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Parallel Mechanisms with Variable Compliance

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PARALLEL MECHANISMS WITH VARIABLE COMPLIANCE By HYUN KWON JUNG 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 2006

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Copyright 2006 by Hyun Kwon Jung

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This dissertation is dedicated to my wife, Eyun Jung Lee and son, Sung Jae.

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iv ACKNOWLEDGMENTS I would like express my thanks to Dr. Carl D. Crane III, my academic advisor and committee chair, for his continual support and guidance throughout this work. I would also like to thank the other members of my supervisory committee, Dr. John C. Ziegert, Dr. John K. Schueller, Dr. A. Antonio Arroy o, and Dr. Rodney G. Roberts, for their time, expertise, and willingness to serve on my committee. I would like to thank all of the personnel of the Center for Intelligent Machines and Robotics for their support and expertise. I al so would like to thank other friends of mine for providing plenty of advice and diversions. Last but not least, I would like to thank to my parent, parents-in-law, my wife, and son for their unwavering support, love, and sacrifice. This research was performed with funding from the Department of Energy through the University Research Program in Robotics (URPR), grant number DE-FG0486NE37967.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT....................................................................................................................... .x CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Motivation..........................................................................................................1 1.2 Literature Review...............................................................................................2 1.3 Problem Statement.............................................................................................6 2 STIFFNESS MAPPING OF PLANAR COMPLIANT MECHANISMS....................8 2.1 Spring in a Line Space.......................................................................................8 2.2 A Derivative of Planar Spring Wrench Joining a Moving Body and Ground.11 2.3 A Derivative of Spring Wrench Joining Two Moving Bodies........................15 2.4 Stiffness Mapping of Planar Complia nt Parallel Mechanisms in Series.........22 2.5 Stiffness Mapping of Planar Compliant Parallel Mechanisms in a Hybrid Arrangem ent....................................................................................................27 3 STIFFNESS MAPPING OF SPATIAL COMPLIANT MECHANISMS..................33 3.1 A Derivative of Spatial Spring Wrench Joining a Moving Body and Ground.............................................................................................................33 3.2 A Derivative of Spring Wrench Joining Two Moving Bodies........................39 3.3 Stiffness Mapping of Spatial Complia nt Parallel Mechanisms in Series........49 4 STIFFNESS MODULATION OF PLANAR COMPLIANT MECHANISMS.........56 4.1 Parallel Mechanisms with Variable Compliance.............................................56 4.1.1 Constraint on Stiffness Matrix...............................................................56 4.1.2 Stiffness Modulation by Varying Spring Parameters............................60

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vi 4.1.3 Stiffness Modulation by Vary ing Spring Parameters and Displacement of the Mechanism.............................................................64 4.2 Variable Compliant Mechanisms with Two Parallel Mechanisms in Series...69 4.2.1 Constraints on Stiffness Matrix.............................................................69 4.2.2 Stiffness Modulation by using a De rivative of Stiffness Matrix and Wrench....................................................................................................70 4.2.3 Numerical Example...............................................................................74 5 CONCLUSIONS........................................................................................................78 APPENDIX A MATLAB CODES FOR NUMERICAL EXAMPLES IN CHAPTER TWO AND THREE.......................................................................................................................81 B MAPLE CODE FOR DERIVATIVE OF STIFFNESS MATRIX IN CHAPTER FOUR..........................................................................................................................9 8 LIST OF REFERENCES.................................................................................................105 BIOGRAPHICAL SKETCH...........................................................................................108

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vii LIST OF TABLES Table page 2-1 Spring properties of the comp liant couplings in Figure 2-6.....................................25 2-2 Positions of pivot points in terms of the inertial frame in Figure 2-6......................26 2-3 Spring properties of the comp liant couplings in Figure 2-7.....................................30 2-4 Positions of the fixed pivot points of the compliant couplings in Figure 2-7..........30 2-5 Positions and orientations of the coordinates systems in Figure 2-7.......................30 3-1 Spring properties of the mechanism in Figure 3-5...................................................52 3-2 Positions of pivots in ground in Figure 3-5..............................................................53 3-3 Positions of pivots in bottom side of body A in Figure 3-5.....................................53 3-4 Positions of pivots in top side of body A in Figure 3-5...........................................53 3-5 Positions of pivots in body B in Figure 3-5.............................................................53 4-1 Positions of pivot points in bod y E for numerical example in 4.1.2........................63 4-2 Positions of pivot points in bod y A for numerical example in 4.1.2........................63 4-3 Spring parameters with minimu m norm for numerical example 4.1.2....................63 4-4 Given optimal spring parameters for numerical example 4.1.2...............................64 4-5 Spring parameters closest to given spring parameters fo r numerical example 4.1.2.......................................................................................................................... 64 4-6 Positions of pivot point s for numerical example 4.1.3.............................................67 4-7 Initial spring parameters for numerical example 4.1.3.............................................67 4-8 Calculated spring paramete rs for numerical example 4.1.3.....................................68 4-9 Positions of pivot points in body A for numerical example 4.1.3............................68

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viii 4-10 Spring parameters of the compliant couplings for numerical example 4.2.3...........74 4-11 Positions of pivot point s for numerical example 4.2.3.............................................74 4-12 Spring parameters with no cons traint for numerical example 4.2.3.........................75 4-13 Spring parameters with body A fixed for numerical example 4.2.3........................75 4-14 Spring parameters with body A and body B fixed for numerical example 4.2.3.....77 A-1 Matlab function list..................................................................................................81

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ix LIST OF FIGURES Figure page 1-1 Planar robot with variable geometry base platform...................................................4 1-2 Adaptive vibration absorber.......................................................................................4 1-3 Parallel topology 6DOF w ith adjustable compliance.................................................5 2-1 Spring in a line space.................................................................................................9 2-2 Spring arrangements in a line space. (a) parallel and (b) series..............................10 2-3 Planar compliant coupling connecting body A and the ground...............................11 2-4 Small change of pos ition of P1 due to a small twist of body A...............................13 2-5 Planar compliant coupling joining two moving bodies............................................15 2-6 Mechanism having two compliant mechanisms in series........................................23 2-7 Mechanism consisting of four rigid bodies connected to each other by compliant couplings in a hybrid arrangement...........................................................................32 3-1 Spatial compliant coupling joining body A and the ground.....................................34 3-2 Unit vector expressed in a polar coordinates system...............................................34 3-3 Small change of pos ition of P1 due to a small twist of body A...............................36 3-4 Spatial compliant coupling joining two moving bodies...........................................39 3-5 Mechanism having two compliant parallel mechanisms in series...........................52 4-1 Compliant parallel mechanism with N number of couplings...................................58 4-2 Poses of the complia nt parallel mechanism for numerical example 4.1.3...............69 4-3 Poses of the compliant mechanism with body B fixed............................................76 4-4 Poses of the compliant m echanism with no constraint............................................76

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x 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 PARALLEL MECHANISMS WITH VARIABLE COMPLIANCE By Hyun Kwon Jung May 2006 Chair: Carl D. Crane III Major Department: Mechanical and Aerospace Engineering Compliant mechanisms can be considered as planar/spatial springs having multiple degrees of freedom rather than one freedom as line springs have. The compliance of the mechanism can be well described by the stiffness matrix of the mechanism which relates a small twist applied to the mechanism to the corresponding wrench exerted on the mechanism. A derivative of the spring wrench connec ting two moving rigid bodies is derived. By using the derivative of the spring wrench, the stiffness matrices of compliant mechanisms which consist of rigid bodies c onnected to each other by line springs are obtained. It is shown that the resultant co mpliance of two compliant parallel mechanisms that are serially arranged is not the summation of the compliances of the constituent mechanisms unless the external wrench applied to the mechanism is zero. A derivative of the stiffness matrix of planar compliant mechanisms with respect to the twists of the constituent rigid bodies and th e spring parameters such as the stiffness coefficient and free length is obtained. It is shown that the compliance and the resultant

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xi wrench of a compliant mechanism may be controlled at the same time by using adjustable line springs.

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1 CHAPTER 1 INTRODUCTION 1.1 Motivation Robots have been employed successfully in applications that do not require interaction between the robot and the e nvironment but require only position control schemes. For instance, arc welding and paintin g belong to this category of application. There are many other operations involving c ontact of the robot and its environment. A small amount of positional error of the robot system, which is almost inevitable, may cause serious damage to the robot or the obj ect with which it is in contact. Compliant mechanisms, which may be inserted between the end effecter and the last link of the robotic manipulator, can be a solution to this problem. Compliant mechanisms can be considered as spatial springs having multiple degrees of freedom rather than one freedom as line springs have. A small force/torque applied to the compliant mechanism generates a small displacement of the compliant mechanism. This relation is well described by the compliance matrix of the mechanism. RCC (Remote Center of Compliance) devices, developed by Whitney (1982), are one of the most successful compliant mechanisms. They have a unique compliant property at a specific operation point and are mainly used to compensate positional errors during tasks such as inserting a peg into a chamfered hole. Compliant mechanisms can also be employed for force control applications by us ing the theory of Kinestatic Control which was proposed by Griffis (1991). Kinestatic C ontrol varies the position of the last link of

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2 the manipulator to control the position and cont act force of the distal end of the robotic manipulator at the same time with th e compliance of the mechanism in mind. Mechanisms with variable compliance, whic h is the topic of this dissertation, are believed to have severa l advantages over mechanisms having fixed compliance. Since RCC devices typically have a specific operation point, if the length of the peg to be inserted is changed, a different RCC device should be employed to do insertion tasks unless the RCC device has variable compliance. As for force control tasks, each task may have an optimal compliance. With variable compliant mechanisms, several different tasks involving different force ranges can be accomplished without having to physically change the compliant mechanism. Variable compliant mechanisms also can improve the performance of humanoid robot parts such as ankles and wrists, a nd animals are believed to have physically variable leg complian ce and utilize it when running and hopping (see Hurst et al. 2004). Many compliant mechanisms including RCC devices have been designed typically based on parallel kinematic mechanisms. Parallel kinematic mechanisms contain positive features compared to serial mechanisms su ch as higher stiffness, compactness, and smaller positional errors at the cost of a smaller workspace and increased complexity of analysis. In this dissertation mechanisms having two compliant parallel mechanisms in a serial arrangement as well as compliant pa rallel mechanisms are investigated. These mechanisms may have a trade-off of characteristics relative to traditional parallel and serial mechanisms. 1.2 Literature Review The concepts of twists and wrenches were introduced by Ball (1900) in his groundbreaking work A Treatise on the Theory of Screws These concepts are employed

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3 throughout this dissertation to describe a small (or instantane ous) displacement of a rigid body and a force/torque applied to a body (Crane et al. 2006). The compliance of a mechanism can be well described by the stiffness matrix which is a 6 6 matrix for a spatial mechanism and a 3 3 matrix for a planar mechanism. Using screw theory, Dimentberg (1965) studied properties of an elastically suspended body. Loncaric (1985) used Lie groups rather than screw theory to study symmetric spatial stiffness matrices of compliant mechanisms assuming that the springs are in an equilibrium position and derived a constraint that makes the number of independent elements of symmetric 6 6 stiffn ess matrices 20 rather than 21. Loncaric (1987) also defined a normal form of the stiffness matrix in which rotational and translational parts of the stiffne ss matrix are maxi mally decoupled. Griffis (1991) presented a global stiffne ss model for compliant parallel mechanisms where he used the term global to state that the springs are not restricted to an unloaded equilibrium position. Griffis (1991) also s howed that the stiffness matrix is not symmetric when the springs are deflected from the equilibrium positions due to an external wrench. Ciblak and Lipkin (1994 ) decomposed a stiffness matrix into a symmetric and a skew symmetric part and showed the skew symmetric part is negative one-half the externally applied load expre ssed as a spatial cross product operator. Compliant parallel mechanisms have been investigated by a number of researchers to realize desired compliances because of its high stiffness, compactness, and small positional errors. Huang and Schimmels ( 1998) obtained the bounds of the stiffness matrix of compliant parallel mechanisms which consist of simple elastic devices and proposed an algorithm for synthesizing a realiz able stiffness matrix with at most seven

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4 simple elastic devices. Roberts (1999) a nd Ciblak and Lipkin (1999) independently developed algorithms for implementing a realizable stiffness matrix with r number of springs where r is the rank of the stiffness matrix. As for serial robot manipulators, Salis bury (1980) derived the stiffness mapping between the joint space and the Cartesian space. Chen and Kao (2000) showed that the formulation of Salisbury (1980) is only valid in the unloaded equilibrium pose and derived the conservative congr uence transformation for stiffness mapping accounting for the effect of an external force. Figure 1-1. Planar robot with variable geom etry base platform (from Simaan and Shoham 2002). Figure 1-2. Adaptive vibration ab sorber (from Ryan et al. 1994).

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5 Planar/spatial compliant mechanisms are in general constructe d with rigid bodies which are connected to each other by simple springs. The stiffness matrix of the mechanism depends on the geometry of th e mechanism and the properties of the constituent springs such as stiffness coefficient and free length. To realize variable compliant mechanisms, variable geometry or ad justable springs have been investigated. Simaan and Shoham (2002) studi ed the stiffness synthesis problem using a variable geometry planar mechanism. They changed the geometry of the base using sliding joints on the circular base (see Figure 1-1). Ryan et al. (1994) designed a variable spring by changing the effective number of coils of th e spring for adaptive-passive vibration control (see Figure 1-2). Figure 1-3. Parallel topology 6DOF with adjustable compliance (from McLachlan and Hall 1999). Cantilever beam-based variable compliant devices have been studied by a few researchers. Under an external force, a cantilever beam deflects and its deflection

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6 depends on the length of the beam and the Young’s modulus of the material. Henrie (1997) investigated a cantilever beam which is filled with magneto-rheological material and changed the Young’s modulus by changing the magnetic field. McLachlan and Hall (1999) devised a programmable pa ssive device by changing the length of the cantilever beam as shown in Figure 1-3. Hurst et al. (2004) presented an actuator with physically variable stiffness by using two motors and analyzed it for application to legged locomotion. 1.3 Problem Statement Planar/spatial compliant mechanisms consisting of rigid bodies which are connected to each other by adjustable complia nt couplings are investigated. For spatial mechanisms, each adjustable compliant coupling is assumed to have a spherical joint at each end and a prismatic joint with an adjustable line spring in the middle. For planar cases, spherical joints are replaced with revolute joints. Mechanisms having two compliant parallel mechanisms that are serially arranged are mainly investigated. The compliant mechanisms are not restricted to be in unloaded equilibrium configuration and this makes the analysis of the mechanism more complicated. Firstly a stiffness mapping of a line spring connecting two moving bodies is derived for planar and spatial cases. The line spring is assumed to have a fixed stiffness coefficient and free length at this stage. Th is stiffness mapping lead s to the derivation of the stiffness matrix of compliant mechanisms consisting of rigid bodies connected to each other by line springs. A derivative of the stiffness matrix of a compliant mechanism with respect to the twists of the constituent rigid bodies and the sp ring properties such as spring constant and free length is obtained. Since the compliant mechanism is assumed initially in static

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7 equilibrium under an external wrench, changing the spring constants and the free lengths of the constituent springs may result in the change of the resultant wrench and it may change the position of the compliant mechan ism. Stiffness modulation methods, which utilize adjustable line springs and vary the position of the robot where the compliant mechanism is attached, are investigated to realize a desired compliance and to regulate the position of the compliant mechanism.

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8 CHAPTER 2 STIFFNESS MAPPING OF PLANAR COMPLIANT MECHANISMS When a rigid body supported by a compliant coupling moves, the deflection and/or the directional change of the coupling may lead to a change of the force. In this chapter, a planar stiffness mapping model which maps a small twist of the body into the corresponding wrench variation is studied. To describe a small (or instantaneous) displacement of a rigid body and a force/torque applied to a body, the concepts of twist and wrench from screw theory are used throughout this dissertation (see Ball 1900 and Crane et al. 2006). Further, the notations of Kane and Levinson are also employed (see Kane and Levinson 1985) to describe spatial motions of rigid bodies. Specifically, as part of the notation, th e position of a point P embedded in body B measured with respect to a reference system embedded in body A will be written as AB Pr. The derivative of the displacement of this point P (embedded in body B in terms of a reference coordinate system embedded in body A) is denoted as AB Pr. The derivative of an angle of body B with respect to a body A is denoted by AB and its magnitude is denoted by AB. The twist of a body B with respect to a body A will be denoted by ABD. 2.1 Spring in a Line Space The analysis of rigid bodies which are constrained to move in a line space and connected to each other by line springs is presented because it is simple and intuitive and a similar approach can be applied for planar and spatial compliant mechanisms. Figure

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9 2-1 illustrates a line spring connecting body A to ground. Body A is allowed to move only on a line along the axis of the spring. The spring has a spring constant k and a free length o x The position of body A can be expressed by a scalar x and the force from the spring by a scalar f Figure 2-1. Spring in a line space. The spring force can be written as ()o f kxx=Š (2.1) The relation between a small change of the position of body A and the corresponding small force variation can be obtained by taking a derivative of Eq. (2.1) as f kx=. (2.2) When springs are arranged in parallel as shown in Figure 2-2 (a), the resultant spring constant Rk may be derived as Eq. (2.3). 12 R f kxkxkx==+ 12 Rkkk=+ (2.3) A x k

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10 (a)(b) Figure 2-2. Spring arrangements in a line space. (a) parallel and (b) series. For a serial arrangement as shown in Figure 2-2 (b), the resultant spring constant Rk which maps a small change of position of body B into a small force variation upon body B may be written as Eq. (2.4). ()12RBABA f kxkxkxx===Š 2 12 ABk x x kk= + 111 12 12 12 RRkk korkkk kkŠŠŠ==+ + (2.4) It is obvious from Eqs. (2.3) and (2.4) that th e resultant spring constant of springs in parallel is the summation of each spring constant and that the resultant compliance of springs in series is the summation of each spring compliance. This statement is valid for springs in a line space. A B x 1k B 2k A x A 1k 2k x

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11 2.2 A Derivative of Planar Spring Wrench Joining a Moving Body and Ground In this section, a derivative of the planar spring wrench joining a moving body and ground, which was presented by Pigoski (1993) and led to the stiffness mapping of a planar parallel mechanism, is restated. Fi gure 2-3 illustrates a rigid body connected to ground by a compliant coupling. The compliant coupling has a revolute joint at each end and a prismatic joint with a spring in the middle part. Body A can translate and rotate in a planar space. Figure 2-3. Planar compliant coup ling connecting body A and the ground. The force which the spring exerts on body A can be written as ()okll =Š f$ (2.5) where k, l, and ol are the spring constant, current spring length, and spring free length of the compliant coupling, respectively. Also $ represents the unitized Plcker coordinates of the line along the compliant coupling which may be written as 01 EEEA PP == SS $ rSrS (2.6) A P0 P1 S

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12 where S is the unit vector along the compliant coupling and 0 EE Pr and 1 EA Pr are the position of the pivot point PO in the ground body and that of P1 in body A, respectively, measured with respect to a reference coordina te system attached to ground. To obtain the stiffness mapping, a small twist EAD is applied to body A and the corresponding change of the spring force will be obtained. The twist EAD may be written in axis coordinates as 0 EA EA EA = r D (2.7) where EA or is the differential of the position of point O in body A which is coincident with the origin of the inertial frame E measured with respect to the inertial frame. In addition EA is the differential of the angle of body A with respect to the inertial frame. Taking a derivative of Eq. (2.5) with the consideration that $ is a function of in planar cases yields () (1)o oklkll l klkl l =+Š =+Š f$$ $ $ (2.8) where 0EE P = S $ S r (2.9) and where S is a unit vector perpendicular to S.

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13 Figure 2-4. Small change of position of P1 due to a small twist of body A. Using screw theory, the variation of position P1 can be written as 11 AAEA EEEA PoP=+ rr r (2.10) It may be decomposed into two perpendicular vectors, one along S and one along S. These vectors correspond to the change of the spring length l and the change of the direction of the spring l as shown in Figure 2-4. The change of the position of point P1 may thus also be written as () 111EAEAEA PPPll =+ =+ SS rrSSr S S (2.11) From Eqs. (2.10), (2.11), (2.6) and (2.7) expressions for land l may be obtained as 11 1 EAEAEAEA PoP EAEAEA oP TEAl ==+ =+ = rSrS rS rS rS $D (2.12) P1 l l l E P 0 1 EA Pr

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14 11 1 EAEAEAEA PoP EAEEA oP T EAl ==+ =+ = SSS rr r SS r r $ D (2.13) and where 1 EA P = S $ S r. (2.14) All terms of Eq. (2.14) are known. From Eqs. (2.8), (2.12), and (2.13), a derivativ e of the spring force may be written as [] (1) (1)o T TEAEA o EA Fl klkl l l kk l K =+Š =+Š = $ f$ $$ $$DD D (2.15) where [] (1)T T o Fl Kkk l =+Š $$ $$. (2.16) []F K is the stiffness matrix of a planar compliant coupling and maps a small twist of body A into the corresponding variation of th e wrench. The first term of Eq. (2.16) is always symmetric and the second term is not. When the spring deviates from its equilibrium position due to an external wren ch, the second term of Eq. (2.16) doesn’t vanish and it makes the stiffness matrix asymmetric.

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15 2.3 A Derivative of Spring Wrench Joining Two Moving Bodies Figure 2-5. Planar compliant coupling joining two moving bodies. In this section a derivative of the spring wrench joining two moving bodies is derived, which supersedes the result of the pr evious section and is essential to obtain a stiffness mapping of springs in complicated arrangement. Figure 2-5 illustrates two rigid bodies connected to each other by a compliant coupling with a spring constant k, a free length ol, and a current length l. Body A can move in a planar space and the co mpliant coupling exerts a force f to body B which is in equilibrium. The spring force may be written by ()okll =Š f$ (2.17) where 12EAEB PP == SS $ rSrS (2.18) P1 S P2 Body B Body A E

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16 and where S is a unit vector along the compliant coupling and 1EA Pr and 2EB Pr are the position vector of the point P1 in body A and that of point P2 in body B, respectively, measured with respect to the refere nce system embedded in ground (body E). A small twist of body B with respect to an inertial frame E EBD is applied and it is desired to find the corresponding chan ge of the spring force. The twist EBD may be written as EBEAAB=+ DDD (2.19) where EB o EB EB = r D (2.20) EA o EA EA = r D (2.21) AB o AB AB = r D (2.22) and where the notation from Kane and Levins on (1985) is employed as stated in the beginning of this chapter. For example, EB or is the differential of the coordinates of point O, which is in body B and coincident with the origin of the inertial frame, measured with respect to the inertial frame and EA is the differential of angle of body A with respect to the inertial frame. The derivative of the spring force, Eq. (2.17), can be written as ()EE oklkll=+Š f$$. (2.23)

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17 From the twist equation, the variation of the position of point P2 in body B with respect to body A can be expressed as 22 BBBAB AAA PoP=+ rr r (2.24) where B 2 P Ar is the position of P2, which is embedded in body B, measured with respect to a coordinate system embedded in body A which at this instant is coincident and aligned with the reference system attached to gr ound. It can also be decomposed into two perpendicular vectors along S and A S which is a known unit vector perpendicular to S. These two vectors correspond to the change of the spring length l and the directional change of the spring l in terms of body A in a way that is analogous to that shown in Figure 2-4. Thus the variation of position of point P2 in body B in terms of body A can be written as () 222 AA ABABAB PPP All =+ =+ SS rrSSr S S (2.25) where 1 A A A AA P = S $ S r. (2.26) From Eqs. (2.24) and (2.25), l and l can be obtained as 22 2 ABABABAB PoP ABABAB oP TABl ==+ =+ = rSrS rS rS rS $D (2.27)

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18 22 2 AAA ABABABAB PoP AA ABABAB oP AT ABl ==+ =+ = SSS rr r SS r r $ D (2.28) where 2 A A A AB P = S $ S r. (2.29) It is important to note that screw A $ has the same direction as A $ but has a different moment term. Only E$ is unknown in Eq. (2.23). It is a derivative of the unit screw along the spring in terms of the inertial frame and may be written as 11 E E EAEAE PP = + S $ rSrS. (2.30) Using an intermediate frame attached to body A, a derivative of the direction cosine vector may be written as EAEA=+ SS S. (2.31) Then, E$ may be decomposed into three screws as

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19 () () 11 11 1 1 1 E E EAEAE PP AEA EAEAAEA PP EA A EA EAA EAEA P P P = + + = ++ =++ S $ rSrS S S rSrS S S 0 S rS rS r S. (2.32) Since S is a function of alone from the vantage of body A and l is already described in Eq. (2.28), the first sc rew in Eq. (2.32) can be written as 1 111A A AAAT AB EAA A P EA Pl ll === S S $$$ D rS S r. (2.33) As for the second screw in Eq. (2.32), EA S has the same direction with A S and a magnitude of EA and thus may be written as A EA EA = S S. (2.34) Then the second screw in Eq. (2.32) can be expressed as () [] 1 1001A EA EA A EAEA EA P PEA AA EA EA = == S S S r S r $$ D. (2.35) As to the third screw in Eq. (2.32), 1 EA Pr can be decomposed into two perpendicular vectors along S and A S respectively and may be written as

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20 () 11 11 EAEAEAEA PoP AA EAEA PP =+ =+rr r SS rSSr (2.36) where 11 1 EAEAEAA PoP EAEAA oP TEA =+ =+ = rSrS rS rS rS $D (2.37) 11 1 AAA EAEAEAA PoP AA EAEAA oP AT EA =+ =+ = SSS rr r SS r r $ D. (2.38) By combining Eqs. (2.36), (2.37), and (2.38) 1 EA Pr can be written as () () 111 AA EAEAEA PPP ATA TEAEA =+ =+ SS rrSSr $S $DSD. (2.39) The third screw in Eq. (2.32) can now be written as () [] 10 0001 1ATA EA TEAEA P ATA AT EA EA T ATA EAEA = + == Š =Š=Š 0 0 $S rS $DSDS 0 0 $S $ DS D $$ DD (2.40)

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21 since ()1A =Š S Sk. Among all unknowns in Eq. (2.23), lwas obtained in Eq. (2.27) and all the terms of E$ were obtained through Eqs. (2.33), (2.35), and (2.40). Hence the derivative of the spring force can be rewritten as [] [] [] [] () 0 1 ()0010 1 0 (1)()0010 1EE o AATAAT TABABEAEA o T AATAA TABEA o o ABE FMklkll kkll l l kkkll l KK =+Š =+Š+Š =+Š+ŠŠ =+ f$$ $$$$ $$DDDD $$$$ $$DD DDA (2.41) where [] (1)AAT T o Fl Kkk l =+Š $$ $$ (2.42) [] [] []()001001T AA MoKkll =ŠŠ $$ (2.43) It is important to note that []M K is a function of the external wrench. To prove it, Eq. (2.18) is explicitly expressed in a planar coordinate system and 1T EA Pxypp=r to yield c s csxy p p = Š $ (2.44)

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22 s c scA xy p p Š = ŠŠ $ (2.45) where ()cosc= and ()sins=. By substituting Eq. (2.45) for A $ in Eq. (2.43), []M K can be expressed as []0000 ()0000 00 y Mox yx s f K kllcf scff ŠŠ =Š= ŠŠ (2.46) where T xyzffm = f is the initial spring wrench. As shown in Eq. (2.41), the derivative of the spring wrench joining two rigid bodies depends not only on a relative twist between two bodies but also on the twist of the intermediate body, in this case body A, in terms of the inertial frame. []F K which maps a small twist of body B in terms of body A into the corresponding change of wrench upon body B is identical to the stiffness matrix of the spring assuming the body A is stationary. []M K is newly introduced from this research and results from the motion of the base frame, in this case body A, and is a function of the initial external wrench. 2.4 Stiffness Mapping of Planar Compliant Parallel Mechanisms in Series The derivative of the spring wrench derive d in the previous section is applied to obtain the stiffness mapping of compliant parallel mechanisms in series as shown in Figure 2-6.1 Body A is connected to ground by three compliant couplings and body B is connected to body A in the same way. Each compliant coupling has a revolute joint at 1 Figure 2-6 shows a coordinate system attached to each of three bodies for illustration purposes. In this analysis, the three coordinate systems are assumed to be coincident and aligned at each instant.

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23 each end and a prismatic joint with a spring in the middle part. It is assumed that an external wrench extw is applied to body B and that bot h body B and body A are in static equilibrium. The positions and orientations of bodies A and B and the spring constants and free lengths of all constituent springs are given. The stiffness matrix [] K which maps a small twist of body B with respect to the ground EBD into a small wrench variation extw is desired to obtain. 1k4k5k6k3k2k Figure 2-6. Mechanism having two compliant mechanisms in series. The static equilibrium equation of bodies B and A can be written by 123 456ext=++ =++ wfff fff (2.47) where if are the forces from the compliant couplings.

