarian t (LTI) control methods. The nonlinear effects of electrostatic actuation are pe rhaps most evident for parallelpl ate actuator systems. Lu and Fedder used a linearized plant model for a para llelplate 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 21. 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 ParallelPlate 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 ParallelPlate Electrostatic No [74] [Miathripala, et al., 2003] Nonlinear Stability ParallelPlate Electrostatic No [73] [Sane, 2006] Nonlinear Increase Stability, Position Tracking ParallelPlate Electrostatic No [77] [Horsley, et al., 1999] Classical (PD, Phaselead) Position Control ParallelPlate 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] StateFeedback Position Control Electrostatic Lateral Comb Drive Yes, Capacitve [70], [71] [Lu, Fedder, 2002, 2004] Classical (P) Increase Stability, Position Tracking ParallelPlate Electrostatic Yes, Capacitve [15] [Chu, et al., 2005] StateFeedback 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 (MuSynthesis) Position Tracking DualStage Disk Drive Yes, unspecified [27] [Kim, et al., 2004] Adaptive control, Robust (HInfinity) 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 DualStage 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 closedloop 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 realtime 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 bulkmicromachining. 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) Hinfinity control and ad aptive control resulted in good disturbance rejection of bandlimited noi se and the Hinfinity cont roller improved performance by eliminating steadystate drift and reducing noise [26]. There are few examples of robust control de sign methods such as Hinfinity and musynthesis that have been applied to MEMS systems. In addition to the use of Hinfinity control PAGE 39 39 demonstrated by Kim, et al. for a nonMEMS mi cromirror system [26], musynthesis controller design was applied to a dualstage actuator system for trackfollowing in a harddisk drive [82]. The controller design was successf ul in simulations, but no experi mental work has been done so far. The application of musynthesis 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 parallelplate 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 parallelplate 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 discretetime SMC for a dualstage ac tuator for harddisk 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 pullin instability of twoaxis 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 nanoscale 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 multiloop 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 closedloop 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 surfacemicromachining 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 signaltonoise 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 pullin 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 pullin 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 pullin is addressed by introducing a novel design technique called the progre ssivelinkage 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, piecewise 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 pullin 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 pullin. 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 parallelplate 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 combdrive 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 singleinput/multipleoutput (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 31 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 31. 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 31 and 32. This information was ga thered through diagnosti c process testing as described in [98]. Table 31 gi ves the mean and standard deviat ions of the thic knesses of the layers of polysilicon and silicon di oxide. Table 32 gives values fo r variations in the dimensions of the line widths of the device design. Figure 32 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 31. 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 32. 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 32. 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 24pin dual inline package (DIP) in Figure 33. 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 34 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 onedegree of freedom actuated electrostatically and are shown schematically in Figure 35. The electrostati c micromirror arrays have a ground plane and a A B Figure 33. Images of the micromirror array. A) Packaged device. B) Micrograph of the surface of the array. Figure 34. 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 llelplate 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 35. 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 2D crosssection view of a unit cell (figure not to scale). Figure 36. 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 35(a). Not shown in the drawing is a design constraint that restrict s the motion of the fixedend of the mirror plate from moving a large distance in the Zdirection. While some motion may occur, the assumption is made that this device acts in one degreeoffreedom by ro tating about the xaxis. Figure 36 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 lldocumented phenomenon of electrostatic pullin. The electrostatic force is nonlinear, as it is inversely proportional to the square of the electrode gap. Pullin, sometim es called snapdown, 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 pullin instability is that once the mirror has pulledin, the voltage required to maintain the pullin position is lower than the pullin voltage. The mirror will not return from this position until the actuating voltage has been reduced below the holdingvoltage. The result of this holding effect is hysteresis. This section will examine the modeling of the electrostaticmechanical 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 37, 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 37. Schematic of a parall el plate electrostatic actuator modeled as a massspringdamper system. The equation of motion for this massspringdamper system is derived by the balance of the forces on the system from Newtons second law x m F (31) where m is the mass of the moving plate. When th e top plate is displaced in the positive xdirection, shown in Figure 37, 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 (32) The damping force is assumed to be linearly proportional to the velocity by a factor of b the damping coefficient. x b Fb (33) 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 (34) where 0 is the permittivity of free space, 8.854 x1012 F/m. The electric field is given by A Q E0 (35) where Q is the electric charge. The charge, Q, can be written as CV Q (36) 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 (37) Equation 34 can be rewritten as ) ( 2 2 10 2 0 2x x A V CV U (38) 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 (39) The force balance for the system yields the equation of motion. emFmxbxkx (310) 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 (311) PAGE 53 53 Equation 311 can be interprete d to show the relationship between the voltage and the displacement, as plotted in Figure 38 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 pullin instability. It turns out that the soluti ons in the lower portion of Figure 38 are stable solutions and the solutions in the upper porti on are unstable. The maximum voltage value corresponds to the actuation voltage at which pullin occurs, and the maximum stable position for parallel plate actuator occurs at onethird the gap between the electrodes. To further explore the pullin phenomena, the static relationship in Equation 311 can be examined graphically, by plotting the electrostatic force and the mechanical force separately in Figure 39. 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 38, 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 pullin. At the pullin 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 pullin 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 311. 