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Page i Page ii Dedication Page iii Acknowledgement Page iv Table of Contents Page v Page vi List of Tables Page vii List of Figures Page viii Page ix Page x Abstract Page xi Page xii 1. Introduction Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 2. Spatial geometry and statics Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 3. The optimum quality index for a spatial redundant 44 inparallel manipulator Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 4. The kinematic analysis of the spatial redundant 44 inparallel manipulator Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 5. The optimum quality index for a spatial redundant 48 inparallel manipulator Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 6. The forward kinematic analysis of the spatial redundant 48 inparallel manipulator Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 7. The optimum quality index for a spatial redundant 88 inparallel manipulator Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 8. The forward kinematic analysis of the spatial redundant 88 inparallel manipulator Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Conclusions Page 121 Page 122 Appendix A. Constants for the forward kinematic analysis of the redundant 44 inparallel manipulator Page 123 Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Appendix B. Constants for the forward kinematic analysis of the redundant 88 inparallel manipulator Page 130 Page 131 Page 132 Page 133 Page 134 Page 135 Page 136 Page 137 Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 Page 145 Page 146 Page 147 List of references Page 148 Page 149 Page 150 Page 151 Biographical sketch Page 152 Page 153 Page 154 Page 155 

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QUALITY INDEX AND KINEMATIC ANALYSIS OF SPATIAL REDUNDANT INPARALLEL MANIPULATORS By YU ZHANG 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 2000 Copyright 2000 by Yu Zhang To my wife, Ying, and our parents. ACKNOWLEDGMENTS I want to express my deep and sincere gratitude to Dr. Joseph Duffy, my supervisor during my Ph.D. study, for providing me with the opportunity to complete my study under his exceptional guidance. Without his untiring patience, constant encouragement, guidance and knowledge this work would not have been possible. I would also like to thank my supervisory committee members, Dr. Carl D. Crane, Dr. Gloria J. Wiens, Dr. Ali A. Seirig, and Dr. Ralph Selfridge. I am grateful for their willingness to serve on my committee, providing me help whenever needed and for reviewing this dissertation. I especially thank Professor Chonggao Liang of Beijing University of Posts and Telecommunications for educating me on the various aspects of mechanism analysis and design. Also, I would like to thank all my colleagues in the Center for Intelligent Machines and Robotics for their help and support. Finally, I would like to thank my lovely wife, Ying Zhu. Her love, support and encouragement has had made my life rich and complete. I am grateful to my parents and parentsinlaw for their constant support and encouragement throughout my educational endeavors. TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................. iv LIST O F TAB LES ........................................................................ .................................... vii LIST O F FIG U RES....................................................................................................viii A B ST R A C T ....................................................................................................................... xi 1. INTRODUCTION..................................................................................................... 1 1.1 Redundant Parallel Manipulators ..................................... ............................ 1 1.2 Q quality Index................................................................................................... 6 1.3 Outline of Dissertation .................................................................................. 9 2. SPATIAL GEOMETRY AND STATICS ........................................ .......................... 12 2.1 Plicker Line Coordinates............................................................................ 12 2.2 Statics of a Rigid Body................................................................................ 15 2.3 The Statics of a Parallel Manipulator.......................................................... 20 3. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 44 INPARALLEL MANIPULATOR .............................................................. 24 3.1 Determination of /detJmJ. .............................................................................25 3.2 Im plem entation.............................................................................................. 30 4. THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 44 INPARALLEL MANIPULATOR .............................................................. 43 4.1 Inverse Kinematic Analysis..............................................................................44 4.2 Forward Kinematic Analysis....................................................................... 47 4.2.1 Introduction ............................................................................................ 47 4.2.2 Coordinate Transformations.................................................................48 4.2.3 Constraint Equations ............................................................................ 51 4.2.4 The Solution ........................................................................................... 52 4.2.5 Numerical Verification......................................................................... 55 5. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 48 INPARALLEL MANIPULATOR ................................................................... 58 5.1 Determination of /detJ.J .....................................................................59 5.2 Implementation.............................................................................................. 64 6. THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 48 INPARALLEL MANIPULATOR................................................ ................ 79 6.1 Forward Kinematic Analysis............................... .............................................. 79 6.2 Numerical Verification.................................................................................. 82 7. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 88 INPARALLEL MANIPULATOR ............................. ............................ 85 7.1 Determination of 4detJ.J. ...................................................................... 85 7.2 Im plem entation.............................................................................................. 94 8. THE FORWARD CINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 88 INPARALLEL MANIPULATOR .................................................................... 110 8.1 C oordinate System s........................................................................................... 110 8.2 Constraint Equations ................. ......................................................................... 113 8.3 Equation Solution .............................................................................................. 114 8.4 N um erical V erification...................................................................................... 118 9. CO N CLU SIO N S ................................................................................................... 121 APPENDIX A: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE REDUNDANT 44 INPARALLEL MANIPULATOR ................................. 123 APPENDIX B: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE REDUNDANT 88 INPARALLEL MANIPULATOR ................................. 130 LIST OF REFEREN CES ................................................................................................ 148 BIOGRAPHICAL SKETCH..................................................................................... 152 LIST OF TABLES Table Page Table 4.1: Numerical results of the redundant 44 inparallel manipulator .................. 56 Table 4.2: A numerical example for the special case of the redundant 44 inparallel m anipulator..................................................... ................................... 57 Table 6.1: Numerical results of the redundant 48 inparallel manipulator .................. 83 Table 6.2: A numerical example for the special case of the redundant 48 inparallel m anipulator..................................................... ................................... 84 Table 8.1: Numerical results of the redundant 88 inparallel manipulator .................... 119 Table 8.2: A numerical example for the special case of the redundant 88 inparallel m anipulator.............................................................................................. 119 LIST OF FIGURES Figure Page Figure 1.1: A planar parallel xy manipulator with one redundant actuator.................... 2 Figure 1.2: A 2DoF planar parallel manipulator....................................................... 3 Figure 1.3: A redundant 2DoF planar parallel manipulator......................... .............. 3 Figure 1.4: Planar view of spatial nonredundant 44 inparallel manipulators.................. 4 Figure 1.5: Selfdeployable space structure ............................................. ................ 11 Figure 2.1: Determination of a line ........................................................................... 13 Figure 2.2: Pliicker line coordinates............................................................................... 15 Figure 2.3: Representation of a force on a rigid body................................... .......... .. 16 Figure 2.4: Dyname and wrench ................................................. .......................... 19 Figure 2.5: A 66 inparallel manipulator ................................................................. 21 Figure 3.1: A redundant 44 inparallel manipulator .................................... ............ 24 Figure 3.2: Plan view of the redundant 44 inparallel manipulator ............................. 24 Figure 3.3: Plan view of the optimal configuration of the redundant 44 inparallel manipulator with the maximum quality index ................................... 30 Figure 3.4: Quality index for platform vertical movement .............................................31 Figure 3.5: Quality index for platform horizontal translation ........................................... 34 Figure 3.6: Platform rotations about the y'axis........................................ ............ .... 35 Figure 3.7: Quality index for platform rotations about the x' and y'axes..................... 37 Figure 3.8: Platform rotations about the zaxis ........................................... ........... ... 38 Figure 3.9: Quality index for platform rotation about the zaxis ....................................40 Figure 3.10: Plan view of the singularity position of the redundant 44 inparallel m anipulator when z = 900................................................................ 41 Figure 4.1: Coordinate systems of a redundant 44 inparallel manipulator................... 45 Figure 4.2: Coordinate transformations..................................................................... 49 Figure 5.1: A redundant 48 inparallel manipulator ................................... ............. 58 Figure 5.2: Plan view of the redundant 48 inparallel manipulator ............................. 58 Figure 5.3: Plan view of the optimal configuration of the redundant 48 inparallel manipulator with the maximum quality index ................................... 63 Figure 5.4: Compatibility between the redundant 44 and the 48 parallel manipulators. 64 Figure 5.5: Quality index for platform vertical movement ............................................. 65 Figure 5.6: Quality index for platform horizontal translation with different values of f. 70 Figure 5.7: Platform rotation about the y'axis ........................................... .............. 71 Figure 5.8: Quality index for platform rotations about the x' and y'axes..................... 73 Figure 5.9: Platform rotation about the zaxis............................................ ............... 