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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.......... ...... OutofPlane 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 OutofPlane 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 31 Solution for the reverse analysis, case 1. ............. ...............39..... 41 Mechanical and thermal properties for aluminum and silicone dioxide. ........................50 42 Properties of polymide HD8000. .........._.... ...............58..__... .... 43 Main dimensions of the device. ........._._... ...............63.._.__. .... LIST OF FIGURES Figure page 11 Prismatic tensegrity structure with 6 struts. ......___ ........_._ ...._._ ..........1 12 Sequence of motions for the rising of the structure. ................ ............... ........ ...16 13 Configuration of a bimetallic actuator. .............. .....................17 14 Bending of a bimetallic actuator. ........................... ........17 15 Scheme of the device ................ .......... .................. .......................18 21 Device in a general position ................. ...............19............... 22 Arbitrary forces acting on the platform. ............. ...............20..... 23 Moment of a force ................. ...............22........... ... 24 Nomenclature for the forward analysis............... ...............23 25 Coordinates of the free ends of the actuators ................. ...............27........... . 26 Initial position of the system ................. ...............27............... 27 Device in the evaluated equilibrium position. ............. ...............30..... 31 Normal vector to the moving platform. ............. ...............31..... 32 Path of the free end. ................ ...............32............... 33 Parameters for the reverse analysis, case 1. ............. ...............33..... 3.4 Location of the local reference systems for the reverse analysis ................. ................. 34 35 Distributions of points E. ................ ............. .................. .................38 36 Solution for the reverse analysis case 1. ............. ...............39..... 37 Prescribed vertical component of point P1 ......_......._.__........._ ..........4 38 Nomenclature for the reverse analysis, case 2. .............. ...............40.... 39 Device in its initial position. ............. ...............46..... 310 Device for the example of reverse analysis, case 2. ........._._ .... .__ ...............48 41 Maximum deflection of a cantilever beam. ............. ...............5......1 42 Deflection of the free end for several conditions ................. ...............52........... . 43 Maximum elevation of the free end of the beam. ................ ...............53............. 44 Bending after release from sub state ................. ...............53........... .. 45 Positions for minimum and maximum deformation of the springs. ................ ...............55 46 Maximum deformation of the springs............... ...............56 47 Maximum force in the spring............... ...............57. 48 Possible geometries for the spring. ............. ...............58..... 49 Geometry of a segment of the spring ................. ...............59.............. 410 Segment of the spring. ................ ...............60.......... .... 411 Parameters for the stress analysis of a spring. ............. ...............60..... 412 Deflection in the spring............... ...............62. 413 Stress in the spring. .............. ...............62.... 51 Silicone substrate. ............. ...............64..... 52 First layer of silicone dioxide. ............. ...............64..... 53 Layer of chrome. .............. ...............65.... 54 Resistor. ............. ...............65..... 55 Second layer of silicone dioxide. .............. ...............66.... 56 Aluminum layer. ............. ...............66..... 57 Etching of areas in the aluminum corresponding to the actuators and platform. ........._....66 58 Etching of silicone dioxide .............. ...............67.... 59 Polymide layer. ............. ...............67..... 510 Polymide springs ................. ...............68........... .... 511 Backside etch. ............. ...............68..... 512 Section view of the device. ................ ...............68............... 513 Deep reactive ion etching............... ...............69 514 Isotropic etch ................. ...............69........... .... B1 Mask used in the previous work. ............. ...............74..... B2 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 threedimensional 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. OutofPlane Motion Devices Outof plane actuators can convert input signals into displacements normal to the surface of a substrate. Threedimensional microdevices are useful for different tasks as for example, object positioning, micromanipulators, optical scanners, tomographic imaging, optical switches, microrelays, adjustable lenses and bioMEMS applications. To obtain outofplane motion is a challenging problem and several approaches based have been proposed. Usually outofplane 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 outofplane 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 pullin voltage, see Bronson and Wiens [4]. Combination of TiNi and Si cantilever or other substrates such as SU8 or polymide have been used to create outofplane 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 outofplane 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 microscanner. 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 outofplane 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. Outofplane 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 11). Figure 11. 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 12 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 12. 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 bilayer 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 CMOScompatible. Therefore, control circuits can be integrated with the device on the same chip. A bimetallic actuator is illustrated in Figure 13. Figure 13. Configuration of a bimetallic actuator. If a, > a, the structure bends with an increase of temperature as shown in Figure 14A. If a, < a, the structure bends like in Figure 14B. 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 14. 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. Regularcoiled 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 15 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 12. 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 15. 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 21 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 21. 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. (21) The other important concept is this: a necessary and sufficient condition that three vectors be linearly dependent is that they be coplanar. Figure 22 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 (22) 3 > ~ P, Figure 22. Arbitrary forces acting on the platform. Equation 22 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 24 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 (24) 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 24 are different from zero. From Equation 21 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 24 can be presented as SIx Sx S3x f, +i~ fi + ]= (25) 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 23), 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 (26) Equation 26 can be combined with Equation 25 to obtain, see Duffy [30] Figure 23. 