(ebook pdf) advances in industrial control snake robots modelling mechatronics and control 1st editi

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Advances in Industrial Control

For further volumes: www.springer.com/series/1412

Pål Liljebäck r Kristin Y. Pettersen r Øyvind Stavdahl r Jan Tommy Gravdahl

Snake Robots Modelling, Mechatronics, and Control

Pål Liljebäck Applied Cybernetics SINTEF ICT Trondheim, Norway and Department of Engineering Cybernetics Norwegian University of Science & Technology Trondheim, Norway Kristin Y. Pettersen Department of Engineering Cybernetics Norwegian University of Science & Technology Trondheim, Norway

Øyvind Stavdahl Department of Engineering Cybernetics Norwegian University of Science & Technology Trondheim, Norway Jan Tommy Gravdahl Department of Engineering Cybernetics Norwegian University of Science & Technology Trondheim, Norway

The following images in the book are used under license from Shutterstock.com: Image of snake on page 1: Copyright Angel Simon, 2011. Image in Fig. 1.2b: Copyright RedTC, 2011. Image in Fig. 1.3: Copyright Srdjan Draskovic, 2011. Image in Fig. 1.4: Copyright photoBeard, 2011. Image of snake on page 287: Copyright Steve Bower, 2011. The following images in the book are used under license from Dreamstime.com: Images in Fig. 1.5: Copyright Isselee, 2011. ISSN 1430-9491 ISSN 2193-1577 (electronic) Advances in Industrial Control ISBN 978-1-4471-2995-0 ISBN 978-1-4471-2996-7 (eBook) DOI 10.1007/978-1-4471-2996-7 Springer London Heidelberg New York Dordrecht Library of Congress Control Number: 2012938883 © Springer-Verlag London 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To our families

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies, . . . , new challenges. Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. Over the years the Advances in Industrial Control series has been very fortunate in publishing monographs that were often seminal for the development of new areas in control systems theory and industrial technology. These monographs were often written by young researchers making their way in the industrial control field or were a report of a substantial research project that was now ready for holistic presentation and dissemination. For a monograph series that spans two decades, it is actually quite easy to find examples of this type of motivational text. From the early years of the series, Iterative Learning Control for Deterministic Systems by Kevin L. Moore (ISBN 978-3-540-19707-2, 1992) and Autotuning of PID Controllers by Cheng-Ching Yu (ISBN 978-3-540-76250-8, 1999) are good examples. In more recent years we can cite Control of Fuel Cell Power Systems by Jay T. Pukrushpan, Anna G. Stefanopoulou, and Huei Peng (ISBN 978-1-85233-816-9, 2004), Predictive Functional Control by Jacques Richalet and Donal O’Donovan (ISBN 9781-84882-492-8, 2009) and finally Internet-Based Control Systems by Shuang-Hua Yang (ISBN 978-1-84996-358-9, 2011) as typical Advances in Industrial Control monographs that are studied as key texts for their respective topics. Clearly Snake Robots by Pål Liljebäck, Kristin Y. Pettersen, Øyvind Stavdahl, and Jan Tommy Gravdahl is going to be a much read, studied, and cited monograph in this particular field of robot development. After a truly fascinating introductory chapter that examines among other topics, biological snake motion, the monograph is structured into two parts. Part I investigates and reports on modelling, technology, and control for snake robot locomotion in a planar (flat) environment (Chaps. 2–8). vii

viii

Series Editors’ Foreword

Part II moves on to snake locomotion in a cluttered environment with stationary objects (Chaps. 9–13). This group of chapters introduces and explores the concept of “obstacle-aided” locomotion. So much of mobile robot technology is concerned with obstacle avoidance, so it is interesting to see a robot application that exploits the contact with objects (obstacles) in the environment to aid locomotion. The monograph closes with a concluding chapter, three short technical appendices, a useful Glossary of technical terms, and an exhaustive Index. The authors have succeeded in writing a well-structured text that is both a scientific and an engineering monograph. The structure and contents of the monograph can be accessed in several different ways. For example, the monograph presents mathematical models that elucidate snake motion per se; this describes important fundamental scientific principles in the field. Alternatively, the monograph can be used as a source for up-to-date survey and review material on snake-robotic engineering and technology; apart from the thorough historical review in Chap. 1, each subsequent chapter opens with a section that creates the context and reviews the past literature relating to the work to be presented by the authors. One of the attractive features of the monograph is the way in which the authors use the chapter sequence to work through increasingly complex issues in the understanding, control and technology of snake robots. Such careful attention to structure allows the expert researcher and the researcher new to the field rapidly to assess the importance of the material presented and its relation to past developments. Finally, the monograph presents the authors’ own research and development in the field. This research covers the full spectrum of mathematical modelling, control design, simulation studies and fascinating experimental demonstrator prototypes. Closing the monograph is a chapter on the future research and technological challenges for snake robot locomotion (Chap. 14). The series Editors have no doubt that the control and robotics community will find much of interest in this monograph. The monograph’s progress through the historical record for the field, the proofs and descriptions of fundamental snake robot principles and the practical demonstrations using robot prototypes will ensure this new entry to the Advances in Industrial Control series becomes a key reference and source text for snake-robot locomotion research and development. Industrial Control Centre Glasgow Scotland, UK

M.J. Grimble M.A. Johnson

Preface

The purpose of this book is to present theoretical and practical topics related to snake robots. Snake robots are robotic mechanisms designed to move like biological snakes. The advantage of such mechanisms is their ability to move and operate in challenging environments where human presence is unwanted or impossible. Future applications of these mechanisms include search and rescue operations, inspection and maintenance in industrial process plants, and subsea operations. Research on snake robots has been conducted for several decades. For instance, the world’s first snake robot was developed in Japan already in 1972. There are, however, still many theoretical and practical aspects of snake robot locomotion which have not yet been addressed in the snake robot literature. Current literature is characterised by numerous different approaches to modelling, development, and control of these mechanisms, but a unified theoretical foundation of snake robots has not yet been established. In this book, we attempt to target these limitations of current literature on snake robots. The main goal of the book is to contribute to the mathematical foundation of the control theory of snake robots, and also stimulate and support future research on these fascinating mechanisms. To this end, the book is a complete treatment of snake robotics, with topics ranging from mathematical modelling techniques, mechatronic design and implementation, and control design strategies. In particular, several new approaches to modelling snake robot locomotion are presented. Moreover, numerous properties of snake robot dynamics are derived using nonlinear system analysis tools, and several new control strategies for snake robots are proposed. The book also describes the development of two snake robots that are employed to experimentally validate many of the theoretical results. Whereas previous literature has mainly focused on flat surface locomotion, a distinct feature of the book is the strong focus on locomotion in uneven and cluttered environments. The organisation of the book is detailed in Sect. 1.5. Although the results presented in this book are new and based on recent conference and journal publications, they are presented at an initial level which is accessible to audiences with a standard undergraduate background in control theory or mechatronics. The book is written in a clear and easily understandable manner with numerous figures and pictures which help illustrate and visualise the material. The target audience of this book includes academic researchers and graduate students with an interest in snake robots or underactuated systems in general. The book may ix

x

Preface

also be used for self-study or as a reference by engineers and applied mathematicians, and by anyone who would like to find out more about the exciting field of snake robotics. We believe the book will be particularly useful to new researchers taking on a topic related to snake robotics since the book provides an extensive overview of the snake robot literature and also represents a suitable starting point for research in this area. We are indebted to a number of people who have been integral to the completion of this book. We express our sincere gratitude to Professor Scott David Kelly (University of North Carolina at Charlotte), Professor Shugen Ma (Ritsumeikan University), and Professor Ole Morten Aamo (Department of Engineering Cybernetics at NTNU) for their feedback to the material in this book in conjunction with their participation in the doctoral dissertation of Pål Liljebäck. Furthermore, we gratefully acknowledge all the support we have received from our friends and colleagues at the Department of Engineering Cybernetics at NTNU. In particular, we thank Idar Haugstuen for our cooperation in conjunction with his M.Sc. project on snake robots in 2009/2010. Moreover, we thank Christian Holden for providing useful feedback to the material in this book, and we thank Alexey Pavlov for our talks and his many useful suggestions regarding our research. For their untiring help and efforts with the experimental systems considered in this book, we thank Terje Haugen, Per Inge Snildal, and Glenn Angel at the mechanical workshop of the department. We also thank Stefano Bertelli for his positive spirit and for his help with documenting the experimental results. We owe many thanks to our friends and colleagues at SINTEF Applied Cybernetics for their support and for contributing to a positive and stimulating work environment. In particular, we thank the Research Director, Sture Holmstrøm, for his enthusiasm and his willingness to financially support research on snake robots. We are thankful to Aksel A. Transeth, Sigurd Fjerdingen, and Erik Kyrkjebø for their positivity and for our many interesting discussions related to snake robots over the last years. A special recognition goes to Aksel A. Transeth for contributing to the research underlying this book with his knowledge and expertise of snake robots. We thank Geir Mathisen, Espen Helle, and Knut Vidar Skjersli for their work on circuit boards and software for the experimental systems considered in the book. We are grateful to Anders Beitnes for conceiving the idea of a self-propelled fire hose, which initiated research on snake robots at SINTEF and NTNU. We also thank Wheeko and Kulko for their outstanding and obedient performance during experiments, and for never complaining about late work hours. Finally, we express our deepest gratitude to the Norwegian University of Science and Technology (NTNU) and SINTEF for providing the resources and environment that made it possible to write this book, and to the Research Council of Norway for supporting our research on snake robots. Trondheim, Norway

Pål Liljebäck Kristin Y. Pettersen Øyvind Stavdahl Jan Tommy Gravdahl

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . 1.2 Biological Snakes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Anatomy of Snakes . . . . . . . . . . . . . . . . . . . 1.2.2 The Locomotion of Snakes . . . . . . . . . . . . . . . . . 1.3 Previous Work on Modelling, Mechatronics, and Control of Snake Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Previous Work on Modelling and Analysis of Snake Robots 1.3.2 Previous Work on Implementation of Physical Snake Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Previous Work on Control of Snake Robots . . . . . . . . . 1.4 The Scope of This Book . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 An Analytical Approach . . . . . . . . . . . . . . . . . . . 1.4.2 Snake Robots Without a Fixed Base . . . . . . . . . . . . . 1.4.3 A Planar Perspective . . . . . . . . . . . . . . . . . . . . . 1.4.4 Locomotion Without Sideslip Constraints . . . . . . . . . . 1.4.5 Motion Based on Lateral Undulation . . . . . . . . . . . . 1.5 An Outline of This Book . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Outline of Part I—Snake Robot Locomotion on Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Outline of Part II—Snake Robot Locomotion in Cluttered Environments . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Publications Underlying This Book . . . . . . . . . . . . . . . . .