74 Figure 5.10: Quality index for platform rotation about the zaxis ..................................76 Figure 5.11: Plan view of the singularity position of the redundant 48 inparallel manipulator when Oz = 90 ........................................ .................... ... 77 Figure 6.1: Coordinate systems of a redundant 48 inparallel manipulator................. 80 Figure 6.2: Leg relations ............................................................................................. 81 Figure 7.1: A redundant 88 inparallel manipulator ................................... ............ 86 Figure 7.2: Plan view of the redundant 88 inparallel manipulator ............................. 86 Figure 7.3: Plot of f(a, fl) = 2a 2a 2 +1 = 0 ........................................................ 90 Figure 7.4: Plot of h vs. raand f with a = 1 ...................................................................... 92 Figure 7.5: Plot of /det J.J vs. a with a = 1 .................................... ............... .. 93 Figure 7.6: An example of redundant 88 manipulator in optimal configuration............. 93 Figure 7.7: Quality index for platform vertical movement ............................................... 96 Figure 7.8: Reduction of the size of the redundant 88 inparallel manipulator ............... 96 Figure 7.9: Quality index for platform horizontal translation with different values of al00 Figure 7.10: Platform rotations about the y'axis....................................................... 102 Figure 7.11: Quality index for platform rotations about the x' and y'axes................... 105 Figure 7.12: Platform rotations about the zaxis ........................................................... 106 Figure 7.13: Quality index for platform rotation about the zaxis ................................ 108 Figure 7.14: Plan view of the singularity position of redundant 88 inparallel manipulator when 8z = 900................................................................. 109 Figure 8.1: Coordinate systems of a redundant 88 inparallel manipulator.................. 11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy QUALITY INDEX AND KINEMATIC ANALYSIS OF SPATIAL REDUNDANT INPARALLEL MANIPULATORS By Yu Zhang December 2000 Chairman: Dr. Joseph Duffy Major Department: Mechanical Engineering Parallel manipulators have been the subject of much investigation over the last decade because of their inherent advantages of load carrying capacity and spatial rigidity compared to serial manipulators. Usually they have the same number of actuators as their degree of freedom, but in some cases, it may be interesting to have more actuators than needed and to consider redundant parallel manipulators. Redundancy in actuation can be used to increase dexterity, to reduce or even eliminate singularities, to increase reliability, to simplify the forward kinematics, and to improve load distribution in actuators. The purpose of this work is to design and analyze several spatial redundant parallel manipulators. The proposed quality index will assist a designer to choose the relative dimensions of the fixed and moving platforms, locate joint centers in the fixed and moving platforms, determine an optimum position which would be an 'ideal' location of the workspace center, and determine acceptable ranges of pure translations and pure rotations for which the platform is stable. The quality index for redundant parallel manipulators is defined as a dimensionless ratio that takes a maximum value of 1 at a central symmetrical configuration that is shown to correspond to the maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose. The Jacobian matrix is none other than the normalized coordinates of the leg lines. When the manipulator is actuated so that the moving platform departs from its central configuration, the determinant always diminishes, and, as is well known, it becomes zero when a special configuration is reached (the platform then gains one or more uncontrollable freedoms). It is shown that the quality index 2, for which 012< 1, can be used as a constructive measure of not only acceptable and optimum design proportions but also an acceptable operating workspace (in the static stability sense). We also studied the forward kinematic analysis of the redundant inparallel manipulators to determine the position and orientation of the platform, given the leg lengths. CHAPTER 1 INTRODUCTION Parallel manipulators have been studied extensively over the last decade with their high structural stiffness, position accuracy and good dynamic performance. Usually they have the same number of actuators as their degree of freedom, but in some cases, it may be interesting to have more actuators than needed to overcome disadvantages of the nonredundant parallel manipulators shown by Merlet [25]. 1.1 Redundant Parallel Manipulators A number of redundant parallel manipulators have been studied in literature, for example, the development of a directdrive redundant parallel manipulator for haptic displays by Buttolo and Hannaford [2, 3], the design of a 2DoF parallel manipulator (Figure 1.1) with actuation redundancy for high speed and stiffnesscontrolled operation by Kock and Schumacher [17], and the addition of a redundant (fourth) branch to three branch manipulators for the purpose of uncertainty elimination and assembly mode reduction by Notash and Podhorodeski [28]. Maeda et al. [24] also designed a redundant wiredriven parallel manipulator that is suitable especially for high speed assembling of lightweight objects such as semiconductors. By studying a parallel machining center, O'Brien and Wen [29] examined the effectiveness of singularity modification through redundant actuation and suggested that augmenting the actuation of a mechanism provides a mechanically feasible means of increasing kinematic manipulability. Figure 1.1: A planar parallel xy manipulator with one redundant actuator (Adapted from: Kock and Schumacher, A Parallel Xy Manipulator with Actuation Redundancy for High Speed and Active Stiffness Applications (1998) [17]) LeguayDurand and Reboulet [23] studied a redundant spherical parallel manipulator and showed that actuator redundancy removes singularities and improves dexterity in an enlarged workspace. Using a conditioning measure, they compared the redundant spherical parallel manipulator with an equivalent nonredundant structure and found notably improved uniformity of dexterity for the redundant structure. Similar results were also found by Kurtz and Hayward [19]. Kokkinis and Millies [18] found that actuation redundancy allows the selection of optimal joint torque for a given load. Nakamura and Ghodoussi [27] also showed that the redundant actuation could increase the payload and improve the dynamic response of manipulators. Dasgupta and Mruthyunjaya [6, 7] saw the redundancy of parallel manipulators as the seriesparallel dual part of redundancy in serial manipulators. They proposed the concept of force (static) redundancy for redundancy in parallel manipulators in contrast to kinematic redundancy (widely studied in literature) in serial manipulators. In summary, redundant parallel manipulators have the following advantages: 1. Redundancy in actuation can be used to increase dexterity and reduce or even eliminate singularities of parallel manipulators (Perng and Hsiao [30]). Usually, parallel manipulators have a high stiffness, except in some special positions or postures where the platform has selfmotion and may even collapse. These singular configurations may cause serious damage to the manipulator and/or objects in its environment. Redundant legs can be used to pull out the platform from singularity positions. For example, for the 2DoF planar parallel manipulator shown in Figure 1.2, its singularity positions can be found on the line joining the two fixed pivot positions, and this singularity can be eliminated by adding another leg as shown in Figure 1.3. The new redundant parallel manipulator is entirely free from singularity as long as its three fixed pivots are noncollinear. Figure 1.2: A 2DoF planar parallel Figure 1.3: A redundant 2DoF planar manipulator parallel manipulator Here is another example, considering the two cases of spatial nonredundant 44 inparallel manipulators shown in Figure 1.4. They are in singularity positions when their platforms are parallel to the base. Such singularity is dangerous because it is not immediately obvious from its configuration and if we build such manipulators, they may collapse immediately when their platforms are parallel to the base. One possible solution is to add another two legs to form a redundant 44 inparallel manipulator (Chapter 3) (Figure 3.1). SI "G b_ b (a) (b) Figure 1.4: Planar view of spatial nonredundant 44 inparallel manipulators 2. Redundancy in actuation can be used to increase the reliability of inparallel manipulators (Shin and Lee [33]). That is, even if some of the actuators fail, a manipulator can still operate normally as long as the number of operating actuators is not less than the mobility of the manipulator. Thus such a redundant system has a failure tolerance, which is increasingly important in robotics, especially when robots and manipulators are used in remote or harsh environments such as space, deep sea, nuclear plants and for bomb disposal. Because these environments do not allow immediate human intervention for repair or recovery, the ability of a robot or a manipulator to cope with the failures becomes desirable. 3. The information from the length of the redundant legs can be used to simplify the forward kinematics. When controlling a parallel manipulator, we need to do the forward kinematic analysis, i.e., to determine the configuration of the moving platform given all the leg lengths. This analysis is usually difficult as it involves a set of nonlinear equations and, generally, there is more than one solution. For example, the forward analysis for the general 66 parallel manipulator requires the solution of a 40th degree polynomial (Raghavan [31]) the solution of which is clearly impractical for realtime implementation. The additional information from the redundant legs reduces many uncertainty positions and even can obtain a unique solution to the forward analysis. 4. The actuator forces and joint torques in the redundant parallel manipulators are not uniquely determined. This characteristic can be used to optimize some criteria. For example, the joint torque required for a given motion can be minimized. Accordingly, it is possible to increase the payload of a closedlink mechanism by adding redundant actuators. Some other advantages of using redundant actuators are increasing workspace while improving dexterity, having autonomous calibration, and building variable geometry trusses. Possibilities of redundancy in parallel manipulators and their effective use have not been studied extensively until now. The purpose of this work is to design and analyze several spatial redundant parallel manipulators. 1.2 Quality Index Parallel manipulators have better load carrying capacity and spatial rigidity than serial manipulators. However, the complexity of the kinematics of parallel manipulators makes it more difficult for a designer to determine a set of kinematic and geometry parameters that will efficiently produce prescribed performances. Indeed, the behavior of parallel manipulators is far less intuitive than that of serial manipulators. The geometric properties associated with singularities, for example, may be much more difficult to identify directly (Fichter [10] and Merlet [26]). Therefore, more systematic analysis and optimization tools are needed to make parallel manipulators more accessible to designers. At this time little information is available to assist designers in thefollowing task: (a) Choose the relative sizes of the fixed and moving platforms. (b) Locate the positions of the centers of the spherical joints in the base and the centers in the moving platform. (c) Determine an optimum position that would be an ideal 'center' location of the workspace. (d) Determine acceptable ranges of pure translations of the platform for which the platform is stable (i.e., not too close to a singularity). However, the question "How close is too close?" is often hard to answer. (e) Determine acceptable ranges of pure rotations of the platform for which the platform is stable. (f) Determine the ranges of leg displacements. These considerations are the reasons that the quality index was proposed. The quality index was defined initially for a planar 33 inparallel device by the dimensionless ratio (Lee, Duffy, and Keler [22]) Idet J S= det(1.1) IdetJ.I where J is the threebythree Jacobian matrix of the normalized coordinates of three leg lines. Then it was defined for an octahedral inparallel manipulator by Lee, Duffy, and Hunt [21] and 36, 66 inparallel devices by Lee and Duffy [20]. For these cases J is the sixbysix matrix of the normalized coordinates of the six leg lines. For these fully symmetrical nonredundant parallel manipulators the quality index takes a maximum value of A = 1 at a central symmetrical configuration that corresponds to the maximum value of the determinant of the sixbysix Jacobian matrix (i.e., det J = det Jm) of the manipulator. When the manipulator is actuated so that the moving platform departs from its central configuration, the determinant always diminishes, and, as is well known, it becomes zero when a special configuration is reached (the platform then gains one or more uncontrollable freedoms). In this dissertation, the quality index is extended for redundant manipulators by the dimensionless ratio detJJT (1.2) This makes complete sense because the CauchyBinet theorem detJJT =LA +A, +* .