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 (28) 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 24, 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 24 depicts a scheme including the variables and parameters used for this model . P Figure 24. Nomenclature for the forward analysis. The nomenclature defined here will be used later in the reverse analysis. The elements presented in Figure 24 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 xaxis * p : angle of rotation of the platform with respect to the local xaxis * 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 (29) gE i 1e) 1.P", (210) yE E~ E, (211) The transformation that relates systems A and E is given by Crane [31] AT = Eo 1 e (212) where R E.4E. E (213) 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 (214) Remaining coordinates az, b,,as, b, can be found from the relations az b8 ,=l B EO= _o 3 .4 (215) (216) where ET (T)1 Note that since Q, is located on the Exa axis and thus b, = 0 (217) The problem as depicted in Figure 24 involves the following ten unknowns d d,, d, (218) cos y, cos y, cos y, (d,I d, +( ,I d,) +, d, = (219) 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 24 it is clear that 3, e'"1= Lp e' +32 e'"1 (220) 31 e'1= Lp e;(P~a) +33 e'" (221) 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 214 and 217 the last two equations can be simplified to (d~, + 3,) e"' =~ a d +3)e (222) (d, + ,) e'" =3 a + dz 3 ellv (223) Scalar components of Equations 219 through 223 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 NewtonRaphson method. Once the variables are found, points AP, are evaluated using the transformation API =T ElP (224) where points EP, are given by (Figure 24) P = O+d, .(225) E )2EI) L COS P1 P ,= P + (226) a; E1 + p LSin + &> 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 4q for equilibrium given the free lengths of the ties d, = 20, and the coordinates of points Q, (Figure 25) 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 26), therefore Figure 25. 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 26). ry, = > = 5 ry3 6 6 2 Figure 26. Initial position of the system. It is also necessary to evaluate variables a a b, which depend on values of Oe From Equations 29 through 213 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 215 and 216 the terms a2, a3, and b3 are determined as a, = 168.67 a, 65.87 and b, = 138.09 . Now the NewtonRaphson method can be implemented to solve system of Equations 219 through 223. 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 225 and 227 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 (228) where AA s1 = (229) 1A, A s = (230) s3 3 (231) 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  (232) l( 1 S2 ) Similarly, the intersection of lines passing through P2 2 e and P3 3 eis given by A 23 3 S3S 3S O 2 S2S 3(233) 1(3 s )2 where sl= Aex s (234) SO2 A 2S (235) s3A X S (236) Substituting values of AP, and A 2 into Equations 228 through 236 yields CF =k I0.2208 EF=k0.07 1*104 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 27 displays the device in the evaluated equilibrium position. Figure 27. 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 31). 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 31. 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 32A 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 32B 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 32. 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 33 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 33. 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 xaxis are aligned with the axis of the binary links (Figure 33), 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 (31) 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 34a shows a top view when only the first two transformations of Equation 31 are carried out. Note that angles 7, are constant. Figure 34b illustrates the local reference systems in their final orientation after performing the last transformation involving Bin Equation 31. 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 32 represent the local xaxis expressed in the global system A. In particular the local axis A x, iS obtained by substituting i=2 in Equation 32 as x I cos y, cos 8 Ax, = sin 7, cos8, (33) 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 33 3 rl) z = 0 :.. r? IE = YI n (34) From the geometry of the device (Figure 33) Tz = E, +t, (35) From Figure 34b and considering Equation 33 cos y, cos 8, t, = t Ax, = t sin y cos8, (36) sin 8 The scalar product of Equation 35 with n yields Tz n = E, a +t, n (37) Substituting Equations 34 and 36 into Equation 37 yields 1cos y, cos B E, n =E, n t 2,n n sin y, cos 8,(38 sin 8 Regrouping Equation 38 yields A, cos 8 + B, sin 8 + D = 0 (39) where A, = nx cos y, + n, sin y (310) B2 = n, (311) TI nE, n D, = (312) It is possible to obtain a closed solution for 8, in Equation 39, See Crane [31]. Substituting the value of 6, in Equations 36 and 35, the coordinates of r,, and therefore of point O are determined Similarly, from Figure 33 3E y0 rj y= r, n (313) From the geometry of the device (Figure 33) r3 = E3 + t3 (314) where t, = t xD (315) Unit vector xD XS obtained from the first three terms of the first column of the matrix defined in Equation 32 when i=3, and thus (315) may be written as cos y, cos8, tz= in o 1 (316) sin 8, The scalar product of Equation 314 with n yields '3 = E_3 + t3 (317) Substituting Equations 313 and 316 into Equation 317 yields 1cos y, cos 8, r n=E, nt E2 n, nI sin y cos83 38 sin 8, Regrouping Equation 318 yields A, cos 8, + B, sin 8, + D, = 0 (319) where A, = nx cos y, + n, sin7 y B3 = n z r Y nE3 nY D,= (320) (321) (322) Equation 319 permits one to evaluate 63, then Equations 316 and 315 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 34, 7, = r /3 and 7, = xi / 3 Points Ez E, and E, are evaluated with the aid of Figure 35 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 35. Distributions of points E. From Figure 34, 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 (310) through (312) and (320) through (322). Substituting the coefficients into Equations 310 and 319 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 35 and 36 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 31. Table 31 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 36. The second solution is also an equilibrium position, but the current device cannot reach that position. Figure 36. 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 37). 'I ~c~ L Figure 37. 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 38. Nomenclature for the reverse analysis, case 2. A) Isometric view. B) Plane of the forces. Figure 38a shows the device in an arbitrary position. Figure 38b 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 (32) they can be expressed as t cos y, cos 8, sin y, cos y, sin 8, Ex Oe = 7* T = (323) 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 = (324) ' 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*( (325) 0 sin 8, 0 cos9 0, 1 0 0 0 1 Equations 323 through 325 simplify to tcos y, cos8, + E,, tsin y, cos8, + E, Q = (326) _1 tsin 8, tcos y cos8, + Ex "r2 t smny cos 8 + E, O = (327) t sin 8, tcos7, cos 8 + E,, tsin y, cos 8 + Ez O = '(328) t sin 8, Expressions for distances Q!,Q! between points OC) and OI~ can be obtained from Equations 326 through 328 as follows Q O O O1 (329) Q Q3 ~= O ~ O) (330) The relation between EP, and AP, is given by 1P=ET EP, (332) Transformation AT defines the,,+, relation between, theglbalsyte A and a referenc system E which origin is located at Q lwith its xaxis points from OL) to O ), and for which the zaxis is the unit vector n (Figure 38B), therefore 4R Q o1(3 AT = E 33 where Ri =x 4' E 2 (334) where xE 24 (335) E = n (336) yE X4E4E (3 3 7) Coordinates of EP, can be obtained from Figure 38b, and Equation 332 can be expressed as AFx 11 2 1~3 Aex dz cosly, A~ ~ re r O? d,2 ~1 ~ Usin cy, (338) 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 (33 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 38b can be related to points A 0 using the cosine law QQ3 QQ +QQ 20,,*OQ3*ose(342) where the terms Q,Q, are given by Equations 329 through 331. 