Part I 2

1 1 5 5 7 10 10 16 22 27 27 27 27 28 28 28 29 32 34

Snake Robot Locomotion on Flat Surfaces

A Complex Model of Snake Robot Locomotion on Planar Surfaces . 2.1 The Relation Between This Chapter and Previous Literature . . . . 2.2 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Parameters of the Snake Robot . . . . . . . . . . . . . . . . . 2.4 The Kinematics of the Snake Robot . . . . . . . . . . . . . . . . .

39 40 40 40 42 xi

xii

3

4

5

Contents

2.5 The Ground Friction Models . . . . . . . . . . . . . . . . . . . . 2.5.1 The Friction Models and Their Role in This Book . . . . . 2.5.2 A Coulomb Friction Model . . . . . . . . . . . . . . . . . 2.5.3 A Viscous Friction Model . . . . . . . . . . . . . . . . . . 2.6 The Dynamics of the Snake Robot . . . . . . . . . . . . . . . . . 2.7 Separating Actuated and Unactuated Dynamics . . . . . . . . . . . 2.8 Partial Feedback Linearisation of the Model . . . . . . . . . . . . 2.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 47 48 50 52 54

Development of a Mechanical Snake Robot for Motion Across Planar Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Relation Between This Chapter and Previous Literature . . . . 3.2 The Joint Actuation Mechanism . . . . . . . . . . . . . . . . . . . 3.3 The Passive Wheels . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Power and Control System . . . . . . . . . . . . . . . . . . . 3.5 The Experimental Setup of the Snake Robot . . . . . . . . . . . . 3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 58 59 59 61

Analysis and Synthesis of Snake Robot Locomotion . . . . . . . . . . 4.1 The Relation Between This Chapter and Previous Literature . . . . 4.2 Introduction to Nonlinear Controllability Analysis . . . . . . . . . 4.3 Stabilisability Properties of Planar Snake Robots . . . . . . . . . . 4.4 Controllability Analysis of Planar Snake Robots . . . . . . . . . . 4.4.1 Controllability with Isotropic Viscous Friction . . . . . . . 4.4.2 Controllability with Anisotropic Viscous Friction . . . . . 4.5 Analysis of Propulsive Forces During Snake Locomotion . . . . . 4.6 Synthesis of Propulsive Motion for the Snake Robot . . . . . . . . 4.7 The Gait Pattern Lateral Undulation . . . . . . . . . . . . . . . . . 4.8 The Control System of the Joints . . . . . . . . . . . . . . . . . . 4.8.1 A Simple Joint Controller . . . . . . . . . . . . . . . . . . 4.8.2 An Exponentially Stable Joint Controller . . . . . . . . . . 4.9 Analysis of Turning Motion During Lateral Undulation . . . . . . 4.10 Analysis of Relative Motion Between Consecutive Links During Lateral Undulation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . Path Following Control and Analysis of Snake Robots Based on the Poincaré Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Relation Between This Chapter and Previous Literature . . . . 5.2 Introduction to Poincaré Maps . . . . . . . . . . . . . . . . . . . . 5.2.1 General Description of Poincaré Maps . . . . . . . . . . . 5.2.2 Practical Application of Poincaré Maps . . . . . . . . . . . 5.3 Straight Line Path Following Control of Snake Robots . . . . . . . 5.3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Straight Line Path Following Controller . . . . . . . . 5.4 Stability Analysis of the Path Following Controller Based on the Poincaré Map . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 65 67 68 69 69 74 76 80 81 81 82 82 85 86 89 90 91 91 93 94 94 95 96

Contents

xiii

5.4.1

Converting the Snake Robot Model to a Time-Periodic Autonomous System . . . . . . . . . . . . . . . . . . . . . 97 5.4.2 Specification of the Poincaré Section for the Snake Robot . 97 5.4.3 Stability Analysis of the Poincaré Map . . . . . . . . . . . 98 5.5 Simulation Study: The Performance of the Path Following Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6

A Simplified Model of Snake Robot Locomotion on Planar Surfaces 103 6.1 The Relation Between This Chapter and Previous Literature . . . . 104 6.2 Overview of the Modelling Approach . . . . . . . . . . . . . . . . 104 6.3 The Kinematics of the Snake Robot . . . . . . . . . . . . . . . . . 107 6.4 The Ground Friction Model . . . . . . . . . . . . . . . . . . . . . 109 6.5 The Dynamics of the Snake Robot . . . . . . . . . . . . . . . . . 112 6.5.1 The Translational Dynamics of the Snake Robot . . . . . . 112 6.5.2 The Rotational Dynamics of the Snake Robot . . . . . . . 114 6.6 The Complete Simplified Model of the Snake Robot . . . . . . . . 115 6.7 Discussion of the Simplified Model . . . . . . . . . . . . . . . . . 116 6.7.1 Applications of the Simplified Model . . . . . . . . . . . . 116 6.7.2 Accuracy Issues of the Simplified Kinematics . . . . . . . 116 6.7.3 Accuracy Issues of the Ground Friction Model . . . . . . . 117 6.7.4 Accuracy Issues of the Rotational Dynamics . . . . . . . . 118 6.8 Stabilisability Analysis of the Simplified Model . . . . . . . . . . 118 6.9 Controllability Analysis of the Simplified Model . . . . . . . . . . 119 6.10 Simulation Study: Comparison Between the Complex and the Simplified Model . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.10.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . 122 6.10.2 Relationship Between the Joint Coordinates in the Complex and Simplified Models . . . . . . . . . . . . . . 123 6.10.3 Comparison of Straight Motion . . . . . . . . . . . . . . . 123 6.10.4 Comparison of Turning Motion . . . . . . . . . . . . . . . 126 6.11 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7

Analysis of Snake Robot Locomotion Based on Averaging Theory . . 131 7.1 The Relation Between This Chapter and Previous Literature . . . . 132 7.2 Introduction to Averaging Theory . . . . . . . . . . . . . . . . . . 132 7.3 The Velocity Dynamics During Lateral Undulation . . . . . . . . . 133 7.4 The Averaged Velocity Dynamics During Lateral Undulation . . . 135 7.5 The Steady-State Behaviour of the Velocity Dynamics During Lateral Undulation . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.6 Relationships Between the Gait Parameters and the Forward Velocity During Lateral Undulation . . . . . . . . . . . . . . . . . 138 7.7 Simulation Study: Comparison Between the Original and the Averaged Velocity Dynamics . . . . . . . . . . . . . . . . . . . . 139 7.7.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . 139 7.7.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 140

xiv

Contents

7.8 Simulation Study: Investigation of the Relationships Between Gait Parameters and Forward Velocity . . . . . . . . . . . . . . . 142 7.8.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . 142 7.8.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 143 7.9 Experimental Study: Investigation of the Relationships Between Gait Parameters and Forward Velocity . . . . . . . . . . . . . . . 146 7.9.1 Layout of the Experiment . . . . . . . . . . . . . . . . . . 147 7.9.2 Experimental Results . . . . . . . . . . . . . . . . . . . . 149 7.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8

Path Following Control of Snake Robots Through a Cascaded Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.1 The Relation Between This Chapter and Previous Literature . . . . 154 8.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . 154 8.3 Straight Line Path Following Control of Snake Robots . . . . . . . 157 8.3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . 157 8.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.3.3 Model Transformation . . . . . . . . . . . . . . . . . . . . 158 8.3.4 The Straight Line Path Following Controller . . . . . . . . 160 8.3.5 The Stability Properties of the Path Following Controller . 163 8.3.6 Proof of Theorem 8.2 . . . . . . . . . . . . . . . . . . . . 164 8.4 Path Following Control of Snake Robots Along Curved Paths . . . 167 8.4.1 Comments on the Curved Path Following Controller . . . . 167 8.4.2 The Curved Path Following Controller . . . . . . . . . . . 168 8.5 Waypoint Guidance Control of Snake Robots . . . . . . . . . . . . 169 8.5.1 Description of the Approach . . . . . . . . . . . . . . . . . 169 8.5.2 The Waypoint Guidance Strategy . . . . . . . . . . . . . . 170 8.6 Simulation Study: The Performance of the Straight Line Path Following Controller . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.6.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . 171 8.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 172 8.7 Experimental Study: The Performance of the Straight Line Path Following Controller . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.7.1 Implementation Issues . . . . . . . . . . . . . . . . . . . . 173 8.7.2 Implementation of the Path Following Controller of the Physical Snake Robot . . . . . . . . . . . . . . . . . . . . 174 8.7.3 Experimental Results . . . . . . . . . . . . . . . . . . . . 175 8.8 Simulation Study: The Performance of the Waypoint Guidance Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.8.1 Implementation of the Guidance Strategy with the Simplified Model . . . . . . . . . . . . . . . . . . . . . . 182 8.8.2 Implementation of the Guidance Strategy with the Complex Model . . . . . . . . . . . . . . . . . . . . . . . 182 8.8.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 183 8.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Contents