2 2 2 01 2mC Vk xx (312) PAGE 54 54 Figure 38. Static equilibriu m relationship for the parallel plate electrostatic actuator. Figure 39. Electrostatic force for different voltages and mechan ical force showing pullin for the electrostatic para llel plate actuator. Stable Unstable PAGE 55 55 Substituting Equation 312 into 311 and evalua ting at the pullin position results in the following relationship that is only a function of the capacitance and the pullin position, xPI. 2 2 000 ()()PIPIPI xxxxCC x xxxx (313) Assuming that the restoring springs are linear ly deformed in the range of actuation, the pullin angle is independent of the spring stiffness, and depends only on the angle of rotation. A pullin function, PI(x) is defined to determine the pullin angle, which occurs when PI(x) is equal to zero. 2 00 2 00()() ()() CC PIxxxx x xxx (314) In turn, once the pullin angle is determined, the pullin voltage can be calculated by the following expression, 002 () P ImPI PI xxxkx V C xx (315) The pullin function for the parallel plate electr ostatic actuators is shown in Figure 310, Figure 310. Pullin function for the pa rallelplate electrostatic actuator. PAGE 56 56 and verifies that pullin does o ccur at 1/3 the gap between the pl ates. The pullin voltage is calculated from Equation 315 to be equal to 57.85 V. To further investigate the eff ects of changing the spring consta nt on the pullin, the static equilibrium relationships are plotted for different values of the mechani cal spring constant in Figure 311. This shows that even for a diffe rent spring constant, th e pullin displacement location remains at 1/3 the gap, wh ile the pullin voltage changes. A B Figure 311. 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 312. 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 onedegreeoffreedom massspring damper system of the form (,)meJbkTV (316) PAGE 57 57 Figure 312. 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 (317) where C is the capacitance, is the angle of rotati on about the Xaxis, 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 (318) 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 317 and 318. 21 2mC Vk (319) As was previously shown for parallelplate 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 pullin. At the pullin point, both the electrostatic and mechanical torques are equal in magnitude a nd slope [14], [46]. As was shown previously for the parallelplate actuator, electrostatic pullin can be cons idered as the mechanical and electrostatic torques bein g equal, as in Equation 319, and th eir first derivatives being equal. 2 2 21 2mC Vk (320) Combining Equations 319 and 320 and evaluating at the pullin angle results in the following relationship that is only a function of the capacitan ce and the pullin angle, PI. 02 2 PI PIC CPI (321) Assuming that the restoring springs are linearly deformed in the range of actuation, the pullin angle is independent of the spring stiffness, and depends only on the angle of rotation. A pullin function, PI( ) is defined to determine the pullin angle, which occurs when PI( ) is equal to zero. 2 2) ( C C PI (322) In turn, once the pullin angle is determined, the pullin voltage can be calculated by the following expression, PIC k VPI m PI 2 (323) 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 parallelplate 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 312, the capacitance for a torsion actuator in terms of the angle of rotation about its axis is given as 0 maxmax()ln1ln1mW C (324) where is L3/L2, is L2/L1 and max is H0/L1 [46]. From Equation 324 it is possible to calculate the performance for a torsion actuator. The pullin function for this system is 22 maxmaxmaxmax 0 max 22 2 max maxmax3434 1 ()3ln 1 11mW PI (325) 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 313. The A B Figure 313. Static equilibrium relationships for the torsion actu ator. A) The electrostatic and mechanical torques. B) The static rotationvoltage relationship. PAGE 60 60 displacement as a function of the voltage is also shown in Figure 313. The pullin function from Equation 325 is plotted in Figure 314. From thes e figures, it is found that the pullin for this system occurs at 82% of maximum rotation angle for the system. For the given spring constant, the pullin voltage is 6.91 V. As with the parall el plate actuator, the pullin angle will remain the same despite the mechanical spring constant, but the pullin voltage will change. The pullin angle can change, however, if the system geometry is changed. This is different from the parallelplate actuator, which alwa ys pulls in at 1/3 the gap. Figure 314. Pullin 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 pullin and hysteresis. 3.4.1 Mechanical Model The mechanical spring is shown in Figure 315 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 316 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 316 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 sixdegreesoffreedom at the two anchor points. Figure 315. Drawing of the mechanical spring th at supports the micromirrors and provides the restoring force. PAGE 62 62 Displacement loads are applied in all sixdegr eesoffreedom at th e point indicated in Figure 316 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 317. Assuming Hookes law for the force applied to a linear spring, the spring st iffness in all six degreesoffreedom 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 33, 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 offaxis rotations about the Y or Z. It is the value of qX equal to 612.4 pNm/rad that is used for km in Equation 317. Figure 316. 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 317. 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 33. 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 pNm/rad qZ stiffness 11360 pNm/rad qY stiffness16310 pNm/rad 3.4.2 Electrostatic Model In order to compute the electros tatic torque values in Equation 316, it is necessary to find an expression for the capacitance as a function of the rotation angle. For parallelplate 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, 3D 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 crosssecti on of a unit cell made up of onehalf 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 35(c). The mode l of the geometry in Figure 318 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 Xaxis 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 x1012 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 (325) Using numerical values generated in the electrostatic FEA model, Equation 324 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 leastsquares 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 (326) 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 319 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 34. Table 35 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 (327) 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 pullin behavior. The advantage of using the first order fit is that its derivative which is us ed in Equation 316 is a constant, thus simplifying the plant model to a linear approximation in V2. In order to capture th e nonlinear behaviors of pullin 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 318. Solid model geometry of the unit ce ll used in the electros tatic FEA simulation. PAGE 66 66 Figure 319. Capacitance calculation as a function of rotation angle, calculated using 3D FEA and varying orders of polynomi al curve fit approximations. Table 34. 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 0.0001041 0.000777 0.000078 Table 35. Comparison of polynomial fit for approxima tion of capacitance function Order normr s n r2 4 1.1192E05 0.000185 18 0.999785 3 1.1463E05 0.000185 18 0.999775 2 3.6691E05 0.000185 18 0.997691 1 8.608E05 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 316 to 320. Equations 318 to 320 calculate the electrostatic pullin characteristics of the device. A plot of the pullin function is shown in Figure 320 where pullin occurs when the function equals zero at 16.5 degrees. Using this value in Equation 320, the pullin voltage is 71.06 V. PAGE 67 67 Figure 320. Plot of th e Pullin function PI( ) for the micromirror with the vertical comb drive actuator showing that pullin occurs at 16.5 degrees. The static equilibrium behavior can also be evaluated from Equations 316, and 317, 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 321 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 317. 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 pullin point, whic h corresponds to the calculated values of 16.5, and 71.06 volts. The electrostatic torque curve at the pullin voltage, VPI, is also indicated in Figure 321. The pullin 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 pullin angle. Only the value of the pullin voltage wo uld be affected. This is shown in Figure 322. The pullin 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 pulledin, it is possible to reduce the voltage below the pullin 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 pulledin position, but it will re turn to a position different from the pullin 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 323, including the pullin 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 321. Electrostatic and M echanical torque as a function of rotation angle, theta, and voltage for different voltage values. PAGE 69 69 Figure 322. Torque as a function of rotation angl e, theta, and voltage for different values of mechanical spring constant. Figure 323. Plot of static equi librium behavior, showing pullin 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 324 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 324. 