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24 The stiffness matrix is derived by taking a derivative of the static equilibrium equation, Eq. (2.47), to yield [] 123 456EB extK = =++ =++ wD fff fff. (2.48) The derivatives of spring forces can be written by Eqs. (2.49) and (2.50) since springs 4, 5, and 6 connect body A and ground and spring s 1, 2, and 3 join two moving bodies. [] [] [] [] 456 456 ,EAEAEA FFF EA F RLKKK K ++=++ = fffDDD D (2.49) [] [] [] [] [] [] [] [] 123 123 123 ,,ABABAB FFF EAEAEA MMM ABEA FM RURUKKK KKK KK ++=++ +++ =+ fffDDD DDD DD (2.50) where [][]6 4FF RLi iKK== [][]3 1FF RUi iKK== [][]3 1MM RUi iKK==. From Eqs. (2.49), (2.50), and (2.19) twist EAD can be written as [] [] [] [] ()[] ,,, ,, EAABEA FFM RLRURU EBEAEA FM RURUKKK KK =+ =Š+ DDD DDD (2.51) [][][]()[] 1 ,,,, EAEB FFMF RLRURURUKKKKŠ=+Š DD (2.52)

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25 Substituting Eq. (2.52) for EAD in Eq. (2.49) and comparing it with Eq. (2.48) yields the stiffness matrix as [] [] [][][][]()[] 1 ,,,,, EBEA F RL EB FFFMF RLRLRURURUKK KKKKK Š= =+Š DD D (2.53) [][][][][]()[]1 ,,,,, FFFMF RLRLRURURUKKKKKKŠ=+Š. (2.54) It was generally accepted that the resultant compliance, which is the inverse of the stiffness, of serially connected mechanisms is the summation of the compliances of all constituent mechanisms (see Griffis 1991). Ho wever, the stiffness matrix derived from this research shows a different result. Taking an inverse of the stiffness matrix Eq. (2.54) yields [][][][][][]11111 ,,,,, FFFMF RLRURURURLKKKKKKŠŠŠŠŠ=+Š (2.55) The third term in Eq. (2.55) is newly introdu ced in this research and it does not vanish unless the external wrench is zero. A numerical example is pres ented to support the derived stiffness mapping model. The geometry information, spring properties of the mechanism shown in Figure 2-6, and the external wrench extw are given in Tables 2-1 and 2-2. []0.010.020.03T extNNNcm =Š w Table 2-1. Spring properties of the compliant couplings in Figure 2-6. Spring No. 1 2 3 4 5 6 Stiffness constant k 0.2 0.3 0.4 0.5 0.6 0.7 Free length ol 5.0040 2.2860 4.9458 5.5145 3.1573 5.2568 (Unit: N/cm for k, cm for lo)

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26 Table 2-2. Positions of pivot points in te rms of the inertial frame in Figure 2-6. Pivot points E1 E2 E3 B1 B2 B3 X 0.0000 1.5000 3.0000 0.0903 1.7063 1.9185 Y 0.0000 1.2000 0.5000 9.8612 8.6833 10.6721 (Unit: cm) Table 2-2. Continued. A1 A2 A3 A4 0.9036 2.5318 2.7236 1.6063 4.5962 3.4347 5.4255 5.4659 Two stiffness matrices are obtained. 1[] K is from Eq. (2.54) and 2[] K from the same equation ignoring [], M RUK. 10.0108/0.0172/0.0797 []0.0172/0.3447/0.8351 0.09970.82512.6567 NcmNcmN KNcmNcmN N NNcm ŠŠ =Š Š 20.0111/0.0157/0.0874 []0.0162/0.3462/0.8124 0.09690.81502.6129 NcmNcmN KNcmNcmN N NNcm ŠŠ =Š Š The result is evaluated in the following way: 1. A small wrench Tw is applied in addition to extw to body B and twists 1 EBD and 2 EBD are obtained by multiplying the inverse matrices of the stiffness matrices, []1 K and []2 K respectively, by Tw as of Eq. (2.48). Corresponding positions for body B are then determined, based on the calculated twists 1 EBD and 2 EBD. 2. EAD is calculated by multiplying the inverse matrix of [], F RLK by Tw as of Eq. (2.49). The position of body A is then determined from this twist. 3. The wrench between body B and body A is calculated for the two cases based on knowledge of the positions of bodies A and B and the spring parameters. The change in wrench for the two cases is determined as the difference between the new equilibrium wrench and the original. The changes in the wrenches are named ,1 ABw and ,2 ABw which correspond to the matrices []1 K and []2 K 4. The given change in wrench Tw is compared to ,1 ABw and ,2 ABw.

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27 The given wrench Tw and the numerical results are presented as below. []5100.50.20.4TŠ=w []3 1100.76740.11860.0672T EBŠ=Š D []3 2100.82080.11520.0679T EBŠ=Š D []3100.23500.08980.0330T EAŠ=Š D []5100.50000.19980.3996T EAŠ= w []5 ,1100.50040.19690.3919T ABŠ= w []5 ,2100.56660.23000.0117T ABŠ= w where EAw is the wrench between body A and ground. The unit for the wrenches is []T N NNcm and that of the twists is []Tcmcmrad. The difference between EAw and Tw is small and is due to the fact that the twist was not infinitesimal. The difference between ,1 ABw and Tw is also small and is most likely attributed to the same fact. However, the difference between ,2 ABw and Tw is not negligible. This indicates that the stiffness matrix formula derived in this research produces the proper result and that the term [], M RUK cannot be neglected in Eq. (2.54). 2.5 Stiffness Mapping of Planar Compliant Parallel Mechanisms in a Hybrid Arrangement Figure 2-7 depicts a compliant mechanism having compliant couplings in a serial/parallel arrangement. Each compliant coupling has a revolute joint at each end and a prismatic joint with a spring in the middle part. An external wrench extw is applied to

PAGE 39

28 body T and body T is separately connected to body B, body C, and body D by three compliant couplings. Body B, body C, and body D are connected to ground by two compliant couplings. It is assumed that all bodies are in static equilibrium. It is desired to find the stiffness matrix which maps a small twist of body T in terms of ground ETD to the corresponding wrench variation extw. The stiffness constants and free lengths of all constituent springs and the geometry of the mechanism are assumed to be known. The stiffness matrix of the mechanism can be derived by taking a derivative of the static equilibrium equations. The static equilibrium equations may be written as 789 ext=++ wfff (2.56) 712=+ fff (2.57) 834=+ fff (2.58) 956=+ fff (2.59) where extw is the external wrench and if is the force of the i-th spring. Derivatives of Eqs. (2.56)-(2.59) can be written as [] 789 ext ET RK =++ = wfff D (2.60) 712=+ fff (2.61) 834=+ fff (2.62) 956=+ fff (2.63) where []R K is the stiffness matrix and ETD is a small twist of body T in terms of the inertial frame attached to the ground.

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29 Using Eqs. (2.15) and (2.41), Eq. (2.61) can be rewritten as [] [] [] [] [][]() 7 77 1212 BTEB FM EBEBEB FFFFKK KKKK =+ =+=+fDD DDD. (2.64) where BTD is a small twist of body T in terms of body B and EBD is that of body B in terms of the inertial frame. []F i K and []M iK are the matrices for i-th spring defined by Eqs. (2.42) and (2.43) respectively. The twist of body T can be decomposed as ETEBBT=+DDD. (2.65) From Eqs. (2.64) and (2.65), EBD can be expressed in terms of ETD as Eq. (2.66). [] ()[] [][]() 7712 ETEBEBEB FMFFKKKKŠ+=+ DDDD [][][][]()[] 1 12777 EBET FFFMFKKKKKŠ=++Š DD (2.66) By substituting Eq. (2.66) for EBD in Eq. (2.64), 7f can be expressed in terms of ETD as [][]()[][][][]()[] 1 7 1212777 ET FFFFFMFKKKKKKKŠ=+++Š fD. (2.67) Analogously, 8f and 9f can be written respectively as [][]()[][][][]()[] 1 8 3434888 ET FFFFFMFKKKKKKKŠ=+++Š fD (2.68) [][]()[][][][]()[] 1 9 5656999 ET FFFFFMFKKKKKKKŠ=+++Š fD. (2.69) Finally from Eq. (2.60) and Eqs. (2.67)-(2.6 9), the stiffness matrix can be written as

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30 [][][]()[][][][]()[] [][]()[][][][]()[] [][]()[][][][]()[]1 1212777 1 3434888 1 5656999 FFFFFMF R FFFFFMF FFFFFMFKKKKKKKK KKKKKKK KKKKKKKŠ Š Š=+++Š ++++Š ++++Š. (2.70) A numerical example of the compliant mechanism depicted in Figure 2-7 is presented. The four bodies are identical equilateral triangles whose edge length is 2 cm. Four coordinate systems, B, C, D, and T are attached to body B, C, D, and T, respectively and their positions of origin and orientations in terms of the inertial frame are given in Table 2-5. The spring properties and the positions of the fixed pivot points are given in Table 2-3 and Table 2-4, respectively. The external wrench is given as 0.1 0.1 0.2extN N N cm = w Table 2-3. Spring properties of the compliant couplings in Figure 2-7. Spring No. 1 2 3 4 5 6 7 8 9 Stiffness constant k 0.40 0.43 0.49 0.52 0.58 0.61 0.46 0.55 0.64 Free length ol 2.2547 2.40141.59101.84501.70772.26952.3924 2.22001.8711 ( Unit: /Ncm for k and cm for ol) Table 2-4. Positions of the fixed pivot points of the compliant couplings in Figure 2-7. A1 A2 A3 A4 A5 A6 X 1.6700 4.4600 13.3449 14.6731 8.2300 4.9400 Y 4.4333 1.3964 3.2500 6.8400 14.1400 13.4943 ( Unit: cm) Table 2-5. Positions and orientations of the coordinates systems in Figure 2-7. Bo Co Do To X 4.0746 12.2367 7.2479 8.3174 Y 5.1447 4.4972 12.7430 6.9958 -0.8112 1.2283 3.8876 0.5818 ( Unit: cm for x, y and radians for )

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31 Two stiffness matrices are obtained. 1[] K is from Eq. (2.70) and 2[] K from the same equation ignoring all []M K ’s which are newly introduced in this research. 10.2501/0.0216/1.7651 []0.0216/0.2910/2.6661 1.66512.566138.5180NcmNcmN KNcmNcmN N NNcm Š = Š 20.2463/0.0172/1.7844 []0.0315/0.2888/2.5749 1.61392.573038.2221NcmNcmN KNcmNcmN N NNcm Š = Š To evaluate the result, a small wrench w is applied to body T and the static equilibrium pose of the mechanism is obtained by a numerically iterative method. From the equilibrium pose of the mechanism, the twist of body T with respect to ground ETD is obtained as 40.5 100.2 0.3N N N cmŠ = w 0.0050 0.0058 0.0006ETcm cm rad =Š D Then the twist ETD is multiplied by both of the stiffness matrices to see if they result in the given small wrench w. 4 110.4997 []100.2000 0.3020ETN KN N cmŠ == wD

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32 4 220.4502 []100.2726 0.6622ETN KN N cmŠ == wD The numerical example indicates that 1[] K produces the given wrench w with high accuracy and that 2[] K involves significant errors. B D T extw Figure 2-7. Mechanism consisting of four rigid bodies connected to each other by compliant couplings in a hybrid arrangement.

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33 CHAPTER 3 STIFFNESS MAPPING OF SPATIAL COMPLIANT MECHANISMS Taking a similar approach adopted for planar compliant mechanisms, a stiffness mapping of spatial compliant mechanisms is presented. 3.1 A Derivative of Spatial Spring Wrench Joining a Moving Body and Ground Figure 3-1 depicts a rigid body and a compliant coupling connecting the body and the ground. The compliant coupling has a spherical joint at each end and a prismatic joint with a spring in the middle. Body A can translate and rotate in a spatial space. The wrench which the spring exerts on body A can be written as ()okll =Šw$ (3.1) where k, l, and ol are respectively the spring constant, current spring length, and spring free length of the compliant coupling. Further, $ represents the unitized Plcker coordinates of the line along the compliant coupling which may be written by 01 EEEA PP == SS $ rSrS (3.2) where S is the unit vector along the compliant coupling and 0 EE Pr and 1 EA Pr are the position of the pivot point PO in the ground body and that of P1 in body A, respectively, measured with respect to a reference coordinate system attached to ground.

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34 Figure 3-1. Spatial compliant coupling joining body A and the ground. S 1e 2e 3e Figure 3-2. Unit vector expresse d in a polar coordinates system. A polar coordinates system can be used to express the unit vector S (see Figure 3-2) as sincos sinsin cos = S (3.3) It is obvious from Eqs. (3.2) and (3.3) that $ is a function of and since 0 EE Pr is fixed on ground. Hence a derivative of the spring wrench can be written as () (1)o oklkll l klkll l =+Š =+Š+ w$$ $$ $ (3.4) P0 P1 S E Body A

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35 where 0 EE P = S $ S r (3.5) 0 EE P = S $ S r. (3.6) By taking a derivative of Eq. (3.3), S and S can be explicitly written by sinsin sincos 0 Š = S (3.7) coscos cossin sin = Š S (3.8) Since S is not a unit vector, a unit vector S is introduced as sin cos 0 Š = S (3.9) sin = SS (3.10) Hence Eq. (2.8) can be rewritten as (1)sinol klkll l =+Š+ $$ w$ (3.11) where

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36 0 EE P = S $ S r. (3.12) It is important to note that $ and $ are the unitized Plcker coordinates of the lines perpendicular to S and go through the pivot point P0. sin l l 1 A E Pr l l sin l Figure 3-3. Small change of position of P1 due to a small twist of body A. In Eq. (3.11) l, sinl and l can be considered as the change of the spring length and the changes of the direction of the spring (see Figure 3-3). These values correspond to the projections of the variation of position P1, 1A E Pr, onto the orthonormal vectors S, S and S respectively. Thus 1A E Pr can be rewritten as

PAGE 48

37 () 1111sinEAEAEAEA PPPPlll =++ =++ SSSS rrSSrr SS S (3.13) From the twist equation, the variation of position P1 can be written as 11AAEA EEEA PoP=+rr r (3.14) where 0 EAr is the differential of the position of point O in body A which is coincident with the origin of the inertial frame E measured with respect to the inertial frame. EA is the differential of the angle of body A with respect to the inertial frame. From Eqs. (2.11) and (2.10), l, sinl and l can be expressed as 11 1 EAEAEAEA PoP EAEAEA oP TEAl ==+ =+ =rSrS rS rS rS $D (3.15) 11 1sinEAEAEAEA PoP EAEEA oP T EAl ==+ =+ = SSS rr r SS r r $ D (3.16) 11 1 EAEAEAEA PoP EAEEA oP T EAl ==+ =+ = SSS rr r SS r r $ D (3.17) where

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38 0 EA EA EA = r D (3.18) 1 EA P = S $ S r (3.19) 1 EA P = S $ S r. (3.20) It is important to note that $ and $ are the unitized Plcker coordinates of lines perpendicular to S which pass through the pivot point P1 in body A and EAD is a small twist of body A with respect to ground. Substituting Eqs. (2.12), (3.16), and (2.13) for l, sinl and l in Eq. (3.11) yields [] (1)sin (1)o TT TEAEA o EA Fl klkll l l kk l K =+Š+ =+Š+=$$ w$ $$$$ $$DD D (3.21) where [] (1)TT T o Fl Kkk l =+Š+ $$$$ $$ (3.22) []F K is the stiffness matrix of a spatial compliant coupling and maps a small twist of body A into the corresponding variation of the wrench. The first term of Eq. (2.16) is

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39 always symmetric and the second term is not. When the spring deviates from its equilibrium position due to an external wrench, the second term of Eq. (2.16) doesn’t vanish and it makes the stiffness matrix asymmetric. This result agrees with the works of Griffis (1991). 3.2 A Derivative of Spring Wren ch Joining Two Moving Bodies Figure 3-4. Spatial compliant coupling joining two moving bodies Figure 3-4 illustrates two rigid bodies connected to each other by a compliant coupling with a spring constant k, a free length ol, and a current length l. Both of body A and body B can move in a spatial space and the compliant coupling exerts a wrench w to body B which is in equilibrium. The spring wrench may be written as ()okll =Šw$ (3.23) where 12 EAEB PP == SS $ rSrS (3.24) P1 S Body B Body A E P2

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40 and where S is a unit vector along the compliant coupling and 1 EA Pr and 2 EB Pr are the position vectors of the point P1 in body A and that of point P2 in body B, respectively, measured with respect to the reference system embedded in ground (body E). It is desired to express a derivative of the spring wrench in terms of the twist of body B EBD and that of body A EAD The twist EBD may be expressed as EBEAAB=+DDD (3.25) where EB o EB EB = r D (3.26) EA o EA EA = r D (3.27) AB o AB AB = r D (3.28) and where EB or is the differential of point O, which is in body B and coincident with the origin of the inertial frame, measured with respect to the inertial frame and EB is the differential of angle of body B with respect to the inertial frame. EA or AB or EA and AB are defined in the same way. The derivative of the spring wrench in Eq. (2.17) can be written as ()EE oklkll=+Šw$$ (3.29) and it is required to express l and E$ in Eq. (2.23) in terms of the twists of the bodies. From the twist equation, the variation of position of point P2 in body B with respect to body A can be expressed as

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41 22BBBAB AAA PoP=+rr r (3.30) where B 2 P Ar is the position of P2, which is embedded in body B, measured with respect to a coordinate system embedded in body A which at this instant is coincident and aligned with the reference system attached to ground. It can also be decomposed by projecting it onto the orthonormal vectors S, A S and A S which are defined in a similar way as Eqs. (3.3), (3.9), and (3.8). These three ve ctors correspond to the change of the spring length l and the directional changes of the spring such as sinl and l in terms of body A in a way that is analogous to that shown in Figure 3-3. Thus the variation of position of point P2 in body B in terms of body A can be written as () 2222sinAAAA ABABABAB PPPP AAlll =++ =++ SSSS rrSSrr SS S. (3.31) From Eqs. (2.24) and (2.25), l in Eq. (2.23) can be obtained as 22 2ABABABAB PoP ABABAB oP TABl ==+ =+ = rSrS rS rS rS $D. (3.32) In the same way, sinl and l can be expressed as 22 2sinAAA ABABABAB PoP AA ABABAB oP A ABl ==+ =+ = SSS rr r SS r r $ D (3.33)

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42 22 2 AAA ABABABAB PoP AA ABABAB oP A ABl ==+ =+ = SSS rr r SS r r $ D (3.34) where 2 A A A AB P = S $ S r (3.35) 2 A A A AB P = S $ S r. (3.36) Now in Eq. (2.23), only E$ is yet to be obtained. It is a derivative of the unit screw along the spring in terms of the inertial frame and may be written as 11 E E EAEAE PP = + S $ rSrS. (3.37) Using an intermediate frame attached to body A, a derivative of the unit vector S can be written by EAEA=+SS S. (3.38) Thus E$ may be decomposed into three screws as

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43 () () 11 11 1 1 1 E E EAEAE PP AEA EAEAAEA PP EA A EA EAA EAEA P P P =+ +=++ =++ S $ rSrS S S rSrS S S 0 S rS rS r S. (3.39) Since S is a function of and from the vantage of body A and sinl and l were already described in Eqs. (2.28) and (3 .34), the first screw in Eq. (2.32) can be written as 1 1 1111 sin 1AA A A A EAA AA AA P EA EAEA P PP AA AATAAll ll l + ==+ + =+ =+ SS S S S rS SS SS r rr $$ $$$$ T AB D .(3.40) As to the second screw in Eq. (2.32), EA S can be decomposed onto three orthonormal vectors along S, A S and A S respectively, as () {} () () AAAA EAEAEAEA =++ SSSS S SSS S S (3.41)

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44 From the fact that S, A S and A S are unit vectors and perpendicular to each other (see Figure 3-3), each dot product of Eq. (3.41) can be expressed as () 0EA= SS (3.42) () AAA EAEAEA TT EA o EA AA EA ==Š =Š=Š SSS S S 00 r D SS (3.43) () AAA EAEAEA TT EA o EA AA EA == == SSS S S 00 r D SS (3.44) where []000T= 0. Hence, EA S can be rewritten as T T AA EAEAEA A A =Š+ 0 0 SS SDD S S (3.45) and the second screw in Eq. (2.32) can be expressed as

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45 () 1 1 1 T T AA EAEA A A EA T EAEA T P AA EAEAEA A A P A A EA P Š+ = Š+ =Š 0 0 SS DD S S S r S 0 0 SS rDD S S S S r 1 A T T EAEA A A A EA P T T AA EA A A + =Š+ S 0 0 DD S S S r 0 0 $$ D S S. (3.46) As to the third screw in Eq. (2.32), 1 EA Pr can be decomposed onto three orthonormal vectors along S, A S and A S respectively, as () 11 111 EAEAEAEA PoP AAAA EAEAEA PPP =+ =++ rr r SSSS rSSrr (3.47) The first dot product in Eq. (3.47) can be expressed as 11 1 EAEAEAA PoP EAEAA oP TEA =+ =+ =rSrS rS rS rS $D (3.48) In the same way, the second and third dot products in Eq. (3.47) can be written as

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46 11 1 AAA EAEAEAA PoP AA EAEAA oP T A EA =+ =+ = SSS rr r SS r r $ D (3.49) 11 1 AAA EAEAEAA PoP AA EAEAA oP T A EA =+ =+ = SSS rr r SS r r $ D (3.50) Finally, 1 EA PrS of the third screw in Eq. (2.32) can be expressed as () 1 TT AAAA EATEAEAEA P TT AAAA EAEA TT AAAA EAEA =++ =+ =Š $S$S rS$DSDDS $S$S DSDS $S$S DD (3.51) since S, A S and A S are unit vectors and perpendicular to each other (see Figure 3-3). Substituting Eq. (3.51) for 1 EA PrS of the third screw in Eq. (2.32) yields

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47 1 TT AAAA EA EAEA P TT AA EAEA A A T AA A A = Š =Š =Š 0 0 $S$S rS DD 0 0 $$ DD S S 0 0 $ S S T EA $ D. (3.52) By replacing l and E$ in Eq. (2.23) with Eqs. (2.27) (2.33), (2.35), and (2.40) and sorting it into the twists, the derivative of the spring wrench can be rewritten as [] [] ()EE o ABEA FMklkll KK =+Š =+ w$$ DD (3.53) where [] (1)AATAAT T o Fl Kkk l =+Š+ $$$$ $$ (3.54) [] ()T T T T AA A A AA AA MoKkll =ŠŠ+Š 00 00 $$$$ SS SS. (3.55) It is important to note that []M K is identical to the negative of the spring wrench expressed as a spatial cro ss product operator (see Feathe rstone 1985 and Ciblak and Lipkin 1994). To prove it, all terms in Eq. (2.43) are explicitly expressed in a polar coordinate system and 1 T EA Pxyz p pp = r to yield

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48 [] [][] [][]MK = 0K12 K12K22 (3.56) where []()0css c0sc sssc0okll Š =ŠŠ Š K12 (3.57) []0ssscscc ()sssc0css scccss0xyzx oxyyz zxyzpppp kllpppp pppp ŠŠ+ =ŠŠ+Š ŠŠ+ K22 (3.58) and where []0 is 33 zero matrix, cos()c=, and sin()s=, etc. In the same way the spring wrench can be explicitly written as 1()() ()EA oo P x y z o yz x zx y xy zkllkll sc f ss f c f kll pcpss m pscpc m psspsc m =Š=Š =Š== Š Š Š S w$ rS f m. (3.59) By comparing Eqs. (3.57) and (3.58) with Eq. (3.59) it is obvious that []() 0 0 0zy ozx yxff kllff ff Š =ŠŠ=Š Š K12f (3.60) [] 0 0 0zy zx yxmm mm mm Š =Š=Š Š K22m (3.61) where fand m are skew-symmetric matrices representing vector multiplication.

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49 Then []M K can be expressed as [] [ ] MK Š ==Š ŠŠ 0f w fm (3.62) where w is the spring wrench expressed as a spatial cross product operator (see Featherstone 1985). Finally the derivative of the spring wrench can be written as [] () EABEA FK=Š wDwD. (3.63) As shown in Eq. (3.63), the derivative of the spring wrench joining two rigid bodies depends not only on a relative twist between two bodies but also on the twist of the intermediate body, in this case body A, in terms of the inertial frame unless the initial external wrench w is zero. []F K which maps a small twist of body B in terms of body A into the corresponding change of wrench upon body B is identical to the stiffness matrix of the spring assuming the body A is stationary. 3.3 Stiffness Mapping of Spatial Compliant Parallel Mechanisms in Series The derivative of spring wrench derived in the previous section is applied to obtain the stiffness mapping of the compliant mechanism shown in Figure 3-5. Body A is connected to ground by six compliant couplings and body B is connected to body A in the same way. Each compliant coupling has a spherical joint at each end and a prismatic joint with a spring in the middle. It is assumed that an external wrench extw is applied to body B and that both body B and body A are in static equilibrium. The poses of body A and body B and the spring constants and fr ee lengths of all compliant couplings are known.

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50 The stiffness matrix [] K which maps a small twist of the moving body B in terms of the ground, EBD (written in axis c oordinates), into the corresponding wrench variation, extw (written in ray coordinates), is desired to be derived and this relationship can be written as [] EB extK=wD. (3.64) The stiffness matrix can be derived by taking a derivativ e of the static equilibrium equations of body A and body B which may be written as 612 17extii ii==== www (3.65) 612 17extii ii====www (3.66) where iw is the wrench from i-th compliant coupling. Since springs 1 to 6 join the two moving bodies and springs 7 to 12 connect body A to ground (see Figure 3-5), the derivatives of the spring wrenches can be written as [] () ()[] () 66 11 ABEA iFi i ii ABEA Fext RUK K ===Š =ŠwDwD DwD (3.67) [] [] 1212 77 ABEA iFiF iRL iiKK====wDD (3.68) where [][]12 7 FF RLi iKK== (3.69) [][]6 1 FF RUi iKK== (3.70)

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51 and where extw is the external wrench expressed as a spatial cross product operator. From Eqs. (3.67), (3.68), and (3.25), twist EAD can be written as Eq. (3.72). [] [] () [] () () ,, EAABEA FFext RLRU EBEAEA Fext RUKK K =Š =ŠŠDDwD DDwD (3.71) [][] ()()[] 1 ,,, EAEB FFextF RLRURUKKKŠ=++DwD. (3.72) Substituting Eq. (3.72) for EAD in Eq. (3.68) and comparing it with Eq. (3.64) yield the stiffness matrix as Eq. (3.74). [] [] [][][] ()()[] 1 ,,,, EBEA F RL EB FFFextF RLRLRURUKK KKKK Š= =++DD wD (3.73) [][][][] ()()[]1 ,,,, FFFextF RLRLRURUKKKKKŠ=++w. (3.74) It was generally accepted that the resultant compliance, which is the inverse of the stiffness, of serially connected mechanisms is the summation of the compliances of all constituent mechanisms (see Griffis 1991). Ho wever, the stiffness matrix derived from this research shows a different result. Taking an inverse of the stiffness matrix Eq. (3.74) yields [][][][] ()[]11111 ,,,, FFFextF RLRURURLKKKKKŠŠŠŠŠ=++w. (3.75) The third term in Eq. (3.75) is newly introdu ced in this research and it does not vanish unless the external wrench is zero.

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52 extw 1k2k3k4k5k6k7k8k9k10k11k12k Figure 3-5. Mechanism having two comp liant parallel mechanisms in series.2 A numerical example of the compliant mechanism depicted in Figure 3-5 is presented. The geometry information and sp ring properties of the mechanism shown in Figure 3-5 are presented in Tables 3-1 through 3-5. The external wrench extw is given as [ ]0.30.40.82.31.30.7T ext=ŠŠŠ w. (unit:[N,N,N,Ncm,Ncm,Ncm]) Table 3-1. Spring properties of the mechanism in Figure 3-5 (Unit: N/cm for k, cm for lo). Spring No. 1 2 3 4 5 6 Stiffness coefficient ik 4.6 4.7 4.5 4.4 5.3 5.5 2 The coordinate systems attached to bodies E, A, and B are for illustrative purposes only. In the analysis it is assumed that the three coordinate systems are at this instant coincident and aligned.

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53 Free length oil 1.63051.02764.00981.85921.7591 3.8364 Table 3-1. Continued. Spring No. 7 8 9 10 11 12 Stiffness coefficient ik 4.4 4.9 4.7 4.5 5.1 4.8 Free length oil 4.47181.27605.21492.67802.2712 3.4244 Table 3-2. Positions of pivots in ground in Figure 3-5 (Unit: cm). No. 1 2 3 4 5 6 X 0.0000 1.3000 0.6000 -0.7000 -1.1000 -0.5000 Y 0.0000 1.1000 2.7000 2.6000 1.8000 0.4000 Z 0.0000 0.2000 0.1000 -0.1000 0.3000 0.1000 Table 3-3. Positions of pivots in bottom side of body A in Figure 3-5 (Unit: cm). No. 1 2 3 4 5 6 X 0.2000 1.1833 0.4616 -0.6575 -1.1452 -0.2189 Y 1.2000 2.1235 3.5111 3.3783 2.5652 1.6879 Z 3.2000 3.1843 3.3010 3.1013 3.0704 3.1196 Table 3-4. Positions of pivots in top side of body A in Figure 3-5 (Unit: cm). No. 1 2 3 4 5 6 X 0.2086 1.4860 0.7553 -0.5501 -0.9278 -0.2945 Y 1.2033 2.3329 3.9187 3.7867 2.9797 1.5942 Z 3.2996 3.2514 3.3121 3.2792 3.3590 3.3804 Table 3-5. Positions of pivots in body B in Figure 3-5 (Unit: cm). No. 1 2 3 4 5 6 X -0.3000 0.9216 0.2183 -0.8385 -1.2525 -0.5589 Y 1.6000 2.7822 3.8980 3.9919 2.8972 2.0875 Z 5.5000 5.5000 5.4782 5.8447 5.8317 5.7745 Two stiffness matrices are calculated: []1 K from Eq. (3.74) and []2 K from the same equation but without the matrix w. The numerical results are 10.34290.00770.26610.78531.73780.4076 0.00770.51031.71221.27600.21570.2885 0.26611.712210.510320.00120.75180.2695 [] 0.78532.076019.601254.32221.13481.2570 0.93780.21570.45180.434812.13293.8667 K ŠŠŠŠ ŠŠ ŠŠ = Š Š 0.00760.01150.26950.04301.56670.0798 ŠŠŠŠŠ

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54 20.30390.01090.26410.78581.31340.2770 0.05040.46171.71221.98630.08750.0375 0.43641.622210.663321.78340.71440.5956 [] 1.07881.986220.757459.47361.18742.5822 0.97540.07590.09010.128312.18523.0 K ŠŠŠŠ ŠŠ ŠŠ = ŠŠ ŠŠŠ 399 0.03190.02180.05760.69831.64400.1157 ŠŠŠŠ where, the units of upper left 3 3 sub matr ix is N/cm, that of lower right 3 3 sub matrix is Ncm, and that of remainder is N. The result is evaluated in the following way: 1. A small wrench Tw is applied in addition to extw to body B and twists 1EBD and 2EBD are obtained by multiplying the inverse matrices of the stiffness matrices, []1 K and []2 K respectively, by Tw as of Eq. (3.64). Corresponding positions for body B are then determined, based on the calculated twists 1EBD and 2EBD. 2. EAD is calculated by multiplying the inverse matrix of [], F RLK by Tw as of Eq. (3.68). The position of body A is then determined from this twist. 3. The wrench between body B and body A is calculated for the two cases based on knowledge of the positions of bodies A a nd B and the spring parameters. The change in wrench for the two cases is determined as the difference between the new equilibrium wrench and the original. The changes in the wrenches are named ,1ABw and ,2ABw which correspond to the matrices []1 K and []2 K 4. The given change in wrench Tw is compared to ,1ABw and ,2ABw. The given wrench Tw and the numerical results are presented as below. [ ]4100.50.20.40.30.80.4T TŠ=ŠŠ w [ ]3 1100.35220.30810.09120.01370.04290.0367T EBŠ=ŠŠŠŠ D [ ]3 2100.33540.26820.08450.01320.04040.0365T EBŠ=ŠŠŠŠ D [ ]3100.11130.01000.00670.00810.02670.0650T EAŠ=ŠŠŠŠ D [ ]4100.50000.19950.40170.30350.80000.4000T EAŠ=ŠŠ w

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55 [ ]4 ,1100.49970.19980.40200.30410.80100.4011T ABŠ=ŠŠ w [ ]4 ,2100.54620.06930.35340.78310.23190.0967T ABŠ=ŠŠŠ w where EAw is the wrench between body A and ground. The unit for the wrenches is [N, N, N, Ncm, Ncm, Ncm]T and that of the twists is [cm, cm, cm, rad, rad, rad]T. The difference between EAw and Tw is small and is due to the fact that the twist was not infinitesimal. The difference between ,1ABw and Tw is also small and is most likely attributed to the same fact. However, the difference between ,2ABw and Tw is not negligible. This indicates that the stiffness matrix formula derived in this research produces the proper result and that the term extw cannot be neglected in Eq. (3.74).