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 (343) (d, d,)sin c, + (dZ d,)siny, + (d, d,)sin 3 = 0 (344) The geometry of the system involving relations for the moving platform is the same as found in the forward analysis (Figure 38B) 3,el'" = L,pe'P + 63e'" (345) 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 38B) (d, + 3)e'" = QQ ~e'" + (dZ + 33)el' (347) (d, + 3,)e"'" = QQ e'"~ + (d, + 33)e"" (348) 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 (349) O O n= 0(350) Equations 341 through 350 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 326 through (328). Coordinates xI~,AF, are easily evaluated using Equations 339 and 340 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 (351) 000 1 [ The rotation matrix EAR is given by Equation 334 and with the aid of Figure 38B the following relations are obtained L,cos f A Lp sin f "P, T FP2 FIP (352) 4 Lp sin(P + a) "P_ = T FP3 FP3 = CO@ (353) 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 39 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 341 through 350 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 326 through 328 yield 128 62.23 59.58 0 107.79 102.20 "o 69.14 4L) 55.54 ) 31.98 1 1 1 Equations 339 and 340 permit one to evaluate the remaining coordinates Ap,, AIy. As a result AP, is defined completely. 89.766 P,=1.28 Finally, Equations 351, 352 and 353 yield the values for AP, and "P, 39.55 38.40 76.75 72.30 "4P, = AP 55.08 38.34 1 1 Figure 310 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 109 Le Figure 310. 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 41 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 41 Mechanical and thermal properties for aluminum and silicone dioxide. Material Young modulus, E Thermal exp. coeff, a MPa 1 K 106 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 41). 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 41, is given by L3 So= F, ,where I = wt3 3El 12 then F, Ewt3 (41) 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 41. Maximum deflection of a cantilever beam. Equation 41 can be solved for several length, width and thickness of the actuators (Figure 42). 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 41 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 (42) OutofPlane 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 42. Deflection of the free end for several conditions. From Figure 43B it is clear that the height of the free end is given by Q, = p(1 cosf) (43) where p : radius of curvature and # the angle of p with the vertical. From Figure 43B La = pf (44) Substituting Equation 44 into Equation 43 yields O a ~(1 cos # (45) XS Figure 43. Maximum elevation of the free end of the beam. A) Isometric view. B) Lateral view. Equation 45 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 44). Figure 44. 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 = (46) M~ where Ig 2 = zt + E2t 2 1 Z 2 [: t2 ~,OV) tla (1v )+ 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 46 can be simplified to 8 E p = t (47) 3 G To be useful for numerical evaluation, Equation 47 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 47 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 45A). When beams reach the maximum elevation (Figure 45B), the length of the springs is maximum. L L Figure 45. 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 46 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. 84 0 5 10 15 20 25 30 g, deg Figure 46. 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 47 shows the components of the force acting on the spring. In Equation 42 the maximum admissible perpendicular force was selected as Fp = 5 CIN and then when F, is maximum S=' (48) F F cos q =P : F = (49) F, cosy G Figure 47. Maximum force in the spring. For # = 20", Equation 48 yields r = 80" and from Equation 49, 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 48A). 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 48B is used, it is easier to achieve the required deformations. Figure 48. 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 42 shows some properties of the polyimide HD 8000 from HD MicroSystems. For the following analysis they are considered as isotropic materials. Table 42 Properties of polymide HD8000. Viscosity Thickness Cure Tensile Modulus St Clm oC MPa MPa 3.5 3 to 5 350 122 2500 Figure 49 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 (410) 2r sin ii + L, sin ii = H (411) Squaring Equations 410 and 411 and adding the results yields 4r 2 + Lc2 2 + H2 ... Le = JT2 H24r2 (412) Since Le must be positive, Equation 412 shows that the selection of T and H must fulfill the relation T2 +H2 ) 4r2 (413) Figure 49. Geometry of a segment of the spring. If Equation 410 is multiplied by sin ii and Equation 411 is multiplied by cos ii then 2r sin ii cos ii + L, sin 2 ii = T Sin ii (414) 2r sin iicos i+ L, cos2 ii= H cos i (415) Adding Equations 414 and 415 yields H cos ii+ T sin ii L, = 0 (416) Equation 416 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 410). The maximum tensile stress use occurs at point B (Figure 410). Figure 411 presents the nomenclature to derive the expressions for the circumferential stress in a curved beam following the procedure explained by Boresi [34] F +Mx(ARAm as,= (417) A AR,(RA, A) where A = tb (418) Mx = F +R (419) Figure 410. Segment of the spring. R, = R (420) 2R +b A, = tln (421) "'2R b Le Figure 411i. Parameters for the stress analysis of a spring. Spring Deflection For the element shown in Figure 410 and 411, 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 (422) where M~=F,zcosei (423) From Equations 422 and 423 3, = FLcos2 m, I = tb3 (424) 3El 12 The deflection of the curvilinear segment, 6, is given by Sc=(xRda (425) where Mr = F, [Lc cos u, R sin w + R sin(mL + all (426) From Equations 425 and 426 PR [L cos m R sin a + R sin m + a)l 2da (427) Total deflection per segment 6, is just the superposition of 3, and c,, 6,= 3, + So (428) For n segments the total deflection is 3, = n (429) Spring Dimensions Expressions found for the stress and deflection of the spring can be evaluated for different values of the parameters. Figure 412 presents the results for the deflection of the spring when F, = 10 GIN, T=28 Clm and t3 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 412. 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 413. Stress in the spring. Similarly, Figure 413 presents the results for the stress cr for the same conditions of Figure 411. It is clear from the highlighted value, that the stress is only 19 MPa compared to the admissible value of 122 MPa (Table 42). Dimensions found yield a conservative and reliable device. Table 43 summarizes the results. Some of them are not critical and their calculations were not included. Table 43 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 51). Figure 51. Silicone substrate. Figure 52. 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 52). Then a layer 0.20 Clm of chrome is sputtered over the first silicone dioxide layer (Figure 5 3). Figure 53. Layer of chrome. Using a mask, the chrome is patterned and then plasma etching is used to obtain the shape of the resistor (Figure 54). Figure 54. Resistor. A) General view. B) Detail. A second layer of silicone dioxide with thickness 0.70 Clm is applied using PECVD (Figure 55A). As a result the resistor is isolated, however it is necessary to open a via to be able to apply voltage (Figure 55B). Figure 55. 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 56A). 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 56B). Figure 56. 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 57 illustrates the result when the aluminum has been removed and the second layer of silicone dioxide is exposed. Figure 57. 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 58). Figure 58. Etching of silicone dioxide Springs are created in the next two steps. A layer of polymide is spun on the wafer (Figure 59). Figure 59. Polymide layer. With a mask, the photodefinable polymide is patterned to create the springs. After curing and removing remaining material (Figure 510). Figure 510. 