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 316 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 (328) The linearization in Equation 328 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 (329) The nonlinear torque approximation is reduced to a constant. 00 TdC CV d (330) 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 325 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 325 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 326. The linearized models discussed above are important when considering control design techniques that require a linear transfer function or statespace 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 325. 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 326. 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 fixedpoint solutions [106], [107]. One advantage of evaluating the bifurcati on behavior of the device is that unlike the methods used in Equations 314 and 315, 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 pullin. Here, th e spring constant is still assumed to be linear, and the results may be compared to thos e obtained using Equations 314 and 315. The state space model for the system is 1 2 2 1 2 1211 ()m ex x x x k b x Txxx JJJ (331) Recall that Te is a function of the capacitance expression from Equation 326. In order to capture the nonlinear effects of the syst em, a fourthorder curve fit approximation is used. The fixed points occur at x2 = 0 and PAGE 74 74 0 ) (1 1 x k x Tm e (332) This can be expressed in full as 322 112131411 (432)0 2mNPxPxPxPVkx (333) Equation 333 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 (334) where 123 and eee are the three roots, and the coefficients A, B, C and D are 2 12 A NVP (335) 2 23 2 B NVP (336) 2 3mCNVPk (337) 2 41 2DNVP (338) Further, define 23 39272 54ABCADB q A (339) 3 2 2 23 9 ACB uq A (340) 3squ (341) 3tqu (342) The roots of Equation 334 are 13e B st A (343) PAGE 75 75 213 ()() 232eB ststi A (344) 313 ()() 232eB ststi A (345) The roots of Equation 333 can be found to determine the static voltagedisplacement 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 (346) In Figure 327, 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 nonphysical 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 steadystate solutions disappear. This is another description of the pullin instability caused by the disappearance of all physically possible steadystate 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 331 and retaini ng only the first order terms [107]. PAGE 76 76 Figure 327. 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 ) ( (347) where 22 1 11213 1() 1 (1262) 2eTx NPxPxPV x (348) The Jacobian defined in Equation 347 relates the perturbation of the states from equilibrium as 11 0 22eix x x Dfx x x (349) The stability is determined by evaluating the ma trix in Equation 347 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 (350) Substituting Equation 348 into 350 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 (351) 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 squeezefilm effect, in whic h air that is compressed between very small spaces begins to act as a viscous fluid [3]. Squeezefilm 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 squeezefilm da mping coefficient analytically difficult. For the purpose of this discussion, an approximation is made to consider the squeezefilm damping term for a torsional plate developed by Pan, et al [100]. 5 3 rot L w bK g (352) 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 (353) PAGE 78 78 Table 36 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 pullin 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 pullin voltage. Table 36. List of parameters used for this analysis. Parameter Value density of polysilicon 2331 kg/m3 absolute viscosity of air 1.73e5 Ns/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 pullin angl e, pullin 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 328. Bifurcation diagram for a MEMS tors ion mirror with electrostatic vertical comb drive actuator. Figure 329. 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 31 and 32 respectively, a nd reprinted in this chapter for convenience as Tables 41 and 42. These show th at dimensions can vary by as much as eight percent in layer thickness, and as much as twen tynine 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 41. 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 42. 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 42 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 41. Fabrication tolerances can changes the thickn esses of the layers, resulting in changes in the final geometry dimensions. PAGE 83 83 Figure 42. 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 43. 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 1011 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 43. 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 41, three linewidth variations listed in Table 42, 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 37. The length of this beam and the crosssectional 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 43 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 43. The first entry in Table 43 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 43. 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 pNm % 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 41 and 42. The results are shown in terms of the capacita nce in Figures 44, 45, and 46. Figure 44 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 45 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 46 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 47, 48, and 49 show these results. Figure 47 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 44, 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 48 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 45. Likewise, Figure 49 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 44. Capacitance functions for the electr ostatic model with parametric changes in the layer thickness of the structural polysilicon. Figure 45. 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 46. Capacitance functions for the electr ostatic model with parametric changes in the linewidth error of the structural polysilicon layers. Figure 47. Static displacement re lationships for the micromirror model with parametric changes in the layer thickness of th e structural polysilicon. PAGE 90 90 Figure 48. Static displacement re lationships for the micromirror model with parametric changes in the layer thickness of the Dimple3 backfill and the sacrificial oxide. Figure 49. 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 (41) 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 410 displays the sensitiv ity of the system to changes in line widths. The same analysis for variations in layer thickness is given in Figure 411. 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 timeconsuming 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 410 Sensitivity of voltage with respect to changes in line width for each value of Figure 411. Sensitivity of voltage with respect to changes in layer thickness for each value of PAGE 93 93 from the fabrication data in Tables 41 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 (42) 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 412. 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 412. Gaussian distribution with a mean of 0 and standard deviation of 1. resulting spring constants, km, had a mean of 634.21 pNm and a standard deviation of .17 pNm. 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 pNm 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 43 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 413. Histogram for mechanical stiffness when accounting for variations in thickness of MMPoly1 and Youngs modulus. Figure 414. 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 Zdirection. The capacitance function is affected by both these changes in dime nsions. Material properties do not play a role in the electrostatic analysis. Figure 415 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 44, 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 415. 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 44. 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 4.73E05 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 416 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 416. 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 417 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 418 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 417. 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 418. 