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56 CHAPTER 4 STIFFNESS MODULATION OF PLANAR COMPLIANT MECHANISMS Planar mechanisms with variable compliance, specifically, compliant parallel mechanisms and mechanisms having two compliant parallel mechanisms in a serial arrangement are investigated in this chapte r. The mechanisms consist of rigid bodies joined by adjustable compliant couplings. Each adjustable compliant coupling has a revolute joint at each end and a prismatic joint with an adjustable spring in the middle. The adjustable springs are assumed to be able to change their stiffness coefficient and free length and the mechanisms are in static equilibrium under an external wrench. It is desired to modulate the compliance of the mechanism while regulating the pose of the mechanism. 4.1 Parallel Mechanisms with Variable Compliance 4.1.1 Constraint on Stiffness Matrix Figure 4-1 illustrates a compliant parallel mechanism having N number of compliant couplings. The mechanism is in st atic equilibrium under the external wrench extw and it can be expressed as 1N exti i == wf (4.1) where if is the spring wrench of i-th compliant coupling. By using Eq. (2.15) a derivative of Eq. (4.1) may be written as

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57 [] 1 1 N exti i N EA F i iK = === wf D (4.2) where [] (1)T ii T oi Fiiii i iiil Kkk l =+Š $$ $$. (4.3) ik, oil, il, and i in Eq. (4.3) are the spring constant, the spring free length, the current spring length, and the rising angle of i-th compliant coupling, respectively (see Figure 24). In addition, i$ represents the unitized Plcker coordinates of the line along the ith compliant coupling as in Eq. (2-6) and may be written explicitly as ,,cos sin sincosi i ii Pii xiiyiirr == Š S $ rS (4.4) where ,xir and y ir are the pivot position of i-th compliant coupling in body E. Then Eq. (4.4) leads to ,,sin cos cossini i i i xiiyiirr Š = + $. (4.5) The stiffness matrix of the mechanism [] K can be written from Eq. (4.2) as [][]1 N F i iKK==. (4.6)

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58 A E3k1k2kNk ENE3E2E1A1A2A3……. AN extw Figure 4-1. Compliant parallel mechanism with N number of couplings. Ciblak and Lipkin (1994) showed that the stiffness matrix of compliant parallel mechanisms can be decomposed into a symmetric and a skew symmetric part and that the skew symmetric part is negative one-half the externally applied load expressed as a spatial cross product operator. For planar m echanisms, the skew symmetric part can be written as [] [][] [][] [][]22TT SymmetricSkewSymmetricKKKK K KK+Š =+ =+ (4.7) [] [][] 00 1 00 22 0T y x SkewSymmetric yx f KK K f ff Š ==ŠŠ Š (4.8) where ,,T extxyzffm = w is the external wrench.

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59 It is important to note that no matter how many compliant couplings are connected and no matter how the spring constants and the free lengths of the constituent compliant couplings are changed the stiffness matrix of a compliant parallel mechanism contains only six independent variables and the stiffness matrix may be rewritten as []111213 122232 133233 x yKKK K KKKf KfKK =+ + (4.9) From Eqs. (4.3)-(4.6) the six independe nt elements of the stiffness matrix [] K can be explicitly written as 2 11 1sinN oi iii i il Kkk l= =Š (4.10) 12 1sincosN oi iii i il Kk l= = (4.11) 2 13,,, 1(sin)(sinsincossin)N oi iiiyiiiixiiiyii i il Kklrklrr l= =Š++++ (4.12) 2 22 1cosN oi iii i il Kkk l= =Š (4.13) 2 32,,, 1(cossincos)N oi ixiixiiyiii i il Kkrkrr l= =Š+ (4.14) ,,,, 2 33,,, 1 2 ,,,(cos)(sin) (cossincoscos) (sinsincossin)ixiiixiiyiiiyi N oi ixiiiyiiixii i i oi iyiiixiiiyii ikrlrkrlr l Kkrlrr l l krlrr l = +++ =Š++ Š++ (4.15)

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60 4.1.2 Stiffness Modulation by Varying Spring Parameters In this case it is desired to find an a ppropriate set of spring constants and free lengths of the constituent compliant couplings of the mechanism shown in Figure 4-1 to implement a given stiffness matrix and to regulate the pose of body A under a given external wrench. It is important to note that the stiffn ess matrix contains only six independent variables and the equations for the independent variables are linear in terms of ik ’s and ioikl’s as shown in Eqs. (4.10)-(4.15) since all geometrical terms are constant. In addition to the equations for the stiffness matrix, the system should satisfy static equilibrium equations to regulate the pose of the mechanism and from Eqs. (4.1) and (4.4) it can be written as ()1 ,,cos sin sincosxi N extyiioii i zxiiyiif fkll mrr = ==Š Š w. (4.16) Eq. (4.16) consists of three equations which are also linear in terms of ik ’s and ioikl’s. Since there are nine linear equations to be fulfilled and each adjustable compliant coupling possesses two control variables such as spring constant and free length, at least five adjustable compliant couplings are required. The nine equations may be written in matrix form as [] A=XB (4.17) where 111213223233,,,,,,,,T xyzKKKKKKffm = B (4.18)

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61 121122,,,,,,,T NooNoNkkkklklkl = X (4.19) []1,11,21, 2,12,22, 3,13,23,4,14,24, 5,15,25, 6,16,26,7,17,27, 8,18,28,9,19,29, 1,11,21,2,12,22, 3,13,23,4,14,24, 5,15,25,6,1111 000 111N N NN N NN NN NN NN NGGG GGG GGGGGG GGG GGGGGG A GGGGGG HHHHHH HHHHHH HHHH= 6,26, NHH (4.20) and where 2 1,sini i iG l=Š, 2,sincosii i iG l=, 3,,siniiiyiGlr=ŠŠ 2 ,, 4,sincossin sinxiiiyii ii irr G l + =+, 2 5,cosi i iG l=Š 6,, ixiGr=, 2 ,, 7,cossincosxiiyiii i irr G lŠŠ = 22 8,,,,,(cossin)ixiyiixiiyiiGrrlrr=+++ 2222 ,,,, 9,,,cossin2sincos cossinxiiyiixiyiii ixiiyii irrrr Grr l ++ =ŠŠŠ 1,cosiiiHl=, 2,cosiiH=Š, 3,siniiiHl=, 4,siniiH=Š ()5,,,sincosiixiiyiiHlrr=Š, ()6,,,sincosixiiyiiHrr=ŠŠ. It is important to note that []A X, and B are 9 (2* N ), (2* N ) 1, and 9 1 matrices, respectively, where N denotes the number of the adjustable compliant couplings.

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62 It is required to solve Eq. (4.17) where the number of columns of matrix []A is in general greater than that of rows and the general solution s olX can be written as [] solph pNullA=+ =+XXX XC (4.21) where p X, hX, []NullA and C are the particular solution, the homogeneous solution, the null space of matrix []A and the coefficient column matrix, respectively (see Strang 1988). Once a solution s olX is obtained, oil’s are calculated from ik ’s and ioikl’s in s olX. It is important to note that []NullA is (2* N ) (2* N -9) matrix and C is (2* N -9) 1 column matrix. There might be many strategies to select the matrix C which leads to a specific solution. For instance, if the norm of s olX is desired to be minimized, then by using projection matrix []NullPAŠ (see Strang 1988), the solution can be obtained as [] (). MinsolpNullPpAŠ=+ŠXXX (4.22) where [][][][]()[]1 TT NullPNullNullNullNullAAAAAŠ Š=. (4.23) For another case, we might want the solution closest to a desired solution dX which may be constructed from operation ra nges of adjustable compliant couplings, for instance, minimum and maximum spring constant and free length. Then, the solution can be obtained as [] (). dsolpNullPdpAŠ=+ŠXXXX. (4.24)

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63 Unfortunately, these methods involve mi xed unit problems and do not guarantee a solution consisting of only positive spring constants and free lengths. A numerical example is presented. The external wrench extw and the desired stiffness matrix [] K are given as []1.88322.88053.2851ext N NNcm=ŠŠw 0.0216/2.2483/2.2750 []2.2483/25.3914/60.9800 5.155562.8632270.4409 NcmNcmN KNcmNcmN N NNcmŠ = Š The geometry information of the mechanism s hown in Figure 4-1 is given in Tables 4-1 and 4-2. The mechanism is assumed to have five compliant couplings. Table 4-1. Positions of pivot points in body E for numerical example in 4.1.2. Pivot points E1 E2 E3 E4 E5 X 0.0000 0.6000 2.5000 3.9000 5.3000 Y 0.0000 0.8000 0.3000 0.9000 0.0000 (Unit: cm) Table 4-2. Positions of pivot points in body A for numerical example in 4.1.2. Pivot points A1 A2 A3 A4 A5 X 0.6000 1.4055 2.6736 3.3368 4.7284 Y 4.5000 2.7447 3.3209 3.9614 4.1442 (Unit: cm) The spring parameters which have th e minimum norm and satisfy the given conditions can be obtained by using Eq. (4.22) and these values are shown in Table 4-3. Table 4-3. Spring parameters with mi nimum norm for numeri cal example 4.1.2. Spring No. 1 2 3 4 5 Stiffness constant k 4.6674 7.2485 3.5188 5.0243 6.3280 Free length ol 4.1678 2.1490 6.3995 1.9322 3.9104 (Unit: N/cm for k, cm for lo) The spring parameters which are closest to the given spring parameters as shown in Table 4-4 can be obtained by applying Eq (4.24) and it is shown in Table 4-5.

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64 Table 4-4. Given optimal spring pa rameters for numerical example 4.1.2 Spring No. 1 2 3 4 5 Stiffness constant k 5.0 5.0 5.0 5.0 5.0 Free length ol 3.0 3.0 3.0 3.0 3.0 (Unit: N/cm for k, cm for lo) Table 4-5. Spring parameters closest to gi ven spring parameters for numerical example 4.1.2. Spring No. 1 2 3 4 5 Stiffness constant k 4.8664 6.8783 3.8968 4.8990 6.2974 Free length ol 4.3386 2.3374 5.0230 2.1667 4.0492 (Unit: N/cm for k, cm for lo) Two sets of the spring parameters, one in Ta ble 4-3 and the other in Table 4-5, implement the given wrench and stiffness matrix. 4.1.3 Stiffness Modulation by Varying Spring Parameters and Displacement of the Mechanism In this case, different from the previ ous section, the pose of body A is not constrained as fixed. A change of the pose of body A, which is considered to be in contact with the environment, may be compensated by attaching body E to the end of a robot system and by controlling the position of the robot end effector in a similar manner as described in the Theory of Kinestatic Control proposed by Griffis (1991). As presented in the previous section there are nine values to be fulfilled: six from the stiffness matrix and three from the wrench equations. A typical planar parallel mechanism which has three couplings is investigated since the mechanism has nine control input variables which is same with that of valu es to be fulfilled: six from adjustable compliant couplings and three from the planar displacement between body A and body E. The target variables may be expressed in matrix form as B in Eq. (4.18) and control input variables U may be written in matrix form as []123123,,,,,,,,T oooookkklllxy =U (4.25)

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65 where ik ’s and oil’s are the spring constant and free length of ith compliant coupling, respectively. In addition, o x and oy are the position of point O in body A, which is coincident with the origin of the inertial frame E, and is the rotation angle of body A with respect to ground. The stiffness matrix equations and wrench equations are highly nonlinear in terms of the displacement of the bodies as shown in Eqs. (4.9)-(4.16). In this section a derivative of the target variables B with respect to input variables U is investigated and the derivative is used to obtain the small chan ge of input variables for the desired small change of target values and it may be written as d d=B BU U (4.26) 1d dŠ= B UB U (4.27) where 111213223233[,,,,,,,,]T xyz K KKKKKffm=B (4.28) 123123[,,,,,,,,]T oooookkklllxy =U. (4.29) 11 19 99 19 B B UU d d B B UU = B U (4.30) For instance, 1 B can be written as

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66 1 1 111111111111 123123 123123 111111 ooo ooo oo oodB B d KKKKKK kkklll kkklll KKK xy xy = =+++++ +++ U U. (4.31) In Eqs. (4.9)-(4.16) all elements of B were presented as functions of not U but p U which is defined as []123123123123,,,,,,,,,,,T poookkkllllll=U. (4.32) Hence among Eq. (4.31) 11 o K x 11 o K y and 11 K are not obtained from simple differentiation. Since B is a function of p U, 1 B can also be written as 1 1 111111111111 123123 123123 111111111111 123123 123123 p p ooo ooodB B d KKKKKK kkklll kkklll KKKKKK lll lll = =+++++ ++++++ U U. (4.33) In addition, il’s and i’s in Eq. (4.33) can be substituted with Eqs. (2.12) and (2.13) which is restated here as o TT iiio x ly == $D$ (2.12) ''11TT o ii io iiii x y ll == $$ D. (2.13)

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67 By the above substitution in Eq. (4.33) and collecting the coefficients according to each term of U, 1 B can be written as 1 2 3 1 111111111111111111 1 2 123123 3 1 o o ooooo o o ok k k l KKKKKKKKK B l kkklllxy l x y dB d = =U U.(4.34) All terms of B may be obtained in the same way in which 1 B was obtained and d dB U can be derived by combining all idB dU’s. A numerical example is presented. The mechanism shown in Figure 4-1 is in static equilibrium under the external wrench extw and the geometry information and the spring parameters are given below. The mechanism is assumed to have three compliant couplings. Table 4-6. Positions of pivot poi nts for numerical example 4.1.3. Pivot points E1 E2 E3 A1 A2 A3 X 0.0000 0.6000 2.5000 0.6000 1.4055 2.6736 Y 0.0000 0.8000 0.2000 4.5000 2.7447 3.3209 (Unit: cm) Table 4-7. Initial spring parameters for numerical example 4.1.3. Spring No. 1 2 3 Stiffness constant k 5.5 5.7 5.1 Free length ol 4.8 3.1 2.0 (Unit: N/cm for k, cm for lo)

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68 The external wrench extw and the initial stiffness matrix []I K are calculated from the geometry of the mechanism and the spring parameters. []2.04090.926312.8594ext N NNcm=ŠŠw 0.1679/3.9107/3.9623 []3.9107/14.9590/10.9558 3.036012.996625.9764INcmNcmN KNcmNcmN N NNcm = The desired stiffness matrix []D K is given below. 0.6679/4.3107/4.1823 []4.3107/15.4290/10.8358 3.256012.876626.3764DNcmNcmN KNcmNcmN N NNcm = Since the difference between the desired stiffness matrix and the initial stiffness matrix is not small enough, the difference is divided into a number of small B’s and Eq. (4.27) is applied repeatedly to obtain the spring parameters and the displacement of body B which implement the desired stiffness matrix and the given wrench. The calculated spring parameters and the pos e of body A are shown in Table 4-8 and Table 4-9 and the initial and final pose of the mechanism is shown in Figure 4-2. Table 4-8. Calculated spring para meters for numerical example 4.1.3. Spring No. 1 2 3 Stiffness constant k 6.2563 5.5311 5.1492 Free length ol 5.4810 4.3584 3.1954 (Unit: N/cm for k, cm for lo) Table 4-9. Positions of pivot points in body A for numerical example 4.1.3. Pivot points A1 A2 A3 X 0.8201 1.9165 3.0661 Y 5.2909 3.7010 4.4874 (Unit: cm)

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69 -1 0 1 2 3 4 0 1 2 3 4 5 6 Figure 4-2. Poses of the compliant parallel mechanism for numerical example 4.1.3. 4.2 Variable Compliant Mechanisms with Two Parallel Mechanisms in Series In this section mechanisms having two planar compliant parallel mechanisms that are serially arranged as shown in Figure 2-6 are investigated. 4.2.1 Constraints on Stiffness Matrix The stiffness matrix of the mechanism was derived in Chapter two and restated as [][][][][]()[]1 ,,,,,FFFMF RLRLRURURUKKKKKKŠ=+Š. (2.54) Applying the constraint presen ted by Ciblak and Lipkin (1994), [],F RLK and [],F RUK may be written as []111213 122232 133233 LLL LLL Fx RL LLL yKKK K KKKf KfKK =+ + (4.35) Initial pose Final pose

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70 []111213 122232 133233 UUU UUU Fx RU UUU yKKK K KKKf KfKK =+ + (4.36) where ,,T extxyzffm = w is the external wrench. In addition, [],M RUK is a function of only the external wrench as shown in Eq. (2.46) which is restated as [],00 00 0 y Mx RU yx f K f ff Š = Š (2.46) Plugging in Eqs. (4.35), (4.36), and (2.46) into Eq. (2.54) and carrying out a symbolic operation using Maple software shows [][]00 00 0 y T x yx f K Kf ff Š Š= Š which is the same with Ciblak and Lipki n (1994)’s statement for compliant parallel mechanisms in planar cases. This result indicates that mechanisms having two planar compliant parallel mechanisms in a serial ar rangement also contain only six independent variables. 4.2.2 Stiffness Modulation by using a Deri vative of Stiffness Matrix and Wrench Since the stiffness matrix of the mechanism shown in Figure 2-6 is complicated and nonlinear in term of the spring parameters and the displacement of the rigid bodies, a derivative of the stiffness matrix and the static equilibrium equation is derived and applied for stiffness modulation of the compliant mechanism. The stiffness matrix elements and the wrenches may be written in matrix form as 111213223233,,,,,,,,,,,T AAABBB xyzxyzKKKKKKffmffm = B (4.37)

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71 where ,,T AAAA xyzffm = w and ,,T BBBB xxzffm = w are the wrenches from the compliant couplings connecting body A to ground and from the couplings connecting body B to body A, respectively. The spring parameters and the displacements of the rigid bodies may be written as 123456123456[,,,,,,,,,,,,,,,,,]AAABBBT ooooooooookkkkkkllllllxyxy =U (4.38) where ik ’s and oil’s are the spring constant and free length of ith compliant coupling, respectively. In addition, A o x and A oy are the position of point O in body A which is coincident with the origin of the inertial frame E and A is the rotation angle of body A with respect to ground. B o x B oy, and B are defined in the same way in terms of the inertial frame E. In chapter two the stiffness matrix and the wrenches were presented as functions of p U which is defined as 123456123456123456 123456,1,2,3,1,2,3[,,,,,,,,,,,,,,,,,, ,,,,,,,,,,,]poooooo T xxxyyykkkkkkllllllllllll rrrrrr=U (4.39) where il ’s and i’s are the current spring length and rising angle of i-th compliant coupling. In addition xir and y ir are the pivot positions of i-th compliant coupling in body A. A similar approach taken in the previous s ection is applied to get a derivative of the stiffness matrix and the wrenches d dB U: each element of B is differentiated with respect to p U and the terms not belonging to U are substituted in the terms of U. In other words, il’s, i’s, xir’s, and y ir’s are expressed in terms of the twists of the bodies.

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72 The coefficient of each term of U corresponds to an element of the derivative matrix in an analogous way to Eq. (4.34). Since springs 4, 5, and 6 connect body A and ground, il and i for i=4, 5, 6 can be written as Eqs. (2.12) and (2.13) which are A o TEATA iiio A x ly == $D$ (2.12) ''11A TT o EAA ii io iiii A x y ll == $$ D. (2.13) Springs 1, 2, and 3 join body B and body A and thus il and i for i=1, 2, 3 may be expressed as BA oo TABTBA iiioo BA x x lyy Š ==Š Š $D$ (4.40) ''11BA TT oo ABABAA ii ioo iiii BAxx yy ll Š =+=Š+Š $$ D. (4.41) Lastly xir’s, and y ir’s are the positions of the pivot point in body A and by using the twist equation it can be written as ,, ,,0 0 000A xioxi A y ioyi Arxr ryr =+ (4.42) Now all terms in the differential of B are expressed in terms of U and by writing it in matrix form gives

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73 d d=B BU U (4.43) where d dB U is 12 18 matrix. It is required to obtain the small change of input values U corresponding to a small change of stiffness matrix and wrenches B and since the number of columns of the matrix is greater than that of rows it is a redundant system. There are in general an infinite number of solutions and a variety of constraints may be imposed on the system. Since U is the change from the current values, minimizing the norm of U may be one of reasonable options. Then in a similar way to Eq. (4.22) minU may be obtained as ()1 min. TT p sol psol NullNullNullNulldddd ddddŠ =+Š BBBB UUU UUUU (4.44) where p solU is a particular solution of Eq. (4.43) and Nulld d B U is the null space of matrix d dB U (see Strang 1988). Body B is considered to be in contact with the environment and it may be required to preserve the pose of body B. It indicates that the twist of body B is equal to zero and can be written as 0B ox=, 0B oy=, 0B=.

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74 We can implement it by removing the last three columns of d dB U and the last three rows of U and solving the problems in a similar way to that of the previous problem since the system is still redundant. If both of the bodies are required to be st ationary then the twists of the bodies should be zero and it may be written as 0A ox=, 0A oy=, 0A=, 0B ox=, 0B oy=, 0B=. This can be implemented by removing the last six columns of d dB U and the last six rows of U, and by solving the problem which is not redundant. 4.2.3 Numerical Example The geometry information and spring pa rameters of the mechanism shown in Figure 2-6 and the external wrench extw are given below. []1.72.512.7extNNN =Š w Table 4-10. Spring parameters of the comp liant couplings for numerical example 4.2.3. Spring No. 1 2 3 4 5 6 Stiffness constant k 5.0 5.0 5.0 5.0 5.0 5.0 Free length ol 3.0614 0.6791 2.3608 2.8657 0.7258 1.2732 (Unit: N/cm for k, cm for lo) Table 4-11. Positions of pivot poi nts for numerical example 4.2.3. Pivot points E1 E2 E3 B1 B2 B3 X 0.0000 1.0000 3.0000 -0.8000 0.4453 1.1965 Y 0.0000 0.8000 0.0000 4.5000 3.7726 4.6186 (Unit: cm) Table 4-11. Continued. A1 A2 A3 A4 0.2000 1.1261 2.1646 1.2760 2.3000 1.8179 1.9252 2.6038

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75 The initial stiffness matrix []I K is calculated from the geometry of the mechanism and the spring parameters. 1.5992/1.1571/6.0650 []1.1571/5.7047/7.7521 3.56509.452122.6794INcmNcmN KNcmNcmN N NNcmŠŠ =Š Š The desired stiffness matrix []D K is given below. 1.6442/1.1921/6.0230 []1.1921/5.7417/7.7931 3.52309.493122.6384DNcmNcmN KNcmNcmN N NNcmŠŠ =Š Š Since the difference between the desired stiffness matrix and the initial stiffness matrix is not small enough, the difference is divided into a number of small B ’s and the problem is solved repeatedly to obtain the spring parameters and the displacements of body B and body A which implement the desired stiffness ma trix. Three sets of spring parameters are obtained: one with no constraint on the di splacements of body A and body B, another with body A fixed, and the other with body A and body B fixed. The calculated spring parameters are presented in Table 4-12, Tabl e 4-13, and Table 4-14, respectively. In addition, the initial and final poses of the mechanism are shown in Figures 4-3 and 4-4. Table 4-12. Spring parameters with no constraint for numerical example 4.2.3 Spring No. 1 2 3 4 5 6 Stiffness constant k 5.5786 4.8852 5.5513 5.1865 5.1506 5.1505 Free length ol 2.5502 0.5084 1.9188 2.7939 0.8039 1.3746 (Unit: N/cm for k, cm for lo) Table 4-13. Spring parameters with body B fixed for numerical example 4.2.3 Spring No. 1 2 3 4 5 6 Stiffness constant k 6.3556 4.4317 5.9605 5.3306 5.5194 4.4334 Free length ol 3.0234 0.6588 2.6016 2.8192 0.6220 1.0270 (Unit: N/cm for k, cm for lo)

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76 -1 0 1 2 3 0 1 2 3 4 5 Figure 4-3. Poses of the compliant mechanism with body B fixed. -1 0 1 2 3 0 1 2 3 4 5 Figure 4-4. Poses of the complia nt mechanism with no constraint. Initial pose Final pose Final pose Initial pose

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77 Table 4-14. Spring parameters with bod y A and body B fixed for numerical example 4.2.3. Spring No. 1 2 3 4 5 6 Stiffness constant k 12.3070 10.5719 6.9847 1.9411 4.9528 3.8741 Free length ol 2.6786 1.0769 2.5033 3.7435 0.7229 1.0333 (Unit: N/cm for k, cm for lo) The results indicate that there are greater changes of the spring parameters with more constraints imposed on the bodies. These control methods all require the inverse of d dB U or T NullNulldd dd BB UU depending on the constraint and it may cause a singularity problem. With more constraints the mechan ism is more vulnerable to the singularity problem.

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78 CHAPTER 5 CONCLUSIONS The compliance of mechanisms containing rigid bodies which are connected to each other by line springs was studied. A derivative of the planar spring wrench connecting two moving bodies was obtained and then, through a similar approach, a derivative of the spatial spring wrench which is more general was obtained. It is obvious from Eq. (3.36) that the derivative of the spri ng wrench joining two rigid bodies depends not only on a relative twist between two bodies but also on the twist of the intermediate body in terms of the inertial frame unless the initial external wrench is zero. The derivative of the spring wrench was applied to obtain the resultant stiffness matrix of two compliant parallel mechanisms in a serial arrangement. The resultant stiffness matrix indicates that the resultant compliance, which is the inverse of the resultant stiffness matrix, is not the summation of the compliances of the constituent mechanisms unless the external wrench applied to the mechanism is zero which was generally accepted. The derivative of the spring wrench was also applied to acquire the stiffness matrix of planar springs in a hybrid arrangement and may be applied for mechanisms having an arbitrary number of parallel mechanisms in a serial arrangement. Planar mechanisms with variable compliance were investigated with the knowledge of the stiffness model obtained in the research. Adjustable line springs which can change their spring constants and free lengths were employed to connect rigid bodies in the mechanisms. Ciblak and Lipkin (1994) showed that the stiffness matrix of compliant parallel mechanisms can be decomposed into a symmetric and a skew symmetric part and

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79 that the skew symmetric part is negative one-half the externally applied load expressed as a spatial cross product operator. It was s hown through this research that the same statement is valid for the stiffness matrix of a mechanism having two compliant parallel mechanisms in a serial arrangement. In other words there are only six rather than nine independent variables in the stiffness matrix of planar compliant parallel mechanisms and in the stiffness matrix of a mechanism having two planar compliant parallel mechanisms in a serial arrangement. Derivatives of the stiffness matrices of planar compliant mechanisms with respect to the spring parameters and the twists of the constituent rigid bodies were obtained. It was shown that these derivatives may be utilized to control and regulate the stiffness matrix and the pose of the mechanism respectively at the same time by adjusting the spring parameters of each c onstituent coupling with or without the change of the position of the robot where the compliant mechanism is attached. Several future works are presented. The singularity conditions associated with the resultant stiffness matrix of compliant parallel mechanisms in a serial arrangement needs to be studied. This study will identify under what condition the mechanism collapses even with a small change in the applied wrench. It was required to solve a redundant syst em of linear equations to obtain the changes of the spring parameters and the twists of the bodies corresponding to a small change of the stiffness matrix and the resultant wrench of a compliant mechanism. There are in general an infinite number of solutions and the least square solution was chosen in

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80 this research. A better solution may be selected by considering the following considerations: 1. The inverse of matrix d dB U or Tdd ddBB UU as of Eq. (4.43) is required to solve the linear equations. Solutions close to singular cases of d dB U or Tdd ddBB UU should be avoided. 2. The operation ranges of the spring para meters should be taken into account. 3. Proper consideration should be given to si ngularities of the compliant mechanism.