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 511i. This step determines the depth of the platform. Figure 511. Backside etch. A) General view. B) Detail. The process continues in the front side. Figure 512 shows a section view of the wafer and the detail of the substrate under a spring. The substrate material must be removed. Figure 512. 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 513). 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 513. Deep reactive ion etching. A) General view. B) Detail. Finally isotropic etching is used to remove the portions under the beams and spring (Figure 514). Some undercut is also present in the platform, but it does not affect its strength. Figure 514. 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=(dld0)*cos(psi l)+(d2d0)*cos(psi2)+(d3d0)*cos(psi3) * F2=(d ld0)*sin(psi l)+(d2d0)* 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)*tE2x)^2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*tE2y)^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)*tE3x)^2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma3)*cos(theta3)*tE3y)^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)*tE3x)^2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma3)*cos(theta3)*tE3y)^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)*tE2y)^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 )*tsin(theta3)*t)^'22*(( cos(gammal )*cos(thetal)*t+E Ix+cos(gamma2)*cos(theta2)*tE2x)^`2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*tE2y)^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)*tsin(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)*tE 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)*tE 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)*tE2x)^2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*tE2y)^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 )*tEly)/(( cos(gammal )*cos(thetal )*t+Elx+cos(gamma2)*cos(theta2)*tE2x)^2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*tE2y)^2(sn' hea)*t sin(theta2)*"t)^`2)^(1/2)ny* ( cos(gamma2)*cos(theta2)*t+E2x+cos(gammal )*cos(thetal)*tE Ix)/(( cos(gammal )*cos(thetal )*t+Elx+cos(gamma2)*cos(theta2)*tE2x)^2+( sin(gammal )*cos(thetal )*t+Ely+sin(gamma2)*cos(theta2)*tE2y)^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 B1 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 B1. 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 Q14010 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 B2 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 B2. 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. LIST OF REFERENCES 1. Chen,W., Chien, C. Hsieh, J. and Fang, W., 2003, "A Reliable SingleLayer OutofPlane Micromachined Thermal Actuator," Sensors and Actuators A, 103(12), pp. 4858. 2. Milanovic, V., 2004, "Multilevel Beam SOIMEMS Fabrication and Applications," Journal of Microelectromechanical Systems, 13(1), pp 1930. 3. Lee,C., 2004, "Design and Fabrication of Epitaxial Silicon Micromirror Devices," Sensors and Actuators A, 115, pp 581590. 4. Bronson, J.R., and Wiens, G. J., 2006, "Control 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, "TiNiBased Thin Films in MEMS Applications: a Review," Sensors and Actuators A, 112(23), pp. 395408. 6. Jain, A., Qu, H., Todd, S, Xie, H., 2005, "A Thermal Bimorph Micromirror with Large Bi directional and Vertical Actuation," Sensors and Actuators A, 122(1), pp. 915. 7. Ebefors, T., Mattsson, J., Kalvesten, E. and Stemme, G., 1999, "A Robust Micro Conveyer Realized By Arrayed Polymide Joint Actuators," Proceedings TMI I&/?h IEEE Conference on Micro Electro mechanical Systems, Orlando, FL, January, pp. 576581. 8. Suh, J., Darling, R., Bohringer, K., Donald, B., Baltes, H., and Kovacs, G., 1999, "CMOS Integrated Ciliary Actuator Array as a GeneralPurpose Micromanipulation Tool for Small Objects," Journal of Microelectromechanical Systems, 8(4), pp. 483496. 9. Schweizer, S., Calmes, S., Laudon, M., and Renaud, P., 1999, "Thermally Actuated Optical Microscanner with Large Angle and Low Consumption," Sensors and Actuators A, 76(13), pp. 470477. 10. Jensen K., Howell L., and Lusk K. 2004, "Force Relationships for an XYZ Micromanipulator with Three Translational Degrees of Freedom," Proceedings Design Engineering Technical Conferences, ASME, Salt Lake City, Utah, USA, September. 11. Bamberger, H. and Shoham, M., 2004, "Kinematic Structure of a Parallel Robot for MEMS Fabrication," Proceedings On Advances in Robot Kinematics, Netherlands, pp. 113122. 12. Tung, Y., and Kurabayashi, K., 2005, "A SingleLayer PDMSonSilicon Hybrid Microactuator with MultiAxis OutOfPlane Motion Capabilities Part I: Design and Analysis," Journal of Microelectromechanical Systems, 14(3), pp. 548557. 13. Fuller, R., 1975, Synergetics, Explorations in the Geometry of Thinking, Collier Macmillan, London. 14. Kenner, H., 1976, Geodesic Math and How to Use It, University of Califomnia Press, Berkeley. 15. Calladine, C., 1978, "Buckminster Fuller' s Tensegrity Structures and Clerk Maxwell's Rules for the Construction of Stiff Frames," International Journal of Solids and Structures, 14, pp. 161172. 16. Murakami, H., 2001, "Static and Dynamic Analyses of Tensegrity Structures. Part 1. Nonlinear Equations of Motion," International Journal of Solids and Structures, 38, pp. 35993613. 17. Crane, C., Duffy, J. and Correa, J., 2005, "Static Analysis of Tensegrity Structures," Journal of Mechanical Design, 127(2), pp. 257268. 18. Knight, B.F., 2000, "Deployable Antenna Kinematics using Tensegrity Structure Design," Ph.D. thesis, University of Florida, Gainesville, FL. 19. Sultan, C., and Corless, M., 2000, "Tensegrity Flight Simulator," Journal of Guidance, Control, and Dynamics, 23(6), pp. 10551064. 20. Tibert, A., and Pellegrino S., 2002, "Deployable Tensegrity Reflectors for Small Satellites," Journal of Spacecraft and Rockets, 39(5), pp.701709. 21. Sultan, C., and Skelton, R., 2004, "A Force and Torque Tensegrity Sensor," Sensors and Actuators A, 112(23), pp. 220231. 22. Ingber, D. E., 1993, "'Cellular Tensegrity: Defining New Rules of Biological Design That Govern the Cytoskeleton," Journal of Cell Science, 104, pp. 613627. 23. Furuya, H., 1992, "Concept of Deployable Tensegrity Structures in space Application," Int. J. Space Struct. 7(2), pp. 143151. 24. Sultan, C., and Skelton, R., 2003, "Deployment of Tensegrity Structures," Intemnational Journal of Solids and Structures 40, pp. 46374657. 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 LowVoltage Actuated Micromachined Microwave Switch Using Torsion Springs and Leverage," IEEE Transactions On Microwave Theory and Techniques, 48(12), 25402545. 27. Yang, S., Chen, X., Hasegawa, M., and Motojima S., 2004, "Conformations of Super Elastic Carbon Micro/NanoSprings 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 6768, pp. 574581. 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. PAGE 1 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 PAGE 2 2 2007 Julio Csar Correa Rodrguez PAGE 3 3 To my mother for her infinite generosity PAGE 4 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. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................13 OutofPlane 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 OutofPlane Elevation......................................................................................................... ..51 Springs Design................................................................................................................. .......55 Springs Elongation..........................................................................................................55 Maximum Force Acting on the Spring............................................................................56 Spring Geometry and Material........................................................................................57 Spring Stress.................................................................................................................. ..59 PAGE 6 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 PAGE 7 7 LIST OF TABLES Table page 31 Solution for the reverse analysis, case 1............................................................................39 41 Mechanical and thermal properties fo r aluminum and silicone dioxide............................50 42 Properties of polymide HD8000.......................................................................................58 43 Main dimensions of the device..........................................................................................63 PAGE 8 8 LIST OF FIGURES Figure page 11 Prismatic tensegrity st ructure with 6 struts........................................................................15 12 Sequence of motions for the rising of the structure...........................................................16 13 Configuration of a bimetallic actuator...............................................................................17 14 Bending of a bimetallic actuator........................................................................................17 15 Scheme of the device....................................................................................................... ..18 21 Device in a general position...............................................................................................19 22 Arbitrary forces acting on the platform.............................................................................20 23 Moment of a force.......................................................................................................... ....22 24 Nomenclature for the forward analysis..............................................................................23 25 Coordinates of the free ends of the actuators.....................................................................27 26 Initial position of the system............................................................................................. .27 27 Device in the evalua ted equilibrium position....................................................................30 31 Normal vector to the moving platform..............................................................................31 32 Path of the free end....................................................................................................... .....32 33 Parameters for the reverse analysis, case 1........................................................................33 3.4 Location of the local reference sy stems for the reverse analysis.......................................34 35 Distributions of points E.................................................................................................. ..38 36 Solution for the reverse analysis case 1.............................................................................39 37 Prescribed vertical component of point P1.........................................................................40 38 Nomenclature for the reverse analysis, case 2...................................................................40 39 Device in its initial position............................................................................................. ..46 310 Device for the example of reverse analysis, case 2...........................................................48 41 Maximum deflection of a cantilever beam........................................................................51 PAGE 9 9 42 Deflection of the free end for several conditions...............................................................52 43 Maximum elevation of the free end of the beam...............................................................53 44 Bending after release from substrate..................................................................................53 45 Positions for minimum and maxi mum deformation of the springs...................................55 46 Maximum deformation of the springs................................................................................56 47 Maximum force in the spring.............................................................................................57 48 Possible geometries for the spring.....................................................................................58 49 Geometry of a se gment of the spring.................................................................................59 410 Segment of the spring..................................................................................................... ...60 411 Parameters for the stress analysis of a spring....................................................................60 412 Deflection in the spring.................................................................................................. ....62 413 Stress in the spring...................................................................................................... .......62 51 Silicone substrate......................................................................................................... ......64 52 First layer of silicone dioxide............................................................................................64 53 Layer of chrome............................................................................................................ .....65 54 Resistor................................................................................................................... ...........65 55 Second layer of silicone dioxide........................................................................................66 56 Aluminum layer............................................................................................................. ....66 57 Etching of areas in the aluminum co rresponding to the actuators and platform...............66 58 Etching of silicone dioxide................................................................................................67 59 Polymide layer............................................................................................................. ......67 510 Polymide springs.......................................................................................................... ......68 511 Backside etch............................................................................................................. ........68 512 Section view of the device................................................................................................ .68 513 Deep reactive ion etching................................................................................................. ..69 PAGE 10 10 514 Isotropic etch............................................................................................................ ..........69 B1 Mask used in the previous work........................................................................................74 B2 Manufactured feature....................................................................................................... ..75 PAGE 11 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 threedimensional 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. PAGE 12 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. PAGE 13 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. OutofPlane Motion Devices Outof plane actuators can convert input si gnals into displacements normal to the surface of a substrate. Threedimensional 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 bioMEMS applications. To obtain outofplane motion is a challenging problem and se veral approaches based have been proposed. Usually outofplane 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 outofplane 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 PAGE 14 14 of comb drives is limited and they require a car eful design and control to avoid jumps associated with the pullin voltage, s ee Bronson and Wiens [4]. Combination of TiNi and Si cantilever or othe r substrates such as SU8 or polymide have been used to create outofplane 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 outofplane 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 microscanner. 