kinematic analysis and design of a compliant microplatform · model for their description. two...
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KINEMATIC ANALYSIS AND DESIGN OF A COMPLIANT MICROPLATFORM
By
JULIO CÉSAR CORREA RODRÍGUEZ
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
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© 2007 Julio César Correa Rodríguez
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To my mother for her infinite generosity
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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.
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
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
7
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
8
LIST OF FIGURES
Figure page 1-1 Prismatic tensegrity structure 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 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.........................................................................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
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 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
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5-14 Isotropic etch......................................................................................................................69
B-1 Mask used in the previous work. .......................................................................................74
B-2 Manufactured feature. ........................................................................................................75
11
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 ANALYISIS AND DESIGN OF A COMPLIANT MICROPLATFORM
By
Julio César Correa Rodríguez
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 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.
12
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.
13
CHAPTER 1 INTRODUCTION
Mechanisms formed by rigid links and rigid joints have been the 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 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
14
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. 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 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 [11]. 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.
15
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 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 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
16
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 joints 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 joints.
Bimorph Actuators
The bi-layer electrothermal actuator combines two materials with different coefficients of
thermal expansion (α). 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 joint with
the second material, the entire structure bends, see Pelesko [25]. It is possible to extend and
17
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 21 αα > the structure bends with an increase of temperature as shown in Figure 1-4A. If
21 αα < 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) 21 αα > . B) 21 αα < .
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
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 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
joints. 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 joints, 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.
19
CHAPTER 2 FORWARD ANALYSIS
Figure 2-1 depicts the device in a general position. In the forward analysis the location of
points iQ with respect to a global reference system are given and the objective is to evaluate the
coordinates of points iP 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 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
system.
20
Forces Acting on the Platform
To begin it is important to recall two basic concepts from vector algebra. The n vectors
nuuu ..., 21 are said to be linearly dependent if there exist n real numbers nλλλ ..., 21 not all zero
such that, see Brand [29]
0...2211 =+++ nn uuu λλλ . (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
0321 =++ FFF . (2-2)
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
0332211 =++ sfsfsf (2-3) where
is : unit vector from iP to iQ
if : is the magnitude of the force in each spring.
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
( ) ( ) ( ) 0303202101 =−+−+− sddksddksddk (2-4) where
id : actual length of the springs
0d : 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 21,ss 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 321321 ,,,,, PPPQQQ belong to the same plane and Equation 2-4 can be
presented as
03
33
2
22
1
11 =⎥
⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡
y
x
y
x
y
x
ss
fss
fss
f (2-5)
where ixix ss , : 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 sf is a vector perpendicular to the plane of the forces and whose
magnitude is magnitude is pf * (Figure 2-3), where p is the perpendicular distance between an
arbitrary point V and the line of action of force sf . 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
0332211 =++ pfpfpf . (2-6)
Equation 2-6 can be combined with Equation 2-5 to obtain, see Duffy [30]
22
Figure 2-3. Moment of a force.
0
3
2
1
321
321
321
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
fff
pppssssss
yyy
xxx
(2-7)
Nontrivial solutions for if requires that,
0
321
321
321
=pppssssss
yyy
xxx
(2-8)
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
point.
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.
