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DESIGN AND IMPLEMENTATION OF A 6 DOF PARALLEL MANIPULATOR WITH PASSIVE FORCE CONTROL By BO ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005
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Copyright 2005 by Bo Zhang
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iii ACKNOWLEDGMENTS The author wishes to express his great gr atitude to Dr. Carl D. Crane, III, his dissertation advisor, for sugge sting the topic of this wor k, providing continuous guidance, support and encouragement throughout this re search. Also the au thor would like to acknowledge the efforts of his supervisory co mmittee members: Dr. John C. Ziegert, Dr. Gloria J. Wiens, Dr. A. Antonio Arroyo and Dr. Rodney G. Roberts, whose time and expertise were greatly appreciated. The author wishes to acknowledge the gui dance from Dr. Joseph Duffy; he will be greatly missed. The author would like to thank Dr. Carol Chesney, Shannon Ridgeway, Dr. Michael Griffis, Dr. Bob Bicker and JeanFrancois Kama th for their great help. He also would like to thank his fellow students in the Center for Intelligent Machines and Robotics for sharing their knowledge. This research was performed with funding from the Department of Energy through the University Research Program in Robotics (URPR), grant number DEFG0486NE37967. Finally, the author gives speci al appreciation to his parents, for their love, patience and support.
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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... xi CHAPTER 1 MOTIVATION AND INTRODUCTION....................................................................1 Motivation..................................................................................................................... 1 Introduction................................................................................................................... 4 2 BACKGROUND AND LI TERATURE REVIEW....................................................11 Kinematics Analysis...................................................................................................11 Singularity Analysis....................................................................................................13 Statics and Compliance Analysis................................................................................14 Compliance and Stiffness Matrix...............................................................................17 3 PARALLEL PLATFORM DESIGN..........................................................................20 Design Specification...................................................................................................20 Conceptual Design......................................................................................................22 Prototype Design........................................................................................................28 4 SINGULARITY ANALYSIS.....................................................................................34 5 STIFFNESS ANALYSIS OF COMPLIANT DEVICE.............................................41 Simple Planar Case Stiffness Analysis.......................................................................41 Planar Displacement and Force Representation.........................................................46 Stiffness Mapping for a Planar System......................................................................48 Stiffness Matrix for Spatial Compliant Structures......................................................51
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v 6 FORWARD ANALYSIS FOR SPEC IAL 66 PARALLEL PLATFORM...............65 Forward Kinematic Analysis for a 33 Platform........................................................65 Forward Kinematic Analysis for Special 66 InParallel Platform............................70 7 RESULTS AND CONCLUSION...............................................................................78 Calibration Experiment for the Force Sensor.............................................................78 Individual Leg Calibration Experiment......................................................................80 Parallel Platform Force/Wrench Testing Experiment.................................................85 Determination of Stiffness Matrix at a Loaded Position............................................95 Future Research........................................................................................................106 APPENDIX MECHANICAL DRAW INGS OF THE PARTS FOR PCCFC..............107 LIST OF REFERENCES.................................................................................................114 BIOGRAPHICAL SKETCH...........................................................................................118
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vi LIST OF TABLES Table page 31 Design objective specification.................................................................................21 61 Mappings of angles of sp herical four bar mechanism..............................................69 62 Parameter substitutions............................................................................................75 71 Load cell regression statistics...................................................................................80 72 Forcetorque Sensor Resolution...............................................................................85 73 Repeatability experiment, method 1 (Encoder Counts)...........................................86 74 Repeatability experiment, method 2 (Encoder Counts)...........................................87 75 Forward analysis......................................................................................................88 76 Spring constants.......................................................................................................92 77 Numerical experiment 1encoder counts and leg lengths.......................................93 78 Numerical experiment 2encoder counts and leg lengths.......................................93 79 Numerical experiment 3encoder counts and leg lengths.......................................93 710 Numerical experiment 4encoder counts and leg lengths.......................................93 711 Numerical experiment 5encoder counts and leg lengths.......................................93 712 Numerical experiment 6encoder counts and leg lengths.......................................93 713 Numerical experiment 1 wrench............................................................................94 714 Numerical experiment 2 wrench............................................................................94 715 Numerical experiment 3 wrench............................................................................94 716 Numerical experiment 4 wrench............................................................................94 717 Numerical experiment 5 wrench............................................................................94
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vii 718 Numerical experiment 6 wrench............................................................................95 719 Measured leg connector lengths at positions A through F (all units mm).............100 720 Displacement of leg connectors from unloaded home position at positions A through F (all units are encoder counts).................................................................101
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viii LIST OF FIGURES Figure page 11 Passive compliance device for contact force control.................................................3 12 Serial structure........................................................................................................... .4 13 Inparallel platform....................................................................................................5 14 General parallel platform...........................................................................................9 15 33 parallel platform...................................................................................................9 21 Remote Center Compliance Coupling......................................................................17 31 Plan view of octahedr al 33 parallel platform..........................................................23 32 Special 66 para llel platform....................................................................................26 33 3D model for special 66 parallel mechanism........................................................27 34 Plan view of sp ecial 66 platform............................................................................29 35 Flow chart of the design process..............................................................................30 36 Assembled model for speci al 66 parallel platform.................................................32 37 Photo of the assembled parallel platform.................................................................33 41 Planar singularity......................................................................................................35 42 Plan view of a special 66 parallel platform.............................................................36 43 Plan view of the specia l singularity configuration...................................................37 44 Photo of the parallel platform in singularity............................................................39 51 Planar inparallel springs..........................................................................................42 52 Planar serially c onnected springs system.................................................................43 53 Planar 2 DOF spring.................................................................................................44
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ix 54 Planar compliant system...........................................................................................49 55 Stiffness mapping for a planar compliant system....................................................50 56 3D Model for special 66 parallel mechanism........................................................55 57 Top view of the special 66 configuration...............................................................55 61 3D drawing of the 33 inparallel platform.............................................................67 62 A general spherical fourbar mechanism.................................................................69 63 33 inparallel mechanism with three spherical fourbar linkages...........................70 64 Special 66 platform a nd equivalent 33 platform...................................................72 65 Definition of angles and .....................................................................................73 66 Planar triangle..........................................................................................................74 71 Load cell calibration experiment..............................................................................79 72 Load cell calibration experi ment opposite direction..............................................79 73 Calibration plot for leg1..........................................................................................81 74 Calibration plot for leg2..........................................................................................82 75 Calibration plot for leg3..........................................................................................82 76 Calibration plot for leg4..........................................................................................83 77 Calibration plot for leg5..........................................................................................83 78 Calibration plot for leg6..........................................................................................84 79 Photo of the testing experi ment for the parallel platform........................................85 710 Coordinate systems of the parallel platform............................................................90 711 Compliant Platform Model.......................................................................................96 A1 Strip holder.............................................................................................................10 8 A2 Lower leg part........................................................................................................109 A3 Upper leg part.........................................................................................................110 A4 Leg........................................................................................................................ ..111
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x A5 Top plate.................................................................................................................1 12 A6 Base plate...............................................................................................................11 3
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xi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DESIGN AND IMPLEMENTATION OF A 6 DOF PARALLEL MANIPULATOR WITH PASSIVE FORCE CONTROL By Bo Zhang August 2005 Chair: Carl D. Crane III Major Department: Mechanic al and Aerospace Engineering Parallel mechanism has been studied for se veral decades. It has various advantages such as high stiffness, high accuracy a nd high payload capacity compared to the commonly used serial mechanism. This work presents the design, analysis, and control strategy for a parallel compliance coupler for force control (PCCFC) based on a parallel platform design. The device is installed on the distal end of an indus trial manipulator to regulate the contact wrench experienced when the manipulator comes into contact with objects in its environment. The parallel mechanism is comprised of a top platform and a base platform that are connect ed by six instrumented complia nt leg connectors. The pose of the top platform relative to the base as well as the external wrench applied to the top platform is determined by measuring the displ acements of the individual leg connectors. The serial robot then moves the PCCFC in orde r to achieve the desired contact wrench at the distal end.
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xii A mechanical system has been fabricated with an emphasis placed on minimizing the system size and minimizing friction at th e joints. Each leg has been calibrated individually offline to determine connector properties, i.e., spring constant and free length. The spatial compliance matrix of the PCCF C has been studied to better understand the compliant property of the passive manipulat or. The forward analysis for the special 66 parallel platform as well as the kinematic control is also studied. The outcome of this research will advance the contact force /torque and pose regul ation for incontact operations.
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1 CHAPTER 1 MOTIVATION AND INTRODUCTION Motivation With the development of robotics and c ontrol technologies, many tasks can now be done with higher efficiency than ever befo re. Robotic systems can perform extremely high precision operations in automatic assembly lines as well as perform operations in environments that are too dange rous for humans to work in directly. Simple operations can be completed by position control alone and manipulators are capable of high repeatability which makes these operations eas y to control. More complex operations require the robot to be controlled as it mani pulates a work piece which comes into contact with other objects in its environment. The control of forces during incontact ope rations is a more difficult problem than that which occurs during noncontact operati ons. For example, during an incontact operation it is necessary to monitor the loads applied externally to the end effector to ensure that the forces and torques applied to the object being manipulated and the objects in the environment do not exceed allowable sp ecified values. The manipulator should provide the appropriate loads to the object to function pr operly without exceeding the external load limits of the object. As an ex ample, the manipulator may need to implant a prismshaped part into a corresponding hol e of a precision instrument. During the assembly operation, the part should be carefully fitted with minimal collision so as not to damage the component or the instrument.
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2 Industrial manipulators have been in use since the mid 1970s. Traditional serial industrial robots can be positioned and orie nted very accurately and moved along a desired trajectory. However, without some form of force control, any positional misalignment of the manipulator could cause unexpected yielding interaction with other parts if the manipulated endeffector is rigidly positioned. To satisfy these kinds of requirements, one solution is to integrate a force control scheme into the manipulator controller where measured forces are fed back in a closedloop approach. Load cells that are commerci ally available are capable of measuring a spatial wrench. These devices, however, are quite stiff and the response of the manipulator may not be fast enough to prevent high contact forces. The focus of this dissertation is on the creation a nd analysis of a more compliant spatial device that can be incorporated into the control loop of a ma nipulator in order to successfully accomplish the incontact force control task. Compliance control must be integrated in the robotic control when the manipulator is handling some fragile or da ngerous objects with force regul ation. A properly designed compliance control system has following advantages [Hua98]: 1. Avoiding collision and damage of the object and the environment. 2. Regulating the loads and wrench applied on the manipulator to meet any special external load requirements. 3. Compensating for the inevitable positional inaccuracies that results form rigid position/trajectory control. A simple example is shown in Figure 11. A serial pair of actua ted prismatic joints supports a wheel via a twospring system. The actuated prismatic join ts are controlled so that the wheel could maintain a desired cont act force with a rigid wall [Gri91A, Duf96].
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3 Figure 11. Passive compliance device for contact force control The objective of this simple application is to control the contact force between the wheel and the rigid wall when the wheel is ro lling along the surface. The serial pair of actuated prismatic joints suppor ts the body containing points B1 and B2. This body is connected to the wheel by two compliant connectors (1 B C and2 B C ). The actuator drives the end effector body with pure motion in the i and j directions to sustain the desired contact force between the wh eel and the environment and to move the wheel along the surface as specified. This can be accomplishe d given the compliant propriety of the two connectors (spring constants and free lengt hs) and the geometri cal values of the
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4 mechanism ( 1 and 2). The relationship between a ch ange in the actuator positions (1d and2d ) and the change in contact force can be analytically determined. Introduction Most industrial manipulators are serial in st ructure. The serial structure is an openended structure consisting of several links c onnected one after anothe r as shown in Figure 12. The human arm is a good example of a se rial manipulator. The kinematic diagrams of most industrial manipulators look simila r to that of the G.E. P60 and PUMA 560 industrial robots in that most consist of seven links (i ncluding ground) interconnected by six rotational joints using a sp ecial geometry (such as having three consecutive joint axes being parallel or intersecting) These structures are well constructed, highly developed, and are widely used in the i ndustrial applications. Serial manipulators do not have closed kinematical loops and are actuated at each joint along the serial linkage. Figure 12. Serial structure Accordingly, each actuator can significantl y contribute to the net force/torque that is applied to the end effector link, either to accelerate this link to cau se it to move or to influence the contact force and torque if th e motion is restricted by the environment. Since motions are provided serially, the e ffects of control and actuation errors are
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5 compounded. Further, with serially connected links, the stiffness of the whole structure may be low and it may be very difficult to re alize very fast and highly accurate motions with high stiffness. Compared to the serial ma nipulator, the parallel ma nipulator is a closedloop mechanism in which the endeffecter (mobile platform) is connected to the base by at least two independent kinematic chains (see for example Figure 13). With the multiple closed loops, it can improve the stiffness of the manipulator because all the leg connectors sustain the payload in a distributive manner as long as the device is far from a singular configuration. The probl em of endpoint positioning e rror is also reduced due to no accumulation of errors. Figure 13. Inparallel platform Hence this type of manipulator enjoys the advantages of compact, high speed, high accuracy, high loading capacity, and high stiffn ess, compared to serial manipulators. Disadvantages, however, include a limited work space due to connector actuation limits and interference.
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6 Therefore, the inparallel mechanism is perceived to be a good counterpart and necessary complement to that of the serial mani pulator. It is very useful to combine these two types of structures to utilize both adva ntages and decrease the weaknesses. One of these applications is force control of serial manipulators with an inparallel manipulator, whose leg connectors are compliant and instrume nted, is mounted at the end of the serial arm. This is the approach that will be taken here in this dissertati on; i.e., a commercially available serial industrial ma nipulator will be augmented by attaching a compliant inparallel platform to the end effector so that contact forces can be effectively controlled. Compared to parallel manipul ators, the direct forward pos ition analysis of serial manipulators is quite simple and straightforwar d and the reverse positio n analysis is very complex and often requires the solution of multiple nonlinear equations to obtain multiple solution sets. For the parallel manipul ator, the opposite is the case in that the reverse position analysis is straightforwar d while the forward position analysis is complicated. This kind of phenomenon is usually referred to as seri alparallel duality. There are similar duality propert ies between serial robots an d fully parallel manipulator with regards to instantaneous kinematics and statics. The first parallel spatial indus trial robot is credited to Pollards five degreeoffreedom parallel spray painting manipulator [Pol40] that had three branches. This manipulator was never actually built. In la te 1950s, Dr. Eric Gough invented the first wellknown octahedral hexapod with six stru ts symmetrically forming an octahedron called the universal tiretesting machine to respond to the proble m of aerolanding loads [Gou62]. Then in 1965, Stewart published hi s paper of designing a parallelactuated mechanisms as a sixDOF flight simulator, which is different from the octahedral
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7 hexapod and widely referred to as the Ste wart platform [Ste65]. Stewarts paper gained much attention and has had a great impact on the subsequent development of parallel mechanisms. Since then, much work has been done in the field of parallel geometry and kinematics such as geometri c analysis [Lee00, Hun98, Tsa99], kinematics and statics [Das98A], and parallel dynamics and controls [Gri91B]. Relative works will be discussed in a more detailed manner in the literature review section. In addition to theoretical studies, experi mental works on prototypes of the Stewart platform have also been conducted for studyi ng its property and performance. The fine positioning and orienting capability of the Stew art platform make it very suitable for use in control applications as dexterous wrists and various constructions of Stewart platform based wrists. Because there are different nomenclatur es and definitions used by different researchers in the field of parallel mechan isms, it is quite necessary to define the nomenclature [Mer00] used in current re search. The following terms are defined: 1. Parallel mechanism: A closedloop mechanism in which the endeffecter (mobile platform) is connected to the base by at le ast two independent kinematic chains. It is also called Parallel Platform or a Parallel Kinematic Mechanism (PKM). 2. Inparallel mechanism: A 6DOF parall el mechanism with two rigid bodies connected through six identical leg connectors, such as for example six extensible legs each with spherical joints at both e nds or with a spherical joint at one end and a universal joint at the other. 3. Base Platform: The immovable plate of the pa rallel platform. It is also called the fixed platform. 4. Top Platform: The moving body connected to th e base platform via extensible legs. It is also called moving plat form or mobile platform. 5. Legs: The independent kinematic chains in parallel connecting the top platform and the base platform. Also called a connector. 6. Joint: Kinematic connection between two ri gid bodies providing relative motion.