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 outofplane 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 pullin 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 outofplane measurements of a surf ace. The working principle of the instrument PAGE 101 101 is shown in Figure 419. 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 419. 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 420. 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 420, as well as look at crosssections of the data. Figure 421 shows a cross s ection of the micromirror data Figure 420. 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, outofplane, Zdirection, and the horizontal, inplane Xdirecti 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 421 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 421 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 421. Data records from the SureVision disp lay that show the crosssection profile of the tilt angle measurements. A) An example that clearly shows the crosssectional 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 422, 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 422, 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 423. It is clear that the pullin point for this set of experimental data is similar to the data collected on the arrays, and the pullin angle, 13.87, is at the lower range of the pullin angles for the arrays of mirrors. The pullin 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 422. Micrograph image of a single micromirror. Figure 423. 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 onehalf 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 421, this amount of error in the Zdirect 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 424 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 425, 426, and 427, 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 45 give s a summary of the pullin angle and voltages for the data. The PAGE 106 106 Figure 424. Approximate locations of data collection on all three arrays. average pullin 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 pullin value of 16.5 from the analytical model, the lowest value is within 20 percent. Also, the values listed in Table 46 are averaged values over multiple data sets. From Figures 425 through 427, it is evident that in many cases the mirrors did experience pullin very close to th e predicted angle of 16.5. The pullin voltages Figure 425. Experimental re sults from array 1, area A. PAGE 107 107 Figure 426. Experimental results from array 2, areas D and E. Figure 427. Experimental result s from array 3, areas A and D. Area A Area D First run PAGE 108 108 Table 45. Mean and standard deviation for pullin 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 pullin voltage observed at two different locations on the array. Compared to the predicte d pullin 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 426 shows this occurred for array 2 when multiple sets of data were taken. Figure 428 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 fabricationinduced 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 429, and it is evident that the experiment al values fall mostly within the bounds of the modeled variation results. Figure 428. Nominal model with experimental data. Figure 429. Modelpredicted 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 pullin 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 48 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 418 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 429, 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 noncontact 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 pullin angle, and pullin 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 steadystate 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 pullin and hystere sis behaviors of the openloop system, a case study is presented for a progressivel 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 316 in terms of natural frequency, n, and the damping ratio, m nk J (51) 2mb kJ (52) Written in statespace form, the system is described as follows, 1 2 11 2 22 10 01 1 2 2nnx x xx dC V xx Jdx (53) 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 lumpedparameter model determined from Equation 51 is found to be approximately 188 kHz. As stated previously in Section 3.4.5, the sque ezefilm 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 51 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 51. Openloop 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 (54) 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 (55) where i is the eigenvector representing the ith natural frequency, i is the ith natural frequency (rad/s), and t is time. Substituting Equation 55 into 54 yields 20i iMK (56) Ignoring the trivial solution to Equation 56, which is 0i, then the following expression must be true. 20iKM (57) Equations 56 and 57 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 (58) The vector D describes the excitation direction and is of the form DTe (59) 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 (510) The matrix [T] is 00 00 001000()() 010()0() 001()()0 000100 000010 000001 Z ZYY ZZXX YYXX T (511) 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 52 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 degreesoffreedom 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 51. The ratio of each participation factor to the largest participation factor value for a given direction is also listed in Table 51, 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 x1011 kg. The mass result that was reported in Chapter 4 based on the volume of the moving geometry was 2.34 x1011 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 degreeoffreedom, rotating about the Xaxis (ROTX). It is likewise assumed that the first natural frequency will occur in this rotational directi on and be given by Equation 51. 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 onedegreeoffreedom assumption for the model in Equation 53. The largest participation factor is 6.5E05 for the ROTX direction, and this is an orde r of magnitude larger than the next largest participation factor which occurs in the Zdirecti 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 degreeoffreedom motion and in fact, the first resonant frequency excites motion in both the Xaxis (ROTX) and the Zdirection. The motion that occurs in the Zdire ction will affect the compliance of the system, which will result in a different natural frequency than that predicted using Equation 51, which assumes one degreeoffreedom motion about the Xaxis only. Th e spring stiffness results presented in Table PAGE 118 118 Figure 52. Solid model created for modal analysis. A) View of the top and back. B) View of the bottom showing the comb fingers. Table 51. Modal analysis results for first 10 modes and their natural frequencies, and the participation factors and ra tios for each direction. XDirection YDirection ZDirection Mode Freq. (Hz) Participation Factor Ratio Participation Factor Ratio Participation Factor Ratio 1 84736.51 2.1183E10 0.000087 2.7197E06 0.758918 3.5955E06 1.000000 2 120372.52 9.5920E07 0.394546 8.8566E10 0.000247 4.3162E09 0.001200 3 162970.10 1.3215E07 0.054358 3.5837E06 1.000000 3.0841E06 0.857751 4 164493.10 8.3800E07 0.344691 5.5914E07 0.156023 4.8219E07 0.134109 5 391530.45 8.0827E10 0.000332 1.7035E06 0.475336 9.7970E08 0.252526 6 1208580.00 4.6105E08 0.018964 2.2081E08 0.006161 7.5056E08 0.020875 7 1310412.38 2.4312E06 1.000000 3.6331E09 0.001014 2.7242E09 0.000758 8 1610211.37 3.1366E07 0.129016 5.4173E08 0.015116 3.3755E09 0.000939 9 1696417.45 3.0772E07 0.126574 2.2398E08 0.006250 1.1384E08 0.003166 10 1853628.28 2.9854E07 0.127970 1.9532E07 0.054501 1.2589E08 0.003501 ROTXDirection ROTYDirection ROTZDirection Mode Freq. (Hz) Participation Factor Ratio Participation Factor Ratio Participation Factor Ratio 1 84736.51 6.4831E05 1.000000 8.7467E08 0.000395 4.3414E08 0.000197 2 120372.52 1.9988E08 0.000308 2.2171E04 1.000000 1.5238E05 0.069261 3 162970.10 9.4879E06 0.146350 9.8134E07 0.004426 3.4382E05 0.156274 4 164493.10 1.5072E06 0.232480 7.3266E06 0.033047 2.2001E04 1.000000 5 391530.45 6.3384E06 0.977690 8.7374E08 0.000394 2.8728E07 0.001306 6 1208580.00 1.3954E07 0.002152 9.0970E08 0.000410 1.8579E08 0.000084 7 1310412.38 2.7564E08 0.000425 3.4525E06 0.015572 3.1098E06 0.014135 8 1610211.37 5.5281E09 0.000085 2.5788E07 0.001163 8.3555E06 0.037978 9 1696417.45 4.2753E07 0.006595 1.0185E07 0.000459 3.2656E07 0.001484 10 1853628.28 4.5237E08 0.000698 2.3618E08 0.000107 1.8531E06 0.008423 33 previously show that the spring is very comp liant in the Zdirection 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 Xaxis 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 degreeoffreedom 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 51 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 capgun, 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 53 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 53. 