23
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
• iP : point that define the moving platform
• iQ : free end of the actuator i
• ii ba , : coordinates of point iQ in the local system
• id : current length of the spring i
• iδ : distance between point iP and the intersection point I
• iψ : angle between id and the local x-axis
• β : angle of rotation of the platform with respect to the local x-axis
• pL : length of a side of the equilateral platform
24
• α : internal angle of the moving platform and therefore equal to 3/π
Global system A may be located in any arbitrary position. In this problem statement it is
assumed that the coordinates of points 21,QQ and 3Q are known in system A. With the
knowledge of points iQ , the local system E is defined as follows
12
12
QQx
AA
AA
EA
−
−= (2-9)
( ) ( )( ) ( )
1312
1312
QQQQ
QQQQz
AAAA
AAAA
EA
−×−
−×−= (2-10)
EA
EA
E
A xzy ×= (2-11)
The transformation that relates systems A and E is given by Crane [31]
⎥⎦
⎤⎢⎣
⎡=
10001
QRTAA
EAE (2-12)
where [ ]E
A
E
AE
AAE zyxR =
(2-13)
Coordinates of points iQ in the system E are given by ),( ii ba . Since 1QE is the origin of
system E, then
0,0 11 == ba (2-14)
Remaining coordinates 3322 ,,, baba can be found from the relations
22
2
2
10
QTba
Q AEA
E =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
(2-15)
33
3
3
10
QTba
Q AEA
E =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
(2-16)
25
where ( ) 1−= TT A
EEA
Note that since 2Q is located on the xE axis and thus
02 =b (2-17)
The problem as depicted in Figure 2-4 involves the following ten unknowns
βψψψδδδ
321
321
321
,,,,,, ddd
(2-18)
From Equation 2-4 equilibrium of forces evaluated in system E yields
( ) ( ) ( ) 0sincos
sincos
sincos
3
303
2
202
1
101 =⎥
⎦
⎤⎢⎣
⎡−+⎥
⎦
⎤⎢⎣
⎡−+⎥
⎦
⎤⎢⎣
⎡−
ψψ
ψψ
ψψ
dddddd . (2-19)
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
2121
ψβψ δδ iip
i eeLe += (2-20) ( ) 31
31ψαβψ δδ ii
pi eeLe += + (2-21)
Loops defined by IQQ −− 21 and IQQ −− 31 yield
( ) ( ) 21221211
ψψ δδ iEEi edQQed ++−=+
( ) ( ) 31331311
ψψ δδ iEEi edQQed ++−=+
Considering Equations 2-14 and 2-17 the last two equations can be simplified to
( ) ( ) 2122
211 0
ψψ δδ ii eda
ed ++⎥⎦
⎤⎢⎣
⎡=+ (2-22)
( ) ( ) 3133
3
311
ψψ δδ ii edba
ed ++⎥⎦
⎤⎢⎣
⎡=+ (2-23)
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-Raphson method. Once the variables are found, points
iA P are evaluated using the transformation
iEA
EiA PTP = (2-24) where points i
E P are given by (Figure 2-4)
⎥⎦
⎤⎢⎣
⎡+=
1
1111 sin
cosψψ
dQP EE (2-25)
⎥⎦
⎤⎢⎣
⎡+=
ββ
sincos
12 pEE LPP (2-26)
( )( )⎥⎦
⎤⎢⎣
⎡++
+=αβαβ
sincos
13 pEE LPP (2-27)
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 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
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
44.100
34.78
1QA ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
63.6431.91
72.52
2QA ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
12.4902.8070.46
3QA
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
27
Figure 2-5. Coordinates of the free ends of the actuators.
0321 dddd ===
6cos
32
321πδδδ pL===
0=β
Initial values for ψ are easily obtained from the geometry of the platform in its first
position (Figure 2-6).
61πψ = ,
652πψ = ,
23πψ −=
Figure 2-6. Initial position of the system.
28
It is also necessary to evaluate variables 332 ,, baa which depend on values of i
AQ . From
Equations 2-9 through 2-13 the given values of i
A Q yield
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
−−
=
100045.10939.0123.0321.0
0072.0838.0541.034.78338.0531.0777.0
TAE
When the transformation ( ) 1−= TT A
EEA is evaluated and substituted into Equations 2-15 and
2-16 the terms a2, a3, and b3 are determined as
67.1682 =a , 87.653 =a and 09.1383 =b .