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 outofplane 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. PAGE 15 15 Outofplane 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 11). Figure 11. 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 PAGE 16 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 12 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 12. 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 bilayer 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 PAGE 17 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 CMOScompa tible. Therefore, cont rol circuits can be integrated with the device on the same chip. A bimetallic actuator is illustrated in Figure 13. Figure 13. Configuration of a bimetallic actuator. If 2 1 the structure bends with an increase of temperature as shown in Figure 14A. If 2 1 the structure bends like in Figure 14B. 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 14. 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. Regularcoiled 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 PAGE 18 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 15 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 12. 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 15. 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. PAGE 19 19 CHAPTER 2 FORWARD ANALYSIS Figure 21 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 21. 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. PAGE 20 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 (21) The other important concept is this: a necessa ry and sufficient condition that three vectors be linearly dependent is that they be coplanar. Figure 22 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 (22) Figure 22. Arbitrary forces acting on the platform. Equation 22 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 (23) where is: unit vector from iP to iQ if : is the magnitude of the force in each spring. PAGE 21 21 Since the springs are linear, each force magnitude in Equation 24 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 (24) 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 24 are di fferent from zero. From E quation 21 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 24 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 (25) 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 23), 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 (26) Equation 26 can be combined with E quation 25 to obtain, see Duffy [30] PAGE 22 22 Figure 23. 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 (27) 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 (28) 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 24, 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 24 depicts a scheme includi ng the variables and parameters used for this model. PAGE 23 23 Figure 24. Nomenclature fo r the forward analysis. The nomenclature defined here will be used later in the reverse analysis. The elements presented in Figure 24 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 xaxis : angle of rotation of the platform with respect to the local xaxis pL: length of a side of the equilateral platform PAGE 24 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 (29) 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 (210) E A E A E Ax z y (211) The transformation that relates systems A and E is given by Crane [31] 1 0 0 01Q R TA A E A E (212) where E A E A E A A Ez y x R (213) 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 (214) 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 (215) 3 3 3 31 0Q T b a QA E A E (216) PAGE 25 25 where 1 T TA E E A Note that since 2Qis located on the xE axis and thus 02 b (217) The problem as depicted in Figure 24 involves the following ten unknowns 3 2 1 3 2 1 3 2 1, , d d d (218) From Equation 24 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. (219) 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 24 it is clear that 2 12 1 i i p ie e L e (220) 3 13 1 i i p ie e L e (221) 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 214 and 217 the la st two equations can be simplified to 2 12 2 2 1 10 i ie d a e d (222) 3 13 3 3 3 1 1 i ie d b a e d (223) PAGE 26 26 Scalar components of Equations 219 through 223 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 NewtonRa phson method. Once the variables are found, points i AP are evaluated using the transformation i E A E i AP T P (224) where points i EP are given by (Figure 24) 1 1 1 1 1sin cos d Q PE E (225) sin cos1 2p E EL P P (226) sin cos1 3p E EL P P (227) 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 25) 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 26), therefore PAGE 27 27 Figure 25. 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 26). 61 6 52 23 Figure 26. 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 29 through 213 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 215 and 216 the terms a2, a3, and b3 are determined as 67 1682 a, 87 653aand09 1383 b. Now the NewtonRaphson method can be implemen ted to solve system of Equations 219 through 223. 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 225 and 227 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 (228) where PAGE 29 29 1 1 1 1 1P Q P Q sA A A A (229) 2 2 2 2 2P Q P Q sA A A A (230) 3 3 3 3 3P Q P Q sA A A A (231) 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 (232) 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 (233) where 1 1 01s Q sA (234) 2 2 02s Q sA (235) 3 3 03s Q sA (236) Substituting values of i AP and i AQ into Equations 228 through 236 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 27 displays the device in the evaluated equilibrium position. Figure 27. Device in the ev aluated equilibrium position. PAGE 31 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 31). 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 31. 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 32A 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 32B 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 32. 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 33 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. PAGE 33 33 Figure 33. 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 xax is are aligned with the axis of the binary links (Figure 33), 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 (31) 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 (32) When i=1, 2, 3 references systems B, C and D are obtained. Figure 34a shows a top view when only the first two transformations of E quation 31 are carried out. Note that angles i are constant. Figure 34b illustrates the local reference systems in their final orientation after performing the last transformation involving in Equation 31. PAGE 34 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 32 represent the local xaxis expressed in the global system A. In particular the local axis C Axis obtained by substituting i=2 in Equation 32 as PAGE 35 35 2 2 2 2 2sin cos sin cos cos C Ax (33) 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 33 n r n r n r r 1 2 1 20 (34) From the geometry of the device (Figure 33) 2 2 2t E r (35) From Figure 34b and co nsidering Equation 33 2 2 2 2 2 2sin cos sin cos cos t x t tC A (36) The scalar product of Equation 35 with nyields n t n E n r 2 2 2 (37) Substituting Equations 34 and 36 into Equation 37 yields 2 2 2 2 2 2 1sin cos sin cos cos z y xn n n t n E n r (38) Regrouping Equation 38 yields 0 sin cos2 2 2 2 2 D B A (39) where 2 2 2sin cos y xn n A (310) zn B 2 (311) PAGE 36 36 t n E n r D 2 1 2 (312) It is possible to obtai n a closed solution for 2 in Equation 39, See Crane [31]. Substituting the value of 2 in Equations 36 and 35, the coordinates of 2r, and therefore of point 2QA, are determined Similarly, from Figure 33 n r n r n r r 1 3 1 30 (313) From the geometry of the device (Figure 33) 3 3 3t E r (314) where D Ax t t 3 (315) Unit vector D Ax is obtained from the first three terms of the first column of the matrix defined in Equation 32 when i=3 and thus (315) may be written as 3 3 3 3 3 3sin cos sin cos cos t t (316) The scalar product of Equation 314 with n yields n t n E n r 3 3 3 (317) Substituting Equations 313 and 316 into Equation 317 yields 3 3 3 3 3 3 1sin cos sin cos cos z y xn n n t n E n r (318) Regrouping Equation 318 yields 0 sin cos3 3 3 3 3 D B A (319) where PAGE 37 373 3 3sin cos y xn n A (320) zn B 3 (321) t n E n r D 3 1 3 (322) Equation 319 permits one to evaluate3 then Equations 316 and 315 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 34, 3 /2 and3 /3 Points 1E 2Eand 3Eare evaluated with the aid of Figure 35 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 PAGE 38 38 Figure 35. Distribu tions of points E. From Figure 34, 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 (310) through (31 2) and (320) through (322). Substituting the coefficients into Equations 310 and 319 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 PAGE 39 39 195 3 68 13 2 b b Equations 35 and 36 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 31. Table 31 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 36. The second so lution is also an equilibrium position, but the current device cannot reach that position. A B Figure 36. 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 PAGE 40 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 37). A B Figure 37. 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 38. Nomenclature for the reverse analysis, case 2. A) Isometric view. B) Plane of the forces. PAGE 41 41 Figure 38a shows the device in an arbitrary position. Figu re 38b 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 (32 ) 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 (323) 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 (324) PAGE 42 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 (325) Equations 323 through 325 simplify to 1 sin cos sin cos cos1 1 1 1 1 1 1 1 t E t E t Qy x A (326) 1 sin cos sin cos cos2 2 2 2 2 2 2 2 t E t E t Qy x A (327) 1 sin cos sin cos cos3 3 3 3 3 3 3 3 t E t E t Qy x A (328) Expressions for distances j iQ Q between points i AQand j AQ can be obtained from Equations 326 through 328 as follows 1 2 2 1Q Q Q QA A (329) 2 3 3 2Q Q Q QA A (330) 1 3 3 1Q Q Q QA A (331) The relation between 1PE and 1PA is given by 1 1P T PE A E A (332) Transformation TA Edefines the relation between the global system A and a reference system E which origin is located at 1QAwith its xaxis points from 1QA to 2QA, and for which the zaxis is the unit vector n (Figure 38B), therefore PAGE 43 43 1 0 0 01Q R TA A E A E (333) where E A E A E A A Ez y x R (334) where 1 2 1 2Q Q Q Q xA A A A E A (335) n zE A (336) E A E A E Ax z y (337) Coordinates of 1PE can be obtained from Figure 38b, and Equation 332 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 (338) where the terms ijr depend only on i At this point all the developments required for the mathematical model are obtained. Expression (338) 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 (339) y A y AQ d r d r P1 1 1 22 1 1 21 1sin cos (340) z A z AQ d r d r P1 1 1 32 1 1 31 1sin cos (341) The angle in Figure 38b 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 (342) where the terms j iQ Q are given by Equations 329 through 331. 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 (343) PAGE 44 44 0 sin sin sin3 0 3 2 0 2 1 0 1 d d d d d d (344) The geometry of the system involving relations for the moving platform is the same as found in the forward an alysis (Figure 38B) 2 12 1 i i p ie e L e (345) 3 13 1 i i p ie e L e (346) Geometry relations for the actual lengths of the springs involve the terms 3 1 2 1,Q Q Q Q and (Figure 38B) 2 12 2 0 2 1 1 1 i i ie d e Q Q e d (347) 3 13 3 3 1 1 1 i i ie d e Q Q e d (348) 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 (349) 01 3 n Q QA A (350) Equations 341 through 350 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 326 through (328). Coordinates y A x AP P1 1, are easily evaluated using E quations 339 and 340 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 PAGE 45 45 1 0 0 01P R TA A E A F (351) The rotation matrix RA E is given by Equation 334 and w ith the aid of Figure 38B the following relations are obtained 1 0 sin cos ,2 2 2 p p F F A F AL L P P T P (352) 1 0 sin cos ,3 3 3 p p F F A F AL L P P T P (353) 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 39 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 39. 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 341 through 350 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 326 through 328 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 339 and 340 permit one to evaluate the remaining coordinatesx AP1,y AP1. As a result 1PAis defined completely. 64 28 1 76 891PA Finally, Equations 351, 352 a nd 353 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 310 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. PAGE 48 48 Figure 310. Device for the exampl e of reverse analysis, case 2. PAGE 49 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 41 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. PAGE 50 50 Table 41 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 41). 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 41, is given by a a aF EI L 33 where 312 1 wt I then 3 34a a aL Ewt F (41) 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 PAGE 51 51 Figure 41. Maximum deflec tion of a cantilever beam. Equation 41 can be solved for several length, width and thickness of the actuators (Figure 42). 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 41 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 (42) OutofPlane 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. PAGE 52 52 Figure 42. Deflection of the free end for several conditions. From Figure 43B it is clear that the height of the free end is given by ) cos 1 ( zQ (43) where : radius of curvature and the angle of with the vertical. From Figure 43B aL (44) Substituting Equation 44 into Equation 43 yields cos 1 a zL Q (45) PAGE 53 53 Figure 43. Maximum elevation of the free end of the beam. A) Isometric view. B) Lateral view. Equation 45 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 44). Figure 44. 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. PAGE 54 54 M E Ieff 0 (46) 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 46 can be simplified to 1 1 2 21 1 3 8 E t (47) To be useful for numerical evaluation, Equa tion 47 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 47 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. PAGE 55 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 45A). When beams reach the maxi mum elevation (Figure 45B), the length of the springs is maximum. Figure 45. 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 46 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. PAGE 56 56 Figure 46. 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 47 shows the components of the for ce acting on the spring. In Equation 42 the maximum admissible perpendicular force was selected as 5 pF N and then when sF is maximum 2 (48) cos cosp s s pF F F F (49) PAGE 57 57 Figure 47. Maximum fo rce in the spring. For20 Equation 48 yields 80 and from Equation 49, 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 48A). 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 48B is used, it is easier to achieve the required deformations. PAGE 58 58 Figure 48. 