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 54 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 52 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 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 0100200300400500 Frequency [kHz]Velocity [m /s]186.88 kHz 82.66 kHz 35.78 kHz B 0.0E+00 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 0100200300400500 Frequency [kHz]Velocity [m /s]43.44 kHz 82.19 kHz 140.78 kHz C 0.0E+00 5.0E05 1.0E04 1.5E04 2.0E04 2.5E04 3.0E04 3.5E04 0100200300400500 Frequency [kHz]Velocity [m /s]48.13 kHz 183.28 kHz Figure 54. 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 52. The first three natural frequencies determined from the LDV experiment. Results from the linear model, using Equation 51, 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. 51 188.12 Modal Analysis 84.74 120.37 5.2 OpenLoop Step Response The openloop response of the system is determined by the actuation voltage signal that is given to the micromirrors. For openloop 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 pullin 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 openloop response of the plant model is considered using different values of stiffness, km. Figure 55 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 53 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 56 shows the openloop 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 openloop pl ant will result in steadystate error in the response. In or der to correct for this in openl oop operation, the system must be carefully recalibrated for each device to en sure the proper response is achieved. 5.2.2 Effects of Pullin and Hyster esis on OpenLoop Response Electrostatic instability and hysteresis can also greatly affect the system response in openloop operation. Recall from the discussion in Section 3.3, that pullin 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 105 0 2 4 6 8 Time (sec)Theta (deg) Command Nominal +10% km 10% km Figure 55. Openloop 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 56. Openloop 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 holdingvoltage, causing hysteresis. The effects of pullin and hysteresis for the static response are investigated in Chapters 3 and 4, but there ar e dynamic effects that can affect pullin as well. It is known that pullin 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 pullin 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 pullin angle, PI, the system response sets theta equal to the final pullin position, F. After pullin has occurred, the system response remains pulledin until the voltage drops below the holding voltage, VH. The system then returns to the released position, R. This response is shown in Figure 57 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 pullin angle of 16.5, the response shows pullin 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 pullin point. In the case of large overshoot in the response, the device will pu llin and will not be released as the voltage command for a step input is constant. Figu re 58 shows the openloop 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 pullin as it is greater than the pullin angle. However in this case, overshoot in the response for a step PAGE 126 126 A B Figure 57. Openloop 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 58. Openloop responses to a step co mmand showing overshoot that result in pullin. command of 14 also results in pullin of the response as the overshoot causes the device to move beyond the pullin 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 pullin [111]. Dynamic pullin can result in cases where the velocity of the actuator is high as it approaches the pullin point. This can be caused in the case of applying instantaneous ac tuation voltages, and it PAGE 127 127 can cause the actuator to pullin at a lower voltage than th e static pullin 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 pullin can occur at an 8% lower voltage than the static pullin 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 59 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 59, and as a function of voltage in Figure 510. In this instance, only two of the micromirrors, 1 and 3, experienced pullin and hysteresis, while the other two, 2 and 4, did not. A B Figure 59. Results from dynamic study showing pullin and hysteresis. A) Actuation signal applied for dynamic study. B) Results from applying the actuation signal. PAGE 128 128 A B Figure 510. Results showing the hysteretic behavior of the micromirrors. A) Mirrors 1 and 3 show pullin and hysteresis. B) Mirrors 2 and 4 do not have pullin. 5.3 Hysteresis Case Study: ProgressiveLinkage As discussed in the literature re view Section 2.3, there are ways that researchers have used nonlinear flexure designs to mitigate electrostati c pullin and hysteresis. One such nonlinear flexure design is presented here, called a progres sivelinkage [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 openloop operation. The re sults presented are only th eoretical and have not been fully realized in fabrication. 5.3.1 ProgressiveLinkage 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 pullin 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 fourbar model as depicted in Figure 511. The geometric relationships between th e links are also shown in Figure 511. 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 511. 3 1 0 2r r r r (512) Since the fourbar mechanism is a onedegreeoffreedom 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. (513) In order to determine the angle 3, begin with the relationship 3 3 2 3 2 2 2cos 2d dr r r r r (514) This yields an expression for3 3 2 2 2 3 2 1 32 cosr r r r rd d (515) An expression for d is found from y y z z dr r r r1 0 1 0 1tan (516) The angle 3 is given as 3 3 d. (517) 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 511. Diagram of fourbar 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 fourbar mechanism. For the sake of this discussion it is assumed that this 2D representa tion of the fourbar 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 512, a beam of length L with a rectangular crosssection 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 (518) 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 (519) PAGE 131 131 when t > w For the case of t < w the expression is 34 416 3.361 16312iwttt K ww (520) Figure 512. Cantilever beam with crosssection w x t and length L The resulting static force and moment equa tions can be determined from the free body diagrams in Figure 513. 02 1 1F F Fbar (521) 03 2 2F F Fbar (522) 00 3 3F F Fbar (523) int112120joSeSMTTTrF (524) int223230joSSMTTrF (525) int303300joSSMTTrF (526) 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 515 shows plots of the Figure 513. 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 [pNm] r0 = 8 r0 = 6 r0 = 4 r0 = 10 r0 =12 Figure 514. Progressivelinkage 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 321. Tables 53 and 54 give the link length and joint dimensions used for this progressivelinkage design. The Youngs Modulus is assumed to be 164.3 GPa and the Poissons ratio is 0.22. The linear restoring force from Figure 321 is also included for comparison. The requirements for pullin 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 pullin 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 515. 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 53. Link length dimensions used for progressivelinkage design. Link Length ( m) r0 9.000 r1 8.625 r2 9.953 r3 4.375 Table 54. Joint dimensions used for progressivelinkage 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 516. Static V relationship for micromirror with a progressivelinkage. Recall from Section 3.4.5 that a bifurcation analysis may be used to examine the electrostatic pullin 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 (527) Applying this analysis to the device using the progressive linkage yields the bifurcation diagram shown in Figure 517. 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 518 shows how the bifurcation plot will change as the progressive s tiffness profile increases or decreases by a factor of 2. 5.3.2 OpenLoop Response Using a ProgressiveLinkage Because this device does not experience pullin, 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 pullin. The system openloop response to sinusoidal inputs is shown in Figure 519. Figure 520 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 pullin 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 57and 58. 