Mechanics of Terrestrial Locomotion: With a Focus on Non-pedal Motion Systems

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Mechanics of Terrestrial Locomotion

“This page left intentionally blank.”

Klaus Zimmermann · Igor Zeidis · Carsten Behn

Mechanics of Terrestrial Locomotion With a Focus on Non-pedal Motion Systems

123

Univ.-Prof. Dr.-Ing.habil. Klaus Zimmermann TU Ilmenau Fakultät für Maschinenbau FG Technische Mechanik PSF 100565 98684 Ilmenau/Germany [email protected]

Dr.rer.nat. Igor Zeidis TU Ilmenau Fakultät für Maschinenbau FG Technische Mechanik PSF 100565 98684 Ilmenau/Germany [email protected]

Dr. Carsten Behn TU Ilmenau Fakultät für Maschinenbau FG Technische Mechanik PSF 100565 98684 Ilmenau/Germany [email protected]

ISBN 978-3-540-88840-6 e-ISBN 978-3-540-88841-3 DOI 10.1007/978-3-540-88841-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926965 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Starting with the research of G. Bögelsack in the 1970s, the analysis of biological locomotion and manipulation systems and their technical realization has been an important research field within the Faculty of Mechanical Engineering at the Ilmenau University of Technology. In 1996, the German Research Foundation (DFG) funded the Innovation College “Motion Systems” at the University of Jena in a cooperation with engineers at the Ilmenau University of Technology. Thus, research was able to be intensified and extended. Of course, the whole spectrum of biologically inspired systems is much too wide, so the analysis was still focused on locomotion and manipulation systems. At this stage J. Steigenberger from the Faculty of Mathematics and Natural Sciences at the Ilmenau University of Technology contributed important studies of worm-like locomotion systems with much dedication and technical competence. Moreover, he conceived and carried out a lecture series entitled “Mathematical Basics for Locomotion Systems”, which was based on his evaluation of national and international research developments in this field. I. Zeidis and K. Zimmermann contributed many publications on the mechanics of worm-like locomotion systems based on continuum and rigid-body models as well as asymptotic methods. Since 2004 the German Research Foundation has supported a series of projects led by K. Zimmermann dedicated to biologically inspired robotics. In addition to these activities, the Department of Technical Mechanics and the Department of Computer Application in Mechanical Engineering (M. Weiß) together with masters and doctoral students started the development of mobile robots for the RoboCup Small-Size League in 1998. On the basis of these teaching and research activities, the idea for a manuscript about the mechanics of terrestrial locomotion was born, despite knowing full well that there are no “new” mechanics to be developed for such systems. Rather, specific aspects of mechanics and control theory that are applied in terrestrial locomotion systems were to be prepared methodically in textbook style to be used by students. Exercises for self-study are provided in addition to the theoretical basics.

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Preface

This textbook is mainly aimed at undergraduate students in mechanical engineering, electrical engineering, automation engineering, and mechatronics, but should also be of interest to graduates in engineering, mathematics, or computer science looking for an introduction to the field of mobile robotics from the mechanical point of view. To keep the book self-contained, we summarize some fundamental results and principles from mechanics and control theory that are needed later on in the first two chapters. We decided to keep the mathematical tools used in the book to a level that is digestible for an engineering undergraduate and only require some familiarity with undergraduate linear algebra, elementary differential and integral calculus and vector calculus. In particular, we do not include functional-analytic or differentialgeometric considerations in this book. The authors refer the reader to the excellent books A Mathematical Introduction to Robotic Manipulation by R. Murray et al., and Geometric Control of Mechanical Systems by F. Bullo & A.D. Lewis for these advanced topics.

Ilmenau, November 2008

Klaus Zimmermann Igor Zeidis Carsten Behn

Acknowledgements

The authors are very grateful to many colleagues for their collaboration and for their input to this book, directly or indirectly. Firstly, we would like to thank those colleagues who actively supplemented this textbook with their own contributions: J. Steigenberger (Section 6.2), H. Witte and E. Andrada (Section 5.2), A. Seyfarth (Section 5.3), T. Rieß (Section 4.3.3). Three examples and exercises in Chapter 4 are from Y. Martynenko and his colleagues. We are grateful to R. Blickhan, F. Chernousko, M. Fischer, A. Huba, S. Jatsun, V. Lysenko, P. Maißer, V.T. Minchenya, V. Naletova, and C. Schilling for the collaboration in the field of biomimetic robotics during the last 10 years. We would like to acknowledge B. Smale, G. Miller, K. Berns, H. Ulbrich, W. Pril, S. Haddadin, S. Collins, M. Wisse, N. Michiels, T. Schmalz and E. Kaplan for the permission to use their photographs in this book. Appreciative thanks go to K. Abaza, M. Jahn, V.V. Minchenya, M. Pivovarov, C. Ußfeller, and T. Wagenknecht for their magnificent help in realizing this book. The authors are also grateful for the careful reading of the Chapters “Mechanical Background” and “Mathematical Methods and Elements of Control Theory” by N. Bolotnik and A. Neishtadt. The important financial support from the German Research Foundation was the basis for the development of prototypes under the qualified coordination of H.P. Walkling. We would like to thank H. Sachse, S. Stauche, and D. Voges for the invaluable help with the graphics and photos for the book and H. Kirsten for typing the manuscript in LaTeX. Also, many students contributed to the development of the locomotion systems presented here. In addition, we would like to thank all colleagues in the Department of Technical Mechanics at the Ilmenau University of Technology for the pleasant working discussions and cooperations. B. Percle’s work in improving the manuscript language is greatly appreciated.

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Acknowledgements

Thanks also to the anonymous reviewers of the book and to the staff at SpringerVerlag, especially E. Hestermann-Beyerle. She gave the authors cooperative assistance and friendly support from the first contact during the GAMM meeting in 2007 until the last version of the manuscript. And finally, I would like to express my gratitude towards my wife Jana, whose understanding and patience helped me to write this book together with my colleagues. Klaus Zimmermann in the name of the authors

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Fascination of Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Biologically inspired Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 RoboCup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Summary Remarks and a Diagram of the Book . . . . . . . . . . . . . . . . . .

1 1 2 4 5

2

Mechanical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Kinematics of Multibody Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Kinematics of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Kinematics of Multibody Systems with Open-loop Structures 2.2.3 Holonomic and Non-Holonomic Constraints . . . . . . . . . . . . . 2.3 Dynamics of Multibody Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Synthetic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.1 Principle of Linear Momentum . . . . . . . . . . . . . . . . . 2.3.1.2 Principle of Angular Momentum . . . . . . . . . . . . . . . 2.3.2 Analytical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 D’ALEMBERT’s Principle . . . . . . . . . . . . . . . . . . . . 2.3.2.2 LAGRANGE’s Equations of the 2nd Kind . . . . . . . 2.3.2.3 Multibody Systems with Additional Constraints . . 2.3.2.4 VORONETS’ Equations . . . . . . . . . . . . . . . . . . . . . . 2.3.2.5 APPELL’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Forces Related to Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Force of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Spring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Friction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.1 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2 Viscous Friction Force . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.3 Dry Friction Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.4 An Approach of Mathematical Friction Modeling .

7 7 9 9 11 12 15 15 15 16 19 19 20 23 25 27 29 29 30 33 33 34 36 41

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3

4

Contents

Mathematical Methods and Elements of Control Theory . . . . . . . . . . . 3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis and Modification of the Model – the Role of an Experiment 3.3 Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Stability of Stationary Motions . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Introduction of Dimensionless Variables and Method of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Remarks on Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 3.4 Some Aspects from Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Motivating Example and General Formulations . . . . . . . . . . . 3.4.2 Open- and Closed-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.2 Relative Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.3 Minimum-Phase Condition and Invariant Zeros . . . 3.4.4 Recapitulatory Example and High-Gain Control . . . . . . . . . .

47 47 48 48 48 50 55 59 61 61 63 64 64 65 67 70

Wheeled Locomotion Systems – Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1 Rolling - the Exclusive Engineering Idea for Locomotion . . . . . . . . . 73 4.2 Two-Wheel Planar Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.1 Two-Wheel Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.3.1 APPELL’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.3.2 LAGRANGE’s Equations with Multipliers . . . . . . . 85 4.2.3.3 VORONETS’ Equations . . . . . . . . . . . . . . . . . . . . . . 87 4.2.3.4 Synthetic Method – Basic Theorems of Dynamics . 88 4.2.4 Analysis of the Equations of Motion . . . . . . . . . . . . . . . . . . . . 91 4.3 The Three-Wheel RoboCup Player “Lukas” Utilizing Omnidirectional Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.3 Control of the RoboCup Player “Lukas” . . . . . . . . . . . . . . . . . 102 4.3.3.1 Velocity Control Loop . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.3.2 Positioning “Lukas” . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.3.3 High-Level Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4 Non-Holonomic Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4.1 The Idea and Performance of the Mobile Robot . . . . . . . . . . . 106 4.4.2 Mechanical Model of a Planar Four-Bar Mechanism with Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.3 Kinematics of the Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4.4 Dynamics of the Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.4.5 Dynamic Simulations of the Locomotion of the Planar Four-Bar Mechanism with Wheels . . . . . . . . . . . . . . . . . . . . . . 116

Contents

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4.5

Trimaran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5

Walking Machines – Walking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1 Human Walking and Running: History and General Remarks . . . . . . 125 5.2 Dynamic Models of Walking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.1 Dynamics of Human Trunk Locomotion . . . . . . . . . . . . . . . . . 128 5.2.2 Dynamics of Extremities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2.2.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2.2.2 Mathematical and Physical Pendulum . . . . . . . . . . . 134 5.2.2.3 Double Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.3 Simplest Integrative Model of Walking – Inverted Pendulum 139 5.2.3.1 Inverted Pendulum with Fixed Length . . . . . . . . . . . 139 5.2.3.2 Inverted Pendulum with Variable Length . . . . . . . . . 140 5.2.3.3 Inverted Double Pendulum . . . . . . . . . . . . . . . . . . . . 142 5.2.3.4 Inverted Double Pendulum with a Rigid Body . . . . 144 5.2.4 The Three-Body Model of Walking . . . . . . . . . . . . . . . . . . . . . 146 5.2.4.1 Kinetic Energy of the Upper Body . . . . . . . . . . . . . . 147 5.2.4.2 Kinetic and Potential Energies of the Leg . . . . . . . . 147 5.2.4.3 LAGRANGE’s Equations for the Three-Body Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2.5 Five-Body Model of a Walking Robot . . . . . . . . . . . . . . . . . . . 150 5.2.5.1 Kinetic Energy of a Lower Leg . . . . . . . . . . . . . . . . . 151 5.2.5.2 Potential Energy of the System and the Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.2.5.3 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 155 5.3 Robustness and Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4 Generalization – Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6

Worm-like Locomotion Systems – Crawling . . . . . . . . . . . . . . . . . . . . . . . 161 6.1 Modeling Worm-Like Locomotion Systems (WLLS) . . . . . . . . . . . . . 161 6.2 Straight Discrete Worms with Contact via Spikes, [137] . . . . . . . . . . 163 6.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2.3 Geometric Interpretation of the Results . . . . . . . . . . . . . . . . . . 168 6.3 Straight Discrete Worms with Contact via Dry Friction . . . . . . . . . . . 170 6.4 Worm-Like Locomotion based on Friction with Anisotropic Friction Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.4.1 System of Two Mass Points and a Kinematic Drive . . . . . . . . 171 6.4.2 System of Two Mass Points with a Linear Elastic Element . . 173 6.4.2.1 Mechanical Model and Equations with a Linear Elastic Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.4.2.2 Application of the Averaging Method . . . . . . . . . . . 175 6.4.2.3 Stationary Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.4.2.4 Conditions of Stability . . . . . . . . . . . . . . . . . . . . . . . . 183

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6.4.3

6.5

6.6

7

System with Two Mass Points and a Nonlinear Elastic Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.4.3.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.4.3.2 Stationary Regime and Conditions of Stability . . . . 187 6.4.3.3 Discussion of Results and Graphical Illustrations . . 188 6.4.4 System of n Mass Points with Kinematic Constraints . . . . . . 191 6.4.4.1 Mechanical Model and Equations of Motion . . . . . 191 6.4.4.2 Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . 193 Worm-like Locomotion Based on Periodic Change of Normal Forces200 6.5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.5.2 Vibration-Driven Robot with One Moveable Internal Mass . 200 6.5.2.1 Mechanical Model and Equation of Motion . . . . . . 200 6.5.2.2 Steady-State Motion of the System in the Case of Small Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.5.2.3 Vibration-Driven Robot with One Unbalance Exciter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.5.2.4 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . 211 6.5.3 Vibration-Driven Robot with Two Unbalance Exciters . . . . . 213 6.5.3.1 Mechanical Model and Equations of Motion . . . . . 213 6.5.3.2 Application of the Method of Averaging . . . . . . . . . 216 6.5.3.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Worm-like Locomotion Based on Viscous Friction . . . . . . . . . . . . . . . 226 6.6.1 Two Mass Points Subjected to a Kinematic Constraint . . . . . 226 6.6.1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . 226 6.6.1.2 Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . 228 6.6.1.3 Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.6.2 A Rigid Body Acted upon by a Periodic Force . . . . . . . . . . . . 232 6.6.3 A Rigid Body with a Moving Internal Mass . . . . . . . . . . . . . . 234 6.6.4 Two Bodies Connected by a Spring . . . . . . . . . . . . . . . . . . . . . 235 6.6.4.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . 235 6.6.4.2 Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . 236 6.6.5 System of Three Mass Points with Kinematic Constraints . . 239 6.6.5.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.6.5.2 Smooth Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.6.5.3 Motion with Viscous Friction . . . . . . . . . . . . . . . . . . 243 6.6.5.4 Comparison between Viscous and Dry Friction . . . 244

Adaptive Control Approach to Worm-like Locomotion Systems . . . . . 247 7.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.2 The Worm-like Locomotion System as a Dynamic Control System . 249 7.3 Adaptive High-Gain Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.3.1 Motivation and History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.3.2 Control Objective and Adaptation Law . . . . . . . . . . . . . . . . . . 252 7.3.3 Feedback and Controllers for Systems with Different Relative Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

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7.4

Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.4.1 Simulation using a 2-D COULOMB Model . . . . . . . . . . . . . . 257 7.4.1.1 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.4.1.2 Tracking of a Time-Shifted Sine Signal . . . . . . . . . . 257 7.4.1.3 Tracking an “Optimal” Kinematic Gait . . . . . . . . . . 259 7.4.2 Simulation using a 3-D COULOMB Model . . . . . . . . . . . . . . 261 7.4.2.1 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.4.2.2 Tracking of an “Optimal” Kinematic Gait . . . . . . . . 261 3-D Animations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

7.5 8

Prototypes of Worm-Like Locomotion Systems . . . . . . . . . . . . . . . . . . . . 265 8.1 Worm-like Locomotion System with Two Stepping Motors . . . . . . . . 265 8.2 Locomotion Systems with One Unbalance Rotor System . . . . . . . . . . 267 8.3 Locomotion Systems with Two Unbalance Rotors . . . . . . . . . . . . . . . 268 8.4 Vibration-Driven Robot – “MINCH Robot” . . . . . . . . . . . . . . . . . . . . . 269 8.5 Miniaturized Vibration-Driven Robot with a Piezo Actuator . . . . . . . 273 8.6 Worm-like Locomotion System Based on Smart Materials . . . . . . . . 274

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287


Acronyms

BDF backward differentiation formula CC contractile element DC dissipative element DOF number of degrees of freedom LTI linear time-invariant (system) MBS multibody system MIMO multi input multi output (system) ODE ordinary differential equation PEC parallel elastic (contractile) element SEC serial elastic (contractile) element SISO single input single output (system) WLLS worm-like locomotion system

xv


Symbols

A parameter A matrix in a stability/control problem AS cross-section δ A virtual work a length , index a absolute acceleration ai j elements of a matrix B matrix in a control problem B magnetic field C stiffness matrix , matrix in a control problem C center of mass c stiffness of a linear spring D damping matrix D dissipation function D angular momentum d coefficient of viscous friction E total energy , YOUNG’s modulus E orthonormal rotation matrix e(t) error unit vectors of an inertial frame ei Ei unit vectors of a body-fixed frame Ei j elements of an orthonormal rotation matrix E F force Fr FROUDE number fi (t) excitation f b a coefficient in the equations of constraints g gravitational acceleration H(s) transfer function h geometric parameter

xvii

xviii

h feedback of system information h(a, b, x) HEAVISIDE function I electrical current I unit matrix mass moment of inertia Ja mass moment of inertia Jii product of inertia Jik k coefficient in a control law number of segments ks k(t) time-varying high-gain parameter L LAGRANGE function distances between mass points Li (t) original spring length l0 kinematic gait l j (t) worm length lw M total mass of a system M moment coordinates of a moment Mi m mass , index n number of degrees of freedom , index n unit vector in the normal direction P(u) function (in the context of motion stability of WLLS) p parameter p total momentum of a rigid body Q matrix in a control problem generalized force Qa q vector of generalized coordinates generalized coordinate qa δ qa virtual displacement generalized velocity q˙a generalized acceleration qä R radius r position vector in an inertial frame r˙ velocity r¨ acceleration δr virtual displacement S acceleration energy s LAPLACE variable T kinetic energy , characteristic time t time dimensionless time t∗ cycle time tc U potential energy u(t) control input V matrix in a control problem

Symbols

Symbols

xix

V Vn v Wn w X(s) x xc xi Xi Y (s) y(t) yc yref (t) z(t) zi (t)

volume , generalized potential , stationary velocity part of the velocity (in the context of a WLLS) velocity part of the velocity of the center of mass (in the context of a WLLS) part of the velocity of the center of mass (in the context of a WLLS) input (LAPLACE transform) position vector in a body-fixed frame coordinate of the center of mass cartesian coordinates cartesian coordinates input (LAPLACE transform) system output coordinate of the center of mass reference signal function disturbances (in a stability problem)

α α a2 a1 β β a2 a1 d γ Δi δi j ε ζ η κ λ0 λb μ Πai θi ξ ρ σ (A) τ τ1 ϕ ϕ0 ψ χ Ω ω

exponent , dimensionless parameter coefficient scheme dimensionless parameter coefficient scheme (“anholonomy object”) tuning parameter HURWITZ determinant KRONECKER symbol small parameter coordinate of a body-fixed frame coordinate of a body-fixed frame dimensionless parameter original spring length LAGRANGE multiplier coefficient of dry friction generalized force (in the context of APPELL’s equation) angle coordinate of a body-fixed frame , slow variable angle of the friction cone , density spectrum of a matrix A unit vector in the tangential direction time shift angle phase shift angle , dimensionless parameter dimensionless parameter frequency frequency

xx

Symbols

ω ω ωi ωi j

skew matrix angular velocity coordinates of the angular velocity elements of the skew matrix ω

E3 EUCLIDean space N set of natural numbers R set of real numbers vector space of m-dimensional vectors with real elements Rm o(·) , O(·) BACHMANN-LANDAU-notation

Chapter 1

Introduction

1.1 The Fascination of Locomotion A plane circles on a low-level flight above a town. It will not even land and virtually nobody is aboard, yet 150,000 people in Hamburg want to see it. It is Saturday morning, time to sleep in. But tens of thousands of people are on their way to Frankfurt, just to catch a glimpse of the approach for landing of an airplane in the dense morning fog, see. Fig. 1.1. This time the plane lands, but nobody will board because the Airbus A380 - which is the reason for the fuss in both cases - is still in the testing stage.

Fig. 1.1 The Airbus A380 during touchdown in fog at Frankfurt International Airport [Photo courtesy of Fraport AG Frankfurt]

Locomotion is a source of great fascination for many people. For centuries it has been a major challenge for engineers to imitate the excellence in efficiency, speed and mobility found in nature, such as in flying hummingbirds, running cheetahs, swimming dolphins and the like.

K. Zimmermann et al., Mechanics of Terrestrial Locomotion, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-88841-3 1,

1

2

1 Introduction

The arc of historical progression is drawn from the wing concepts of LEONARDO DA VINCI in his work “Sul volo degli uccelli” (“On the Flight of Birds”) through the parachute drafts by CAYLEY resembling the shape of the fruit of composite plants and LILIENTHAL’s studies of the first sustainable glider to the present. And these are only examples for flying locomotion. But this type of locomotion especially shows how a bionic approach should look. It is not the propulsion and lift of a bird that is achieved by muscle power simultaneously on a pair of wings that is copied and technically realized in airplanes. Rather, the knowledge of the creation of lift as observed on a bird’s wings led to the airfoil principle used in many technical aviation systems. Also, the “winglet”, which is a wingtip extension to reduce unwanted turbulence at the edge of an airfoil, is a design idea inspired by nature. Even today, mankind is trying to find even faster systems for terrestrial and aquatic locomotion - often with large energy demands -to challenge nature’s leading position regarding speed of locomotion.

1.2 Biologically inspired Robotics The urge for individual mobility has led to the development of airplanes, trains and cars, which are much faster locomotion systems than human legs. Nevertheless, pedal locomotion systems and humanoid robots are main focal points of worldwide research in biologically inspired robotics. Biomimetical robots are developed by engineers and scientists in the life sciences by joint integrative analysis (i.e., combining different analytical layers) of the construction and functionality of animal locomotion systems and the transfer of the construction principles to technical fields. Currently, the development of “walking machines,” i.e., pedal locomotion, dominates the research of biologically inspired locomotion systems. The known solutions for “walking machines” range from uniquies for fundamental research to series manufacturing of commercial products for the entertainment industry. From bipedal to octopedal constructions, almost all biological prototypes have been constructed by engineers. Due to the dedication of BERNS of the University of Kaiserslautern, the walking machines catalog (www.walking-machines.org) has given an excellent overview of available walking machines worldwide for many years. In recent years the research focus has been shifting more and more towards legless, or apedal, locomotion systems that are modeled after snakes or worms, among others. The motivation for this research direction is of very different nature. Apart from the developments by HIROSE, which he published in his book Biologically Inspired Robots in 1993, the studies of MILLER are well-known. MILLER’s goal was to make the motion of animated worms look as realistic as possible in feature films, see Fig. 1.2.

1.2 Biologically inspired Robotics

3

Fig. 1.2 Computer animation of an earthworm [Photo courtesy of G. MILLER]

This relates to a recent application field of robotics, the entertainment industry. This area is often considered without the necessary seriousness, but it plays an ever increasing role in its inspiration from and for robotics, not to mention its recent financial success. Currently, the efficient applications expected of worm-like locomotion systems (WLLS) are in the area of the inspection of cable or pipeline systems or for the rescue of people buried after an earthquake. Also planned are applications in medical engineering for diagnosis systems and minimally invasive surgery, see Fig. 1.3.

Fig. 1.3 Robot endoscope by HUBA et al. [68]

Legged or wheeled robots are especially out of place inside the human cardiovascular system. One possible model for mobile robots that can cope with the aforementioned challenges seems to be a completely elastic structure that exhibits peristaltic motion similar to an earthworm. This and other application areas create new

4

1 Introduction

demands regarding dimensions, actuation systems and general functional principles. A main criterion is the preferably autonomous operation of this kind of robot. To efficiently operate in pipeline systems or earthquake regions, information transfer and energy supply should be wireless. The latter is a major problem on its own for all autonomous system developed so far, despite the progress in energy storage techniques and more and more powerful accumulators. Nevertheless, the field of mobile robots that follow biological principles is continually expanding.

1.3 RoboCup RoboCup is a challenge in which teams of mobile robots play soccer against each other. From the scientific point of view, robot soccer is an application of multi-agent systems. Robot soccer players are autonomous, intelligent and flexible agents that can communicate with other players and cooperate to perform a given task. The Robot World Cup Initiative (RoboCup) was founded in 1997 with the goal of creating a standardized environment in which to combine problems from different fields within the engineering sciences. This allows an impartial comparison of the results of different working groups. For that reason an annual RoboCup world championship is held along with an accompanying conference and additional talks. This championship is one of the most important competitive exhibitions in mobile robotics. Moreover, RoboCup provides a wide range of areas for academic education. In addition to the world championship, there are several regional championships, such as those in Europe and Asia. Different leagues with different sets of rules and specifications were founded within the RoboCup framework. The rules and requirements are changed by a commission in cooperation with the teams on a yearly basis to adjust to technical progress and to make the game more interesting and lifelike. Later, one of the leagues will be presented in more detail, as the authors describe taking part in this particular league. The mechanics of the robots used are explained in detail in Chapter 5. In the small-sized league up to five robots per team play on a field measuring 4.9 m by 3.4 m. The base area of each robot has to fit inside a circle with a diameter of 180 mm. The maximum height of a robot is 225 mm (or 150 mm if the global vision system is used). A video camera is installed above the field of play, which both teams can access. Self-localization as well as the localization of the ball and the building of a world model is simplified by this setup. Other league configurations are the simulation league, the mid-sized league, the Sony four-legged league and the humanoid league. The RoboCup championship has not only pushed research in information technology and control theory but has also led to many innovations in mechanics. One of the most interesting developments is the omnidirectional wheel system commonly used in the small-sized league. In contrast to the “classical” wheel these wheels provide movement in the direction of the wheel axis while avoiding side slip, see Fig. 1.4. Together with an intelligent positioning of the wheels on the robot, this

1.4 Summary Remarks and a Diagram of the Book

5

Fig. 1.4 RoboCup Player LUKAS (Ilmenau University of Technology, 2008)

leads to a very high mobility of the system. The effects on the kinematics and dynamics of the robot are discussed in Section 4.3.

1.4 Summary Remarks and a Diagram of the Book Recent developments in microsystems technology and new materials and production technologies make it possible to turn new ideas into reality, particularly on the micro- and nanoscales. On the macroscopic scale, the biggest challenge scientists face is the need to save materials and energy in connection with transportation applications. Locomotion in the nanoscopic or the macroscopic world, as a slow worm or as a high-speed wide-bodied aircraft, based on a biological model or using unnatural wheels, will remain an exciting field of research in the future and is already a route to success. What is needed to take part in this field of research? With this textbook the authors are trying to give an understandable answer from the point of view of mechanics. Consequently, the purpose of Chapters 2 and 3 is to quickly introduce enough basics of multibody dynamics, selected mathematical methods and elements of control theory so that we can describe some examples of wheeled locomotion systems (Chapter 4), walking models (Chapter 5) and, notably, worm-like locomotion systems (Chapter 6). See Fig. 1.5 for a general overview of the textbook structure.

6

1 Introduction Adaptive Control Approach of WLLS Chapter 7 Wheeled Locomotion Systems - Rolling

Walking Machines - Walking

Chapter 4

Prototypes of WLLS

Worm-like Locomotion Systems (WLLS) - Crawling

Chapter 5

Chapter 8

Chapter 6

Mathematical Methods and Elements of Control Theory

Mechanical Background Chapter 2

Introduction

Chapter 3 Chapter 1

Fig. 1.5 Diagram of the textbook

In Chapter 7 the reader will find aspects of adaptive control applied to worm-like locomotion system. The authors then conclude this textbook with an explanation of some prototypes of non-pedal locomotion systems developed at the Department of Technical Mechanics at the Ilmenau University of Technology (Chapter 8). Additionally, these mobile robots are presented in a more impressive video form at www.tu-ilmenau.de/mb/tm-textbook.html. All presented simulations and measurements in the book were done in MATLAB Version 7.1 (R14SP3), MAPLE Version 8.0, LabVIEW Version 8.2, ANSYS Version 11.0, WINanalyze , and alaska Version 4.0 which are registered trademarks of the companies “The MathWorks”, “Waterloo Maple Inc.”, “National Instruments”, “Ansys Inc.”, “Mikromak”, and “Institute of Mechatronics Chemnitz e.V.”, respectively.

Chapter 2

Mechanical Background

2.1 Modeling The multibody system is an appropriate model for the locomotion systems considered in this book. Therefore, the basic kinematics and dynamics of such systems are briefly introduced in this chapter. The starting point is a rigid body as a basis element of a multibody system. Definition 2.1. A multibody system (MBS) is a finite set of rigid bodies that are physically and/or geometrically interconnected with each other and with a ground not belonging to the MBS. Physical coupling is specified by the applied forces/torques (e.g., spring and damper forces). Constraints describe the geometric interconnections (e.g., prismatic and rotational joints).

Fig. 2.1 Example of a multibody system

The MBS is the most commonly used mechanical model for the analysis of mechatronic systems. There are different possibilities to classify MBSs, for example, according to the system’s degrees of freedom or to the type of constraints. One K. Zimmermann et al., Mechanics of Terrestrial Locomotion, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-88841-3 2,

7

8

2 Mechanical Background

of the most important is the classification based on the topology of the system, i.e., based on the alignment of the bodies. a) MBS with a kinematic tree structure The MBS with a kinematic tree structure consists of pairs of bodies, each having only one sequence of linking joints and force coupling elements (without duplication) (also: “open kinematic chain”; example: robot with spherical coordinates).

Fig. 2.2 MBS in the form of an open kinematic chain

b) Constrained mechanical system An MBS with a kinematic tree structure and additional constraints on the bodies is called a “constrained mechanical system” (also: “closed kinematic chain”; example: four-bar linkage).

Fig. 2.3 MBS in the form of a closed kinematic chain

2.2 Kinematics of Multibody Systems

9

2.2 Kinematics of Multibody Systems 2.2.1 Kinematics of Rigid Bodies For the analysis of a rigid body, a body-fixed coordinate system Σ is introduced with the origin at an arbitrary but fixed point Ω of the rigid body. The position of the rigid body in an inertial coordinate system Σ is determined by the set of all position vectors − → (2.1) 0P =r(P) =rΩ +x =rΩ + xi Ei .

C

X2 P

e2

x

G‘

rW

G 0

E3

S

r

e3

x3

e1

x2

E2 E1

x1

X1

X3 Fig. 2.4 A rigid body with an inertial and a body-fixed coordinate system

Remark 2.1 EINSTEIN’s summation convention is used where appropriate, i.e., when an index occurs more than once in the same expression, the expression is implicitly summed over all possible values for that index. According to the definition of the rigid body it holds: xi = const for i = 1, 2, 3 . The movement of the rigid body is given by: rΩ =rΩ (t) ,

Ei = Ei (t) (i = 1 , 2 , 3) .

(2.2)

The velocity vector follows from the time derivative of the position vector: ˙ = r˙Ω + xi E˙i . r(t) The time derivative of the body-fixed unit vectors Ei (i = 1 , 2 , 3) can be obtained by the orthonormal rotation matrix E = (Ei j ) (i, j = 1 , 2 , 3), which describes the orientation of the body-fixed coordinate system Σ relative to the inertial coordinate system Σ . For orthonormal matrices it is given that E−1 = ET and, therefore, it follows that: E˙i = E˙ik ek = E˙ik E jk E j = ω i j E j . (2.3)

10


The angular velocity vector follows from the three main elements of the skew symmetric matrix ω = (ω i j ) = ω 23 E1 + ω 31 E2 + ω 12 E3 ω where

E˙i = ω × Ei ,

(i = 1 , 2 , 3) .

(2.4)

(2.5)

First, two special cases are considered: a) Translation: The body-fixed unit vectors Ei are not time-dependent for i = 1 , 2 , 3. The vector x2 −x1 is a constant vector. The movement of a point P2 is obtained by parallel translation of the movement of another point P1 , see Fig. 2.5.

S

G‘ x1

P1 x2 - x1

x2 0

P2

G

Fig. 2.5 Special case of rigid body motion: translation

b) Rotation: The vectorrΩ is not time-dependent. Then - without loss of generality - the coincidence of the coordinate origins 0 = Ω can be assumed, i.e.,rΩ =0. It holds thatr(t) = xi Ei (t), where xi = const characterizes the rigid body. This means the body performs a (rotatory) movement of the body-fixed frame Ei about the origin Ω. Summarizing, the general movement of a rigid body consists of a translation and a rotation about an arbitrary fixed point Ω of the body. The velocity vector is: ˙ = r˙Ω +x˙ = r˙Ω + ω ×x . r(t)

(2.6)

The vector r˙Ω denotes the translational part, which is the same for all points of the ×x is the rotational part of the velocity, which depends on P. The straight body; ω } is called the instantaneous axis of rotation of the body. The acceleration line {Ω , ω vector ¨ = r¨Ω + ω ˙ ×x + ω × (ω ×x) (2.7) r(t) can be obtained for any point of the rigid body. Exercise 2.1. k for each Using formulas (2.3) and (2.4) find the vector of the angular velocity ω body k = 1 , 2 , 3 , 4 of the robot presented in Fig. 2.10. Show the validity of the


11

k=ω k ,0 = ω k ,k−1 + ω k−1 ,0 ! formula ω i , j means the vector of the angular velocity during the motion of the body Here, ω with number i relative to the body with number j. The body with number 0 is the ground.

2.2.2 Kinematics of Multibody Systems with Open-loop Structures First, an MBS with a kinematic tree structure is considered:

X3

1

1

k

3 2

e3

Rk

xk

1

0

e2

k

k

K

E1 k

Pk

rk

X2

E3

E2

Ck

C1 E 1

1

e1

E3

E2

E3

K

CK E1 K

E2 K

X1 Fig. 2.6 MBS with an open-loop structure and body-fixed cartesian coordinate systems

The rigid bodies of the system are numbered 1 , . . . , K. The inertial coordinate system in the reference body 0 is denoted by {0 ,ei }, the center of mass of a body k (k= 1 , 2 , . . . , K) is denoted by Ck , and a body-fixed coordinate system is denoted by Ck , Ei . This means k

− → 0Pk =rk = Rk +xk = Rk + xi Ei ,

− → 0Ck = Rk = Xi ei ,

k

Ei = Ei j e j .

k

k

(2.8)

k

For each free rigid body Xi , Ei j depends on the 6 parameters x1 , x2 , . . . , x6 : k

k

k

Xi = Xi (x1 , x2 , x3 ) , k

k

k

k

k

k

k

Ei j = Ei j (x4 , x5 , x6 ) , k

k

k

k

(2.9)

k

with x1 , x2 , x3 being for example cylindrical or spherical coordinates and x4 , x5 , x6 k

k

k

k

k

k

for example EULER or CARDAN angles. The geometric constraints of the bodies are given by the equations Z m (x1 , x2 , . . . , x6 ) = 0 , (m = 1 , . . . , M < 6 K ; s = 1 , . . . , 6) , k

k

k

(2.10)

12


with rank

m

∂Z ∂ xs

= M.

They are satisfied using n parameters qa by xs = xs (qa ) , s = 1 , . . . , 6 ; a = 1 , . . . , n ; k = 1 , . . . , K , k

(2.11)

k

which means the following identity holds: Z m xs (qa ) ≡ 0. k

Definition 2.2. If there is a one-to-one mapping from the set of all possible positions at time t to the set of parameters (q1 , q2 , . . . , qn ), it follows that − → 0Pk =rk =rk (xi , q1 , q2 , . . . , qn ,t) .

(2.12)

The number n is the degree of freedom (DOF) of the MBS, and the parameters qa are called generalized coordinates. The position of the MBS is described a) in the EUCLIDean space E3 by: Rk = Xi (qa )ei – position of the centers of mass of each body, k

Ei = Ei j (qa )e j – orientation of the body-fixed frames of each body, k

k

b) in the configuration space Rn by: qa xs = xs (qa ) , s = 1 , . . . , 6 , a = 1 , . . . , n , k = 1 , . . . , K . k

k

The MBS is called scleronomous ifrk =rk (xi , q1 , q2 , . . . , qn ); otherwise, it is called rheonomous, i.e.,rk =rk (xi , q1 , q2 , . . . , qn ,t).

2.2.3 Holonomic and Non-Holonomic Constraints For numerous technical systems it makes sense to use more than n parameters to describe the position of the system, i.e., see (2.12), − → 0Pk =rk =rk (xi , q1 , q2 , . . . , qm ,t) , (m > n) .

(2.13)

Then, these m parameters are not independent, and in mathematical terms this means there are constraints for q1 , q2 , . . . , qm . The classification of the constraints into two classes is of fundamental relevance for the description of the MBS.


13

Definition 2.3. Constraints of the form (2.14) f b (q1 , q2 , . . . , qm ) = f b (q) = 0 , (b = 1 , 2 , . . . , r < m) , b with q = (q1 , q2 , . . . , qm ) and rank ∂∂ qf a = r , (a = 1 , 2 , . . . , m) are called holonomic. Other forms of constraints containing velocities that cannot be transformed into this form are called non-holonomic, i.e., these constraints cannot be integrated to yield constraints in positions. A holonomic constraint is also called geometric because it limits the system’s position. Using equation (2.14) we can find n = m − r independent generalized coordinates qa (a = 1 , 2 , . . . , n = DOF) that completely describe the position of the MBS. Non-holonomic constraints limit the velocities and do not impose restrictions on the position (coordinates) of the system. For locomotion systems that are considered in the following chapters, there are often non-integrable differential (i.e., non-holonomic) constraints in q˙a of the form f b a (q) q˙a = 0 , (a = 1 , 2 , . . . , m ; b = 1 , 2 , . . . , r < m) , rank( f b a ) = r .

(2.15)

These constraints cannot be reduced to the form (2.14) with an integration procedure. Example 2.1 We consider a wheel rolling without slip along a straight line, see Fig. 2.7.

E1

E2

j

E3 C

e3

R

e2 0

e1

M

Fig. 2.7 Rolling wheel on a straight line

The position of the wheel is given by the vector of generalized coordinates q = (q1 , q2 , q3 ) = (xc , yc , ϕ ). The coordinates (xc , yc ) identify the center of mass C. From equation (2.6) and given that the point M is the instantaneous center of rotation, i.e., r˙M = 0, it follows that −→ −→ × MC = ω × MC = (ϕ˙ E3 ) × (Re3 ) = R ϕ˙ e1 , r˙c = x˙c e1 + y˙c e2 = r˙M + ω and for the coordinates of the velocity

14


x˙c = R ϕ˙ ,

y˙c = 0 .

(2.16)

Integration of (2.16) yields xc = R ϕ +C1

yc = C2 ,

(2.17)

with the initial values xc (0) = 0, yc (0) = R, ϕ (0) = 0. We obtain yc − R = 0 xc − R ϕ = 0 , or q1 − R q3 = 0 ,

q2 − R = 0 .

(2.18)

Hence, the system in Fig. 2.7 is holonomic.

Example 2.2 We again discuss the problem of a wheel rolling without slipping, not following any particular curve, i.e., the trajectory of the point of contact is not prescribed, see Fig. 2.8.

E2

E1 j E3 e3

C

R

e2 0

e1

q M

Fig. 2.8 Rolling wheel on a plane

Now, the position and the orientation is determined by the vector of generalized coordinates q = (q1 , q2 , q3 , q4 ) = (xc , yc , ϕ , θ ). The angle θ gives the orientation of the plane of the wheel. Because the point M is again the instantaneous center of rotation, the velocity of the wheel center is −→ × MC = (θ˙ e3 + ϕ˙ E3 ) × (Re3 ) r˙c = x˙c e1 + y˙c e2 = r˙M + ω = θ˙ e3 + ϕ˙ (− sin θ e1 + cos θ e2 ) × Re3 = R ϕ˙ cos θ e1 + R ϕ˙ sin θ e2 . Thus the constraints are

2.3 Dynamics of Multibody Systems

15

x˙c = R ϕ˙ cos θ ,

y˙c = R ϕ˙ sin θ

(2.19)

q˙2 − R sin q4 q˙3 = 0 .

(2.20)

or q˙1 − R cos q4 q˙3 = 0 ,

The relationship between the velocities (2.20) is non-integrable, and thus the system must be classified as non-holonomic.

2.3 Dynamics of Multibody Systems The synthetic and the analytical methods are two methods to classify almost all formalisms for describing the dynamics of MBSs. The synthetic method applies the principle of linear momentum (NEWTON’s second law) and the principle of angular momentum to all free-cut partial bodies in order to derive the differential equations describing the motion. On the other hand, analytical methods use one closed concept of the complete system (for example, the kinetic energy). Some selected analytical methods that are well-suited for efficiently deriving the system equations for locomotion systems are introduced in the following sections.

2.3.1 Synthetic Method 2.3.1.1 Principle of Linear Momentum The mass elements dm are under the action of the forces d F, see Fig. 2.9. The position of the center of mass is given by

r dm

(V )

rc =

dm

(V )

the sum of the acting forces by F =

(V )

d F .

,

(2.21)

16


x3

dF

X2

A

dm

E3 x

r

C

0

e3

E2 E1

rC e1

x2

x1

X1

X3 Fig. 2.9 Rigid body with distributed forces in an inertial coordinate system

The total momentum of the rigid body is defined as

p =

r˙ dm = mr˙c .

(2.22)

(V )

Principle of linear momentum: The center of mass of a rigid body moves as a particle, the mass of which coincides with the mass of the body and which is acted upon by a force equal to the resultant external force applied to the body: p˙ = mr¨c = F .

(2.23)

2.3.1.2 Principle of Angular Momentum The principle of angular momentum describes the rotation of a rigid body analogous to the principle of linear momentum used for translational motion. According to EULER this second principle is an independent one (in general not derivable from the principle of linear momentum). According to Fig. 2.9 the angular momentum and the moment of forces with respect to the origin 0 of the inertial coordinate system are introduced as

D0 =

r ×r˙ dm ,

0 = M

(V )

r × d F .

(2.24)

(V )

With respect to the center of mass C, the corresponding equations are

Dc = (V )

x ×x˙ dm ,

c = M (V )

x × d F .

(2.25)


17

Principle of angular momentum: The total time derivative of the angular momentum vector D0 is equal to the moment 0 of the resulting force acting on the body with respect to the origin 0 of the inertial M coordinate system 0. D˙ 0 = M (2.26) The total time derivative of the angular momentum vector Dc is equal to the moment c of the resulting force F with respect to the body’s center of mass C M c. D˙ c = M

(2.27)

The principle of angular momentum of the form “derivative of angular momentum is A equal to moment” holds true for the center of mass of the body. However, D˙ A = M does not hold in general for an arbitrary point A of the rigid body. With respect to the angular momentum for an arbitrary point A, the principle takes the form: A + mrÄ ×a , D˙ A = M

(2.28)

− → − → withrA = 0A and a = AC, see Fig. 2.9. For practical use the principle of angular momentum is needed in form of vector coordinates. For the angular momentum vector Dc using (2.6) we obtain

Dc =

x ×x˙ dm =

(V )

×x) dm = Jik ωi Ek . x × (ω

(2.29)

(V )

Here, J = (Jik ) is the inertial tensor consisting of the mass moments of inertia J11 =

(x2 )2 + (x3 )2 dm ,

(V )

J22 =

(x1 )2 + (x3 )2 dm ,

(2.30)

(V )

J33 =

(x1 )2 + (x2 )2 dm

(V )

and the products of inertia J12 = J21 = −

x1 x2 dm ,

(V )

J13 = J31 = −

x1 x3 dm ,

(V )

J23 = J32 = −

(V )

x2 x3 dm .

(2.31)

18


Then, the inertia matrix is ⎞ J11 J12 J13 J = (Jik ) = ⎝J21 J22 J23 ⎠ . J31 J32 J33 ⎛

(2.32)

The products of inertia Jik (i = k ; i, k = 1, 2, 3) vanish if the body-fixed coordinate c follow the system Σ : {C, Ei } is a principle axis system. From D˙ c = (Jik ωi Ek )˙= M three equations J1i ω˙ i − J2 j ω j ω3 + J3k ωk ω2 = Mc1 , J2i ω˙ i − J1 j ω j ω3 − J3k ωk ω1 = Mc2 ,

(2.33)

J3i ω˙ i − J1 j ω j ω2 + J2k ωk ωl = Mc3 . Exercise 2.2. For wheeled locomotion systems expressions are needed for the mass moments of inertia for axes parallel and perpendicular to the wheel plane. Assuming the wheel is a thin disk with mass m and radius R, calculate the mass moments of inertia J33 and J11 = J22 for the wheel shown in Fig. 2.7! For the important special case of rotation of a rigid body about a fixed principle axis (without loss of generality assumed to be E3 =e3 ) and a constant mass moment of inertia J33 , the principle of angular momentum takes the form J33 ω˙ = Mc3 .

(2.34)

The following transformation equations of the inertia tensor describe the transition to a new coordinate system {C , Ei } obtained by parallel translational displacement (known as STEINER’s theorem or the parallel axes theorem) or rotation of the original coordinate system {C , Ei }. • Parallel translational displacement (STEINER’s theorem) {C , Ei } → {C , Ei } with the vector a = ai Ei : Ji k = Jik + m (a j a j δik − ai ak ) .

(2.35)

• Rotation {C , Ei } → {C , Ei } with the rotation matrix E = (Ei j ) , (i = 1 , 2 , 3 ; j = 1, 2, 3) : Ji k = Ei i Ek k Jik .

(2.36)

Exercise 2.3. The locomotion system presented in Section 4.4 consists of four bars in a rhombus configuration, see Fig. 4.20. Compute the mass moments of inertia for a slender bar with mass m and length l with respect to axes perpendicular to the axis


19

of the bar and passing through (a) the center of mass and (b) the end of the bar (STEINER’s theorem).

2.3.2 Analytical Method 2.3.2.1 D’ALEMBERT’s Principle The concept of virtual displacement is an essential element in the formulation of the principles of mechanics. According to Sections 2.2.2 and 2.2.3 the position vector − → 0P =r(xi , q1 , q2 , . . . , qm ,t) can be written as a function of the body-fixed coordinates xi of the point P in a rigid body of the MBS, the generalized coordinates qa (a = 1 , 2 , . . . , m) and the time t. For the effective or actual displacement of the position vectorr, it holds that dr =

∂r ∂r dt . dqa + a ∂q ∂t

(2.37)

Definition 2.4. The set of differential changes of the position vectorr to a variation δ qa for fixed time t is called virtual displacement δr

δr =

∂r δ qa . ∂ qa

(2.38)

In contrast to the actual displacement dr, the virtual displacement δr is a possible (i.e., compatible with the constraints) displacement at a fixed time t. The virtual work δ A of a force F can be obtained by the scalar multiplication of F with the virtual displacement δr. The virtual work of the ideal constraint forces F (C) is zero, i.e., F (C) · δr = 0. This statement forms the content of D’ALEMBERT’s principle for rigid bodies:

(d F −r¨ dm) δr = 0 , ∀ δr ,

(2.39)

(V )

where d F is an applied active force at the mass element dm of the rigid body. D’ALEMBERT’s principle is valid for systems with holonomic and non-holonomic constraints. It is not dependent on the choice of the coordinate system since it contains a scalar product in its formulation.

20


2.3.2.2 LAGRANGE’s Equations of the 2nd Kind We consider a MBS with holonomic constraints which can be described by n independent generalized coordinates qa (a = 1 , 2 , . . . , n = DOF). LAGRANGE’s equations of the 2nd kind can be derived from D’ALEMBERT’s principle. ˙ ∂r˙ ∂r d Using the HELMHOLTZ-Identities ∂ q˙a = ∂ qa and dt ∂∂qra = ∂∂qra we obtain for the expression r¨ ∂ra , which is contained in (2.39) ∂q

∂r 1 d r¨ a = ∂q 2 dt

˙2 ∂r 1 ∂r˙ 2 − . ∂ q˙a 2 ∂ qa

(2.40)

Introducing the generalized forces

∂r ∂ qa

(2.41)

r˙ 2 dm ,

(2.42)

d F

Qa = (V )

and the kinetic energy of the rigid body T=

1 2

(V )

from (2.38)

(V )

∂r (d F −r¨ dm) · a δ qa = 0 , ∀ δ qa ∂q

concerning (2.40) - (2.42) it follows that d ∂T ∂T − − Qa δ qa = 0 . dt ∂ q˙a ∂ qa

(2.43)

Because the virtual displacements δ qa are independent for all a, we obtain LAGRANGE’s equations of the 2nd kind in the form ∂T d ∂T − = Qa , (a = 1 , 2 , . . . , n) . (2.44) dt ∂ q˙a ∂ qa The generalized forces Qa can be divided into 4 classes Qa = Qa 1 + Qa 2 + Qa 3 + Qa 4 . • 1st class Qa 1 : All generalized forces Qa 1 = Qa 1 (qa ,t) belong to this class if a potential function U(qa ,t) (so-called potential energy) exists such that


21

Qa 1 = −

∂U . ∂ qa

(2.45)

Qa 1 -forces are potential forces. Example: Spring force U(qa ) =

1 c (qa )2 → Qa 1 = −c qa . 2

• 2nd class Qa 2 : This class consists of forces Qa 2 = Qa 2 (q˙a , qa ,t) that can be calculated by the specification ∂D (2.46) Qa 2 = − a , ∂ q˙ where D(q˙a , qa ,t) is called the dissipation function and Qa 2 -forces are dissipative forces. Example: STOKES friction force D(q˙a ) =

1 d (q˙a )2 → Qa 2 = −d q˙a . 2

• 3rd class Qa 3 : Qa 3 = Qa 3 (q˙a , qa ,t) can be calculated by the formula ∂V d ∂V − a, Qa 3 = dt ∂ q˙a ∂q

(2.47)

(2.48)

using the generalized potential V (q˙a , qa ,t). Example: LORENTZ force acting on an electron with the elementary charge e ˙ ey + z˙ez in a magnetic field B = Bz ez and the velocity r˙ = x ˙ ex + y V (q˙a , qa ,t) = hab qb q˙a =

1 e Bz (y x˙ − x y) ˙ , 2

(2.49)

where hab are the coefficients of a bilinear form in qa and q˙a . • 4th class Qa 4 : All remaining forces that do not belong to classes Qa 1 - Qa 3 are classified as 4th class forces. They can only be obtained from the definition (2.41), i.e., in the special case of N single forces, N ∂r Qa 4 = ∑ Fi a , ∂ q i=1

(2.50)

whereri is the radius vector to the force application point. Example: COULOMB friction force, because of the discontinuity at the point v = 0, see Section 2.4.3.3.

22


Exercise 2.4. Using (2.44) and (2.49) define the trajectory that describes an electron with the elementary charge e in a magnetic field B = Bz ez . The initial conditions are ˙ = v0 ex . r(0) = 0 , r(0) The generalized forces can also be obtained from the virtual work using the formula δ A = Qa δ qa . This way is efficient in particular for Qa 4 -forces. Using the LAGRANGE function L = T −U −V the LAGRANGE’s equations of the 2nd kind can be obtained • for non-conservative systems ∂L ∂D d ∂L − = − a + Qa 4 , dt ∂ q˙a ∂ qa ∂ q˙ • for conservative system (D ≡ 0 , Qa 4 ≡ 0) ∂L d ∂L − = 0, dt ∂ q˙a ∂ qa

(a = 1 , 2 , . . . , n) ,

(a = 1 , 2 , . . . , n) .

(2.51)

(2.52)

Kinetic energy is an important component for describing the dynamics of an MBS as well as being an element of the LAGRANGE’s equations. Using definition (2.42) of the kinetic energy for rigid bodies, a more workable form should be given. Using (2.6) implies T=

1 2

(V )

1 = 2

(V )

1 r˙ 2 dm = 2

×x)2 dm (r˙Ω + ω

(V )

2

r˙Ω dm +

1 ×x) dm + r˙Ω (ω 2

(V )

×x)2 dm (ω

(V )

and finally −−→ 1 1 2 × Ω C) + Jik(Ω ) ω i ω k , T = TTrans + T * + TRot = mr˙Ω + mr˙Ω (ω 2 2

(2.53)

(Ω )

where Jik are the elements of the inertia tensor with respect to the system with = ωi Ei , see Fig. 2.4. In the special case of Ω = C, the origin at the point Ω and ω i.e., the center of mass C is taken for the arbitrary chosen point Ω , the general form (2.53) of the kinetic energy takes the following form: T = TTrans + TRot = which is used for practical applications.

1 ˙2 1 mrc + Jik ω i ω k , 2 2

(2.54)


23

2.3.2.3 Multibody Systems with Additional Constraints In the previous section we only considered systems with holonomic constraints, described by n independent coordinates, i.e., the virtual displacements δ qa are independent for all a. Now, we assume that for the MBS r additional non-holonomic constraints exist of the form f b a (q) q˙a = 0 , (a = 1, 2, ..., n ; b = 1, 2, ..., r < n) , rank ( f b a ) = r

(2.55)

with q = (q1 , q2 , ..., qn ). These constraints are called homogeneous. In practice MBSs also exist with additional constraints in an inhomogeneous form, i.e., f b a (q)q˙a = gb (q) . An example for such an MBS is shown in Fig. 2.10. The robot has 5 primary degrees of freedom and possesses a kitchen knife in its gripper for cutting meat.

Fig. 2.10 Five-axis serial manipulator with additional constraints (left) and a practical example (right) [59]

Exercise 2.5. Formulate the equations for the additional constraints of the robot shown in Fig. 2.10. Assume a knife-edge condition (no-side-slip condition) for the point P moving in the x-y-plane. In the case of additional constraints (2.55) the virtual displacements δ qa are not independent for all a, and the conclusion leading from equation (2.43) to the LAGRANGE equations (2.44) is not possible. However, the idea for drawing a conclusion about the dynamics of the MBS with constraints should be the same as before. Thus, we divide the sum in (2.55) into two parts:

24


f b a1 q˙a1 + f b a2 q˙a2 = 0 , (a1 = 1 , 2 , . . . , n − r ; a2 = n − r + 1 , . . . , n)

(2.56)

or f b a1 dqa1 + f b a2 dqa2 = 0 . Considering the definition of the virtual displacements, it follows that f b a1 δ qa1 + f b a2 δ qa2 = 0 .

(2.57)

Multiplying this relation by the arbitrary coefficients λb , we obtain

λb f b a1 δ qa1 + λb f b a2 δ qa2 = 0 .

(2.58)

Subtracting (2.58) from the sum (2.43)

d dt

∂T ∂ q˙a1

∂T − ∂ qa1

− Qa1

δ qa1 d ∂T ∂T − − Qa2 δ qa2 = 0 + dt ∂ q˙a2 ∂ qa2

we get

d dt

∂T ∂ q˙a1

∂T − ∂ qa1

− Qa1 − λb f a1 δ qa1 d ∂T ∂T b − − Qa2 − λb f a2 δ qa2 = 0 . + dt ∂ q˙a2 ∂ qa2 b

The n − r virtual displacements δ qa1 are independent, and the r ones δ qa2 are dependent (rank( f b a2 ) = r). Next, we choose r arbitrary coefficients λb such that d ∂T ∂T b − a a dt ∂ q˙ 2 ∂ q 2 − Qa2 − λb f a2 = 0. Then, because of the independency of δ qa1 , LAGRANGE’s equations have the following form: ∂T ∂T d − = Qa1 + λb f b a1 , (a1 = 1 , 2 , . . . , n − r) , dt ∂ q˙a1 ∂ qa1 (2.59) d ∂T ∂T b − = Q + λ f , (a = n − r + 1 , . . . , n) a2 2 b a2 dt ∂ q˙a2 ∂ qa2 or

d dt

∂T ∂ q˙a

∂T − ∂ qa

= Qa + λb f b a , (a = 1 , 2 , . . . , n) .

(2.60)

The terms Ra = λb f b a represent the reaction forces due to the r constraints (2.55). The unknown functions in (2.60) are n generalized coordinates qa and r so-called Lagrangian multipliers λb . These unknowns can be determined by n equations (2.60) and r constraints (2.55).


