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Kinematic Analysis and Design of a Compliant Microplatform

Permanent Link: http://ufdc.ufl.edu/UFE0021208/00001

Material Information

Title: Kinematic Analysis and Design of a Compliant Microplatform
Physical Description: 1 online resource (79 p.)
Language: english
Creator: Correa, Julio C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: compliant, mechanism, mems, platform, spatial
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our research addresses the kinematics and the design of a three-dimensional device at the micro level. The device is formed by three actuators that transmit the motion to a central platform. Techniques used to manufacture microelectromechanical systems (MEMS) have severe limitations and they cannot permit the construction of complex joints. To solve this problem compliant joints are used in this device to connect the platform and actuators. At the MEMS level they offer significant advantages compared to their counterparts at the macro level. The device is able to provide complex motions that require an elaborate mathematical model for their description. Two kinematic issues are presented: the forward and reverse analyses. The forward formulation allows for the determination of the location of the moving platform given the position of the actuators, while the reverse analysis finds the location of the actuators for a desired position of the platform. The models are based on a Newtonian approach and are subjected to several assumptions to simplify the formulation. The Newtonian approach is preferred because it relates in a natural way the forces and the geometry of the device. Examples and verifications of the models are provided. Actuators consist of two beams with different thermal expansion coefficients and a resistor between them. This configuration allows for the bending of the beam when temperature increases. Springs are formed of a compliant and photodefinable material. Issues associated with the selection of dimensions and materials as well as the manufacturing process that permits to build the device are presented. The combination of compliant beams and compliant joints exhibits important advantages at the MEMS level and also poses challenging kinematic problems. The principles presented here will be useful for the generation of more complex devices.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Julio C Correa.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Crane, Carl D.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021208:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021208/00001

Material Information

Title: Kinematic Analysis and Design of a Compliant Microplatform
Physical Description: 1 online resource (79 p.)
Language: english
Creator: Correa, Julio C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: compliant, mechanism, mems, platform, spatial
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our research addresses the kinematics and the design of a three-dimensional device at the micro level. The device is formed by three actuators that transmit the motion to a central platform. Techniques used to manufacture microelectromechanical systems (MEMS) have severe limitations and they cannot permit the construction of complex joints. To solve this problem compliant joints are used in this device to connect the platform and actuators. At the MEMS level they offer significant advantages compared to their counterparts at the macro level. The device is able to provide complex motions that require an elaborate mathematical model for their description. Two kinematic issues are presented: the forward and reverse analyses. The forward formulation allows for the determination of the location of the moving platform given the position of the actuators, while the reverse analysis finds the location of the actuators for a desired position of the platform. The models are based on a Newtonian approach and are subjected to several assumptions to simplify the formulation. The Newtonian approach is preferred because it relates in a natural way the forces and the geometry of the device. Examples and verifications of the models are provided. Actuators consist of two beams with different thermal expansion coefficients and a resistor between them. This configuration allows for the bending of the beam when temperature increases. Springs are formed of a compliant and photodefinable material. Issues associated with the selection of dimensions and materials as well as the manufacturing process that permits to build the device are presented. The combination of compliant beams and compliant joints exhibits important advantages at the MEMS level and also poses challenging kinematic problems. The principles presented here will be useful for the generation of more complex devices.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Julio C Correa.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Crane, Carl D.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021208:00001


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KINEMATIC ANALYSIS AND DESIGN OF A
COMPLIANT MICROPLATFORM





















By

JUL~IO CESAR CORREA RODRIGUEZ


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 Julio Cesar Correa Rodriguez


































To my mother for her infinite generosity









ACKNOWLEDGMENTS

I thank my supervisory committee members: Dr. Carl Crane, Dr. Gloria Wiens, Dr Hiukai

Xie and Dr. John Schueller for their valuable suggestions on my thesis. I extend a special thanks

to professor Carl Crane, my academic advisor, for his continuous support and encouragement

throughout my graduate study here. I thank professor Hiukai Xie for his guidance and his help

with microsystem technology.












TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ................ ...............7............ ....


LIST OF FIGURES .............. ...............8.....


AB S TRAC T ............._. .......... ..............._ 1 1..


CHAPTER


1 INTRODUCTION ................. ...............13.......... ......


Out-of-Plane Motion Devices ................. ...............13........... ....
Tensegrity Structures ................. ...............15.................
Bimorph Actuators............... ...............1
Elastic Joints ................. ...............17.......... .....
The Device ................. ...............18.................


2 F ORW ARD AN ALYSIS ............... ............... 1


Forces Acting on the Platform ................. ...............20........... ...
Mathematical Model ................. ...............22.......... ......
Numerical Example .............. ...............26....


3 REVERSE ANALYSIS ................. ...............31........... ....


Replacement of Compliant Beams .............. ...............3.....1
Reverse Analysis Case 1............... ...............32...
Numerical Example .............. ...............37....
Reverse Analysis Case 2............... ...............39...
Numerical Example .............. ...............45....


4 DEVICE DE SIGN ................. ...............49................


Actuator Design ................. ...............49.................
M material s ................. ...............49.......... .....
Width of the Beam ................. .......... ......... ........ ......... ...............50
Deflection and Length of the Beams ................ ....................... ................50
Out-of-Plane Elevation ................ ...............51.................

Springs Design ................. ...............55.................
Springs Elongation .................. .. ...............55.
Maximum Force Acting on the Spring ................. ...............56...............
Spring Geometry and Material .............. ...............57....
Spring Stress ................. ...............59.................












Spring Deflection............... ...............6
Spring Dimensions .............. ...............61....

5 MANUFAC TURING PROCES SS ................. ...............64.......... .....


6 CONCLUSIONS .............. ...............70....


APPENDIX


A REVERSE ANALYSIS EQUATIONS ................. ...............72................


B PREVIOUS WORK............... ...............74..


LIST OF REFERENCES ................. ...............76........... ....


BIOGRAPHICAL SKETCH .............. ...............79....










LIST OF TABLES

Table page

3-1 Solution for the reverse analysis, case 1. ............. ...............39.....

4-1 Mechanical and thermal properties for aluminum and silicone dioxide. ........................50

4-2 Properties of polymide HD-8000. .........._.... ...............58..__... ....

4-3 Main dimensions of the device. ........._._... ...............63.._.__. ....











LIST OF FIGURES


Figure page

1-1 Prismatic tensegrity structure with 6 struts. ......___ ........_._ ...._._ ..........1

1-2 Sequence of motions for the rising of the structure. ................ ............... ........ ...16

1-3 Configuration of a bimetallic actuator. .............. .....................17

1-4 Bending of a bimetallic actuator. ........................... ........17

1-5 Scheme of the device ................ .......... .................. .......................18


2-1 Device in a general position ................. ...............19...............

2-2 Arbitrary forces acting on the platform. ............. ...............20.....

2-3 Moment of a force ................. ...............22........... ...


2-4 Nomenclature for the forward analysis............... ...............23

2-5 Coordinates of the free ends of the actuators ................. ...............27........... .

2-6 Initial position of the system ................. ...............27...............

2-7 Device in the evaluated equilibrium position. ............. ...............30.....

3-1 Normal vector to the moving platform. ............. ...............31.....

3-2 Path of the free end. ................ ...............32...............


3-3 Parameters for the reverse analysis, case 1. ............. ...............33.....

3.4 Location of the local reference systems for the reverse analysis ................. ................. 34

3-5 Distributions of points E. ................ ............. .................. .................38

3-6 Solution for the reverse analysis case 1. ............. ...............39.....

3-7 Prescribed vertical component of point P1 ......_......._.__........._ ..........4

3-8 Nomenclature for the reverse analysis, case 2. .............. ...............40....

3-9 Device in its initial position. ............. ...............46.....

3-10 Device for the example of reverse analysis, case 2. ........._._ .... .__ ...............48

4-1 Maximum deflection of a cantilever beam. ............. ...............5......1











4-2 Deflection of the free end for several conditions ................. ...............52........... .


4-3 Maximum elevation of the free end of the beam. ................ ...............53.............


4-4 Bending after release from sub state ................. ...............53........... ..


4-5 Positions for minimum and maximum deformation of the springs. ................ ...............55


4-6 Maximum deformation of the springs............... ...............56


4-7 Maximum force in the spring............... ...............57.


4-8 Possible geometries for the spring. ............. ...............58.....


4-9 Geometry of a segment of the spring ................. ...............59..............


4-10 Segment of the spring. ................ ...............60.......... ....


4-11 Parameters for the stress analysis of a spring. ............. ...............60.....


4-12 Deflection in the spring............... ...............62.


4-13 Stress in the spring. .............. ...............62....

5-1 Silicone substrate. ............. ...............64.....


5-2 First layer of silicone dioxide. ............. ...............64.....


5-3 Layer of chrome. .............. ...............65....

5-4 Resistor. ............. ...............65.....


5-5 Second layer of silicone dioxide. .............. ...............66....


5-6 Aluminum layer. ............. ...............66.....


5-7 Etching of areas in the aluminum corresponding to the actuators and platform. ........._....66


5-8 Etching of silicone dioxide .............. ...............67....


5-9 Polymide layer. ............. ...............67.....


5-10 Polymide springs ................. ...............68........... ....

5-11 Backside etch. ............. ...............68.....


5-12 Section view of the device. ................ ...............68...............


5-13 Deep reactive ion etching............... ...............69












5-14 Isotropic etch ................. ...............69........... ....


B-1 Mask used in the previous work. ............. ...............74.....


B-2 Manufactured feature. .............. ...............75....









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

KINEMATIC ANALYSIS AND DESIGN OF A
COMPLIANT MICROPLATFORM

By

Julio Cesar Correa Rodriguez

August 2007

Chair: Carl Crane
Major: Mechanical Engineering

Our research addresses the kinematics and the design of a three-dimensional device at the

micro level. The device is formed by three actuators that transmit the motion to a central

platform. Techniques used to manufacture microelectromechanical systems (MEMS) have severe

limitations and they cannot permit the construction of complex j points. To solve this problem

compliant j points are used in this device to connect the platform and actuators. At the MEMS

level they offer significant advantages compared to their counterparts at the macro level.

The device is able to provide complex motions that require an elaborate mathematical

model for their description. Two kinematic issues are presented: the forward and reverse

analyses. The forward formulation allows for the determination of the location of the moving

platform given the position of the actuators, while the reverse analysis finds the location of the

actuators for a desired position of the platform.

The models are based on a Newtonian approach and are subj ected to several assumptions

to simplify the formulation. The Newtonian approach is preferred because it relates in a natural

way the forces and the geometry of the device. Examples and verifieations of the models are

provided.









Actuators consist of two beams with different thermal expansion coefficients and a resistor

between them. This configuration allows for the bending of the beam when temperature

increases. Springs are formed of a compliant and photodefinable material. Issues associated with

the selection of dimensions and materials as well as the manufacturing process that permits to

build the device are presented.

The combination of compliant beams and compliant j points exhibits important advantages at

the MEMS level and also poses challenging kinematic problems. The principles presented here

will be useful for the generation of more complex devices.









CHAPTER 1
INTTRODUCTION

Mechanisms formed by rigid links and rigid j points have been the obj ect of extensive

studies for the theory of mechanisms. These kind of devices are well suited to work at the

macroworld, however when the dimensions of the systems are on the order of microns,

limitations due to manufacturing processes impose severe limitations, and the generation of

motion requires alternative approaches.

Devices for microelectromechanical systems are basically planar devices. This is due the

current manufacturing techniques that are derived from the IC industry. Creating 3D structures

at the micro level is a difficult task. Most of the motion of MEMS devices is constrained to the

plane. Some works have been made to create spatial motion.

Out-of-Plane Motion Devices

Out-of -plane actuators can convert input signals into displacements normal to the surface

of a substrate. Three-dimensional microdevices are useful for different tasks as for example,

object positioning, micromanipulators, optical scanners, tomographic imaging, optical switches,

microrelays, adjustable lenses and bio-MEMS applications.

To obtain out-of-plane motion is a challenging problem and several approaches based have

been proposed. Usually out-of-plane actuators are multilayer structures, although single layer

devices have been reported by Chen [1]. Generally speaking current solutions are based on

vertical comb drives, on the deformation of the materials or on the assembly of basic linkages.

The following references report out-of-plane motion devices.

Vertical comb drives are formed by an array of capacitors. When a voltage is applied, the

movable components of the capacitors rise out of the plane. They are combined with torsion

mirrors to tilt micromirrors as it is described by Milanovic [2] and Lee [3]. The vertical motion









of comb drives is limited and they require a careful design and control to avoid jumps associated

with the pull-in voltage, see Bronson and Wiens [4].

Combination of TiNi and Si cantilever or other substrates such as SU-8 or polymide have

been used to create out-of-plane motion devices. Fu [5] reports several devices based on a TiNi

film which is actuated when a current is applied to the electrode.

A micromirror having a large vertical displacement has been presented by Jain and Xie [6].

The mirror plate is attached to a rigid silicon frame by a set of aluminum/silicon dioxide bimorph

beams. A polysilicon resistor is embedded within the silicon dioxide layer to form the heater for

thermal bimorph actuation.

Ebefors [7] and Suh [8] implemented conveyors systems for out-of-plane motion able to

perform complex manipulations. They are based on arrays of structures that can deflect out of the

plane due to different coefficients of thermal expansion. Obj ects that are placed on the array can

be moved according to the deflection of each actuator.

Schwizer [9] reports a monolithic silicon integrated optical micro-scanner. The device

consists of a mirror located on the tip of a thermal bimorph actuator beam and it is able to

achieve large scan angles.

The other alternative to achieve out-of-plane motion is the assembly of planar linkages. A

platform described by Jensen [10] has three degrees of freedom and the top platform remains

horizontal throughout the device's motion. A proposal for a three degree of freedom parallel

robot is presented by Bamberger [1l]. The device uses only rigid revolute joints. Both revolute

actuators are located at the base during the manufacturing process, making the device suitable for

MEMS fabrication.









Out-of-plane motion has also been realized through the use of elastic elements. A device

actuated by comb drives is presented by Tung [12]. Drives are connected to a platform made of

polydimethilsiloxane (PDMS) via thin flexural microj points.

Previous works suggest that compliant links and elastic j points may be a feasible alternative

to create mechanical devices at the microlevel. There are many configurations based on these

simple elements, one example of which are tensegrity structures and they illustrate another way

to obtain spatial motion.

Tensegrity Structures

The word tensegrity is a contraction of tension and integrity and refers to structures formed

by rigid and elastic elements that maintain their shape due only to their configuration. Rigid

elements do not touch one another and they do not require external forces to maintain their

unloaded position (Figure 1-1).














Figure 1-1. Prismatic tensegrity structure with 6 struts.

Tensegrity structures were developed by architects in the middle of the last century.

Research began with Fuller [13]. First contributions were made by Kenner [14] and Calladine

[15]. Static and dynamic analysis studies have been made Murkami [16] and Correa [17].

Proposed applications include antennas, Knight [18], flight simulators, Sultan [19], deployable









structures, Tibert [20], and force and torque sensors, Sultan [21]. Tensegrity has been also

proposed by Ingber [22], to explain the deformability of cells.

Due to the presence of elastic ties, tensegrity structures are foldable. If in the folded

position external constraints are released, they can recover suddenly their original shape by

themselves. The deployment can be also achieved in a controlled way using telescopic struts,

see Furuya [23] or controlling the elastic ties, see Sultan [24].

Figure 1-2 shows the same principle but in this case links are not rigid but rather are

compliant. When the radius of curvature is changed, the whole structure is able to move in 3D

following a complex path.


Figure 1-2. Sequence of motions for the rising of the structure.

Although the device seems feasible, the manufacture of the required j points is very complex

at the MEMS level, however it is possible to modify its constitutive elements to reach the same

result in a simpler way. Before presenting the idea to be developed in this research it is important

to consider in more detail the requirements for the actuators and the j points.

Bimorph Actuators

The bi-layer electrothermal actuator combines two materials with different coefficients of

thermal expansion (a). The layers are joined along a common interface and the entire device is

heated. Since one material tries to expand more than the other but is restrained by the j oint with

the second material, the entire structure bends, see Pelesko [25]. It is possible to extend and










contract the beam by controlling the temperature of the beam via the use of a resistor embedded

in the beam. The electrothermal actuators have the advantages of low operation voltage, a

simple fabrication process, and are CMOS-compatible. Therefore, control circuits can be

integrated with the device on the same chip. A bimetallic actuator is illustrated in Figure 1-3.










Figure 1-3. Configuration of a bimetallic actuator.

If a, > a, the structure bends with an increase of temperature as shown in Figure 1-4A. If

a, < a, the structure bends like in Figure 1-4B. It is usual that the bending of the beam take

place out of the plane, but there is not any restriction to bend the beam in the plane.










Figure 1-4. Bending of a bimetallic actuator. A) a, > a, B) a, < a .

Elastic Joints

The functionality of the device is intimately related to the elastic elements located at the

ends of the beams.

The development of torsion springs at the microlevel has been achieved and presented by

Hah [26]. However the development of linear springs is less frequent. Regular-coiled carbon

fibers have been obtained by Yang [27], using chemical procedures,. Also, the design of a

vertical linear conical microspring attached to the substrate is reported by Hata [28]. None of










these ideas are appropriate for a 3D device and for the purpose of this work it is necessary to find

an alternative.

The decision about the material and the shape and process must include the following

considerations: material with low Young modulus, applicable through spinning and be

photodefinable, resistant to heat to avoid future complications due to the actuation of the beams,

and compatible with the other processes involved in the tensegrity based MEMS device.

The Device

Figure 1-5 shows a scheme of the device that was addressed in our research. It can be

considered as a simplification of the tensegrity system presented in Figure 1-2. The system

maintained its shape due to the upward deflections of the beams. It was formed by three sets of

bimorph actuators which transmitted their motion to the central platform through compliant

j points. The moving platform could be described by an equilateral triangle. The fixed ends of the

actuators were distributed along the vertexes of an equilateral triangle.














Figure 1-5. Scheme of the device.

The position of the device is influenced by the stiffness and free lengths of the ties, the

location and nature of the j points, and the length and the current curvature of the beams. The

presence of elastic elements increases the complexity of the mathematical model that describes

the relations between internal forces and the positions of the beams.









