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Abstract Structural Synthesis and Analysis Of ABSTRACT Title of dissertation: STRUCTURAL SYNTHESIS AND ANALYSIS OF PLANAR AND SPATIAL MECHANISMS SATISFYING GRUEBLER’S DEGREES OF FREEDOM EQUATION Rajesh Pavan Sunkari Doctor of Philosophy, 2006 Dissertation directed by: Dr. Linda Schmidt Department of Mechanical Engineering Design of mechanisms is an important branch of the theory of mechanical design. Kinematic structural studies play an important role in the design of mech- anisms. These studies consider only the interconnectivity pattern of the individual links and hence, these studies are unaffected by the changes in the geometric prop- erties of the mechanisms. The three classical problems in this area and the focus of this work are: synthesis of all non-isomorphic kinematic mechanisms; detection of all non-isomorphic pairs of mechanisms; and, classification of kinematic mecha- nisms based on type of mobility. Also, one of the important steps in the synthesis of kinematic mechanisms is the elimination of degenerate or rigid mechanisms. The computational complexity of these problems increases exponentially as the num- ber of links in a mechanism increases. There is a need for efficient algorithms for solving these classical problems. This dissertation illustrates the successful use of techniques from graph theory and combinatorial optimization to solve structural kinematic problems. An efficient algorithm is developed to synthesize all non-isomorphic planar kinematic mechanisms by adapting a McKay-type graph generation algorithm in combination with a degeneracy testing algorithm. This synthesis algorithm is about 13 times faster than the most recent synthesis algorithm reported in the literature. There exist efficient approaches for detection of non-isomorphic mechanisms based on eigenvalues and eigenvectors of the adjacency or related matrices. However these approaches may fail to detect all cases. The reliability of these approaches is established in this work. It is shown, for the first time, that if the number of links is less than 15, the eigenvector approach detects all non-isomorphic mechanisms. A matrix is also proposed whose characteristic polynomial detects non-isomorphic mechanisms with a higher reliability than the adjacency or Laplace matrix. An erroneous assumption often found in structural studies is that the graph of a planar kinematic chain is a planar graph. It is shown that all the existing algorithms for degeneracy testing and mobility type identification, except those by Lee and Yoon, have this error. Further, Lee and Yoon’s algorithms are heuristic in nature and were not rigorously proved. Several structural results and implicit assumptions for planar kinematic chains are proved in this work without relying on the erroneous assumption. These new results provide the mathematical justification for Lee and Yoon’s algorithms, thereby validating the adoption of the Lee and Yoon’s algorithms for practical applications. A polynomial-time algorithm based on combinatorial optimization techniques is proposed for degeneracy testing. This polynomial-time algorithm is the first degeneracy testing algorithm that works for both planar and spatial kinematic mechanisms with different types of joints. STRUCTURAL SYNTHESIS AND ANALYSIS OF PLANAR AND SPATIAL MECHANISMS SATISFYING GRUEBLER’S DEGREES OF FREEDOM EQUATION by Rajesh Pavan Sunkari Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2006 Advisory Committee: Dr. Linda Schmidt, Chair/Advisor Dr. Shapour Azarm Dr. Peter Sandborn Dr. Greg Schultz Dr. Lawrence Washington c Copyright by Rajesh Pavan Sunkari 2006 DEDICATION To my parents, for their guidance, support, love and patience. ii ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Dr. Linda Schmidt, for her guidance and support in both academic and nonacademic aspects during my graduate study. Over the past five years she has played the role of a mentor as well as a parent and a friend. Her motivational speeches helped me succeed at several fronts during my dissertation work. I would like to thank the committee members, Dr. Shapour Azarm, Dr. Pe- ter Sandborn, Dr. Greg Schultz and Dr. Lawrence Washington for their inputs and comments. I would also like to thank Dr. Sergey Novikov and Dr. Niran- jan Ramachandran for their efforts spent explaining several intricate mathematical concepts. I would like to thank the Department of Mechanical Engineering, University of Maryland, College Park, for providing the financial support during my graduate study. I would like to thank Dr. Sami Ainane for his advice on teaching and for giving me the valuable opportunity to be his teaching assistant. I wish to thank my colleagues, past and present, from the Designer Assistance and