25

2.3.2.4 VORONETS’ Equations For special applications when the reaction forces are not of interest, it makes sense to eliminate the r unknown multipliers λb in (2.60). This tedious procedure results in the VORONETS’ equations, which are applicable to locomotion systems as well. From equations (2.56) we find for the last r generalized velocities (assuming a regular matrix f b a2 ) (2.61) q˙a2 = α a2 a1 q˙a1 , with α a2 a1 = α a2 a1 (q1 , q2 , . . . , qn ). Then, equations (2.57) takes the form

α a2 a1 δ qa1 − δ qa2 = 0 and multiplying this relation (analogous to Section 2.3.2.3) by an arbitrary coefficient λa2 , we find λa2 α a2 a1 δ qa1 − λa2 δ qa2 = 0 . In this case LAGRANGE’s equations (2.59) are d ∂T ∂T − = Q a 1 + λa 2 α a 2 a 1 dt ∂ q˙a1 ∂ qa1 d ∂T ∂T − = Q a 2 − λa 2 . dt ∂ q˙a2 ∂ qa2

(2.62)

Using (2.61) we consider the kinetic energy of the MBS as a function of n generalized coordinates qa (a = 1 , 2 , . . . , n) and n − r generalized velocities q˙ a1 (a1 = 1 , 2 , . . . , n − r): T = T t , q1 , q2 , . . . , qn , q˙1 , q˙2 , . . . , q˙n (2.63) = T t , q1 , q2 , . . . , qn , q˙1 , q˙2 , . . . , q˙n−r , q˙n−r+1 , . . . , q˙n = T t , q1 , q2 , . . . , qn , q˙1 , q˙2 , . . . , q˙n−r . We define the derivatives of the “new” kinetic energy T

∂T ∂T ∂ T ∂ q˙a2 ∂T ∂T = + = a + a α a2 a1 , a a a a 1 1 2 1 1 ∂ q˙ ∂ q˙ ∂ q˙ ∂ q˙ ∂ q˙ ∂ q˙ 2 d d ∂T ∂T ∂ T d α a2 a1 d ∂T a2 . = + α + a 1 dt ∂ q˙a1 dt ∂ q˙a1 dt ∂ q˙a2 ∂ q˙a2 dt With respect to (2.62), it follows that

26

d dt

∂T


∂ q˙a1 ∂T ∂T ∂ T d α a2 a1 = a + Qa1 + λa2 α a2 a1 + a α a2 a1 + Qa2 α a2 a1 − λa2 α a2 a1 + a ∂q 1 ∂q 2 ∂ q˙ 2 dt a 2 ∂T ∂T ∂ T d α a1 . = a + Qa1 + a α a2 a1 + Qa2 α a2 a1 + a ∂q 1 ∂q 2 ∂ q˙ 2 dt

It remains to eliminate the term

∂T ∂ qa

with (a = a1 , a2 ):

∂T ∂T ∂ T ∂ q˙c ∂T ∂ T ∂ α cd d = a+ c = a+ c q˙ , a a ∂q ∂q ∂ q˙ ∂ q ∂q ∂ q˙ ∂ qa ∂T ∂ T ∂ T ∂ α cd d = − q˙ , (c = n − r + 1 , . . . , n ; d = 1 , . . . , n − r) . ∂ qa ∂ qa ∂ q˙c ∂ qa Now, we have ∂T ∂T d ∂T = a + Qa1 + a α a2 a1 + Qa2 α a2 a1 a dt ∂ q˙ 1 ∂q 1 ∂q 2 ∂ T d α a2 a1 ∂ T ∂ α a2 d d ∂ T ∂ α a2 d d − a + a q ˙ − q ˙ α c a1 ∂ q˙ 2 dt ∂ q˙ 2 ∂ qa1 ∂ q˙a2 ∂ qc ∂T ∂T = a + Qa1 + a α a2 a1 + Qa2 α a2 a1 ∂q 1 ∂q 2 a a ∂ T d α 2 a1 ∂ α 2 d ∂ α a2 d c − + a + α a1 q˙d ∂ q˙ 2 dt ∂ qa1 ∂ qc (2.64) with

∂ α a2 a1 ∂ α a2 a1 ∂ ql ∂ α a 2 a 1 l1 ∂ α a 2 a 1 l2 d α a2 a1 = + = 0 + q˙ + q˙ , dt ∂t ∂ ql ∂ t ∂ ql1 ∂ ql2 (l1 = 1 , . . . , n − r ; l2 = n − r + 1 , . . . , n) . Taking into account (2.61) q˙l2 = α l2 l1 q˙l1 with new indices l1 and l2 , which take the same set of values as a1 and a2 , we obtain for

d α a2 a1 dt

∂ α a 2 a 1 l1 ∂ α a 2 a 1 l2 l1 d α a2 a1 = q˙ + α l1 q˙ = dt ∂ ql1 ∂ ql2

the formula

∂ α a2 a1 ∂ α a2 a1 c + α d ∂ qc ∂ qd

q˙d ,

where the indices l1 and l2 were changed to d and c, respectively. We put this formula into equation (2.64). The expression in the square brackets in (2.64) takes the form


27

d α a2 a1 ∂ α a2 d ∂ α a2 d c − + α a1 q˙d dt ∂ qa1 ∂ qc a ∂ α 2 a1 ∂ α a2 a1 c ∂ α a2 d ∂ α a2 d c d + α − − α = a1 q˙ . d ∂ qc ∂ qa1 ∂ qc ∂ qd

Let β a2 a1 d denote the coefficient of q˙d on the right-hand side, i.e.,

∂ α a2 a1 ∂ α a2 a1 c ∂ α a2 d ∂ α a2 d c + α − − α a1 , d ∂ qc ∂ qa1 ∂ qc ∂ qd (a1 , d = 1 , . . . , n − r and a2 , c = n − r + 1 , . . . , n) .

β a2 a1 d =

(2.65)

We note that the coefficients β a2 a1 d satisfy the relation: β a2 lk = −β a2 kl . Finally, we obtain the VORONETS’ equations of the form ∂T ∂ T a2 ∂T d ∂T − − α a1 = Qa1 + Qa2 α a2 a1 + a β a2 a1 d q˙d . dt ∂ q˙a1 ∂ qa1 ∂ qa2 ∂ q˙ 2

(2.66)

In the special case when the kinetic energy T and the coefficients in the constraints α a2 a1 do not depend on the generalized coordinates qa2 , using (2.66) we arrive at CHAPLYGIN’s equations: ∂T ∂ T ∂ α a2 a1 ∂ α a2 d d ∂T a2 − q˙d . (2.67) = Q + Q α + − a1 a2 a1 dt ∂ q˙a1 ∂ qa1 ∂ q˙a2 ∂ qa1 ∂ qd 2.3.2.5 APPELL’s Equations To obtain APPELL’s equations we use D’ALEMBERT’s principle (2.39) in the form

r¨ dm δr =

(V )

d F δr ,

∀δr .

(2.68)

(V )

Writing (2.61) in the form of variations

δ qa2 = α a2 a1 δ qa1

(2.69)

and taking into account equation (2.38), we find

∂r ∂r ∂r δr = a δ qa = a δ qa1 + a δ qa2 = 1 ∂q ∂q ∂q 2

∂r ∂r + a α a2 a1 a 1 ∂q ∂q 2

δ qa1 .

Introducing the vectors ga1 = ga1 (xi , q1 , . . . , qn ,t) =

∂r ∂r a2 + α a1 , (a1 = 1 , 2 , . . . , n − r) ∂ qa1 ∂ qa2

(2.70)

28


it holds that

δr = ga1 δ qa1 .

(2.71)

The velocity of a mass element dm of the MBS takes the form ∂r ∂r a ∂r ∂r a1 ∂r a2 ∂r ∂r ∂r a2 ˙ r = + + + q˙ = q˙ + a q˙ = + α a1 q˙a1 , ∂ t ∂ qa ∂ t ∂ qa1 ∂q 2 ∂t ∂ qa1 ∂ qa2 and with (2.70) it follows that

∂r +ga1 q˙a1 , (a1 = 1 , 2 , . . . , n − r) . r˙ = ∂t

(2.72)

After differentiating r˙ with respect to time, the acceleration of the mass element dm is expressed by d r¨ = dt

∂r ∂t

+

dga1 a1 q˙ +ga1 q¨ a1 dt =

∂ 2r ∂ 2r a ∂ga1 a1 ∂ga1 c a1 q˙ + + q˙ + q˙ q˙ +ga1 q¨ a1 . (2.73) ∂ t 2 ∂ t ∂ qa ∂t ∂ qc

From equation (2.73) we obtain

∂r¨ = ga1 . ∂ qä1

(2.74)

With respect to (2.71) and (2.74), the left-hand-side of equation (2.68) takes the form

∂r¨ (2.75) r¨ dm δr = r¨ dmga1 δ qa1 = r¨ a dm δ qa1 . ∂q 1 (V )

(V )

(V )

Using (2.71) the right-hand-side of equation (2.68) can be expressed by

d F δr =

(V )

d Fga1 δ qa1 .

(2.76)

(V )

Introducing the acceleration energy S=

1 2

r¨ 2 dm ,

S = S q1 , . . . , qn , q˙1 , . . . , q˙ n−r , q¨1 , . . . , q¨ n−r

(2.77)

(V )

and the generalized forces

Πa1 =

d Fga1 (V )

(2.78)

2.4 Forces Related to Locomotion

29

from D’ALEMBERT’s principle (2.68) taking into account (2.75) to (2.78) we obtain the equation ∂S Πa1 − δ qa1 = 0 . (2.79) ∂ q¨ a1 Since the virtual displacements δ qa1 are independent, relation (2.79) implies APPELL’s equations ∂S = Πa1 , (a1 = 1 , 2 , . . . , n − r) . (2.80) ∂ q¨ a1

2.4 Forces Related to Locomotion 2.4.1 Force of Gravity The force of gravity (Fig. 2.11) is the force that causes a massively large object to attract another object towards itself. All objects on Earth are under the influence of a gravitational force directed towards the center of the earth. Near the surface (h R), where the locomotion systems considered in the following chapters operate, the force of gravity is FG = mg , (2.81) where m is the mass of the object and g is the gravitational acceleration (|g| = 9.80665 sm2 ).

Fig. 2.11 Force of gravity

Of course, the force of gravity is only one of many forces acting on a locomotion system on Earth. Nevertheless, it is of interest in this context that some walking machines have been developed for which gravity is the only motion-generating force,

30


so-called passive walkers. McGEER [93] has shown that a simple planar mechanism with two legs can walk down a slight slope with no other energy input or control. This system consists of two coupled pendula. The pivot leg acts as an inverted pendulum, and the swinging leg acts as a free pendulum attached to the pivot leg at the hip. Given sufficient mass at the hip, the system will have a stable limit cycle; that is, a nominal trajectory that repeats itself and will return to this trajectory even if perturbed slightly. After McGEER’s pioneering work on passive dynamic walking, other researchers have developed new prototypes, including systems with knees, see for example passive walkers from the Delft University of Technology [45], [159], see Fig. 2.12.

Fig. 2.12 A passive-dynamics-based walker (left) and the passive walker “MIKE” with pneumatic McKIBBEN muscles (right), Delft University of Technology [Photos courtesy of S. COLLINS and M. WISSE]

2.4.2 Spring Force The spring force is the force exerted by a compressed or stretched spring upon any object attached to it. For most springs under consideration, the magnitude of the force is directly proportional to the amount of stretch or compression; therefore, spring forces take the form Fc = −c (λ − λ0 )ex = −c xex ,

(2.82)

with the constant stiffness c, the actual spring length λ and the original spring length λ0 , see Fig. 2.13.


31

Fig. 2.13 Spring force

Technicians have developed various forms of springs for different applications, see Fig. 2.14. For motion generation most pedal and non-pedal systems follow the principle of undulatory locomotion, see Section 6.1. For this type of locomotion, a periodic deformation of the shape of the system is needed. Thus, the spring representing the elasticity in the locomotion system is an important element in vibrationdriven systems.

Fig. 2.14 Various forms of springs [Photo courtesy of WAFIOS AG Reutlingen]

In biomechanics the spring is a typical element for modeling the muscle-ligament complex of humans and animals. BLICKHAN and SEYFARTH used spring-mass models to describe the long jump, see Fig. 2.15. Knowing the velocity, the leg stiffness, and the angle of attack, a very precise prediction of jump length is possible.

32


Fig. 2.15 Spring-mass model describing jumping [127]

WITTE et al. [162] explain the mechanism responsible for energy storage during the locomotion of horses with simplified models, including various springs. In these models, the extremities move like pendula and are compressed like springs during contact with the ground, see also Section 5.2.2.

Fig. 2.16 Vibration models with various springs describing the gaits of a horse [162]

The muscle is of great interest to engineers since it is effectively a linear actuator. Muscles have many features and an outstanding strength-to-weight ratio, which is useful in practical actuators. Because of its specific characteristics and parameters, the performance of muscle is significantly different from the performance of current artificial actuators. Different mechanical models have been developed in order to understand the working principles of muscles in biomechanics. The dynamic behavior of a muscle was first described using mass-damper-spring models by HILL [65] and then by HUXLEY [69], ZAJAC et al. [166]. SCHMALZ [125] considered various models for the muscle-tendon complex, which consists of elastic elements (springs) both in series (SEC) and in parallel (PEC) with a contractile element (CC), see Fig. 2.17.


33

Fig. 2.17 Muscle-tendon models with serial and parallel elastic elements [125]

Biological and technical locomotion systems also use springs as an energy storage. A biologically inspired artificial exoskeleton that supports a walking human is shown in Fig. 2.18.

Fig. 2.18 Exoskeleton supporting walking motion [Photo courtesy of B. SMALE]

2.4.3 Friction Forces 2.4.3.1 General Notes Friction simultaneously plays a paramount yet antagonistic role in connection with motion. It ultimately causes the standstill of all types of motion both in the nature and in technology. Therefore, engineers generally take great pains to overcome

34


friction for in order to maintain motion. Also, evolution has created remarkable solutions to reduce friction as much as possible. A well-known biological paradigm is the scale structure of the shark skin, which is now being used in a technical setting to decrease the flow resistance of aircrafts. On the other hand, locomotion becomes impossible without the existence of friction forces. According to NEWTON’s second law an external force is essential for the motion of the center of mass, which in the case of walking, crawling, rolling, etc. is the static friction force. Friction of rest, predominantly considered as a hindering effect, can also support movements. Numerous prototypes of artificial worms, presented in Chapter 6, are based on the targeted creation of different static and sliding friction forces between the worm segments and the environment. When active forces are capable of creating relative motion of a body having contact with a surface or an environment, other forces that resist the motion of the body and have the opposite direction of the velocity vector will appear. These forces are called friction forces. Friction forces can also appear when the body is at rest. The question about the direction of such forces will be considered below. The derivation of the friction forces is a very confusing question; therefore, we will discuss the properties of these forces and their mathematical description methods (without considering their physical nature). Many studies have been done regarding the mathematical description of the different laws of friction, see for exam¨ et al. [109] or AWREJCEWICZ & OLEJNIK [13]. Here, ˚ ple OLSSON, ASTR OM we shall briefly consider the laws of friction where the friction force (during motion) depends only on the size and direction of the relative velocity of the body. The force law can be different for the different directions despite the fact that the direction of the friction force is always opposite to the velocity vector (anisotropy of the size of the friction force). In that case the friction force can be determined using the following expression for v = 0 : Ff r = −F(v) v . |v|

(2.83)

Here, F(v) is a scalar function of the vector argument and continuous for all v = 0. In the case of isotropy (directional independence), the value of the friction force is only a function of the magnitude of velocity and F(v) = F(|v|). Friction, which depends on the velocity, can be conditionally divided into viscous and dry. Physically, this classification is caused by the size of the force that is necessary to bring the body out of rest. Mathematically, this is evidenced by the different behavior of the function used to assign the friction force for v = 0.

2.4.3.2 Viscous Friction Force With viscous friction the motion begins at any size of the applied (active) force, no matter how small. Mathematically, this means that the function F(v) in (2.83) is continuous for allv (also forv =0) and zero forv =0, i.e., F(0) = 0. Thus, equation


35

(2.83) can be written in the following form: ⎧ ⎨ −F(v) |vv| , v = 0 Ff r = 0 , ⎩ v = 0 .

(2.84)

In the case of an object’s motion along the x-axis with the velocity x˙ = v, the expression for the isotropic friction force (identical during the motion “back and forth”) takes the form: Ff r = −F |v| sign(v)ex , F(0) = 0 . (2.85) The dependence F(|v|) is usually described by the power of dependency with the coefficient of viscous friction d and the exponent α , i.e., F(|v|) = d |v|α . In this case the expression (2.85) becomes Ff r = −d |v|α sign(v)ex , α > 0 .

(2.86)

This law describes well the deceleration of bodies in fluids. With not very high motion velocities the friction forces can be described by the linear function of velocity (α = 1). With sufficiently high velocities the friction force is proportional to the square of speed (α = 2, see Fig. 2.19). Ffr a=1

a=2

1 -1

0

1

v

-1

Fig. 2.19 Viscous friction force vs. velocity

If the value of the viscous friction force depends on the direction of motion, then the friction force becomes anisotropic (asymmetric). Figure 2.20 shows the

36


asymmetric dependence between the force of a viscous friction and the velocity (d− > d+ ). This dependence is piecewise linear (α = 1), which gives a satisfactory approximation of the real physical phenomenon. Analytically, this dependence can be represented in the following form: −d− v , v < 0 Ff r = −d+ v , v > 0 . If v = 0, then the coefficient is arbitrary since Ff r (0) = 0. Let us note that this dependence is piecewise linear, i.e., generally speaking, non-linear.

- d-v

Ffr

v

0

-d +v

Fig. 2.20 Anisotropic viscous friction force vs. velocity

2.4.3.3 Dry Friction Force The main difference between dry and viscous friction is the fact that the motion with dry friction cannot start with any value of the applied (acting) force. Therefore, an acting force must exceed a finite value to initiate the motion. The mathematical meaning is that the friction force, according to (2.83), immediately varies its direction but not necessarily its modulus while crossing the zero-velocity axis. The most established classical dry friction model is from COULOMB. He studied the friction force by the slow mutual displacement of contacted bodies and proposed a simple empirical law: The friction force during motion does not depend on the velocity, it only depends on the direction of motion (and is always directed against the motion). According to COULOMB’s law the friction force is proportional to the normal load FN during motion (v = 0). It is described by the relationship, see Fig. 2.21: (2.87) Ff r = −μ FN sign(v) .


37

Ffr

+ m FN

v

0 - m FN

Fig. 2.21 COULOMB’s friction model

The coefficient of proportionality μ can be determined experimentally and is called the coefficient of kinetic friction. The friction model in (2.87) does not specify the friction force in the case of zero velocities. Obviously, this is a drawback, but because of its simplicity it is often used in the first steps of friction modeling. A detailed discussion of this fact, see also the uncertainty of the expression (2.83) with v = 0, leads to the consideration of such concepts as the static friction force (also called the friction force at rest or stiction). This idea was introduced by MORIN [99]. It follows from the force equilibrium that the static friction force is directed against the resultant vector of the active forces which are trying to force the body to slide. A geometric interpretation of the equilibrium of the forces in the case of stiction is given using the friction cone, see Fig. 2.22. If a body is located on a surface, then the full reaction acting on the body from the surface consists of the normal reaction FN and the friction force Ff r . The angle of friction ρ is connected to the coefficient of static friction μS and expressed through the value of the maximal static friction force Ff r max = Fs (static friction force):

μS = tan ρ =

Fs . FN

The coefficient of static friction μS can also be determined experimentally.

38


Fig. 2.22 Friction cone

We can construct a cone at the point of contact between the body and the surface. If the cone’s axis is directed along the normal of the surface and the angle between the generatrix and the axis is equal to ρ , then the reaction force due to the contact in the state of equilibrium will always be located inside the cone, which is then called the friction cone. In order to illustrate the meaning of the friction at rest, we can consider a simple experiment. The object is to try to move a body by pulling it with a rope connected to a spring dynamometer, see Fig. 2.23. x0

x1 0

Fa

x

Fig. 2.23 Experiment illustrating friction of rest

The body does not move from a small displacement of the end of the rope. This case means that the friction force fully compensates the applied force. We gradually increase the force Fa and, therefore, the displacement of the end of the spring. At a certain moment the body will start moving. The indication of the dynamometer registered at this moment is usually called the (maximum) static friction force. If we continue to slowly pull the rope, the body will move over the surface. It appears that the indications of the dynamometer registered during the motion will not be as high as the reading at the moment of the motion began. Usually, the friction force


39

during sliding is less than the force necessary for to initiate the motion. Thus, the static friction force, generally speaking, differs from the friction force of motion (kinetic friction force). In Fig. 2.24 the displacement of the force application point x0 is plotted versus the displacement of the body x1 . Sliding only occurs when a certain force is exceeded, meaning the body will start moving at a certain x0 (point A). x1

C

D E B

F A

0

x0

Fig. 2.24 Displacement of the force application point vs. displacement of the mass [118]

In accordance with COULOMB’s law, the force necessary to overcome the static friction and to move one body over the surface of another only depends on the normal component FN of the reaction force. Now, we consider the one-dimensional case with anisotropic friction, a theme which is discussed extensively in Chapter 6. The motion dynamics of the body under the action of the force Fa (the sum of all acting forces except for the dry friction force) and the friction force Ff r can be described by the differential equation x˙ = v ,

m v˙ = Fa + Ff r (v, Fa ) .

(2.88)

The extended COULOMB law (including stiction) takes the form: ⎧ F− = μ− FN , ⎪ ⎪ ⎪ ⎪ ⎪ − F+ = −μ+ FN , ⎪ ⎪ ⎨ − Fa , Ff r (v, Fa ) = ⎪ ⎪ ⎪ F− , ⎪ ⎪ ⎪ ⎪ ⎩ − F+ ,

v 0

v = 0 and

⎧ −F ≤ Fa ≤ Fs+ , (2.89) ⎪ ⎨ s− Fa < −Fs− , ⎪ ⎩ Fa > Fs+ .

Now, we will assume that v(t) = x(t) ˙ is a piecewise continuously differentiable function with respect to the time. This assumption is sufficient for describing the physically realizable motions. In the one-dimensional case for anisotropic dry friction, the corresponding force can be represented in the following, more general form than formula (2.89):

40


⎧ F− (v) , ⎪ ⎪ ⎪ ⎪ ⎪ − F+ (v) , ⎪ ⎪ ⎨ − Fa , Ff r (v, Fa ) = ⎪ ⎪ ⎪ F− , ⎪ ⎪ ⎪ ⎪ ⎩ − F+ ,

v 0

v =0

and

⎧ −F ≤ Fa ≤ Fs+ , ⎪ ⎨ s− Fa < −Fs− , ⎪ ⎩ Fa > Fs+ .

(2.90)

Here, the functions F− (v) and F+ (v) are continuous and have non-negative values, and it is usually assumed that F− (−v) − F+ (v) = const. The parameters Fs± are the bounds of the static friction force. Let us pay attention to the limiting values F− and F+ of the functions F− (v) and F+ (v) (i.e., F− = limv→−0 F− (v) , F+ = limv→+0 F+ (v)) which can be less than the values of Fs− and Fs+ . In this case the instantaneous decrease of the friction force after the start of motion will be reflected. If the corresponding limits are equal to the values of Fs− and Fs+ , then the friction force will be decreased continuously after the start of motion, but possibly with a high gradient, see Fig. 2.25. Ffr FsF-

0

v

-F+ -Fs+

Fig. 2.25 Dry friction force vs. velocity

Exercise 2.6. What is the fundamental mathematical difference between the dependencies of the friction force Ff r on the velocity v for dry and viscous friction? Some biological subjects, for example worms, have special attachments to create an asymmetry of friction, see Fig. 2.26.


41

Fig. 2.26 “Spikes” on an earthworm [Photo courtesy of N. Michiels]

The influence of the dry friction law is essential for the analysis of worm-like locomotion systems, see Chapter 6. Thus, let us finally go more into detail for the conditions of expression (2.90) which correspond to the value of dry friction for v = 0. First, we shall pay attention to the last two conditions. Let us suppose that the velocity during some moment of time t is v(t ) = 0 and that one of the two last conditions of expression (2.90) for the friction force is fulfilled. Because of these conditions and because of the equation of motion (2.88) it holds that v(t ˙ ) = 0. Because of the piecewise continuity v(t) ˙ = 0 in a certain neighborhood of the point t (possibly one-sided), and the sign is preserved in this area, i.e., v(t) decreases (increases) in this neighborhood and the function v(t) is negative (positive) accordingly. Thus, the fulfilment of the condition v = 0 with the simultaneous fulfilment of one of the two last conditions (2.90) is possible only at isolated points that do not affect the result of the integration of the equation (2.88), that is, the motion of the system. The first condition, corresponding to v = 0 in (2.90), is connected either with a full stop or with motion interrupted with stops if, for example, Fa is a periodic function of time. This regime is called “stick-slip”, which is characteristic of systems with dry friction when the velocity is zero in a finite interval of time. In the next subsection we focus on an adequate mathematical model of COULOMB dry friction which explicitly incorporates friction of rest and describes stickslip motion.

2.4.3.4 An Approach of Mathematical Friction Modeling In this subsection we go into more details with respect to some aspects concerning the mathematical handling of static and kinetic friction forces in numerical simulations. Well-known descriptions of friction models were considered in the previous sections. We would now like to emphasize that the modeling makes friction a function of two arguments. The dependent variables are the velocity v and the sum of all

42


acting forces Fa (except for the force of dry friction). We will give a short introduction to this mathematical friction model and emphasize some figures of 3-D friction graphs that obviously arise in modeling Ff r = Ff r (v, Fa ). The friction values, especially at v = 0, are only roughly known because of measurement uncertainty and a lack of knowledge of what happens inside the thin layer between contacting surfaces. An adequate mathematical model of friction will be necessary to achieve good stick-slip behavior. In order to achieve a satisfactory handling of the COULOMB rules on the computer, we replace v = 0 by −Δ < v < Δ with some small Δ (obviously, Δ ≈ 10−12 might be seen to model a computer accuracy). This method is based on an idea from KARNOPP to deal with a Δ blow-up interval to numerically accomodate zero velocity. Summarizing, these facts lead us to a friction model that depends on two variables and consists of a KARNOPP structure. Simulations and experiments do have to be changed, but, for example, with respect to later adaptive tracking problems of artificial worms (ground contact modeled via dry friction with stiction), we have to check some assumptions to get a well-defined solution since Ff r = Ff r (v, Fa ). Hence, control inputs might be parts of Fa ; therefore, we may arrive at more complicated dynamics of the systems. We again consider a mechanical system with DOF = 1, which is described by equation (2.88). Following COULOMB’s description Ff r compensates the force Fa if and only if the velocity v is zero and Fa does not exceed certain limit values, whereas Ff r takes constant values if v = 0. These values are characteristic of the contacting surfaces. They may depend on the orientation of the motion (anisotropic friction), and often they are supposed to be proportional to the normal force FN acting between the surfaces with constant friction coefficient μ , see equation (2.89). The description above qualifies Ff r to be a physically given force during motion v = 0 (kinetic friction force), whereas Ff r appears as a constraint force with given bounds as long as v = 0 (static friction force), according to HAMEL [62]. In connection with the dynamics above, the physical model of friction possesses the form shown in equation (2.91). The model uncertainty mentioned at v = 0 might allow the following relaxation of the above model with some small Δ > 0: ⎧ v < −Δ F− , ⎪ ⎪ ⎪ ⎪ ⎪ − F+ , v >Δ ⎪ ⎪ ⎨ − Fa , Ff r (v, Fa ) = ⎪ ⎪ ⎪ F− , |v| ≤ Δ and ⎪ ⎪ ⎪ ⎪ ⎩ − F+ ,

⎧ −F ≤ Fa ≤ Fs+ , ⎪ ⎨ s− Fa < −Fs− , ⎪ ⎩ Fa > Fs+ .

(2.91)

Using the HEAVISIDE function h, or more precisely the so-called “boxcar”function: 1, a ≤ x < b , h(a, b, x) = (2.92) 0, else ,


43

and ignoring some inconsistencies at the limit points a and b (with respect to the later smooth approximation this is not a problem) we write h in the form h(a, b, x) =

1 sign(x − a) + sign(b − x) , 2

(2.93)

then Ff r can be given in the form (disregarding its values at v = ±Δ ) ⎫ Ff r (v, Fa ) = −Fa h(−Δ , Δ , v) h(−Fs− , Fs+ , Fa ) ⎪ ⎬ +F− h(−∞ , −Δ , v) + h(−Δ , Δ , v) h(−∞ , −Fs− , Fa ) (2.94) ⎪ ⎭ −F+ h(Δ , +∞ , v) + h(−Δ , Δ , v) h(Fs+ , +∞ , Fa ) . In order to avoid computational difficulties caused by the afore-mentioned jumps in the h function, we turn to a smooth mathematical model (in the sense of an approximation). Basically, with respect to later numerical simulations and illustrations, we use a tanh approximation of the sign function sign(x) ≈ tanh(A x) with some sufficiently large A 1. The smooth mathematical model is now Ff r (v, Fa ) = −Fa H(−Δ , Δ , v) H(−Fs− , Fs+ , Fa ) +F− H(−∞ , −Δ , v) + H(−Δ , Δ , v) H(−∞ , −Fs− , Fa ) (2.95) −F+ H(Δ , +∞ , v) + H(−Δ , Δ , v) H(Fs+ , +∞ , Fa ) , where H(a, b, x) =

1 tanh A (x − a) + tanh A (b − x) 2

(2.96)

is the smooth approximation of h(a, b, x). The following Fig. 2.27 sketches the blow-up and approximation procedure:

Fig. 2.27 Blow-up and approximation procedure.

44


a) the original dry (COULOMB) friction with stiction, b) the Δ -blow-up interval around zero (non-smooth), and c) the smooth approximation with the HEAVISIDE function H. In fact, Fig. 2.27 shows the projection of the graph of Ff r to a plane Fa = const. This graph (for some data) is given in Fig. 2.28, somewhat distorted by the coarse coordinate grid.

Fig. 2.28 Graph of friction force Ff r (v, Fa )

Now, we present two examples and simulations of applying this friction law to a DOF = 1 mechanical system. We notice that, for the illustration of the method, we choose arbitrary parameters of the models. Further, if we investigate real prototypes, we take into account the concrete parameter values of the locomotion systems. For approximating h(−Δ , Δ , v) the value of A to be chosen will be strongly influenced by the value of Δ to be used. The combination Δ = 0.0005 and A = 105 has proved suitable in calculations. The following friction values are taken into account: Fs± = 6, F± = 3. Example 2.3 x˙ = v,

m v˙ = −bt + Ff r v(t), −bt ,

x(0) = 0 ,

v(0) = 0 ,

(2.97)

with m = 1 and b = 1. Figure 2.29 shows the solutions x(t) and v(t) on a given time interval. The drive Fa has to grow until the friction Ff r reaches Fs− , only then does the system begin to slide.


45

Fig. 2.29 The solutions x(t) and v(t)

The corresponding space curve t → v(t) , Fa (t) , Ff r v(t) , Fa (t) lies on the graph of the friction function, see Fig. 2.30.

Fig. 2.30 The space curve (thick) on the friction graph

Example 2.4 x(t) ˙ = v(t), m v˙ = − d x(t) + 4 bt + Ff r v(t), − d x(t) + 4 bt x(0) = 0 , v(0) = 0,

(2.98)

46


with m = 1, b = 1 and d = 2. The stick-slip effect can clearly be seen in Fig. 2.31.

Fig. 2.31 The solutions x(t) and v(t)

The corresponding space curve t → v(t) , Fa (t) , Ff r Fa (t) , v(t) lies on the graph of the friction function, see Fig. 2.32.

Fig. 2.32 The space curve (thick) on the friction graph

In using the afore-mentioned friction law in the later adaptive tracking problem, we must mind the fact that this function not only depends on the velocity but also on the visco-elastic forces (and the control inputs).

Chapter 3

Mathematical Methods and Elements of Control Theory

3.1 Modeling A description and model of any class of phenomena can be expressed by mathematical means with a certain degree of accuracy. The analysis of a mathematical model makes it possible to reveal the essence of the studied phenomena and to switch over to prediction and control problems. The process of mathematical simulation can be divided into four stages: • Formulation of the laws that connect the elements of the model. This main stage involves creating a record of the qualitative representations of the connections between the elements of the model in a mathematical form. • The study of tasks or problems that lead to the mathematical model, with the aid of various mathematical methods. • Answering the following question: Does the accepted hypothetical model satisfy the experimental data? • Subsequent analysis and improvement of the model. The next few sections will cover these stages in detail. The most common method of model construction is the use of the fundamental laws of nature in a concrete situation. These laws have been repeatedly confirmed by experience and their validity is generally not in doubt, even in situations in which the use of laws of mechanics may encounter a number of difficulties. The model should allow us to deduce the object’s primary mechanisms of functioning; the problem then consists of revealing and describing these mechanisms. Frequently, with the attempt to develop a model, it is difficult to indicate the fundamental laws to which the model is subordinated, or more generally, there is no confidence in the existence of similar laws that allow a mathematical formulation. One of the possible approaches in such cases is an analogy with phenomena already studied.


47

48

3 Mathematical Methods and Elements of Control Theory

3.2 Analysis and Modification of the Model – the Role of an Experiment First of all, it is necessary to directly compare the model to experimental data in order to establish the model’s adequacy. Significant divergence from the experimental data must be analyzed closely to determine the cause, and it should be verified that the laws governing the expected effects are sufficiently accounted for. For example, failing to factor in the irreversibility of processes (friction, loss of warmth, etc.) can lead to significant deviations from physical data. In this case the mathematical model should be completely rejected and a new model constructed. However, the cause might be that the experimental data is insufficient for a direct comparison, in which case a careful analysis of the model is necessary. The constructed mathematical model should be studied by all available methods, including mutually checking various approaches. It is necessary to make corresponding limiting transitions in the analytical model to be convinced that it works in special cases. Often, the modification of a model leads to a complexity that it is no longer suitable for analytical investigations. Therefore, it is necessary to use computational methods for finding the approximate solution of the stated problem with the aid of computer technology. These methods make it possible to conduct the comprehensive numerical investigation of the initial model to confirm the computational simulation in order to carry out the analysis of the model in different situations and to obtain exhaustive information about the studied object. Numerical experimentation does not simply mean specifying the quantitative characteristics of the model but also allows the study of the qualitative characteristics of objects whose behavior can be rather complex and unexpected. Numerical modeling has its specific difficulties (see Subsection 2.4.3.4), which will be discussed briefly in another subsection. Additionally, it should be noted that these problems are not reduced in proportion to the appearance of increasingly powerful computers.

3.3 Mathematical Methods 3.3.1 Perturbation Methods Unfortunately, only a few problems (especially linear ones) may have an exact analytical solution or description. Here, we will focus on those problems that are linear except for small nonlinear additive terms. These problems are called (nonlinearly) perturbed problems. Special methods, appropriately called perturbation methods, have been developed for solving them. These methods are effective and make it possible to describe the perturbed motion with sufficient completeness. The delineation of processes bearing an oscillatory character leads to the need for investigating the behavior of the system over a large or even infinite time interval. For example, the averaging method is based on the idea of separating the motion

3.3 Mathematical Methods

49

into a smooth progression (slow) and frequent oscillations about the trend (rapid), with the “slow” and “rapid” variables corresponding to these motions. Usually, we are basically interested in the behavior represented by the slow variables. The supposition, then, is that averaging leads to the rejection of large oscillations, and small oscillation terms are superimposed onto the model of the average system. This is the reason that slight perturbations of the system lead to the appearance of small additions in the motion equations and the reason for adding the small parameter ε into the different second-order equations that describe the dynamics of the system. The equations describing oscillating processes can be given in many instances with the aid of the appropriate variables in the following form: y˙ = ε f (y , ϕ , ε ) ,

ϕ˙ = ω + ε g(x , ϕ , ε ) .

(3.1)

In this system of equations the variable y is slow since the change of y over time is proportional to ε (with ε = 0 we have y˙ = 0, for all t). ϕ is rapid (fast) since the derivative ϕ˙ is proportional to the value ω plus a small additional term. The functions f and g are 2 π -periodic in ϕ . As a result of applying the averaging method for the approximate description y(t) in the time interval [0, ε1 ], the perturbed system is substituted by the system averaged through ϕ : x˙ = ε F(x) ,

(3.2)

where F(x) = 21π 02 π f (x , ϕ , 0) d ϕ . The variablex is introduced here in order to em-

phasize that system (3.2) describes a slow motion of the averaged system. The following mathematical assertion serves as the substantiation of the method described: with sufficiently general assumptions about f and g, the difference between the slow motion y(t) = (y1 , . . . , yn )T in the exact system and x(t) = (x1 , . . . , xn )T in the averaged system remains small during the time interval [0, ε1 ]; this means for some constant C, xi (t) − yi (t) < C ε ,

if xi (0) = yi (0) ,

1 , 0≤t ≤ ε

(i = 1, . . . , n) .

There are mathematical theorems that considerably generalize the given assertion, including a theorem under specific conditions which determines the size of the differences between the solutions of the actual and averaged system over an infinite time interval. The justification in applying the described method is given in [49]. The form of the set of nonlinear differential equations with the parameter ε as well as the character of its appearance are (very) different. But, in most cases this system of equations can be transformed into (3.1). In general the system of averaged equations (3.2) is nonlinear as well. However, these equations (3.2) are basically simpler in handling and in an easier form to gain deeper analytical insight. In the context of the locomotion systems under consideration in this book, we particularly focus on the stationary solution of (3.2), which can be determined by zeroing the right-hand-side of (3.2).

50


Let us first consider the following equation as an illustrative example for the meaning of a stationary solution: dx = F(x) , dt

(3.3)

where x denotes a physical variable (e.g., an amplitude). The following is an analysis of the behavior of the solution x(t) depending on the properties of the function F(x). Let us assume that a positive real number x∗ does not exist that fulfills the criterion F(x) > 0 for all x > x∗ . If such a number were to exist, we would be able to conclude, in choosing an initial condition x0 = x(0) > x∗ , from (3.3) the unboundedness of x(t), i.e., x(t) → ∞ as t → ∞. Hence, the assumption presented is reasonable from a physical point of view. The stationary solution xs is the solution of F(x) = 0. From (3.3) we conclude that if x0 = xs , x(t) is monotonically decreasing (if F(x0 ) < 0) or monotonically increasing (if F(x0 ) > 0) towards the stationary solution xs . This process is called the transition process or transient behavior. In the very special case that F(x) ≡ 0, we have no transition and every solution is a stationary solution. Summarizing, we conclude that x(t) → xs as t → ∞. This fact points out the special character of the stationary regime. In the case of high-frequency oscillating processes, there is a small transition to the stationary solution because of the very small oscillation periods. Therefore, these kinds of oscillations are often considered stationary. For systems of differential equations, the problem discussed above is more complicated. But, the particular relevance of the stationary solution for systems of differential equations remains as well.

3.3.2 Stability of Stationary Motions In practice, only stable motion is physically realizable. Therefore, it is important to find out whether the motion is stable. First, stability must be formally defined. Let us consider a mechanical system described by the following set of differential equations d xi = Xi (x1 , . . . , xn ,t) , i = 1, . . . , n , (3.4) dt or written in a vector form dx = X(x,t) . (3.5) dt Here, xi are functions related to the motion, e.g., coordinates, velocities, amplitudes, phases, etc. The functions Xi (x1 , . . . , xn ,t) are real functions smoothly dependent on the input variables, providing that a unique solution exists with the initial conditions xi (t0 ) = xi0 over the time interval t ≥ t0 . If the vector function X does not depend explicitly on time, the system is called autonomous; otherwise, the system is non-autonomous.


51

Let xi (t) be a particular solution of system (3.4), which we will call undisturbed motion. Other solutions are called disturbed motion. Definition 3.1. The undisturbed motion xi (t) is stable if for every positive ε there exists a positive δ (ε ) such that for all disturbed motions yi (t), satisfying yi (t0 ) − xi (t0 ) < δ , we have for all t > t0 :

yi (t) − xi (t) < ε .

Otherwise, the motion is called unstable. If the motion is stable and in addition limt→∞ (yi (t) − xi (t)) = 0, then the undisturbed motion is called asymptotically stable. Determining the stability of the system in deviations (variations) can become very transparent (the limiting factor being of course our ability to visualize Rn ) if we consider a cylindrical tube of radius ε whose axis is coincident with the t-axis. In the cross-section t = t0 of this tube, a δ -vicinity of xi (t0 ) can be defined in which all the solutions that leave this vicinity do not extend beyond the limits of the tube for all t > t0 , see Fig. 3.1. Next, we will examine the stability of the stationary motion described by the following systems of equations: d xi = Xi (x1 , . . . , xn ) , dt

i = 1, . . . , n .

(3.6)

n

R

x1 . . .

V (l)

e d

z

V (l)

z

xn t0

t

l

Fig. 3.1 Graphical interpretation of the term “stability”

Let us consider disturbances (deviations) of the form zi = yi −xi . Using equations (3.6), we get the disturbance equations zi : d zi = Zi , dt

i = 1, . . . , n ,

(3.7)

52


with Zi = Xi (z1 + x1 , . . . , zn + xn ) − Xi (x1 , . . . , xn ) . Now the trivial solution z1 = . . . = zn = 0 of system (3.7) corresponds to undisturbed motion. This allows us to reduce the question of the stability of the undisturbed motion in (3.6) to the question of stability of the null solution (3.7). We expand the right-hand sides of the disturbed motion in a power series in zi . Since with z1 = . . . = zn = 0 the right-hand sides of equations (3.7) are equal to zero, the power series does not contain free members. As a result we obtain d zi = ai1 z1 + . . . + ain zn + Zi∗ , dt

i = 1, . . . , n .

Here, Zi∗ is a combination of the higher-order members (relative to zi ). In stability problems the equation solutions are considered for small initial values of the quantities zi . Hence, it is to be expected that the behavior of the solutions is defined in general by the linear members; that is, it is enough to consider the system of linear equations (system of equations of first approximation): d zi = ai1 z1 + . . . + ain zn , dt

i = 1, . . . , n ,

(3.8)

or in matrix form: ⎛ ⎞ z1 dz ⎜ .. ⎟ = Az , z = ⎝ . ⎠ , dt zn

⎛

a11 ⎜ .. A=⎝ .

⎞ . . . a1n .. .. ⎟ . . . ⎠

(3.9)

an1 . . . ann

Our consideration is valid for the stationary case. Therefore, all the elements ai j of the matrix A are constant. To solve the linear system with constant coefficients, we need to find the roots of the corresponding characteristic polynomial P(λ ): detA − λ In = (−1)n P(λ ) . Here, In is the n-dimensional unity matrix, and P(λ ) = a0 λ n + a1 λ n−1 + . . . + an−1 λ + an ,

a0 = 1 .

(3.10)

To answer the question about the relationship between the stability of the nonlinear system (3.7) and the stability of the linear system (3.10), the following theorem should be used: Theorem 3.1. If the real parts of all the roots λi of the characteristic equation of the first (linear) approximation are negative, then the undisturbed motion is asymptotically stable, irrespective of the higher-order members. If there is at least one root among those of the characteristic polynomial that has a positive real part, then the undisturbed motion is unstable, also irrespective of the members of the higher order according to smallness. So-called critical cases are those in which the characteristic polynomial has roots with real parts equal to zero. In such cases the stability criteria cannot be stated using


53

the first-approximation equation (3.9). Therefore, it is important to formulate necessary and sufficient conditions for the case in which all roots of the characteristic polynomial (3.10) of A have negative real parts (i.e., are located in the open left-half complex plane). To formulate these criteria, we present the following determinants (HURWITZ determinants) using the coefficients of the characteristic polynomial: a1 a0 0 . . . 0 a3 a2 a1 . . . 0 a1 a0 , Δn = a5 a4 a3 . . . 0 = an Δn−1 . Δ1 = a1 , Δ2 = (3.11) a3 a4 .. .. .. . . . . . . . .. 0 0 0 . . . an There are coefficients with odd indices starting from a1 in the first column of the determinant. The elements of each subsequent column are formed from the corresponding elements of the previous column by decreasing of the respective indices by 1. Moreover, if i > n or i < 0, then ai = 0. The criterion emphasized here (LIENARD criterion) is formulated as follows: Theorem 3.2. For a polynomial to have only roots with negative real parts, it is necessary and sufficient that all coefficients of the polynomial (3.10) are positive, i.e., ai > 0 , (i = 1, . . . , n) , and the following conditions for the HURWITZ determinants are fulfilled: Δn−1 > 0 , Δn−3 > 0 , . . . . Let us describe these conditions in detail up to n = 4, taking into account that a0 = 1. n = 1 , a1 > 0 n = 2 , a1 > 0 n = 3 , a1 > 0 n = 4 , a1 > 0

, , a2 > 0 , , a1 a2 − a3 > 0 , a3 > 0 , (3.12) 2 , a3 > 0 , a3 (a1 a2 − a3 ) − a1 a4 > 0 , a4 > 0 .

Notice that in the last two groups of relationships, the condition a2 > 0 is fulfilled automatically. Stability in the sense of LYAPUNOV provides for an investigation of the system behavior over an infinite time interval. Furthermore, we apply the stability criteria formulated for the stationary solutions of the averaged system (3.2). System (3.2) itself is an approximation. It describes the solutions of the original system (3.1) with a certain precision (on the order of ε ) and over the finite time interval [0 , ε1 ] . Hence, in the common case one cannot consider the stability of the solutions of the original system (3.1) on the basis of the consideration of the stability of the averaged system (3.2). However, as long as the asymptotic method is applied in cases in which the behavior of the system is considered over the finite (although large) time interval, the information about the stability of the averaged system (3.2) is enough to solve a majority of practical, relevant problems. There are theorems

54


which state that for certain conditions one can draw a conclusion about the stability of the solutions to the original equations on the basis of the stability of the stationary solutions to the averaged equations. Example 3.1 Let us consider the motion of a mass point with mass m under the action of an active force Fa = av and a NEWTON friction force proportional to the square of the velocity, i.e., Ff r = −b v2 (a, b > 0). The initial conditions are x(0) = 0 , v(0) = v0 . The goal is to solve the following problems: • • • •

obtain the equation of motion, find the stationary solutions vs , investigate the stability of the stationary solutions, and analyze the exact solution, considering different initial conditions for v(0).

Using the principle of linear momentum (2.23), we find the differential equation of motion m v˙ = a v − b v2 or where α =

v˙ = α v − β v2 , a m

(3.13)

, β = mb . Separating the variables and considering that 1 1 1 α 1 ,γ= , =− − 2 α v−β v α v−γ v β

we obtain the solution of equation (3.13) in the form v=

v0 γ . v0 − (v0 − γ )e−α t

(3.14)

Stationary solutions of equation (3.13) are vs = 0 and vs = γ . The variational equations take the form

δ v˙ = −β (2 vs − γ ) · δ v , δ v = v − vs .

(3.15)

For vs = 0 from equation (3.15), we find that δ v˙ = β γ · δ v, and the stationary solution vs = 0 is unstable because α > 0. For vs = γ from (3.15), it follows that δ v˙ = −β γ · δ v, and the solution vs = γ is stable. Figure 3.2 (left) shows solution (3.14) with the initial condition v0 = 0.01 and for the parameters α = 2 and γ = 2. In spite of the fact that v0 = 0.01 is close to the (unstable) solution vs = 0, this solution tends toward the stable solution vs = γ . The solution with the initial condition v0 = 6 is presented in Fig. 3.2 (right).


55

2,5

7

6 2

5 1,5

4

V

V 3

1

2 0,5

1

0

0 0

2

4

6

8

10

0

1

t

2

3

4

5

t

Fig. 3.2 Solution v(t) for v0 = 0.01 (left) and v0 = 6 (right)

3.3.3 Introduction of Dimensionless Variables and Method of Dimensions Some mathematical methods, including the perturbation methods mentioned above, are based on the existence of small parameters that describe the behavior of the system. The question, then, is what constitutes a small parameter in the equation. Hence, the concept of the “small parameter” is relative - in other words the value need only be small in comparison with other values of the same dimensionality. Dimensional quantities are those whose numerical values depends on the units of measurement. To compare kilograms with seconds, meters with degrees, etc. is meaningless. Therefore, in order to conduct an analysis of the order of the values in terms of the equations, dimensionless variables should be entered and the calculated non-dimensional combinations examined. This procedure we will call the introduction of dimensionless variables. The calculated non-dimensional equation parameters should now be compared to the unit of a small parameter in order to determine if it is noticeably smaller than the unit itself. This property for the parameter ε is written in the form ε 1. Additionally, we notice that the calculation of any function such as sin , exp , ln , arctan etc. (except polynomial functions) is only possible with dimensionless arguments. The dimensional analysis of model parameters is one of the methods to deduce the symmetry properties of the phenomenon under study, which makes it possible to simplify the mathematical model. The choice of measurement units is sufficiently arbitrary. For example, energy can be measured with joules, calories, tons of coal

56


(which must be burnt in order to obtain the given amount of energy) and with other measurement units. The corresponding numbers that give the value of energy substantially depend on the choice of measurement units. Values for which the measurement units have been chosen by experience and with the aid of standards are called basic measurement units. Thus, different fundamental measurement units have been chosen for length, time, mass, etc. Let us now consider the structure of the functional dependence (interconnections) between the dimensional values that express physical laws. Such laws should not depend on the system of units being used. The laws of nature have an objective character and do not vary according to the measurement units - for example, units of distance: angstroms, meters, or light years. The discussion, here, can deal only with the convenience of a specific unit. In this case the functional dependence expressing a law is invariant as far as the chosen system of measurement is concerned. In classical dynamics the correct use of dimensionless variables has the advantage of leading to compact derivations and expressions. The method of dimensions shows its real deductions and effectiveness in the mechanics of continua.

Example 3.2 Let the motion of a mass point be accomplished along a straight line starting from the state of rest under the action of a force f (t) = a + b cos ω t and assuming the absence of friction. Let us compare the precise and averaged equations of motion of a mass point and find expressions for its velocity in both cases. The motion equation of the mass point with the coordinate x takes the form: mx¨ = a + b cos ω t. For example, a represents an active force and b a (small) amplitude of a perturbation. With x˙ = v, where v is the velocity and the variable ϕ = ω t, we rewrite the equation in the following form: mv˙ = a + b cos ϕ ,

ϕ˙ = ω .

(3.16)

Now, we introduce dimensionless variables, bringing the time t in relation to a characteristic period T and the velocity v in relation to a characteristic velocity U: t v , v∗ = . t∗ = T U The asterisk indicates dimensionless variables. Inserting the expressions for dimensional variables into equation (3.16), we obtain the equation in dimensionless variables (asterisks are omitted): m or

U v˙ = a + b cos(ω T · t) , T

1 ϕ˙ = ω T


57

" b Ta! 1 + cos(ω T · t) , mU a We choose T and U such that

ϕ˙ = ω T .

v˙ =

Ta = 1, mU

ω T = 1,

i.e., 1 a , U= . ω mω Finally, equation (3.16) in dimensionless variables takes the form T=

v˙ = 1 +

b cost , a

ϕ˙ = 1 .

(3.17)

The dimensionless value ba characterizes the relationship between the amplitude of the oscillatory component of the applied force and its constant part. We assume that this quotient is small and denote it by ε = ba 1. In order to transform equation (3.17) into the form (3.1), we introduce a new variable u = v−t and obtain the form

u˙ = ε cost .

According to the procedure for averaging, it holds that u˙ = 0 because 1 2π cost dt = 0 . 2π 0 Returning to the variable v, we find the solutions for the exact and the averaged equations, respectively, see Fig. 3.3: v = t + ε sint ,

V =t,

(3.18)

with the initial conditions v(0) = V (0) = 0 . From equation (3.18) it can easily be seen that |v −V | = ε | sint| ≤ ε ,

∀t .

(3.19)

This result corresponds to the accuracy of the averaging method. Using dimensional variables the solution (3.18) takes the form v=

b a t+ sin ω t , m mω

V=

a t. m

(3.20)

58


Fig. 3.3 Solutions of the exact (dashed) and the averaged (solid) equations (ε = 0.25)

Exercise 3.1. Assume that the mass point in Example 3.2 is being acted upon by the force f (t) = a + b sin ω t. Friction is negligible. Perform the procedure of averaging on the equation of motion and compare the solution of the exact equation with the averaged ones.

Example 3.3 In Example 3.2 we find the displacement of the mass point in the period T = 2ωπ . Since 0T sin ω t dt = 0, the displacements in one period, found from the exact and the averaged equations, are identical:

T

s= 0

at a T2 a π2 dt = =2 . m m 2 m ω2

Let us note that if a = 0, we have no displacement of the mass point, because the small external force is symmetrical. Actually, to achieve locomotion an asymmetry is necessary, either in the applied force or in the contact conditions. The asymmetry of the contact conditions is evidenced in the asymmetry of the friction force. In our case friction is absent.


59

3.3.4 Remarks on Numerical Methods We have presented some analytical methods such as averaging or methods using TAYLOR series. The aim is to obtain an analytical term of the solution or of one approximation of it. We will especially focus on these analytical methods in Chapter 6. But, it is obvious that these methods are not all-encompassing because of certain nonlinear properties of the equations. Hence, the analytical approach is not the only way even if averaging methods might be helpful tools. We also need numerical methods, for instance, to compare exact solutions with approximated ones. This subsection gives a very short introduction in numerical methods, especially numerical integrations and the numerical solving of ordinary differential equations (ODEs). There are two main reasons to do numerical integration: analytical integration may be impossible or infeasible, sin(x2 ) dx or exp(x2 ) dx, or we have to integrate tabulated data rather than known functions. Here, we focus on the first problem and show a first simple idea in the following to numerically solve ODEs. We analytically deduce an iteration method from the integration task x˙ = f (t, x). An approximation of the derivative is the difference quotient x(t ˙ 1) =

x(t2 ) − x(t1 ) x(tn+1 ) − x(tn ) , or in general x(t ˙ n) = . t2 − t1 tn+1 − tn

We conclude = x(t ˙ n ) = f tn , x(tn ) ⇔ x(tn+1 ) − x(tn ) = (tn+1 − tn ) f tn , x(tn ) ⇔ x(tn+1 ) = x(tn ) + (tn+1 − tn ) f tn , x(tn ) . x(tn+1 )−x(tn ) tn+1 −tn

With h = tn+1 − tn (step size) and xn = x(tn ) for all n ∈ N we have xn+1 = xn + h f (tn , xn ) = ψ (h,tn , xn ) .

(3.21)

This first and simple method (3.21) was found by EULER, hence it is called the EULER or forward EULER method. This ODE solver is the default one in MAPLE, for example. If we have the initial values t0 and x0 (initial value problem) and also the step size h, we can derive t1 , x1 and so on.