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81 APPENDIX A MATLAB CODES FOR NUMERICAL EXAMPLES IN CHAPTER TWO AND THREE Matlab codes for the numerical examples in chapter two and three are presented in this appendix. Table A-1 contains the function list and functions NuEx_21, NuEx_22, and NuEx_31 are main functions, and other functions are called inside these main functions. For instance, to get the result of numerical example 2-1, it is needed to call function NuEx_21 and other functions are pla ced in the same folder where function NuEx_21 is located. Table A-1. Matlab function list. Function Name Description NuEx_21 Main function for numerical example 2-1 NuEx_22 Main function for numerical example 2-2 NuEx_31 Main function for numerical example 2-3 StaticEq21 Static equilibrium equation for numerical example 2-1 StaticEq22 Static equilibrium equation fo r numerical example 2-2 StiffMatrix Computes matrix [KF] GetKM Computes matrix [KM] SpringWrench Computes spring wrench GetPLine Computes Pl ucker line coordinates GetOriginVel Computes origin velo city from velocity of a point and angular velocity of a rigid body GetVelP2D Computes displacement of a point from twist GetGlobalPos2D Computes position of a point in terms of inertial frame in planar space GetGlobalPos3D Computes position of a point in terms of inertial frame in spatial space function NuEx_21 % Numerical example 2-1 % Test Stiffness Matrix of planar parallel mechanisms in series

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82 %============================================================== %====== GIVEN VALUES %============================================================== %Wext : External wrench %A : Coordinates of fixed pivot points %lb: Local coordinates of ve rtices of middle platform % the first 3 columns are coordinates of pivot points connected to lower % parts and the second 3 columns for upper parts %lt: Local coordinates of vertices of top triangle %lo : Free lengths of springs %k : Spring constants %============================================================== global Wext A lb lt lo k %External wrench Weq=[0.01, -0.02, 0.03]'; %Coordinates of fixed points A = [ 0.0, 1.5, 3.0 0.0, 1.2, 0.5 ]; %Local coordinates of vertices of triangles lb = [0, 1.0, 2.0, 0.0, 1.0, 2.0 0, -1.7321, 0.0, 0.0, 0.5, 0.0]; lt = [0, 1.0, 2.0 0, -1.7321, 0.0]; %Free lengths of springs lo = [5.0040, 2.2860, 4.9458, 5.5145, 3.1573, 5.2568]'; %Spring constants k = [0.2, 0.3, 0.4, 0.5, 0.6, 0.7]'; %Initial guess of positions and orientatins of the local coordinate systems Xo=zeros(6,1); Bo=[0.2, 5.0, 10.8*pi/180]'; To=[-0.2, 10.5, 3.4*pi/180]'; Xo=[Bo;To]; %============================================================== % Find Xo %============================================================== options=optimset('fsolve'); optionsnew=optimset(options, 'MaxFunEvals ',1256); % Option to display output

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83 Wext=Weq; [X error] = fsolve(@StaticEq_21, Xo, optionsnew); Xo=X; %Calculate the position of vertices of triangles Bo = X(1:3); To = X(4:6); B = GetGlobalPos2D(Bo, lb); T = GetGlobalPos2D(To, lt); %Calculate initial wrenches W_L_I = zeros(3,1); W_L_I = W_L_I + SpringWrench(A(:,1), B(:,1), k(1), lo(1)); W_L_I = W_L_I + SpringWrench(A(:,2), B(:,2), k(2), lo(2)); W_L_I = W_L_I + SpringWrench(A(:,3), B(:,3), k(3), lo(3)); W_U_I = zeros(3,1); W_U_I = W_U_I + SpringWrench(B(:,4), T(:,1), k(4), lo(4)); W_U_I = W_U_I + SpringWrench(B(:,5), T(:,2), k(5), lo(5)); W_U_I = W_U_I + SpringWrench(B(:,6), T(:,3), k(6), lo(6)); %============================================================== % Calculate stiffness matrix %============================================================== k1=StiffMatrix(A(:,1), B(:,1), k(1), lo(1)); k2=StiffMatrix(A(:,2), B(:,2), k(2), lo(2)); k3=StiffMatrix(A(:,3), B(:,3), k(3), lo(3)); KF_L = k1+k2+k3; k4=StiffMatrix(B(:,4), T(:,1), k(4), lo(4)); k5=StiffMatrix(B(:,5), T(:,2), k(5), lo(5)); k6=StiffMatrix(B(:,6), T(:,3), k(6), lo(6)); KF_U = k4+k5+k6; k4_2 = GetKM(B(:,4), T(:,1), k(4), lo(4)); k5_2 = GetKM(B(:,5), T(:,2), k(5), lo(5)); k6_2 = GetKM(B(:,6), T(:,3), k(6), lo(6)); KM_U = k4_2+k5_2+k6_2; K1 = KF_L*inv(KF_L+KF_U-KM_U)*KF_U; K2 = KF_L*inv(KF_L+KF_U)*KF_U; %============================================================== %============================================================== % Verify the results %============================================================== dW = [0.000005,0.000002,0.000004]';

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84 % 1.Calculate twists D_EB_1 = inv(K1)*dW; D_EB_2 = inv(K2)*dW; D_EA = inv(KF_L)*dW; % 2.Get the position of the bodies Bo_W = GetVelP2D(D_EA, Bo(1:2))+ Bo(1:2); Bo_W(3)= Bo(3)+D_EA(3); B_F = GetGlobalPos2D(Bo_W, lb); To_W_1 = GetVelP2D(D_EB_1, To(1:2))+ To(1:2); To_W_1(3)= To(3)+D_EB_1(3); T_F_1 = GetGlobalPos2D(To_W_1, lt); To_W_2 = GetVelP2D(D_EB_2, To(1:2))+ To(1:2); To_W_2(3)= To(3)+D_EB_2(3); T_F_2 = GetGlobalPos2D(To_W_2, lt); % 3.Calculate wrenches W_L_F = zeros(3,1); W_L_F = W_L_F + SpringWrench(A(:,1), B_F(:,1), k(1), lo(1)); W_L_F = W_L_F + SpringWrench(A(:,2), B_F(:,2), k(2), lo(2)); W_L_F = W_L_F + SpringWrench(A(:,3), B_F(:,3), k(3), lo(3)); dW_L_F = W_L_F Weq; W_U_F_1 = zeros(3,1); W_U_F_1 = W_U_F_1 + SpringWrench(B_F(:,4), T_F_1(:,1), k(4), lo(4)); W_U_F_1 = W_U_F_1 + SpringWrench(B_F(:,5), T_F_1(:,2), k(5), lo(5)); W_U_F_1 = W_U_F_1 + SpringWrench(B_F(:,6), T_F_1(:,3), k(6), lo(6)); dW_U_F_1 = W_U_F_1 Weq; W_U_F_2 = zeros(3,1); W_U_F_2 = W_U_F_2 + SpringWrench(B_F(:,4), T_F_2(:,1), k(4), lo(4)); W_U_F_2 = W_U_F_2 + SpringWrench(B_F(:,5), T_F_2(:,2), k(5), lo(5)); W_U_F_2 = W_U_F_2 + SpringWrench(B_F(:,6), T_F_2(:,3), k(6), lo(6)); dW_U_F_2 = W_U_F_2 Weq; %============================================================== %============================================================== % Display the result %============================================================== K1, K2, D_EB_1, D_EB_2, D_EA, dW, dW_L_F, dW_U_F_1, dW_U_F_2 %============================================================== function NuEx_22

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85 % Numerical Example 2-2 % Test Stiffness Matrix of planar parallel mechanisms in hybird %============================================================== %====== GIVEN VALUES %============================================================== %Wext : External wrench %A : Coordinates of fixed pivot points %lb, lc, ld : Local coordinates of vertices of intermediate triangle %lt: Local coordinates of vertices of top triangle %lo : Free lengths of springs %k : Spring constants %============================================================== global Wext A lb lc ld lt lo k %External wrench Weq=[0.1, 0.1, 0.2]'; %Coordinates of fixed points A = [ 1.67, 4.46, 13.3449, 14.6731, 8.23, 4.94 4.4333, 1.3964, 3.25, 6.84, 14.1400, 13.4943 ]; %Local coordinates of vertices of triangles lb = [0, 2, 1 0, 0, 1.7321]; lc = lb; ld = lb; lt = lb; %Spring constants k = [0.40 0.43 0.46 0.49 0.52 0.55 0.58 0.61 0.64]; %Spring free lengths lo = [2.2547,2.4014,2.3924,1.5910,1.8450,2.2200,1.7077,2.2695,1.8711]'; %Initial guess of positions and orientatins of the local coordinate systems Xo=zeros(12,1); Bo=[3.8, 4.8,-39.8*pi/180]'; Co=[12.4, 4.5,76.5*pi/180]'; Do=[8.1, 12.4,202.4*pi/180]'; To=[7.0, 7.5,-23.4*pi/180]'; Xo=[Bo;Co;Do;To];

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86 %============================================================== % Find Xo %============================================================== Wext = Weq; options=optimset('fsolve'); optionsnew=optimset(options, 'MaxFunEvals ',1256); % Option to display output [X error] = fsolve(@StaticEq_22, Xo, optionsnew); Xo=X; %Calculate the position of vertices of triangles Bo = X(1:3) Co = X(4:6) Do = X(7:9) To = X(10:12) B = GetGlobalPos2D(Bo, lb); C = GetGlobalPos2D(Co, lc); D = GetGlobalPos2D(Do, ld); T = GetGlobalPos2D(To, lt); %============================================================== %Calculate stiffness matrix using stiffness equation %============================================================== k1=StiffMatrix(A(:,1), B(:,1), k(1), lo(1)); k2=StiffMatrix(A(:,2), B(:,2), k(2), lo(2)); k3=StiffMatrix(B(:,3), T(:,1), k(3), lo(3)); k3_2 = GetKM(B(:,3), T(:,1), k(3), lo(3)); KK1_1 = (k1+k2)*inv(k1+k2+k3-k3_2)*k3; KK1_2 = (k1+k2)*inv(k1+k2+k3)*k3; k4=StiffMatrix(A(:,3), C(:,1), k(4), lo(4)); k5=StiffMatrix(A(:,4), C(:,2), k(5), lo(5)); k6=StiffMatrix(C(:,3), T(:,2), k(6), lo(6)); k6_2 = GetKM(C(:,3), T(:,2), k(6), lo(6)); KK2_1 = (k4+k5)*inv(k4+k5+k6-k6_2)*k6; KK2_2 = (k4+k5)*inv(k4+k5+k6)*k6; k7=StiffMatrix(A(:,5), D(:,1), k(7), lo(7)); k8=StiffMatrix(A(:,6), D(:,2), k(8), lo(8)); k9=StiffMatrix(D(:,3), T(:,3), k(9), lo(9)); k9_2 = GetKM(D(:,3), T(:,3), k(9), lo(9)); KK3_1 = (k7+k8)*inv(k7+k8+k9-k9_2)*k9; KK3_2 = (k7+k8)*inv(k7+k8+k9)*k9; K1 = KK1_1+KK2_1+KK3_1; K2 = KK1_2+KK2_2+KK3_2; %==============================================================

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87 %============================================================== % Verify the result %============================================================== dD = zeros(3,1); dW = [0.00005,0.00002,0.00003]'; Wext=Weq+dW; [X error] = fsolve(@StaticEq_22, Xo, optionsnew); dDP = X(10:12)-To; dD(1:2) = GetOriginVel(dDP, To(1:2)); dD(3)=dDP(3); K1_dD = K1*dD; K2_dD = K2*dD; %============================================================== %============================================================== % Display the result %============================================================== K1, K2, dW, dD, K1_dD, K2_dD %============================================================== function NuEx31 % Numerical example 3.1 % Test Stiffness Matrix of spatial parallel mechanisms in serial %============================================================== %====== GIVEN VALUES %============================================================== %Wext : External wrench %A : Coordinates of fixed pivot points %lb_L, lb_U : Local coordinates of vertices of middle platform %lt: Local coordinates of vertices of top triangle %lo_U, lo_L : Free lengths of springs %k_U, k_L : Spring constants %============================================================== %External wrench Weq=[ -0.3, 0.4, 0.8, -2.3, -1.3, 0.7 ]'; %Coordinates of fixed points A = [ 0.0, 1.3, 0.6, -0.7, -1.1, -0.5 0.0, 1.1, 2.7, 2.6, 1.8, 0.4 0.0, 0.2, 0.1, -0.1, 0.3, 0.1 ];

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88 %Local coordinates of vertices of triangles lb_L = [ 0.0, 1.0, 0.3, -0.8, -1.3, -0.4 0.0, 0.9, 2.3, 2.2, 1.4, 0.5 0.0, 0.1, 0.2, -0.1, -0.2, -0.1 ]; lb_U = [ 0.0, 1.3, 0.6, -0.7, -1.1, -0.5 0.0, 1.1, 2.7, 2.6, 1.8, 0.4 0.1, 0.2, 0.25, 0.1, 0.12, 0.15 ]; lt = [ 0.0, 1.2, 0.5, -0.6, -1.0, -0.3 0.0, 1.2, 2.3, 2.4, 1.3, 0.5 0.0, 0.1, -0.1, 0.1, 0.1, 0.2 ]; %Spring constants k_U = [4.6, 4.7, 4.5, 4.4, 5.3, 5.5]'; k_L = [4.4, 4.9, 4.7, 4.5, 5.1, 4.8]'; %Positions and orientatins of the local coordinate systems %Rotation angles are Euler angles (3-2-1) Xo=zeros(12,1); B_Po=[0.2, 1.2, 3.2]'; B_R=[1.2*pi/180, 5.0*pi/180, -1.8*pi/180]'; T_Po=[-0.3, 1.6, 5.5]'; T_R=[-0.4*pi/180, 8.5*pi/180, 3.8*pi/180]'; Xo=[B_Po;B_R;T_Po;T_R]; Xo_I = Xo; %Convert local coord. to global coord. B_L = GetGlobalPos3D(B_Po, B_R, lb_L) B_U = GetGlobalPos3D(B_Po, B_R, lb_U) T = GetGlobalPos3D(T_Po, T_R, lt) %============================================================== % Plucker line coordinates(or Jac obian) and lengths of all springs %============================================================== JS_U=zeros(6,6); JS_L=zeros(6,6); l_U=zeros(6,1); l_L=zeros(6,1); lo_U=zeros(6,1); lo_L=zeros(6,1); F_U=zeros(6,1); F_L=zeros(6,1);

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89 for i=1:6 [JS_U(:,i), l_U(i)] = GetPLine(B_U(:,i), T(:,i)); [JS_L(:,i), l_L(i)] = GetPLine(A(:,i), B_L(:,i)); end %============================================================== % spring forces F_U = inv(JS_U)*Weq; F_L = inv(JS_L)*Weq; %============================================================== % spring free lengths for static equilibrium of the platform for i=1:6 lo_U(i)=l_U(i)-F_U(i)/k_U(i); lo_L(i)=l_L(i)-F_L(i)/k_L(i); end lo_I = [lo_U; lo_L] %============================================================== % Check wrench %============================================================== W_L_I = zeros(6,1); W_U_I = zeros(6,1); for i=1:6 W_L_I = W_L_I + SpringWrench(A(:,i), B_L(:,i), k_L(i), lo_L(i)); W_U_I = W_U_I + SpringWrench(B_U(:,i), T(:,i), k_U(i), lo_U(i)); end %============================================================== %============================================================== %Calculate stiffness matrix using stiffness equation %============================================================== KF_U = zeros(6,6); KF_L = zeros(6,6); KM_U = zeros(6,6); for i=1:6 KF_U = KF_U+StiffMatrix(B_U(:,i), T(:,i), k_U(i), lo_U(i)); KF_L = KF_L+StiffMatrix(A(:,i), B_L(:,i), k_L(i), lo_L(i)); KM_U = KM_U+GetKM(B_U(:,i), T(:,i), k_U(i), lo_U(i)); end K1 = KF_L*inv(KF_L+KF_U-KM_U)*KF_U; K2 = KF_L*inv(KF_L+KF_U)*KF_U;

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90 %============================================================== %============================================================== % Check the wrench %============================================================== dW = 10^(-5)*[5,-2,4,3,-8,4]'; % 1.Calculate twists D_ET_1 = inv(K1)*dW; D_ET_2 = inv(K2)*dW; D_EB = inv(KF_L)*dW; % 2.Get the position of the bodies for i=1:6 B_L_1(:,i) = B_L(:,i)+ D_EB(1:3) + cross(D_EB(4:6), B_L(:,i)); B_U_1(:,i) = B_U(:,i)+ D_EB(1:3) + cross(D_EB(4:6), B_U(:,i)); T_1(:,i) = T(:,i)+ D_ET_1(1: 3) + cross(D_ET_1(4:6), T(:,i)); T_2(:,i) = T(:,i)+ D_ET_2(1: 3) + cross(D_ET_2(4:6), T(:,i)); end % 3.Calculate wrenches W_L_F = zeros(6,1); W_U_F_1 = zeros(6,1); W_U_F_2 = zeros(6,1); for i=1:6 W_L_F = W_L_F + SpringWrench(A(:,i), B_L_1(:,i), k_L(i), lo_L(i)); W_U_F_1 = W_U_F_1 + SpringWrench(B _U_1(:,i), T_1(:,i), k_U(i), lo_U(i)); W_U_F_2 = W_U_F_2 + SpringWrench(B _U_1(:,i), T_2(:,i), k_U(i), lo_U(i)); end dW_L = W_L_F Weq; dW_U_1 = W_U_F_1 Weq; dW_U_2 = W_U_F_2 Weq; %============================================================== %============================================================== % Display the result %============================================================== K1, K2, D_EB, D_ET_1, D_ET_2, dW, dW_L, dW_U_1, dW_U_2 %============================================================== function f = StaticEq21(x) %============================================================== % Global variables

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91 %============================================================== %Wext : External wrench %A : Coordinates of fixed pivot points %lb,lt: Local coordinates of vertices of triangles %lo : Free lengths of springs %k : Spring constants %============================================================== global Wext A lb lt lo k %Bo,Co,Do,To : Postions and Orienta tions of local coordinate systems Bo = x(1:3); To = x(4:6); %Get the position of pivot points B = GetGlobalPos2D(Bo, lb); T = GetGlobalPos2D(To, lt); %Resultant wrench for the middle platform RW_1 =zeros(3,1); for i=1:3 RW_1 = RW_1 + SpringWrench(A(:,i), B(:,i), k(i), lo(i)); end %Resultant wrench for the middle platform RW_2 =zeros(3,1); for i=1:3 RW_2 = RW_2 + SpringWrench(B(:,i+3), T(:,i), k(i+3), lo(i+3)); end f = [ RW_1-RW_2; RW_2-Wext ]'; function f = StaticEq_22(x) %============================================================== % Global variables %============================================================== %Wext : External wrench %A : Coordinates of fixed pivot points %lb,lc,ld,lt : Local coordinates of vertices of triangles %lo : Free lengths of springs %k : Spring constants %============================================================== global Wext A lb lc ld lt lo k %Bo,Co,Do,To : Postions and Orienta tions of local coordinate systems Bo = x(1:3);

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92 Co = x(4:6); Do = x(7:9); To = x(10:12); B = GetGlobalPos2D(Bo, lb); C = GetGlobalPos2D(Co, lc); D = GetGlobalPos2D(Do, ld); T = GetGlobalPos2D(To, lt); %spring forces fA1B1 = SpringWrench(A(:,1), B(:,1), k(1), lo(1)); fA2B2 = SpringWrench(A(:,2), B(:,2), k(2), lo(2)); fB3T1 = SpringWrench(B(:,3), T(:,1), k(3), lo(3)); fA3C1 = SpringWrench(A(:,3), C(:,1), k(4), lo(4)); fA4C2 = SpringWrench(A(:,4), C(:,2), k(5), lo(5)); fC3T2 = SpringWrench(C(:,3), T(:,2), k(6), lo(6)); fA5D1 = SpringWrench(A(:,5), D(:,1), k(7), lo(7)); fA6D2 = SpringWrench(A(:,6), D(:,2), k(8), lo(8)); fD3T3 = SpringWrench(D(:,3), T(:,3), k(9), lo(9)); RW_1 = fA1B1+fA2B2-fB3T1; RW_2 = fA3C1+fA4C2-fC3T2; RW_3 = fA5D1+fA6D2-fD3T3; RW_4 = fB3T1+fC3T2+fD3T3-Wext; f = [ RW_1; RW_2; RW_3; RW_4 ]; function K = StiffMatrix(aa, bb, k, lo) %Calculate Stiffness matrix for 2D and 3D %Check whether it is planar or spatial case and make it spatial a=zeros(3,1); b=zeros(3,1); if(size(aa,1)==2) a(1:2)=aa; b(1:2)=bb; else a=aa; b=bb; end N = b a; %non-unitized directional vector l = norm(N,2); %length rho = lo/l;

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93 S = N/l; %unit direction vector So = cross(a, S); w = [S;So]; K1 = k*(w*w'); alpha = atan2(S(2),S(1)); beta = atan2(sqrt(S(1)^2+S(2)^2), S(3)); dSdB_S =[cos(beta)*cos(alpha), cos(beta)*sin(alpha), -sin(beta)]'; dSdB_So = cross(a,dSdB_S); dSdB = [dSdB_S; dSdB_So]; dSdB_So_2 = cross(b,dSdB_S); dSdB_2 = [dSdB_S; dSdB_So_2]; K2 = k*(1-rho)*(dSdB*dSdB_2'); udSdA_S = [-sin(alpha), cos(alpha), 0.0]'; udSdA_So = cross(a, udSdA_S); dSdA = [udSdA_S; udSdA_So]; dSdA_So_2 = cross(b, udSdA_S); dSdA_2 = [udSdA_S; dSdA_So_2]; K3 = k*(1-rho)*(dSdA*dSdA_2'); K = K1+K2+K3; if(size(aa,1)==2) K(:,3:5)=[]; K(3:5,:)=[]; end function KM = GetKM(aa, bb, k, lo) a=zeros(3,1); b=zeros(3,1); if(size(aa,1)==2) a(1:2)=aa; b(1:2)=bb; else a=aa; b=bb; end N = b a; %non-unitized directional vector l = norm(N,2); %length S = N/l; %unit direction vector

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94 alpha = atan2(S(2),S(1)); beta = atan2(sqrt(S(1)^2+S(2)^2), S(3)); dSdB_S =[cos(beta)*cos(alpha), cos(beta)*sin(alpha), -sin(beta)]'; dSdB_So = cross(a,dSdB_S); dSdB = [dSdB_S; dSdB_So]; dSdB_So_2 = cross(b,dSdB_S); dSdB_2 = [dSdB_S; dSdB_So_2]; udSdA_S = [-sin(alpha), cos(alpha), 0.0]'; udSdA_So = cross(a, udSdA_S); dSdA = [udSdA_S; udSdA_So]; dSdA_So_2 = cross(b, udSdA_S); dSdA_2 = [udSdA_S; dSdA_So_2]; K1=[0;0;0;dSdB_S]*dSdA_2'; K2=dSdB_2*[0;0;0;udSdA_S]'; KM = k*(l-lo)*(K1+K2-(K1+K2)'); if(size(aa,1)==2) KM(:,3:5)=[]; KM(3:5,:)=[]; End function w = SpringWrench(lp1, lp2, k, lo) %Calculate spring wrench %p1,p2 : pivot points %k : spring constant %l0 : spring free length if size(lp1,1) == 2 p1=[lp1;0]; p2=[lp2;0]; else p1=lp1; p2=lp2; end %Current spring lengths l=norm(p2-p1, 2); rho =lo/l;

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95 w = k*(1-rho)*[p2-p1; cross(p1,p2-p1)]; if size(lp1,1) == 2 w(3:5)=[]; end function [w, l] = GetPLine(lp1, lp2) % Get Plucker line coordinate s and length of spring %p1,p2 : pivot points %Convert into spatial vector if size(lp1,1) == 2 p1=[lp1;0]; p2=[lp2;0]; else p1=lp1; p2=lp2; end %Magnitude l=norm(p2-p1, 2); %Unitize w = 1/l*[p2-p1; cross(p1,p2-p1)]; if size(lp1,1) == 2 w(3:5)=[]; end function Vo = GetOriginVel(lVp, lrp) % Calculate origin velocity from veloci ty of a point and angular velocity % of rigid body Vp = zeros(3,1); W = zeros(3,1); rp = zeros(3,1); if size(lVp,1)==3 Vp(1:2) = lVp(1:2); W(3) = lVp(3); rp(1:2) = lrp; else Vp=lVp(1:3); W=lVp(4:6);

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96 rp = lrp; end Vo = Vp cross(W, rp); if size(lVp,1)==3 Vo(3)=[]; end function [dRp] = GetVelP2D(dD, lRp) %Calculate small displacement of point P from twist RM = [ cos(dD(3)), -sin(dD(3)) sin(dD(3)), cos(dD(3)) ]; dRp= dD(1:2) + RM*lRp lRp; function GP = GetGlobalPos2D(Po, LP) % Calculate Global coordina tes for local coordinates % Po(1:2) : position of origin of local coord. % Po(3) : rotation angle of local coord. % LP : local coord. num=size(LP, 2); GP = zeros(2, num); R_GL = [cos(Po(3)), -sin(Po(3)) sin(Po(3)), cos(Po(3))]; for i=1:num GP(:,i) = Po(1:2) + R_GL*LP(:,i); end function GP = GetGlobalPos3D(Po, ER, LP) %Convert coordinates using Po(Position of origin) and ER(Euler angles) num=size(LP, 2); GP = zeros(3, num); gamma = ER(1); beta = ER(2); alpha = ER(3);

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97 RM = [cos(gamma)*cos(beta), sin(gamma)*cos(alpha)+cos(ga mma)*sin(beta)*sin(alpha), sin(gamma)*sin(alpha)+cos(gamma)*sin(beta)*cos(alpha) sin(gamma)*cos(beta), cos(gamma)* cos(alpha)+sin(gamma)*sin(beta)*sin(alpha), cos(gamma)*sin(alpha)+sin(ga mma)*sin(beta)*cos(alpha) -sin(beta), cos(beta)*sin(alpha), cos(beta)*cos(alpha)]; for i=1:num GP(:,i) = Po + RM*LP(:,i); end

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98 APPENDIX B MAPLE CODE FOR DERIVATIVE OF STIFFNESS MATRIX IN CHAPTER FOUR Maple code to compute the matrix d dB U shown in Eq. (4.43) is presented in this appendix. This code creates a text file called “dBdU.map” and writes symbolic equations for the derivative in it. The computation of the derivative is quite complicated and thus the size of the created file exceeds two megabytes. Then this file may be converted to a Matlab file with a little modification. Since the matrix d dB U is quite complicated, symbolic equations for matrices d dB P, d dP Q, and d dQ U are obtained and column vectors P and Q are defined as 111213223233 111213223233[,,,,,,,,, ,,,,,,,,]UUUUUUUUU xyz LLLLLLLLLT xyz K KKKKKffm KKKKKKffm=P 126126126 111111121212131313 126126126 222222323232333333 126126126[,,,,,,,,,,,, ,,,,,,,,,,,, ,,,,,,,,,,,]T xxxyyyzzzKKKKKKKKK K KKKKKKKK ffffffmmm = Q Please reference chapter four for detailed description of each term. Then d d B U can be computed as d ddd dddd=Q BBP UPQU.

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99 > restart; > with(LinearAlgebra): > fopen("dBdU.map",WRITE): ===================================== Get dXdP ===================================== >B:=[KR[1,1],KR[1,2],KR[1,3],KR[2,2],KR[3,2],KR[3,3],Fx_U,Fy _U,Mz_U,Fx_L,Fy_L,Mz_L]; := B [ ] ,,,,,,,,,,, K R, 11 K R, 12 K R, 13 K R, 22 K R, 32 K R, 33 F x_U F y_U M z_U F x_L F y_L M z_L >F_L:=<||>; := KF_L k11_Lk12_Lk13_L k12_Lk22_L + k32_LFx_L + k13_LFy_Lk32_Lk33_L >KF_U:=<||>; := KF_U k11_Uk12_Uk13_U k12_Uk22_U + k32_UFx_U + k13_UFy_Uk32_Uk33_U > KM_U:=<<0,0,Fy_U>|<0,0,-Fx_U>|<-Fy_U,Fx_U,0>>; := KM_U 00 Š F y_U 00 Fx_U Fy_U Š Fx_U 0 > CM:=KF_L+KF_U-KM_U; := CM + k11_Lk11_U + k12_Lk12_U + + k13_Lk13_U F y_U + k12_Lk12_U + k22_Lk22_U + + k32_LFx_Lk32_U + + k13_LFy_Lk13_U + + k32_Lk32_UFx_U + k33_Lk33_U > KR:=KF_L.MatrixInverse(CM).KF_U: > P:=[k11_U,k12_U,k13_U,k22_U,k32_U,k33_U,Fx_U,Fy_U,Mz_U, k11_L,k12_L,k13_L,k22_L,k32_L,k33_L,Fx_L,Fy_L,Mz_L]; P k11_Uk12_Uk13_Uk22_Uk32_Uk33_U F x_U F y_U M z_Uk11_Lk12_L ,,,,,,,,,,, [ := k13_Lk22_Lk32_Lk33_LFx_LFy_LMz_L ,,,,,,] > nrow:=nops(B); := nrow 12 > ncol:=nops(P); := ncol 18 > dBdP:=Matrix(nrow,ncol): > for i from 1 to nrow do

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100 for j from 1 to ncol do dBdP[i,j]:=diff(B[i], P[j]); end do: end do: ===================================== Write dBdP on file "dBdU.map" ===================================== > for i from 1 to nrow do for j from 1 to ncol do fprintf("dBdU.map","dBdP(%d,%d)=%s;\n\n",i,j,convert(d BdP[i,j],string)); end do: end do: > fprintf("dBdU.map","\n\n\n"): ===================================== Get dPdQ ===================================== > k11_U:=k11[1]+k11[2]+k11[3]: > k12_U:=k12[1]+k12[2]+k12[3]: > k13_U:=k13[1]+k13[2]+k13[3]: > k22_U:=k22[1]+k22[2]+k22[3]: > k32_U:=k32[1]+k32[2]+k32[3]: > k33_U:=k33[1]+k33[2]+k33[3]: > Fx_U:=fx[1]+fx[2]+fx[3]: > Fy_U:=fy[1]+fy[2]+fy[3]: > Mz_U:=mz[1]+mz[2]+mz[3]: > k11_L:=k11[4]+k11[5]+k11[6]: > k12_L:=k12[4]+k12[5]+k12[6]: > k13_L:=k13[4]+k13[5]+k13[6]: > k22_L:=k22[4]+k22[5]+k22[6]: > k32_L:=k32[4]+k32[5]+k32[6]: > k33_L:=k33[4]+k33[5]+k33[6]: > Fx_L:=fx[4]+fx[5]+fx[6]: > Fy_L:=fy[4]+fy[5]+fy[6]: > Mz_L:=mz[4]+mz[5]+mz[6]: > > Q:=[seq(k11[i], i=1..6)]: > Q:=[op(Q),seq(k12[i], i=1..6)]: > Q:=[op(Q),seq(k13[i], i=1..6)]: > Q:=[op(Q),seq(k22[i], i=1..6)]: > Q:=[op(Q),seq(k32[i], i=1..6)]: > Q:=[op(Q),seq(k33[i], i=1..6)]:

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101 > Q:=[op(Q),seq(fx[i], i=1..6)]: > Q:=[op(Q),seq(fy[i], i=1..6)]: > Q:=[op(Q),seq(mz[i], i=1..6)]; Qk111k112k113k114k115k116k121k122k123k124k125k126k131k132,,,,,,,,,,,,,, [ := k133k134k135k136k221k222k223k224k225k226k321k322k323k324k325,,,,,,,,,,,,,,, k326k331k332k333k334k335k336fx1fx2fx3fx4fx5fx6fy1fy2fy3fy4fy5,,,,,,, ,,,,,,,,,,, fy6mz1mz2mz3mz4mz5mz6,,,,,,] > nrow:=nops(P); := nrow 18 > ncol:=nops(Q); := ncol 54 > dPdQ:=Matrix(nrow,ncol): > for i from 1 to nrow do for j from 1 to ncol do dPdQ[i,j]:=diff(P[i], Q[j]); end do: end do: =================================== Write dPdQ on file "dBdU.map" =================================== > for i from 1 to nrow do for j from 1 to ncol do fprintf("dBdU.map","dPdQ(%d,%d)=%s;\n\n",i,j,convert(d PdQ[i,j],string)); end do: end do: > fprintf("dBdU.map","\n\n\n"): =================================== Get dQdU =================================== > for i from 1 to 6 do k11[i]:=k[i]-k[i]*lo[i]/l[i]*sin(theta[i])^2; k12[i]:=k[i]*lo[i]/l[i]*sin(theta[i])*cos(theta[i]); k13[i]:=-k[i]*(l[i]*sin(theta[i])+y[i]) +k[i]*lo[i]/l[i]*(l[i]*sin(theta[i])+x[i]*sin(theta[i])*c os(theta[i])+y[i]*sin(theta[i])^2); k22[i]:=k[i]-k[i]*lo[i]/l[i]*cos(theta[i])^2;