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 42 Properties of polymide HD8000. Viscosity St Thickness m Cure C Tensile MPa Modulus MPa 3.5 3 to 5 350 122 2500 Figure 49 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 (410) H L rc sin sin 2 (411) Squaring Equations 410 and 411 and adding the results yields 2 2 2 24H T L rc 2 2 24 r H T Lc (412) Since CL must be positive, Equation 412 shows that the selection of T and H must fulfill the relation 2 2 24 r H T (413) PAGE 59 59 Figure 49. Geometry of a segment of the spring. If Equation 410 is multiplied by sin and Equation 411 is multiplied by cos then sin sin cos sin 22T L rc (414) cos cos cos sin 22H L rc (415) Adding Equations 414 and 415 yields 0 sin cos cL T H (416) Equation 416 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 410) The maximum tensile stress occurs at point B (Figure 410). Figure 411 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 (417) where tb A (418) R H F Ms x2 (419) PAGE 60 60 Figure 410. Segment of the spring. 2 b R Ri (420) b R b R t Am 2 2 ln (421) Figure 411. Parameters for the stress analysis of a spring. Spring Deflection For the element shown in Figure 410 and 411, the total deflection can be considered as the superposition due to the de flection of the linear segment and the deflection due to the PAGE 61 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 (422) where cos z F Ms (423) From Equations 422 and 423 3 2 312 1 cos 3 2 tb I EI L Fc s r (424) The deflection of the curvilinear segment, c is given by w x x cRda F M E M (425) where a R R L F Mc s x sin sin cos (426) From Equations 425 and 426 da a R R L EI PRc2sin sin cos (427) Total deflection per segment n is just the superposition of r andc c r n (428) For n segments the total deflection is n sn (429) Spring Dimensions Expressions found for the stress and deflection of the spring ca n be evaluated for different values of the parameters. Figure 412 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 PAGE 62 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 412. Deflection in the spring. Figure 413. Stress in the spring. PAGE 63 63 Similarly, Figure 413 presents the results for the stress for the same conditions of Figure 411. It is clear from the hi ghlighted value, that the stress is only 19 MPa compared to the admissible value of 122 MPa (Table 42). Dimensions found yield a conservative and reliable device. Tabl e 43 summarizes the results. Some of them are not critical and their calculations were not included. Table 43 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 PAGE 64 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 51). Figure 51. Silicone substrate. Figure 52. First layer of silicone dioxide. PAGE 65 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 52). Then a layer 0.20 m of chrome is sputtered over the first silicone di oxide layer (Figure 53). Figure 53. Layer of chrome. Using a mask, the chrome is patterned and then plasma etching is used to obtain the shape of the resistor (Figure 54). A B Figure 54. Resistor. A) General view. B) Detail. A second layer of silicone dioxide with thic kness 0.70 m is applied using PECVD (Figure 55A). As a result the resistor is isolated, however it is necessary to open a via to be able to apply voltage (Figure 55B). PAGE 66 66 A B Figure 55. 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 56A). 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 56B). A B Figure 56. 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 57 illustrates the result when the aluminum has been removed and th e second layer of silicone dioxide is exposed. A B Figure 57. Etching of areas in the aluminum corresponding to the actuators and platform. A) General view. B) Detail. PAGE 67 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 58). Figure 58. Etching of silicone dioxide Springs are created in the next two steps. A layer of polymid e is spun on the wafer (Figure 59). Figure 59. Polymide layer. With a mask, the photodefinable polymide is patterned to crea te the springs. After curing and removing remaining material (Figure 510). PAGE 68 68 A B Figure 510. 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 511. This step determines the depth of the platform. A B Figure 511. Backside etch. A) General view. B) Detail. The process continues in the fr ont side. Figure 512 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 512. Section view of the de vice. A) General view. B) Detail. PAGE 69 69 The exposed parts of the substrate are removed using deep reactive ion plasma etch (Figure 513). 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 513. 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 514). Some undercut is also pr esent in the platform, but it doe s not affect its strength. A B Figure 514. Isotropic etch. A) General view. B) Detail. PAGE 70 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. PAGE 71 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. PAGE 72 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=(d1d0)*cos(psi1)+(d2d0)*c os(psi2)+(d3d0)*cos(psi3) F2=(d1d0)*sin(psi1)+(d2d0)*s in(psi2)+(d3d0)*sin(psi3) F3=(d1+delta1)*cos(psi1)( d2+delta2)*cos(psi2)((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*tE2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*tE2y)^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)*tE3x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma3 )*cos(theta3)*tE3y)^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)*tE3x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma3 )*cos(theta3)*tE3y)^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)*tE2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*tE2y)^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)*tsin(theta3)*t)^22*((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*tE2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*tE2y)^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)*tsin(theta3 )*t)^2)^(1/2)*cos(epsilon)(cos(gamma2)*cos(theta2)*t+E2x+cos(gamma3)*cos(theta3)*tE3x)^2( PAGE 73 73 sin(gamma2)*cos(theta2)*t+E2y+sin(gamma3 )*cos(theta3)*tE3y)^2(sin(theta2)*tsin(theta3)*t)^2 F12=(cos(gamma2)*cos(theta2)*t+E2x+ cos(gamma1)*cos(theta1)*tE1x)*nx+(sin(gamma2)*cos(theta2)*t+E2y+sin(gamma1 )*cos(theta1)*tE1y)*ny+(sin(theta2)*tsin(theta1)*t)*nz F13=(cos(gamma3)*cos(theta3)*t+E3x+ cos(gamma1)*cos(theta1)*tE1x)*nx+(sin(gamma3)*cos(theta3)*t+E3y+sin(gamma1 )*cos(theta1)*tE1y)*ny+(sin(theta3)*tsin(theta1)*t)*nz F14=P1z(sin(theta2)*tsin(theta1)*t)/((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*tE2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*tE2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2 )*d1*cos(psi1)(nx*(sin(gamma2)*cos(theta2)*t+E2y+sin(gamma1)*cos(theta1)*tE1y)/((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*tE2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*tE2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2)ny*(cos(gamma2)*cos(theta2)*t+E2x+co s(gamma1)*cos(theta1)*tE1x)/((cos(gamma1)*cos(theta1)*t+E1x+cos(gamma2)*cos(theta2)*tE2x)^2+(sin(gamma1)*cos(theta1)*t+E1y+sin(gamma2 )*cos(theta2)*tE2y)^2+(sin(theta1)*tsin(theta2)*t)^2)^(1/2))* d1*sin(psi1)sin(theta1)*t PAGE 74 74 APPENDIX B PREVIOUS WORK A process using four masks was tested. Figur e B1 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 B1. 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 Q14010 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. PAGE 75 75 Figure B2 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. 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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 LowVoltage Actuated Micromachined Microwave Switch Using Torsion Springs and Leverage, IEEE Transactions On Microwave Theory and Techniques, 48(12), 25402545. 27. Yang, S., Chen, X., Hasegawa, M., and Motojima S., 2004, Conformations of SuperElastic Carbon Micro/NanoSpri 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. PAGE 78 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. PAGE 79 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. 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