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PAGE 1 1 THREEDIMENSIONAL RIGID BODY GUIDAN CE USING GEAR CONNECTIONS IN A ROBOTIC MANIPULATOR WITH PARALLEL CONSECUTIVE AXES By JAVIER AGUSTIN ROLDAN MCKINLEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 Javier Agustn Roldn Mckinley PAGE 3 3 To the memories of Joseph Duffy and Ali Seireg PAGE 4 4 ACKNOWLEDGMENTS My greatest gratitude in the attainment of th is goal is toward my supervisory committee chair, Dr. Carl Crane III. This work was possibl e thanks to his continuous advisership, patience and support in both the academ ic and personal aspects. I also thank my committee memb er Dr. David Dooner. His trus t in my work made possible that I became part of CIMAR Lab. His guidanc e, support and encouragement throughout this research will always be greatly appreciated. The effort and support of my supervisory committee members: Dr. Antonio Arroyo, Dr. William Hager, Dr. John Schueller, and Dr Gloria Wiens are also appreciated. I want also to remark on the support of cl ose friends and colleagues, namely Jaime Bestard, Jahan Bayat, and JeanFrancois Kamath. I gratefully acknowledge the financial support provided by the Department of Energy via the University Research Program in Robo tics (URPR), grant number DEFG0486NE37967. Finally, I express special deference to my family and parents for their undying love, patience and support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................13 1.1 Preliminary Spatial Mechanisms..................................................................................13 1.1.1 Kinematic Link.................................................................................................14 1.1.2 Revolute Joint...................................................................................................15 1.1.3 Mobility Criterion in Spatial Mani pulators and ClosedLoop Mechanisms.....16 1.2 Preliminary NonCircular Gears...................................................................................17 1.3 State of the Art..............................................................................................................18 1.4 Motivation.....................................................................................................................19 1.5 Previous Work...............................................................................................................21 1.6 Objective and Scope......................................................................................................23 1.7 Overview of the Manuscript.........................................................................................24 2 SPATIAL KINEMATIC BASIC CONCEPTS......................................................................26 2.1 Standard Link Coordinate System................................................................................26 2.2 Transformation Matrix between Standard Systems......................................................27 2.3 Forward Kinematic Analysis of a 6Li nk 6R Open Loop Spatial Mechanism.............30 2.4 Reverse Kinematics Problem Statement for 6L6R Manipulator.................................31 2.5 ClosetheLoop Solution Technique.............................................................................32 2.5.1 Determination of the ClosetheLoop Parameters.............................................34 2.5.1.1 Twist Angle 71...................................................................................34 2.5.1.2 Joint Angle 7......................................................................................35 2.5.1.3 Angle 1..............................................................................................36 2.5.1.4 Distances a71, S7 and S1......................................................................37 2.5.2 Special Cases for Closedt heLoop Solution Technique..................................39 2.5.2.1 S1 and S7 are Parallel or Antiparallel..................................................39 2.5.2.2 S1 and S7 are Collinear........................................................................40 2.6 Spherical ClosedLoop Mechanisms.............................................................................40 2.7 Mobility of Spherical Mechanisms and Classification of Spatial Mechanisms............42 2.8 Joint Vectors Expressions in the Spherical Mechanism...............................................42 2.9 Link Vectors Expressions in the Spherical Mechanism................................................45 2.10 Orientation Requirements Specifi cation: XYZ Fixed Angles....................................48 PAGE 6 6 3 NONCIRCULAR GEARS FUNDAMENTALS..................................................................49 3.1 Pitch Curve and Pitch Point..........................................................................................49 3.2 Input/Output NonCircular Gear Relationship..............................................................50 3.2.1 Input/Output Relationship with the Connecting Link Grounded......................51 3.2.2 Input/Output Relationship with All Links Movable.........................................51 3.3 Complete Profile Synthesis: NonCircular Relationships for the Entire Rotation........52 3.4 Pitch Curve Coordinates...............................................................................................57 4 REVERSE KINEMATIC ANALYSIS OF THE 1DOF SPATIAL MOTION GENERATOR...................................................................................................................... ..61 4.1 Mobility Analysis of the Spatial Motion Generator......................................................62 4.2 Grouping of the 7L7R Spatial Mechanis m with Consecutive Axes Parallel: Equivalent Spherical Mechanism..................................................................................63 4.3 Reverse Kinematic of the 6L6R OpenLoop Mechanism with Consecutive Axes Parallel....................................................................................................................... ...65 4.3.1 Problem Statement............................................................................................65 4.3.2 Solution of the Equivalent Spherical Quadrilateral..........................................66 4.3.2.1 Solving for 1Q....................................................................................67 4.3.2.2 Solving for 2Q....................................................................................68 4.3.2.3 Solving for 3Q....................................................................................69 4.3.3 Vector Loop Equation of the 7L 7R Parallel Axes Mechanism.......................69 4.3.4 TanHalfAngle Solution for 5........................................................................74 4.3.5 Solution for 3...................................................................................................81 4.3.6 Solution for 2...................................................................................................83 4.3.7 Solution for 1, 4 and 6...................................................................................83 4.3.8 Solution Tree.....................................................................................................83 4.4 Reverse Kinematics Analysis for the Co mplete Spatial Motion Generator..................84 5 SYNTHESIS OF THE NONCIRCULAR PITCH PROFILES.............................................89 5.1 I/O NonCircular Gear Relationships for the Spatial Motion Generator......................89 5.2 Sequence Parameter Approach.....................................................................................91 5.2.1 Polynomial Interpolation of Joint Angl es: Discarded First and Last Points.....92 5.2.2 I/O Relationship Expressions from Sequence Parameter Approach.................94 6 RESULTS: ILLUSTRATIVE EXAMPLES..........................................................................96 6.1 Reverse Kinematics for a Single Position of the End Effector: Single Position, Half Mechanism Case...................................................................................................96 6.2 Motion Generation along Discrete Path a nd Orientation Require ments: Complete Mechanism Case.........................................................................................................100 7 CONCLUSION AND FUTURE WORK.............................................................................111 7.1 Conclusion..................................................................................................................111 7.2 Future Work................................................................................................................112 PAGE 7 7 APPENDIX A JOINT AND LINK VECTORS EXPRESSIONS IN THE SPHERICAL MECHANISM..114 B POSITION AND ORIENTATION SPECI FICATION IN EXAMPLE 6.2.........................118 LIST OF REFERENCES.............................................................................................................120 BIOGRAPHICAL SKETCH.......................................................................................................123 PAGE 8 8 LIST OF TABLES Table page 41 Direction cosines of a closedloop 7L7 R mechanism expressed in a coordinate system were the x and z axes are aligned with the vectors a23 and S3, respectively..........71 61 Constant mechanism parame ters for numerical example..................................................96 62 Desired position and orientation requirements..................................................................96 63 Calculated closetheloop parameters................................................................................97 64 Sums of consecutive joint angles obtained from the equivalent s pherical quadrilateral solution....................................................................................................................... ........97 65 Joint angles corresponding to the sixt een solutions of the numerical example.................98 66 Constant mechanism parameters for the closedloop example........................................102 B1 Continuous position and orient ation needs specification.................................................118 PAGE 9 9 LIST OF FIGURES Figure page 11 A kinematic link, the simplest representation of a spatial link..........................................14 12 Revolute joint and joint angle............................................................................................16 13 Noncircular gears......................................................................................................... .....18 14 Robotic manipulators used in industry..............................................................................18 15 1DOF planar motion genera tor using noncircular gears.................................................19 16 1DOF spatial motion generator implementing noncircular gears...................................24 21 Standard ith coordinate sy stem attached to the link ij......................................................27 22 Coordinate systems of two successive links ij and jk........................................................29 23 Given information in the reverse analysis.........................................................................32 24 Hypothetical closure link.................................................................................................. .33 25 Vector loop to find the distances a71, S7 and S1.................................................................38 26 Spherical quadrilateral.................................................................................................... ...41 31 Pitch point and pitch curves of two bodies in mesh...........................................................50 32 Noncircular input and output bodies in mesh...................................................................51 33 Arbitrary working and nonworking secti ons and angles in a complete noncircular gear........................................................................................................................... ..........53 34 Boundary conditions for the whole synt hesis of the Input/Output relationship................54 35 Rectangular coordinate systems xyo and xyi for a gear pair in mesh...............................58 36 Local coordinate systems xyi and xyo in the synthesis of th e pitch point with input, output and connecting bodies movable..............................................................................59 41 1DOF spatial motion generator........................................................................................61 42 Labeling of the spatial motion generator for mobility analysis. Parts with the same number label are one single link........................................................................................63 43 Kinematic labeling of the 6L6R openloop mechanism...................................................64 PAGE 10 10 44 Equivalent spherical mechanism........................................................................................68 45 Solution tree of the spherical quadrilateral........................................................................69 46 Sixteen solutions tree for 7L7R closed loop mechanism with consecutive pairs of parallel joint axes............................................................................................................ ...84 47 Kinematic labeling of the 6L6R openloop mechanism 112 7.....................................86 48 Transformation of the fixed coordinate systems xF1yF1zF1 and xF2yF2zF2.....................87 49 Disposition of the two sixth coordinate systems................................................................87 51 Noncircular gear connections labeling............................................................................90 52 Labeling of the h nonci rcular gear connection.................................................................90 53 Introduction of the sequence parameter u to discretely express the motion generation requirements along {[xo,xf],[yo,yf],[zo,zf]}.........................................................................93 61 Real solutions AH in Table 65, corr esponding to the A solution of the spherical quadrilateral in Table 64...................................................................................................99 62 Real solutions IP in Table 65, corresponding to the B solution of the spherical quadrilateral in Table 64.................................................................................................100 63 Motion needs............................................................................................................... .....102 64 Motion needs versus the sequence parameter in each fixed coordinate system..............103 65 Joint angles for the closedloop mechanism....................................................................105 66 Gear synthesis results for c onnection 234 in half mechanism 01 6.............................106 67 Gear synthesis results for c onnection 456 in half mechanism 01 6.............................107 68 Centrode, output angle a nd Input/Output gear relationshi p versus the input angle in half mechanism 011 6..................................................................................................108 69 Closedmechanism at two differe nt motion needs in Table B.1......................................110 PAGE 11 11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THREEDIMENSIONAL RIGID BODY GUIDAN CE USING GEAR CONNECTIONS IN A ROBOTIC MANIPULATOR WITH PARALLEL CONSECUTIVE AXES By Javier Agustn Roldn Mckinley August 2007 Chair: Carl Crane III Major: Mechanical Engineering Robot manipulators are often employed in industry to perform repetitive motions. Examples are pickandplace operations that occur during packaging processes and positioning operations that occur during assembly. Of ten, six degreeoffreedom openloop robotic manipulators are used in these operations. A disadvantage of using these manipulators for repetitive tasks is twofold. First, the hardware associated with th e six actuators that are required for a six degreeoffreedom chain is expensive. Second, the coordinate d control of these six actuators that is needed for precision motion is complicated. In our research, a new approach is presente d to address repetitive motion operations such that they can be accomplished by a one degreeoffree dom device. The end effector that is to be positioned and oriented along a specified path (or th rough a set of specified poses) is attached to the middle link of a spatial closedloop mechan ism that is comprised of 12 links (including ground) that are interconnected serially by 12 revolu te joints. Five pairs of noncircular gears are then designed and inserted into the mechanism to reduce the degreeoffreedom of the device to one. To make the approach more practical, the geometry of the closedloop mechanism was chosen so that pairs of adjacent