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102 k32[i]:=k[i]*x[i]-k[i]*lo[i]/l[i]*(x[i]*cos(theta[i])^2 +y[i]*sin(theta[i])*cos(theta[i])); k33[i]:=k[i]*x[i]*(l[i]*cos(theta[i])+x[i])+k[i]*y[i]*(l[ i]*sin(theta[i])+y[i])-k[i]*lo[i]/l[i]*x[i] *(l[i]*cos(theta[i])+y[i]*sin(theta[i])*cos(theta[i]) +x[i]*cos(theta[i])^2)-k[i]*lo[i]/l[i]*y[i] *(l[i]*sin(theta[i])+x[i]*sin(theta[i])*cos(theta[i]) +y[i]*sin(theta[i])^2); fx[i]:=k[i]*(l[i]-lo[i])*cos(theta[i]); fy[i]:=k[i]*(l[i]-lo[i])*sin(theta[i]); mz[i]:=k[i]*(l[i]-lo[i])*(x[i]*sin(theta[i])y[i]*cos(theta[i])); end do: > > Z:=[seq(k[i], i=1..6)]: > Z:=[op(Z),seq(lo[i], i=1..6)]: > Z:=[op(Z),seq(theta[i], i=1..6)]: > Z:=[op(Z),seq(l[i], i=1..6)]: > Z:=[op(Z),seq(x[i], i=1..3)]: > Z:=[op(Z),seq(y[i], i=1..3)]; Z k 1 k 2 k 3 k 4 k 5 k 6lo1lo2lo3lo4lo5lo6123456 l 1 l 2 l 3 l 4 l 5 l 6 x 1,,,,,,,,,,,,,, ,,,,,,,,,,, [ := x2x3y1y2y3,,,,] > nrow:=nops(Q); := nrow 54 > ncol:=nops(Z); := ncol 30 > dQdZ:=Matrix(nrow,ncol): > for i from 1 to nrow do for j from 1 to ncol do dQdZ[i,j]:=diff(Q[i], Z[j]); end do: end do: > > dQdU:=Matrix(nrow,18): > for i from 1 to nrow do for j from 1 to 12 do dQdU[i,j]:=dQdZ[i,j]; end do: end do:

PAGE 114

103 > > #column 13: dAxo, 14:dAyo, 15:dAt > #column 16: dBxo, 17:dByo, 18:dBt > for i from 1 to nrow do # For theta:1,2,3 and l:1,2,3 for j from 1 to 3 do dQdZ[i,12+j]:=1/l[j]*( Lp[j]*(dxo_b-dxo_a)+Mp[j] *(dyo_b-dyo_a)+Rp[j]*(dth_b-dth_a) + l[j]*dth_a) *dQdZ[i,12+j]; dQdZ[i,18+j]:=( L[j]*(dxo_b-dxo_a)+M[j]*(dyo_b-dyo_a) +R[j]*(dth_b-dth_a) )*dQdZ[i,18+j]; end do; # For theta:4,5,6 and l:4,5,6 for j from 4 to 6 do dQdZ[i,12+j]:=1/l[j]*(Lp[j]*dxo_a+Mp[j]*dyo_a +Rp[j]*dth_a)*dQdZ[i,12+j]; dQdZ[i,18+j]:=(L[j]*dxo_a+M[j]*dyo_a+R[j]*dth_a) *dQdZ[i,18+j]; end do; # For x:1,2,3 and y:1,2,3 for j from 1 to 3 do dQdZ[i,24+j]:=(dxo_a-y[j]*dth_a)*dQdZ[i,24+j]; dQdZ[i,27+j]:=(dyo_a+x[j]*dth_a)*dQdZ[i,27+j]; end do; temp:=0; for j from 13 to 30 do temp:=temp+dQdZ[i,j]; end do; dQdU[i,13]:=coeff(temp,dxo_a); dQdU[i,14]:=coeff(temp,dyo_a); dQdU[i,15]:=coeff(temp,dth_a); dQdU[i,16]:=coeff(temp,dxo_b); dQdU[i,17]:=coeff(temp,dyo_b); dQdU[i,18]:=coeff(temp,dth_b); end do: > > for i from 1 to 3 do

PAGE 115

104 L[i]:=cos(theta[i]); M[i]:=sin(theta[i]); R[i]:=x[i]*sin(theta[i])-y[i]*cos(theta[i]); Lp[i]:=-sin(theta[i]); Mp[i]:=cos(theta[i]); Rp[i]:=x[i]*cos(theta[i])+y[i]*sin(theta[i])+l[i]; end do: =================================== Write dQdU on file "dBdU.map" =================================== > for i from 1 to nrow do for j from 1 to 18 do fprintf("dBdU.map","dQdU(%d,%d)=%s;\n\n", i,j,convert(dQdU[i,j],string)); end do: end do: > fprintf("dBdU.map","\n\n\n"): > > fclose("dBdU.map"):

PAGE 116

105 LIST OF REFERENCES Ball, R. S., 1900, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, UK. Chen, S., and Kao, I., 2000, “Conservativ e Congruence Transformation for Joint and Cartesian Stiffness matrices of Robotic Hands and Fingers,” International Journal of Robotic Research, Vol. 19, No. 9, pp. 835-847. Ciblak, N., and Lipkin, H., 1994, “Asymmetri c Cartesian Stiffness for the Modeling of Compliant Robotic Systems,” Proc. ASME 23rd Biennial Mech. Conf., Des. Eng. Div., Vol. 72, pp. 197-204, New York, NY. Ciblak, N., and Lipkin, H., 1999, “Synthes is of Cartesian Stiffness for Robotic Applications,” Proceedings of the IEEE Inter national Conference on Robotics and Automation, pp. 2147–2152, Detroit, MI. Crane, C. D., and Duffy, J., 1998, Kinematic Analysis of Robot Manipulators, Cambridge University Press, Cambridge, UK. Crane, C. D., Rico, J. M., and Duffy, J., 2006, Screw Theory and Its Application to Spatial Robot Manipulators, Cambridge University Press, Cambridge, UK. Craig, J. J., 1989, Introduction to Robotics: Mechanics and Control, Addison Wesley, Reading, MA. Dimentberg, F. M., 1965, The Screw Calculus and its Applications in Mechanics. Foreign Technology Division, Wright-Patterson Ai r Force Base, Ohio. Document No. FTDHT-23-1632-67. Duffy, J., 1996, Statics and Kinematics with Applications to Robotics, Cambridge University Press, Cambridge, UK. Featherstone, J., 1985, Robot Dynamics Algorithms, Kluwer Academic Publishers, Boston, MA. Griffis, M., 1991, “A Novel Theory for Simultaneously Regulating Force and Displacement,” Ph.D. dissertation, Univ ersity of Florida, Gainesville, FL. Henrie, A. M., 1997, “Variable Compliance vi a Magneto-Rheological Materials,” M.S. Thesis, Brigham Young University, Provo, UT.

PAGE 117

106 Huang, S., 1998, “The Analysis and Synthesis of Spatial Compliance, ” Ph.D. dissertation, Marquette University, Milwaukee, WI. Huang, S., and Schimmels, J. M., 1998, “The Bounds and Realization of Spatial Stiffness Achieved with Simple Springs Connected in Parallel,” IEEE Transactions on Robotics and Automation, Vol. 14, No. 3, pp. 466-475. Hurst, J. W., Chestnutt, J., and Rizzi, A., 2004, “An Actuator with Physically Variable Stiffness for Highly Dynamic Legged Locomotion,” Proceedings of the 2004 International Conference on Robotics and Automation, pp. 4662-4667, New Orleans, LA. Kane, T. R., and Levinson, D. A., 1985, Dynamics: Theory and Applications, McGraw, New York. Loncaric, J., 1985, “Geometrical Analysis of Compliant Mechanisms in Robotics,” Ph.D. Dissertation, Harvard University, Cambridge, MA. Loncaric, J., 1987, “Normal Forms of Stiffness and Compliance Matrices,” IEEE Journal of Robotics and Automation, Vol. 3, No. 6, pp. 567–572. McCarthy, J. M., 1990, Introduction to Theo retical Kinematics, MIT Press, Cambridge, MA. McLachlan, S., and Hall, T., 1999, “Robust Forward Kinematic So lution for Parallel Topology Robotic Manipulator,” 32nd ISATA Conference – Track : Simulation, Virtual Reality and Supercompu ting Automotive Applications, pp 381-388, Vienna, Austria. Peshkin, M., 1990, “Programmed Complia nce for Error Corrective Assembly,” IEEE Transactions on Robotics and Automation, Vol. 6, No. 4, pp. 473-482. Pigoski, T., 1993, “An Introductory Theoretical Analysis of Planar Compliant Couplings,” M.S. Thesis, University of Florida, Gainesville, FL. Ryan, M. W., Franchek, M.A., and Bernha rd, R., 1994. Adaptive-Passive Vibration Control of Single Frequency Exc itation Applied to Noise Control. Noise-Con Proceedings, pp. 461-466, Fort Lauderdale, FL. Roberts, R. G., 1999, “Minimal Realization of a Spatial Stiffness Matrix with Simple Springs Connected in Parallel,” IEEE Transactions on Robotics and Automation, Vol. 15, No. 5, pp. 953-958. Salisbury, J. K., 1980, “Active Stiffness Control of a Manipulator in Cartesian Coordinates,” Proc. 19th IEEE Conference on Decision and Control, pp. 87-97, Albuquerque, NM.

PAGE 118

107 Simaan, N., and Shoham, M., 2002, “Stiffness Sy nthesis of a Variable Geometry Planar Robots,” the 8th International Symposium on Ad vances in Robot Kinematics (ARK 2002), Kluwer Academic Publisher, Caldes de Malavella, Spain. Simaan, N., and Shoham, M., 2003, “Geometric Interpretation of the Derivatives of Parallel Robot's Jacobian Matrix with Application to Stiffness Control,” ASME Journal of Mechanical Design, Vol. 125, pp. 33-42. Strang, G., 1988, Linear Algebra and Its Applications, Harcourt Brace Jovanovich, New York. Whitney, D. E., 1982, “Quasi-static Assembly of Compliantly Supported Rigid Parts,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 104, pp. 6577.

PAGE 119

108 BIOGRAPHICAL SKETCH Hyun Kwon Jung was born in Chunju, South Korea, in 1971. He attended Hanyang University, Seoul, where he received Bachelor of Science and Master of Science degrees in precision mechanical en gineering in 1994 and 1996 respectively. He then worked for five years at Samsung Electronics, Suwon, South Korea, and that was the complement to mandatory milita ry service for every Korean man. In 2002 he enrolled in graduate school at the University of Florida for a Ph.D. degree in mechanical engineering and in 2003 he started working with Dr. Carl D. Crane III at the Center for Intelligent Machines and Robotics as a research assistant. His fields of interest include kinematics, control, and realtime programming.


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PARALLEL MECHANISMS WITH VARIABLE COMPLIANCE


By

HYUN KWON JUNG













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


2006


































Copyright 2006

by

Hyun Kwon Jung
































This dissertation is dedicated to my wife, Eyun Jung Lee and son, Sung Jae.















ACKNOWLEDGMENTS

I would like express my thanks to Dr. Carl D. Crane III, my academic advisor and

committee chair, for his continual support and guidance throughout this work. I would

also like to thank the other members of my supervisory committee, Dr. John C. Ziegert,

Dr. John K. Schueller, Dr. A. Antonio Arroyo, and Dr. Rodney G. Roberts, for their time,

expertise, and willingness to serve on my committee.

I would like to thank all of the personnel of the Center for Intelligent Machines and

Robotics for their support and expertise. I also would like to thank other friends of mine

for providing plenty of advice and diversions.

Last but not least, I would like to thank to my parent, parents-in-law, my wife, and

son for their unwavering support, love, and sacrifice.

This research was performed with funding from the Department of Energy through

the University Research Program in Robotics (URPR), grant number DE-FG04-

86NE37967.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ............... ................. ............ .......................... vii

LIST O F FIGU RE S ......... ........................................ ........ ........... ix

A B ST R A C T ...................................... ................................ ...................... x

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1 .1 M o tiv atio n .............................. ................................................... ............... 1
1.2 L literature R eview ........... .......................................................... ......... .... .2
1.3 Problem Statem ent .................. ............................ .... .... .. ........ .. ..

2 STIFFNESS MAPPING OF PLANAR COMPLIANT MECHANISMS ....................8

2 .1 Spring in a L in e Sp ace ............................................ .. ....... ........................8
2.2 A Derivative of Planar Spring Wrench Joining a Moving Body and Ground. 11
2.3 A Derivative of Spring Wrench Joining Two Moving Bodies ........................15
2.4 Stiffness Mapping of Planar Compliant Parallel Mechanisms in Series .........22
2.5 Stiffness Mapping of Planar Compliant Parallel Mechanisms in a Hybrid
A rrangem ent .....................................................................27

3 STIFFNESS MAPPING OF SPATIAL COMPLIANT MECHANISMS..................33

3.1 A Derivative of Spatial Spring Wrench Joining a Moving Body and
G rou n d ....................................... .... ............. ........... ............... 3 3
3.2 A Derivative of Spring Wrench Joining Two Moving Bodies ......................39
3.3 Stiffness Mapping of Spatial Compliant Parallel Mechanisms in Series ........49

4 STIFFNESS MODULATION OF PLANAR COMPLIANT MECHANISMS.........56

4.1 Parallel M echanisms with Variable Compliance .................. .... ............56
4.1.1 Constraint on Stiffness M atrix .......................................... ......... ......56
4.1.2 Stiffness Modulation by Varying Spring Parameters ..........................60









4.1.3 Stiffness Modulation by Varying Spring Parameters and
Displacement of the M echanism ............................................................ 64
4.2 Variable Compliant Mechanisms with Two Parallel Mechanisms in Series...69
4.2.1 Constraints on Stiffness M atrix ............................................. ........69
4.2.2 Stiffness Modulation by using a Derivative of Stiffness Matrix and
W rench .............. ...... ............................... 70
4.2.3 N um erical Exam ple ........................................ .......................... 74

5 CON CLU SION S .................................. .. .......... .. ............78

APPENDIX

A MATLAB CODES FOR NUMERICAL EXAMPLES IN CHAPTER TWO AND
T H R E E ............................................................................. 8 1

B MAPLE CODE FOR DERIVATIVE OF STIFFNESS MATRIX IN CHAPTER
F O U R ........................................................................... 9 8

LIST OF REFEREN CES ........................................................... .. ............... 105

BIOGRAPHICAL SKETCH ............................................................. ............... 108
















LIST OF TABLES


Table pge

2-1 Spring properties of the compliant couplings in Figure 2-6............... ................ 25

2-2 Positions of pivot points in terms of the inertial frame in Figure 2-6 ....................26

2-3 Spring properties of the compliant couplings in Figure 2-7 ...................................30

2-4 Positions of the fixed pivot points of the compliant couplings in Figure 2-7. .........30

2-5 Positions and orientations of the coordinates systems in Figure 2-7. ...................30

3-1 Spring properties of the mechanism in Figure 3-5............. ............ ...............52

3-2 Positions of pivots in ground in Figure 3-5......................................................53

3-3 Positions of pivots in bottom side of body A in Figure 3-5. ................ ............53

3-4 Positions of pivots in top side of body A in Figure 3-5. .........................................53

3-5 Positions of pivots in body B in Figure 3-5. .........................................................53

4-1 Positions of pivot points in body E for numerical example in 4.1.2......................63

4-2 Positions of pivot points in body A for numerical example in 4.1.2......................63

4-3 Spring parameters with minimum norm for numerical example 4.1.2. .................63

4-4 Given optimal spring parameters for numerical example 4.1.2 ............................64

4-5 Spring parameters closest to given spring parameters for numerical example
4 .1 .2 ........................................ ..................................... 6 4

4-6 Positions of pivot points for numerical example 4.1.3.........................................67

4-7 Initial spring parameters for numerical example 4.1.3...........................................67

4-8 Calculated spring parameters for numerical example 4.1.3................. ............68

4-9 Positions of pivot points in body A for numerical example 4.1.3 ............................68









4-10 Spring parameters of the compliant couplings for numerical example 4.2.3...........74

4-11 Positions of pivot points for numerical example 4.2.3.........................................74

4-12 Spring parameters with no constraint for numerical example 4.2.3.......................75

4-13 Spring parameters with body A fixed for numerical example 4.2.3 ......................75

4-14 Spring parameters with body A and body B fixed for numerical example 4.2.3.....77

A -1 M atlab function list. ...................................... ........... .... ............ 1
















LIST OF FIGURES


Figure pge

1-1 Planar robot with variable geometry base platform ................................................ 4

1-2 A daptive vibration absorber. .............................................. ............................ 4

1-3 Parallel topology 6DOF with adjustable compliance......................... ..........5

2-1 Spring in a line space. .................................... .. .. .... .... ............ .9

2-2 Spring arrangements in a line space. (a) parallel and (b) series............................10

2-3 Planar compliant coupling connecting body A and the ground. .............................11

2-4 Small change of position of Pl due to a small twist of body A..............................13

2-5 Planar compliant coupling joining two moving bodies................ ..................15

2-6 Mechanism having two compliant mechanisms in series. .....................................23

2-7 Mechanism consisting of four rigid bodies connected to each other by compliant
couplings in a hybrid arrangement. ........................................ ....................... 32

3-1 Spatial compliant coupling joining body A and the ground...............................34

3-2 Unit vector expressed in a polar coordinates system. ............................................34

3-3 Small change of position of Pl due to a small twist of body A..............................36

3-4 Spatial compliant coupling joining two moving bodies ............ .....................39

3-5 Mechanism having two compliant parallel mechanisms in series .........................52

4-1 Compliant parallel mechanism with N number of couplings..............................58

4-2 Poses of the compliant parallel mechanism for numerical example 4.1.3 ..............69

4-3 Poses of the compliant mechanism with body B fixed. ................. .................76

4-4 Poses of the compliant mechanism with no constraint. ........................................76















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

PARALLEL MECHANISMS WITH VARIABLE COMPLIANCE

By

Hyun Kwon Jung

May 2006

Chair: Carl D. Crane III
Major Department: Mechanical and Aerospace Engineering

Compliant mechanisms can be considered as planar/spatial springs having multiple

degrees of freedom rather than one freedom as line springs have. The compliance of the

mechanism can be well described by the stiffness matrix of the mechanism which relates

a small twist applied to the mechanism to the corresponding wrench exerted on the

mechanism.

A derivative of the spring wrench connecting two moving rigid bodies is derived.

By using the derivative of the spring wrench, the stiffness matrices of compliant

mechanisms which consist of rigid bodies connected to each other by line springs are

obtained. It is shown that the resultant compliance of two compliant parallel mechanisms

that are serially arranged is not the summation of the compliances of the constituent

mechanisms unless the external wrench applied to the mechanism is zero.

A derivative of the stiffness matrix of planar compliant mechanisms with respect to

the twists of the constituent rigid bodies and the spring parameters such as the stiffness

coefficient and free length is obtained. It is shown that the compliance and the resultant









wrench of a compliant mechanism may be controlled at the same time by using adjustable

line springs.














CHAPTER 1
INTRODUCTION

1.1 Motivation

Robots have been employed successfully in applications that do not require

interaction between the robot and the environment but require only position control

schemes. For instance, arc welding and painting belong to this category of application.

There are many other operations involving contact of the robot and its environment. A

small amount of positional error of the robot system, which is almost inevitable, may

cause serious damage to the robot or the object with which it is in contact. Compliant

mechanisms, which may be inserted between the end effecter and the last link of the

robotic manipulator, can be a solution to this problem.

Compliant mechanisms can be considered as spatial springs having multiple

degrees of freedom rather than one freedom as line springs have. A small force/torque

applied to the compliant mechanism generates a small displacement of the compliant

mechanism. This relation is well described by the compliance matrix of the mechanism.

RCC (Remote Center of Compliance) devices, developed by Whitney (1982), are one of

the most successful compliant mechanisms. They have a unique compliant property at a

specific operation point and are mainly used to compensate positional errors during tasks

such as inserting a peg into a chamfered hole. Compliant mechanisms can also be

employed for force control applications by using the theory of Kinestatic Control which

was proposed by Griffis (1991). Kinestatic Control varies the position of the last link of









the manipulator to control the position and contact force of the distal end of the robotic

manipulator at the same time with the compliance of the mechanism in mind.

Mechanisms with variable compliance, which is the topic of this dissertation, are

believed to have several advantages over mechanisms having fixed compliance. Since

RCC devices typically have a specific operation point, if the length of the peg to be

inserted is changed, a different RCC device should be employed to do insertion tasks

unless the RCC device has variable compliance. As for force control tasks, each task

may have an optimal compliance. With variable compliant mechanisms, several different

tasks involving different force ranges can be accomplished without having to physically

change the compliant mechanism. Variable compliant mechanisms also can improve the

performance of humanoid robot parts such as ankles and wrists, and animals are believed

to have physically variable leg compliance and utilize it when running and hopping (see

Hurst et al. 2004).

Many compliant mechanisms including RCC devices have been designed typically

based on parallel kinematic mechanisms. Parallel kinematic mechanisms contain positive

features compared to serial mechanisms such as higher stiffness, compactness, and

smaller positional errors at the cost of a smaller workspace and increased complexity of

analysis. In this dissertation mechanisms having two compliant parallel mechanisms in a

serial arrangement as well as compliant parallel mechanisms are investigated. These

mechanisms may have a trade-off of characteristics relative to traditional parallel and

serial mechanisms.

1.2 Literature Review

The concepts of twists and wrenches were introduced by Ball (1900) in his

groundbreaking work A Treatise on the Theory of Screws. These concepts are employed









throughout this dissertation to describe a small (or instantaneous) displacement of a rigid

body and a force/torque applied to a body (Crane et al. 2006).

The compliance of a mechanism can be well described by the stiffness matrix

which is a 6 x 6 matrix for a spatial mechanism and a 3 x 3 matrix for a planar

mechanism. Using screw theory, Dimentberg (1965) studied properties of an elastically

suspended body. Loncaric (1985) used Lie groups rather than screw theory to study

symmetric spatial stiffness matrices of compliant mechanisms assuming that the springs

are in an equilibrium position and derived a constraint that makes the number of

independent elements of symmetric 6 x 6 stiffness matrices 20 rather than 21. Loncaric

(1987) also defined a normal form of the stiffness matrix in which rotational and

translational parts of the stiffness matrix are maximally decoupled.

Griffis (1991) presented a global stiffness model for compliant parallel mechanisms

where he used the term global to state that the springs are not restricted to an unloaded

equilibrium position. Griffis (1991) also showed that the stiffness matrix is not

symmetric when the springs are deflected from the equilibrium positions due to an

external wrench. Ciblak and Lipkin (1994) decomposed a stiffness matrix into a

symmetric and a skew symmetric part and showed the skew symmetric part is negative

one-half the externally applied load expressed as a spatial cross product operator.

Compliant parallel mechanisms have been investigated by a number of researchers

to realize desired compliances because of its high stiffness, compactness, and small

positional errors. Huang and Schimmels (1998) obtained the bounds of the stiffness

matrix of compliant parallel mechanisms which consist of simple elastic devices and

proposed an algorithm for synthesizing a realizable stiffness matrix with at most seven









simple elastic devices. Roberts (1999) and Ciblak and Lipkin (1999) independently

developed algorithms for implementing a realizable stiffness matrix with r number of

springs where r is the rank of the stiffness matrix.

As for serial robot manipulators, Salisbury (1980) derived the stiffness mapping

between the joint space and the Cartesian space. Chen and Kao (2000) showed that the

formulation of Salisbury (1980) is only valid in the unloaded equilibrium pose and

derived the conservative congruence transformation for stiffness mapping accounting for

the effect of an external force.




r-yr



Variable
slder geometry
base
S= prismatic o = revolute

Figure 1-1. Planar robot with variable geometry base platform (from Simaan and Shoham
2002).


Gear Spring

all Ncois






f D.C. motor and Absorbera
geair box


Figure 1-2. Adaptive vibration absorber (from Ryan et al. 1994).










Planar/spatial compliant mechanisms are in general constructed with rigid bodies

which are connected to each other by simple springs. The stiffness matrix of the

mechanism depends on the geometry of the mechanism and the properties of the

constituent springs such as stiffness coefficient and free length. To realize variable

compliant mechanisms, variable geometry or adjustable springs have been investigated.

Simaan and Shoham (2002) studied the stiffness synthesis problem using a variable

geometry planar mechanism. They changed the geometry of the base using sliding joints

on the circular base (see Figure 1-1). Ryan et al. (1994) designed a variable spring by

changing the effective number of coils of the spring for adaptive-passive vibration control

(see Figure 1-2).


Thumb Screw moves Beam Deflection
fulcrum along and Movement
Cantilever of Ball Joint










Cantilever Beam
Built-In at
Centre Hexagon
Legs Mounted
onto Bottom Plate
(Remote Centre \with Ball Jomt



Figure 1-3. Parallel topology 6DOF with adjustable compliance (from McLachlan and
Hall 1999).

Cantilever beam-based variable compliant devices have been studied by a few

researchers. Under an external force, a cantilever beam deflects and its deflection









depends on the length of the beam and the Young's modulus of the material. Henrie

(1997) investigated a cantilever beam which is filled with magneto-rheological material

and changed the Young's modulus by changing the magnetic field. McLachlan and Hall

(1999) devised a programmable passive device by changing the length of the cantilever

beam as shown in Figure 1-3. Hurst et al. (2004) presented an actuator with physically

variable stiffness by using two motors and analyzed it for application to legged

locomotion.

1.3 Problem Statement

Planar/spatial compliant mechanisms consisting of rigid bodies which are

connected to each other by adjustable compliant couplings are investigated. For spatial

mechanisms, each adjustable compliant coupling is assumed to have a spherical joint at

each end and a prismatic joint with an adjustable line spring in the middle. For planar

cases, spherical joints are replaced with revolute joints. Mechanisms having two

compliant parallel mechanisms that are serially arranged are mainly investigated. The

compliant mechanisms are not restricted to be in unloaded equilibrium configuration and

this makes the analysis of the mechanism more complicated.

Firstly a stiffness mapping of a line spring connecting two moving bodies is

derived for planar and spatial cases. The line spring is assumed to have a fixed stiffness

coefficient and free length at this stage. This stiffness mapping leads to the derivation of

the stiffness matrix of compliant mechanisms consisting of rigid bodies connected to each

other by line springs.

A derivative of the stiffness matrix of a compliant mechanism with respect to the

twists of the constituent rigid bodies and the spring properties such as spring constant and

free length is obtained. Since the compliant mechanism is assumed initially in static






7


equilibrium under an external wrench, changing the spring constants and the free lengths

of the constituent springs may result in the change of the resultant wrench and it may

change the position of the compliant mechanism. Stiffness modulation methods, which

utilize adjustable line springs and vary the position of the robot where the compliant

mechanism is attached, are investigated to realize a desired compliance and to regulate

the position of the compliant mechanism.














CHAPTER 2
STIFFNESS MAPPING OF PLANAR COMPLIANT MECHANISMS

When a rigid body supported by a compliant coupling moves, the deflection and/or

the directional change of the coupling may lead to a change of the force. In this chapter,

a planar stiffness mapping model which maps a small twist of the body into the

corresponding wrench variation is studied. To describe a small (or instantaneous)

displacement of a rigid body and a force/torque applied to a body, the concepts of twist

and wrench from screw theory are used throughout this dissertation (see Ball 1900 and

Crane et al. 2006). Further, the notations of Kane and Levinson are also employed (see

Kane and Levinson 1985) to describe spatial motions of rigid bodies.

Specifically, as part of the notation, the position of a point P embedded in body B

measured with respect to a reference system embedded in body A will be written as Ar .

The derivative of the displacement of this point P (embedded in body B in terms of a

reference coordinate system embedded in body A) is denoted as Adr The derivative of

an angle of body B with respect to a body A is denoted by A60B and its magnitude is

denoted by A 0 The twist of a body B with respect to a body A will be denoted by

AoJDB


2.1 Spring in a Line Space

The analysis of rigid bodies which are constrained to move in a line space and

connected to each other by line springs is presented because it is simple and intuitive and

a similar approach can be applied for planar and spatial compliant mechanisms. Figure









2-1 illustrates a line spring connecting body A to ground. Body A is allowed to move

only on a line along the axis of the spring. The spring has a spring constant k and a free

length xo. The position of body A can be expressed by a scalar x and the force from the

spring by a scalar f.



A



k x








Figure 2-1. Spring in a line space.

The spring force can be written as

f =k(x-xo) (2.1)

The relation between a small change of the position of body A and the corresponding

small force variation can be obtained by taking a derivative of Eq. (2.1) as

8 f = kx. (2.2)

When springs are arranged in parallel as shown in Figure 2-2 (a), the resultant

spring constant kR may be derived as Eq. (2.3).

8f = kRx = k,5x+k,2


kR = k +k 2


(2.3)

























(a) (b)

Figure 2-2. Spring arrangements in a line space. (a) parallel and (b) series.

For a serial arrangement as shown in Figure 2-2 (b), the resultant spring constant

kR which maps a small change of position of body B into a small force variation upon

body B may be written as Eq. (2.4).

8f = kRgxB = klgxA = k, (gx 3xA)


x,A = k2 XB


kR kk- or kR-1 = k1-1 + k- (2.4)
k, +k,

It is obvious from Eqs. (2.3) and (2.4) that the resultant spring constant of springs in

parallel is the summation of each spring constant and that the resultant compliance of

springs in series is the summation of each spring compliance. This statement is valid for

springs in a line space.