Part II 9

xv

Snake Robot Locomotion in Cluttered Environments

Introduction to Part II . . . . . . . . . . . . . . . . . . . . . . . . . . 189

10 A Hybrid Model of Snake Robot Locomotion in Cluttered Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.1 The Relation Between This Chapter and Previous Literature . . . 194 10.2 Hybrid Dynamical Systems and Complementarity Systems . . . . 195 10.2.1 Modelling of Hybrid Dynamical Systems . . . . . . . . . 195 10.2.2 Complementarity Systems . . . . . . . . . . . . . . . . . 196 10.3 The Dynamics of the Snake Robot Without Obstacles . . . . . . . 197 10.3.1 The Ground Friction Model . . . . . . . . . . . . . . . . 198 10.3.2 The Equations of Motion Without Obstacles . . . . . . . 199 10.4 Overview of the Contact Modelling Approach . . . . . . . . . . . 200 10.5 Detection of Obstacle Impacts and Detachments . . . . . . . . . 203 10.6 The Continuous Dynamics of the Snake Robot During Constrained Motion . . . . . . . . . . . . . . . . . . . . . . . . 204 10.6.1 The Unilateral Constraints from the Obstacles . . . . . . 205 10.6.2 The Constrained Dynamics of the Snake Robot Without Obstacle Friction . . . . . . . . . . . . . . . . . . . . . . 206 10.6.3 The Constrained Dynamics of the Snake Robot with Obstacle Friction . . . . . . . . . . . . . . . . . . . . . . 209 10.7 The Discontinuous Dynamics of the Snake Robot During Obstacle Impacts and Detachments . . . . . . . . . . . . . . . . 211 10.7.1 The Discontinuous Dynamics of the Snake Robot During Obstacle Impacts . . . . . . . . . . . . . . . . . . . . . . 211 10.7.2 The Discontinuous Dynamics of the Snake Robot During Obstacle Detachments . . . . . . . . . . . . . . . . . . . 212 10.8 The Complete Hybrid Model of the Snake Robot in an Obstacle Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.8.1 The Jump Set . . . . . . . . . . . . . . . . . . . . . . . . 213 10.8.2 The Jump Map . . . . . . . . . . . . . . . . . . . . . . . 214 10.8.3 The Flow Set . . . . . . . . . . . . . . . . . . . . . . . . 214 10.8.4 The Flow Map . . . . . . . . . . . . . . . . . . . . . . . 215 10.8.5 Summary of the Complete Hybrid Plant . . . . . . . . . . 215 10.9 Simulation Study: Comparison of the Hybrid Model with Previous Experimental and Simulation Results . . . . . . . . . . 215 10.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . 216 11 Development of a Mechanical Snake Robot for Obstacle-Aided Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.1 The Relation Between This Chapter and Previous Literature . . . 221 11.2 Overview of the Snake Robot Design . . . . . . . . . . . . . . . 222 11.3 The Exterior Gliding Surface . . . . . . . . . . . . . . . . . . . . 223 11.4 The Contact Force Measurement System . . . . . . . . . . . . . 224 11.4.1 Assumptions Underlying the Sensor System . . . . . . . 224

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11.5

11.6

11.7 11.8 11.9

11.4.2 The Sensor System Setup . . . . . . . . . . . . . . . . . 225 11.4.3 Calculation of Contact Forces . . . . . . . . . . . . . . . 227 The Power and Control System . . . . . . . . . . . . . . . . . . 228 11.5.1 The Power System . . . . . . . . . . . . . . . . . . . . . 228 11.5.2 The Control System . . . . . . . . . . . . . . . . . . . . 230 The Performance of the Snake Robot . . . . . . . . . . . . . . . 231 11.6.1 Experimental Validation of the Contact Force Measurement System . . . . . . . . . . . . . . . . . . . . 231 11.6.2 Demonstration of Motion Patterns . . . . . . . . . . . . . 232 The Experimental Setup of the Snake Robot . . . . . . . . . . . . 233 An Alternative Approach for Measuring External Contact Forces 235 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . 237

12 Hybrid Control of Obstacle-Aided Locomotion . . . . . . . . . . . . 239 12.1 The Relation Between This Chapter and Previous Literature . . . 240 12.2 Preliminary Note on Hybrid Controllers . . . . . . . . . . . . . . 241 12.3 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . 242 12.4 Notation and Basic Assumptions . . . . . . . . . . . . . . . . . . 242 12.5 The Hybrid Controller for Obstacle-Aided Locomotion . . . . . . 244 12.5.1 The Leader-Follower Scheme . . . . . . . . . . . . . . . 244 12.5.2 The Jam Detection Scheme . . . . . . . . . . . . . . . . 246 12.5.3 The Jam Resolution Scheme . . . . . . . . . . . . . . . . 246 12.5.4 The Joint Angle Controller . . . . . . . . . . . . . . . . . 248 12.5.5 The Complete Hybrid Controller . . . . . . . . . . . . . . 248 12.6 Summary of the Closed-Loop System . . . . . . . . . . . . . . . 251 12.7 Simulation Study: The Performance of the Hybrid Controller . . . 252 12.7.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . 252 12.7.2 Attempting Lateral Undulation in Open-Loop in a Structured Obstacle Environment . . . . . . . . . . . . . 253 12.7.3 Hybrid Controller in an Obstacle Environment . . . . . . 253 12.8 Experimental Study: The Performance of the Hybrid Controller . 255 12.8.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . 256 12.8.2 Experimental Results . . . . . . . . . . . . . . . . . . . . 256 12.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . 263 13 Path Following Control of Snake Robots in Cluttered Environments 265 13.1 The Relation Between This Chapter and Previous Literature . . . 266 13.2 A Controller Framework for Snake Robot Locomotion . . . . . . 266 13.3 Straight Line Path Following Control in Cluttered Environments . 268 13.3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . 268 13.3.2 Notation and Basic Assumptions . . . . . . . . . . . . . . 269 13.3.3 The Body Wave Component . . . . . . . . . . . . . . . . 271 13.3.4 The Environment Adaptation Component . . . . . . . . . 272 13.3.5 The Heading Control Component . . . . . . . . . . . . . 273 13.3.6 The Joint Angle Controller . . . . . . . . . . . . . . . . . 274 13.3.7 Summary of the Path Following Controller . . . . . . . . 274

Contents

13.4 13.5

13.6

13.7

xvii

Waypoint Guidance Control in Cluttered Environments . . . . . . 275 Simulation Study: The Performance of the Path Following Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 13.5.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . 276 13.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . 277 Experimental Study: The Performance of the Environment Adaptation Strategy . . . . . . . . . . . . . . . . . . . . . . . . 279 13.6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . 280 13.6.2 Experimental Results . . . . . . . . . . . . . . . . . . . . 280 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . 285

14 Future Research Challenges of Snake Robot Locomotion . . . . . . . 287 14.1 Control Design Challenges . . . . . . . . . . . . . . . . . . . . . 287 14.2 Hardware Design Challenges . . . . . . . . . . . . . . . . . . . 289 Appendix A Proof of Lemma 8.2 . . . . . . . . . . . . . . . . . . . . . . 293 Appendix B

Proof of Lemma 8.3 . . . . . . . . . . . . . . . . . . . . . . 295

Appendix C Low-Pass Filtering Reference Models . . . . . . . . . . . . 297 C.1 A 2nd-Order Low-Pass Filtering Reference Model . . . . . . . . 297 C.2 A 3rd-Order Low-Pass Filtering Reference Model . . . . . . . . 298 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Chapter 1

Introduction

1.1 Background and Motivation A snake robot is a robotic mechanism designed to move like a biological snake. Inspired by the robustness and stability of biological snake locomotion, snake robots carry the potential of meeting the growing need for robotic mobility in unknown and challenging environments. These mechanisms typically consist of many serially connected joint modules capable of bending in one or more planes. The many degrees of freedom of snake robots make them difficult to control but provide potential locomotion skills in cluttered and irregular environments which surpass the mobility of more conventional wheeled, tracked, and legged robots. P. Liljebäck et al., Snake Robots, Advances in Industrial Control, DOI 10.1007/978-1-4471-2996-7_1, © Springer-Verlag London 2013

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Introduction

Fig. 1.1 The water hydraulic snake robot Anna Konda

Before we motivate research on snake robots in general, let us begin by describing how the authors became involved in and excited about this research activity. Research on snake robots at the Norwegian University of Science and Technology (NTNU) has spawned from a research project at SINTEF.1 The project was initiated in 2003 after several major city fires in Trondheim, which launched an initiative to bring the fire department in closer relation with the research community in Trondheim to stimulate efforts that would improve fire safety. A specific idea which spurred from this initiative was the vision of a self-propelled fire hose as a robotic tool to aid human firefighters. This idea is clever in that the high-pressure water inside the hose can be employed as a hydraulic medium in the propulsion mechanism, a fire extinguishing medium, and a cooling medium for cooling the robot in environments with extreme temperatures. The resulting system would be a robotic fire hose that could move in extreme environments with the agility of a biological snake or, in other words, a water hydraulic snake robot. The Applied Cybernetics department at SINTEF was brought in to investigate this idea further and so began the research activity on snake robots at SINTEF and NTNU. Researchers at SINTEF and NTNU quickly realised that developing such a mechanism represents a highly interdisciplinary task with challenges ranging from heatresistant materials and water hydraulic joint actuation to control design and human– machine interaction. To show the feasibility of the concept, it was decided to develop a simple technology demonstrator in the form of a water hydraulic snake robot. The robot, which was named Anna Konda, is shown in Fig. 1.1 and is described in more detail in Liljebäck et al. (2006), Transeth et al. (2010). Anna Konda can move over relatively flat surfaces and can spray water through nozzles in its head. The robot is, however, far from ready for operating in harsh environments. 1 SINTEF is a Norwegian research organisation which is tightly coupled with NTNU both geographically and through joint research activities.