joint axes are paralle l. For example, the first joint axis is parallel PAGE 12 12 to the second, the third is parallel to the fourth, and so on. The r eason for this is that planar noncircular gears can be designed to be inserted into the mechanism. A kinematic analysis of this analysis is presented and it is shown that an openloop chain comprise d of six joints where adjacent joint axes are parallel (in effect half of the closedloop mechanism) have a total of sixteen possible solution configura tions that will position and orient the distal link as specified. The kinematic analysis of this special geometry and the design of the working and nonworking sections of the gear pairs are the main contribut ions of our research. Numerical examples are presented. PAGE 13 13 CHAPTER 1 INTRODUCTION Robotic systems and mechanisms exist to pe rform two kinematical tasks: function and motion generation. In both cases a body, namely the coupler, or a poi nt of this body, must occupy certain positions. In motion generation, the coupler must traverse a particular path and orientation with respect to a certain reference frame, while function ge neration requires that a point of the coupler describes a cer tain path and the orientation of this body is of no interest. If the task is performed in the twodimensional space then it is referred to as a planar case, while a task performed in threedimensional space is referred to as a spatial case. Some authors consider closed loop linkages as mechanisms, while open loop multidegree of freedom (DOF) manipulators serial manipulators with progra mmable motionare considered as robots or robotic systems. In this work, bot h open and closed loop linkages are considered indistinctively as mechanisms. The thrust of this research is the integration of noncircular gear pair s into a serial closed chain spatial mechanism to fulfill motion genera tion requirements through a single degree of freedom mechanism in threedimensional space. In order to present the idea and further its analysis, some preliminary con cepts about spatial mechanisms and noncircular gearing are considered. Later in this chapter is the justificat ion of this project, the objective statement, and a complete review of related previous work. 1.1 Preliminary Spatial Mechanisms Basic concepts related to spatial mechanisms are presented here: mob ility or degrees of freedom, kinematic link, revolute jo int, joint axis, li nk axis, and joint a nd link distances. For further information about, see Crane and Duffy [1]. PAGE 14 14 1.1.1 Kinematic Link In this work, the spatial mechanism of interest is to be formed by a se ries of links (assumed to be rigid bodies) and joints forming a clos ed chain, where one link is connected to ground. These links are to be represented in their simp lest form: kinematic links, which can have in reality many equivalent configurations. Figure 11 shows a kinematic link with his joint vectors, Si and Sj, which are unit vectors along the two consecutive joint axes i and j. Figure 11. A kinematic link, the simplest representation of a spatial link. The relative position of the skew joint axes is defined by the link length aij and the twist angle ij, depicted in Figure 11. The link distance aij is the mutual perpendicular distance between the skew joint ax es, and the twist angle ij is the angle between the vectors Si and Sj. The vector aij is defined as the cross product of the vectors Si and Sj, defined as Six Sj = aij sin ij, then the direction and sense of aij is defined by the right hand rule applied to the vectors Si and Sj. In the same way, the sense of the angle ij is obtained by applying th e right hand rule, with the PAGE 15 15 thumb pointing along aij and the fingers sweeping from the direction of Si to the direction of Sj to form the angle ij. It is evident that the direction of the vectors Si and Sj presented in Figure 11 is arbitrary because either Si or Sj can be drawn in the opposite direction; however aij will always be determined here by the cross product of Si and Sj according to their direction. 1.1.2 Revolute Joint The term joint was used without any previous de finition in the last s ection. In general, a joint is the connection existing between a pair of successive links that determines the nature of their relative motion. The revolute joint, denoted by the letter R, is the simplest and most common joint. The revolute joint presented in Figur e 12 connects the link ij with th e link jk. Link jk can rotate about the joint vector Sj, relative to the link ij. As a must, the vector Sj of link ij and the vector Sj of the link jk are to be para llel and not antiparallel. The link jk presented in Figur e 12 can only rotate about Sj relative to the link ij, thus the link jk has one degree of freedom with respect to link ij. A new concept is presented here, the joint angle j, which allows measurement of the relativ e motion of link jk relative to link ij. From Figure 12, the joint angle j is the angle between the unit vectors aij and ajk, measured in a righthand sense with th e thumb pointing along the joint vector Sj, defined as aijx ajk= Sjsin j. A joint distance associated with the joint vector is also defined here; this is the joint or offset distance Sj. As depicted in Figure 12, Sj is the mutual perpendicular distance between the vectors aij and ajk. For the revolute joint, the offset distance Sj is a constant because the link jk cannot translate along the link ij, it can only rotate about it changing only the joint angle j. PAGE 16 16 Figure 12. Revolute joint and joint angle. 1.1.3 Mobility Criterion in Spatial Ma nipulators and ClosedLoop Mechanisms The mobility or degrees of freedom (DOF) of a mechanism is the number of independent inputs required to determine the position of all links of the mechanism with respect to ground. A shorter concept for the DOF in a mechanism is th e number of inputs that need to be provided in order to define the output. Complete inform ation about mobility in planar and spatial mechanisms can be found in DiezMartnez et al. [2]. The Kutzbach mobility equation for spatial li nkages where one link is connected to ground is1 5 4 3 2 1f f 2 f 3 f 4 f 5 1) 6(n M (11) 1 Some particular geometry may exist in special mech anisms where the real mobility does not match the value obtained by Equation 11. PAGE 17 17 where: M: mobility or DOF n: number of spatial links or connected bodies fi: number of joints with i re lative degrees of freedom. A more compact format of Equation 11 for cl osed spatial mechanisms is presented as n 1 i i6 f M (12) where: M: mobility or DOF n: number of spatial links or connected bodies fi: number relatives degrees of freedom permitted by the ith joint. The purpose of this subsection is to introduce th e mobility concept. A mobility analysis is later presented in Section 2.1. 1.2 Preliminary NonCircular Gears Noncircular gears are toothed bodies whose pitch curves (or primitive curves) are not common circles. These primitive curves can ha ve any functional shape, derived from the necessity of varying output pos ition, velocity, and/or acceler ation. Figure 13A shows a noncircular gear profile, with the teeth and pitch pr ofiles. Figure 13B presen ts a spur noncircular gear pair in mesh. For detailed information, see Dooner and Seireg [3]. PAGE 18 18 Figure 13. Noncircular gears. A) Noncircular teet h and pitch profiles. B) Spur noncircular gear pair in mesh. Courtesy of Dr. David Dooner. 1.3 State of the Art The spatial path and motion generation task s are currently fulfilled by openloop robot manipulators, being the PUMA 560, GE P60 (see Figure 14) and the Cincinnati Milacron T3 the most used configurations. Further information about the reverse kinematic of these industrial manipulators can be found in Crane and Duffy [1]. Figure 14. Robotic manipul ators used in industry. A) Puma robot. B) GE P60. Courtesy of Dr. Carl Crane. PAGE 19 19 Current applications of nonci rcular gears include printing presses, packaging machines, conveyors, and low speed precision instruments; their combined us e with linkages to accomplish motion generation tasks is currently limited to the planar case. The 1DOF sixlink mechanism motion generator depicted in Figure 15 incorp orates two noncircular gear pairs and was proposed to achieve any path and orientation requirements in th e planar scenario, subject to feasibility of the noncircular gear profiles. For further informati on, see Roldn Mckinley et al. [4]. There are no known references related to 3D2 spatial motion generation using noncircular gears. Figure 15. 1DOF planar motion ge nerator using noncircular gears. 1.4 Motivation There are applications in industry where it is necessary for a body in a mechanism to repeatedly traverse a particular path and orientation with respect to a certain reference frame. Planar and spatial applications include automotiv e suspension systems, garbage truck delivery, automotive welding and painting processes, a nd shoe testing machin es. The path and/or orientation requirements can be met th rough function and motion generation. 2 Three dimensional or spatial scenario. PAGE 20 20 Currently, these spatial kinematic tasks are fu lfilled by open and closedloop mechanisms. Figure 14A and 14B are examples of open l oop mechanisms and Figure 15 shows a closedloop planar mechanism. The inverse kinematics i nvolved in three dimensi onal kinematic tasks is solved via analytical methods (in some cases co mbined with numerical it erative solutions). One degreeoffreedom threedimensional closedloop linkages can be designed to satisfy a finite number of values for function, path, and motion generation. Robot manipulators are versat ile 3D motion generators. Th ese open loop manipulators are widely accepted in industry; however, because of their mobility or degreesoffreedom, specification of the actuators is necessary. Typical industrial ma nipulators such as the PUMA and GE robots possess six degreesoffreedom, wh ere six actuators must be synchronized for a single motion of the working tool. Typically, actua tion is achieved via moto rs (stepper or servo with controls). The cost and complexity associ ated with achieving this coordinated computer control can be eliminated if a 1DOF mechanis m can generate the desired repetitive motion. A single actuator can be used and the mech anism response is exactly quantified. To elaborate on the last statement, the cost of the mechanism can be reduced due to the elimination of five actuators and their corresponding electronic equipment. This becomes the main saving when comparing the 6DOF and th e 1DOF mechanisms. Although there is a cost associated with the 1DOF spatial mechanism building, such as the manufacture of the noncircular gears, the cost of the electronic equi pment in a six degreeoffreedom mechanism is anticipated to be higher than the noncircular gear manufacture and assembling costs. The elimination of actuators in the system can also have additional benefits: less maintenance and lower operation costs. PAGE 21 21 To summarize, the implementation of a 1DOF spatial motion generator is suited for repetitive tasks and can reduce the fabricati on, installation, progr amming, operation, and maintenance costs of multidegreeoffreedom robotic manipulators. 1.5 Previous Work The fourbar spatial mechanism was the firs t linkage used to accomplish path and motion generation tasks in the three dimensional scenario. Better results were obtained in just path generation due to some limitations of this four link mechanism. Suh [5 ] proposed the synthesis of an RSSR3 function generator through a generalized matrix for the description of finite functional displacement, based on a kinematic inversion technique. Suh [6] also worked on the rigid body guidance through finitely separated multi positions by the synthesi s of the RSSR fourbar and the RSSRSS sixbar spatial mechanisms, including again a genera lized matrix derived from the constraint equations. Spatial path and motion generation have been traditionally solved by parallel and serial robot manipulators. Significant contributions in this field we re done by Rooney and Duffy [7] by proposing the closure of spatial linkages as a dire ct method to solve the displacement analysis. The same authors, Duffy and Rooney [8], pres ented their landmark work in robotics by developing a unified procedure for th e analysis of spatial fourlink, fivelink, sixlink, and sevenlink mechanisms, by stating that there are on ly three fundamental loop equations for any spherical polygon: sine, sinec osine, and cosine laws. Later, Duffy and Rooney [911] expressed the inputoutput displa cement equation as a degree eight polynomial in the halftangent of the output angular displacement for spatial sixlink 4RPC mechanisms; they realized it for the RCRPRR, R CRRPR, and RRRPCR inversions. 3 R: revolute joint, P: prismatic joint, C: cylindrical joint, and S: spherical joint. PAGE 22 22 Duffy and Rooney [12] also deduced the inputoutput displacement equation of degree 16 for spatial sixlink 5RC mechanisms, in partic ular for the RRRRCR and RRCRRR inversions. The displacement spatial seven link 5R2P mechanism was also solv ed by Duffy [13]; an eight degree inputoutput displacement equation was derived for the RPPRRRR, RRRPPRR, RPRRRPR, RPRRPRR, and RPRPRRR inversions. Separately, Sandor, Kohli, and Zhuang [14] considered a RSSRSRR spatial mechanism for motion generation with prescr ibed crank rotations, limited to four precision points. A singleactuator RSSRRSS adjustable spatial moti on generator was synthesized by Sandor et al. [15], where the authors also presented the advantages of this mechanism in comparison with the multidegreeoffreedom parallel robotic manipulators when performing hi ghly repetitive tasks. As a disadvantage, the mechanism was limited to only two exact prescribed positions. Lee and Liang [16] also performed the displacement analysis of spatial mechanisms: the inversions RRPRRRR, RRRPRRR, and RRRRRPR of the spatial 7link 6RP linkage. The inputoutput equations of degree 16 in the halftanangl e of output angular displacem ents were presented. Additional contributions are now presented. Sandor, Weng, and Xu [17] synthesized an RPCPRR spatial mechanism for mo tion generation without branching defect. Premkumar, Dhall, and Kramer [1821] developed path and functio n generation with dive rse spatial fourbar mechanisms using the selective precision synthesis method. Duffy and Crane [22] obtained a 32nd degree inputout put equation in the tanhalfangle of the output angular displacement for 7link 7R mechanism. The Mount Everest of kinematic problems 4 was later presented by Lee and Liang [23], based on [22], as a 16th degree polynomial inputoutput equation in the tanhalfa ngle of the output angular displacement. The 4 As the General Spatial 7Link 7R Mechanism was described by Professor Ferdinand Freudenstein. PAGE 23 23 inverse kinematic analytical solution of the General Spatial 7Link 7R Mechanism is of particular interest for this project. Noncircular gears have not b een included into spatial motion generation tasks, only into planar path and motion generation. Dooner [24] proposed utilizing a 1DOF eightlink mechanism and optimized noncircular gear elemen ts with application to automotive steering to fulfill path generation requirements. Later, Roldn Mckinley [25] developed the planar 1DOF sixlink mechanism with two pairs of noncircular gears to achieve path and orientation meets, and finally Roldn Mckinley et al. [4] obtained noncircular gear pitch profiles for a motion with a path that was approximated as a quadratic function. More recently, Mundo, Liu and Yan [26] utilized noncircular profile synthesis to integrate cams and linkages to obtain a precise planar path generator. Gatti and Mundo [27] also proposed a sixlink mechanism incorporati ng two prismatic joints and two cams to achieve planar rigid body guidance. 1.6 Objective and Scope The objective of this work is to develop a 1DOF spatial mechanism, depicted in Figure 16, to achieve generalized threedimensional motion generation. This mechanism will allow satisfying any generalized path f(x,y,z) of point P in link 7 and orientation ( x, y, z) of end effector link 7. To this end, it is first necessary to solve the inverse kinematic of the 1DOF spatial mechanism involving parallel axes in th e geared transmissions. Next, the noncircular pitches curves are synthesized. PAGE 24 24 Figure 16. 1DOF spatial motion generator im plementing noncircular gears. A) Position requirement f(x,y,z. B) Orientation requirements ( x, y, z). In addition to the attainment of link lengths, joint offset distances, link angles, and joint angles required to synthesize the spatial mechanism, attainment of the pitch profiles for the noncircular gears are of interest in the present project. 