2.2 A Derivative of Planar Spring Wrench Joining a Moving Body and Ground
In this section, a derivative of the planar spring wrench joining a moving body and

ground, which was presented by Pigoski (1993) and led to the stiffness mapping of a

planar parallel mechanism, is restated. Figure 2-3 illustrates a rigid body connected to

ground by a compliant coupling. The compliant coupling has a revolute joint at each end

and a prismatic joint with a spring in the middle part. Body A can translate and rotate in

a planar space.




P1


0\ A
S


P0
PO


Figure 2-3. Planar compliant coupling connecting body A and the ground.
The force which the spring exerts on body A can be written as

f =k(l-l)$ (2.5)

where k, 1, and / are the spring constant, current spring length, and spring free length

of the compliant coupling, respectively. Also $ represents the unitized Plticker

coordinates of the line along the compliant coupling which may be written as

rS rS
$= AXS (2.6)
PO P1









where S is the unit vector along the compliant coupling and Er Eo and Er are the

position of the pivot point PO in the ground body and that of Pl in body A, respectively,

measured with respect to a reference coordinate system attached to ground. To obtain the

stiffness mapping, a small twist EJDA is applied to body A and the corresponding change

of the spring force will be obtained. The twist E5DA may be written in axis coordinates

as


EDA = E (2.7)



where ErA is the differential of the position of point O in body A which is coincident

with the origin of the inertial frame E measured with respect to the inertial frame. In

addition E5&pA is the differential of the angle of body A with respect to the inertial frame.

Taking a derivative of Eq. (2.5) with the consideration that $ is a function of 0 in planar

cases yields

f = kl$ + k(l-l8o)S
/ ) (2.8)
1 Bo

where



]$ = (2.9)
ao E E aS
PO


as
and where is a unit vector perpendicular to S.
ao















lse ',, V "*
H A


1(5


0

E PO


Figure 2-4. Small change of position of Pl due to a small twist of body A.

Using screw theory, the variation of position P1 can be written as

E A E A E (2.10)
Er&A = E8r& + E(IA xE r p. (2.10)


as
It may be decomposed into two perpendicular vectors, one along S and one along
a0

These vectors correspond to the change of the spring length 61 and the change of the

direction of the spring 180 as shown in Figure 2-4. The change of the position of point

P1 may thus also be written as

AA.s ) sas) as

(2.11)
= 1 S +1680


From Eqs. (2.10), (2.11), (2.6), and (2.7) expressions for 81 and 180 may be obtained as

=E Es A rS+ E AEr .S


= ST EDA









_S aS as
18 = E=e Er + EA E 1.

E A. +S E A (2.13)
Sa -
S$T ESDA


and where


as' !. (2.14)
a0 EHA aS


All terms of Eq. (2.14) are known.

From Eqs. (2.8), (2.12), and (2.13), a derivative of the spring force may be written as

/o as
sf = kl$+k(1-l ) 1/6
1 Bo

Sk$$T EDA+k(l ) E-DA (2.15)
I ao ao
= [KF]ESDA

where

$T a$ a$'2
[KF]= k$$ +k(- ) (2.16)
/ ao ao

[KF] is the stiffness matrix of a planar compliant coupling and maps a small twist

of body A into the corresponding variation of the wrench. The first term of Eq. (2.16) is

always symmetric and the second term is not. When the spring deviates from its

equilibrium position due to an external wrench, the second term of Eq. (2.16) doesn't

vanish and it makes the stiffness matrix asymmetric.









2.3 A Derivative of Spring Wrench Joining Two Moving Bodies




Body B
P2








P1
Body A



E


Figure 2-5. Planar compliant coupling joining two moving bodies.

In this section a derivative of the spring wrench joining two moving bodies is

derived, which supersedes the result of the previous section and is essential to obtain a

stiffness mapping of springs in complicated arrangement.

Figure 2-5 illustrates two rigid bodies connected to each other by a compliant

coupling with a spring constant k, a free length lo, and a current length 1. Body A can

move in a planar space and the compliant coupling exerts a force f to body B which is in

equilibrium. The spring force may be written by

f =k(/-/)$ (2.17)

where


(2.18)


S S
$ = x = xS
E A x E Bx
P1 -P2









and where S is a unit vector along the compliant coupling and E r'1 and E r2 are the

position vector of the point P1 in body A and that of point P2 in body B, respectively,

measured with respect to the reference system embedded in ground (body E).

A small twist of body B with respect to an inertial frame E EDB is applied and it

is desired to find the corresponding change of the spring force. The twist EDB may be

written as

EJDB = EDA + AJDB (2.19)

where


E= B B (2.20)



_E~AA (2.21)



AB =AsB (2.22)


and where the notation from Kane and Levinson (1985) is employed as stated in the

beginning of this chapter. For example, Er B is the differential of the coordinates of

point O, which is in body B and coincident with the origin of the inertial frame, measured

with respect to the inertial frame and E105A is the differential of angle of body A with

respect to the inertial frame.

The derivative of the spring force, Eq. (2.17), can be written as

Esf_ = k 1 $ + k (1-l/o) .E_$. (2.23)









From the twist equation, the variation of the position of point P2 in body B with respect

to body A can be expressed as

A& B A8AB + A-85B X A B
r r + &x r P2 (2.24)

where A r2 is the position of P2, which is embedded in body B, measured with respect to

a coordinate system embedded in body A which at this instant is coincident and aligned

with the reference system attached to ground. It can also be decomposed into two

AS
perpendicular vectors along S and which is a known unit vector perpendicular to S.
a0

These two vectors correspond to the change of the spring length 81 and the directional

change of the spring 180 in terms of body A in a way that is analogous to that shown in

Figure 2-4. Thus the variation of position of point P2 in body B in terms of body A can

be written as

Age Age Age)B AaS AaS
A 2B (A SB rA8B A S>'
(2.25)
AaS
= 1 S+18 -


where

AaS
A 5 .Aa (2.26)



From Eqs. (2.24) and (2.25), 81 and 180 can be obtained as

81 = r S = ArB S + A x ArB B S

r= A r S+ A9B ArB xS (2.27)
= $s AWDB









AaS AS4
1.5 = A5 B S A gB -A+ AB XA r S
A rB AS AASrB AS B BASS
= Ar- A.J B P A2 (2.28)
0
A bTA B
a0

where

AaS
S AB xaS (2.29)

4P2 x 4

Aa$' Aa$
It is important to note that screw has the same direction as -but has a different
ao ao

moment term.

Only EA$ is unknown in Eq. (2.23). It is a derivative of the unit screw along the

spring in terms of the inertial frame and may be written as


E rA xS+Er x ESS (2.30)
P1 -P1

Using an intermediate frame attached to body A, a derivative of the direction cosine

vector may be written as

ES = AS + E.CA S. (2.31)


Then, E'$ may be decomposed into three screws as










rP1 -xS+ ErA AxSSE

A S + E5 xS S



SrxASS] + E& x( AxS) r 1x


Since S is a function of 0 alone from the vantage of body A and 180 is already

described in Eq. (2.28), the first screw in Eq. (2.32) can be written as


iAS aS 90 Aa$ Aa$ A a'T
SASA A
PI X5] ,s1= 1 ==1 ao I I ao a.


(2.32)












(2.33)


AaS
As for the second screw in Eq. (2.32), E5 AA xS has the same direction with and a

magnitude of E 5A and thus may be written as

AaS
E S=A XS = EA (2.34)
ao

Then the second screw in Eq. (2.32) can be expressed as



SA A XS AaE A
LrP E i~ xS)- Er X EA (2.35)

Aas Aas
0A =- [0 0 1] EDA
O0 E0 -

As to the third screw in Eq. (2.32), ESrA can be decomposed into two perpendicular

An S
vectors along S and Arespectively and may be written as
O06









E & 1A _.5r A + A E A
(=Es~ C sj~+A as AaS


where


-P1 o S -P1
S- S = ErA A S+ E mA Ax S

= + DA rl S
= $T H(WA


SA aS
1 0


AOS AOS
_- E 0 A- A30E

AOS AOS
Eg A E+ A A x
0 0 -P 30


AT$r
(a EH DA
30 ~

By combining Eqs. (2.36), (2.37), and (2.38) 'EA can be written as


ErA (E .A s)s+ E A1
kL1.!P1 -P1


AaS AaS
As} ds
A^J S


=(sT EDA )SK+ E -A


The third screw in Eq. (2.32) can now be written as


0
E AxS
P'r -x_


S07
AWT Aa$
= 0 E A [0


0 1]2 EDA


(2.36)


(2.37)


(2.38)


(2.39)


EDA


(2.40)








AS
since xS= -1(k).
a0 -

Among all unknowns in Eq. (2.23), 81 was obtained in Eq. (2.27) and all the terms of

E'5$ were obtained through Eqs. (2.33), (2.35), and (2.40). Hence the derivative of the

spring force can be rewritten as

Ef = kl$ +k(l-lo)E S

1$$T AB I Aa$ Aa$'T A.5I)B+a$[O 0 1]E.5--A A$ H(-DA
A 1 'r$ A$ 'T a$ 1 A $T
+k(1-) JD + [0 0 1] J- 0 EJ4




=[KF] AB +[KM]EDA
(2.41)

where
/ U$ S' + _$_ (2.42)
[K, F]=k$$T +k(l-) (2.42)0
I a0 a0

[Km]=k( )0 [0 0 1]- [0 0 1] (2.43)


It is important to note that [K,] is a function of the external wrench. To prove it,

Eq. (2.18) is explicitly expressed in a planar coordinate system and Erip = [p ppyI to

yield

F co
$= so (2.44)
Co P 1 So Py









-so
0 c, (2.45)
soPx Py

where c, = cos (0) and s, = sin (0).

A )
By substituting Eq. (2.45) for in Eq. (2.43), [K,] can be expressed as
30

0 0 -so 0 0 -f
[K,]=k(l-l) 0 0 c = 0 (2.46)
s, -c, O f_ -f 0

where f=[f\ f m, is the initial spring wrench.

As shown in Eq. (2.41), the derivative of the spring wrench joining two rigid bodies

depends not only on a relative twist between two bodies but also on the twist of the

intermediate body, in this case body A, in terms of the inertial frame. [KF] which maps

a small twist of body B in terms of body A into the corresponding change of wrench upon

body B is identical to the stiffness matrix of the spring assuming the body A is stationary.

[K,] is newly introduced from this research and results from the motion of the base

frame, in this case body A, and is a function of the initial external wrench.

2.4 Stiffness Mapping of Planar Compliant Parallel Mechanisms in Series

The derivative of the spring wrench derived in the previous section is applied to

obtain the stiffness mapping of compliant parallel mechanisms in series as shown in

Figure 2-6.1 Body A is connected to ground by three compliant couplings and body B is

connected to body A in the same way. Each compliant coupling has a revolute joint at

1 Figure 2-6 shows a coordinate system attached to each of three bodies for illustration purposes. In this
analysis, the three coordinate systems are assumed to be coincident and aligned at each instant.









each end and a prismatic joint with a spring in the middle part. It is assumed that an

external wrench wex, is applied to body B and that both body B and body A are in static

equilibrium. The positions and orientations of bodies A and B and the spring constants

and free lengths of all constituent springs are given. The stiffness matrix [K] which

maps a small twist of body B with respect to the ground E)DB into a small wrench

variation w ext is desired to obtain.


S02


01


Figure 2-6. Mechanism having two compliant mechanisms in series.

The static equilibrium equation of bodies B and A can be written by

Wext=f,+f,+f
=where f are the forces from the compliant couplings.

where f are the forces from the compliant couplings.


(2.47)








The stiffness matrix is derived by taking a derivative of the static equilibrium equation,

Eq. (2.47), to yield


8wx, =[K] E}JB
= f_, + f_,+s (2.48)
= f4 -f 5 + 6f6

The derivatives of spring forces can be written by Eqs. (2.49) and (2.50) since springs 4,

5, and 6 connect body A and ground and springs 1, 2, and 3 join two moving bodies.

f4 + 4f5- + 8f6 = [KF] E A +[KF E A +[KF ] EDA

= [KF]R,L EJDA

{_fl + f2 + _f3 = [KF]1 ADB +[KF] ADB +[K]3 A3DB
+[KM ]1 E A + [K ] E--DA +[KM ] E DA (2.50)
=[KF]R U A5DB +[K], ER DA

where
6
[KF]R,L = [KF ]
1=4
3
[KF]R,U =Z [KF ],

3
l=1

[KM]R,U =Z[KM.


From Eqs. (2.49), (2.50), and (2.19) twist ED A can be written as

[KF]R,L EDA = [KF]R,U AD +[K]R,U DA
(2.51)
= [K]R (E DB E DA)+[K]RU E DA

E DA =([KF]RL +[KFRU -[K ]R,U ) [KF R, E DB (2.52)









Substituting Eq. (2.52) for E DA in Eq. (2.49) and comparing it with Eq. (2.48) yields the

stiffness matrix as



[K] EoDB = [KF]R,L EDDA
S F (2.53)
= [KFIR,L ([KF R,L +[KF R,U [KM ], )-1[KF R, E B (25


[K] = [KF R,L ([KF]R,L + [KF R,U -[KMIR,)- [KF R,U (2.54)

It was generally accepted that the resultant compliance, which is the inverse of the

stiffness, of serially connected mechanisms is the summation of the compliances of all

constituent mechanisms (see Griffis 1991). However, the stiffness matrix derived from

this research shows a different result. Taking an inverse of the stiffness matrix Eq. (2.54)

yields

[K] = [KF]RL +[KF]RV --[KF]R, [KMIR,C [KF R,L1 (2.55)

The third term in Eq. (2.55) is newly introduced in this research and it does not vanish

unless the external wrench is zero.

A numerical example is presented to support the derived stiffness mapping model.

The geometry information, spring properties of the mechanism shown in Figure 2-6, and

the external wrench wt are given in Tables 2-1 and 2-2.

wext=[0.01N -0.02 N 0.03 Ncm]

Table 2-1. Spring properties of the compliant couplings in Figure 2-6.
Spring No. 1 2 3 4 5 6
Stiffness constant k 0.2 0.3 0.4 0.5 0.6 0.7
Free length l/ 5.0040 2.2860 4.9458 5.5145 3.1573 5.2568
(Unit: N/cm for k, cm for lo)









Table 2-2. Positions of pivot points in terms of the inertial frame in Figure 2-6.
Pivot points El E2 E3 Bl B2 B3
X 0.0000 1.5000 3.0000 0.0903 1.7063 1.9185
Y 0.0000 1.2000 0.5000 9.8612 8.6833 10.6721
(Unit: cm)


Table 2-2. Continued.
Al A2 A3 A4
0.9036 2.5318 2.7236 1.6063
4.5962 3.4347 5.4255 5.4659


Two stiffness matrices are obtained. [K,] is from Eq. (2.54) and [K,] from the same

equation ignoring [KM ]R,


0.0108 N/cm
[K,]= -0.0172 N/cm
-0.0997 N


0.0111 N/cm
-0.0162 N/cm
-0.0969 N


-0.0172 N/cm
0.3447 N/cm
0.8251 N

-0.0157 N/cm
0.3462 N/cm
0.8150 N


-0.0797 N
0.8351 N
2.6567 Ncm

-0.0874 N
0.8124 N
2.6129 Ncm


The result is evaluated in the following way:

1. A small wrench 3wT is applied in addition to wext to body B and twists E)D' and
EoDB are obtained by multiplying the inverse matrices of the stiffness matrices,
[K]1 and [K]2, respectively, by 3w, as of Eq. (2.48). Corresponding positions
for body B are then determined, based on the calculated twists EoDB and EoDB.

2. E'DA is calculated by multiplying the inverse matrix of [K ]R,L by Sw1 as of Eq.
(2.49). The position of body A is then determined from this twist.

3. The wrench between body B and body A is calculated for the two cases based on
knowledge of the positions of bodies A and B and the spring parameters. The
change in wrench for the two cases is determined as the difference between the new
equilibrium wrench and the original. The changes in the wrenches are named
SwAB1 and SwAB,2 which correspond to the matrices [K]1 and [K]2.

4. The given change in wrench 5wr is compared to SwAB,1 and wAB,2.


[K21=









The given wrench ow, and the numerical results are presented as below.

Swr = 10'-x[0.5 0.2 0.4]

EoD=10-3x[0.7674 -0.1186 0.0672]

ESD-=103x[0.8208 -0.1152 0.0679]

ESA =10- 3x[0.2350 -0.0898 0.0330]f


SwEA =10-5 [0.5000 0.1998 0.3996]T

w AB, =10-5x[0.5004 0.1969 0.3919]T

SWAB,2 10-' [0.5666 0.2300 0.0117]T

where w ,EA is the wrench between body A and ground. The unit for the wrenches is

[N N Ncm]T andthatofthe twists is [cm cm rad] The difference between

SwE and 5w, is small and is due to the fact that the twist was not infinitesimal. The

difference between Sw AB, and 3wT is also small and is most likely attributed to the

same fact. However, the difference between SwAB,2 and Sw, is not negligible. This

indicates that the stiffness matrix formula derived in this research produces the proper

result and that the term [KM ]R,U cannot be neglected in Eq. (2.54).

2.5 Stiffness Mapping of Planar Compliant Parallel Mechanisms in a Hybrid
Arrangement

Figure 2-7 depicts a compliant mechanism having compliant couplings in a

serial/parallel arrangement. Each compliant coupling has a revolute joint at each end and

a prismatic joint with a spring in the middle part. An external wrench wex, is applied to









body T and body T is separately connected to body B, body C, and body D by three

compliant couplings. Body B, body C, and body D are connected to ground by two

compliant couplings. It is assumed that all bodies are in static equilibrium. It is desired

to find the stiffness matrix which maps a small twist of body T in terms of ground E 6T

to the corresponding wrench variation 5wext. The stiffness constants and free lengths of

all constituent springs and the geometry of the mechanism are assumed to be known.

The stiffness matrix of the mechanism can be derived by taking a derivative of the

static equilibrium equations. The static equilibrium equations may be written as

ext = f7 +8 +9 (2.56)

f7 = f +f2 (2.57)

f8 = 3 +4 (2.58)

f9 =f5 +6 (2.59)

where wext is the external wrench and f, is the force of the i-th spring.

Derivatives of Eqs. (2.56)-(2.59) can be written as

ex 7 + 4f8 -f9
(2.60)
= [K]R EDT

Sf7 = sf1 +sf2 (2.61)

f8 = f3 + f_4 (2.62)

Sf9 = f 5 + Sf (2.63)

where [K]R is the stiffness matrix and E'DT is a small twist of body T in terms of the

inertial frame attached to the ground.






29

Using Eqs. (2.15) and (2.41), Eq. (2.61) can be rewritten as

f_7 = [KF]7 B _D +[KM]7 E3DB
(2.64)
= [KF]I 1E B +[KF]2 EbDB =([KF]1 +[KF]2) EDB(

where B)DT is a small twist of body T in terms of body B and E1DB is that of body B in

terms of the inertial frame. [KF, and [K, ] are the matrices for i-th spring defined by

Eqs. (2.42) and (2.43) respectively.

The twist of body T can be decomposed as
ESDT = E-DB + BDT (2.65)

From Eqs. (2.64) and (2.65), ED B can be expressed in terms of E1DT as Eq. (2.66).

[KF 7(EDT E )+[KM]7 EDB =([KF]1+[KF]2) E B

EDB = ([KF ] + [KF ]2 + [KF 7 -[KM] ) [KF 7 E.DT (2.66)

By substituting Eq. (2.66) for E5DB in Eq. (2.64), Sf7 can be expressed in terms of

EJDT as

s7 = [KF ] +[KF 2)([KF ] +[KF]2 +[KF7 -K,]) [KF]7 ED (2.67)

Analogously, Sf8 and Sf 9 can be written respectively as

sf8 = ([KF] 3+[[KF 4)([KF]3+[KF]4+[KF8 -[KM]8) [KF]8E3DT (2.68)

S= ([KF5 +[KF]6[KF] 5 +[KF +[KF 9-[K ]9) [KF] ESD (. (2.69)

Finally from Eq. (2.60) and Eqs. (2.67)-(2.69), the stiffness matrix can be written as










[K]R = ([KF +[KF ]2)([KF ] +[KF 2 +[KF ]7 [KM ] ) [KF ]7

+([KF]3+[KF]4)([KF]3+[KF]4+[KF]8-[KM],)l [KF] (2.70)

+([KF]5 +[KF6 )([KF5 +[KF 6 +[KF9 -[K ]9 -1[KF 9

A numerical example of the compliant mechanism depicted in Figure 2-7 is

presented. The four bodies are identical equilateral triangles whose edge length is 2 cm.

Four coordinate systems, B, C, D, and T are attached to body B, C, D, and T, respectively

and their positions of origin and orientations in terms of the inertial frame are given in

Table 2-5. The spring properties and the positions of the fixed pivot points are given in

Table 2-3 and Table 2-4, respectively. The external wrench is given as

0.1 N
w-ext 0.1 N
0.2 Ncm

Table 2-3. Spring properties of the compliant couplings in Figure 2-7.
Spring No. 1 2 3 4 5 6 7 8 9
Stiffness
0.40 0.43 0.49 0.52 0.58 0.61 0.46 0.55 0.64
constant k
Free length 1l 2.2547 2.4014 1.5910 1.8450 1.7077 2.2695 2.3924 2.2200 1.8711
(Unit: N cm for k and cm for lo)

Table 2-4. Positions of the fixed pivot points of the compliant couplings in Figure 2-7.
Al A2 A3 A4 A5 A6
X 1.6700 4.4600 13.3449 14.6731 8.2300 4.9400
Y 4.4333 1.3964 3.2500 6.8400 14.1400 13.4943
(Unit: cm )

Table 2-5. Positions and orientations of the coordinates systems in Figure 2-7.
Bo Co Do To
X 4.0746 12.2367 7.2479 8.3174
Y 5.1447 4.4972 12.7430 6.9958
0 -0.8112 1.2283 3.8876 0.5818
(Unit: cm for x, y and radians for D)









Two stiffness matrices are obtained. [K, ] is from Eq. (2.70) and [K,] from the same

equation ignoring all [K,] 's which are newly introduced in this research.

0.2501 N/cm 0.0216 N/cm -1.7651 N
[K,]= 0.0216 N/cm 0.2910 N/cm 2.6661 N
-1.6651 N 2.5661 N 38.5180 Ncm

0.2463 N/cm 0.0172 N/cm -1.7844 N
[K2]= 0.0315 N/cm 0.2888 N/cm 2.5749 N
-1.6139 N 2.5730 N 38.2221 Ncm

To evaluate the result, a small wrench Sw is applied to body T and the static

equilibrium pose of the mechanism is obtained by a numerically iterative method. From

the equilibrium pose of the mechanism, the twist of body T with respect to ground E5DT

is obtained as

0.5 N
w = 10-4 0.2 N
0.3 Ncm

0.0050 cm
E-DT = -0.0058 cm
0.0006 rad

Then the twist E'DT is multiplied by both of the stiffness matrices to see if they result in

the given small wrench Sw .

0.4997 N
Sw, =[K1] E.DT =10-4x 0.2000 N
0.3020 Ncm











Sw2 = [K2 ] EDT = 10-4


0.4502 N
0.2726 N
0.6622 Ncm


The numerical example indicates that [K,] produces the given wrench 45w with

high accuracy and that [K2] involves significant errors.


d3 \W
\ ext

k9 \


\ k3


A2 t


Figure 2-7. Mechanism consisting of four rigid bodies connected to each other by
compliant couplings in a hybrid arrangement.














CHAPTER 3
STIFFNESS MAPPING OF SPATIAL COMPLIANT MECHANISMS

Taking a similar approach adopted for planar compliant mechanisms, a stiffness

mapping of spatial compliant mechanisms is presented.

3.1 A Derivative of Spatial Spring Wrench Joining a Moving Body and Ground

Figure 3-1 depicts a rigid body and a compliant coupling connecting the body and

the ground. The compliant coupling has a spherical joint at each end and a prismatic

joint with a spring in the middle. Body A can translate and rotate in a spatial space. The

wrench which the spring exerts on body A can be written as

w = k(l-/)$ (3.1)

where k, 1, and / are respectively the spring constant, current spring length, and spring

free length of the compliant coupling. Further, $ represents the unitized Plticker

coordinates of the line along the compliant coupling which may be written by


E EA (3.2)
$= xEE = Er XS
po P 1

where S is the unit vector along the compliant coupling and ErE0o and Er A are the

position of the pivot point PO in the ground body and that of Pl in body A, respectively,

measured with respect to a reference coordinate system attached to ground.










Body A


S




/ PO



Figure 3-1. Spatial compliant coupling joining body A and the ground.






13 S



e2

------------ ---



Figure 3-2. Unit vector expressed in a polar coordinates system.

A polar coordinates system can be used to express the unit vector S (see Figure 3-2) as

sin/ cos a
S =sinj lsina (3.3)
Scos/

It is obvious from Eqs. (3.2) and (3.3) that $ is a function of a and / since Er E is

fixed on ground. Hence a derivative of the spring wrench can be written as

Sw = k,/$+ k(l- lo),$
io) a+ af/// (3.4)
/ =ka a$+k(1-8 a I









where

as

SLaaS (3.5)





as E E as
POr a







ssas
By taking a derivative of Eq. (3.3), r a and as can be explicitly written by



a -sin sin a
a sin cos a (3.7)
0,


aa
/cossin (3.8)

sin/


Since t is not a unit vector, a unit vector is introduced as

-sin a
= sin cos a (3.7)














as
0

= cosin a (3.10)


Hence Eq. (2.8) can be rewritten as


w = k6I$+k(1- 0) sin--a+ (.1
Saa as
S= cos (3.9)


where










/ as
as aa (3.12)
E E
r 0x s



It is important to note that and are the unitized Plicker coordinates of the lines
aa a,/

perpendicular to S and go through the pivot point PO.


1sini 65a EP1


----- ------- J ^

I sin,


1 / 3"/3







Figure 3-3. Small change of position of Pl due to a small twist of body A.

In Eq. (3.11) 61, 1sin/3 ca, and I 8/3 can be considered as the change of the

spring length and the changes of the direction of the spring (see Figure 3-3). These

values correspond to the projections of the variation of position P1, Erpl, onto the

as as A
orthonormal vectors S, and -, respectively. Thus E 1rp1 can be rewritten as










=A(Eb .S)S+S BJS S A S
1I + I 1
(3.13)
Js Js
=5 S+lsin &as +160as


From the twist equation, the variation of position P1 can be written as

ErI E + EqA XE (3.14)


where EdrA is the differential of the position of point O in body A which is coincident

with the origin of the inertial frame E measured with respect to the inertial frame. E .A

is the differential of the angle of body A with respect to the inertial frame.

From Eqs. (2.11) and (2.10), 61, sin 6 Sa, and I /3 can be expressed as
l1 = EreA S = ErA. S+ EPAX ErAp S
-P o -P1
=Er A S+ E PA Er x S (3.15)
= $1 EJDA
ASTESE4


lsin/3a= Er A aS Er S + EAErA as
= P a -0 aa aa

c*A ras as
o A a + HE Erp1 p aX- (3.16)




/ l3 E 1rA E 6A E_ rpElA
aa -

as as


= ErA. + X (3.17)

a &TEDA
a,# -


where










EDA_ (3.18)
Eg9A


S as
(3.19)
aa E as'




as a)3 (3.20)
ErA x



as as
It is important to note that and are the unitized Plicker coordinates of lines


perpendicular to S which pass through the pivot point P1 in body A and ED)A is a small

twist of body A with respect to ground. Substituting Eqs. (2.12), (3.16), and (2.13) for

81, /sin/3 6a, and /I 8 in Eq. (3.11) yields

)a as$
w = kl$ + k(1- ) I sinfl + 3a+ l


=k$$T EDA +k(1 +r EA (3.21)

SK]EDA

where


[KF ]= k$$ +k(1-L+ (3.22)


[KF ] is the stiffness matrix of a spatial compliant coupling and maps a small twist of

body A into the corresponding variation of the wrench. The first term of Eq. (2.16) is









always symmetric and the second term is not. When the spring deviates from its

equilibrium position due to an external wrench, the second term of Eq. (2.16) doesn't

vanish and it makes the stiffness matrix asymmetric. This result agrees with the works of

Griffis (1991).

3.2 A Derivative of Spring Wrench Joining Two Moving Bodies




P2/ -Body B


S Body A


P1








Figure 3-4. Spatial compliant coupling joining two moving bodies

Figure 3-4 illustrates two rigid bodies connected to each other by a compliant

coupling with a spring constant k, a free length o, and a current length 1. Both of body

A and body B can move in a spatial space and the compliant coupling exerts a wrench w

to body B which is in equilibrium. The spring wrench may be written as

w = k(/-)$ (3.23)

where

$ rr S (
A B (3.24)
P1 P2









and where S is a unit vector along the compliant coupling and Er'1 and E r2 are the

position vectors of the point P1 in body A and that of point P2 in body B, respectively,

measured with respect to the reference system embedded in ground (body E). It is

desired to express a derivative of the spring wrench in terms of the twist of body B E DB

and that of body A EDAA. The twist ED)B may be expressed as

EJDB = EDA + AJDB (3.25)

where

ErB
EDB = (3.26)


A
EDA = A (3.27)



D = AB (3.28)


and where Er B is the differential of point O, which is in body B and coincident with

the origin of the inertial frame, measured with respect to the inertial frame and E6 B is

the differential of angle of body B with respect to the inertial frame. Ero ArB ,

E5A and A35B are defined in the same way.

The derivative of the spring wrench in Eq. (2.17) can be written as

Esw = k 1 $ + k ( -lo) ES$ (3.29)

and it is required to express 81 and ES$ in Eq. (2.23) in terms of the twists of the bodies.

From the twist equation, the variation of position of point P2 in body B with respect to

body A can be expressed as









A B A B + AB X A B
rp2 r + AB rP2 (3.30)

where A r2 is the position of P2, which is embedded in body B, measured with respect to

a coordinate system embedded in body A which at this instant is coincident and aligned

with the reference system attached to ground. It can also be decomposed by projecting it

AaS AaS
onto the orthonormal vectors S, and which are defined in a similar way as
aa a,8

Eqs. (3.3), (3.9), and (3.8). These three vectors correspond to the change of the spring

length 81 and the directional changes of the spring such as /sin/3 &a and I 53 in terms

of body A in a way that is analogous to that shown in Figure 3-3. Thus the variation of

position of point P2 in body B in terms of body A can be written as


A ABA a S A aS' A AS A aS
B- Ar B 2.
2 + 4-P2 2 aa + P 2
(3.31)
A aS' A aS
=1lS+1sin #Sa +d1 -
aa P ,

From Eqs. (2.24) and (2.25), 81 in Eq. (2.23) can be obtained as
81 = A S = ArBS+ AYB A B S
2P -o -P2
SAr B. S+ A9B BArBp2 xS (3.32)
= ST ADB

In the same way, Isin/ &a and /I 8 can be expressed as

AaS AaS AaS





Aa$
lsinJ/J3 =A8< AB A rB A- AB +ArB,


A$B AA BA AB (3.33)



iaa









f=.r AB S B"S A S
AA rB2 AS A B A AB A B AS
/ S,"~3S +A~jj + pA .

