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Use of Continuation Methods for Kinematic Synthesis and Analysis Thiagaraj Subbian Iowa State University

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Use of Continuation Methods for Kinematic Synthesis and Analysis Thiagaraj Subbian Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1990 Use of continuation methods for kinematic synthesis and analysis Thiagaraj Subbian Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mechanical Engineering Commons Recommended Citation Subbian, Thiagaraj, "Use of continuation methods for kinematic synthesis and analysis " (1990). Retrospective Theses and Dissertations. 9898. https://lib.dr.iastate.edu/rtd/9898 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. 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University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9110571 Use of continuation methods for kinematic synthesis and analysis Subbian, Thiagariy, Ph.D. Iowa State Univenity, 1090 UMI 300N.ZeebRd Ann Aibor, MI 48106 Use of continuation methods for kinematic synthesis and analysis by Thiagaraj Subbian A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Mechanical Engineering Signature was redacted for privacy. Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1990 Copyright © Thiagaraj Subbian, 1990. All rights reserved. TABLE OF CONTENTS INTRODUCTION 1 PART I. SYNTHESIS OF MOTION GENERATING MECHANISMS BY A CONTINUATION METHOD 6 INTRODUCTION 7 SOLVING n EQUATIONS IN n UNKNOWNS 9 SOLVING n EQUATIONS IN (n+1) UNKNOWNS 10 DEVELOPMENT OF EQUATIONS FOR THE MOTION GENERATION PROBLEM 11 FOUR POSITION SYNTHESIS 14 Determination of initial points on the curve 16 To generate the Burmester curves 18 Four position synthesis example 18 FIVE POSITION SYNTHESIS 22 Five position synthesis example 22 REFERENCES 27 PART II. FOUR-BAR PATH GENERATION SYNTHESIS BY A iii CONTINUATION METHOD 28 INTRODUCTION 29 CONTINUATION METHOD 32 DEVELOPMENT OF EQUATIONS FOR PATH GENERATION SYN­ THESIS 34 THE FIVE POSITION SYNTHESIS PROBLEM 39 Example 1 42 Example 2 44 REFERENCES 53 PART III. FIVE POSITION TRIAD SYNTHESIS WITH APPLI­ CATIONS TO FOUR- AND SIX-BAR MECHANISMS 55 INTRODUCTION 56 APPLICATIONS 58 DISPLACEMENT EQUATIONS FOR A TRIAD 61 DISPLACEMENT EQUATIONS FOR A DYAD 64 FIVE POSITION TRIAD SYNTHESIS USING CONTINUATION METH­ ODS 65 To solve 4 equations in 4 unknowns 65 To solve 4 equations in 5 unknowns 68 EXAMPLES 69 Synthesis of a four-bar function generator 69 Synthesis of a six-bar motion generating mechanism with prescribed timing 71 IV REFERENCES 79 PART IV. SIX AND SEVEN POSITION TRIAD SYNTHESIS US­ ING CONTINUATION METHODS 81 INTRODUCTION 82 DEVELOPMENT OF EQUATIONS FOR TRIAD SYNTHESIS 84 SIX POSITION SYNTHESIS 88 Example: Synthesis of an eight-bar mechanism 91 SEVEN POSITION SYNTHESIS 96 Example: Synthesis of a geared five-bar mechanism 98 REFERENCES 103 PART V. USE OF CONTINUATION METHODS FOR KINEMATIC SYNTHESIS 105 INTRODUCTION 106 THE CONTINUATION METHOD 108 SEVEN POSITION TRIAD SYNTHESIS 110 Development of equations 110 Number of paths 113 Projective transformation 114 Path tracking 116 Start system 116 Solutions for random displacements 116 Secant homotopy 117 V Examples 118 REVOLUTE-SPHERICAL (R-S) DYAD SYNTHESIS 122 Development of equations 122 Number of paths 124 Projective transformation 125 Start system 126 Solutions for random displacements 127 Parameter homotopy 128 Examples 128 REFERENCES 132 PART VI. ROBOT TRAJECTORY PLANNING BY A CONTIN­ UATION METHOD 135 INTRODUCTION 136 CONTINUATION METHOD 138 DEVELOPMENT OF EQUATIONS FOR THE 3-R MANIPULATOR . 