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
07.351 =d 93.681 =δ 389.01 =ψ 065.0=β 49.412 =d 98.402 =δ 643.22 =ψ 73.363 =d 48.663 =δ 272.13 −=ψ
Equations 2-25 and 2-27 permit one the evaluation of points iA P as
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−−
=
114.4876.4495.35
,
144.5597.54
95.34
,
156.2242.606.46
321 PPP AAA
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
( ) ( ) ( ) 303202101 sddksddksddkF −+−+−=Σ (2-28) where
29
11
111
PQ
PQs
AA
AA
−
−= (2-29)
22
222
PQ
PQs
AA
AA
−
−= (2-30)
33
333
PQ
PQs
AA
AA
−
−= (2-31)
The intersection point of the lines passing through points 11 QP − and 22 QP − is given by
(Crane, C., Rico, J., Duffy, J., Screw Theory for Spatial Robot Manipulators, Cambridge
University Press, In Preparation)
( ) ( )( )221
220110212102212 1 ss
ssssssssssrA
⋅−⋅×+×⋅−×
= (2-32)
Similarly, the intersection of lines passing through 22 QP − and 33 QP − is given by
( ) ( )( )232
330220323203323 1 ss
ssssssssssrA
⋅−⋅×+×⋅−×
= (2-33)
where 1101 sQs A ×= (2-34)
2202 sQs A ×= (2-35)
3303 sQs A ×= (2-36)
Substituting values of iA P and
i
A Q into Equations 2-28 through 2-36 yields
41008.007.0
22.0−∗
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=Σ kF
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
36.4606.1939.17
12rA and ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
36.4606.1939.17
23rA
30
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.
31
CHAPTER 3 REVERSE ANALYSIS
In the reverse analysis the objective 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 vector n (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
32
angle θ 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 iE 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 iQ can be
defined by the vectors ir in a global reference system.
33
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
),(*),(*)( iiiAL yRotationzRotationEnTranslatioT θγ=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
10000cos0sin00100sin0cos
10001000cossin0sincos
10000100
010001
ii
ii
ii
ii
iy
ix
AL
EE
Tθθ
θθγγγγ
(3-1)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
10000cos0sin
sinsincoscossinsincossincoscos
1
ii
iyiiii
ixiiiii
AL
EE
Tθθθγγθγθγγθγ
(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 Equation 3-1 are carried out. Note that angles iγ are
constant. Figure 3-4b illustrates the local reference systems in their final orientation after
performing the last transformation involvingθ in Equation 3-1.
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 CA x is obtained by substituting i=2
in Equation 3-2 as
35
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
2
22
22
sincossincoscos
θθγθγ
CA x (3-3)
Without lost of generality assume that the free end whose position is given as1
QA , 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 Manipulators, Cambridge University Press, In
Preparation), and Figure 3-3
( ) nrnrnrr ⋅=⋅∴=⋅− 1212 0 (3-4)
From the geometry of the device (Figure 3-3)
222 tEr += (3-5)
From Figure 3-4b and considering Equation 3-3
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=−=
2
22
22
2
sincossincoscos
θθγθγ
txtt CA (3-6)
The scalar product of Equation 3-5 with n yields
ntnEnr ⋅+⋅=⋅ 222 (3-7)
Substituting Equations 3-4 and 3-6 into Equation 3-7 yields
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−⋅=⋅
2
22
22
21
sincossincoscos
θθγθγ
zyx nnntnEnr (3-8)
Regrouping Equation 3-8 yields
0sincos 22222 =++ DBA θθ (3-9) where
222 sincos γγ yx nnA += (3-10)
znB −=2 (3-11)
36
tnEnrD ⋅−⋅
= 212 (3-12)
It is possible to obtain a closed solution for 2θ in Equation 3-9, See Crane [31].