CHAPTER 2
FORWARD ANALYSIS

Figure 2-1 depicts the device in a general position. In the forward analysis the location of

points Q. with respect to a global reference system are given and the obj ective is to evaluate the

coordinates of points withh respect to the global system. Despite the simplicity of the

mechanism, the answer to this question is not trivial due to the presence of the compliant

elements. To simplify the problem the following assumptions are made:

* The moving platform is massless.

* The stiffness of the compliant elements are linear and they are the same for all the springs.

* Deflections of actuators due to the spring forces are minimal and they do not affect the
motion of the platform.















Figure 2-1. Device in a general position.

The solution can be performed using a Newtonian approach or energy approach.

Newtonian is preferred here because it gives a better understanding of the geometry of the

sy stem.










Forces Acting on the Platform

To begin it is important to recall two basic concepts from vector algebra. The n vectors

u,,u,...u,, are said to be linearly dependent if there exist n real numbers Az,, A ,...il, not all zero

such that, see Brand [29]

ig, +Al,u, +...+Al,u4, = Q. (2-1)

The other important concept is this: a necessary and sufficient condition that three vectors

be linearly dependent is that they be coplanar. Figure 2-2 shows the forces acting on the platform

in a general position. Since the platform is massless, the equilibrium of forces yields

F, + F, + F, = 0 (2-2)




3 > ~






P,





Figure 2-2. Arbitrary forces acting on the platform.

Equation 2-2 can be expressed more conveniently in terms of the magnitude and direction

of each force as



where
s : unit vector from (q to Q,
J;: is the magnitude of the force in each spring.









Since the springs are linear, each force magnitude in Equation 2-4 can be expressed as a function

of its stiffness and its deformation as follows

k(d, d,,)s1 + k(d, d,,)s, + k(d, dil)sj = Q (2-4)
where
d, : actual length of the springs
d,: free length of the springs

When the platform is working, the current lengths are always greater than the free lengths,

and then the coefficients in Equation 2-4 are different from zero. From Equation 2-1 it is clear

that vectors s,,s, and s, are linearly dependent. In addition, since they are linearly dependant,

they are also coplanar. From the definition ofs,, this result implies that despite the space motion

of the platform, points Qe,Qe,,Q3,4,P,, P, belong to the same plane and Equation 2-4 can be

presented as

SIx Sx S3x
f, +i~ fi + ]= (2-5)
where
s,,,sx: rectangular components of the unit vectors s, expressed in terms of a coordinate
system whose z axis is normal to the plane

The moment of the force fs is a vector perpendicular to the plane of the forces and whose

magnitude is magnitude is f p (Figure 2-3), where p is the perpendicular distance between an

arbitrary point V and the line of action of force f4 Equilibrium of forces establishes that

summation of moments with respect to the arbitrary point V must be zero, then for the forces

acting on the moving platform

f, p1 + f, p + fs p = 0 (2-6)

Equation 2-6 can be combined with Equation 2-5 to obtain, see Duffy [30]






















Figure 2-3. Moment of a force.

S~x S~x S I3x i =o 2
s,, s,, s,, f =0(27
P1 Pt Ps I

Nontrivial solutions for J; requires that,

SIx Sx Sx
sz sy sz, = 0 (2-8)
Pi Pt P3

This situation occurs when the forces are concurrent or parallel. For the configuration of

the current device it is not possible for the forces to be parallel and therefore they must meet in a

pomnt.

Before leaving this section, another interesting fact is that from Equation 2-4, the stiffness

of the springs vanish since they are assumed to all have the same stiffness value, k, and therefore

knowledge of their actual values is not necessary for purposes of obtaining the equilibrium pose.

Mathematical Model

There are several ways to solve the forward analysis problem, according with the selected

variables. Figure 2-4 depicts a scheme including the variables and parameters used for this

model .














P


Figure 2-4. Nomenclature for the forward analysis.

The nomenclature defined here will be used later in the reverse analysis. The elements

presented in Figure 2-4 have the following meaning:

* Coordinate system A: global reference system

* Coordinate system E: local reference system (origin at point Q1, point Q2 On x axis, and z
axis perpendicular to plane)

* I: point of intersection of the line of action of the forces acting on the platform

* 14 : point that define the moving platform

* Q, : free end of the actuator i

* a,, bl : coordinates of point Q, in the local system

* d, : current length of the spring i

* 3, : distance between point (q and the intersection point I

* ry7 : angle between d, and the local x-axis

* p : angle of rotation of the platform with respect to the local x-axis

* L,: length of a side of the equilateral platform









*a : internal angle of the moving platform and therefore equal to ir /3

Global system A may be located in any arbitrary position. In this problem statement it is

assumed that the coordinates of points Q,, Q, and Q3 are known in system A. With the

knowledge of points Q,, the local system E is defined as follows


x E -2 4- (2-9)


gE i 1e) 1.P", (2-10)


yE E~ E, (2-11)

The transformation that relates systems A and E is given by Crane [31]


AT = Eo 1 e (2-12)

where
R E.4E. E (2-13)

Coordinates of points Q, in the system E are given by (a,, b, ). Since Ee 1is the origin of

system E, then

az = 0, b, = 0 (2-14)


Remaining coordinates az, b,,as, b, can be found from the relations




az

b8 ,=l B


EO= _o 3 .4


(2-15)





(2-16)









where ET (T)-1

Note that since Q, is located on the Exa axis and thus

b, = 0 (2-17)

The problem as depicted in Figure 2-4 involves the following ten unknowns

d d,, d,

(2-18)





cos y, cos y, cos y,
(d,I -d, +( ,I -d,) +, -d, = (2-19)
sin ry, sin ry sin ry,

Since the forces are concurrent, equilibrium of moments does not give any new

information. Further equations must be developed based on the kinematics of the device. From

Figure 2-4 it is clear that

3, e'"1= Lp e' +32 e'"1 (2-20)
31 e'1= Lp e;(P~a) +33 e'" (2-21)

Loops defined by Q, Q, I and Q, e, I yield

(d, +3,) e"'/' = EO -EO 1+(d +S3) e"'/
(d~,+3,) e'"l = EO -EO 1+(d ?+3j ) e

Considering Equations 2-14 and 2-17 the last two equations can be simplified to


(d~, + 3,) e"' =~ a d +3)e (2-22)

(d, + ,) e'" =3 a + dz 3 ellv (2-23)









Scalar components of Equations 2-19 through 2-23 form a nonlinear system with ten

equations that can be solved for the ten unknowns using numerical methods. A program to solve

the mathematical model for the forward analysis was implemented. The program takes advantage

of a function that implements the Newton-Raphson method. Once the variables are found, points

AP, are evaluated using the transformation

API =T ElP (2-24)
where points EP, are given by (Figure 2-4)

P = O+d, .(2-25)


E )2EI) L COS P1
P ,= P + (2-26)

-a; -E1 + p LSin + &> 2-7


Numerical Example

A numerical example is provided to demonstrate the mathematical model. To simplify the

presentation of the numerical data, it is understood that lengths are given in consistent units and

angles in radians.

Find the coordinates of points 4q for equilibrium given the free lengths of the ties d, = 20,

and the coordinates of points Q, (Figure 2-5) in a global reference system A

-78.34 52.72 46.70
Q ~ ~ = Q 9.1 ,O = 80.02
1 3
10.44 64.63 49.12

The numerical solution of the system requires a guess for the initial values. An easy way to

obtain them is from the device when it is in the planar position. In that location springs are not

stretched and the platform is not rotated yet, and therefore, point I coincides with the intersection

of the heights of the platform (Figure 2-6), therefore





Figure 2-5. Coordinates of the free ends of the actuators.

d, =d2 =d3 do
2 xi
3, = 32 63 L, cos -
3 6
p=0

Initial values for ry are easily obtained from the geometry of the platform in its first


position (Figure 2-6).


ry, = > = 5 ry3
6 6 2


Figure 2-6. Initial position of the system.










It is also necessary to evaluate variables a a b, which depend on values of Oe From


Equations 2-9 through 2-13 the given values of Qe yield

0.777 0.531 -0.338 -78.34
-0.541 0.838 0.072 0
AT =
E0.321 0.123 0.939 10.45
0 00 1


When the transformation ETj. ET)-1 is OValuated and substituted into Equations 2-15 and

2-16 the terms a2, a3, and b3 are determined as

a, = 168.67 a, 65.87 and b, = 138.09 .

Now the Newton-Raphson method can be implemented to solve system of Equations 2-19

through 2-23. The solution to the 10 unknowns yields

d', = 35.07 3, = 68.93 ry, = 0.389 p = 0.065
d', = 41.49 6, = 40.98 ry, = 2.643
d, =36.73 6, 66.48 ry, = -1.272

Equations 2-25 and 2-27 permit one the evaluation of points P, as

46.06 34.95 35.95
6.42 54.97 44.76
"4P, =, APZ AP3
22.56 55.44 48.14
1 1 1

One way to verify the validity of the results is to check if they satisfy equilibrium

equations and if the lines of action of the forces intersect at the same point, when they are

evaluated in the global system A, instead of the local system E.

The equilibrium condition in the global system can be written as

EF = k(dl, d, s, + k~d d, s, +k(dz d, sz (2-28)
where









AA
s1 = (2-29)
-1A, A-

s = (2-30)



s3 3 (2-31)


The intersection point of the lines passing through points P, Q, and P2 2 eis given by

(Crane, C., Rico, J., Duffy, J., Screw Theory for Spatial Robot Manipulators, Cambridge

University Press, In Preparation)

AS2 X SO2 (S1 -S2 SI XSO2 (S X SoI S2 S2
r -- (2-32)
l( 1 S2 )

Similarly, the intersection of lines passing through P2 2 e and P3 3 eis given by


A 23 3 S3-S 3S O 2 S2-S 3(2-33)
1-(3 -s )2
where
sl= Aex s (2-34)

SO2 A 2S (2-35)

s3A X S (2-36)


Substituting values of AP, and A 2 into Equations 2-28 through 2-36 yields

CF =k I0.2208

EF=k-0.07 1*10-4


-17.39 -17.39
A 2=19.06 an 3=19.06
-46.36 -46.36










It is clear that the solution satisfies the equilibrium condition and that the lines of action of

the forces intersect at the same point. Figure 2-7 displays the device in the evaluated equilibrium

position.


Figure 2-7. Device in the evaluated equilibrium position.









CHAPTER 3
REVERSE ANALYSIS

In the reverse analysis the obj ective is to find the location of the actuators in order to

obtain a desired output. Since there is not an external wrench, it is not possible to achieve an

arbitrary location and orientation of the platform, however it is feasible to constrain the moving

platform to be perpendicular to a given vectorn (Figure 3-1). The evaluation of the actuator

positions required to reach the desired orientation is not intuitive for this kind of mechanisms and

a mathematical model is necessary. The reverse problem for this device admits different

formulations, considering which parameters are considered as given and which must be

evaluated.











A B

Figure 3-1. Normal vector to the moving platform. A) Isometric view. B) Lateral view.

Replacement of Compliant Beams

The motion of the free end of the actuator is the result of bending the bimorph beam due to

the increase of temperature, which is in turn, a function of the thermal resistance and the applied

voltage. Figure 3-2A shows the path of the free end for several positions of the beam. Lowell

[32] has been shown that for the purpose of analysis, compliant elements can be replaced by

hypothetical rigid binary links. Figure 3-2B shows how the original path of the free end can be

approximated for a link whose center lies on the horizontal axis and with a radius t forming an










angle B with the horizontal. The path of the free end may be obtained experimentally and the

center and radius of the hypothetical link adjusted by fitting the curve.












Figure 3-2. Path of the free end. A) Original path. B) Approximated path.

Since all the beams are equal, the radius t is equal for all the actuators and the location ofE

with respect to G is also the same for all the actuators. In the following developments it will be

assumed that points El and radius t are already evaluated.

Reverse Analysis Case 1

This case may be stated as follows:

Given :

* The position of the free end of one of the actuators.

* A unit vector perpendicular to the moving platform.

Find:

* The position of the free ends of the remaining actuators.

Figure 3-3 shows the plane that contains the moving platform and the actuators represented

as binary links. Unit vector n is perpendicular to this plane and positions of points Q. can be

defined by the vectors r, in a global reference system.
























Figure 3-3. Parameters for the reverse analysis, case 1.

One sequence of transformations that relates the global system and any of the local

systems located at the fixed pivots of the binary links and whose x-axis are aligned with the axis

of the binary links (Figure 3-3), is

AT =- Trans~lation(E) \*Ro~tationz,7,) \*Ro~tatio;,O,)
1 0 0 E,, cos y, sin y, 0 cos8, O sin 8, O
T = lI (3-1)
L 0 0 1 0 0 0 1 -iO oO
0 00 1 0 0 01 0 0 0 1
cos ylcos8, sin y, cos y, sin 8, E,,
sin y, cos 8, cos y, sin y, sin 8, E,
AT = v(3 -2)
L Sin 8, 0 cos8, O
0 0 0 1

When i=1, 2, 3 references systems B, C and D are obtained. Figure 3-4a shows a top view

when only the first two transformations of Equation 3-1 are carried out. Note that angles 7, are

constant. Figure 3-4b illustrates the local reference systems in their final orientation after

performing the last transformation involving Bin Equation 3-1.











y,=7d3



~
i


c B

Figure 3.4. Location of the local reference systems for the reverse analysis. A) First rotation. B)
Second rotation.

The first 3 elements of the first column of Equation 3-2 represent the local x-axis


expressed in the global system A. In particular the local axis A x, iS obtained by substituting i=2

in Equation 3-2 as


x
I









cos y, cos 8
Ax, = sin 7, cos8, (3-3)
sin 8,


Without lost of generality assume that the free end whose position is given as" Q is

known. The vector r, is then also known. From the equation of a plane (Crane, C., Rico, J.,

Duffy, J., Screw Theory for Spatial Robot Manipulators, Cambridge University Press, In

Preparation), and Figure 3-3

3 rl) z = 0 :.. r? IE = YI n (3-4)

From the geometry of the device (Figure 3-3)

Tz = E, +t, (3-5)

From Figure 3-4b and considering Equation 3-3

cos y, cos 8,
t, = -t Ax, = -t sin y cos8, (3-6)
sin 8

The scalar product of Equation 3-5 with n yields

Tz n = E, a +t, n (3-7)

Substituting Equations 3-4 and 3-6 into Equation 3-7 yields


1cos y, cos B
E, -n =E, -n -t 2,n n sin y, cos 8,(38
sin 8

Regrouping Equation 3-8 yields

A, cos 8 + B, sin 8 + D = 0 (3-9)
where
A, = nx cos y, + n, sin y (3-10)
B2 = -n, (3-11)









TI -n-E, -n
D, =-


(3-12)


It is possible to obtain a closed solution for 8, in Equation 3-9, See Crane [31].

Substituting the value of 6, in Equations 3-6 and 3-5, the coordinates of r,, and therefore of


point O are determined

Similarly, from Figure 3-3

3-E -y0 rj y= r, n (3-13)

From the geometry of the device (Figure 3-3)

r3 = E3 + t3 (3-14)
where
t, = -t xD (3-15)

Unit vector xD XS obtained from the first three terms of the first column of the matrix

defined in Equation 3-2 when i=3, and thus (3-15) may be written as

cos y, cos8,

tz=- in o 1 (3-16)
sin 8,

The scalar product of Equation 3-14 with n yields

'3 = E_3 + t3 (3-17)

Substituting Equations 3-13 and 3-16 into Equation 3-17 yields


1cos y, cos 8,
r -n=E, -n-t E2 n, nI sin y cos83 3-8
sin 8,

Regrouping Equation 3-18 yields

A, cos 8, + B, sin 8, + D, = 0 (3-19)
where










A, = nx cos y, + n, sin7 y
B3 = -n z
r Y -n-E3 -nY
D,=


(3-20)
(3-21)

(3-22)


Equation 3-19 permits one to evaluate 63, then Equations 3-16 and 3-15 yield r, and

therefore" O) The reverse analysis for this case is completed.

Numerical Example

A numerical example is provided to demonstrate the solution process for the reverse

analysis, case 1. Angles are in radians and lengths in consistent units.

Given: the position of point Q, (as defined by the elevation of rigid link 1 0, = 0.6109 ), the

length of a side of the moving platform Lp = 23, the free lengths of the springs d, = 5 the

length of the binary rigid links t = 26 and the unit normal vector to the platform

n = 0O 0.2588 0.9659] ex ressed in the global reference s stem. Find the coordinates of

points Oz O, P,, P, and P, expressed in the global system.

From Figure 3-4, 7, = -r /3 and 7, = xi / 3 Points Ez E, and E, are evaluated with the aid

of Figure 3-5 which shows the device when it is at the plane level.

cos(-ri) 44.279
E,= L~os~/6+d,+tsin(-ri) =
0 0

cos(-r/ 3) 22.139
Eu Lcosr/6+d, +tsin(-r/~l3) =1 -38.34


cos(ri/3) 22.139
Ez Lcosr/6+d, +tsin(ri/3) 1 38.34

































Figure 3-5. Distributions of points E.

From Figure 3-4, coordinates of Q! are given by the vector r, as follows

rl = l + t = 1
cos y, cos 8, 22.981
E, E +(3= tsin y cos O,1 =
sin 8, 14.9130


From the given data

nx = 0, n, = 0.2588 and n = 0.9659

Now it is possible to evaluate coefficients (3-10) through (3-12) and (3-20) through (3-22).

Substituting the coefficients into Equations 3-10 and 3-19 yields

0.2241cos8, 0.9659 sin 8, + 0.93 58 = 0
0.2241cos8, 0.9659 sin 3 + 0. 1723 = 0

These last equations yield two sets of solutions for 6, and 8, that will yield equilibrium

configurations. Each pair is selected considering the equilibrium conditions. The solutions are

8, = 1.0056, 8,, = 0.4027










02b = 1.68, 63b = 3.195

Equations 3-5 and 3-6 permit one to evaluate points O! and O! for both solutions of 8, and


03 and then, following the procedure presented in the forward analysis, it is possible to evaluate

pointslP,, P, andP,. The results are summarized in Table 3-1.

