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Legged Robots That Balance

The MIT Press Series in Artificial Intelligence Edited by Patrick Henry Winston and Michael Brady Artificial Intelligence: An MIT Perspective, Volume I: Expert Problem Solving, Natural Language Understanding, Intelligent Computer Coaches, Representation and Learning edited by Patrick Henry Winston and Richard Henry Brown, 1979 AritiHcial Intelligence: An MIT Perspective, Volume II: Understanding Vision, Manipulation, Computer Design, Symbol Manipulation edited by Patrick Henry Winston and Richard Henry Brown, 1979 NETL: A System for Representing and Using Real- World Knowledge by Scott Fahlman, 1979 The Interpretation of Visual Motion by Shimon Ullman, 1979 A Theory of Syntactic Recognition for Natural Language by Mitchell P. Marcus, 1980 Turtle Geometry: The Computer as a Medium for Exploring Mathematics by Harold Abelson and Andrea diSessa, 1981 From Images to Surfaces: A Computational Study of the Human Early Visual . System by William Eric Leifur Grimson, 1981 Robot Manipulators: Mathematics, Programming and Control by Richard P. Paul, 1981 Computational Models of Discourse edited by Michael Brady and Robert C. Berwick, 1982 Robot Motion: Planning and Control by Michael Brady, John M. Hollerbach, Timothy Johnson, Tomas Lozano-Perez, and Matthew T. Mason, 1982 In-Depth Understanding: A Computer Model of Integrated Processing for Narrative Comprehension by Michael G. Dyer, 1983 Robotics Research: The First International Symposium edited by Michael Brady and Richard Paul, 1984 Robotics Research: The Second International Symposium edited by Hideo Hanafusa and Hirochika Inoue, 1985 Robot Hands and the Mechanics of Manipulation by Matthew T. Mason and J. Kenneth Salisbury, Jr., 1985 The Acquisition of Syntactic Knowledge by Robert C. Berwick, 1985 The Connection Machine by W. Daniel Hillis, 1985 Legged Robots that Balance by Marc H. Raibert, 1986 Legged Robots That Balance Marc H. Raibert The MIT Press Cambridge, Massachusetts London, England

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the author. To Ivan Sutherland Library of Congress Cataloging-in-Publication Data Raibert, Marc H. Legged robots that balance. (The MIT Press series in artificial intelligence) Bibliography: p. Includes index. 1. Robotics. 2. Artificial intelligence. I. Title. II. Series. TJ211.R35 1986 ISBN 0-262-18117-7 629.8'92 85-23888 The MIT Press, Cambridge, Massachusetts Copyright © 1986 by Marc H. Raibert All rights reserved. Published 1986 Printed in the United States of America

Contents Series Foreword Preface Chapter 1. Introduction Why Study Legged Machines? Dynamics and Balance Improve Mobility Research on Legged Machines Research on Active Balance Introduction to Running Machines Additional Readings . . . . . . Chapter' 2. Hopping on One Leg in the Plane A Planar Machine That Hops on One Leg . . Control of Running Decomposed into Three Parts Hopping Experiments . Improvements and Limitations Summary . . . . . . . . . . . . . Chapter 3. Hopping in Three Dimensions Balance in Three Dimensions . . 3D One-Legged Hopping Machine Control System for 3D One-Legged Machine . . . . . . . . Xl Xlll 1 1 4 6 11 14 26 29 30 37 48 52 55 57 58 62 67

vm Chapter . . . . Hopping Experiments in Three Dimensions Summary . . . . . . . . . . . . . . . . . . . Appendix 3A. Kinematics of 3D One-Legged Machine Chapter 4. Biped and Quadruped Running . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Foot Gaits Virtual Legs . . . . . Quadruped Trotting Experiments Using Virtual Legs Discussion of Quadruped Experiments Summary . . . . . . . . . . . . Appendix 4A. Equations for Virtual Leg Appendix 4B. Kinematics for Four-Legged Machine Chapter 5. Symmetry in Running Mechanics of Symmetry . . . Symmetry in Animal Running Scissor Symmetry . . . . . What Does Symmetry Mean? Summary Appendix 5A. Equations of Motion for Planar Systems Appendix 5B. Proof of Symmetric Leg Motion . . . . . . . . . . . . . Contents ix Chapter 8. Research on Animals and Vehicles Experiments in Animal Locomotion The Development of Useful Legged Robots Running Is Like Juggling . . . . . . . Do Locomotion and Manipulation Have a Common Ground? Bibliography Index 189 189 196 198 200 203 229 71 78 80 83 84 92 96 102 105 107 109 115 117 124 130 134 136 138 140 Chapter 6. Alternatives for Locomotion Control 143 . . . More on the Control of Bouncing 143 An Alternative Three-Part Control . . . 154 Taxonomy of Control for Quadruped Running 160 Summary . . . 170 Appendix 6A. Equations of Motion for Planar One-Legged Model 171 . . . . . . . . . . . . . . . . . . Chapter 7. Tabular Control of Running Background on the Use of Tables for Control A Tabular Method for Control . . Polynomial Approximations to Tabular Data Conclusions . . Appendix 7 A. Performance Index Minimization for Table Appendix 7B. Performance Index Minimization for Polynomial . . . . . . . . . . . . . . . . 173 173 176 180 184 186 187