+A , has geometrical meaning. Here, each A (1 i m = () is simply the determinant of the 6x6 submatrices of J which is a 6xn matrix. This is clear when n = 6, (1.2) reduces to (1.1). It has been shown by Lee et al. [21] that by using the GrassmannCayley algebra (White and Whiteley [35]), for a general octahedron, when the leg lengths are not normalized, detJ has dimension of (volume)3 and it is directly related to the products of volumes of tetrahedra that form the octahedron. In this way detJ and VdetJJT have geometrical meaning. We mention in passing the work of Cox [4] and Duffy [8], both of which cover special configurations of planar motion platforms. Hunt and McAree [14] go into considerable detail regarding the general octahedral manipulator. Its special configurations are described in the context of other geometrical properties. A few papers were published on the optimal design of nonredundant parallel manipulations (see for example Gosselin and Angeles [11, 12], Zanganeh and Angeles [36]). Zanganeh and Angeles [36] point out problems with quantities such as condition number due to the inherent inhomogeneity of the columns of the Jacobian, J. This is precisely why equations (1.1) and (1.2) are adopted as an index of quality rather than other wellestablished methods (found in books on theory of matrices and linear algebra) that lead (via norms, diagonalization and singular values, etc.) to properties that relate to 'conditioning'. All such methods are based implicitly on the presumption that a column vector (say, of a sixbysix matrix) can be treated as a vector in 96. However, the six elements in the column of a typical robot Jacobian are the normalized coordinates of a screw (almost always of zero pitch; i.e., a line); in a metrical coordinate frame three of them are dimensionless and three have dimension [length], such a length being the measure of the moment about a reference point of a unit force. The column generally comprises two distinct vectors (each of them in 9%3). For the legs of the nonredundant and redundant manipulators it is not possible to remove all the length dimensions from their coordinates. Even the adoption of some artificial length unit fails, simply because a moment can never be converted to a pure force. Moreover, any index of quality derived from such textbook techniques is likely to vary according to the coordinate frame in which the Jacobian is formulated. Our method works for two reasons: first, the determinant of a (square) Jacobian of line coordinates depends solely on the configuration in 9t3 of the actuated axes and not on the coordinate frame in which the line coordinates are determined. The second reason is that equations (1.1) and (1.2) are dimensionless ratios, and our quality indices are always independent of the choice of units of length measurement. Unlike the case of a mechanism designed for a specific task, the tasks to be performed by a manipulator are varied. Hence, there should not be any preferred general orientation for which the manipulator would have better properties. It suggests that the manipulator should be symmetrical. Such symmetrical configurations may not always exist, of course. However, except for unusual applications (and there will undoubtedly be some where for example unusual loads must be sustained) we are safe in seeking centrally symmetrical designs to which we can assign the highest quality index A= 1, or close to it. For these cases, contours of quality index help to determine a realistic workspace volume that is free from singularities. Therefore, we are concerned primarily with symmetrical redundant parallel manipulators in this dissertation. 1.3 Outline of Dissertation A simple introduction to the screw theory is presented in Chapter 2 to provide insight into how a screwbased Jacobian matrix of a parallel manipulator is determined. In Chapters 3 and 4 a spatial redundant 44 inparallel manipulator is studied first. The device consists of a square platform and a square base connected by eight actuated legs. As in Chapter 3, the quality index for the redundant 44 parallel manipulator is determined. To achieve the maximum quality index for a redundant 44 inparallel manipulator with platform side a, the base has side Za and the perpendicular distance between the platform and the base is a The kinematic analysis of the redundant 44 inparallel manipulator is studied in Chapter 4. The derivation of forward kinematic equations for position and orientation of the platform is described. Chapters 5 and 6 extend the study to a redundant 48 inparallel manipulator with a square platform and an octagonal base. The octagonal base is formed by separating from each vertex of a square by a small distance. The quality index for this manipulator is determined in Chapter 5. The compatibility between the redundant 44 and the 48 parallel manipulators also is discussed in this Chapter. Chapter 6 solves the forward kinematics of the redundant 48 parallel manipulator by transferring the problem to the corresponding redundant 44 case. Finally, in Chapters 7 and 8, a redundant 88 inparallel manipulator is studied. The device has an octagonal platform and a similar octagonal base connected by eight legs. Such arrangement avoids using doublespherical joints because they can produce serious mechanical interference. However, by using the quality index determined in Chapter 7, the best design can be obtained when the pair of separated joints in the base and top platform are as close as possible. In Chapter 8, the kinematic analysis of the redundant 88 parallel manipulator is performed. The forward analysis gives a much simpler solution than that of the nonredundant case. Using quality index, variable motions are investigated for which a moving platform rotates about a central axis or moves parallel to the base. The quality index can be used as a constructive measure not only of acceptable and optimum design proportions but also of an acceptable operating workspace (in the static stability sense). Moreover, analysis of these redundant inparallel manipulators can be used to model and design a selfdeployable space structure that has a pair of flexible antenna platforms in the base and top platform as shown in Figure 1.5 (Duffy et al. [9] and Knight et al. [16]). I> Figure 1.5: Selfdeployable space structure CHAPTER 2 SPATIAL GEOMETRY AND STATICS Chapter 1 showed that the quality index of parallel manipulators is based on the Jacobian matrix. This chapter, which is mostly a general background in screw theory (Ball [1]), provides insight into how the Jacobian matrix of parallel manipulators is determined. Firstly, we review some basic concepts of spatial geometry and screw theory. 2.1 Plucker Line Coordinates Two distinct points ri(xl, Yi, zi) and r2(x2, Y2, Z2) can be connected by a line in space. The vector S whose direction is along the line can be written in the form S = r2 r. (2.1) Alternatively this may be expressed as S = Li + Mj + Nk (2.2) where L=x2xI, M=y2yi, N=z2Z1 (2.3) are defined as the direction ratios of the line and they are related to the distance ISI between the two points by L2 +M2 + N2 = IS12 (2.4) where the notation 1I denotes absolute magnitude. Often L, M, and N are expressed in the form X2 X1 Y2 yl Z2 Z1 L = 2N M=Y2., N=, (2.5) ISI ISI ISI which consists of unit direction ratios of the line, and (2.4) reduces to L2 +M2 +N2 = 1. (2.6) If r represents a vector from the origin to any general point on the line (Figure 2.1), then the vector rr1 is parallel to S and therefore the equation of the line can be written as (r ri) x S = 0 (2.7) and in the form r x S = So (2.8) where So = ri x S (2.9) is the moment of the line about origin O and is clearly origin dependent. Further, because So=rlXS, the vectors S and So are perpendicular and as such satisfy the orthogonality condition SSo = 0. (2.10) Figure 2.1: Determination of a line The coordinates of a line are written as [S; So]' and are referred to as the Pliicker coordinates of the line [13]. The coordinates [S; So] are homogeneous since from (2.8) the coordinates [kS; kSo] (k is a nonzero scalar) determine the same line. Expanding (2.9) yields i jk So = xi y, zl (2.11) LM N which can be expressed in the form So = Pi + Qj + Rk (2.12) where P= yN zM, Q= zL xN, (2.13) R= xM y1L. From (2.2) and (2.12) the orthogonality condition SSo0 can be expressed in the form LP + MQ + NR= 0. (2.14) The Pliicker coordinates of the line [S; So] now can be written in terms of their components as [L, M, N; P, Q, R], which are known as the ray coordinates for a line (Figure 2.2). Unitized coordinates for a line can be obtained by imposing the constraint that ISI=1. The Pliicker coordinates thus must satisfy equations (2.6) and (2.14) and hence only four of the six scalars L, M, N, P, Q, and R are independent. It follows that there are 4 2 04 lines in space2. SThe semicolon is introduced to signify that the dimension of ISI is different from ISol. 2 Systems of lines and their properties, o' (line series), 02 (congruence), _3 (complex), are described by Hunt [13] which contains an extensive bibliography on the subject. R R S YKN ^'L Zl y Figure 2.2: PlUcker line coordinates A straightforward method to obtain the Plucker coordinates was given by Grassmann (Hunt [13]) by expressing the coordinates of the points rl(xi, yi, zl) and r2(x2, Y2, Z2) in the array X1 Yl Zl 1 x, y, z, x1 2 Y2 Z2 and by expanding the sequence of 2x2 determinants 1 x, 1= z y2 2 x, Y2 Z2 Z2 X2 P= ,z Q z X, 1 z2 N= , 1 z2 R= x1 Yi x2 Y2 2.2 Statics of a Rigid Body The concepts developed in the previous section now can be applied directly to the statics of a rigid body. A line $ with ray coordinates [S; So] (where ISI = 1) can be used to (2.15) (2.16) express the action of a force upon a body (Figure 2.3). Because the body is rigid, the point of application can be moved anywhere along the line. Z f=f r Figure 2.3: Representation of a force on a rigid body As illustrated by Figure 2.3, a force f can be expressed as a scalar multiple JS of the unit vector S that is bound to the line $. The moment of the force f about a reference point O is mo which can be written as mo = rxf where r is a vector to any point on the line $. This moment can also be expressed as a scalar multiple fSo where So is the moment vector of the line $ (i.e., So = rxS). The action of the force upon the body thus can be expressed elegantly as a scalar multiple f$ of the unit line vector, and the coordinates for the force are given by f$ =f[S; So = [f; mo] (2.17) where SS=l and SSo=0. Clearly, when the reference point O is coincident with A, then mo=0 and the coordinates of the force are [f; 0]. Therefore, f is a line bound vector that is invariant with a change of coordinate systems while mo is origin dependent. An important special case is [0; mo] which can be considered as the resultant of a pair of equal and opposite forces with coordinates [f; mo1] =f[S; Sol] and [f; m02] =f[S; S02], where ISI=1. The coordinates of the resultant [0; ma]= [0; mol+m2] =f[0; Sol+S02] are not a line bound vector, but a pure couple. The couple can be considered as equivalent to a force of of infinitesimal magnitude (16Sf+0) acting along a line that is parallel to the lines of action of the pair of parallel forces. The line of action