Robot and Multibody Dynamics Abhinandan Jain Robot and Multibody Dynamics Analysis and Algorithms 123 Abhinandan Jain Ph.D. Jet Propulsion Laboratory 4800 Oak Grove Drive Pasadena, California 91109 USA [email protected] ISBN 978-1-4419-7266-8 e-ISBN 978-1-4419-7267-5 DOI 10.1007/978-1-4419-7267-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938443 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) In memory of Guillermo Rodriguez, an exceptional scholar and a gentleman. To my parents, and to my wife, Karen. Preface “It is a profoundly erroneous truism, repeated by copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle – they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.” Alfred North Whitehead Robotics, and its need for real-time control, has expanded research in multibody dynamics beyond modeling and systematic formulations, to computational and con- trol issues. While significant progress has been made in these areas, the mathemat- ical complexity of the system dynamics, and the paucity of analytical concepts and mathematical tools to manage the complexity, present imposing challenges. The body of work known as the spatial operator algebra (SOA) provides such analytical and mathematical techniques that address these challenges. The SOA pro- vides spatial operator mathematical constructs, that concisely describe the system dynamics and enable dynamics analysis in a unified and general setting. That the equations of multibody dynamics can be described by an algebra of spa- tial operators is certainly of mathematical interest. However, the SOA also provides the means to manipulate at a high level of abstraction the equations describing multi- body behavior. The significance of this goes beyond the mathematics and is useful in a practical sense. At any stage of the manipulation of equations, spatially recur- sive algorithms to implement the operator expressions can be readily obtained by inspection. Therefore, the transition from abstract operator mathematics to practical implementation is straightforward, and often requires only a simple mental exercise. The focus of this book is on providing a comprehensive and coherent description of the SOA methodology and techniques, and of the interplay between mathematical analysis, structural properties, and computational algorithms. The material in this book serves as a point of departure for further exploration of topics in multibody dynamics. Thus, while the text is written in a style that is acces- sible to a newcomer, it would be best appreciated by practitioners and researchers in the field. vii viii Preface The book is organized into three parts. Part 1, consisting of Chaps. 1–7, is a study of the dynamics of rigid-link, serial-chain systems. These are the simplest examples of multi-link systems. They serve to introduce the notion of spatial operators, and to develop the concepts of mapping operator expressions into low-order computa- tional algorithms. In Part 2, consisting of Chaps. 8–13, the operator techniques are generalized to include the broader class of multibody systems. Graph theory ideas are used to describe the structure of the operators at a level of abstraction that makes them independent of the size, the topological structure, and the detailed properties of the linkages. Part 3 is an exploration of the application of SOA techniques to advanced topics. These include methods for partitioning and sub-structuring multi- body models, constraint embedding for transforming closed-chain systems into tree systems, the dynamics of under-actuated systems and free-flying systems, derivation of analytical sensitivity expressions, and diagonalized dynamics formulations. The text is interspersed with exercises that further explore a topic or present a related idea. All solutions are provided at the end of the book. Key results are summarized in lemmas with self-contained assumptions and conclusions. Rarely is there a single approach or solution that works for the variety of prob- lems encountered when working with robot and multibody system dynamics. The emphasis of this book therefore is on deepening the theoretical and mathematical understanding of the operator tools, and on the rich tapestry of inter-relationships among them. A single topic is thus sometimes explored from different angles, and themes and concepts cutting across different contexts are highlighted. Acknowledgments My introduction to the fields of robotics and multibody dynamics began when I joined the Jet Propulsion Laboratory (JPL). Guillermo Rodriguez had recently dis- covered the rich parallels between the mathematical theory underlying optimal fil- tering and estimation theory, and the structure of the dynamical properties of robotic systems that formed the basis of the SOA. I had the good fortune of joining in this pioneering research. and, until his untimely passing, had the incredible personal experience of collaborating and working with Guillermo. Over the years, I have benefited in no small measure from the continuous inter- actions and support from many exceptional colleagues at JPL. Ken Kreutz-Delgado introduced me to the early developments of the SOA methodology. Guy Man’s vi- sion helped initiate the “real-world” application of the SOA techniques to solve challenging dynamics modeling problems for NASA’s space missions. I am grateful to my colleagues at JPL’s DARTS Lab, including Bob Balaram, Jonathan Cameron, Christopher Lim, Marc Pomerantz, Hari Nayar, Jeff Biesiadecki and many others who have continued to advance the methodology and make possible its application to challenges in space robotics. It has also been a pleasure to work with Nagarajan Vaidehi in the non-engineering application of the SOA ideas to the molecular dy- namics arena. I would like to thank John Wen and Rich Volpe for their persuasive Preface ix encouragement to work on and to publish this book. Last, but not least, I would like to thank my family, Karen, Aarti, and Nikhil, who valiantly helped with figures, proof-reading, and stylistic suggestions for the text, and for their incredible patience and forbearance through the months of its preparation. Contents Part I Serial-Chain Dynamics 1 Spatial Vectors ................................................. 3 1.1 Homogeneous Transforms ................................... 3 1.2 Differentiation of Vectors .................................... 5 1.2.1 Vector Derivatives in Rotating Frames ................... 5 1.2.2 Rigid Body Vector Derivatives ......................... 6 1.3 Spatial Vectors ............................................. 8 1.3.1 Six-Dimensional Spatial Notation ...................... 9 1.3.2 The Cross-Product for Spatial Vectors ................... 9 1.4 The Rigid Body Transformation Matrix φ(x,y) ................. 11 1.4.1 Spatial Velocity Transformations ....................... 12 1.4.2 Properties of φ(·) .................................... 12 1.5 Spatial Forces ............................................. 15 2 Single Rigid Body Dynamics..................................... 17 2.1 Spatial Inertia and Momentum of a Rigid Body ................. 17 2.1.1 The Spatial Inertia ................................... 17 2.1.2 The Parallel-Axis Theorem for Spatial Inertias ............ 20 2.1.3 Spatial Inertia of a Composite Assemblage of Rigid Bodies .21 2.1.4 The Spatial Momentum of a Rigid Body ................. 21 2.2 Motion Coordinates ........................................ 22 2.2.1 Generalized Coordinates and Velocities .................. 23 2.2.2 Generalized Forces ................................... 24 2.2.3 Generalized Accelerations ............................. 24 2.3 Equations of Motion with Inertial Frame Derivatives ............. 25 I 2.3.1 Equations of Motion with βI = V(C) .................. 25 I 2.3.2 Equations of Motion with βI = V(z) .................. 27 2.4 Equations of Motion with Body Frame Derivatives............... 29 2.5 Equations of Motion with an Inertially Fixed Velocity Reference Frame.................................................... 31 2.6 Comparison of the Different Dynamics Formulations ............. 33 xi xii Contents 3 Serial-Chain Kinematics ........................................ 35 3.1 Serial-Chain Model ......................................... 35 3.2 Hinge Kinematics .......................................... 36 3.2.1 Hinge Generalized Coordinates