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8 7. PCCFC: Parallel Compliance Coupler for Fo rce Control. A parallel mechanism whose leg connectors are compliant and instrumented. 8. Platform Configuration: The combined positions and orientations of all leg connectors and the top platform. 9. Direct Analysis: Given the kinematic prope rties of the legs, determine the position and orientation of the top platform. Also called the forward analysis. 10. Reverse Analysis: Given the position and or ientation of the top platform, determine the kinematic properties of the leg actuators. 11. MN PKM: A PKM with M joint connecti on points on the base platform and N joint connection points on the top platform. 12. DOF: An abbreviating for degree of freedom. A generalized parallel platform is shown in Figure 14 [Rid04]. In this example the base platform and top platform are connect ed by six extensible legs. The joints connecting the legs to the top platform and base platform are spherical joints, which would introduce an additional tr ivial rotation of legs about their axes. These additional freedoms will not affect the overall system performance and could be eliminated by replacing the spherical joint on one end of each connector with a universal joint. The above parallel platform has six legs connecting the base pl atform with the top platform and there are six separate joint poi nts on the base platform and top platform respectively. This geometric arrangement it is called a 66 PKM or 6 platform according to the above nomenclature. The si x joint points on the base platform are not necessarily located on one plane, but it is often the case that all joint points lie on a single plane and are arranged in some symmetric pattern.
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9 Figure 14. General para llel platform [Rid02]. Besides the general 66 parall el platform, there are some other configurations. One common configuration of a para llel mechanism is the 33 parallel manipulator as shown in Figure 15. Figure 15. 33 parallel platform. Base Platfor m To p Pla t for m Le g connecto r Joint
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10 The 33 parallel platform also has six conn ector legs, but each leg shares one joint point with another leg on the base platform and similarly on the top platform. The three shared joint points form a tria ngle on both the planar top platform and the planar base platform. A literature review of these types of devices will be presented in the following chapter.
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11 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW Although it was Dr. Eric Gough [Gou62] who invented the first variablelengthstrut octahedral hexapod in England in the 1950 s, his parallel mechanism, also called the universal tiretesting machine, did not dr aw much public attention. Then in 1965, Stewart presented his paper on the design of a flight simulator based upon a 6 DOF parallel platform [Ste65]. This work had a great impact on subsequent developments in the field of parallel mechanisms. Since th en, many researchers have done much work on parallel mechanisms and both theoretical anal yses and practical applications have been studied. Kinematics Analysis The forward kinematic analysis of parall el mechanisms was one of the central research interests in this field in the 1980s and 1990s. While the reverse kinematics analysis, which is to calculate connector pr operties based on the top platform position and orientation information, is quite straightfo rward, the forward kinematic analysis is comparatively difficult to solve. The problem is to determine the position and orientation information of the top platform (mobile plat form) based on the connect or properties, i.e., typically the connector lengths Usually the legs are composed of actuated prismatic joints that are connected to th e top platform and base platfo rm by a spherical joint at one end and either a spherical or universal joint at the other. The connector properties are generally referred to as the connector length or leg length as this parameter is usually measured as part of a closedloop feedback scheme to control the prismatic actuator.
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12 It has been observed [Das00]that th e closedform solution for the forward kinematics problem could be simplified by regul ating the connector joint locations. The simplest case is the 33 platform, which is degenerated by grouping the connector joints into three pairs for both the top platform a nd base platform; each pair of connectors shares one joint point. Alt hough it is difficult to design a ph ysical octahedral structure with the concentric doublesphe rical joints that would have sufficient loading capacity and adequate workspace due to collisions of the leg connectors themselves, the forward analysis is the simplest for this class [Hun98]. The forwar d analysis of this device was first solved by Griffis and Duffy [Gri89] w ho showed how the position and orientation of the top platform can be determined with respec t to the base when given the lengths of the six connectors as well as the geometry of th e connection points on the top and base. This solution is a closedform solution and is based on the analysis of the input/output relationship of a series of three spherical four bar mechanisms. The resulting solution yielded a single eighth degree po lynomial in the square of one variable which resulted in a total of sixteen distinct solution poses fo r a given set of connector lengths. Eight solutions were reflected about the plane fo rmed by the base connector points. The solution technique was extended to solve other special configurations such as the 44 and 45 platforms with extended methods [L in90, Lin94]. Other methods [Inn93, Mer92] used constraints to solve the problem. First, part of the structure is ignored so that the locus curves of the connector joints could be determined according to the connector and joint properties. Then the removed part toge ther with the angular and distance constraints is added back for further determinations of the solution for th e forward kinematics analysis.
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13 Some theoretical analyses [Rag93, Wen94] found that there are as many as 40 solutions for the forward kinematic equations Hunt and Primrose used a geometrical method to determine the maximum number of the assembly [Hun93]. In 1996, Wampler [Wam96], found that the maximum number of possible configurations for a general Stewart platform is forty by using a continuation method. Singularity Analysis A singular configuration is some special configuration in which the parallel mechanism gains some uncontrollable freedom. For parallel manipulators, there are different types of singularity c onditions based on the analysis of the Jacobian matrix that is formed from the lines of acti on of the six leg connectors [Zha04]. Both Tsai [Tsa99] and Merlet [Mer00] poi nt out in their books that there are 3 types of singularities based on the Jacobian matrix analysis: the inverse kinematic singularities where the manipulat or loses one or more degree s of freedom and can resist external loads in some dire ctions; the direct kinematic singularities where the top platform gains additional degree(s) of freedom and the parallel platform cannot sustain external loads in certa in direction (or just uncontrollable); and th e combined singularities which could exhibit the features of both th e direct kinematic singularities and reverse kinematic singularities. Other researchers [Dan95, Ma91] fo und an interesting phenomenon called architecture singularities, which means that some parallel manipulators with particular configurations exhibit conti nuous motion capabilities in a re latively large portion of the whole workspace with all actuators fixed. Th is kind of singularity should be avoided in the early stage of design.
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14 One of the reasons to study singularities is to avoid or restri ct any singularities from occurring within the effective worksp ace. Dasgupta [Das98B] developed a method to plan paths with good performance in th e workspace of the manipulator for some special cases although there is not solid evidence to prove that the method is applicable to all beginposetoendpose path planning. Statics and Compliance Analysis The forward static analysis is very st raightforward as the end effector (top platform) force can be directly mapped fr om the connector forces, as described by [Fic86]. Duffy [Duf96] perf ormed stiffness mapping analyses when considering the relationship between the twist (instantaneous motion of the top platform) in response to a change in the externally applied wrench. The mapping matrix (compliance matrix) is related to the stiffness properties of the c onnectors as well as the geometric dimension values. The static equilibrium equation for a para llel mechanism with six leg connectors can be written as = { f ; m } = { f1 ; m01} + { f2 ; m02} + + { f6 ; m06} (21) where is the applied external wrench applied to the top platform which is comprised of a force f applied through the origin point of the reference coordinate system and a moment m (dyname interpretation). The terms fi and mi, i = 1..6, represen t the pure force that is applied along each of the leg connectors, i.e., fi represents the force applied along connector i and m0i represents the moment of the fo rce along connector i measured with respect to the origin of the reference frame. This equation may also be written as = f{ S ; S0} = f1{ S1 ; S0L1} + f2{ S2 ; S0L2} + + f6{ S6 ; S0L6} (22)
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15 where f is the magnitude of the external wrench, { S ; S0} are the coordinates of the screw associated with the wrench, fi represents the magnitude of the force along connector i, and { Si ; S0Li} are the Plcker line coordinates of the line along connector i [Cra01]. This equation can also be written in matrix format as 1 2 1234563 00102030405064 5 6 f f f f f f f SSSSSSS SSSSSSS (23) where the 6 matrix formed by the Plcker coordinates of the lin es along the six leg connectors comprises the Jacobian matrix that relates the force ma gnitudes along each leg to the externally applied wrench. A derivative of the above equation will yi eld a relationship between the changes in individual leg forces to the change in the externally applied wrench. Griffis [Gri91A] extended this analysis to show how the change in the externally applied wrench could be mapped to the instantaneous motion of the top platform. According to the static force analysis presented above, it is straightforward to design a compliant parallel plat form to detect forces and torques. Knowledge of the spring constant and free length of each conn ector and measurement of the current leg lengths allows for the computation of the external wrench, once a forward position analysis is completed to determine the curren t pose. The first sensor of this type was developed by Gaillet [Gai83] based on the 33 pa rallel octahedral structure. Griffis and Duffy [Gri91B] then introduced the kinestatic control theory to simultaneously control force and displacement for a certain constrai ned manipulator based on the general spatial
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16 stiffness of a compliant coupling. In their th eoretical analysis of the compliant control strategy, a model of a passive Stewart platform with compliant legs was utilized to describe the spatial stiffness of the pa rallel platform based wrench sensor. Compared to the open loop Remote Center Compliance Modules (RCC) that are commercially available (see Figure 21), the St ewartplatformbased forcetorque sensor can provide additional information about external wrench to assist force/ position control. Dwarakanath and Crane [Dwa 00], and Nguyen and Antrazi [Ngu91] studied a Stewart platform based force sensor, using a LVDT to measure the compliant deflection for wrench detection. Some researchers [Das94, Sv i95] also considered the optimality of the condition number of the force transformation matrix for isotropy and stability problems. It is important that any wrench sens ing device be far from a singularity configuration. Lee [Lee94] defined the problem of closeness to a singularity measure by defining what is known as quality index (QI) for planer inparalle l devices. Lee et al. [Lee98] extended the definition of quality inde x to spatial 33 inparallel devices. The quality index is the ratio of the determinan t of the matrix formed by the Plcker line coordinates of the six connector legs of the platform in some arbitrary position to the maximum value of the determinant that is possible for the inparallel mechanism. The index ranges from 0 to 1 and the maximum valu e of 1 refers to the optimal configuration. Lee [Lee00] provides a detailed quality index presentation on several typical inparallel structures.
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17 Figure 21. Remote Center Compliance Coupling Compliance and Stiffness Matrix The theory of screws is a very powerfu l tool to investigate the compliance or stiffness characteristic of a compliant device. Ball [Bal00] first introduced the theory and used it to describe the general motion of a rigid body. He presented the idea of the principal screw of a rigid body, where a wrench applied along a principal screw of inertia generates a twist on the same screw. The principal screw can be found by solving the spatial eigenvector problem. Dimentberg [Dim65] analyzed the static and dynamic properties of a spring suspended system using screw theory. Patterson and Lipkin [Pat90] studied the n ecessary and sufficient conditions for the existence of compliant axes and showed a new classification of general compliance matrix based on the numbers of the compliant axes produced.
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18 Loncaric [Lon87] also did research on unl oaded compliant systems for the spatial stiffness matrix and normal form analysis by using Lie groups. Then Loncaric [Lon91A] used a geometric approach to define and an alyze the elastic system s of rigid bodies and described how to choose coordinates to simp lify the interpretation of the compliance and stiffness matrix. Lipkin and Duffy [Lip88] presented a hybrid control theory for twist and wrench control of a rigid body, and showed that it is based on the metric of elliptic geometry and is noninvariant with the cha nge of Euclidean unit length and change of basis. Duffy [Duf96] analyzed plan er parallel spring systems theoretically. Griffis and Duffy [Gri91B] studied the nonsymmetric sti ffness behavior for a special octahedron parallel platform with 6 springs as the c onnectors, and the stiffness mapping could be represented by a 6 matrix [K]. Patte rson and Lipkin [Pat92] also found that a coordinate transformation changes the stiffness matrix into an asymmetric matrix but the eigenvectors and eigenvalues are invariant under the coordinate tr ansformation, and the eigenvalue problems for compliance matrix and stiffness matrix are shown to be equivalent. Lipkin [Lip92A] introduce a geometric decomposition to diagonalize the 6 compliance matrix and the diagonal elements are the linear and rotational compliance and stiffness values. It is also proved that th e decomposition always ex ists for all cases. Another paper [Lip92B] presents several geom etric results of the 6 compliance matrix via screw theory. Huang and Schimmels [Hua00] also studi ed the decomposition of the stiffness matrix by evaluating the rank1 matrices that compose a spatial stiffness matrix by using
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19 the stiffnesscoupling index. The decom position, which is also called eigenscrew decomposition, is shown to be invariant in co ordinate transformation with the concept of screw springs. Roberts [Rob02] points out that the normal s tiffness matrix with a preload wrench is asymmetric. However when only a small twist from the equilibrium pose is considered, the spatial stiffness matrix is a symmetric positive semidefined matrix and the normal form could be derived from it. Ciblak and Lipkin [Cib94] have shown that for a preload system with relative large deflection from its original pos ition, the stiffness matrix is no longer symmetric. In a more recent study, Ciblak and Lipkin [Cib99] present a systematic approach to the synthesis of Cartesian stiffness by springs us ing screw theory, and shown that a stiffness matrix with rank n can be synthesized by at least n springs. Researchers [Cho02, Hua98, Lon91B, Rob00] also work on synthesis for a predefined spatial stiffness matrix.