60


x

xn+1

x2 x1

x3 xn

x0

t0

exact solution

t1

t2

t3

tn

tn+1

t

Fig. 3.4 The successive construction of the solution via (3.21)

Therefore, we can numerically solve the integration problem by means of a successive construction of the solution function, see Fig. 3.4, but it is only an approximation. There are a lot of improvements of this method: • controlling the step size – This is important for stiff ODEs, where a stiff ODE is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. A precise definition of stiffness is very difficult and not unique in the literature. But, the main idea is that the equation includes some terms that can lead to rapid variation in the solution. • new iteration formulas (HEUN, RUNGE-KUTTA, ...) • explicit and implicit methods – A method is called explicit if the iteration formula (recurrence relation) can be used directly to determine xn+1 ; on the other hand, a method can be implicit, and the recurrence relation is an equation for xn+1 which has to be solved. A simple example is the backward EULER method xn+1 = xn + h f (tn+1 , xn+1 ). • single-step and multi-step methods – The presented EULER method is a classical one-step method because the right-hand side of the iteration formula consists only of the value of the predecessor: ψ (h,tn , xn ). On the other hand, the function ψ of an r−step method (e.g., ADAMS–BASHFORTH methods, ADAMSMOULTON methods, backward differentiation formulas (BDFs), etc.) consists of more than only the predecessor, namely, r predecessors:

ψ (h,tn , xn−r , xn−r+1 , . . . , xn ) . The primary purpose of referring to several previous predecessor values is to achieve greater accuracy. In the context of this book it is impossible to give a general survey about all existing integration methods, as there are very numerous sources. For further details we

3.4 Some Aspects from Control Theory

61

refer the reader to DEUFLHARDT ([46]), HAIRER & WANNER ([60] and [61]), STREHMEL ([139]) and STUART ([140]). However, we want to point out that only numerical results are not enough to found a good (analytical) theory. We close this subsection with a quote from R.W. HAMMING [63]: “The purpose of computing is insight not numbers.”

3.4 Some Aspects from Control Theory In this section we present some facts and properties from control theory. It is not the authors’ intention to give a complete introduction into the theory of control systems as there are already numerous complete works on the subject, for example, SONTAG ([132]), ZABCZYK ([165]) and ISIDORI ([76] and [77]). Rather, we present some aspects of control theory relevant to the material in subsequent chapters.

3.4.1 Motivating Example and General Formulations Let us consider the following mechanical model of a spring-mass-damper system with permanent excitation of the spring, see Fig. 3.5.

Fig. 3.5 Spring-mass-damper system

The equations of motion are derived by using the principle of linear momentum (NEWTON’s second law), see (2.23) in Subsection 2.3.1.1: m x¨ + d x˙ + c x = c f (t) + u(t) ,

˙ = x˙0 . x(0) = x0 , x(0)

(3.22)

The constants m, c, d > 0 are the mass, the spring stiffness, and the damping coefficient. The function f (t) is an external excitation acting on the system and can be thought as a permanent disturbance of the system. The function u(t) is an impressed force and can be thought as a control input. It is the only way for us in this example to influence the system’s behavior from the outside in order to achieve a desired

62


system behavior, for example, that the (measured) output y(t) = x(t) follows a given reference trajectory. The control force is a general force of the class Qa 4 , see Subsection 2.3.2.2. Investigating (3.22), we deal with a control system that can be transformed into the equivalent form 0 1 0 x1 x1 0 d = + f (t) + c 1 u(t) , dt x2 x2 − mc − md m m (3.23) x1 y(t) = 1 0 . x2 Assuming f ≡ 0 we have a general system of first-order linear differential equations: ˙ = Ax(t) + Bu(t) , x(t0 ) =x0 , x(t) (3.24) y(t) = Cx(t) , ˙ is the vector of the velocities Here, x(t) is an n-dimensional state vector, and x(t) at time t. The right-hand side of (3.24) gives a functional interrelationship between the velocity, the state x(t), and the input defined by the function u(t). In this special case we have a linear dependency via the matrices A and B. The corresponding output is described by the function y(t), which depends on the state x(t). In this introduction we do not want to consider the case in which the output y(t) directly depends on the input u(t) as well. The output characterizes the physically measurable information about the system. The input u(t) describes effects or influences the system is exposed to. In [82] it is mentioned that there are two different types of influences or, more simply, two kinds of inputs: completely and incompletely known variables. The former are called control (or controlled) inputs or control variables. The latter describe uncontrolled, (partly) unknown influences also acting on the system and are be called external disturbances, see Fig. 3.6.

control

system output y (t)

input u (t) possible external disturbances

(described by state x (t))

Fig. 3.6 Model of a control system

In Fig. 3.6 one can clearly see the interaction of the system with its environment. First, the environment acts via the input variables u(t) on the system. As mentioned


63

above, some components of u(t) could represent non-influenceable disturbances. But additionally, some control variables exist that allow some system control of the system. On the other hand, the system affects the environment via the output y(t) as well, [82].

3.4.2 Open- and Closed-Loop Control Control theory deals with the formulation of control systems/problems and solutions. The main point is to what extent a system can be affected in such a way that its behavior is close to a desired one. Solving these control problems needs two main (mathematical) disciplines: system theory and control theory, where system theory forms the basis for control theory. The former is concerned with the analysis of dynamic systems and the study of properties such as stability, observability, and controllability. Control theory deals with the synthesis of controls for dynamic systems. By choosing controls and control variables u(t), respectively, new dynamic systems arise, which have to be analyzed with elements of the system theory. There are two main but different strategies in control theory: open-loop and closed-loop control, [165]. • “Open-loop” control involves the use of an arbitrary function u(t), such that the system (3.24) has a well-defined solution. • “Closed-loop”control uses a function h, such that the system (3.24) with the input u(t) = h x(t) has a well-defined solution. The function h x(t) is called feedback. Applying an open-loop control strategy to a control system yields only an open action flow, see Fig. 3.7.

Fig. 3.7 Open-loop control scheme

The primary problem arising during the consideration of an open-loop controller is the fact that it is susceptible to even (small) disturbances occurring in almost all control systems. This means that open-loop control systems quickly become uncompetitive against other strategies because the open-loop controller tend to exhibit low accuracy with respect to a control objective due to the unknown system response to disturbances. On the other hand, a closed-loop control strategy possessed a closed action flow, see Fig. 3.8, which allows the system to react to disturbances. This is usually a significant advantage over open-loop controllers.

64


Fig. 3.8 Closed-loop control scheme

Using feedback the behavior (e.g., the state of the system) can be controlled because the response of the system to a disturbance is measured or observed. Applying a closed loop-control to a control system yields a so-called closed-loop system.

3.4.3 Output Feedback 3.4.3.1 Introduction A widely studied and comfortable concept in closed-loop control is state feedback, see Fig. 3.9.

Fig. 3.9 State feedback scheme

However, practical realization of state feedback in applications is not suitable because the entire system state at each time t would need to be measured. In most cases this would be unrealistic and prohibitively expensive. One of the most important terms used in this book in connection with adaptive control (as well as being an integral part of modern control theory in general) is output feedback. Output feedback is a very important and powerful tool in designing controllers to achieve a pre-specified system behavior. Obviously, output feedback, see Fig. 3.10, the focus of this section, does not make use of the whole system state, but only necessary information with respect to the output.


65

Fig. 3.10 Output feedback scheme

The following example shows the application of classical stabilizing output feedback. Example 3.4 Let us consider a linear scalar system x(t) ˙ = ax(t) + bu(t) ,

x(0) = x0 ,

y(t) = cx(t) ,

with known system parameters a , b , c , x0 . Furthermore, let us assume c b > 0. The aim is to find an output feedback function such that limt→∞ y(t) = 0. Applying the feedback formula u(t) = −ky(t), k ∈ R, to the linear scalar system yields a closed-loop system of the form x(t) ˙ = ax(t) − kby(t)

c=0

⇒

cx(t) ˙ = cax(t) − ckby(t) ⇔ y(t) ˙ = (a − kcb)y(t) .

Since c b > 0, we have to choose a k > k∗ = cab . Then, the system is asymptotically stable, moreover exponentially stable, because we have y(t) = e(a−k c b)t y0 . Since t → ∞, the desired system property has been achieved. When c = 0, it can be concluded that limt→∞ x(t) = 0. Here, one can clearly see that this linear system is exponentially stable if the gain parameter k is large enough. This is called high-gain stabilization. Next, we will explain some system properties such as the relative degree and the minimum phase.

3.4.3.2 Relative Degree For later investigations, for example normal forms and control strategies, it is necessary to divide the presented system class (3.24) into various subclasses depending on the relative degree of the corresponding system. The relative degree (sometimes also called order) of a system defines the strength of the influence of the input u(t) on the output y(t) or on one of its derivatives y(i) (t). The relative degree is a number that determines the amount of differentiation of the output y(t) that is required for

66


us to observe a direct influence from the input u(t). The following algorithm may visualize this: • Step 1: y = Cx → y˙ = Cx˙ = CAx + CBu →

˙ CB = 0 → u acts directly on y. CB = 0 → y˙ = CAx → Step 2

• Step 2: y˙ = CAx → y¨ = CAx˙ = CA2x + CABu ¨ CAB = 0 → u acts directly on y. → CAB = 0 → y¨ = CA2x → Step 3 .. . • Step N: y(N−1) = CAN−1x → y(N) = CAN−1x˙ = CANx + CAN−1 Bu CAN−1 B = 0 → u acts directly on the Nth derivative of y. → CAN−1 B = 0 → y(N) = CAN−1x → . . . If, at the point where the algorithm is left, the matrix is invertible, than we can explicitly determine the control vector u. For example, after the Nth step we have u = (CAN−1 B)−1y(N) − (CAN−1 B)−1 CANx , if the matrix CAN−1 B is invertible. This algorithm, together with the invertibility condition of the matrix, leads us to the following definition of the strict relative degree (in the time domain), similar to [47]. Definition 3.2. A linear system (3.24) has strict relative degree N if there exists an integer N such that CAi B = 0 , holds.

i = 0, 1, . . . , N − 2 ,

and

det(CAN−1 B) = 0

Here, we have introduced and defined the relative degree for linear systems. In the literature Definition 3.2 is automatically a result of defining the strict relative degree for nonlinear systems, see for example ISIDORI [76], NIJMEIJER [106], SASTRY [123], and SONTAG [132], where the authors define it for both onedimensional in- and output systems (“single input, single output system”, SISO system) and multi-dimensional in- and output systems (“multi-input, multi-output system”, MIMO-system). The motivating algorithm (including the invertibility condition) was presented for SISO systems in connection with the invertibility of linear systems in [26] and [35], and in [122] and [129] for MIMO systems. The finiteness of the algorithm is presented in [26].


67

Exercise 3.2. Determine the relative degree of the linear SISO system # x˙ = Ax +b u y = cT x , with

⎡

⎤ ⎛ ⎞ 1 0 2 −4 A = ⎣−1 1 3 ⎦ , b = ⎝−3⎠ , cT = 1 −2 2 . 4 −1 −3 −1

3.4.3.3 Minimum-Phase Condition and Invariant Zeros Control design is often affected by the fact that the designer is not always able to see the direct influence of the input on the output. Transformations are usually the necessary step to remedy this problem. Example 3.5 Let us consider a system (3.24). This system is not in normal form because we are not able to see the direct influence of u(t) on y(t). By means of a coordinate transformation depending on the relative degree (see for example [47] or [72]), the direct influence can be brought to the forefront by splitting the state variables. Assuming a relative degree of 1, it can be determined that: ˙ = Ax(t) + Bu(t) , # x(t) system y(t) = Cx(t) ( ( ( )

coordinate transformation x

⎫ • ⎪ CB A1 A2 y(t) y(t) ⎬ + u(t) , ⎪ = 0 A3 A4 z(t) z(t) normal form ⎪ T + * ⎪ ⎭ T T y(t) = Im 0 y(t) z(t)

y z

68


Exercise 3.3. Consider the linear MIMO system x˙ = Ax + Bu

#

y = Cx , with (A , B , C) ∈ Rn×n × Rn×m × Rm×n and det(CB) = 0. Furthermore, let V ∈ Rn×(n−m) be a basis matrix * of Ker(C).−1 + (n−m)×n . Define N := (VT V)−1 VT In − * B(CB)−1 C+ ∈ Rn×n Considering the matrix Q := B(CB) V ∈ R , show that: +T * (a)Q−1 = CT NT ; (b)y = [Im 0] Q−1 x; (c)the MIMO system is equivalent to y˙ = A1 y + A2z + CBu z˙ = A3 y + A4z , with A1 ∈ Rm×m , A2 , AT3 ∈ Rm×(n−m) and A4 ∈ R(n−m)×(n−m) , and specify these matrices. Another example should demonstrate some problems in the system structure. If we want to control this system, the system behavior will be significantly influenced by the dynamics of the z-component; more precisely, the system behavior wholly depends on the matrix A4 . Example 3.6 Consider the following two linear scalar systems with a strict relative degree of 1 and with the initial conditions y(0) = y0 and z(0) = z0 = 0:

• y(t) 1 1 y(t) 1 = + u(t) z(t) 0 −1 z(t) 0

•

y(t) z(t)

=

11 01

y(t) 1 + u(t) z(t) 0

Applying an output feedback function of the form u(t) = −k y(t) with k ∈ R, it can be shown that limt→∞ y(t) = 0. The closed-loop systems are: • y(t) 1−k 1 y(t) = z(t) 0 −1 z(t)

•

y(t) z(t)

=

1 − k 1 y(t) 0 1 z(t)

Considering the differential equation in y, we have: y(t) ˙ = (1 − k) y(t) + z(t) with an exponentially diminishing disturbance z(t) = e−t z0 ,

y(t) ˙ = (1 − k) y(t) + z(t) with an unbounded disturbance z(t) = et z0 .


69

Choosing k > 1, the solution converges to zero in the first case. In the second case we have an unbounded disturbance as t → ∞. We cannot expect convergence to zero. Conclusion: To ensure at least boundedness of the solution in z(t), we have to claim (in case of relative degree 1) that the matrix A4 is a HURWITZ matrix, i.e., all eigenvalues have negative real parts. These matrix eigenvalues, which must have an a priori structure, are called invariant zeros as in the following definition [47]. Definition 3.3. The invariant zeros of system (3.24) are the n − N m complex solutions s of the equation sIn − A B = 0. det C 0 From this definition it follows that these zeros are invariant with respect to any sIn − A B transformations that preserve the rank of the matrix . The invariant C 0 zeros of (3.24) are invariant under the following transformations [47]: ˆ x, • change of coordinates in the state space from x to x = T ˆ • feedback transformations of the form u = Fx +v. These transformations are very important in designing controls in this book. Definition 3.4. A system (3.24) is called minimum-phase if sIn − A B det = 0 C 0

for every complex s with non-negative real part.

Therefore, we can conclude that, here, the minimum-phase condition is equivalent to the fact that the invariant zeros have strict negative real parts, or stable invariant zeros for short. The differential equation of the corresponding z-component involved with the invariant zeros possesses “zero dynamics” provided y = 0, see [76], [106]. Exercise 3.4. Consider the system x˙ = Ax + Bu ,

x(0) =x0 ∈ R

#

y = Cx , with (A , B , C) ∈ Rn×n × Rn×m × Rm×n and det(CB) = 0. Prove the equivalence of the following two assertions:

70


s In − A B = 0 for all s ∈ C with ℜ(s) ≥ 0 and C 0 • all eigenvalues of A4 (see Exercise 3.3) have strict negative real parts. • det

3.4.4 Recapitulatory Example and High-Gain Control Consider the motivating example (3.22) and (3.23) again. With the output y(t) = x(t) one can clearly see the direct influence of u(t) on the second derivative of the output; therefore, this is a system with a relative degree 2. Now we check the relative degree with Definition 3.2: 0 =0 CB = 1 0 c

σ (CAB) = σ

10

m

0 1 − md − mc

0 c m

=σ

01

0 c m

=

c > 0. m

We conclude that the system has strict relative degree 2. The matrix CAB, arising here in this special case, in general CAN−1 B, is called the high-frequency gain matrix, or simply high-frequency gain, in the literature, see for example [72], [91], [103] or [124]. Furthermore, consider that with f ≡ 0, ⎞ ⎛ ⎛ ⎞ 0 1 s −1 0 0 sI2 − A B sI − 1 ⎠ = det ⎝ mc s + md m1 ⎠ = det ⎝ 2 * − mc+ − md det m C 0 10 1 0 0 0 d 1 −1 0 = − −s− , = det s + md m1 m m the invariant zeros (see Definition 3.3) are the solutions of s+

1 d + = 0. m m

With s = − m1 (1 + d) < 0 the system is minimum-phase, see Definition 3.4. Now, we can apply the following theorem: Theorem 3.3. Stabilizing high-gain output feedback: For every linear system (3.24) (with strict relative degree 2, with positive high-frequency gain, and which fulfills the minimum-phase condition), there exists a k∗ ∈ R such that the closed-loop system ˙ = Ax(t) + Bu(t) ,y(t) = Cx(t) ,u(t) = −ky(t) − ky(t) ˙ x(t) is exponentially stable for all k ≥ k∗ .


71

A different version of a proof of this well-known result can be found in [15]. The term “high-gain” is based on the kind of use of output feedback. The assertion of the theorem is fulfilled if the gain parameter (also, tuning parameter) k is sufficiently large. We use k as the gain variable in accordance with [72]. In some sources, ε1 (0 < ε 1) is used instead (for example in [47]).


Chapter 4

Wheeled Locomotion Systems – Rolling

4.1 Rolling - the Exclusive Engineering Idea for Locomotion All biological forms of locomotion have been copied in technical locomotion systems. The only form of locomotion scarcely found in biological systems but dominating in technology is rolling, see Fig. 4.1.

Fig. 4.1 Biological and technical locomotion and the placement of rolling in engineering [Photos courtesy of Herrenknecht AG, Transrapid International GmbH & Co. KG, TETRA Gesellschaft für Sensorik, Robotik und Automation mbH, and D. VOGES]


73

74

4 Wheeled Locomotion Systems – Rolling

Organisms do not possess wheels, but some plants and animals use wheel-like motion for transport - that is to say, they roll. The golden wheel spider in the Namib Desert of western Africa curls into a ball when threatened and can then roll for up to 100 meters with a speed of 44 revolutions per second. Also, the well-known dung beetle Scarabeus sacer rolls up dung, the primary food source for itself and its offspring, into large balls, see Fig. 4.2. More precisely, it traps the ball between its hind legs and, going backwards, rolls them until it finds a suitable place to bury them.

Fig. 4.2 Dung beetle [Photo courtesy of D. VOGES]

Today, we know that the invention of the wheel was the start of rapid technological developments, leading to vastly increased mobility. Going forward, we can begin to understand and estimate the consequences. From a mechanical point of view “rolling” means first and foremost “stiction”. For example, a lot of engineering know-how is invested in developing tires to achieve the goal of zero speed at the point of contact. In practice, however, one always encounters slip, and the instantaneous center of rotation lies above the point of tire contact. In wheels-on-tracks systems a conic wheel profile is used to ensure permanent contact between the wheel and the railway. This conic profile provides the so-called “sine-like run,” which allows the wheel set to center itself on long straight runs or in curves with a sufficiently large radius. In addition, the wheel flange, a bulge on the inside of the wheel, keeps the wheel safely on the track. Despite the 6000 years of wheel development, there is still space for improvement. Recently, omnidirectional wheels have attracted a lot of interest, not least because of their successful operation in the RoboCup, see Fig. 4.3. For these wheels the classical no side-slip condition r˙C · E3 = 0, see Fig. 2.7, does not exist and, consequently, motion in the wheel’s axial direction is possible. Obviously, this leads to a highly increased maneuverability of robots using such wheels.

4.2 Two-Wheel Planar Mobile Robot

75

Fig. 4.3 Omnidirectional wheel

Besides their maneuverability the main advantage of wheeled locomotion systems is their performance in terms of power consumption, velocity, and available payload. This chapter describes in detail the kinematics and dynamics of selected wheeled locomotion systems using methods presented in Chapter 2.

4.2 Two-Wheel Planar Mobile Robot The differential drive is the most commonly used form of wheeled locomotion system used in robots because it is the simplest and easiest to implement. It has freemoving castor wheels in the front accompanied by left and right drive wheels. This principle is used as the basic concept in a wheelchair for the physically disabled, see Fig. 4.4. In mobile robots the two wheels are normally driven by different motors. The independent drives and the difference in friction make it difficult for straightline motion.

Fig. 4.4 Computer simulation of the dynamic behavior of a wheelchair under consideration of the interaction between the human and the machine (with the software alaska 4.0, Institute of Mechatronics Chemnitz)

76


4.2.1 Two-Wheel Vehicle Model Let us consider the motion of a two-wheel vehicle on a plane, see Fig. 4.5. For this case we will derive and compare the equation of motion with all the methods considered in the previous chapter: APPELL’s equations, LAGRANGE’s equations with multipliers, VORONETS’ equations and the basic theorems of dynamics (laws of linear momentum and angular momentum). 3

1

P1 C1 4

l

C

r

y P2

0 l

y

x

C2

2

Fig. 4.5 Model of a wheelchair with geometric parameters

The vehicle represents a mechanical system consisting of the left (1) and the right (2) wheels positioned on one common axis and a body (3). The mass of each wheel is designated as m1 , the wheel radius R, the axis length 2l, and the mass of the body m0 . The center of the body is located on the straight line perpendicular to the wheel axis and goes through the center of the axis. The distance OC is equal to ρ . The mass moment of inertia of the body about the vertical axis, which goes through the center of mass of the body, is equal to JC . Wheels 1 and 2 roll without slip. There are two wheels with an additional vertical axis of rotation in the points P1 and P2 . The system is actuated by the torques M1 and M2 , applied to the left and the right wheel, respectively, see Fig. 4.6. The position of the system is defined by five coordinates: • Angle q1 = ψ between the straight line perpendicular to the wheels’ axis of rotation and abscissa of the fixed coordinate system. • Angles of rotation q2 = ϕ1 and q3 = ϕ2 of the left and the right wheels, respectively.


77

• Coordinates q4 = x0 and q5 = y0 of the point 0, the center of the axis of rotation of the wheels.

4.2.2 Kinematics The coordinates of the centers of mass C1 of the left and C2 of the right wheels are x1 = x0 − l sin ψ , y1 = y0 + l cos ψ , x2 = x0 + l sin ψ , y2 = y0 − l cos ψ . The coordinates of the body’s center of mass C are: xC = x0 + ρ cos ψ , yC = y0 + ρ sin ψ . The velocities of the mass centers of the wheels and the body: x˙1 = x˙0 − l ψ˙ cos ψ , y˙1 = y˙0 − l ψ˙ sin ψ , x˙2 = x˙0 + l ψ˙ cos ψ , y˙2 = y˙0 + l ψ˙ sin ψ ,

(4.1)

x˙C = x˙0 − ρ ψ˙ sin ψ , y˙C = y˙0 + ρ ψ˙ cos ψ . The accelerations of the mentioned points are x¨1 = x¨0 − l ψ¨ cos ψ + l ψ˙ 2 sin ψ , y¨1 = y¨0 − l ψ¨ sin ψ − l ψ˙ 2 cos ψ , x¨2 = x¨0 + l ψ¨ cos ψ − l ψ˙ 2 sin ψ , y¨2 = y¨0 + l ψ¨ sin ψ + l ψ˙ 2 cos ψ ,

(4.2)

x¨C = x¨0 − ρ ψ¨ sin ψ − ρ ψ˙ 2 cos ψ , y¨C = y¨0 + ρ ψ¨ cos ψ − ρ ψ˙ 2 sin ψ . The equations of constraint follow from the condition of vanishing the velocity vector of the wheel-plane contact point, as was shown in Section 2.2.3:

78


x˙0 cos ψ + y˙0 sin ψ − l ψ˙ = R ϕ˙ 1 , x˙0 cos ψ + y˙0 sin ψ + l ψ˙ = R ϕ˙ 2 , −x˙0 sin ψ + y˙0 cos ψ = 0 .

(4.3)

We rewrite the constraint equations (4.3) as follows: x˙0 = l ψ˙ cos ψ + R ϕ˙ 1 cos ψ , y˙0 = l ψ˙ sin ψ + R ϕ˙ 1 sin ψ , 2l ϕ˙ 2 = ψ˙ + ϕ˙ 1 . R

(4.4)

We see that there are two non-holonomic constraints and one holonomic constraint. If the functions ϕ1 (t) and ϕ2 (t) are given, we are able to integrate the equations (4.4). The integration of these equations under the following assumptions

ϕi = ωi t , ϕ˙ i = ωi = const , (i = 1, 2) ,

R (ω2 − ω1 ) = ψ˙ = Ω = const 2l

and with the initial conditions x0 (0) = y0 (0) = 0 and ψ (0) = 0 leads to the results:

ω1 + ω2 sin Ω t , 2Ω ω1 + ω2 y0 (t) =R (1 − cos Ω t) . 2Ω x0 (t) =R

Finally, excluding parameter t one can derive the motion trajectory x02 + (y0 − b)2 = b2 , with b = R ω12+Ωω2 .

4.2.3 Dynamics The applied and the reaction forces are needed in order to analyze the dynamic behavior of the two-wheel planar robot. These forces are defined using the method of 2y of the resultant internal 1y and M section, see Fig. 4.6 and 4.7. The components M moments between the vehicle body and the wheels are the given driving moments, i.e., M1y = M1 (t) and M2y = M2 (t).


79

Fig. 4.6 Model of the main body with forces and moments

Fig. 4.7 Model of wheel 2 (right wheel) with forces and moments

4.2.3.1 APPELL’s Equations The generalized coordinates are q1 = ψ , q2 = ϕ1 , q3 = ϕ2 , q4 = x0 , q5 = y0 . To derive the acceleration energy we represent it as a sum of the acceleration energies of the mass point (the mass of which is equal to the total mass of the system and which is placed at the center of mass of the system) and the acceleration energy of the relative motion of the system around its mass center, i.e., S = STrans + SRot .

80


Exercise 4.1. Determine the formula for the acceleration energy in an analogous way to the kinetic energy, see equation (2.54). For the investigation of locomotion systems with wheels, the acceleration energy of one wheel is a basic element. Therefore, in the next few steps we define this energy, starting with the relation for the acceleration of an arbitrary point of a wheel.

Fig. 4.8 Velocities and accelerations of an arbitrary point M on a wheel

Let 01 x1 y1 z1 be the coordinate system with its origin at the center of the wheel. The coordinate axes are directed parallel to the inertial coordinate system. The origin of the moving coordinate system 01 ξ η ζ is the point 01 . Axis 01 ζ is directed along axis 01 z1 , axis 01 η lies in the plane of the wheel, and axis 01 ξ is perpendicular to the plane of the wheel. We consider an arbitrary point M on the wheel located a −−→ distance r = |01 M| from the center of mass of the wheel 01 , see Fig. 4.8. The absolute acceleration a of point M consists of the relative acceleration ar , the acceleration of transportation motion ae , and the CORIOLIS acceleration ac : a = ar +ae +ac .


81

Acceleration ar relative to coordinate system 01 ξ η ζ is represented by a sum of two components: normalarnr (directed toward the center of the wheel) and tangential arτr (lies in the plane of the wheel and directed perpendicular to the radius) ar nr = r ϕ˙ 2 ,

ar τr = r ϕ¨ .

The acceleration of transportation motion ae of point M is represented also as a sum of two components: normal aene (perpendicular to axis 01 z1 ) and aeτe (perpendicular to the plane of the wheel) ae ne = h θ˙ 2 ,

ae τe = h θ¨ .

Here, h is the distance from point M to axis 01 z1 . The CORIOLIS acceleration is defined by the equation e ×vr ) , ac = 2 (ω e is directed along axis where the angular velocity of transportation of the wheel ω 01 z1 . Relative velocity vr is directed perpendicular to the radius in the plane of the wheel. CORIOLIS acceleration ac is directed parallel to axis 01 ξ , we get π − ϕ = 2 rϕ˙ θ˙ cos ϕ . ωe = θ˙ , vr = r ϕ˙ , ac = 2 ωe vr sin 2 In the non-inertial frame 01 ξ η ζ the projections of vectorac are given as follows: acξ = −2 r ϕ˙ θ˙ cos ϕ ,

acη = 0 ,

acζ = 0 .

The projections of vector ar on the axes of the non-inertial frame 01 ξ η ζ are ar ξ = 0 ,

arη = −ar nr sin ϕ + ar τr cos ϕ ,

ar ζ = ar nr cos ϕ + ar τr sin ϕ .

The projections of vector ae on the axes of the non-inertial frame 01 ξ η ζ are ae ξ = −h θ¨ ,

ae η = −h θ˙ 2 ,

ae ζ = 0 .

Finally, the projections of the absolute acceleration a are aξ = −2 r ϕ˙ θ˙ cos ϕ − h θ¨ , aη = r ϕ¨ cos ϕ − r ϕ˙ 2 sin ϕ − h θ˙ 2 ,

(4.5)

aζ = r ϕ¨ sin ϕ + r ϕ˙ cos ϕ . 2

The squared value of vector a is a2 =r2 ϕ¨ 2 + h2 θ¨ 2 − 2 r h ϕ¨ θ˙ 2 cos ϕ + 4 r h θ¨ ϕ˙ θ˙ cos ϕ + r2 ϕ˙ 4 + h2 θ˙ 4 + 4 r2 ϕ˙ 2 θ˙ 2 cos2 ϕ − 2 r h ϕ˙ 2 θ˙ 2 sin ϕ .

(4.6)

82


We need to introduce the acceleration energy of the wheel (the mass-weighted integral of an element dm accelerations squared), which is determined by the same relation as kinetic energy: SRot

1 = 2

1 a dm = Δ ρ 2

2π R

2

a2 r dr d ϕ ,

0 0

(V )

where Δ is the wheel thickness, R the radius of the wheel, ρ the material density. Let us now calculate the afore-mentioned integral for every component of equation (4.6). It is given that for all the points of the wheel, the angular velocities ϕ˙ and θ˙ as well as angular accelerations ϕ¨ and θ¨ are the same. For the first component we have

2π R m1 R2 R4 = J1 ϕ¨ 2 , Δρ r3 ϕ¨ 2 dr d ϕ = 2 π Δ ρ ϕ¨ 2 = 4 2 0 0 2

where m1 = ρπ R2 Δ is the mass of the disk and J1 = m12R is the mass moment of inertia with respect to the axis of rotation of the wheel. For the second component, taking into account h = r | sin ϕ |,

Δρ

2 π R

R4 m1 R2 ¨ 2 r3 θ¨ 2 sin2 ϕ dr d ϕ = π Δ ρ θ¨ 2 = θ = J2 θ¨ 2 , 4 4

0 0 2

where J2 = m14R is the mass moment of inertia according to the axis, which lies on the plane of the wheel and goes through its center. If we recall that 02 π | sin ϕ | cos ϕ d ϕ = 0, then obviously the integrals of the third and the fourth components are equal to zero. The integral result of the other components, which contain no second-order derivatives, are represented as Φ1 (ϕ˙ , θ˙ ). Therefore, it follows for SRot : SRot =

J1 2 J2 ¨ 2 ϕ¨ + θ + Φ1 (ϕ˙ , θ˙ ) . 2 2

(4.7)

Next, we apply APPELL’s equation (2.80), where the partial derivatives ∂∂q¨Sa1 are to be calculated. Hence, the components of acceleration energy that do not depend on the second derivatives can be dropped. Consequently, formula (4.7) is the expression for the acceleration energy of the wheel in the result of the rotation of the wheel in the vertical plane, i.e., we obtain one (the most complicated) part of the wheel’s acceleration energy. The part of the acceleration energy concerning the motion of the center of mass (x01 , y01 ) of the wheel takes the form STrans =

1 2 m x¨01 + y¨201 . 2

(4.8)


83

In the next few steps, we will come back to the acceleration energy for the twowheel planar mobile robot. First, one can write: x¨12 + y¨21 =x¨02 + y¨20 + l 2 ψ¨ 2 − 2 l ψ¨ (x¨0 cos ψ + y¨0 sin ψ ) + 2 l ψ˙ 2 (x¨0 sin ψ − y¨0 cos ψ ) + l 2 ψ˙ 4 , x¨22 + y¨22 =x¨02 + y¨20 + l 2 ψ¨ 2 + 2 l ψ¨ (x¨0 cos ψ + y¨0 sin ψ ) − 2 l ψ˙ 2 (x¨0 sin ψ − y¨0 cos ψ ) + l 2 ψ˙ 4 , x¨C2 + y¨C2 =x¨02 + y¨20 + ρ 2 ψ¨ 2 − 2 ρ ψ¨ 2 (x¨0 sin ψ − y¨0 cos ψ ) + 2 l ψ˙ 2 (x¨0 cos ψ + y¨0 sin ψ ) + l 2 ψ˙ 4 , from which one readily finds that x¨12 + y¨21 + x¨22 + y¨22 = 2 x¨02 + y¨20 + l 2 ψ¨ 2 + 2 l 2 ψ˙ 4 . Therefore, taking into account equations (4.7) and (4.8) the relation for the energy sum of the right and the left wheels can be given as follows: J1 2 ϕ¨ + ϕ¨ 22 + . . . , S1 + S2 = m1 x¨02 + y¨20 + l 2 ψ¨ 2 + J2 ψ¨ 2 + 2 1

(4.9)

2

J1 = m12R is the mass moment of inertia of the disk relative to the axis that is perpen2

dicular to the plane of the disk and that goes through its center of mass. J2 = m14R is the mass moment of inertia of the thin disk relative to the axis that is perpendicular to the plane of the disk and that goes through its center of mass. Since in APPELL’s equations there are only derivatives of the acceleration energies over second-order derivatives of the generalized coordinates, the members of the equation containing first-order derivatives are unimportant and are therefore omitted. The energy of acceleration of the body S0 has the following form: S0 =

JC m0 2 x¨0 + y¨20 + ρ 2 ψ¨ 2 + ψ¨ 2 − m0 ρ ψ¨ (x¨0 sin ψ − y¨0 cos ψ ) 2 2 − m0 ρ ψ˙ 2 (x¨0 cos ψ + y¨0 sin ψ ) + . . . (4.10)

Finally, taking into account (4.9) and (4.10), one can readily obtain the relation for the acceleration energy of the whole system: J3 m 2 J1 2 x¨0 + y¨20 + ψ¨ 2 + ϕ¨ 1 + ϕ¨ 22 S = S0 + S1 + S2 = 2 2 2 − m0 ρ ψ¨ x¨0 sin ψ − y¨0 cos ψ − m0 ρ ψ˙ 2 x¨0 cos ψ + y¨0 sin ψ + . . . , (4.11) with m = 2 m1 + m0 ,

J3 = J0 + 2 J2 + 2 m1 l 2 ,

J0 = JC + m0 ρ 2 .

84


We have five generalized coordinates and three constraint equations. Let us take q1 = ψ and q2 = ϕ1 as independent variables, and the others are excluded using the constraint equations. From the constraint equations (4.4) we obtain: x¨0 = l ψ¨ cos ψ + R ϕ¨ 1 cos ψ − (l ψ˙ + R ϕ˙ 1 ) ψ˙ sin ψ , y¨0 = l ψ¨ sin ψ + R ϕ¨ 1 sin ψ + (l ψ˙ + R ϕ˙ 1 ) ψ˙ cos ψ , which yields: x¨02 + y¨20 = l 2 ψ¨ 2 + R2 ϕ¨ 12 + 2 l R ψ¨ ϕ¨ 1 + (l ψ˙ + R ϕ˙ 1 )2 ψ˙ 2 , x¨0 sin ψ − y¨0 cos ψ = −l ψ˙ 2 − R ϕ˙ 1 ψ˙ , x¨0 cos ψ + y¨0 sin ψ = l ψ¨ + R ϕ¨ 1 .

(4.12)

The acceleration energy (4.11), taking into account (4.12), takes the form: S=

J4 2 J5 2 l ψ¨ + ϕ¨ 1 + J5 ψ¨ ϕ¨ 1 + m0 ρ R ψ¨ ψ˙ ϕ˙ 1 − m0 ρ R ϕ¨ 1 ψ˙ 2 + . . . , 2 2 R

where

l2 J1 + m l 2 , J5 = 2 J1 + m R2 . R2 One can write the left-hand sides of APPELL’s equations as J4 = J3 + 4

l ∂S = J4 ψ¨ + J5 ϕ¨ 1 + m0 ρ R ψ˙ ϕ˙ 1 , ∂ ψ¨ R l ∂S = J5 ϕ¨ 1 + J5 ψ¨ − m0 ρ R ψ˙ 2 . ∂ ϕ¨ 1 R

(4.13)

To derive the right-hand side of APPELL’s equations, we calculate the virtual work on virtual displacements of the independent coordinates

δ A = M1 δ ϕ1 + M2 δ ϕ2 . Since

δ ϕ2 =

2l δ ψ + δ ϕ1 , R

then

2l M2 δ ψ + (M1 + M2 ) δ ϕ1 . R Taking into account (4.13) and (4.14) we obtain APPELL’s equations

δA=

(4.14)

l 2l J5 ϕ¨ 1 + m0 ρ R ψ˙ ϕ˙ 1 = M2 , R R l 2 J5 ϕ¨ 1 + J5 ψ¨ − m0 ρ R ψ˙ =M1 + M2 , R

(4.15)

J4 ψ¨ +

or in another form:


l J ψ¨ + m0 ρ ψ˙ (l ψ˙ + R ϕ˙ 1 ) = (M2 − M1 ) , R J ϕ¨ 1 − m0 ρ ψ˙ (α R ψ˙ + l ϕ˙ 1 ) =α M1 + β M2 .

85

(4.16)

Here, l2 l2 l2 J5 = J3 + 2 J1 2 = 2 m1 l 2 + 2 J2 + J0 + 2 J1 2 , 2 R R R l2 2 J0 + 4 R2 J1 + 2 J2 + (m + 2m1 ) l J4 α= = , J5 2 J1 + m R2 J4 l2 J0 + 2 J2 − m0 l 2 β = −2 2 = . J5 R 2 J1 + m R2

J = J4 −

4.2.3.2 LAGRANGE’s Equations with Multipliers To derive LAGRANGE multipliers we calculate the kinetic energy of the system first. From (4.1) it follows x˙12 + y˙21 = x˙02 + y˙20 + l 2 ψ˙ 2 − 2 l ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) , x˙22 + y˙22 = x˙02 + y˙20 + l 2 ψ˙ 2 + 2 l ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) , x˙C2 + y˙C2 = x˙02 + y˙20 + ρ 2 ψ˙ 2 − 2 ρ ψ˙ (x˙0 sin ψ + y˙0 cos ψ ) . Hence, the sum of the kinetic energies of the wheels T1 + T2 and the body T0 will be J1 2 T1 + T2 = m1 x˙02 + y˙20 + l 2 ψ˙ 2 + ϕ˙ 1 + ϕ˙ 22 + J2 ψ˙ 2 , 2 J0 2 m1 2 2 x˙ + y˙0 + ψ˙ − m0 ρ ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) . T0 = 2 0 2 Then, the kinetic energy of the whole system can be defined as T=

J3 m 2 J1 x˙0 + y˙20 + ψ˙ 2 + (ϕ˙ 12 + ϕ˙ 22 ) − m0 ρ ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) , (4.17) 2 2 2

with m = 2m1 + m0 , J3 = J0 + 2 J2 + 2 m1 l 2 , J0 = JC + m0 ρ 2 . Furthermore,

86


d ∂T = m x¨0 − m0 ρ ψ¨ sin ψ − m0 ρ ψ˙ 2 cos ψ , dt ∂ x˙0 d ∂T = m y¨0 + m0 ρ ψ¨ cos ψ − m0 ρ ψ˙ 2 sin ψ , dt ∂ y˙0 d ∂T = J3 ψ¨ − m0 ρ (x¨0 sin ψ − y¨0 cos ψ ) − m0 ρ ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) , dt ∂ ψ˙ d ∂T = J1 ϕ¨ 1 , dt ∂ ϕ˙ 1 d ∂T = J1 ϕ¨ 2 , dt ∂ ϕ˙ 2 ∂T ∂T ∂T ∂T = = = = 0, ∂ x0 ∂ y0 ∂ ϕ1 ∂ ϕ2 ∂T = −m0 ρ ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) . ∂ψ Now, one can write LAGRANGE’s equations with multipliers: m x¨0 − m0 ρ ψ¨ sin ψ − m0 ρ ψ˙ 2 cos ψ = λ1 cos ψ + λ2 cos ψ − λ3 sin ψ , m y¨0 + m0 ρ ψ¨ cos ψ − m0 ρ ψ˙ 2 sin ψ = λ1 sin ψ + λ2 sin ψ + λ3 cos ψ , J3 ψ¨ − m0 ρ (x¨0 sin ψ − y¨0 cos ψ ) = λ1 l − λ2 l , J1 ϕ¨ 1 = M1 − λ2 R , J1 ϕ¨ 2 = M2 − λ1 R .

(4.18)

The constraint equations should be added in the form (4.4): x˙0 = l ψ˙ cos ψ + R ϕ˙ 1 cos ψ , y˙0 = l ψ˙ sin ψ + R ϕ˙ 1 sin ψ , 2l ϕ˙ 2 = ψ˙ + ϕ˙ 1 . R Excluding λ3 and deriving λ1 and λ2 from the first three equations of system (4.18), we obtain

λ1 =

ρ 1 ! J3 ψ¨ − m0 ρ ψ˙ 2 − m0 (x¨0 sin ψ − y¨0 cos ψ ) 2 l l

λ2 =

1 ! J3 ρ − ψ¨ − m0 ρ ψ˙ 2 + m0 (x¨0 sin ψ − y¨0 cos ψ ) 2 l l

" + m (x¨0 cos ψ + y¨0 sin ψ ) , " + m (x¨0 cos ψ + y¨0 sin ψ ) .

(4.19)


87

If one introduces relations (4.19) for λ1 and λ2 in the last two equations of system (4.18) and excludes x¨0 , y¨0 and ϕ¨ 2 using constraint equations, then the resulting relationships will fully agree with equations (4.15) or (4.16).

4.2.3.3 VORONETS’ Equations The generalized coordinates are q1 = ψ , q2 = ϕ1 , q3 = ϕ2 , q4 = x0 , q5 = y0 . The kinetic energy T of the system determined using equation (4.17) is T=

J3 J1 m 2 (x˙ + y˙20 ) + ψ˙ 2 + (ϕ˙ 12 + ϕ˙ 22 ) − m0 ρ ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) . 2 0 2 2

To derive the relationship for kinetic energy T , we exclude ϕ˙ 2 , x˙0 and y˙0 using the constraint equations. As a result, we obtain T =

1 1 l J4 ψ˙ 2 + J5 ϕ˙ 12 + J5 ψ˙ ϕ˙ 1 . 2 2 R

Since the relationship for kinetic energy T and the non-holonomic constraint equations do not depend on coordinates q3 = ϕ2 , q4 = x0 and q5 = y0 , the equations of motion take on CHAPLYGIN’s form: a ∂ α 2 a1 ∂ α a2 d ∂T ∂T d ∂T a2 − q˙d , (4.20) = Q + Q α + − a1 a2 a1 dt ∂ q˙a1 ∂ qa1 ∂ q˙a2 ∂ qa1 ∂ qd with a1 = 1, 2, a2 = 3, 4, 5, d = 1, 2. Let us calculate the work of the generalized forces on virtual displacements, i.e., the virtual work δ A:

δ A = Q1 δ ψ + Q2 δ ϕ1 + Q3 δ ϕ2 + Q4 δ x0 + Q5 δ y0 = M1 δ ϕ1 + M2 δ ϕ2 . Therefore, Q2 = M1 ,

Q3 = M2 ,

Q1 = Q4 = Q5 = 0 .

The derivatives of kinetic energies T and T are:

88


∂T ∂ q˙1 ∂T ∂ q˙2 ∂T ∂ q1 ∂T ∂ q˙3 ∂T ∂ q˙4 ∂T ∂ q˙5

= = = = = =

∂T

l = J4 ψ˙ + J5 ϕ˙ 1 , ∂ψ R l ∂T = J5 ϕ˙ 1 + J5 ψ˙ , ∂φ R ∂T = 0, ∂ q2 ∂T = J1 ϕ˙ 2 , ∂ ϕ˙ 2 ∂T = m x˙0 − m0 ρ ψ˙ sin ψ , ∂ x˙0 ∂T = m y˙0 + m0 ρ ψ˙ cos ψ . ∂ y˙0

(4.21)

Now, we calculate the relationship in parentheses on the right-hand side of equation (4.20):

∂ α13 ∂ α13 ∂ α23 ∂ α23 ∂ α14 ∂ α24 ∂ α15 ∂ α25 = = = = = = = = 0, ∂ q1 ∂ q2 ∂ q1 ∂ q2 ∂ q2 ∂ q2 ∂ q2 ∂ q2 (4.22) ∂ α15 ∂ α25 ∂ α14 ∂ α24 = −l sin ψ , = −R sin ψ , = l cos ψ , = R cos ψ . ∂ q1 ∂ q1 ∂ q1 ∂ q1 Introducing equations (4.21) and (4.22) into equation (4.20), we readily obtain the VORONETS’ equations in the form: l 2l J5 ϕ¨ 1 = M2 + m R ϕ˙ 1 (x˙0 sin ψ − y˙0 cos ψ ) − m0 ρ R ψ˙ ϕ˙ , R R l J5 ϕ¨ 1 + J5 ψ¨ =M1 + M2 − m R ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) + m0 ρ R ψ˙ 2 . R J4 ψ¨ +

If we exclude x˙0 and y˙0 using constraint equations, we obtain the equations of motion in the form of (4.15) or (4.16).

4.2.3.4 Synthetic Method – Basic Theorems of Dynamics To use the basic theorems of dynamics, we first make the system free of constraints. This is done by introducing the reaction forces instead. Then, considering each of the free bodies of the system (left and right wheels and vehicle body), one can apply following principles: • the principle of linear momentum and • the principle of angular momentum with respect to the axes parallel to the axes of the motionless frame with the origin in the center of mass of each body. Of course, vector equations expressing these principles can be projected onto any coordinate axis. To derive such a system of equations for the wheels, we choose the


89

following coordinate system: the origin is located in the center of wheel C2 , axis C2 x2 lies in the wheel plane, and is parallel to the horizontal plane, axis C2 y2 is directed perpendicular to the wheel plane, and axis C2 z2 is directed upwards, see Fig. 4.7. In contact point A the reaction force F2A prevents slipping along the wheel plane as well as perpendicular to it. For the right wheel the link between the wheel and the vehicle body is replaced with the reaction force F2C , which is applied at the 2x , which provides the center of the wheel. In addition there are two moments: M 2z , which is related to the rotation of the perpendicular position of the wheel, and M wheel about the vertical axis, see Fig. 4.6. The equations incorporating the vertical components of the reaction forces are of no use in our case and will not be derived. Let us consider the right wheel first (index 2). The principle of linear momentum, involving projections onto axes C2 x2 and C2 y2 , yields C A + F2x , m1 (x¨2 cos ψ + y¨2 sin ψ ) =F2x C A m1 (−x¨2 sin ψ + y¨2 cos ψ ) =F2y − F2y .

(4.23)

Taking into account equation (4.2), we rewrite equations (4.23) in the form: C A + F2x , m1 (x¨0 cos ψ + y¨0 sin ψ + l ψ¨ ) =F2x C A m1 (−x¨0 sin ψ + y¨0 cos ψ + l ψ˙ 2 ) =F2y − F2y .

(4.24)

The principle of angular momentum, involving projections onto axes C2 x2 , C2 y2 and C2 z2 of the coordinate system C2 x2 y2 z2 , yields A R + M2x , (J2 − J1 ) ψ˙ ϕ˙ 2 = −F2y A J1 ϕ¨ 2 = −F2x R + M2 , J2 ψ¨ = M2z .

(4.25)

We exclude the components of the reaction forces in point A from equation (4.24), using the first two equations of (4.25), yielding following system of equations: C R + (J2 − J1 ) ϕ˙ 2 ψ˙ − M2x , m1 R (−x¨0 sin ψ + y¨0 cos ψ + l ψ˙ 2 ) =F2y C R − J1 ϕ¨ 2 + M2 , m1 R (x¨0 cos ψ + y¨0 sin ψ + l ψ¨ ) =F2x J2 ψ¨ =M2z .

(4.26)

By analogy, for the left wheel, taking into account (4.2), we have C R + (J2 − J1 ) ϕ˙ 1 ψ˙ − M1x , m1 R (−x¨0 sin ψ + y¨0 cos ψ − l ψ˙ 2 ) =F1y C R − J1 ϕ¨ 1 + M1 , m1 R (x¨0 cos ψ + y¨0 sin ψ − l ψ¨ ) =F1x ¨ J2 ψ =M1z .

(4.27)

The coordinate system C x0 y0 z0 is linked to the vehicle body with the origin at the center of mass. Axis C x0 is directed along the symmetry axis of the vehicle body,

90


axis C y0 is parallel to the axis of rotation of the wheels, and axis C z0 is directed vertically upwards, see Fig. 4.6. The theorem regarding the motion of the center of mass, involving projections onto axes C x0 and C y0 , yields C C − F1x , m0 (x¨C cos ψ + y¨C sin ψ ) = −F2x C C m0 (−x¨C sin ψ + y¨C cos ψ ) = −F2y − F1y .

(4.28)

Taking into account equation (4.2), we rewrite equations (4.28) in the form: C C − F1x , m0 (x¨0 cos ψ + y¨0 sin ψ − ρ ψ˙ 2 ) = − F2x C C m0 (−x¨0 sin ψ + y¨0 cos ψ + ρ ψ¨ ) = − F2y − F1y .

(4.29)

The principle of angular momentum with respect to axis C z0 yields: C C C C + F2y ) + l (F1x − F2x ) − M2z − M1z . JC ψ¨ = ρ (F1y

(4.30)

Combining equations (4.26), (4.27), (4.29), and (4.30) (we will not rewrite equations that include unknown moments M2x and M1x ), we obtain the following system of equations: C C − F1x , m0 (x¨0 cos ψ + y¨ sin ψ − ρ ψ˙ 2 ) = −F2x

m0 (−x¨0 sin ψ + y¨ cos ψ + ρ ψ¨ ) = −FyC , C m1 R (x¨0 cos ψ + y¨0 sin ψ + l ψ¨ ) = F2x R − J1 ϕ¨2 + M2 , J2 ψ¨ = Mz ,

(4.31)

C m1 R (x¨0 cos ψ + y¨0 sin ψ − l ψ¨ ) = F1x R − J1 ϕ¨1 + M1 , C C JC ψ¨ = ρ FyC + l (F1x − F2x ) − 2 Mz , C + F C , M = M = M . Relationships (4.31) represent a system of with FyC = F1y z 1z 2z 2y C , F C , F C , M . We six equations with nine unknown variables: ψ , ϕ1 , ϕ2 , x0 , y0 , F1x z 2x y extend this under-determined system by 3 additional equations. We can define only C and F C , which are directed along one straight line. Excluding a sum of reactions F1y 2y C , F C , F C and moment M from equation system (4.31), the unknown reactions F1x z y 2x we obtain

l l J1 (ϕ¨ 2 − ϕ˙ 1 ) = (M2 − M1 ) , R R m R (x¨0 cos ψ + y¨0 sin ψ ) − m0 ρ R ψ˙ 2 + J1 (ϕ¨ 2 + ϕ¨ 1 ) =M2 + M1 . J3 ψ¨ − m0 ρ (x¨0 sin ψ − y¨0 cos ψ ) +

From the constraint equations we find:

(4.32)


91

x¨0 sin ψ − y¨0 cos ψ = − l ψ − R ϕ˙ 1 ψ˙ , x¨0 cos ψ + y¨0 sin ψ =l ψ¨ + R ϕ¨ 1 , 2l ϕ¨ 2 = ψ¨ + ϕ¨ 1 . R ˙2

(4.33)

Now, introducing equations (4.33) into (4.32), we obtain relation (4.16).

4.2.4 Analysis of the Equations of Motion Let us analyze the equations of motion (4.16): l J ψ¨ + m0 ρ ψ˙ (l ψ˙ + R ϕ˙ 1 ) = (M2 − M1 ) , R J ϕ¨ 1 − m0 ρ ψ˙ (α R ψ˙ + l ϕ˙ 1 ) =α M1 + β M2 .

(4.34)

Let us suppose that at time t = 0 we have

ψ (0) = ϕ1 (0) = 0 ,

ψ˙ (0) = ϕ˙ 1 (0) = 0 .

After finding ψ (t) and ϕ1 (t), the functions x0 (t), y0 (t) and ϕ2 (t) can be derived from the constraint equations. If in that initial moment in time x0 (0) = y0 (0) = 0, ϕ2 (0) = 0, then x0 (t), y0 (t) and ϕ2 (t) are defined by the constraint equations x0 (t) = l sin ψ + R

t

ϕ˙ 1 cos ψ dt ,

0

y0 (t) = l (1 − cos ψ ) + R

t

ϕ˙ 1 sin ψ dt ,

(4.35)

0

2l ϕ2 (t) = ψ (t) + ϕ1 (t) . R If the moments applied to the wheels are the same (M1 (t) = M2 (t) = M(t)), then the only solution of system (4.34) that satisfies these initial conditions is

ψ (t) = 0 , ϕ¨ 1 (t) = ϕ¨ 2 (t) =

α +β M(t) , J

α +β R M(t) , x¨0 (t) = J y0 (t) = 0 ,

(4.36)

92

4 Wheeled Locomotion Systems – Rolling 2

with α + β =

2 J0 +4 l 2 J1 +4 J2 +4 m1 l 2 R

2 J1 +m R2 acceleration α +J β R M(t).

. In this case the vehicle moves along the abscissa

with the If the center of mass of the system is located on the wheels’ rotation axis (ρ = 0), and if the moments M1 , M2 applied to the left and right wheels are constant, then from system (4.34) it follows that l (M2 − M1 ) , R J ϕ¨ 1 = α M1 + β M2 . J ψ¨ =

In this case we have a 2 t , 2 b ϕ1 = t 2 , 2

ψ=

with a =

l J R (M2 − M1 ),

b = 1J (α M1 + β M2 ). From equations (4.35) it follows that

ϕ2 =

2l

t2

a+b

, R 2 b sin ψ , x0 (t) = l + R a b (1 − cos ψ ) . y0 (t) = l + R a

Finally, excluding ψ , one can derive the equation for the motion trajectory in the x-y-plane ! b "2 b 2 = l +R . x02 + y0 − l + R a a This means that the trajectory of the point 0(x0 , y0 ) is a circle with radius l + R ba and its center at point Q(0, l + R ba ). The following figures present the results of the numerical simulation with parameters given in the caption of Fig. 4.9. Also, the angles of rotation ϕ1 and ϕ2 for the left and right wheel are shown in this figure. The orientation ψ of the two-wheel planar robot and the trajectory of the motion is presented in Fig. 4.10.


93

Fig. 4.9 Angles ϕ1 (left) and ϕ2 (right) vs. time t for the following parameters: m0 = 80 kg, m1 = 3.0 kg, l = 0.5 m, R = 0.3 m, M1 = 0.5 Nm, M2 = 5.0 Nm

Fig. 4.10 Angle ψ vs. time t (left) and trajectory y0 (x0 ) (right)

Exercise 4.2. Assuming the driving moments have the form M1 (t) = M2 (t) = M0 sin2 ω t, calculate the trajectory of motion of the two-wheel planar mobile robot. The moment M0 and the circular frequency ω are constant.