1.7 Overview of the Manuscript The aim of this research was presented in this chapter along w ith basic introductory concepts. This dissertation has seven chapters, wh ere Chapters 2 and 3 include basic concepts in PAGE 25 25 spatial kinematics; and noncircular gearing and screw theory, respec tively. Chapter 4 details the reverse kinematic analysis of th e mechanism utilized in this re search. The synthesis of the noncircular pitch profiles is presented in Chapter 5, based on gearing concepts previously presented in Chapter 3. The results and comments section is presented in Chapter 6. This dissertation ends with a conclusion and description of future work in Chapter 7. PAGE 26 26 CHAPTER 2 SPATIAL KINEMATIC BASIC CONCEPTS The reverse kinematic solution technique used in this work is based on the unified theory proposed by Crane and Duffy [1] to solve spatia l robotic manipulators. In this chapter, the fundamentals to apply the unified theory are presen ted. The totality of the topics here presented Standard coordinate systems a nd their transformation matrices are presented followed by the forward analysis. The loop closure technique is also explai ned, where an open loop manipulator is transformed into a closed loop mechanism by in troducing a virtual closur e link. Next, based on the closed loop mechanism, equivalent spherica l mechanisms are introduc ed. The solution of the equivalent spherical mechanism reverse kinematic s satisfies the solution of the spatial closed loop mechanism. The chapter ends with the fixed angles specification for the orientation of a rigid body with respect to a fixed reference fram e, which will be used when specifying the orientation needs for th e robotic manipulator. 2.1 Standard Link Coordinate System This section describes the methodology proposed by Crane and Duffy [1]. For the analysis of robot manipulators, it is comm on to attach a coordinate system to each link. A standardized approach will be used to select these coordinate systems. First, the coordinate system will have its origin located at the intersection of the link and joint unit vectors, see Fi gure 21. If the link ij is to be considered, then the origin of its coor dinate system is located at the intersection of aij and Si. Second, the direction of the x axis is parallel to the link unit vector aij. Finally, the direction of the z axis is parallel to the joint unit vector Si. PAGE 27 27 The direction of the y axis is to be determined through a cross product of the vectors i and k (or aij and Si): kxi = j where i, j and k are unit vectors along the x, y, and z axes. It is important to note that the vectors aij and Sj are unit vectors. Figure 21. Standard ith coordinate system attached to the link ij. 2.2 Transformation Matrix between Standard Systems This section describes the methodology proposed by Crane and Duffy [1]. In a general sense, spatial kinematics is based on the tran sformations of coordina tes between standard coordinate systems attached to the mechanism links. The objective here is to find the mathematical expression that relates two successi ve standard coordinate systems. To start, consider two consecutive links, the links ij and jk, as shown in Figure 22. The transformation matrix that relates the standard coordinate systems i (xiyizi) and j (xjyjzj) can be obtained as: 1) Start with the two coordinate system aligned. 2) Translate the ith coordinate system the distance aij along the xi axis. The translation matrix is PAGE 28 28 1 0 0 0 0 1 0 0 0 0 1 0 a 0 0 1ij 1M (21) 3) Rotate the ith coordinate system an angle ij about the xi axis. The rotation matrix is 1 0 0 0 0 c s 0 0 s c 0 0 0 0 1ij ij ij ij 2M (22) where cij=cos( ij) and sij=sin( ij). 4) Translate the ith coordinate system a distance Sj along the zj axis. The translation matrix is given by 1 0 0 0 S 1 0 0 0 0 1 0 0 0 0 1j 3M (23) 5) Rotate the ith coordinate an angle j about the zj axis. The rotation matrix is 1 0 0 0 0 1 0 0 0 0 c s 0 0 s cj j j j 4M (24) where cj=cos( j) and sj=sin( j). 6) Systems xiyizi and xjyjzj are aligned. PAGE 29 29 Figure 22. Coordinate systems of two successive links ij and jk. It is possible to relate Equations 21 to 24 to obtain the transformation matrix that relates the standard coordinate systems i and j. This matrix is 4 3 2 1 i jM M M M T (25) The explicit expression for this transformation matrix is 1 0 0 0 S c c s c s s S s s c c c s a 0 s cj ij ij ij j ij j j ij ij ij j ij j ij j j i jT (26) The inverse of this transforma tion can be readily obtained as, 1 0 0 0 S c s 0 a s s c c c s a c s s c s cj ij ij ij j ij j ij j j ij j ij j ij j j j iT (27) One more transformation matrix is needed for completeness; this is the matrix that relates the fixed coordinate system (ground) and the firs t link, link 12. The ground coordinate system is selected with its origin coincident with that of the coordinate system attached to the first link. PAGE 30 30 The z axis of the ground coordinate system is parallel to the first joint axis. Thus the transformation matrix th at relates the first and ground coordi nate system can be obtained as a rotation of an angle 1 about the z axis, 1 0 0 0 0 1 0 0 0 0 cos sin 0 0 sin cos T1 1 1 1 F 1 (28) where 1 is the angle measured from the x axis of the ground coordinate system to the x axis of the coordinate system attached to the first m oving link, measured in a righthand sense about S1. 2.3 Forward Kinematic Analysis of a 6Link 6R Open Loop Spatial Mechanism This section describes the methodology propos ed by Crane and Duffy [1]. The forward analysis is the inverse process of the reverse ki nematic analysis. Because this analysis is much simpler than the reverse kinematic analysis, it will be presented first as follows. Given the manipulator joint angles, i.e. 1 through 6, and the position of the tool point (attached to the end effector) measured in the coordinate syst em attached to the sixth (distal) link, 6Ptool, determine the position of the tool point a nd the orientation of the sixth link measured with respect to ground. The orientation of the last link is defined by the vectors Fa67 and FS6 which are the directions of the x and z axes of the sixth coordi nate system measured with respect to the fixed coordinate system. The position of the tool point measured with respect to the fixed coordinate system is obtained as tool 6 5 6 4 5 3 4 2 3 1 2 F 1 tool FP T T T T T T P (29) PAGE 31 31 The orientation of the sixth link as measured in the fixe d coordinate system can be obtained as the first and third column of the ro tation matrix (upper left 3 submatrix) of the product T T T T T T5 6 4 5 3 4 2 3 1 2 F 1. 2.4 Reverse Kinematics Problem Statement for 6L6R Manipulator This section describes the methodology propos ed by Crane and Duffy [1]. The reverse analysis for the 6L6R determines all possible sets of the six joint angles that satisfy any position and orientation of the end effector, given the desi red coordinates of the to ol in the fixed and 6th coordinate system, the geometry of the openlo op mechanism (link lengths, twist angles, and joint offsets), and the desired orientation of the end effector as defined by the vectors FS6 and Fa67. Figure 23 summarizes the reverse analysis gi ven information and based on this the reverse kinematic problem statement can be written as Given: 1) The constant mechanism parameters: link lengths a12 to a56, twist angles 12 to 56, and offset distances S2 to S5 2) Offset distance S6 and the direction of the a67 relative to the vector S6 3) Position and orientati on of the end effector: FPtool, FS6 and Fa67 4) Location of the tool point in the 6th coordinate system: 6Ptool Find: The angle between the fixed coordinate system and the first standard coordinate system, 1, and the joints angles 2 to 6. The transformation matrix that relates the sixth coordinate system and ground is given by PAGE 32 32 1 0 0 0orig 6 F F 6 F 6P R T (210) where: 6 F 67 F 6 F 67 F F 6S a S a R (211) 6 F tool 6 67 F 6 F tool 6 67 F tool 6 tool F orig 6 FS k P a S j P a i P P P (212) The analytical solution to this problem involves two main steps. The first step is to find the hypothetical closure link that connects an imaginary jo int axis with the first joint axis. This step is presented in the immediate section. Th e second step is obtaining the angles 1 to 6 via analysis of the equivalent s pherical closedloop mechanism. Figure 23. Given information in the reverse analysis. 2.5 ClosetheLoop Solution Technique This section describes the methodology proposed by Crane and Duffy [1]. The closetheloop solution technique will introduce a hypothetic al closure link that connects an imaginary joint axis with the first joint ax is. For this work, there are six links in the openloop mechanism, then the hypothetical clos ure link will connect an imaginary joint axis S7 with the first joint axis PAGE 33 33 S1. The direction and location of the joint axis S7 must be selected first, and it is required that the vector S7 is perpendicular to the link vector a67. An arbitrary value for the link angle 67 will be set as 90 degrees, which will define the direction of S7. After this conscious selection of 67, S7 is calculated as 6 F 67 F 7 FS a S (213) The vector FS7 is selected to pass through the origin of the standard sixth coordinate system, where from the link distance a67=0. An interesting result de rived from the closedform solution technique is that the offset distance S1 is introduced as the perpendicular distance between the vectors a71 and a12 measured along S1. Figure 24 shows the h ypothetical joint axis S7 and the hypothetical link 71. Also the offset distance S1 is depicted along with all the closetheloop parameters to be determined: a71, S7, S1, 71, 7, 1. As shown in the figure, the angle 1 is the angle swept from a71 to the x axis of the fixed coordinate system as measured in a righthand sense about S1. It is apparent that the relationship between 1, 1, and 1 is 1 = 1+ 1. Figure 24. Hypotheti cal closure link. PAGE 34 34 2.5.1 Determination of the ClosetheLoop Parameters The problem statement for the closetheloop parameters is as follows. Given: Joint vectors FS1, FS7 and FP6orig, the coordinates of the origin of the sixth coordinate system measured in the fixed system, Find: Joint distances S1, S7; link distance a71; joint angle 7; twist angle 71, angle 1 First the direction of the vector a71 measured with respect to th e fixed coordinate system is to be found. By definition, a71 is perpendicular to S7 and S1, then 1 F 7 F 1 F 7 F 71 FS S S S a (214) The vector FS7 is known from Equation 213, and the vector FS1 is parallel to the z axis of the fixed coordinate system, then 1 0 01 FS (215) Substituting Equations 213 and 215 into Equation 214, Fa71 leaves 1 0 0 1 0 06 F 67 F 6 F 67 F 71 FS a S a a. (216) 2.5.1.1 Twist Angle 71 The general definitions for the sine and cosine of the twist angle ij are PAGE 35 35 ij j i ijsa S S (217) j i ijcS S (218) The twist angle 71 can be obtained by knowing its sine and cosine, by substituting Equations 213, 215 and 216 into E quations 217 and 218, to obtain 1 0 0 c6 F 67 F 71S a (219) 1 0 0 1 0 0 1 0 0 s6 F 67 F 6 F 67 F 6 F 67 F 71S a S a S a (220) 2.5.1.2 Joint Angle 7 The sine and cosine expressions for the joint angle j are j jk ij js S a a (221) jk ij jc a a (222) The joint angle 7 can be found by substituting Equations 213 and 216 into Equations 221 and 222, to obtain 6 F 67 F 6 F 67 F 6 F 67 F 67 F 71 0 0 1 0 0 s S a S a S a a (223) PAGE 36 36 1 0 0 1 0 0 c6 F 67 F 6 F 67 F 67 F 7S a S a a (224) 2.5.1.3 Angle 1 The sine and cosine of 1 can be determined based on the definition of j. The angle 1 is the angle between the vector a71 and the x axis of the fixed coor dinate system (as depicted in Figure 24, Equations 221 and 222 give 1 F F 71 F 1sin S x a (225) x aF 71 F 1cos (226) The vector x denoted as Fx is by definition 0 0 1Fx (227) The substitution of Equations 215, 216 and 227 into Equations 225 and 226 gives the final expressions for the sine and cosine of 1 1 0 0 0 0 1 1 0 0 1 0 0 sin 6 F 67 F 6 F 67 F 1S a S a (228) PAGE 37 37 0 0 1 1 0 0 1 0 0 cos 6 F 67 F 6 F 67 F 1S a S a (229) 2.5.1.4 Distances a71, S7 and S1 Figure 25 shows a vector loop st arting at the origin of the fixed coordinate system and involving the unit vectors FS7, Fa71 and FS1. The vector loop in Figur e 25 is expressed as 0 S a S P 1 F 1 71 F 71 7 F 7 orig 6 FS a S. (230) The only unknowns in Equation 230 are the distances S7, a71 and S1. The distance S7 is obtained by forming a cross pr oduct of Equation 230 with FS1 0 S a S S S P 1 F 71 F 71 1 F 7 F 7 1 F orig 6 Fa S. (231) From Equation 218 71 F 71 1 F 7 Fs a S S (232) Substituting Equation 232 into Equation 231, a nd forming now a scalar (dot) product with Fa71, it can be solved for S7 to obtain 71 71 F orig 6 F 1 F 7s Sa P S (233) PAGE 38 38 Figure 25. Vector loop to find the distances a71, S7 and S1. Similarly, the distance a71 is obtained by first forming a dot product of Equation 231 with FS7 to obtain 0 a7 F 1 F 71 F 71 7 F 1 F orig 6 F S S a S S P (234) The scalar triple product in the a71 term, can be expressed as 1 F 7 F 71 F 7 F 1 F 71 F 7 F 1 F 71 FS S a S S a S S a (235) Equation 232 can be again substitu ted into Equation 235 to obtain 71 7 F 1 F 71 Fs S S a. (236) The substitution of Equation 236 into Equation 234 allows solving for a71, to obtain 71 7 F 1 F orig 6 F 71s a S S P (237) Finally, the expression for S1 is obtained by forming a cro ss product of Equation 230 with FS7, to obtain PAGE 39 39 0 S S S a S P 7 F 1 F 1 7 F 71 F 71 7 F orig 6 FS a. (238) Next, a scalar product on E quation 238 is taken with Fa71 to obtain 0 S71 F 7 F 1 F 1 71 F 7 F orig 6 F a S S a S P (239) Equations 232 and 235 are substituted in to Equation 239 and the expression for S1 is 71 71 F 1 F orig 6 F 1s Sa S P (240) 2.5.2 Special Cases for ClosedtheLoop Solution Technique Equations 233, 237 and 240 become useless when the twist angle 71 is 90 degrees, this occurs when S1 and S7 are collinear, parallel or antiparallel. 2.5.2.1 S1 and S7 are Parallel or Antiparallel When the joint vectors S1 and S7 are parallel or antiparallel, the choice S7=0 gives a solution to the problem. Substituting S7=0 into Equation 230 and following a similar procedure to obtain S1 and a71, it is found 1 F orig 6 F 1SS P (241) 1 F 1 orig 6 F 71S a S P (242) The vector Fa71 is also unknown in this case, and its unique value is give by 71 1 F 1 orig 6 F 71 Fa SS P a (243) The angles 7 and 1 are found using Equations 223 a nd 224, and Equations 228 and 229, respectively. PAGE 40 40 2.5.2.2 S1 and S7 are Collinear When a71=0 is obtained from Equati on 242, the joint vector S1 and S7 are collinear. In this case, the angle 7 is chosen as zero, ma king parallel the vectors a71 and a67. The angle 1 is calculated from Equations 228 and 229. The closedtheloop solution t echnique ends here. Pointing to the reverse kinematic problem, the angle 1 was obtained. The other parameters obtained here will be used when finding 1 to 6 by solving the resulting cl osedloop mechanism. This is explained in the next sections of this chapter. 2.6 Spherical ClosedLoop Mechanisms This section describes the methodology proposed by Crane and Duffy [1]. In the last section, it was shown how any openloop serial mechanism can be transformed into a closedloop spatial mechanism by adding a hypothetical closure link. In this section, a new equivalent spherical mechanism is formed from the closed loop spatial mechanism in order to generate equations relating the joint angles and twist angl es that will assist in the reverse kinematic analysis problem. The mechanism in this research involves only revolute joints, and thus the creation of the equivalent spherical mechanism will be focused on this type of joint. To facilitate the explanation, a spatial closedloop quadrilateral w ill be considered. The first step to create the equivalent spherical mechanism is to translate all the joint vectors (uni t vectors by definition), Si (i=1,,4), of the revolute joints so that their ta il is coincident with a common point O, as shown in Figure 26. PAGE 41 41 Figure 26. Spherical qu adrilateral. Courtesy of Dr. Carl Crane. The second step consists in dr awing a unit sphere centered at the point O. The unit vectors Si (i=1,,4) will meet the sphere at a sequence of points labeled 1,,4, respectively. This was expected because the joint vectors have unit norm Because the joint vectors have maintained the same direction, the angles be tween them, the twist angles, ij (ij=12, 23, 34, 41), remain unaltered. Finally the equivalent spherical mech anism is formed by connecting adjacent links ( 1223, 2334, 3441, 4112) with the joints. If an original joint in the spatial mechanism was a revolute joint, the equivalent joint in the sphe rical mechanism is also a revolute joint. Figure 26 presents a complete spherical quadrilateral equivalent to a 4L4R spatial closedloop mechanism. According to the way as the adjacent links have been connected, the link angles ij and the joint angles j are defined to be the same for th e closedloop spatial mechanism and the equivalent spherical quadrilateral. As a conseque nce, any equation that relates twist and joint angles of the equivalent spherical mechanism is also valid for the corresponding closedloop spatial mechanism. PAGE 42 42 2.7 Mobility of Spherical Mechanisms and Classification of Spatial Mechanisms This section describes the methodology propos ed by Crane and Duffy [1]. A link of a spherical mechanism has three degrees of freedom. Only three angles are needed to completely specify the position and orientat ion of the spherical link in the unit sphere. By analogy to Equation 12, the mobility of a spherical mechanism with one link fixed to ground is 3 f Mj 1 i i S (244) where: Ms: mobility of the equivalent spherical mechanism fi: relative degrees of freedom of the ith joint j: number of revolute joints. Knowing the mobility of the equivalent spherical mechanism will allow for classification of the closedloop spatia l mechanism. All one degree of freed om spatial closedloop mechanisms are classified based on the mobility of their equivalent spherical mechanism. It has been shown that the complexity of the solution of a one de gree of freedom spatial mechanism (where one joint angle is given) is directly related to th e degree of mobility of its equivalent spherical mechanism. One degree of freedom mechanisms with the same e quivalent spherical mechanism mobility have similar solution techniques for solution of the reverse kinematics. 