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20 CHAPTER 3 PARALLEL PLATFORM DESIGN The parallel mechanism offers a straight forward method to determine the external wrench applied to the top platform based on measured leg lengths and knowledge of the system geometry (location of joint points in top and base platforms), connector spring constants, and connector spring free lengths. Because of this, the Passive Compliant Coupler for Force Control (PCCFC) can be ut ilized in force feedback applications for serial robots by employing it as a wrist elemen t that is between the distal end of the manipulator and the environment. In this ch apter, a prototype for the compliant parallel platform has been designed and tested in order to study the position/wrench control with the PKM device. Design Specification The design process could roughly be composed by 3 stages [Tsa99]: 1. Product specification and planning stage, 2. Conceptual design stage, 3. Physical design stage. During the first stage, the functionalit y, dimension, and other requirements are specified for the development of the produc t. In the second stage, several rough conceptual design alternatives are develope d based on the product specification and the one with best overall performance is chosen to be developed. In the last stage the dimensional and functional design, analysis and optimization, assembly simulation, material selection, and engineering docume ntation are completed. Although the third
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21 stage usually is the most timeconsuming stage, the first two stages have great impact on design results and cost control. In the PCCFC specification and planning stage, the products geometrical and functional requirements are specified as follows: SP1: The mechanism should be compliant in nature to measure contact loads; SP2: The mechanism should have 6 degrees of freedom; SP3: The mechanism should be a spatial parallel structure; SP4: The mechanism should be ability to measure loads without large errors; SP5: the structure should have a relatively compact size. As mentioned in the previous chapter, the Stewart platform is a promising structure for such spatial compliant devices. The Stew art platform structure satisfies the SP2 and SP3 criteria. With each connector being co mposed of a prismatic joint and compliant component, the SP1 and SP4 criteria could also be satisfied. One of the main advantages of the Stewart platform is that it can with stand a large payload (compared to serial geometries) with relative compact size. As a result, the dimensi onal requirement SP5 could be met during the conceptual design and ph ysical design stage. The detailed design specifications are presented in Table 31. Note that in this table, the direction of the z axis that is referred to is perpendicu lar to the plane of the base platform. Table 31. Design objective specification Specification Range Individual Connector Deflection : Magnitude 5mm Maximum perpendicular load: Magnitude 50N Motion range in Z direction: Magnitude 4mm Motion range in X,Y direction: Magnitude 3.5mm
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22 Table 31. Continued Specification Range Rotation range about Z axis: Magnitude 6 Rotation range about X,Y axis: Magnitude 4 Position Measuring Resolution: 0.01mm Conceptual Design In the conceptual design stage, a best conceptual design is selected from among alternatives. The 33 octahedral structure was the first to be considered since it is the simplest structure of its kind. The mobility of general spatial mechanisms can be calculated from the Grbler Criterion [Tsa99]: j 1 i i) f ( ) 1 n ( M, (31) where: M: Mobility, or number of degr ee of freedom of the system. : Degrees of freedom of the space in which a mechanism is intended to function, for the spatial case, = 6 n: Number of links in the mech anism, including the fixed link j : Numbers of joints in a mechanism, a ssuming that all the joints are binary. i f : Degrees of relative motion permitted by joint i. The octahedral 33 parallel platform is shown in Figure 31. It is the simplest geometry for a spatial parallel platform. Fo r the 33 parallel platform, the octahedron has a top platform and a base platform, 6 connector s, and 6 concentric s pherical joints, three on the base platform and three on the top platform. Each conn ector is treated as 2 bodies
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23 connected by a prismatic joint. Thus there are a total of 14 bodies; 2 for each of the 6 legs, the top platform, and the base platform. Each spherical joint has 3 relative degrees of freedom and thus there are a total of 12 joints that each has 3 degrees of relative freedom. Figure 31. Plan view of octa hedral 33 parallel platform For this case, the mobility equation becomes 12 30 36 78 ) 1 6 ( ) 3 6 ( ) 1 14 ( 6 M6 1 i 12 1 i (32) Six out of the 12 degrees of fr eedom are trivial in that e ach leg connector can rotate about its own axis. Eliminating these from c onsideration results in a 6 degree of freedom device. The forward kinematic analysis was first performed by Griffis and Duffy [Gri89]. They discovered that a maximum of sixteen pose configurations can exist for this device (eight above the base platform and eight more reflections below the base platform) when given a set of leg connector lengths. Hunt does a comprehe nsive study of the octahedral
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24 33 parallel platform [Hun98]. L ee [Lee94] analyzed the singul ar configurations of this device and defined the problem of closeness to a singularity by de fining what is known as the quality index (QI) for planer inpara llel devices. Lee et al [Lee98] extended the definition of quality index to the spatial 33 inparallel device. The quality index is the ratio of the determinant of the matrix formed by the Plcker line coordinates of the six connector legs of the platform in some arbitrary position to the maximum value of determinant that is possible for the inparalle l mechanism. This index is a dimensionless value: max    J J (33) where J is the matrix of the line coordinates. The quality index has a maximum value of 1 at an optimal configuration that is show n to correspond to the maximum value of the determinant. As the top platform departs from this configuration relative to the base, the determinant would decrease, and at some positions and orientations it can reach zero which indicates an uncontrollable singularity state. As previously stated, the mechanism has been designed such that the determinant of the matrix J will be near the maximum at the unloaded home position. Lee [Lee00] showed that for a 33 platform with a top tr iangle of side length a and a bottom triangle of side length b, that the maximum value of th e determinant of J would occur when the top platform was parallel to the base and was rotated as depict ed by the plan view shown in Figure 31. The determinant of the matrix J was shown to be given by 3 2 2 2 3 3 3h 3 b ab a 4 h b a 3 3 J (34)
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25 where h is the separation distance or height of the top platform with respect to the base. Lee proved that the maximum value for the determinant of J would occur when 221 () 3 haabb (35) and 2 ba (36) From (35) and (36), it is easy to obtain 22 bah (37) Although the geometric configur ation of the 33 parallel platform has the simplest geometry with regards to the forward analysis it is difficult to design and implement in practice due to the pair of concentric spherical joints or other type of universal joints that connect each leg to the base and top platform These kind of concentric joint pairs are not only very difficult to manuf acture but also very hard to have sufficient payload capacity. This configuration also induces unwanted interference be tween moving legs. One other alternative is the 63 platform which has six points of connection on the base platform and three points of connec tion on the top platform. While this 63 configuration has less concentric joints pair in the base platform, it still has the same difficulty and complicated arrangement of joint connections on the top platform. In order to completely eliminate the need for coincide nt connections, the 66 parallel platform was studied. The forward analysis for a general 66 platform, however, is very complicated requiring lengthy computation and resulting in a maximum of forty configurations. Griffis and Duffy [Gri93] developed a sp ecial 66 platform geometry, however, where the forward analysis was comparable in difficulty to that of the 33 platform. As
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26 opposed to a general 66 parallel platform, th e special 66 platform has six leg connector points in a plane on the top a nd base platforms. The arrangement of these connecting points is such that three conn ecting points of the platform de fine a triangle as the apices and the other three connecting points lie on the sides of this triangle. There are a few configurations of the leg connection: one configuration, known as the apex to apex, has three of the legs attaching to the corners or apices of the triangle of the top platform and the base platform, and the remaining legs are attached to the sides or edges of the platform. The other pref erred configuration is known as apex to midline as shown in Figure 32. Figure 32. Special 66 parallel platform.
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27 In order to eliminate the six additional ro tational freedoms, the six spherical joints connecting the top platform and leg connectors are replaced by universal joints. Equation (31) is utilized here to ca lculate the mobility of the special 66 parallel platform 6 30 24 18 78 ) 1 6 ( ) 2 6 ( ) 3 6 ( ) 1 14 ( 6 M6 1 i 6 1 i 6 1 i (38) The legs 16 are arranged in such a way that each of them is connecting to a corner connecting point on either the t op or base platform while the other connecting point is in the midline of the triangle of the opposing platform. For example, leg 6 is connecting the top platform 19 on the corner connecting point 18 with the base platform 20 on the midline connecting point 12. Griffis noted that this particular configuration is singular Figure 33. 3D model for speci al 66 parallel mechanism If the midline connecting points are not exac tly on the middle of the triangle side and for each side of the triangle, and the poi nts are distributed symmetrically, there are two different configurations which are us ually called clockwise configuration and
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28 anticlockwise (the rotation direction of the top platform when pressed) configuration based on the plan view of the structure. Th ere is no significant analytical difference between the two configurations for the kine matic analysis and c ontrol, so only one configuration is used to bu ild the prototype. One computer generated model of these configurations is s hown in Figure 33. Prototype Design After choosing the desired configurati on of the prototype, more detailed and specific information about the parallel platform should be studied in order to build the physical mechanism. One plan view of the configuration of the parallel platform is shown in Figure 34. The dimensional relationship between the top/base platform and the free length of the legs was studied by Lee [Lee00] based on the optimal dimensional restriction for the special 66 parallel platform stated in the previous section. Th e free lengths of the connectors are calculated by the following equations: (39) As shown in Figure 34: l: free length of the legs, divi ded into two groups based on th e fact that there are two different values for free length of the legs. a: refers to the functional triangl e edge length of the top form b: refers to the functional triangl e edge length of the base form. : refers to the rotation angle of th e top platform about the zaxis. 2222 22221 3(331)(cos3(21)sin) 3 1 3(cos3(21)sin)(331) 3long shortlhaabb lhaabb
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29 a,b: refer to the distance between adjacent jo int points on the base platform and top platform. Figure 34. Plan view of special 66 platform The design process is show n in the Figure 35 below
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30 Figure 35. Flow char t of the design process Consider the requirements of feasibility with regards to dimensional limits for manufacturing parts, encoders available in the market, a nd the need for a continuous working space with no interf erence of moving parts, a 3D modeling and assembly simulation resulted in the following choices fo r the size of the base and top platforms and the lengths of the two sets of conn ectors at the unloaded home position:
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31 a = 60 mm, b =120 mm, = 28/120 = 0.2333, h = a= 60mm, Lshort = 68.021 mm, Llong = 80.969 mm. The elastic component is a crucial elem ent for the passive compliant parallel manipulator prototype. Preci sion linear springs were selected for use as the elastic component to provide elongation and compression of the leg connectors. After carefully studying and comparing different springs, a grou p of cylindrical precision springs with different spring constants and the same high linear coefficient ( 5%) were selected to use as the elastic component in the legs. The spring rates were respectively 8.755 N/mm, 2.627 N/mm and 0.8755 N/mm (or 50 lb/in, 15lb/in and 5 lb/in). By using different groups of springs, the stiffness matrix of the parallel platform could be changed by replacing the spring and thus the device can be applied to different applications where the expected force ranges are different. The free length variation of each connector is measured by a linear optical encoder. Each leg connector is comprised of two parts that translate with respect to each other and are interconnected by the spring. The encoder read head is attached to one of the leg parts and the encoder linear scale is attach ed to the other. The two leg parts are constrained by a ball spline to ensure that they translate re lative to one another and to help maintain alignment of the encoder read head and linear scale.
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32 The initial free length of each connector is measured by microphotography with a specially designed clamp appara tus. Since the six connectin g points on the base platform are in the same plane, the centers of the ps eudospherical joints ar e also in the same virtual plane. The free length of the connect ors is defined as the distance between the two centers of the joints, which connect the c onnector with the top platform and the base platform. Figure 36 shows renderings of the protot ype device. Figure 37 presents a photograph of the prototype. A B Figure 36. Assembled model for special 66 parallel platform. A) Solid model, B) Frame model.
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33 Figure 37. Photo of the a ssembled parallel platform
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34 CHAPTER 4 SINGULARITY ANALYSIS In this chapter, a singula rity analysis is presente d and a special singularity configuration for a special 66 parallel platform is analyzed. It is important singular ity configurations, which are identified as the case when the Plcker line coordinates of the six le g connectors become linearly dependent, be avoided throughout the effective workspace of th e device. Singularities can occur in both planar and spatial mechanisms. As an example, a planar structure is shown in Figure 41. The structure is in a singular configuration when point S is at location (0, 0). At this instant, considering the displacement of point S as the input and the displacement of point D as the output, it is not possible to obtain a user desired velocity for point D, no matter what the velocity of the input is. This c ondition where point S is at (0, 0) is called a singular point as distinct from other simple points, which are also called regular points [Hun78]. For this group of mechanism, a singularity only occurs when the mechanism comes to some discrete special configuration while for all other point s or configurations the mechanism is normal or controllable. Partic ularly for parallel st ructures, it could be said that singularity configurations exist in the workspace at those configurations where the parallel structure gains some uncontrollable freedom.
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35 Figure 41. Planar singularity For spatial parallel manipulators, there ar e different types of singularity conditions based on the analysis of the Jacobian matrix th at is formed from the lines of action of the six leg connectors. For example, a parallel manipulator may be able to resist external loads in some directions with zero actuator forces at an inverse kinematical singular configuration. Some manipulat ors can also have a different kind of singularity condition, in which with all the actuators are lock ed, the top platform could still move infinitesimally in some direction. Further, there are combined singul arities that occur for some special kinematic architecture wh en both conditions mentioned above occur [Tsa99]. All the singularity conditions described a bove are temporary and conditional. This means that only when the special geometrica l configuration occurs, the mechanism will function differently from the norm. The special 66 parallel platform is considered here and the joint arrangement is shown in Figure 42. Lee [Lee00] showed that when the dimensional ratio p=q=1/2, where p and q represent the offset percentage of the joints in the base and top plat form respectively, the determinant of the Jacobian matrix is zero and the platform is in a singularity. He also D S (0, 0)
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36 identified the singularity case that occu rs when the top platform rotates 90 around the Zaxis. Figure 42. Plan view of a special 66 parallel platform Certain geometries exist, however, that ar e always singular. One planar example is the parallelogram. A parallelogram with four revolute joints is normally unstable i.e., it could not sustain external load s with internal angles unchange d. Compared to the very stable triangle, such special geometries ar e often referred to as unstable as opposed to being described as being in a singular state [Tsa 99]. Examples of unstable spatial mechanisms can also be found. Figure 43 shows the particular case that was identified in this research. This special geometry is always unstable, or in a singularity condition. It will be shown that the top platform will have continuous mobility when the lengths of the six leg connectors are fixed. For this pb qa
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37 mechanism, both the top and the base platform s are equilateral triangles. The following notation is used for this speci al singularity configuration: a: the length of the side of the top equilateral triangle, b: the length of the side of the base equilateral triangle, q,p: a dimensionless number in the range of 0 to 0.5 that defines the offset distance of the connection points (for example, the distan ce between points Ec1 and Ec2 is pb and the distance between points B1 and B2 is qa), h: the vertical distance from the geometric cen ter of the base plate to the center of top plate. Figure 43. Plan view of the special singularity configuration In order to simplify the process, here the structure will be analyzed only in an initial configuration where the two plates ar e parallel and the relative rotation angle is zero. The analysis process is the same for other positions and orientations. The matrix of the line coordinates along the connectors for this configurat ion is written as qa
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38 (1)(21)(12)(1) 222222 3[(31)]3()3()3[(31)]3[(23)2]3[2(23)] 666666 3333(31)233(23) 666666 (21)(1) 0 22222 33 6 qabapbqabapbqapb aqbbabaapbaqbapb hhhhhh bhbhbhpbhbhpbh bhpbhbhpbhpbh qabp 3333 66666 abqabpabqabpab (41) The determinant of the Jacobian matrix is ,(42) It was shown by Lee [Lee00] when analyzing the mechanism depicted in Figure 42 that when p=q=0.5, the determinant of the J acobian matrix equals zero and the structure is unstable. For all other values for p and q in the range of 0 to 0.5, the structure is stable. For the structure shown in Figure 43, th e result is quite different. Clearly the determinant of J is zero when p=q. Under the constraints 0 q, p 0.5, the term 2()3(1)(1) p qpqpq is clearly nonnegativ e and is zero precisely when p=q=0. Consequently, we have that under the natura l physical constraints 0
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39 A photo of the platform model is shown in Figure 44. From th e picture it can be seen that the structure has collapsed due to th e gravity effect until leg interference occurs. The structure could sustain the gravity of the top plate in the shown state only because of collisions of adjacent leg connectors which st ops it from moving downwards further. The top plate could easily be moved upwards with little upwards force applied. Figure 44. Photo of the para llel platform in singularity Although singular conditions are typically so mething to be avoided, there is one application where this particular geometry w ould be of use. Platforms that incorporate tensegrity must be in a state that when no exte rnal load is applied to the top platform, the lines of action of the six leg connectors mu st be linearly dependent. Tensegrity based platforms incorporate compliance such as spri ngs to prestress certa in leg connectors in tension which will place the other connectors in compression. The internal forces in the
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40 leg connectors can only sum to zero when the lines of acti on are linearly dependent. Thus, the platform geometry discussed here could have particul ar application when tensegrity is incorporated in the platform.
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41 CHAPTER 5 STIFFNESS ANALYSIS OF COMPLIANT DEVICE For a compliant component, such as a spring, the basic property of it is its stiffness. Theoretically, no object is infinitely stiff; the only difference between a so called stiff component and compliant device is that comparing with the compliant component, the stiff component has a relati vely much higher rigidity or stiffness. The spring is the most commonly used compliant component. The stiffness property of a spring is usually quantified by a ratio called the spring consta nt, also known as the el astic coefficient or stiffness coefficient and is used to describe compliant devices. In this chapter, simple analyses for planar compliant devices are introduced together with an example of a compliant 2D contact force control system. Then the stiffness analysis of a 3D system is studied using Screw theory. The stiffness matrix for the special 66 parallel manipul ator is developed and a numer ical example is provided for the specific design. Simple Planar Case Stiffness Analysis Hookes Law describes the fundamental rela tionship between an external force and the compliant displacement in static equilibrium 0()s F klkll (51) where s F is the force exerted on th e spring (or more generally, the compliant component), k is the spring constant, l is the current length of the spring, l0 is the free length, and l is the displacement from the equilibrium position.