Table 3-1 Solution for the reverse analysis, case 1.
Solution a Solution b

X V Z X V Z

O, -22.98 0 14.91 -22.98 0 14.91

O, 15.18 -26.28 21.95 23.56 -40.80 25.85

Oj 10.18 17.63 10.19 3_5.12 60.83 -1.39

P, -12.16 -3.57 15.87 -0.54 7.29 12.96

P_, 7.72 -14.73 18.86 19.36 -3.84 15.94

P_3 7.79 7.48 12.90 19.39 18.38 9.99


Results are shown in Figure 3-6. The second solution is also an equilibrium position, but

the current device cannot reach that position.












Figure 3-6. Solution for the reverse analysis case 1. A) Solution a. B) Solution b.

Reverse Analysis Case 2

One could desire to include information about the location of points in the moving

platform in the reverse analysis. Since there are no external forces acting on the









nx~ n?, n ,mechanism, it is not possible to specify a general position for one point of the

platform. However it is feasible to specify in addition to the orientation of the platform given by

the vector perpendicular to its plane, the height with respect to the horizontal plane of one of the

points of the platform. Any point is equally appropriate, for this case the point A P, is selected

(Figure 3-7).


'I


~c~- L


Figure 3-7. Prescribed vertical component of point P1. A) Isometric view. B) Lateral view.

In case 1 it was possible to obtain a closed solution easily because it did not involve any

information regarding the location of the points on the moving platform. In the new situation, the

mathematics are more involved and requires a numerical technique for its solution.















Figure 3-8. Nomenclature for the reverse analysis, case 2. A) Isometric view. B) Plane of the
forces.









Figure 3-8a shows the device in an arbitrary position. Figure 3-8b shows the variables

located on the plane of the moving platform. Positions of points A are unknown and depend


on angles 8, Angle E is also an unknown as well as the x and y coordinates of point AP,

(coordinates of AP, and AP3 can be found once the model is solved). Therefore, in addition to

the 10 variables used in the forward analysis and enumerated in (2. 18), here there are 6 new

unknowns: 8,,O 8, 8,e, Ex and FY The solution requires 16 equations. The reverse analysis for

this case may be posed as follows

Given :

n : normal vector perpendicular to the moving platform.

A, scalar component : of the vector P

Find:

Q ,, O :~%~ : location of the free ends of the binary links.


x,, 4,~: scalar components xe and y of the vector P, .

SPt P3 : location of the vertexes of the moving platform.

Points Qe depend on 6, and using transformation (3-2) they can be expressed as

t cos y, cos 8, sin y, cos y, sin 8, Ex

Oe = 7* T = (3-23)
-1 0 sin 8, 0 cos 9, O
1 0 0 0 1
t cos y cos 8, sin y, cos y sin 0, Ex

O~ = T T = (3-24)
-' 0 sin 8, 0 cos 8, O
1 0 0 0 1









tcos y, cos 8, sin y cos y, sin 8, Ex

"O = T*( (3-25)
0 sin 8, 0 cos9 0,
1 0 0 0 1

Equations 3-23 through 3-25 simplify to

tcos y, cos8, + E,,
tsin y, cos8, + E,
Q = (3-26)
_1 tsin 8,

tcos y cos8, + Ex

"r2 t smny cos 8 + E,
O = (3-27)
t sin 8,

tcos7, cos 8 + E,,
tsin y, cos 8 + Ez
O = '(3-28)
t sin 8,



Expressions for distances Q!,Q! between points OC) and OI~ can be obtained from

Equations 3-26 through 3-28 as follows

Q O O O1 (3-29)

Q Q3 ~= O ~- O) (3-30)



The relation between EP, and AP, is given by

1P=ET EP, (3-32)
Transformation AT defines the,,+, relation between, theglbalsyte A and a referenc


system E which origin is located at Q lwith its x-axis points from OL) to O ), and for which

the z-axis is the unit vector n (Figure 3-8B), therefore










4R Q o1(-3
AT = E 3-3

where
Ri =x 4' E 2 (3-34)

where

xE 24- (3-35)

-E = n (3-36)
yE X4E4E (3 -3 7)

Coordinates of EP, can be obtained from Figure 3-8b, and Equation 3-32 can be expressed

as

AFx 11 2 1~3 Aex dz cosly,
A~ ~ re r O? d,2 ~1 ~ Usin cy,
(3-38)
A~~Z '1 '2 '33 1:z
1 0 0 0 1 1
where the terms ra,depend only on 8, .

At this point all the developments required for the mathematical model are obtained.

Expression (3-3 8) yields 3 scalar equations, one of them involving the prescribed value A,, then

APx= r zd, cosry, + r, d, sin r,+ Qz(339
4) = r zd, cos 7, +r I;d, sin y,+ Q (-40
= r zd, cos y, + r, d, sin r,+ Qg(341

The angle E in Figure 3-8b can be related to points A 0 using the cosine law


QQ3 QQ +QQ -20,,*OQ3*ose(3-42)
where the terms Q,Q, are given by Equations 3-29 through 3-31.

Equilibrium conditions can be expressed in the plane of the moving platform as it was

done in the forward analysis:

(d, d,)cosc, + (d, d,~)cos y + (dj d,)cos y3 = 0 (3-43)









(d, d,)sin c, + (dZ d,)siny, + (d, d,)sin 3 = 0 (3-44)

The geometry of the system involving relations for the moving platform is the same as

found in the forward analysis (Figure 3-8B)

3,el'" = L,pe'P + 63e'" (3-45)
3,e"/' = LIe'Cl~ 8 + 3,e'Y (3 -46)

Geometry relations for the actual lengths of the springs involve the terms Q ,e, Q 03

and E (Figure 3-8B)

(d, + 3)e'" = QQ ~e'" + (dZ + 33)el' (3-47)
(d, + 3,)e"'" = QQ e'"~ + (d, + 33)e"" (3-48)

Points O e and O e with respect to the location of point Q ) must be perpendicular to

vector n. To assure that, two more relations are required

O O)-n= 0 (3-49)
O O -n= 0(3-50)

Equations 3-41 through 3-50 form a system of 14 equations and 14 unknowns that can be

solved for 8,, 0,,8,e, dz, d,, ds,36,,33,3,ry, ry,,7 and P. Appendix A presents the set of

equations in extended form.

Once the solution is obtained, it is possible to evaluate Qe using Equations 3-26 through

(3-28). Coordinates xI~,AF, are easily evaluated using Equations 3-39 and 3-40 which

determines point Pl .

To complete the reverse analysis for the current case it is necessary to evaluate p

and P3. A coordinate system F is defined as parallel to system E and located at P,, then









T b' = Ep (3-51)
000 1 [


The rotation matrix EAR is given by Equation 3-34 and with the aid of Figure 3-8B the

following relations are obtained

L,cos f
A Lp sin f
"P, T FP2 FIP (3-52)



4 Lp sin(P + a)
"P_ = T FP3 FP3 = CO@ (3-53)



Numerical Example

Given the following parameters and the prescribed values for the device where lengths are

in consistent units and angles in radians

do 30
t = 200
L, = 150


Y 0.1330094
n=0.1116


1:Z 64

Find points O 0 and P, .

Figure 3-9 shows the device in its initial position. This position permits the evaluation of

the initial values required for the numerical solution and the coordinates of points E, Lines









through P, O e intersect at the point of intersection of the heights of the equilateral triangle,

there is no stretching and no rotation of the platform, then

4 = dz = d3 = d

2 xi

p, =0~~ 3, pCS
3 3





t3


















6 6 2

Sne =l ,h correspaeondt the angle between pointse,,

B~o= R, -1, =0 0 =O

=ro Rh cos(-ry / 6)e sin(-orm / 6) s 0 is os




= R ~cos(- / 6) sin(-l / 6) 0r










where R, = 3, + d, + t

The solution to system of equations given by 3-41 through 3-50 yields

8, = 0.3530 yr, = 0.4832 p = -0.0150
6, = 0.2841 ry, = 2.7090 E = 1.0122
8, 0. 1606 ry, = -1.6673
d, = 39.52 3, = 76.70
d, = 38.44 3, = 90.38
d, =38.00 6, = 93.55

Equations 3-26 through 3-28 yield

-128 62.23 59.58
0 -107.79 102.20
"o 69.14 4L) 55.54 -) 31.98
1 1 1


Equations 3-39 and 3-40 permit one to evaluate the remaining coordinates Ap,, AIy. As a

result AP, is defined completely.

-89.766

P,=-1.28



Finally, Equations 3-51, 3-52 and 3-53 yield the values for AP, and "P,

39.55 38.40
76.75 72.30
"4P, = AP
55.08 38.34
1 1


Figure 3-10 illustrates the device in its final position. For verification of the results

equilibrium condition in the global system and the point of intersection of forces are evaluated

using equations (2.28) through (2.36). It results yield that summation of forces is zero and the

points of intersection of forces are identical, with a maximum deviation of 1 x 10-9













Le


Figure 3-10. Device for the example of reverse analysis, case 2.









CHAPTER 4
DEVICE DESIGN

Once the kinematics of the system are understood, the next step is to design the device.

This includes the selection of the dimensions and materials for all the elements to obtain a

reliable platform.

The critical elements in the device are the actuators and the springs that join the actuators

to the moving platform. Their design requires assumptions to simplify the analysis, procedures

and analysis techniques from the area of strength of materials, and information from previous

experiences for the issues where analytical approaches cannot be implemented. The forward

analysis that was previously presented will be used to obtain the maximum deflections associated

with the springs.

Actuator Design

Bimorph beams can provide motion out of the plane when they are heated if the difference

between the thermal expansion coefficients of the material forming the beams is significant.

Also, since the kinematic model assumes that the links are rigid, it is important to limit the

deflections at the end of the beams that would result from the forces applied at the end of the

beam by the spring element.

Materials

A pair of materials that can be used for the beams are aluminum and silicone dioxide.

Table 4-1 summarizes from Senturia [33], some material properties for the beams. There is an

appreciable difference in their coefficients of thermal expansion. In addition these materials are

very common in MEMS microfabrication, therefore there are well established procedures for

their deposition and etching processes.











Table 4-1 Mechanical and thermal properties for aluminum and silicone dioxide.
Material Young modulus, E Thermal exp. coeff, a
MPa 1
K 10-6
SO0 70000 0.7
Al 69000 23.1

Width of the Beam

For details associated with the manufacturing process, it is convenient to avoid large

values for the width of the beam to make it easier to release the beam using isotropic etching. A

width of 7 Clm is recommended.

Deflection and Length of the Beams

Deflection of the end of the beam is strongly influenced for the length of the beam. For an

initial estimation of the deflection it can be assumed that the beams are made of one material.

This assumption is justified in the fact that for both aluminum and SiO2, their Young's modulus

is almost the same (Table 4-1). If it is assumed that the beams are straight, from strength of

materials the deflection produced by a force at the free end of a cantilever beam, see Boresi [34]

and Figure 4-1, is given by

L3
So= F, ,where I = -wt3
3El 12
then
F, Ewt3
-(4-1)
6,4L3,
where
6,: deflection at the free end of the beam
F/I : force applied at the free end of the beam
La : length of the beam
E: Young's modulus
w: width of the beam
t: thickness of the beam











Fa


Figure 4-1. Maximum deflection of a cantilever beam.

Equation 4-1 can be solved for several length, width and thickness of the actuators (Figure

4-2). From the point of view of manufacturing, length of the beams is not a constraint, and the

larger beams, the higher motion out of the plane, but at the same time to avoid that deflections

increase dramatically, the thickness must increase. Large thickness are difficult to obtain

therefore they are limited by the manufacturing process. Guided for these reasons the length of

the beam is selected as 200 Clm and the total thickness (this is aluminum and silicon dioxide) as 3

Clm, then from Equation 4-1 and with w=7 Clm

S0.41


The current selection establishes that if the maximum deflection is limited to 1 Clm, then

the vertical force acting in each beam is Fa= 0.41 LN. If each actuator has 12 beams, then the

maximum perpendicular force that can be applied to the system is

F, =12*, = 5 CLN (4-2)

Out-of-Plane Elevation

The design of the spring requires knowing its maximum deflection and one of the factors

that influences this parameter is the elevation of the free end of the beam. The maximum

deformation of the spring is obtained when the beams reach their maximum height (Figure 4-

3A). This situation occurs at the end of the manufacturing process, when the beams are released

from the substrate.















0alOW




1.5



0. La 200


_1I

L~a 300



0.

2 2 5 3 3.5 4
t, pum


Figure 4-2. Deflection of the free end for several conditions.

From Figure 4-3B it is clear that the height of the free end is given by

Q, = p(1- cosf) (4-3)
where
p : radius of curvature and # the angle of p with the vertical.

From Figure 4-3B


La = pf (4-4)

Substituting Equation 4-4 into Equation 4-3 yields


O a ~(1- cos # (4-5)



















XS





Figure 4-3. Maximum elevation of the free end of the beam. A) Isometric view. B) Lateral view.

Equation 4-5 evidences the dependence of Q, on #, however the value of # is difficult to

obtain analytically. At the end of the manufacturing process and before releasing from the

substrate, the beams are in the plane but intrinsic stresses are present. They appear because the

materials are deposited at a higher temperature and after the etching process the two materials

cool to ambient temperature. Once they are released from the substrate, the beam curves up to

release the stresses (Figure 4-4).











Figure 4-4. Bending after release from substrate.

The initial value of the radius of curvature p and therefore the initial value of angle #

depend on the geometry and material properties of the bimorph beam. Liu [3 5] presents the

following equations that should permit the evaluation of the initial radius of curvature of a

bimorph beam.










IgfEo
p = (4-6)
M~
where



Ig 2 = zt + E2t 2 1 Z 2

[: t2 ~,OV) tla (1-v )+ t2 2 ( 2):


2 Ezt, +E2 2

11' Eztz2 +1t)+E2 1 2
z=2
Ezt, +E2 2
I fEo : is the effective flexural stiffness
M~: bending moment required to bend the cantilever beam
z : the position of the neutral axis
t: thickness of the layers of the bimorph
E: Young's modulus
w: width of the bimorphs
a : intrinsic stress in the layers after deposition
v : Poisson's ratio for the layers
and subindex 1 is for the material at the bottom and subindex 2 is for material on the top.

If t, = t2 = t and E, = E2 = E, Equation 4-6 can be simplified to

8 E
p =- t (4-7)
3 G

To be useful for numerical evaluation, Equation 4-7 requires the knowledge of the intrinsic

stresses a, and 62 They depend strongly on all the conditions for the manufacturing process and

also on the thickness of the layers. For the same reason, very few values are referenced, and in

the best of the cases only ranges of values varying from negative to positive can be obtained. In

the absence of better information, Equation 4-7 cannot be applied and the estimation of angle

theta must be done from previous experiences. Xie [6] reports that from a beam of similar length

an angle # of 17o has been observed. For purposes of this design, angle # is assumed to be 200.









The exact value is not of interest as well as the spring be able to yield the maximum deformation

without excessive stress that may damage it.

Springs Design

The spring must be able to provide the maximum deformation required for the mechanism

and stand the stresses generated by this deformation. The first step it to evaluate the maximum

elongation that the springs must provide.

Springs Elongation

When the device is on the horizontal plane the length of the springs correspond to the free

length d,, (Figure 4-5A). When beams reach the maximum elevation (Figure 4-5B), the length of

the springs is maximum.








L L




Figure 4-5. Positions for minimum and maximum deformation of the springs. A) Initial position.
B) Maximum deformation position.

Following the procedures presented in chapter 2 it is possible to evaluate the position of

the platform given the points Q, and therefore the maximum elongation of the springs, dmax d,,,

for several values of the parameters of the device. A set of results is presented in Figure 4-6 for a

length of the actuator L, = 200 Clm, free length of the spring d, = 80 Clm, and the length of the

side of the platform L, = 120 lm.


















8-4










0 5 10 15 20 25 30
g, deg

Figure 4-6. Maximum deformation of the springs.

Values of do cannot be very small to avoid exaggerated stresses. For the current values


oft,, L, do and for the already selected value of 200 for95, the maximum deformation is 4.0

lm. This is the elongation that the spring must provide.

Maximum Force Acting on the Spring

Force acting on the spring must be limited. If this force is excessive its vertical

components acting on the beams may generate a deflection larger than the design value.

Figure 4-7 shows the components of the force acting on the spring. In Equation 4-2 the

maximum admissible perpendicular force was selected as Fp = 5 CIN and then when F, is

maximum


S=' (4-8)

F F
cos q =P : F = (4-9)
F, cosy























G--





Figure 4-7. Maximum force in the spring.

For # = 20", Equation 4-8 yields r = 80" and from Equation 4-9, F, = 28.8 IN.

In summary, to get a maximum deformation of 4.0 Clm the maximum force cannot exceed

28.8 GIN, in order to keep the component perpendicular of the force to GQ less than 5 IN.

Spring Geometry and Material

The simplest geometry for the spring is a bar with rectangular section, similar to a common

tie (Figure 4-8A). However, to obtain significant deflections with that geometry requires a

material like rubber that is able to deform with low external forces. Some tests were performed

in the laboratory using different kinds of silicone rubber. However due to the fact that the

dimensions of the ties are very small compared with the other components of the device there

was no evidence that the silicone filled the channels. Another undesirable aspect with this

material is the difficulty to etch it with conventional plasma. Since ties are essential to this work

it was necessary to look for other alternatives. A simple one is to change the geometry of the

spring. If instead of a simple bar, a shape like the presented in Figure 4-8B is used, it is easier to

achieve the required deformations.





















Figure 4-8. Possible geometries for the spring. A) Bar. B) By segments.

Although it is possible to create the spring using the same materials for the beam, this is

aluminum and silicone dioxide, their stiffness is still high. One alternative is to use

photodefinable polymides. They have low modulus, and in addition can be patterned easily,

which is a very important advantage. Table 4-2 shows some properties of the polyimide HD-

8000 from HD MicroSystems. For the following analysis they are considered as isotropic

materials.

Table 4-2 Properties of polymide HD-8000.
Viscosity Thickness Cure Tensile Modulus
St Clm oC MPa MPa
3.5 3 to 5 350 122 2500

Figure 4-9 shows the geometrical parameters for a segment of the spring. If T and H are

given, angle ii and length Le can be evaluated as follows

2r cos i+ L, sin ii= T (4-10)
2r sin ii + L, sin ii = H (4-11)

Squaring Equations 4-10 and 4-11 and adding the results yields

4r 2 + Lc2 2 + H2 ... Le = JT2 H2-4r2 (4-12)

Since Le must be positive, Equation 4-12 shows that the selection of T and H must fulfill

the relation

T2 +H2 ) 4r2 (4-13)

























Figure 4-9. Geometry of a segment of the spring.