2.8 Joint Vectors Expression s in the Spherical Mechanism This section describes the methodology proposed by Crane and Duffy [1]. Expressing the direction cosines of each unit joint vector, S1 to S7, in each of the standard coordinate systems is important in the analysis of both spherical and sp atial mechanism. In th is section, the joint vectors S1 to S3 will be expressed in the first standard coordinate system. Later, useful recursive notations derived from the joint v ectors expressions are presented. PAGE 43 43 All the joint vectors are un it vectors aligned with the z axis of its associated standard coordinate system, where for example 1 0 01 1S (245) 1 0 02 2S (246) From the result obtained in Equation 26, the rotational part of the transformation matrix that relates the first and second coordinate system s will allow expressing the second joint vector presented in Equation 246 in the firs t standard coordinate system as 12 12 12 12 2 12 2 12 12 2 12 2 2 2 2 1c s 0 1 0 0 c c c s s s c c c s 0 s c S (247) Exchanging the subscripts in Equation 247, th e expression for the third joint axis in the second standard coordinate syst em can be obtained by making 1 2 and 2 3, it is found 23 23 3 2c s 0 S. (248) Now, 1S3 can be obtained as 2 23 12 23 12 2 23 12 23 12 2 23 23 23 12 12 2 12 2 12 12 2 12 2 2 2 3 2 1 2 3 1c s s c c c s c c s s s c s 0 c c c s s s c c c s 0 s c R S S. (249) A recursive shorthand notation is introduced in Equation 249 PAGE 44 44 2 23 12 23 12 2 23 12 23 12 2 23 2 2 2 3 1c s s c c c s c c s s s Z Y X S (250) And the recursive expressions for the terms in Equation 250 are j jk js s X (251) j jk ij jk ij jc s c c s Y (252) j jk ij jk ij jc s s c c Z (253) The next expression to be found is the joint vector S4 in terms of the first standard coordinate system, 1S4. First, the term 2S4 is obtained by exchanging subscripts in the first two terms of Equation 250 3 3 3 4 2Z Y X S (254) 1S4 is obtained by 3 12 2 3 2 3 12 3 12 2 3 2 3 12 2 3 2 3 3 3 3 12 12 2 12 2 12 12 2 12 2 2 2 3 2 1 2 4 1Z c ) c Y s X s Z s ) c Y s X c s Y c X Z Y X c c c s s s c c c s 0 s c R S S (255) From Equation 255, another recursive notati on can be obtained. The objective of this section is not to deduce all the expressions, which is an extremely lengthy process. The objective is just to introduce the recursive notations re sulting from expressing the joint vectors in a standard coordinate syst em different to the system attached to that joint. The summary for the joint vectors expressed in the first standard coordinate system is 1 0 01 1S ; 12 12 2 1c s 0 S ; 2 2 2 3 1Z Y X S ; 32 32 32 4 1Z Y X S ; 2 ,..., 2 1 2 ,..., 2 1 2 ,..., 2 1Z Y Xn n n n n n n 1S (256) PAGE 45 45 The result obtained in Equation 256 can be extended for expressing the joint vectors in any standard coordinate system (n=2,,7). Al ong the process of finding the results in Equation 256, Crane and Duffy [1] found recursive nota tions when expressing the joint vectors in different standard coordinate systems, which are useful for solving consciously the reverse kinematic problem. The expressions are presente d in Appendix A, Equations A1 to A38. 2.9 Link Vectors Expressions in the Spherical Mechanism This section describes the methodology proposed by Crane and Duffy [1]. Expressing the direction cosines of each link vector, a12 to a67, in each of the standard coordinate systems is important in the analysis of both spherical and spatial mechanisms. In this section, the joint vectors a12 to a34 are expressed in the first standard coor dinate system. Later, recursive notation derived from the link vectors expressions is presented. All the link vectors are unit vect ors aligned with the x axis of its associated standard coordinate system, and thus 0 0 112 1a (257) 0 0 123 2a (258) From the result obtained in Equation 26, the rotational part of the transformation matrix that relates the first and second coordinate systems will allow expressing the link vector a23, presented in Equation 258, in the fi rst standard coordinate system as 2 1 2 12 2 2 12 12 2 12 2 12 12 2 12 2 2 2 3 2 1s s c s c 0 0 1 c c c s s s c c c s 0 s c a (259) PAGE 46 46 Exchanging the subscripts in Equation 259, the expression for the link vector a34 in the second standard coordinate syst em can be obtained, by making 1 2, 2 3 and 3 4, it is found 3 2 3 23 3 3 4 3 2s s c s c a. (260) Now, 1a34 can be obtained as 23 3 23 3 3 12 12 2 12 2 12 12 2 12 2 2 2 4 3 2 1 2 4 3 1s s c s c c c c s s s c c c s 0 s c R a a 23 3 2 3 2 12 23 3 12 23 3 2 3 2 12 23 3 12 23 3 2 3 2 34 1c s c c s s s s c c s c c s c s s s c s s c c a. (261) A recursive shorthand notation is introduced in Equation 261 23 3 2 3 2 12 23 3 12 23 3 2 3 2 12 23 3 12 23 3 2 3 2 321 321 32 34 1c s c c s s s s c c s c c s c s s s c s s c c U U W a. (262) The next expression to be found is the joint vector a45 in terms of the first standard coordinate system, 1a45. First, the term 2a45 is obtained by exchanging subscripts in the first two terms of Equation 262 432 432 43 45 2U U W a. (263) The link vector 1a45 is obtained by 432 432 43 12 12 2 12 2 12 12 2 12 2 2 2 45 2 1 2 45 1U U W c c c s s s c c c s 0 s c R a a PAGE 47 47 and a new recursive notation is identified as 1 432 1 432 2 43 43 2 432 2 2 1 432 2 1 43 2 432 2 2 1 432 2 1 43 2 432 2 45 1U U W W s U c s U c W s U c c U s W c U s a (264) By following the same procedure, expressions for all the link vectors can be expressed in the first standard coordinate system, to obtain 0 0 112 1a ; 2 1 2 2 1 2 2 23 1s s c s c a ; 1 32 1 32 32 34 1U U W a; 1 32 4 1 32 4 432 45 1U U W a; 1 32 54 1 32 54 5432 56 1U U W a; 1 32 654 1 32 654 65432 67 1U U W a. (265) The results presented in Equation 265 can be extended for finding th e link vectors in any standard coordinate system. This is done by properly exchanging the subscripts in Equation 265. Crane and Duffy [1] found recurs ive notations when expressing the link vectors in different standard coordinate systems, which are useful for solving co nsciously the reverse kinematic problem. The recursive notations are presente d in Appendix A, Equations A39 to A60. The joint and link vectors can be expressed in any coordinate system, then there is a set consisting of seven joint vectors ( S1 to S7) and six link vectors ( a12 to a67) expressed in each of the standard coordinate systems based on the recursiv e equations presented, seven sets in total. In addition, there are sets that are combinations of the standard coordinate sy stems: the x and z axes can be aligned with the a and S axes or two different standard coordinate systems. All the sets for the closed loop mechanisms can be obtained at the appendix section of Crane and Duffy [1]. A useful characteristic of the not ation here presented is that when the X, Y, or Z notation terms are fully expanded, they only contain the sine or cosine of the joint angles whose PAGE 48 48 subscripts appear in the shorthand notation. This will be used when solving the reverse kinematic of the mechanism in Chapter 4. 2.10 Orientation Requirements Speci fication: XYZ Fixed Angles The desired orientation of the body to which the e nd effector is attached is specified by the angles x, y and z. In this method, two frames, [A: xAyAzA] and [B: xByBzB], are initially aligned. Frame [B] is then rotated an angle x about xA, then it is rotated an angle y about yA and finally it is rotated an angle z about the axis zA. The transformation matrix resulting from the successive rotations is obtained by 33 32 31 23 22 21 13 12 11 A z A y A xr r r r r r r r r ) z ( ) y ( ) x ( R R R (266) where the righthand side matrix in Equati on 266 is given by Equation 211. Given the resulting transformation matrix, the angles can be obtained through the atan2 function as 2 21 2 11 31 yr r r 2 tan a (267) y 1 1 y 1 2 xcos / r cos / r 2 tan a (268) y 33 y 32 zcos / r cos / r 2 tan a (269) Further information about representation of th e orientation of rigid bodies, see Craig [28]. PAGE 49 49 CHAPTER 3 NONCIRCULAR GEARS FUNDAMENTALS This chapter introduces noncircular gear f undamentals that will s upport the noncircular gear synthesis proposed in Chapter 5. Gears are b odies in direct contact that have been toothed to ensure motion transmission without slipping. Gear pairs are used to transmit motion either from a rotating shaft to a body which rotates or fro m a rotating shaft to a body which translates which is considered to be rotating about an axis at infinity. This chapter presents basic gearing concepts such as the pitch point, pitch curve, Input/Output5 noncircular gear relationship, and input and output angles. Geometri c synthesis of the pitch profile in Cartesian coordinates closes this chapter. 3.1 Pitch Curve and Pitch Point Figure 31 shows a directcontact threelink mechanism. The line NN is the common normal to the contacting surfaces an d it intersects the centers line O2O3 at point P. The points O2 and O3 are instant centers and the line NN is called line of action. The ArnoldKennedy theorem states: Any three bodies having plane motion relative to one anot her have exactly three instant centers, and they lie on the same straight line. As a consequence of this theorem, the point P is the third instant center (See Dooner and Seireg [3] and Erdman, Sandor and Kota [29]). An instant center is (1) a point in one body about which some other body is rotating either permanently or at the instant, or (2) a common point to two planar bodies in motion which has the same velocity in both magnitude and direction (e. g., s ee Martin [30]). Uniform motion transmission between two parall el axes is only possible if the line of action passes through a fixed point, known as the pitch point Dooner and Seireg [3], this is the point P in Figure 31. Now, for each different pos ition of the links 2 and 3, the locus of pitch 5 Also written as I/O. PAGE 50 50 point determines the pitch circle. The path of the moving instant center pitch pointgenerates the pitch curve or centrode. In the ca se of circular gears the distances O2P2 and O3P3 are constant. In the case of nonci rcular gears, the distances O2P2 and O3P3 vary. Figure 31. Pitch point and pitch curves of two bodies in mesh. 3.2 Input/Output NonCircula r Gear Relationship The transmission function for two bodies in me sh is defined as the relationship between the angular position of the input element and the corresponding angular position of the output element. From this concept, the instantaneous ge ar ratio, g, is written as the ratio between the infinitesimal displacement of the output body an d the corresponding infinitesimal displacement of the input body. Accordingl y, it can be written as i o d d g (31) where: g: instantaneous gear ratio or I/O relationship PAGE 51 51 d o: instantaneous angular displacement of the output body d i: instantaneous angular displacement of the input body. 3.2.1 Input/Output Relationship wi th the Connecting Link Grounded When the connecting link is fixed to ground, as presented in Figure 32A, the I/O relationship is given directly by Equation 31. The I/O relationship g, in Equation 31 is defined as positive according to the senses of the input and output angles, i and o, depicted in Figure 32A. Figure 32. Noncircular input and output bodies in mesh. A) Grounded connecting link. B) All links moving. 3.2.2 Input/Output Relationship with All Links Movable Figure 32B depicts the geared li nks with the links j1, j, a nd j+1 all movable which is the general case for two bodies in mesh. For noncircul ar gears in mesh att ached to movable links, the recurrent equation for the I/O noncircular re lationship is given by Eq uation 32. The angles j1, j, and j+1 are the net angular displacements. These an gles are measured with respect to the fixed x axis. ) d( ) d( g1 j j j 1 j j (32) PAGE 52 52 where: gj: I/O relationship of the transmission dj: instantaneous angular displacement of the jth link. If the connecting link in Figur e 32B is fixed to ground, E quations 31 and 32 are the same since j becomes zero. For the case where the connecting link is grounded, dj in Equation 32 is zero, then the expression for gj in Equation 32 looks to be negative. This apparent difference is explained if it is no ticed that in Figures 32A and 32B, the input and output angles, i and o, are measured respect to the x fixed axis in clockwise and counterclockwise sense, respectively. On the other hand, all the link angles in Figure 32B are measured in clockwise sense; this explains the appare nt difference between Equations 31 and 32. In the current work, the recurrent expression presented in Equation 32 will be used and referred to as the local coordinates approach model, in which the reference system xy is perpendicular to the rotation axes of the links (bodies) in mesh pointing out of the page in Figure 32A and 32B. 3.3 Complete Profile Synthesis: NonCircula r Relationships for the Entire Rotation Some real applications of t oothed bodies are fulfilled by par tial circular or noncircular gears, where the covered angle is less than 2 However, a complete profile is sometimes required for manufacturing purposes in particular for the simula tion of the cutter, Dooner and Seireg [3]. To this end, working and nonwork ing sections of the cen trode are now introduced. The working section of the gear is used to pe rform a task. The nonworking section converts the working slice into a complete disk. Figure 33 sh ows a complete noncircular gear with arbitrary working and nonworking sections. PAGE 53 53 Figure 33. Arbitrary working a nd nonworking sections and angl es in a complete noncircular gear. If the working section is circular, then the complete profile becomes a conventional circular gear. If the working se ction is noncircular, additional ge ometric considerations must be taken into account to achieve a co mplete noncircular pitch profile. The I/O noncircular relationship correspondi ng to the working sect ion is written as gw and its attaintment was presented in Equations 31 an d 32. Similarly, the I/O relationship for the nonworking section is labeled by gnw. The geometric considerations to obtain the nonworking section are discussed below. When the I/O relationship is not specified for the complete rotation of the input gear, the nonworking I/O relationship can be achieved by intr oducing a polynomial in the input angle, whose order and coefficients are derived from th e geometry constraints of the gearing. Figure 34 illustrates four constraints required to guarantee a continuous profile where int makes reference to the final value of the in put angle for the working section. PAGE 54 54 Figure 34. Boundary conditions for the whole synt hesis of the Input/Output relationship. A) g boundary conditions. B) Slope boundary conditions. The I/O noncircular gear rela tionship must satisfy another particular constraint to successfully synthesize a complete profile. The in tegral shown in Equation 33 must be always rational; otherwise, the output gear cannot maintain an indefinite number of cycles with the desired functional relationship (D ooner and Seireg [3]). Therefore 2 0 i d g 2 (33) where: g: instantaneous gear ratio or I/O relationship di: instantaneous angular displacement of the input body. PAGE 55 55 From Figure 34 and the constr aint imposed by Equation 33, the five boundary conditions are ) ( g ) ( gint nw int w (34) ) ( g' ) ( g'int nw int w (35) ) 2 ( g ) 0 ( gnw w (36) ) 0 ( g' ) 0 ( g'nw w (37) 2 0 i w i nwint intd g 2 d g (38) where in general, id dg g' This is the derivative of the I/ O relationship respect to the input angle, for both working and nonworking sections. Equation 38 is not evident from Figure 34 and its attainment is explained as follows. Figure 33 presents the working and nonworking a ngles for an arbitrary noncircular gear. From Figure 34 is apparent that work ing and nonworking angles satisfy 2 nw w (39) Equation 39 is valid for both i nput and output angles. If Equation 39 is used for the output angle output body, it becomes 2 nwo wo (310) From Equation 31, it can be solved for d o to obtain i od g d (311) where integration gives the solution for the output angle o. This integral is divided into two terms: the working and the nonworking sections whose integration limits are evident from Figure 34. The working and nonworki ng output angles are, respectively PAGE 56 56 int 0 i w wo d g (312) 2 i nw nwoint d g (313) Substituting Equations 312 and 313 into Equation 310 gives 2 d g d g 2 i nw 0 i wint int 0 (314) which is the same as Equation 38. In summary Equation 38 is derived from Equation 33 and the linearity principle for the integral. A conscious selection of the degree of th e polynomial in the i nput angle of the nonworking profile is degree 4. There are five co efficients that can be obtained from the five Equations 34 to 38. In a ddition, this selection of gnw makes the integral in Equation 38 always rational and guarantees an indefi nite number of cycles. The 4th degree polynomial in the velocity or g domain is given by 0 i 1 2 i 2 3 i 3 4 i 4 i nwc c c c c ) ( g (315) where: i: input angle gnw: nonworking I/O gear relationship ci: fourthdegree polynomials coefficients. The derivative of the polynomial is 1 i 2 2 i 3 3 i 4 i nwc c 2 c 3 c 4 ) ( g' (316) The substitution of Equations 315 and 316 into Equations 34 to 38 forms a set of five linear equations representing th e five unknown coefficients c0,,c4 of the polynomial presented in Equation 315. The solution fo r the coefficients can be written in the matrix format as PAGE 57 57 C1 (317) where: 0 1 2 3 4c c c c c C, int 0 i i w w int w int wd ) ( g 2 (0) g 2 ) ( g ) ( g H, int 2 int 2 3 int 3 4 int 4 5 int 5 2 3 2 3 4 int 2 int 3 int int 2 int 3 int 4 int 2 2 ) (2 3 ) (2 4 ) (2 5 ) (2 0 1 ) 2(2 ) 3(2 ) 4(2 1 ) (2 ) (2 ) (2 ) (2 0 1 2 3 4 1 (318) At this point, a (not unique ) g has been completely derived for the nonworking profile. For additional information about the nonworking gear relationship synthesis, see Dooner [24]. In the next section, a method to obtain the Cartesia n coordinates of the pitc h point is presented. 