Substituting the value of 2θ in Equations 3-6 and 3-5, the coordinates of 2r , and therefore of
point2
QA , are determined
Similarly, from Figure 3-3
( ) nrnrnrr ⋅=⋅∴=⋅− 1313 0 (3-13)
From the geometry of the device (Figure 3-3)
333 tEr += (3-14) where
DA xtt −=3 (3-15)
Unit vector DA x 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
33
33
3
sincossincoscos
θθγθγ
tt (3-16)
The scalar product of Equation 3-14 with n yields
ntnEnr ⋅+⋅=⋅ 333 (3-17)
Substituting Equations 3-13 and 3-16 into Equation 3-17 yields
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−⋅=⋅
3
33
33
31
sincossincoscos
θθγθγ
zyx nnntnEnr (3-18)
Regrouping Equation 3-18 yields
0sincos 33333 =++ DBA θθ (3-19) where
37
333 sincos γγ yx nnA += (3-20)
znB −=3 (3-21)
tnEnrD ⋅−⋅
= 313 (3-22)
Equation 3-19 permits one to evaluate 3θ , then Equations 3-16 and 3-15 yield 3r and
therefore3
QA . 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 1Q (as defined by the elevation of rigid link 1 6109.01 =θ ), the
length of a side of the moving platform 23=pL , the free lengths of the springs 50 =d , the
length of the binary rigid links 26=t and the unit normal vector to the platform
[ ]Tn 9659.02588.00= expressed in the global reference system. Find the coordinates of
points 2
Q , ,3
Q 1P , 2P and 3P expressed in the global system.
From Figure 3-4, 3/2 πγ −= and 3/3 πγ = . Points 1E 2E and 3E are evaluated with the aid
of Figure 3-5 which shows the device when it is at the plane level.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
⎟⎠⎞
⎜⎝⎛ ++=
00279.44
0)sin()cos(
)6/cos(32
01 ππ
π tdLE p
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
⎟⎠⎞
⎜⎝⎛ ++=
034.38
139.22
0)3/sin()3/cos(
)6/cos(32
02 ππ
π tdLE p
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛ ++=
034.38139.22
0)3/sin()3/cos(
)6/cos(32
03 ππ
π tdLE p
38
Figure 3-5. Distributions of points E.
From Figure 3-4, coordinates of 1
Q are given by the vector 1r as follows
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+=+=
9130.140
981.22
sincossincoscos
1
11
11
1111
θθγθγ
tEtEr
From the given data
2588.0,0 == yx nn and 9659.0=zn
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
09358.0sin9659.0cos2241.0 22 =+−− θθ 01723.0sin9659.0cos2241.0 33 =+−− θθ
These last equations yield two sets of solutions for 2θ and 3θ that will yield equilibrium
configurations. Each pair is selected considering the equilibrium conditions. The solutions are
4027.0,0056.1 32 == aa θθ
39
195.3,68.1 32 == bb θθ
Equations 3-5 and 3-6 permit one to evaluate points 2
Q and 3
Q for both solutions of 2θ and
3θ and then, following the procedure presented in the forward analysis, it is possible to evaluate
points 1P , 2P and 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 solution 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
40
[ ]zyx nnn 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 1PA is selected
(Figure 3-7).
A B 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.
41
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 i
AQ are unknown and depend
on angles iθ . Angle ε 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 solved). Therefore, in addition to
the 10 variables used in the forward analysis and enumerated in (2.18), here there are 6 new
unknowns: xP1321 ,,,, εθθθ and yP1 . 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.
zAP1 : scalar component z of the vector i
A P .
Find:
1QA ,
2QA ,
3QA : location of the free ends of the binary links.
yx PP 11 , : scalar components x and y of the vector iA P .
2PA , 3PA : location of the vertexes of the moving platform.