Exercise 4.3. Formulate the equations of motion of a sand yacht [92] using the LAGRANGE

94


equation with multipliers (2.60). The pressure of the wind on the sail is modeled by the force F (in point D) acting on the sail in normal direction. The line of action of the force F goes through point A. The normal vector n includes an − → angle γ away from axis Ax1 . We denote the geometric parameter by |AC| = a and the mass moment of inertia of the sand yacht with respect to axis z1 by JC . At point H a castor (assumed to be massless) is connected to the body by an ideal frictionless joint, see Fig. 4.11.

x1 H

y1 y

E

B C y

g

D

n

F

A

x 0 Fig. 4.11 Model of a sand yacht with parameters [92]

4.3 The Three-Wheel RoboCup Player “Lukas” Utilizing Omnidirectional Wheels

95

4.3 The Three-Wheel RoboCup Player “Lukas” Utilizing Omnidirectional Wheels 4.3.1 Kinematics

Fig. 4.12 RoboCup player “Lukas”

Figure 4.12 shows the actual RoboCup player and Fig. 4.13 a detailed description with its essential kinematical parameters. It is possible to idealize this robot with 4 bodies (3 wheels with radius R and the vehicle body).

l1

y j

2

2

j

1

E1

E2

C

0 (x , y ) 0

y

1

0

3

ey

l2

j3

ex Fig. 4.13 Model of the RoboCup player “Lukas” with parameters

x

96


Based on a planar motion we choose 4 generalized coordinates (xi , yi , ϕi , ψ ) for each wheel and 3 coordinates (x0 , y0 , ψ ) for the vehicle body. Hence, we are dealing with 15 parameters, which are interdependent. There are 6 geometric constraints derived from the relationship between the position of the vehicle center of mass and the wheel center points. The coordinates and the velocities of the wheel center points are: Wheel 1:

y1 = y0 + l1 sin ψ , x1 = x0 + l1 cos ψ , x˙1 = x˙0 − l1 ψ˙ sin ψ , y˙1 = y˙0 + l1 ψ˙ cos ψ .

Wheel 2:

y2 = y0 + l2 cos ψ , x2 = x0 − l2 sin ψ , x˙2 = x˙0 − l2 ψ˙ cos ψ , y˙2 = y˙0 − l2 ψ˙ sin ψ . (4.37)

Wheel 3:

y3 = y0 − l2 cos ψ , x3 = x0 + l2 sin ψ , x˙3 = x˙0 + l2 ψ˙ cos ψ , y˙3 = y˙0 + l2 ψ˙ sin ψ .

Furthermore, the wheels offer additional constraints of the form (2.55). From the rolling conditions we conclude:

Wheel 2:

v1 · E2 = −R ϕ˙ 1 v2 · E1 = R ϕ˙ 2

Wheel 3:

v3 · E1 = R ϕ˙ 3 .

Wheel 1:

Vectors ex and ey are fixed unit vectors of the inertial coordinate system, and E1 and E2 represent an orthonormal basis of the vehicle-fixed coordinate system with E1 = cos ψ ex + sin ψ ey ,

E2 = − sin ψ ex + cos ψ ey .

These considerations imply the following the constraints of the form: −x˙0 sin ψ + y˙0 cos ψ + l1 ψ˙ = − R ϕ˙ 1 , x˙0 cos ψ + y˙0 sin ψ − l2 ψ˙ =R ϕ˙ 2 , x˙0 cos ψ + y˙0 sin ψ + l2 ψ˙ =R ϕ˙ 3 .

(4.38)

If functions ϕ1 (t), ϕ2 (t), and ϕ3 (t) are known, we can integrate equations (4.38). Thus, we obtain the angular velocity about the z-axis of the robot after subtraction of the last two equations of (4.38):

ψ˙ =

R (ϕ˙ 3 − ϕ˙ 2 ) 2 l2

(4.39)

and the orientation of the robot (rotation angle ψ about the z-axis) after its integration, respectively: R ψ= (ϕ3 − ϕ2 ) (4.40) 2 l2


97

with ψ (0) = ϕ2 (0) = ϕ3 (0) = 0. The relationship between the velocities of the vehicle body x˙0 and y˙0 follows from the multiplication of the first two equations of (4.38) with sin ψ and cos ψ , and subsequent addition and subtraction, respectively: x˙0 = (l1 ψ˙ + R ϕ˙ 1 ) sin ψ + (l2 ψ˙ + R ϕ˙ 2 ) cos ψ , y˙0 = (l2 ψ˙ + R ϕ˙ 2 ) sin ψ − (l1 ψ˙ + R ϕ˙ 1 ) cos ψ .

(4.41)

Integration of these equations is done under the following assumptions:

ϕi = ωi t , ϕ˙ i = ωi = const (i = 1, 2, 3) ,

R (ω3 − ω2 ) = ψ˙ = Ω = const 2 l2

and the following initial conditions x0 (0) = 0 and y0 (0) = 0:

ω1 ω2 + ω3 ω1 + l1 ) cos Ω t + R sin Ω t + R + l1 , Ω 2Ω Ω

(4.42)

ω2 + ω3 ω1 ω2 + ω3 cos Ω t − (R + l1 ) sin Ω t + R 2Ω Ω 2Ω

(4.43)

x0 (t) = − (R y0 (t) = − R

with ω2 = ω3 . From equations (4.42) and (4.43), we obtain (x0 − a)2 + (y0 − b)2 = a2 + b2 ; where a = R ωΩ1 + l1 and b = R ω22+Ωω3 . The case ϕ˙ i = ωi = const with ω2 = ω3 always yields a circle path of the robot, see Fig. 4.14.

Fig. 4.14 The path of the robot for ω2 = ω3 [53]

In the case of ω2 = ω3 (Ω = 0), equations (4.42) and (4.43) are not defined. This restriction results from the reduction of the DOF for the vehicle. Under this

98


condition the vehicle will exhibit translational motion, as rotation does not occur in this case, see (4.39). This special case has to be considered in the first two equations of (4.38), where the robot follows rectilinear paths by setting ψ˙ = 0, see Fig. 4.15.

Fig. 4.15 The path of the robot for ω2 = ω3 = ω1 = 0

4.3.2 Dynamics As previously stated, the robot being considered here (and shown in Fig. 4.13) consists of a body (mass m0 ) and three omnidirectional wheels (mass m1 ). Wheels 2 and 3 are located along the same axis, whereas the axis of wheel 1 is orthogonal to the other two. The robot body’s center of mass C lies on the axis of symmetry, which is also wheel 1’s axis of rotation. The point 0(x0 , y0 ) is the intersection point of the two axes of rotation, ψ is the angle between the axis of symmetry of the body and the x-axis of the inertial frame. The distance from point 0 to the center of mass of wheel 1 is l1 , and the distance to the centers of mass of the wheels 2 and 3 is l2 . Let ϕ1 , ϕ2 , and ϕ3 be the angles of rotation of the wheels. Furthermore, we assume that the driving moments M1 , M2 , and M3 act on each of the corresponding wheels, respectively. The position of the system is determined by the generalized coordinates q1 = x0 , q2 = y0 , q3 = ψ , q4 = ϕ2 , q5 = ϕ3 , q6 = ϕ1 . As mentioned above, these coordinates are coupled via the equations of nonholonomic constraints (4.39). To obtain the equations of motion, we use the LAGRANGE’s equations with multipliers (2.60). The kinetic energy of the system is a sum of four parts T = TC + T1 + T2 + T3 . For the body we obtain TC =

m0 2 1 (x˙ + y˙C2 ) + JC ψ˙ 2 , 2 C 2


99

where JC is the mass moment of inertia of the body relative to an axis passing through the center of mass and perpendicular to the body plane. Since xC = x0 + ρ cos ψ ,

yC = y0 + ρ sin ψ ,

we can rewrite the expression for TC in the form: TC =

m0 2 1 (x˙ + y˙02 ) + J0 ψ˙ 2 − m0 ρ ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) , 2 0 2

with J0 = JC + m0 ρ 2 . Now, we calculate the kinetic energy of each wheel. The translational energy of the left wheel, using expression (4.37), takes the form T2 Trans =

m1 2 m1 2 (x˙2 + y˙22 ) = (x˙ + y˙02 + l22 ψ˙ 2 ) − m1 l2 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) . 2 2 0

The rotational energy of the wheel is T2 Rot = where J1 =

m1 R2 2

1 1 J1 ϕ˙ 22 + J2 ψ˙ 2 , 2 2

is the mass moment of inertia relative to an axis lying in the 2

wheel plane, and J2 = m14R is the mass moment of inertia relative to an axis passing through the center of mass and perpendicular to the body plane. So the total kinetic energy of the left wheel can be expressed as T2 =

m1 2 1 1 (x˙ + y˙02 ) + J3 ψ˙ 2 + J1 ϕ˙ 22 − m1 l2 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) , 2 0 2 2

with J3 = J2 + m1 l2 2 . Analogously, the expression for the energy of the right wheel is m1 2 1 1 (x˙0 + y˙02 ) + J3 ψ˙ 2 + J1 ϕ˙ 32 + m1 l2 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) T3 = 2 2 2 and the front wheel T1 =

m1 2 1 1 (x˙ + y˙02 ) + J4 ψ˙ 2 + J1 ϕ˙ 12 − m1 l1 ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) , 2 0 2 2

with J4 = J2 + m0 l1 2 . Thus, the kinetic energy of the complete robot is T=

m 2 1 1 (x˙ + y˙02 ) + J5 ψ˙ 2 + J1 (ϕ˙ 12 + ϕ˙ 22 + ϕ˙ 32 ) 2 0 2 2 − (m0 ρ + m1 l1 ) ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) .

Here, m = m0 + 3 m1 is the total mass of the system and J5 = J0 + 2 J3 + J4 .

100


The generalized forces Qa (a = 1, 2, . . . , 6) can be found from the expression of the virtual work δ A = M1 δ ϕ1 + M2 δ ϕ2 + M3 δ ϕ3 in the form Q1 = Q2 = Q3 = 0 ,

Q4 = M2 ,

Q5 = M3 ,

Q6 = M1 .

Comparing the equations of non-holonomic constraints x˙0 cos ψ + y˙0 sin ψ − l2 ψ˙ − R ϕ˙ 2 = 0 , x˙0 cos ψ + y˙0 sin ψ + l2 ψ˙ − R ϕ˙ 3 = 0 , −x˙0 sin ψ + y˙0 cos ψ + l1 ψ˙ + R ϕ˙ 1 = 0

(4.44)

with the general form (2.15), it follows for the coefficients fa b (q), (a = 1, 2, . . . , 6 , b = 1, 2, 3) f1 1 = cos ψ , f2 1 = sin ψ , f3 1 = −l2 , f4 1 = −R , f5 1 = f6 1 = 0 , f1 2 = cos ψ , f2 2 = sin ψ , f3 2 = l2 , f4 2 = f6 2 = 0 , f5 2 = −R , f1 3 = − sin ψ , f2 3 = cos ψ , f3 3 = l1 , f4 3 = f5 3 = 0 , f6 3 = R . Hence, the motion equations are m x¨0 − (m0 ρ + m1 l1 ) ψ¨ sin ψ − (m0 ρ + m1 l1 ) ψ˙ 2 cos ψ = λ1 cos ψ + λ2 cos ψ − λ3 sin ψ , m y¨0 + (m0 ρ + m1 l1 ) ψ¨ cos ψ − (m0 ρ + m1 l1 ) ψ˙ 2 sin ψ = λ1 sin ψ + λ2 sin ψ + λ3 cos ψ , J5 ψ¨ − m2 l1 (x¨0 sin ψ − y¨0 cos ψ ) = −λ1 l2 + λ2 l2 + λ3 l1 ,

(4.45)

J1 ϕ¨ 2 = M2 − λ1 R , J1 ϕ¨ 3 = M3 − λ2 R , J1 ϕ¨ 1 = M1 + λ3 R . We find a system of 6 equations for the definition of 9 unknown functions x0 , y0 , ψ , ϕ1 , ϕ2 , ϕ3 , λ1 , λ2 , λ3 . Consequently, we have to add the equations of constraint (4.44). Using the last 3 equations from (4.45) with the help of (4.44) we define the Lagrangian multipliers λ1 , λ2 , λ3 as follows


M2 J1 − 2 (x¨0 cos ψ + y¨0 sin ψ − l2 ψ¨ − x˙0 ψ˙ sin ψ + y˙0 ψ˙ cos ψ ) , R R M3 J1 − 2 (x¨0 cos ψ + y¨0 sin ψ + l2 ψ¨ − x˙0 ψ˙ sin ψ + y˙0 ψ˙ cos ψ ) , λ2 = R R M1 J1 + 2 (x¨0 sin ψ − y¨0 cos ψ − l1 ψ¨ + x˙0 ψ˙ cos ψ + y˙0 ψ˙ sin ψ ) . λ3 = − R R

101

λ1 =

(4.46)

Combining the first two equations in (4.45) and keeping the third equation in the previous form, we get m x¨0 cos ψ + m y¨0 sin ψ − (m0 ρ + m1 l1 ) ψ˙ 2 = λ1 + λ2 , m x¨0 sin ψ − m y¨0 cos ψ − (m0 ρ + m1 l1 ) ψ¨ = −λ3 , J5 ψ¨ − (m0 ρ + m1 l1 ) (x¨0 sin ψ − y¨0 cos ψ ) = (λ2 − λ1 ) l2 + λ3 l1 . Substituting the multipliers λ1 , λ2 , λ3 in these equations with (4.46), it holds that m+

m+

2 J1 2 J1 2 J1 x¨0 cos ψ + m + 2 y¨0 sin ψ − 2 ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) 2 R R R 1 − (m0 ρ + m1 l1 )ψ˙ 2 = (M2 + M3 ) , R ρ J1 J1 J1 x¨0 sin ψ − m + 2 y¨0 cos ψ − l1 m0 + m1 + ψ¨ 2 R R l1 R2 J1 M1 , + 2 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) = R R ρ l2 2 l1 2 J1 x¨0 sin ψ + J1 2 ψ¨ − l1 m0 + m1 + 2 R R l1 R2 ρ J1 J1 y¨0 cos ψ − l1 2 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) + l1 m0 + m1 + l1 R2 R l1 l2 = (M3 − M2 ) − M1 . (4.47) R R

J5 + 2 J1

Noting that J6 = J1 + m R2 , J8 = J1 + m0

J7 = J1 + 2 m R2 ,

ρ 2 R + m1 R2 , l1

l 2 l 2 2 1 J9 = J5 + J1 2 2 + 2 , R R

the equation (4.47) takes the form J6 (x¨0 cos ψ + y¨0 sin ψ ) − 2 J1 ψ˙ (x˙0 sin ψ − y˙0 cos ψ ) − (m0 ρ + m1 l1 ) R2 ψ˙ 2 = R (M2 + M3 ) ,

102


J7 (x¨0 sin ψ − y¨0 cos ψ ) − J8 l1 ψ¨ + J1 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) = R M1 ,

J8 (x¨0 sin ψ − y¨0 cos ψ ) − J9

R2 ψ¨ + J1 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) l1 = R M1 + R

l2 (M2 − M3 ) . l1

We add the corresponding initial conditions to the last equation and calculate x0 , y0 , ψ as functions of time. Then, the angles ϕ1 , ϕ2 , and ϕ3 follow from the constraint equations.

4.3.3 Control of the RoboCup Player “Lukas” We derived the equations that describe the motion of “Lukas” motion in the previous section. This section shows how these equations can be used for controlling the robot to perform a variety of tasks [78] , [86] , [143]. For its real-world application as a “soccer robot”, “Lukas” has to be controlled via commands that are transmitted to it. In a typical small-size league RoboCup environment, the commands are sent via a radio transmitter from a computer that is also connected to a visual system that allows the control application to know the position of every robot as well as the ball on the field. The knowledge of all these positions allows the control application to perform high-level tasks, such as chasing the ball or shooting at the goal, while avoiding collision with other robots.


103

Fig. 4.16 RoboCup environment with information flow

“Lukas” has a fixed set of commands, a subset of which allows the operator to control the angular velocity of each individual wheel. Recall that equations (4.40), (4.42), and (4.43) allow us to compute the position (cartesian coordinates x0 , y0 ) and the orientation (angle ψ ) of the robot for given wheel velocities and initial conditions. Hence, in theory, to perform a particular maneuver, it would suffice to compute the functions ϕ˙ 1 (t), ϕ˙ 2 (t), and ϕ˙ 3 (t) for the angular velocities of the individual wheels and send them to the robot. In practice, however, the outcome of such an open-loop control would be poor, due to the limited accuracy of the motors and the visual system and due to other sources of perturbation (such as friction and slipping). Moreover, such an approach would require a static setting; therefore, it is better to use a closed-loop control, in which the current state of the robot and the positions of all relevant objects are permanently fed back into the control loop. To further increase the accuracy, we do not use the angular velocity of the individual wheels; rather, we transmit the desired translational and rotational velocity vector to the robot. Section 4.3.3.1 gives an overview of the velocity control loop that is actually implemented in the robot itself and that controls the motors in such a way ˙ ˙ y, ˙ ψ˙ )T . In Section 4.3.3.2 we that the robot moves at a certain target speed θr = (x, demonstrate how a control loop for a more complicated task is set up, namely the problem of positioning the robot at a target position (typically behind the ball facing the target to be aimed for). Finally, Section 4.3.3.3 contains some ideas on how the robot’s basic abilities can be used to perform high-level tasks, such as chasing the ball while avoiding obstacles.

104


4.3.3.1 Velocity Control Loop

robot dynamics

..

.

qr

qr

..

fr=Dqr

fr

+ I

open - loop u force control III

II

drive IV

f .. =C f q

V

.

..

q

q

q

VII

VI

Fig. 4.17 Velocity control loop

Figure 4.17 shows the velocity control loop for the direct-current motors that is implemented via a microprocessor in the robot itself. The basic idea is to use the ˙ ˙ difference between the desired and the current velocities, i.e., the error, e = θr − θ to determine an acceleration vector using a proportional-integral or PI controller (I): ⎛ ⎞ θ¨r = k p ⎝e + 1 TN

t

e(τ )d τ ⎠ .

0

This acceleration vector is then used by a decoupling controller (II) to compute the necessary force vector fr , which in turn, if applied to the robot, results in the desired acceleration. The open-loop force control (III) does the actual computation of the necessary armature voltagesu to create the desired force vector fr . This computation uses the actual motor characteristics and also takes slip restrictions into account. Application of the armature voltage u to the motors (IV) results in an actual force ¨ ¨ vector f and an actual acceleration vector θ (V). Integration of θ yields the velocity ˙ feedback θ and the output position vector θ . Overall, this control loop effectively decouples translational and rotational motion of the robot while providing quick reaction times and maximum accuracy.

4.3.3.2 Positioning “Lukas” Figure 4.18 shows a closed-loop control that can be used to move “Lukas” to a target position, which also includes the correct orientation. This control loop is implemented in the controlling computer and generates the commands that are sent to the robot.


105

Fig. 4.18 RoboCup environment with information flow

The polar control (I) generates the needed robot velocities x, ˙ y˙ (translational) and ψ˙ (angular) by computing the difference between the target position and the actual robot position and multiplying them by individual proportionality factors kx , ky , and kψ . These robot velocities are transmitted to the robot, where the velocity control (II) applies the correct voltages to the motors and thus moves the robot (as explained in Section 4.3.3.1). The actual robot position and orientation represent the control loop feedback, the implementation of which is done by a visual recognition system that identifies the robot position and its orientation on the field, see [87].

4.3.3.3 High-Level Tasks Many high-level tasks can be performed based on the positioning control loop shown above, for instance, aiming and shooting. Figure 4.19 shows a typical situation on the field, with the robot position denoted by 0, the ball position by B, and the goal by G.

Fig. 4.19 Problem of aiming

106


Position 0r denotes the target position and orientation of the robot in order to shoot the ball at the goal. This target position and orientation can be achieved by −→ −→ simple computations involving the vector GB and the angle ∠(E1 , GB). Using this target position the positioning control loop moves the robot to the correct position, and as soon as the robot reaches the target position, the shooting mechanism can be activated to shoot the ball at the goal. Note that the positioning control loop does not include any collision avoidance; therefore, it cannot be used on its own for a real RoboCup match. One possible setup for a more realistic robot control is to use a path-planning algorithm in combination with a fast-reacting gradient method. This system consists of an obstacle-avoiding gradient control loop similar to the positioning control loop for short paths, while the path-planning algorithm provides collision avoidance on a large scale. The details of the system are beyond the scope of this text, see [142] for more information.

4.4 Non-Holonomic Mobile Robot 4.4.1 The Idea and Performance of the Mobile Robot The development of the locomotion system shown in Fig. 4.20 is based on the utilization of the classical knife-edge conditions (no side slip) on a wheel. We can realize two-dimensional movement of the system with only one driving force via a special arrangement of four wheels on a parallel-crank mechanism. The driving force created by pneumatic cylinders varies the diagonal length of the rhombus. This happens by controlling the contact of the four vertices with the ground.

Fig. 4.20 The non-holonomic mobile robot by STEIGENBERGER (left) and its principle of locomotion (right)

4.4 Non-Holonomic Mobile Robot

107

The system works as follows: One of the vertices is lifted by the support system so that the nearest two wheels lose contact with the ground. Hence, the system is now supported by this vertex and the other two wheels. When one of the vertices is lifted, the distance from this vertex to the opposite one is changed. This change is caused by two forces that are developed along the rhombus’s diagonal. The angle between the plane of the rhombus and the ground is negligible; these planes are considered as parallel in our case. We also neglect the friction in the joints.

4.4.2 Mechanical Model of a Planar Four-Bar Mechanism with Wheels The model consists of four bars in a rhombus configuration along with a mass. The bars are coupled together by means of revolute joints, see Fig. 4.21. Each bar (of mass m0 and length l) is also the axis of rotation of a wheel. The wheels (each of mass m1 and radius R) are located in the middle of the respective bars. The vertices of the rhombus also has the function of point supports.

Fig. 4.21 Mechanical model of the locomotion system with geometric parameters

The motion of the mechanism is considered relative to the inertial coordinate system with origin 0. Axes 0 x and 0 y are located in a horizontal plane, and axis 0 z is vertical. The origin of the coordinate system 0 x y z is located in the center of the rhombus, with axes 0 x and 0 y directed along the diagonals of the rhombus and axis 0 z being upright. Let (x0 , y0 ) be the coordinates of the origin 0 in the inertial

108


−→ coordinate system, i.e., 0 0 = x0 ex + y0 ey . The angle between diagonal L R and the side of the rhombus is designated as α and the angle between diagonal L R and axis 0 x as ψ , see Fig. 4.21. Let ϕ1 and ϕ2 be the angles of rotation of the wheels that are in contact with the ground. The position of the system is fully determined by six generalized coordinates: q1 = ψ , q2 = α , q3 = x0 , q4 = y0 , q5 = ϕ1 , q6 = ϕ2 .

4.4.3 Kinematics of the Mechanism The coordinates of the centers of mass of wheels 1, 2, 3, and 4 in the coordinate system 0 x y z are l l l l cos α , x2 = − cos α , x3 = − cos α , x4 = cos α , 2 2 2 2 l l l l y1 = sin α , y2 = sin α , y3 = − sin α , y4 = − sin α , 2 2 2 2 z1 = 0 , z2 = 0 , z3 = 0 , z4 = 0 . x1 =

Following the coordinate transformation equations from system 0 x y z to the system 0 x y z x = x0 + x cos ψ − y sin ψ , y = y0 + x sin ψ + y cos ψ , z = z , and assuming z = 0, we can write l l x1 = x0 + (cos α cos ψ − sin α sin ψ ) = x0 + cos(ψ + α ) , 2 2 l l x2 = y0 + (cos α sin ψ + sin α cos ψ ) = y0 + sin(ψ + α ) , 2 2 l l x2 = x0 + (− cos α cos ψ − sin α sin ψ ) = x0 − cos(ψ − α ) , 2 2 l l y2 = y0 + (− cos α sin ψ + sin α cos ψ ) = y0 − sin(ψ − α ) , 2 2 l l x3 = x0 + (− cos α cos ψ − sin α sin ψ ) = x0 − cos(ψ + α ) , 2 2 l l y3 = y0 + (− cos α sin ψ + sin α cos ψ ) = y0 − sin(ψ + α ) , 2 2 l l x4 = x0 + (cos α cos ψ + sin α sin ψ ) = x0 + cos(ψ − α ) , 2 2 l l y4 = y0 + (cos α sin ψ − sin α cos ψ ) = y0 + sin(ψ − α ) . 2 2

(4.48)


109

The velocities of the wheel centers with respect to (4.48) are defined as l x˙1 = x˙0 − (ψ˙ + α˙ ) sin(ψ + α ) , 2 l y˙1 = y˙0 + (ψ˙ + α˙ ) cos(ψ + α ) , 2 l x˙2 = x˙0 + (ψ˙ − α˙ ) sin(ψ − α ) , 2 l y˙2 = y˙0 − (ψ˙ − α˙ ) cos(ψ − α ) , 2 l x˙3 = x˙0 + (ψ˙ + α˙ ) sin(ψ + α ) , 2 l y˙3 = y˙0 − (ψ˙ + α˙ ) cos(ψ + α ) , 2 l x˙4 = x˙0 − (ψ˙ − α˙ ) sin(ψ − α ) , 2 l y˙4 = y˙0 + (ψ˙ − α˙ ) cos(ψ − α ) . 2

(4.49)

The conditions of rolling without slip (vP =0) yield the non-holonomic constraints: For wheel 1 l x˙0 − (ψ˙ + α˙ ) sin(ψ + α ) − R ϕ˙ 1 sin(ψ − α ) = 0 , 2 l y˙0 + (ψ˙ + α˙ ) cos(ψ + α ) + R ϕ˙ 1 cos(ψ − α ) = 0 . 2

(4.50)

l x˙0 + (ψ˙ − α˙ ) sin(ψ − α ) − R ϕ˙ 2 sin(ψ + α ) = 0 , 2 l y˙0 − (ψ˙ − α˙ ) cos(ψ − α ) + R ϕ˙ 2 cos(ψ + α ) = 0 . 2

(4.51)

l x˙0 + (ψ˙ + α˙ ) sin(ψ + α ) − R ϕ˙ 3 sin(ψ − α ) = 0 , 2 l y˙0 − (ψ˙ − α˙ ) cos(ψ + α ) + R ϕ˙ 3 cos(ψ − α ) = 0 . 2

(4.52)

For wheel 2

For wheel 3

For wheel 4

110


l x˙0 − (ψ˙ − α˙ ) sin(ψ − α ) − R ϕ˙ 4 sin(ψ + α ) = 0 , 2 l y˙0 + (ψ˙ − α˙ ) cos(ψ − α ) + R ϕ˙ 4 cos(ψ + α ) = 0 . 2

(4.53)

We have to investigate four contact regimes between the wheels and the ground. If wheels 1 and 4 are in contact with the ground, then it follows from equations (4.49) and (4.53) that l l (ψ˙ − α˙ ) , ϕ˙ 4 = (ψ˙ + α˙ ) , 2R 2R l l x˙0 = (ψ˙ − α˙ ) sin(ψ − α ) + (ψ˙ + α˙ ) sin(ψ + α ) , 2 2 l l y˙0 = − (ψ˙ − α˙ ) cos(ψ − α ) − (ψ˙ + α˙ ) cos(ψ + α ) . 2 2

ϕ˙ 1 =

(4.54)

If wheels 2 and 3 are in contact with the ground, then equations (4.51) and (4.52) yield l l (ψ˙ + α˙ ) , ϕ˙ 3 = − (ψ˙ − α˙ ) , 2R 2R l l x˙0 = − (ψ˙ − α˙ ) sin(ψ − α ) − (ψ˙ + α˙ ) sin(ψ + α ) , 2 2 l l y˙0 = (ψ˙ − α ) cos(ψ − α ) + (ψ˙ + α˙ ) cos(ψ + α ) . 2 2

ϕ˙ 2 = −

(4.55)

If the motion is arranged so that wheels 1 and 2 are in contact with the ground, then with (4.50) and (4.51) we have l l (ψ˙ − α˙ ) , ϕ˙ 2 = (ψ˙ + α˙ ) , 2R 2R l l x˙0 = − (ψ˙ − α˙ ) sin(ψ − α ) + (ψ˙ + α˙ ) sin(ψ + α ) , 2 2 l l y˙0 = (ψ˙ − α˙ ) cos(ψ − α ) − (ψ˙ + α˙ ) cos(ψ + α ) . 2 2

ϕ˙ 1 = −

(4.56)

Finally, if wheels 3 and 4 are in contact with the ground, then it follows from equations (4.52) and (4.53) that l a (ψ˙ − α˙ ) , ϕ˙ 4 = − (ψ˙ + α˙ ) , 2R 2R l l x˙0 = (ψ˙ − α˙ ) sin(ψ − α ) − (ψ˙ + α˙ ) sin(ψ + α ) , 2 2 l l y˙0 = − (ψ˙ − α˙ ) cos(ψ − α ) + (ψ˙ + α˙ ) cos(ψ + α ) . 2 2

ϕ˙ 3 =

(4.57)


111

4.4.4 Dynamics of the Mechanism In order to derive the equations of motion, we will apply APPELL’s equations for non-holonomic systems (2.80). The acceleration energy for the wheel again consists of the acceleration energy S1Trans of the wheel’s center of mass and the acceleration energy of the wheel S1Rot relative to the non-inertial frame calculated above: S1 = S1Trans + S1Rot . The value of S1Rot for each wheel is defined by equation (4.7). Value S1Trans is given as follows: m1 2 (x¨ + y¨2 ) , S1Trans = 2 where x and y are the coordinates of the center of the respective wheel. Using equations (4.49) one can readily obtain for wheels 1 and 4 l l x¨ = x¨0 − θ¨ sin θ − θ˙ 2 cos θ , 2 2

l l y¨ = y¨0 + θ¨ cos θ − θ˙ 2 sin θ , 2 2

with θ = ψ + α for wheel 1 and θ = ψ − α for wheel 4. Value S1Trans is S1Trans

m1 2 l2 x¨0 + y¨20 + θ¨ 2 − l x¨0 θ¨ sin θ = 2 4

l2 ˙ 4 2 2 ˙ ¨ ˙ − l x¨0 θ cos θ + l y¨0 θ cos θ − l y¨0 θ sin θ + θ . 4

For wheels 2 and 3 we have l l x¨ = x¨0 + θ¨ sin θ + θ˙ 2 cos θ , 2 2

l l y¨ = y¨0 − θ¨ cos θ + θ˙ 2 sin θ , 2 2

with θ = ψ − α for wheel 2 and θ = ψ + α for wheel 3. Value S1Trans is S1Trans =

m1 2 l2 x¨0 + y¨20 + θ¨ 2 + l x¨0 θ¨ sin θ 2 4 + l x¨0 θ˙ 2 cos θ − l y¨0 θ¨ cos θ + l y¨0 θ˙ 2 sin θ +

l2 ˙ 4 θ . (4.58) 4

Analogously, we obtain the energy of acceleration of the four bars, which represent the axis of rotation of the wheels. The energy of acceleration S2 of each bar consists of two addends: the energy of acceleration of the bar’s center of mass S2Trans (assuming that the mass of the bar m0 is concentrated in the center of mass) and the acceleration energy of the bar S2Rot with respect to the non-inertial reference frame S2 = S2Trans + S2Rot .

112


Because of the coincidence of the center of mass of the wheel and the corresponding bar, the energy S2Trans follows from (4.58), changing the mass of the wheel m1 by the mass of the bar m0 : S2Trans =

m0 2 l2 x¨0 + y¨20 + θ¨ 2 + l x¨0 θ¨ sin θ 2 4 + l x¨0 θ˙ 2 cos θ − l y¨0 θ¨ cos θ + l y¨0 θ˙ 2 sin θ +

l2 ˙ 4 θ ). 4

The acceleration of an arbitrary point M of the bar moving along a circle is the sum of the normal and tangential components. Thus, for the squared acceleration a we obtain a 2 = h2 θ¨ 2 + h2 θ˙ 4 , where h is the distance between point M and the axis of rotation. By integrating the density ρ and the area of cross section AS over the length of the bar, we find the energy S2Rot :

S2Rot

1 = 2

V

l

1 a dm = ρ AS 2 2

2

a 2 dh =

J3 ¨ 2 θ + Φ1 (θ˙ ) . 2

− 2l

2

0l is the mass moment of inertia of a slender bar about an axis going Here, J3 = m12 through the center of mass and perpendicular to the plane of rotation of the bar. With only two wheels in contact with the ground when the mechanism is in motion, only two wheels roll. Hence, the equation for the acceleration energy takes the form:

m l2 J1 2 + 2 J2 + 2 J3 ψ¨ 2 + α¨ 2 + ϕ¨ 1 + ϕ¨ 22 + Φ2 (ψ˙ , α˙ ) , S = 2 m x¨02 + y¨20 + 2 2 (4.59) with m = m0 +m1 as the sum of masses of one wheel and a bar (the mass of the entire system is 4 (m0 + m1 ) ), and ϕ¨ 1 and ϕ¨ 2 are angular accelerations of the wheels that are in contact with ground. Following equations (4.54) – (4.57) ϕ¨ 12 + ϕ¨ 22 is the same for all pairs of wheels, which means equation (4.59) can be rewritten as follows: , 2 m l2 J1 l 2 2 + 2 J2 + 2 J3 + (ψ¨ 2 + α¨ 2 ) + Φ2 (ψ˙ , α˙ ) . S = 2 m x¨0 + y¨0 + 2 2 R2 Obviously, it is necessary to calculate the relationship x¨02 + y¨20 . For wheel pair 1-4 and satisfying equations (4.54), we have


113

l l x¨0 = (ψ¨ − α¨ ) sin(ψ − α ) + (ψ˙ − α˙ )2 cos(ψ − α ) 2 2 l l + (ψ¨ + α¨ ) sin(ψ + α ) + (ψ˙ + α˙ )2 cos(ψ + α ) , 2 2 l l y¨0 = − (ψ¨ − α¨ ) cos(ψ − α ) + (ψ˙ − α˙ )2 sin(ψ − α ) 2 2 l l − (ψ¨ + α¨ ) cos(ψ + α ) + (ψ˙ + α˙ )2 sin(ψ + α ) . 2 2 For wheel pair 2-3 and using (4.55), we obtain l l x¨0 = − (ψ¨ − α¨ ) sin(ψ − α ) − (ψ˙ − α˙ )2 cos(ψ − α ) 2 2 l l − (ψ¨ + α¨ ) sin(ψ + α ) − (ψ˙ + α˙ )2 cos(ψ + α ) , 2 2 l l y¨0 = (ψ¨ − α¨ ) cos(ψ − α ) − (ψ˙ − α˙ )2 sin(ψ − α ) 2 2 l l + (ψ¨ + α¨ ) cos(ψ + α ) − (ψ˙ + α˙ )2 sin(ψ + α ) , 2 2 from which for both cases we have ! + x¨02 + y¨20 = l 2 ψ¨ 2 cos2 α + α¨ 2 sin2 α + sin 2α ψ˙ 2 α¨ + α˙ 2 α¨ − 2 ψ˙ α˙ ψ¨ + Φ3 (ψ˙ , α˙ ) , finally obtaining S=

! J 2 (ψ¨ + α¨ 2 ) + 2 ml 2 ψ¨ 2 cos2 α + α¨ 2 sin2 α 2

" + sin 2α (ψ˙ 2 α¨ + α˙ 2 α¨ − 2 ψ˙ α˙ ψ¨ ) + Φ4 (ψ˙ , α˙ ) ,

2 with J = ml 2 + 4 J2 + 4 J3 + J1 2lR2 = 43 m0 l 2 + m1 R2 + 54 l 2 . From equation (4.56) for wheels 1 and 2 we obtain l l x¨0 = − (ψ¨ − α¨ ) sin(ψ − α ) − (ψ˙ − α˙ )2 cos(ψ − α ) 2 2 l l + (ψ¨ + α¨ ) sin(ψ + α ) + (ψ˙ + α˙ )2 cos(ψ + α ) , 2 2 l l y¨0 = (ψ¨ − α¨ ) cos(ψ − α ) − (ψ˙ − α˙ )2 sin(ψ − α ) 2 2 l l − (ψ¨ + α¨ ) cos(ψ + α ) + (ψ˙ + α˙ )2 sin(ψ + α ) , 2 2 and for the pair 3-4 from (4.57)

114


l l x¨0 = (ψ¨ − α¨ ) sin(ψ − α ) + (ψ˙ − α˙ )2 cos(ψ − α ) 2 2 l l − (ψ¨ + α¨ ) sin(ψ + α ) − (ψ˙ + α˙ )2 cos(ψ + α ) , 2 2 l l y¨0 = − (ψ¨ − α¨ ) cos(ψ − α ) + (ψ˙ − α˙ )2 sin(ψ − α ) 2 2 l l + (ψ¨ + α¨ ) cos(ψ + α ) − (ψ˙ + α˙ )2 sin(ψ + α ) . 2 2 For these two cases we obtain ! x¨02 + y¨20 = l 2 ψ¨ 2 sin2 α + α¨ 2 cos2 α

" − 2 sin 2α (ψ˙ 2 α¨ + α˙ 2 α¨ − 2 ψ˙ α˙ ψ¨ ) + Φ5 (ψ˙ , α˙ ) .

Hence, the relationship for the acceleration energy takes on the following form S=

! J 2 (ψ¨ + α¨ 2 ) + 2 ml 2 ψ¨ 2 sin2 α + α¨ 2 cos2 α 2

" − sin 2α (ψ˙ 2 α¨ + α˙ 2 α¨ − 2 ψ˙ α˙ ψ¨ ) + Φ6 (ψ˙ , α˙ ) . (4.60)

Now, we derive the generalized forces. It is provided that two forces with equal magnitude and opposite directions are applied to the opposite vertices L and R of the rhombus while the mechanism is in motion. The forces are directed along the diagonal connecting the vertices. The force applied to vertex L(xL , yL ) is defined as follows: FL = FLx , FLy T = (F cos ψ , F sin ψ )T . The force at the vertex R(xR , yR ) is FR = FRx , FRy T = (−F cos ψ , −F sin ψ )T . The value of F can be time-dependent and can be either positive or negative. The work done by these forces to cause possible displacements is

δ A = FLx δ xL + FLy + FRx δ xR + FRy δ yR = F cos ψ (δ xL − δ xR ) + F sin ψ (δ yL − δ yR ) , with xL = x0 − l cos α cos ψ , xR = x0 + l cos α cos ψ , and therefore

yL = y0 − l cos α sin ψ , yR = y0 + l cos α sin ψ

(4.61)


δ xL = δ x0 + l sin α δ yL = δ y0 + l sin α δ xR = δ x0 − l sin α δ yR = δ y0 − l sin α

115

cos ψ δ α + l cos α sin ψ δ ψ , sin ψ δ α − l cos α cos ψ δ ψ , cos ψ δ α − l cos α sin ψ δ ψ , sin ψ δ α + l cos α cos ψ δ ψ .

(4.62)

Substituting equations (4.62) into (4.61), we obtain

δ A = Q1 δ ψ + Q2 δ α = 2 l F sin α δ α , from which it follows that Q1 = 0, Q2 = 2 l F sin α . Consequently, APPELL’s equations of motion (2.80) takes on the following form: Wheel pairs 1-4 and 2-3: J ψ¨ + 4 m l 2 ψ¨ cos2 α − 4 m l 2 sin 2α ψ˙ α˙ = 0 , J α¨ + 4 m l 2 α¨ sin2 α + 2 m l 2 sin 2α (ψ˙ 2 + α˙ 2 ) = 2 l F sin α .

(4.63)

Wheel pairs 1-2 and 3-4: J ψ¨ + 4 m l 2 ψ¨ cos2 α + 4 m l 2 sin 2α ψ˙ α˙ = 0 , J α¨ + 4 m l 2 α¨ sin2 α − 2 m l 2 sin 2α (ψ˙ 2 + α˙ 2 ) = 2 l F sin α .

(4.64)

We will solve the system (4.63) and (4.64) with the following initial conditions: π ˙ , α˙ (0) = 0 . ψ (0) = ψ0 , ψ (0) = 0 , α (0) = α0 0 < α0 < 2 It follows from the first equations of both (4.63) and (4.64) that ψ (t) = ψ0 satisfies these systems, i.e., for the initial conditions mentioned the diagonals of the rhombus move progressively. The dependence of angle α on time in this case for wheel pairs 1-4 and 2-3 is defined by the equation (J + 4 m l 2 sin2 α ) α¨ + 2 m l 2 sin 2α α˙ 2 − 2 l F sin α = 0

(4.65)

and for wheel pairs 1-2 and 3-4 (J + 4 m l 2 sin2 α ) α¨ − 2 m l 2 sin 2α α˙ 2 − 2 l F sin α = 0

(4.66)

with the initial conditions α (0) = α0 and α˙ (0) = 0. The time function of the coordinates of the center of the rhombus x0 (t) and y0 (t) can be found from equations (4.54) – (4.57). Without loss of validity, we are able to assume ψ0 = 0, i.e., at the beginning of the motion, the diagonals of the rhombus are parallel to the axes of the inertial coordinate system. In this case it follows from (4.54) that for the velocities of center of the rhombus (with active wheels 1 and 4) x˙0 = l α˙ sin α ,

y˙0 = 0 .

116


We propose that the center of mass has the coordinates (x0 0 , y0 0 ) at time t = 0. According to (4.64) we find for the coordinate x0 (t) = x0 0 − l (cos α − cos α0 ) ,

y0 (t) = y0 0 .

(4.67)

Analogously, we consider the other three cases. Using (4.55) for active wheels 2 and 3, the velocities are x˙0 = −l α˙ sin α ,

y˙0 = 0 ,

and for the coordinate of point O , we get x0 (t) = x0 0 + l (cos α − cos α0 ) ,

y0 (t) = y0 0 .

(4.68)

y0 (t) = y0 0 − l (sin α − sin α0 ) ,

(4.69)

For active wheels 1 and 2 we obtain from (4.55) x˙0 = 0 , 0

x0 (t) = x0 ,

y˙0 = −l α˙ cos α ,

and if wheels 3 and 4 are in contact with the ground (4.56): x˙0 = 0 , 0

x0 (t) = x0 ,

y˙0 = l α˙ cos α , y0 (t) = y0 0 + l (sin α − sin α0 ) .

(4.70)

For each phase of motion, if a pair of wheels is in contact with the ground, the initial conditions (α0 , x0 , y0 ) are identical to those achieved in the final stage of the former phase of motion.

4.4.5 Dynamic Simulations of the Locomotion of the Planar Four-Bar Mechanism with Wheels The following parameters are used in the numerical simulations: l = 20 cm, R = 2 cm, m0 = 0.2 kg, m1 = 0.4 kg, F = 0.1 N. We assume that the first phase of motion starts with contact of wheel pair 1-4. Figure 4.23 (left) presents the dependence of angle α on time t, calculated using equation (4.65) with F > 0 and α0 = 0.1. The function x0 (t) for this phase, obtained from equation (4.67) with x0 0 = y0 0 = 0, is presented in Fig. 4.24 (left). In the second phase of motion, after a complete stop of the robot, wheel pair 1-2 is in contact with the ground. Figure 4.23 (right) shows the dependence of angle α on time t, calculated using equation (4.66), and Fig. 4.24 (right) shows the function y0 vs. time t for the second phase of motion, using equation (4.69). The initial condition coincides with the final values of the parameters at the end of the first phase of motion, namely F < 0.


117

The trajectory of motion y0 (x0 ) of the mechanism in the horizontal plane is shown in Fig. 4.25. The third phase of motion is marked by wheel pair 1-4 being in contact with the ground (F > 0), and in the fourth phase wheels 3 and 4 are on the ground. Phase 5 again has wheels 1 and 4 in contact (F > 0), and finally, in phase 6 wheel pair 3-4 is in contact with the horizontal plane, and the motion can be described by equations (4.66) and (4.70), with F < 0. The Fig. 4.22 gives an overview of the control regime in a compact form. Motion phase Active wheel pair 1 1-4 2 1-2 3 1-4 4 1-2 5 1-4 6 3-4

Time t [s] x0 [m] y0 [m] sign(F) 0 ≤ t < 3.5 0 ≤ x0 ≤ 0.13 y0 = 0 + 3.5 ≤ t < 6.4 x0 = 0.13 0 ≤ y0 ≤ 0.17 6.4 ≤ t < 9.9 0.13 ≤ x0 ≤ 0.26 y0 = 0.17 + 9.9 ≤ t < 12.8 x0 = 0.26 0.17 ≤ y0 ≤ 0.43 12.8 ≤ t < 16.3 0.26 ≤ x0 ≤ 0.39 y0 = 0.34 + 16.3 ≤ t < 19.2 x0 = 0.39 0.17 ≤ y0 ≤ 0.43 -

Fig. 4.22 Control regime of the mobile robot moving along the trajectory shown in Fig. 4.25

1,2

1,2 1

1 0,8

0,8

alpha 0,6

alpha 0,6

0,4

0,4

0,2

0,2

0

0,5

1

1,5

2

2,5

3 t

3,5

3,5

4

4,5

5 t

Fig. 4.23 Angle α vs. time t during the first (left) and the second (right) step

5,5

6

118


0,16

y0 0,12

0,12

x0

0,1

0,08

0,08

0,06

0,04

0,04

0,02

0

0 0

0,5

1

1,5

2

2,5

3

3,5

3,5

4

4,5

5

5,5

6

t

t

Fig. 4.24 Coordinates x0 (t) (left, 1st phase of motion) and y0 (t) (right, 2nd phase of motion) vs. time t

0,4

0,3

y0 0,2

0,1

0 0

0,1

0,2

0,3

0,4

x0

Fig. 4.25 Trajectory of the non-holonomic mobile robot in the x-y-plane

4.5 Trimaran As an example for using VORONETS’ equations for dynamic analysis, we consider a system consisting of a disk 0 (mass m0 , radius R) and two counterbalances 1 and 2 (mass m1 for each part), see Fig. 4.26 (left) [92]. When outfitted as a sea-going vessel (with a third hull replacing the wheel in the center), this mechanical system is called a “trimaran”, see Fig. 4.26 (right).

4.5 Trimaran

119

Fig. 4.26 A trimaran-like locomotion system with a wheel in place of the center hull (left) and, for comparison, an actual trimaran (right) [Photo courtesy of E. KAPLAN]

The system moves on a horizontal plane under the influence of a moment M acting on the disk. The centers of mass of the disk and the counterbalances are located on a straight line. The distance between the center of mass is l, the radius of inertia of each counterbalance relative to a vertical axis passing through the center of mass is ρ . The disk rolls over the horizontal plane without slipping. Friction between the counterbalances and the plane is neglected. We formulate the mathematical model using VORONETS’ equations. The position of the trimaran is described by four parameters, see Fig. 4.27: • angle of rotation ϕ of the disk, • angle ψ between the disk plane and the axis 0x (orientation of the systems), and • coordinates (x0 , y0 ) of the center of mass C0 . The generalized coordinates are q1 = ϕ , q2 = ψ , q3 = x0 , q4 = y0 . At time t = 0 the trimaran is at rest, and the disk’s center of mass is at the origin of the coordinate system. Axis Ox lies in the disk plane.

120


y

y1

1 C1

z1

x1

M

y

y

1

M

0 j

C0

C0

2

C2

x1

P x

0

Fig. 4.27 The system with four generalized coordinates x, y, ϕ , ψ

We obtain the equations of constraint using the instantaneous center of rotation P, i.e., r˙P = 0, see Fig. 4.27 (right). Thus, in the same way as in Section 2.2.3, the non-holonomic constraints become x˙0 = R ϕ˙ cos ψ ,

(4.71)

y˙0 = R ϕ˙ sin ψ .

According to the designations in Section 2.2.3, we have n = 4 and r = 2, and there are n − r = 2 independent variables. Referring to formula (2.61), the coefficients α a2 a1 with (a1 = 1, 2; a2 = 3, 4) are as follows:

α 3 1 = R cos ψ = R cos q2 , α 4 1 = R sin ψ = R sin q2 ,

α 32 = 0 , α 42 = 0 .

(4.72)

The kinetic energy of the trimaran consists of three parts: T = T0 + T1 + T2 , where T0 is kinetic energy of the disk and T1 and T2 the kinetic energies of the counterbalances. As is true for the whole system, each subsystem can move in a generalized way, i.e., T = TTrans + TRot . The energy of the disk can be calculated as T0 =

m0 R2 2 m0 R2 2 m0 2 x˙0 + y˙20 + ϕ˙ + ψ˙ . 2 4 8

For the counterbalances we get T1 = T2 =

m1 2 m1 2

x˙12 + y˙21 + m21 ρ 2 ψ˙ 2 , 2 x˙2 + y˙22 + m21 ρ 2 ψ˙ 2 .

(4.73)

4.5 Trimaran

121

The velocities of the center of mass following from the time derivative of the coordinates, see Fig. 4.27

are

x1 = x0 − l sin ψ ,

y1 = y0 + l cos ψ ,

x2 = x0 + l sin ψ ,

y2 = y0 − l cos ψ

x˙1 = x˙0 − l ψ˙ cos ψ ,

y˙1 = y˙ − l ψ˙ sin ψ ,

x˙2 = x˙0 + l ψ˙ cos ψ ,

y˙2 = y˙2 + l ψ˙ sin ψ .

Finally, we obtain for energies T1 and T2 2 cos ψ − 2 l y˙ ψ˙ sin ψ + l 2 ψ˙ 2 + m12ρ ψ˙ 2 , 2 T2 = m21 x˙02 + y˙20 + 2 l x˙ ψ˙ cos ψ + 2 l y˙ ψ˙ sin ψ + l 2 ψ˙ 2 + m12ρ ψ˙ 2 .

T1 =

m1 2 2 ˙ 2 x˙0 + y˙0 − 2 l x˙ ψ

(4.74)

The kinetic energy of the trimaran then takes the form T=

m0 R2 2 1 m 2 x˙0 + y˙20 + ϕ˙ + J0 ψ˙ 2 , 2 4 2

(4.75)

2

with m = m0 + 2 m1 and J0 = 2 m1 (l 2 + ρ 2 ) + m04R . Substituting the expressions for x˙0 and y˙0 from (4.71) into (4.75), we obtain an expression for the kinetic energy T : T =

1 1 J1 ϕ˙ 2 + J0 ψ˙ 2 , 2 2

(4.76)

2 0 with J1 = ( 3 m 2 + 2 m1 ) R . We then find the generalized forces Q1 , Q2 , Q3 , and Q4 , corresponding to the coordinates q1 = ϕ , q2 = ψ , q3 = x0 , q4 = y0 , respectively, using the virtual work δ A = Q1 δ ϕ + Q2 δ ψ + Q3 δ x0 + Q4 δ y0 = M δ ϕ . We obtain

Q1 = M ,

Q2 = Q3 = Q4 = 0 .

(4.77)

As can be seen from (4.76), in our case the kinetic energy T does not depend on the generalized coordinates q1 = ϕ and q2 = ψ ; therefore, ∂∂ Tq1 = ∂∂ Tq2 = 0. VORONETS’ equations now take the form ∂ T ∂T d = Qa1 + Qa2 α a2 a1 + a β a2 a1 d q˙d , a1 = 1, 2 . (4.78) d t ∂ q˙a1 ∂ q˙ 2 From (4.76) we have d dt

and using (4.75) we get

∂ T ∂ q˙1

= J1 ϕ¨ ,

d dt

∂ T ∂ q˙2

= J0 ψ¨ ,

(4.79)

122


∂T ∂T = = m x˙0 , ∂ q˙3 ∂ x˙0

∂T ∂T = = m y˙0 . ∂ q˙4 ∂ y˙0

(4.80)

Finally, we calculate the system of coefficients with three indices β a2 a1 d (see formula (2.65)) for a1 , d = 1, 2 and a2 = 3, 4. The coefficients with the same indices are equal to zero, i.e., β 3 11 = β 3 22 = β 4 11 = β 4 22 = 0. For the other coordinates we find the characteristics β 3 12 = −β 3 21 , β 4 12 = −β 4 21 . Keeping in mind formula (4.72), the unknown coefficients take the form

β 3 12 = −β 3 21 = −R sin ψ ,

(4.81)

β 4 12 = −β 4 21 = R cos ψ .

Thus, putting all expressions into formula (4.78), we obtain VORONETS’ equations as J1 ϕ¨ + m R (x˙0 sin ψ − y˙0 cos ψ ) ψ˙ = M , (4.82) J0 ψ¨ − m R (x˙0 sin ψ − y˙0 cos ψ ) ϕ˙ = 0 . The consideration of the constraints (4.71) yields the final form: J1 ϕ¨ = M ,

ψ¨ = 0 .

(4.83)

Equations (4.83) and (4.71) constitute a determined system describing the dynamic behavior of the mechanical system. We can compute the motion of the system for a constant value M of the moment and with all initial conditions equal to zero except ψ˙ (0) = ω = 0. From (4.83) it follows that

ϕ˙ =

M J1

t,

ψ˙ = ω ,

ϕ=

M 2 2 J1 t ,

ψ = ωt.

Using (4.71) we obtain x˙0 = y˙0 =

MR J1 t MR J1 t

cos ω t , sin ω t ,

*

+

MR 1 , J1 ω t sin ω t − ω 1 − cos ω t * + MR 1 y0 = − J1 ω t cos ω t − ω 1 − sin ω t

x0 =

.

Functions x0 (t) and y0 (t), which define the motion of the center of mass of the disk 0, and the trajectory y0 (x0 ) of the trimaran in the plane are shown in Fig. 4.28 and 4.29, respectively.

4.5 Trimaran

123 15

10 10 5 5 0 0

2

4

6

8

10

12

0

14

0

t

2

4

6

8

10

12

14

t

-5

-5

-10

-10

Fig. 4.28 Functions x0 (t) (left) and y0 (t) (right)

15

10

5

0 -10

0

-5

5

10

-5

-10

Fig. 4.29 Trajectory of the center of mass of the disk 0 in the x-y-plane

The trimaran moves along a spiral curve around the origin of the coordinate system when the angular velocity ψ˙ (0) at time t = 0 is not equal to zero. Exercise 4.4. 1 A tricycle is moving along a horizontal plane due to the the driving moment M arising from the cyclist stepping on the pedals of wheel 1, with radius R. Steering 2 acting on the tricycle handlebar. of the tricycle is done by the control moment M Formulate the equations of motion of the tricycle using the LAGRANGE equation with multipliers (2.60). The mass of the system (tricycle plus cyclist) is m,

124


and the mass moment of inertia with respect to the z-axis is JC . Geometric param− → − → −→ eters are denoted by |AC| = a , |AB| = b and |BD| = h. The wheels roll without slipping, and their mass is negligible, see Fig. 4.30.

y1

y2

y

M2

D

x1

B

1

1

b

-M2 y

1

x2

z2

c

1

c

A

1

C

1

1

M1

.

j

1

D x2 P

0 Fig. 4.30 Model of a tricycle with parameters [92]

x

Chapter 5

Walking Machines – Walking

As mentioned in Chapter 1, locomotion with legs is not within the scope of this book. But in order to show the difference between rolling and walking from the point of view of mechanics (as is our goal), simple models of walking are discussed in this chapter. Also, the problem of the robustness and adaptivity of walking is briefly examined. The high mobility aspect makes legged machines attractive from the commercial point of view. RAIBERT in 1986 demonstrated the superiority of legged robots compared to wheeled locomotion systems in unstructured environments [120]. He determined that the mobility of a legged robot is limited by the best available footholds in the reachable terrain, whereas a wheeled vehicle is limited by the worst terrain. Research on legged vehicles should help us to understand how animals and humans move. Thus, all biological models from bipedal to octopedal constructions have been realized by engineers in one form or another, i.e., from commercial systems in series up to a small number of prototypes for research. BERNS maintains an online catalogue of walking machines built around the world over the last 20 years, see at http://www.walking-machines.org/. A detailed description of the history of walking machines and the state of art is given by PFEIFFER & INOUE [115]. In this book, we anthropocentrically concentrate on human bipedalism as the best analyzed realization of walking.