3.4 Pitch Curve Coordinates Equations that relate the input angles, output angles, and the I/O relationships in order to obtain the pitch curve co ordinates are presented here. Figure 35 introduces the center distance, E, as an additional parameter. This plot also de fines two rectangular coor dinate systems used to describe the pitch curves of th e input and the output bodies. The i nput and output radii are related by E r ri o (319) The I/O relationship is introdu ced to add one more equation. For the two bodies in mesh shown in Figure 35, g becomes PAGE 58 58 o ir r g (320) Equations 319 and 320 form a system of two equations in the two unknowns, ro and ri. The radii expressions are g 1 E ro (321) g 1 g E ri (322) Finally, the Cartesian coordina tes for the output and input pi tch profiles (deduced from Figure 35) are, respectively, ) sin r cos (r Ro o o o o (323) ) sin r cos (r Ri i i i i. (324) Figure 35. Rectangular coordinate systems xyo and xyi for a gear pair in mesh. In this work, noncircular g ear transmissions with movabl e links are considered, as presented earlier in Figure 32(b). If this is th e case, the pitch profile Cartesian coordinates are easily obtained by attaching the abscissa of the rect angular coordinate system in Figure 35 to the connecting link j in Figure 32(b) as presented in Figure 36. PAGE 59 59 Expressions for the input and out put angle are now presented. For the working section, the output and input angles can be directly obtained from the numerator and denominator of Equation 32, to obtain j 1 j wo (325) 1 j j wi (326) Figure 36. Local coordinate systems xyi and xyo in the synthesis of th e pitch point with input, output and connecting bodies movable. In reference to the nonworking se ction, from the abscissas in Figure 3.4, the input angle is set as the portion from the end of the working profile, int, to the end of the whole disk: 2 The input angle of the nonworking section can be represented as the interval ,2 int nwi (327) The output angle of the nonworking profile is found by us ing Equation 311 for the nonworking section, to obtain nwi nw nwod g d (328) PAGE 60 60 A substitution of the polynomial expression for g in the nonworking section, Equation 315, into Equation 328 will give nwi i 0 i 1 2 i 2 3 i 3 4 i 4 nwod c c c c c d (329) where nwi is varied from Equation (327). After integrating Equation 321, the compact fo rmat equation for the output angle at the nonworking profile is 4 0 i 1 i 1 i i nwo ) (2 1 i c (330) where: ci: coefficients in polynomial of Equation 315, found in Equation 317 : nonworking input angle gi ven by the interval pr esented in Equation 327. Here end the gearing considerations for th e synthesis of the n oncircular profiles. Input/Output noncircular gear re lationship, and output and input angles synthesis have been presented for both working and nonworking profile sections. PAGE 61 61 CHAPTER 4 REVERSE KINEMATIC ANALYSIS OF TH E 1DOF SPATIAL MOTION GENERATOR The closedloop spatial mechanis m previously presented in Figure 17 can be separated into two open loop spatial mechanisms. Cons ider the two fixed coordinates systems xF1yF1zF1 and xF2yF2zF2 as depicted in Figure 41. Two paths from the origins of the new coordinate systems, OF1 and OF2, to the point P can be easily identified, they are 1234567 and 1121110987, respectively. Figure 41. 1DOF spatial motion generator. Each path defines an openloop mechanism fo rmed by six links (not counting ground) and six revolute joints. The revers e kinematic analysis of the cl osedloop spatial motion generator will be solved by performing the reverse kinematic analysis for each of the two openloop mechanisms separately. In this way the joint angl es for the left and rightside open loop chains can be determined for a specific position of point P in link 7 and for the orientation of link 7. In this chapter, the reverse kinematic soluti on of an openloop mechanism comprised of six links and six revolute joints with consecutive pairs of joint axes parallel is presented. A closed PAGE 62 62 form solution technique is used where a hypothetical link that c onnects the first and sixth joint axes is introduced which converts the ope nloop chain to a closedloop mechanism. Before starting the reverse kinematic analysis a brief analysis of the overall mechanism mobility is presented; followed by the introduction of the standard coordinate systems, forward kinematic analysis, reverse kinematic problem st atement, closetheloop technique, and spherical mechanisms. For detailed explanatio n about the topics presented in this chapter, see Crane and Duffy [1]. The solution of the complete closedloop mechanism is obtained by combining the result obtained in the solution of the openloop mech anism with a coordinate transformation. 4.1 Mobility Analysis of the Spatial Motion Generator Figure 42 presents a detailed labeling of the spatial motion generator. The noncircular gears do not represent extra links because they are attached to the kinematical links. Links 1 (fixed link), 3, 5, 7, 9 and 11 have noncircular gears attached to them. The mobility equation for this closedloop mechanism can be written as k 1 j j n 1 i ig 6 f M (41) where: M: mobility or DOF n: number of spatial links or connected bodies fi: number of relative degrees of freedom permitted by the ith joint gj: relative degrees of freedom that removes the jth modification k: number of modifications introduced to the mechanism. PAGE 63 63 Figure 42. Labeling of the spatial motion genera tor for mobility analysis. Parts with the same number label are one single link. When Equation 41 is applied to the spatia l motion generator in Figure 42, there are twelve links (n=12); every joint is a revolute joint with one relative DOF (fi=1; i=1,,12); there are five geared connections or modifications (k =5); and each modification allows one relative DOF (gj=1; j=1,). Thus the mobility of the mechanism is one (1DOF). 4.2 Grouping of the 7L7R Spatial Mechanis m with Consecutive Pairs of Parallel Joint Axes: Equivalent Spherical Mechanism The 6L6R6openloop spatial mechanism forms a 7L7R closedloop spatial mechanism after the closetheloop technique is used. Figure 43 presents the kinematic labeling for the openloop spatial mechanism. A special geometry is used whereby the twist angles 12, 34 and 56 are all zero. This results in the pair of axes S1S2, S3S4 and S5S6 being parallel. 6 Mechanism comprised by six links and six revolute joints. PAGE 64 64 Figure 43. Kinematic labeling of the 6L6R ope nloop mechanism. A) Joint angles. B) Link angles. C) Link and joint distances. PAGE 65 65 In Section 2.7 it was explained how the equiva lent spherical mechanism is formed by first translating the joint vectors to the center of a uni t sphere. If two joint vectors are parallel, then they will be coincident on the sphere. As a result, the joint vectors S1 and S2 will be coincident, similarly for the pairs of vectors S3S4 and S5S6. This implies that the equivalent spherical mechanism associated with the 7L7R parallel axes is a spherical quadrilateral with only four revolute joints, as presente d in Figure 26, where S1, S2, S3 and S4 in the quadrilateral represent the joint vectors S1S2, S3S4, S5S6 and S7, respectively. The mobility equation of the spherical mechanisms was presented in Equation 244. There are four revolute joints in the spherical quadrilateral, from Equation 244 the mobility of this mechanism is one. This result is interesting b ecause it classifies this 7L7R mechanism with parallel axes as Group 1, diffe rent to the conventional 7L7R mechanism which is Group 4, which means that two spatial mechanisms with th e same number of links and revolute joints are to be solved by differe nt solution techniques. The next section presents the solution of the equivalent spherical quadrilateral, which is the step following the closuretheloop technique for solving the reverse kinematic problem. 4.3 Reverse Kinematic of the 6L6R OpenLoop Mechanism with Consecutive Pairs of Parallel Joint Axes 4.3.1 Problem Statement It is assumed that the closetheloop parameters are already found (closetheloop technique was extensively explained in Section 2.5) and are known parameters at this stage. The problem statement is reduced to Known 1) The constant mechanism parameters: link lengths a12 to a56, twist angles 12 to 56, and offset distances S2 to S5. PAGE 66 66 2) Offset distance S6 and the direction of the vector a67 relative to the vector S6 3) Position and orientati on of the end effector: FPtool, FS6 and Fa67 4) Location of the tool point in the 6th coordinate system: 6Ptool 5) Close the loop parameters: Offset distances S1, S7; link distance a71; joint angle 7; angle 1. Find The joint angles 1,, 6 of the closedloop 7L7R spatial mechanism. Because the equivalent spherical mechanism is a quadrilateral, solving this spherical quadrilateral will be the first st ep. The second part of the revers e kinematics is to express the joint and link axes of the vector loop equation (for the 7L7R) in a set of joint and link axes expressed in a convenien t coordinate system. 4.3.2 Solution of the Equi valent Spherical Quadrilateral Figures 44A and 44B present the spherical qua drilateral with the quadrilateral labeling and heptagon labeling, respectively, where the subscript Q stands for quadrilateral and the subscript H stands for heptagon. Comparing Figures 44A and 44B it is apparent that H 7 Q 4 (42) H 67 H 56 Q 34 (43) H 71 Q 41 (44) It was stated that this is th e second stage of the solution. Th e first step was the closetheloop procedure, where the joint angles 7H, 71H and 71H were found. From Equations 42 to 44, the joint angle 4Q and the link angles 34Q and 41Q are known parameters. The angle 4Q is chosen as the input angle for being a known para meter. When comparing Figures 44A and 44B is also evident that PAGE 67 67 H 23 H 12 Q 12 (45) H 45 H 34 Q 23 (46) The twist angles 23H and 45H are known parameters which depend on the geometry of the spatial mechanism, see Problem statement in Section 2.4, then the angles 12Q and 23Q are also known parameters from Equations 45 and 46. At this point, the only unknowns in the spherical mechanism ar e the joint angles 1Q, 2Q and 3Q. Crane and Duffy [1] easily solved the sphe rical quadrilateral by writing an appropriate spherical cosine law which contains the angle 1Q as the only unknown. This solution is presented next. 4.3.2.1 Solving for 1Q The spherical cosine law Z41=c23 is used (defined in Equation A6) 23 4 12 Q 1 4 Q 1 4 12c Z c ) c Y s (X s (47) Rearranging terms in Equation 47 yields 0 D Bs AcQ 1 Q 1 (48) where: 4 12Y s A 4 12X s B 23 4 12c Z c D X4, Y4 and Z4 are expanded using Equations 257 to 259, respectively. The trigonometric solution of Equation 48 gives 2 2 Q 1B A D cos (49) where a unique value for is determined from 2 2B A B sin and 2 2B A A cos PAGE 68 68 Because two values satisfy cos( 1) in Equation 49, then two values of 1Q will also be obtained, 1QA and 1QB. Figure 44. Equivalent spherical mechanism. A) Spherical notation. B) Heptagon notation. 4.3.2.2 Solving for 2Q In order to find a unique value for 2Q, the sine and the cosine expressions are obtained from Q 2 23 41s s X (410) Q 2 23 41c s Y (411) Substituting Equations 260 and 262 into Equations 410 and 411, it is solved for the sine and cosine as 23 Q 1 4 Q 1 4 Q 2s s Y c X s (412) 23 4 12 Q 1 4 Q 1 4 12 Q 2s Z s ) c Y s (X c c (413) where X4, Y4 and Z4 are expanded using Equations A4 to A6, respectively. PAGE 69 69 A unique value for 2Q is obtained for each value of 1Q ( 1QA and 1QB), they are 2QA and 2QB. 4.3.2.3 Solving for 3Q A suitable choice for finding th e sine and cosine of the 3Q is Q 3 23 14s s X (414) Q 3 23 14c s Y (415) Substituting Equations 264 and 266 into Equati ons 414 and 415 and solving for sine and cosine of 3Q yields 23 Q 4 1 Q 4 1 Q 3s s Y c X s (416) 23 1 34 Q 4 1 Q 4 1 34 Q 3s Z s ) c Y s X ( c c (417) where 1X, 1Yand 1Z are obtained from Equations 251 to 253, respectively. A unique value is for 3Q each value of 4Q ( 4QA and 4QB), they are 3QA and 3QB. A solution tree of the spherical quadrilatera l solution obtained angl es is Figure 45. Figure 45. Solution tree of the spherical quadrilateral. 4.3.3 Vector Loop Equation of the 7L7R Parallel Axes Mechanism Some known parameters will be first rewrit ten. According to the geometry of the mechanism, the values of the twist angles are 0 12 (418) PAGE 70 70 0 34 (419) 0 56 (420) The zero offset distances are 0 S3 (421) 0 S5 (422) The twist angle 67 and the link distance a67 are free choices that were chosen before the closetheloop procedure respectively as o 6790 (423) 0 a67 (424) On the other hand, when comparing Figures 44A and 44B, as a direct consequence of the parallel axes geometry, it is apparent that Q 1 H 2 H 1 (425) Q 2 H 4 H 3 (426) Q 3 H 6 H 5 (427) From now on, the notation will be simplified, the joint angles, 1H,, 6H will be denoted by 1,, 6. Equation 428 is the vector loop equation of the closedloop mechanism 0 a S a S a S a S a S a S a S 71 71 7 7 67 67 6 6 56 56 5 5 45 45 4 4 34 34 3 3 23 23 2 2 12 12 1 1a S a S a S a S a S a S a S (428) The first reduction of Equation 428 is made by substituting Equations 421, 422 and 424, to yield 0 a S S a a S a a S a S 71 71 7 7 6 6 56 56 45 45 4 4 34 34 23 23 2 2 12 12 1 1a S S a a S a a S a S (429) PAGE 71 71 Table 41. Direction cosines of a closedloop 7L7R mechanism expressed in a coordinate system were the x and z axes are aligned with the vectors a23 and S3, respectively. Joint vectors Link vectors S3 (0, 0, 1) a23 (1, 0, 0) S2 (0, s23, c23) a12 (c2, s2c23, U23) S1 (X2, Y2, Z2) a71 (W12, U* 123, U123) S7 (X12, Y12, Z12) a67 (W712, U* 7123, U7123) S6 (X712, Y712, Z712) a56 (W6712, U* 67123, U67123) S5 (X6712, Y6712, Z6712) a45 (W56712, U* 567123, U567123) S4 (X56712, Y56712, Z56712) a34 (c3, s3, 0) Crane and Duffy [1]. Equation 429 is conveniently expressed in a par ticular coordinate syst em where the xaxis is aligned with the link vector a23 and the zaxis is aligned with the joint vector S3 (See Table 41), to obtain 0 0 0 U U W a Z Y X S Z Y X S U U W a U U W a Z Y X S 0 s c a 0 0 1 a c s 0 S U c s c a Z Y X S123 123 12 71 12 12 12 7 712 712 712 6 67123 67123 6712 56 567123 567123 56712 45 56712 56712 56712 4 3 3 34 23 23 23 2 23 23 2 2 12 2 2 2 1. (430) Three scalar equations are extracted from Equa tion 430 for the x, y and z projection of the vector loop equation onto Se t 13. These equations are 0 W a X S X S W a W a X S c a a c a X S12 71 12 7 712 6 6712 56 56712 45 56712 4 3 34 23 2 12 2 1 (431) 0 U a Y S U a U a Y S s a s S c s a Y S* 123 71 712 6 67123 56 567123 45 56712 4 3 34 23 2 23 2 12 2 1 (432) 0 U a Z S Z S U a U a Z S c S U a Z S123 71 12 7 712 6 67123 56 567123 45 56712 4 23 2 23 12 2 1 (433) The theory of analysis developed in robot manipulators also allows expressing some recurrent terms in different forms; with different subscripts by using funda mental and