of of is infinitely distant with coordinates [0; mo], such that Ipl= o where p is the vector from the origin perpendicular to the line of action of sf, and the moment of the force 8f about the origin is pxSf=mo. A pure couple thus can be represented as a scalar multiple of a line at infinity. The problem of determining the resultant of an arbitrary system of forces with coordinates [fl; moI], [f2; mo2], ..., [fn; mon] acting on a rigid body is essentially the determination of the quantity ^v = [f ; mo], (2.18) where n n f=If, and mo= m,. (2.19) i=1 i=i It is assumed at the outset that a reference point 0 was chosen so that the forces acting on the rigid body were translated to point 0 and so that moments mol, mo2, ..., mon were introduced to yield an equivalent system of forces and torques that act on the rigid body. Therefore, the line of action of the resultant force f passes through point O and the resultant moment mo is a couple [0; me]. In general f and mo are not perpendicular (i.e., fmoe0). The new quantity with coordinates ^v' = [f; mo] therefore is not a force and was defined as a dyname by Plicker. Because in general f mo # 0, it is not possible to translate the line of action of force f through some point other than point O and to have the translated force produce the same net effect on the rigid body as the original dyname. The moment mo, however, can be resolved into two components, ma and mt, which are respectively parallel and perpendicular to f (Figure 2.4a) and mo = ma + mt. (2.20) The moment ma can be determined as m. = (mo S) S (2.21) where S is a unit vector in the direction of the resultant force f. The moment mt is then determined as mt = mo ma. (2.22) The line of action of force f now can be translated so that the force with coordinates [f; mt] plus the moment [0; a] (Figure 2.4b) is equivalent to the dyname [f;mo]. Therefore, the dyname is represented uniquely by a force f acting on the line [S; Sot] mi (where Sot= ) and a parallel couple ma. This parallel forcecouple combination was f called a wrench by Ball [1]. From (2.18) and (2.20), the wrench which is equivalent to the dyname [f; mo] can be expressed in the form 19 ^v = [f; mo] = [f; mt + ma] = [f; mnt + [0 ; ma]. Clearly, [f; mt] is a pure force because fmt=0. 9 1A ma r rxf=mt y x (a) Dyname, [f; mo] (b) Wrench, [f; mt] + [0; na] Figure 2.4: Dyname and wrench Further, because ma is parallel to f, then ma = hf where h is a nonzero scalar which is called the pitch of the wrench. From (2.20) f mo = f ma (2.24) (2.25) and from (2.24) and (2.25), the pitch h is given by h= f m= f m f f f f (2.26) Substituting (2.24) into (2.23), together with (2.22) allows the wrench 1v to be written as wV = [f ; mo hf] + [0; hfJ. (2.23) (2.27) Thus, the coordinates for the line of action of the wrench are [f; mohf] and from (2.27) the equation for the line is r x f = mo hf. (2.28) In the same way as the action of a force can be expressed as a scalar multiple of a unit line vector, a wrench can be expressed elegantly as a scalar multiple of a unit screw $ where $ = [S ; Sol (2.29) and where S.S=1. From (2.26), the pitch of the screw is given by h = S So. (2.30) Further, from (2.2) and (2.12), h =LP + MQ + NR. (2.31) Therefore, Ball [1] defined a screw as "a line with an associated pitch". Following (2.28), the Plicker coordinates for the screw axis are [S; SohS] and the equation of the axis is r x S = So hS. (2.32) 2.3 The Statics of a Parallel Manipulator Figure 2.5 illustrates a nonredundant 663 inparallel manipulator. The device has a moving platform and a fixed base connecting by six legs each of which is the same kinematic chain. The prismatic joint in each leg is actuated and the moving platform has six degrees of freedom. 3 These numbers indicate the number of connecting points in the top and base platform respectively. $5 moving $1 6 $ platform base Figure 2.5: A 66 inparallel manipulator Consider that the six leg forces with magnitudes fi,f2, ...,f6 are generated in each of the lines $1, $2, ..., $6. The resultant wrench ^ =[f; mo] acting upon the moving platform due to these six leg forces is given by *v = [fi; moi] + [f2; mo2] + ... + [f6; mos], (2.33) or in the alternative form, wv =f [S1; Soil +f2[S2; S02] + ... +f6[S6; Soi] (2.34) where [Si; Soil (ISil=l, i=1...6) are the Piicker line coordinates of the six legs. Further, (2.34) can be expressed in the matrix form i = JF (2.35) where = [f, m,] and F = [fi, /f, f4, f5, f6, f are 6x1 column vector. J is a 6x6 matrix of line coordinates given as S, S, S3 S4 Ss S6 J = (2.36) .S01 So2 S03 S04 S05 S06 and is called the Jacobian matrix, or simply Jacobian, which enables us to determine the resultant wrench *^ = [f; mo] produced by six actuator forces generated in the legs. It should be noted that for redundant parallel manipulators the Jacobian matrix is not square. The transpose of the Jacobian matrix relates the infinitesimal displacements 81i in each leg to the infinitesimal displacement twist4 of the platform and 61 = JTSD (2.37) where 1 =[ [81, 312,..., S16]T and 8D = [x, Sy, 8z; &P, 8(Py, ypz ]T. Here, 8x, Sy, and 8z are the infinitesimal displacement of a point in the moving platform coincident with a reference point O which is chosen to be the origin of a fixed coordinate system on the fixed base. The quantities 8x, Spy, and 8iz are infinitesimal rotations of the moving platform about the axes of the fixed reference coordinate system. In summary, the Jacobian matrix of parallel manipulators serves two distinct purposes. In its ordinary form the columns of which are the coordinates of the actuator lines (normalized), it enables us to obtain from actuated force inputs the wrench at the end effector platform. In its transposed form the Jacobian can give the relative speeds required at each actuator that corresponds to a given twist to be executed by the platform. The first of these gives the instantaneous solution to a problem of static equilibrium; the second, the solution of first order kinematic compatibility. When the Jacobian matrix is 4 An infinitesimal twist is also a scalar multiple of a unit screw, as the scalar is an infinitesimal rotation with unit of radian. 23 singular (i.e., its rank is less than six) the actuators (i) cannot equilibrate a general wrench applied to the platform and (ii) cannot on their own prevent a transitory uncontrollable movement of the platform. This latter phenomenon is associated with the platform's gaining one or more freedoms when all the actuators are locked. The platform is then in a singularity position. CHAPTER 3 THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 44 INPARALLEL MANIPULATOR A spatial redundant 44 inparallel manipulator is shown in Figure 3.1. The device has a square platform and a square base connected by eight legs. Figure 3.2 shows a plan view of this manipulator, where the moving platform is symbolically represented by four pairs of concentric spherical joints located at points A, B, C, and D, and the fixed base is represented by another four pairs of concentric spherical joints located at points E, F, G, and H. This manipulator is said to be redundant since the platform and the base are connected by eight actuated legs. Figure 3.1: A redundant 44 inparallel manipulator Figure 3.2: Plan view of the redundant 44 inparallel manipulator 3.1 Determination of detJ.JJ. The moving platform of the redundant 44 parallel manipulator shown in Figure 3.2 is located at its central symmetrical configuration and is parallel to the base with a distance h. At this configuration, the manipulator is fully symmetric and each leg has the same length. Clearly, at such position the platform is most stable from the geometric static point of view. When the platform departs from this central symmetric position, the platform will lose its geometric symmetry and the eight leg lengths will be different. Therefore, it is reasonable to assume that at the central symmetric configuration shown in Figure 3.2, it is possible to determine the values of square base side b and height h based on square platform side a so that a maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose, i.e., detJ.J. may be obtained. Firstly, the coordinates of the points A, B, C, and D on the platform and E, F, G, and H on the base are determined with the origin of a fixed coordinate system placed at the center of the square base as shown in Figure 3.2, and A a h B 0 h} CO0 O a h), D ( a 0 h 2 2 2 2 (3.1) E b 0 o, F(b b 0, G 0 H o b o0. (2 2 2 2 2 2 2 2 Then, using the Grassmann method described in Chapter 2 to calculate the Pliicker line coordinates of the eight leg lines, i.e., counting the 2x2 determinants of the various arrays of the joins of the pairs of points EA, FA, FB, GB, GC, HC, HD, and ED. For example, the coordinates of the line $1 are obtained using the coordinates of points E and A in (2.15) to form the array b 2 1 0 b 2 2a 2 and using (2.16) to yield V2a b bh h; 2 2 i h bh 42ab 2 4 Similarly from points F and A, Sr = b V2ab 2 2 bh bh F2ab h; 2 4 2 2 4 From points F and B, S3 b 2 b bh 2 h; 2~ 2 F2ab 4 From points G and B, S4 Va 2 b 2' bh bh h; 2 2 2 From points G and C, S5 = 2 2 Va b 2 h; 2 bh bh 2 2 From points H and C, s, b 2 V2ab 2 bh bh 2ab 2 2' 4 From points H and D, (3.2) (3.3) (3.4) bh 2 (3.5) 4Vab 4 (3.6) 2ab] 4 (3.7) (3.8) S b 2 [ ab b 7=[2 2' hbh bh ijab 2 2 4 (3.9) From points E and D, Sbh h; 2 bh FJab] 2 4 (3.10) It should be noted that the above Plicker line coordinates are not normalized and each line must be divided by 1, = S,I (i = 1, 2, ..., 8). Hence, the normalized Jacobian matrix of the eight leg lines (now all reduced to unit length) can be expressed as 1 3T T T T T 17 8 1= 12 13 14 15 16 17 18 (3.11) Since the device is in a symmetrical position, the normalization divisor is the same for each leg, namely li = (i = 1, 2, ..., 8), and for every leg = L2 +M2+N2 = a2 2ab+b2+2h2). From (3.3) to (3.10), the Jacobian matrix in (3.11) becomes b d, di 2 d, di b d, di 2 2 h h h h h h h h (3.12) bh 2 bh 2 bh bh 2 2 rab 2jab 4 4 bh 2 42ab bh 2 v2ab 4 4 bh bh 2 2 j2ab /jab 4 4 J I 1 bh 2 bh 2 ,2ab 4 ii (3.13) bh 2 bh 2 42ab 4 Ss = ,ab b 2 2 28 where d 2a b d, = 2 Using equation (3.13), the determinant of the product J J T can be expressed in the form d2 0 0 0 d3 0 0 d2 0 d3 0 0 1 0 0 8h2 0 0 0 detJJ 2(3.14) d3 0 0 0 2b2h2 0 0 0 0 0 0 a2b2 where d2 =2(a 2 2ab + b2), d3 = (a2a2b)bh. Expanding (3.14) and using (3.12), then extracting the square root yields / v 32ia'b'hi Vdet JJ = 322a. (3.15) (a2 ab +b + 2h2 Y Assuming the top platform size a is given, now taking the partial derivative of (3.15) with respect to h and b respectively and equating to zero yield 96F2 a3b3h2(a2 aab+b2 2h2) (3.16) (a2 ab+b2+ 2h2)4 and 962a3b2h3(a2 b2 +2h2) (3.17) (a2 Fab+b2 +2h)4 7) When a, b, and h are not equal to zero, equations (3.16) and (3.17) give a2 4ab+b22h2 =0, (3.18) a2 b2 + 2h2 =0. (3.19) Adding (3.18) and (3.19) yields 2a2 2ab=0. (3.20) Solving the above equation, we obtain b = 2a (3.21) Further, substituting (3.21) into equation (3.19) yields a2 2h2 =0. (3.22) There are two solutions for h in the above equation, here we only take the positive solution (the negative solution is simply a reflection through the base) h= (3.23) Finally, substituting (3.21) and (3.23) into (3.15) we get vdetJ,Jt = (det J ), = 4Va3 (3.24) where Jm denotes the Jacobian matrix for the configuration at which the 44 redundant parallel manipulator has a maximum quality index. This optimum configuration is shown in Figure 3.3. Figure 3.3: Plan view of the optimal configuration of the redundant 44 inparallel manipulator with the maximum quality index 3.2 Implementation From the definition of quality index (see (1.2)) and (3.24), the quality index for the redundant 44 parallel manipulator shown in Figure 3.1 becomes dettjj AVa= (3.25) The variation of the quality index now is investigated for a number of simple motions of the top platform. Here, an optimal redundant 44 parallel manipulator with platform side a = 1, and thus base side b = 2 is taken as an example. First, consider a pure vertical translation of the platform from the central symmetric position shown in Figure 3.2 along the zaxis while remaining parallel to the base. For such movement, from (3.15) and (3.25), the quality index is given by 31 8b3h3 8b 3 h (3.26) (a2 2.Iab 2 + 2h2 With a = 1 and b = /2, this reduces to 16/2h3 S= 2h (3.27) (1+ 2h 2)3 a and is plotted in Figure 3.4 as a function of h. It shows that at height h = = the T2 2 quality index of the redundant 44 parallel manipulator has a maximum value, A = 1. 