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To Download the PDF File NOTE TO USERS This reproduction is the best copy available. UMJ Algebraic Screw Pairs by James D. Robinson A Dissertation submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Ottawa-Carleton Institute for Mechanical and Aerospace Engineering Department of Mechanical and Aerospace Engineering Carleton University Ottawa, Ontario, Canada May 2012 Copyright © 2012 - James D. Robinson Library and Archives Bibliotheque et Canada Archives Canada Published Heritage Direction du 1+1 Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-93679-5 Our file Notre reference ISBN: 978-0-494-93679-5 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distrbute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. Canada Abstract The algebraic screw pair, or A-pair, represents a novel class of kinematic pair that al­ gebraically couples relative curvilinear translation with rotation about a single reference translation curve. The A-pair is a generalization of the well known helical screw pair, or H-pair, that linearly couples rotation with translation along the axis of rotation where the pitch of the screw is the linear constant of proportionality. The particular kind of A-pair examined in this thesis generates a sinusoidal coupling of relative rotation and linear trans­ lation along the axis of rotation between two adjacent links in a kinematic chain. The novelty of the A-pair requires that a full kinematic and dynamic analysis be performed prior to the implementation of the A-pair into useful kinematic chains (A-chains) and the determination of the advantages and disadvantages of A-pairs when compared to other kine­ matic constraints, in particular revolute pairs (R-pairs). This dissertation establishas the full kinematic and dynamic analysis of A-pairs and A-chains. The existing position level kinematic analysis of A-chains is revised for application to general A-chains. This includes revising the application of the Denavit-Hartenburg param­ eter convention to general A-chains, adapting the direct kinematics to this new definition, and determining the constraint varieties of 2A-chains using kinematic mapping techniques for use in the inverse kinematics algorithm. The joint limits that result from the colli­ sions between the legs of the A-pair are found and a novel algorithm for approximating the reachable workspace of serial manipulators is used to obtain the workspaces of 2A- and 4A-chains. The velocity level kinematics are addressed by obtaining the Jacobian matrices for nA-chains by adapting standard methods to account for the coupled A-pair motion. The dynamic analysis provides dynamic equations of motion for A-chains. The single To Jessie and Olivia, my amazing family, for their love and support. Acknowledgments I have many people I would like to thank for their help and support throughout my PhD studies and throughout my entire student career at Carleton. To begin with, John Hayes has taken me under his wing since my fourth year of undergrad and has guided my throughout the course of my graduate studies. He has provided much encouragement and support on and off the ice and his proficient editing skills and financial support have helped send me to many exotic conferences (in Montreal). Also, his financial support (and that of NSERC) have put the “professional” in professional student for many years. I must also acknowledge Owen Roberts who did an amazing job with the construction of the prototype 4A-chain. I owe much thanks to the people at the Institute for Geometric und CAD at Leopold- Franzens-Universitat Innsbruck for their help and support through both my master’s and PhD. Manfred Husty, Martin Pfumer, and Hans-Peter Shrocker have helped me understand the math and theory behind the kinematics and Dominic Walter has been amazingly helpful in the area of constraint varieties. The secretaries in the MAAE office have been very helpful over the years. Nancy, Christie, Irene, and Marlene (who will be missed, go Habs go!). Also providing invaluable support in the areas of sounding boards and procrastination are my fellow gad students, including Sean, Ali, Heather, Calvin, Wes, Derek, Peter, Alanna, and many others. Thanks to my parents, Gwen and Andy McDougall, and Don and Karen Robinson for their support over my many years in school. And also to my in-laws, Brian Simon and Amy Simon, who have welcomed me into their family, and are certain to be among the only ones who call me “Doctor” when I am done. I appreciate everything my entire family has done for me, but I must certainly acknowledge my brother Tom Robinson who has helped me through a lot of Maple problems, even as he is working on his owii PhD. Last, but most importantly I want to thank my wonderful wife Jessie and our beautiful baby girl Olivia. Jessie has been so patient as I strive to become Dr. Dad, and Olivia has been the best distraction in the whole world. Table of Contents Abstract ii Acknowledgments v Table of Contents vii List of Tables xi List of Figures xii 1 Introduction 1 1.1 Organization of this Dissertation .......................................................................... 3 1.2 Statement of Originality and Contributions ......................................... 4 2 Background Theory and Literature Review 6 2.1 Representation of Displacements ......................................................................... 6 2.1.1 Homogeneous Coordinates ......................................................................... 6 2.1.2 Displacements in Euclidean Space, E% ................................................... 8 2.1.3 G r o u p s ......................................................................................................... 10 2.1.4 Quaternions and Representation of Rotations in E3 ........................... 10 2.1.5 Dual Quaternions and Representing Displacements in E3 ................ 12 2.1.6 Kinematic M a p p in g .................................................................................. 16 2.1.7 Study Parameters ....................................................................................... 16 2.1.8 Effect of Transformations in E$ on Points in P 7 ..................................... 18 vii 2.2 Manipulator Basics ........................................... ................. ..................................... 21 2.2.1 Classes of M anipulators .............................................................................. 22 2.2.2 Denavit-Hartenburg Parameters ............................................................. 24 2.2.3 Direct Kinematics of Serial M an ip u lato rs.............................................. 27 2.2.4 Inverse Kinematics of Serial Manipulators ....................................... 28 2.2.5 Manipulator Workspace A nalysis ............................................................. 30 2.3 Constraint Varieties ................................................................................................ 35 2.3.1 A New Method for Obtaining Constraint Varieties of Serial Chains . 36 2.4 The Inverse Kinematics of General 6R-Chains Using Kinematic Mapping . 39 2.5 The Algebraic Screw P a i r ....................................................................................... 41 2.5.1 Overview of Griffis-Duffy Platforms ....................................................... 42 2.5.2 Self-motions of G D P s ................................................................................. 44 2.5.3 The Griffis-Duffy Platform as a Kinematic P a i r .................................... 48 2.5.4 Preliminary Assignment of DH-Parameters to A-Chains ....... 50 2.5.5 The Direct Kinematics of A-Chains ....................................................... 50 2.5.6 The Inverse Kinematics of A-chains ....................................................... 53 2.6 Obtaining the Jacobian Matrix of
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