Points i
AQ depend on iθ and using transformation (3-2) they can be expressed as
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
∗=
10000cos0sin
sinsincoscossinsincossincoscos
,
100
11
111111
111111
1 θθθγγθγθγγθγ
y
x
AB
AB
A EE
T
t
TQ (3-23)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
∗=
10000cos0sin
sinsincoscossinsincossincoscos
,
100
22
222222
222222
2 θθθγγθγθγγθγ
y
x
AC
AC
A EE
T
t
TQ (3-24)
42
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
∗=
10000cos0sin
sinsincoscossinsincossincoscos
,
100
33
333333
333333
3 θθθγγθγθγγθγ
y
x
AD
AD
A EE
T
t
TQ (3-25)
Equations 3-23 through 3-25 simplify to
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡+−+−
=
1sincossincoscos
1
111
111
1 θθγθγ
tEtEt
Q y
x
A (3-26)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡+−+−
=
1sincossincoscos
2
222
222
2 θθγθγ
tEtEt
Q y
x
A (3-27)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡+−+−
=
1sincossincoscos
3
333
333
3 θθγθγ
tEtEt
Q y
x
A (3-28)
Expressions for distances jiQQ between points i
AQ and j
AQ can be obtained from
Equations 3-26 through 3-28 as follows
1221 QQQQ AA −= (3-29)
2332 QQQQ AA −= (3-30)
1331 QQQQ AA −= (3-31)
The relation between 1PE and 1PA is given by
11 PTP EAE
A = (3-32)
Transformation TAE defines the relation between the global system A and a reference
system E which origin is located at 1
QA with its x-axis points from 1
QA to 2
QA , and for which
the z-axis is the unit vector n (Figure 3-8B), therefore
43
⎥⎦
⎤⎢⎣
⎡=
10001
QRTAA
EAE (3-33)
where [ ]E
A
E
AE
AAE zyxR = (3-34) where
12
12
QQx
AA
AA
EA
−
−= (3-35)
nz EA = (3-36)
EA
EA
E
A xzy ×= (3-37)
Coordinates of 1PE can be obtained from Figure 3-8b, and Equation 3-32 can be expressed
as
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
10
sincos
10001
11
11
1333231
1232221
1131211
1
1
1
ψψ
dd
QrrrQrrrQrrr
PPP
zA
yA
xA
zA
yA
xA
(3-38)
where the terms ijr depend only on iθ .
At this point all the developments required for the mathematical model are obtained.
Expression (3-38) yields 3 scalar equations, one of them involving the prescribed value zAP1 , then
xA
xA QdrdrP 1111211111 sincos ++= ψψ (3-39)
yA
yA QdrdrP 1112211211 sincos ++= ψψ (3-40)
zA
zA QdrdrP 1113211311 sincos ++= ψψ (3-41)
The angle ε in Figure 3-8b can be related to points i
A Q using the cosine law
εcos2 31212
312
212
32 ∗∗−+= QQQQQQQQQQ (3-42)
where the terms jiQQ 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:
( ) ( ) ( ) 0coscoscos 303202101 =−+−+− ψψψ dddddd (3-43)
44
( ) ( ) ( ) 0sinsinsin 303202101 =−+−+− ψψψ dddddd (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)
2121
ψβψ δδ iip
i eeLe += (3-45) ( ) 31
31ψβαψ δδ ii
pi eeLe += + (3-46)
Geometry relations for the actual lengths of the springs involve the terms 3121 , QQQQ
andε (Figure 3-8B)
( ) ( ) 2122
02111
ψψ δδ iii edeQQed ++=+ (3-47) ( ) ( ) 31
333111ψεψ δδ iii edeQQed ++=+ (3-48)
Points 2
QA and 3
QA with respect to the location of point 1
QA must be perpendicular to
vector n . To assure that, two more relations are required
( ) 012
=⋅− nQQ AA (3-49)
( ) 013
=⋅− nQQ AA (3-50)
Equations 3-41 through 3-50 form a system of 14 equations and 14 unknowns that can be
solved for 321321321321 ,,,,,,,,,,,, ψψψδδδεθθθ ddd andβ . Appendix A presents the set of
equations in extended form.