subsidiary PAGE 72 72 equations for a spherical and polar heptagon, furt her information see Crane and Duffy [1]. It was explained before that the joint angles associated with the subscripts in the shorthand notation also appear in the expansion of that term, then some substitutions are going to be made in order to obtain expressions that involve the known values, so far the sums of the angles ( 1+ 2), ( 3+ 4), ( 5+ 6), and the closure angle 7. Based on this, the xprojection written in Equation 431 can be rewritten if some fundame ntal and subsidiary equa tions are used, they are 3 56712X X (434) 43 56712W W (435) 543 6712W W (436) The substitution of Equation 434 to 436 into Equation 431 allows expressing the projection onto a23 in Set 13 as 0 W a X S X S W a W a X S c a a c a X S12 71 12 7 712 6 543 45 43 45 3 4 3 34 23 2 12 2 1 (437) Similarly, the substitutions 3 34 56712c s Y (438) 43 567123V U (439) 543 67123V U (440) into Equation 432 will convert the yaxis of Set 13 into 0 U a Y S Y S V a V a c s S s a s S c s a Y S* 123 71 12 7 712 6 543 56 43 45 3 34 4 3 34 23 2 23 2 12 2 1 (441) For the projection onto S3 of Set 13, the substitutions 34 56712c Z (442) 43 567123U U (443) PAGE 73 73 543 67123U U (444) into Equation 433 yield 0 U a Z S Z S U a U a c S c S U a Z S123 71 12 7 712 6 543 56 43 45 34 4 23 2 23 12 2 1 (445) Equations 437, 441 and 445 are going to be transformed by using the recursive notation presented in Appendix A. Consider firs t Equation 437; the expansions for X2, 3X, W43, W543, X712, X12 and W12 are going to be taken from Equations A4, A1, A44, A48, A15, A7 and A41, respectively. Next, the substitution of Equations 418 to 420 the link angles 12, 34 and 56and 423 67into Equation 237 yields 0 c a s s S ) c c s s (c S c a a s c s a c c a c a c a2 1 71 71 2 1 7 7 71 2 1 7 2 1 6 4 3 45 23 5 45 4 3 56 5 4 3 56 3 34 2 12 (446) The shorthand notation ci+j=cos icos jsin isin j and si+j=sin icos jcos isin j has been used in Equation 446 so that the terms ( 1+ 2), ( 3+ 4), and ( 5+ 6) (previously found in the spherical quadrilateral solution) have been introduced. Similarly, Equations 418 to 420 and 423 are substituted into Equation 441and 445. Next, the expansion of the shorthand terms in Equations 441 and 445 using the recursive notation presented in Appendix A, yields for the yprojection and S3 projections on Set 13, respectively 0 S ) c s s c s s c c c (c c s a s a S ) s c c c (s s c c c s s S s S c s a s c c a s a s c a6 23 7 2 1 7 71 23 23 71 2 1 7 23 2 1 71 4 3 45 7 71 2 1 23 71 23 71 2 1 23 71 23 23 2 23 1 5 4 3 56 5 45 4 3 56 3 34 2 23 12 (447) 0 ) c c s c s ( S s s a S c S c S ) c s c s s s c s c c ( S s s a s s a71 23 71 2 1 23 7 23 2 1 71 4 23 2 23 1 7 71 23 23 7 2 1 71 23 7 2 1 6 5 45 56 2 23 12 (448) In the next section, Equati ons 446 to 448 are solved for the three joint angles 2, 3 and 5. PAGE 74 74 4.3.4 TanHalfAngle Solution for 5 Equations 446 to 448 are conveniently rewritten as 0 A s A c A c A c A5 5 4 5 3 3 2 2 1 (449) where: 12 1a A 34 2a A 4 3 56 3c a A 45 4 3 56 4c s a A 2 1 71 71 2 1 7 7 71 2 1 7 2 1 6 4 3 45 23 5c a s s S ) c c s s (c S c a a A 0 B s B c B s B s B5 5 4 5 3 3 2 2 1 (450) where: 23 12 1c a B 34 2a B 4 3 56 3s a B 45 4 3 56 4c c a B 6 7 71 23 23 7 2 1 71 2 1 7 23 2 1 71 4 3 45 7 71 2 1 23 71 23 23 2 23 1 5S ) c s s c ] s s c c ([c c s a s a S ) s c c c (s s S s S B 0 D s D s D3 5 2 2 1 (451) where: 23 12 1s a D 45 56 2s a D PAGE 75 75 ) c c s c s ( S s s a S c S c S ) c s c s ] s s c c c ([ S D71 23 71 2 1 23 7 23 2 1 71 4 23 2 23 1 7 71 23 23 7 2 1 71 7 2 1 6 3 The purpose of this section is to strategically manipulate Equations 449 to 451 so that a single equation where only s5 and c5 appears, and later substituting the tanhalfangle identities. From Equation 449 it is solved for c3 2 5 5 4 5 3 2 1 3A A s A c A c A c (452) The same can be done from Equation 450 for s3 to obtain 2 5 5 4 5 3 2 1 3B B s B c B s B s (453) The next step is squaring both sides of E quations 452 and 453, adding term to term, applying the trigonometric identity 1 c s2 3 2 3 and multiply by 2 2 2 2B A in order to eliminate the denominator, to obtain 0 E s E s E c E c E5 2 4 2 2 3 2 2 2 2 1 (454) where: 2 2 2 1 1B A E 3 2 5 2 2 5 21 2E s E c E E where: 2 2 3 1 1 2B A A 2 E 2 2 4 1 2 2B A A 2 E 2 2 5 1 3 2B A A 2 E 2 1 2 2 3B A E 43 5 42 5 41 4E s E c E E PAGE 76 76 where: 3 1 2 2 1 4B B A 2 E 4 1 2 2 2 4B B A 2 E 5 1 2 2 3 4B B A 2 E 6 5 5 55 5 54 5 5 53 2 5 52 2 5 51 5E s E c E s c E s E c E E where: 2 2 2 3 2 3 2 2 1 5B A B A E 2 4 2 2 2 2 2 4 2 5B A B A E ) B A A B B 2(A E2 2 4 3 4 3 2 2 3 5 ) B B A B A 2(A E5 3 2 2 2 2 5 3 4 5 ) B B A B A 2(A E5 4 2 2 2 2 5 4 5 5 2 5 2 2 2 2 2 2 2 2 2 5 6 5B A B A B A E The strategy now is to eliminate the terms s2 and c2 in Equation 454 in order to have the single unknown 5, in the terms s5 and c5. To this end, first Equa tion 454 is rewritten as 2 2 5 2 4 2 2 3 2 2 1c E E s E s E c E (455) Next, Equation 455 is squa red both sides to obtain 0 c s E E 2 s c E E 2 s E E 2 s E c E s E E 2 s ) E E E (2 c ) E E E (2 E2 2 2 4 1 2 2 2 2 3 1 3 2 4 3 4 2 2 3 4 2 2 1 2 5 4 2 2 2 4 5 3 2 2 2 2 5 1 2 5 (456) On the other hand, from Equation 451 it is solved for s2 1 3 5 2 2D D s D s (457) Using the basic trigonometric identity 22 22c1s from Equation 457 it is obtained for c2 2 PAGE 77 77 2 1 3 5 2 2 2D D s D 1 c (458) The substitution of Equations 457 and 458 into Equation 456 will arise a trigonometric equation involving only the sine and cosine of the joint angle 5. The substitution of the tanhalfangle identities 2 5 5 5x 1 x 2 s (459) 2 5 2 5 5x 1 x 1 c (460) into the trigonometric equation in sine and cosine of 5 yield an 8th degree polynomial in x5, which after being divided by 4 2 5) x (1 is expressed as 0 C x C x C x C x C x C x C x C x C05 5 15 2 5 25 3 5 35 4 5 45 5 5 55 6 5 65 7 5 75 8 5 85 (461) where: C05 = [2E1E562E51E542E1E542E1E51+2E23E21E54 2+E23 2E1 22E51E562E54E56E56 2E51 2+E21 2]D1 4+(2E41E56+2E43E56+2E51E43+2E1E43+2E51E41+2E54E43+2E1E41+2E54E41)D3D1 3+[E43 22E3E562E54E32E41E43+2E1E54+2E1 22E1E3+2E1E51E41 2E23 2E21 2+2E1E562E51E32E23E21]D3 2D1 2+(2E3E432E1E43+2E3E412E1E41)D3 3D1+[E3 2+2E1E3E1 2]D3 4 C15 = [4E1E554E53E564E54E554E51E53+4E22E214E51E554E55E564E1E53+4E22E234E53E54]D1 4+{[4E43E56+4E1E41+4E1E43+4E54E43+4E41E56+4E54E41+4E51E41+4E51E43]D2+ (4E53E43+4E43E55+4E54E42+4E41E55+4E42E56+4E1E42+4E53E41+4E51E42)D3}D1 3+{[4E43 28E51E34E21 24E23 24E41 2+8E1E568E1E38E3E568E41E43+8E1E548E54E38E23E21+8E1E51+8E1 2]D3D2+(4E22E21+4E1E534E22E234E41E42+4E1E554E3E554E53E34E42E43) D3 2}D1 2+{[12E1E41+12E3E43+12E3E4112E1E43]D3 2D2+(4E3E424E1E42)D3 3}D1+(8E1 2+16E1E38E3 2)D3 3D2 PAGE 78 78 C25 = [8E51E528E52E548E1E564E54E568E53E55+4E51E54+4E23 24E1 2+4E23E214E1E544E56 2+4E51 24E53 24E55 2+4E22 28E1E528E52E56]D1 4+{[8E41E55+8E43E55+8E53E43+8E54E42+8E51E42+8E53E41+8E42E56+8E1E42]D2+(4E1E41+8E43E56+4E54E43+4E41E56+ 8E1E43+8E42E554E51E41+8E52E41+8E53E42+8E52E43)D3}D1 3+{[4E43 28E51E34E21 24E23 24E41 2+8E1E568E1E38E3E568E41E43+8E1E548E54E38E23E21+8E1E51+8E1 2]D2 2+(16E1E5316E22E2116E53E316E41E42+16E1E5516E42E4316E22E2316E3E55)D3D2+[8E1E524E42 24E41E434E54E34E22 28E1E3+8E1 24E43 28E52E34E23 2+4E1E54+8E1E568E3E564E23E21]D3 2}D1 2+{[24E3E4324E1E4324E1E41+24E3E41]D3D2 2+(24E3E4224E1E42)D3 2D2+ [4E1E41+4E3E41+8E3E438E1E43]D3 3}D1+[24E1 2+48E1E324E3 2]D3 2D2 2+(4E3 2+8E1E34E1 2)D3 4 C35 = [16E52E5512E1E554E53E56+4E51E5512E55E5616E52E53+12E51E53+4E22E214E1E53+12E22E23+4E53E544E54E55]D1 4+{[12E43E56+16E52E41+4E41E56+16E42E55+4E54E43+16E53E4212E51E41+16E52E43+12E1E43+4E1E414E51E434E54E41]D2+(12E43E55+12E42E56+4E53E43+12E1E42+4E41E554E51E424E53E41+16E52E42+4E54E42)D3}D1 3+{[16E1E5316E22E2116E53E316E41E42+16E1E5516E42E4316E22E2316E3E55]D2 2+(24E1 2+8E1E548E54E3+4E41 232E52E3+24E1E568E23E2124E3E5616E22 212E43 2+32E1E5216E42 212E23 28E1E51+4E21 2+8E51E324E1E38E41E43)D3D2+[12E1E5512E22E2312E3E5512E42E43+4E1E534E41E424E53E34E22E21]D3 2}D1 2+{[16E3E41+16E3E4316E1E4316E1E41]D2 3+ (48E1E42+48E3E42)D3D2 2+[12E3E41+36E3E4336E1E4312E1E41]D3 2D2+ (12E1E42+12E3E42)D3 3}D1+[32E3 2+64E1E332E1 2]D3D2 3+(24E1 2+48E1E324E3 2)D3 3D2 PAGE 79 79 C45 = [16E1E52+16E51E528E55 2+4E51E5616E52 2+2E54 2+8E22 2+6E23 26E1 2+4E1E5112E1E566E56 26E51 22E21 2+8E53 216E52E56]D1 4+{[16E51E42+32E52E4216E53E41+16E1E42+16E43E55+16E42E56]D2+(16E52E434E51E434E54E41+12E43E56+16E42E55+12E1E43)D3}D1 3+{[8E23 2+8E41 232E52E3+32E1E528E43 2+8E21 2+16E1E5616E1E316E3E5616E1E51+16E51E316E22 216E42 2+16E1 2]D2 2+ (32E42E4332E22E2332E3E55+32E1E55)D3D2+[6E43 28E42 2+2E21 2+12E1E564E1E5112E3E5612E1E3+2E41 28E22 216E52E3+16E1E52+12E1 26E23 2+4E51E3]D3 2}D1 2+{[32E3E4232E1E42]D2 3+(48E3E4348E1E43)D3D2 2+[48E1E42+48E3E42]D3 2D2+(12E3E4312E1E43)D3 3}D1+[32E1E316E3 216E1 2]D2 4+(48E1 248E3 2+96E1E3)D3 2D2 2+[12E1E36E3 26E1 2]D3 4 C55 = [16E52E55+4E51E55+4E54E55+4E1E53+12E22E23+4E53E54+4E53E5612E51E53+16E52E5312E55E564E22E2112E1E55]D1 4+{[12E51E4116E52E41+12E43E564E54E414E51E434E41E5616E53E42+12E1E43+16E42E554E54E43+16E52E434E1E41]D2+(12E42E56+12E43E55+16E52E424E54E424E51E424E53E43+12E1E424E41E554E53E41)D3}D1 3+{[16E42E4316E1E53+16E22E2116E3E5516E22E23+16E1E55+16E53E3+16E41E42]D2 2+(4E41 2+32E1E5232E52E316E22 2+24E1E5624E1E324E3E568E1E5112E23 2+8E41E4316E42 212E43 2+24E1 2+4E21 2+8E51E3+8E54E3+8E23E218E1E54)D3D2+[12E42E4312E22E234E1E53+4E41E42+4E22E21+4E53E3+12E1E5512E3E55]D3 2}D1 2+{[16E1E4116E3E4116E1E43+16E3E43]D2 3+(48E1E42+48E3E42)D3D2 2+[12E3E41+36E3E43+12E1E4136E1E43]D3 2D2+(12E1E42+12E3E42)D3 3}D1+[32E3 2+64E1E332E1 2]D3D2 3+ (24E1 2+48E1E324E3 2)D3 3D2 PAGE 80 80 C65 = [4E1E544E23E21+8E53E558E52E568E1E528E1E56+8E52E54+4E23 24E1 28E51E52+4E54E564E56 2+4E51 24E53 2+4E22 24E55 24E51E54]D1 4+{[8E1E428E53E438E54E42+8E53E41+8E51E42+8E42E56+8E43E558E41E55]D2+(8E53E42+8E43E564E41E56+8E1E434E1E41+8E42E55+4E51E418E52E41+8E52E434E54E43)D3}D1 3+{[8E1E51+8E1 2+8E23E214E43 2+8E41E434E41 28E1E38E3E564E23 28E1E54+8E54E38E51E34E21 2+8E1E56]D2 2+(16E42E4316E1E53+16E22E2116E3E5516E22E23+16E1E55+16E53E3+16E41E42)D3D2+[4E23E218E52E38E1E34E23 2+8E1E56+4E54E34E42 24E1E54+8E1 24E43 2+8E1E52+4E41E438E3E564E22 2]D3 2}D1 2+{[24E1E41+24E3E4324E1E4324E3E41]D3D2 2+(24E3E4224E1E42)D3 2D2+ [4E3E41+4E1E418E1E43+8E3E43]D3 3}D1+[24E1 2+48E1E324E3 2]D3 2D2 2+(8E1E34E3 24E1 2)D3 4 C75 = [4E1E53+4E22E23+4E54E55+4E53E56+4E51E534E53E544E55E564E22E214E1E554E51E55]D1 4+{[4E51E434E41E56+4E54E414E51E414E54E43+4E1E434E1E41+4E43E56]D2+ (4E41E55+4E1E42+4E42E564E53E434E54E42+4E51E42+4E53E41+4E43E55)D3}D1 3 +{[8E1E51+8E1 2+8E23E214E43 2+8E41E434E41 28E1E38E3E564E23 28E1E54+8E54E38E51E34E21 2+8E1E56]D3D2+(4E53E34E42E43+4E22E21+4E1E554E1E534E22E23+4E41E424E3E55)D3 2}D1 2+{[12E3E43+12E1E4112E3E4112E1E43]D3 2D2+(4E3E424E1E42)D3 3}D1+[8E1 28E3 2+16E1E3]D3 3D2 C85 = [2E51E542E51E562E1E562E23E21+E23 2E1 22E1E51E51 2E56 2+E21 2E54 2+2E54E56+2E1E54]D1 4+(2E51E43+2E43E562E51E41+2E54E412E1E412E54E43+2E1E432E41E56)D3D1 3+[E21 22E1E3E23 2+2E1E56E41 2+2E54E32E51E3E43 2+2E1 2+2E41E432E3E562E1E54+2E23E21+2E1E51]D3 2D1 2+(2E1E412E1E432E3E41+2E3E43)D3 3D1+[E1 2E3 2+2E1E3]D3 4. PAGE 81 81 To obtain the joint angle 5, the roots of the polynomial in x5 presented in Equation 461 are first found. Next, Equations 459 and 460 are used to obtain the sine and cosine of the angle, then a unique value for 5 is obtained for each root of x5. The coefficients C05 to C85 are function of the known pa rameters and the new sum of angles found in the spherica l quadrilateral solution: ( 1+ 2), ( 3+ 4), and ( 5+ 6). According to the solution tree for the spherical quadrilateral presented in Figure 45, and according to Equations 425 to 427, there are two solutions for the spherical quadrilateral, they are ( 1+ 2)A, ( 3+ 4)A, and ( 5+ 6)A; and ( 1+ 2)B, ( 3+ 4)B, and ( 5+ 6)B, which implies that there is a total of sixteen solutions for 5, eight for A solution and eight for the B solution. 4.3.5 Solution for 3 At this point the joint angle 5 is known, which makes possible to write Equations 449 to 451, respectively, as 6 3 2 2 1A c A c A (462) where: ) A s A c (A A5 5 4 5 3 6 6 3 2 2 1B s B s B (463) where: ) B s B c (B B5 5 4 5 3 6 4 2 1D s D (464) where: ) D s (D D3 5 2 4 Equation 462 is initially rearranged as 3 2 6 2 1c A A c A (465) PAGE 82 82 and both sides are squared to yield 3 6 2 2 3 2 2 2 6 2 2 2 1c A A 2 c A A c A (466) which after substituting the trigonometric identity 2 i 2 is 1 c for i=2,3 yields 3 6 2 2 3 2 2 2 6 2 2 2 1c A A 2 ) s (1 A A ) s (1 A (467) On the other hand, from Equation 463 is solved for the sine of 3 2 2 1 6 3B s B B s (468) Substituting Equation 468 into Equation 467 and solving for c3 yields 6 2 2 2 2 1 2 2 2 1 6 2 2 2 6 3A A 2 1) (s A B s B B 1 A A c (469) From Equation 464 it is so lved for the sine of 2 1 4 2D D s (470) Substituting Equation 470 into Equation 469 will allow expressing c3 in terms of known parameters, 2 2 1 2 2 2 1 2 4 2 1 2 2 1 2 1 2 2 2 6 3] B [D B ] D [D A A F D B A c (471) where: 2 4 2 1 4 1 6 1 2 6 2 1 2 1 2 2 1D B D D B B 2 B D D B F The value for the sine of 3 is to be obtained in orde r to define a unique value for 3, s3 is found by substituting Equation 470 into Equation 468 to yield 1 2 4 1 1 6 3D B D B D B s (472) PAGE 83 83 Knowing the cosine as sine values for 3 from Equations 471 and 472, respectively, defines a unique angle 3 for each value of 5.The solutions A and B were obtained for 5, then also A and B solutions are obtained for 3. 4.3.6 Solution for 2 Once the values of 3 and 5 are known, the sine and cosine values that determine a unique value for 2 are easily obtained. The value for s2 was already defined in Equation 470. The value for c2 is obtained directly form Equation 462 1 3 2 6 2A c A A c (473) The values for s2 and c2 defined by Equations 462 and 473, respectively, define a unique angle 2 for each set of angles 3 and 5. A and B solutions stand for 2. 4.3.7 Solution for 1, 4 and 6 Directly from Equations 425 to 427, it is solved for 1, 4 and 6 to obtain 2 Q 1 1 (474) 3 Q 2 4 (475) 5 Q 3 6 (476) 4.3.8 Solution Tree The closure of the solution of the spatial 7L7R closedloop w ith three pairs of parallel axes is a solution tree as presented in Figure 46 (the Q subscript stands for quadrilateral), to illustrate the way as the sixteen solutions were obtained. PAGE 84 84 Figure 46. Sixteen solutions tree for 7L7R cl osedloop mechanism with consecutive pairs of parallel joint axes. The reverse kinematic analysis of the half of the closedloop motion generator has been performed. In the next section, the closed mech anism reverse kinematic analysis is completed. 4.4 Reverse Kinematics Analysis for the Complete Spatial Motion Generator So far, the twelvelink spatial closedloop m echanism has been divided into two open loop sixlink mechanisms (see Figure 41) whose revers e position kinematic analysis has been solved. The two resulting openloop sixlink mechanisms ar e not identical but are similar in geometry. Based on this similarity, a simple transformation of coordinates is presented for completing the PAGE 85 85 reverse kinematic analysis of th e twelvelink closedloop mechanism. To this end, consider the two fixed coordinate systems xF1yF1zF1 and xF2yF2zF2, attached to the two fixed joint axes as presented in Figure 41. The solution of the position reverse kine matic analysis of the half mechanism corresponding to links 1121110987 in Figu re 41 starts by presenting the respective kinematic labeling in Figure 47. The subscript (2 ) in the nomenclature is used for distinguishing these parameters from the ones previously used in solving the half mechanism1234567 (i.e., S1(2) makes reference to the link distance S1 in the fixed coordinate system x2y2z2 and 7(2) makes reference to the joint angle that closes the loop with the part of th e link 1 attached to the x2y2z2 coordinate system). The problem statement is Given 1) The constant mechanism parameters: link lengths a1,12 to a87, twist angles 1,12 to 87, and offset distances S1(2) to S9. 2) Offset distance S8 and the direction of the vector a87 relative to the vector S8 3) Position and orientati on of the end effector: F2Ptool, F2S8 and F2a87. 4) Location of the tool point in the 8th coordinate system: 8Ptool. 