1.0.......... 0 0.0zniii 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Height h Figure 3.4: Quality index for platform vertical movement We now derive an expression for detJJT when the platform of the redundant 44 parallel manipulator is translated away from its central location while remaining parallel to the base at height h. Assume the center of the moving platform to move to point (x, y, h), then the coordinates of the points A, B, C, and D on the platform become Ax y h, 2 C x y+ a h\ 2 B +DiF y h , D x1 y h. 2 The coordinates of points E, F, G, and H on the base can be found from (3.1). Thus, the Plicker line coordinates for each of the eight leg lines can be determined as Sb .V2ab ,= x+, y 2 , 2 2 b J2ab 2 = x y a b 2 2 S3=[x,2a ab +b S54= x+ , y b S [ b i2ab S =x, y+ 2 2 2' S b 42ab S= x+a, y+ , 7 = 2ab b = x +, y + , 2 2' $ = ab b =x2 y +2' bh bh b(2x 2y+ V2a) 2' 2' 4 J' h; bh bh b(2x + 2y 2a) 2' 2' 4 bh bh b(2x+2y+J2a) 2' 2' 4 bh bh b(2x 2y+ ia) 2' 2' 4 Sbh bh b(2x 2y f2a) 2' 2' 4 Sbh bh b(2x+2y + 2a)] 2 2' 4 Sbh bh b(2x + 2y a)1 2' 2 4 bh bh b(2x 2y ]2a) Sbh bh b(2x2y4 2a) 2 2 4 (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) The above coordinates are not normalized and each row must be divided by the corresponding leg length. The Jacobian matrix J then can be constructed by using (3.11). Further, substituting and expanding VdetJJr yields det J =a3b3h32(1l +12 +I2 +l 2)(l +142 +12 +12) S1112131415161718 (3.37) where the leg lengths are I b 2 b 2 2, 11 = x+ + y +h 2, , = x + y a b +h 2, 7 ab 2 b 2 = x + y +h 2 2 1If ,b2F2a b I 2, 2 2 S2ab2 14 = X+ + y +h2, 18= x b+ y+ +h 2 l2 f 8=x + y+ +h' 2 4 +2. vJ With a = 1, b = 2, and h = , from (3.25) and (3.37), the quality index A becomes 2 S detJJT x2 +y2 +1 det jT (x4+2xy2+y4 + 22+1) (4+2x2y24 +2y2+1) (3.38) which is plotted in Figure 3.5(a) as a function of x and y. Figure 3.5(b) shows the contours of the quality index for this platform horizontal movement. The contours are labeled with values of constant quality index and they are close to being concentric circles of various radii. When x ory is infinite, A=0, and when x=y=0, A= 1. 1 1 1.0 0.5 0.0 0.5 1.0 (a) (b) Figure 3.5: Quality index for platform horizontal translation To illustrate the variation of the quality index during some simple rotations of the platform, a new coordinate system x'y'z' is attached to the square platform. The origin of the new coordinate system is located at the center of the top platform and the coordinate system is oriented such that its x'axis is passing through vertex B, its y'axis then is passing vertex C, and its z'axis is normal to the square platform ABCD. Thus, when the platform locates at its initial central position shown in Figure 3.2, the x' and y'axes are parallel to the x and yaxes on the base respectively. Figure 3.6 illustrates a side view of the redundant 44 parallel manipulator when the square platform ABCD is rotated by an angle 0y about the y'axis from its initial position. For such platform rotation, the coordinates of the vertices A, B, C, and D become Af0 2a h BF cosO 0 h+ sinO , 2 2 I 2 C 0 V2a h D acosoy 0 h sin m 2 2 2 \i /~ \1 /(~cs, o _~ i (3.39) Figure 3.6: Platform rotations about the y'axis It should be noted that the positions of line $1, $2, $5, and $6 do not change during this movement and their corresponding Plicker line coordinates can be obtained from (3.3), (3.4), (3.7), and (3.8) respectively. The Plicker coordinates for the line $3, $4, $7, and $s are now given by S, _2acosb, b b asiny +2h 2 2' 2 (3.40) b(j2a sin O + 2h) b(12a sin 9O + 2h) F2abcos OY 4 4 4 [2acos0, b S4 = 2 b(Jfa sin O, + 2h) 4 7 4/acosO b 2 b(r2a sin 9, 2h) 4 8 2acos9 b 2 b([2a sin 09 2h) 4 b 2asin O +2h 2 2 b(r2a sin Y + 2h) 4 b 2 (3.41) V2abcos , 4 J2a sin y 2h Y l b(2a sin 0Y 2h) 4 b 2' (3.42) F2ab cos 0, 4 V/2asin 0 2h 2 b(f2a sin 9, 2h) 4 V2ab cos O0 4 Since the configuration of the manipulator keeps symmetric about the xaxis during the platform rotation about the y'axis, from Figure 3.6, we have 1 = l2 = 15 = 16 = = 4 and 1, = 1g. The corresponding Jacobian matrix then can be determined by (3.11), and further detJJ" becomes det JJ = N2ab (((4h2 +(2a2 22ab+b2)sin20 )(12 +12) 41 12 3 4h(2a b)(l2 12) sin ,O + 2(4h2 cos2 O, + b2 sin2 6 )12) (h2(4h2 cos2 y +b2 sin2 0y)(l2 +l) 4bh3(l2 12)cosO, sinOy + ((4a2h2 cos2 0y + a2b2sin 20 + 2b2h2 8N2abh2 cosOr)sin2 O + 8h4 cosO ,)12)) (3.44) where (3.43) 37 13= ]b(bacos b +b'+(2 asin, + 2h)2 1, = ia cos9y b) + b2 + (,a sin 2h)2, and I can be found from (3.12). With a =1, b = 2, and h= from (3.25) and (3.44), the quality index 2 becomes /detJJT J (3cosO,)(6cos30 cos20, 7cosy +4) V(3A =(3.45) /det J. J 2(2cos2 O, 4cosOy +3) Since the redundant 44 inparallel manipulator is fully symmetric at its central configuration shown in Figure 3.2, the same result can be obtained when the platform is rotated about the x'axis. From (3.45), the variation of the quality index for rotations about the x' and y' axes is drawn in Figure 3.7. 1.0 .. .....I.......... . . ...... ................ I 0.6 . ........ .......... ........ ........... . 0.2 0,0 .i0i i 90 60 30 0 30 60 90 Rotation Angle 0 (degree) Figure 3.7: Quality index for platform rotations about the x' and y'axes Figure 3.8 illustrates a plan view of the redundant 44 parallel manipulator with the moving platform ABCD rotated 6a about the zaxis. The x and y coordinates of the vertices A, B, C, and D then become xA = rsin0., YA =rcosOZ, Xg =rcos90, yB = rsin 0, (3.46) xc =rsin0z, Yc = rcosO, XD = rcos z, Yo = r sin 0z a where r= From (3.46) XA + XB + xC + XD = 0, (3.47) YA + Y + Yc + YD =0. The complete set of coordinates of points A, B, C, and D are therefore A(xA YA h), B(x, YB h), C(xc yc h), D(xD Y, h) (3.48) where h is the height of the moving square ABCD above the base square EFGH. Yi I Figure 3.8: Platform rotations about the zaxis Then the coordinates for the corresponding lines $1, $2, ..., $8 are given by S= [X + b, s,6 . + [ b 2 b 92 r A b 2 9= xc +, S b 2 7 = xD + , S 2' 9, = x, +b, =[ b 2 b YA + 2, 2 b YA + b Ys + 2 b YB 2' b b Yc 2 b Yo + 2 Sbh h; 2 bh h; 2 bh h; 2 2 bh b(xA Y)] 2 2 bh 2' bh 2 ' bh bh h; 2" 2 bh h; 2 bh 2 b(x, + y ) 2 J b(x, + y ) 2 J b(x y)] b(xc yc) 2 Sbh bh b(xc + Yc) 2 2 2 bh h; ~ 2 bh b(x, + y,) 2 2 J bh bh b(x, y,) 2 2 2 It is apparent from Figure 3.8 that =13 = 15 = 1, and 12 =4 =16 = The corresponding Jacobian matrix can be determined by (3.11). Furthermore, calculating detJJT yields 42a3'b3h'Icos90, detJJ'= a3b1 1 2 (3.57) where (3.49) (3.50) (3.51) (3.52) (3.53) (3.54) (3.55) (3.56) 40 = 2asin0 +b + fV2acos b 2 ,2 I = +h,2 2 2 a I sin0, b r2 2acos9 b 2 12 ..2 2 + h . J2 With a = 1, b = N2, and h = from (3.25) and (3.57), the quality index becomes 2 SJdet JT Icos0 (3 Vdet J mJ (2cos2 0o 4cosO, +3) This is plotted in Figure 3.9 and it shows how the quality index varies as the platform is rotated about the vertical zaxis through its center. The eight legs are adjusted in length to keep the platform parallel to the base at a distance h. It is shown in the figure that the manipulator has the highest quality index A = 1 when 0~ = 0', and A = 0 (singularity) when 0= 90. 1.0 0.8 .. 0.6 .  0.4 ........ ......... ............. .... ..... 0.2  0.0 iI 90 60 30 0 30 60 90 Rotation Angle 0 (degree) Figure 3.9: Quality index for platform rotation about the zaxis It is interesting to note that the redundant 44 parallel manipulator shown in Figure 3.1 always becomes singular when its platform rotates z=900 about the zaxis from its central symmetric position. This can be seen from (3.57), Jdet JJ = 0 when 60 = 900. Figure 3.10 illustrates the singularity position of the redundant 44 parallel manipulator when a6 = 900. It is not immediately obvious from the figure why the eight connecting legs are in a singularity position. This kind of singularity has been discussed in detail by Hunt and McAree [14]. They explain that at such position, even when all eight leg actuators are locked, the connectivity between the base and moving platform is one. The moving platform can move instantaneously on a screw reciprocal5 to the eight leg forces on the zaxis with pitch hz, i.e., a screw with coordinates [0, 0, 1; 0, 0, he]. (3.59) y H G Figure 3.10: Plan view of the singularity position of the redundant 44 inparallel manipulator when Oz = 900 s When a wrench acts on a rigid body in such a way that it produces no work while the body is undergoing an infinitesimal twist, the two screws are said to be reciprocal. Now when az=90, from (3.49) through (3.56), the component of moments about the zaxis for each of the eight legs are all the same Viab N, =h and R, 2 = (i = 1, 2,...,8). (3.60) 4 The coordinates for the eight legs become = L, M,, h; Pi, Q,, (3.61) Hence, from (3.59) and (3.61), Nliab Jiab hhh =0 or hz =T (3.62) 4 4h It follows that all eight legs lie on a linear complex, which is a threeparameter system of linearly dependent lines (Hunt [13]). CHAPTER 4 THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 44 INPARALLEL MANIPULATOR The kinematic analysis of inparallel manipulators deals with the study of the platform motion determined by the leg displacements. Two problems can be distinguished for the kinematic aspects: inverse kinematics and forward kinematics. The inverse kinematics problem, i.e., finding the leg lengths for a given location (position and orientation) of the mobile platform (a difficult problem for serial manipulators), is straightforward for parallel manipulators. On the other hand, the forward kinematics problem, i.e., finding the platform location from a given set of leg lengths, is much more difficult. In general, this problem has more than one solution for nonredundant cases. As an example, the forward analysis for the general 66 platform requires the solution of a 40th degree polynomial (Raghavan [31]), the solution of which is clearly impractical for realtime implementation. A lot of methods have been presented to solve various types of nonredundant parallel manipulators as summarized by Dasgupta and Mruthyunjaya [7]. However, few works have been done on the kinematic analysis of redundant parallel manipulators, particularly the forward kinematics. A similar problem for determining a unique position and orientation of the platform of a general geometry parallel manipulator by using seven linear transducers has been solved by Innocenti [15]. He constructed a 146x147 constant matrix for solving the problem, which makes the computation time still larger than real time. Also, Innocenti's method produces only one solution for a general geometry parallel manipulator, but when the fixed base of a manipulator is in a plane, there should be two solutions instead of one since the manipulator can have two reflection configurations through the base according to the same set of readings from linear transducers no matter how many transducers are used. Therefore, his method may not be correct to solve the problem when the base of a parallel manipulator is in a plane. Zhang, Crane and Duffy [37] have performed the forward kinematic analysis on a special redundant parallel mechanism whose platform and base are connected by a ball andsocket joint with four legs to determine the orientation of the platform. In this chapter the kinematic analysis of the redundant 44 inparallel manipulator shown in Figure 3.1 is performed. From here on, the notations from Crane and Duffy [5] are used to describe the coordinates of point and the transformation matrix. For example, the notation 'PA is used to indicate the coordinates of a point A as measured in a coordinate system I and 2R is defined as the orientation of the coordinate system 2 relative to the coordinate system 1. 