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42 Consider two springs that are connected inparallel, with spring constants1k and2k respectively. An external force is applie d on the right end of th e two springs and the force direction is along the axes of the springs. Figure 51. Planar inparallel springs The force in each spring can be calculated as fs1 = k1 l1 fs2 = k2 l2 where in this case l1 = l2 = l The total force applied to the across the two springs can be written as fs = f1 + f2 (52) The equivalent mapping of stiffness of the model is derived by: 112212()()s e f klklkkl kl (53) where ek is the equivalent stiffness/sp ring constant of the inpar allel connected springs. It is the sum of the individual spring consta nts. This shows that when compliant components are connected inparallel, the overa ll equivalent stiffness is much higher than the stiffness of any individual compliant component. s f 1 s f 2 s f 1k 2k
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43 Figure 52. Planar serially connected springs system Now consider two springs connected in se ries, as shown in Figure 52 above. The axes of the two springs are collinear and they are connected end to e nd. The direction of the external force applied on them is also al ong the axes of the springs. The derivation is quite simple, so only the result is provided here as 12()sss f klkll (54) where 12111skkk The term s k is the equivalent stiffness/spring c onstant of the serially connected springs. Its reciprocal value is the sum of the reciprocal values of the individual spring constants. This means that when compliant components are connected serially, the overall equivalent stiffness is less than the stiffness of individual compliant component. If 12kkk then 2ek k and if12kk, then1ekk This shows that serially connected components with wi dely different spring consta nts, have an equivalent stiffness, which is more dependent on the component with the smaller spring constant. According to this result, if one compliant component is connected to a relatively stiff bar, the equivalent stiffness of this twocomponent system is very close to the stiffness of the compliant component. The systems discussed above have one degreeoffreedom (DOF). A more general planar case would have two or more DOF. s f 1 s f 2 s f 1k 2k
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44 X Figure 53. Planar 2 DOF spring Figure 53 shows a planar 2 DOF spring case. In this spring system, two translational springs are connect ed at one end P and grounded se parately at pivot points A and B respectively. Here translational springs behave like prismatic joints in revoluteprismaticrevolute (RPR) serial chains. In the XY plane, two such springs form the simple compliant coupling as shown in Figure 53. The twospring compliant coupling system is equivalent to a planar twodimensional spring. The spring is twodime nsional because two independent forces act in the translational spring, and it is plan ar since the forces remain in a plane. The external force applied at point P is in static equilibrium with the forces acting in the springs. The twodimensional spring remains in quasi static equilibrium as the point P move gradually. In order to analyze the twodimensiona l force/displacement relationship, or mapping of stiffness, it is necessary to decompose both the external force and displacements into standard Cartes ian coordinate vectors such that xy f fxfy A P B fx fy K2 K1 2 1 Y
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45 The locations of points A, B, and P, the initial and current lengths of AP, BP, and the angles 12, are all known. The free length of AP is 01l and the free length of BP is02l The spring constants are1k and2k The current length of AP and BP are 1l and 2l respectively. To simplify the e quations, dimensionless parameters 1011ll and 2022ll are introduced. By definition, these two scalar values are always positive, and no negative spring lengths are allowed. When i >1, the corresponding spring is elongated, and if i is less than 1, then it is compressed. The external force applied on the spring system is given by: 12111 12222(1) (1)x yf cckl f sskl (55) where cos()iic andsin()iis Differentiating the above equation will result in the following equation x yf x k f y (56) where [k] is the mapping of the stiffness of the system which can be written as [Gri91A] 111212111 212212222 121111 1222220 [] 0 (1)0 0(1) kkcckcs k kksskcs ssksc ccksc (57) The spring matrix [k] can be written in the form (1)TT iiikjkjjkj (58) where [j] is the static Jacobian matrix of the system, [ j ] is the differential matrix with respect to 1 and2 and ik, 1iik are 2 2 diagonal matrices. In general12
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46 as if the two angles are equal, then th e spring matrix expression is singular and meaningless. For that case the two springs are also parall el and equation (53) can be applied instead. Planar Displacement and Force Representation Screw theory is utilized in the analysis of more complex compliant systems, such as the structure shown in Figure 11. In screw theory [Bal00, Duf96], a line in the XY plane is written using Plcker coordinates as a triple of real numbers {L,M; R}. L and M represent the direction of the line in the XY plane and as such are dimensionless. R represent the moment of the line about the Z axis and has units of length. For this analysis the direction of th e line will be represented by a unit vector and thus 1 22 2()LM =1. The Plcker coordinates of a line are often written as } ; { 0S S s (59) where S= (Li + Mj) and S0= (Rk). Although Plcker coordinates are homogeneous, i.e., { S ; S0} describe the same line for all man zero values for it will be assumed throughout this analysis that S is a unit vector. The subscript 0 is introduced to indicate that the moment of the line about the Z ax is will change with a translation of the reference coordinate system. A force is represented by a line multiplied by a force magnitude. A twist is represented by a line multiplied by an angular ve locity. It is interesting to note that a pure moment of magnitude m is written as {0, 0; m} and a pure translation of magnitude v is write as {0, 0; v} which represent lines at infinity that are multiplied by the moment magnitude and the velocity magnitude respectively.
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47 The resultant of an arbitrary set of planar forces fi {Si; S0i}, i=1..n, can always be represented by a wrench which in the planar case reduces to a particular linebound force which may be calculated as } ; { f } ; { fi 0 i n 1 i i 0S S S S w (510) This resultant is often written as } ; { 0m f w (511) where f = f S and m0 = f S0. Now, consider that three forces in the XY plane of magnitude 123,, f ff are applied to a rigid body. The equivalent resu ltant force can be determined by the sum of the force vectors as 3 12 112233 3 12 1 000123 23 123wfsfsfs SSS f+f+f SSS off ff mm mm (512) Equation (512) can be further transf ormed with the following expression 123 1123 12323ccc wfs+fs+fs p pp (513) where si and ci represent the sine and cosine of an orientation angle of line i measured with respect to the X axis and pi represent the moment of line i Equation (513) may be writt en in matrix form as wj (514) where
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48 123 123 123ccc jsss p pp 1 2 3 f f f Given the magnitude of the external forces and their coordinates of each line of action, the coordinates of the resultant force (both magnitude and the line of action) can be determined by the above equation. For stat ic equilibrium [Duf96], there must be one corresponding external force also applied on the rigid body with equal magnitude and opposite direction. Stiffness Mapping for a Planar System The general mapping of stiffness is a one toone correspondence that associates a twist describing the relative displacement between the bodies with the corresponding resultant wrench which interacts between them Considering a simple 2D planar system shown in Figure 54, the rigid moving body is connected to ground by three RPR compliant serial chains. A linear spring is located inside the prismatic joint and axial spring forces are applied both on the top movi ng stiff body (to simplify, it is also called top platform) and the ground. According to equation (513), the re sultant force can be calculated. In order to maintain equilibri um, an external force with the exact same magnitude f and opposite coordinatesw much be applied on the top platform.
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49 Figure 54. Planar compliant system 112233 110112202222022 ()()()wfsfsfs kllskllsklls (515) where twist wcontains the magnitude and geometrical information of the resultant force, 1 f ,2 f and 3 f are the magnitudes of the sp ring forces correspondingly, 12 ,ss and 3sare the line coordinate of the axis of the connectors, 1k,2kand 3kare the spring constants and 0(),(1,2,3)iilli is the change in length between the current connector length and the initial length correspondingly. The detailed derivation process can be f ound in [Duf96], and only the result of the stiffness mapping is provided here as wKD (516)
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50 where the stiffness matrix [K] is (1)T TKjkjBkC (517) where 1 2 300 00 00 k kk k 11 22 33(1)00 [(1)]0(1)0 00(1) k kk k and 0/iiill 123 123 123 ccc jsss p pp 123sss, 123 111123 123BBB BBBsss B sssccc qqq and 123 123123 123CCC CCCsss Csssccc qqq /iBiisdsd are the coordinates of the line that is perpendicular to is that passes though a fixed pivot i B icsare the coordinates of the line iCS that is parallel toiBSand through pivot iC, as shown in Figure 55, Figure 55. Stiffness mapping fo r a planar compliant system
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51 Stiffness Matrix for Spatial Compliant Structures The analysis of stiffness matrix for the planar structure was presented in the previous sections. Now the spatial stiffness property will be explored In this section, necessary concepts of projective geometry fo r point and linevector are provided briefly, followed by the analysis of a spatial 33 oc tahedron and the desired special 66 spatial parallel manipulator. In order to perform the analysis, it is esse ntial to expand some of the concepts from the 2D plane to 3D space. In the planar case, the Cartesian coor dinate for a point is defined by two dimensionless scalar values: X and Y, which are also dependent on the reference point of the selected coordina te system. In Screw theory [Cra01], homogeneous coordinates (w; x, y, z) are us ed to describe the location of a point and iii i i x yz r w i j k refers to the position vector from th e reference point O to the point A with Cartesian coordinate(,,)iii x yz. Usually it is assumed that w=1 so that the ratios of x/w, y/w, z/w are equal to x, y and z. This homogeneous coordinate system also allows for a point at infinity. When w=0, the point A is at infinity in the direction parallel to () x yz i j k For points not at infini ty, w is always a nonzero value. To simplify the expressions in this work, the coordinate of a point will also be expressed as (x, y, z). It is clear that in the planar case, the equation of a line can be expressed by three numbers L, M, and R, which are first introdu ced by Plcker, and ar e called Plcker line coordinates. In 3D space, the ray coordina te of a line is defined by any two distinct points on the line. The coordina te of point 1 and point 2 are 1111(,,) x yzr and
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52 2222(,,) x yzr respectively. The vector S whose direction is para llel to the line could be written as: () 21Srr (518) or LMN Si j k (519) So vector S provides directional information fo r the line segment composed by the two points. Now cons ider another vector r from the reference origin to any general point 111(,,) x yzon the line, the cross product of the two vectors defines a new vector which is perpendicular to both of the vectors. This is written as OL SrS (520) The vector OLS is the moment of the line about th e origin O and is clearly origin dependent, while the vector S provides the directional information for the line. Thus the coordinate of a line is written as {}OLS;S and this is also called the Plcker coordinates of the line. The semicolon in the coordi nates indicates that the dimensions of S andOLS are different, i.e., S is dimensionless while S0L has units of length. From equation (520), the moment vector OLS is: 111OL x yzPQR LMN ijk Si j k, (521) where 11 11 11PyNzL QzLxN R xMyL (522) The Plcker coordinates for the line jo ining two points with coordinates 1111;,, x yzand 2221;,, x yzwas expressed by Grassmann by the six 2 determinants of the array:
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53 111 2221 1 x yz x yz (523) as 1 21 1 x L x 1 21 1 y M y 1 21 1 z N z and 11 22yz P yz 11 22zx Q zx 11 22 x y P x y The Plcker line coordinates {}{,,;,,}OLLMNPQR S;S are homogeneous since nonzero scalar multiples of all the coordinates still determine the sa me line [Cra01]. The semicolon inside the coordinate separates the (L, M, N) and (P, Q, R) because their dimensions are different. From equation (519) the direction ratios (L, M, N) are related to the distance  S  by 2222LMN S (524) It is useful to unitize these directional va lues of these homogeneous coordinates to simplify the application; 212121,,  x xyyzz LMN SSS (525) which are also called direction co sines of the lines or unit dir ection ratios. Therefore, the new set of L, M, and N has the new restriction 2221 LMN (526) Using the Plcker line coordinates, a force fis expressed as a scalar multiple f S. The reference point is selected in such a way that the moment of the force fabout this reference point, 0m, can be expressed as a scalar multiple OL f Swhere OLSis the moment vector of the line. So the action of the fo rce applied on the body can be expressed as a scalar multiple of the standard Plucker line coordinates and the coordinates of the force are:
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54 {}{,}OLwff 0SS;Sfm (527) where {}OLS;Sis the Plucker line coordinates and 1 S. It is clear that fis a line bound vector and coordinate inde pendent, while the moment 0mis origin dependent. For spatial parallel mechanisms, the forward static analysis consists of computing the resultant wrench {}wf;m due to linearly independent fo rces generated in the legs acting upon the moving platform. The resultan t wrench could be simply expressed: 101202303 110122023303 {}{}{}{}... {;}{;}{;}...w fSSfSSfSS f;mf;mf;mf;m (528) Equation (528) may be written as w j (529) where j is the Jacobian matrix of the structure. The columns of the matrix are composed of the Plcker coordinates of the lin es of the leg connector forces as 123 010203... ... SSS SSS j. (530) is the column vector with the scal ar of the forces as its elements. 1 2 3... f f f (531)
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55 Figure 56. 3D Model for speci al 66 parallel mechanism Figure 57. Top view of th e special 66 configuration
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56 Now consider the proposed special 66 para llel passive structure, its 3D model and plan view are shown in Figures 56 and 57. This special 66 spatial parallel manipulat or has a moveable top platform connected to the fixed platform (ground) by six translati onal springs acting inpar allel. Each leg has a conventional spring as the compliant component and as acting in the prismatic joint of sphericalprismaticspherical se rial chain. The parallel manipulator with six such compliant legs as well as one top platform a nd one base platform thus has six degrees of freedom. The top platform is connected to the base platform by the initially unloaded compliant coupling which restricts any rela tive spatial motion between the bodies. The corresponding stiffness mapping is thus a one toone correspondence that associates a twist describing the relative displacement between the top and base platform with the corresponding resultant wrench prov ided by the springs between them. The basic shapes of the top platform a nd the base platform of this special configuration are equilateral triangles with th e top and base triangle side lengths being a and b respectively while pa and pb are the join ts separation distance between the pivots. The distance h between the geometrical center of the top platform and the geometrical center of the base platform is selected in such a way that it is qualify the following expression based on the analysis in the previous chapters: 2b ha (532) The top platform has 6 spherical pivot points1,1,1,1,1,1ABCDEF located along its triangle sides and the base platform has 6 spherical pivot points ,,,,,ABCDEF along its triangle sides. The six springs are connected respectively and in order to simplify the
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57 process of the analysis, it is necessary to establish the geometry of the system. The following are position vectors that define th e directions of the six legs and their magnitudes define the lengths of the six legs: 111211 311411 511611, , lAAOAOAlBBOBOB lCCOCOClDDODOD lEEOEOElFFOFOF (533) One coordinate system is attached to th e base platform, which is fixed to ground, with the z axis pointing upwards to the top plat form. The origin of the coordinate system is located right on the geometrical cent er of the base triangle platform. Now considering some external wrench is applied on the top platform and the whole system is in static equilibrium w ith the six spring forces as the moving top platform twists relative to the fixed base platform. Here a wrench w 0f;mcan be thought of as a force f;0acting though the origin togeth er with a general couple 00;m, adding these sets of coordinates together re produces the original wr ench coordinates. Similar to the wrench, a twist (w ritten in axial coordinates) can be thought of as a rotation 0, about a line through the origin toge ther with a general translation X,0, adding these sets of coordinates together reprodu ces the original six twist coordinates: DX, .[Gri91A] In order to maintain the system equilibrium the top platform moves as the external wrench changes. The incremental change of wrench may be written as ; w fm, and the form of the desired stiffness mapping of the compliant structure may be written as
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58 wKD (534) where Kis the 6 stiffness matrix, which relates the incremental twist ( ,DX ) of the top platform relative to th e fixed base platform/ground to the incremental change of the wrench. As it is specified in the previous chapters, the geometrical properties of the platform are known: the leg lengths are meas ured, while the dimensional values of the top and base triangles are inva riant and predefined. The i ndividual spring constants of the springs are also known. In order to determine the mapping of (534), it is necessary to first differentiate (529) [Pig98], which could also be written as 1101 2202 3303 4404 5505 6606() () () () () () kll kll kll w kll kll kll 0f j j m, (535) where ikis the individual sp ring constant of the thispring, and 0,iill are the current length and free length of the thispring respectively, so that0()iiikll is the force in thethileg due to the extension of the thispring. The Jacobian matrix j is: 123456 123456 SSSSSS j OASOBSOCSODSOESOFS (536) where,iSis the direction cosine of the axial line of the thileg. These direction cosines could be calculated a nd unitized via (525):
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59 123456 123456111111 ,,,,,llllllAABBCCDDEEFF SSSSSS The lower three rows of the 6 matrix in (536) are the moments of the six lines relative to the referenc e point O. Because OA, OB, OC, OD, OE and OF are all fixed line vectors on the base platfo rm triangle, and are constant, this matrix is solely a function of the direction cosines of the six legs. The differential of equation (535) is expressed as 11 22 12345633 12345644 55 66 1101 2202 1234563303 123456() () ( kl kl SSSSSSkl OASOBSOCSODSOESOFSkl kl kl kll kll SSSSSSkll OASOBSOCSODSOESOFS o f m4404 5505 6606) () () () kll kll kll (537) Hereil is related toD by the following expression: TlD j (538) where j is the Jacobian matrix of the parallel st ructure. The column of this 6 matrix is the line coordinates of the thileg [Gri91A]. Griffis demonstrated that in order to relate iS with D each leg needs to have two derivatives, which are perpendicula r to the axis of that leg at its base point, to describe the iS Because each leg is individually connected to the base platform and top platform via a ball joint and a hook joint, it only possess two degrees of freedom: two
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60 rotations whose rotation axis are mutual perpendicular. Griffis also provides the stiffness mapping analysis for a 33 octahedral spatia l platform: the analysis is based on Screw theory and utilizes some geometrical restric tions and configurations of the octahedral structure. Three examples include that two le gs sharing one concentr ic ball joint at one fixed pivot, four legs sharing one triangle side and the pivots are located on the vertexes of the triangles. Although the analysis resu lt is exclusively fo r the 33 octahedral mechanism, the conceptual approach is applicab le to the general spatial stiffness analysis. The stiffness mapping of the special 66 para llel manipulator is determined by following the same procedure. The general expression of the stiffness matrix for the special 66 parallel platform is presented as (1)(1) (1)(1)TTT iiiii TT iiiiKkkk kVkV jjjjjj jj, (539) where 123456 123456 SSSSSS j OASOBSOCSODSOESOFS (540) 1 2 3 4 5 600000 00000 00000 00000 00000 00000ik k k k k k k 11 22 33 44 55 66(1)00000 0(1)0000 00(1)000 (1) 000(1)00 0000(1)0 00000(1)iik k k k k k k (541)
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61 and where0/iiill are dimensionless ratios incorporat ed to simplify the expressions. Further, 123456 123456 SSSSSS j OASOBSOCSODSOESOFS(542) and 123456 123456 SSSSSS j OASOBSOCSODSOESOFS, (543) wherei S, i Sare all unitized di rection cosine vectors for the derivatives of the iS vector; the three vectors are mutually perpendicular at the pivot point. Next, six unit vectors are defi ned as intermediate variable s describing the directional information of the base platform triangle sides as ,,,,, 123456EAACACECECEA uuuuuu EAACACECECEA. (544) iV is given by the follow expression: i i i i iuS V uS (545) ii SV and iii SVS (546) 112233445566llllll 000000 V VVVVVV (547) 111222333444555666llllll 000000 V SVSVSVSVSVSV (548) where 0 is a zero vector. Substituting all the necessary components in equation (545) yields the global stiffness matrix of the special 66 parallel passive mechanism via the stiffness mapping analysis.