5.1 Human Walking and Running: History and General Remarks Scientific research on human movement from the biomechanical and physiological point of view goes back to the year 1836. At that time brothers ERNST and WIL¨ HELM WEBER published Uber die Mechanik der menschlichen Gehwerkzeuge (“On the Mechanics of Human Walking Tools”), giving an analysis of human bipedalism [153]. Their studies were supported by the Prussian government in order to improve the performance of walking soldiers and thus of the whole army. That K. Zimmermann et al., Mechanics of Terrestrial Locomotion, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-88841-3 5,

125

126

5 Walking Machines – Walking

publication represents the first time that the periodic transfer of locomotion energy between kinetic and potential energy is described. Potential energy may be stored in the gravitational field, which leads to the concept that the extremities form pendula during the swing phases of walking and running. This concept was later taken by VUKOBRATOVIC [152], HEMAMI [64] and other. The dynamic principles of human walking were first formulated by BRAUNE & FISCHER, based on LAGRANGE’s equations, at the end of the 19th century [34]. A first attempt to solve the inverse dynamic problem of human locomotion was done in 1970 by CHOW & JACOBSON [43]. They computed driving moments in the hip and knee using a variation principle. Starting with BERNSTEIN in the 1930s, scientists such as GRITSENKO and MOREINIS and BELETZKI and OCHOZIMSKI of the Russian School of Biomechanics developed mathematical models of human locomotion in order to improve walking machines and prosthetics [25], [107]. Another well-known school in Europe was founded by the afore-mentioned VUKOBRATOVIC. The Yugoslav group solved the inverse problem of anthropomorphic locomotion, based on the principles of analytical mechanics [149], [150], [151]. His “Zero Moment Point (ZMP) concept” is still a basic principle in control strategies of walking machines. For example Honda’s well known “Asimo robot” uses ZMP control. “Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure” was the title of a paper by CAVAGNA and co-workers in 1977 [41]. They emphasized that vertebrate animals and humans realize two simple principles for saving energy during walking: (1) the periodic exchange between kinetic and potential energy as that which takes place in a suspended pendulum and (2) the additional exchange of elastically stored energy (like in a bouncing ball). They as well as MOCHON and McMAHON used models of inverted pendula for the analysis of bipedal motion [98]. Even if the human leg does not strictly speaking represent a genuine pendulum (primarily because the hip joint is not fixed and is even accelerated), the explanatory value of this model is quite high. WITTE considered the motions of human legs, arms, and trunk as a common vibrational system [160]. The use of the legs as springs with defined stiffness leads to less power consumption during walking. The underlying “spring-mass model” was mathematically formulated by BLICKHAN [28]. Together with SEYFARTH, FRIEDRICH, and WANK he identified optimal take-off parameters for long jump, based on simple spring-mass models, and discussed the problem of self-stabilizing “robust” mechanisms [127]. Today, the mechanical paradigms of the stance phase during walking and running mostly used are thus the (1) stiff-legged inverted pendulum and (2) the compliant spring-mass model. The current trend in biomechanics due to the rapid development of computing technology is to investigate ever more complex mechanical models with higher degrees of freedom. However, the control of these systems becoming increasingly ambitious. PFEIFFER, WEIDEMANN, et al. tend to collapse the degrees of freedom to make control possible and transparent [116]. They used neurobiological findings ¨ by CRUSE & BASSLER with respect to the control of walking stick insects for designing their first walking machine “MAX”. The same idea of close cooperation

5.1 Human Walking and Running: History and General Remarks

127

between biologists and engineers (and of mechanics and control) was pursued by DILLMANN, ILG, and BERNS in developing the four-legged machine “BISAM”, see Fig. 5.1 [75].

Fig. 5.1 Four-legged walking machine “BISAM” (Universität Karlsruhe) [Photo courtesy of K. BERNS]

The behavior-based control of their machine follows biological findings (see FISCHER & WITTE [50], [52]. At present, these approaches focus on bipedal humanoid robots such as “JOHNNIE” and “LOLA”, see Fig. 5.2.

Fig. 5.2 Biped walking and running robots “JOHNNIE” (left) and “LOLA” (right) (Technische Universität München) [Photo courtesy of H. ULBRICH]

128


Current studies show that spring-mass models are not only able to describe fast locomotion such as running and jumping, but also are applicable to slow locomotion such as human walking (GEYER & SEYFARTH [56]). FISCHER & LEHMANN were intrigued by comparative analyses showing that the phylogenetic ancestors of humans in legged locomotion realize(d) up to 50% of spatial gain in pedal locomotion by mechanisms of their trunks [51]. Also, the functional integration of extremities and trunk in locomotion even in humans is of growing interest. Since trunks without extremities exist and are able to move (snakes), but not vice versa, we also start our description of models for human locomotion with analyses of trunk motion under the hypothesis that trunks are the driving elements in vertebrate locomotion.

5.2 Dynamic Models of Walking 5.2.1 Dynamics of Human Trunk Locomotion In the following basic model of trunk usage in walking, the body stem is divided into two elliptical cylinders, see Fig. 5.3. Body 1 represents the mass and the inertial properties of the pelvic girdle and the lower half of abdomen (“pelvis”), while body 2 brings together the upper half of the abdomen, thorax, shoulder girdle, arms, neck, and head (“thorax”). Body 1 is linked via a cylindrical joint to the ground and via a revolute joint to body 2. The system, therefore, possesses three degrees of freedom: vertical translation and axial rotation of each body. The reaction forces acting on the hip joint are reduced to the vertical and the anterior and posterior components. In this simple model, the resultant moment about the vertical axis is modeled as a torsional spring-damper system (c1 , d1 ) and the visco-elastic properties of the muscles and soft tissues between pelvis and thorax again as a torsional spring-damper system (c2 , d2 ). We want to find the response of the thorax reacting to the ground reaction forces.

129 J ,k

5.2 Dynamic Models of Walking

g

m2 , J2

q2

y2

m

C2

d2 J ,k

1

c2 c

m1 , J1

q1

C1 h

y1

h y d1 x

c

1

c1 F1

0

z

F2

Fig. 5.3 Trunk model with parameters

Applying the principle of linear momentum, i.e., NEWTON’s second law (2.23), the equations for the translational motions of the bodies are: m1 y¨1 = −c1 y1 − d1 y˙1 + c2 (y2 − y1 ) + d2 (y˙2 − y˙1 ) + F1y + F2y − m1 g , m2 y¨2 = −c2 (y2 − y1 ) − d2 (y˙2 − y˙1 ) − m2 g .

(5.1)

With the definition of the coordinate yc of the center of mass yc =

m1 y1 + m2 y2 , m1 + m2

we find m y¨c = Fy − m g ,

(5.2)

with Fy = F1y + F2y − c1 y1 − d1 y˙1 and m = m1 + m2 . This equation gives only the time-dependent vertical position of the bodies as a function of the force Fy but not any information about the angle variation. In order to obtain the equations governing the rotational motions, we apply the principle of angular momentum (2.27). Between two bodies, stiffness and damping are defined by the skew-symmetric matrices c −c d −d C= , D= (5.3) −c c −d d and the motion equations in matrix form are:

130


J1 0 0 J2

d + d2 −d2 θ¨1 θ˙1 + 1 ¨ −d2 d2 θ2 θ˙2 c1 + c2 −c2 h F1z (t) − h F2z (t) θ1 M(t) + = = . (5.4) −c2 c2 θ2 0 0

The forces F1 (t) and F2 (t) are the anterior and posterior components of the forces applied to the hip joint and can be modeled as sine functions Fi (t) = A sin(ω t + αi ). In this system we define one input M(t) = h F1z (t) − h F2z (t) and one output θ2 (t), i.e., a SISO system. In order to obtain the angle θ2 (t) as a function of M(t), we apply transfer function analysis. In its most simple form for continuous-time input signal x(t) and output y(t), the transfer function will be the linear mapping of the LAPLACE transform of the input X(s) to the output Y (s): Y (s) = H(s) X(s)

⇒

H(s) =

Y (s) , X(s)

(5.5)

where H(s) is the transfer function of the linear time-invariant (LTI) system. First, equation (5.4) should be linearized around θ1 = θ2 = 0. We assume that θ1 = 0 + φ1 and θ2 = 0 + φ2 (φ1 and φ2 are small angular deviations from the zero position). After linearization the two motion equations become J1 φ¨1 + d1 φ˙1 + d2 (φ˙1 − φ˙2 ) + c1 φ1 + c2 (φ1 − φ2 ) = u , J2 φ¨2 + d2 (φ˙2 − φ˙1 ) + c2 (φ2 − φ1 ) = 0 ,

(5.6)

where u represents the input. To obtain the transfer function of the linearized system analytically, we must first take the LAPLACE transform of the system equations: J1 Φ1 (s) s2 + d1 Φ1 (s) s + d2 Φ1 (s) s − d2 Φ2 (s) s + c1 Φ1 (s) + c2 Φ1 (s) − c2 Φ2 (s) = U(s) ,

(5.7)

J2 Φ2 (s) s2 + d2 Φ2 (s) s − d2 Φ1 (s) s + c2 Φ2 (s) − c2 Φ1 (s) = 0 . Since we are looking for Φ2 as an output, we have to solve the second equation for Φ1 (s), J2 s2 Φ1 (s) = Φ2 (s) +1 (5.8) d2 s + c2 and substitute that into the first equation:


131

J2 s2 J2 s2 + 1 s2 + d1 Φ2 (s) +1 s d2 s + c2 d2 s + c2 J2 s2 J2 s2 + d2 Φ2 (s) + 1 s − d2 Φ2 (s) s + c1 Φ2 (s) +1 d2 s + c2 d2 s + c2 J2 s2 + c2 Φ2 (s) + 1 − c2 Φ2 (s) = U(s) . (5.9) d2 s + c2

J1 Φ2 (s)

Re-arranging and carrying out the necessary algebra, the transfer function is: ! Φ2 (s) = (d2 s + c2 ) c1 c2 + (d1 c2 + c1 d2 ) s + . . . U(s) . . . + (J1 c2 + J2 c1 + J2 c2 + d1 d2 ) s2 + (J1 d2 + J2 d1 + J2 c2 ) s3 + J1 J2 s4

"−1

.

(5.10) The undamped (natural frequency), lightly damped, and critically damped openloop response of the thorax is shown in Fig. 5.4. The system properties were set as follows: mass moments of inertia J1 = J2 = 0.11 kg m2 , stiffness c = 10 Nm rad−1 .

1.5

1.0

0.5

0

-0.5

-1.0

-1.5 0

1

2

3

Fig. 5.4 Impulse response for different damping parameters

4

5

6

132


The open-loop response is normally used to assess whether the system is stable relative to external perturbations. In the case of instability, the system will not regain its equilibrium position. To understand the physical meaning . of the responses presented above, one has to first introduce the quantities ω0 = Jc and ζ = 2√dc J , known as the natural frequency and damping factor, respectively. The natural frequency is a measure of the amount of stiffness in relation to mass or inertia and is related to the potential energy, whereas the damping factor is a measure of the energy dissipation in the system. Energy dissipation is caused by internal friction as well as non-conservative forces, such as those generated by a dashpot. When ζ = 0, the motion is referred to as undamped. The nature of the motion is harmonic, repeating itself in cycles, see Fig. 5.4 (“Nat. freq.”). When 0 < ζ < 1, the motion is called lightly damped, and the motion is in the form of a decaying sinusoidal, with an exponential decay envelope, see Fig. 5.4 (“Light damping”). When ζ = 1, the system is called critically damped and regains its equilibrium position in the shortest possible time without any oscillations. The motion is in the form of a decaying exponential, see Fig. 5.4 (“Critical damping”). There are two other cases of damping, the first being when ζ > 1. With this value range the system response is in the form of a decaying exponential and is not periodic. The second is when ζ < 0 with an exponentially growing solution and instability. Let us return to our example, since we have observed that the forces applied at the hip are sine functions; therefore, our system is subjected to harmonic excitations. In these kinds of systems, the determination of the natural frequency plays an essential role due to the resonance phenomenon, i.e., the driving frequency becomes very close to the natural frequency or frequencies. But, while in engineering machines are designed such that resonant frequencies are avoided in order to protect them from possible physical damage, humans and animals use resonance velocities during locomotion in order to reduce energy expenditure. Exercise 5.1. Determine the natural frequencies associated with the body stem system discussed in this section for the values J1 = J2 = J, c1 = c2 = c, d1 = d2 = 0.

Exercise 5.2. Find the motion equations for the torsion of the body stem in Fig. 5.5.

133 J ,k


J4 q4

m

C4

c3

d3

J3

c

1

M4

d2

M3

c

J ,k

1

c2

q3

J ,k

C3

q2

d1

M2

c

1

c1

C1

J ,k

m

C2

J2

J1

q1

M1 Fig. 5.5 Enhanced trunk model with parameters

5.2.2 Dynamics of Extremities 5.2.2.1 Introductory Remarks As mentioned in the Section 5.1, human walking can be described as an interaction between the swing of suspended pendula in the swing phase and inverted pendula in the stance phase. During the swing phase of walking, human legs and arms form suspended pendula with distances between pivot and center of mass of 37 cm and 35 cm, respectively [160]. Using this concept we derive the equation of motion for several pendula as basic models of walking, see Fig. 5.6.

134


gait phases during walking stance phase

swing phase

mechanical models - pendula s

s

s

s

s

complex mechanical models - multibody systems

Fig. 5.6 Swing and stance phase during walking modeled as suspended and inverted pendula (adapded from WITTE et al. [163])

5.2.2.2 Mathematical and Physical Pendulum We consider the mathematical pendulum with parameters as shown in Fig. 5.7. There are many possibilities to obtain the motion equation of a mathematical pendulum, but the following derivation is performed by applying the principle of conservation of energy. In an idealized pendulum the energy is continuously transformed from potential into kinetic, and vice versa, the total energy E = T + U remaining constant.


135

g

0

s

s

q

U=0

l

h

C m

Fig. 5.7 Model of a mathematical pendulum

From Fig. 5.7 we have U = m g h = m g (l − l cos θ ) = m g l (1 − cos θ ) ,

(5.11)

and the total energy 1 E = m g l (1 − cos θ ) + m v2 = const . 2

(5.12)

Recalling that v = l θ˙ equation (5.12) becomes 1 E = m g l(1 − cos θ ) + m l 2 θ˙ 2 = const . 2

(5.13)

By means of time derivation of equation (5.13) we obtain m g l θ˙ sin θ + m l 2 θ˙ θ¨ = 0 .

(5.14)

Finally, dividing this equation by m l 2 θ˙ = 0, we obtain the well-known equation of motion g θ¨ + sin θ = 0 , (5.15) l with the solution for small angle θ (i.e., θ 1) of / g t (5.16) θ = θ0 cos l for the initial conditions θ (0) = θ0 and θ˙ (0) = 0. Therefore, the period T and the frequency f of the pendulum are 0 / 1 1 l g , f= = . (5.17) T = 2π g T 2π l

136


Exercise 5.3. Applying the principle of angular momentum (2.26), determine the equation of motion M ga θ¨ + sin θ = 0 (5.18) J0 for the physical pendulum (with mass M and mass moment of inertia J0 ) shown in Fig. 5.8. g s

0

s

q

y

a C

x

Mg

Fig. 5.8 Model of a physical pendulum

Comparing motion equation (5.18) to that of the mathematical pendulum (5.15), it becomes obvious that the oscillation of the physical pendulum is equivalent to that of a mathematical pendulum with length l = MJ0a (reduced length of pendulum). 5.2.2.3 Double Pendulum Using the equation presented above, one can construct a mathematical double pendulum whose segmental oscillations represent those of thigh and shank during the swing phase. Let us set li = MJiiai (i = 1 , 2). Note that by using a mathematical pendulum as a paradigm, we neglect the influence of inertia except for small amplitudes and accelerations as they occur in humans during normal walking, making this simplification acceptable.


137

g s

0

s

q1

y

l1 m1 (x1 , y1 ) s

q2

l2

m2 (x2 , y2 )

x Fig. 5.9 Model of a double pendulum

The mathematical double pendulum shown in Fig. 5.9 is a two-degree-of-freedom system. Therefore, we need two generalized coordinates q1 = θ1 and q2 = θ2 in order to derive the equations of motion. We use LAGRANGE’s equations of the 2nd kind (2.52). The position of the two mass points with masses m1 and m2 are described by the cartesian coordinates x1 = l1 cos θ1

y1 = l1 sin θ1

x2 = l1 cos θ1 + l2 cos θ2

y2 = l1 sin θ1 + l2 sin θ2 .

(5.19)

The kinetic energy can be calculated as 1 1 T = m1 x˙12 + y˙21 + m2 x˙22 + y˙22 ) 2 2 + 1 * = m2 l12 θ˙12 + l22 θ˙22 + 2 l1 l2 θ˙1 θ˙2 cos(θ1 − θ2 ) (5.20) 2 and the potential energy is * + U = m1 g [l1 + l2 − l1 cos θ1 ] + m2 g l1 + l2 − (l1 cos θ1 + l2 cos θ2 ) .

(5.21)

Realizing the necessary derivations of the LAGRANGE function L = T − U using (2.52) for a = 1 , 2, we get the equations of motion in the form

138


(m1 + m2 ) l12 θ¨1 + m2 l1 l2 θ˙2

cos(θ1 − θ2 )

+ m2 l1 l2 θ˙22 sin(θ1 − θ2 ) + (m1 + m2 ) g l1 sin θ1 = 0 , (5.22) m2 l22 θ¨2 + m2 l1 l2 θ¨1 cos(θ1 − θ2 ) − m2 l1 l2 θ˙12 sin(θ1 − θ2 ) + m2 g l2 sin θ2 = 0 . In humans, the mass and the mass moment of inertia of the thigh are approximately two times those of the shank, while the distance from the proximal segment (hip and knee, resp.) to the center of mass for each segment is almost the same (0.245 × body length × 0.433) (for more information see WINTER [157]). Taking this into account, we can rewrite equation (5.22), substituting l1 = l2 = l, m1 = m, and m2 = m2 . Equations (5.22) become

m+

m 2 ¨ m l θ1 + l 2 θ¨2 cos(θ1 − θ2 ) 2 2 m m 2 ˙2 g l sin θ1 = 0 , + l θ2 sin(θ1 − θ2 ) + m + 2 2 m m 2¨ l θ2 + l 2 θ¨1 cos(θ1 − θ2 ) 2 2 m m 2 ˙2 − l θ1 sin(θ1 − θ2 ) + g l sin θ2 = 0 . 2 2

(5.23)

Linearizing for small amplitudes in around the equilibrium position, cos θi = 1, 2 sin θi ≈ θi , and θi θ˙j ≈ 0 ,(i, j = 1, 2) the leg’s motion equations for the swing phase of walking can be reduced to 3 l θ¨1 + l θ¨2 + 3 g θ1 = 0 l θ¨2 + l θ¨1 + g θ2 = 0 .

(5.24)

Recalling the solutions θ1 = A1 sin ω t and θ2 = A2 sin ω t, equations (5.24) can be written in the form 3 (g − l ω 2 ) A1 − l ω 2 A2 = 0 l ω 2 A1 + (g − l ω 2 ) A2 = 0 .

(5.25)

For the non-trivial solution the determinant of the coefficients must vanish, leading to the following angular frequencies √ √ 3− 3 3+ 3 2 2 g and ω2 = g. (5.26) ω1 = 2l 2l A more realistic model of the lower limbs is presented in Fig. 5.10. The leg is represented by a physical double pendulum, where mi , Li , Ji , and ri are, respectively,


139

the mass, length, mass moment of inertia, and length from the proximal pivot to the center of mass of the ith segment. M1 and M2 are, respectively, externally applied moments to the hip and knee.

g M1

s

s

0 r1 q1

y

l1

m1 , J1

C1

s

q2

r2 x

M2

l2 C2

m2 , J 2

Fig. 5.10 Model of a physical double pendulum

In this case we obtain the equations of motion (again following LAGRANGE) in the form (m1 r12 + m2 l12 + J1 ) θ¨1 + m2 l1 r2 θ¨2 cos(θ1 − θ2 ) + m2 l1 r2 θ˙22 sin(θ1 − θ2 ) + (m1 r1 + m2 l1 ) g sin θ1 = M1 − M2 , (m2 r22 + J2 ) θ¨2 +m2 l1 r2 θ¨1 cos(θ1 − θ2 ) − m2 l1 r2 θ˙12 sin(θ1 − θ2 ) + m2 g r2 sin θ2 = M2 . (5.27)

5.2.3 Simplest Integrative Model of Walking – Inverted Pendulum 5.2.3.1 Inverted Pendulum with Fixed Length Let us consider a model consists of a weightless bar of length l, which rotates in the vertical plane around horizontal axis 0z, see Fig. 5.11. There is a mass point m on the free upper end of the bar. Let M be a moment of the forces that acts at joint 0. Angle θ is formed by the bar and the vertical axis and is positive if the rotation is counterclockwise.

140


y

g

m

s

l q M x z

0

Fig. 5.11 Model of an inverted pendulum

To derive the equation of motion, we apply the principle of angular momentum (2.26) with respect to the axis 0z. With Dz = m l 2 θ˙ 2 and Mz = m g l sin θ − M, the equation of motion takes the form m l 2 θ¨ = m g l sin θ − M .

(5.28)

If the length of the pendulum is a function of time l(t), the principle of angular momentum (2.26) yields d (m l 2 θ˙ ) = m g l sin θ − M , dt i.e.,

m l 2 θ¨ + 2 m l l˙θ˙ − m g l sin θ = −M .

(5.29)

With l = const equation (5.29) reduces to equation (5.28). If we consider a solid body with a physical weight (inverted physical pendulum) instead of a mass point, the equation of motion will be similar to the equation of pendulum (5.28), namely J θ¨ = m g l sin θ − M , where J is the mass moment of inertia of the solid body about the axis of rotation. In the case considered here, the position of the solid body is defined by angle θ between the plane, which is formed by the horizontal axis of rotation and the center of mass, and a vertical straight line.

5.2.3.2 Inverted Pendulum with Variable Length Let us consider a more complicated case of the model of leg when the leg is formed by the inverted pendulum of variable length. Moreover, the length of the leg dynamically changes in time under the influence of a definite force F. The force is


141

directed along the leg towards the axis of rotation. Hence, the leg length change is acts in joint 0, see Fig. 5.12. The mechaniunknown. It is provided that moment M cal system has two degrees of freedom. To derive the equations of motion, we apply LAGRANGE’s equations of the 2nd kind. The generalized coordinates are the angle of deviation of the pendulum the from vertical line θ (t) and the length of the pendulum l(t), or q1 = θ , q2 = l.

y

y1

g s

s

m x1

q

F

M x 0

Fig. 5.12 Model of an inverted pendulum with variable length

Let us compute the kinetic energy T . The moving frame 0 x1 y1 is connected with the pendulum and the origin is located on the rotation axis. Axis 0y1 is directed along the pendulum. The velocity of the mass point is equal to the sum of the relative velocity vr with respect to the coordinate system 0x1 y1 and the velocity of transportation motion ve : v =vr +ve . ˙ The mass point moves along a straight line Oy1 , so that vr = l(t). Since vectors vr and ve are orthogonal, v2 = vr 2 + ve 2 = l˙2 + l 2 θ˙ 2 ; then, the kinetic energy is

m ˙2 2 ˙ 2 l +l θ . 2 The potential energy is U = m g h = m g y, where y = l cos θ . Therefore, the LAGRANGE function L = T −U yields T=

L=

m ˙2 2 ˙ 2 l + l θ − m g l cos θ . 2

To obtain the generalized forces, we use the equation for the virtual work of nonconservative forces acting on the system:

δ A = Q1 δ θ + Q2 δ l = −M δ θ + F δ l . Hence, Q1 = −M and Q2 = F. LAGRANGE’s equations (2.44) take the form

142


m l 2 θ¨ + 2 m l l˙θ˙ − m g l sin θ = −M , m l¨ − m l θ˙ 2 + m g cos θ = F .

5.2.3.3 Inverted Double Pendulum Let us consider an inverted double pendulum, which consists of two weightless bars 0B and BA with lengths l1 and l2 , respectively. There are two mass points A and B with masses m1 and m2 . The bars are connected with the help of a joint at point B, see Fig. 5.13. The position of the doubled pendulum is defined by angles ϕ and θ . They are formed by corresponding bars and a vertical line. The angle is positive 2 act at points 0 and B, 1 and M if the bar rotates counterclockwise. Moments M respectively. The complete mechanical system possessed two degrees of freedom.

A

m2

y

g s

l2

q

s

M2

B m1

l1 j M1

x

0 Fig. 5.13 Model of an inverted double pendulum

To derive the equations of motion, we again apply LAGRANGE’s equations of the 2nd kind. This time, the generalized coordinates are the angles of the pendulum’s deviation from the vertical line, ϕ (t) and θ (t): q1 = ϕ and q2 = θ . The coordinates of points B and A are given as follows: xB = −l1 sin ϕ , xA = −l1 sin ϕ − l2 sin θ ,

yB = l1 cos ϕ , yA = l1 cos ϕ + l2 cos θ ,

and the velocities are: x˙B = −l1 ϕ˙ cos ϕ , x˙A = −l1 ϕ˙ cos ϕ − l2 θ˙ cos θ , Now, we calculate the kinetic energy T :

y˙B = l1 ϕ˙ sin ϕ , y˙A = −l1 ϕ˙ sin ϕ − l2 θ˙ sin θ .


143

1 1 T = m1 x˙B2 + y˙2B + m2 x˙A2 + y˙2A 2 2 + 1 1 * = m1 l1 2 ϕ˙ 2 + m2 l2 2 ϕ˙ 2 + l2 2 θ˙ 2 + 2 l1 l2 cos(ϕ − θ ) . 2 2 With the potential energy U = m1 g yB + m2 g yA , the LAGRANGE function L = T −U takes the form + 1 * 1 L = m1 l12 ϕ˙ 2 + m2 l1 2 ϕ˙ 2 + l2 2 θ˙ 2 + 2 l1 l2 ϕ˙ θ˙ cos(ϕ − θ ) 2 2 − m1 g l1 cos ϕ − m2 g (l1 cos ϕ + l2 cos θ ) 1 1 = (m1 + m2 ) l1 2 ϕ˙ 2 + m2 l2 2 θ˙ 2 + m2 l1 l2 ϕ˙ θ˙ cos(ϕ − θ ) 2 2 − (m1 + m2 ) g l1 cos ϕ − m2 g l2 cos θ . Next, we consider the equation for the virtual work of non-conservative forces that act on the system with respect to the virtual displacements δ ϕ and δ θ :

δ A = −M1 δ ϕ + M2 (δ ϕ − δ θ ) = (M2 − M1 ) δ ϕ − M2 δ θ . Hence, Q1 = M2 − M1 and Q2 = −M2 . Finally, LAGRANGE’s equations (2.44) become (m1 + m2 ) l1 2 ϕ¨ + m2 l1 l2 θ¨ cos(ϕ − θ ) + m2 l1 l2 θ˙ 2 sin(ϕ − θ ) − (m1 + m2 ) g l1 sin ϕ = M2 − M1 , m2 l2 2 θ¨ + m2 l1 l2 ϕ¨ cos(ϕ − θ ) − m2 l1 l2 ϕ˙ 2 sin(ϕ − θ ) − m2 g l2 sin θ = −M2 . The model of the inverted double pendulum forms the basis for modeling the walking process of a “passive walker”, see Section 2.4.1. The thighs and shanks are considered to be connected by revolute joints for simulation purposes. The same holds for the connection of the thighs and the hips, see Fig. 5.14.

144


Fig. 5.14 Model of a “passive walker” (Software alaska 4.0), [42]

Buckling of the knees is avoided by the fact that the foot is in front of the leg. In this way, the direction of the angular momentum in the knee of the supporting leg inhibits the knee. This fundamental aspect is described in McGEER [93]. An planar inclination of 0.2 rad is sufficient to guarantee steady walking. The following Fig. 5.15 shows simulations of the four angles (hip joint, knee joint) of the model describe above.

Fig. 5.15 Angles of thigh and shank (left, right) of a “passive walker”, [42]

5.2.3.4 Inverted Double Pendulum with a Rigid Body We consider a system in which a rigid body with mass m is joined to the end of an inverted weightless bar-pendulum. The position of the system is defined by two


145

parameters. First, we have q1 = ϕ formed by the bar and a vertical line. Second, we have q2 = θ between the line that connects joint O with mass center C and the 2 act at points O and A, respectively, see Fig. 5.16. 1 and M vertical line. Moments M

Jy

g s

s

s

C q r A

M2 m, l j M1

x

0 Fig. 5.16 Model of an inverted pendulum with a rigid body

We calculate the kinetic energy T = TTrans + TRot following (2.54). Let ρ be the distance between the center of mass C of the body and joint A. JC is a mass moment of inertia of the body with respect to the center of mass C. The coordinates of the center of mass xC , yC are given as follows: xC = −l sin ϕ − ρ sin θ ,

yC = l cos ϕ + ρ cos θ .

The kinetic energy T takes the form T=

m 2 1 m 1 (x˙ + y˙C2 ) + JC θ˙ 2 = l 2 ϕ˙ 2 m l ρ ϕ˙ θ˙ cos(ϕ − θ ) + (m ρ 2 + JC )θ˙ 2 . 2 C 2 2 2

The term m ρ 2 + JC constitutes the mass moment of inertia JA of the body about point A. Finally, the relation for the kinetic energy takes the form T=

m 2 2 1 ˙2 l ϕ˙ + JA θ + m l ρ ϕ˙ θ˙ cos(ϕ − θ ) . 2 2

The potential energy is U = m g yC = m g (l cos ϕ + ρ cos θ ). Thus, the LAGRANGE function L can be written as L=

m 2 2 1 l ϕ˙ + JA ϕ˙ 2 + m l ρ ϕ˙ θ˙ cos(ϕ − θ ) − m g (l cos ϕ + ρ cos θ ) . 2 2

The equation for the virtual work of non-conservative forces on possible displacements δ ϕ and δ θ is

δ A = −M1 δ ϕ + M2 (δ ϕ − δ θ ) = (M2 − M1 ) δ ϕ − M2 δ θ ,

146


and hence, Q1 = M2 − M1 and Q2 = −M2 . Finally LAGRANGE’s equations (2.44) can be derived, namely m l 2 ϕ¨ + m l ρ θ¨ cos(ϕ − θ ) + m l ρ θ˙ 2 sin(ϕ − θ ) − m g l sin ϕ = M2 − M1 , JA θ¨ + m l ρ ϕ¨ cos(ϕ − θ ) − m l ρ ϕ˙ 2 sin(ϕ − θ ) − m g ρ sin θ = −M2 .

5.2.4 The Three-Body Model of Walking Let us consider an anthropomorphic three-body mechanism. The system consists of an upper body connected with two legs by means of rotary joints (hip joints), see Fig. 5.17. Friction in the joints is negligible. Given that the distance between the joints is also negligible (both joints can be reduced to a common point O), the mechanism can be considered a planar system.

g s

s

y 0

M1 , M 20

0

C1

y

j j

2

C

2

1

R2

R1 x E1

E2

Fig. 5.17 Model of walking with three rigid bodies

The mechanism is located within a vertical plane in a gravitational field. The position of the three-body model is fully described by four parameters: • coordinates of the joint 0(x, y), • angle ψ between y-axis and the straight line which connects joint 0 and the center of mass of the upper body, and • angles ϕ1 and ϕ2 between y-axis and straight lines that pass through joint 0 and the fulcrum of the left and right legs, respectively.


147

The angle is positive when a bodily rotation occurs counterclockwise. The center of mass of each leg is located on the straight line that connects the ends of the leg. The equations of motion are derived using LAGRANGE’s equations of the 2nd kind. The generalized coordinates are q1 = x, q2 = y, q3 = ψ , q4 = ϕ1 , q5 = ϕ2 . LAGRANGE’s equations of the 2nd kind (2.51) in this case (D ≡ 0) are: ∂L d ∂L − = Qa 4 , (a = 1, 2, . . . , 5) . (5.30) dt ∂ q˙a ∂ qa To derive the LAGRANGE function L = T −U, we need to find the kinetic energy T first. The kinetic energy of the whole mechanism is equal to the sum of the kinetic energies of the upper body T3 and the two legs T1 , T2 , i.e., T = T3 + T1 + T2 .

5.2.4.1 Kinetic Energy of the Upper Body The designations are: m3 – mass of the upper body; ρ3 – distance between the center of mass of the body and joint 0; and J3C – mass moment of inertia of the upper body about the center of mass. The coordinates of the center of mass of the upper body x3 , y3 are given as follows: y3 = y0 + ρ3 cos ψ . (5.31) x3 = x0 − ρ3 sin ψ , The velocities of the center of the mass (derivatives of the coordinates) are x˙3 = x˙0 − ρ3 ψ˙ cos ψ ,

y˙3 = y˙0 − ρ3 ψ˙ sin ψ .

(5.32)

The kinetic energy of the upper body can be calculated as m3 2 x˙3 + y˙23 + J3C ψ˙ 2 2 m3 2 1 x˙0 + y˙0 2 − m3 ρ3 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) + m3 ρ3 2 + J3C ψ˙ 2 . (5.33) = 2 2

T3 =

Here, m3 ρ3 2 + J3C is the mass moment of inertia J3 0 of the upper body with respect to the point 0. Finally, equation (5.33) have the form T3 =

m3 2 1 x˙0 + y˙20 − m3 ρ3 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) + J3 0 ψ˙ 2 . 2 2

(5.34)

5.2.4.2 Kinetic and Potential Energies of the Leg This time, the designations are: m – mass of the leg; ρ – distance between the center of mass of the leg Ci (xi , yi ) , i = 1, 2 and joint 0; JC – mass moment of inertia of the leg about the center of mass Ci , see Fig. 5.18.

148


The coordinates of the center of mass of the leg xi , yi are as follows (index i is omitted): y = y0 − ρ cos ϕ . (5.35) x = x0 + ρ sin ϕ , The velocities of the center of the mass are x˙ = x˙0 + ρ ϕ˙ cos ϕ ,

y˙ = y˙0 + ρ ϕ˙ sin ϕ .

(5.36)

Here, ϕ must be substituted with ϕ1 for the left leg and ϕ2 for the right one. In Fig. 5.17 ϕ1 is negative (in counter direction to the positive angle ϕ2 ). The kinetic energy of each leg T is (index i is omitted): m 2 x˙ + y˙2 + JC ϕ˙ 2 2 m 2 1 x˙0 + y˙20 + m ρ ϕ˙ (x˙0 cos ϕ + y˙0 sin ϕ ) + m ρ 2 + JC ϕ˙ 2 . = 2 2

T=

(5.37)

Equation J 0 = m ρ 2 + JC represents the mass moment of inertia of a leg about point 0. Finally, equation (5.37) takes the form T=

1 m 2 x˙0 + y˙20 + m ρ ϕ˙ (x˙0 cos ϕ + y˙0 sin ϕ ) + J 0 ϕ˙ 2 . 2 2

(5.38)

Using (5.34) and (5.38), we can obtain a kinetic energy of the whole three-body mechanism: 1 M 2 x˙0 + y˙20 + J3 0 ψ˙ 2 − m ρ ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) 2 2 * + 1 + J 0 ϕ˙ 12 + ϕ˙ 22 + m ρ ϕ˙ 1 (x˙0 cos ϕ1 + y˙0 sin ϕ1 ) + ϕ˙ 2 (x˙0 cos ϕ2 + y˙0 sin ϕ2 ) , 2 (5.39) T=

where M = m3 + 2m is the mass of the entire mechanism. The potential energy of the mechanism is written as follows: U = m3 g y3 + m g y1 + y2 , where y1 and y2 are the coordinates of the centers of the mass of the left and right legs, respectively. Taking into account (5.31) and (5.35), we have U = M g y0 + m3 g ρ3 cos ψ − m g ρ (cos ϕ1 + cos ϕ2 ) .

(5.40)

5.2.4.3 LAGRANGE’s Equations for the Three-Body Model Let us consider the non-conservative forces acting on the mechanism, see Fig. 5.17. 0 at joint 0 (hip joint), which are imparted using 0 and M There are two moments M 1 2


149

two respective drives. Two reaction forces R1 (R1x , R1y ) and R2 (R2x , R2y ) act from the ground on the left and the right legs, respectively.

C

r

3

0

r

r l

l C1

C2

Right (2)

Left (1) Fig. 5.18 Mechanical model with geometric parameters

What is needed to derive the equations of motion is to evaluate the generalized forces Qa , a = 1, . . . , 5, first, that are related to the five generalized coordinates. The virtual work δ A of all the forces that act on the mechanism is

δ A = R1x δ xE 1 + R2x δ xE 2 + R1y δ yE 1 + R2y δ yE 2 + M1 0 (δ ϕ1 − δ ψ ) + M2 0 (δ ϕ2 − δ ψ ) , (5.41) with E1 (xE 1 , yE 1 ) and E2 (xE 2 , yE 2 ) as the ends of the left and right legs, respectively. The coordinates of the ends are xE = x0 + l sin ϕ ,

δ xE = δ x0 + l cos ϕ δ ϕ ,

yE = y0 − l cos ϕ ,

δ yE = δ y0 + l sin ϕ δ ϕ .

(5.42)

Here, ϕ must be substituted with ϕ1 for the case of the left leg and ϕ2 for the right one, l is the length of one leg. Inserting equation (5.41) into (5.42), the expression for δ A yields:

150


δ A = (R1x + R2x ) δ x0 + (R1y + R2y ) δ y0 − (M1 0 + M2 0 ) δ ψ + (M1 0 + R1x l cos ϕ1 + R1y l sin ϕ1 ) δ ϕ1 + (M2 0 + R2x l cos ϕ2 + R2y l sin ϕ2 ) δ ϕ2 . Now the generalized forces can be found as follows: Q1 =R1x + R2x , Q2 = R1y + R2y , Q3 = −(M1 0 + M2 0 ) , Q4 =M1 0 + l (R1x cos ϕ1 + R1y sin ϕ1 ),

(5.43)

Q5 =M2 0 + l (R2x cos ϕ2 + R2y sin ϕ2 ) . Upon substituting equations (5.39), (5.40), and (5.43) into LAGRANGE’s equations (5.30), the equations of motion of the three-body mechanism can be derived: M x¨0 − m3 ρ3 ψ¨ cos ψ + m3 ρ3 ψ˙ 2 sin ψ + m ρ ϕ¨ 1 cos ϕ1 − m ρ ϕ˙ 12 sin ϕ1 + m ρ ϕ¨ 2 cos ϕ2 − m ρ ϕ˙ 22 sin ϕ2 = R1x + R2x , (5.44) M y¨0 − m3 ρ3 ψ¨ sin ψ − m3 ρ3 ψ˙ 2 cos ψ + m ρ ϕ¨ 1 sin ϕ1 + m ρ ϕ¨ 12 cos ϕ1 + mρ ϕ¨ 2 sin ϕ2 + m ρ ϕ˙ 22 cos ϕ2 − M g = R1y + R2y , (5.45) J3 0 ψ¨ − m3 ρ3 x¨0 cos ψ − m3 ρ3 y¨0 sin ψ − m3 g ρ3 sin ψ = −M1 0 − M2 0 ,

(5.46)

J 0 ϕ¨ 1 + m ρ x¨0 cos ϕ1 + m ρ y¨0 sin ϕ1 + m g ρ sin ϕ1 = M1 0 + l (R1x cos ϕ1 + R1y sin ϕ1 ) , (5.47) J 0 ϕ¨ 2 + m ρ x¨0 cos ϕ2 + m ρ y¨0 sin ϕ2 + m g ρ sin ϕ2 = M2 0 + l (R2x cos ϕ2 + R2y sin ϕ2 ) . (5.48) The definition of the forces R1 and R2 depends on the mechanism’s type of walk.

5.2.5 Five-Body Model of a Walking Robot In this new model the mechanism has more complex legs consisting of two bodies each: a thigh and a shank. The left and right thighs are coupled by means of joints B1 and B2 to the respective shanks. The thighs are also coupled with the upper body


151

as in the previous model by means of joint 0. Friction in the joints can be neglected. The center of mass of each shank is located on a straight line that connects the ends of the shank, see Fig. 5.19. The generalized coordinates are q1 = x, q2 = y, q3 = ψ , q4 = ϕ1 , q5 = ϕ2 , q6 = θ1 , and q7 = θ2 .

g s

s

y 0

M1 , M20 0 B

M2

j

2

B2

j

1

B

M1

y

B1 E1

q1

q2

E2

x Fig. 5.19 Five-body model of the walking robot

LAGRANGE’s equations (2.51) take the form ∂L d ∂L − = Qa 4 , (a = 1, 2, . . . , 7). dt ∂ q˙a ∂ qa

(5.49)

The kinetic energy of the mechanism can be evaluated by addition of the kinetic energy of left and right shanks to equation (5.39).

5.2.5.1 Kinetic Energy of a Lower Leg In this sub-model m2 is the mass of a shank, ρ2 the distance between the center of mass of the shank S(xS , yS ) and joint B, and J2 S the mass moment of inertia of the shank about the center of mass S.

152


C

r

3

0

l1

r1

r

1

C1

l2

l1

C2

r

r2

2

S1

l2

S2

Right (2)

Left (1) Fig. 5.20 Five-body model with geometric parameters

The coordinates and the velocities of the center of mass of each shank xS , yS are given as follows: xS = x0 + l1 sin ϕ + ρ2 sin θ , yS = y0 − l1 cos ϕ − ρ2 cos θ ,

x˙S = x˙0 + l1 ϕ˙ cos ϕ + ρ2 θ˙ cos θ , y˙S = y˙0 + l1 ϕ˙ sin ϕ + ρ2 θ˙ sin θ .

(5.50)

Here ϕ and θ must be substituted with ϕ1 and θ1 for the case of the left leg, ϕ2 and θ2 for the right one. The length of the thigh is l1 . The kinetic energy of the shank T2 :


153

m2 2 m2 2 m2 2 2 (x˙ + y˙2S ) + J2 S θ˙ 2 = (x˙ + y˙20 ) + l1 ϕ˙ 2 S 2 0 2 + m2 l1 ϕ˙ (x˙0 cos ϕ + y˙0 sin ϕ ) + m2 ρ2 θ˙ (x˙0 cos θ + y˙0 sin θ )

T2 =

1 + m2 l1 ρ2 ϕ˙ θ˙ (cos ϕ cos θ + sin ϕ sin θ ) + (m2 ρ2 2 + J2 S ) θ˙ 2 . (5.51) 2 The term m2 ρ2 2 + J2 S represents the mass moment of inertia of a shank about the points B1 and B2 , which we designate with J2 B . Finally, the relationship in (5.51) for the kinetic energy takes the form T2 =

m2 2 m2 2 2 (x˙ + y˙20 ) + l1 ϕ˙ + m2 l1 ϕ˙ (x˙0 cos ϕ + y˙0 sin ϕ ) 2 0 2

1 + m2 ρ2 θ˙ (x˙0 cos θ + y˙0 sin θ ) + m2 l1 ρ2 ϕ˙ θ˙ cos(ϕ − θ ) + J2 B θ˙ 2 . (5.52) 2 To derive the kinetic energy of the five-body mechanism, it is necessary to add the kinetic energy of the right and left shanks (5.52) to the kinetic energy of the threebody mechanism (5.39), yielding T=

1 M 2 (x˙0 + y˙20 ) + J3 0 ψ˙ 2 − m3 ρ3 ψ˙ (x˙0 cos ψ + y˙0 sin ψ ) 2 2 1 1 + (J1 0 + m2 l1 2 ) (ϕ˙ 12 + ϕ˙ 22 ) + J2 B (θ˙12 + θ˙22 ) 2 2 * + + (m1 ρ1 + m2 l1 ) ϕ˙ 1 (x˙0 cos ϕ1 + y˙0 sin ϕ2 ) + ϕ˙ 2 (x˙0 cos ϕ2 + cos y0 sin ϕ2 ) * + + m2 ρ2 θ˙1 (x˙0 cos θ1 + y˙0 sin θ1 ) + θ˙2 (x˙0 cos θ2 + y˙0 sin θ2 ) * + + m2 l1 ρ2 ϕ˙ 1 θ˙1 cos(ϕ1 − θ1 ) + ϕ˙ 2 θ˙2 cos(ϕ2 − θ2 ) . (5.53)

Here, m1 is the mass of the thigh, J1 0 is the mass moment of inertia of a thigh about point 0. The mass of the whole mechanism is M = m3 + 2 m1 + 2 m2 .

5.2.5.2 Potential Energy of the System and the Generalized Forces The equation for the potential energy of the five-body mechanism is obtained by adjoining a member m2 g (yS 1 + yS 2 ) to the equation (5.40) for the potential energy of the three-body mechanism. Here, yS 1 and yS 2 are coordinates of the centers of mass of the left and the right shanks, respectively. Taking into account (5.50), we have U = M g y0 + m3 g ρ3 cos ψ − g (m1 ρ1 + m2 l1 ) (cos ϕ1 + cos ϕ2 ) − m2 g ρ2 (cos θ1 + cos θ2 ) . (5.54)

154


In comparison to the three-body mechanism, two moments which act in joints B B act on joints B and M are added to the non-conservative forces. Hence, moments M 1 2 B1 and B2 , see Fig. 5.19. They are imparted by corresponding drives. To derive the equations of motion, it is first necessary to evaluate the generalized forces Qa , a = 1, . . . , 7 (related to the seven generalized coordinates). The virtual work δ A of all the forces that act within the mechanism is

δ A = R1x δ xE 1 + R2x δ xE 2 + R1y δ yE 1 + R2y δ yE 2 + M1 0 (δ ϕ1 − δ ψ ) + M2 0 (δ ϕ2 − δ ψ ) + M1 B (δ θ1 − δ ϕ1 ) + M2 B (δ θ2 − δ ϕ2 ) , (5.55) where E1 (xE 1 , yE 1 ) and E2 (xE 2 , yE 2 ) are the ends of the left and the right legs, respectively. The coordinates and the virtual displacements of the ends in this case are xE = x0 + l1 sin ϕ + l2 sin θ ,

δ xE = δ x0 + l1 cos ϕ δ ϕ + l2 cos θ δ θ ,

yE = y0 − l1 cos ϕ − l2 cos θ

δ yE = δ y0 + l1 sin ϕ δ ϕ + l2 sin θ δ θ .

(5.56)

Here, ϕ and θ must be substituted with ϕ1 and θ1 for the case of the left leg, ϕ2 and θ2 for the right one. The length of the shanks is l2 . Substituting equation (5.56) into (5.55), we obtain

δ A =(R1x + R2x ) δ x0 + (R1y + R2y ) δ y0 − (M1 0 + M2 0 ) δ ψ + (M1 0 − M1 B + R1x l1 cos ϕ1 + R1y l1 sin ϕ1 ) δ ϕ1 + (M2 0 − M2 B + R2x l1 cos ϕ2 + R2y l1 sin ϕ2 ) δ ϕ2

(5.57)

+ (M1 B + R1x l2 cos θ1 + R1y l2 sin θ1 ) δ θ1 + (M2 B + R2x l2 cos θ2 + R2y l2 sin θ2 ) δ θ2 . Hence, the generalized forces can be found as follows: Q1 = R1x + R2x , Q2 = R1y + R2y , Q3 = −(M1 0 + M2 0 ) , Q4 = M1 0 − M1 B + l1 (R1x cos ϕ1 + R1y sin ϕ1 ) , Q5 = M2 0 − M2 B + l1 (R2x cos ϕ2 + R2y sin ϕ2 ) , Q6 = M1 B + l2 (R1x cos θ1 + R1y sin θ1 ) , Q7 = M2 B + l2 (R2x cos θ2 + R2y sin θ2 ) .

(5.58)


155

The definition of the forces R1 and R2 depends on the phases of walking of the system.

5.2.5.3 The Equations of Motion Substituting equations (5.53), (5.54), and (5.58) into LAGRANGE’s equations (5.30), we have M x¨0 − m3 ρ3 ψ¨ cos ψ + m3 ρ3 ψ˙ 2 sin ψ + (m1 ρ1 + m2 l1 ) ϕ¨ 1 cos ϕ1 − (m1 ρ1 + m2 l1 ) ϕ˙ 12 sin ϕ1 + (m1 ρ1 + m2 l1 ) ϕ¨ 2 cos ϕ2 − (m1 ρ1 + m2 l1 ) ϕ˙ 22 sin ϕ2 + m2 ρ2 θ¨1 cos θ1

(5.59)

− m2 ρ2 θ˙12 sin θ1 + m2 ρ2 θ¨2 cos θ2 − m2 ρ2 θ˙22 sin θ2 = R1x + R2x , M y¨0 − m3 ρ3 ψ¨ sin ψ − m3 ρ3 ψ˙ 2 cos ψ + (m1 ρ1 + m2 l1 ) ϕ¨ 1 sin ϕ1 + (m1 ρ1 + m2 l1 ) ϕ˙ 12 cos ϕ1 + (m1 ρ1 + m2 l1 ) ϕ¨ 2 sin ϕ2 + (m1 ρ1 + m2 l1 ) ϕ˙ 22 cos ϕ2 + m2 ρ2 θ¨1 sin θ1 + m2 ρ2 θ˙12 cos θ1

(5.60)

+ m2 ρ2 θ¨2 sin θ2 + m2 ρ2 θ˙22 cos θ2 + M g = R1y + R2y , J3 0 ψ¨ − m3 ρ3 x¨0 cos ψ − m3 ρ3 y¨0 sin ψ − m3 g ρ3 sin ψ = −M1 0 − M2 0 ,

(5.61)

(J1 + m2 l1 )ϕ¨ 1 + (m1 ρ1 + m2 l1 )x¨0 cos ϕ1 0

2

+ (m1 ρ1 + m2 l1 )y¨0 sin ϕ1 + m2 l1 ρ2 θ¨1 cos (ϕ1 − θ1 ) + m2 l1 ρ2 θ˙12 sin (ϕ1 − θ1 ) + (m1 ρ1 + m2 l1 )g sin ϕ1

(5.62)

= M1 0 − M1 B + l1 (R1x cos ϕ1 + R2y sin ϕ1 ), (J1 0 + m2 l1 2 ) ϕ¨ 2 + (m1 ρ1 + m2 l1 ) x¨0 cos ϕ2 + (m1 ρ1 + m2 l1 ) y¨0 sin ϕ2 + m2 l1 ρ2 θ¨2 cos(ϕ2 − θ2 ) + m2 l1 ρ2 θ˙22 sin(ϕ2 − θ2 ) + (m1 ρ1 + m2 l1 ) g sin ϕ2 = M2 0 − M2 B + l1 (R2x cos ϕ2 + R2y sin ϕ2 ) ,

(5.63)

156


J2 θ¨1 + m2 ρ2 x¨0 cos θ1 + m2 ρ2 y¨0 sin θ1 + m2 l1 ρ2 ϕ¨ 1 cos(ϕ1 − θ1 ) B

− m2 l1 ρ2 ϕ˙ 12 sin(ϕ1 − θ1 ) + m2 g ρ2 sin θ1

(5.64)

= M1 B + l2 (R1x cos θ1 + R1y sin θ1 ) , J2 B θ¨2 + m2 ρ2 x¨0 cos θ2 + m2 ρ2 y¨0 sin θ2 + m2 l1 ρ2 ϕ¨ 2 cos(ϕ2 − θ2 ) − m2 l1 ρ2 ϕ˙ 22 sin(ϕ2 − θ2 ) + m2 g ρ2 sin θ2

(5.65)

= M2 B + l2 (R2x cos θ2 + R2y sin θ2 ) . Definition of the forces R1 and R2 depends on the type of the walk of the mechanism. During the walking there are two phases of motion, namely “two supporting legs” (both legs are on the ground) and “one supporting leg” (one leg is carried above the ground). They appear subsequently. The equations (5.44) - (5.48) for three-body mechanism or (5.59) - (5.65) for five-body mechanism can be used for description of both phases of motion. These equation constitute second order nonlinear ordinary differential equations with five and seven unknowns, respectively. They can be solved numerically.

5.3 Robustness and Adaptivity Until a few years ago, the robotic scene was primarily influenced by the idea that locomotion is mostly subordinated to neuronal or computational intelligence. Such a line of thinking neglects the influence that an intelligent and compliant mechanical design produces in the final behavior of an embodied agent (e.g., emerging indispensable functionalities like adaptivity). Following the same line of thinking, biomechanical approaches used to explain the behavior observed in biological limbs have postulated a lot of mostly different optimization criteria, taking into account physiological, energetic or metabolic aspects. Today, the insight is of gaining importance in the robotic community that both control and mechanical systems are targets for robot’s design and should therefore be treated with an equal emphasis. Such a general paradigm, which is expected to be an indispensable concept for building adaptive agents in the future, was derived from a change in the understanding of human and animal motion: FISCHER & WITTE [50] gave insights into the intrinsic evolutive mechanical properties of the locomotor system, which diminish the necessity of control. Further studies have revealed the importance of self-stabilization for legged systems (GARCIA et al. [55], KUO [84]), and the visco-elastic properties of muscles and soft tissues for elucidating trunk and body stem kinematics during locomotion (ANDRADA [7], ANDRADA & WITTE [8]). These findings explicitly indicate that some kind of computation has been embedded in the morphology, “embodiment” as postulated by BROOKS in [36] and [37]. This means that a certain amount of “computation” for generating locomotion

5.3 Robustness and Adaptivity

157

or different tasks is produced by the mechanical system. This concept is now known as intelligence in mechanics (BLICHHAN et al. [30]) or morphological computation (PFEIFER et al. [113]). Three main elements can be described as key elements for intelligence in mechanics: • materials, • architecture, and • compliance (elasticity). Human lower limbs still retain an ancestral three-segmented leg architecture. The proportions of these segments have been reformed and adapted during evolution to the necessities of endurant walking (WITTE et al. [161]). During human gait, loading of the leg is associated with knee and ankle flexion. The overall compli¨ ant limb function can be localized at elastically operating leg joints (GUNTHER & BLICKHAN [58], KYROLINEN et al. [85]). A proper distribution of joint stiffness is required to guarantee the stable bending of the segmented leg (SEYFARTH et al. [126]). The required stiffness ratio between the knee and ankle largely depends on the ratio of the outer segment lengths (thigh and foot) and the nominal configurations of both joints. For instance, the extended knee configuration reduces the stiffness requirements at this joint. Surprisingly, even with an optimal setting of knee and ankle stiffness, homogenous bending of both joints may fail due to the nonlinear characteristics of segmented legs (BLICKHAN et al. [29]). In a given leg configuration joint bending at one joint might result in the extension of the other joint. This mechanical instability was first identified based on theoretical investigations (SEYFARTH et al. [126]) but might even be useful for human walking, namely when transitioning from stance to swing phase, see Fig. 5.21 (A). During this period, leg rotation rapidly changes from leg retraction to protraction. The folding of the leg late in the stance phase of walking is initiated by knee flexion and catapult-like ankle extension according to ISHIKAWA et al. [70]. Supported by elastic structures spanning the leg joints, this transition, which is based on a transient mechanical instability due to knee bending, could facilitate the short swing phases as required in human walking (GEYER et al. [56]). Elastic structures within the calf muscles spanning the ankle joint are important to quickly accelerate the leg forward in preparation of the following swing phase.

158


Fig. 5.21 Leg kinematics in walking and running: (A) Stick figures of the stance (solid lines) and swing legs (gray lines). (B) Tracings of knee and ankle angles and patterns of the vertical ground reaction force (GRF). Gray areas indicate ground contact. In walking, thigh protraction is initiated in late contact (A) by taking advantage of opposite operation of knee and ankle joint (B). [128]

In summary, the function of the human leg during locomotion is largely influenced by leg segmentation and compliant joint function. Fast repulsive movements (such as running or hopping) require synchronous joint loading. However, out-ofphase joint function is found in human walking and may facilitate fast leg protraction during the pre-swing phase.