4.1 Inverse Kinematic Analysis The objective of the inverse kinematic analysis of the redundant 44 parallel manipulator is to find the eight leg lengths for a given position and orientation of the square moving platform. In Figure 4.1, coordinate systems 1 and 2 have been attached to the fixed base and the moving platform respectively. The origins of the coordinate systems 1 and 2 are located at points E and A receptively. The coordinates of points E, F, G, and H on the base are known in terms of the coordinate system 1 and 0 b b 0 PE = 0 'P = 0, P = b, 'PH = b, (4.1) while the coordinates of points A, B, C, and D on the platform are known in terms of the coordinate system 2 and 0 a a 0 = (4.2) 2 A = 0 2 PPB = 0 2C = a 2 D = a (4.2) Figure 4.1: Coordinate systems of a redundant 44 inparallel manipulator It is well known that the location of a rigid body in space can be described by the position and orientation of a coordinate system attached to the body with respect to a fixed reference frame. Thus, for the inverse analysis, the position and orientation of the coordinate system 2 is considered known, and can be given by the position vector 'PA, which defines the position of the origin A of the system 2 relative to the origin E of the fixed frame 1, and a rotation matrix 2R, which is a 3x3 matrix whose columns are the unit vectors along the coordinate axes of the system 2 as measured in the system 1. Then the coordinates of points B, C, and D in terms of the coordinate system 1 become 'PB = 'PA + 2R2pB, 'Pc =PA + A R2Pc, (4.3) 'PD= 'PA+ R2 PD. Finally, the eight leg lengths can be calculated by 11 =('PA'PE)('PA'PE) 12 = PA'PF'PA'PF) 13 =('PB'PF 'PB' PF) 14 = ('PBPG ) PB'PG) (4.4) 15 =( Pc' P) 'Pc'P). 16 =('Pc'PH 'Pc'PH), 17 =('PD'PH (PD'PH) e e (' PDuct of two vects. where represents the dot product of two vectors. Hence, for a given location of the moving platform, there is only one possible solution for each leg length. 4.2 Forward Kinematic Analysis The objective of the forward kinematic analysis of the redundant 44 parallel manipulator is to find the location of the moving platform given the actuator displacements of all the eight legs. Thus, the coordinates of points A, B, C, and D measured in the coordinate system 1 shown in Figure 4.1 need to be determined for a given platform side a, base side b, and eight leg lengths 1i (i= 1, 2,..., 8). 4.2.1 Introduction The forward analysis is performed in detail in this section and thus provides a unique solution for the location of the moving platform above the base platform together with a reflected solution through the base for an arbitrarily specified set of eight leg lengths. However, extreme care must be taken in applying this analysis since what appears to be an arbitrary set of leg lengths may well be special and the solution will fail. For such cases, the constraint equations employed in the analysis presented here become linearly dependent in one way or another. A class of special cases has been reported by Selfridge [32] where he obtained a pair of assembly configurations (as opposed to a unique solution) above the base and a corresponding pair of reflected solutions through the base. It is interesting to note that one class of solutions reported by Selfridge [32] occurs when the platforms are parallel, the odd leg lengths are all equal and 1, = = 1 = 1 = 1 = Further, the even leg lengths are all equal and 12 =4 1 = 1 = 1'. This class of solution embraces the workspace generated by a rotation of the top platform about the zaxis (Figure 3.8). While this does not raise a problem with the quality index analysis it is important to recognize that these are in fact a pair of assembly configurations above the base platform. A numerical example is present in section 4.2.5. All this of course raises the issue of other classes of special cases that are worth further investigation. 4.2.2 Coordinate Transformations First, the coordinates of any point in coordinate system 2 need to be transferred to coordinate system 1. To do so, coordinate system 2 may be obtained by initially aligning it with coordinate system 1 and then introducing the following transformations: 1. Rotate the coordinate system 2 by an angle 01 about the xaxis until the yaxis is in the plane defined by points A, E, and F, and the scalar product of the yaxis with the vector S1 is positive as shown in Figure 4.2. 2. Translate the origin from point E to F along the positive xaxis. 3. Rotate by an angle i about its current zaxis, which causes the xaxis to point along the vector S2. 4. Translate the origin from F to A along the negative xaxis. 5. Rotate the coordinate system about its current xaxis by an angle 0z until the yaxis is in the plane defined by points A, F, and B, and the scalar product of the yaxis with the vector S3 is positive. 6. Rotate by an angle 2 about its current zaxis, which causes the xaxis to point along the vector S3. 7. Rotate by an angle 03 about its current xaxis until the yaxis points along the vector S4. S4A Y2 DL_ Figure 4.2: Coordinate transformations The coordinates of points A, B, C, and D may now be expressed in the coordinate system 1 as PA= R, [T, + R3(T4+RRRR 2A, IPB = R, [T2 +R,(T4 +R,RR 2PB), 'Pc = R, T + R3(T4 +R5R6R7 2)], (4.5) SPD= R, [T2 +R3(T4 +RRR,72P)], where 1 0 0 b cos sin 1 0 1, R,= 0 cos sin8, T2= 0, R3= sin cos 0 T4= 0 , 0 sin cos8, 0 0 0 1 0 1 0 0 cosO2 sin 0 1 0 0 R,= 0 cosO2 sin2n R6 = sin2 cosQ2 0 R7= 0 cos03 sinO3 , 0 sin 2 cos2 J 0 0 1 0 sin 0 cos03 and the coordinates of points A, B, C, and D are known in terms of the coordinate system 2 and are written as 2P^, 2pB' 2Pc, and 2P. The angles A and A are shown in Figure 4.2 as the inner angles of the triangles AEF and AFB, respectively. Therefore, the angles 0i and 0 are constrained to lie in the range of 0 to n. The cosines of 01 and 0 may be determined from a planar cosine law as b2 +122 l2 Cos 0 = 2bl2 (4.6) a2 +12 12 2 _ cos, 22 3 2al2 and the values of 0 and 0 are determined as the inverse cosine value in the range of 0 to 7E. The coordinates of points A, B, C, and D as measured in the coordinate system 1 have been written as a function of the parameters 01, 6O, and 03. The objective now is to determine these parameters that will locate points A, B, C, and D such that they satisfy the distance constraints with points E, F, G, and H. 4.2.3 Constraint Equations Since the eight leg lengths, ii (i= 1, 2, ..., 8), have been given for the forward analysis, the distance between points A, B, C, and D on the platform and points E, F, G, and H on the base must satisfy these leg lengths as shown in Figure 4.2. The coordinates of points A, B, C, D, E, F, G, and H all have been expressed in terms of the coordinate system 1 (see (4.5) and (4.1)), and the distance between these points may be expressed in the coordinate system 1 as ('PB'P,)(IPB PG)= 1 (4.7) (' Pc PG).(PC''G ) = 5 (4.8) ('PC1 P)'(lP'P,)=12, (4.9) ('PDoPH)(' PD'PH)= 1, (4.10) ('PD PE) IPD PE) = (4.11) Note that three constraint equations are not written for the distance between points A and E, A and F, and B and F. The distance between these points will be equal to 11, 12, and 13, respectively. These three leg lengths have been used in the transformation of coordinate systems, which relates the coordinate systems 1 and 2 included rotation angles 1 and 2 and origin translation distance 12. Equations (4.7) through (4.11) may be expanded and factored into the form 2abs s s 2bc, (as cc2 + s (ach 12))+a2 +b2 2ac 1 +12 12 =0, (4.12) 2abs, (s2 s2 + c0 s2c3 + c2s3) + 2bc (ac. s2s3 acA s c2 a(cA c, 2 S s )c3  s (ac 12))+2a2 +b22ac 12+1 lI =0, (4.13) 2absi (s s2 + ch s2c3 + cs3) + 2bc, (ac, (s2s3 s c2) + a(c, cc,2 + s5A ss )c3  s. (ac, 12))+ 2abs (s2s3 s c2) 2a(bs cO c2 + (bc, 1)s )c, + (4.14) 2a2 + 2b2 + 2abc ch 2bc 12 2ac 12 +12 l =0, 2abs, (c s2c3 + c2s3)+ 2bc, (acA s2s3 a(c c c2 s s0 )c3 + l2so )+ (4.15) 2abs s2s3 2a(bs c c +(bc l2)s )c +a2 +2b2 2bcA 1 +12 l2 =0, 2abs s2s3 2a(bs c c2 +bc sh 12sh)c +a2 +b2 2bcl2 +l1 12 = 0 (4.16) after recognizing that s2 + c2 =1 and s2 + c2 =1, and where si, ci, (i= 1, 2, 3) represent the sine and cosine of 9i and s c j, (j= 1, 2) represent the sine and cosine of O. The objective now is to determine values for tO, 02, and 03 which will simultaneously satisfy the five equations represented by (4.12) through (4.16). 4.2.4 The Solution To solve equations (4.12) (4.16), we consider sl, cl, s2, C2, 53, and c3 as independent variables. It gives us three more equations since sin2 O + sin2 i9 = s2 + c = (i= 1, 2,3). (4.17) Now equations (4.12) (4.16) will be manipulated to eliminate si, ci, s3, and c3 first. The algebra to achieve this is what follows. Adding equations (4.12) and (4.15), and then subtracting (4.14) yields kc +k2c2 + k3 =0, (4.18) where k1, k2, and k3 are known constants and are defined in Appendix A. Similarly, an equation that is linear in c2 and c3 is generated by subtracting (4.14) from the sum of (4.13) and (4.16): k2c2+k4c3 + k =0. (4.19) Solving equations (4.18) and (4.19) for ci and c3 respectively yields k2 k3 c, = C2 (4.20) k, kI c3 = 2 (4.21) k4 k4 Now substituting the expressions for cl and c3 into equations (4.12) and (4.16) produces k6sIs2 + k7c + kc2 + k = 0, (4.22) k6s2s3 + klos + kc2 + k12 =0. (4.23) Solving the above two equations for sl and s3 respectively yields k7c 2 + kSc2 + k9 s, = (4.24) k6s2 s3 = 1 (4.25) k6 s2 Finally, substituting the expressions for si, s3, cl, and c3 into equations (4.15) and s, + c2 = 1 for i = 1 and 3 produces three equations in two variables, s2 and c2. Further, replacing s2 with 1 c2, it is interesting to note that s2 cancels from these equations. This leaves following three equations in only one unknown, c2 4 Eq,(c2)= IM,c =0 (i = 1,2,3) (4.26) j=0 where the constants Mi are defined in Appendix A. The objective now is to determine value for c2 that simultaneously satisfy the three equations represented by (4.26). Multiplying the three equations in (4.26) by c2, we obtain three additional equations. Thus, a total of six equations in the unknown c2 are obtained. These equations can be written in matrix form as My = 0 (4.27) where M,4 M13 M12 M11 M10 0 c M24 M23 M22 M21 M20 0 C2 M34 M33 M32 M31 M30 0 C M= y= 0 M14 M,3 M12 M,, M,1o 0 M24 M23 M22 M21 M20 2 0 M34 M33 M32 M31 M30 Here, we treat c2, c2, c2, c2, c2, and 1 as unknowns and thus equation (4.27) can be regarded as a homogeneous linear system in six unknowns. The trivial solution of y=0 is not feasible, since the last element of y must equal 1. Solutions other than the trivial solution exist only if the homogeneous equations are linearly dependent, and as such the determinant of the matrix M must equal zero. Evaluating this determinant and seeing how close it is to zero will provide an indication of the quality of the measured data (i.e., the platform side a, base side b, and the joint positions) and the sensed data (i.e., the eight measured displacements 11, 12, ..., Is). The issue of how close to zero is satisfactory is not addressed in this dissertation. The six equations represented by (4.27) may now be rearranged into the form Ux = v (4.28) where M14 M13 M12 M11 M1 0 5 c2 AM24 M23 22 M21 M20 0 4 C2 MM M33 M32 M31 M30 0 U= x= c v= 0 MI4 M13 M12 M11 M10 2 C2 0 M24 M23 M22 M21 M20 c2J 0 M34 M33 M32 M31. M30 Equation (4.28) represents six linear equations in five unknowns. The vector x may be solved for by selecting any five of those equations. The term c2 is the fifth component of the vector x and unique value for this term is thereby determined. However, it should be noted that since 92 is in the range of 0 to 2x, there are actually two solutions of 92 for a value of C2. Thus, the manipulator has two configurations for a given set of leg lengths. These two configurations are due to a reflection through the base plane. For each value of 02, corresponding values for cl, C3, si, and s3 can be calculated from (4.20), (4.21), (4.24), and (4.25) respectively. Then, values for 01 and 03 can be determined. Finally, the coordinates of points A, B, C, and D in terms of the coordinate system 1 can be obtained by substituting A9, 02 and 93 into (4.5). 