5) Close the loop parameters calculated in the x2y2z2 coordinate system: Offset distances S1(2), S7(2); link distance a71(2); joint angle 7(2); angle 1(2). Find The joint angles 1(2), 12, 11, 10, 9 and 8 of the closedloop 7L7 R spatial mechanism 11211109871. PAGE 86 86 Figure 47. Kinematic labeling of the 6L6R openloop mechanism 112 7. Solution In the above problem statement it was assume d that the close the loop parameters are already known. This problem statement is identical to the problem stated in Section 4.3.1 for the half mechanism 1234567, so the solution is taken in the x2y2z2 fixed coordinate system. It is possible to obtain the transformation matrix that relates the coordinate systems xF1yF1zF1 and xF2yF2zF2 (see Figure 48), as a designers c onvenient choice based on the physical space available. This transfor mation matrix is defined as T1 F 2 F, which contains the translation and the rotation information that relates the second fixed coordinate system (xF2yF2zF2) as seen in the first fixed coordinate system (xF1yF1zF1). This transformation matrix is defined as 1 0 0 02 OF 1 F 1 F 2 F 1 F 2 FP R T (477) where: R1 F 2 F: rotation information that relates the 1 and 2 fixed coordinate systems PAGE 87 87 2 OF 1 FP: position vector of the origin of the second fixed coordinate system OF2 measured with respect to the first coordinate system. Figure 48. Transformation of th e fixed coordinate systems xF1yF1zF1 and xF2yF2zF2. The tool coordinates as seen in the xF2yF2zF2 fixed coordinate system need to be obtained as 1 1tool 1 F 2 F 1 F tool 2 FP T P. (478) The tool coordinates as seen in the 6th coordinate system are transformed based on Figure 49, where the orientation of the z axes of the each coordinate system are chosen to be antiparallel to ensure the configuration o the sixt h link as shown in the same figure, where it is remarked that F2S8 = F2S6(2) and F2a87=F2a6(2). Figure 49. Disposition of the two sixth coordinate systems. PAGE 88 88 From Figure 49, the transformati on matrix that relates the two 6th coordinate systems is obtained from a rotation of 180 degrees about the x axes which are collinear, to obtain 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1) 1 ( 6 ) 2 ( 6T. (479) The transformation matrix that relates the orientation of the x and z axes of the 6th coordinate system as seen in the xF1yF1zF1 coordinate system is obtained by 1 0 0 0 0 0 06 1 F 67 1 F 6 1 F 67 1 F 1 F ) 1 6(S a S a T. (480) Combining Equations 477, 479 and 480, yields the vectors F2S8 and F2a87, T T T S a S a) 1 ( 6 ) 2 ( 6 1 F ) 1 6( 1 1 F 2 F 8 2 F 87 2 F 8 2 F 87 2 F1 0 0 0 0 0 0 (481) One more transformation matrix is needed to ex press the tool point co ordinates as seen in the new introduced coordinate system xF2yF2zF2, 1 1 1tool ) 1 ( 6 (2) 6 (1) 6 tool ) 2 ( 6 tool 8P T P P. (482) Known parameters F2Ptool, F2S8F2a87 and 8Ptool have been specified in the fixed xF2yF2zF2 coordinate system, in Equations 478, 481 and 482, respectively. The solution presented in Section 4.3 is now appropriate for solving th e half mechanism compri sed for the links 78 1. Here ends the position reverse kinematic solution of the spatial manipulator. PAGE 89 89 CHAPTER 5 SYNTHESIS OF THE NONCIRCULAR PITCH PROFILES A numerical approach for finding the working section centrodes of the noncircular gear connections are presented in this chapter. Th is approach introduces a sequence parameter to define the instantaneous joint angl es required to calculate the I/O noncircular gear relationship. First, the position and orienta tion specification for each positi on are associated with a unique sequence parameter. Next, each joint angle is associated with its respective parameter and numerical derivatives of the joint angles are obt ained. Finally, the ratios of the numerical derivatives yield the expressions for the I/O gear relationships. 5.1 I/O NonCircular Gear Relationship s for the Spatial Motion Generator Figure 51 depicts the noncircular gear connections labeling of the spatial motion generator. The five noncircular gear relations hips are distinguished from each other by the subscript associated with the connecting link: g3, g5, g7, g9 and g11. Figure 52 presents four consecu tive links f, g, h, and i; wh ere links g and i have the noncircular gears attached to them, and the joint axes Sh and Si are parallel ( hj=0) or antiparallel ( hj=180). When using Equation 32 to find the I/ O relationship, it must be recognized that every joint angle is, by definition, taken relative to th e link vector of th e previous link (as defined in Figure 12). Based on this, the expr ession for the I/O relationship for the geared connection in Figure 52, with connecting link h, is given by h i hd d g (51) where: d i: infinitesimal change of the i joint angle d h: infinitesimal change of the h joint angle. PAGE 90 90 Figure 51. Noncircular g ear connections labeling. Figure 52. Labeling of the h noncircular gear connection. Equation 51 is used to find the instantaneous speed ratio of the noncircular gears depicted in Figure 51. For the half mechanism 0123456, the I/O relationships for the connecting links 3 and 5 are 3 4 3d d g (52) PAGE 91 91 5 6 5d d g (53) respectively, where the output body is the link after the connecting link, following the sequence 01 6. Analogously, following the sequence 011 6 for the half mechanism 011109876, the output bodies are links 10, 8 and 6 and the I/O relationships are, respectively, ) 2 ( 1 10 11d d g (54) 9 8 9d d g (55) 7 6 7d d g (56) where 1(2) is the angle between the x axes of the s econd fixed coordinate system and the first standard coordinate system. In the next sections, two approaches to calculate the value of the instantaneous gear ratio g in Equation 51 are presented. 5.2 Sequence Parameter Approach The path requirements f(x,y,z) can be di scretely represented as a set of points {(xj,yj,zj)/j=1,,n} along the inters ection of the intervals [x1,xn];[y1,yn]; and [z1,zn], depicted in Figure 53(a). The discrete representation of the path requirements allows one to introduce a nonnegative integer sequence parameter u, associated with each point (xj,yj,zj), see Figure 53(b). Thus there is a single sequence parameter a ssociated with each position giving a total of n parameters: uj=j (j=1,,n). Figures 53(c) and 53(d) depi ct the discrete representation of the position and orientation needs, respectively, related to the introduced se quence parameter. On the other hand, due to the solution technique used to solv e the reverse kinematic position an alysis (Chapter 4), the motion PAGE 92 92 variables x, y, z, x, y and z are not explicitly presented in th e expressions obtained for the joint angles; i.e., the joint angle 5 is obtained from the roots of an 8th degree polynomial in the tan half angle. However, the numerical value of eac h joint angle depends on the numerical value of the six motion generation variables written above. Ba sed on this, it is possible to relate each joint angle with the sequence parameter u, as presented in the next section. 5.2.1 Polynomial Interpolation of Joint Angles: Discarded First and Last Points Section 3.1.5 presented the polynomial to ap proximate the centrode corresponding to the nonworking section. In reference to the work ing sections centrode, its smoothness is going to depend on the interpolation method used to relate the joint angles with th e sequence parameter. In order to evaluate the first derivatives of the I/O relationship in Equations 35 and 37 that guarantee infinite number of cycles for the gears in mesh, the order of the polynomial approximation used to relate the joint angles and the sequence parameter is to be at least quadratic. This is apparent from Equation 51. A cubic polynomial interpolation would also guarantee the smoothness of the entire profile. Cu bic spline functions ar e proven to be smooth functions with which to fit data, and when used for interpolation they do not have the oscillatory behavior that is characteristic of highdegree polynomial interpolation, Atkinson [31]. Based on this, a natural cubic spline is suitable for the piecewise interpolation. The last reference can be consulted for informa tion about cubic spline interpolation. PAGE 93 93 Figure 53. Introduction of the se quence parameter u to discretely express the motion generation requirements along {[xo,xf],[yo,yf],[zo,zf]}. A) Original path requirements {x,y,z}. B) Sequence parameter, u, along the path. C) E quivalent representation of the x, y and z coordinates of the path versus the sequ ence parameter u. D) Extension of this representation to the orientation requirements x, y and z. PAGE 94 94 The lack of information about the slope of the end points (first and la st points) would lead to the assumption of reasonable slope values in order to calculate the derivatives of the joint angles with respect to the sequence parameter at those points. If there is no information, any assumption made is a speculation that might affe ct the smoothness of the centrode. To preserve the smoothness of the pitch curve, the first and la st points are discarded after the natural cubic spline interpolation is pe rformed. The first derivative of the joint angles with respect to the sequence parameters is numerically approximated as u u u u lim du d 0 u (57) where: : joint angle approximated using c ubic natural spline interpolation u: sequence parameter u: infinitesimal increment of u u : value of approximated joint angle evaluated at u u u : value of the approximated joint angle evaluated at u+ u d /du: numerical derivative of the joint angl e respect to the sequence parameter u. The approximation described above in Equati on 57 is also applicable for the angle 1. 5.2.2 I/O Relationship Expression s from Sequence Parameter Approach Once the joint angles are parameterized usi ng spline polynomials in terms of the sequence parameter u, the value of the I/O relationshi p can be found by introdu cing the unit ratio du/du (chain rule) into Equation 51 to yield du d du d gh i h (58) PAGE 95 95 where the subscript h makes reference to the conn ecting link in Figure 52 and the derivatives of the joint angles respect to u ar e obtained through Equation 57. Equa tions 52 to 56 are modified by the unit ratio, to yield du d du d g3 4 3 (59) du d du d g5 6 5 (510) du d du d g) 2 ( 1 10 11 (511) du d du d g9 8 9 (512) du d du d g7 6 7. (513) PAGE 96 96 CHAPTER 6 RESULTS: ILLUSTRATIVE EXAMPLES 6.1 Reverse Kinematics for a Single Position of the End Effector: Single Position, Half Mechanism Case The objective of this example is to illustrate the reverse kinematic analysis presented in Chapter 4. The constant mechanism parameters fo r the example case are presented in Table 61. The twist angle 67 and the link length a67 were selected as 90 and 0 in respectively. The free choice values for the offset S6, the hypothetical link length a67, and the hypothetical twist angle 67 are presented in Table 61. Table 61. Constant mechanism parameters for numerical example. Offset Distance [in] Link Lengths [in] Twist Angles [deg] S2 = 3.4947 a12 = 14.2368 12 = 0 S3 = 0 a23 = 0.7411 23 = 59.2992 S4 = 1.3465 a34 = 12.9009 34 = 0 S5 = 0 a45 = 6.1349 45 = 76.8924 S6* = 6.0 a56 = 10.3782 56 = 0 a67* = 0 67* = 90 Free choice. The desired position and orientati on of the end effector are summarized in Table 62. In this example, the orientation requirements are specified by the x and z axes of the sixth coordinate system (attached to th e end effector) as seen in the fi xed coordinate system, i.e. the vectors Fa67 and FS6. The closetheloop parameters obtai ned for this case are summarized in Table 63. The solution for the equivale nt spherical quadrilateral for the sums 1+ 2, 3+ 4, and 5+ 6 is presented in Table 64. Table 62. Desired position a nd orientation requirements. Fa67 [0.4771 0.5994 0.6428] FS6 [ 0.7393 0.6692 0.0752] [ x, y, z][deg]* [135.1142, 47.6716, 63.5884] FPtool [in]** [10.1041 8.0151 0.5516] 6Ptool [in] [5 7 8] XYZ Fixed Angles ** [x, y, z] position needs PAGE 97 97 Table 63. Calculated closetheloop parameters. Distance [in] Angle [degrees] a71 = 4.2174 71 = 40.3277 S7 = 0.8123 7 = 96.6756 S1 = 9.119 1 = 132.7510 Table 64. Sums of consecutive joint angles obtain ed from the equivalent spherical quadrilateral solution. Sum of Angles Solution A [degrees] Solution B [degrees] 1+2 7.5924 162.2098 3+4 87.2244 87.2244 5+6 80.4042 160.6752 Fourteen real solutions and two complex solu tions were found for this numerical example for the joint angles 1 through 6. Figures 61 and 62 illustrate the real configurations for the openloop manipulator in this numerical exampl e. Figures 61 and 62 were generated using MatLab. A forward analysis was performed as a check for all 16 solutions, including the complex solutions D and E. Each solution positio ned and oriented the end effector as desired. PAGE 98 98 Table 65. Joint angles corresponding to the sixteen solutions of the numerical example. Sol 1[deg] 2[deg] 3[deg] 4[deg] 5[deg] 6[deg] A 119.6877 5.4709 177.7643 95.0113 177.7795 97.3753 B 281.5373 156.3786 6.1702 81.0542 145.8062 133.7896 C 155.4960 30.3374 175.0900 87.8656 136.4939 143.1019 5.6494 130.8063 31.5585 118.7799 127.9873 47.5841 D +31.5413 i 31.5413 i +42.3301 i 42.3301 +39.4481 39.4481 5.6494 130.8063 31.5585 118.7799 127.9873 47.5841 E 31.5413 i +31.5413 i 42.3301 i +42.3301 39.4481 +39.4481 F 174.3520 54.1905 114.2585 158.5171 115.1178 34.7136 G 173.63906 48.4804 149.8232 122.9524 79.5779 159.9822 H 262.0020 136.8434 47.4029 134.6273 64.8370 145.2412 I 129.0418 158.5006 19.6492 67.5752 150.2720 10.4032 J 185.5112 145.0300 52.55648 34.6679 140.0509 59.2739 K 2.4264 31.88519 172.7676 85.5432 136.2312 24.4440 L 21.5798 51.0386 136.5833 136.1922 83.5839 77.0913 M 100.3648 129.8236 47.9943 135.2187 79.1644 81.5108 N 270.2382 60.3030 159.7512 72.5268 89.8907 109.4341 O 273.2697 57.2715 172.7296 85.5052 75.1938 124.1310 P 171.1326 159.4086 43.9788 131.2032 21.9573 177.3675 PAGE 99 99 Figure 61. Real solutions AH in Table 65, corresponding to the A solution of the spherical quadrilateral in Table 64. PAGE 100 100 Figure 62. Real solutions IP in Table 65, corresponding to th e B solution of the spherical quadrilateral in Table 64. 6.2 Motion Generation along Discrete Path and Orientation Requirements: Complete Mechanism Case Consider a hypothetical manufacturing pro cess where a tool traverses 3D space as illustrated by the position and orientation specifications shown in Figure 63A and 63B. PAGE 101 101 Discrete data corresponding to these figures ar e given in Table B1 (Appendix B), where the fixed coordinate system xF2yF2zF2 is selected to be the reference system on which the data in Table B1 is given. The origin of the fixed coordinate system xF1yF1zF1 was selected to be located a distance of 300.0mm along the z axis, a nd the coordinate axes are parallel to the xF2yF2zF2 coordinate system. The transformation matrix T2 F 1 F that relates the two fixed coordinate systems xF1yF1zF1 and xF2yF2zF2 is 1 0 0 0 0 300 1 0 0 0 0 1 0 0 0 0 12 F 1 FT. (61) The transformation matrix given in Equati on 61 and the motion information in the xF2yF2zF2 coordinate system is used to expr ess the motion needs as seen in the xF1yF1zF1 coordinate system by using methodologies in Sections 2.10 a nd 4.4. The orientation requirements are given in Figure 63B, where since the fixed reference systems xF1yF1zF1 and xF2yF2zF2 are parallel, the XYZ FixedAngles representations for the orientation needs are the same in both fixed coordinate systems. Representations of the posi tion and orientation requirements as a function of the sequence parameters are presented in Figures 64A and 64B, respectively. From now on, the nomenclature for links and joint angles follo ws the labeling in Figure 51. The mechanism parameters and free choices for the closedl oop mechanism are summarized in Table 66. The position angles obtained from the reverse kinematic stage are summarized in Figure 65. For the half mechanism 01 6, solution I was chosen as th is configuration was able to position and orient the end effector at every pose along the path with real solutions for the joint angles. These real joint angles are presented in Figure 65A. On the other hand, real joint angles PAGE 102 102 corresponding to solution P sa tisfied the motion needs for the half mechanism 011 6; they are shown in Figure 65B. Figure 63. Motion needs. A) Position re quirements. B) Orientation requirements. Table 66. Constant mechanism parameters for the closedloop example. Offset Distance [mm] Link Lengt hs [mm] Twist Angles [deg] S2 = 0 a12 = 277.44 12 = 0 S3 = 0 a23 = 203.20 23 = 67.75 S4 = 168.95 a34 = 331.5 34 = 0 S5 = 0 a45 = 171.37 45 = 108.78 S7 = 0 a56 = 349.22 56 = 0 S8 = 50 a11,10 = 100.00 11,10 = 0 S9 = 0 a10,9 = 200.00 10,9 = 30.00 S10 = 0 a98 = 105.00 98 = 0 S6(1)* = 70.00 a76 = 95.00 87 = 0 S6(2)* = 60.00 a67(1)* = 0 67(1)* = 90.00 a67(2)* = 0 67(2)* = 90.00 Free choice PAGE 103 103 Figure 64. Motion needs versus the sequence parameter in each fixed coordinate system. A) Position needs. B) Orientation needs. The value of the Input/Output relationships along the working profile section can be obtained since the joint angles are known at this stage. The gear synthesis results regarding the half mechanism 01 6 are depicted in Figures 66 and 67 for gear connections 234 and 456, respectively. These figures show the variati on of the Input/Output re lationship and the output angle versus the input angle along the working section. Th e values of the Input/Output relationship for the working section in Figures 66B and 67B were calculated using the sequence parameter approximation in Section 5.3. PAGE 104 104 Feasibility of the gear connections in this work is limited to externalexternal gears in mesh configuration. Feasible externalexternal pitch profiles were not obtained for the working sections, since the Input/Output re lationship presents asymptotic be havior as depicted in Figures 66B and 67B. At different positions of the mo tion needs, both Input/Out put gear relationships behaved asymptotically. The asymptotic behavior lead to a sign change in both cases, therefore the gear