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62 It is clear that the stiffness matrix can be written as the sum of five different matrices, the first three matrices are sy mmetric, while the last two matrices are asymmetric, so the overall stiffness matrix is asymmetric. It is obvious that the stiffness matrix is dependent on the selection of th e coordinate system, so by using different coordinate systems, it might be possible to have a shorter or longe r expression, but the change of coordinate system does not change the stiffness properties of the system, such as the rank of the matrix, or the eigenvector s and eigenvalues of the stiffness matrix (which are also called corre sponding eigenscrew of sti ffness and eigenstiffness) [Gri91B, Sel02]. It is also intuitive that the stiffness property of a given compliant manipulator should not change just by using a different c oordinate system, and neither could it be changed to be a symmetric matr ix by choosing a different coordinate system In the five matrices constituting the global stiffness matrix, only the first matrix does not have the diagonal matrix of (1)iik When the parallel compliant manipulator sustains a relatively small de flection (and a correspondi ngly small twist), and each leg also endures a relatively small amount of axial force, therefore each leg length is close to its initial length. Then because 00 01ii i iill ll all the other four matrices components who contain the diagonal matrix of (1)iik are so close to the zero matr ix that their effects can be neglected and only the first part of the global stiffness matrix is left. With these simplifications, the stiffness matrix is a positive definite symmetric matrix. Consider an example parallel passive plat form with the special 66 configuration shown in Figure 57) that ha s the following specifications:
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63 a = 60 mm, b =120 mm, p=0.233, 20ik lb/ in = 3.5003 N/mm. The stiffness matrix of the system will be calculated for the case when it is sustaining a small external wrench in static equilibrium and the top platform is slightly deflected from its original position. The corresponding Jacobian matrix of the structure is calculated as 0.654570.235290.197610.235290.456960.47059 0.149740.407540.641740.407540.4920 0.741020.882350.741020.882350.741020.88235 25.679.169751.3439.73525.6730.566 44.46140.588012.35344.46128.235 13.6j 9116.30213.69116.30213.69116.302 (549) where each column is the Plcker line coordina tes of the axis for each leg, the upper three numbers are unit less, the lower th ree numbers have the unit of mm. The corresponding stiffness ma trix is calculated as 3.53010022.256242.750 03.53010242.7522.2560 0013.9420044.511 22.256242.7502293000 242.7522.25600229300 0044.511004758.7 K (550) In this matrix, the units are also differe nt. The four 3 submatrices have the following units / NmmN NNmm Introduce the equation (538) into the stiffness mapping (534):
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64 TwKDKj l (551) For a given set of minute deflections of th e legs, the above equation can be used to calculate the applied external wrench. 2.29120.82360.691680.82361.59951.6472 0.524141.42652.24631.42651.72220 2.59383.08852.59383.08852.59383.0885 89.85232.097179.7139.0989.852106.99 155.63142.07043.239155.6398.832 47.92157.06147w1 2 3 4 5 6.92157.06147.92157.061l l l l l l (552) If the top platform of the special 66 pa rallel passive platform is deflected much from its unladed home position, then the compliance matrix can be calculated from equation (539).
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65 CHAPTER 6 FORWARD ANALYSIS FOR SPECIAL 66 PARALLEL PLATFORM The kinematic forward analysis is very impor tant to control applications. In this chapter, the forward kinematic analysis for pa rallel 33 manipulator is introduced. Then a geometrical method, patented by Duffy and Griffis [Gri93], of determining the equivalent 33 parallel structure for a special 66 configuration mechanism is described. Forward Kinematic Analys is for a 33 Platform It is necessary to determine the position a nd orientation of the end effector of the manipulator. Given the geomet rical properties of the para llel platform, including leg length, connector pivots locations and etc., th e forward kinematic analysis determines the relation between the end effector pose and it geometrical properties. As stated in the previous chapters, the forward kinematic analysis is complicat ed compared to the reverse kinematic analysis. This problem is also geometrically equivalent to the problem of finding out ways to place a rigid body such th at six of its given points lie on six given spheres. During the late 80s and early 90s of la st century, many researchers had worked on the forward kinematic analysis of parallel ma nipulators. Many differe nt approaches such as closedform solutions of special cases, num erical schemes, and analytical approaches were considered for both special cases a nd general 66 configur ations. Hunt [Hun98] studied the geometry and mobility of the 33 inparallel manipulator. Wen and Liang [Wen94] used an analytical approach and solv ed the problem for the 66 Stewart platform with a planar base and platform by reducing of the kinematic equations into a univariate
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66 polynomial and concluded that the upper bound of the solutions for forward kinematic problems is 40 for this class. Merlet [Mer 92, Mer00] had studied direct kinematic solutions for general 66 parallel platform and special cases with additional sensors. Various researchers have stated that for a general 66 inparallel platform and with a given set of fixed leg lengths, it is possible to assemble it in 40 di fferent configurations [Das00, Hun98]. It is not likely that all of the 40 configurations are real and applicable. Due to the difficulty of the forward kinematic analysis for the general 66 inparallel platform, it is worthwhile to consider some special cases with reduced complexity. The simplest form of this class is first analyzed. Although there are different configurations a nd designs for a simple spatial parallel platform, the symmetric 33 parallel platfo rm shown in Figure 61 is among the most important and widely analyzed structures. This 33 parallel platform is also called an octahedron. It has eight triangular faces, six vertices and twelve edges. Four edges are concurrent at each vertex. Every vertex is contained in four faces. The devi ce has all six degrees of freedom, ignoring the trivial freedom of each leg rotating about its own axis. The forward kinematic analysis for the 33 inparallel manipulator was first solved by Griffis and Duffy [Gri89] who showed th at the position and orientation of the top platform can be determined with respect to the base when given the lengths of the six connectors as well as the geometry of the connection points on the top and base. The solution is a closedform solution and is based on the analysis of the input/output relationship of three spherical four bar m echanisms. The three spherical four bar mechanisms (see Figure 63) result in three e quations in the tanhalf angle of the three
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67 unknowns x, y, and z. These tanhalf angles are simply named x, y, and z. Elimination of the variables y and z resu lts in the following polynomial wh ich contains the variable x as its only unknown [Gri89]. Figure 61. 3D drawing of the 33 inparallel platform 2 124 (61) where
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68 222222 332123312331123321 22 3312123311233221 22 3312331112331212 2222 33112332213312 222 331213321332424 244 822 448 242ABaccABcbACaccACcb AEaaccAEacbAEacb AEbbADccbbBCaacc BCacbBCacbBCbb BEaacBEabBD 2112 222 331223312331212 2222 33121231212312 222222222 312312312 331233123312 331233123312 2 1111 2 2222242 2 4 4acbb CEaacCEabCDacbb DEaabbDaaccAcc BcaCacEaa AEbbADccDEaa BCbbBDcaCDac bac bac 22 11111111 22 22222222,0.5, ,0.5,aAxCbDxcBxE aAxBbDxcCxE The coefficients ,...,,1,2,3iiAEi are expressed in terms of known quantities. Their values for the generic spherical four bar mechanism shown in Figure 62 may be written as 12413412413441341223124134 12413412413441341223124134 12413412413441341223124134 1234 124134124134413412231241344Ascssscssccccc B scssscssccccc Cscssscssccccc Dss Escssscssccccc where sin()ijijs and cos()iic For this problem, the coefficients are obtained by replacing the vertex as the or igin of the three spherical f ourbar linkages (shown in Figure 63) with points o, p, and q respectively. The corresponding mapping of the angles of the spherical fourbar linkages is shown in table 61.
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69 Figure 62. A general spherical f ourbar mechanism, Griffis [Gri89] Table 61. Mappings of angles of spherical four bar mechanism. Origin o p Q Output: 12a qor ops p qr Coupler: 23a ros spt tqr Input: 34a sop tpq rqo Ground: 41a p oq p qo oqp The resulting solution shown above is an ei ghth degree polynomial in the square of one variable which yields a maximum of si xteen solution poses for any given set of connector lengths and basetop tr iangles lengths. Eight soluti ons were reflected about the
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70 plane formed by the base connector points. There can be up to 16 real solutions and therefore, there can be 0,2,4,6 or 8 pairs of real, reflected solutions. Figure 63. 33 inparallel mechanism with three spherical fourbar linkages, Griffis [Gri89] Forward Kinematic Analysis for Special 66 InParallel Platform The configuration of the special 66 inpar allel platform is carefully chosen such that the forward kinematic anal ysis can use a method similar to the method for the 33 inparallel platform. This section will show the geometric relationship between the special 66 inparallel platform and an equivalent 33 platform and how to solve the direct kinematic analysis problem by using this conv ersion. The relationship between a Special 66 platform and its equivalent 33 platform was discovered by Griffis and Duffy [Gri93] and is presented here for completeness as the fo rward analysis is an important part of the control algorithm for the fo rce control device develope d in this dissertation.