5.3.3 Parametric Sensitivity of the ProgressiveLinkage 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 ogressivelinkage will add to the effects of this sensitivity. The following discussion will exam ine the effects of fabrication tolerances on the progressivelinkage 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 517. Bifurcation diagram for micromirror using a progressivelinkage to avoid pullin behavior. Figure 518. Bifurcation diagram for the micromi rrors using a progressivelinkage to avoid pullin behavior for different valu es of mechanical stiffness. PAGE 137 137 A B Figure 519. Openloop 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 520. Openloop response to a step i nput for device using a progressivelinkage. 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 53 and 54 gave numbers for the progres sivelinkage 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 54, 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 55 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 fourbar 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 55. Uncertainties in the joint dimensi ons for a proposed progressivelinkage 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 0.00231 2.501.00 MMPoly1 1.02 0.00232 2.253.00 MMPoly4 2.29 0.00630.07 0.05 3 2.253.50 MMPoly3 2.36 0.00990.24 0.05 The sensitivity of the progressivelinkage desi gn is examined when only one variable is altered at a time. Figure 521 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 torquetheta profile are shown in Figure 522. It is striking to see the very la rge effects of these very small parametric perturbations, and from a qualitat ive pointofview, 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 ProgressiveLinkage 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 523, 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 524 shows a micrograph image of the progressivelinkage and the micromirror device. Because of the pl anar fabrication requirements of surface micromachining, the diagonal top member of the fourbar device, 2r, can be acquired via a kinematically equivalent Lshaped beam, shown in Figure 524. 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 timeconstra ints with the available fabrication run, indepth analysis of the device performance was not conducted before the final design was submitted for PAGE 140 140 Figure 521. Results of parametric analysis for individual errors in joint fabrication of the progressivelinkage. Figure 522. Fifty Monte Carlo simulation results for varying the joint fabrication parameters for the progressivelinkage 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 timetab les of foundry services and available project funding. It also gives a good example of the co nsequences of incomplete a priori analysis. Figure 523. Schematic drawing of the prototype progressive linkage spri ng. A) Progressive linkage spring. B) Spring attached to th e micromirror. C) Drawing of Lshaped equivalent beam. Figure 524. 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 525 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 progressivelinkage 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 pullin 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 progressivelinkage 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 progressivelinkage is modeled in ANSYS using beam189 elements [99]. As the structure is displaced about the Xaxis, 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 fourbar model, but is instead deflecting in the positive Zdirecti on, outofplane. This Zdirection deflection prevents the joints, which are fabricated as thin beams, from rotating as they are intended. Figure 526 shows the results of the FEA analysis of the pr ototype design for both linear deflection analysis, and nonlinear, largedeflectio 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 progressivelinkage 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 pullin 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 fourbar linkage design principles. This remains as future work. Figure 525. Experimental data collected fo r the prototype of the micromirror with the progressivelinkage Figure 526. Results from FE A of the prototype progressiv elinkage design for linear and nonlinear deflection analysis shows the prototype progressi velinkage 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 degreeoffreedom massspringdamper 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 degreeoffreedom. 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 onedegreeoffreedom model assumption. However, it is clear that motion in other di rections, namely the Zdirection 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 Zaxis 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 Xaxis 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 degreeoffreedom 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 pullin, as demonstrat ed previously, but also the point at which the mirrors will release from pullin 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 pullin and hysteresis also have the ability to negatively affect the dynamic response for actuation signals that occur below the pullin 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 pullin, 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 steadystate 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 steadystate 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, pullin 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 singleinput/multipleoutput (SIMO) system and will discuss the feedback signals available by considering two different kinds of optical sensors: position detecting sensors (PSD) and chargecoupled devices (CCD). The performance of these sensors is considered as well as the impact th ey will have on implementation of closedloop control system on the array of micromirrors. Figure 61. Basic block diagram with unity feedback. PAGE 147 147 6.1 PID Control ProportionalIntegralDerivative (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 secondorder dynamics. 6.1.1 PID Control Theory The general form of the transfer function for a PID controller is 11 ()1CPIDPd iGsKKKsKTs sTs (61) where P i IK T K (62) and D d P K T K (63) The block diagram of this system is shown in Figure 62. The closedloop transfer function for this block diagram, with the plan t modeled as a linear second order system is ()() () ()1()()()CP CPGsGs Cs R sGsGsHs (64) Assuming unity feedback, that is H(s) = 1, and substituting Equation 61 into 64 gives an expression for the closedloop transfer function. 2 322() () ()(2)()DPI nDPnIsKsKK Cs R ssKsKsK (65) PAGE 148 148 Figure 62. 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 steadystate error in the system. The derivative term, as seen in the denominator of Equation 65, 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 closedloop 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 steadystate 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 steadystate 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 63. 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 closedloop system has no overshoot, wh ich is important in electrostatic systems that experience pullin. 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 64 shows the effects of model uncertainty for the nonlinear plant response, including mode l variations of % variation in the spring stiffness, km. Openloop analysis in Chapter 5 presen ted the openloop 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 65 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 llin point is known to vary for different PAGE 150 150 A B Figure 63. Step responses for PI D controller. A) Linear plant model. B) Nonlinear plant model. Figure 64. Closedloop 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 openloop hysteretic behavi or of the micro devi ces when subject to electrostatic pullin. Here, this same scenario is considered for the case of the closedloop system with a PID controller in place. The resu lt is that if the mirror is commanded to an unstable position, and subsequently pullsin, th e controller will see this position error and PAGE 151 151 Figure 65. Closedloop 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 pullin. 