If Equation 4-10 is multiplied by sin ii and Equation 4-11 is multiplied by cos ii then

2r sin ii cos ii + L, sin 2 ii = T Sin ii (4-14)
2r sin iicos i+ L, cos2 ii= H cos i (4-15)

Adding Equations 4-14 and 4-15 yields

H cos ii+ T sin ii L, = 0 (4-16)

Equation 4-16 can be solved for ii using a trigonometric method.

Spring Stress

Since the geometry is simple, it is interesting to find analytical relations for the stress and

the deflection of one segment of the spring (Figure 4-10). The maximum tensile stress use occurs

at point B (Figure 4-10). Figure 4-11 presents the nomenclature to derive the expressions for the

circumferential stress in a curved beam following the procedure explained by Boresi [34]

F +Mx(A-RAm
as,= (4-17)
A AR,(RA, A)
where
A = tb (4-18)

Mx = F -+R (4-19)


























Figure 4-10. Segment of the spring.


R, = R- (4-20)

2R +b
A, = tln (4-21)
"'2R b








Le











Figure 4-11i. Parameters for the stress analysis of a spring.

Spring Deflection

For the element shown in Figure 4-10 and 4-11, the total deflection can be considered as

the superposition due to the deflection of the linear segment and the deflection due to the









curvilinear segment. Moreover, when the relation R/b>2, the effect of shear and normal forces

can be neglected. For this case the deflection of the straight element 6,, is given by


32=2dZ (4-22)

where
M~=F,zcosei (4-23)

From Equations 4-22 and 4-23


3,- = FLcos2 m, I = tb3 (4-24)
3El 12

The deflection of the curvilinear segment, 6, is given by


Sc=(xRda (4-25)
where
Mr = F, [Lc cos u, R sin w + R sin(mL + all (4-26)

From Equations 4-25 and 4-26

PR [L cos m R sin a + R sin m + a)l 2da (4-27)


Total deflection per segment 6, is just the superposition of 3, and c,,

6,= 3, + So (4-28)

For n segments the total deflection is

3, = n (4-29)

Spring Dimensions

Expressions found for the stress and deflection of the spring can be evaluated for different

values of the parameters. Figure 4-12 presents the results for the deflection of the spring when

F, = 10 GIN, T=28 Clm and t-3 lm. From then is clear that for the parameters H=50 Clm, b=7 Clm

and R=12 Clm, a force of 10 GIN, just a third of the admissible force, is enough to deflect one











segment of the spring 2 lm. If 3 elements are used to create the spring, the resultant deflection

provides more than the required deflection of 4 lm.

Fq=10pY; t=3 pm, T=28 pmn


R=10 R=12







b~b=
5

b=5=







40 50 60 40 50 60
H, pm~


Figure 4-12. Deflection in the spring.

F(=10prN, t=3 pmn
40
R=10 /I R=I2

35
b=-y I b=5

30



25n I =

20


15~ =i ~


40 50 60 40 50 60
H, pm~


40 50 60


40 50 60


Figure 4-13. Stress in the spring.










Similarly, Figure 4-13 presents the results for the stress cr for the same conditions of

Figure 4-11. It is clear from the highlighted value, that the stress is only 19 MPa compared to the

admissible value of 122 MPa (Table 4-2).

Dimensions found yield a conservative and reliable device. Table 4-3 summarizes the

results. Some of them are not critical and their calculations were not included.

Table 4-3 Main dimensions of the device.
Parameters Beams Parameters Spring
Length, La : 200 Clm Width, b: 7 Clm
Widt, w:7 pmThickness, 2t: 3 Clm
Thickness of each lawyer, t: 1.5 pmn Free length, do: 80 Clm
Number of beams: 12 H: 50 Clm
Material beams: aluminum and silicone dioxide T: 28 Clm
R: 11.5 Cim

Parameters Platform Parameters resistor
Lp : 120 Clm Material: chrome
Width: 5 Clm
Thickness: 0.2 Clm









CHAPTER 5
MANUFACTURING PROCESS

This chapter presents a sequence to manufacture the device. The device is formed by three

materials: silicone dioxide and aluminum for the bimorph beams and between them chrome for

the resistor. Processes are common and they do not involve any strange requirements. Some

experiments were performed at the University of Florida nanofacilities. The manufacturing

process presented here takes advantage of that experience, which is described in Appendix B.

Five masks are used to define all the features of the system. In the following figures depth

dimensions are magnified to assist in the visualization of the geometry of each step.

The substrate for the device is silicon. The wafer does not require any particular electrical

or mechanical properties, and the crystal orientation does not affect the process (Figure 5-1).


Figure 5-1. Silicone substrate.


Figure 5-2. First layer of silicone dioxide.









A first layer of 0.80 Clm of silicone dioxide is deposited over the wafer using plasma

enhanced chemical vapor deposition (PECVD) process (Figure 5-2).

Then a layer 0.20 Clm of chrome is sputtered over the first silicone dioxide layer (Figure 5-

3).


Figure 5-3. Layer of chrome.

Using a mask, the chrome is patterned and then plasma etching is used to obtain the shape

of the resistor (Figure 5-4).


Figure 5-4. Resistor. A) General view. B) Detail.

A second layer of silicone dioxide with thickness 0.70 Clm is applied using PECVD (Figure

5-5A). As a result the resistor is isolated, however it is necessary to open a via to be able to apply

voltage (Figure 5-5B).




















Figure 5-5. Second layer of silicone dioxide. A) General view. B) Detail.

Aluminum with thikness1.5 Clm is sputtered to complete the materials for the device

(Figure 5-6A). To be able to apply voltage to the resistors it is necessary to create isolated areas

called the pads. For this purpose an additional mask is required (Figure 5-6B).












Figure 5-6. Aluminum layer. A) General view. B) Detail.

The next step is to obtain the shape of the cantilever beams and the moving platform. A

new mask is required to avoid damages in the already created pads. Figure 5-7 illustrates the

result when the aluminum has been removed and the second layer of silicone dioxide is exposed.












Figure 5-7. Etching of areas in the aluminum corresponding to the actuators and platform. A)
General view. B) Detail.










The photoresist required for this process is still over the aluminum, but it is not presented

to simplify the visualization. Between the central platform and the actuators appear some free

areas that will be used for the j points.

A new etching process, but the same mask, is required to remove both layers of silicone

dioxide. The process stops when the substrate is reached. At this moment the photoresist is

stripped (Figure 5-8).


Figure 5-8. Etching of silicone dioxide

Springs are created in the next two steps. A layer of polymide is spun on the wafer (Figure

5-9).


Figure 5-9. Polymide layer.

With a mask, the photodefinable polymide is patterned to create the springs. After curing

and removing remaining material (Figure 5-10).





















Figure 5-10. Polymide springs. A) General view. B) Detail.

The main elements of the mechanisms are ready. The obj ective of the next steps is to

release the beams, springs and platform. A backside etch on the substrate is illustrated in Figure

5-11i. This step determines the depth of the platform.













Figure 5-11. Backside etch. A) General view. B) Detail.

The process continues in the front side. Figure 5-12 shows a section view of the wafer and

the detail of the substrate under a spring. The substrate material must be removed.


Figure 5-12. Section view of the device. A) General view. B) Detail.










The exposed parts of the substrate are removed using deep reactive ion plasma etch (Figure

5-13). Since the etching process is much more aggressive on silicone than on aluminum or

polymide, a new mask is not required. At this moment all the elements are still attached to the

wafer by small portions of silicone.












Figure 5-13. Deep reactive ion etching. A) General view. B) Detail.

Finally isotropic etching is used to remove the portions under the beams and spring (Figure

5-14). Some undercut is also present in the platform, but it does not affect its strength.


Figure 5-14. Isotropic etch. A) General view. B) Detail.









CHAPTER 6
CONCLUSIONS

Our research addresses the study of a device formed by compliant links and compliant

joints. At the MEMS level the study of devices with compliant links has been studied extensively

but the configuration proposed in this research is a different approach to the design of MEMS

devices.

Forward and reverse kinematic analyses were performed. They provide the basis for

control of the platform since they permit determination of the position of the system or to know

the inputs to get a desired output.

The manufacture process suggested is feasible. Dimensions for the components of the

mechanism assure a reliable system.

The reverse analysis shows that three actuators provide limited mobility to the platform. It

is possible to increase the mobility adding more actuators, which requires a new mathematical

model, however the basis of the manufacturing process remains the same.

All the information required for the manufacturing of the device is provided and therefore

it is straightforward to create the masks required for the construction of the mechanism.

Experimental evaluation of the final results will provide insight about how to enhance the design.

Compliant j points offers interest possibilities at the MEMS level since rigid j points are

difficult to implement. They can be used to create not only simple platforms, but also to

implement more complex devices such arrays of actuators able to move in the plane or in the

space. In any case, the required kinematics is a complex topic and issues like that associated with

closed solutions are still open. There is a large field of applications related with kinematics

theory of devices intended to work at the MEMS level.









Achievement of new devices requires both theoretical and experimental work. It is

necessary to explore materials that permit better manufacturing and performance of the

compliant joints.

The analysis presented in this research is basically a static analysis. Future works should

include dynamic behavior which is essential to the vibration and control issues.

This work was the first of its kind in the Center for Intelligent Machines and Robotics

(CIMAR) lab at the University of Florida. It shows a way to involve the extensive knowledge

acquired in the design of mechanisms to the MEMS level.









APPENDIX A
REVERSE ANALYSIS EQUATIONS

This appendix presents the long form of the equations derived for the second case of the

reverse analysis.

* Fl=(dl-d0)*cos(psi l)+(d2-d0)*cos(psi2)+(d3-d0)*cos(psi3)

* F2=(d l-d0)*sin(psi l)+(d2-d0)* sin(psi2)+(d3 -d0)* sin(psi3)

* F3=(dl+deltal)*cos(psi l)-(d2+delta2)*cos(psi2)-((-
cos(gammal )*cos(thetal )*t+Elx+cos(gamma2)*cos(theta2)*t-E2x)^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*t-E2y)^2(sn' hea)*t-
sin(theta2)*"t)^`2)^(1/2)

* F4=(d l+deltal)*sin(psil1)-(d2+delta2)* sin(psi2)

* F5=(dl+deltal)*cos(psil1)+1/2*((-
cos(gammal )*cos(thetal )*t+Elx+cos(gamma3)*cos(theta3)*t-E3x)^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma3)*cos(theta3)*t-E3y)^2(sn' hea)*t-
sin(theta3)*t)^'2)^'(1/2)-(d3+delta3)*cos(psi3)

* F6=(dl+deltal)*sin(psil1)+1/2*((-
cos(gammal )*cos(thetal )*t+Elx+cos(gamma3)*cos(theta3)*t-E3x)^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma3)*cos(theta3)*t-E3y)^2(sn' hea)*t-
sin(theta3)*t)^'2)^'(1/2)*3^`(1/2)-(d3+delta3)* sin(psi3)

* F 7=deltal1 "cos(p sil1)-Lp "cos(b eta)-delta2*"cos(p si2)

* F 8=deltal1 "sin(psi l)-Lp "sin(b eta)-delta2*"sin(psi2)

* F9=deltal~cos(psi l)-Lp~cos(beta+alfa)-delta3*"cos(psi3)

* F10=deltal~sin(psi l)-Lp~sin(beta+alfa)-delta3*"sin(psi3)

* F11l=(-cos(gamma)csteal)*t+c xco(gamma2)*cotteta2)*-Eo x^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*t-E2y)^2(sn' hea)*t-
sin(theta2)*t)^2+(-cos(gammal )*cos(thetal )*t+Elx+cos(gamma3)*cos(theta3)*t-
E3x)^'2+(-sin(gammal )*cos(thetal)*t+E ly+sin(gamma3)*cos(theta3)*t-
E3y)^'2+(sin(thetal )*t-sin(theta3)*t)^'2-2*((-
cos(gammal )*cos(thetal)*t+E Ix+cos(gamma2)*cos(theta2)*t-E2x)^`2+(-
sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*t-E2y)^2(sn' hea)*t-
sin(theta2)*t)^'2)^( 1/2)*((-cos(gammal )*cos(thetal)*t+E Ix+cos(gamma3)*cos(theta3)*t-
E3x)^'2+(-sin(gammal )*cos(thetal)*t+E ly+sin(gamma3)*cos(theta3)*t-
E3y)^'2+(sin(thetal)*t-sin(theta3)*t)^)(/'2)*o pio)(
cos(gamma2)*cos(theta2)*t+E2x+cos(gamma3)*o~ht3*-~)2(









sin(gamma2)*cos~theta2)*t+E2y+sin(gamma3)cstea*-Ey^(inht2*-
sin(theta3)*"t)^`2

* F l2= (-cos(gamma2)*cos(theta2)*t+E2x+cos(gaml*othal)-Exn+(
sin(gamma2)*cos(theta2)*t+E2y+sin(gammal )*cos(thetal)*t-E ly)*ny+(sin(theta2)*t-
sin(thetal )*t)*nz

* F1 3=(-cos(gamma3)*cos(theta3)*t+E3x+cos(gama)cstel*-Exn+(
sin(gamma3)*cos(theta3)*t+E3y+sin(gammal )*cos(thetal)*t-E ly)*ny+(sin(theta3)*t-
sin(thetal )*t)*nz

* Fl4=Pl z-(sin(theta2)*"t- sin(theta l)*t)/((-
cos(gammal )*cos(thetal )*t+Elx+cos(gamma2)*cos(theta2)*t-E2x)^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*t-E2y)^2(sn' hea)*t-
sin(theta2)*t)^'2)^(1/2)*dl "cos(psi l)-(nx*(-
sin(gamma2)*cos(theta2)*t+E2y+sin(gammal )*cos(thetal )*t-Ely)/((-
cos(gammal )*cos(thetal )*t+Elx+cos(gamma2)*cos(theta2)*t-E2x)^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*t-E2y)^2(sn' hea)*t-
sin(theta2)*"t)^`2)^(1/2)-ny* (-
cos(gamma2)*cos(theta2)*t+E2x+cos(gammal )*cos(thetal)*t-E Ix)/((-
cos(gammal )*cos(thetal )*t+Elx+cos(gamma2)*cos(theta2)*t-E2x)^2+(
sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*t-E2y)^2(sn' hea)*t-
sin(theta2)*t)^'2)^(1/2))*dl "sin(psi l)-sin(thetal)*t









APPENDIX B
PREVIOUS WORK

A process using four masks was tested. Figure B-1 shows the layout of one of the masks

used and a detail of one of the features. The goals of this set of experiments were to gain

experience about the basic micromanufacturing techniques, to study the behavior of silicone

rubber as material for the compliant elements and find if the whole process could be done from

one side of the wafer. The tests were made at the University of Florida Nanofacilities.


















Figure B-1. Mask used in the previous work. A) Outline. B) Detail of a feature.

Compliant j points made of rubber exhibit a good performance in macro devices. It seemed

appropriate to implement a similar solution for the microdevice, and in this way, simplify the

design of the springs. For this purpose the references Q1-4010 and JCR 6122 from the brand of

encapsulants and led materials from Dow Corning were selected. According with the information

of the manufacturer, they are flowable, cure to a flexible elastomer, operate in a wide interval of

temperature (-450 to 200 oC) and exhibit minimal shrinkage. Information about elastic modulus is

not always available, but some data suggest values less than 200 MPa. Their original viscosities

are in the range from 300 to 800 centipoises.










Figure B-2 depicts one of the features manufactured at the lab. Black areas are holes.

Silicone rubber should fill the gaps and then rubber should be removed from selected areas in

order to form the ties that connect the central platform and the actuators. However, there was no

evidence that the material filled the trenches, even though when the viscosity was reduced using

a solvent. Oxygen plasma was used to etch the rubber, but for practical purposes the material was

insensitive to this procedure. As a conclusion from these results, the material for the ties must be

photodefinable and still provide enough deformation. Polymides are a good alternative for this

purpose.





















Figure B-2. Manufactured feature.

Other set of tests were performed trying to release the central platform through the use of

isotropic etch and working from the front side only. The procedure showed that it is not feasible

to release the platform in this way and etching from the back side is necessary.

No additional constraints were found from the experiments. The design presented in

chapter 5 includes these considerations.










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25. Pelesko, J. A., and Bernstein, D. H., 2002, M\~odelingM\~EM~S andNEM~S, CRC Press, Boca
Raton, FL.

26. Hah, D., Yoon, E., and Hong, S., 2000, "A Low-Voltage Actuated Micromachined
Microwave Switch Using Torsion Springs and Leverage," IEEE Transactions On
Microwave Theory and Techniques, 48(12), 2540-2545.

27. Yang, S., Chen, X., Hasegawa, M., and Motojima S., 2004, "Conformations of Super-
Elastic Carbon Micro/Nano-Springs and their Properties," Proceedings of the International
Conference on M~EMS, NANO and Smart Systems, Alberta, CA, August.

28. Hata, S., Kato, T., Fukushige, T., and Shimokohbe, A., 2003, "Integrated Conical Spring
Linear Actuator," Microelectronic Engineering 67-68, pp. 574-581.










29. Brand, L., 1947, Vector and' Tensor Analysis, Wiley, New York.

30. Duffy, J., 1996, Statics and' Kinematics n ithr Applications to Robotics, Cambrid ge
University Press, New York.

3 1. Crane, C., and Duffy, J., 1998, Kinematic Analysis of Robot Manipulators, Cambrid ge
University Press, USA.

32. Howell, L. L., 2001, Compliant M~echanisms, Wiley, New York

33. Senturia, S. D. 2001, M~icrosystem Design, Kluwer, Boston.

34. Boresi, A. P., and Schmidt, R.J., 2003, Advanced2~echanics of2aterals, Wiley, New
York.

35. Liu, C., 2006, Found'ations ofM~EMS, Prentice, Upper Saddle River, NJ.









BIOGRAPHICAL SKETCH

Julio Correa completed his master' s degree with Dr. Joseph Duffy in 2001 at the

University of Florida. He worked on kinematics, a topic he enjoyed. He returned to Colombia

his native country after completion of his master' s degree.