Once the solution is obtained, it is possible to evaluate i
A Q using Equations 3-26 through
(3-28). Coordinates yA
xA PP 11 , are easily evaluated using Equations 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
45
⎥⎦
⎤⎢⎣
⎡=
10001PRT
AAEA
F (3-51)
The rotation matrix RAE is given by Equation 3-34 and with the aid of Figure 3-8B the
following relations are obtained
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
==
10sincos
, 222
ββ
p
p
FFAF
A LL
PPTP (3-52)
( )( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡++
==
10
sincos
, 333
αβαβ
p
p
FFAF
A LL
PPTP (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
3/3/,3/,
150200
30
321
0
παπγπγπγ
==−==
===
pLtd
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
9848.01116.01330.0
n
641 =zAP
Find points i
A Q and iA 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 iA E . Lines
46
through iA P -
i
AQ intersect at the point of intersection of the heights of the equilateral triangle,
there is no stretching and no rotation of the platform, then
00,30,20,1 dddd ===
6cos
32
0,30,20,1πδδδ pL===
00 =β
Figure 3-9. Device in its initial position.
Since all the links are on the horizontal plane
00,1 =θ , 00,2 =θ , 00,3 =θ
From the geometry of the platform in its first position
60,1πψ = ,
650,2πψ = ,
20,3πψ −=
30πε = , correspond to the angle between points 213 QQQ .
[ ]TgA RE 0011 −=
[ ]TgA RE 0)6/sin()6/cos(2 ππ −−=
[ ]TgA RE 0)6/sin()6/cos(3 ππ=
47
where tdRg ++= 00,1δ
The solution to system of equations given by 3-41 through 3-50 yields
3530.01 =θ 4832.01 =ψ 0150.0−=β 2841.02 =θ 7090.22 =ψ 0122.1=ε 1606.03 =θ 6673.13 −=ψ
52.391 =d 70.761 =δ 44.382 =d 38.902 =δ 00.383 =d 55.933 =δ
Equations 3-26 through 3-28 yield
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
198.3120.10258.59
,
154.55
79.10723.62
,
114.69
0128
321QQQ AAA
Equations 3-39 and 3-40 permit one to evaluate the remaining coordinates xAP1 , y
AP1 . As a
result 1PA is defined completely.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
=64
28.176.89
1PA
Finally, Equations 3-51, 3-52 and 3-53 yield the values for 2PA and 3PA
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
134.3830.7240.38
,
108.5575.76
55.39
32 PP AA
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 9101 −× .
48
Figure 3-10. Device for the example of reverse analysis, case 2.
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 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.
50
Table 4-1 Mechanical and thermal properties for aluminum and silicone dioxide. Material Young modulus, E
MPa Thermal exp. coeff, α
6101
−K
2OSi 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 µm 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
aa
a FEIL
3
3
=δ , where 3
121 wtI =
then
3
3
4 aa
a
LEwtF
=δ
(4-1)
where aδ : deflection at the free end of the beam
aF : force applied at the free end of the beam
aL : length of the beam E: Young’s modulus w: width of the beam t: thickness of the beam
51
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 µm and the total thickness (this is aluminum and silicon dioxide) as 3
µm, then from Equation 4-1 and with w=7 µm
mNF
a
a
μμ
δ41.0=
The current selection establishes that if the maximum deflection is limited to 1 µm, then
the vertical force acting in each beam is 41.0=aF μN. If each actuator has 12 beams, then the
maximum perpendicular force that can be applied to the system is
5*12 ≈= ap FF μN (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.
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
)cos1( φρ −=zQ (4-3) where ρ : radius of curvature and φ the angle of ρ with the vertical.
From Figure 4-3B
ρφ=aL (4-4)
Substituting Equation 4-4 into Equation 4-3 yields
( )φφ
cos1−= az
LQ (4-5)
53
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 zQ onφ , however the value of φ 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 ρ and therefore the initial value of angle φ
depend on the geometry and material properties of the bimorph beam. Liu [35] presents the
following equations that should permit the evaluation of the initial radius of curvature of a
bimorph beam.