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 66, resulting in a fast switching behavior until the commanded positi on of the mirror returns to the stable range of motion. In Figure 66, 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 pullin 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 pullin 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 ivelinkage that can be utilized to eliminate the effects of pullin 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 pullin 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 closedloop contro l is still necessary. Figure 67 PAGE 153 153 shows the closedloop PID step responses for the micromirrors with the progressive linkage, and they are in fact able to achieve stable rotatio n above the pullin 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 66. Closedloop PID response to a commanded position in the unstable region. A B Figure 67. Closedloop 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 statespace 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 fullstate 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 68, assumes fullstate feedback. Cases without fullstate feedback will require the use of an estimator and will be discussed in section 6.2.1. Figure 68. General block diagra m for LQR controller problem The plant is modeled as a continuous time, lin ear system described by a set of statespace equations x AxBu y CxDu (66) PAGE 155 155 It is desired to find a controller that minimizes a cost function, J 0min()KJxQxuRudt (67) 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 positivedefinite 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 (68) Substituting Equation 68 into Equation 66 gives () x AxBKxABKx (69) Thus, Equation 67 becomes 00()() JxQxxKRKxdtxQKRKxdt (610) The following relationship sets a condition that restricts K to be finite. ()()d x QKRKxxPx dt (611) where P is a positivedefinite Hermitian or real symm etric matrix. Evaluating the right hand side of Equation 611 and substituting in Equation 69 yields x PxxPxxABKPPABKx (612) Equation 612 must hold true for any x, therefore ()()()ABKPPABKQKRK (613) If ABK is a stable matrix, there exists a positivedefinite matrix P that satisfies Equation 613. In order to determine this matrix P, evaluate the cost function J. PAGE 156 156 0 0() JxQKRKxdtxPx (614) Since ABK is stable, all of the eigenvalues are assu med to have negative real components and 0 x Equation 614 becomes (0)(0)JxPx (615) Because R is defined as a positivedefinite Hermitian matrix, it can be written in terms of a nonsingular matrix, T. R TT (616) Substitute Equation 616 into 613 to get 111()() [()][()]0 AKBPPABKQKTTK APPATKTBPTKTBPPBRBPQ (617) The minimization of the cost function J with respect to K requires the minimization of 11[()][()] x TKTBPTKTBPx (618) with respect to K. This expression is nonnegative and the minimum occurs when it is zero, or when 1()TKTBP (619) Hence, the optimal matrix K is found by 111()KTTBPRBP (620) The matrix P in Equation 620 must satisfy the reducedmatrix Riccati equation 10APPAPBRBPQ (621) Equation 621 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 620 to find the optimal gain matrix K that is used in the control law of Equation 68. 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 (622) 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 (623) For this problem, the cost function J can be defined in term s of the error signal. 0()eeJeQeuRudt (624) Equation 624 can be rewritten as follows 0(2) JxQxuRuxNudt (625) with ()()eQHAFHQHAFH (626) ee R BHQHBR (627) ()eNHAFHQHB (628) ,eeQIRI (629) 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 (630) The solution of Equation 630 results in the matrix P such that the controller is described as 11[][] f bffKRBPRNKK (631) where Kfb and Kff are the feedback and feedfoward controll er gains, respectively. The control law is thus written as ()fbffuKxKe (632) The use of this LQR control law for tracking a reference command with zero steadystate 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 typeone 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 feedforward gain, Kff, and a feedback gain, Kfb, as in Equation 632. In this case, the error signal is the Figure 69. 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 fullstate feedback is available for this sy stem for the feedback loop. In cases where fullstate feedback is not ava ilable, stateestimation 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 fullstate 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 610. 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 610. Block diagram of LQ R controller using a stateestim 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 (633) Let x represent the vector of the estimated stat es. The control law of Equation 632 becomes ()fbffuKxKe (634) PAGE 160 160 The mathematical model for the estimator is similar to the plant model of Equation 66 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 (635) where the A, B, C, D are the matrices of the plant model from Equation 66, 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 closedloop poles of the estimat or at desired closedloop pole locations, Ackermanns formula is 1 10001()nLBABABA (636) for an arbitrary integer n. The term A is the characteristic polynomial of matrix A. 12 121()nnn nnAAAAAI (637) The coefficients have a relationship with th e roots of the polynomial, which are also the closedloop pole locations. 12 12121()()()nnn nnnsssssss (638) 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 stateestimator 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 (639) 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 openloop poles of the linear plant model of the micromirrors are p = 307489.41 977754.00 i Therefore, the desired closedloop pole locations for the stateestimator 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 openloop poles on the real axis of the Splane. 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 611. The closedloop system response for the LQR A B Figure 611. Step responses for LQR controller. A) Linear plan t model. B) Nonlinear plant model. PAGE 162 162 Figure 612. Closedloop 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 closedloop system when the micromirror is commanded to an unsta ble position, and thus experiences electrostatic pullin 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 ateestimation 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 progressivelinkage. Based on the similarity between the responses from the PID and LQR cont rollers, it safe to assume that the closedloop LQR performance for the system with a progressivelinkage will be very similar to that of Figure 67. Figure 613 shows the closedloop LQR step responses for the micromirrors with the progressive linkage, and they are able to achieve st able rotation above the pullin 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 613. Closedloop 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 singleinput/singl eoutput (SISO) system. In reality, these micromirrors are part of an array that is single input/multipleoutput (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 argecoupled 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 dystate 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 614 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 615 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 openloop 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 614. Schematic of modeling an ar ray of mirrors as a SIMO system. PAGE 166 166 Figure 615. 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 chargec 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 616, the case of light fr om one array of mirrors reflecting onto a 1D 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 616 also shows the 1D 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 1D array is only able to measure the CG in terms of one direction (ya xis shown), the offaxis 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 1D PSD entirely, also affecting the location of the CG. For a single micromirror array being meas ured by a 2D 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 617, e rrors in spacing can shift the CG in the ydirec tion, but linearity Figure 616. Illustration of the measured center of gravity (CG) on a 1D PSD when there are errors in the spacing and lineari ty of the micromirrors. Figure 617. Illustration of the measured center of gravity (CG) on a 2D 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 xdirection. 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 metaloxidesemiconduc 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 ydirection displacement measurements for each spot and compare that with the desired positions. This is illustrated in Figure 618 for light from two arrays in which the dark spot indicates the actual position of the Figure 618. 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 619 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 620. In general, the angle, between two vectors, a and bas shown in Figure 620, can be calculated using the dot product relationship. cosabab (640) 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 (641) 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 621 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 (642) cos0 0 2 B Ar (643) 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 619. Schematic of the beam steeri ng experiment with only one micromirror. PAGE 172 172 A B Figure 620. 