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71 A Special 66 platform is defined as one which is geometrically reducible to an equivalent 33 platform Figure 64 depicts a perspective and plan view of a 66 platform where the leg connector points R0, S0, and T0 lie along the edges of the triangle defined by points O0, P0, and Q0 and the leg connector points O1, P1, and Q1 lie along the edges of the triangle defined by the points R1, S1, and T1. The objective here is to determine the distance between the pairs of points O0 R1, O0 S1, P0 S1, P0 T1 Q0 T1, and Q0 R1. These distances represent the leg connector lengths for an equivalent 33 platform. Throughout this analys is, the notation minj will be used to represent the distance between the two points Mi and Nj and the notation minj will represent the vector from point Mi to Nj. Using this notation, the probl em statement can be written as: given: o0o1, p0p1, q0q1, r0r1, s0s1, t0t1 connector lengths fo r Special 66 platform o0p0, p0q0, q0o0, o0s0, p0t0, q0r0 base triangle parameters r1s1, s1t1, t1r1, r1o1, s1p1, t1q1 top triangle parameters find: o0r1, o0s1, p0s1, p0t1, q0t1, q0r1 connector lengths for equivalent 33 platform Obviously it is the case that s0p0 = o0p0 o0s0 ; t0q0 = p0q0 p0t0 ; r0o0 = q0o0 q0r0 o1s1 = r1s1 r1o1 ; p1t1 = s1t1 s1p1 ; q1r1 = t1r1 t1q1 (62 ) The six leg connectors of the equivalent 33 platform de fine an octahedron, i.e., there are eight triangular f aces; the top and bottom platform triangles and six faces defined by two intersecting connectors and an e dge of either the base or top platform. The solution begins by first defining angles and which, for each of the six faces
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72 defined by intersecting connectors, defines the angle in the plane between a connector of the Special 66 platform and an edge of either the top or base platform as appropriate. A B Figure 64. Special 66 platform and equivalent 33 pl atform, A) Perspective view, B) Plan view Figure 65 shows the angles 1 and 1. The angle 2 is defined as the angle between r0r1 and r0o0 and the angle 3 is defined as the angle between s0s1 and s0o0. Similarly, the O0 O1 P0 P1 Q0 Q1 R1 R0 S1 S0 T1 T0 O0 O1 P0 P1 Q0 Q1 R1 R0 S1 S0 T1 T0
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73 angle 2 is defined as the angle between q1t1 and q1q0 and the angle 3 is defined as the angle between o1s1 and o1o0. Figure 65. Defi nition of angles and Figure 66 shows a planar triangle that is defined by the vertex points G1, G2, and G3. A point G0 is defined as a point on the line defined by G1 and G2 and the angle is shown as the angle between g0g2 and g0g3. The cosine law for the triangle defined by points G0, G2, and G3 yields c b 2 c b cos2 b 2 2 (63) Next consider that the triangle lies in a plane defined by u and v coordinate axes. The coordinates of any point Gi may then be written as (ui, vi). The distance a may then be expressed as P0 Q0 O0 S1 R1 T1 P1 T0 1 1
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74 cos c a 2 c a cos c a 2 a ) cos (sin c ) sin c ( ) cos c a ( ) v v ( ) u u (2 2 2 2 2 2 2 2 2 1 3 2 1 3 2 a (64) Figure 66. Planar triangle Substituting (63) into (64) and simplifying gives ) a c a b a c b a b ( b 1 ) c b ( b a c a2 b 2 2 2 2 2 b 2 2 2 2 2 a (65) Regrouping this equation gives 222222()()()abbaabcabababcab (66) The result of equation (66) can be applie d to each of the six side faces of the octahedron defined by the leg connectors of th e equivalent 33 platform. The parameter substitutions are shown in Table 62. G1 G2 G0 G3 a b c ab u v
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75 Table 62. Parameter substitutions G0 G1 G2 G3 a b c a b 1 P1 T1 S1 P0 p1t1 p1s1 p0p1 p0t1 p0s1 2 Q1 R1 T1 Q0 q1r1 t1q1 q0q1 q0r1 q0t1 3 O1 S1 R1 O0 s1o1 o1r1 o0o1 o0s1 o0r1 1 T0 P0 Q0 T1 p0t0 t0q0 t0t1 p0t1 q0t1 2 R0 Q0 O0 R1 q0r0 r0o0 r0r1 q0r1 o0r1 3 S0 O0 P0 S1 o0s0 s0p0 s0s1 o0s1 p0s1 Using Table 62 to perform appropriate parameter substitutions in (A5) will result in six equations that can be written in matrix form as A q = M (67) where 6 5 4 3 2 1 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1M M M M M M r o s o t q r q s p t p 0 p s 0 0 s o 0 r q 0 0 o r 0 0 0 0 t p 0 0 q t o s r o 0 0 0 0 0 0 r q q t 0 0 0 0 0 0 t p s p M q A (68) where M1 = (p1t1+p1s1) (p0p12 + p1t1 p1s1), M2 = (q1r1+t1q1) (q0q12 + q1r1 t1q1), M3 = (s1o1+o1r1) (o0o12 + s1o1 o1r1), M4 = (p0t0+t0q0) (t0t12 + p0t0 t0q0), M5 = (q0r0+r0o0) (r0r12 + q0r0 r0o0), M6 = (o0s0+s0p0) (s0s12 + o0s0 s0p0). (69)
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76 Then these equations are manipulated su ch that dividing the first rows of A and M by p1t1+p1s1= s1t1, the second rows by q1r1+t1q1= r1t1, and so on to obtain AqM (610) where 0 0 0 0 2 1 0 0 0 0 0 2 1 0 0 0 0 0 2 1 0 1 1 1 1 2 1 0 1 1 1 1 2 1 0 1 1 1 1 2 1 00 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 p s s o s s o r r q r r q t t p t t r o o s o o q t r q q q s p t p p p s s r r t t w w q q p p M A with 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1p o o s p s o s o s s s r r o s o r o r o w o q q r o r q r q r r t r t q r q t q t q q q p p t q t p t p t t t s s p t p s p s p p Note that 1 , , 0 s r t w q p The matrix A is dimensionless and depends only on the ratios of where the connections occur along the sides of the upper and lower bases. Matrix A is also independent of the shapes of the upper and lo wer triangles. Now all the terms in matrix A are known as they are dimensionless ratios expressed in terms of given distances of the top and bottom platform. All the terms in vector M are also known as they are expressed in terms of given dimensions as well as the square of the lengths of the six leg connectors of the Special 66 platform. Thus the square of the lengths of the connectors for the equivalent 33 platform may be determined as:
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77 1qAM (611) Further more, the determinant of A is: pqwtrs s r t w q p ) 1 )( 1 )( 1 )( 1 )( 1 )( 1 ( ) det( A (612) It is interesting to note that the matrix A is singular if and only if the following condition is satisfied: 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0t p t p s o s o r q r q r o r o q t q t p s p s (613) One case where this could happen would be if all the middle connector pointes S0, T0, R0, O1, P1, and Q1 are located at the midpoints of the sides of the upper and lower triangles. We can also conclude that for A to be singular, if some of the parameters s r t w q p , , are less than then some of the other parameters must be greater than Thus we immediately conclude th at if all of the parameters ar e less than (as is the case in Figure A1), then A is nonsingular. The same statemen t of course is true if all the parameters are greater than Once the leg length dimensions of the equi valent 33 platform are determined, the forward analysis of the 33 device as descri bed in Section 6.1 is used to determine the position and orientation of the top platform relative to the base
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78 CHAPTER 7 RESULTS AND CONCLUSION In this chapter, the individual components of the designed manipulator are tested to validate the machine design and theoretical anal ysis presented in the previous chapters. In order to calibrate each legs stiffness, firs t the force sensor is calibrated. An optical encoder measures the compliant displacement of the leg and the lo ad cell records the applied force. The forcedisplacement relationship is provided for each leg and the results are analyzed. The parallel platform is then assembled for the 6 DOF wrench measuring testing. Several experiments are presented and the results are analyzed. At the end of this chapter, a summary of the work is followed by conclusions and future work suggestions. Calibration Experiment for the Force Sensor In order to get the force/ displacement re lationship of the compliant connectors, the detected force signal and the corresponding displacement signal should be recorded simultaneously during the individua l leg calibration experiment. A load cell is used to measure the axial force applied on the individual leg. The rated capacity of the lo ad cell is 5 lb, (22.246 N) and th e output of the load cell is an analog voltage signal, ranging from 5~+ 5 Vdc. The rated output is 2mV/V20%. It is necessary to calibrate the load cell first to ensure the validity and accuracy of the following experiments results.
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79 0 5 10 15 20 25 1 0 1 2 3 4 5 6 In p ut Force ( Ns ) Output Voltage (Volts)Load cell linearity calibration experiment plot Figure 71. Load cell calibration experiment 0 2 4 6 8 10 12 14 0.5 0 0.5 1 1.5 2 2.5 3 3.5 Input Force (Ns)Output Voltage (Volts)Load cell linearity calibration experiment plot: Fine plot Figure 72. Load cell calibration experiment opposite direction
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80 The load cell output analog signal is conn ected to an A/D port of a multifunctional I/O board and the value of the voltage is recorded with the co rresponding known force, which is provided by a set of standard weights for accurate force calibration. The mapping of the force/voltage relationship is shown in Figure 71 and Figure 72. Five additional sets of expe rimental data were analyzed and the plots are very similar, so only one plot is shown here. Th e linear equation that relates the applied force to the measured voltage can be written as 1ykx (71) where the coefficient k1 is determined to equal 0.00225 Volt/gram. The linear regression statistics are shown in Table 71. Table 71. Load cell regression statistics Regression Statistics Multiple R 0.999115584 R Square 0.998231949 Adjusted R Square 0.966981949 Standard Error 0.059572297(V) From the table and the figure above, the lo ad cell shows very high linearity and is adequate to be used in the following experiments. Individual Leg Calibration Experiment After calibrating the load cell, it is necessa ry to calibrate the compliant connectors and identify the stiffness property for each of them. During the experiment, the compliant connector is fixed vertically and is attached to the load cell in such a way that the applied external force causing either elongation or compression can be detected properly. The two physical quantities, force and displacement, are detected by the load
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81 cell and the optical encoder. The optical enc oder is attached on the compliant leg both for the calibration experiment and for the designed operation. Axial force is applied on the top of the co mpliant leg with variable magnitude and direction in order to get the stiffness mapping of the leg for both compression and elongation. The resolution of the optical encoder is 1000 counts/inch. The individual plot of the stiffness ma pping for each of the six leg connectors is shown in Figures 73 through 78. 2 1 0 1 2 3 4 5 5 0 5 10 15 Leg 1 Calibration: Force Vs. Displacement X Dis p lacement ( mm ) Y Force (N) Figure 73. Calibration plot for leg1
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82 2 1 0 1 2 3 4 5 6 4 2 0 2 4 6 8 10 12 14 Leg 2 Calibration: Force Vs. Displacement X Displacement (mm)Y Force (N) Figure 74. Calibration plot for leg2 2 1 0 1 2 3 4 5 5 0 5 10 15 Leg 3 Calibration: Force Vs. Displacement X Dis p lacement ( mm ) Y Force (N) Figure 75. Calibration plot for leg3
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83 2 1 0 1 2 3 4 5 6 4 2 0 2 4 6 8 10 12 14 Leg 4 Calibration: Force Vs. Displacement X Displacement (mm)Y Force (N) Figure 76. Calibration plot for leg4 2 1 0 1 2 3 4 5 6 6 4 2 0 2 4 6 8 10 12 14 Leg 5 C alibration: Force Vs. Displacement X Dis p lacement ( mm ) Y Force (N) Figure 77 Calibration plot for leg5
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84 2 1 0 1 2 3 4 5 6 4 2 0 2 4 6 8 10 12 14 Leg 6 Calibration: Force Vs. Displacement X Displacement (mm)Y Force (N) Figure 78 Calibration plot for leg6 In the calibration plots, the displacement is measured in mm and force measured in Newtons. In the original data file, the disp lacement at each sample time is recorded as encoder counts and the force is recorded as digitized voltage value. Then the units are converted by the load cell ca libration experiment results together with the following conversion equations: 125.4 453.64.448 inmm lbgram (72) Then, the data in the plots is used to build six individual lookup tables of displacement vs. force for each leg. Duri ng operation, the relative displacement is measured and the corresponding force value is obtained from the lookup table. These calibration tables can also be replaced by linea r regression of the data for lessaccurate operations, and thus a linear spring constant is assigned to each leg.
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85 Parallel Platform Force/Wrench Testing Experiment The force/torque sensor has been factory ca librated, so only the resolution is listed here. The resolutions for force and torque are shown in the table below Table 72. Forcetorque Sensor Resolution The parallel platform is attached on the t op of the force/torque sensor as shown in Figure 79 and its mass center is carefully balanced such that its weight is directly over the origin of the sensors refe rence coordinate system. This coordinate system is based on the force/torque sensor, so there is no mome nt introduced. This zdirection bias force is easily eliminated in the sampling program, so it will not be discussed further. Figure 79. Photo of the testing e xperiment for the parallel platform Fx Fy Fz Tx Ty Tz 0.1 (N) 0.1 (N) 0.2 (N) 5 (N mm) 5 (Nmm) 5 (Nmm)
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86 The home position refers to the position of th e platform when it is attached with the force/torque sensor and placed horizontally with no other force applied on the top platform or any leg. The home position is also called the initial position/ orientation, and it can be specified by its relative displacem ent from the encoders indexed position. Before use, it is necessary to home the device. This is done by applying some small force/ wrench to the top platform, so each le g pass through its index position. Thus after homing, the current position and orientation can be accurately measured even when it is not in its home position. Repeatability test It is important to know the repeatabi lity of the manipulator. The previous prototype, built by Tyler [Dwa00], did not have good repeatability. Significant friction in the spherical joints was noted when an external force was applied and then removed. This friction impacted the ability of the top platform to return repeatedly to the same home position when external loading was removed. In order to test its repeatability, tw o sets of experiment are performed. 1. Apply random force/wrench to the top pl ate and then remove it, measure the unloaded position/ orientation via the leg lengths measurements 2. Apply a weight of 500 grams on the top plate several times, and compare the measured leg lengths. Table 73. Repeatability expe riment, method 1 (Encoder Counts) Test Number Leg 1 Leg 2 Leg 3 Leg 4 Leg 5 Leg 6 #1 1 0 0 0 0 1 #2 0 0 1 1 2 1 #3 6 0 4 6 1 3 #4 5 2 0 4 3 1 #5 3 3 1 1 2 1 #6 2 0 1 0 2 2 Average 2.83333 0.16667 1.16667 1.66667 1 0.83333 STD 2.316607 1.602082 1.47196 2.73252 1.788854 1.602082
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87 Table 74. Repeatabil ity experiment, method 2 (Encoder Counts) Test Number Leg 1 Leg 2 Leg 3 Leg 4 Leg 5 Leg 6 #1 200 64 5 103 101 98 #2 202 65 5 104 101 100 #3 207 66 5 104 101 102 #4 197 63 4 100 100 97 #5 203 66 5 102 101 98 #6 202 65 4 103 100 101 Average 201.833 64.8333 4.66667 102.667 100.6667 99.33333 STD 3.311596 1.169045 0.516398 1.505545 0.516398 1.966384 The experiment results are shown in Tabl es 73 and 74. The numbers in the column shows the encoder counts when there is no external load a pplied (for method 1) and when the external load is applied on the platform (for method 2). The overall repeatability for the parallel platform is reasonable. Forward analysis verification The forward kinematic analysis was presen ted in Chapter 5. An experiment is shown here to validate the analysis result and al so to provide for the static analysis process. The parallel platform has some external wrench applied on the top plate. The counts of the relative displacement for each le g is then recorded. Table 75 shows the lengths of the six leg connectors, both in units of encoder counts from the unloaded position and in units of mm, with the external wrench applied. The table also shows the calcu lated lengths of the six c onnectors of the equivalent 33 platform. After performing the forward an alysis of the equivalent 33 platform, a reverse analysis of the special 66 platform is conducted to calculate the leg lengths and to verify the forward analysis result.
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88 Table 75. Forward analysis Leg 1 Leg 2 Leg 3 Leg 4 Leg 5 Leg 6 Encoder Counts 15 10 2 13 26 19 Leg length for special 66 (measured) (mm) 80.588 67.746 80.918 67.669 80.308 67.517 Leg length for the equivalent 33 platform (mm) 84.595 84.829 84.722 84.145 84.469 80.588 Leg length for the 66 platform (reverse analysis) (mm) 80.587 67.746 80.918 67.669 80.309 67.517 From the table, it is clear that the forwar d analysis is very accurate, the leg lengths calculated from the reverse analysis, which ar e also based on the re sults of the forward kinematic analysis, are very close to the leg length values me asured by the optical encoder. The coordinate systems of the parallel plat form are shown in Figure 710. The base points are 00,, OP and0Q the origin of the base coor dinate system is at point0O 0P is on the X axis, and 0Q is on the XY plane. This coordinate system is fixed in the base. The top plate corner points are111,, R ST the origin of the top platfo rm coordinate system is at point1 R 1S is on the x axis, and 1T is in the xy plane. After finding the leg lengths of the equiva lent 33 platform by the geometrical conversion, the position and orie ntation of the platform can be calculated by the forward kinematic analysis. The detail of this anal ysis is presented in Chapter 6, so only the results are presented here.
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89 For the measured leg lengths listed in Ta ble 75, the forward analysis has only 8 solutions.1 Four are above the base and four ar e reflected through the base. The four configurations above the base are discussed he re. Each solution can be represented by a 44 transformation matrix. The first three elemen ts in the fourth column have units of length (mm) and all other elements are dimens ionless. From now on, the units for the element in the transformation matrices are omitted. The four calculated transformation matrices that relate the top a nd base coordinate systems are base top10.51050.10950.852929.2029 0.85980.07180.505552.4117 0.00590.99140.130859.4408 0001 T base top20.50210.86480.000729.7107 0.86480.50200.000752.1185 0.00580.00420.999959.4469 0001 T base top30.49870.86680.006129.9149 0.12110.06270.990752.0006 0.85830.49470.136359.4478 0001 T (73) base top40.35260.37570.857080.9928 0.36700.78690.496022.5108 0.86080.48940.13968.1510 0001 T Although all four solutions ar e real and satisfy the geometrical conditions, there is only one solution that corresponds to the actual current c onfiguration. With these transformation matrices, the coordinates of all points of the 33 structure can be determined. Then drawings of these confi gurations (or the numeri cal determination of 1 In general, the forward analysis of the special 66 platform will yield 16 possible solutions, 8 above the base and 8 reflections through the base. However for the measured set of leg lengths, only 8 real solutions exist, 4 above the base and 4 reflections through the base.