1.1 Background and Motivation

3

The work on the Anna Konda robot helped us identify several major research challenges. The critical and most significant research challenge was, and still is, the serpentine propulsion mechanism of this system. After the development of Anna Konda, the research on snake robots at NTNU and SINTEF has therefore targeted snake robot locomotion in general without concern about the specific application of the robot. Although fire fighting was the initial motivation behind this research, the scope of the current research activities extends beyond merely fire intervention tasks since snake robots can potentially be used in many other applications where robust robotic mobility is required. A very relevant and important application of future snake robots involves search and rescue missions in earthquake areas. In particular, the earthquake in Haiti in January 2010 and the Tsunami which struck Japan in March 2011 are natural disasters in recent time which illustrate the need for technology that can be employed to efficiently locate survivors inside the debris of collapsed buildings. Other potential applications of snake robots include inspection and intervention operations in hazardous environments of industrial plants, manipulator tasks in tight spaces, and space and subsea operations. Figure 1.2 illustrates some of these applications. In a global perspective, research on snake robots has been conducted for several decades. The research field was pioneered about 40 years ago by Professor Shigeo Hirose at Tokyo Institute of Technology, who developed the world’s first snake robot as early as 1972 (see Hirose 1993). The robot, which is shown in Fig. 1.7, was equipped with passive wheels mounted tangentially along its body. The wheels enabled the robot to travel forward on a flat surface by controlling the joints according to a periodic body wave motion similar to the body waves displayed by biological snakes. In the decades following the pioneering research by Professor Hirose, several agile and impressive snake robots have been developed by research communities around the world in efforts to mimic the motion capabilities of their biological counterpart. However, the locomotive capabilities of current snake robots are still limited to fairly simple and controlled lab environments, and the world has not yet seen practical applications of snake robot locomotion. Nonetheless, researchers working with snake robots are motivated by the vision of a robotic propulsion mechanism with robust and agile mobility in challenging environments. Development and control of snake robots is generally quite challenging for two primary reasons. First of all, a snake robot has many degrees of freedom, which means that the physical mechanism will contain a complex interconnection of sensors, actuators, and control logic. Moreover, the many degrees of freedom represent complex nonlinear dynamics which is challenging to analyse from a control design perspective. Second, the dependence on environment interaction is more complicated for a snake robot than for more conventional mobile robots. In particular, the propulsion mechanism of a wheeled, tracked, or legged robot is achieved with a separate and dedicated part of the robot. A snake robot, on the other hand, has no separate part which is dedicated to propulsion. Being essentially a smooth and flexible manipulator arm, the propulsion mechanism of a snake robot is rather an integrated part of the entire body, which means that propulsion requires synchronised motion of the entire robot in order to produce appropriate environment interaction forces.

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Introduction

Fig. 1.2 Examples of future applications of snake robots

Motion based on such environment interaction is challenging both with respect to control design and mechanical implementation. This book targets some of the challenges discussed above, and it is motivated by the long-term goal of developing snake robots which can move in unknown and challenging environments in order to support human intervention tasks. The main goal of the book is to create a foundation for future research by presenting the fun-

1.2 Biological Snakes

5

damentals of snake robot locomotion. To this end, the focus of the book is primarily directed towards control design. Efficient control strategies are vital to future applications of snake robots, and they are also instrumental in the development of these mechanisms. In particular, a control strategy with a solid mathematical foundation will immediately reveal what sensory capabilities, ground friction properties, actuator forces, etc., which are required in the physical robot to achieve a specific control objective. It is our hope that the results presented in this book will stimulate and support future research on these fascinating mechanisms.

1.2 Biological Snakes This book is inspired by the robust motion capabilities of biological snakes. These amazing creatures are optimal in the sense that they have emerged through millions of years of evolution. In the following, we present aspects of biological snakes that we consider relevant to modelling, mechatronics, and control of snake robots. The material is based on Bauchot (1994), Hirose (1993), Hu et al. (2009), Mattison (2002).

1.2.1 The Anatomy of Snakes The typical appearance of the skeletal structure of a snake is shown in Fig. 1.3 and consists of vertebrae, ribs, and a skull. Snakes can have between 130 and 500 vertebrae, with ribs attached to each one. The vertebrae constitute a column of movable joints that run through the body of the snake, and protects the spinal cord, which runs through a channel along the top of the vertebral column. The ribs attached to each side of a vertebra protect the internal organs. The mechanical interconnection of the vertebrae is interesting. As illustrated in Fig. 1.4, two vertebrae are connected in a ball and socket arrangement. The magnitude of the relative rotational motion between two vertebrae is quite limited. In particular, the relative rotation between two vertebrae about the vertical axis ranges between 10° and 20°, while the relative rotation about the horizontal axis is limited to only a few degrees. These limitations may appear contradictory to the flexibility that snakes are known for, but this flexibility is, in fact, produced by the sum of the small movements of many vertebrae. Moreover, limiting the range of the relative movements leads to increased strength in the connection between the vertebrae. To prevent damage to the spinal cord due to twisting of the vertebrae about the axis tangential to the body, each vertebra has a number of wing-like projections that interlock loosely with their counterparts on the adjacent vertebrae. This limits the amount of twisting. The body shape of a snake is changed with the help of muscles that are arranged diagonally along each side of the snake. The ends of these muscles are attached to

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Introduction

Fig. 1.3 The skeleton of a snake consisting of vertebrae, ribs, and a skull Fig. 1.4 Close-up of vertebrae from a snake

ribs, sometimes joining adjacent ribs, but mostly joining ribs that are some distance apart. The pattern of contraction and relaxation of these muscles determines the type of locomotion that is performed. For instance, if muscles on one side of the snake are contracted at the same time as the equivalent muscles on the other side are relaxed, then the body will be bent. If, on the other hand, opposite sets of muscles are contracted or relaxed simultaneously, then the snake will, to some extent, be able to shorten or extend its body at this location.

1.2 Biological Snakes

7

Fig. 1.5 The skin of a snake is completely covered by scales, which are formed from thickened areas of the skin

The skin of a snake is completely covered with scales. The scales are formed from thickened areas of the skin and are therefore integrated with the skin. The typical appearance of snake scales is shown to the left of Fig. 1.5, while the right shows the skin when it is stretched, thereby pulling the scales apart. The areas of skin between the scales allow the snake to flex its body while maintaining a smooth coverage of the scales. An important purpose of the scales is to form a physical protection from general wear and tear when the snake moves across rough surfaces. At the same time, the use of small units of armour allows greater flexibility than would large bony plates. Another feature of the scales is that they give the snake anisotropic ground friction properties, i.e. the scales give the snake a larger friction coefficient in the transversal direction of the snake body compared to in the tangential direction. Studies of biological snakes and simulation studies have indicated that this difference in the friction coefficients is important during forward gliding motion. The assumption on the importance of this friction property is proved in this book (Sect. 4.4).

1.2.2 The Locomotion of Snakes Snakes are almost unique among the terrestrial vertebrates in their lack of legs. However, the lack of legs do not appear to have placed restrictions on the ability of snakes to move around. On the contrary, snake locomotion is stable, robust, and versatile. The speed of snake locomotion is, however, relatively slow, although certain species can move at speeds up to 11 km/h. Some snakes display specialised forms of motion. For instance, certain snakes can jump to heights of up to 1 m by curving their body into a vertical S-shape to serve as a spring, and then jump by stretching their body. Other snakes are able to glide through the air by throwing themselves from trees and forming their body in an aerodynamically favourable manner. In the following, the four most common types of biological snake locomotion are presented.

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Introduction

Lateral Undulation Lateral undulation, also called serpentine crawling, is the fastest and most common form of snake locomotion and is illustrated in Fig. 1.6(a). During lateral undulation, continuous waves are propagated backwards along the snake body from head to tail. During this wave motion, the sides of the snake body push against irregularities in the surface, thereby pushing the snake forward. This form of locomotion is therefore not suitable on slippery surfaces. As the snake progresses, every point along the body passes the same point on the ground, and there is never any static contact between the ground and any point along the body. During swimming, the same wave motion is produced, but the body then pushes against the resistance of the water. The weight distribution of a snake during lateral undulation is not uniform, but rather distributed so that the peaks of the body wave curve are slightly lifted from the ground. Concertina Locomotion Concertina locomotion, which is illustrated in Fig. 1.6(b), is often employed in narrow spaces where the available range of motion is limited. The motion is carried out by first extending the front part of the body forward, while the back part is curved several times to provide an anchor against the narrow environment. Once the head and front part of the body are fully extended, they are subsequently used to provide an anchor in the same way so that the back part of the body can be drawn up. The sequence is then repeated. The principle behind concertina locomotion relies on the difference between the large static friction forces at the anchor points and the low kinetic friction forces in the part of the body which is extended. The motion pattern is not very efficient in terms of energy consumption, but is often needed in order to traverse tight spaces. Rectilinear Crawling Rectilinear crawling is a slow form of locomotion often employed by heavy-bodied snakes. Also snakes in the final stages of stalking their pray use rectilinear crawling to avoid alerting their intended victim. During rectilinear crawling, the snake uses the edges of the scales on its underside as anchor points to pull itself forward in a more or less straight line. The operation consists of stretching forward and hooking the edges of the scales over small irregularities, then pulling the body up to this point. Alternate parts of the body will be stretching and pulling at the same time. The motion pattern is illustrated in Fig. 1.6(c). Sidewinding Sidewinding is a form of locomotion which is usually employed by snakes that live in areas of loose sand, e.g. desert snakes. The motion resembles concertina motion

1.2 Biological Snakes

9

Fig. 1.6 Different forms of biological snake locomotion

in that one part of the body acts as an anchor, while another part is moved forward. Starting from a resting position, the head and neck are raised off the ground and thrown sideways, while the rest of the body provides an anchor against the ground. Once the head and fore part of the body are again on the ground, they in turn act as an anchor while rest of the body repeats the same motion. The snake moves at about 45° with respect to its heading and leaves a trail of characteristic markings in the sand, as illustrated in Fig. 1.6(d).