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 622, 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 (644) 00xxx B RBtx (645) 00yyy B RBty (646) 00zzz B RBtz (647) 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 (648) This value for t calculated from Equation 636 may be s ubstituted back into Equations 645, 646 and 647 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 621. Calculating the reflected ray of light. PAGE 174 174 Figure 622. The intersection of the line from B0 to R and the plane C occurs at point P. 6.4.1 PSD Response A 1D 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 (649) 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 623 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 2D 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 623. 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 2D view of the reflected light on the sensor plane wi th the CG marked in the red star. Figure 624. 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 625 shows the resulting openlo 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 624. Calibration of the PSD fo r ideal case of five micromirrors. A B Figure 625. Openloop 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 626 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 pullin 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 627. To illustrate the eff ects of using a PSD as the sensor for this A B Figure 626. Openloop response for system w ith one broken mirror and 4 ideal mirrors, measured by a PSD. A) Step response. B) Position error. Figure 627. Incorporating feedb ack control into array model. micromirror array system, the closed loop resp onse is determined. Figure 628 shows the response for a PID controller, and likewise, Figure 629 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 628. Controlled PID step response usi ng PSD sensor. A) Step response. B) Position error. A B Figure 629. 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 closedloop control in the case of a device in which one mirror in the array is not functioning. Shown in Figure 630 is the closedloop 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 630. Closedloop 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 631. 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 632 in which the PID cl osedloop 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 631. 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 632. Closedloop 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 closedloop 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 openl 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 voltagedisplacement relationships and the electrostatic pullin 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 ressivelinkage 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 Xdirection (ROTX), which corresponds to the onedegreeof freedom model assumption. However, it is clear that motion in other direct ions, namely the Zdirection (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 degreeoffreedom 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 openloop dynamics, while exhibiting some overshoot behavior in the transient respons e, operates on a very fast timescale, 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 closedloop control techniques will enable both cost reduction as the devices will no longer require extensive calibration for openloop performance, as well as improved performance and reliability. The impact of this work is not limited to the appli cation of micromirror or microoptics 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 onedegreeoffreedom 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 pullin. In order to design a robust microsystem th at can be deployed in a wide variety of scenarios, the device should have onchip 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 onchip 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 closedloop 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 A1 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 XY plane, and extruding the thickness in the positive Zaxis. Figure A1. 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. Fivehundred 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 B1. 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 B2. 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 B3. 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 B4. 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 B5. 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 B6. 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 B7. 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 B8. 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 B9. 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 B10. 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 B11. 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 B12. 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 B13. 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 1011 0 2 4 6 8 10 12 14 Mass [kg]Frequency of Occurance Figure B14. Histogram of the mass values calc ulated from the parametric variation data. Figure B15. 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 B16. Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the linewidth erro r of MMPoly2. Figure B17. Histogram of values from the Mont e Carlo simulations for the layer thickness of MMPoly1. PAGE 198 198 Figure B18. Static displacement curves from the Monte Carlo simulations that indicate the effect of large variations in the thickness of MMPoly1. Figure B19. Histogram of values from the Mont e Carlo simulations for the layer thickness of Sacox4. PAGE 199 199 Figure B20. 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 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]44.06 kHz 81.41 kHz 186.88 kHz Figure C1. Magnitude of FFT results for device 1, trial 1. 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 3.50E04 4.00E04 4.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]47.03 kHz 81.41 kHz 187.81 kHz Figure C2. Magnitude of FFT results for device 1, trial 2. PAGE 201 201 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 3.50E04 4.00E04 4.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]42.03 kHz 81.71 kHz 187.03 kHz Figure C3. Magnitude of FFT results for device 1, trial 3. 0.00E+00 1.00E04 2.00E04 3.00E04 4.00E04 5.00E04 6.00E04 7.00E04 8.00E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]186.88 kHz 82.66 kHz 35.78 kHz Figure C4. Magnitude of FFT results for device 1, trial 4. PAGE 202 202 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]187.19 kHz 43.75 kHz 51.25 kHz 81.56 kHz 123.75 kHz Figure C5. Magnitude of FFT results for device 1, trial 5. 0.00E+00 1.00E04 2.00E04 3.00E04 4.00E04 5.00E04 6.00E04 7.00E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]43.44 kHz 82.19 kHz 140.78 kHz Figure C6. Magnitude of FFT results for device 2, trial 1. PAGE 203 203 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 3.50E04 4.00E04 4.50E04 5.00E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]43.13 kHz 139.53 kHz 85.63 kHz Figure C7. Magnitude of FFT results for device 2, trial 2. 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 050100150200250300350400450500Frequency [kHz]Velocity [m/s]43.44 kHz 85.31 kHz 118.59 kHz 25.63 kHz 136.56 kHz Figure C8. Magnitude of FFT results for device 2, trial 3. PAGE 204 204 0.00E+00 2.00E05 4.00E05 6.00E05 8.00E05 1.00E04 1.20E04 1.40E04 1.60E04 1.80E04 2.00E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]27.34 kHz 43.59 kHz 92.03 kHz 137.03 kHz Figure C9. Magnitude of FFT results for device 2, trial 4. 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 3.50E04 4.00E04 4.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]25.63 kHz 41.88 kHz 136.10 kHz 85.31 kHz Figure C10. Magnitude of FFT results for device 2, trial 5. PAGE 205 205 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]42.03 kHz 105.31 kHz 80.94 kHz 50.16 kHz 182.34 kHz Figure C11. Magnitude of FFT results for device 3, trial 1. 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]31.25 kHz 116.56 kHz 46.88 kHz 90.31 kHz 183.91 kHz Figure C12. Magnitude of FFT results for device 3, trial 2. PAGE 206 206 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]112.97 kHz 81.41 kHz 42.34 kHz 132.66 kHz 180.63 kHz Figure C13. Magnitude of FFT results for device 3, trial 3. 0.00E+00 5.00E05 1.00E04 1.50E04 2.00E04 2.50E04 3.00E04 3.50E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]48.13 kHz 183.28 kHz Figure C14. Magnitude of FFT results for device 3, trial 4. PAGE 207 207 0.00E+00 2.00E05 4.00E05 6.00E05 8.00E05 1.00E04 1.20E04 1.40E04 1.60E04 1.80E04 2.00E04 050100150200250300350400450500 Frequency [kHz]Velocity [m/s]83.28 kHz 57.34 kHz 43.44 kHz 106.56 kHz 183.28 kHz Figure C15. 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. 688690, 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. 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Electrical Computer Syst., vol. 27, no. 4, pp. 195200, 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 closedloop 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. 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