He came back to University of Florida to pursue and complete the PhD program under

supervision of Professor Carl Crane in 2004.

He returned to Universidad Pontificia Bolivariana in Medellin, Colombia after completion

of his PhD program to conduct research on kinematics of microdevices and do teaching.





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1 KINEMATIC ANALYSIS AND DESIGN OF A COMPLIANT MICROPLATFORM By JULIO CSAR CORREA RODRGUEZ 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

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2 2007 Julio Csar Correa Rodrguez

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3 To my mother for her infinite generosity

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4 ACKNOWLEDGMENTS I thank my supervisory committee members: Dr. Carl Crane, Dr. Glor ia Wiens, Dr Hiukai Xie and Dr. John Schueller for thei r valuable suggestions on my th esis. I extend a special thanks to professor Carl Crane, my academic advisor, for his continuous support and encouragement throughout my graduate study here. I thank profe ssor Hiukai Xie for his guidance and his help with microsystem technology.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................13 Out-of-Plane Motion Devices.................................................................................................13 Tensegrity Structures.......................................................................................................... ....15 Bimorph Actuators.............................................................................................................. ....16 Elastic Joints................................................................................................................. ..........17 The Device..................................................................................................................... .........18 2 FORWARD ANALYSIS........................................................................................................19 Forces Acting on the Platform................................................................................................20 Mathematical Model............................................................................................................. ..22 Numerical Example.............................................................................................................. ..26 3 REVERSE ANALYSIS..........................................................................................................31 Replacement of Compliant Beams.........................................................................................31 Reverse Analysis Case 1........................................................................................................ .32 Numerical Example.............................................................................................................. ..37 Reverse Analysis Case 2........................................................................................................ .39 Numerical Example.............................................................................................................. ..45 4 DEVICE DESIGN..................................................................................................................49 Actuator Design................................................................................................................ ......49 Materials...................................................................................................................... ....49 Width of the Beam...........................................................................................................50 Deflection and Length of the Beams...............................................................................50 Out-of-Plane Elevation......................................................................................................... ..51 Springs Design................................................................................................................. .......55 Springs Elongation..........................................................................................................55 Maximum Force Acting on the Spring............................................................................56 Spring Geometry and Material........................................................................................57 Spring Stress.................................................................................................................. ..59

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6 Spring Deflection.............................................................................................................60 Spring Dimensions..........................................................................................................61 5 MANUFACTURING PROCESS...........................................................................................64 6 CONCLUSIONS....................................................................................................................70 APPENDIX A REVERSE ANALYSIS EQUATIONS..................................................................................72 B PREVIOUS WORK................................................................................................................74 LIST OF REFERENCES............................................................................................................. ..76 BIOGRAPHICAL SKETCH.........................................................................................................79

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7 LIST OF TABLES Table page 3-1 Solution for the reverse analysis, case 1............................................................................39 4-1 Mechanical and thermal properties fo r aluminum and silicone dioxide............................50 4-2 Properties of polymide HD-8000.......................................................................................58 4-3 Main dimensions of the device..........................................................................................63

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8 LIST OF FIGURES Figure page 1-1 Prismatic tensegrity st ructure with 6 struts........................................................................15 1-2 Sequence of motions for the rising of the structure...........................................................16 1-3 Configuration of a bimetallic actuator...............................................................................17 1-4 Bending of a bimetallic actuator........................................................................................17 1-5 Scheme of the device....................................................................................................... ..18 2-1 Device in a general position...............................................................................................19 2-2 Arbitrary forces acting on the platform.............................................................................20 2-3 Moment of a force.......................................................................................................... ....22 2-4 Nomenclature for the forward analysis..............................................................................23 2-5 Coordinates of the free ends of the actuators.....................................................................27 2-6 Initial position of the system............................................................................................. .27 2-7 Device in the evalua ted equilibrium position....................................................................30 3-1 Normal vector to the moving platform..............................................................................31 3-2 Path of the free end....................................................................................................... .....32 3-3 Parameters for the reverse analysis, case 1........................................................................33 3.4 Location of the local reference sy stems for the reverse analysis.......................................34 3-5 Distributions of points E.................................................................................................. ..38 3-6 Solution for the reverse analysis case 1.............................................................................39 3-7 Prescribed vertical component of point P1.........................................................................40 3-8 Nomenclature for the reverse analysis, case 2...................................................................40 3-9 Device in its initial position............................................................................................. ..46 3-10 Device for the example of reverse analysis, case 2...........................................................48 4-1 Maximum deflection of a cantilever beam........................................................................51

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9 4-2 Deflection of the free end for several conditions...............................................................52 4-3 Maximum elevation of the free end of the beam...............................................................53 4-4 Bending after release from substrate..................................................................................53 4-5 Positions for minimum and maxi mum deformation of the springs...................................55 4-6 Maximum deformation of the springs................................................................................56 4-7 Maximum force in the spring.............................................................................................57 4-8 Possible geometries for the spring.....................................................................................58 4-9 Geometry of a se gment of the spring.................................................................................59 4-10 Segment of the spring..................................................................................................... ...60 4-11 Parameters for the stress analysis of a spring....................................................................60 4-12 Deflection in the spring.................................................................................................. ....62 4-13 Stress in the spring...................................................................................................... .......62 5-1 Silicone substrate......................................................................................................... ......64 5-2 First layer of silicone dioxide............................................................................................64 5-3 Layer of chrome............................................................................................................ .....65 5-4 Resistor................................................................................................................... ...........65 5-5 Second layer of silicone dioxide........................................................................................66 5-6 Aluminum layer............................................................................................................. ....66 5-7 Etching of areas in the aluminum co rresponding to the actuators and platform...............66 5-8 Etching of silicone dioxide................................................................................................67 5-9 Polymide layer............................................................................................................. ......67 5-10 Polymide springs.......................................................................................................... ......68 5-11 Backside etch............................................................................................................. ........68 5-12 Section view of the device................................................................................................ .68 5-13 Deep reactive ion etching................................................................................................. ..69

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10 5-14 Isotropic etch............................................................................................................ ..........69 B-1 Mask used in the previous work........................................................................................74 B-2 Manufactured feature....................................................................................................... ..75

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11 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 KINEMATIC ANALYSI S AND DESIGN OF A COMPLIANT MICROPLATFORM By Julio Csar Correa Rodrguez August 2007 Chair: Carl Crane Major: Mechanical Engineering Our research addresses the kinematics and the design of a three-dimensional device at the micro level. The device is formed by three act uators that transmit the motion to a central platform. Techniques used to manufacture microele ctromechanical systems (MEMS) have severe limitations and they cannot permit the construction of complex joints. To solve this problem compliant joints are used in this device to c onnect the platform and actuators. At the MEMS level they offer significant advantages compared to their counterparts at the macro level. The device is able to provide complex motions that require an elaborate mathematical model for their description. Two kinematic issu es are presented: the forward and reverse analyses. The forward formulation allows for th e determination of the location of the moving platform given the position of the actuators, while the reverse analysis finds the location of the actuators for a desired po sition of the platform. The models are based on a Newtonian approach and are subjected to several assumptions to simplify the formulation. The Newtonian approach is preferred because it relates in a natural way the forces and the geometry of the device. Examples and verificati ons of the models are provided.

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12 Actuators consist of two beams with different thermal expansi on coefficients and a resistor between them. This configuration allows fo r the bending of the beam when temperature increases. Springs are formed of a compliant a nd photodefinable material. I ssues associated with the selection of dimensions and materials as we ll as the manufacturing process that permits to build the device are presented. The combination of compliant beams and compliant joints exhibits important advantages at the MEMS level and also poses challenging kinema tic problems. The principles presented here will be useful for the generation of more complex devices.

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13 CHAPTER 1 INTRODUCTION Mechanisms formed by rigid links and rigid joints have been th e object of extensive studies for the theory of mechanisms. These kind of devices are well suited to work at the macroworld, however when the dimensions of the systems are on th e order of microns, limitations due to manufacturing processes impo se severe limitations, and the generation of motion requires alternative approaches. Devices for microelectromechanical systems ar e basically planar devices. This is due the current manufacturing techniques th at are derived from the IC i ndustry. Creating 3D structures at the micro level is a difficult ta sk. Most of the motion of MEMS devices is constrained to the plane. Some works have been made to create spatial motion. Out-of-Plane Motion Devices Out-of -plane actuators can convert input si gnals into displacements normal to the surface of a substrate. Three-dimensional microdevices ar e useful for different tasks as for example, object positioning, micromanipulators, optical sc anners, tomographic imag ing, optical switches, microrelays, adjustable lenses and bio-MEMS applications. To obtain out-of-plane motion is a challenging problem and se veral approaches based have been proposed. Usually out-of-plane actuators are multilayer structures, although single layer devices have been reported by Chen [1]. Ge nerally speaking current solutions are based on vertical comb drives, on the deformation of the ma terials or on the assembly of basic linkages. The following references report out-of-plane motion devices. Vertical comb drives are formed by an array of capacitors. When a voltage is applied, the movable components of the capacito rs rise out of the plane. They are combined with torsion mirrors to tilt micromirrors as it is describe d by Milanovic [2] and Lee [3]. The vertical motion

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14 of comb drives is limited and they require a car eful design and control to avoid jumps associated with the pull-in voltage, s ee Bronson and Wiens [4]. Combination of TiNi and Si cantilever or othe r substrates such as SU-8 or polymide have been used to create out-of-plane motion devices. Fu [5] reports several devices based on a TiNi film which is actuated when a curr ent is applied to the electrode. A micromirror having a large vert ical displacement has been pres ented by Jain and Xie [6]. The mirror plate is attached to a rigid silicon frame by a set of aluminum/silicon dioxide bimorph beams. A polysilicon resistor is embedded within the silicon dioxide layer to form the heater for thermal bimorph actuation. Ebefors [7] and Suh [8] implemented conveyor s systems for out-of-plane motion able to perform complex manipulations. They are based on arra ys of structures that can deflect out of the plane due to different coefficien ts of thermal expansion. Objects that are placed on the array can be moved according to the deflection of each actuator. Schwizer [9] reports a monolithic silicon integrated optical micro-scanner. The device consists of a mirror located on the tip of a th ermal bimorph actuator beam and it is able to achieve large scan angles. The other alternative to achieve out-of-plane motion is the assembly of planar linkages. A platform described by Jensen [10] has three de grees of freedom and the top platform remains horizontal throughout the de vices motion. A proposal for a th ree degree of freedom parallel robot is presented by Bamberger [11]. The device uses only rigid revolute joints. Both revolute actuators are located at the base during the manuf acturing process, making the device suitable for MEMS fabrication.

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15 Out-of-plane motion has also been realized th rough the use of elastic elements. A device actuated by comb drives is presented by Tung [12]. Drives are connected to a platform made of polydimethilsiloxane (PDMS) via thin flexural microjoints. Previous works suggest that compliant links and elastic joints may be a feasible alternative to create mechanical devices at the microlevel. There are many configurations based on these simple elements, one example of which are tensegrity structures and they illustrate another way to obtain spatial motion. Tensegrity Structures The word tensegrity is a contra ction of tension and integrity a nd refers to structures formed by rigid and elastic elements that maintain their shape due only to their configuration. Rigid elements do not touch one another and they do no t require external forces to maintain their unloaded position (Figure 1-1). Figure 1-1. Prismatic tensegrity structure with 6 struts. Tensegrity structures were developed by arch itects in the middle of the last century. Research began with Fuller [13]. First contri butions were made by Kenner [14] and Calladine [15]. Static and dynamic analysis studies have been made Murkami [16] and Correa [17]. Proposed applications include an tennas, Knight [18], flight s imulators, Sultan [19], deployable

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16 structures, Tibert [20], and force and torque se nsors, Sultan [21]. Tensegrity has been also proposed by Ingber [22], to explain the deformability of cells. Due to the presence of elastic ties, tensegrity structures are foldable. If in the folded position external constraints are released, they can recover suddenly th eir original shape by themselves. The deployment can be also achie ved in a controlled way using telescopic struts, see Furuya [23] or controlling th e elastic ties, see Sultan [24]. Figure 1-2 shows the same principle but in th is case links are not rigid but rather are compliant. When the radius of cu rvature is changed, the whole stru cture is able to move in 3D following a complex path. Figure 1-2. Sequence of motions for the rising of the structure. Although the device seems feasible the manufacture of the requi red joints is very complex at the MEMS level, however it is possible to mo dify its constitutive elements to reach the same result in a simpler way. Before presenting the idea to be developed in this re search it is important to consider in more detail the require ments for the actuators and the joints. Bimorph Actuators The bi-layer electrothermal actuator combines two materials with different coefficients of thermal expansion ( ). The layers are joined along a co mmon interface and the entire device is heated. Since one material tries to expand more th an the other but is restrained by the joint with the second material, the entire structure bends, s ee Pelesko [25]. It is possible to extend and

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17 contract the beam by controlling the temperature of the beam via the use of a resistor embedded in the beam. The electrothermal actuators ha ve the advantages of low operation voltage, a simple fabrication process, and are CMOS-compa tible. Therefore, cont rol circuits can be integrated with the device on the same chip. A bimetallic actuator is illustrated in Figure 1-3. Figure 1-3. Configuration of a bimetallic actuator. If 2 1 the structure bends with an increase of temperature as shown in Figure 1-4A. If 2 1 the structure bends like in Figure 1-4B. It is usual that the bending of the beam take place out of the plane, but there is not any restriction to bend the beam in the plane. A B Figure 1-4. Bending of a bimetallic actuator. A)2 1 B)2 1 Elastic Joints The functionality of the device is intimately re lated to the elastic el ements located at the ends of the beams. The development of torsion springs at the mi crolevel has been ach ieved and presented by Hah [26]. However the developm ent of linear springs is less frequent. Regular-coiled carbon fibers have been obtained by Ya ng [27], using chemical procedures,. Also, the design of a vertical linear conical microspring attached to th e substrate is reported by Hata [28]. None of

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18 these ideas are appropriate for a 3D device and for the purpose of this work it is necessary to find an alternative. The decision about the material and the sh ape and process must include the following considerations: material w ith low Young modulus, applicable through spinning and be photodefinable, resistant to heat to avoid future complications due to the actuation of the beams, and compatible with the othe r processes involved in the te nsegrity based MEMS device. The Device Figure 1-5 shows a scheme of the device that was addressed in our research. It can be considered as a simplification of the tensegri ty system presented in Figure 1-2. The system maintained its shape due to the upward deflections of the beams. It was formed by three sets of bimorph actuators which transmitted their moti on to the central platform through compliant joints. The moving platform could be described by an equilateral triangle. The fixed ends of the actuators were distributed along the ve rtexes of an equilateral triangle. Figure 1-5. Scheme of the device. The position of the device is influenced by th e stiffness and free lengths of the ties, the location and nature of the joints, and the length and the current curvature of the beams. The presence of elastic elements increases the comple xity of the mathematical model that describes the relations between internal forces and the positions of the beams.

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19 CHAPTER 2 FORWARD ANALYSIS Figure 2-1 depicts the device in a general positio n. In the forward analysis the location of points iQ with respect to a global reference system ar e given and the objective is to evaluate the coordinates of pointsiP with respect to the global system. Despite the simplicity of the mechanism, the answer to this question is not trivial due to the presence of the compliant elements. To simplify the problem the following assumptions are made: The moving platform is massless. The stiffness of the compliant elements are lin ear and they are the same for all the springs. Deflections of actuators due to the spring forces are minimal and they do not affect the motion of the platform. Figure 2-1. Device in a general position. The solution can be performed using a Newtonian approach or energy approach. Newtonian is preferred here because it gives a better understanding of the geometry of the system.

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20 Forces Acting on the Platform To begin it is important to recall two basi c concepts from vector algebra. The n vectors nu u u ... ,2 1 are said to be linearly dependent if there exist n real numbers n ... ,2 1 not all zero such that, see Brand [29] 0 ...2 2 1 1 n nu u u (2-1) The other important concept is this: a necessa ry and sufficient condition that three vectors be linearly dependent is that they be coplanar. Figure 2-2 shows the forces acting on the platform in a general position. Since the platform is massless, the eq uilibrium of forces yields 03 2 1 F F F (2-2) Figure 2-2. Arbitrary forces acting on the platform. Equation 2-2 can be expressed more convenientl y in terms of the magnitude and direction of each force as 03 3 2 2 1 1 s f s f s f (2-3) where is: unit vector from iP to iQ if : is the magnitude of the force in each spring.

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21 Since the springs are linear, each force magnitude in Equation 2-4 can be expressed as a function of its stiffness and its deformation as follows 03 0 3 2 0 2 1 0 1 s d d k s d d k s d d k (2-4) where id : actual length of the springs 0d : free length of the springs When the platform is working, th e current lengths are always gr eater than the free lengths, and then the coefficients in Equation 2-4 are di fferent from zero. From E quation 2-1 it is clear that vectors 2 1, s s and 3s are linearly dependent. In addition, since they are linearly dependant, they are also coplanar. From the definition of is, this result implies that despite the space motion of the platform, points 3 2 1 3 2 1, , P P P Q Q Q belong to the same plane and Equation 2-4 can be presented as 03 3 3 2 2 2 1 1 1 y x y x y xs s f s s f s s f (2-5) where ix ixs s ,: rectangular components of the unit vectors is expressed in terms of a coordinate system whose z axis is normal to the plane The moment of the force s fis a vector perpendicular to the plane of the forces and whose magnitude is magnitude isp f (Figure 2-3), where p is the perpendicular distance between an arbitrary point V and the line of action of force s f. Equilibrium of forces establishes that summation of moments with resp ect to the arbitrary point V must be zero, then for the forces acting on the moving platform 03 3 2 2 1 1 p f p f p f (2-6) Equation 2-6 can be combined with E quation 2-5 to obtain, see Duffy [30]

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22 Figure 2-3. Moment of a force. 03 2 1 3 2 1 3 2 1 3 2 1 f f f p p p s s s s s sy y y x x x (2-7) Nontrivial solutions for if requires that, 03 2 1 3 2 1 3 2 1 p p p s s s s s sy y y x x x (2-8) This situation occurs when the forces are conc urrent or parallel. For the configuration of the current device it is not possible for the forces to be parallel and therefore they must meet in a point. Before leaving this section, another interesting fact is that from E quation 2-4, the stiffness of the springs vanish since they are assumed to a ll have the same stiffness value, k, and therefore knowledge of their actual values is not necessary for purposes of obtaining the equilibrium pose. Mathematical Model There are several ways to solve the forward an alysis problem, according with the selected variables. Figure 2-4 depicts a scheme includi ng the variables and parameters used for this model.