4.2.5 Numerical Verification In this section, a numerical example is presented for a redundant 44 parallel manipulator to verify the analysis. The dimensions of the manipulator are measured in an arbitrary length unit and given as follows: platform side a = 10, base side b = 15. A set of leg lengths are given as 1, =13.62421, 12 =10.40411, 13=14.47201, 14=11.16409, 15 = 16.34095, 16= 17.59696, 17 =16.22984, 18 = 15.92500. The numerical results are presented in Table 4.1 and two configurations are shown to be reflecting through the base plane. Thus, a unique configuration may be easily determined by checking the sign of z coordinate of one of the platform joints. In order to verify these results, an inverse kinematic analysis was performed. All solutions reproduced the correct leg lengths. Table 4.1: Numerical results of the redundant 44 inparallel manipulator No. O (deg.) 62 (deg.) 03 (deg.) P PB 'PC PD 10.079 16.119 8.921 2.881 1 105.534 133.523 27.872 2.455 10.327 15.045 7.173 8.832 10.077 15.168 13.923 10.079 16.119 8.921 2.881 2 105.534 133.523 27.872 2.455 10.327 15.045 7.173 8.832 10.077 15.168 13.923 In the following example the above solution failed because equation (4.7) through (4.11) become linearly dependent while equation (4.9) through (4.11) are redundant for the system. Using the same dimensions as the above example, the leg lengths now become: S=13 1=17 ==18 and 12 =14 =16 = 1'= 16. The numerical results are presented in Table 4.2. It is apparent that there are now two configurations above the base plane and further two solutions reflected through the base. Table 4.2: A numerical example for the special case of the redundant 44 inparallel manipulator No. h I PA 1 PB 'PC IPD 2.267 6.698 2.267 6.698 1 5.199 6.698 2.267 6.698 2.267 5.199 5.199 5.199 5.199 2.267 6.698 2.267 6.698 2 15.099 6.698 2.267 6.698 2.267 15.099 15.099 15.099 15.099 2.267 6.698 2.267 6.698 3 5.199 6.698 2.267 6.698 2.267 5.199 5.199 5.199 5.199 2.267 6.698 2.267 6.698 4 15.099 6.698 2.267 6.698 2.267 15.099 15.099 15.099 15.099 CHAPTER 5 THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 48 INPARALLEL MANIPULATOR A redundant 48 parallel manipulator is shown in Figure 5.1, which is derived simply by separating the double ballandsocket joints in the base of the redundant 44 manipulator shown in Figure 3.1. A plan view of the redundant 48 parallel manipulator is shown in Figure 5.2. The device has a square platform of side a and an octagonal base formed by 4 pairs of joints ED and EA, FA and FB, GB and Gc, and He and HD. Each pair of joints is separated from a vertex of a square of side b by a distance fb for which 1 1 Ol< . Clearly the platform is degenerate when f =  2 2 $ / GF E EA b B Figure 5.1: A redundant 48 inparallel Figure 5.2: Plan view of the redundant 48 manipulator inparallel manipulator 5.1 Determination of det JJT The moving platform of the redundant 48 parallel manipulator shown in Figure 5.2 is located at its central symmetrical configuration and is parallel to the base with a distance h. By analogy with the redundant 44 parallel manipulator, a maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose, i.e., /detJ.JT may be obtained for this configuration. A fixed coordinate system is placed at the center of the octagonal base as shown in Figure 5.2. Then, the coordinates of the points A, B, C, and D on the platform are 2 2 2 2 Af0 ]a h, BL 0 hJ], C0 Vi h}, D( /a 0 hJ) (5.1) The coordinates of the points EA, FA, FB, GB, Gc, Hc, Ho, and ED on the base are EAd, _ o10 L,F d o1 F d4 o0, GB( d4 0), 2 2 2 2 ) ^ ) (5.2) Gc d4 b o Hc d4 b 0 H _ d4 0o, E d4 0 where (1 2/)b d4 = 2 Counting the 2x2 determinants of the various arrays of the joins of the pairs of points EAA, FAA, ..., EoD yields the Plicker line coordinates of the eight leg lines. That is, from points EA and A, J =d4. J2ab bh; d4h, (5.3) 2 2 2 From points FA and A, $2 = Id4, bh iad4 h; , d4h,  2 2 From points FB and B, 3 = 2a b, d4 2 bh h; d4h, , 2 Viad4 2 From points GB and B, 4 ab S,2a b 2 bh d4, h; d4h, , 2 V2ad4 2 From points Gc and C, s= d4, From points Hc and C, $6 = d4, bh _2ad4 h; h, d4h, 2 2 From points HD and D, 7 =[a b I 2ab 2 bh d4, h; d4h, , 2' From points ED and D, S2a ab 2 d4, h; bh i2ad4 d4h,  2 2 The normalized Jacobian matrix of the eight leg lines can be expressed in the form i1 1 3 ^T ^5T T 6T 7T 18 1 12 13 14 15 16 17 I1 (5.11) 5a b 2 (5.4) (5.5) (5.6) bh h; , 2 d4h ad 2 VZab 2 5ab 2 (5.7) (5.8) V2ad4 2 (5.9) (5.10) 61 where 11, 12, ..., 1 are the leg lengths and since the device is in a symmetrical position, 11 = 12 = = 1 = I, and l= L2 +M +N2 = [a2 l ab+(2f2 2 p+1)b2 +2h2. From (5.3) to (5.10), (5.11) becomes h h (5.12) J I bh 2 bh 2 d4h d4h bh d4h d4h 2 if2ad4, 2ad4 l2ad4 S2 2 2 where same as (3.13), di bh 2 i2ad4 2 d4h d4h ,2ad4 2 2lad4 2 bh 2 I2ad4 2 _2ab 2 From equation (5.13), the determinant of the product JJ T becomes 4(d 2+d2) 0 0 0 2(2d2bd,)h 0 4(d2 +d'2) 0 2(bd,2d2)h 0 8h2 detJJT= 1 12 0 2(bd,2d2)h 0 (b2+4d2 )h2 2(2d2bd )h 0 0 0 (b2+4d2)h2 0 0 0 4a2d2 Expanding (5.14) and using (5.12), then extracting the square root, we obtain d4h d4h (5.13) bh 2 iFad4 2 . (5.14) d4 d, jjT = 322(1 2#)3 a 3b3h3 vdet JJ = ,  f. (5.15) [a2 2ab+(222 p +)b2+2h 2' Assuming the top platform size a is given and taking the partial derivative of (5.15) with respect to h and b respectively and equating to zero yield 96,f2(1 2f)3'a3b3h2[a 2_ab+(2'2 2p + )b2 2h2]= (5.16)   & = U (5. 16) [a2 2ab+(2/f2 2l + l)b2 +2h24 and 9642(1+2)3 a3b 2h [a2 +(2f 2 2f +1)b2 2h2] [a2 2ab+(2f2 2,f+l)b2 +2h2 (517) 1 Note that we already assumed f , then when a, b, and h are not equal to zero, from equations (5.16) and (5.17), we get a2 2ab+(2fi2 2f + l)b2 2h2 =0, (5.18) a2 +(2fl2 2 + 1)b2 2h2 =0. (5.19) Subtracting (5.19) from (5.18) yields 2a 2 ab = 0, (5.20) and thus b = 42a. (5.21) Substituting (5.21) into (5.19) gives (4 82 4f +1)a 2 2h2 =0. (5.22) The above equation yields two solutions for h, here we only take the positive solution h = (1 2f)a. (5.23) V2 Therefore, when b = 2a and h= (1 2f)a, the redundant 48 parallel manipulator is at the maximum quality index as shown in Figure 5.3, and from (5.15) qdetJ.J =( detJJ') = 44a' (5.24) where Jm denotes the Jacobian matrix of this configuration. It is interesting to note that this maximum value of det JJ T is independent to the value of f. Figure 5.3: Plan view of the optimal configuration of the redundant 48 inparallel manipulator with the maximum quality index From (3.24) and (5.24), it shows that both the redundant 44 and the 48 parallel manipulators have the same maximum value of [det JJ. Figure 5.4 illustrates the compatibility of these two results. It can be observed that as the distance between the pairs of separation points of the double ballandsocket joints E, F, G and H of the original 44 manipulator increases, the height h at which the manipulator has the ED___ EEA d a" FA maximum quality index decreases (see (5.23)) from h 7= ( = 0, concentric balland 2 socket joints) to h = 0 (f/ = , platform is degenerate). $ $6 $ 4 H G B $8 $1 $2 $3 EA b FA E F Figure 5.4: Compatibility between the redundant 44 and the 48 parallel manipulators (1h. 2f)a (h, =) 5.2 Implementation From (1.2) and (5.24), the quality index for the redundant 48 parallel manipulator shown in Figure 5.1 can be expressed as (5.25) Sdet JJ 4,r= . In this section, a redundant 48 parallel manipulator with a = 1 and b = 22 is taken as an example for the investigation of the variation of the quality index A during a number of platform movements. The first platform movement we studied is a pure vertical translation of the platform along the zaxis that passes through the center of the platform. From (5.15) and (5.25), the quality index for this movement becomes 8(1 2f)3b3h3' [a2 Nab+(2p2 2f +l)b2 +2h2]3 With a = 1 and b =2 this reduces to S16r2(1 23)3 h3 Al= ( ,)h 3 (5.27) [(12f)2 +2h2]3 and is plotted in Figure 5.5(a) as a function of h and ft. Figure 5.5(b) plots the variation curves of the quality index for several different values of fl. From these figures, we can see the height (hm) at which the manipulator has the maximum quality index is reduced as p increases. Each value of f designates the distance between the separation points in the base and is a first design parameter. Clearly, f= 0 is the best overall design. h 1.0 .. =0 0.8 . 0. x= 1/8 1 0.6 ......o.. ............. .................... 0 0. 0. .. ........... .................... ~ 11.5 0.3 h" 2 .2/5 2.5 0.1 0.0 30 0 1 2 3 Height h (a) (b) Figure 5.5: Quality index for platform vertical movement 66 The second platform movement is a pure horizontal translation of the platform away from its initial location at height h. To derive an expression for detJJr we assume the center of the platform move to point (x, y, h), then the coordinates of the points A, B, C, and D on the platform become Ax C x 2ia 2 ya h*} y+ h h, 2 B x+  2 D x 2 y h y hJ. (5.28) The coordinates of points EA, FA, FB, GB, Gc, He, Ho, and ED on the octagonal base can be found from (5.2). Calculating the Pliicker line coordinates for each of the eight leg lines yields S2= x d, S S3 = x+ S4 = X+ V S = xd, S, = x+d4, Via b y . 2 Jab y 2 a b 2 ab 2 bh h; , 2' bh h;  2 y+d4, h; d4h, yd4, h; d4h,  Viab y+ 2 Via b 2 bh , h; , 2 bh h; , 2 dh, bx+d,4(Va2y) d~h,  2 bxd4(I2a2y) dh, bh d4 (2a+2x)+by 2' 2 bh d4(,2a+2x)by 2' 2 d4h, bxd4(Via+2y) d4h, 2, 2 bx+d4(2a + 2y)] d4h, , 2 (5.29) (5.30) (5.31) (5.32) (5.33) (5.34) 7 X Fa b 7 =x a b 2 yd4, h; d4h, bh 2 d4 (2a 2x) by 2 S8 = I 42ab 2 d4, h;d4h, bh d,(F2a2x)+by y+d,, h; dh, , 2' 2 (5.36) (1 2f)b where same as (5.2), d4  2 The Jacobian matrix can then be constructed by using (5.11). Further, det JJ becomes detJJT = a25 + + 2 2 +12 +12) lll2131415161718 (5.37) where the leg lengths are 11= (x+d42 + Y ab +h2, 2 =, j .22 2a b +h2 12= (xd4)2+ y +h , 2,e f +(y+ d4)2 +h2, 13= (x+ 2ab I 15= (xd4)+ 2 ab +h2 Sb 2 ,2 17= x +(yd4 )+h2 , 2 14 = (x+ 2a b 2 +(yd4)2 +h2, 16 = x+d4)2++ i2ab +h2, 2 Viabx2 18 x 2a b 2 +(y+d4)2 +h2. (12f8)a = 12,8 With a = 1, b = V2, and h = hm = 2(1 from (5.25) and (5.37), the quality 2 Vf2 index becomes (5.35) VdetJJ _(12f1)6(l +1+2+13+1 ) t = 2222 (5.38) detJmJ 411 12134 where 11 = 1, = x2 +2(12)x+ y +(12f)2, 12 =l5 =Vx2 v(l2f)x+ y2 +(12i)2 , 13 =18 = x2 +f(12fi)y+y2 +(12f)2, 14 =17 = Vx2 2(12f8)y+ y2 +(12f#)2. In Figure 5.6, the quality index and its contours as the platform is translated away from the central location while remaining parallel to the base at hm, are drawn for various values of jf. It should be noted that when fl=0, the 48 manipulator becomes the 44 manipulator and its corresponding quality index is drawn in Figure 3.5. Comparing Figure 5.6(a)(d) and Figure 3.5, it is clear that the smaller f, the larger workspace area of the platform is with high quality index. Now we attach a new coordinate system x'y'z' to the square platform. This coordinate system may be obtained from the platform configuration shown in Figure 5.2 by initially aligning it with the xyz coordinate system on the base and then raising it by a distance h along the zaxis to the top platform. Thus, the x' and y'axes are parallel to the x and yaxes respectively when the platform locates at its initial central position shown in Figure 5.2. We are interested in deriving detJJr when the platform rotates about the x' and y'axes from its central position. Here, we only derive the platform rotation about the 69 y'axis. But the result to be derived is the same for the platform rotation about the x'axis since the redundant 48 parallel manipulator is fully symmetric. 1.0 0.5 A 0.0 0.5 1.0 1.0 0.5 0.0 x 0.5 1.0 2 N1 (a) = at h,= = 0.14 5 10 1 1 0.5  1.0  1.0 0.5 0.0 0.5 1.0 1 5 (b) = at h, = 6 0.24 3 6 S............... i . ......... ............. i .......... . .  .. . . ."............. 4 ........... i ..............  . ........... . . . . . .. . . . . . . . .......... ............... ............................... .."............... ;................ .. ............. ............... .. . .. ............. ........... ........................ i 1.0  0.5  ~ 0.0  0.5  1.0  1 1 1 (c) = at hm 4 1.0 0.5 0.0 0.5 1.0 = = 0.35 4 1.0  0.5  p0.0 0.5  1.0  I 1 1.0 0.5 0.0 0.5 1.0 1 a (d) #= at h, = = 0.53 8 8 Figure 5.6: Quality index for platform horizontal translation with different values of 6 ............... .............. b Figure 5.7 illustrates a side view of the moving platform ABCD rotated 0y about the y'axis. The coordinates of the vertices A, B, C, and D become A f l2a h, B cos9 0 h+ sinej, 2 2 2 (5.39) C 0 a h D  ca os9, 0 ha sin 2 2 ( 2 and the coordinates of vertices EA, FA, FB, GB, Gc, Hc, Ho, and ED on the base can be found from (5.2). Figure 5.7: Platform rotation about the y'axis Note that the positions of line $1, $2, $5, and $6 do not change during this platform rotation and their corresponding Plicker line coordinates can be obtained from (5.3), (5.4), (5.7), and (5.8) respectively. The Plucker coordinates for the line $3, $4, $7, and $8 are now given by [V acos0 b S3 = 2ac 2Oyb d4, 2 d4 (V2a sin ,y + 2h) 2 2a sin y + 2h 2 (5.40) b(V2a sin Oy + 2h) 4 F2ad4 cos O 2 ^IFacosOy b V2asin OY +2h 24 2 d4 2 $ (5.41) d4 (V2a sin Oy + 2h) 2 7 2acos Oyb 2 d4,(2asin Oy 2h) 2 b(2a sin 0y + 2h) 4 i4 a sin 0 2h 2 b( 2a sin O, 2h) 4 2ad4 cos ,0 2 (5.42) V2ad4 cos Oy 2 Viacos6/b 8 /2a cos =Y b S2 d4 ( a sin 0y 2h) 2 F2a sin 0y 2h 2 b(VF2a sin 0, 2h) 4 (5.43) i2ad4 cos Oy 2 (12fl)b with d4 = 2 From Figure 5.7, we have 1, = 12 = l5 = 6 = 1, 13 14, and 17 = 1. The corresponding Jacobian matrix can now be obtained by (5.11), and further detJJT can be determined. With a =1, b = 2, and h = h= 2 from (5.25), the quality index = rm 52) teqait ne then becomes 73 detJJ (12i)4 212 + 2+12 = = 2(12) 4213 +17 (2(5(42 4f8 +2)(cosO, + VdetJ.Jr 41413 cos 30,) + (16 84 32l3 +24/82 8 2)(cos20, +1)) + (l2 + 72)(14/2  6f +1+ 8f3' (f 2)(cos20 + 1) + 2P8(5f# 1)cos20y) + (l3 l 7)(8f8'  12w2 + 6 l)sin 202 where (5.44) = 12fi, 13 = 4f2 4f, + 2 cos0Y (2f 1)sin 0,, 17 = 4fi2 4/f+2cos9 +(2/i1)sin9y. Figure 5.8(a) plots the quality index as a function of y4 and ft. Figure 5.10(b) presents the change of the quality index for several different values of f. ...I......... I .................. 