connection changed from externalexterna l to externalinternal configuration at this point. This is due to the doublevalue shape that the out put angle presents, as seen in Figures 66A and 67A. PAGE 105 105 Figure 65. Joint angles for the closedloop mechanism. A) Half mechanism 01 6. B) Half mechanism 011 6. PAGE 106 106 Figure 66. Gear synthesis results fo r connection 234 in half mechanism 01 6. A) Output angle versus the input angle. B) Input/Output relationshi p versus the input angle. Better results were obtained for the remaining gear connections. Figures 68A, 68B and 68C show the gear synthesis results corresponding to the half mechanism 011 6, for gear connections 10110, 8910 and 678, respectively. Th e values of the Input/Output relationship along the working section were obtained using the sequence parameter approach. Feasible externalexternal pitch profile s were obtained since the Input /Output relationships neither reached negative values nor presented changes of sign. The nonworking profile approximation model also would allow the simulation of the cutt er, wherefrom the synthesi s of gear connections 10110, 8910 and 678 were successful. Threedime nsional animations that show the gear connections in Figure 68 in mesh is availabl e upon request at the CI MAR Laboratory at the University of Florida. PAGE 107 107 Figure 67. Gear synthesis results fo r connection 456 in half mechanism 01 6. A) Output angle versus the input angle. B) Input/Output relationshi p versus the input angle. PAGE 108 108 ACentre distance a11,10Centre distance a98BCentre distance a76C Figure 68. Centrode, out put angle and Input/Output gear rela tionship versus the input angle in half mechanism 011 6. A) Connection 10110. B) Connection 8910. C) Connection 678. PAGE 109 109 Finally, two views of the closedloop mechan ism are presented in Figure 69, where the Figures 69A and 69B correspond to positions 15 and 35 in Table 66, respectively. A rigid body has been attached to the end effector link. A MatLab animation of the example here presented is available upon request to the CIMAR Laboratory at the University of Florida. Here ends the successive points illustrative example. PAGE 110 110 Figure 69. Closedmechanism at tw o different motion needs in Tabl e B.1. A) At position 5. B) At position 30. PAGE 111 111 CHAPTER 7 CONCLUSION AND FUTURE WORK 7.1 Conclusion In this research, the reverse kinematics and the gear connections s ynthesis of a spatial closedloop manipulator mechanism comprised of 12 links, 12 revolute joints, and 5 noncircular gear connections were analyzed. Two similar openloop mechanisms comprised of 6 links and 6 revolute joints with consecutive pairs of joint axes parallel were derived from the original mechanism. The first joint axis is parallel to the second, the third is parallel to the fourth, and so on. A closedloop mechanism comprised of 7 link s and 7 revolute joints was obtained after the hypothetical closure loop was inserted in each openloop mechanism. The equivalent closedloop spherical mechanism possesses a single degreeoffreedom. The reverse kinematic analysis was reduced to the solution of an openloop r obotic manipulator comprised of 6 links and 6 revolute joints where consecutive pairs of axes ar e parallel. A transforma tion of coordinates that relates the fixed coordinate systems, and the position and orientation needs of each openloop mechanism completed the kinematic analysis. It was deduced that for the 6 link 6 revolute joint openloop mechanism that for a single position and orientation of the end effector there are 16 possible c onfigurations that satisfy the motion needs. This finding was one of the contri butions of this research as the particular manipulator geometry had not been presented in the prior literature. One example was included to illustrate the degree of the solution, supported by the graphical display of 14 real solutions. A forward analysis of the remaining two imaginary joint angles solutions positioned and oriented the end effector as desired. In theory, for the closedloop 12 link12 revolute joints mechanism there are 256 possible configuratio ns that result when combining the degrees of the solutions of each openloop mechanism. PAGE 112 112 The reverse kinematic analysis was also succes sfully proven on a discrete set of specified successive position and orientation needs to be tr aversed by the end effector. One single solution (out 16 possible) was found for each openloop mechan ism in which real joint angles satisfy the successive position and orientation needs. This case was presented as a second illustrative example and was supported by a computer animation. A methodology of synthesis of the pitch curves was also tested in the second example. Each motion requirement was associated to a monotonic sequence parameter, and the joint angles were numerically related to this paramete r in order to numerically approximate the gear relationship values for each gear connection. Th e pitch curves of three gear connections were successfully synthesized with feasible pitch pr ofiles (externalexternal gears in mesh). Numerical values of the remaining gear conne ctions were obtained; however, the geometric synthesis was unsuccessful since negative values a nd asymptotic behavior were present. As a result, the resulting mechanism was not a one degr eeoffreedom as desired, but a three degreeoffreedom that would still require three actuators to work. As a disadvantage of this numerical approach, initial and final motions needs are di scarded after approximati ons are performed. Extra points in the vicinity of the end needs should be included. This work is intended to be the starting platform for future analysis th at would lead to the implementation of this technology in highly repeti tive tasks manufacture indus try. Nonfeasible gear connections and intersecti on of the links were some of the unresolved problems in this research that point to future considerations They are presented in the next section. 7.2 Future Work Subjects that deserve future an alysis in order to implement this project into industry are presented here. Some of the proposed problems were difficulties faced along this research and others are complimentary studies that would make the project marketable. PAGE 113 113 The implementation of a routine that combines the selection of the mechanism parameters (link lengths, link angles, and jo int distances) and evaluate the gear connections feasibility, other than just randomly checking and discarding would be the ne xt step in this research. The evaluation of the links intersection is also another algorithm to implement. The shortest distance between two nonconsecutive links must be evaluated ac cording to the real shape of the links. Also the intersection of the lin ks with the gear sections must be considered. The effect of the relative positi on and orientation of the two fixed coordinates systems in the number of real solutions obtained in each half openloop mechanism is to be evaluated. The physical space available would be the main constraint when evaluating this condition. The design of generic gear trains that replace the single externalexternal gear connections when negative values and asymptotic behaviors arise should be explored as a solution to the unfeasibility of the gear connections. To minimize the centre distance on the gear c onnections in order to reduce the area of the gears and the shaking moment is another aspect to be considered. Also, the minimization of the noncircularity of the pitch profiles by bounding the value of the gear relationship is necessary. A more general and elaborated approach in order to obtain th e values of the noncircular gear ratios along the working profile should be co nsidered. This additional method is currently under investigation based on screw theory analysis. A general methodology to evaluate the economi cal feasibility of this technology that indicates in what specific cases the implementation of this te chnology is advantageous according to the market conditions, is required in order to implement this technolog y in the industry. The number of tasks to be performed, the fabricati on costs, the maintenance costs, and the operation costs would be the main aspects to consider. PAGE 114 114 APPENDIX A JOINT AND LINK VECTORS EXPRESSIONS IN THE SPHERICAL MECHANISM (CRANE AND DUFFY [1]) Joint Vectors Expressions: Recursive Notations for One Subscript j jk js s X (A1) j jk ij jk ij jc s c c s Y (A2) j jk ij jk ij jc s s c c Z (A3) j ij js s X (A4) j ij jk ij jk jc s c c s Y (A5) j ij jk ij jk jc s s c c Z (A6) X, Y and Z Recursive Notations for Two Ascending Subscripts j i j i ijs Y c X X (A7) j i j i ijc Y s X X (A8) i jk ij jk ijZ s X c Y (A9) i jk ij jk ijZ c X s Z (A10) X, Y and Z Recursive Notations for Two Descending Subscripts j k j k kjs Y c X X (A11) j k j k kjc Y s X X (A12) k ij kj ij kjZ s X c Y (A13) k ij kj ij kjZ c X s Z (A14) X, Y and Z Recursive Notations for Three Ascending Subscripts PAGE 115 115 k ij k ij ijks Y c X X (A15) k ij k ij ijkc Y s X X (A16) ij kl ijk kl ijkZ s X c Y (A17) ij kl ijk kl ijkZ c X s Z (A18) X, Y and Z Recursive Notations for Three Descending Subscripts i kj i kj kjis Y c X X (A19) i kj i kj kjic Y s X X (A20) kj hi kji hi kjiZ s X c Y (A21) kj hi kji hi kjiZ c X s Z (A22) X, Y and Z Recursive Notations for Four Ascending Subscripts k hij k hij hijks Y c X X (A23) k hij k hij hijkc Y s X X (A24) hij kl hijk kl hijkZ s X c Y (A25) hij kl hijk kl hijkZ c X s Z (A26) X, Y and Z Recursive Notations for Four Descending Subscripts h kji h kji kjihs Y c X X (A27) h kji h kji kjihc Y s X X (A28) kji gh kjih gh kjihZ s X c Y (A29) kji gh kjih gh kjihZ c X s Z (A30) X, Y and Z Recursive Notations for Five Ascending Subscripts PAGE 116 116 l hijl l hijk hijkls Y c X X (A31) l hijk l hijk hijklc Y s X X (A32) hijk lm hijkl lm hijklZ s X c Y (A33) hijk lm hijkl lm hijklZ c X s Z (A34) X, Y and Z Recursive Notations for Five Descending Subscripts h lkji h lkji lkjihs Y c X X (A35) h lkji h lkji lkjihc Y s X X (A36) lkji gh lkjih gh lkjihZ s X c Y (A37) lkji gh lkjih gh lkjihZ c X s Z (A38) Link Vectors Expressions: U, V and W Recursive Notations for Two Ascending Subscripts ij i ijs s U (A39) ij i j i j ijc s c c s V (A40) ij i j i j ijc s s c c W (A41) U, V and W Recursive Notations for Two Descending Subscripts ij j jis s U (A42) ij j i j i jic s c c s V (A43) ij j i j i jic s s c c W (A44) U, V and W Recursive Notations for Th ree Ascending and Descending Subscripts jk ij jk ij ijks V c U U (A45) PAGE 117 117 jk ij jk ij ijkc V s U U (A46) ij k ijk k ijkW s U c V (A47) ij k ijk k ijkW c U s W (A48) U, V and W Recursive Notations for F our Ascending and Descending Subscripts jk hij jk hij hijks V c U U (A49) jk hij jk hij hijkc V s U U (A50) hij k hijk k hijkW s U c V (A51) hij k hijk k hijkW c U s W (A52) U, V and W Recursive Notations for Fi ve Ascending and Descending Subscripts jk ghij jk ghij ghijks V c U U (A53) jk ghij jk ghij ghijkc V s U U (A54) ghij k ghijk k ghijkW s U c V (A55) ghij k ghijk k ghijkW c U s W (A56) U, V and W Recursive Notations for Si x Ascending and Descending Subscripts jk fghij jk fghij fghijks V c U U (A57) jk fghij jk fghij fghijkc V s U U (A58) fghij k fghijk k fghijkW s U c V (A59) fghij k fghijk k fghijkW c U s W (A60) PAGE 118 118 APPENDIX B POSITION AND ORIENTATION SPECI FICATION IN EXAMPLE 6.2 Table B1. Continuous position and or ientation needs specification. Sequence x [mm] y [mm] z [mm] x [rad] y [rad] z [rad] 1 705.9128 16.8861119.67881.2993050.146000.215781 2 704.4675 2.304591124.36911.2964160.177470.258972 3 702.3611 21.38499129.02851.2930710.208900.302415 4 699.6003 40.32938133.65461.2892520.240300.346144 5 696.1929 59.11239138.24501.2849400.271660.390194 6 692.1478 77.70905142.79731.2801130.302960.434604 7 687.4749 96.09491147.30901.2747460.334190.479414 8 682.1852 114.2460151.77791.2688100.365340.524667 9 676.2905 132.1390156.20171.2622730.396400.570409 10 669.8039 149.7511160.57811.2550980.427350.616690 11 662.7392 167.0603164.90501.2472450.458190.663563 12 655.1112 184.0451169.18031.2386680.488890.711085 13 646.9355 200.6849173.40191.2293170.519430.759319 14 638.2286 216.9599177.56771.2191360.549810.808332 15 629.0077 232.8510181.67571.2080610.579990.858199 16 619.2907 248.3399185.72411.1960230.609950.909000 17 609.0965 263.4095189.71101.1829440.639670.960822 18 598.4444 278.0433193.63461.1687380.669131.013763 19 587.3542 292.2257197.49311.1533090.698281.067927 20 575.8465 305.9423201.28491.1365520.727091.123430 21 563.9424 319.1795205.00841.1183500.755531.180399 22 551.6633 331.9248208.66201.0985750.783551.238971 23 539.0311 344.1666212.24431.0770850.811111.299296 24 526.0681 355.8945215.75381.0537270.838141.361537 25 512.7970 367.0988219.18931.0283340.864581.425870 26 499.2404 377.7713222.54941.0007290.890371.492482 27 485.4216 387.9044225.83290.9707200.915431.561571 28 471.3637 397.4920229.03870.9381120.939671.633345 29 457.0901 406.5286232.16580.9027010.962991.708013 30 442.6241 415.0100235.21320.8642900.985281.785783 31 427.9891 422.9331238.17990.8226921.006431.866850 32 413.2087 430.2957241.06520.7777441.026311.951386 33 398.3059 437.0968243.86820.7293241.044772.039522 34 383.3040 443.3360246.58830.6773691.061662.131330 35 368.2260 449.0145249.22490.6218971.076842.226801 36 353.0945 454.1340251.77730.5630301.090142.325822 37 337.9321 458.6974254.24520.5010131.101422.428157 38 322.7609 462.7084256.62810.4362281.110532.533433 39 307.6028 466.1718258.92570.3692001.117362.641136 40 292.4791 469.0933261.13770.3005771.121822.750625 41 277.4108 471.4792263.26400.2311121.123842.861158 42 262.4185 473.3369265.30440.1616151.123422.971936 43 247.5221 474.6745267.25880.0929011.120573.082151 44 232.7412 475.5009269.12740.0257421.115363.092142 45 218.0946 475.8259270.91010.0391801.107892.985244 PAGE 119 119 46 203.6006 475.6597272.60710.101311.098282.880888 47 189.2769 475.0135274.21870.160221.086692.779483 48 175.1406 473.8989275.74500.215641.073262.681300 49 161.2081 472.3282277.18650.267411.058182.586481 50 147.4949 470.3144278.54350.315491.041602.495059 51 134.0161 467.8708279.81650.359921.023692.406979 52 120.7859 465.0113281.00600.400801.004602.322121 53 107.8177 461.7504282.11250.438300.984462.240322 54 95.12436 458.1027283.13670.472580.963412.161389 55 82.71782 454.0834284.07920.503850.941572.085113 56 70.60926 449.7079284.94080.532310.919042.011282 57 58.80912 444.9922285.72220.558150.895921.939686 58 47.32703 439.9522286.42420.581570.872291.870120 59 36.17186 434.6041287.04770.602740.848241.802390 60 25.35166 428.9644287.59360.621830.823841.736316 61 14.87375 423.0497288.06280.639000.799141.671727 62 4.744633 416.8766288.45630.654390.774211.608467 63 5.029950 410.4620288.77510.668130.749101.546391 64 14.44500 403.8225289.02020.680340.723851.485366 65 23.49630 396.9751289.19270.691130.698511.425271 66 32.18040 389.9363289.29370.700590.673121.365994 67 40.49460 382.7231289.32440.708810.647721.307432 68 48.43670 375.3519289.28580.715870.622341.249492 69 56.00550 367.8391289.17920.721840.597011.192087 70 63.20040 360.2013289.00570.726780.571771.135139 71 70.02140 352.4543288.76660.730750.546641.078574 72 76.46910 344.6143288.46300.733800.521641.022326 73 82.54500 336.6969288.09620.735980.496820.966333 74 88.25090 328.7175287.66750.737330.472190.910536 75 93.58940 320.6912287.17820.737890.447780.854884 76 98.56360 312.6331286.62940.737690.423600.799327 77 103.1770 304.5575286.02240.736770.399690.743819 78 107.4340 296.4788285.35860.735150.376070.688318 Points to be discarded after numerical approximations. 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Mech. Des., 128(8), pp. 12531260. [27] Gatti, G., and Mundo, D., 2007, Optimal Synthesis of SixBar CammedLinkages for Exact Rigid Body Guidance, Mech. Mach. Theory, 42(9), pp. 10691081. [28] Craig, J., 1989, Introduction toRobotics, Mechanics and Control 2 ed., Addison Wesley, USA. [29] Erdman, A., Sandor, G., and Kota, S., 2001, Mechanism DesignAnal ysis and Synthesis Vol. I, Prentice Hall, USA. [30] Martin, G. H., 1982, Kinematics and Dynamics of Machines, McGrawHill, USA. [31] Atkinson, K., 1989, An Introduction to Numerical Analysis 2 ed., John Wiley & Sons, USA. PAGE 123 123 BIOGRAPHICAL SKETCH Javier Roldn Mckinley studied to become a mechanical engineer at his home town school, University of Atlntico at BarranquillaColombia, with machine design as his area expertise. He earned his MSc degree from the University of Puer to Rico at Mayagez in the field of planar kinematics. Javier continued his research at the University of Florida where he earned his PhD degree in 2007 in the area of robotics and spatial kinematics. His research interests also include the design of noncircular gear connections, com puter graphics, and computer animations of robotic manipulators. During hi s experience at the corrugated board industry in his home country, Javier participated and led projects in diverse areas of mechan ical engineering. The attainment of the Professional Engineer (PE) license with a concentration in mechanical engineering is his immediate professional goal. 