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90 the configuration closest to the most previ ous pose) can be used to choose the correct solution set. Figure 710. Coordinate syst ems of the parallel platform Here the geometrical meanings of the 44 transformation matrices are used to make the selection. By using homogeneous coordinates, the 44 transformation matrix can be represented as 00001AA BB A B RP T, (74) where 0A BPis the location of the origin of the B c oordinate system measured with respect to the A coordinate system, AAAA BBBB RXYZ, its columns vectors are the orientation of the B coordinate system measured with respect to the A coordinate system. O0 O1 P0 P1 Q0 Q1 R1 R0 S1 S0 T1 T0 X Z Y z x y
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91 So by comparing the columns of the transfor mation matrices, the right solution can be properly selected. The transformation matrix for the home position is given below as base tophome0.50560.8628029.6651 0.86280.5056052.1547 00159.9987 0001 T (75) Clearly, the second solution is the correct one since physical ly the Z axis of the top coordinate system remained relatively parallel to the Z axis of the base coordinate system after the loading was applied. For all othe r solutions, the coordinate systems have too much rotation relative to the base coordinates system. Wrench and force testing. After choosing the correct transformation ma trix, the coordinates of all points are known and the Jacobian matrices of the structur e at any instant be ca lculated. Using the equilibrium condition, the follo wing equation can be used to calculate the applied force/ wrench as stated in the previous chapters. w j (76) Numerical example: In order to verify the accuracy of the PCCFC, several numerical experiments have been done. For each set of experiment data the external wrench applied on the top platform of the PCCFC is measured by the force/torque sensor on which the device is mounted for this experiment and compared to the calculated wrench based on the sensed leg lengths. The force/torque sensor date is reported in te rms of the sensors coordinate system and the calculated wrench is calculated in terms of the base coordinate system. The wrenches are compared by converting the calculated wrench to the sensor coordinate
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92 system. It is important to note that the for ce/torque sensor has a lefthanded coordinate system. That is its Zaxis and Xaxis are parallel to the corres ponding axes of the base coordinate system of the PCCFC, but the Yaxes are antiparallel. The sensor coordinate system is used as the common reference system and the coordinates of the external wrench det ected by the PCCFC are transformed by the following equation: 221111 00{;}{;} SSSSpS (77) where superscript 1 and 2 refer to the Screw co ordinates in the former coordinate system and new coordinate system respectively, 1p refers the coordinates of origin of the second coordinate system in terms of the first coordinate system. This transformation will determine the calculated wrench in terms of a righthanded coordi nate system whose origin is at the origin of the sensor coordinate system. The x and z axes will be parallel to the x and z axes of the sensor coordinate system but the y axis will be antiparallel to that of the sensor coordinate system. For each set of experiment data, 0, SS will be different, but 1p will remain the same as 1[60,203,19]p mm. Several numerical expe riments are conducted here. The spring constants are shown in Table 76. Table 76. Spring constants Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Spring Constant(N/mm) 3.0914 3.0469 3.1003 3.1047 2.9757 3.038 Numerical experiments: Six different loads were applied to the top platform. The encoder counts and the corresponding leg lengths for the six cases are shown in Tables 77 through 712.
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93 Table 77. Numerical experiment 1encoder counts and leg lengths Table 78. Numerical experiment 2encoder counts and leg lengths Table 79. Numerical experiment 3encoder counts and leg lengths Table 710. Numerical experiment 4encoder counts and leg lengths Table 711. Numerical experiment 5encoder counts and leg lengths Table 712. Numerical experiment 6encoder counts and leg lengths Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Counts 15 15 12 73 61 8 Leg length(mm) 80.59 67.62 80.66 66.15 79.42 68.20 Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Counts 76 13 13 4 5 87 Leg length(mm) 79.04 68.33 80.64 67.90 80.84 65.79 Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Counts 6 80 63 6 20 2 Leg length(mm) 80.82 65.97 79.37 67.85 80.46 68.05 Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Counts 74 30 37 54 21 83 Leg length(mm) 82.85 67.24 80.03 66.63 80.44 65.89 Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Counts 46 31 37 47 22 67 Leg length(mm) 82.14 67.21 80.03 66.81 80.41 66.30 Leg_1 Leg_2 Leg_3 Leg_4 Leg_5 Leg_6 Counts 43 101 27 66 59 35 Leg length(mm) 82.06 65.43 80.28 66.32 82.47 67.11
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94 The corresponding wrench information for each of the six cases is shown in Tables 713 through 718. The sensor data are shown in the first row, the calculated data from PCCFC in terms of its base coordinate system are shown in the second row. Table 713. Numerical experiment 1 wrench Fx(N) Fy(N) Fz(N) Tx(Nmm)Ty(Nmm) Tz(Nmm) F/T sensor 0.1 0.3 9.9 56.5 192.1 8.5 PCCFC (base coord system) 0.1 0.4 10.4 429.2 821.1 36.2 PCCFC (Transformed) 0.1 0.4 10.4 60.0 193.7 14.0 Table 714. Numerical experiment 2 wrench Fx(N) Fy(N) Fz(N) Tx(Nmm) Ty(Nmm) Tz(Nmm) F/T sensor 0.1 0 9.8 107.3 240.1 0 PCCFC (base coord system) 0.1 0.1 10.6 499.7 380.9 4.8 PCCFC (Transformed) 0.1 0.1 10.6 130.6 257.4 3.1 Table 715. Numerical experiment 3 wrench Fx(N) Fy(N) Fz(N) Tx(Nmm) Ty(Nmm) Tz(Nmm) F/T sensor 0.2 0.1 10.5 237.3 31.1 5.6 PCCFC (base coord system) 0.2 0.3 10.7 115.8 691.0 29.8 PCCFC (Transformed) 0.2 0.3 10.7 263.9 41.5 14.9 Table 716. Numerical experiment 4 wrench Fx(N) Fy(N) Fz(N) Tx(Nmm) Ty(Nmm) Tz(Nmm) F/T sensor 4.3 3.6 11.0 420.4 519.7 189.2 PCCFC (base coord system) 5.3 4.0 10.8 674.3 1051.6 163.0 PCCFC (Transformed) 5.3 4.0 10.8 379.3 507.7 223.2 Table 717. Numerical experiment 5 wrench Fx(N) Fy(N) Fz(N) Tx(Nmm) Ty(Nmm) Tz(Nmm) F/T sensor 3.1 2.5 10.8 310.7 401.1 144.0 PCCFC (base coord system) 3.9 2.8 10.8 576.3 951.2 133.7 PCCFC (Transformed) 3.9 2.8 10.8 256.7 379.5 166.7
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95 Table 718. Numerical experiment 6 wrench Fx(N) Fy(N) Fz(N) Tx(Nmm) Ty(Nmm) Tz(Nmm) F/T sensor 1.8 1.9 9.4 420.8 254.2 276.8 PCCFC (base coord system) 2.0 2.3 9.9 29.6 794.8 540.9 PCCFC (Transformed) 2.0 2.3 9.9 417.9 239.0 327.8 The different signs between Fy and Ty of the two set of data are caused by the different coordinate systems. Considering the resolution of the sensor and environmental noise, the results are very good. Determination of Stiffness Ma trix at a Loaded Position In this section the stiffness matrix for th e prototype device when it is at a loaded position will be determined in two ways i.e., analytically and experi mentally, and the two results will be compared. Figure 711 show s a model of the compliant platform with coordinate system 0 attached to the base a nd coordinate system 1 attached to the top platform. The constant platform dimensions are given as mm 14 L L L mm 60 L L L mm 28 L L L mm 120 L L L1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0Q T P S O R R T T S S R R Q T P S O O Q Q P P O (78) where the notation j iL is used to represent the distance between points i and j. A load was applied to the top platform and the leg lengths were measured as mm 9492 67 L mm 2032 68 L mm 8570 66 L mm 8928 80 L mm 8928 80 L mm 1054 80 L1 0 1 0 1 0 1 o 1 0 1 0T T S S R R Q Q P P O O (79)
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96 Figure 711. Compliant Platform Model At this position, a forward displacement analysis was conducted and four possible real positions and orientations of the t op platform were determined. During the experiment, the z axis of the top platform rema ined close to parallel to the z axis of the bottom platform and the solution which most closely satisfied this constraint was selected. The 4 transformation matrix that relates the top and base coordinate systems was determined to be 1 0 0 0 5839 58 9996 0 0133 0 0256 0 6518 51 0154 0 5021 0 8647 0 1091 30 0243 0 8647 0 5017 00 1T (710) the units of the terms in the first three colu mns are dimensionless and the units of the top three terms of the fourth column are millimeters. At this position, the force in each of the six leg conn ectors was determined. The displacement of each of the legs from the unloaded home position measured in terms of encoder counts was as follows:
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97 2 L 8 L 45 L 3 L 3 L 34 L1 0 1 0 1 0 1 o 1 0 1 0T T S S R R Q Q P P O O (711) The resolution of the encoder is 1000 c ounts/inch (39 counts/mm). The spring constant of each leg was previously determined experimentally as m m N 1047 3 k mm N 0469 3 k mm N 0380 3 k mm N 9757 2 k mm N 1003 3 k mm N 0914 3 k1 0 1 0 1 0 1 o 1 0 1 0T T S S R R Q Q P P O O (712) The force in each of the legs is thus calculated as 1577 0 6191 0 4724 3 2267 0 2362 0 6697 21 0 1 0 1 0 1 1 0 1 0N f N f N f N f N f N fT T S S R R Q Q P P O Oo (713) The external wrench applied at position 1, 1w is calculated as the sum of the individual forces as
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98 7457 2 5602 179 1979 263 9436 4 1829 0 0263 01w (714) where the first three terms have units of Newt ons and the last three terms have units of Newtonmillimeters. The stiffness matrix for the device at this position will now be computed analytically and experimentally. Analytical Determinatio n of Stiffness Matrix In chapter 5, the spatial stiffness matrix for PCCFC was studied and presented. The spatial stiffness matrix can be calcul ated analytically ba sed on the geometrical dimension values, spring constants of the compliant devices and the current transformation matrix. Equation (539) is used to determine the spatial stiffness matrix. For the specified configur ation discussed above, its stiffness matrix can be calculated as 3.05050.04260.023921.406211.79101.96 0.04263.03650.0265205.7821.241180.95 0.0240.026512.101412.15732.3842.647 21.406210.72411.97337772489510396 216.7321.241732.36248986451911169 101.7818aK 0.9342.647102261090618475 (715) where stands for analytically determined. As stated in Chapter 5, the four 3 submatrices have the following units /NmmN NNmm
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99 Experimental Determination of Stiffness Matrix An additional external load was used to displace the platform slightly from the position defined by (710) (to be referred to as position 1) and the leg lengths were recorded. This procedure was repeated for a to tal of six times and these six cases will be identified by the letters A through F. Fo r each of these cases it was necessary to determine the change in the applied wrench (to be written in ray coordinates) and the twist that describes the motion of the platfo rm from position 1 to the new position (to be written in axis coordinates). The wrench and twist will both be evaluated in terms of the coordinate system attached to the base platform. As previously stated, the stiffness matrix K relates the change in the applied wrench to the instantaneous mo tion of the top platform as D K w ] [ (716) By evaluating the change in the wrench and the twist six times, (716) can be written in matrix format as ] [ ] [ ] [6 6 6 6 D K w (717) where ] [6 6 w and ] [6 6D are 6 matrices defined as ] [ ] [F 1 E 1 D 1 C 1 B 1 A 1 6 6 w w w w w w w (718) ] [ ] [F 1 E 1 D 1 C 1 B 1 A 1 6 6 D D D D D D D (719) The columns of ] [6 6w are the changes in the extern al wrench as the platform moved from position 1 to position A, position 1 to position B, etc. The columns of ] [6 6D are the twists that represent the motion of the platform from position 1 to position
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100 A, from position 1 to position B, etc. Once the matrices ] [6 6 w and ] [6 6D are determined, the stiffness matrix can be calculated as 1 6 6 6 6] ][ [ ] [ D w K (720) Prior to calculating ] [6 6w and ] [6 6D it is necessary to document the measured leg lengths at the six new poses of the top plat form. Table 719 presents this information. Table719. Measured leg connector lengt hs at positions A through F (all units mm) 1 0O OL 1 0P PL 1 0Q QL 1 0R RL 1 0S SL 1 0T TL A 80.5118 79.3688 79.9530 66.9586 69.5748 68.3048 B 79.9530 80.2578 80.9182 66.9840 67.3142 68.0508 C 79.9784 80.7912 80.5372 66.9840 68.2032 67.1364 D 80.4102 81.3246 81.3500 66.4760 67.5428 67.7206 E 81.2738 79.6228 80.6134 66.5014 69.2192 67.7206 F 79.3434 80.4356 82.2390 67.4158 68.0762 67.4412 The position and orientation of the top platform was determined via a forward displacement analysis. The transforma tion matrices were determined as 1 0 0 0 5063 58 9995 0 0222 0 0225 0 0301 50 0084 0 4998 0 8661 0 4078 30 0305 0 8659 0 4993 0 1 0 0 0 7887 58 9997 0 0064 0 0217 0 0331 52 0154 0 5099 0 8601 0 7774 31 0165 0 8602 0 5097 0 1 0 0 0 6251 58 9996 0 0181 0 0222 0 4777 .52 0109 0 4782 0 8782 0 4274 30 0265 0 8781 0 4778 0 1 0 0 0 6024 58 9997 0 0036 0 0259 0 4415 51 0242 0 5020 0 8645 0 0147 30 0099 0 8648 0 5019 0 1 0 0 0 6874 58 9997 0 0239 0 0061 0 5888 51 0068 0 5040 0 8637 0 0674 30 0237 0 8633 05041 0 1 0 0 0 5788 58 9997 0 0072 0 0241 0 0055 51 0164 0 5394 0 8419 0 8315 30 0191 0 8420 0 5391 00 0 0 0 0 0 T T T T T TF E D C B A (721) The matrix ] [6 6w is relatively simple to calcula te. For example, the wrench at position 1, 1 w, is calculated as the sum of the fo rces along the leg connector lines at
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101 position 1. Similarly, th e wrench at position A, A w, is also calculated as the sum of the forces along the leg connector lines at position A. The change in the wrench, A 1w is simply calculated as the difference between these two wrenches as 1 A A 1 w w w (722) Similar calculations are performed for the other five legs. The displacements of each leg measured in encoder counts from the unloaded home position are listed in the table below. Table 720. Displacement of leg connectors from unloaded home position at positions A through F (all units are encoder counts) 1 0O OL 1 0P PL 1 0Q QL 1 0R RL 1 0S SL 1 0T TL A 18 63 40 41 62 12 B 40 28 2 40 27 2 C 39 34 17 40 8 17 D 22 14 15 60 18 11 E 12 53 14 59 48 11 F 64 21 50 23 3 22 The calculated wrenches at each of the six positions are presented as 0032 329 4282 212 7469 125 0937 5 9762 4 3890 0 7374 47 3909 600 4532 420 8078 4 0262 2 8796 5 9454 88 5356 156 3332 257 7095 5 1190 0 6523 0 1008 63 1178 465 3918. 370 1272 8 5456 0 3539 0 4402 21 1519 363 6900 223 4476 8 4166 0 3449 0 8671 138 2140 508 9854 433 8295 4 2219 2 7730 4 F E D C B Aw w w w w w (723)