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Introduction

The Control System of Snakes The employed locomotion method of snakes sometimes depends on the size of the snake and sometimes on the substrate over (or through) which it is moving. In fact, an interesting difference between snake locomotion and legged forms of locomotion is that the basic repeating motion that leads to propulsion of legged animals to a large extent depends on the progression speed of the animal. On the other hand, the basic repeating motion that leads to propulsion of snakes largely depends on the environment, and not on the speed. Considering the large number of muscles involved in the motion of a snake and also the large number of contact points that are sensed by its nervous system, it is fair to say that the coordination of snake movements is both impressive and complex. Investigations of the electrical activity that accompanies the muscular contraction during movement show that the motor response is segmentary. Nerve impulses are propagated backwards along the snake body through the bone marrow. These impulses successively activate local muscle groups, which bend the snake body. Musculature is, in other words, successively, and not simultaneously, active, and only for a few elements at a time. The bending motion at a point along the snake body is also influenced by the sensory information transmitted by the skin. Simply speaking, the snake produces a relatively simple motor command which is modulated by local reflexes. This explains how every point in the body is able to follow the same trajectory.

1.3 Previous Work on Modelling, Mechatronics, and Control of Snake Robots In the following, we provide an overview of previous literature on snake robot locomotion. The review is structured according to the title of this book by first considering research efforts related to modelling and analysis of snake robots, followed by research on physical implementation of these mechanisms, and finally considering previous control design efforts for snake locomotion. The scope of this book, which we present in Sect. 1.4, is motivated and justified based on this literature review.

1.3.1 Previous Work on Modelling and Analysis of Snake Robots Previous literature on modelling and analysis of snake robot locomotion is summarised in Table 1.1. The table separates works that consider snake locomotion from a planar (2D) perspective and works that also include three-dimensional aspects of the motion. A more detailed description of this literature is presented in the following.

1.3 Previous Work

11

Table 1.1 Previous work on modelling and analysis of snake robots Biomechanical studies of biological snakes 2D perspective

Gray (1946), Moon and Gans (1998), Ma (1999)

3D perspective

Hirose (1993), Hu et al. (2009)

Flat surface locomotion with sideslip constraints 2D perspective

Hirose (1993), Krishnaprasad and Tsakiris (1994), Kelly and Murray (1995), Ostrowski (1996), Ostrowski and Burdick (1998), Ishikawa (2009), Hatton and Choset (2009a), Prautsch and Mita (1999), Ute and Ono (2002), Matsuno and Mogi (2000), Matsuno and Sato (2005)

3D perspective

Ma et al. (2003), Tanaka and Matsuno (2008b), Date and Takita (2005)

Flat surface locomotion without sideslip constraints 2D perspective

Ma (2001), Ma and Tadokoro (2006), Saito et al. (2002), Li and Shan (2008), Kane and Lecison (2000), Grabec (2002), Hicks (2003), Mehta et al. (2008), Chernousko (2005), Nilsson (2004), Hu et al. (2009)

3D perspective

Shapiro et al. (2007), Ma et al. (2004), Transeth et al. (2008a)

Robotic fish and eel-like mechanisms 2D perspective

McIsaac and Ostrowski (2003a), Kanso et al. (2005)

3D perspective

Boyer et al. (2006), Zuo et al. (2008), Morgansen et al. (2001, 2002, 2007), Vela et al. (2002a)

Locomotion in environments with obstacles 2D perspective

Shan and Koren (1993), Bayraktaroglu and Blazevic (2005), Date and Takita (2007)

3D perspective

Chirikjian (1992), Chirikjian and Burdick (1995), Yamada and Hirose (2006a), Shan and Koren (1995), Tanev et al. (2005), Transeth et al. (2008b)

Biomechanical Studies of Biological Snakes A complete treatment of previous studies of biological snakes is beyond the scope of this book. The biomechanical studies that we consider to be most relevant to this book are summarised in the following. One of the earliest analytical studies of snake locomotion was given in Gray (1946), where mathematical descriptions of the forces acting on a snake are proposed and used to derive properties of snake locomotion. One of Gray’s conclusions was that forward motion of a planar snake requires the existence of external forces acting in the direction normal to the snake body. Hirose (1993) studied biological snakes and modelled the snake body as a continuous curve that could not move sideways (sideslip constraints). A well-known result by Hirose is the formulation of the serpenoid curve, which is a mathematical description of lateral undulation (the most common form of snake locomotion). This mathematical description is elaborated in Sect. 4.7. Hirose also investigated adaptive functions of biological snakes (i.e. sinus-lifting, the α-adaptive principle, and the l-adaptive principle) and proposed mathematical descriptions of how external factors, such as ground friction and temperature, affect the shape of a snake during

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Introduction

locomotion. Furthermore, Hirose investigated locomotion efficiency inside a maze, i.e. when the snake touches a wall on each side. An alternative description of lateral undulation, named the serpentine curve, was proposed in Ma (1999), where a mathematical model of the muscle characteristics of snakes is employed to derive the resulting form of the body shape during lateral undulation. Ma showed that snake locomotion according to the serpentine curve has a higher locomotive efficiency than locomotion according to the serpenoid curve. The locomotive efficiency during slip-free motion was defined as the ratio between the tangential and normal direction friction forces on the snake body. Other interesting studies of snake locomotion include the work in Moon and Gans (1998), which considers the mechanism by which muscular activity of a snake produces curvature and propulsion. In particular, the muscular activity is studied as a snake interacts with pegs in order to push itself forward. A more recent study given in Hu et al. (2009) investigates the frictional properties of snake skin both mathematically and experimentally. In particular, the study shows that the friction coefficient of a snake in the transversal direction of the body is larger than the friction coefficient in the tangential direction. This property is important during forward gliding motion. The study also shows that the weight distribution of a snake during lateral undulation is not uniform, but rather distributed so that the peaks of the body wave curve are slightly lifted from the ground. This is often referred to as sinus-lifting.

Modelling of Flat Surface Locomotion with Sideslip Constraints As noted in e.g. Gray (1946), each part of a biological snake conducting lateral undulation follows the path traced out by the head. This phenomenon is partially explained by the frictional anisotropy of snake skin studied in e.g. Hu et al. (2009) but is also caused by irregularities on the surface that provide grip and enable the snake to glide forward without slipping sideways. To mimic this motion, many models of snake robots have been developed under the explicit assumption that the body cannot move sideways (sideslip constraints). This assumption introduces nonholonomic constraints (Bloch et al. 2003) in the equations of motion of the robot. In practice, such conditions are usually achieved by installing passive wheels along the body of the snake robot. Several works attack the motion control problem of wheeled snake robots with tools from differential geometry. Early approaches of such a form are presented in Kelly and Murray (1995), Krishnaprasad and Tsakiris (1994), which model the kinematics of wheeled snake robots and analyse the relationship between body shape changes and the resulting displacement of the robot. These works also assess the controllability of such mechanisms. Similar approaches are considered in Ostrowski and Burdick (1998), Ostrowski (1996), where also the dynamics of wheeled snake robots is considered, and where system symmetries are utilised to arrive at reduced forms of the model. Modelling and controllability analysis of the kinematics of a three-linked wheeled snake robot is also considered in Ishikawa (2009). Furthermore, Hatton and Choset (2009a) introduce the concept of a body velocity integral

1.3 Previous Work

13

in order to easily approximate the net displacement of a snake robot during a gait. The method requires that the system coordinates are properly chosen. A model of the 2D dynamics of a wheeled snake robot is developed in Prautsch and Mita (1999) from Lagrange’s equations of motion, and in Ute and Ono (2002) from first principles. The works in Matsuno and Mogi (2000), Matsuno and Sato (2005) present models of the 2D kinematics and dynamics of snake robots, respectively, where some, but not all, of the links are wheeled. The wheel-less links correspond to links that are lifted from the ground. Lifting some of the wheeled links is sometimes desirable from a control perspective to make the motion of the robot less constrained. A model of the 3D kinematics of a snake robot that describe the lifting of the links more accurately is presented in Ma et al. (2003). Furthermore, Tanaka and Matsuno (2008b) present a model of the 3D dynamics of a snake robot consisting of a grounded base part and a lifted head part (for manipulation purposes), where some, but not all, of the links in the base part are wheeled. Continuum models of snake robot dynamics, where the snake is treated as a continuous curve that cannot move sideways, are presented in Date and Takita (2005), Hirose (1993). The model in Hirose (1993) is planar, while the model in Date and Takita (2005) considers the 3D dynamics of the continuous snake robot.