140 INVERSE KINEMATIC SOLUTION AT THE START OF THE TRAJEC­ TORY 145 TRAJECTORY PLANNING FOR THE 3-R ROBOT 147 Example for configuration 1: 148 Example for configuration 2: 149 TRAJECTORY PLANNING FOR A 6-R PUMA TYPE ROBOT 153 Example 3: 6-R PUMA type robot 154 REFERENCES 158 vi CONCLUSIONS 161 REFERENCES 164 APPENDIX A: MULTI-HOMOGENEOUS STRUCTURE AND PRO­ JECTIVE TRANSFORMATION 167 Bezout's theorem 168 Example 1: 2-homogeneous system 168 Example 2: 3-homogeneous system 169 Projective transformation 170 APPENDIX B: COEFFICIENT, SECANT AND PARAMETER HO- MOTOPIES 172 Coefficient homotopy 173 Parameter homotopy 174 Secant homotopy 174 ACKNOWLEDGEMENTS 176 1 INTRODUCTION Synthesis and analysis of mechanisms have been accomplished in the past by graphical and analytical methods. Graphical methods are mainly useful for planar mechanisms. The success of these methods is limited to a few types of mechanisms involving only a few precision points. Furthermore, all possible solutions may not be obtained graphically. Closed-form analytical solutions are possible but are difficult to obtain for more than a few precision points. Due to the above reasons, application of numerical methods is inevitable. Newton-Raphson, steepest descent and generalized reduced gradient are the com­ monly used numerical methods for kinematic synthesis and analysis. These proce­ dures, due to their inadequacies, limit the number of precision points or the type of mechanism to be synthesized. In the above methods, the user has to come up with initial solution estimates, and the reliability of the procedure depends on the choices made. This need imposes a particularly difficult requirement in the synthesis problem because there is usually no logical basis for estimating starting values for link dimensions. Furthermore, when multiple solutions exist, these methods do not offer a methodical procedure for finding them. Therefore, this calls for a numerical procedure that does not need initial guesses and provides all the solutions. Continuation methods fall in this category for polynomial equations. The kine­ 2 matic analysis and synthesis problems can be easily expressed in polynomial form and so the procedure is particularly useful for these applications. Also, the method can be used to solve n equations in (n+1) unknowns, which could be used to generate the Burmester curves in kinematic synthesis, and the (n + l)th variable can be used as an independent variable for analysis. Dimensional synthesis is often classified as function generation, path generation, motion generation, path generation with timing, or motion generation with timing. To carry out the synthesis tasks, different techniques, such as precision position syn­ thesis [Erdman, 1981], selective precision synthesis [Kramer and Sandor, 1975], and least squares [Sarkisyan et al., 1973] method have been applied. In this dissertation, the above synthesis tasks are carried out for precision position synthesis using contin­ uation. One popular approach to kinematic synthesis is to break up the linkage into dyads and triads and to design them independently [Sandor and Erdman, 1984, Suh and RadclifFe, 1978, and Erdman, 1981]. Here, four-bar function generation, five-bar path generation, and six- and eight-bar motion generation with prescribed timing mechanisms are designed using this approach. Application of a type of continuation for kinematic synthesis was first carried out by Roth and Freudenstein [1963]. They termed their technique the bootstrap method. Only recently has further effort been focused on continuation methods for mechanism synthesis. Researchers such as Morgan and Sommese [1989], as well as Garcia and Zangwill [1981], have made continuation methods more efficient and easier to implement. These advancements combined with rapid developments in computer technology have made the application of continuation methods for kinematic synthesis feasible and highly desirable. 3 A detailed description of homotopy continuation methods is provided in Morgan [1986a and 1987], and Garcia and Zangwill [1981]. The number of paths to be tracked in solving a system of polynomial equations is given by Morgan
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