SERIES FOREWORD Artificial intelligence is the study of intelligence using the ideas and meth ods of computation. Unfortunately, a definition of intelligence seems im possible at the moment because intelligence appears to be an amalgam of so many information-processing and information-representation abilities. Of course psychology, philosophy, linguistics, and related disciplines offer various perspectives and methodologies for studying intelligence. For the most part, however, the theories proposed in these fields are too in complete and too vaguely stated to be realized in computational terms. Something more is needed, even though valuable ideas, relationships, and constraints can be gleaned from traditional studies of what are, after all, impressive existence proofs that intelligence is in fact possible. Artificial intelligence offers a new perspective and a new methodology. Its central goal is to make computers intelligent, both to make them more useful and to understand the principles that make intelligence possible. That intelligent computers will be extremely useful is obvious. The more profound point is that artificial intelligence aims to understand intelligence using the ideas and methods of computation, thus offering a radically new and different basis for theory formation. Most of the people doing arti ficial intelligence believe that these theories will apply to any intelligent information processor, whether biological or solid state. There are side effects that deserve attention, too. Any program that will successfully model even a small part of intelligence will be inherently massive and complex. Consequently, artificial intelligence continually con fronts the limits of computer science technology. The problems encountered have been hard enough and interesting enough to seduce artificial intelli gence people into working on them with enthusiasm. It is natural, then, that there has been a steady flow of ideas from artificial intelligence to computer science, and the flow shows no sign of abating. The purpose of this MIT Press Series in Artificial Intelligence is to provide people in many areas, both professionals and students, with timely, detailed information about what is happening on the frontiers in research centers all over the world. Patrick Henry Winston Michael Brady Preface I first became interested in legged locomotion in 1974 when I was a graduate student at MIT. Berthold Horn and Mitch Weiss had been thinking about legs being like the spokes of a wheel, so they took the rim off of a spoked wheel and rolled it down a ramp to see what it would do. It didn't do very much, but it got me thinking about legs and legged locomotion. One problem with the rimless wheel was that the spokes were too stiff, so they did not stay on the floor long enough to give the hub a chance to turn. I had learned from Emilio Bizzi that the springy characteristics of muscle and tendon played an important role in controlling animal limb movement, and it seemed to me that the spokes would work better if they were somewhat springy too. Another problem with the rimless wheel was the lack of a means for keeping it upright, so it fell over after one or two steps-it needed a mechanism for balance. It was not until 1979 that I had an opportunity to pursue the subject further. I was teaching a robotics course as a visitor at Caltech when Ivan Sutherland encouraged me to start a robotics project in his department. I mentioned to him that a computer controlled pogostick would be a good model for learning about control and balance, and that it might lead to a fundamental understanding of legged locomotion. To my astonishment he took the idea seriously. With money from one of Sutherland's slush funds and substantial help from Elmer Szombathy in the machine shop, I built a preliminary machine that was to hop and balance like a pogostick on a single springy leg (see photograph). From the moment Sutherland and I showed the machine to Craig Fields in 1980-we had carried it to Washington in an old suitcase-it took less than a month for the Defense Advanced Research Projects Agency to sup port the project and to initiate a national research program on legged vehicles. In 1981 I relocated to Pittsburgh and established the Leg Lab oratory at Carnegie-Mellon University where my colleagues and I finished the preliminary machine and went on to build several others. This book

Pref ace xin They have made this work possible through their support, and the support of their institutions, the Defense Advanced Research Projects Agency and the System Development Foundation. Many colleagues and students have helped by reading all or parts of the manuscript. They include: Ben Brown, Nancy Cornelius, Matt Mason, Ken Goldberg, Ivan Sutherland, and especially Michael Chepponis and Jessica Hodgins. Ivor Durham's PLOT program generated the graphs used in the text. Ivor made several additions and changes to PLOT specifically for this book, for which I am particularly grateful. The technical illustrations were drawn by Steve Talkington. Michael Ullner wrote the typesetting macros and Roberto Minio helped with the formatting. Sylvia Brahm contributed in many ways. Finally, I must thank my family, Nancy, Matthew, and Linda, for their loving support throughout this project. Pittsburgh, Pennsylvania October 1985 M.H.R. xii Preface An early one-legged hopping machine (right) as shown to Craig Fields, together with a competitor. This picture was taken in the Spring of 1981 when neither legged system was yet operational. is about those machines and the progress we have made in using them to understand the fascinating problem of legged locomotion. Unfortunately, the behavior of a dynamic legged system is difficult to convey through the printed word and still photograph. To compensate for this limitation, I have compiled a videotape that forms an appendix to this book. It includes material from the running machines described in the text and from several of the computer simulations. The tape runs about fifteen minutes and copies are available from the MIT Press. It is not possible to thank individually all those who have contributed to this book and to the work it reports. I am particularly grateful to the members of the Leg Laboratory who have made contributions of all sorts, especially Ben Brown and Michael Chepponis. Ivan Sutherland helped get this research started and continues to be a rich source of inspiration and good ideas. Stimulating and provocative discussions with Matt Mason cleared up several technical issues. I am indebted to Craig Fields, Clint Kelly, and Charlie Smith who let the idea of a legged technology capture their imaginations and vision.

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