Modelling of Flat Surface Locomotion Without Sideslip Constraints In addition to the many models of snake robots with sideslip constraints, there are also many models that do not enforce such constraints, but instead only assume that the links exhibit anisotropic ground friction properties similar to biological snakes. With anisotropic ground friction properties, the friction coefficients describing the friction force in the tangential and normal direction of a link, respectively, are different. Models based on such ground friction properties are generally more complex to analyse than models based on sideslip constraints since there is no longer a direct connection between the body shape changes and the resulting displacement of the robot. Ma (2001) employs the Newton–Euler formulation to develop a 2D model of the dynamics of a snake robot with anisotropic ground friction properties. The ground friction model includes both static and dynamic Coulomb ground friction forces. The model of the robot is formulated in two ways, where the first form gives the propulsion of the robot and the joint torques based on knowledge of the body shape changes, whereas the second form gives the propulsion and body shape changes of the robot based on knowledge of the joint torques. The model is extended in Ma and Tadokoro (2006) to also describe snake locomotion on a slope. Another model of planar wheel-less snake robot dynamics is developed in Saito et al. (2002) from first principles. The model considers both viscous and Coulomb ground friction forces. Simulations with the model are carried out to derive properties of snake robot dynamics. The model from Saito et al. (2002) is employed in Li and Shan (2008) to study the controllability of the joints of a snake robot under

14

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Introduction

the assumption that one joint is passive. However, the analysis does not consider the position of the robot. Models of planar snake robot dynamics with anisotropic viscous ground friction are presented in Grabec (2002), Hicks (2003), Kane and Lecison (2000). The work Hicks (2003) exploits symmetries in the system (cyclic coordinates) to transform the model to a reduced form where the shape dynamics is decoupled from the displacement dynamics of the snake robot and investigates general requirements for the propulsion of a three-linked snake robot. A friction model that includes both viscous and Coulomb friction forces is proposed and analysed in Mehta et al. (2008). A model that considers isotropic Coulomb ground friction forces (both static and dynamic friction) is presented in Chernousko (2005). Isotropic ground friction is also assumed in Nilsson (2004), where a continuum approach along with energy arguments are employed to analyse planar snake locomotion under isotropic friction conditions. Shapiro et al. (2007) model the frictional contact forces between a snake robot and a compliant surface. The dynamics of planar snake locomotion is described in terms of a continuum model in Hu et al. (2009), where the snake is treated as a continuous curve influenced by Coulomb friction forces from the ground. The model is employed to study the effect of anisotropic ground friction properties on the propulsion of snakes. The 3D dynamics of a snake robot during locomotion across flat surfaces is considered in Ma et al. (2004), Transeth et al. (2008a). The model in Ma et al. (2004) is developed from the Newton–Euler formulation and includes both static and dynamic Coulomb ground friction forces. The model is employed to study sinus-lifting during lateral undulation. Transeth et al. (2008a) model snake robot dynamics by use of the framework of nonsmooth dynamics. The model, which represents a hybrid system, describes the normal direction contact forces from the ground and the Coulomb ground friction forces by use of set-valued force laws.

Modelling of Robotic Fish and Eel-Like Mechanisms Research on robotic fish and eel-like mechanisms is relevant to research on snake robots since these mechanisms are very similar. A complete treatment of robotic underwater locomotion is beyond the scope of this review. However, a representative part of previous research related to modelling of such mechanisms is presented in the following. A model of eel-like motion is developed in McIsaac and Ostrowski (2003a) based on tools from differential geometry that were also considered in some of the works concerning wheeled snake robots described above. However, the model does not place sideslip constraints on the robot. Instead, the eel-like mechanism is propelled by hydrodynamic forces modelled by a viscous friction model. The dynamics of eel-like motion is also considered in Kanso et al. (2005), where a model reduction is proposed to allow the net motion of the robot to be described as a sum of geometric and dynamic phases over closed curves in the shape space, in Boyer et al. (2006), where a continuum model is formulated based on beam theory, and in Zuo et al.

1.3 Previous Work

15

(2008), where first principles are employed to model the dynamics of a swimming snake robot. The works in Morgansen et al. (2001, 2002, 2007), Vela et al. (2002a), model the dynamics of a robotic fish influenced by lift and drag forces in an inviscous fluid. The controllability of the fish-like mechanism is also assessed in these works. Modelling of Locomotion in Environments with Obstacles In Chirikjian and Burdick (1995), Chirikjian (1992), the kinematics of snake robots is modelled in terms of a continuous backbone curve that captures the macroscopic geometry of the robot. Gaits for the backbone curve, which determine the shape of the snake robot, are specified with respect to environment constraints and the desired locomotion trajectory of the robot. The approach is original in that the problem of locomotion in cluttered environments is attacked at a purely kinematic level. The work by Chirikjian and Burdick is extended in Yamada and Hirose (2006a), where a continuum kinematics model is presented that explicitly handles the case of backbone curves that can be bent, but not twisted. This condition is in line with most physical snake robots, which are generally able to bend but not twist their body. The kinematic constraints imposed on a snake robot due to external obstacles are modelled in Shan and Koren (1993, 1995). These works also analyse how obstacles around a snake robot affect its degrees of freedom. The only known works that consider the dynamics of snake robots in environments with obstacles (i.e. where obstacle contact forces are considered) are presented in Bayraktaroglu and Blazevic (2005), Date and Takita (2007), Tanev et al. (2005), Transeth et al. (2008b). In Bayraktaroglu and Blazevic (2005), a dynamic simulation software called WorkingModel is used to simulate a planar snake robot interacting with circular obstacles. Contact forces are calculated from a springdamper approximation. A similar approach is employed in Tanev et al. (2005), where the simulation software Open Dynamics Engine (ODE) is used to model a snake robot interacting with various forms of obstacles. Date and Takita use the multi-body dynamics simulation software Autolev to study the motion of a snake robot during contact with a single peg, where the contact with the peg is modelled as a spring-damper system. The works in Bayraktaroglu and Blazevic (2005), Date and Takita (2007), Tanev et al. (2005) do not provide the equations underlying the dynamics of the snake robot due to the use of general-purpose simulation software. On the other hand, the model proposed in Transeth et al. (2008b) is, to our best knowledge, the only work which explicitly presents the equations of motion underlying the obstacle interaction dynamics of a snake robot. The model, which represents a hybrid system, is formulated within the framework of nonsmooth dynamics. A timestepping method is used to simulate the dynamics of the robot, which means that the system equations are discretised with a time step determined by a fixed error criterion, and trajectories of the system are approximated without tracking events (i.e. obstacle impacts). Transeth et al. (2008b) also introduce the term obstacle-aided locomotion, which involves using external obstacles as push points to aid the propulsion instead of avoiding them.

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Introduction

Table 1.2 Previous work on implementation of physical snake robots Snake robots without contact force sensors With passive wheels

Endo et al. (1999), Togawa et al. (2000), Ma et al. (2001), Wiriyacharoensunthorn and Laowattana (2002), Mori and Hirose (2002), Miller (2002), Ye et al. (2004b, 2007), Yamada et al. (2005), Crespi and Ijspeert (2008), Yu et al. (2008, 2009), Kamegawa et al. (2009)

Without passive wheels

Yim (1994), Yim et al. (2002), Worst and Linnemann (1996), Dowling (1997, 1999), Nilsson (1998), Ohno and Hirose (2001), Saito et al. (2002), Brunete et al. (2006), Chen et al. (2007), Wright et al. (2007), Kuwada et al. (2008), Yamada and Hirose (2008, 2009), Ohashi and Hirose (2010)

With active propulsion

Kimura and Hirose (2002), Yamada and Hirose (2006b), Taal et al. (2009), Fjerdingen et al. (2009), Kamegawa et al. (2004), Masayuki et al. (2004), Granosik et al. (2006), Gao et al. (2008), McKenna et al. (2008), Ijspeert et al. (2007), Hara et al. (2007)

Snake robots with contact force sensors With passive wheels

Hirose (1993), Chen et al. (2008)

Without passive wheels

Bayraktaroglu (2008), Andruska and Peterson (2010), Liljebäck et al. (2006), Fjerdingen et al. (2008)

With active propulsion

Taal et al. (2009)

1.3.2 Previous Work on Implementation of Physical Snake Robots Previous literature that considers development of physical snake robots is summarised in the following. The review is structured according to the focus of this book on snake robot locomotion based on measurements of environment contact forces, which we consider important for body shape adaptation in cluttered environments. In particular, we have chosen to separate the works that consider snake robots with contact force sensors from the works that do not include such sensor capabilities in the robot design. This partitioning illustrates, as seen in Table 1.2, that previous research on environment sensing for snake robots is limited. The referred works are summarised in Table 1.2, which separates between snake robots with passive wheels, which are advantageous during motion across flat surfaces, snake robots without such passive wheels, and snake robots equipped with active propulsion.

Snake Robots Without Contact Force Sensors Hirose developed the world’s first snake robot as early as 1972 (Hirose 1993). The robot, which is shown in Fig. 1.7, was equipped with passive wheels to realise the anisotropic ground friction property that enables forward locomotion on flat surfaces.

1.3 Previous Work

17

Fig. 1.7 The snake robot ACM III, which was the world’s first snake robot developed by Prof. Shigeo Hirose in 1972. Courtesy of Tokyo Institute of Technology

Fig. 1.8 The snake robot ACM R3 developed at Tokyo Institute of Technology. The robot is covered with passive wheels. Courtesy of Tokyo Institute of Technology

Several other snake robots with passive wheels have been proposed over the years, such as the robots presented in Endo et al. (1999), Togawa et al. (2000), Ma et al. (2001), Wiriyacharoensunthorn and Laowattana (2002), Mori and Hirose (2002) (see Fig. 1.8), Miller (2002) (see Fig. 1.9), Ye et al. (2004b, 2007), Yamada et al. (2005) (see Fig. 1.10), Crespi and Ijspeert (2008), Yu et al. (2008, 2009), and Kamegawa et al. (2009). Some of the robots can only display planar motion, while other robots can move their links both horizontally and vertically. Some robots have shielded joint modules that enable motion in environments with e.g. mud and dust, and even motion under water (see Yamada et al. 2005 and Fig. 1.10), while other robots have modules with exposed electronic components which only allow them to move in clean lab environments. A common feature of these mechanisms, however, is that they are generally only able to move across relatively flat surfaces since passive wheels do not move very well in a cluttered environment. Such mechanisms are therefore suitable for motion control experiments on relatively flat surfaces, but not for practical applications of snake robots in more challenging environments.