Tools Labora- tory. I extend my thanks to the faculty and staff of the Department of Mechanical Engineering who over the years have helped me in many ways. I would like to express my deepest gratitude to my parents for their guidance, support, love and patience. Without their support and encouragement throughout iii my life, I would not have achieved this important milestone in my career. I would like to thank Gayatri Cuddalorepatta for patiently proof reading my technical drafts and providing critical commentary. I would also like to specially thank her for providing motivation and helping me stay focused on my dissertation during difficult times. I would like to thank Niranjan Balachandran for patiently explaining some of the subtle concepts of graph theory and also for his insightful comments on the dissertation. I would like express my gratitude to my housemates Joe (Joseph Varghese), Mak (Anil Maddulapalli), Mungi (Kiran Panchamgam), Raju (Rajasekhar Jam- malamadaka), Reddy (Prabhakar Gudla), Sasi (Sasidhar Karavadi) and my good friends Harish Dontineni, Krishna Kaipa, Nagendra Cuddalorepatta and Neeraja Balachander, for providing strong moral support during my graduate study. Regrettably, it is impossible to list all those who deserve credit and I apologize to those that I might have inadvertently left out. iv TABLE OF CONTENTS List of Tables vii List of Figures viii 1 Introduction 1 1.1 Motivation................................. 1 1.2 Research Theme and Objectives . 5 1.3 Methodology ............................... 6 1.4 OrganizationoftheDissertation . 10 2 Background 11 2.1 KinematicChainsandMechanisms . 11 2.2 Representations of Kinematic Chains . 14 2.3 Mobility type of Kinematic Chains . 16 3 Literature Review of Structural Studies 20 3.1 Introduction................................ 20 3.2 StructuralSynthesis ........................... 21 3.2.1 LinkAssortment ......................... 23 3.2.2 Building up from Simpler Chains . 26 3.2.3 OtherMethods .......................... 27 3.3 StructuralAnalysis ............................ 28 3.3.1 StructuralIsomorphism . 32 3.3.2 Degeneracy Testing . 37 4 Structural Synthesis - Adapting a McKay-Type Algorithm 39 4.1 Introduction................................ 39 4.2 Background on Generation of Non-Isomorphic Combinatorial Struc- tures.................................... 40 4.2.1 Homomorphism Principle-Type Algorithms . 41 4.2.2 Read/Faradzev-Type Algorithms . 43 4.2.3 McKay-Type Algorithms . 45 4.3 Approach ................................. 50 4.4 MethodologyandResults. 51 4.5 Discussion................................. 56 4.6 Summary ................................. 57 5 Efficiency and Reliability of Spectral Methods 59 5.1 Introduction................................ 59 5.2 Reliability of the characteristic polynomial method for isomorphism detection.................................. 60 5.3 Reliability of the eigenvector method for isomorphism detection . 62 5.4 Efficiencies of the spectral methods for isomorphism . 67 v 5.5 Implementation .............................. 73 5.6 Discussion................................. 75 5.7 Summary ................................. 77 6 Laplace and Extended Adjacency Matrices for Isomorphism Detection 78 6.1 Introduction................................ 78 6.2 Characteristicpolynomialmethods . 78 6.3 Laplacematrix .............................. 80 6.4 Extendedadjacencymatrix . 82 6.5 Discussion................................. 89 6.6 Summary ................................. 94 7 Reevaluation of degeneracy testing and mobility type identification algo- rithms 95 7.1 Introduction................................ 95 7.2 Evaluation of existing degeneracy testing algorithms . ....... 96 7.2.1 Basic degeneracy testing algorithm . 96 7.2.2 Hwang and Hwang’s degeneracy testing algorithm . 99 7.2.3 Lee and Yoon’s degeneracy testing algorithm . 105 7.2.4 Proposition 7.10 and its proof . 106 7.3 Support for Lee and Yoon’s mobility type identification algorithms . 109 7.4 Future directions for generalizing structural analysis algorithms . 112 7.5 Summary .................................114 8 Polynomial-time Degeneracy Testing Algorithm 115 8.1 Introduction................................115 8.2 Formulation of Degeneracy Testing as an Optimization Problem . 118 8.3 NetworkFlows ..............................122 8.4 MinimumCutsand0-1Optimization . .125 8.5 Solution of Degeneracy Testing . 127 8.6 ResultsandDiscussion . .130 8.7 Summary .................................134 9 Contributions and Future Work 136 9.1 SummaryofResults ...........................136 9.2 Relevance of Current Research . 139 9.3 Contributions ...............................140 9.4 FutureWork................................141
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