54
MEIeff 0=ρ (4-6)
where
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+++⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+=
2
12
22
22
21
21
110 212212zttttEztttEwEIeff
( ) ( ) ( )+⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+−−−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
2211
222111111
21 1112 tEtE
ttEtwM νσνσνσ
( ) ( ) ( )⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+−−−⎟⎟
⎠
⎞⎜⎜⎝
⎛ +
2211
222111222
2122 111
2 tEtEttEtttw νσνσνσ
( )2211
212222
2112
1
tEtE
ttEtEtEz
+
++=
0EIeff : 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 σ : intrinsic stress in the layers after deposition ν : 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 ttt == 21 and EEE == 21 , Equation 4-6 can be simplified to
( ) ( )1122 1138
νσνσρ
−−−=
Et (4-7)
To be useful for numerical evaluation, Equation 4-7 requires the knowledge of the intrinsic
stresses 1σ and 2σ . 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 φ of 17º has been observed. For purposes of this design, angle φ is assumed to be 20º.
55
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 0d (Figure 4-5A). When beams reach the maximum elevation (Figure 4-5B), the length of
the springs is maximum.
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, 0max dd − ,
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 200=aL µm, free length of the spring 800 =d µm, and the length of the
side of the platform 120=pL µm.
56
Figure 4-6. Maximum deformation of the springs.
Values of 0d cannot be very small to avoid exaggerated stresses. For the current values
of aL , pL 0d and for the already selected value of 20º forφ , the maximum deformation is 4.0
µm. 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 5=pF µN and then when sF is
maximum
2φπη −
= (4-8)
ηη
coscos p
ss
p FF
FF
=∴= (4-9)
57
Figure 4-7. Maximum force in the spring.
For o20=φ , Equation 4-8 yields o80=η and from Equation 4-9, 8.28=sF µN.
In summary, to get a maximum deformation of 4.0 µm the maximum force cannot exceed
28.8 µN, in order to keep the component perpendicular of the force to GQ less than 5 µN.
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.
58
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 St
Thickness µm
Cure ºC
Tensile MPa
Modulus 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 λ and length CL can be evaluated as follows
TLr c =+ λλ sincos2 (4-10) HLr c =+− λλ sinsin2 (4-11)
Squaring Equations 4-10 and 4-11 and adding the results yields
22224 HTLr c +=+ 222 4rHTLc −+=∴ (4-12)
Since CL must be positive, Equation 4-12 shows that the selection of T and H must fulfill
the relation
222 4rHT ⟩+ (4-13)
59
Figure 4-9. Geometry of a segment of the spring.
If Equation 4-10 is multiplied by λsin and Equation 4-11 is multiplied by λcos then
λλλλ sinsincossin2 2 TLr c =+ (4-14) λλλλ coscoscossin2 2 HLr c =+− (4-15)
Adding Equations 4-14 and 4-15 yields
0sincos =−+ cLTH λλ (4-16)
Equation 4-16 can be solved for λ 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 θθσ 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]
( )( )ARAAR
ARAMAF
mi
mixs
−−
+=θθσ (4-17)
where tbA = (4-18)
⎟⎠⎞
⎜⎝⎛ += RHFM sx 2
(4-19)
60
Figure 4-10. Segment of the spring.
2bRRi −= (4-20)
bRbRtAm −
+=
22ln (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 deflection of the linear segment and the deflection due to the
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 element rδ , is given by
zs
Ld
FM
EIMc
∂∂
= ∫02 2δ (4-22)
where ωcoszFM s= (4-23)
From Equations 4-22 and 4-23
323
121,cos
32 tbI
EILF cs
r == ωδ (4-24)
The deflection of the curvilinear segment, cδ is given by
∫−
∂∂
=ωπ
δw
xxc Rda
FM
EM (4-25)
where ( )[ ]aRRLFM csx ++−= ωωω sinsincos (4-26)
From Equations 4-25 and 4-26
( )[ ]∫−
++−=ωπ
ωωωωδ daaRRL
EIPR
c2sinsincos (4-27)
Total deflection per segment nδ is just the superposition of rδ and cδ ,
crn δδδ += (4-28)
For n segments the total deflection is
ns n δδ ∗= (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
10=sF µN, T=28 µm and t=3 µm. From then is clear that for the parameters H=50 µm, b=7 µm
and R=12 µm, a force of 10 µN, just a third of the admissible force, is enough to deflect one
62
segment of the spring 2 µm. If 3 elements are used to create the spring, the resultant deflection
provides more than the required deflection of 4 µm.