5.4 Generalization – Scaling The “dynamic similarity” theory (ALEXANDER [2]) accounts for an invariant interchange between mechanical energies and states that two geometrically similar bodies whose motions are based on the exchange of kinetic and potential energy will be dynamically similar if they move at the same FROUDE number Fr. The FROUDE number is named after William FROUDE, a naval engineer who introduced it in the late 19th century to predict the dynamic behavior of real vessels from smaller models. The FROUDE number was applied to locomotion in order to compare the velocity of different-sized individuals and is defined as the ratio of the inertial force of the mass point to the gravity force to the leg direction, see Fig. 5.22.

5.4 Generalization – Scaling

159

Fig. 5.22 To the definition of the FROUDE number

From Fig. 5.22 we have 2

m vl v2 v2 , = ≈ Fr = m g cos θ g l cos θ gl

(5.66)

where v represents the velocity along the circle trajectory of the center of mass, l is the leg length, and m is the total mass. θ represents the angle of the stance leg to the slope normal, and g is the gravitational acceleration. The FROUDE number is particularly useful when comparing moving bodies of different sizes (at the same g) or the same body in different g’s. In particular, given the maximum swing velocity v1 of a pendulum of length l1 , we can predict the maximum velocity of a pendulum of different length l2 starting to swing from the same angle: Fr1 =

v21 g l1

and

Fr2 =

v22 . g l2

(5.67)

Therefore, for the same FROUDE number / v2 = v1

l2 . l1

(5.68)

Alternately, if we focus on equivalent velocities for the same pendulum in different gravitational environments (e.g., different planets similar to Earth), the equation for dynamic similarity is: / g2 . (5.69) v2 = v1 g1 A FROUDE number Fr = 1 is the dimensionless limit velocity of walking (ALEXANDER [3]). Recall that with Fr > 1, the centrifugal force acting on the body becomes larger than the gravity force, meaning contact with the ground is lost.


Chapter 6

Worm-like Locomotion Systems – Crawling

6.1 Modeling Worm-Like Locomotion Systems (WLLS) This chapter deals with mathematical models for WLLS. The investigations are aimed at gaining insight into how such systems move and at finding hints for the implementation as a hardware object (artificial worm) and its control. Contrasting with other investigations published in the literature and based on complex models, the present explorations are confined to simple worm-like systems “residing” along a straight line. This enables us to develop simple theories for both discrete and continuous models without the need for numerics in utilization. Also, locomotion can be done by a kinematic drive. This means that the dynamics of the internal actuators are not required as part of the theory. The actuators are viewed as hidden controlling devices that immediately achieve the desired internal deformations. In fact, the shape of the system is controlled off-line, and the (generally reciprocating) change of shape is transformed into locomotion via the non-symmetric interaction with the ground. In principle, two possibilities exist for describing this interaction. First, in Section 6.2 we assume the idealized interaction between the system and the ground that mimics a scaly or bristly surface of the worm: in sliding forward the friction forces are zero, while in the opposite direction the scales dig in and cause a large amount of friction and ultimately restricting the sign of one of the velocity coordinates. From the mechanical point of view, this is dealt with using differential constraints in the form x˙ ≥ 0. This entails another advantage of the present theory: an essential and utilizable part of it is pure kinematics and, at least for worms that are not too large, exact results can be obtained in a formal way. Second, in Section 6.3 the interaction of the worm system with the ground is modeled as non-symmetric (i.e., orientationdependent) dry friction. The following (taken from [138]) is used as the basis of our theory: (a)A worm is a terrestrial locomotion system with one dominant linear dimension and no active legs nor wheels.


161

162

6 Worm-like Locomotion Systems – Crawling

(b)Global displacement is achieved by a (periodic) change of shape (such as local strain) and interaction with the environment. (c)The model body of a worm is a one-dimensional continuum that serves as the support of various fields. The continuum in (c) is merely the interval of a body-fixed coordinate. The most important parameters are: mass, continuous (with a density function) or discrete distribution (chain of point masses), actuators (devices that produce internal displacements or forces, thus mimicking muscles), and surface structure (cause the interaction with the environment). In observing the locomotion of worms, one first recognizes the surface contact with the ground. It is well-known that, if there is contact between two bodies (worm and ground), some form of friction must occur that depends on the physical properties of the surfaces of the bodies. In particular, the friction may be anisotropic (depends on the orientation of the relative displacement). This interaction (mentioned in (b)) could emerge from a surface texture as asymmetric Coulomb friction or from a surface endowed with scales or bristles (which we will call “spikes”), preventing backward displacements. This type of surface is responsible for the conversion of (mostly periodic) internal and internally driven motions into a change of external position (undulatory locomotion) (OSTROWSKI et al. [111]), see STEIGENBERGER [134] and ZIMMERMANN et al. [171]. One of the first works in the context of worms, snakes and scales is from MILLER [95]. He considers massspring systems with scales aiming at modeling virtual worms and snakes and their animations within the context of computer graphics, see Fig. 6.1.

Fig. 6.1 Legless animals are modeled as mass-spring systems following MILLER [95]

To summarize, we consider a chain of mass points in a straight line (a discrete straight worm) as a physical model of finite DOF of a WLLS; these points are connected consecutively by linear or nonlinear elastic or linear visco-elastic elements, see for example [22], [171], and [172]. Consecutive mass points are connected by massless links endowed with hidden (muscle-like) actuators. The actuators are controllable and determine the lengths of the links directly. Such a control will be called kinematic drive in the following. Each mass point contacts the ground through spikes (of common orientation), preventing velocities from changing sign and possibly causing friction while in motion.

6.2 Straight Discrete Worms with Contact via Spikes, [137]

163

6.2 Straight Discrete Worms with Contact via Spikes, [137] 6.2.1 Kinematics As mentioned above, the goal of this subsection is to consider the case in which the mass points are equipped with spikes, which makes the friction orientationdependent, i.e., the friction forces while sliding forward are minimal, while in opposite direction the spikes dig in and cause significant friction, see [134] and [137].

Fig. 6.2 WLLS with a kinematic drive

Let the mass points be labeled 1 , . . . , n and suppose equal masses m. If the lengths l j , j = 1 , . . . , n − 1 were not time-dependent, the system would have n degrees of freedom. However, in applying the kinematic drive, the controls l j (t), j = 1, . . . , n − 1, are given (accomplished by suitable actuators, e.g., linear stepping motors), and the system is reduced to one degree of freedom. In order to avoid difficulty in setting up a mathematical model containing accelerations, we assume the controls to be piecewise functions continuously differentiable to the second order (i.e., the first derivatives l˙j are continuous, while the second derivatives l¨j may have isolated jumps). Now, the system takes on the rheonomic holonomic constraints x j+1 − x j − l j (t) = 0 ,

j = 1,...,n−1.

(6.1)

This entails the successive representation xi = xn − Li (t) ,

n−1

Li =

∑ lj ,

i = 1,...,n−1

(6.2)

j=i

thus, xn may serve as a system coordinate showing the DOF to be equal to 1. Furthermore, the spikes give rise to the differential constraints: x˙i (t) ≥ 0 ,

i = 1,...,n,

∀t .

(6.3)

Now, we exploit the kinematics of the system, i.e., we search for results in (6.1) and (6.3). Differentiating (6.2) with respect to t and observing (6.3), we obtain x˙i = x˙n − L˙ i ≥ 0 ,

i = 1,...,n−1.

Thus, x˙n (t) ≥ 0, x˙n (t) ≥ L˙ 1 , x˙n (t) ≥ L˙ 2 etc., which means

(6.4)

164


x˙n (t) ≥ max 0 , L˙ 1 , L˙ 2 , . . . , L˙ n−1 = Vn .

(6.5)

The lower velocity bound Vn is a function of time t, uniquely determined by the controls l j . Now, (6.2) implies that x˙n = Vn + w ,

(6.6)

where w(t) ≥ 0 is a common part of the velocities, which describes a rigid motion of the system. Here, w remains undetermined in kinematics, as it only follows from the dynamics of the system. Finally, (6.4) gives us x˙i = Vn − L˙ i + w ,

i = 1,...,n.

(6.7)

In this representation w plays the part of the generalized velocity. It describes a superimposed rigid motion in the positive x-direction of the system. The non-negative part Vn − L˙ i of the velocity x˙i is purely kinematically determined. Following (6.2), the center of mass of the system is at x = xn − L ,

L=

1 n−1 1 n−1 L = i ∑ ∑ jlj ; n i=1 n j=1

(6.8)

Wn = Vn − L˙ .

(6.9)

its velocity is then x˙ = v = Wn + w ,

Exercise 6.1. Check that Wn (t) = 0 is equivalent to the requirement l˙j (t) = 0, j = 1 , . . . , n − 1. Equations (6.7) and (6.9) entail the representation x˙i = v + L˙ − L˙ i ,

i = 1,...,n,

(6.10)

which now distinguishes v as the generalized velocity of the system. An interesting consideration here is which mass points are at rest during the motion of the system (i.e., active spikes!). From (6.6) and (6.7) it immediately follows that x˙n = 0 ⇔ w = 0 ∧ L˙ k ≤ 0 , k = 1 , . . . , n − 1 , x˙ j = 0 ⇔ w = 0 ∧ Vn = L˙ j , j = 1 , . . . , n − 1 .

(6.11)

Until now, the rigid motion part w of the velocities x˙i has been arbitrary (it is determined using dynamics, which is where it plays its primary role). Therefore, one may be tempted to set w = 0 and come up with a kinematic theory of the worm system. However, the one DOF is locked in this theory, making the system a guided mechanism that gains its locomotion through the constraints in (6.3) that appear in Vn . The theory can be summarized as follows:


165

1. Prescribe l j (t) > 0 , j = 1 , . . . , n − 1 . 2. Vn = max 0 , l˙1 , l˙1 + l˙2 , . . . , l˙1 + l˙2 + . . . + l˙n−1 . t

n−1

0

k= j

3. xn (t) = Vn (s) ds and x j (t) = xn (t) − ∑ lk (t) ,

(6.12) j = 1,...,n−1.

The set of functions l j (t) > 0, j = 1 , . . . , n − 1 is called a kinematic gait. Any kinematic gait generates a forward locomotion. Note that during locomotion of this type, not all x˙i , i = 1 , . . . , n, can be simultaneously positive since a joining of all the negated second components in the right-hand side of (6.11) would cause a contradiction. It is obvious that at least one spike must be active in order to achieve any locomotion at all. One could attempt to find optimal gaits with respect to some goal, such as fast motion or many active spikes. In particular, the gait could be a contraction wave going “up the worm”, l j (t) = l1 (t − τ j ), j = 2 , . . . , n, with shifts τ j ≤ τ j+1 . Example 6.1 We consider a worm system with n = 5 links using sinusoidal drives with different phase shifts. The graphs show the lengths l j vs. time t (left) and the corresponding movement of the worm (right). In the first row the shift is τ = [0.6, 0.4, 0.2, 0], in the second row τ = [0.2, 0.6, 0.4, 0]. Clearly, the motions become periodic with period T = 1 after demonstrating transient behavior over the time interval (0, 0.6). This is particularly evident in the third row, which shows Vn vs. t for the second shift. Then, the average velocities per period are v¯ = 1.15 and v¯ = 1.06, respectively.

166


Fig. 6.3 Worm system n = 5 using sinusoidal drives following kinematic theory


167

6.2.2 Dynamics Now, let us consider the dynamics of the system (with a kinematic drive). From the principle of linear momentum of the mass points, we obtain the equations of motion m xï = Fi + FSi − FSi−1 + FRi ,

i = 1,...,n,

(6.13)

where Fi are external, physically given forces, FSi are internal forces (FS0 = FSn = 0), and FRi are the reaction forces corresponding to the one-sided constraint (6.3). Therefore, the complementary slackness conditions hold: x˙i ≥ 0 , FRi ≥ 0 , x˙i · FRi = 0 ,

i = 1,...,n.

(6.14)

Equation (6.14) means that FRi must be zero when the corresponding inequality constraint is strict (“has slack”), x˙i ≥ 0, while FRi may have arbitrary non-negative values as long as x˙i = 0 (reaction force acting on a resting spike). If Fi + FSi − FSi−1 = Φi , then for any motion (6.14) will be satisfied by the “controller”: 1 (6.15) FRi = − 1 − sign(x˙i ) 1 − sign(Φi ) Φi . 2 Now, let us suppose a kinematic drive, i.e., the actuators are assumed to precisely determine the mutual distances l j as functions of time, see for example Fig. 6.3. The kinematic drive implies the holonomic constraint (6.1), with FSi as respective reactions. L˙ i and Vn are now given functions of time, and the system is reduced to one degree of freedom. We confine the external forces to Fi = −k x˙i − Γ , i.e., besides the viscous friction force we allow for a constant force acting upon each mass point (e.g., a weight component along an inclined x-axis). Then, summing up all principles of linear momentum (6.13) while observing x˙i = Vn − L˙ i + w, there follows an algebraic differential equation for w, with R =

1 n

n

∑ FRi ,

i=1

m w˙ + k w + σ (t) = R , w ≥ 0 , R ≥ 0 , w R = 0 , n−1

σ = m W˙ n + kWn + Γ , Wn = Vn − 1n ∑ L˙ i .

(6.16)

i=1

In mode 1 with {w > 0 , R = 0} no mass point is at rest, whereas in mode 2 with {w = 0 , R > 0} at least one mass point does not move (active spike). If we set w = 0, then all that follows becomes our kinematic theory. This theory is easy to deduce. If σ (t) > 0 for all t, then the motion is always in mode 2. However, if at any time at least one spike is active, the motion is welldetermined by kinematics. Example 6.2 We consider a system of mass points with n = 3. The actuators are chosen so as to

168


generate l1 and l2 as T -periodic piecewise cosine functions. This given gait will be reconsidered in Subsection 7.4.1.3. Figure 6.4 shows l1 and l2 vs. t/T (left) and the corresponding worm motion (right, t−axis upward) with chosen some system data m , k , Γ .

Fig. 6.4 Gait and worm motion governed by kinematic theory

We must keep in mind that one spike is active (resting mass point) at any time. Hence, this gait might be suitable for motion “in the plane”. An uphill gait (two resting mass points an any time) could easily be constructed as well. To ensure w = 0 and a sufficiently small R (to prevent the spikes from breaking down), certain restrictions for T and Γ must be obeyed, see [137] and [138].

6.2.3 Geometric Interpretation of the Results We give here a geometric interpretation of the motion of two mass points with contact via spikes described above, assuming a linear spring between the mass points, no damping, and both masses are equal, see Fig. 6.5.

c m

m x1 Fig. 6.5 Two-mass model

x2


169

The energy integral exists for x˙1 ≥ 0, x˙2 ≥ 0: 1 c 1 m1 x˙12 + m2 x˙22 + (x2 − x1 )2 = const . 2 2 2 At time t = 0 there is x˙1 (0) = x˙2 (0) = 0 and x2 (0) − x1 (0) = L, which means that * +2 x˙12 + x˙22 + ω (x2 − x1 ) = (ω L)2 , where ω 2 = mc . The last equation is the equation of a sphere in space x˙1 , x˙2 , ω (x2 − x1 ). There are no external forces for x˙1 > 0 , x˙2 > 0; for this reason, one more integral exists: m

x˙1 + x˙2 = const or x˙1 + x˙2 = C . 2

This is the equation of a plane in space x˙1 , x˙2 , ω (x2 − x1 ), see Fig. 6.6.

Fig. 6.6 Geometric interpretation of the motion of two masses with differential constraints

Hence, the motion trajectory is the intersection of a sphere and a plane in the area x˙1 ≥ 0 , x˙2 ≥ 0.

170


A – initial point: x˙1 = x˙2 = 0, x2 − x1 = L. AB, BC – the first mass moves: x˙1 > 0, the second mass is standing still: x˙2 = 0, the spring contracts: x˙2 − x˙1 < 0 but is stretched out relative to the non-deformed state: x2 − x1 > 0. CD – the first mass moves: x˙1 > 0, the second mass is standing still: x˙2 = 0, the spring contracts: x˙2 − x˙1 > 0 and is contracted relative to the non-deformed state: x2 − x1 < 0. DE – both masses move: x˙1 > 0, x˙2 > 0, the centers of mass move uniformly: x˙1 + x˙2 = C, the spring contracts: x˙2 − x˙1 < 0 and is contracted relative to the non-deformed state: x2 − x1 < 0. EF – both masses move: x˙1 > 0, x˙2 > 0, the centers of mass move uniformly: x˙1 + x˙2 = C, the spring stretches: x˙2 − x˙1 > 0, but is contracted relative to the non-deformed state: x2 − x1 < 0. FG – the first mass is standing still: x˙1 = 0, the second mass moves: x˙2 > 0, the spring stretches: x˙2 − x˙1 > 0, but is contracted relative to the non-deformed state: x2 − x1 < 0. GH – the first mass is standing still: x˙1 = 0, the second mass moves: x˙2 > 0, the spring stretches: x˙2 − x˙1 > 0 and is stretched relative to the non-deformed state: x2 − x1 > 0. HI – both masses move: x˙1 > 0, x˙2 > 0, the centers of mass moves uniformly: x˙1 + x˙2 = C, the spring stretches: x˙2 − x˙1 > 0, and is stretched relative to the non-deformed state: x2 − x1 > 0. IJ – both masses moves: x˙1 > 0, x˙2 > 0, the centers of mass move uniformly: x˙1 + x˙2 = C, the spring contracts: x˙2 − x˙1 < 0, but is stretched relative to the non-deformed state: x2 − x1 > 0.

6.3 Straight Discrete Worms with Contact via Dry Friction Beginning with this chapter we describe straight discrete worms with contact via dry friction. The interaction of the worm system with the ground has been modeled as an anisotropic (i.e., orientation-dependent) dry friction force. This excludes a simple theory based on a kinematic drive and requires dynamics from the very beginning. Results were obtained either numerically or by means of perturbation and averaging methods. The two possibilities to attain the anisotropic characteristics follow from the formula of dry friction “friction coefficient times normal force”. In Section 6.4 we discuss problems with anistotropic (i.e., non-symmetric) friction coefficients, and locomotion systems based on the periodical change of normal forces are investigated in Section 6.5.

6.4 Worm-Like Locomotion based on Friction with Anisotropic Friction Coefficients

171

6.4 Worm-Like Locomotion based on Friction with Anisotropic Friction Coefficients 6.4.1 System of Two Mass Points and a Kinematic Drive We consider a system of two mass points connected by a drive element, which we use to control the distance between the masses, see Fig. 6.7. A piezoelectric element is one such component that can be used for this purpose, see Section 8.5.

l(t) m

m x

0

x1

x2

Fig. 6.7 Mechanical model for the system with a kinematic drive

The distance between mass points with the masses m and coordinates x1 and x2 varies according to the relationship: x2 (t) − x1 (t) = l(t) = l0 + b sin ω t .

(6.17)

˙ a simpler We use (2.89) to describe anisotropic COULOMB friction Ff r = m F(x)in form: ⎧ F , x˙ < 0 , ⎪ ⎨ − (6.18) F(x) ˙ = −F0 , x˙ = 0 , ⎪ ⎩ −F+ , x˙ > 0 , where F− ≥ F+ ≥ 0 are fixed, whereas F0 may take any value in the interval (−F− , F+ ). We assume that the friction force is small and motion occurs without sticking, i.e., the velocity is not identical to zero over any finite time interval. This question will be considered in detail in Section 6.5. Exercise 6.2. Express the equations of motion using the principle of linear momentum for each mass point and the kinematic constraint equation (6.17). The equations of motion takes the form: + 1* 1¨ − F(x˙2 ) + F(x˙1 ) = 0 , x¨1 + l(t) 2 2 + 1* 1¨ − F(x˙2 ) + F(x˙1 ) = 0 . x¨2 − l(t) 2 2

(6.19)

172


Let V = 12 (x˙1 + x˙2 ) denote the velocity of the center of mass. We obviously get from the equations in (6.19) the following equation for V : 1 1 1 F V + l˙ + F V − l˙ . (6.20) V˙ = 2 2 2 We can then introduce the following dimensionless variables (denoted here with the asterisk): l V F l∗ = , t ∗ = t ω , V ∗ = , F∗ = . (6.21) b bω Fm Here, Fm is some characteristic value of the friction force acting on the mass points (for example, Fm = F− ). By expressing dimensional variables with dimensionless variables and substituting into equation (6.20), we obtain the following equation in dimensionless variables: 1 ˙ 1 ˙ 1 ˙ V = ε F V + l +F V − l , 2 2 2 (6.22) Fm ε= . b ω2 At the large values of frequency ω and/or for small friction values Fm , the parameter ε can be considered as small. Equation (6.22) has a standard form, with which we can to use the procedure of time averaging (BOGOLJUBOV and MITROPOLSKI, [31]): 1 ε V˙ = 4π

1 1 F V + cost + F V − cost dt . 2 2

2π

(6.23)

0

Finally, we obtain the equation: 1 1 (F+ − F− ) + (F− + F+ ) arcsin 2V . V˙ = −ε 2 π

(6.24)

A steady-state motion with constant velocity V arises, where V is determined from the expression π F −F 1 − + · . (6.25) V = sin 2 2 F− + F+ or using dimensional variables: V=

π F −F bω − + sin · . 2 2 F− + F+

(6.26)

The result of the numerical solution of system (6.19) is given in Fig. 6.8. The following values of parameters were taken: b = 0.1 , ω = 10 , F− = 1 , F+ = 0.2. The purpose of the calculations presented is to illustrate the efficiency of the method. For this reason we use arbitrary parameter values (dimension units are not specified).


173

Further on, when concrete prototypes are considered, the parameter values used will correspond to the actual prototypes.

Fig. 6.8 Time dependence of the velocity of the center of mass for the system with a kinematic drive

Formula (6.26) gives the value for the velocity V of the center of mass as: V = = 0.43. Let us consider the situation in which friction F+ = 0. In this case the velocity of the center of mass is at its maximum and can be found from the formula V = b2ω . Figure 6.8 shows the result of the numerical solution of the exact equation (6.22) for ε = 0.1. For comparison, the stationary velocity V = 0.43 is presented as well (see the horizontal line). √ bω 3 2 2

6.4.2 System of Two Mass Points with a Linear Elastic Element 6.4.2.1 Mechanical Model and Equations with a Linear Elastic Element Let us consider a WLLS consisting of two identical mass points of mass m at the coordinates x1 and x2 and moving along a straight line. The mass points are connected to each other by an elastic element with stiffness c, see Fig. 6.9.

G(t) m

m c 0

x1

Fig. 6.9 Model of a WLLS consisting of two mass points

x2

x

174


The corresponding elastic force can be expressed by the formula (see (2.82)): Fc (x) = −c (x2 − x1 − l0 ) .

(6.27)

Here, l0 = λ0 is the distance between the mass points without a deformation of the ˙ ε 1, affects the elastic element. A small dry COULOMB friction force ε m F(x), system. We take the force F(x) ˙ again in the form (6.18): ⎧ F , x˙ < 0 , ⎪ ⎨ − F(x) ˙ = −F0 , x˙ = 0 , ⎪ ⎩ −F+ , x˙ > 0 .

(6.28)

Here, F− ≥ F+ ≥ 0 are fixed, and the value F0 can be found somewhere in the interval (−F− , F+ ). Perturbation is accomplished by the small harmonic force G(t): G(t) = ε m b sin ψ ,

ψ = νt,

ε 1.

(6.29)

We assume that the friction and the excitation forces are small in comparison to the spring force. Unlike the kinematic drive, the spring force is defined by a change in length of a spring (amplitude), which is not known in advance but instead defined during problem-solving. This aspect will be covered in more detail in Section 6.4.2. The motion equations of the system described take the following form: m x¨1 + c (x1 − x2 + l0 ) = −ε m b sin ψ + ε m F(x˙1 ) , m x¨2 + c (x2 − x1 − l0 ) = +ε m b sin ψ + ε m F(x˙2 ) . Designating ω 2 = form:

c m

and substituting x2 − l0 for x2 yields system equations in the * + x¨1 + ω 2 (x1 − x2 ) = ε F(x˙1 ) − b sin ψ , (6.30) * + x¨2 + ω 2 (x2 − x1 ) = ε F(x˙2 ) + b sin ψ .

Figure 6.10 shows the dependency of the velocity of the center of mass V (t) vs. time t 1 V (t) = (x˙1 + x˙2 ) . 2


175

Fig. 6.10 Velocity of the center of mass as calculated from (6.30)

The graph is obtained as a result of the numerical integration of system (6.30) with the following system parameter values: ε = 0.01, ω = 1, F− = 2, F+ = 1, b = 10, ν = 1.5 and the initial conditions: x1 (0) = x2 (0) = 0, x˙1 (0) = x˙2 (0) = 0. It is evident from the graph that, beginning at a certain point in time (approximately t = 350), the velocity of the system’s center of mass shows rapid oscillations around a certain slower motion with a constant velocity. The velocity of this slower motion is of greatest interest.

6.4.2.2 Application of the Averaging Method The system (6.30) is nonlinear because the function F(x) ˙ suffers a discontinuity at x˙ = 0. We are interested in the “slow” motion of the system and the stationary regimes of this motion. As we already indicated, the “stick-slip” regime is possible in systems with friction. This regime has time intervals with resting mass points (one or both mass points) during system motion. An example of the numerical solution of system (6.30) when the stick-slip zones are present is given in Fig. 6.11. The horizontal sections of the curves x1 (t) and x2 (t) in Fig. 6.11 correspond to the condition of rest of one or both masses. Here, we are interested in regimes near the resonance values (when vibrations are close to harmonious); therefore, motion with stick-slip regimes is not considered.

176


Fig. 6.11 V , x1 and x2 vs. time t in the stickslip case with the following parameters: ε = 0.01, ω = 1.0, F− = 9.0, F+ = 8.0, b = 10, ν = 1.5

We apply a standard procedure to indicate the slow variables and introduce new variables, namely the velocity of the center of mass V and deviations regarding the center of mass: x2 − x1 x˙1 + x˙2 , z= . (6.31) V= 2 2 From this, we find x˙2 = V + z˙ . (6.32) x˙1 = V − z˙ , Adding and subtracting the equations of system (6.30), we arrive at a system with new variables: + ε* F(V + z˙) + F(V − z˙) , V˙ = 2 (6.33) * + ε 2 F(V + z˙) − F(V − z˙) + 2 b sin ψ , z¨ + Ω z = 2 √ with Ω = 2 ω . We can now search for the function z in the form: z˙ = −a Ω sin ϕ , (6.34) z = a cos ϕ , √ where ϕ = Ω t + ϑ , Ω = 2 ω . The functions a = a(t) and ϕ = ϕ (t) (or ϑ = ϑ (t)) here are unknown time functions for which we wish to determine the corresponding equations. We note that the second equation (6.34) is not the result of the differentiation of the first one since the unknown function z has been replaced by two unknowns a and ϕ . Hence, an additional condition is required to determine these functions. The second equation (6.34) is this condition.


177

Now, differentiating the first equation (6.34) and comparing the result with the second equation (6.34), the connection between a and ϑ becomes clear: a˙ cos ϕ = a ϑ˙ sin ϕ .

(6.35)

Again differentiating the second equation (6.34) and substituting the result into the second equation of system (6.33) and excluding some conditions with the help of equation (6.35), we obtain the differential equation for the amplitude a: a˙ =

* + ε sin ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) − 2 b sin ψ . 2Ω

(6.36)

From equation (6.35) it follows that

ϑ˙ =

* + ε cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) − 2 b sin ψ . 2aΩ

(6.37)

Thus, we have the following system: + ε* F(V + a Ω sin ϕ ) + F(V − a Ω sin ϕ ) , V˙ = 2 * + ε sin ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) − 2 b sin ψ , a˙ = 2Ω (6.38) * + ε ϕ˙ = Ω + cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) − 2 b sin ψ , 2aΩ

ψ˙ = v . We further investigate the system near the main resonance value, assuming that the frequency of the perturbed force is close to the natural frequency of the vibration, i.e., v = Ω + ε Δ . For this purpose let us introduce the new slow variable ξ = ψ − ϕ . Then, ξ˙ = v − ϕ˙ = Ω − ϕ˙ + ε Δ , and finally, system (6.38) takes the form: + ε* F(V + a Ω sin ϕ ) + F(V − a Ω sin ϕ ) , V˙ = 2 * ε a˙ = sin ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) − 2 b sin(ξ + ϕ )] , 2Ω * ε ξ˙ = − cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) 2aΩ − 2 b sin(ξ + ϕ )] + ε Δ ,

ϕ˙ = Ω +

(6.39)

* ε cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) 2aΩ + − 2 b sin(ξ + ϕ ) .

Thus, we have three slow variables V (t), a(t) and ξ (t) and the fast frequency ϕ (t). Now, it is possible to take advantage of the theorem formulated in Section 3.3.1 and to apply the procedure of averaging (BOGOLJUBOV and MITROPOLSKI,

178


[31]). Replacing the right parts of system (6.39) with their averaged values in terms of the fast variable ϕ yields the expression 21π 02 π (. . .) d ϕ . We then calculate the variable I1 = 21π 02 π F(V + a Ω sin ϕ ) d ϕ . If V ≥ a Ω , then I1 = −F+ . But if 0 ≤ V < a Ω , then it is necessary to determine the intervals over which V + a Ω sin ϕ has the specific sign. The result of solving the corresponding inequalities is given in Fig. 6.12. Taking these principles into account, we get , 1 I1 = 2π =−

π +arcsin aVΩ

2 π −arcsin aVΩ

(−F+ ) d ϕ +

F− d ϕ +

π +arcsin

0

2 π −arcsin

V aΩ

-

2 π

(−F+ ) d ϕ V aΩ

V F− − F+ F− + F+ . arcsin + π aΩ 2

Similarly, it can be easily determined that I2 = 21π 02 π F(V − a Ω sin ϕ ) d ϕ = I1 . Now, let us calculate I3 = 21π 02 π sin ϕ F(V + a Ω sin ϕ ) d ϕ . If V ≥ a Ω , then I3 = 0, but if 0 ≤ V < a Ω , then , 1 I3 = 2π

π +arcsin aVΩ

2π −arcsin aVΩ

(−F+ ) sin ϕ d ϕ +

π +arcsin

0

F− sin ϕ d ϕ V aΩ

-

2π

(−F+ ) sin ϕ d ϕ

+ 2π −arcsin

=− and I4 =

F− + F+ cos π

-

, arcsin

1 2π 2π 0 sin ϕ F(V

V aΩ

=−

F− + F+ π

− a Ω sin ϕ ) d ϕ = −I3 .

/ 1−

V aΩ

V2 a2 Ω 2

,


179

F+

0 2p

p

p + arcsin V aW

V 2p - arcsin aW -F-

Fig. 6.12 Determination of the sign of V + a Ω sin ϕ

Further, we can calculate I5 = 21π 02π cos ϕ F(V + a Ω sin ϕ ) d ϕ and I6 = 21π 02 π cos ϕ F(V + a Ω sin ϕ ) d ϕ . As a result it appears, that I5 = I6 = 0. Also, we can determine that I7 = 21π 02 π sin ϕ sin(ϕ + ξ ) d ϕ = 12 cos ξ and I8 = 1 2π 1 2 π 0 cos ϕ sin(ϕ + ξ ) d ϕ = 2 sin ξ . Finally, the averaged system takes the form (preserving previous designations): F− +F+ F− −F+ V , if 0 ≤ V < a Ω , ε arcsin − − π a Ω 2 V˙ = if V ≥ a Ω , −ε F+ , 1 2 + (6.40) 1 − a2VΩ 2 + b2 cos ξ ) , if 0 ≤ V < a Ω , − Ωε ( F− +F π a˙ = if V ≥ a Ω , −ε 2bΩ cos ξ , b ξ˙ = ε sin ξ + Δ . 2aΩ The averaged system also remains nonlinear. Figures 6.13 to 6.15 show the results of the numerical integration of this system for the same parameter values used for the integration of the exact system (6.30) and given in Fig. 6.11. The value of Δ is selected as being equal to 10, which corresponds to the value v = Ω + ε Δ = 1.5, with which the exact system (6.30) has been solved. The initial conditions were assigned in the form: V (0) = 0, ξ (0) = 0, a(0) = 0.1. Thus, the condition V < a Ω is satisfied. From the graph given in Fig. 6.13, it follows that beginning from a certain point in time (approximately t = 350), oscillations are absent and the presence of the stationary regime is obvious, i.e., the center of mass moves at a constant velocity. Amplitude a(t) and phase ξ (t) also remain constant (Figs. 6.14 and 6.15).

180


Fig. 6.13 Dependence of velocity V on time t as a result of the√numerical integration of system (6.40) with the following parameters: F+ = 1.0, F− = 2.0, Ω = 2, b = 10.0, Δ = 10.0, ε = 0.01

Fig. 6.14 Dependence of a on t as a result of the numerical integration of system (6.40)


181

Fig. 6.15 Dependence of ξ on t as the result of the numerical integration of system (6.40)

The averaged system allows us to allocate the stationary regime and to define the corresponding velocity of the center of mass, the amplitude, and the phase.

6.4.2.3 Stationary Regime Since we are interested in the motion of the system with a constant velocity, we seek the solution of system (6.40) in the form V˙ = 0. In this case and according to the first equation of system (6.40), we find: V = a Ω sin Φ ,

Φ=

π F− − F+ · . 2 F− + F+

(6.41)

Regarding equation (6.41), a = const since V = const. That means that from the second and the third equations of system (6.40), we get: b cos ξ = −E cos Φ , 2

E=

F− + F+ . π

(6.42)

It follows that ξ = const and from the third equation of system (6.40), we find: b sin ξ = −a Δ Ω . 2 Furthermore, excluding ξ from equations (6.42) and (6.43), we find: / 1 b2 − E 2 cos2 Φ , a= Δ = 0 . Ω |Δ | 4 Thus, sin Φ V= |Δ |

/

b2 − E 2 cos2 Φ 4

(6.43)

(6.44)

(6.45)

182


and the expression for phase ξ has the form:

Δ 2 sin ξ = − |Δ | b

2 cos ξ = − E cos Φ , b

/

b2 − E 2 cos2 Φ . 4

(6.46)

The necessary condition for the existence of a stationary regime is: b ≥ 2 E cos Φ .

(6.47)

Physically, this condition means that in order for the stationary regime to exist, the amplitude of the perturbed force should exceed a certain value defined by the friction force and given by expression (6.47). Figure 6.13 depicts the dependence of the amplitude a on the value Δ in accordance with equation (6.44). Since frequency v is connected with the value of Δ using expression v = Ω + ε Δ , the curve represented by the equation is a resonance curve.

5

4

3 a 2

1

0 -20

-10

0

10

20

Delta

Fig. 6.16 Resonance curve (F− = 2 , F+ = 1 , b = 10 , Ω =

√ 2)

The discontinuity of the resonance curve at Δ = 0 is connected with the linearity of the elastic element and the absence of viscous damping. The infinite increase of the amplitude at Δ = 0 can be removed by the introduction of a nonlinear spring or a viscous damping element (or both). Since the stable stationary regimes can only be realized physically, the stability of stationary solutions (6.44) to (6.46) obtained are investigated in the next section.


183

6.4.2.4 Conditions of Stability To investigate the stability of stationary solutions (6.44) to (6.46) of the system, we first write down the equations in variations, which can be obtained from the equation system (6.40) (the index “s” is related to the stationary solutions in (6.40)). V = Vs + δ V ,

a = as + δ a ,

ξ = ξs + δ ξ .

Neglecting the smaller members (of higher order) with respect to δ V , δ a and δ ξ , we obtain the system of equations in variations (the index “s” is omitted in the stationary amplitude). E E · δ V + ε tan Φ · δ a , a Ω cos Φ a E E δ a˙ = ε tan Φ · δ V − ε tan Φ sin Φ · δ a − ε Δ a · δ ξ , aΩ2 aΩ Δ E δ ξ˙ = ε · δ a − ε cos Φ · δ ξ . a aΩ

δ V˙ = −ε

(6.48)

Furthermore, it is possible to use the stability criterion of LYAPUNOV, which makes it possible to judge the stability of the solutions of system (6.40) by using the sign of the real parts of the characteristic system polynomial in variations (6.48). The characteristic polynomial is obtained as a result of the expansion of the determinant: −ε a Ω Ecos Φ − λ E −ε (−1)3 P(λ ) = ε a Ω 2 tan Φ 0

tan Φ sin Φ − λ −ε Δ a Δ E εa −ε a Ω cos Φ − λ

ε E aΩ

E d

tan Φ

0

or P(λ ) = λ 3 + λ 2 ε

2E a Ω cos Φ 2 E E Δ2 . (6.49) + λ ε 2 2 2 (1 + sin2 Φ ) + Δ 2 + ε 3 a Ω Ω cos Φ a

The question about the signs of the roots of the characteristic polynomial can be answered using the criterion of HURWITZ (see Section 3.3.2). The stability condition for the third-degree polynomial in (6.49) can be written in the form:

184


2E > 0, a Ω cos Φ 3 2 2 Δ E 2 E2 > 0, (1 + sin2 Φ ) + Ω cos Φ a3 Ω 2 a

(6.50)

E Δ2 > 0. Ω cos Φ a Since E > 0 and 0 ≤ Φ < π2 (with F+ = 0), the conditions (6.50) are satisfied with all values of the input parameters, and the stationary regime of motion with a constant velocity is stable. A three-dimensional phase portrait for the cases presented in Figs. 6.13 to 6.15 is shown in Fig. 6.17. The phase portrait is the spiral of decreasing radius approaching a stable focal point.

Fig. 6.17 Phase portrait

We then note that the value of the stationary velocity V calculated with the parameter values √ ω = 1 (Ω = 2ω ) , F− = 2 , F+ = 1 , b = 10 , Δ = 10 / sin π6 102 32 π − 2 cos2 = 0.25 . V= 10 4 π 6 The comparison of this result with that from the exact system (6.30), given in Fig. 6.10, shows that expression (6.45) gives a very precise value of the system velocity “on average”. is


185

6.4.3 System with Two Mass Points and a Nonlinear Elastic Element 6.4.3.1 Equation of Motion Now, we consider a model of two mass points connected to each other by a nonlinear spring with a small cubic nonlinearity.

G(t) m

m c,d 0

x

x2

x1

Fig. 6.18 Two mass points with a nonlinear spring

In this case the elastic force is expressed by the formula: Fc (x) = −c (x − l0 ) − ε m d (x − l0 ) 3 ,

d > 0,

ε 1.

(6.51)

Assuming the remaining parameters of the prior model and the same friction force ε m F(x) ˙ and driving force G(t) = ε m b sinψ , ψ = ν t, we obtain the system: m x¨1 + c (x1 − x2 + l0 ) + ε m d (x1 − x2 + l0 )3 = −ε m b sin ψ + ε m F(x˙1 ) , m x¨2 + c (x2 − x1 − l0 ) + ε m d (x2 − x1 − l0 )3 = +ε m b sin ψ + ε m F(x˙2 ) . Designating ω 2 =

c m

and substituting x2 − l0 for x2 yields the form:

* + x¨1 + ω 2 (x1 − x2 ) = ε F(x˙1 ) − d (x1 − x2 )3 − b sin ψ , * + x¨2 + ω 2 (x2 − x1 ) = ε F(x˙2 ) − d (x2 − x1 )3 + b sin ψ .

(6.52)

Now, we change system (6.52) to the standard form using the method described above, by entering the velocity of the center of mass V and the deviation of the mass points relative to the center of mass z: V=

x˙1 + x˙2 , 2

z=

x2 − x1 . 2

(6.53)

We determine the value z as z = a cos(Ω t + ϑ ) , z˙ = −a Ω sin(Ω t + ϑ ) , (6.54) √ where Ω = 2 ω . Replacing V and z from expressions (6.53) and (6.54), system (6.52) can be written as follows:

186


" ε! V˙ = F(V + a Ω sin ϕ ) + F(V − a Ω sin ϕ ) , 2 ! ε sin ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) a˙ = 2Ω

" + 2 d a3 cos3 ϕ − 2b sin ψ ,

! ε ϕ˙ =Ω + cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) 2aΩ

(6.55)

" + 2 d a3 cos3 ϕ − 2 b sin ψ ) ,

ψ˙ =ν . Here, V and a are slow variables. We investigate system (6.55) in the vicinity of the main resonance value ν = Ω + ε Δ . For this purpose we introduce a new slow variable ξ = ψ − ϕ and exclude the fast variable ψ . As a result we obtain the system (6.55) in the form: " ε! V˙ = F(V + a Ω sin ϕ ) + F(V − a Ω sin ϕ ) , 2 ! ε sin ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) a˙ = 2Ω " + 2 d a3 cos3 ϕ − 2 b sin(ξ + ϕ ) ,

ξ˙ = −

! ε cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) 2aΩ

ϕ˙ =Ω +

(6.56)

" + 2 d a3 cos3 ϕ − 2 b sin(ξ + ϕ ) + εΔ ,

! ε cos ϕ F(V + a Ω sin ϕ ) − F(V − a Ω sin ϕ ) 2aΩ

" + 2 d a3 cos3 ϕ − 2 b sin(ξ + ϕ ) .

After averaging system (6.56) with the fast variable ϕ , we get: F− +F+ F− −F+ V − , if 0 ≤ V < a Ω , ε arcsin − π aΩ 2 V˙ = if V ≥ a Ω , −ε F+ , 1 V2 b + , if 0 ≤ V < a Ω , 1 − + cos ξ − Ωε F− +F 2 2 π 2 a Ω a˙ = b if V ≥ a Ω , −ε 2 Ω cos ξ , b 3 ξ˙ = ε sin ξ + Δ − d a2 . 2aΩ Ω If d = 0 equation (6.57) is identical to equation (6.40).

(6.57)


187

6.4.3.2 Stationary Regime and Conditions of Stability We are interested in approximate steady-state motion as a single unit. Therefore, we seek the solution V˙ = 0 of the system of equations in (6.57): V = a Ω sin Φ ,

Φ=

π F− − F+ · . 2 F− + F+

(6.58)

Since V = const, it follows from equation (6.58) that a = const and further from system (6.57) that ξ = const. Then, the second and the third equations of system (6.57) can be written as follows: b cos ξ = −E cos Φ , 2

E=

F− + F+ , π

b sin ξ = 3 a3 d − a Δ Ω . 2

(6.59)

Then, eliminating the value ξ from equations (6.59), we obtain the equation for the stationary amplitude a: / 2 b2 − E 2 cos2 Φ . a 3 a d − Δ Ω = (6.60) 4 Also, a stationary regime can only exist if the the condition b ≥ 2 E cos Φ is fulfilled. Exercise 6.3. Find the equations in variations for system (6.57). In order to investigate the stability of the stationary amplitudes defined by equation (6.60), we consider the conditions of stability. The characteristic polynomial P(λ ) for the system in variations for system (6.57) is P(λ ) = λ 3 + λ 2 ε +λ ε

2

2E a Ω cos Φ

E2 a4 d 2 a2 d 2 (1 + sin Ω ) + 27 2 + Δ Δ − 12 a2 Ω 2 Ω Ω E a3 d 2 Δ a2 d 3 +ε 27 2 + . (6.61) Δ − 12 Ω cos Φ Ω a Ω

For the polynomial given by the formula (6.61), the HURWITZ criterion can be written in the form:

188


2E > 0, a Ω cos Φ 2 E2 a3 d 2 Δ a2 d E 2 > 0, (1 + sin Φ ) + 27 2 + Δ − 12 Ω cos Φ a3 Ω 2 Ω a Ω a3 d 2 Δ a2 d E 27 2 + > 0. Δ − 12 Ω cos Φ Ω a Ω

(6.62)

Conditions (6.62) can then be reduced to a single condition: 27

d2 4 dΔ 2 a − 12 a +Δ2 > 0. 2 Ω Ω

(6.63)

6.4.3.3 Discussion of Results and Graphical Illustrations Figure 6.19 shows the velocity V of the center of mass vs. time t, obtained by numerically integrating the exact equations of motion in (6.52) with the following parameters: ε = 0.01, ω = 1.0, F+ = 1.0, F− = 2.0, b = 10.0, d = 10.0. We retain the values of these parameters in all further calculations. For the case considered here (εΔ = −0.1), the equation for stationary amplitudes (6.60), obtained from the system of averaged equations (6.57), has only one stable solution: a = 0.28. The corresponding value for the velocity V of the center of mass calculated with formula (6.58) is V = 0.20.

Fig. 6.19 Velocity V vs. time t (Δ = −10.0)

Figure 6.20 shows the dependence of stationary amplitude a on the value Δ , given by the equation (6.60). Since frequency ν of the driving force is connected with Δ by the formula ν = Ω + ε Δ , the curve in the Fig. 6.20 is a resonance curve.


189

Fig. 6.20 Stationary amplitude a vs. value ε Δ

0

For 3 Δ< Ω

3

3 d 4

b2 − E 2 cos2 Φ 4

(6.64)

there is only one stable stationary amplitude a1 . For 0 2 b 3 3 3 2 2 d − E cos Φ Δ> Ω 4 4

(6.65)

there are three stationary amplitudes a1 , a2 , a3 (ascending order), 1 only two of which

are stable, namely the least and the greatest. The value Δ = Ω3 3 34 d( b4 − E 2 cos2 Φ ) corresponds to the point of the vertical tangent to the resonance curve. Let us consider some characteristic points on the resonance curve. Taking Δ = −10, then ε Δ = −0.1. For the parameter values set as below, only the stable stationary amplitude a1 = 0.28 exists. The condition in (6.64) is satisfied because this point is placed left to the vertical tangent. The point considered next corresponds to the positive value Δ = 10, ε Δ = 0.1. Again, there is a single stable stationary amplitude a1 = 0.8. Finally, let us consider Δ = 15. In this case condition (6.65) is satisfied because the point εΔ = 0.15 is to the right of the vertical tangent. There are three stationary amplitudes, only two of which are stable. Figures 6.21, 6.22 and 6.23 show velocity V , the stationary amplitude a, and the phase ξ as functions of time t, obtained from the solution of the averaged equation system (6.57) for various initial conditions. For the curves in Fig. 6.21, the initial condition for V0 , a0 , and ξ0 is chosen such that the motion takes place with a stable stationary amplitude a1 = 0.27. The corresponding value of the stationary velocity of the center of mass is V1 = 0.18. 2

190


Fig. 6.21 Values of V , a, and ξ vs. time t (initial conditions: V0 = 0.14, a0 = 0.2, ξ0 = 0, Δ = 15)

For the curves in Fig. 6.22 the initial conditions for the amplitude are chosen close to the unstable stationary amplitude a2 = 0.65. The graphs reveal that the unstable stationary velocity V2 = 0.45 corresponding to this amplitude drops toward the stable stationary velocity V1 = 0.18.

Fig. 6.22 Values of V , a, and ξ vs. time t (initial conditions: V0 = 0.4, a0 = 0.6, ξ0 = 4.7, Δ = 15)

The curves in Fig. 6.23 demonstrate that the initial conditions are set up such that motion occurs with the maximum stable stationary amplitude a3 = 0.95. The value of the maximal stable stationary velocity of the center of mass is therefore V1 = 0.67.


191

Fig. 6.23 Values of V , a, and ξ vs. time t (initial conditions: V0 = 0.71, a0 = 1.0, ξ0 = 0, Δ = 15)

Figure 6.24 presents the three-dimensional phase portrait for the case shown in Fig. 6.21. The circles mark the equidistant time intervals. The phase portrait represents a helix of decreasing radius, which converges to a stable focal point.

Fig. 6.24 Phase portrait for V , a and ξ

6.4.4 System of n Mass Points with Kinematic Constraints 6.4.4.1 Mechanical Model and Equations of Motion Let us consider the system of n mass points with coordinates xi (i = 1 , . . . , n) and mass m which are moving along a straight line and connected to each other by the kinematic constraints li (t), see Fig. 6.25.

192


ln-1(t)

l1(t) 0

...

x xn-1

x2

x1

xn

Fig. 6.25 System with n mass points

We assume that an asymmetrical COULOMB friction force F(x˙i ) acts on the mass points. The values of the friction coefficients μ+ and μ− depend on the motion direction, i.e., we use (2.89), putting it in the form: ⎧ μ− m g , x˙i < 0 , ⎪ ⎨ −F0 , x˙i = 0 , (6.66) F(x˙i ) = ⎪ ⎩ −μ+ m g , x˙i > 0 . Here, the normal force coincides with the weight of the mass point m g, −m g μ− ≤ F0 ≤ m g μ+ , i = 1 , . . . , n. Let V be the velocity of the center of mass of the system. Then, 1 V = (x˙1 + . . . + x˙n ) . n

(6.67)

The motion equation of such a system has the following form: n · m V˙ = F(x˙1 ) + . . . + F(x˙n ) .

(6.68)

Taking into account n − 1 kinematic constraints: xi+1 − xi = li (t) ,

i = 1,...,n−1.

(6.69)

Let us study the case in which li (t) = l(t) = l0 + b sin ω t ,

˙ = b ω cos ω t , l(t)

i = 1,...,n−1.

(6.70)

Taking (6.68) into account, we find from (6.69) the expressions for velocity x˙i of the mass points through the velocity of the center of mass V : n+1 ˙ , i = 1,...,n. − i l(t) (6.71) x˙i = V − 2 Switching to dimensionless variables (denoted by asterisks), we get: xi ∗ ˙∗ F(x˙i ) , F (xi ) = , i = 1,...,n, b m g μ− g μ− μ+ ≤ 1, ε = . t∗ = t ω , μ = μ− b ω2

xi∗ =


193

Retaining the previous designations for the dimensionless variables, equation (6.67) has the following form: 1 F(x˙1 ) + . . . + F(x˙n ) , V˙ = ε n and the expression of the friction forces in (6.66): ⎧ 1 , x˙i < 0 , ⎪ ⎨ F(x˙i ) = −F0 , x˙i = 0 , −1 ≤ F0 ≤ μ , i = 1 , . . . , n , ⎪ ⎩ −μ , x˙i > 0 , x˙i = V −

n+1 − i cos t , i = 1 , . . . , n . 2

Substituting expression (6.74) into equation (6.71) yields: n+1 1 n ˙ − i cos t . V =ε ∑F V − n i=1 2

(6.72)

(6.73)

(6.74)

(6.75)

Equation (6.75) is a nonlinear, non-autonomous first-order differential equation.

6.4.4.2 Asymptotic Approximation For the asymptotic solution we consider the parameter ε small. Applying the procedure of averaging to equation (6.75), we get: 1 V˙ = ε 2π n

n

∑

i=1

2 π

F V−

n+1 − i cos t dt . 2

(6.76)

0

To calculate the corresponding integrals, the time intervals must be defined over which the argument sign of the function F is determined. After designating these intervals and setting τ+ and τ− , we find

2 π

F dt = −μ · τ+ + 1 · τ− ,

τ + + τ− = 2 π .

(6.77)

0

Thus, it is important to know the sign of the expression V − ( n+1 2 − i) cos t. We define the conditions for which V − ( n+1 − i) cos t > 0 (or < 0), i.e., ( n+1 2 2 − i) cos t < n+1 V (> V ). If i < 2 , then the following inequality holds: cos t < θ (or > θ ) , θ =

2V . n+1−2i

(6.78)

194


Now, if θ > 1, then τ+ = 2 π , τ− = 0. If θ < −1, then τ+ = 0, τ− = 2 π , and if |θ | ≤ 1, the solutions of the corresponding inequalities are t+ ∈ (arccos θ , 2 π − arccos θ ) and t− ∈ (0 , arccos θ ) ∪ (2 π − arccos θ , 2 π ) as well as τ+ = 2 (π − arccos θ ) and τ− = 2 arccos θ . Similarly, if i > n+1 2 , we obtain the following inequality: 2V . (6.79) cos t > θ1 (or < θ1 ) , θ1 = n+1−2i If θ1 > 1, then τ+ = 0, τ− = 2 π , and if θ1 < −1, then τ+ = 2 π , τ− = 0. In the case of |θ1 | ≤ 1 we have τ+ = 2 arccos θ1 and τ− = 2 (π − arccos θ1 ). For the values i (equidistant from the ends of the chain), the expression (6.78) for θ and 2V (6.79) for θ1 differ only by the sign. Actually, let k = i < n+1 2 , then θ = n+1−2 k . 2V For i = n − k + 1 the value θ1 = −n−1+2 k , i.e., θ1 = −θ . Then the condition θ1 > 1 is reduced to θ < −1, and the condition θ1 < −1 is reduced to θ > 1. Further, since arccos θ1 = π − arccos θ , then τ+ = 2 arccos θ1 = 2 (π − arccos θ ) and τ− = 2 (π − arccos θ1 ) = 2 arccos θ . Half of the paired summands on the right side of the equation can be remove and the result doubled. For any odd elements there is a averaged element i = n+1 2 that corresponds to the mass point coinciding with the centers of mass of the system. In this case the corresponding element, which is equal to 2 π F(V ), needs to be added to the right side of equation (6.76). On the basis of what has been presented thus far, it is possible to limit the equation to the terms whose magnitudes satisfy the condition i < n+1 2 . In this case if n+1 V > n+1 − i, then τ = 2 π , τ = 0, if V < −( − i) then τ = 0, τ− = 2 π , and if + − + 2 2 n+1 2V 2V |V | ≤ 2 − i, then τ+ = 2 (π − arccos n+1−2 i ) and τ− = 2 arccos n+1−2 i. n+1 Let the condition |V | ≤ 2 − i be satisfied up to i = p and not satisfied starting from p + 1. If the condition is not satisfied, it is necessary to add the product of these elements to the right side, either −2 π · μ when V > 0 or 2 π ·1 when V < 0. As V = 0 here, it is possible to describe both these situations by the product of the number of members on 2 π · F(V ). Then, the quantity of these members for an even n equals n π 2 − p. Furthermore, as arccos x = 2 − arcsin x, we obtain the following expression: , p 2V 1 2 ∑ μ · 2 π − arccos V˙ = −ε 2π n n+1−2i i=l p n 2V − ∑ 1 · 2 arccos −2π − p F(V ) n+1−2i 2 i=l , p 1 2 2V = −ε + (μ − 1) p − (n − 2 p) F(V ) , (μ + 1) ∑ arcsin n π n+1−2i i=l (6.80) with

n+1 n+1 − (p + 1) < |V | ≤ − p, 2 2

n p = 1,..., . 2

(6.81)


195

For an odd n the quantity of members not satisfying the condition |V | ≤ n+1 2 −i n−1 equals 2 − p. Using an averaged element yields: , p p 2V 2V 1 ˙ 2 ∑ μ · 2 π − arccos − ∑ 1 · 2 arccos V = −ε 2π n n + 1 − 2 i n + 1−2i i=l i=l n−1 1 − p F(V ) − · 2 π F(V ) +2π 2 2 , p 2V 1 2 − (1 − μ ) p − (n − 2 p) F(V ) , = −ε (μ + 1) ∑ arcsin n π n+1−2i i=l (6.82)

with

n+1 n+1 − (p + 1) < |V | ≤ − p, 2 2

p = 1,...,

n−1 . 2

(6.83)

Since equations (6.80) and (6.82) coincide, we can combine the conditions (6.81) and (6.83). For this, we set p = 1 , . . . , N, where N = n2 if n is even and N = n−1 2 if n is odd. Assuming N = n2 , it is possible to consider both cases (. . . is floor function, i.e., the whole-number part of the value). Having designated that

Φ (V, μ ) =

p 2 2V − (1 − μ ) p − (n − 2 p) F(V ) , (μ + 1) ∑ arcsin π n+1−2i i=1

(6.84)

we finally obtain the following equation

V˙ =

⎧ ⎪ ⎨ ⎪ ⎩

ε, −ε 1n Φ (V, μ ) , −ε μ ,

V < − n−1 2 , n+1 2

− (p + 1) < |V | ≤ V>

n−1 2

n+1 2

− p, p = 1,...,N ,

(6.85)

.