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23 Figure 2-4. Nomenclature fo r the forward analysis. The nomenclature defined here will be used later in the reverse analysis. The elements presented in Figure 2-4 ha ve the following meaning: Coordinate system A : global reference system Coordinate system E: local refe rence system (origin at point Q1, point Q2 on x axis, and z axis perpendicular to plane) I : point of intersection of the line of ac tion of the forces acting on the platform iP : point that define the moving platform iQ : free end of the actuator i i ib a ,: coordinates of point iQ in the local system id : current length of the spring i i : distance between point iP and the intersection point I i : angle between id and the local x-axis : angle of rotation of the platform with respect to the local x-axis pL: length of a side of the equilateral platform

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24 : internal angle of the moving pl atform and therefore equal to 3 / Global system A may be located in any arbitrary pos ition. In this problem statement it is assumed that the coordinates of points 2 1, Q Q and 3Q are known in system A. With the knowledge of pointsiQ, the local system E is defined as follows 1 2 1 2Q Q Q Q xA A A A E A (2-9) 1 3 1 2 1 3 1 2Q Q Q Q Q Q Q Q zA A A A A A A A E A (2-10) E A E A E Ax z y (2-11) The transformation that relates systems A and E is given by Crane [31] 1 0 0 01Q R TA A E A E (2-12) where E A E A E A A Ez y x R (2-13) Coordinates of points iQ in the system E are given by) (i ib a. Since 1QE is the origin of system E, then 0 01 1 b a (2-14) Remaining coordinates 3 3 2 2, ,b a b a can be found from the relations 2 2 2 21 0Q T b a QA E A E (2-15) 3 3 3 31 0Q T b a QA E A E (2-16)

PAGE 25

25 where 1 T TA E E A Note that since 2Qis located on the xE axis and thus 02 b (2-17) The problem as depicted in Figure 2-4 involves the following ten unknowns 3 2 1 3 2 1 3 2 1, , d d d (2-18) From Equation 2-4 equilibr ium of forces evaluated in system E yields 0 sin cos sin cos sin cos3 3 0 3 2 2 0 2 1 1 0 1 d d d d d d. (2-19) Since the forces are concurrent, equili brium of moments doe s not give any new information. Further equations must be develope d based on the kinematics of the device. From Figure 2-4 it is clear that 2 12 1 i i p ie e L e (2-20) 3 13 1 i i p ie e L e (2-21) Loops defined by I Q Q 2 1and I Q Q 3 1 yield 2 12 2 1 2 1 1 i E E ie d Q Q e d 3 13 3 1 3 1 1 i E E ie d Q Q e d Considering Equations 2-14 and 2-17 the la st two equations can be simplified to 2 12 2 2 1 10 i ie d a e d (2-22) 3 13 3 3 3 1 1 i ie d b a e d (2-23)

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26 Scalar components of Equations 2-19 through 2-23 form a nonlinear system with ten equations that can be solved for the ten unknowns using numerical methods. A program to solve the mathematical model for the forward analysis was implemented. The program takes advantage of a function that implements the Newton-Ra phson method. Once the variables are found, points i AP are evaluated using the transformation i E A E i AP T P (2-24) where points i EP are given by (Figure 2-4) 1 1 1 1 1sin cos d Q PE E (2-25) sin cos1 2p E EL P P (2-26) sin cos1 3p E EL P P (2-27) Numerical Example A numerical example is provided to demonstrat e the mathematical model. To simplify the presentation of the numerical data it is understood that lengths ar e given in consistent units and angles in radians. Find the coordinates of points iP for equilibrium given the free lengths of the ties 200 d, and the coordinates of points iQ (Figure 2-5) in a global reference system A 44 10 0 34 781QA, 63 64 31 91 72 522QA, 12 49 02 80 70 463QA The numerical solution of the system requires a guess for the initial values. An easy way to obtain them is from the device when it is in th e planar position. In that location springs are not stretched and the platform is not rotated yet, and therefore, point I coincides with the intersection of the heights of the platform (Figure 2-6), therefore

PAGE 27

27 Figure 2-5. Coordinates of the free ends of the actuators. 0 3 2 1d d d d 6 cos 3 23 2 1 pL 0 Initial values for are easily obtained from the geometry of the platform in its first position (Figure 2-6). 61 6 52 23 Figure 2-6. Initial position of the system.

PAGE 28

28 It is also necessary to evaluate variables 3 3 2, ,b a a which depend on values of i AQ. From Equations 2-9 through 2-13 the given values of i AQ yield 1 0 0 0 45 10 939 0 123 0 321 0 0 072 0 838 0 541 0 34 78 338 0 531 0 777 0TA E When the transformation 1T TA E E Ais evaluated and substituted into Equations 2-15 and 2-16 the terms a2, a3, and b3 are determined as 67 1682 a, 87 653aand09 1383 b. Now the Newton-Raphson method can be implemen ted to solve system of Equations 2-19 through 2-23. The solution to the 10 unknowns yields 07 351d 93 681 389 01 065 0 49 412d 98 402 643 22 73 363d 48 663 272 13 Equations 2-25 and 2-27 permit one the evaluation of points i AP as 1 14 48 76 44 95 35 1 44 55 97 54 95 34 1 56 22 42 6 06 463 2 1P P PA A A One way to verify the validity of the result s is to check if they satisfy equilibrium equations and if the lines of action of the forces intersect at the same point, when they are evaluated in the global system A, instead of the local system E. The equilibrium condition in the gl obal system can be written as 3 0 3 2 0 2 1 0 1s d d k s d d k s d d k F (2-28) where

PAGE 29

29 1 1 1 1 1P Q P Q sA A A A (2-29) 2 2 2 2 2P Q P Q sA A A A (2-30) 3 3 3 3 3P Q P Q sA A A A (2-31) The intersection point of th e lines passing through points 1 1Q P and 2 2Q P is given by (Crane, C., Rico, J., Duffy, J., Screw Theory for Spatial Robot Manipulators, Cambridge University Press, In Preparation) 2 2 1 2 2 01 1 02 1 2 1 02 2 121 s s s s s s s s s s s s rA (2-32) Similarly, the intersection of lines passing through 2 2Q P and 3 3Q P is given by 2 3 2 3 3 02 2 03 2 3 2 03 3 231 s s s s s s s s s s s s rA (2-33) where 1 1 01s Q sA (2-34) 2 2 02s Q sA (2-35) 3 3 03s Q sA (2-36) Substituting values of i AP and i AQ into Equations 2-28 through 2-36 yields 410 08 0 07 0 22 0 k F 36 46 06 19 39 1712rA and 36 46 06 19 39 1723rA

PAGE 30

30 It is clear that the solution satisfies the equi librium condition and that the lines of action of the forces intersect at the same point. Figure 2-7 displays the device in the evaluated equilibrium position. Figure 2-7. Device in the ev aluated equilibrium position.

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31 CHAPTER 3 REVERSE ANALYSIS In the reverse analysis the objective is to fi nd the location of the act uators in order to obtain a desired output. Since ther e is not an external wrench, it is not possible to achieve an arbitrary location and orientation of the platform, however it is f easible to constrain the moving platform to be perpendicular to a given vector n (Figure 3-1). The eval uation of the actuator positions required to reach the desired orientation is not intuitive for this kind of mechanisms and a mathematical model is necessary. The revers e problem for this device admits different formulations, considering which parameters ar e considered as given and which must be evaluated. A B Figure 3-1. Normal vector to the moving platform. A) Isom etric view. B) Lateral view. Replacement of Compliant Beams The motion of the free end of th e actuator is the result of bend ing the bimorph beam due to the increase of temperature, which is in turn, a function of the thermal resistance and the applied voltage. Figure 3-2A shows the path of the fr ee end for several positions of the beam. Lowell [32] has been shown that for the purpose of an alysis, compliant elements can be replaced by hypothetical rigid binary links. Figu re 3-2B shows how the original path of the free end can be approximated for a link whose center lies on the horizontal axis and with a radius t forming an

PAGE 32

32 angle with the horizontal. The path of the free end may be obtained experimentally and the center and radius of the hypothetical link adjusted by fitting the curve. Figure 3-2. Path of the free end. A) Original path. B) Approximated path. Since all the beams are equal, the radius t is equal for all the actuators and the location of E with respect to G is also the same for all the actuators. In the following developments it will be assumed that points iEand radius t are already evaluated. Reverse Analysis Case 1 This case may be stated as follows: Given: The position of the free end of one of the actuators. A unit vector perpendicular to the moving platform. Find: The position of the free ends of the remaining actuators. Figure 3-3 shows the plane that contains the moving platform and the actuators represented as binary links. Unit vector n is perpendicular to this plane and positions of points iQ can be defined by the vectors ir in a global reference system.

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33 Figure 3-3. Parameters for the reverse analysis, case 1. One sequence of transformations that relate s the global system and any of the local systems located at the fixed pivots of the binary links and whose x-ax is are aligned with the axis of the binary links (Figure 3-3), is ) ( ) ( ) (i i i A Ly Rotation z Rotation E n Translatio T 1 0 0 0 0 cos 0 sin 0 0 1 0 0 sin 0 cos 1 0 0 0 1 0 0 0 cos sin 0 sin cos 1 0 0 0 0 1 0 0 0 1 0 0 0 1i i i i i i i i iy ix A LE E T (3-1) 1 0 0 0 0 cos 0 sin sin sin cos cos sin sin cos sin cos cos1i i iy i i i i ix i i i i i A LE E T (3-2) When i=1, 2, 3 references systems B, C and D are obtained. Figure 3-4a shows a top view when only the first two transformations of E quation 3-1 are carried out. Note that angles i are constant. Figure 3-4b illustrates the local reference systems in their final orientation after performing the last transformation involving in Equation 3-1.

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34 A B Figure 3.4. Location of the local reference systems for the reverse analysis. A) First rotation. B) Second rotation. The first 3 elements of the first column of Equation 3-2 represent the local x-axis expressed in the global system A. In particular the local axis C Axis obtained by substituting i=2 in Equation 3-2 as

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35 2 2 2 2 2sin cos sin cos cos C Ax (3-3) Without lost of generality assume that the free end whose position is given as 1QA, is known. The vector 1r is then also known. From the equation of a plane (Crane, C., Rico, J., Duffy, J., Screw Theory for Spatial Robot Ma nipulators, Cambridge University Press, In Preparation), and Figure 3-3 n r n r n r r 1 2 1 20 (3-4) From the geometry of the device (Figure 3-3) 2 2 2t E r (3-5) From Figure 3-4b and co nsidering Equation 3-3 2 2 2 2 2 2sin cos sin cos cos t x t tC A (3-6) The scalar product of Equation 3-5 with nyields n t n E n r 2 2 2 (3-7) Substituting Equations 3-4 and 3-6 into Equation 3-7 yields 2 2 2 2 2 2 1sin cos sin cos cos z y xn n n t n E n r (3-8) Regrouping Equation 3-8 yields 0 sin cos2 2 2 2 2 D B A (3-9) where 2 2 2sin cos y xn n A (3-10) zn B 2 (3-11)

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36 t n E n r D 2 1 2 (3-12) It is possible to obtai n a closed solution for 2 in Equation 3-9, See Crane [31]. Substituting the value of 2 in Equations 3-6 and 3-5, the coordinates of 2r, and therefore of point 2QA, are determined Similarly, from Figure 3-3 n r n r n r r 1 3 1 30 (3-13) From the geometry of the device (Figure 3-3) 3 3 3t E r (3-14) where D Ax t t 3 (3-15) Unit vector D Ax is obtained from the first three terms of the first column of the matrix defined in Equation 3-2 when i=3 and thus (3-15) may be written as 3 3 3 3 3 3sin cos sin cos cos t t (3-16) The scalar product of Equation 3-14 with n yields n t n E n r 3 3 3 (3-17) Substituting Equations 3-13 and 3-16 into Equation 3-17 yields 3 3 3 3 3 3 1sin cos sin cos cos z y xn n n t n E n r (3-18) Regrouping Equation 3-18 yields 0 sin cos3 3 3 3 3 D B A (3-19) where

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373 3 3sin cos y xn n A (3-20) zn B 3 (3-21) t n E n r D 3 1 3 (3-22) Equation 3-19 permits one to evaluate3 then Equations 3-16 and 3-15 yield 3r and therefore 3QA. The reverse analysis for this case is completed. Numerical Example A numerical example is provided to demons trate the solution process for the reverse analysis, case 1. Angles are in radi ans and lengths in consistent units. Given: the position of point 1Q(as defined by the elevation of rigid link 1 6109 01 ), the length of a side of the moving platform 23 pL, the free lengths of the springs 50d, the length of the binary rigid links 26 t and the unit normal vector to the platform Tn9659 0 2588 0 0 expressed in the global referen ce system. Find the coordinates of points 2Q, ,3Q 1P, 2Pand 3Pexpressed in the global system. From Figure 3-4, 3 /2 and3 /3 Points 1E 2Eand 3Eare evaluated with the aid of Figure 3-5 which shows the device when it is at the plane level. 0 0 279 44 0 ) sin( ) cos( ) 6 / cos( 3 20 1 t d L Ep 0 34 38 139 22 0 ) 3 / sin( ) 3 / cos( ) 6 / cos( 3 20 2 t d L Ep 0 34 38 139 22 0 ) 3 / sin( ) 3 / cos( ) 6 / cos( 3 20 3 t d L Ep

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38 Figure 3-5. Distribu tions of points E. From Figure 3-4, coordinates of 1Qare given by the vector 1r as follows 9130 14 0 981 22 sin cos sin cos cos1 1 1 1 1 1 1 1 1 t E t E r From the given data 2588 0 0 y xn nand 9659 0zn Now it is possible to evaluate coefficients (3-10) through (3-1 2) and (3-20) through (3-22). Substituting the coefficients into Equations 3-10 and 3-19 yields 0 9358 0 sin 9659 0 cos 2241 02 2 0 1723 0 sin 9659 0 cos 2241 03 3 These last equations yield two sets of solutions for 2 and 3 that will yield equilibrium configurations. Each pair is selected consider ing the equilibrium conditions. The solutions are 4027 0 0056 13 2 a a

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39 195 3 68 13 2 b b Equations 3-5 and 3-6 permit one to evaluate points 2Qand 3Qfor both solutions of 2 and 3 and then, following the procedure presented in th e forward analysis, it is possible to evaluate points 1P, 2Pand 3P. The results are summarized in Table 3-1. Table 3-1 Solution for the reverse analysis, case 1. Solution a Solution b x y z x y z 1Q -22.98 0 14.91 -22.98 0 14.91 2Q 15.18 -26.28 21.95 23.56 -40.80 25.85 3Q 10.18 17.63 10.19 35.12 60.83 -1.39 1P -12.16 -3.57 15.87 -0.54 7.29 12.96 2P 7.72 -14.73 18.86 19.36 -3.84 15.94 3P 7.79 7.48 12.90 19.39 18.38 9.99 Results are shown in Figure 3-6. The second so lution is also an equilibrium position, but the current device cannot reach that position. A B Figure 3-6. Solution for the reverse analysis case 1. A) Solution a. B) Solution b. Reverse Analysis Case 2 One could desire to include information about the location of points in the moving platform in the reverse analysis. Since there are no external forces acting on the

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40 z y xn n nmechanism, it is not possible to specif y a general position for one point of the platform. However it is feasible to specify in ad dition to the orientation of the platform given by the vector perpendicular to its plane, the height w ith respect to the horizontal plane of one of the points of the platform. Any point is equa lly appropriate, for this case the point 1PA is selected (Figure 3-7). A B Figure 3-7. Prescribed vert ical component of point P1. A) Isometric view. B) Lateral view. In case 1 it was possible to obt ain a closed solution easily be cause it did not involve any information regarding the location of the points on the moving platform. In the new situation, the mathematics are more involved and requires a numerical technique for its solution. Figure 3-8. Nomenclature for the reverse analysis, case 2. A) Isometric view. B) Plane of the forces.