Figure 4-12. Deflection in the spring.
Figure 4-13. Stress in the spring.
63
Similarly, Figure 4-13 presents the results for the stressσ , 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, aL : 200 µm Width, w: 7 µm Thickness of each layer, t: 1.5 µm Number of beams: 12 Material beams: aluminum and silicone dioxide
Width, b: 7 µm Thickness, 2t: 3 µm Free length, 0d : 80 µm H: 50 µm T: 28 µm R: 11.5 µm
Parameters Platform Parameters resistor pL : 120 µm Material: chrome
Width: 5 µm Thickness: 0.2 µm
64
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.
65
A first layer of 0.80 µm of silicone dioxide is deposited over the wafer using plasma
enhanced chemical vapor deposition (PECVD) process (Figure 5-2).
Then a layer 0.20 µm 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).
A B Figure 5-4. Resistor. A) General view. B) Detail.
A second layer of silicone dioxide with thickness 0.70 µm 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).
66
A B Figure 5-5. Second layer of silicone dioxide. A) General view. B) Detail.
Aluminum with thikness1.5 µm 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).
A B 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.
A B Figure 5-7. Etching of areas in the aluminum corresponding to the actuators and platform. A)
General view. B) Detail.
67
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 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 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).
68
A B Figure 5-10. Polymide springs. A) General view. B) Detail.
The main elements of the mechanisms are ready. The objective of the 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 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.
A B Figure 5-12. Section view of the device. A) General view. B) Detail.
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 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.
A B Figure 5-14. Isotropic etch. A) General view. B) Detail.
70
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 joints offers interest possibilities 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 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.
71
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.
72
APPENDIX A REVERSE ANALYSIS EQUATIONS
This appendix presents the long form of the equations derived for the second case of the
reverse analysis.
• F1=(d1-d0)*cos(psi1)+(d2-d0)*cos(psi2)+(d3-d0)*cos(psi3)
• F2=(d1-d0)*sin(psi1)+(d2-d0)*sin(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)*t-sin(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)*t-sin(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)*t-sin(theta3)*t)^2)^(1/2)*3^(1/2)-(d3+delta3)*sin(psi3)
• F7=delta1*cos(psi1)-Lp*cos(beta)-delta2*cos(psi2)
• F8=delta1*sin(psi1)-Lp*sin(beta)-delta2*sin(psi2)
• F9=delta1*cos(psi1)-Lp*cos(beta+alfa)-delta3*cos(psi3)
• F10=delta1*sin(psi1)-Lp*sin(beta+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)*t-sin(theta2)*t)^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)*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)*t-sin(theta2)*t)^2)^(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)*t-sin(theta3)*t)^2)^(1/2)*cos(epsilon)-(-cos(gamma2)*cos(theta2)*t+E2x+cos(gamma3)*cos(theta3)*t-E3x)^2-(-
73
sin(gamma2)*cos(theta2)*t+E2y+sin(gamma3)*cos(theta3)*t-E3y)^2-(sin(theta2)*t-sin(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)*t-sin(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)*t-sin(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)*t-sin(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)*t-sin(theta2)*t)^2)^(1/2)-ny*(-cos(gamma2)*cos(theta2)*t+E2x+cos(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)*t-sin(theta2)*t)^2)^(1/2))*d1*sin(psi1)-sin(theta1)*t
74
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.
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 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 (-45º 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.
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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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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 Medellín, Colombia after completion
of his PhD program to conduct research on kinematics of microdevices and do teaching.