We are interested in the stationary solution of equation (6.85). It is clear that the n−1 stationary solution does not exist when V < − n−1 2 . When V > 2 the stationary solution is possible only when μ = 0 and then such solution will be any V = Vs > n−1 2 . Now, we search for the solution of the equation Φ (V, μ ) = 0 with the condition (6.81) or (6.83): p 2V 2 − (1 − μ ) p − (n − 2 p) F(V ) = 0 . (μ + 1) ∑ arcsin π n + 1−2i i=1

(6.86)

Next, we investigate equation (6.86). First assuming that V < 0, since the inequal2V n ity i ≤ p ≤ N is correct, then in this case all arcsin n+1−2 i < 0. Furthermore, p ≤ 2 , the coefficient μ ≤ 1 and keeping in mind condition (6.73), the value F(V ) = 1. Thus, Φ (V, μ ) < 0. In other words, equation (6.86) has no stationary solution V < 0,

196


i.e., the center of mass cannot move in the direction of the greater friction “on average” (under the assumption that the coefficient of friction μ+ does not exceed the coefficient of friction μ− , i.e., μ = μμ+− ≤ 1). So, we are interested in the non-negative solutions of equation (6.86) under the following conditions: n+1 − N when p = N 2 and N − (p + 1) < V ≤ N − p when p = 1 , . . . , N − 1 . 0≤V ≤

Now, we separately consider the case when V = 0. In this case p = N. If n is even, then p = n2 , and from equation (6.86) it follows that

Φ (0, μ ) = −(1 − μ ) p = 0 . Then, the solution V = 0 exists only when μ = 1. If n is odd, then p = from equation (6.86) it follows that F(0) = −F0 = −(1 − μ )

n−1 . 2

n−1 2 ,

and

(6.87)

In view of the remark made above, the condition for the existence of the stationary solution V = 0 as follows from the equation of motion is equivalent to fulfilling the inequality −1 ≤ F0 ≤ μ , which means the relationship in (6.87) is reduced to the inequality: n−1 ≤μ. (1 − μ ) 2 From here n−1 ≤ μ ≤ 1. (6.88) n+1 Letting V > 0, F(V ) = −μ and expression (6.88) for the function Φ (V, μ ) has the following form:

Φ (V, μ ) =

p 2 2V − p + μ (n − p) , (μ + 1) ∑ arcsin π n + 1−2i i=1

(6.89)

n+1 − N when p = N 2 n+1 n+1 − (p + 1) < V ≤ − p when p = 1 , . . . , N − 1 . and 2 2 Setting n to a concrete value, it follows from expression (6.88) that for each p: 0≤V ≤

ΦV =

p 1 2 1 (μ + 1) 2 ∑ π n + 1 −2i i=1

1−

1

2 2V n+1−2 i

> 0.

(6.90)

n+1 That is, the function over the intervals 0 < V ≤ n+1 2 −N (p = N) and 2 −(p+1) < n+1 V ≤ 2 − p (p = 1 , . . . , N − 1) increases and the equation Φ (V, μ ) = 0 has only one


197

root. In order for the number of roots to be exactly one over the specified intervals, the function Φ (V, μ ) must accept the value of the different signs at the ends of the corresponding interval: n+1 − N, μ ≥ 0 , p = N , Φ (0, μ ) < 0 , Φ 2 n+1 n+1 − (p + 1), μ < 0 , Φ − p, μ ≥ 0 , p = 1 , . . . , N − 1 . Φ 2 2 These conditions can be reduced to the following: p

N π N − 2 ∑ arcsin n+1−2 n+1−2 i i=1

p

N π (n − N) + 2 ∑ arcsin n+1−2 n+1−2 i

N , for p = N , n−N

≤μ
μ FN ,

(6.105)

FN = M g − Φy ,

FN ≥ 0 .

(6.106)

Formally, expressions (6.104) - (6.106) describe the motion of a mass point M along a horizontal straight line under the action of the control forces Φx and Φy , ˙ The force Φx acts the dry friction force Ff r , and the viscous friction force −d x. along the horizontal straight line on which the mass point is moving, and Φy acts in the upwards direction, see Fig. 6.27 (right). By controlling the force Φy , we can change the normal force using (6.106) and, as a consequence, regulate the force of dry friction. The control forces Φx and Φy can be created using different methods. In principle, in order to obtain any variational relationship of these forces, it is sufficient

6.5 Worm-like Locomotion Based on Periodic Change of Normal Forces

203

to have the internal body moving relative to the body along a curvilinear trajectory parametrically represented by the equations ξ1 = ξ1 (t) and η1 = η1 (t). Choosing the functions ξ1 (t) and η1 (t) in an appropriate way, we can implement the desired functional of the control forces Φx (t) = −m1 ξ¨1 (t) and Φy (t) = −m1 η¨ 1 (t). Alternatively, we can use two masses, m1 and m2 , one of which moves relative to the robot body in the horizontal direction and produces the force Φx , while the other moves vertically and produces the force Φy . In what follows, we consider the important particular case in which the control forces are harmonic time-dependent functions with the same frequency but shifted in phase, namely

Φx = Fx sin ω t ,

Φy = −Fy sin(ω t + ϕ0 ) ,

(6.107)

where Fx and Fy are the amplitudes of the corresponding forces, ω is the frequency of oscillations of these forces, and ϕ0 is the difference in phase between the forces Φx and −Φy . The quantities Fx , Fy , ω , and ϕ0 are constant parameters, with Fx > 0, Fy ≥ 0, and ω > 0. To satisfy inequality (6.106), we set Fy ≤ M g. In this case, relations (6.104) - (6.106) have the form M x¨ = Fx sin ω t − d x˙ + Ff r ,

(6.108)

⎧ −μ FN signx˙ , if x˙ = 0 , ⎪ ⎨ Ff r = −Fx sin ω t , if x˙ = 0 and |Fx sin ω t| ≤ μ FN , ⎪ ⎩ −μ FN sign(sin ω t) , if x˙ = 0 and |Fx sin ω t| > μ FN , FN = M g + Fy sin(ω t + ϕ0 ),

Fy ≤ M g .

(6.109)

(6.110)

We now introduce the dimensionless variables x∗ and t ∗ and parameters ε , α , and κ using the formulas x∗ =

M ω2 x, Fx

t∗ = ω t ,

ε=μ

Mg , Fx

α=

Fy , Mg

κ =d

Fx . (6.111) μ M2 g ω

Using these variables, equations (6.108) - (6.110) take the form (with the asterisks for dimensionless variables omitted) x¨ = sint − ε κ x˙ + fC , ⎧ − f signx˙ , if x˙ = 0 , ⎪ ⎨ fC = − sint , if x˙ = 0 and | sint| ≤ f , ⎪ ⎩ − f sign(sint) , if x˙ = 0 and | sint| > f , f = ε (1 + α sin(t + ϕ0 )) ,

α ≤ 1.

(6.112)

(6.113)

(6.114)

The parameter ε characterizes the order of magnitude of the ratio of the maximum possible value of the dry friction force (Ff r max = μ FN max = μ (Mg + Fy ) ≤

204


2 μ M g) to the amplitude Fx of the force Φx . Similarly, the product ε κ = Mdω reflects the ratio of the period of oscillations of the exciting force Te ∼ ω1 to the characteristic time of damping caused by the viscous friction Td ∼ Md . Thus, the smallness of the parameter ε shows that value of the dry friction force is relatively small compared to the amplitude of the “driving” force Φx , and the smallness of the parameter ε κ demonstrates that the kinetic energy is weakly dissipated by the viscous friction for the period of oscillations of the exciting force.

6.5.2.2 Steady-State Motion of the System in the Case of Small Friction Differential equation (6.112) is a first-order equation relative to the velocity x, ˙ since its right side only depends on time t and the velocity x˙ and does not contain the coordinate x. Here, we introduce a new variable u connected with the velocity x˙ by the relation x˙ = − cos t + u . (6.115) This relation divides the velocity of the mass point into a harmonic component − cos t, caused by the action of the harmonic exciting force and the component u, the change of which is caused by friction. In the absence of friction (ε = 0), the velocity x˙ corresponding to the solution of equation (6.112) is determined by the expression (6.116) x˙ = − cos t + u0 , where u0 is a constant component that depends on the initial conditions. Substituting (6.115) into (6.112), we obtain an equation for the variable u: u˙ = −ε κ (u − cos t) + fC .

(6.117)

Suppose that the parameters ε and ε ν are small quantities of the same order, i.e., ε 1 and κ ∼ 1. This means that the friction is small, and the effects of the components of COULOMB and viscous friction are comparable. Formally applying the averaging method to equation (6.117), in which fC is replaced by ε (1 + α sin(t + ϕ0 )) sign(u − cos t), we can assume that the movement occurs without sticking, i.e., the velocity is not identical to zero over any time interval of finite duration. Remark 6.1 We recall that in systems with COULOMB friction, motion with sticking (stick–slip motion) is possible. However, for an asymptotically small coefficient of dry friction, the total duration of the intervals of sticking is much less than the duration of the time interval on which the motion is considered, and, in the first approximation, we can neglect it. A more thorough substantiation of this assumption follows. In the case under consideration, equation (6.117) can be represented in the standard form


205

!

" u˙ = −ε κ (u − cos t) + 1 + α sin(t + ϕ0 ) sign(u − cos t) .

(6.118)

Averaging the right side of equation (6.118) relative to the variable t over the period 2 π , we obtain ⎧ κ u + 1 , if u > 1 , ⎪ ⎪ ⎨ √ u˙ = −ε κ u + π2 arcsin u − α sin ϕ0 1 − u2 , if |u| ≤ 1 , (6.119) ⎪ ⎪ ⎩ κ u − 1 , if u < −1 . Next, we find a stationary solution u = us of equation (6.119), corresponding to the steady-state motion of the system under consideration. A stationary solution makes the right side of differential equation (6.119) vanish. It can be seen directly from this equation that its right side is not zero for |u| > 1 and, consequently, all possible stationary values of the variable u are zeros of the function P(u , ϕ0 , α , κ ) = κ u +

. 2 arcsin u − α sin ϕ0 1 − u2 , π

(6.120)

considered for fixed values of the parameters ϕ0 , α , and κ . In what follows, as a rule, these parameters will be omitted in the list of arguments to reduce the notation. The derivative Pu of the function P(u , ϕ0 , α , κ ) is positive for |u| < 1 2 (1 + α u sin ϕ0 ) √ > 0 , |u| < 1 , α ≤ 1 , (6.121) π 1 − u2 meaning that this function monotonically increases over the interval −1 < u < 1. At the ends of this interval, we have P(−1 , ϕ0 , α , κ ) = −(1 + κ ) < 0 and P(1 , ϕ0 , α , κ ) = 1 + κ > 0. This implies that the function P(u , ϕ0 , α , κ ) has a zero in the interval −1 ≤ u ≤ 1, and this zero is unique. The form of the function P(u , ϕ0 , α , κ ) implies that, if us is its zero corresponding to the parameter ϕ0 , then the zero corresponding to the parameter −ϕ0 is −us . Let us be the desired zero of function (6.120). Then, in the considered approximation by (6.115), the velocity of the mass point in steady-state motion is determined by the expression (6.122) x˙ = − cos t + us , |us | ≤ 1 . Pu = κ +

This implies that us is the average velocity of the mass point in steady-state motion. On average, the mass point moves forward (i.e., in the positive direction on the axis x) if us > 0 and backward if us < 0. The steady-state motion is asymptotically stable relative to the velocity experiencing small perturbations. To prove this, we write the variational equation for equation (6.119) in the neighborhood of the stationary solution u = us

δ u = −ε Pu (us , ϕ0 , α , κ ) δ u .

(6.123)

This equation and the fact that the derivative Pu (us , ϕ0 , α , κ ) (see (6.121)) is positive imply that the motion with the average velocity u = us is asymptotically stable.

206


Since us depends on the angle ϕ0 that characterizes the phase shift between the force Φx acting in the horizontal direction and the force Φy acting in the vertical direction, we can control the value and direction (sign) of the average velocity of the mass point by changing this angle. For sin ϕ0 > 0, the mass point moves in the forward direction, whereas for sin ϕ0 < 0, it moves in the backward direction. This can be seen from the relationships P(0 , ϕ0 , α , κ ) = − π2 α sin ϕ0 , P(−1 , ϕ0 , α , κ ) < 0, and P(1 , ϕ0 , α , κ ) > 0 since it has been shown above that the function P monotonically increases in u over the interval −1 ≤ u ≤ 1. If sin ϕ0 > 0, then the value of P is negative over the interval −1 ≤ u ≤ 0, and, consequently, us > 0. When sin ϕ0 < 0, the function P is positive over the interval 0 ≤ u ≤ 1, and, consequently us < 0. If sin ϕ0 = 0, then us = 0. Let us investigate the quantity us on the parameters ϕ0 , α , and κ . We find the derivatives of the quantity us with the specified parameters using the implicit function theorem. As a result, we obtain Pϕ 2 α cos ϕ0 (1 − u2s ) ∂ us . = − 0 = , ∂ ϕ0 Pu π κ 1 − u2s + 2 (1 + α us sin ϕ0 ) P 2 sin ϕ0 (1 − u2s ) ∂ us . = − α = , ∂α Pu π κ 1 − u2s + 2 (1 + α us sin ϕ0 ) . π us 1 − u2s P ∂ us . = − κ = − . ∂κ Pu π κ 1 − u2s + 2 (1 + α us sin ϕ0 )

(6.124)

(6.125)

(6.126)

Without loss of generality, we can assume that −π ≤ ϕ0 ≤ π . It follows from (6.124) that the quantity us , considered as a function of the parameter ϕ0 , decreases over the intervals −π ≤ ϕ0 < −π /2 and π2 < ϕ0 ≤ π and increases over the interval − π2 < ϕ0 < π2 . At the point ϕ0 = − π2 , the quantity us takes a minimum value, and it has a maximum at the point ϕ0 = π2 . It has been shown above that us < 0 for ϕ0 = − π2 and us > 0 for ϕ0 = π2 . Thus, when ϕ0 = π2 , the mass point moves forward at a maximum average velocity, whereas for ϕ0 = − π2 , it moves backward at a maximum average velocity. It follows from (6.125) that the quantity us considered as a function of the parameter α decreases over the interval −π < ϕ0 < 0, and increases over the interval 0 < ϕ0 < π . Since 0 ≤ α ≤ 1, for α = 1 the minimum of the quantity us is attained if −π < ϕ0 < 0, and the maximum is attained if 0 < ϕ0 < π . Since us < 0 for ϕ0 < 0 and us > 0 for ϕ0 > 0, the value α = 1 corresponds to the maximum absolute value of the average velocity, irrespective of the direction of the motion. Remark 6.2 It follows from (6.126) that the absolute value of the average velocity of the steady-state motion decreases as the coefficient of viscous friction increases. It is interesting to note that, in this approximation using small friction, the average velocity of the steady-state motion does not depend on the coefficient of dry friction.


207

Example 6.3 We apply the results obtained to a vibration-driven robot with two internal masses. Let the forces Φx and Φy , varying in accordance with equation (6.107), be caused by the harmonic oscillations of two internal masses, m1 and m2 , relative to the robot body. Mass m1 oscillates along a horizontal line according to the function ξ1 = a sin ω t , a > 0 , (6.127) and mass m2 oscillates along the vertical axis according to

η2 = −b sin(ω t + ϕ0 ) ,

b ≥ 0.

(6.128)

From (6.103), these vibrations cause the forces

Φx = m1 a ω 2 sin ω t ,

Φy = −m2 b ω 2 sin(ω t + ϕ0 ) ,

(6.129)

which coincide in form with (6.107) for Fx = m1 a ω 2 ,

Fy = m2 b ω 2 .

(6.130)

In this case, the dimensionless parameters ε , α , and κ are

ε=μ

Mg m1 a ω 2

,

α=

m2 b ω 2 Mg

,

κ = d μm1Ma2ωg ,

(6.131)

with M = m + m1 + m2 , and m is the mass of the robot body. Assume for simplicity that there is no viscous friction, i.e., d = 0. It is worth noting that, under this method of motion excitation, the key role is played by dry friction. If dry friction is absent, the average velocity of the steady-state motion is equal to zero. For example, this follows from equation (6.108) for Ff r = 0. The general solution to this equation for velocity x˙ has the form x˙ = Ae−d M + t

Fx (d sin ω t − M ω cos ω t) , d 2 + M2 ω 2

(6.132)

where A is an arbitrary constant determined by the initial conditions. It can be seen that the velocity of the motion will settle, after the transient process (described by an exponential function) has decayed (for d > 0), a harmonic function with zero mean value. Let us = us (ϕ0 , α ) be the dimensionless average velocity of the steady-state motion computed by the method described above. Denote the same quantity by Vs but express it in terms of the initial dimensional variables. From formulas (6.111), which relate the dimensional and dimensionless variables, we have Fy Fx . (6.133) us ϕ0 , Vs = Mω Mg In the particular case when Fx and Fy have the form of (6.129), we obtain

208


m1 a ω m2 b ω 2 us ϕ0 , . Vs = M Mg

(6.134)

It was shown above that the maximum average velocity is attained for ϕ0 = and α = 1. The latter condition in this case is equivalent to the relation m2 b ω 2 = 1, Mg

π 2

(6.135)

which can be ensured, for example, by an appropriate adjustment of the amplitude of the oscillations of mass m2 for the frequency ω : b=

Mg = 1. m2 ω 2

(6.136)

Using this adjustment, the maximum average velocity of the harmonically excited vibration-driven robot, calculated as m1 a ω π us ,1 (6.137) Vs = M 2 is proportional to the excitation frequency and may be sufficiently large. Note g that for high excitation frequencies, the parameter ε = mμ M 2 is small even for 1 aω relatively large values of the friction coefficient μ , which supports the method of approximate computation of the robot motion based on the averaging method.

6.5.2.3 Vibration-Driven Robot with One Unbalance Exciter One of the simplest methods to excite harmonic oscillations employs an unbalance vibration exciter, which is a rotor with the center of mass shifted relative to the axis of rotation. The action of an unbalance exciter is equivalent to the action of a mass point m1 (equal to the mass of the rotor) moving on a circumference whose radius a is equal to the distance from the axis of rotation to the center of mass of the rotor. In this case, the coordinates of the mass point in the coordinate system 0 ξ η attached to the robot body can be represented by the expressions

ξ1 = a sin ϕ ,

η1 = −a cos ϕ ,

(6.138)

where ϕ is the angle between the position vector of the mass m1 relative to the point 0 coinciding with the rotation center and the negative semi-axis of the axis η , see Fig. 6.28.


209

y

m

h

x

0 j

m1 x

0 Fig. 6.28 Vibration-driven robot with one unbalance exciter

We consider the uniform rotation of the rotor, i.e., ϕ = ω t, where ω = const is the angular velocity. In this case we have

ξ1 = a sin ω t ,

η1 = −a cos ω t .

(6.139)

Substituting (6.139) into (6.103) for n = 1, we obtain

Φx = m1 a ω 2 sin ω t ,

π , Φy = −m1 a ω 2 sin ω t + 2

M = m + m1 .

(6.140)

Expressions (6.140) for Φx and Φy coincide with (6.107) for Fx = Fy = m1 a ω 2 ,

ϕ0 =

π . 2

(6.141)

From equation (6.141) it follows that the problem with a vibration exciter being considered is a simple realization of the phase shift ϕ0 = π2 , which leads to the maximum average velocity. The dimensionless parameters ε and α , see (6.111), are

ε=μ

Mg , m1 a ω 2

α=

m1 a ω 2 . Mg

(6.142)

The expression for the robot’s average velocity, similar to (6.134) but for the case when the oscillation is excited by a single-shaft, single-rotor unbalance vibration exciter, has the form π m1 a ω 2 m1 a ω us , , M = m + m1 . (6.143) Vs = M 2 Mg The following is a numerical example for clarity. Suppose that the mass of the body m = 0.255 kg, the mass of the rotor m1 = 0.03 kg, the distance from the axis of rotation to the center of mass of the rotor a = 0.017 m, the angular velocity of the ro-

210


tor ω = 50 s−1 , the coefficient of dry friction μ = 0.1, and the viscous friction is zero (d = 0). The parameters ε and α of (6.142) are 0.219 and 0.456, respectively. The presented characteristics correspond to the prototype of the vibration-driven robot shown in Chapter 8. Substituting these numerical values into (6.143), we obtain π , 0.456 = 0.036 . (6.144) Vs = 0.09 us 2 The value us π2 , 0.456 = 0.405 was obtained by numerical solving the transcendental equation P(u , π2 , 0.456 , 0) = 0, where the function P(u , ϕ0 , α , κ ) was defined in (6.120). In this case α = 0.456, but the maximum average velocity corresponds to α = 1. To attain this 1value, it is sufficient to increase the angular velocity of the rotor by

1 = 1.481, which gives ω ≈ 74 s−1 . Since the coefficient of us a factor of 0.456 in (6.143) is proportional to ω and, consequently, monotonically increases as the angular velocity grows, the value ω ≈ 74 s−1 corresponds to the maximum possible velocity of a robot of this design equal to Vs = 0.13 · 0.167 ms = 0.087 ms . Note that the parameter ε , which characterizes the relative smallness of the dry friction force, will then be equal to 0.1, which reduces the computational error of the averaging method compared with the case ω = 50 s−1 , in which ε = 0.219. Figure 6.29 presents the numerical solution to equation (6.119) for the initial condition u(0) = 0 with ε = 0.1, κ = 0, ϕ0 = π2 , and α = 1. This solution demonstrates the evolution of the nonharmonic component of velocity of the robot body in equation (6.115) and characterizes the transient process.

Fig. 6.29 Component of the velocity u vs. time t


211

6.5.2.4 Analysis of the Results The initial system of equations (6.112) - (6.114) was analyzed numerically together with the asymptotic analysis of the system behavior for small ε based on averaged equation (6.119). The equation of motion of the robot (6.112) is then integrated numerically for the initial conditions x(0) = 0 and x(0) ˙ = 0. The numerical solution shows that, at a certain time after the beginning of the motion, the motion of the robot settles into a periodic velocity, and the mean velocity in the steady-state mode is nonzero in general. To illustrate this, Figs. 6.30 and 6.31 show the dependence of the velocity of the robot x˙ (Fig. 6.30) and its coordinate x (Fig. 6.31) on time t for ε = 0.219, κ = 0 (viscous friction is absent), α = 0.456, and ϕ0 = π2 . These parameters correspond to the robot model considered at the end of the previous section. The plot for the velocity shows that the periodic mode settles in approximately 20 dimensionless time units after the beginning of the motion. For this time, the vibrating internal masses perform about three oscillations. From (6.111) for the dimensionless equations as a dimensionless time unit, we take the inverse value of the circular frequency of oscillations of the internal masses. The period of oscillations under this choice of time scale is 2 π .

.

Fig. 6.30 Velocity of the robot x˙ vs. time t

212


Fig. 6.31 Coordinate x vs. time t

Figure 6.32 shows the dependence of the average velocity of the steady-state motion us on the angle of the phase shift ϕ0 between the horizontal and vertical oscillations of the internal masses for α = 0.456 and κ = 0. Using (6.120), this curve is implicitly given by the equation 1 (6.145) arcsin us − α sin ϕ0 1 − u2s = 0 .

Fig. 6.32 Average velocity of the steady-state motion us vs. angle of the phase shift ϕ0


213

Exercise 6.5. The horizontal excitation force, acting on the mass point m can also be realized by an additional mass point m1 , which is coupled to the first one by means of a kinematic constraint in the form l(t) = l0 + b sin ω t. By replacing the constraint with the reaction force, find the equation of motion of the system shown in Fig. 6.33. What conclusion can be made by analyzing the form of the obtained equation?

Fy(w t +j0 ) l (wt) m 0

m1

x1

x2

x

Fig. 6.33 Vibration-driven robot with an additional mass

6.5.3 Vibration-Driven Robot with Two Unbalance Exciters 6.5.3.1 Mechanical Model and Equations of Motion Now, we will discuss a locomotion system composed of two identical modules connected by a spring, see Fig. 6.34. Each module consists of a rigid body and a singleshaft unbalance vibration exciter. The bodies of both modules are based on a rough horizontal plane and can realize a translational motion along the same straight line. The axes of rotation of the vibration exciter rotors are parallel to each other and perpendicular to the vertical plane passing through the line of motion of the two bodies. The centers of mass of both vibration exciters are assumed to lie in this plane. When the vibration exciters have been actuated, the entire system is able to perform a plane parallel motion. In the plane of motion, we introduce an inertial coordinate system Oxy, with the x-axis passing through the rough plane contacted by the bodies and the y-axis pointing vertically upward. We assume that the rotors of the vibration exciters rotate in the same direction with constant angular velocities. The values of these velocities can be different in general. Isotropic COULOMB dry friction with the friction coefficient μ acts between the supporting plane and the bodies.

214


w1

l

l

w2

m

m

c M

M x1

x2

Fig. 6.34 Vibration-driven robot with two unbalance vibration exciters

We introduce the following notation: M is the mass of bodies 1 and 2, m is the mass of the each vibration rotor, l is the distance between the center of mass of the exciter and its axis of rotation, c is the stiffness of the spring, ω1 and ω2 are the angular velocities of the rotors of the vibration exciters attached to bodies 1 and 2, respectively; x1 and x2 are the coordinates measuring the displacements of bodies 1 and 2 along the x axis (the zero points for the coordinates x1 and x2 are shifted by the length of the unstrained spring); ϕ1 and ϕ2 are the angles (measured counterclockwise) between the x axis and the perpendiculars dropped from the centers of mass of the rotors of the respective exciters onto the axes of their rotation; μ is the coefficient of friction of bodies 1 and 2 with the rough plane; g is the gravitational acceleration. Without loss of generality, we assume the rotors of the vibration exciters to rotate counterclockwise. The motion of the system is described by the differential equations (M + m) x¨1 + c (x1 − x2 ) = m l ω12 cos ϕ1 + Ff r1 , (M + m) x¨2 + c (x2 − x1 ) = m l ω22 cos ϕ2 + Ff r2 ,

ϕ1 = ω1 t,

(6.146)

ϕ2 = ω2 t + ϕ0 .

In these equations, ϕ0 is the initial shift between the phases of rotation of the vibration exciters, and Ff ri , i = 1 , 2, is the COULOMB friction force acting on body i. According to COULOMB’s law, see (2.89), ⎧ −μ FNi sign x˙i , if x˙i = 0 , ⎪ ⎨ if x˙i = 0 and |Fi | ≤ μ FNi , (6.147) Ff ri = −Fi , ⎪ ⎩ −μ FNi sign Fi , if x˙i = 0 and |Fi | > μ FNi , where


215

F1 = ml ω12 cos ϕ1 − c (x1 − x2 ) , F2 = ml ω22 cos ϕ2 − c (x2 − x1 ) , FNi = (M + m) g − m l ωi 2 sin ϕi ,

(6.148) i = 1,2.

The quantity FNi in (6.147) and (6.148) is the normal force exerted by the supporting plane on body i. Contact between the bodies of the system and the supporting plane represents a unilateral constraint, since the plane resists being penetrated by the bodies but does not resist the loss of contact. Therefore, FNi ≥ 0 while body i keeps contact with the plane. To ensure this inequality, we assume m l ωi2 ≤ 1, (M + m) g

i = 1,2.

(6.149)

To reduce the number of the parameters that characterize the system, we introduce the dimensionless variables / / c M+m xi ∗ ∗ , νi = ωi , xi = , t = t L M+m c (6.150) mcl μ (M + m) g α , α= , ε= β = , cL (M + m)2 g μ where L is the scale of length used for the non-dimensionalization. The choice of this scale will be discussed later. We proceed to the dimensionless variables in equations (6.146) - (6.148) and then omit the asterisks, identifying the variables xi∗ and t ∗ , to obtain x¨1 + x1 − x2 = ε β ν12 cos ϕ1 + r1 , x¨2 + x2 − x1 = ε β ν22 cos ϕ2 + r2 ,

ϕ1 = ν1 t , where

(6.151)

ϕ2 = ν2 t + ϕ0 ,

⎧ −ε ni sign x˙i , if x˙i = 0 , ⎪ ⎨ if x˙i = 0 and | fi | ≤ ε ni , ri = − f i , ⎪ ⎩ −ε ni sign fi , if x˙i = 0 and | fi | > ε ni ,

(6.152)

f1 = ε β ν12 cos ϕ1 − x1 + x2 , f2 = ε β ν22 cos ϕ2 − x2 + x1 ,

(6.153)

ni = 1 − α νi2 sin ϕi . Inequalities (6.149) become

ανi2 ≤ 1 ,

i = 1,2.

(6.154)

216


6.5.3.2 Application of the Method of Averaging The system now being considered consists of two vibration-driven modules elastically connected by a spring. Two bodies connected by a spring form an oscillatory system characterized by its natural frequency. It is reasonable to anticipate characteristic qualitative features in the behavior of the system in the vicinity of the resonance, when the vibration excitation frequency is close to the frequency of natural oscillations. It is the near-resonant behavior of the system that is the subject matter in this section. We will investigate analytically the steady-state motion of the system in the case where the force of friction and the excitation force are small compared with the maximum elastic force developed in the spring. In this investigation, we will use the method of averaging to show that, unlike the single-module system investigated in Section 6.5.2.3, the motion of the two-module system can be inverted by passing from the pre-resonant mode of excitation to the post-resonant mode and that the average speed of this motion can be controlled by changing the phase shift between the rotations of the exciters. To be able to use the method of averaging, we assume

ε 1,

β νi2 ∼ 1 ,

|x2 − x1 | ∼ 1 .

(6.155)

According to the definition of (6.150), the first two relations of (6.155) can be rewritten as m ωi2 l μ (M + m) g 1, ∼ 1. (6.156) cL μ (M + m) g The relative motion of the system’s bodies is described by the equation η¨ + 2 η = ε β ν22 cos(ν2 t + ϕ0 ) − ν12 cos ν1 t + r2 − r1 , η = x2 − x1 . (6.157) This equation is obtained by subtracting the first relation of (6.151) from the second relation. To estimate the order of magnitude of the quantity |x2 − x1 |, we will ignore the dry friction terms r1 and r2 in (6.157) and construct the particular solution of the resulting linear nonhomogeneous equation that corresponds to the forced √ √ oscillations. If ν1 = 2 and ν2 = 2, this solution has the form ν22 ν12 η =εβ cos(ν2 t + ϕ0 ) − cos ν1 t . (6.158) 2 − ν22 2 − ν12 √ √ In the resonant cases in which ν1 = 2 or ν2 = 2, a bounded solution of equation (6.157) with r1 = 0 and r2 = 0 does not exist. The desired estimate can be given by the maximum of the absolute value of the right-hand side of this relation. In what follows, we will consider the case where ν1 = ν2 = ν , which means that the vibration exciters rotate with the same speed. For that particular case the expression for η is reduced to

η =−

2 ε β ν2 ϕ0 ϕ0 sin ν t + . sin 2 2−ν 2 2

(6.159)


217

The maximum of the absolute value of the amplitude of the bodies’ relative oscillations is given by 2 ε β ν 2 ϕ0 (6.160) |η |max = sin . |2 − ν 2 | 2 Substituting the expressions of (6.150) for ε , β , and ν into (6.160), we obtain |η |max =

ϕ 2 m l ω2 0 sin , L |2 c − (M + m) ω 2 | 2

(6.161)

where ω is the common angular velocity of both exciters. To provide the relation |x2 − x1 | ∼ 1 in dimensionless units, we choose the length scale L such that |η |max = 1, i.e., ϕ 2 m l ω2 0 (6.162) L= sin . |2 c − (M + m) ω 2 | 2 With this choice, the conditions of (6.156) (which validate the use of the method of averaging) become (M + m) μ g |2 c − (M + m) ω 2 | 1, 2 m l c ω 2 sin ϕ20

m ω2 l ∼ 1. μ (M + m) g

(6.163)

The natural (resonant) frequency corresponds to the relative oscillations of the system’s bodies in the absence of friction (ε = 0) and excitation (νi = 0). These oscillations are governed by (6.157) with zero on the right-hand side. The general solution of this equation has the form √ η = A cos( 2t + θ ) , (6.164) where A and θ are arbitrary constants, namely the amplitude and the initial phase of the oscillations. The natural oscillations occur with the frequency √ (6.165) νn = 2 . We assume that the difference of the excitation frequencies ν1 and ν2 from the resonant frequency νn has an order of magnitude of ε , i.e., √ √ ν1 = 2 + ε Δ 1 , ν2 = 2 + ε Δ 2 , (6.166) where the constant parameters Δ1 and Δ2 have an order of unity. To apply the method of averaging, we will use the general solution of the unperturbed system (6.151), with a right-hand side of zero, as the generating solution. This solution can be represented as x2 = X + a cos ϕ , x1 = X − a cos ϕ , √ X = B +V t , ϕ = 2t + θ ,

(6.167)

218


2 where B, V , a, and θ are arbitrary constants. The variable X = x1 +x is the absolute 2 coordinate of the center of mass of the unperturbed system. Since the unperturbed system is not acted upon by any external force, the center of mass moves with constant velocity V , in accordance with (6.167). The variable ϕ coincides with the argument of the cosine in equation (6.164) and, consequently, represents the current phase of the relative oscillations of bodies 1 and 2. The coefficients A of equation (6.164) and a of equation (6.167) are related by A = 2 a. With reference to the obtained solution, we proceed in the basic equations (6.151) from the phase variables x1 , x˙1 , x2 , and x˙2 to the phase variables X, V , a, ξ1 , and ξ2 using the following relations: √ √ X˙ = V , x˙1 = V + a 2 sin ϕ , x˙2 = V − a 2 sin ϕ , (6.168) ξ1 = ϕ1 − ϕ , ξ2 = ϕ2 − ϕ ,

where, in accordance with (6.151) and (6.166), √ √ ϕ1 = 2t + ε Δ1 t , ϕ2 = 2t + ε Δ2 t + ϕ0 .

(6.169)

In terms of the new variables, the equations of motion (6.151) acquire the following structure: X˙ = V , √ ϕ˙ = 2 + ε fϕ (V , ϕ , a , ξ1 , ξ2 ) , V˙ = ε fV (V , ϕ , a , ξ1 , ξ2 ) , (6.170) a˙ = ε fa (V , ϕ , a , ξ1 , ξ2 ) , ξ˙1 = ε f (V , ϕ , a , ξ1 , ξ2 ) , ξ1

ξ˙2 = ε fξ2 (V , ϕ , a , ξ1 , ξ2 ) . The functions on the right-hand side of these equations are 2 π -periodic in ϕ . The full expressions for these functions will be given below. Equations (6.170) have a standard form in terms of the method of averaging. The quantities X and ϕ are the fast variables, while V , a, ξ1 , and ξ2 are the slow variables. By averaging the right-hand sides of these equations with respect to the fast variable ϕ we obtain the averaged system in the form X˙ = V , √ ϕ˙ = 2 + ε f¯ϕ (V , a , ξ1 , ξ2 ) , V˙ = ε f¯V (V , a , ξ1 , ξ2 ) , a˙ = ε f¯a (V , a , ξ1 , ξ2 ) , ξ˙1 = ε f¯ (V , a , ξ1 , ξ2 ) , ξ1

ξ˙2 = ε f¯ξ2 (V , a , ξ1 , ξ2 ) .

(6.171)


219

The last four equations of (6.171) form a self-consistent subsystem that can be considered independently of the first two equations. It is this subsystem that will be of major importance for us. In what follows, we are interested in the velocity of the steady-state motion of the vibration-driven system “as a whole”. The variable V will be used to characterize the velocity of the entire system. Through steady-state motion, we understand the behavior of the system that is established after the transient processes have decayed. This concept is similar to that of the forced oscillations in the vibration theory. The steady-state motion will be investigated on the basis of the averaged system. In terms of this system, the steady-state motion is defined by the relations V = const and a = const. Thus, the analysis of the steady-state motion is reduced to the solution of the differential algebraic system of equations f¯V (V , a , ξ1 , ξ2 ) = 0 , f¯a (V , a , ξ1 , ξ2 ) = 0 ,

ξ˙1 = ε f¯ξ1 (V , a , ξ1 , ξ2 ) , ξ˙2 = ε f¯ξ2 (V , a , ξ1 , ξ2 ) .

(6.172)

In the expanded form, the functions fV , fa , fξ1 , and fξ2 are represented as follows: * + fV = 12 2 β cos(ϕ + ξ2 ) + cos(ϕ + ξ1 ) ! √ − 1 − 2 α sin(ϕ + ξ1 ) sign(V + a 2 sin ϕ ) " √ + 1 − 2 α sin(ϕ + ξ2 ) sign(V − a 2 sin ϕ ) + O(ε ) , * + 1 cos( fa = − 2 √ 2 sin ϕ β ϕ + ξ ) − cos( ϕ + ξ ) 2 1 2 ! √ + 1 − 2 α sin(ϕ + ξ1 ) sign(V + a 2 sin ϕ ) " √ − 1 − 2 α sin(ϕ + ξ2 ) sign(V − a 2 sin ϕ ) + O(ε ) , (6.173) * + 1√ fξ1 = a 2 2 cos ϕ 2 β cos(ϕ + ξ2 ) − cos(ϕ + ξ1 ) ! √ + 1 − 2 α sin(ϕ + ξ1 ) sign(V + a 2 sin ϕ ) " √ − 1 − 2 α sin(ϕ + ξ2 ) sign(V − a 2 sin ϕ ) + Δ1 + O(ε ), * + 1√ fξ2 = a 2 2 cos ϕ 2 β cos(ϕ + ξ2 ) − cos(ϕ + ξ1 ) ! √ + 1 − 2 α sin(ϕ + ξ1 ) sign(V + a 2 sin ϕ ) " √ − 1 − 2 α sin(ϕ + ξ2 ) sign(V − a 2 sin ϕ ) + Δ2 + O(ε ).

220


When forming these functions, we used the expressions ri = −ε ni sign x˙i for the dry friction force acting on the system’s modules instead of (6.152). This replacement ignores the possibility of stick-slip motion of the modules. The difference of the solution of the averaged equations (6.171) from that of equations (6.170), subject to the same initial conditions, has an order of magnitude of ε over the time interval of an order of ε1 , see Section 3.3.1. The terms O(ε ) in the functions of (6.173) add terms of an order of ε to the solution of the averaged equations on the time interval O( ε1 ) and, consequently, can be omitted without loss of accuracy. Averaging the functions of (6.173) with the terms O(ε ) being omitted over the fast variable ϕ , we have ⎧ 1, u < −1 , ⎪ ⎪ ⎨ ! " . 2 (6.174) f¯V = − arcsin u + α (cos ξ2 − cos ξ1 ) 1 − u2 , |u| ≤ 1 , ⎪ ⎪ ⎩ π −1 , u > 1, ⎧ " 1 ! ⎪ ⎪ √ β (sin ξ2 − sin ξ1 ) + α (cos ξ2 − cos ξ1 ) , u < −1 , ⎪ ⎪ 2 2 ⎪ ! ⎪ . ⎪ ⎪ ⎨ − √1 4 1 − u2 − π β (sin ξ2 − sin ξ1 ) ¯fa = 2 2π (6.175) " √ ⎪ 2 ⎪ , |u| ≤ 1 , α (cos ξ − cos ξ ) arcsin u − u 1 − u +2 ⎪ 2 1 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ √ [β (sin ξ2 − sin ξ1 ) − α (cos ξ2 − cos ξ1 )] , u > 1, 2 2 ⎧ ! " 1 ⎪ √ π β (cos ξ2 − cos ξ1 ) − α (sin ξ2 − sin ξ1 ) + Δ1 , u < −1 , ⎪ ⎪ ⎪ 2 2π a ⎪ ⎪ ⎪ ⎪ ⎨ √1 π β (cos ξ2 − cos ξ1 ) + 2 α (sin ξ2 − sin ξ1 ) f¯ξ1 = 2 2 π a " √ ⎪ 2 ⎪ · arcsin u + u + Δ1 , |u| ≤ 1 , 1 − u ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ √ [π β (cos ξ2 − cos ξ1 ) − α (sin ξ2 − sin ξ1 )] + Δ1 , u > 1 , 2 2π a (6.176) ⎧ ! " 1 ⎪ √ π β (cos ξ − cos ξ ) − α (sin ξ − sin ξ ) ⎪ 2 1 2 1 + Δ 2 , u < −1 , ⎪ ⎪ 2 2π a ⎪ ⎪ ⎪ ⎪ ⎨ √1 π β (cos ξ2 − cos ξ1 ) + 2 α (sin ξ2 − sin ξ1 ) f¯ξ2 = 2 2 π a " √ ⎪ ⎪ · arcsin u + u 1 − u2 + Δ2 , |u| ≤ 1 , ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ √ [π β (cos ξ2 − cos ξ1 ) − α (sin ξ2 − sin ξ1 )] + Δ2 , u > 1 , 2 2π a (6.177)


where

V u= √ . a 2

221

(6.178)

From the definition of the steady-state motion, the variables V and a are constant according to the averaged equation and, hence, u = const. From the first two equations of (6.171) and expressions (6.174) and (6.175), it follows that . (6.179) arcsin u − 2 α sin χ sin ψ 1 − u2 = 0 , . . 2 1 − u2 − π β sin χ cos ψ + 2 α sin χ sin ψ arcsin u − u 1 − u2 = 0 , (6.180) where

ξ1 + ξ2 ξ2 − ξ1 , χ= . (6.181) 2 2 For u = const, relations (6.179) and (6.180) imply χ = const and ψ = const and, with reference to (6.181), ξ1 = const and ξ2 = const. Therefore, ξ˙1 = 0 and ξ˙2 = 0; accordingly, f¯ξ1 = 0 and f¯ξ2 = 0. From the last two relations and expressions (6.175) and (6.176), it follows that ψ=

Δ1 = Δ2 = Δ ,

(6.182)

where

Δ=√

" . 1 ! π β sin χ sin ψ − 2 α sin χ cos ψ arcsin u + u 1 − u2 . (6.183) 2π a

This relations implies an important corollary that the steady-state motion of the system under consideration occurs only if both vibration exciters have the same off-resonance frequency detuning. Using the definitions in (6.168) for ξ1 and ξ2 , (6.169) for ϕ1 and ϕ2 , (6.181) for χ , and the relation Δ1 = Δ2 = Δ , we obtain

χ=

ϕ2 − ϕ1 ϕ0 = . 2 2

(6.184)

We are mostly√ interested in the steady-state velocity V . In accordance with (6.178), V = u a 2 and, consequently, only the steady-state values of u and a are needed to calculate V . It is possible to eliminate the variable ψ from relations (6.179), (6.180), and (6.183) and to obtain two equations for the desired variables. To that end, we use relations (6.179) and (6.180) to express sin ψ and cos ψ as functions of u. Then, we substitute the resulting functions into equation (6.183) to express a in terms of u and use the identity sin2 ψ + cos2 ψ = 1 to obtain an equation for u. These relations can be represented as follows: ! "2 √ π 2 arcsin2 u + 4 μ 2 2 (1 − u2 ) − arcsin u arcsin u − u 1 − u2 4 π 2 (1 − u2 )

= γ 2 , (6.185)

222


a=

1 π arcsin u 2 (1 − u2 ) 2 μ " . 2μ ! 2 2 3 2 − (1 − u ) (2 + u ) arcsin u + 2 u 1 − u − arcsin u , (6.186) π

πΔ

.

where

ϕ0 . (6.187) 2 Based on equations (6.185) and (6.186), one can draw two important conclusions regarding the behavior of the system, at least for the case of low friction and week excitation (ε 1). Equation (6.185), which defines the magnitude of u, depends of the parameter γ and, therefore, |u| depends on γ . From equation (6.186) it follows that the absolute value of a is uniquely defined for given u. Thus, the magnitude of the steady-state velocity |V | depends on the parameter γ , which, in view of (6.187), depends on the excitation phase shift ϕ0 . Hence, the magnitude of the steady-state velocity of the system can be controlled by changing the angle between the vectors from the axes of rotation to the centers of mass of the exciters. From equations (6.185) and (6.186), it follows that if (u , a) is a solution of these equations for the set of parameters (μ , Δ , γ ), then (−u , a)√is a solution for the set of parameters (μ , −Δ , γ ). In view of the relation V = a u 2, this implies that the steady-state velocity changes in sign when the off-resonance detuning Δ changes in sign. Therefore, the direction of the steady-state motion of the two-module system under consideration can be controlled by changing between pre-resonant and postresonant excitation modes. The reverse of the direction of rotation of the exciters is not required. γ = α sin

6.5.3.3 Numerical Example The numerical calculations are performed for the prototype of the vibration-driven robot shown in Fig. 8.9. It has the following parameters: M = 0.12 kg, m = 0.012 kg, N . The natural frequency of this system is l = 0.015 m, c = 130 m /

ωn =

2c = 44.4 s−1 . M+m

(6.188)

For the numerical calculations, we define the coefficient of friction and the excitation frequency (both exciters rotate with the same angular velocity) as follows:

ω = 50 s−1 ,

μ = 0.1 .

(6.189)

Then, the length scale L of equation (6.162) and the dimensionless parameters ε , νi , α , and β of equation (6.150) are given by


ϕ 0 L = 0.013 sin m , 2

0.077 , sin ϕ20

ε =

ν1 = ν2 = ν = 1.6 ,

α = 0.14 ,

223

β = 1.4 .

Accordingly, with respect to (6.166),

Δ1 = Δ2 = Δ =

√ ϕ ν− 2 0 = 2.5 sin . ε 2

(6.190)

To verify the conditions of (6.149) and (6.155), we calculate αν 2 and β ν 2 and 2 2 the cited conditions are satisfied if obtain ϕ αν ≈ 0.36, β ν ≈ 3.48. Consequently, sin 0 is not too small. For small sin ϕ0 , the condition ε 1 may be violated. In 2 2 what follows, the cases of ϕ0 = π2 (ε = 0.11) and ϕ0 = π (ε = 0.077) will be considered. The maximum value of sin ϕ20 at which the solution of equation (6.185) fails to exist is approximately equal to 0.45, which corresponds to ϕ0 ≈ 54◦ . Figure 6.35 shows the absolute value of the phase shift angle |ϕ 0 | as a function of |u|. This curve was plotted on the basis of equation (6.185). For given |u|, this relation uniquely defines the quantity sin2 ( ϕ20 ). It suffices to consider the values of |ϕ0 | in the range from 0 to π .

Fig. 6.35 Absolute value of the phase shift angle |ϕ 0 | vs. |u|

Figure 6.36 represents the parameter a as a function of |u| in accordance with equation (6.186), the right-hand side of which is an odd function of u. From the cited equation, it follows that u and a coincide in sign for Δ > 0 and differ in sign for Δ < 0. Using this observation and relation (6.178), we obtain the expression

224


V=

√

2 |u| a signΔ .

(6.191)

According to this expression, for Δ > 0, which corresponds to the post-resonant excitation mode, the steady-state velocity V is positive. This means that the system shown in Fig. 6.34 moves to the right when the exciters rotate counterclockwise. For the pre-resonant mode (Δ < 0), the system moves to the left.

Fig. 6.36 Dependence a vs. |u|

Figures 6.35 and 6.36 allow the steady-state velocity V to be calculated for a given ϕ0 . To that end, one should first calculate |u| using Fig. 6.35. After that, Fig. 6.36 should be utilized to calculate a for the resulting |u|. Finally, V is calculated in accordance with equation (6.191). We performed the calculation for ϕ0 = π2 and ϕ0 = π . For ϕ0 = π2 , the calculation yields the dimensionless values |u| ≈ 0.15 ,

a ≈ 0.33 ,

V ≈ 0.07 .

(6.192)

In dimensional units,

m . s For ϕ0 = π , the calculation results in the dimensionless values V ≈ 0.02

|u| ≈ 0.24 ,

a ≈ 0.38 ,

V ≈ 0.13 ,

(6.193)

(6.194)

which in dimensional units corresponds to V ≈ 0.05

m . s

(6.195)


225

We also calculated the motion of the system by integrating the exact equations of motion (6.151) subject to zero initial conditions x1 (0) = 0 ,

x2 (0) = 0 ,

x˙1 (0) = 0 ,

x˙2 (0) = 0 .

(6.196)

Figures 6.37 and 6.38 show the dependence of the dimensionless variable V = and coordinates x1 , x2 on time t for ϕ0 = π2 and ϕ0 = π , respectively.

x˙1 +x˙2 2

Fig. 6.37 Dependence of dimensionless V (left) and x1 , x2 (right) on time t for the phase shift ϕ0 = π2

Fig. 6.38 Dependence of dimensionless V (left) and x1 , x2 (right) on time t for the phase shift ϕ0 = π

It is apparent from these plots that after a transient period, the motion converges to the mode with an almost-constant average value of the variable V , equal approximately to 0.1 for ϕ0 = π2 and 0.15 for ϕ0 = π , which is in good agreement with the values of equation (6.192) and (6.194) obtained based on the averaged equations.

226


6.6 Worm-like Locomotion Based on Viscous Friction From the mechanical point of view, it is interesting to compare locomotion systems under the action of dry and viscous friction. Also looking at some applications, such as mobile robots moving in fluid-filled tubes for the realization of inspection tasks, it makes sense to discuss the problem of locomotion based on viscous friction. Remark 6.3 The authors are aware that, although these problems are being investigated under a chapter heading of “Worm-like Locomotion Systems – Crawling”, the actual motion in this context is closer to “swimming” than to “crawling”. The main difference between viscous and dry friction is that the function characterizing the dependence of the viscous friction force Ff r is continuous at the point v = 0, although not necessarily smooth. We will consider here the so-called asymmetric viscous friction defined by the relation: Ff r (v) = −d(v) v ,

(6.197)

where the coefficient of viscous friction d(v) depends on the direction of motion, see also Fig. 2.20: d− , if v < 0 , (6.198) d(v) = d+ , if v > 0 . The value d(0) can be chosen arbitrarily since Ff r (0) = 0 for any d. It is important to note that the function Ff r (v) that defines the function for asymmetric linear friction is nonlinear (piecewise linear). To be specific, we assume that the coefficient of friction for the forward motion (v > 0) does not exceed the coefficient of friction for the backward motion (v < 0), i.e., d− ≥ d+ ≥ 0 ,

d− = 0 .

(6.199)

We will consider several models of objects, the motion of which is influenced by a viscous friction force defined by expression (6.197).

6.6.1 Two Mass Points Subjected to a Kinematic Constraint 6.6.1.1 Statement of the Problem Consider a system of two mass points moving along the same straight line Ox, see Fig. 6.39. Let m denote the mass of each mass point and let x1 and x2 denote the coordinates of the mass points.

6.6 Worm-like Locomotion Based on Viscous Friction

0

227

m

m

x1

x2

x

Fig. 6.39 Two mass points with kinematic constraints

The mass points are subjected to a kinematic constraint that specifies the time history of the distance 2l between the mass points, i.e., x2 − x1 = 2 l(t) .

(6.200)

We assume the quantity l(t) changes harmonically: l(t) = l0 + b sin ω t .

(6.201)

Here, l0 is the distance from the center of mass of the system to the center of mass of each of the mass points at the initial time instant, b is the amplitude, and ω is the frequency of the variation of the distance. The mass points are acted upon by the viscous friction forces F(x˙1 ) and F(x˙2 ), respectively, defined by relations (6.197) and (6.198). The law of motion of the center of mass for the system under consideration gives (6.202) 2 m x¨C = F(x˙1 ) + F(x˙2 ) , where xC = 12 (x1 + x2 ) is the coordinate of the center of mass of the system. Let V denote the velocity of the center of mass of the system, i.e., V = 12 (x˙1 + x˙2 ). Then, with reference to the constraint equation (6.200), the equation of motion (6.202) can be represented as ˙ + F(V + l) ˙. 2 m V˙ = F(V − l) (6.203) Introducing the dimensionless variables (denoted by an asterisk), we obtain t∗ = ω t ,

V∗ =

V , bω

d ∗ (V ∗ ) =

d(V ) . d−

(6.204)

In terms of the dimensionless variables, equation (6.203), with reference to relation (6.201), becomes " 1 d− ! d(V − cos t) · (V − cos t) + d(V + cos t) · (V + cos t) . V˙ = − 2 mω

(6.205)

In this equation, we preserve the notation used for the primary dimensional variables for the respective dimensionless quantities. We then subject equation (6.205) to the initial condition

228


V (0) = V0 .

(6.206)

The dimensionless coefficient of friction d(V ), in accordance with (6.198), (6.199), and (6.204), takes the form 1 , if V < 0 , (6.207) d(V ) = κ , if V > 0 , where κ =

d+ d− ,

0 ≤ κ ≤ 1. Note that

ε=

d− . mω

(6.208)

To clarify the physical meaning of the parameter ε , we represent it as ε = md−bbωω2 . The product bω , occurring in the numerator, is the maximum magnitude of the velocity of the motion of a mass point about the system’s center of mass and, thus, the numerator characterizes the maximum magnitude of the viscous friction force acting on the mass point. The product b ω 2 in the denominator is the maximum magnitude of the motion acceleration of a mass point relative to the center of mass of the system, and the denominator itself characterizes the maximum magnitude of the excitation force. Therefore, the parameter ε characterizes the ratio of the maximum magnitude of the viscous friction force to the maximum excitation force acting on each of the mass points in the motion relative to the center of mass. In what follows, we assume the parameter ε to be small, i.e., ε 1. The parameter ε may be small even for fairly large values of the viscous friction coefficient d− , if, for example, the frequency ω is high. This is an additional support for considering the case of small ε .

6.6.1.2 Asymptotic Approximation Equation (6.205) subject to the initial condition (6.206) is a standard form equation in terms of the method of averaging (BOGOLJUBOV and MITROPOLSKI, [31]). We apply the procedure of averaging to this equation, i.e., average the right-hand side of equation (6.205) with respect to the fast variable t, thus replacing this equation by 1 1 V˙ = − ε 2 2π

2π

d(V −cos t)·(V −cos t)+d(V +cos t)·(V +cos t) dt . (6.209)

0

We preserve the letter V in the averaged equation to denote the velocity of the center of mass of the system. To calculate the integrals on the right-hand side of equation (6.209), it is necessary to find the time intervals over which the coefficient of viscous friction d is constant. For the first integral


2 π

I1 =

229

d(V − cos t) · (V − cos t) dt

(6.210)

0

it is necessary to determine the intervals over which the expression V − cos t does not change in sign. With reference to relation (6.207), expression (6.210) can be rewritten as

α

I1 =

(V − cos t) dt + κ

0

2

π −α

(V − cos t) dt +

α

2 π

(V − cos t) dt ,

2 π −α

where α = arccosV . The integration leads to the expression ⎧ 2πV , if V < −1 , ⎪ ⎨ √ 2 I1 = 2 π κ V + 2 (1 − κ ) (V arccosV − 1 −V ) , if |V | ≤ 1 , ⎪ ⎩ if V > 1 . 2π κ V , Exercise 6.6. Calculate I2 = 02 π d(V + cos t) · (V + cos t) dt to verify that I2 = I1 . Finally, equation (6.209) is reduced to the equation ⎧ if V < −1 , −ε V , ⎪ ⎪ ⎨ ! " √ 1 V˙ = −ε κ V + π (1 − κ ) (V arccosV − 1 −V 2 ) , if |V | ≤ 1 , ⎪ ⎪ ⎩ −ε κ V , if V > 1 .