90 60 30 0 30 60 9 Rotation Angle 6 (degree) Figure 5.8: Quality index for platform rotations about the x' and y'axes 1. 0. 0.2. 0400.1 60040 0. 460 0.1 80 0 A plan view of the redundant 48 parallel manipulator with the moving platform ABCD rotated 06 about the zaxis is shown in Figure 5.9. The x and y coordinates of the vertices A, B, C, and D become A = r sin O, YA = r cos 0, x, =r cos0 9, y = rsin 0, (5.45) xc =rsin O, Yc = rcos0~, XD = r cos 0z, Yo = r sin Oz a where r =  The complete set of coordinates of points A, B, C, and D are therefore A(xA YA h), B(xB y, h), C(xc Yc h), D(x, yD h) (5.46) where h is the height of the moving square platform above the octagonal base. Figure 5.9: Platform rotation about the zaxis 75 The coordinates for the corresponding lines $1, $2, ... $8 are then given by A S, = xA +d4, [ 2 =XA d4, b 2 b 94 = xB [ 2 5 = xc d4, 9 = [XD + 2 b 98 = xD + ^ b y, + , 2 b YA +, 2 bh h; , 2 bh h; , 2 bh y, +d4, h; d4h, , 2 bh yB d4, h; d4h, 2 b Yc  2 b Yc , 2 bh h; , 2 bh h; , 2 bh YD d4, h; d4h, , 2 Y +d4, h; b(x, y + 2py,)] d4h, 2 2 b(x, + Y, 2fAYA ) ] dh, 2, b(x, 2pxB +y) 2b(x b(xB 2xB yB) 2J b(xc Yc + 2fpyc)  d4h,  b(xc + yc 2pfyc) d4h, 2 b(x, 2fpx + y,) 2 b(x, 2fpx y,) 2 bh d4h, 2' 2 From Figure 5.9, we have I1 = l1 = l = 17 and 12 = 14 = 16 = I. The corresponding Jacobian matrix can now be determined by (5.11), and further ddetJJT becomes de 4[2a3b3h3(1 28)3'cos 0 =detJJ = , 33 (5.55) where I, =l (iasinO, +b2b))2 +(Jacosz b +4h2 12 =V(a sin b+ 2+ 2) +(,iacosO, b +4h2 (5.47) (5.48) (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) 76 With a =1, b = and h = = 2, from (5.25) and (5.55), the quality index becomes A, = (1 2)'6 ICS (56) (2(2f2 2fi+1)(cosO, 2)cosOz +16f84 328' +28fl2 12,+3) This is plotted in Figure 5.10(a) as a function of f and O. It shows how the quality index varies as the platform is rotated about the vertical zaxis through its center. The eight legs are adjusted in length to keep the platform parallel to the base at a distance hm. Figure 5.10(b) illustrates the variation of the quality index for several different values of f. It is shown in these figures that the manipulator has the highest quality index A = 1 when Oz=0, and A = 0 (singularity) when 0= 900. As can be seen in Figure 5.10(b), a slight change of 6z under a large f has a much greater impact on the quality index than that of the same change under small 8. (a) u.u 0 I i I 90 60 30 0 30 60 90 Rotation Angle 0 (degree) (b) Figure 5.10: Quality index for platform rotation about the zaxis ................... ..... . ......... 7 ......... =1/8 1/4...... I II LI Again, from Figures 5.8 and 5.10 we can see clearly that better designs are obtained as f/ reduces to zero. Hence the best 48 parallel manipulator design is obtained when the pair of base joints are as close as possible. Since from (5.55), detJJ =0 when z=900, a redundant 48 parallel manipulator always becomes singular when its platform rotates a=900 about zaxis from its central symmetric position. Figure 5.11 illustrates the singularity position when z=90. y He Gc G HD $ GB $6 B$4 C x ED, FB EA FA F Figure 5.11: Plan view of the singularity position of the redundant 48 inparallel manipulator when 6O = 900 In complete analogy with the redundant 44 parallel manipulator presented in Chapter 3, when 09=900, the moving platform of the redundant 48 parallel manipulator can move instantaneously on a screw reciprocal to the eight leg forces on the zaxis with J2ab pitch h = T This is because for OL=900, from (5.47) through (5.54), the 4h 78 component of moments about the zaxis for each of the eight legs all are equal to 4ab 4 CHAPTER 6 THE FORWARD CINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 48 INPARALLEL MANIPULATOR The kinematic analysis of the redundant 44 parallel manipulator has been performed in Chapter 4. It is shown that the inverse kinematics is straightforward for parallel manipulators while the forward kinematics is difficult. In this chapter the forward kinematics of the redundant 48 parallel manipulator shown in Figure 5.1 is studied. It will be shown how this problem can be easily solved by transferring it to the corresponding redundant 44 case which then can be solved by using the method presented in Chapter 4. 6.1 Forward Kinematic Analysis In Figure 6.1, coordinate systems 1 and 2 have been attached to the fixed base and the moving platform of a redundant 48 parallel manipulator, respectively. The origins of the coordinate systems 1 and 2 are located at points E and A receptively. The objective of the forward kinematic analysis of the redundant 48 parallel manipulator is to find the position and orientation of the moving platform given all the eight leg lengths. Eight dash lines connecting the moving platform and the base are drawn in Figure 6.1. If we connect the platform with the base by legs along these dash lines to replace the original legs, we obtain a redundant 44 parallel manipulator with the platform location same as the original 48 manipulator. Thus, if we can determine the distances, loi (i= 1, 2, ..., 8), between the platform and the base along the dash lines, the forward kinematic problem will have been solved by using the method presented in Chapter 4 for the redundant 44 parallel manipulator. The objective now is to determine values for 10o, 102, .... s08 from the original leg lengths i, 12, ..., Is. XI Figure 6.1: Coordinate systems of a redundant 48 inparallel manipulator Observing Figure 6.1, we find 11, 12, 101, and 102 are in the same plane defined by points A, EA, and FA while 13, 4, 103, and l04 are in the plane defined by points B, FB, and GB, 15, 16, los, and 1o6 are in the plane defined by points C, Gc, and Hc, and 17, 18, 107, and los are in the plane defined by points D, HD, and ED. Thus, loi can be determined from li in the same plane. For example, in the plane defined by points A, EA, and FA as shown in Figure 6.2, we have 81 11o = J2b2 + 1 2fb cos( ,, o2 = V/(1 )2 b2 + 2(1 f)bl cos4p,, where (1 2)2b2 +12 12 cos = 2b(  2b(1 2,)1, Figure 6.2: Leg relations Similarly, the other leg lengths can be obtained l03 = 4f2b2 +1 3 2f0bl3 cos( p2), 104= V(1 _f)2 b2 +12 2(1f)bl3cosqp,, lo = 2b +12 2fbl, cos( r ), (6.1) (6.2) (6.3) (6.4) (6.5) lI = 4(1 f)2b2 +12 2(1/ )bl, cos(p3, (6.6) S= J2b2 +12 2fbl7 cos('r (4), (6.7) ls = 4(l f)2b2 +12 2(1 f)bl7 cos 4 (6.8) where (12p)2b2 +2 122 cos =3 2b(l 2f)l3 (1 2,)2b2 +l1l2 cos P =  2b(1 2 f)1l (12)b2 +12 _12 COS V4 2b(1 2f)17 Using the values of lot, 102, ..., lo8 as input leg lengths to the forward kinematic analysis presented in Chapter 4, the position and orientation of the moving platform will be determined. 6.2 Numerical Verification A numerical example is presented for a redundant 48 parallel manipulator to verify the analysis. The dimensions of the manipulator are measured in an arbitrary length unit and given as follows: 1 platform side a = 10, base side b = 15, =  8 A set of leg lengths are given as I, = 12.21787, 12 = 9.15596, 13 =12.83105, 14 = 7.52035, 1, =13.47917, 16 =13.13367, 17 =13.88865, 1 = 14.04687. Thus, the input leg lengths for the forward analysis of the corresponding redundant 44 parallel manipulator are obtained from (6.1) through (6.8) and 10, = 13.59387, 102= 9.87590, 03 = 9.87590, 14 = 7.94680, 15 = 14.41631, 1 = 13.98464, 107 = 14.72302, 10 = 14.92181. The numerical results are presented in Table 6.1. It has been verified by an inverse kinematic analysis that all solutions reproduced the correct leg lengths. Table 6.1: Numerical results of the redundant 48 inparallel manipulator No. 91 (deg.) 9 (deg.) 0 (deg.) PA 'PB 'PC 'PD 10.409 14.940 7.091 2.560 1 112.939 122.570 28.524 3.408 12.304 16.592 7.696 8.052 7.475 11.948 12.525 10.409 14.940 7.091 2.560 2 112.939 122.570 28.524 3.408 12.304 16.592 7.696 8.052 7.475 11.948 12.525 Similar to the redundant 44 parallel manipulator, there is a special solution when the platform rotates about the zaxis (Figure 5.9). For example, when the leg lengths now become 1, =13 =15 =1=1=18 and 12 =14 =6 = = '=16 for the redundant 48 parallel manipulator in the first example, two configurations above the base plane with another two reflected through the base are obtained as shown in Table 6.2. Table 6.2: A numerical example for the special case of the redundant 48 inparallel manipulator No. h IPA I PB P IPD 3.022 6.393 3.022 6.393 1 7.498 6.393 3.022 6.393 3.022 7.498 7.498 7.498 7.498 3.022 6.393 3.022 6.393 2 15.748 6.393 3.022 6.393 3.022 15.748 15.748 15.748 15.748 3.022 6.393 3.022 6.393 3 7.498 6.393 3.022 6.393 3.022 7.498 7.498 7.498 7.498 3.022 6.393 3.022 6.393 4 15.748 6.393 3.022 6.393 3.022 15.748 15.748 15.748 15.748 CHAPTER 7 THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 88 INPARALLEL MANIPULATOR A redundant 88 inparallel manipulator is shown in Figure 7.1, which is derived by separating the double ballandsocket joints in the base and the top platform of a 44 manipulator shown in Figure 3.1. A plan view of the redundant 88 inparallel manipulator is shown in Figure 7.2. The device has eight legs connecting an octagonal platform and a similar octagonal base. The octagonal top platform is formed by 4 pairs of joints Al and A2, BI and B2, C1 and C2, and D1 and D2. Each pair of joints is separated I from a vertex of a square of side a by a distance ca for which O octagonal base is formed by 4 pairs of joints ED and EA, FA and Fa, GB and Gc, He and HD, and each of them is separated from a vertex of a square of side b by a distance fib for I which 0:<5 <. This design has the distinct advantage that it completely avoids the 2 mechanical interference problem associated with the design of double spherical joints. 7.1 Determination of .det J., . The moving platform of the redundant 88 parallel manipulator shown in Figure 7.2 is located at its central symmetrical configuration and is parallel to the base with a distance h. It will be shown in this chapter that a maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose, i.e., Jdet J J , may be obtained from this symmetric configuration. Figure 7.1: A redundant 88 inparallel manipulator Figure 7.2: Plan view of the redundant 88 inparallel manipulator First, a fixed coordinate system is placed at the center of the octagonal base as shown in Figure 7.2. Then the coordinates of the points A,, A2, B1, B2, C1, C2, Di, and D2 on the platform are V2a d5 2 d5 h, A2(d5 i2a d,5 2 B2NFd5 d5 h 2 (7.1) Va d5 2  2a 2  d 2 2ia d, 2 d5 h) where d5 2aa ds5= A, d, fB, fd 2 C(ds DC ds D, d, C2 d5 ,V2a d5 i 2 The coordinates of the points EA, FA, FB, GB, Gc, Hc, Ho, and ED on the base are EA d4 0 o1, d4 0 FB d4 o0 GB d4 0, 2 2 b2 ) ,) (7.2) G d 02 H d4 b H b d4 0, ED( d4 0 . (1 2 2)b Same as (5.2), d4 = ( 2 Now the Pliicker line coordinates of the eight leg lines of the redundant 88 parallel manipulator can be obtained by counting the 2x2 determinants of the various arrays of the joins of the pairs of points EAAI, FAA2, ..., EoD2. From points EA and A1, we get = d4d5, d5 d,, h; , d4h, d6 (7.3) [ 2bh where d6 = (2afl 2a 2fl +)ab and d,  4 2 Similarly, from points FA and A2, 2= [d5d4, d5d1, h; bh, d4h, d (7.4) Sb2 From points FB and B1, S3= d,d5, d4d,. h; d4hd, d6] (7.5) From points GB and B2, 4 =dds, d5d4, h; d4h, , d6 (7.6) From points Gc and C1, bh 1 S= d,d, d, d5, h; , d4h, d6. (7.7) From points Hc and C2, Si =d4 d, d,d5, h; ,b d4h, d (7.8) Sbh2 From points HD and DI, S = [d5d, d5d4, h; d4h,2 d6]. (7.9) From points Eo and D2, bh 1 S8=[ ddd, d4ds, h; d4h, , d6 (7.10) The above coordinates are not normalized and each leg line needs to be reduced to unit length. Then, the normalized Jacobian matrix of the eight leg lines can be expressed in the form "or Sl T T g T T gr 1 1 j_4 5 7 8(7.11) 1 12 13 14 15 16 17 18 where i, 12, ..., 18 are leg lengths. Here, the device is in a symmetrical position so that the normalization divisor is the same for each leg, namely 11 = 12 = ... = I = 1, and for every leg 1= L +M2 +N2 = 2[(2a22a+l)a2 + i(2aI)ab+(2'l22i+l)b2 +2h2]. (7.12) From (7.3) to (7.10), (7.11) becomes 