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102 where the first three components of these vector s have units of Newtons and the last three components have units of Newtonmillimeters. From (721) and (714) the matrix ] [6 6w is calculated as 75 331 483 50 691 91 847 65 186 24 12 136 868 41 82 429 025 14 56 294 59 192 65 337 94 388 26 157 8646 5 19 107 508 39 79 170 1501 0 1358 0 7659 0 1836 3 5041 3 1141 0 1591 5 8434 1 0639 0 7285 0 5994 0 0391 2 3628 0 8533 5 6785 0 3277 .0 3712 0 7467 4 ] [6 6w (724) The determination of the twists as the top platform moves from position 1 to each of the six other positions is slightly more complicated. The determination of the twist A 1D will be presented. The twist A 1D is comprised of six components. When written in axis coordinates, the last three repres ent the instantaneous angular velo city of the moving body and the first three represent the instan taneous linear velocity of a poi nt in the moving body that is coincident with the origin of the refere nce frame. In this case, both of these instantaneous quantities must be approximated from the finite motion of the platform as it moves between pose 1 and pose A. The angular velocity was approximated by cal culating the axis and angle of rotation that would rotate the top coordi nate system from its orientatio n at pose 1 to its orientation at pose A. This was accomplished by first evaluating the net rotation matrix that relates the A pose to the 1 pose as R R R0 A T 0 1 1 A (725)
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103 where the 3 rotation matrices R0 1 and R0 A are obtained as the upper left 3 elements of the transformation matricesT0 1 and T0 A respectively. For this case the rotation matrix R1 A was evaluated as 0000 1 0049 0 0020 0 0050 0 999 0 0437 0 0018 0 0438 0 999 01RA (726) From [Cra98] the angle of ro tation and axis of rotation th at would rotate coordinate system 1 to be aligned with coordi nate system A are calculated as =2.526 degrees and 1m = [0.1124, 0.0431, 0.9927]T. The superscript associated with the axis vector m is used to indicate that the rotation axis that was determined from (726) was expressed in terms of the coordinate system attached to th e top platform at position 1. The direction of the rotation axis can be expressed in the ground, 0, coordinate system via m R m1 0 1 0 (727) Evaluating the axis of rotation in the 0 coordinate system yielded 0m = [0.0433, 0.1342, 0.9900]T. Finally, the last three components of A 1 D are approximated by ( 0m) and will be dimensionless (radians). As previously stated, the first three components of A 1D represent the instantaneous linear velo city of a point in the moving body that is coincident with the origin of the reference frame. This quantity is estimated by first determining the coordinates of the origin point of the ground, 0, coordinate system in the 1 coordinate system, i.e., 1P0orig. This quantity is determined as orig 1 0 T 0 1 orig 0 1P R P (728)
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104 where orig 1 0P is obtained as the top three terms of the fourth column of T0 1 which was numerically determined in (710). For infini tesimal motion, the coordinates of the origin of the ground coordinate system should be th e same when measured in the 1 or the A coordinate systems. Thus it is assumed that orig 0 1 orig 0 AP P where the notation 0'orig is introduced. This point is then transf ormed to the 1 coordinate system as orig 0 A 1 A orig 0 1P T P (729) where T T T0 A 1 0 1 1 A (730) The translation of the origin point of coor dinate system 0 as seen from coordinate system 1 can be written as the net displacement of the point as orig 0 1 orig 0 1 orig 0 1P P v (731) Lastly, this translation vector can be ev aluated in the 0 coordinate system as orig 0 1 0 1 o 0v R v (732) For this particular case, the translation of the point in the top platform that is coincident with the origin of the reference coordinate system, for the case where the top platform moves from pose 1 to pose A, was evaluated as mm 0052 0 6462 0 7223 0o 0 v (733) Thus for the case where the top platform moves from position 1 to position A, the instantaneous twist written in axis coordinates is estimated as
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105 04364 0 00592 0 00191 0 0052 0 6462 0 7223 0 0 o 0 A 1m v D (734) where the first three components have units of mm and the last three components have units of rad. The process can be repeated to determine the twists associated with the motion from pose 1 to pose B, pose 1 to pose C, pose 1 to pose F. The resulting twists were determined as 0273 0 0025 0 0053 0 0411 0 8259 0 3183 0 0001 0 0144 0 0088 0 0185 0 2102 0 0944 0 0027 0 0006 0 0222 0 1035 0 6291 0 04179 0 1 1 1 D C BD D D (735) 0025 0 0062 0 0071 0 0776 0 6217 1 2987 0 0091 0 0079 0 0001 0 2048 0 3813 0 6683 1 1 1F ED D Lastly, substituting these values into (719) and then substituting (719) and (724) into (717) yields the stiffness matrix for the device at pose 1. This matrix was determined as
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106 2.90450.04991.781267.167136.5397.152 0.02392.96370.8046167.0556.028181.38 0.20850.030811.182364.17693.5332.81 8.5556205.44326.45293902112210796 189.5329.633524.65144595573310174 98.136179.K 7118.616113959997.318204 (736) Comparison of analytical res ult and experimental result Comparing the stiffness matrix determined analytically with the stiffness matrix calculated experimentally, it is clear that bo th matrices are nonsymmetric and they are very close to each other. This result also vali dates the previous stiffness analysis as well as the force/ twist approximation. Future Research The passive parallel platform has been de signed, fabricated, and tested. Future experiments should be applied to test its dynamic properties such as the effects of vibration, velocity, and acceler ation. The performance of kinematic control depends not only on the stiffness matrix of the complia nt component but also on the whole robot system. The system stiffness is dependent on the configuration of the robot system, including the serial i ndustrial robot, passive parallel platform, a gri pper or tool, and any permanently attached compliant devices. Th e ability of accurately modeling the system stiffness ensures the control perfor mance for displacement and force. There are ways to improve the design of the PCCFC. The theoretical analysis presented in this work can be applied dire ctly to the future project. A noncontact displacement laser sensor can be used to re place the optical encoder to greatly reduce the size of the structure. Also a plastic or rubber cover can be designed to protect the connectors for use of the device in a far ranging set of applications.
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APPENDIX MECHANICAL DRAWINGS OF THE PARTS FOR PCCFC
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108 Figure A1. Strip holder
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109 Figure A2. Lower leg part.
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110 Figure A3. Upper leg part
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111 Figure A4. Leg
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112 Figure A5. Top plate.
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113 Figure A6. Base plate.
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114 LIST OF REFERENCES [Bal00] Ball, Robert Stawell, Treatise on the Theory of Screws, Cambridge University Press, Cambridge, UK, 1900. [Cho02] Choi, K., and Jiang, S., Spatial Sti ffness Realization with Parallel Springs Using Geometric Parameters, IEEE Tran s. on Robotics and Automation, Vol. 18 (3), pp. 274284, 2002. [Cib99] Ciblak, N., and Lipki n, H., Synthesis of Cartesian Stiffness for Robotic Applications, in Proc. IEEE Int. Conf Robotics and Automation, Detroit, MI, May 1999 pp. 2147, 1999. [Cra98] Crane, C.D., and Duffy, J., Kinematic Analysis of Robot manipulators, Cambridge University press, Cambridge, UK, 1998. [Cra01] Crane, C.D., and Duffy, J.,Class Notes: Screw Theory and its Application to Spatial Robot Manipulators, MAE, University of Florida, Gainesville, FL, 2001. [Dan95] Daniali H.R.M., and ZsomborMurray P.J., Singularity Analysis of a General Class of Planar Parallel Manipulators, In IEEE In t. Conf. on Robotics and Automation, Nagoya Japan, pp. 15471552, 1995. [Das94] Dasgupta B., Reddy S., and Mruthyunjay a, T.S., Synthesis of a ForceTorque Sensor Based on the Stewart Platform M echanism, Proc. National Convention of Industrial problems in Machines and M echanisms, Bangalore, India, pp.14~23, 1994. [Das98A] Dasgupta, B., and Mruthyunjaya, T. S., A NewtonEuler Formulation for the Inverse Dynamics of the Stewart Platform Manipulator, Mech. Mach. Theory, Vol. 33(8), pp. 11351152, 1998. [Das98B] Dasgupta, B. and Mrut hyunjaya, T.S. Singularit yFree Path Planning for the Stewart Platform Manipulator, Mechanism and Machine Theory, Vol. 33(6), pp.711725, 1998. [Das00]Dasgupta, B., and Mruthyunjaya, T.S ., The Stewart Platform Manipulator: A Review, Mechanism and Machine Theory, Vol. 35, pp. 15~40, 2000. [Dim65] Dimentberg, F.M., The Screw Calc ulus and its Application in Mechanics, Foreign Technology Division, Wrightpatte rson Air Force Base, Ohio. Document No. FTDHT23163267, 1965.
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115 [Duf96] Duffy, J, Statics and Kinematics with Applications to Robotics, Cambridge University press, Cambridge, UK, 1996. [Dwa00]Dwarakanath, T., and Crane, C., I nParallel Passive Co mpliant Coupler for Robot Force Control, Proceedings of the ASME Mechanisms Conference, Baltimore MD, pp. 214221, 2000. [Fic86] Fichter, E.F., A Stewart Platfo rmBased Manipulator: General Theory and Practical Construction, Int. Journal of Robotics Research, Vol.5 Issue 2, pp. 157~181, 1986. [Gai83] Gaillet, A and Reboulet, C., An Is ostatic Six Component Force and Torque Sensor, Proc. 13th Int. Symposium on Industrial Robotics, Chicago, IL 1983. [Gou62] Gough, V.E. and Whitehall, S.G., "Uni versal Tyre Test Machine," Proceedings of the FISITA Ninth Int. Technical Congress, London, UK, pp. 117137, 1962. [Gri89] Griffis, M., and Duffy, J., A Fo rward Displacement Analysis of a Class of Stewart Platforms, Journal of Robotic Systems, Vol. 6(6), pp. 703720, 1989. [Gri91A] Griffis, M., Kinestatic control: A Novel Theo ry for Simultaneously Regulating Force and Displacement, Ph.D. Dissertation, University of Florida, Gainesville, FL, 1991. [Gri91B] Griffis, M., and Duffy, J., K inestatic Control: A Novel Theory for Simultaneously Regulating Force and Disp lacement, Trans. ASME Journal of Mechanical Design, Vol. 113, No. 4, pp. 508515, 1991. [Gri93] Griffis, M., and Duffy, J., Method and Apparatus for Controlling Geometrically Simple Parallel Mechanisms with Distinctive Connections, United States Patent, Patent Number 5,179,525, Jan.12, 1993. [Hua98] Huang, S., The Analysis and Synthesis of Spatial Compliance, PhD dissertation, Marquette Univ., Milwaukee, WI, 1998. [Hun93] Hunt K.H. and Primrose E.J.F., Asse mbly Configurations of Some InParallel Actuated Manipulators, Mechanism and Machine Theory, Vol. 28(1), pp. 3142, 1993. [Hun98] Hunt, K.H., and McAree, P. R., The Octahedral Manipulator: Geometry and Mobility, The Int. Journal of Robotics Research, Vol. 17, No. 8, pp. 868885, 1998. [Inn93] Innocenti C. and Parenti C. V., Close dForm Direct Position Analysis of a 55 Parallel Mechanism, ASME Journal of Mechanical Design, Vol. 115(3), pp. 515521, 1993.
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116 [Lee94] Lee, J., An Investigation of A Qu ality Index for The Stability of InParallel Planar Platform Devices, Master's Thesis, University of Florida, Gainesville, FL, 1996. [Lee98] Lee, J., Duffy, J., and Hunt K., A Practical Quality Index Based On the Octahedral Manipulator, Int. Journal of Robotic Research, Vol.17 (10), pp. 10811090, 1998. [Lee00] Lee, J., Investigation of Quality Indices of InParallel Platform Manipulators and Development of Web Based Analysis Tools, PhD Dissertation, University of Florida, Gainesville, FL, 2000. [Lin 90] Lin, W., Duffy, J., a nd Griffis, M., "Forward Displacement Analysis of the 44 Stewart Platforms," in Proc. 21st Biennial Mech. Conf. ASME, Chicago, IL, Vol. DE25, pp. 263269, 1990. [Lin94] Lin, W., Crane, C.D., and Duffy J. Closed Form Forward Displacement Analysis of the 45 InParallel Platform s, ASME Journal of Mechanical Design, Vol. 116, pp. 4753, 1994. [Lon87] Loncaric, J., Normal Forms of S tiffness and Compliance Matrices,, IEEE Journal of Robotics and Automati on, Vol. 3(6), pp. 567572, 1987. [Ma91] Ma O. and Angeles J, Architecture Si ngularities of Platform Manipulator, In. IEEE Int. Conf. on Robotics and Automation, Sacramento, CA, USA, pp. 15421547, 1991. [Mer92] Merlet, J,, Direct Kinematics and A ssembly Models of Pa rallel Manipulators, Int. Journal of Robotics Research, Vol. 11(2), pp.150162, 1992. [Mer00] Merlet, J., Parallel Robots, Kluwer Academic Publishers, Boston, MA, 2000. [Ngu91] Nguyen C.C, and Antr azi, S.S. Analysis and Implementation of a Stewart PlatformBased Force Sensor for Passiv e Compliant Robotic Assembly, In IEEE Proc. of the Southeast Conf. 91, W illiamsburg, VA, pp. 880884, Williamsburg, 1991 [Pig98] Pigoski, T., Griffis, M., and Duffy, J., Stiffness Mappings Employing Different Frames of Reference, Mech. Mach. Theory Vol.33, No.6. pp. 825838, 1998. [Pol40] Pollard, W.L.G., "Spray Painting Machine," US Patent No. 2,213,108, August 26, 1940. [Rag93] Raghavan, M., The Stewart Pl atform of General Geometry has 40 Configurations, Trans ASME J Mech. Trans. Automat Des Vol. 115, pp. 277~282, 1993.
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117 [Rid02] Ridgeway, S., Research Proposal for Parallel Manipulator, MAE, University of Florida, Gainesville, FL, 2002. [Rob00] Roberts, R.G., Minimal Realization of an Arbitrary Spatial Stiffness Matrix with a Parallel Connectio n of Simple and Complex Springs, IEEE Trans. on Robotics and Automation, Vol. 16(5), pp. 603608, 2000. [Rob02] Roberts, R.G., A Note on the Normal Form of a Spatial Stiffness Matrix, IEEE Trans. on Robotics and Automation, Vol. 17(6), pp.968972, 2001. [Sel02] Selig, J.M. and Ding, X., Structure of the Spatial Stiffness Matrix, Int. Journal of Robotics and Automation, Vol. 17(1), pp.116, 2002. [Ste65] Stewart, D., "A Platform with Six Degrees of Freedom," Proc of the IMechE, Vol. 180, Pt. 1, No. 15, pp. 371385, 196566. [Svi95] Svinin, M.M., and Uchiyama, M, Optimal Geometric Structures of Force/Torque Sensors, Int. journal of Robotics Research, Vol. 14(6), pp.560~573, 1995. [Tsa99] Tsai, L.W., Robot Analysis, the Mechanics of Serial and Para llel Manipulators, John Wiley& Sons, New York, USA, 1999. [Wam96] Wampler, C.W., Forward Displacem ent Analysis of General SixInParallel SPS (Stewart) Platform Manipulators Usi ng soma coordinates, Mechanism and Machine Theory, Vol. 31(3), pp.331337,1996. [Wen94] Wen F. and Liang C., Displacem ent Analysis of the 66 Platform Mechanisms, Mechanism and Machine Theory, Vol. 29(4), pp. 547557, 1994. [Zha04] Zhang, B, and Crane, C., Special Singularity Analysis for a 6DOF Parallel Platform, 10th Int. Conference on Robotics & Remote Systems for Hazardous Environments, Gainesville, FL, USA, 2004.
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118 BIOGRAPHICAL SKETCH Bo Zhang was born in Xian, China, 1975. He received his Bachelor of Science degree in mechanical engin eering from XiDian University in his home town in 1997. Then he went to Beijing to continue his st udy and received his Master of Science degree in mechanical engineering from Tsinghua University in 2000. He found his strong interest in robotics and joined the Department of Mechanical and Aerospace Engineering, University of Florida, for his PhD degree in mechanical engineering. Since then, he has continued his studies and worked as a research assistant with Dr. Carl D. Crane III at the Center for Intelligent Machines and Robotics.