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Introduction

Fig. 1.9 The snake robot S5 developed by Dr. Gavin Miller. The robot has passive wheels on its underside. Courtesy of Dr. Gavin Miller

Snake robots without passive wheels, i.e. robots that basically consist of straight links interconnected by motorised joints, are presented in Yim (1994), Yim et al. (2002), Worst and Linnemann (1996), Dowling (1997, 1999), Nilsson (1998), Ohno and Hirose (2001), Saito et al. (2002), Brunete et al. (2006), Chen et al. (2007), Wright et al. (2007), Kuwada et al. (2008), Yamada and Hirose (2008, 2009), Ohashi and Hirose (2010) (see Fig. 1.11). Despite its lack of wheels, the snake robot in Saito et al. (2002) maintains an anisotropic ground friction property since the underside of each link has edges, or grooves, that run parallel to the link. This robot can therefore move forward by lateral undulation through purely planar motion. Robots whose ground friction properties are isotropic and, on the other hand, can move forward during lateral undulation by resorting to sinus-lifting, i.e. by slightly lifting the peaks of the body wave curve from the ground (see e.g. Ohno and Hirose 2001; Yamada and Hirose 2008). However, robots with isotropic friction are mostly used for studying gaits other than lateral undulation, such as gaits based on sidewinding, inchworm motion, or lateral rolling. A notable feature of the works presented in Wright et al. (2007) (see Fig. 1.12) and Yamada and Hirose (2009) (see Fig. 1.13) is the focus on development of small, light-weight, and strong joint actuation mechanisms, which are important for many future applications of snake robots.

1.3 Previous Work

19

Fig. 1.10 The snake robot ACM R5 developed at Tokyo Institute of Technology. The robot is covered by passive wheels and can swim under water. Courtesy of Tokyo Institute of Technology

Fig. 1.11 The snake robot ACM R7 developed at Tokyo Institute of Technology. The robot can move by curving its body into a loop and rolling forward like a wheel. Courtesy of Tokyo Institute of Technology

There are also works that consider active propulsion along the body of a snake robot, for example by equipping each link with motorised wheels (Kimura and Hirose 2002; Taal et al. 2009; Yamada and Hirose 2006b; Fjerdingen et al. 2009), or by installing tracks along the body of the snake robot (Gao et al. 2008; Granosik et al. 2006; Kamegawa et al. 2004; Masayuki et al. 2004; McKenna et al. 2008), or by installing legs on the links of the robot (Ijspeert et al. 2007), or by employing a screw drive mechanism (Hara et al. 2007). The snake robots with active propulsion presented in Granosik et al. (2006), McKenna et al. (2008), and Ijspeert et al. (2007) are shown in Figs. 1.14, 1.15, and 1.16, respectively.

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Introduction

Fig. 1.12 The snake robot Uncle Sam developed at Carnegie Mellon University. The robot has a strong and compact joint mechanism and can climb up poles. Courtesy of Carnegie Mellon University

Fig. 1.13 A snake robot with a miniature joint mechanism developed at Tokyo Institute of Technology. Courtesy of Tokyo Institute of Technology

Snake Robots with Contact Force Sensors Previous research on environment sensing for snake robots is limited. The wheeled snake robot developed by Hirose already in 1972 (Hirose 1993) was equipped with contact switches, which enabled the robot to demonstrate lateral inhibition with respect to external obstacles. Snake robots with cylindrical modules covered by force sensors are proposed in Fjerdingen et al. (2008), Liljebäck et al. (2006). The force sensor systems in these works are able to detect and, to some extent, assess the magnitude of external forces applied at certain areas of the joint modules. A snake robot with active wheels, where each wheel axis is equipped with a three-axial force sensor, is presented in Taal et al. (2009). The force sensor measures the translational forces on the wheel axis based on optical range measurements. Bayraktaroglu (2008) presents a wheel-less snake robot with contact switches and presents experimental results where the robot is propelled forward by pushing against pegs that are detected by the contact switches. A snake robot with passive wheels and strain gauge sensors is proposed in Chen et al. (2008), where the strain gauge sensors are shown to successfully measure the constraint forces on the wheels. Ideas related

1.3 Previous Work

21

Fig. 1.14 The OmniTread snake robot developed at the University of Michigan. The robot has pneumatic joints and is covered by motorised tracks. Courtesy of the University of Michigan

to environment sensing for snake robots are considered in Andruska and Peterson (2010), where the preliminary design of a capacitive contact sensor is proposed that can be wrapped around each module of a snake robot.

Fig. 1.15 A snake robot with a skin drive propulsion system developed at Carnegie Mellon University. A motor drives the outer skin backwards along the snake body in order to propel the robot forward. Courtesy of Carnegie Mellon University

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Introduction

Fig. 1.16 A salamander-like snake robot with motorised legs that propel the robot forward. The robot can operate under water. Courtesy of the Biologically Inspired Robotics Group at Ecole Polytechnique Fédérale de Lausanne (EPFL)

1.3.3 Previous Work on Control of Snake Robots In the following, we provide an overview of previous control design efforts for snake robots. The review is structured according to Tables 1.3 and 1.4, which summarise all papers referred to in this section. The two tables separate between works that present gait patterns without explicitly controlling the position or heading of the snake robot and works that present gait patterns along with position and/or heading controllers. The review focuses on controllers based on lateral undulation, which is the most common form of snake robot locomotion and which is also most relevant to the results presented in this book. Remark 1.1 Stability analysis of control laws for snake robots is challenging due to the complexity of existing models of these mechanisms. For this reason, applications Table 1.3 Previous work on control of snake robot locomotion (1 of 2) Flat surface locomotion with sideslip constraints Without position or heading control

Shan and Koren (1993), Kelly and Murray (1995), Ostrowski and Burdick (1998), Date and Takita (2005), Tanaka and Matsuno (2009), Ute and Ono (2002), Sato et al. (2010), Wang et al. (2010)

With position and/or heading control

Prautsch et al. (2000), Date et al. (2000, 2001a, 2001b), Yamakita et al. (2003), Matsuno and Mogi (2000), Ma et al. (2003), Matsuno and Suenaga (2003), Ye et al. (2004a), Matsuno and Sato (2005), Tanaka and Matsuno (2008a, 2008b), Wiriyacharoensunthorn and Laowattana (2002), Watanabe et al. (2008), Ishikawa (2009), Ishikawa et al. (2010), Paap et al. (1999), Linnemann et al. (1999), Murugendran et al. (2009)

Flat surface locomotion without sideslip constraints Without position or heading control

Dowling (1997, 1999), Ma (2001), Ma et al. (2004), Saito et al. (2002), Chernousko (2003, 2005), Transeth et al. (2007b), Burdick et al. (1995), Gonzalez-Gomez et al. (2007), Yu et al. (2008), Chirikjian and Burdick (1995), Poi et al. (1998), Yim (1994), Yim et al. (2002), Ohno and Hirose (2001), Rincon and Sotelo (2003), Hatton and Choset (2010), Yamada and Hirose (2010), Mori and Hirose (2002), Chen et al. (2004), Ohashi and Hirose (2010)

With position and/or heading control

Hicks (2003), Hicks and Ito (2005)

1.3 Previous Work

23

Table 1.4 Previous work on control of snake robot locomotion (2 of 2) Robotic fish and eel-like mechanisms Without position or heading control With position and/or heading control

Morgansen et al. (2001), Melli et al. (2006), Crespi and Ijspeert (2008) McIsaac and Ostrowski (2003a, 2003b), Andruska and Peterson (2002, 2007), Vela et al. (2002a)

Locomotion in environments with obstacles Without position or heading control

Hirose (1993), Andruska and Peterson (2008), Kuwada et al. (2008), Kulali et al. (2002), Greenfield et al. (2005), Kamegawa et al. (2009), Zarrouk et al. (2010), Nilsson (1997), Chen et al. (2007), Lipkin et al. (2007), Hatton and Choset (2009b)

With position and/or heading control

Bayraktaroglu and Blazevic (2005), Bayraktaroglu (2008), Date and Takita (2007), Sfakiotakis and Tsakiris (2007)

of formal stability analysis tools in previous snake robot literature are very limited. Simulations and experimental investigations are instead the common approach in the literature for providing support of proposed control strategies.

Control of Flat Surface Locomotion with Sideslip Constraints A majority of previous control design efforts for snake robots has focused on locomotion where the links are subjected to nonholonomic constraints, i.e. where each link is constrained from moving sideways. Shan and Koren (1993) consider a snake robot that uses solenoids for attachment to the environment and proposes gaits for forward and turning motion of this mechanism. Tools from differential geometry are employed in Kelly and Murray (1995), Ostrowski and Burdick (1998) to demonstrate that sinusoidal shape inputs to wheeled snake robots lead to propulsion. A position and path following controller for a wheeled snake robot is proposed in Prautsch et al. (2000), where also Lyapunov analysis is employed to analyse the controller. The work also considers approaches for preventing the snake robot from attaining a straight shape, which is singular with respect to propulsion. The works Date et al. (2000, 2001a, 2001b) propose path following controllers for wheeled snake robots aimed at minimising the lateral constraint forces on the wheels during lateral undulation. The controllers are based on a measure of dynamic manipulability, which describes the ability of the robot to generate propulsive force. A similar approach is employed in Yamakita et al. (2003), which proposes a gait pattern aimed at minimising the lateral constraint forces on the wheels, and in Date and Takita (2005), which formulates and solves an optimisation problem in order to minimise the torque input. The optimisation problem is solved using a 3D continuum model of the snake robot. In Ma et al. (2003), Matsuno and Mogi (2000), Matsuno and Sato (2005), Matsuno and Suenaga (2003), position and path following controllers are proposed for the case where some, but not all, of the snake robot links are wheeled. The wheelless links correspond to links that are lifted from the ground, which give the system