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41 Figure 3-8a shows the device in an arbitrary position. Figu re 3-8b shows the variables located on the plane of the moving platform. Positions of points i AQ are unknown and depend on anglesi Angle is also an unknown as well as the x and y coordinates of point 1PA (coordinates of 2PA and 3PA can be found once the model is solv ed). Therefore, in addition to the 10 variables used in the forward analysis an d enumerated in (2.18), here there are 6 new unknowns: xP1 3 2 1, andyP1. The solution requires 16 equations. The reverse analysis for this case may be posed as follows Given: n: normal vector perpendicula r to the moving platform. z AP1: scalar component z of the vector i AP. Find: 1QA, 2QA, 3QA: location of the free ends of the binary links. y xP P1 1,: scalar components x and y of the vector i AP. 2PA, 3PA: location of the vertexes of the moving platform. Points i AQ depend on i and using transformation (3-2 ) they can be expressed as 1 0 0 0 0 cos 0 sin sin sin cos cos sin sin cos sin cos cos 1 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y x A B A B AE E T t T Q (3-23) 1 0 0 0 0 cos 0 sin sin sin cos cos sin sin cos sin cos cos 1 0 02 2 2 2 2 2 2 2 2 2 2 2 2 2 2 y x A C A C AE E T t T Q (3-24)

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42 1 0 0 0 0 cos 0 sin sin sin cos cos sin sin cos sin cos cos 1 0 03 3 3 3 3 3 3 3 3 3 3 3 3 3 3 y x A D A D AE E T t T Q (3-25) Equations 3-23 through 3-25 simplify to 1 sin cos sin cos cos1 1 1 1 1 1 1 1 t E t E t Qy x A (3-26) 1 sin cos sin cos cos2 2 2 2 2 2 2 2 t E t E t Qy x A (3-27) 1 sin cos sin cos cos3 3 3 3 3 3 3 3 t E t E t Qy x A (3-28) Expressions for distances j iQ Q between points i AQand j AQ can be obtained from Equations 3-26 through 3-28 as follows 1 2 2 1Q Q Q QA A (3-29) 2 3 3 2Q Q Q QA A (3-30) 1 3 3 1Q Q Q QA A (3-31) The relation between 1PE and 1PA is given by 1 1P T PE A E A (3-32) Transformation TA Edefines the relation between the global system A and a reference system E which origin is located at 1QAwith its x-axis points from 1QA to 2QA, and for which the z-axis is the unit vector n (Figure 3-8B), therefore

PAGE 43

43 1 0 0 01Q R TA A E A E (3-33) where E A E A E A A Ez y x R (3-34) where 1 2 1 2Q Q Q Q xA A A A E A (3-35) n zE A (3-36) E A E A E Ax z y (3-37) Coordinates of 1PE can be obtained from Figure 3-8b, and Equation 3-32 can be expressed as 1 0 sin cos 1 0 0 0 11 1 1 1 1 33 32 31 1 23 22 21 1 13 12 11 1 1 1 d d Q r r r Q r r r Q r r r P P Pz A y A x A z A y A x A (3-38) where the terms ijr depend only on i At this point all the developments required for the mathematical model are obtained. Expression (3-38) yields 3 s calar equations, one of them involving the prescribed valuez AP1, then x A x AQ d r d r P1 1 1 12 1 1 11 1sin cos (3-39) y A y AQ d r d r P1 1 1 22 1 1 21 1sin cos (3-40) z A z AQ d r d r P1 1 1 32 1 1 31 1sin cos (3-41) The angle in Figure 3-8b can be related to points i AQ using the cosine law cos 23 1 2 1 2 3 1 2 2 1 2 3 2 Q Q Q Q Q Q Q Q Q Q (3-42) where the terms j iQ Q are given by Equations 3-29 through 3-31. Equilibrium conditions can be expressed in the plane of the moving platform as it was done in the forward analysis: 0 cos cos cos3 0 3 2 0 2 1 0 1 d d d d d d (3-43)

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44 0 sin sin sin3 0 3 2 0 2 1 0 1 d d d d d d (3-44) The geometry of the system involving relations for the moving platform is the same as found in the forward an alysis (Figure 3-8B) 2 12 1 i i p ie e L e (3-45) 3 13 1 i i p ie e L e (3-46) Geometry relations for the actual lengths of the springs involve the terms 3 1 2 1,Q Q Q Q and (Figure 3-8B) 2 12 2 0 2 1 1 1 i i ie d e Q Q e d (3-47) 3 13 3 3 1 1 1 i i ie d e Q Q e d (3-48) Points 2QA and 3QA with respect to the location of point 1QA must be perpendicular to vector n. To assure that, two mo re relations are required 01 2 n Q QA A (3-49) 01 3 n Q QA A (3-50) Equations 3-41 through 3-50 form a system of 14 equations and 14 unknowns that can be solved for 3 2 1 3 2 1 3 2 1 3 2 1, , , d d d and Appendix A presents the set of equations in extended form. Once the solution is obtained, it is possible to evaluate i AQ using Equations 3-26 through (3-28). Coordinates y A x AP P1 1, are easily evaluated using E quations 3-39 and 3-40 which determines point 1PA. To complete the reverse analysis for the current case it is necessary to evaluate 2PA and 3PA. A coordinate system F is defined as parallel to system E and located at 1PA, then

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45 1 0 0 01P R TA A E A F (3-51) The rotation matrix RA E is given by Equation 3-34 and w ith the aid of Figure 3-8B the following relations are obtained 1 0 sin cos ,2 2 2 p p F F A F AL L P P T P (3-52) 1 0 sin cos ,3 3 3 p p F F A F AL L P P T P (3-53) Numerical Example Given the following parameters and the prescrib ed values for the device where lengths are in consistent units a nd angles in radians 3 / 3 / 3 / 150 200 303 2 1 0 pL t d 9848 0 1116 0 1330 0 n 641z AP Find points i AQ and i AP. Figure 3-9 shows the device in its initial positio n. This position permits the evaluation of the initial values required for the numeri cal solution and the c oordinates of points i AE. Lines

PAGE 46

46 through i APi AQ intersect at the point of intersection of the heights of the equilateral triangle, there is no stretching and no ro tation of the platform, then 0 0 3 0 2 0 1d d d d 6 cos 3 20 3 0 2 0 1 pL 00 Figure 3-9. Device in its initial position. Since all the links are on the horizontal plane 00 1 00 2 00 3 From the geometry of the platform in its first position 60 1 6 50 2 20 3 30 correspond to the angle between points2 1 3Q Q Q. T g AR E0 0 11 T g AR E0 ) 6 / sin( ) 6 / cos(2 T g AR E0 ) 6 / sin( ) 6 / cos(3

PAGE 47

47 where t d Rg 0 0 1 The solution to system of equati ons given by 3-41 through 3-50 yields 3530 01 4832 01 0150 0 2841 02 7090 22 0122 1 1606 03 6673 13 52 391 d 70 761 44 382 d 38 902 00 383 d 55 933 Equations 3-26 through 3-28 yield 1 98 31 20 102 58 59 1 54 55 79 107 23 62 1 14 69 0 1283 2 1Q Q QA A A Equations 3-39 and 3-40 permit one to evaluate the remaining coordinatesx AP1,y AP1. As a result 1PAis defined completely. 64 28 1 76 891PA Finally, Equations 3-51, 3-52 a nd 3-53 yield the values for 2PA and 3PA 1 34 38 30 72 40 38 1 08 55 75 76 55 393 2P PA A Figure 3-10 illustrates the device in its final position. For verification of the results equilibrium condition in the global system and the point of inters ection of forces are evaluated using equations (2.28) through (2. 36). It results yield that summa tion of forces is zero and the points of intersection of forces are identical, with a ma ximum deviation of910 1.

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48 Figure 3-10. Device for the exampl e of reverse analysis, case 2.

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49 CHAPTER 4 DEVICE DESIGN Once the kinematics of the system are understood the next step is to design the device. This includes the selection of the dimensions and materials for all the elements to obtain a reliable platform. The critical elements in the de vice are the actuators and the sp rings that join the actuators to the moving platform. Their design requires as sumptions to simplify th e analysis, procedures and analysis techniques from th e area of strength of materials, and information from previous experiences for the issues where analytical a pproaches cannot be implemented. The forward analysis that was previously presented will be used to obtain the maximum deflections associated with the springs. Actuator Design Bimorph beams can provide motion out of the pl ane when they are heated if the difference between the thermal expansion coefficients of the material forming the beams is significant. Also, since the kinematic model assumes that the links are rigid, it is important to limit the deflections at the end of the beams that would result from the forces applied at the end of the beam by the spring element. Materials A pair of materials that can be used for the beams are aluminum and silicone dioxide. Table 4-1 summarizes from Senturia [33], some material properties for the beams. There is an appreciable difference in their coefficients of th ermal expansion. In addition these materials are very common in MEMS microfabrication, theref ore there are well established procedures for their deposition and etching processes.

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50 Table 4-1 Mechanical and thermal propert ies for aluminum and silicone dioxide. Material Young modulus, E MPa Thermal exp. coeff, 610 1K 2O Si 70000 0.7 Al 69000 23.1 Width of the Beam For details associated with the manufacturing process, it is conve nient to avoid large values for the width of the beam to make it easie r to release the beam using isotropic etching. A width of 7 m is recommended. Deflection and Length of the Beams Deflection of the end of the beam is strongly influenced for the length of the beam. For an initial estimation of the deflection it can be assu med that the beams are made of one material. This assumption is justified in the fact that for both aluminum and SiO2, their Youngs modulus is almost the same (Table 4-1). If it is assu med that the beams are straight, from strength of materials the deflection produced by a force at th e free end of a cantilever beam, see Boresi [34] and Figure 4-1, is given by a a aF EI L 33 where 312 1 wt I then 3 34a a aL Ewt F (4-1) where a : deflection at the free end of the beam aF : force applied at the free end of the beam aL : length of the beam E : Youngs modulus w : width of the beam t : thickness of the beam

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51 Figure 4-1. Maximum deflec tion of a cantilever beam. Equation 4-1 can be solved for several length, width and thickness of the actuators (Figure 4-2). From the point of view of manufacturing, length of the b eams is not a constraint, and the larger beams, the higher motion out of the plane, but at the same time to avoid that deflections increase dramatically, the thickness must increase. Large thickness are difficult to obtain therefore they are limited by the manufacturing pr ocess. Guided for these reasons the length of the beam is selected as 200 m and the total th ickness (this is aluminum and silicon dioxide) as 3 m, then from Equation 4-1 and with w =7 m m N Fa a 41 0 The current selection establishes that if the maximum deflection is limited to 1 m, then the vertical force acting in each beam is 41 0 aF N. If each actuator ha s 12 beams, then the maximum perpendicular force that ca n be applied to the system is 5 12 a pF F N (4-2) Out-of-Plane Elevation The design of the spring requi res knowing its maximum deflection and one of the factors that influences this parameter is the eleva tion of the free end of the beam. The maximum deformation of the spring is obtained when the beams reach their maximum height (Figure 43A). This situation occurs at th e end of the manufacturing process, when the beams are released from the substrate.

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52 Figure 4-2. Deflection of the free end for several conditions. From Figure 4-3B it is clear that the height of the free end is given by ) cos 1 ( zQ (4-3) where : radius of curvature and the angle of with the vertical. From Figure 4-3B aL (4-4) Substituting Equation 4-4 into Equation 4-3 yields cos 1 a zL Q (4-5)

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53 Figure 4-3. Maximum elevation of the free end of the beam. A) Isometric view. B) Lateral view. Equation 4-5 evidence s the dependence of zQon however the value of is difficult to obtain analytically. At the end of the manufacturing process a nd before releasing from the substrate, the beams are in the plane but intrinsi c stresses are present. They appear because the materials are deposited at a higher temperature and after the etching process the two materials cool to ambient temperature. Once they are releas ed from the substrate, the beam curves up to release the stresses (Figure 4-4). Figure 4-4. Bending after re lease from substrate. The initial value of the radius of curvature and therefore the initial value of angle depend on the geometry and material properties of the bimorph beam. Liu [35] presents the following equations that should pe rmit the evaluation of the initial radius of curvature of a bimorph beam.

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54 M E Ieff 0 (4-6) where 2 1 2 2 2 2 2 2 1 2 1 1 1 02 12 2 12z t t t t E z t t t E w E Ieff 2 2 1 1 2 2 2 1 1 1 1 1 1 2 11 1 1 2 t E t E t t E t w M 2 2 1 1 2 2 2 1 1 1 2 2 2 2 1 2 21 1 1 2 t E t E t t E t t t w 2 2 1 1 2 1 2 2 2 2 2 1 12 1 t E t E t t E t E t E z 0E Ieff: is the effective flexural stiffness M : bending moment required to bend the cantilever beam z : the position of the neutral axis t: thickness of the layers of the bimorph E: Youngs modulus w: width of the bimorphs : intrinsic stress in the layers after deposition : Poissons ratio for the layers and subindex 1 is for the material at the botto m and subindex 2 is for material on the top. If t t t 2 1and E E E 2 1, Equation 4-6 can be simplified to 1 1 2 21 1 3 8 E t (4-7) To be useful for numerical evaluation, Equa tion 4-7 requires the knowle dge of the intrinsic stresses 1 and2 They depend strongly on all the conditi ons for the manufacturing process and also on the thickness of the layers. For the same reason, very few values are referenced, and in the best of the cases only ranges of values vary ing from negative to positive can be obtained. In the absence of better informati on, Equation 4-7 cannot be applie d and the estimation of angle theta must be done from previous experiences. Xie [6] reports that from a beam of similar length an angle of 17 has been observed. For purposes of this design, angle is assumed to be 20.

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55 The exact value is not of interest as well as th e spring be able to yield the maximum deformation without excessive stress that may damage it. Springs Design The spring must be able to provide the ma ximum deformation required for the mechanism and stand the stresses generated by this deformati on. The first step it to evaluate the maximum elongation that the springs must provide. Springs Elongation When the device is on the horiz ontal plane the leng th of the springs correspond to the free length0d (Figure 4-5A). When beams reach the maxi mum elevation (Figure 4-5B), the length of the springs is maximum. Figure 4-5. Positions for minimu m and maximum deformation of th e springs. A) Initial position. B) Maximum deformation position. Following the procedures presented in chapter 2 it is possible to evaluate the position of the platform given the points Q, and therefore the maximu m elongation of the springs, 0 maxd d for several values of the parameters of the device. A set of results is presented in Figure 4-6 for a length of the actuator200 aL m, free length of the spring 800 d m, and the length of the side of the platform 120 pL m.

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56 Figure 4-6. Maximum deform ation of the springs. Values of 0d cannot be very small to avoid exaggerated stresses. For the current values ofaL pL 0d and for the already selected value of 20 for the maximum deformation is 4.0 m. This is the elongation that the spring must provide. Maximum Force Acting on the Spring Force acting on the spring must be limited. If this force is excessive its vertical components acting on the beams may generate a deflection larger than the design value. Figure 4-7 shows the components of the for ce acting on the spring. In Equation 4-2 the maximum admissible perpendicular force was selected as 5 pF N and then when sF is maximum 2 (4-8) cos cosp s s pF F F F (4-9)

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57 Figure 4-7. Maximum fo rce in the spring. For20 Equation 4-8 yields 80 and from Equation 4-9, 8 28 sF N. In summary, to get a maximum deformation of 4.0 m the maximum force cannot exceed 28.8 N, in order to keep the compon ent perpendicular of the force to GQ less than 5 N. Spring Geometry and Material The simplest geometry for the spring is a bar with rectangular section, similar to a common tie (Figure 4-8A). However, to obtain significant deflections with that geometry requires a material like rubber that is able to deform with low external forces. Some tests were performed in the laboratory using different kinds of silicone rubber. However due to the fact that the dimensions of the ties are very small compared with the other components of the device there was no evidence that the silicone filled the ch annels. Another undesirable aspect with this material is the difficulty to etch it with conventional plasma. Since ties are essential to this work it was necessary to look for other alternatives. A simple one is to change the geometry of the spring. If instead of a simple bar, a shape like the presented in Figure 4-8B is used, it is easier to achieve the required deformations.

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58 Figure 4-8. Possible geomet ries for the spring. A) Bar. B) By segments. Although it is possible to create the spring using the same materials for the beam, this is aluminum and silicone dioxide, their stiffne ss is still high. One alternative is to use photodefinable polymides. They have low modulus and in addition can be patterned easily, which is a very important advantage. Table 42 shows some properties of the polyimide HD8000 from HD MicroSystems. For the following an alysis they are considered as isotropic materials. Table 4-2 Properties of polymide HD-8000. Viscosity St Thickness m Cure C Tensile MPa Modulus MPa 3.5 3 to 5 350 122 2500 Figure 4-9 shows the geomet rical parameters for a segment of the spring. If T and H are given, angle and length CL can be evaluated as follows T L rc sin cos 2 (4-10) H L rc sin sin 2 (4-11) Squaring Equations 4-10 and 4-11 and adding the results yields 2 2 2 24H T L rc 2 2 24 r H T Lc (4-12) Since CL must be positive, Equation 4-12 shows that the selection of T and H must fulfill the relation 2 2 24 r H T (4-13)

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59 Figure 4-9. Geometry of a segment of the spring. If Equation 4-10 is multiplied by sin and Equation 4-11 is multiplied by cos then sin sin cos sin 22T L rc (4-14) cos cos cos sin 22H L rc (4-15) Adding Equations 414 and 4-15 yields 0 sin cos cL T H (4-16) Equation 4-16 can be solved for using a trigonometric method. Spring Stress Since the geometry is simple, it is interesting to find analytical relations for the stress and the deflection of one segmen t of the spring (Figure 4-10) The maximum tensile stress occurs at point B (Figure 4-10). Figure 4-11 presents the nomenclature to derive the expressions for the circumferential stress in a curved beam follo wing the procedure explained by Boresi [34] A RA AR A R A M A Fm i m i x s (4-17) where tb A (4-18) R H F Ms x2 (4-19)

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60 Figure 4-10. Segment of the spring. 2 b R Ri (4-20) b R b R t Am 2 2 ln (4-21) Figure 4-11. Parameters for the stress analysis of a spring. Spring Deflection For the element shown in Figure 4-10 and 4-11, the total deflection can be considered as the superposition due to the de flection of the linear segment and the deflection due to the

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61 curvilinear segment. Moreover, when the relation R/b >2, the effect of shear and normal forces can be neglected. For this case the deflection of the straight elementr is given by z s Ld F M EI Mc 0 22 (4-22) where cos z F Ms (4-23) From Equations 4-22 and 4-23 3 2 312 1 cos 3 2 tb I EI L Fc s r (4-24) The deflection of the curvilinear segment, c is given by w x x cRda F M E M (4-25) where a R R L F Mc s x sin sin cos (4-26) From Equations 4-25 and 4-26 da a R R L EI PRc2sin sin cos (4-27) Total deflection per segment n is just the superposition of r andc c r n (4-28) For n segments the total deflection is n sn (4-29) Spring Dimensions Expressions found for the stress and deflection of the spring ca n be evaluated for different values of the parameters. Figure 4-12 presents the results for the deflection of the spring when 10 sF N, T =28 m and t =3 m. From then is clear that for the parameters H =50 m, b =7 m and R =12 m, a force of 10 N, just a third of th e admissible force, is enough to deflect one

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62 segment of the spring 2 m. If 3 elements are us ed to create the spring, the resultant deflection provides more than the required deflection of 4 m. Figure 4-12. Deflection in the spring. Figure 4-13. Stress in the spring.