S+ A r2 (3.34)

Aa$



where

A aS'
AA




a 'A as
r P2 X



A
A B a (3.36)



Now in Eq. (2.23), only E5$ is yet to be obtained. It is a derivative of the unit screw

along the spring in terms of the inertial frame and may be written as



$_=: rA E AS Ax EJS1_ (3.37)

Using an intermediate frame attached to body A, a derivative of the unit vector S can be

written by

ECS = ASS+ EC5pA xS. (3.38)


Thus ES$ may be decomposed into three screws as










r_ = xS+Er x SS
AgS+ E5A xS
=ErA XS+ Hr A x(AS+ pXS (3.39)

APS E A S 0
=rE X A S E1A X(E x S) 1r xS

Since S is a function of a and / from the vantage of body A and Isinm & a and 1/,8

were already described in Eqs. (2.28) and (3.34), the first screw in Eq. (2.32) can be

written as



A Sa+ as A as A a
I IS I oa a, 5 + o8a
Eri A S A 'Sa+ A', _rA x S Ar_ x gA
P1, I a A 2 r j I (


a lsinJS6a+ P .(3.40)
aa / aft


I a a + aA a


As to the second screw in Eq. (2.32), E053A x S can be decomposed onto three

AS ABS
orthonormal vectors along S, and respectively, as


E xS {( AxS).s} +(E& AxS). As jA AaxS)
v -- -' aa \a a8 I


(3.41)









AaS AaS
From the fact that S, and are unit vectors and perpendicular to each other

(see Figure 3-3), each dot product of Eq. (3.41) can be expressed as

(E5Ax S) S = 0 (3.42)

AaS AaS _E AaS
(E xS).- = A. Sx -=EA
-aa aa -

f o T g L o 0 (3.43)
=- "as =- as _I
-IABS A \ S-- AAS E jDA



a a
AaS AaS AaS
(E3 AxS) = E pAISx = E A

0 o -T (3.44)
SEgr A7
ELJ^A
L AS A = A ]S E


where 0 =[0 0 0].

Hence, E0C5A xS can be rewritten as

0 T 0 T
E A xS= AaS AS E A S EA A (3.45)

and the second screw in Eq. (2.32) can be expressed a
and the second screw in Eq. (2.32) can be expressed as












E~A xS

E X(EA x xS)


1 0 -
aa
L- A )S 3


E A
r P


a0
AaS
a)3


JDA +; AasAo Si

aa


ED A


E DA E DA

aa


0o r j S r o0
as + AaS
H A AP1 -


A$-
AaS


(3.46)


A Ofr EDA
=A8$ H
Lf Ea ]


As to the third screw in Eq. (2.32), E ~r can be decomposed onto three orthonormal


A S A. S
vectors along S, a, and respectively, as
E 1aaE P

Er&A = E&A + E3 A AEr
=P _0 T Pi


(ESrA, )S+


r Al -d -d + E3rA
__PA AS A S
E6 1 E6 ~1


AaS S aS
a3) aJ3


The first dot product in Eq. (3.47) can be expressed as

E rA S = ErA. S+E & xr. S
-P1 o -P1
EI ^ 4oS+ E 44xS.
SErA S + EPArA XS.
EDP1
= $T H 4DA


(3.47)


(3.48)


In the same way, the second and third dot products in Eq. (3.47) can be written as


EoDA


K











0 E A A-S OaS

A$ S AAS
= + EA rx -


A CE A
Oa


ErA AaS E A ABS
rP1 =-=o -
a3? -0


+ A rA AS
P1 a,


AOS A OS
= E~ A + E A A rA


A Eq$ A
afy


Finally, E8rAp xS of the third screw in Eq. (2.32) can be expressed as


A (xS= $(TTEDA)S+ Aa$
.5r P"=
S I (saa


A $"T

aa

Aa$"T

aa


EDA AS
aa


-DA -
K


EJDA AaS
aa


AdS
since S, and
aa


AaS
- are unit vectors and perpendicular to each other (see Figure
a,8


3-3).


Substituting Eq. (3.51) for E rA xS of the third screw in Eq. (2.32) yields


E 1 A S A
-r" )


(3.49)


(3.50)


A a$T

ap


As {xS
"aJ


aS
M


A "T E
xS+ A D


A $"I

a8


(3.51)


A AaS
Daa








0
0 LaX IT H AS A I T AaS'

P I aa3 3, 3 a"


0 T 0 A (3.52)
= AaS EJJA AaI EgDA
aS

0 A a $T 0 A T
= S Ea 4


By replacing 51 and ES$ in Eq. (2.23) with Eqs. (2.27), (2.33), (2.35), and (2.40) and

sorting it into the twists, the derivative of the spring wrench can be rewritten as

Esw = kSl$ +k(l-l ) ES
= [KF] ADB + [KM] EDA (

where


[KF k= +k(1- ) + (3.54)


0 A TT A 0 A 0 0 L
[Kat ]=s k(s-lo) an r+ -
[Km k( AaS A aS + AS -. AaS


(3.55)

It is important to note that [KM] is identical to the negative of the spring wrench

expressed as a spatial cross product operator (see Featherstone 1985 and Ciblak and

Lipkin 1994). To prove it, all terms in Eq. (2.43) are explicitly expressed in a polar

coordinate system and E rP1 p \py p I to yield









[0] [K12]
[KM]= [K2] [K22] (3.56)
L[K12] [K22]

where

0 c s8 S,
[K12]=k(1-/o) -c, 0 s, c, (3.57)
S_ Sa -S Ca 0

0 P. s Sa-Py S C, -pzs, Ca+ P c
[K22]=k(1-lo) -psss,+pySp, 0 py c'-ps s (3.58)
SP SP ca- Px c1 -Py C + z Si sa 0

and where [0] is 3x3 zero matrix, c, = cos(a), and s, = sin(a), etc.

In the same way the spring wrench can be explicitly written as


w= k(/-/o)$=k(/-)E A x
P1i -
s 8ca f~
s sa f. (3.59)

=k(-l = ) = -
PYC/ Pzspsa mx Lm
ps ce --Pxcy my
Pxs /iy PySfcar m_

By comparing Eqs. (3.57) and (3.58) with Eq. (3.59) it is obvious that

0 fZ -f;
[K12]=k(1-1o) -f, 0 f, =-fx (3.60)
f, -f 0

0 m, -my
[K22]= -mz 0 m, =-mx (3.61)
mw -mi 0

where fx and mx are skew-symmetric matrices representing vector multiplication.









Then [K,] can be expressed as

[o1 -fx
[KM=o] -f =-wx (3.62)
-fx -mx

where w x is the spring wrench expressed as a spatial cross product operator (see

Featherstone 1985).

Finally the derivative of the spring wrench can be written as

=[KF AB E-(w ) EDA (3.63)

As shown in Eq. (3.63), the derivative of the spring wrench joining two rigid bodies

depends not only on a relative twist between two bodies but also on the twist of the

intermediate body, in this case body A, in terms of the inertial frame unless the initial

external wrench w is zero. [KF] which maps a small twist of body B in terms of body

A into the corresponding change of wrench upon body B is identical to the stiffness

matrix of the spring assuming the body A is stationary.

3.3 Stiffness Mapping of Spatial Compliant Parallel Mechanisms in Series

The derivative of spring wrench derived in the previous section is applied to

obtain the stiffness mapping of the compliant mechanism shown in Figure 3-5. Body A

is connected to ground by six compliant couplings and body B is connected to body A in

the same way. Each compliant coupling has a spherical joint at each end and a prismatic

joint with a spring in the middle. It is assumed that an external wrench wex, is applied to

body B and that both body B and body A are in static equilibrium. The poses of body A

and body B and the spring constants and free lengths of all compliant couplings are

known.










The stiffness matrix [K] which maps a small twist of the moving body B in terms


of the ground, ESDB (written in axis coordinates), into the corresponding wrench

variation, Swex (written in ray coordinates), is desired to be derived and this relationship

can be written as

Swe = [K] EDB (3.64)

The stiffness matrix can be derived by taking a derivative of the static equilibrium

equations of body A and body B which may be written as

6 12
wext = w = w, (3.65)
1=1 1=7

6 12
t = w, = w, (3.66)
1=1 i=7

where w, is the wrench from i-th compliant coupling.

Since springs 1 to 6 join the two moving bodies and springs 7 to 12 connect body A to

ground (see Figure 3-5), the derivatives of the spring wrenches can be written as

6 6
Zsw, =I ([]KF] A] -B (W, X) ) EDA)
,=1 ==1 (3.67)

= [KF ]R,u A.DB (W't X) EDA

12 12
ZSw, = [KF ], A =DB=[KF RL EDA (3.68)
1=7 1=7

where

12
[KFR,L =Z[KF], (3.69)
1=7

6
[KF]R,U =-[MKF], (3.70)
7=1








and where wex x is the external wrench expressed as a spatial cross product operator.

From Eqs. (3.67), (3.68), and (3.25), twist ESDA can be written as Eq. (3.72).

[KF]R,L EDA = [KF ]R,U A DB (Wext X) EDA
(3.71)
= [KF R,U (EoDB EDA)-(We, X) E A)

EoDAA =([KFR, +[KF R+(ex X)) [KF RU B (3.72)

Substituting Eq. (3.72) for ESDA in Eq. (3.68) and comparing it with Eq. (3.64) yield the

stiffness matrix as Eq. (3.74).

[K] EoDB = [K ]RL E DA
(3.73)
=[KF]R,L([ KFR,L +[KF R,U+( WetX))- [KF R, EJDB (3


[K] =[KF R,L (KF R,L +KF R,U +(Wet )) [KF R,U. (3.74)

It was generally accepted that the resultant compliance, which is the inverse of the

stiffness, of serially connected mechanisms is the summation of the compliances of all

constituent mechanisms (see Griffis 1991). However, the stiffness matrix derived from

this research shows a different result. Taking an inverse of the stiffness matrix Eq. (3.74)

yields

[K]- = KF]RL- + [KF]RU + [KF]R,U (1Wext )[KF] ,L (3.75)

The third term in Eq. (3.75) is newly introduced in this research and it does not vanish

unless the external wrench is zero.









































Figure 3-5. Mechanism having two compliant parallel mechanisms in series.2

A numerical example of the compliant mechanism depicted in Figure 3-5 is

presented. The geometry information and spring properties of the mechanism shown in

Figure 3-5 are presented in Tables 3-1 through 3-5. The external wrench w,, is given as


wex = [-0.3 0.4 0.8 -2.3 -1.3 0.7]T. (unit:[N,N,N,Nc,Ncm,Ncm,Nc])


Table 3-1. Spring properties of the mechanism in Figure 3-5 (Unit: N/cm for k, cm for lo).
Spring No. 1 2 3 4 5 6
Stiffness coefficient k, 4.6 4.7 4.5 4.4 5.3 5.5

2 The coordinate systems attached to bodies E, A, and B are for illustrative purposes only. In the analysis it
is assumed that the three coordinate systems are at this instant coincident and aligned.










Free length 1.6305 1.0276 4.0098 1.8592 1.7591 3.8364

Table 3-1. Continued.
Spring No. 7 8 9 10 11 12
Stiffness coefficient k 4.4 4.9 4.7 4.5 5.1 4.8
Free length lo 4.4718 1.2760 5.2149 2.6780 2.2712 3.4244

Table 3-2. Positions of pivots in ground in Figure 3-5 (Unit: cm).
No. 1 2 3 4 5 6
X 0.0000 1.3000 0.6000 -0.7000 -1.1000 -0.5000
Y 0.0000 1.1000 2.7000 2.6000 1.8000 0.4000
Z 0.0000 0.2000 0.1000 -0.1000 0.3000 0.1000

Table 3-3. Positions of pivots in bottom side of body A in Figure 3-5 (Unit: cm).
No. 1 2 3 4 5 6
X 0.2000 1.1833 0.4616 -0.6575 -1.1452 -0.2189
Y 1.2000 2.1235 3.5111 3.3783 2.5652 1.6879
Z 3.2000 3.1843 3.3010 3.1013 3.0704 3.1196

Table 3-4. Positions of pivots in top side of body A in Figure 3-5 (Unit: cm).
No. 1 2 3 4 5 6
X 0.2086 1.4860 0.7553 -0.5501 -0.9278 -0.2945
Y 1.2033 2.3329 3.9187 3.7867 2.9797 1.5942
Z 3.2996 3.2514 3.3121 3.2792 3.3590 3.3804

Table 3-5. Positions of pivots in body B in Figure 3-5 (Unit: cm).
No. 1 2 3 4 5 6
X -0.3000 0.9216 0.2183 -0.8385 -1.2525 -0.5589
Y 1.6000 2.7822 3.8980 3.9919 2.8972 2.0875
Z 5.5000 5.5000 5.4782 5.8447 5.8317 5.7745

Two stiffness matrices are calculated: [K]1 from Eq. (3.74) and [K]2 from the same

equation but without the matrix w x. The numerical results are

0.3429 -0.0077 -0.2661 -0.7853 1.7378 -0.4076
-0.0077 0.5103 1.7122 1.2760 0.2157 -0.2885
-0.2661 1.7122 10.5103 20.0012 0.7518 -0.2695
[K], =
-0.7853 2.0760 19.6012 54.3222 1.1348 1.2570
0.9378 0.2157 0.4518 0.4348 12.1329 -3.8667
-0.0076 0.0115 -0.2695 -0.0430 -1.5667 -0.0798









0.3039 -0.0109 -0.2641 -0.7858 1.3134 -0.2770
-0.0504 0.4617 1.7122 1.9863 -0.0875 0.0375
-0.4364 1.6222 10.6633 21.7834 -0.7144 0.5956
[K]2 =
-1.0788 1.9862 20.7574 59.4736 -1.1874 2.5822
0.9754 0.0759 -0.0901 -0.1283 12.1852 -3.0399
-0.0319 0.0218 -0.0576 0.6983 -1.6440 -0.1157

where, the units of upper left 3 x 3 sub matrix is N/cm, that of lower right 3 x 3 sub

matrix is Ncm, and that of remainder is N.

The result is evaluated in the following way:

1. A small wrench Sw, is applied in addition to wex to body B and twists E3DB and
EoDB are obtained by multiplying the inverse matrices of the stiffness matrices,
[K]1 and [K]2, respectively, by Sw, as of Eq. (3.64). Corresponding positions
for body B are then determined, based on the calculated twists EDB and E1DB.

2. E'DA is calculated by multiplying the inverse matrix of [KF]R,L by 5Sw1 as of Eq.
(3.68). The position of body A is then determined from this twist.

3. The wrench between body B and body A is calculated for the two cases based on
knowledge of the positions of bodies A and B and the spring parameters. The
change in wrench for the two cases is determined as the difference between the new
equilibrium wrench and the original. The changes in the wrenches are named
SwAB, and wAB,2 which correspond to the matrices [K]1 and [K]2.

4. The given change in wrench Sw, is compared to SwAB,1 and SwAB,2

The given wrench Sw, and the numerical results are presented as below.

w, = 10-4x[0.5 -0.2 0.4 0.3 -0.8 0.4]

E lB =10-3 x[0.3522 -0.3081 0.0912 -0.0137 -0.0429 -0.0367]


E(2B =10-3x[0.3354 -0.2682 0.0845 -0.0132 -0.0404 -0.0365]

E D"A=10-3x[0.1113 -0.0100 -0.0067 0.0081 -0.0267 -0.0650]O


SwEA =10-4x[0.5000 -0.1995 0.4017 0.3035 -0.8000 0.4000]









wAB,1 =10-4x[0.4997 -0.1998 0.4020 0.3041 -0.8010 0.4011]


wAB,2 =10-4x[0.5462 -0.0693 0.3534 -0.7831 -0.2319 0.0967]

where w EA is the wrench between body A and ground. The unit for the wrenches is [N,

N, N, Ncm, Ncm, Ncm]T and that of the twists is [cm, cm, cm, rad, rad, rad]T. The

difference between uSwE and Swr, is small and is due to the fact that the twist was not

infinitesimal. The difference between SwAB,1 and Sw, is also small and is most likely

attributed to the same fact. However, the difference between SwAB,2 and Sw, is not

negligible. This indicates that the stiffness matrix formula derived in this research

produces the proper result and that the term wext x cannot be neglected in Eq. (3.74).















CHAPTER 4
STIFFNESS MODULATION OF PLANAR COMPLIANT MECHANISMS

Planar mechanisms with variable compliance, specifically, compliant parallel

mechanisms and mechanisms having two compliant parallel mechanisms in a serial

arrangement are investigated in this chapter. The mechanisms consist of rigid bodies

joined by adjustable compliant couplings. Each adjustable compliant coupling has a

revolute joint at each end and a prismatic joint with an adjustable spring in the middle.

The adjustable springs are assumed to be able to change their stiffness coefficient and

free length and the mechanisms are in static equilibrium under an external wrench. It is

desired to modulate the compliance of the mechanism while regulating the pose of the

mechanism.

4.1 Parallel Mechanisms with Variable Compliance

4.1.1 Constraint on Stiffness Matrix

Figure 4-1 illustrates a compliant parallel mechanism having N number of

compliant couplings. The mechanism is in static equilibrium under the external wrench

w ex and it can be expressed as

N
wext = f, (4.1)
1=Y


where f, is the spring wrench of i-th compliant coupling. By using Eq. (2.15) a

derivative of Eq. (4.1) may be written as









N

=1 (4.2)
= i[K EDA


where

Sa$, a$'T
[KF] =k $ +k(1 -) (4.3)
ae ao o

k,, lo, /l, and 0, in Eq. (4.3) are the spring constant, the spring free length, the current

spring length, and the rising angle of i-th compliant coupling, respectively (see Figure 2-

4). In addition, $, represents the unitized Plicker coordinates of the line along the ith

compliant coupling as in Eq. (2-6) and may be written explicitly as



$ r sin0, (4.4)
r S sin, ry, cos 0


where r,, and y,, are the pivot position of i-th compliant coupling in body E. Then Eq.

(4.4) leads to

-sin 60
cos (4.5)
,, cos 0, + y,, sin 0,

The stiffness matrix of the mechanism [K] can be written from Eq. (4.2) as

N
[K]= [KF]l. (4.6)
z=1









we
\ ext


. . .


E3


Figure 4-1. Compliant parallel mechanism with N number of couplings.

Ciblak and Lipkin (1994) showed that the stiffness matrix of compliant parallel

mechanisms can be decomposed into a symmetric and a skew symmetric part and that the

skew symmetric part is negative one-half the externally applied load expressed as a

spatial cross product operator. For planar mechanisms, the skew symmetric part can be

written as


[K][K]+[K] [K]-[K]
2 2
[K]ymmetnc + [K]skew Symmetrc


[K]-[K]T 1 0 0
[K]Skew Symmetnc 20 -
-f, fx 0


(4.7)


(4.8)


where wext = [fxy fy' m ]T is the external wrench.









It is important to note that no matter how many compliant couplings are connected and

no matter how the spring constants and the free lengths of the constituent compliant

couplings are changed the stiffness matrix of a compliant parallel mechanism contains

only six independent variables and the stiffness matrix may be rewritten as

11n K12 13
[K]= K12 K22 K32 (4.9)
K13 + fy K32 K33

From Eqs. (4.3)-(4.6) the six independent elements of the stiffness matrix [K] can be

explicitly written as


K11 = = k -k sin2 ~ (4.10)



K12 = I k sin 0,cos0, (4.11)


N /
K13 = -k,(, sin0, +ry,,)+k, (1, sin0 +r, sin0, cosO,+ry, sin2 ) (4.12)
=1


K22 = k-k, cos2 (4.13)


N I(
K32 k,,, k (,, cos2 + ry, sin cos (4.14)
K =1



kr,,1(, cos 0, + r, )+kr),(1, sin 0, + ry,)

33 = -k r,,,(l, cosO0 +rsy, sin0, cos0, +r,, cos2 0) (4.15)
1 /1

-k, ry,1(/ sin 0 + r,, sin 0 cos0, + ry, sin2 20)
11 ,









4.1.2 Stiffness Modulation by Varying Spring Parameters

In this case it is desired to find an appropriate set of spring constants and free

lengths of the constituent compliant couplings of the mechanism shown in Figure 4-1 to

implement a given stiffness matrix and to regulate the pose of body A under a given

external wrench.

It is important to note that the stiffness matrix contains only six independent

variables and the equations for the independent variables are linear in terms of k's and

k,o, 's as shown in Eqs. (4.10)-(4.15) since all geometrical terms are constant. In

addition to the equations for the stiffness matrix, the system should satisfy static

equilibrium equations to regulate the pose of the mechanism and from Eqs. (4.1) and

(4.4) it can be written as

f cose 0
Wex= f, = ki l-/o sin 0, (4.16)
m ~ rx, sin 0, ry, cos 0,

Eq. (4.16) consists of three equations which are also linear in terms of k's and klo, 's.

Since there are nine linear equations to be fulfilled and each adjustable compliant

coupling possesses two control variables such as spring constant and free length, at least

five adjustable compliant couplings are required.

The nine equations may be written in matrix form as

[A]X=B (4.17)

where

B = K11, K12, K13, K22 K 32, K33, f fy, z ]T (4.18)










S= [k,, k?, 2 k'", kl1,o


1 1 *** 1
0 0 ** 0

G3,1 G3,2 *** G3,N
1 1 *** 1

G6,1 G6,2 *** G6,N
G8,1 G8,2 G8,N
HL H1,2 *** Hl,N
H3,1 H3,2 H3,N
H5,1 H5,2 HS,N


k2o2, ..., ,owN


G1,1 G1,2 .
G2,1 G2,2 ..
G4,1 G4,2 .
G5,1 G5,2 .
G7,,1 G7,2
G9,1 G9,2 "
H2,1 H2,2
H4,1 H4,2
H6.1 H6,2


sin2 6,
GI, = G2, =


sin 0 cos -
,--- G3, sin -r y,


Sx,, sin0, cos0 ++/, sin2 2
G4,r = sin 0, +- G, =-


cos2 2
~T


-r,, cos2 60 -r, sin 6 cos 6


G8,, = r,2 + y,2 +, (r,, cos + r, sin )


r 2 cos 2 + 2 sin2 20 + 2 sin 0 cos ,
G9, = -, cos 0, y, Sin 0,


H1I, =/ cos 0, H2,, = -cos,, H3, = sin H4, = -sin0,


Hs,, = 1 (r,, sin 0, ri,, cos 0, ), H6,, = (r,, sin 0, y,, cos 0, ).


It is important to note that [A], X, and B are 9 x (2*N), (2*N) x 1, and 9 x 1 matrices,

respectively, where N denotes the number of the adjustable compliant couplings.


(4.19)


[A] =


G,N
G2,N
G4,N
G,N
G7,N
G9,N
H2,N
H4,N
H6 N


(4.20)


and where


G6,1 = x, 7,1 =









It is required to solve Eq. (4.17) where the number of columns of matrix [A] is in

general greater than that of rows and the general solution Xs,, can be written as

Xsol = Xp + Xh
Xp+[AN]C_ (4.21)
= x,+[AN11I ]C

where Xp, Xh, [ANU1 ], and C are the particular solution, the homogeneous solution, the

null space of matrix [A], and the coefficient column matrix, respectively (see Strang

1988). Once a solution Xsol is obtained, o, 's are calculated from k 's and ko, 's in Xsol.

It is important to note that [ANUW,] is (2*N) x (2*N-9) matrix and C is (2*N-9) x 1

column matrix.

There might be many strategies to select the matrix C which leads to a specific

solution. For instance, if the norm of Xsol is desired to be minimized, then by using

projection matrix [ANlpl ] (see Strang 1988), the solution can be obtained as

X Min=X p +[ANull-P ](-Xp) (4.22)

where

[ANu-] = [A Nul([A ] [ANu [ANu] (4.23)

For another case, we might want the solution closest to a desired solution Xd which may

be constructed from operation ranges of adjustable compliant couplings, for instance,

minimum and maximum spring constant and free length. Then, the solution can be

obtained as

Xd o = X +[ANullP](Xd -X ). (4.24)









Unfortunately, these methods involve mixed unit problems and do not guarantee a

solution consisting of only positive spring constants and free lengths.

A numerical example is presented. The external wrench wext and the desired

stiffness matrix [K] are given as


wex = [-1.8832 N -2.8805 N 3.2851Ncm]

0.0216 N/cm 2.2483 N/cm -2.2750 N
[K]= 2.2483 N/cm 25.3914 N/cm 60.9800 N
-5.1555 N 62.8632 N 270.4409 Ncm

The geometry information of the mechanism shown in Figure 4-1 is given in Tables 4-1

and 4-2. The mechanism is assumed to have five compliant couplings.

Table 4-1. Positions of pivot points in body E for numerical example in 4.1.2.
Pivot points El E2 E3 E4 E5
X 0.0000 0.6000 2.5000 3.9000 5.3000
Y 0.0000 0.8000 0.3000 0.9000 0.0000
(Unit: cm)

Table 4-2. Positions of pivot points in body A for numerical example in 4.1.2.
Pivot points Al A2 A3 A4 A5
X 0.6000 1.4055 2.6736 3.3368 4.7284
Y 4.5000 2.7447 3.3209 3.9614 4.1442
(Unit: cm)

The spring parameters which have the minimum norm and satisfy the given

conditions can be obtained by using Eq. (4.22) and these values are shown in Table 4-3.

Table 4-3. Spring parameters with minimum norm for numerical example 4.1.2.
Spring No. 1 2 3 4 5
Stiffness constant k 4.6674 7.2485 3.5188 5.0243 6.3280
Free length o 4.1678 2.1490 6.3995 1.9322 3.9104
(Unit: N/cm for k, cm for lo)

The spring parameters which are closest to the given spring parameters as shown in

Table 4-4 can be obtained by applying Eq. (4.24) and it is shown in Table 4-5.









Table 4-4. Given optimal spring parameters for numerical example 4.1.2
Spring No. 1 2 3 4 5
Stiffness constant k 5.0 5.0 5.0 5.0 5.0
Free length l 3.0 3.0 3.0 3.0 3.0
(Unit: N/cm for k, cm for lo)

Table 4-5. Spring parameters closest to given spring parameters for numerical example
4.1.2.
Spring No. 1 2 3 4 5
Stiffness constant k 4.8664 6.8783 3.8968 4.8990 6.2974
Free length l 4.3386 2.3374 5.0230 2.1667 4.0492
(Unit: N/cm for k, cm for lo)

Two sets of the spring parameters, one in Table 4-3 and the other in Table 4-5, implement

the given wrench and stiffness matrix.

4.1.3 Stiffness Modulation by Varying Spring Parameters and Displacement of the
Mechanism

In this case, different from the previous section, the pose of body A is not

constrained as fixed. A change of the pose of body A, which is considered to be in

contact with the environment, may be compensated by attaching body E to the end of a

robot system and by controlling the position of the robot end effector in a similar manner

as described in the Theory ofKinestatic Control proposed by Griffis (1991). As

presented in the previous section there are nine values to be fulfilled: six from the

stiffness matrix and three from the wrench equations. A typical planar parallel

mechanism which has three couplings is investigated since the mechanism has nine

control input variables which is same with that of values to be fulfilled: six from

adjustable compliant couplings and three from the planar displacement between body A

and body E. The target variables may be expressed in matrix form as B in Eq. (4.18)

and control input variables U may be written in matrix form as


U= [k,, k2, k3, o2l o2 Y3 o, O2


(4.25)









where k's and lo 's are the spring constant and free length of ith compliant coupling,

respectively. In addition, xo and y, are the position of point O in body A, which is

coincident with the origin of the inertial frame E, and 0 is the rotation angle of body A

with respect to ground.

The stiffness matrix equations and wrench equations are highly nonlinear in terms

of the displacement of the bodies as shown in Eqs. (4.9)-(4.16). In this section a

derivative of the target variables B with respect to input variables U is investigated and

the derivative is used to obtain the small change of input variables for the desired small

change of target values and it may be written as

dB
SB =B U (4.26)
dU -


SU_ d B) =SB (4.27)


where

SB = [K,, 1K, 1, 1K13, K 22, 8K32, K,33, 8f, f ]m ] (4.28)

sU = [8k,, 8k,, k,3, 61,, 81o2, 5 1o, 3x", 8y0, 8]T (4.29)

aB, aBj


d (4.30)
3U iB9 'B9


For instance, 5B, can be written as









dB
dB1 =d 1 U
dU -

= 1 gk, + gk2+ 113k3+ 13 + 1 113 2 + 11o3 (4.31)
ak, ak ak33 o2 o3
+ 113x, + K y + a11
ax, ay, a>

In Eqs. (4.9)-(4.16) all elements of B were presented as functions of not U but Up

which is defined as

U, = [k,,k2, k3, 1o1, o2, ,/ 1, o 2, /3, 01, 0r2 3 ]T. (4.32)

aKH1 aKl 1 anKj
Hence among Eq. (4.31) and are not obtained from simple
axo ayo a

differentiation.

Since B is a function of Up, dB1 can also be written as

dB1
sB = s,
dU -

K 1Sk + + S k-- 11k3+ a1 l + aK 11 o2+ 11 o3 (4.33)
1ak, ak2 ak 3 1 a,2 3o3
+ )K11051 al1- K 12 +2 aKl 13 +l / 1 a 001 +- l a 52-1 2+ aK {9g 3
+ 811 + "12 13 1 61 2+ 1 13, 1139
all a12 a3 a1 a02 a 3

In addition, /8, 's and 0,'s in Eq. (4.33) can be substituted with Eqs. (2.12) and (2.13)

which is restated here as

F3x
l1, = (D o $ (2.12)



go Tx0 T 7
3D 1 (2.13)
/,ae, Is, y









By the above substitution in Eq. (4.33) and collecting the coefficients according to each

term of U 5B1 can be written as

5k,
3k2



SB =A11 a11 11 11 1 11 1 11
ak, ak2 ak3 1 o2 S3 o o 1 .(4.34)



3y,
-yo

SdBi 1
dU -

dB
All terms of AB may be obtained in the same way in which 8B, was obtained and -
dU

dB
can be derived by combining all d 's.
dU

A numerical example is presented. The mechanism shown in Figure 4-1 is in static

equilibrium under the external wrench wex and the geometry information and the spring

parameters are given below. The mechanism is assumed to have three compliant

couplings.