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1

Introduction

more degrees of freedom that can be utilised to follow a trajectory while simultaneously maintaining a high manipulability. Similar approaches are considered in Tanaka and Matsuno (2008a, 2008b, 2009), where also strategies for sinus-lifting during lateral undulation are proposed. Ute and Ono (2002) propose a gait based on a self-excitation principle where joint angle information determines the winding motion of a snake robot. Directional control during lateral undulation is considered in Wiriyacharoensunthorn and Laowattana (2002), Ye et al. (2004a). Watanabe et al. (2008) propose a position controller for a wheeled snake robot that takes ground friction forces into account. A similar approach is employed in Sato et al. (2010), where deviations of the joint angles from their setpoints are used to modify the oscillatory joint motion, thereby enabling the snake robot to automatically adapt its motion to variations in the ground friction conditions. The works in Ishikawa (2009), Ishikawa et al. (2010) propose position and path following controllers for three-linked and four-linked wheeled snake robots based on Lie bracket calculations and controllability analysis results. The concept of passive creeping is considered in Wang et al. (2010), which involves adjusting the motion of a snake robot based on a measure of the dissipated energy, thereby achieving adaptation of the motion to different surface conditions. Local orbital stability of state trajectories during the motion is concluded based on recurrence plots. A snake robot with active wheels is considered in Linnemann et al. (1999), Paap et al. (1999), where an optimisation scheme is employed to make the robot follow the path that minimises energy dissipation due to friction forces. Active wheels are also assumed in Murugendran et al. (2009), where a path following controller for such snake robots is proposed on a kinematic level. Remark 1.2 The works Date et al. (2000, 2001b), Ma et al. (2003), Matsuno and Mogi (2000), Matsuno and Sato (2005), Prautsch et al. (2000), which were described above, all employ a common approach for motion control in that the nonholonomic constraints on the links are used to establish an explicit connection between body shape changes and propulsion, which allows the control input to be specified directly in terms of the desired propulsion of the robot. Such approaches are, to our best knowledge, the only known approaches for motion control of wheeled snake robots which infer some formal and model-based conclusions on the propulsion of the robot.

Control of Flat Surface Locomotion Without Sideslip Constraints The works Dowling (1997, 1999) employ Fourier series to specify periodic functions representing the gait patterns of a wheel-less snake robot. The parameters of the Fourier series are determined using certain learning techniques. In Ma (2001), computer simulations are employed to study properties of lateral undulation related to the optimality of the motion. Ma et al. (2004) propose a control strategy for sinus-lifting during lateral undulation by solving a quadratic optimisation problem.

1.3 Previous Work

25

Saito et al. (2002) consider snake robots influenced by anisotropic ground friction and optimise the gait parameters of lateral undulation based on simulations. The work also proposes a forward velocity controller for wheel-less snake robots. The works Chernousko (2003, 2005) consider several elementary motions for planar snake robots and derive conditions for the feasibility of these motions, such as required actuator strength. In Hicks and Ito (2005), Hicks (2003), methods based on numerical optimal control are considered for determining optimal gaits during positional control of snake robots influenced by anisotropic viscous ground friction. Gonzalez-Gomez et al. (2007) use Open Dynamics Engine (ODE) to propose and simulate various 3D gaits for translational and turning motion of snake robots, including a gait for rotation with very little displacement. Transeth et al. (2007b) propose a controller for the joints of a planar snake robot influenced by anisotropic Coulomb ground friction and prove that the resulting translational and rotational velocity of the robot is bounded. The following works consider other gait patterns than lateral undulation, and the gaits are carried out in open loop without explicitly controlling the position and orientation of the snake robot. Gaits for sidewinding motion, which is a sideways rolling type of motion, are proposed in Burdick et al. (1995), Gonzalez-Gomez et al. (2007), Hatton and Choset (2010), Yu et al. (2008). Inchworm locomotion gaits are proposed in Chirikjian and Burdick (1995), Gonzalez-Gomez et al. (2007), Ohno and Hirose (2001), Poi et al. (1998), Rincon and Sotelo (2003), Yamada and Hirose (2010), Yim (1994), Yim et al. (2002). Lateral rolling, which is achieved by continuously forming the snake body into a vertical U-shape that tips over, is considered in Chen et al. (2004), Dowling (1997), Gonzalez-Gomez et al. (2007), Mori and Hirose (2002), Ohno and Hirose (2001). Furthermore, gaits for loop forming motion are proposed in Ohashi and Hirose (2010), Yim (1994), Yim et al. (2002), where the head and tail of the snake robot are connected to turn the robot into a rolling wheel (see Fig. 1.11). Remark 1.3 To our best knowledge, previous literature has not presented any formal mathematical proofs regarding the propulsion of wheel-less snake robots.

Control of Robotic Fish and Eel-Like Mechanisms A complete treatment of robotic underwater locomotion is beyond the scope of this review. However, we consider the following works to be representative of previous research related to control of such mechanisms. Eel-like motion is considered in McIsaac and Ostrowski (2003a, 2003b), where controllers for tracking straight and curved trajectories are proposed. The works Morgansen et al. (2001, 2002, 2007), Vela et al. (2002a) consider motion control of robotic fish. Lie bracket calculations based on the dynamics of the robotic fish are used to derive gaits for forward motion and various forms of turning motion. Algorithms for closed-loop heading and depth control are also considered. Melli et al.

26

1

Introduction

(2006) propose open-loop gaits for a robotic fish based on curvature plots of the mechanical connection between the shape space motion and the overall displacement of the robot. A swimming snake robot is considered in Crespi and Ijspeert (2008), where a gradient-free optimisation method is employed to adjust the gait parameters online, i.e. while the robot is moving, in order to maximise the forward velocity.

Control of Locomotion in Environments with Obstacles Only a few works in previous literature consider control strategies for snake robots where the surface is no longer assumed to be flat (i.e. in environments with obstacles). To our best knowledge, the works in Bayraktaroglu and Blazevic (2005), Bayraktaroglu (2008), Hirose (1993) are the only works in previous literature which present control strategies for snake robots that employ explicit contact sensing in the feedback loop. Hirose (1993) proposes a strategy for lateral inhibition that modifies the shape of a snake robot based on contact force sensing along the snake body in order to avoid obstacles. Bayraktaroglu and Blazevic (2005) propose an inverse dynamics approach by formulating and numerically solving an optimisation problem in order to, for a given set of obstacle contacts, calculate the contact forces required to propel the robot in a desired direction. A strategy for calculating the actual torque inputs to the joints from the desired contacts was, however, not presented. A kinematic approach is proposed in Bayraktaroglu (2008), where a curve fitting procedure is used to determine the shape of the robot with respect to the detected obstacles. Subsequently, this shape is propagated backwards along the snake body under the assumption that this will push the robot forward. Sensing the environment of a snake robot must not necessarily involve contact force sensing since the environment can be indirectly sensed through the joint angle measurements and/or the actuator torques. This approach is considered in Date and Takita (2007), where the joint torques of a snake robot are specified solely in terms of the measured joint angles to achieve motion through a winding corridor, in Andruska and Peterson (2008), which presents a control strategy that uses motor current measurements to adjust the shape of a snake robot moving through an elastically deformable channel, and in Kuwada et al. (2008), where the deviations of the joint angles from their setpoints are used to adapt the body shape of a snake robot moving inside pipe structures. The remaining works presented in the following consider controllers aimed at locomotion in environments that are not flat but do not appear to involve sensing of the interaction between the snake robot and its environment. Kulali et al. (2002) employ a fuzzy logic controller to switch between various predefined gaits during motion in an obstacle environment. The goal of the motion controller is to avoid the obstacles. In Greenfield et al. (2005), an algorithm is presented that takes contact constraints on a snake robot into account in order to compute the joint torques that produce the desired motion. The algorithm is applied to achieve climbing motion with a snake robot. A gait for climbing motion is also proposed in Kamegawa et al. (2009). Sfakiotakis and Tsakiris (2007) use range sensor measurements to

1.4 The Scope of This Book

27

centre a crawling snake robot between the walls of a corridor. Zarrouk et al. (2010) analyse the efficiency of earthworm-like motion on compliant surfaces motivated by biomedical applications of worm robots. Moreover, various gaits aimed at motion in cluttered and uneven environments, including climbing gaits, are proposed in Chen et al. (2007), Hatton and Choset (2009b), Lipkin et al. (2007), Nilsson (1997).

1.4 The Scope of This Book The work underlying this book has been carried out with the following scope.

1.4.1 An Analytical Approach This book is motivated by practical applications of snake robots. However, the book is primarily a theoretical study, although experimental investigations are also considered. In our opinion, there are many aspects related to control of snake robots that have not yet been addressed. Moreover, even though research on snake robots has been conducted for several decades, our understanding of snake locomotion so far is largely based on empirical studies of biological snakes and simulation-based synthesis of properties of snake robots. In this book, we therefore take an analytical approach in an attempt to provide a basic understanding of snake robot locomotion. We hope that this approach will contribute to the mathematical foundation of the control theory of snake robots.

1.4.2 Snake Robots Without a Fixed Base There are many works in the literature which consider the use of snake robots for manipulation purposes, where one end of the robot is fixed. In such applications, the snake robot is essentially a fixed robotic manipulator arm with many degrees of freedom (see e.g. Chirikjian 2001; Jones and Walker 2006; Transeth et al. 2008c). In this book, however, we only consider snake robots intended for locomotion purposes, i.e. snake robots without a fixed base.

1.4.3 A Planar Perspective This book considers planar snake robot locomotion in the horizontal plane. Of course, snake locomotion is inherently a three-dimensional phenomenon, and a snake robot capable of strict planar motion will generally not be able to operate