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63 Similarly, Figure 4-13 presents the results for the stress for the same conditions of Figure 4-11. It is clear from the hi ghlighted value, that the stress is only 19 MPa compared to the admissible value of 122 MPa (Table 4-2). Dimensions found yield a conservative and reliable device. Tabl e 4-3 summarizes the results. Some of them are not critical and their calculations were not included. Table 4-3 Main dimensions of the device. Parameters Beams Parameters Spring Length, aL: 200 m Width, w : 7 m Thickness of each layer, t : 1.5 m Number of beams: 12 Material beams: aluminum and silicone dioxide Width, b : 7 m Thickness, 2t : 3 m Free length,0d: 80 m H : 50 m T : 28 m R : 11.5 m Parameters Platform Parameters resistor pL: 120 m Material: chrome Width: 5 m Thickness: 0.2 m

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64 CHAPTER 5 MANUFACTURING PROCESS This chapter presents a sequence to manufact ure the device. The device is formed by three materials: silicone dioxide and aluminum for th e bimorph beams and between them chrome for the resistor. Processes are common and they do not involve any strange requirements. Some experiments were performed at the University of Florida nanofacilitie s. The manufacturing process presented here takes advantage of that experience, which is described in Appendix B. Five masks are used to define all the features of the system. In the following figures depth dimensions are magnified to assist in the visualization of the ge ometry of each step. The substrate for the device is silicon. The wa fer does not require any particular electrical or mechanical properties, and the crystal orient ation does not affect the process (Figure 5-1). Figure 5-1. Silicone substrate. Figure 5-2. First layer of silicone dioxide.

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65 A first layer of 0.80 m of silicone dioxide is deposited over the wafer using plasma enhanced chemical vapor depositi on (PECVD) process (Figure 5-2). Then a layer 0.20 m of chrome is sputtered over the first silicone di oxide layer (Figure 53). Figure 5-3. Layer of chrome. Using a mask, the chrome is patterned and then plasma etching is used to obtain the shape of the resistor (Figure 5-4). A B Figure 5-4. Resistor. A) General view. B) Detail. A second layer of silicone dioxide with thic kness 0.70 m is applied using PECVD (Figure 5-5A). As a result the resistor is isolated, however it is necessary to open a via to be able to apply voltage (Figure 5-5B).

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66 A B Figure 5-5. Second layer of silicone dioxide. A) General view. B) Detail. Aluminum with thikness1.5 m is sputtere d to complete the materials for the device (Figure 5-6A). To be able to apply voltage to the resistors it is necessary to create isolated areas called the pads. For this purpose an additional mask is required (Figure 5-6B). A B Figure 5-6. Aluminum layer. A) General view. B) Detail. The next step is to obtain the shape of th e cantilever beams and the moving platform. A new mask is required to avoid damages in the al ready created pads. Figu re 5-7 illustrates the result when the aluminum has been removed and th e second layer of silicone dioxide is exposed. A B Figure 5-7. Etching of areas in the aluminum corresponding to the actuators and platform. A) General view. B) Detail.

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67 The photoresist required for this process is sti ll over the aluminum, but it is not presented to simplify the visualization. Between the central platform and the actuators appear some free areas that will be used for the joints. A new etching process, but the same mask, is required to remove both layers of silicone dioxide. The process stops when the substrate is reached. At th is moment the photoresist is stripped (Figure 5-8). Figure 5-8. Etching of silicone dioxide Springs are created in the next two steps. A layer of polymid e is spun on the wafer (Figure 5-9). Figure 5-9. Polymide layer. With a mask, the photodefinable polymide is patterned to crea te the springs. After curing and removing remaining material (Figure 5-10).

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68 A B Figure 5-10. Polymide springs. A) General view. B) Detail. The main elements of the mechanisms are r eady. The objective of th e next steps is to release the beams, springs and platform. A backside etch on the substrate is illustrated in Figure 5-11. This step determines the depth of the platform. A B Figure 5-11. Backside etch. A) General view. B) Detail. The process continues in the fr ont side. Figure 5-12 shows a s ection view of the wafer and the detail of the substrate under a spring. Th e substrate material must be removed. A B Figure 5-12. Section view of the de vice. A) General view. B) Detail.

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69 The exposed parts of the substrate are removed using deep reactive ion plasma etch (Figure 5-13). Since the etching process is much more aggressive on silicone than on aluminum or polymide, a new mask is not required. At this moment all the elements are still attached to the wafer by small portions of silicone. A B Figure 5-13. Deep reactive ion etch ing. A) General view. B) Detail. Finally isotropic etching is us ed to remove the portions under the beams and spring (Figure 5-14). Some undercut is also pr esent in the platform, but it doe s not affect its strength. A B Figure 5-14. Isotropic etch. A) General view. B) Detail.

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70 CHAPTER 6 CONCLUSIONS Our research addresses the study of a devi ce formed by compliant links and compliant joints. At the MEMS level the st udy of devices with compliant links has been studied extensively but the configuration proposed in this research is a different approach to the design of MEMS devices. Forward and reverse kinematic analyses were performed. They provide the basis for control of the platform since they permit determ ination of the position of the system or to know the inputs to get a desired output. The manufacture process suggest ed is feasible. Dimensions for the components of the mechanism assure a reliable system. The reverse analysis shows that three actuators provide limited mobility to the platform. It is possible to increase the mobility adding more actuators, which requires a new mathematical model, however the basis of the manufacturing process remains the same. All the information required fo r the manufacturing of the de vice is provided and therefore it is straightforward to create the masks requi red for the construction of the mechanism. Experimental evaluation of the final results will provide insight about how to enhance the design. Compliant joints offers interest possibilitie s at the MEMS level since rigid joints are difficult to implement. They can be used to create not only simple platforms, but also to implement more complex devices such arrays of act uators able to move in the plane or in the space. In any case, the required kinematics is a comp lex topic and issues like that associated with closed solutions are still open. There is a larg e field of applications related with kinematics theory of devices intended to work at the MEMS level.

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71 Achievement of new devices requires both th eoretical and experimental work. It is necessary to explore materials that permit better manufacturing and performance of the compliant joints. The analysis presented in this research is basically a static analysis. Future works should include dynamic behavior which is essent ial to the vibration and control issues. This work was the first of its kind in the Center for Intelligent Machines and Robotics (CIMAR) lab at the University of Florida. It shows a way to involve the extensive knowledge acquired in the design of mechanisms to the MEMS level.

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72 APPENDIX A REVERSE ANALYSIS EQUATIONS This appendix presents the l ong form of the equations derived for the second case of the reverse analysis. F1=(d1-d0)*cos(psi1)+(d2-d0)*c os(psi2)+(d3-d0)*cos(psi3) F2=(d1-d0)*sin(psi1)+(d2-d0)*s in(psi2)+(d3-d0)*sin(psi3) F3=(d1+delta1)*cos(psi1)-( d2+delta2)*cos(psi2)-((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*t-E2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*t-E2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2) F4=(d1+delta1)*sin(psi1)-(d2+delta2)*sin(psi2) F5=(d1+delta1)*cos(psi1)+1/2*((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma3)*cos(theta3)*t-E3x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma3 )*cos(theta3)*t-E3y)^2+(sin(theta1)*tsin(theta3)*t)^2)^(1/2)-(d3+delta3)*cos(psi3) F6=(d1+delta1)*sin(psi1)+1/2*((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma3)*cos(theta3)*t-E3x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma3 )*cos(theta3)*t-E3y)^2+(sin(theta1)*tsin(theta3)*t)^2)^(1/2)*3^(1/2)-(d3+delta3)*sin(psi3) F7=delta1*cos(psi1)-Lp*co s(beta)-delta2*cos(psi2) F8=delta1*sin(psi1)-Lp*sin( beta)-delta2*sin(psi2) F9=delta1*cos(psi1)-Lp*cos(be ta+alfa)-delta3*cos(psi3) F10=delta1*sin(psi1)-Lp*sin(be ta+alfa)-delta3*sin(psi3) F11=(-cos(gamma1)*cos(theta1)*t+E1x+ cos(gamma2)*cos(theta2)*t-E2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*t-E2y)^2+(sin(theta1)*tsin(theta2)*t)^2+(-cos(gamma1)*cos(the ta1)*t+E1x+cos(gamma3)*cos(theta3)*tE3x)^2+(-sin(gamma1)*cos(theta1) *t+E1y+sin(gamma3) *cos(theta3)*tE3y)^2+(sin(theta1)*t-sin(theta3)*t)^2-2*((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*t-E2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*t-E2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2)*((-cos(gamma1)*cos (theta1)*t+E1x+cos(ga mma3)*cos(theta3)*tE3x)^2+(-sin(gamma1)*cos(theta1) *t+E1y+sin(gamma3) *cos(theta3)*tE3y)^2+(sin(theta1)*t-sin(theta3 )*t)^2)^(1/2)*cos(epsilon)-(cos(gamma2)*cos(theta2)*t+E2x+cos(gamma3)*cos(theta3)*t-E3x)^2-(-

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73 sin(gamma2)*cos(theta2)*t+E2y+sin(gamma3 )*cos(theta3)*t-E3y)^2-(sin(theta2)*tsin(theta3)*t)^2 F12=(-cos(gamma2)*cos(theta2)*t+E2x+ cos(gamma1)*cos(theta1)*t-E1x)*nx+(sin(gamma2)*cos(theta2)*t+E2y+sin(gamma1 )*cos(theta1)*t-E1y)*ny+(sin(theta2)*tsin(theta1)*t)*nz F13=(-cos(gamma3)*cos(theta3)*t+E3x+ cos(gamma1)*cos(theta1)*t-E1x)*nx+(sin(gamma3)*cos(theta3)*t+E3y+sin(gamma1 )*cos(theta1)*t-E1y)*ny+(sin(theta3)*tsin(theta1)*t)*nz F14=P1z-(sin(theta2)*t-sin(theta1)*t)/((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*t-E2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*t-E2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2 )*d1*cos(psi1)-(nx*(sin(gamma2)*cos(theta2)*t+E2y+sin(gamma1)*cos(theta1)*t-E1y)/((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*t-E2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*t-E2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2)-ny*(cos(gamma2)*cos(theta2)*t+E2x+co s(gamma1)*cos(theta1)*t-E1x)/((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*t-E2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*t-E2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2))* d1*sin(psi1)-sin(theta1)*t

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74 APPENDIX B PREVIOUS WORK A process using four masks was tested. Figur e B-1 shows the layout of one of the masks used and a detail of one of the features. The goa ls of this set of experiments were to gain experience about the basic microm anufacturing techniques, to st udy the behavior of silicone rubber as material for the compliant elements a nd find if the whole process could be done from one side of the wafer. The tests were made at the University of Florida Nanofacilities. A B Figure B-1. Mask used in the previous work. A) Outline. B) Detail of a feature. Compliant joints made of rubber exhibit a good performance in macro devices. It seemed appropriate to implement a similar solution for the microdevice, and in this way, simplify the design of the springs. For this purpose the refe rences Q1-4010 and JCR 6122 from the brand of encapsulants and led materials fr om Dow Corning were selected. According with the information of the manufacturer, they are flowable, cure to a fl exible elastomer, operate in a wide interval of temperature (-45 to 200 C) and exhibit minimal shrinkage. Information a bout elastic modulus is not always available, but some data suggest valu es less than 200 MPa. Th eir original viscosities are in the range from 300 to 800 centipoises.

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75 Figure B-2 depicts one of the features manuf actured at the lab. Black areas are holes. Silicone rubber should fill the gaps and then rubb er should be removed from selected areas in order to form the ties that conne ct the central platform and the actuators. However, there was no evidence that the material fille d the trenches, even though when the viscosity was reduced using a solvent. Oxygen plasma was used to etch the ru bber, but for practical pu rposes the material was insensitive to this procedure. As a conclusion from these results, the material for the ties must be photodefinable and still provide en ough deformation. Polymides ar e a good alternat ive for this purpose. Figure B-2. Manufactured feature. Other set of tests were performed trying to re lease the central platform through the use of isotropic etch and working from the front side on ly. The procedure showed that it is not feasible to release the platform in this way and etching from the back side is necessary. No additional constraints were found from the experiments. The design presented in chapter 5 includes these considerations.

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76 LIST OF REFERENCES 1. Chen,W., Chien, C. Hsieh, J. and Fang, W., 2003, A Reliable Single-Layer Out-of-Plane Micromachined Thermal Actuator Sensors and Actuators A, 103(1-2), pp. 48-58. 2. Milanovic, V., 2004, Multi level Beam SOI-MEMS Fabric ation and Applications, Journal of Microelectro mechanical Systems, 13(1), pp 19-30. 3. Lee,C., 2004, Design and Fabrication of Ep itaxial Silicon Micromirror Devices, Sensors and Actuators A, 115, pp 581-590. 4. Bronson, J.R., and Wiens, G. J., 2006, C ontrol of Micro Mirrors for High Precision Performance, Proceedings of the Florida Conference on Recent Advances in Robotics Miami, FL, May. 5. Fu Y., Du H., Huang W., and Hu, M ., 2004, TiNi-Based Thin Films in MEMS Applications: a Review, Sensors and Actuators A, 112(2-3), pp. 395. 6. Jain, A., Qu, H., Todd, S, Xie, H., 2005, A Thermal Bimorph Micromirror with Large Bidirectional and Vertical Actua tion, Sensors and Actuators A, 122(1), pp. 9-15. 7. Ebefors, T., Mattsson, J., Kalvesten, E. and Stemme, G., 1999, A Robust Micro Conveyer Realized By Arrayed Polymide Joint Actuators, Proceedings Twelfth IEEE Conference on Micro Electro Mechanical Systems Orlando, FL, January, pp. 576-581. 8. Suh, J., Darling, R., Bohringer, K., Dona ld, B., Baltes, H., and Kovacs, G., 1999, CMOS Integrated Ciliary Actuator Array as a Gene ral-Purpose Micromanipulation Tool for Small Objects, Journal of Microelectromechanical Systems, 8(4), pp. 483-496. 9. Schweizer, S., Calmes, S., Laudon, M., and Renaud, P., 1999, Thermally Actuated Optical Microscanner with Larg e Angle and Low Consumption, Sensors and Actuators A, 76(1-3), pp. 470. 10. Jensen K., Howell L., and Lusk K. 2004, Force Relationships for an XYZ Micromanipulator with Three Tran slational Degrees of Freedom, Proceedings Design Engineering Technical Conferences ASME, Salt Lake City, Utah, USA, September. 11. Bamberger, H. and Shoham, M., 2004, K inematic Structure of a Parallel Robot for MEMS Fabrication, Proceedings On Advances in Robot Kinematics Netherlands, pp. 113-122. 12. Tung, Y., and Kurabayashi, K., 2005, A Single-Layer PDMS-on-Silicon Hybrid Microactuator with Multi-Axi s Out-Of-Plane Motion Capab ilities Part I: Design and Analysis, Journal of Microe lectromechanical Systems, 14(3), pp. 548-557. 13. Fuller, R., 1975, Synergetics, Explorations in the Geometry of Thinking Collier Macmillan, London.

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77 14. Kenner, H., 1976, Geodesic Math and How to Use It University of California Press, Berkeley. 15. Calladine, C., 1978, Buckminster Fullers Tensegrity Structures and Clerk Maxwells Rules for the Construction of S tiff Frames, International Jour nal of Solids and Structures, 14, pp. 161. 16. Murakami, H., 2001, Static and Dynamic Anal yses of Tensegrity Structures. Part 1. Nonlinear Equations of Motion, Internat ional Journal of Solids and Structures, 38, pp. 3599-3613. 17. Crane, C., Duffy, J. and Correa, J., 2005, Static Analysis of Tensegrity Structures, Journal of Mechanical Design, 127(2), pp. 257-268. 18. Knight, B.F., 2000, Deployable Antenna Kine matics using Tensegrity Structure Design, Ph.D. thesis, University of Florida, Gainesville, FL. 19. Sultan, C., and Corless, M., 2000, Tensegr ity Flight Simulator, Journal of Guidance, Control, and Dynamics, 23(6), pp. 1055-1064. 20. Tibert, A., and Pellegrino S., 2002, Dep loyable Tensegrity Reflectors for Small Satellites, Journal of Spacecraft and Rockets, 39(5), pp.701-709. 21. Sultan, C., and Skelton, R., 2004, A Force and Torque Tensegrity Sensor, Sensors and Actuators A, 112(2-3), pp. 220. 22. Ingber, D. E., 1993, Cellular Tensegrity: Defining New Rules of Biological Design That Govern the Cytoskeleton, Journal of Cell Science, 104, pp. 613. 23. Furuya, H., 1992, Concept of Deployable Te nsegrity Structures in space Application, Int. J. Space Struct. 7(2), pp. 143. 24. Sultan, C., and Skelton, R., 2003, Deploymen t of Tensegrity Struct ures, International Journal of Solids and Structures 40, pp. 4637. 25. Pelesko, J. A., and Bernstein, D. H., 2002, Modeling MEMS and NEMS CRC Press, Boca Raton, FL. 26. Hah, D., Yoon, E., and Hong, S., 2000, A Low-Voltage Actuated Micromachined Microwave Switch Using Torsion Springs and Leverage, IEEE Transactions On Microwave Theory and Techniques, 48(12), 2540-2545. 27. Yang, S., Chen, X., Hasegawa, M., and Motojima S., 2004, Conformations of SuperElastic Carbon Micro/Nano-Spri ngs and their Properties, Proceedings of the International Conference on MEMS, NA NO and Smart Systems Alberta, CA, August. 28. Hata, S., Kato, T., Fukushige, T., and Sh imokohbe, A., 2003, Integrated Conical Spring Linear Actuator, Micr oelectronic Engineering 67, pp. 574.

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78 29. Brand, L., 1947, Vector and Tensor Analysis Wiley, New York. 30. Duffy, J., 1996, Statics and Kinematics with Applications to Robotics Cambridge University Press, New York. 31. Crane, C., and Duffy, J., 1998, Kinematic Analysis of Robot Manipulators Cambridge University Press, USA. 32. Howell, L. L., 2001, Compliant Mechanisms Wiley, New York 33. Senturia, S. D. 2001, Microsystem Design Kluwer, Boston. 34. Boresi, A. P., and Schmidt, R.J., 2003, Advanced Mechanics of Materials Wiley, New York. 35. Liu, C., 2006, Foundations of MEMS Prentice, Upper Saddle River, NJ.

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79 BIOGRAPHICAL SKETCH Julio Correa completed his masters de gree with Dr. Joseph Duffy in 2001 at the University of Florida. He worked on kinematics, a topic he enjoyed. He returned to Colombia his native country after comple tion of his masters degree. He came back to University of Florida to pursue and complete the PhD program under supervision of Professor Carl Crane in 2004. He returned to Universidad Pontificia Bolivariana in Medelln, Colombia after completion of his PhD program to conduct research on kinematics of microdevices and do teaching.


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