Table 4-6. Positions of pivot points for numerical example 4.1.3.
Pivot points El E2 E3 Al A2 A3
X 0.0000 0.6000 2.5000 0.6000 1.4055 2.6736
Y 0.0000 0.8000 0.2000 4.5000 2.7447 3.3209
(Unit: cm)

Table 4-7. Initial spring parameters for numerical example 4.1.3.
Spring No. 1 2 3
Stiffness constant k 5.5 5.7 5.1
Free length o 4.8 3.1 2.0
(Unit: N/cm for k, cm for lo)












The external wrench we and the initial stiffness matrix [K]I are calculated from the

geometry of the mechanism and the spring parameters.

wex =[-2.0409 N -0.9263 N 12.8594 Ncm]


0.1679 N/cm
[K] = 3.9107 N/cm
3.0360 N


3.9107 N/cm
14.9590 N/cm
12.9966 N


The desired stiffness matrix [K]D is given below.

0.6679 N/cm 4.3107 N/cm
[K]D= 4.3107 N/cm 15.4290 N/cm
3.2560 N 12.8766 N


3.9623 N
10.9558 N
25.9764 Ncm




4.1823 N
10.8358 N
26.3764 Ncm


Since the difference between the desired stiffness matrix and the initial stiffness matrix is

not small enough, the difference is divided into a number of small 5B 's and Eq. (4.27) is

applied repeatedly to obtain the spring parameters and the displacement of body B which

implement the desired stiffness matrix and the given wrench.

The calculated spring parameters and the pose of body A are shown in Table 4-8 and

Table 4-9 and the initial and final pose of the mechanism is shown in Figure 4-2.

Table 4-8. Calculated spring parameters for numerical example 4.1.3.
Spring No. 1 2 3
Stiffness constant k 6.2563 5.5311 5.1492
Free length l 5.4810 4.3584 3.1954
(Unit: N/cm for k, cm for lo)

Table 4-9. Positions of pivot points in body A for numerical example 4.1.3.
Pivot points Al A2 A3
X 0.8201 1.9165 3.0661
Y 5.2909 3.7010 4.4874
(Unit: cm)











6


5 Final pose


S -- ---- -- --- ----- -' -- tl ^ 0
4 -
Initial pose

3



2



3 --- -i -

1-"

0
-1 0 1 2 3 4


Figure 4-2. Poses of the compliant parallel mechanism for numerical example 4.1.3.

4.2 Variable Compliant Mechanisms with Two Parallel Mechanisms in Series

In this section mechanisms having two planar compliant parallel mechanisms that

are serially arranged as shown in Figure 2-6 are investigated.

4.2.1 Constraints on Stiffness Matrix

The stiffness matrix of the mechanism was derived in Chapter two and restated as


[]=[KF R,L ([KF R,L +[KF R,U-KM]R,U ) KF R,U. (2.54)


Applying the constraint presented by Ciblak and Lipkin (1994), [KF]RL and [KF R,
may be written as
11 12 13
[KF]R,L = K2 K K +f (4.35)
KL +f, K3 K3









KU
KF]4 KU
[KF R,U =
13 + fy


Kf K+
'2 K3
22 K32 + f
KU KU
32 33


where wex = [fx, f, m is the external wrench. In addition, [KM ]R,U is a function of

only the external wrench as shown in Eq. (2.46) which is restated as

0 0 -//
[K ]R, 0 0 / (2.46)
f -f 0

Plugging in Eqs. (4.35), (4.36), and (2.46) into Eq. (2.54) and carrying out a symbolic

operation using Maple software shows

0 0 -fy
[K]-[Kr]= 0 /f
f -f o

which is the same with Ciblak and Lipkin (1994)'s statement for compliant parallel

mechanisms in planar cases. This result indicates that mechanisms having two planar

compliant parallel mechanisms in a serial arrangement also contain only six independent

variables.

4.2.2 Stiffness Modulation by using a Derivative of Stiffness Matrix and Wrench

Since the stiffness matrix of the mechanism shown in Figure 2-6 is complicated and

nonlinear in term of the spring parameters and the displacement of the rigid bodies, a

derivative of the stiffness matrix and the static equilibrium equation is derived and

applied for stiffness modulation of the compliant mechanism.

The stiffness matrix elements and the wrenches may be written in matrix form as

B = [K11,K, K1, K22,K K 3, K 3, m, f mfY," ] (4.37)


(4.36)









where wA = [f, fyA, mA and wB = [ ff, m are the wrenches from the

compliant couplings connecting body A to ground and from the couplings connecting

body B to body A, respectively.

The spring parameters and the displacements of the rigid bodies may be written as

U= [k, k, k,, k3, k4, ks, k6,1o2, o 3, 14 ,5 o6, x, ,A o xfB, YB OB T (4.38)

where k's and /, 's are the spring constant and free length of ith compliant coupling,

respectively. In addition, xA and yA are the position of point O in body A which is

coincident with the origin of the inertial frame E and OA is the rotation angle of body A

with respect to ground. x yB and qB are defined in the same way in terms of the

inertial frame E.

In chapter two the stiffness matrix and the wrenches were presented as functions of

U which is defined as

Up = [k,, k,, k2 k4 k4 2ol o2 o3 o4 o5 1o6' 11 2' 13, 4,5, 6
1, 2, 03, 04, 5, 6, 9 r r 3, 2,r3](4.39)


where /, 's and 0, 's are the current spring length and rising angle of i-th compliant

coupling. In addition ,, and r,, are the pivot positions of i-th compliant coupling in

body A.

A similar approach taken in the previous section is applied to get a derivative of the

dB
stiffness matrix and the wrenches --: each element of B is differentiated with respect
dU

to Up and the terms not belonging to MU are substituted in the terms of U In other

words, 681 's, 60, 's, 8r, 's, and 8r, 's are expressed in terms of the twists of the bodies.









The coefficient of each term of MU corresponds to an element of the derivative matrix in

an analogous way to Eq. (4.34).

Since springs 4, 5, and 6 connect body A and ground, 81/ and 80, for i=4, 5, 6 can

be written as Eqs. (2.12) and (2.13) which are


1, = $EA = T yf (2.12)



,T EDA T XOA
1 0 a,= 1 l Ayf (2.13)
1, a8 1, 8 0, A


Springs 1, 2, and 3 join body B and body A and thus /81 and 368 for i=l, 2, 3 may be

expressed as

F3x0B 3xA
,l =$ A _DB T yoB yoA (4.40)


3x A xf
O, = A3DB +A ,yI B- yo + A (4.41)
1, 8, 0, B8 ,


Lastly rg,, 's, and 8r,, 's are the positions of the pivot point in body A and by using the

twist equation it can be written as

~r 8x ~ 0 r~
= l yf + 0 x L (4.42)
0 0 8g 0

Now all terms in the differential of B are expressed in terms of MU and by writing it in

matrix form gives









dB
B= 05U (4.43)
dU

dB
where is 12 x 18 matrix.
dU

It is required to obtain the small change of input values MU corresponding to a

small change of stiffness matrix and wrenches oB and since the number of columns of

the matrix is greater than that of rows it is a redundant system. There are in general an

infinite number of solutions and a variety of constraints may be imposed on the system.

Since MU is the change from the current values, minimizing the norm of MU may

be one of reasonable options. Then in a similar way to Eq. (4.22) Umin. may be obtained

as

SdB dB dB dB
Umin =L dU L dU (-Up) (4.44)
n -- Null Null U Null Null


F dB 7
where MUp sol is a particular solution of Eq. (4.43) and Null is the null space of


dB
matrix (see Strang 1988).
dU

Body B is considered to be in contact with the environment and it may be required

to preserve the pose of body B. It indicates that the twist of body B is equal to zero and

can be written as


XoB =0, yoB =0, 0B =0.









dB
We can implement it by removing the last three columns of and the last three rows
dU

of UT and solving the problems in a similar way to that of the previous problem since

the system is still redundant.

If both of the bodies are required to be stationary then the twists of the bodies

should be zero and it may be written as

xfo =0, Syo =0, OA = 0,

xoB =0, SyB, =0, 6B =0.

dB
This can be implemented by removing the last six columns of and the last six rows
dU

of UT, and by solving the problem which is not redundant.

4.2.3 Numerical Example

The geometry information and spring parameters of the mechanism shown in

Figure 2-6 and the external wrench wex are given below.

wet =[-1.7N 2.5N 12.7N]

Table 4-10. Spring parameters of the compliant couplings for numerical example 4.2.3.
Spring No. 1 2 3 4 5 6
Stiffness constant k 5.0 5.0 5.0 5.0 5.0 5.0
Free length lo 3.0614 0.6791 2.3608 2.8657 0.7258 1.2732
(Unit: N/cm for k, cm for lo)

Table 4-11. Positions of pivot points for numerical example 4.2.3.
Pivot points El E2 E3 B1 B2 B3
X 0.0000 1.0000 3.0000 -0.8000 0.4453 1.1965
Y 0.0000 0.8000 0.0000 4.5000 3.7726 4.6186
(Unit: cm)

Table 4-11. Continued.
Al A2 A3 A4
0.2000 1.1261 2.1646 1.2760
2.3000 1.8179 1.9252 2.6038










The initial stiffness matrix [K]I is calculated from the geometry of the mechanism and

the spring parameters.

1.5992 N/cm -1.1571 N/cm -6.0650 N
[K] = -1.1571 N/cm 5.7047 N/cm 7.7521 N
-3.5650 N 9.4521 N 22.6794 Ncm

The desired stiffness matrix [K]D is given below.

1.6442 N/cm -1.1921 N/cm -6.0230 N
[K]D= -1.1921 N/cm 5.7417 N/cm 7.7931 N
-3.5230 N 9.4931 N 22.6384 Ncm

Since the difference between the desired stiffness matrix and the initial stiffness matrix is

not small enough, the difference is divided into a number of small 8B 's and the problem

is solved repeatedly to obtain the spring parameters and the displacements of body B and

body A which implement the desired stiffness matrix. Three sets of spring parameters

are obtained: one with no constraint on the displacements of body A and body B, another

with body A fixed, and the other with body A and body B fixed. The calculated spring

parameters are presented in Table 4-12, Table 4-13, and Table 4-14, respectively. In

addition, the initial and final poses of the mechanism are shown in Figures 4-3 and 4-4.

Table 4-12. Spring parameters with no constraint for numerical example 4.2.3
Spring No. 1 2 3 4 5 6
Stiffness constant k 5.5786 4.8852 5.5513 5.1865 5.1506 5.1505
Free length lo 2.5502 0.5084 1.9188 2.7939 0.8039 1.3746
(Unit: N/cm for k, cm for lo)

Table 4-13. Spring parameters with body B fixed for numerical example 4.2.3
Spring No. 1 2 3 4 5 6
Stiffness constant k 6.3556 4.4317 5.9605 5.3306 5.5194 4.4334
Free length lo 3.0234 0.6588 2.6016 2.8192 0.6220 1.0270
(Unit: N/cm for k, cm for lo)
























- Initial pose






Final pose


-1 0 1 2 3


Figure 4-3. Poses of the compliant mechanism with








4 ------ -- -- -- -
5




4




3




2



1 -




0-----------


-1 0 1 2 3


body B fixed.


- Initial pose


Figure 4-4. Poses of the compliant mechanism with no constraint.









Table 4-14. Spring parameters with body A and body B fixed for numerical example
4.2.3.
Spring No. 1 2 3 4 5 6
Stiffness constant k 12.3070 10.5719 6.9847 1.9411 4.9528 3.8741
Free length lo 2.6786 1.0769 2.5033 3.7435 0.7229 1.0333
(Unit: N/cm for k, cm for lo)

The results indicate that there are greater changes of the spring parameters with

more constraints imposed on the bodies. These control methods all require the inverse of

dB dB dB
d- or -[ -] depending on the constraint and it may cause a singularity
dU dU d ,,

problem. With more constraints the mechanism is more vulnerable to the singularity

problem.














CHAPTER 5
CONCLUSIONS

The compliance of mechanisms containing rigid bodies which are connected to

each other by line springs was studied. A derivative of the planar spring wrench

connecting two moving bodies was obtained and then, through a similar approach, a

derivative of the spatial spring wrench which is more general was obtained. It is obvious

from Eq. (3.36) that the derivative of the spring wrench joining two rigid bodies depends

not only on a relative twist between two bodies but also on the twist of the intermediate

body in terms of the inertial frame unless the initial external wrench is zero.

The derivative of the spring wrench was applied to obtain the resultant stiffness

matrix of two compliant parallel mechanisms in a serial arrangement. The resultant

stiffness matrix indicates that the resultant compliance, which is the inverse of the

resultant stiffness matrix, is not the summation of the compliances of the constituent

mechanisms unless the external wrench applied to the mechanism is zero which was

generally accepted. The derivative of the spring wrench was also applied to acquire the

stiffness matrix of planar springs in a hybrid arrangement and may be applied for

mechanisms having an arbitrary number of parallel mechanisms in a serial arrangement.

Planar mechanisms with variable compliance were investigated with the knowledge

of the stiffness model obtained in the research. Adjustable line springs which can change

their spring constants and free lengths were employed to connect rigid bodies in the

mechanisms. Ciblak and Lipkin (1994) showed that the stiffness matrix of compliant

parallel mechanisms can be decomposed into a symmetric and a skew symmetric part and









that the skew symmetric part is negative one-half the externally applied load expressed as

a spatial cross product operator. It was shown through this research that the same

statement is valid for the stiffness matrix of a mechanism having two compliant parallel

mechanisms in a serial arrangement. In other words there are only six rather than nine

independent variables in the stiffness matrix of planar compliant parallel mechanisms and

in the stiffness matrix of a mechanism having two planar compliant parallel mechanisms

in a serial arrangement.

Derivatives of the stiffness matrices of planar compliant mechanisms with respect

to the spring parameters and the twists of the constituent rigid bodies were obtained. It

was shown that these derivatives may be utilized to control and regulate the stiffness

matrix and the pose of the mechanism respectively at the same time by adjusting the

spring parameters of each constituent coupling with or without the change of the position

of the robot where the compliant mechanism is attached.

Several future works are presented.

The singularity conditions associated with the resultant stiffness matrix of

compliant parallel mechanisms in a serial arrangement needs to be studied. This study

will identify under what condition the mechanism collapses even with a small change in

the applied wrench.

It was required to solve a redundant system of linear equations to obtain the

changes of the spring parameters and the twists of the bodies corresponding to a small

change of the stiffness matrix and the resultant wrench of a compliant mechanism. There

are in general an infinite number of solutions and the least square solution was chosen in






80


this research. A better solution may be selected by considering the following

considerations:


1. The inverse of matrix


dB dB dB
-- or -
dU dU dU


as of Eq. (4.43) is required to solve the

dR dRT d R


linear equations. Solutions close to singular cases of or should be
dU dU dU
avoided.

The operation ranges of the spring parameters should be taken into account.

Proper consideration should be given to singularities of the compliant mechanism.














APPENDIX A
MATLAB CODES FOR NUMERICAL EXAMPLES IN CHAPTER TWO AND
THREE

Matlab codes for the numerical examples in chapter two and three are presented in

this appendix. Table A-i contains the function list and functions NuEx_21, NuEx_22,

and NuEx 31 are main functions, and other functions are called inside these main

functions. For instance, to get the result of numerical example 2-1, it is needed to call

function NuEx_21 and other functions are placed in the same folder where function

NuEx 21 is located.

Table A-1. Matlab function list.
Function Name Description
NuEx_21 Main function for numerical example 2-1
NuEx_22 Main function for numerical example 2-2
NuEx_31 Main function for numerical example 2-3
StaticEq21 Static equilibrium equation for numerical example 2-1
StaticEq22 Static equilibrium equation for numerical example 2-2
StiffMatrix Computes matrix [KF]
GetKM Computes matrix [KM]
SpringWrench Computes spring wrench
GetPLine Computes Plucker line coordinates
GetOriginVel Computes origin velocity from velocity of a point and
angular velocity of a rigid body
GetVelP2D Computes displacement of a point from twist
GetGlobalPos2D Computes position of a point in terms of inertial frame in
planar space
GetGlobalPos3D Computes position of a point in terms of inertial frame in
spatial space


function NuEx 21

% Numerical example 2-1
% Test Stiffness Matrix of planar parallel mechanisms in series










%=========================================================
%====== GIVEN VALUES

%Wext: External wrench
%A : Coordinates of fixed pivot points
%lb: Local coordinates of vertices of middle platform
% the first 3 columns are coordinates of pivot points connected to lower
% parts and the second 3 columns for upper parts
%lt: Local coordinates of vertices of top triangle
%lo : Free lengths of springs
%k : Spring constants
%=========================================================

global Wext A lb It lo k

%External wrench
Weq=[0.01, -0.02, 0.03]';

%Coordinates of fixed points
A [ 0.0, 1.5, 3.0
0.0, 1.2, 0.5 ];

%Local coordinates of vertices of triangles
lb= [0, 1.0, 2.0, 0.0, 1.0,2.0
0, -1.7321, 0.0, 0.0, 0.5, 0.0];

t= [0, 1.0, 2.0
0, -1.7321, 0.0];

%Free lengths of springs
lo = [5.0040, 2.2860, 4.9458, 5.5145, 3.1573, 5.2568]';

%Spring constants
k= [0.2, 0.3, 0.4, 0.5, 0.6, 0.7]';

%Initial guess of positions and orientatins of the local coordinate systems
Xo=zeros(6,1);
Bo=[0.2, 5.0, 10.8*pi/180]';
To=[-0.2, 10.5, 3.4*pi/180]';
Xo=[Bo;To];


% Find Xo
%=========================================================
options=optimset('fsolve');
optionsnew=optimset(options, 'MaxFunEvals',1256); % Option to display output









Wext=Weq;
[X error] = fsolve(@StaticEq_21, Xo, optionsnew);
Xo=X;
%Calculate the position of vertices of triangles
Bo = X(1:3);
To = X(4:6);
B = GetGlobalPos2D(Bo, lb);
T = GetGlobalPos2D(To, It);

%Calculate initial wrenches
W L I= zeros(3,1);
W L I = WL_I + SpringWrench(A(:,l), B(:,1), k(1), lo(1));
W L I = WL_I + SpringWrench(A(:,2), B(:,2), k(2), lo(2));
WLI = WL_I + SpringWrench(A(:,3), B(:,3), k(3), lo(3));

W U I= zeros(3,1);
W UI = W U I + SpringWrench(B(:,4), T(:,1), k(4), lo(4));
W UI = W U I + SpringWrench(B(:,5), T(:,2), k(5), lo(5));
WUI = W _UI + SpringWrench(B(:,6), T(:,3), k(6), lo(6));

%===========================================
% Calculate stiffness matrix

kl=StiffMatrix(A(:,1), B(:,1), k(1), lo(1));
k2=StiffMatrix(A(:,2), B(:,2), k(2), lo(2));
k3=StiffMatrix(A(:,3), B(:,3), k(3), lo(3));
KFL = kl+k2+k3;

k4=StiffMatrix(B(:,4), T(:,1), k(4), lo(4));
k5=StiffMatrix(B(:,5), T(:,2), k(5), lo(5));
k6=StiffMatrix(B(:,6), T(:,3), k(6), lo(6));
KF U = k4+k5+k6;

k4_2 = GetKM(B(:,4), T(:,1), k(4), lo(4));
k5_2 = GetKM(B(:,5), T(:,2), k(5), lo(5));
k6_2 = GetKM(B(:,6), T(:,3), k(6), lo(6));
KM U = k4 2+k5 2+k6 2;

K1 = KFL*inv(KF_L+KF_U-KMU)*KFU;
K2 = KFL*inv(KFL+KF U)*KFU;
%===========================================


% Verify the results
%======================================= [.000005,.000002,.000004]';
dW = [0.000005,0.000002,0.000004]';










% 1.Calculate twists
D_EB_1 = inv(Kl)*dW;
D_EB_2 = inv(K2)*dW;
D_EA = inv(KFL)*dW;

% 2.Get the position of the bodies
Bo_W = GetVelP2D(DEA, Bo(1:2))+ Bo(1:2);
BoW(3)= Bo(3)+DEA(3);
B_F = GetGlobalPos2D(Bo W, lb);


To W
To W
TF1

To W
To W
TF2


1 = GetVelP2D(DEB_1, To(1:2))+ To(1:2);
1(3)= To(3)+D_EB_1(3);
= GetGlobalPos2D(To W_l, It);

2 = GetVelP2D(DEB_2, To(1:2))+ To(1:2);
2(3)= To(3)+DEB_2(3);
= GetGlobalPos2D(To W 2, It);


% 3.Calculate wrenches
W L F = zeros(3,1);
W L F = W L F + SpringWrench(A(:,l), B
W LF = W L F + SpringWrench(A(:,2), B
W LF = W L F + SpringWrench(A(:,3), B
dW LF = W L Weq;


F(:,1), k(1), lo(1));
F(:,2), k(2), lo(2));
F(:,3), k(3), lo(3));


WUF 1
WUF1=
WUF 1=
WUF1
dW U F 1


W UF 2
W UF 2=
WUF 2=
WUF 2=
dWU F 2


zeros(3,1);
W U F_1 + SpringWrench(B_
W U F_1 + SpringWrench(B_
W U F_1 + SpringWrench(B_
= WU F_1 Weq;

zeros(3,1);
W U F 2 + SpringWrench(B_
W U F 2 + SpringWrench(B_
W U F 2 + SpringWrench(B_
= WU F2 Weq;


F(:,4), T
F(:,5), T
F(:,6), T



F(:,4), T
F(:,5), T
F(:,6), T


_1(:,1), k(4), lo(4));
1(:,2), k(5), lo(5));
1(:,3), k(6), lo(6));



2(:,1), k(4), lo(4));
2(:,2), k(5), lo(5));
2(:,3), k(6), lo(6));


%=====================================================
% Display the result

K1, K2, D EB_1, DEB_2, D_EA, dW, dW L F, dW U F1, dWUF_2
%=====================================================


function NuEx 22










% Numerical Example 2-2
% Test Stiffness Matrix of planar parallel mechanisms in hybird

%===================================================
%====== GIVEN VALUES

%Wext: External wrench
%A : Coordinates of fixed pivot points
%lb, Ic, Id : Local coordinates of vertices of intermediate triangle
%lt: Local coordinates of vertices of top triangle
%lo : Free lengths of springs
%k : Spring constants


global Wext A lb Ic Id It lo k

%External wrench
Weq=[0.1, 0.1, 0.2]';

%Coordinates of fixed points
A= [ 1.67, 4.46, 13.3449, 14.6731, 8.23, 4.94
4.4333, 1.3964, 3.25, 6.84, 14.1400, 13.4943 ];

%Local coordinates of vertices of triangles
lb = [0, 2, 1
0, 0, 1.7321];
Ic = lb;
Id = lb;
It = lb;

%Spring constants
k=[0.40 0.43 0.46 0.49 0.52 0.55 0.58 0.61 0.64];

%Spring free lengths
lo = [2.2547,2.4014,2.3924,1.5910,1.8450,2.2200,1.7077,2.2695,1.8711]';

%Initial guess of positions and orientatins of the local coordinate systems
Xo=zeros(12,1);
Bo=[3.8, 4.8,-39.8*pi/180]';
Co=[12.4, 4.5,76.5*pi/180]';
Do=[8.1, 12.4,202.4*pi/180]';
To=[7.0, 7.5,-23.4*pi/180]';
Xo=[Bo;Co;Do;To];









%=========================================================
% Find Xo

Wext = Weq;
options=optimset('fsolve');
optionsnew=optimset(options, 'MaxFunEvals',1256); % Option to display output
[X error] = fsolve(@StaticEq_22, Xo, optionsnew);
Xo=X;
%Calculate the position of vertices of triangles
Bo = X(1:3)
Co = X(4:6)
Do = X(7:9)
To= X(10:12)

B = GetGlobalPos2D(Bo, lb);
C = GetGlobalPos2D(Co, Ic);
D = GetGlobalPos2D(Do, Id);
T = GetGlobalPos2D(To, It);

%=========================================================
%Calculate stiffness matrix using stiffness equation

kl=StiffMatrix(A(:,1), B(:,1), k(1), lo(1));
k2=StiffMatrix(A(:,2), B(:,2), k(2), lo(2));
k3=StiffMatrix(B(:,3), T(:,1), k(3), lo(3));
k3_2 = GetKM(B(:,3), T(:,1), k(3), lo(3));
KK1_1 = (kl+k2)*inv(kl+k2+k3-k3_2)*k3;
KK12 = (kl+k2)*inv(kl+k2+k3)*k3;

k4=StiffMatrix(A(:,3), C(:,1), k(4), lo(4));
k5=StiffMatrix(A(:,4), C(:,2), k(5), lo(5));
k6=StiffMatrix(C(:,3), T(:,2), k(6), lo(6));
k6_2 = GetKM(C(:,3), T(:,2), k(6), lo(6));
KK2_1 = (k4+k5)*inv(k4+k5+k6-k6_2)*k6;
KK2 2 = (k4+k5)*inv(k4+k5+k6)*k6;

k7=StiffMatrix(A(:,5), D(:,1), k(7), lo(7));
k8=StiffMatrix(A(:,6), D(:,2), k(8), lo(8));
k9=StiffMatrix(D(:,3), T(:,3), k(9), lo(9));
k9_2 = GetKM(D(:,3), T(:,3), k(9), lo(9));
KK31 = (k7+k8)*inv(k7+k8+k9-k9_2)*k9;
KK3 2 = (k7+k8)*inv(k7+k8+k9)*k9;

K1 =KK1 1+KK2 1+KK3 1;
K2 =KK1 2+KK2 2+KK3 2;












% Verify the result
%============================================
dD = zeros(3,1);
dW = [0.00005,0.00002,0.00003]';
Wext=Weq+dW;
[X error] = fsolve(@StaticEq_22, Xo, optionsnew);
dDP = X(10:12)-To;
dD(1:2) = GetOriginVel(dDP, To(1:2));
dD(3)=dDP(3);

K1 dD= Kl*dD;
K2 dD = K2*dD;
%============================================


% Display the result

K1, K2, dW, dD, K1 dD, K2 dD
%============================================

function NuEx31

% Numerical example 3.1
% Test Stiffness Matrix of spatial parallel mechanisms in serial

%============================================
%====== GIVEN VALUES

%Wext: External wrench
%A : Coordinates of fixed pivot points
%lbL, lb_U: Local coordinates of vertices of middle platform
%lt: Local coordinates of vertices of top triangle
%loU, loL : Free lengths of springs
%kU, kL : Spring constants
%============================================

%Extemal wrench
Weq=[ -0.3, 0.4, 0.8, -2.3, -1.3, 0.7 ]';

%Coordinates of fixed points
A =[ 0.0, 1.3, 0.6, -0.7, -1.1, -0.5
0.0, 1.1, 2.7, 2.6, 1.8, 0.4
0.0, 0.2, 0.1, -0.1, 0.3, 0.1];









%Local coordinates of vertices of triangles
lbL = [ 0.0, 1.0, 0.3, -0.8, -1.3, -0.4
0.0, 0.9, 2.3, 2.2, 1.4, 0.5
0.0, 0.1, 0.2, -0.1, -0.2, -0.1 ];

lb_U= [ 0.0, 1.3, 0.6, -0.7, -1.1, -0.5
0.0, 1.1, 2.7, 2.6, 1.8, 0.4
0.1, 0.2, 0.25, 0.1, 0.12, 0.15 ];

It = [0.0, 1.2, 0.5, -0.6, -1.0, -0.3
0.0, 1.2, 2.3, 2.4, 1.3, 0.5
0.0, 0.1, -0.1, 0.1, 0.1, 0.2];

%Spring constants
k_U = [4.6, 4.7, 4.5, 4.4, 5.3, 5.5]';
k_L = [4.4, 4.9, 4.7, 4.5, 5.1, 4.8]';

%Positions and orientatins of the local coordinate systems
%Rotation angles are Euler angles (3-2-1)
Xo=zeros(12,1);
B_Po=[0.2, 1.2, 3.2]';
B_R=[1.2*pi/180, 5.0*pi/180, -1.8*pi/180]';

T_Po=[-0.3, 1.6, 5.5]';
T_R=[-0.4*pi/180, 8.5*pi/180, 3.8*pi/180]';

Xo=[B_Po;BR;T_Po;T_R];
Xo I = Xo;

%Convert local coord. to global coord.
B_L = GetGlobalPos3D(BPo, B_R, lbL)
B_U = GetGlobalPos3D(B Po, B_R, lb U)
T = GetGlobalPos3D(TPo, T R, It)


% Plucker line coordinates(or Jacobian) and lengths of all springs
%==============================================
JSU=zeros(6,6);
JSL=zeros(6,6);

l_U=zeros(6,1);
l_L=zeros(6,1);
loU=zeros(6,1);
loL=zeros(6,1);
F_U=zeros(6,1);
FL=zeros(6,1);











for i=1:6
[JS U(:,i), 1_U(i)] = GetPLine(B U(:,i), T(:,i));
[JSL(:,i), l_L(i)] = GetPLine(A(:,i), B_L(:,i));
end


% spring forces
F_U = inv(JS U)*Weq;
F_L = inv(JSL)*Weq;

%=================================================
% spring free lengths for static equilibrium of the platform
for i=1:6
lo U(i)=l U(i)-F U(i)/k U(i);
loL(i)=1 L(i)-F L(i)/k L(i);
end

loI = [loU; loL]

%=================================================
% Check wrench

W_L I= zeros(6,1);
W U I= zeros(6,1);
for i=1:6
WLI = WL_I + SpringWrench(A(:,i), B_L(:,i), kL(i), loL(i));
WUI = W U I + SpringWrench(B U(:,i), T(:,i), k U(i), lo U(i));
end
%=================================================


%Calculate stiffness matrix using stiffness equation

KFU = zeros(6,6);
KFL = zeros(6,6);
KMU = zeros(6,6);

for i=1:6
KFU = KFU+StiffMatrix(BU(:,i), T(:,i), k U(i), lo U(i));
KFL = KFL+StiffMatrix(A(:,i), BL(:,i), kL(i), loL(i));
KMU = KMU+GetKM(BU(:,i), T(:,i), k U(i), lo U(i));
end

K1 = KFL*inv(KF_L+KF_U-KMU)*KFU;
K2 = KFL*inv(KF L+KF U)*KF U;