(6.211)

subject to the initial condition V (0) = V0 .

6.6.1.3 Steady-State Solution As has already been mentioned, any unsteady motion approaches a steady-state motion as time passes. For that reason, we are interested in steady-state motions, i.e., in motions with a constant velocity. In fact, the velocity of the system will be constant “on average”, since small high-frequency vibrations will be imposed on the constant velocity component. The “average” here refers to the steady-state velocity obtained from the averaged system, rather than the average velocity over the period T , defined by V¯ = T1 0T V (t) dt. Let . 1 (6.212) P(V , κ ) = κ V + (1 − κ ) (V arccosV − 1 −V 2 ) . π Then, the issue of the existence of steady-state solutions of equation (6.211) for κ = 0 is reduced to the issue of the existence of real roots of the equation P(V , κ ) = 0,

230


where P(V , κ ) is defined by expression (6.212). Calculate the derivative PV of the function P(V , κ ) with respect to V for |V | < 1 to obtain PV = κ +

1 (1 − κ ) arccosV > 0 . π

(6.213)

Therefore, for fixed κ , the function P(V , κ ) increases over the interval |V | < 1. In addition, since the values P(−1 , κ ) = −1 and P(0 , κ ) = − π1 (1 − κ ) are negative for 0 < κ < 1, while the value P(1 , κ ) = κ is positive, the equation P(V , κ ) = 0 has a unique root V = Vs for a fixed κ from the interval 0 < κ < 1, and this root lies in the interval 0 < Vs < 1. Now let κ = 1, which corresponds to the symmetric linear (rather than piecewise linear) viscous friction (d− = d+ ). Then, the equation P(V , κ ) = 0 has the unique root Vs = 0, as follows from expression (6.212). This result is quite natural and expected, since in the case of the symmetric friction, preference cannot be given to any of the directions of motion. We now assume that there is no friction for the forward motion (d+ = 0), i.e., κ = 0. In this case, it follows from equation (6.211) that if V0 ≤ 1 in the initial condition (6.206), then the steady-state solution of equation (6.211) is defined by the root of the equation P(V , 0) = 0. In view of condition (6.212), this root is unique and is given by Vs = 1. If V0 > 1, then, as follows from the last condition in equation (6.211), any solution is a steady-state solution to this equation. To investigate the dependence of the steady-state velocity V = Vs on κ , we will use the relation P (Vs , κ ) d Vs . (6.214) = − κ dκ PV (Vs , κ ) We differentiate the function P(V , κ ) of (6.212) with respect to κ to obtain the expression for Pκ for |V | < 1: Pκ =

" . 1! V (π − arccosV ) + 1 −V 2 > 0. π

(6.215)

Based on the equations (6.213) - (6.215), we conclude that ddVκs < 0 and, hence, the value of the steady-state velocity decreases as κ increases. We further investigate the stability of the steady-state solution found. The variational equation for 0 ≤ Vs < 1 has the form

δ V˙ = −PV (Vs , κ ) δ V . Since PV > 0 for 0 ≤ Vs < 1 according to (6.213), the steady-state solution V = Vs is stable. For κ = 0, the stability of the steady-state solution follows from inequality (6.213) for |V | ≤ 1 and, in addition, from the fact that equation (6.211) for V > 1 has the form V˙ = 0. Figure 6.40 shows the result of the numerical solution of the exact equation (6.205) for ε = 0.1 and κ = 0.5. The steady-state solution of the averaged system (6.211) found from the equation P(V, κ ) = 0 is given by Vs = 0.217. It is apparent from Fig. 6.40 that there is rather good agreement between the exact solution and the steady-state solution of the averaged system.


231

Fig. 6.40 Numerical solution of the exact equation of motion

In conclusion, we consider the case where the coefficients of viscous friction d− and d+ are close to each other, i.e., δ = 1 − κ 1. In this case, it is possible to obtain an asymptotic expression for the velocity Vs as a function of κ . To that end, we seek the solution V = Vs of the equation P(V , κ ) = 0 in the form of a series in terms of powers of δ : Vs = Vs0 + δ Vs1 + δ 2 Vs2 + . . . .

(6.216)

By substituting the expansion of (6.216) into expression (6.212) for P(V, κ ) and expanding the functions that occur in this expression into a power series, we find Vs0 = 0, Vs1 = π1 , Vs2 = 21π . Finally, we obtain Vs =

1 (1 − κ ) (3 − κ ) + o(1 − κ )2 . 2π

(6.217)

Exercise 6.7. Perform the necessary calculations to verify equation (6.217). Figure 6.41 shows a plot of the functions Vs (κ ) obtained from the equation P(V , κ ) = 0 (curve 1) and from asymptotic expression (6.217) (curve 2). For 0.6 ≤ κ ≤ 1, the plots almost coincide. Even for κ = 0.5 expression (6.217) gives Vs = 0.199, which is close to the steady-state solution Vs = 0.217 found from the equation P(V, κ ) = 0.

232


Fig. 6.41 Dependence Vs vs. κ

We will now consider a number of other problems that can be reduced to the problem just solved.

6.6.2 A Rigid Body Acted upon by a Periodic Force We consider a rigid body of mass m that moves along a straight line Ox, see Fig. 6.42. The body is acted upon by the force of viscous friction F(v) defined by equation (6.197), and the exciting harmonic force Φ (t) with amplitude B and frequency ω : Φ (t) = B sin ω t .

F (t)

0

m

x

Fig. 6.42 Rigid body with a periodic force

Then, the velocity V of the center of mass is governed by the equation


233

m V˙ = F(V ) + B sin ω t .

(6.218)

Introducing the dimensionless variables (denoted by the asterisk), we obtain t∗ = ω t ,

V∗ =

V , Lω

d ∗ (V ∗ ) =

d(V ) , d−

L=

B . m ω2

In terms of the dimensionless variables, equation (6.218) becomes V˙ = −ε d(V ) + sin t .

(6.219)

The parameter ε = md−ω coincides with that of equation (6.208). This parameter can be represented as ε = md−ω = md−LLωω2 = d−BL ω and, hence, characterizes the ratio of the maximum magnitude of the viscous friction force to the maximum magnitude of the excitation force. We introduce the change of variables as u = V + cos t

(6.220)

u˙ = −ε d(u − cos t) · (u − cos t) .

(6.221)

to represent equation (6.220)

In what follows, we again assume the parameter ε to be small, which means that the force due to viscous friction is small as compared to the excitation force. Equation (6.221) is a standard form equation, to which the procedure of averaging can be applied. The structure of equation (6.221) coincides with that of equation (6.205). The right-hand side of equation (6.205) is multiplied by the coefficient ε2 but contains the sum of two terms that have identical averages. Therefore, one can make use of the result of the averaging of equation (6.205), in which it should be replaced by u, to obtain ⎧ u, if u < −1 , ⎪ ⎨ √ u˙ = −ε κ u + π1 (1 − κ ) (u arccos u − 1 − u2 ) , if |u| ≤ 1 , ⎪ ⎩ κ u, if u > 1 .

(6.222)

All conclusions of the previous section regarding the steady-state solution of equation (6.211) remain valid for equation (6.222). When returning to the variable V in accordance with expression (6.220), we obtain V = us − cos t, where us is the steady-state solution of equation (6.222). Hence, us is the average velocity of the body in the steady-state motion, since the average of the function cos t over a period is equal to zero.

234


6.6.3 A Rigid Body with a Moving Internal Mass Consider a rigid body of mass m0 (the main body) that moves along a straight line Ox. The body is acted upon by the force of viscous friction applied by the environment. The friction is expressed by equation (6.197). Inside the main body, there is an internal body of mass m1 that interacts with the main body and moves relative to it along a straight line parallel to the axis Ox. Introducing the axis Cξ attached to the main body that passes through the center of mass C of this body and is parallel to the axis Ox and denoting the coordinate of the center of mass of the main body relative to a inertial reference frame as x, the coordinate identifying the position of the internal body relative to the main body as ξ , and the coordinate of the center of mass of the system of two bodies relative to the fixed frame as xC , see Fig. 6.43.

m1 C

x

m0

0

x

Fig. 6.43 A rigid body with a moving internal mass

The motion equation of the internal body relative to the main body is assumed to be specified by a function ξ (t). The equation of motion of the center of mass of the system has the form ˙ , (6.223) m x¨C = F(x) where m = m0 +m1 is the total mass of the system. We substitute the expression xC = m0 x+m1 (x+ξ ) for the coordinate of the center of mass of the system into equation m (6.223) to obtain (6.224) m x¨ = F(x) ˙ − m1 ξ¨ . Let the motion equation of the internal body be specified as follows:

ξ (t) = b sin ω t .

(6.225)

x¨ = F(x) ˙ + m1 b ω 2 sin ω t .

(6.226)

Then equation (6.224) becomes

By introducing the notation V = x˙ and B = m1 b ω 2 equation (6.226) is reduced to the relation that coincides with equation (6.218) and governs the motion of a body under the action of a harmonic force.


235

6.6.4 Two Bodies Connected by a Spring Previously, we dealt with single-degree-of-freedom mechanical systems. We now consider a model that represents a mechanical system with two degrees of freedom.

6.6.4.1 Statement of the Problem Let two bodies, each of mass m, move along a straight line Ox, see Fig. 6.44. The bodies are connected by a spring of stiffness c. Each of the bodies is acted upon by the viscous friction force F(v) defined by equation (6.197).

M(t) m

m c

x1

x2

Fig. 6.44 Two bodies connected by a spring

The motion of the system is excited by a harmonic force Φ (t) with amplitude B and frequency ω acting between the bodies:

Φ (t) = B sin ω t . The equations governing the motion are: m x¨1 + c (x1 − x2 ) = −F(x˙1 ) + B sin ω t , m x¨2 + c (x2 − x1 ) = −F(x˙2 ) − B sin ω t ,

(6.227)

where x1 and x2 denote the coordinates of the centers of mass of the bodies relative to a inertial reference frame. Introducing dimensionless variables (again using an asterisk), we obtain / / / xi ∗ d(x˙i ) d− B c m m ∗ ∗ ∗ , d (x˙i ) = ,ε=√ ,β= , , ν =ω xi = , t = t L m d− c mc κ− L c where L is a unit of length. Considerations that suggest the choice of the quantity L are similar to those discussed previously for the case of dry friction. Exercise 6.8. What is the physical meaning of the parameter ε ?

236


In terms of the dimensionless variables, the equations of motion become x¨1 + x1 − x2 = −ε d(x˙1 ) x˙1 + ε β sin ν t , x¨2 + x2 − x1 = −ε d(x˙2 ) x˙2 − ε β sin ν t . Here,

d(x˙i ) =

where κ =

d+ d−

1 , if x˙i < 0 , κ , if x˙i > 0 ,

(6.228)

(6.229)

, 0 ≤ κ ≤ 1 and i = 1 , 2.

6.6.4.2 Asymptotic Approximation In what follows, we assume that ε is a small parameter, while the quantity β is on the order of unity. To enable the method of averaging to be applied to system (6.228), we reduce this system to a standard form. To that end, we introduce the change of variables that has already been utilized: x1 = X − a cos ϕ ,

x2 = X + a cos ϕ , √ √ ˙ X = V , x˙1 = V + a 2 sin ϕ , x˙2 = V − a 2 sin ϕ , √ ϕ = 2t + θ ,

(6.230)

where X is the coordinate of the center of mass, V its velocity, and ϕ , a, θ are functions of time. As was the case previously, we will consider √ the behavior of the system in the neighborhood of the resonance, assuming ν = 2 + ε Δ . For that, we introduce the new slow variable ξ :

ξ = νt −ϕ ,

ξ˙ = ν − ϕ˙ = −θ˙ + ε Δ .

(6.231)

Using relations (6.230) and (6.231), we represent system (6.228) in a standard form: √ √ ε* V˙ = − d (V − a 2 sin ϕ ) · (V − a 2 sin ϕ ) 2 √ √ + + d (V + a 2 sin ϕ ) · (V + a 2 sin ϕ ) , √ √ * ε a˙ = √ sin ϕ d (V − a 2 sin ϕ ) · (V − a 2 sin ϕ ) 2 2 √ √ + − d (V + a 2 sin ϕ ) · (V + a 2 sin ϕ ) + 2 β sin(ξ + ϕ ) , √ √ * ε √ cos ϕ d (V − a 2 sin ϕ ) · (V − a 2 sin ϕ ) 2a 2 √ √ + − d (V + a 2 sin ϕ ) · (V + a 2 sin ϕ ) + 2 β sin(ξ + ϕ ) + ε Δ .

ξ˙ = −

(6.232)


237

Assuming ε 1, we average the right-hand side of system (6.232) with respect to the fast variable ϕ to obtain √ ⎧ −ε a u 2 , u < −1 , ⎪ ⎪ ⎨ ! " √ V˙ = − √ε a u (1 + κ ) − π2 (1 − κ ) (u arcsin u + 1 − u2 ) , |u| ≤ 1 , 2 ⎪ ⎪ √ ⎩ −ε κ a u 2 , u > 1, ⎧ β ε √ ⎪ u < −1 , ⎪ 2 (−a + 2 cos ξ ) , ⎪ ⎪ ! ⎪ √ ⎪ ⎨ ε a (1 − κ ) (arcsin u + u 1 − u2 ) 2 π " a˙ = (1+κ ) β ⎪ √ − , |u| ≤ 1 , a + cos ξ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ β ⎩ ε (−a κ + √ cos ξ ) , u > 1, 2

ε ξ˙ = − 2

(6.233)

2

β √ sin ξ − 2 Δ a 2

,

where u = a V√2 . A similar averaging has already been performed many times previously. We are interested in the solutions that correspond to the motion of the entire system with a constant velocity V . Then, it follows from system (6.233) that the amplitude a and the phase ξ are also constant. The steady-state values of the variables u and a are defined (after eliminating ξ ) from the system of transcendental equations . 2 (1 − κ ) (u arcsin u + 1 − u2 ) = 0 , π π β |u| . a= . 2 2 (1 − κ ) (1 − u2 )3 + 8 π 2 Δ 2 u2

P(u, κ ) = u (1 + κ ) −

(6.234)

Differentiating the function P(u, κ ) of (6.234) with respect to u for |u| < 1 yields Pu = 2 κ +

2 (1 − κ ) arccos u > 0 . π

For a fixed κ , the function P(u , κ ) increases over the interval |u| ≤ 1. It follows from relations (6.234) that the values P(−1 , κ ) = −2 and P(0 , κ ) = − π2 (1 − κ ) are negative for 0 < κ < 1, while P(1 , κ ) = 2 κ is positive for these values of κ . Hence, equation P(u , κ ) = 0 has a unique root u = us , 0 < us < 1, see Fig. 6.45 (curve 1). For κ = 1, the equation P(u, κ ) has the unique root u = 0.

238


S

1 2

Fig. 6.45 Dependence of us on κ

The dependence us = us (κ ) derived from the first equation of (6.234) is shown in Fig. 6.45 (curve 1). Based on this us , we determine the stationary amplitude a from the second equation of (6.234). The dependence as = as (us ) is presented in Fig. 6.46. first find us = 0.22 from (6.234), and then as = as (us ). This In the case κ = 0.5 we √ yields Vs = 0.33 · 0.22 · 2 = 0.10.

Fig. 6.46 Dependence of as on u

Thus, having found a unique value u = us from the first equation of (6.234), we use the second equation to determine the amplitude a = as and then find a unique √ value of the steady-state velocity Vs = as us 2.


239

Figure 6.47 shows the results of the numerical solution of the exact equation (6.228) for ε = 0.1, κ = 0.5, Δ = 1, and β = 1.

Fig. 6.47 Results of the numerical solution of the exact equation

Figure 6.47 shows a good agreement of the stationary velocity value Vs = 0.10 (from the averaged system equations) with the numerical solution of the exact equation. Exercise 6.9. Find the equations in variations for system (6.233).

Exercise 6.10. Consider the case (important for practice) where the difference between the coefficients of friction d− and d+ is small, i.e., δ = 1 − κ 1. The result is shown in Fig. 6.45 (curve 2).

6.6.5 System of Three Mass Points with Kinematic Constraints 6.6.5.1 Equations of Motion Now, we consider the motion of a system of three mass points with the coordinates xi (i = 1 , 2 , 3) and equal masses m, connected by kinematic constraints along an axis Ox, see Fig. 6.48.

240


l1 (t) m 0

x1

l2 (t) m

m

x2

x3

x

Fig. 6.48 Chain of three mass points with kinematic constraints

The motion of the system is excited by the kinematic constraints setting the distances l1 (t) and l2 (t) between mass points l1 (t) = l0 + a1 (t), l2 (t) = l0 + a2 (t) ,

(6.235)

where a1 (t) and a2 (t) are periodic functions with period T and a1 (0) = a2 (0) = 0. The type of functions a1 (t) and a2 (t) and the friction force F(vi ), vi = x˙i , i = 1 , 2 , 3, acting on each mass point from the surface will be discussed later. The velocity of the center of mass of the system can be expressed by 1 V = (x˙1 + x˙2 + x˙3 ) . 3 The equation of the motion of the center of mass is: 3 m V˙ = F(x˙1 ) + F(x˙2 ) + F(x˙3 ) .

(6.236)

Here, we have x2 (t) − x1 (t) = l1 (t) ,

x3 (t) − x2 (t) = l2 (t) .

(6.237)

By substituting expressions (6.235) and (6.237) into (6.236), the equation of the motion (6.236) takes the form 1 2 3 m V˙ = F V − a˙1 − a˙2 3 3 1 1 2 1 + F V + a˙1 − a˙2 + F V + a˙1 + a˙2 . (6.238) 3 3 3 3 We assume that in the initial moment t = 0, the velocity of the center of mass V (0) = 0. Introducing dimensionless variables (with asterisks), we now have xi T (i = 1 , 2 , 3) , V ∗ = V , L L L ) F(V a i T a∗i = , F ∗ (V ∗ ) = . L Fs xi∗ =

t∗ =

t , T

(6.239)


241

Here, L is a characteristic linear dimension (for example, the greatest value a1 (t) or a2 (t) in period T ), Fs is a characteristic value of the friction force. Hereafter, we use dimensionless variables. Introducing the dimensionless variables in equation (6.238) and denoting u1 (t) = a˙1 (t), u2 (t) = a˙2 (t), we rewrite equation (6.238) using dimensionless variables (yet keeping the old symbols) dV ε 2 1 = F V − u1 − u2 dt 3 3 3 1 1 2 1 , (6.240) + F V + u1 − u2 + F V + u1 + u2 3 3 3 3 2

sT where ε = FmL . We can now notice that since a1 (t) and a2 (t) are periodic functions with period T , u1 (t) = a˙1 (t) and u2 (t) = a˙2 (t) are also periodic functions with period T and have an average value of zero. Furthermore, we assume everywhere that ε 1. The smallness of the parameter ε shows that the value of the friction force Fs is . Equation (6.240) has small compared to the amplitude of the “driving” force mL T2 a standard form. Averaging the right side of the equation (6.240) relative to the variable t in the period 1, we obtain

ε dV = G(V ) , dt 3

(6.241)

where

1

2 1 1 1 1 2 F V − u1 − u2 + F V + u1 − u2 + F V + u1 + u2 dt. 3 3 3 3 3 3

G(V ) = 0

Now, it is necessary to define the functions u1 (t), u2 (t) and the friction.

6.6.5.2 Smooth Control Let us consider the functions a1 (t) and a2 (t), composed from the parabolas and shown in Fig. 6.49, and also accordingly marked as a solid and as a dotted line.

242


Fig. 6.49 The functions a1 (t) (solid line) and a2 (t) (dashed line)

These functions have continuous derivatives u1 (t) and u2 (t), shown in Fig. 6.50 and also marked as a solid and as a dashed line. The control equations have the form ⎧ 0 , 0 ≤ t ≤ 13 , ⎪ ⎪ ⎪ ⎨ 2 (3t − 1) , 1 < t ≤ 1 , 3 2 u1 (t) = 1 5 ⎪ −2 (3t − 2) , < t ≤ ⎪ 2 6, ⎪ ⎩ 5 6 (t − 1) , 6 < t ≤ 1 . (6.242) ⎧ 6t , 0 ≤ t ≤ 16 , ⎪ ⎪ ⎪ ⎨ −2 (3t − 1) , 1 < t ≤ 1 , 6 2 u2 (t) = 1 2 ⎪ 2 (3t − 2) , < t ≤ ⎪ 2 3, ⎪ ⎩ 2 0, 3 < t ≤ 1.

Fig. 6.50 The functions u1 (t) (solid line) and u2 (t) (dashed line)

These control functions are equal to zero over an interval of length by one time unit relative to each other.

1 3

and shifted


243

6.6.5.3 Motion with Viscous Friction We assume that the force of viscous friction is a power function of velocity, see (2.86). In our case, the expression for the dimensionless friction force is F ∗ (V ∗ ) = −|V ∗ |α signV ∗ .

(6.243)

The value Fs expressed in formulas used for the transition to dimensionless variables α (6.239) is Fs = d TL α , α > 0. Here, d is the coefficient of viscous friction. After substituting expression (6.242) and (6.243) into equation (6.241), we obtain V˙ =

! " ε 1+α 1+α 1+α 4 (1 − 3V ) , − (3V + 2) − 3 |3V | 2 (1 + α ) 31+α

1 2 − 0 and if V ≥ 13 , then V˙ < 0. Consequently, the stationary solution is only possible over the interval − 23 < V < 13 . In a linear viscous friction environment (α = 1) the chain of mass points remains in rest on average. For another α a stationary velocity V = 0 can be found from the equation G1 (V ) = 4 (1 − 3V )1+α − (3V + 2)1+α − 3 |3V |1+α = 0 .

(6.244)

The roots of equation (6.244) can be calculated as G1 − 23 = 4 · 31+α − 3 · 21+α > 0 , G1 (0) = 4 − 21+α , G1 13 = −31+α − 3 < 0 .

(6.245)

From (6.245) it follows that G1 (0) for α > 1 and G1 (0) > 0 for 0 < α < 1. The first derivative of the function G1 (V ) for V = 0 has the form ! " d G1 = −3 (1 + α ) 4 (1 − 3V )α + (3V + 2)α + 3 |3V |α signV . dV Over the interval 0 < V < 13 the derivative of the function G1 (V ) is negative, which means that this function monotonically decreases over this interval. For α > 1 the value G1 (0) is negative and the function G1 (V ) has values of identical signs at the ends of the interval. Thus, the function G1 (V ) has no positive roots for α > 1 in the interval − 23 < V < 13 . For α > 1 at the ends of the interval − 23 < V < 0, the function G1 (V ) has values of different signs. This implies that the function G1 (V ) has at least one negative root for α > 1 on the interval − 23 < V < 13 . The results of the numerical integration of the exact and the averaged equations for the case of viscous friction with α = 2 and ε = 0.3 are shown in Fig. 6.51.

244


Fig. 6.51 Solutions of the exact and the averaged equations, assuming viscous friction

Finally, we compare the results obtained above with the case in which a dry friction force is acting upon each mass point.

6.6.5.4 Comparison between Viscous and Dry Friction We assume that COULOMB dry friction is acting on each mass point i (i = 1 , 2 , 3), following equation (2.89).

l1 (t) m 0

x1

l2 (t) m

m

x2

x3

x

Fig. 6.52 Chain of three mass points with kinematic constraints and COULOMB dry friction

The expression for the friction force in dimensionless variables takes the form (i = 1 , 2 , 3): ⎧ 1 , x˙i < 0 , ⎪ ⎨ (6.246) F(x˙i ) = −μ0 , x˙i = 0 , ⎪ ⎩ −μ , x˙i > 0 . Here, the value Fs in formulas (6.239) is Fs = F− . The value F− is the magnitude of the friction force for motion in the negative direction. We have

μ=

F+ μ+ = ≥ 0, F− μ−

μ0 ∈ [−1 , μ ] .

(6.247)

Notice that for given controls li (t), i = 1 , 2, the velocity of each mass point cannot be equal to zero over a finite time interval. Consequently, the “stick-slip” effect is absent.


245

After substituting expressions (6.242) and (6.246) into equation (6.241), we obtain ⎧ 3 , V ≤ − 23 , ⎪ ⎪ ⎪ ⎪ 1+μ 2 ε ⎨ 2 − μ − 3V 2 , − 3 < V ≤ 0 , dV = (6.248) dt 3⎪ 2 − μ − 6V (1 + μ ), 0 < V ≤ 13 , ⎪ ⎪ ⎪ ⎩ −3 μ , V > 13 . Now, we consider the solution of equation (6.248) with the initial condition V (0) = 0. If μ = 2 (friction in the positive direction is double the friction in the negative direction), the system remains in rest. If μ < 2 the chain moves to the right with the velocity " 2−μ ! 1 − e−2 ε (1+μ )t V= 6 (1 + μ ) and tends to stationary value Vs =

2−μ . 6 (1 + μ )

1 In the case of isotropic friction, we find Vs = 12 . Thus, using these control algorithms, motion is possible in the case of isotropic friction as well as in the direction of the greater friction in the case of non-isotropic friction. If μ > 2 the chain moves to the left with the velocity

V=

" ε 2 (2 − μ ) ! 1 − e− 2 (1+μ )t 3 (1 + μ )

and tends to stationary value Vs =

2 (2 − μ ) . 3 (1 + μ )

The results of the numerical integration of the exact and averaged equations in the case of symmetric friction (μ = 1) and for ε = 0.3 are shown in Fig. 6.53.

V

t Fig. 6.53 Solutions of the exact and the averaged equations, assuming dry friction

246


We can conclude that using periodic control algorithms, the system is able to move in general given isotropic dry friction. Also, motion is possible in the direction of the greater friction in the case of anisotropic dry friction. Otherwise, locomotion is impossible when assuming a linear friction model without a phase shift in the control functions.

Chapter 7

Adaptive Control Approach to Worm-like Locomotion Systems

7.1 Introductory Remarks The main focus of this chapter comprises several control aspects, whereas friction modeling only plays a minor role. The main problem is that the COULOMB friction model in Section 6.3, which is used instead of the theory of spikes in Section 6.2, consists of COULOMB sliding friction and friction of rest. The latter kind of friction causes the whole friction model to become a set-valued function model, see (2.89) in Section 2.4.3.3 or confer survey articles such as [11], [40], [13], or [109]: Ffr

+ m FN

0

v

- m FN

Fig. 7.1 The extended COULOMB friction model (incl. stiction)

In considering this friction model we have to deal with dynamic equations with a discontinuous right-hand side. This leads some authors to the theory of differential inclusions. We do not want to give an introduction into this theory; rather, we want to find an easy way out by focussing on a simple control approach.


247

248

7 Adaptive Control Approach to Worm-like Locomotion Systems

First, we want to consider only COULOMB sliding friction, which is able to be represented as a function, a step function in (2.87). Since this is an initial simplification of friction and friction modeling is of a minor interest in this chapter, we deal with another simplification (additionally looking toward a later mathematical proof and numerical simulation) by approximating the signum function by means of a tanh function, see also Section 2.4.3.4.: Ffr

Ffr

+ m FN

0 - m FN

Fig. 7.2 COULOMB sliding friction

+ m FN

v

0

v

- m FN

Fig. 7.3 Approximation

At the end of this chapter we present some simulations using the friction model given in Section 2.4.3.4, but also approximated. First, though, we introduce the WLLS as a dynamic control system in the next section. Exercise 7.1. Investigate dry COULOMB sliding friction within the context of WLLS with MAPLE and analyze different numerical methods for solving the arising ODEs. Consider the following model:

As mentioned at the beginning of this chapter, COULOMB sliding friction is given by FR1 (v) = −μ |FN | sign(v), (v = 0). This function carries some poten-

7.2 The Worm-like Locomotion System as a Dynamic Control System

249

tial mathematical difficulties (jumps, discontinuities). Consider also two approximations of this friction force model given by FR2 (v) = − π2 |FN | arctan(A v) and FR3 (v) = −|FN | tanh(A v) with A = 105 . • given: g, A, x0 , v0 , μ • numerical values: x0 = 0 m, v0 = 1 ms , μ = 0.1, g = 9.81 sm2 • find: -

the equations of motion in the general form an analytical solution and a sketch of v(t) the classification of the numerical methods for solving ODEs in MAPLE numerical studies of the model with the three friction laws and all numerical methods (particular focus on accuracy and computing time)

7.2 The Worm-like Locomotion System as a Dynamic Control System Based on Section 6.2 we consider a straight-line chain of mass points (a discrete straight worm) connected consecutively by linear elastic elements like springs and viscous damping elements. Interaction with the ground is modeled as an (approximated) asymmetric COULOMB dry friction force (different in magnitude depending on the direction of the motion of each mass point). We do not assume external force inputs. Rather, we consider internal inputs as actuator-like massless linear springs of constant stiffnesses c1 , c2 and controllable original spring lengths l1 (t) , l2 (t), see Fig. 7.4.

Fig. 7.4 WLLS with actuators as controlled original springs lengths

Remark 7.1 For later simulation we restrict the number of mass points here to k = 3, but we point out that the later adaptive control approach is valid for a fixed but arbitrary k ∈ N, see [15] and [22].

250


The dynamics of the worm system are formulated using NEWTONS’s second law (principle of linear momentum, see (2.23)) for each of the mass points. This yields: m1 x¨1 (t) = −c1 x1 (t) − x2 (t) + F1 x˙1 (t) − d1 x˙1 (t) − x˙2 (t) − u1 (t) , m2 x¨2 (t) = −c2 x2 (t) − x3 (t) − c1 x2 (t) − x1 (t) − d2 x˙2 (t) − x˙3 (t) (7.1) −d1 x˙2 (t) − x˙1 (t) + F2 x˙2 (t) + u1 (t) − u2 (t) , m3 x¨3 (t) = −c2 x3 (t) − x2 (t) + F3 x˙3 (t) − d2 x˙3 (t) − x˙2 (t) + u2 (t) , with x1 (0) = x10 , x˙1 (0) = x11 , x2 (0) = x20 , x˙2 (0) = x21 , x3 (0) = x30 and x˙3 (0) = x31 (all initial values are real numbers). Setting u1 = c1 l1

and

u2 = c2 l2 ,

(7.2)

ui is in fact a control of the original spring length. We deal with a WLLS as a dynamic control system. The outputs of this system could be the actual distances of the mass points y1 = x2 − x1

and

y2 = x3 − x2 .

(7.3)

Fi are the modeled (approximated) dry friction forces. System (7.1) has the following properties: • • • • •

multi-input u(t), multi-output y(t) control system, continuous, strict relative degree of two, nonlinearly perturbed because of the functions Fi (xi (t)), the spectrum of the “high-frequency gain” C A B lies in the open right-half complex plane, and • minimum phase provided Fi (xi (t)) ≡ 0. Exercise 7.2. Rewrite system (7.1) using a matrix-vector representation with matrices A, B, and C and nonlinearity vector F. Then, prove that the given properties are fulfilled, especially the ones concerning assertions about the matrices.

The goal (control objective) is now to simply control this system in order to track a given reference trajectory (e.g., a kinematic gait presented in Subsection 6.2) for the outputs in order to achieve movement of the system. If all parameters are known exactly (worm system parameter, friction force, actuator data), then we are able to derive an actuator input ui (t) which can control the system in such a way as to track a preferred motion pattern constructed in kinematic theory, see Subsection 6.2.1. However, as a rule, the actuator data are not known exactly. In rough terrain unknown or changing friction coefficients can lead to uncertain systems. Then, an adaptive control scheme is required that is able to, despite

7.3 Adaptive High-Gain Control

251

this drawback, at least approximately track a kinematic gait. The goal then is to design a controller that generates the necessary input forces on its own to track a prescribed kinematic gait. This will lead us to an adaptive controller (learning controller), presented in the next section.

7.3 Adaptive High-Gain Control 7.3.1 Motivation and History A typical problem in control theory is to design a controller for an exactly known system, such that the feedback system achieves a pre-specified control objective. The essential difference between this approach and that of adaptive control (discussed here) is that the system is not known exactly. Many actual running systems show a tremendous lack of knowledge of their system parameters. Moreover, these systems are exposed to different disturbances (for examples in WLLS and their environments, see [22] and for other biologically inspired sensors excited externally from the ground, see [17]). Only structural properties such as relative degree or minimum phase condition are available. For those reasons these configurations must be dealt with as uncertain systems. Basically, two different approaches exist: adaptive control and robust control. This book focuses on the former. ´ SASTRY, IOANNOU) deals Adaptive control (e.g., see books by KOKOTOVIC, with the problem of designing a controller (consisting of feedback laws and parameter adaptation rules) that can be used to control any system belonging to a certain system class. The controller has to be designed in such a way that it learns from the behavior of the system (e.g., by using measured output values) and adjusts its ¨ [12]. ˚ parameters based on this information, for a survey we refer to ASTR OM We will focus on a special field of adaptive control wherein no parameter estimation and parameter identification algorithms are considered, the development of which began toward the end of the 1970s. The work of both MORSE and WILLEMS & BYRNES in the beginning of the 1980s encompassed the study of adaptive controllers for dynamic systems whose adaptation strategy did not invoke any identification mechanisms mentioned above. Further, we focus here on the “non-identifierbased” controllers as well. We therefore do not attempt to identify or estimate certain parameters of the system. Instead, we design adaptive controllers that are not based on any parameter identification or estimation algorithms. The objective is not to obtain information about the system, but simply to control the unknown, uncertain system with a simple adaptive controller, in which we use the high-gain property of the system, see [71], [72] and [124]. In particular, the first adaptive controller not based on parameter identification and being useful for certain SISO systems was presented by FEUER & MORSE in 1978 [48]. The first very simple adaptive controller using the high-gain ap-

252


proach goes back to MORSE [100] and WILLEMS & BYRNES [155]. A simple gain adaptation was introduced by ILCHMANN & RYAN [73] and extended by ¨ ALLGOWER & ILCHMANN [5]. This approach has been successful for many applications, for example: • the control of chemical reactions, see [6], [5], [38] or [74]; • the tracking of favorable kinematic gaits for WLLS (this chapter, [22]); and • the application in a biologically inspired sensor for adaptive compensations of unknown ground excitations [17]. Remark 7.2 In this book we have already presented a high-gain output feedback result (see Example 3.4), but the control strategy need not to be an adaptive one, since the system data were known a priori and the controller was designed on the basis of that system information. For example, the gain parameter must always be larger than a constant value k∗ , which is determined using the system parameters. But, if these parameters are not known at all or are known insufficiently, a suitable value for k cannot be chosen. Therefore, an adaptive solution must be used when systems such as this are under consideration. We follow the high-gain approach in choosing a time-varying gain parameter in contrast to the previous constant one. This new parameter can be calculated using an adaptation law. The adaptation of the high-gain parameter k allows the system to approach a necessary value (large enough or growing sufficiently large to provide high-gain control) depending on some of the measured values of the system. Since the calculation of the sufficiently large gain parameter includes values of the system output, the controller must adjust its gain parameter relative to the system parameters - it learns from the system behavior. The system reacts to disturbances from the environment, reflected in the system’s output, and the adaptation law requires this output information in order to calculate the gain parameter. Both the gain parameter and the output are then fed back into the system according to the feedback laws.

7.3.2 Control Objective and Adaptation Law Recalling the goal as formulated above, the control objective is to track given reference trajectories for the outputs of a system in order to achieve desired motion. Here, we do not focus on exact tracking because we are dealing with nonlinear systems, in which we interpret this nonlinearity as a disturbance - we are tolerant of reasonable tracking errors. We instead pay particular attention to the λ -tracking control objective, which is to determine a universal adaptive controller to track a given reference trajectory with a prescribed accuracy λ . More precisely, given λ > 0 a control strategyy → u is sought which, when applied to the system, achieves λ -tracking for every reference signal yref (t), or to put it another way: (i) every solution of the closed-loop system is defined and bounded for all t ≥ 0, and


253

(ii)the output y(t) tracks yref (t) with asymptotic accuracy quantified by λ > 0 in the sense that max{0, y(t) −yref (t) − λ } → 0 as t → ∞, where · denotes the EUCLIDean norm. The last condition means that the errore(t) =y(t) −yref (t) is forced, via an adaptive feedback mechanism, towards a tube around the point zero and with an arbitrarily small, pre-specified radius λ > 0, see Fig. 7.5.

Fig. 7.5 The λ -tube along the reference signal

The simple control objective of λ -tracking (inexact tracking with prescribed tolerance) and the non-identifier-based approach of adaptive control allows us to construct very simple gain adaptation laws, see [73], which can be applied to each system because the gain adaptation law does not depend on the relative degree of a system. Remark 7.3 However, we will later focus on the relative degree of the WLLS to be controlled while designing appropriate feedback laws. Let us consider the following gain adaptation law for λ -tracking: 4 2 4 ˙ = max 0, 4e(t)4 − λ , k(t)

k(0) = k0 ,

(7.4)

where e(t) = y(t) −yref (t) is the error between the system’s output and the desired reference signal. This 4 gain4 adaptation law uses a dead zone as follows: If the norm of the tracking error 4e(t)4 is greater than λ , the gain k(t) has to increase by a power of 2; if the norm is smaller than λ , i.e., the output is close to the reference signal, the gain must remain constant. This adaptation exhibits one crucial characteristic. Using (7.4) we can derive

t

k(t) = k(0) + 0

2 4 4 max 0, 4e(s)4 − λ d s ,

(7.5)

254


and the gain parameter will only be monotonically increasing (possibly unbounded), which is not desirable in some applications. Remark 7.4 Assertion (i) of λ -tracking (boundedness of the solution) is of special interest. It sometimes happens that the solution of the closed-loop system (via an adaptive control strategy) is not bounded for all t ≥ 0. This often occurs in connection with the design of the gain adaptation parameter. If a solution is not bounded for all t ≥ 0, then it exhibits a finite escape time. The following example should illustrate this. Consider the initial value problem x˙ = x2 , x(0) = 1. Separation of variables yields 1 , which is defined over (−∞, 1) and maps to [0, ∞): the solution x(t) = 1−t

Fig. 7.6 Solution xmax with finite escape time

Figure 7.6 clearly shows the finite escape time of the solution.

Exercise 7.3. Consider the system y˙ = y1+ε + u ,

y(0) = y0 ∈ R

with ε > 0 and the feedback u = −k y k˙ = y2 , k(0) = k0 ∈ R .

#

Find values for y0 and k0 such that the closed-loop system has a finite escape time. Graph the solution (y , k) for different initial values using MAPLE or MATLAB.


255

7.3.3 Feedback and Controllers for Systems with Different Relative Degrees Now, the only problem remaining is to find an appropriate feedback law that is dependent on the relative degree of a system. This section will cover several different feedback laws. Feedback for systems with a relative degree of 1: The simplest feedback scheme for control systems with a relative degree one is u(t) = −k(t)e(t), see [71], but we will not focus on this type since WLLS have a relative degree two. Feedback for systems with relative degree 2: Control systems with relative degree two require adaptive controllers, in which ˙ the feedback law consists of the system output y(t) and its derivative y(t), see [132] for an example. In any case the parameter k(t) can be determined using adaptation equation (7.4). • The first controller uses the derivative of the output, see [71] (SISO) and [22] (MIMO): d k(t)e(t) . u(t) = − k(t)e(t) + dt

(7.6)

• The second one includes a dynamic compensator similar to the controller developed by MILLER & DAVISON in [96]. This is done to avoid the drawbacks from ˙ the explicit use of the derivative of the error e(t). The controller has the form: # u(t) = −k(t) θ (t) − ddt k(t) θ (t) , (7.7) θ˙ (t) = −k(t)2 θ (t) + k(t)2e(t) . • A third type, which also avoids the usage of the derivative and which has fewer dimensions (the number of variables used in the internal differential equations), is presented in [15] and applied in [16]: u(t) = −k(t)e(t).

(7.8)

Actually, this controller is for systems with a relative degree one. However, it can still be utilized for systems relative degree two with certain restrictions, as explained in [15]. Summarizing, the following two adaptive λ -tracking control strategies arise: ⎫ e(t) = y(t) −yref (t) , ⎪ ⎪ ⎪ ⎬ u(t) = − k(t)e(t) + ddt k(t)e(t) , (7.9) ⎪ 2 4 ⎪ 4 ⎪ ⎭ ˙ = max 0 , 4e(t)4 − λ , k(t)

256


with k(0) = k0 ∈ R, λ > 0, u(t) ,e(t) ∈ Rm and k(t) ∈ R, and ⎫ e(t) = y(t) −yref (t) , ⎪ ⎪ ⎪ ⎪ d ⎪ u(t) = −k(t) θ (t) − d t k(t) θ (t) , ⎪ ⎬ θ˙ (t) = −k(t)2 θ (t) + k(t)2e(t) , ⎪ ⎪ ⎪ ⎪ 2 4 ⎪ 4 ⎪ ⎭ ˙ = max 0 , 4e(t)4 − λ , k(t)

(7.10)

with θ (0) = θ0 , k(0) = k0 > 0, λ > 0, u(t), e(t), θ (t) ∈ Rm and k(t) ∈ R. The reference signal yref (t) belongs to the set of differentiable and bounded functions with continuous and bounded derivatives up to the second order. Exercise 7.4. Provide a proof that controller (7.10) does not invoke any derivatives. The structure of the feedback law and the simple adaptation law of the controllers already exist in the literature, but they were only applied to systems with a relative degree one. The WLLS under consideration has a relative degree two (see (7.1) and the direct influence of the control input on the second derivative of the output). Therefore, the novelty is the application of the controller in these systems with relative degree two, [47], [164], [96]. Both controllers are simple in their design, rely only on structural properties of the system (and not on the system’s parameters), and do not invoke any estimation or identification mechanism (our goal is not to identify worm system parameters, simply to control this system). The controller consists only of a feedback strategy and a simple parameter adaptation law. Both controllers applied to the presented WLLS (7.1) achieve λ -tracking of kinematic gaits to achieve movement of the WLLS without knowledge of system parameters. We refer to [15] for the mathematical proofs, with further support by the simulations in the next section.

7.4 Simulations In this section we apply the simple adaptive λ -tracking control strategies presented earlier to our WLLS in order to track the desired reference signals. The numerical simulations will demonstrate and illustrate that the adaptive controllers work successfully and effectively. In the following simulations we consider system (7.1) with (7.9) and (7.10), respectively. The arising closed-loop systems are integrated by means of MATLAB. We use the values of non-dimensional variables and parameters according to the procedure presented in Section 3.3.3. For convenience the old notations are used to denote these non-dimensional variables.

7.4 Simulations

257

7.4.1 Simulation using a 2-D COULOMB Model Because some tracking results already exist in the literature (see [15], [22], [24]), we choose the same (dimensionless) parameters as in those sources to obtain comparable simulations of both controllers.

7.4.1.1 Simulation Data We use the following data for all simulations. Any additional data needed for simulations shall be given on the spot: • worm system: m1 = m2 = m3 = 1, c1 = c2 = 10, d1 = d2 = 5; • controller: k(0) = 0, λ = 0.2, θ (0) = 0, γ = 100 (a tuning parameter multiplied with the right-hand side of the adaptation function); • friction data: sign(v) ≈ tanh(100 v) and F+i = 1, F−i = 10, i = 1 , 2 , 3 for the approximated COULOMB sliding friction. We point out that the adaptive nature of the controllers is expressed by the arbitrary choice of the system parameters. Obviously, numerical simulation needs fixed (and known) system data, but the controllers adjust their gain parameters to each set of system data.

7.4.1.2 Tracking of a Time-Shifted Sine Signal In this first simulation section we try to track a time-shifted sin(t)-signal 2 + sin(t) yref (t) = 2 + sin(t + 2) by using controllers (7.9) and then (7.10). The results of the simulation of (7.9) are shown in the following plots.

Fig. 7.7 Output and λ -strip

Fig. 7.8 Gain parameter k(t)

(7.11)

258


u

u

Fig. 7.9 Necessary control inputs

Fig. 7.10 Motion of the worm

Figure 7.7 shows the outputs of the system and the corresponding λ -strips. The reference signal is tracked very quickly with this controller. Figure 7.8 shows the convergence of the gain parameter, Fig. 7.9 the necessary control inputs, and Fig. 7.10 the corresponding motion of the worm. The simulation of the system using (7.10) (see also [24]) is shown in the following diagrams:



u

u



7.4 Simulations

259

Figure 7.11 shows the outputs of the system and the corresponding λ -strips. The reference signal is not tracked very quickly in comparison to controller (7.9). In Fig. 7.11 the outputs have not yet been captured. The gain parameter, shown in Fig. 7.12, increases as long as the outputs are outside the λ -strips. Figure 7.13 shows the necessary control inputs and Fig. 7.14 the corresponding motion of the worm. It can be seen that controller (7.9) works more effectively than controller (7.10) because more information is fed back regarding the output derivative than in (7.10), which has to estimate the derivative. Hence, in the simulation with controller (7.10), the outputs are not captured within the specified time interval and the gain parameter is still increasing. Figure 7.8 clearly shows the convergence of the gain parameter in the simulation with controller (7.9). Exercise 7.5. Figure 7.8 shows the monotonic increase of k(t) towards a limit k∞ . But, if some perturbation were to repeatedly cause the output to leave the λ -strip, then k(t) would repeatedly take larger values because the chosen adaptation law causes only monotonic increases in k (7.5). Therefore, develop improved adaptation laws, using for instance [18] and [19] for assistance as well as [20] for direct applications to WLLS.

7.4.1.3 Tracking an “Optimal” Kinematic Gait Let us consider the “fast” gait in [137] (reference signal), which for t ∈ [0, 1] is: ⎛⎧ l ε cos(3π t) + l0 − l0 ε ⎪ ⎨ 0 ⎜ ⎜ l0 − 2 l0 ε ⎜⎪ ⎜⎩ yref (t) = ⎜ −l0 ε cos(3π t) + l0 − l0 ε ⎜ ⎜ ⎝ −l0 ε cos(3π t) + l0 + l0 ε l0

, t ∈ [0, 13 )

⎞

⎟ , t ∈ [ 13 , 23 ) ⎟ ⎟ , t ∈ [ 23 , 1) ⎟ ⎟, ⎟ 2 ⎟ , t ∈ [0, 3 ) ⎠

(7.12)

, t ∈ [ 23 , 1)

where l0 is the original length (dimensionless, chosen as 2 units) and 2 ε = 0.3 is the elongation. This gait is periodically repeated. Again, we apply both λ -trackers and receive the following simulations. Simulation using controller (7.9):

260




u

u



Simulation using controller (7.10):



7.4 Simulations

261

u

u



Figures 7.15 and 7.19 show the outputs of the respective systems and the corresponding λ -strips. The reference signal is tracked very quickly with controller (7.9) as well, whereas in Fig. 7.19 the outputs have not been captured yet. The gain parameters, shown in Figs. 7.16 and 7.20, increase as long as the outputs are outside the λ -strips. Figures 7.17 and 7.21 show the necessary control inputs, and Figs. 7.18 and 7.22 the corresponding motions of the worm. We can draw the same conclusion as in the previous subsection: controller (7.9) works more effectively than controller (7.10).

7.4.2 Simulation using a 3-D COULOMB Model In this subsection we focus on the problem of tracking reference signal (7.12) with controller (7.9), which turned out to be the best one when assuming friction model presented in Subsection 2.4.3.4.

7.4.2.1 Simulation Data We choose the same data as in Section 7.4.1.1 with the addition of the following new data required for the “new” friction model (2.95): F−i = 10 ,

F+i = 1 ,

Fs−i = 13 ,

Fs+i = 3 , i = 1, 2, 3 ,

7.4.2.2 Tracking of an “Optimal” Kinematic Gait The simulations produced the following results:

Δ = 10−4 .

262

7 Adaptive Control Approach to Worm-like Locomotion Systems u

u



Comparing the worm motion resulting from the old and new friction models yields the following data:

Fig. 7.25 Motion of the worm with old fric- Fig. 7.26 Motion of the worm with new friction tion

The system moves significantly faster because the worm as affected by the new friction model is not able to slide backward as easily (because of the stiction present in the new model but absent in the old one). This observation is reflected in the velocities as well:

7.5 3-D Animations

Fig. 7.27 Velocities with old friction.

263

Fig. 7.28 Velocities with new friction.

Negative velocities occur far less often, which tells us that worm is not sliding backwards as much. This means that the forward progress of the worm is quicker under the new friction model.

7.5 3-D Animations The software package SIMULINK , also from “The MathWorks”, allows the user to make an animation of simulation data using the “virtual reality toolbox”. The following graphics show a few screenshots from the animation of the first worm movement in Subsection 7.4.1.3.

Fig. 7.29 Starting position

Fig. 7.30 Moving

264

Fig. 7.31 Contraction


Fig. 7.32 Extension

The following two figures show screenshots from the animation of an artificial worm consisting of 10 mass points. One can clearly see a traveling extension wave from back to front.

Fig. 7.33 Shot 1

Fig. 7.34 Shot 2

Exercise 7.6. Derive the equations of motion of an WLLS consisting of 10 mass points and do further simulations using MATLAB.

Exercise 7.7. Design kinematic gaits of the WLLS in (7.1) and track these movements dynamically using the controllers presented (see also Chapter 6).

Chapter 8

Prototypes of Worm-Like Locomotion Systems

8.1 Worm-like Locomotion System with Two Stepping Motors The prototypes presented in this section are designed in order to • justify the above-mentioned theory and • perform experimental tests of new principles of non-pedal locomotion. The prototype shown in Fig. 8.1 consists of two stepping motors and a dummy to produce a three mass point worm system [1]. Each stepping motor can travel separately along a threaded rod in both directions with different controllable velocities to generate l˙i .

Fig. 8.1 Three-point-mass worm system with stepping motors

Additionally, a special bristle structure has been integrated to prevent the point masses from slipping backwards (Fig. 8.2).


265

266

8 Prototypes of Worm-Like Locomotion Systems

Fig. 8.2 Bristle structure

To compare the theoretical results with the experimental data, a laser measuring device (see Fig. 8.3) is used with the help of LabVIEW and a DAQ device to detect the actual motion properties. Figure 8.4 presents the control inputs l˙1 (t) and l˙2 (t) and Fig. 8.5 shows the calculated motion x1 (t) and x2 (t) as well as the measured coordinates for x1 and x2 .

Fig. 8.3 Experimental setup for motion analysis

Fig. 8.4 Control inputs l˙1 (t) and l˙2 (t)

8.2 Locomotion Systems with One Unbalance Rotor System

267

x [cm] 8 7 6 5 4 3 2 1 0 3

6

9

t [s]

Fig. 8.5 Comparison of the theoretical results (solid line) for x1 (t) and x2 (t) with the experimental data (dashed line)

In a further stage of development, a set of wheels with a self-locking mechanism is used instead of the bristle structure to prevent backward motion. This construction nearly perfectly ensures the fulfillment of the differential constraints x˙i ≥ 0.

Fig. 8.6 An alternative version for realizing the differential constraints x˙i ≥ 0: wheels with a selflocking mechanism

8.2 Locomotion Systems with One Unbalance Rotor System The prototype, shown in Fig. 8.7, is a vibration-driven locomotion system with one unbalance rotor drive [67]. This drive system consists of two exciters rotating with the same angular velocity; their centers of rotation are located on one vertical axis. The sum of the horizontal components of the centrifugal forces vanishes due to their opposite directions of rotation. A special bristle structure, which realizes the asymmetric friction, is used similarly to the setup described in Section 8.1.

268


Fig. 8.7 Prototype with one unbalance rotor drive

As a virtual prototyping step the vibration-driven robot was designed and optimized using the MBS software tool alaska.

z y x x2 x3 x1

Fig. 8.8 Computer simulation of the dynamic behavior of the vibration-driven robot with one unbalance rotor system

The simulation results in Fig. 8.8 show that the velocity of the locomotion system’s center of mass is about V ≈ 0.02 ms on “average”, which agrees with the measured data.

8.3 Locomotion Systems with Two Unbalance Rotors This vibration-driven locomotion system consists of two bodies connected by a spring. The motion is excited by two unbalance rotors attached to the respective bodies. Control of the two DC-motors is done using a LabVIEW program running on a notebook with data being transferred via USB to a control unit.

8.4 Vibration-Driven Robot – “MINCH Robot”

269

Fig. 8.9 Vibration-driven locomotion system with two unbalance rotors

The theoretical background of this vibration-driven locomotion system is discussed in detail in Section 6.5.3. The benefit of this developed prototype lies in the exact knowledge of the system parameters, which led to good agreement between • the theoretically obtained results based on analytical methods (method of averaging), • numerical integration of the exact equations of motion, and • measured results in the experiment. The velocity can be controlled by changing the initial value of the phase shift between the rotations of the rotors. The direction of the motion can be changed by varying the difference between the natural frequency of the system and the angular velocities of the rotors in sign (without changing the direction of rotation of the exciters).

8.4 Vibration-Driven Robot – “MINCH Robot” As mentioned in Section 6.1 undulatory locomotion is a basic principle for a WLLS. Only this micro-robot shows this fundamental principle of locomotion in a perfect way. The internal bending vibration of the body is converted into locomotion of the whole system through the interaction between the robot and the environment. Expressed in configuration variables and following [111], the cyclic variation of the shape variables induces the monotonous variation of the position variables.

270


Fig. 8.10 Vibration-driven locomotion system with two unbalance rotors

The prototype “MINCH Robot” consists of a piezo actuator as well as two “legs” and a tail. These legs are not classical (active) legs with drives in the hip and in the knee joint, but rather passive members; they are only called legs for simplification. The undulatory locomotion is produced through transmission of the high frequency vibrations of its body (piezo actuator) on the legs. Thus, complex spatial trajectories of the leg endpoint can be produced. The asymmetry of the two legs plays a substantial role in the control this prototype. This purposefully realized asymmetry between the right and left legs is the cause for a shift of the resonance ranges of the legs, see Fig. 8.11. The weight of the robot is 1.7 g and the maximum velocity is about 0.5 ms .

Fig. 8.11 Amplitude frequency response for describing the control principle of the micro-robot

Thus, the direction of the robot becomes controllable. The “MINCH Robot” realizes the operating principle of the actuator system by mimicking the functionality

8.4 Vibration-Driven Robot – “MINCH Robot”

271

of living organisms performing their movements in the mode of “controlable resonance”. A control mode like this is of great interest for the actuators of microrobots as it possible to use an efficient piezoelectric driver. Accurate synchronization between natural frequencies allows small control actions yield large amplitudes. A micro-robot in controlable resonance mode can adapt its movements quickly to changing environmental conditions by keeping the resonance of compliant elements at the frequencies necessary for the control motions. Two fundamental principles are necessary for undulatory locomotion of the micro-robot: • non-symmetrical contact forces between the legs and the surface and • periodical deformations of the body (or alternatively, a periodic driving force). A realistic simulation of such a system can be obtained by a finite element analysis, see Fig. 8.12.

Fig. 8.12 FEM simulation of the eigenforms of the micro-robot using ANSYS software

Also, on a much higher abstraction level, an investigation of the dynamic behavior of the micro-robot is possible based on a very simple system with three mass points. Figure 8.13 shows the mechanical model with three masses A, B, and D, on which one driving force FD (t) acts. The result of the numerical simulation in form of motion sequences is presented in Fig. 8.14.

272


Fig. 8.13 Simple three-mass-point model of the micro-robot

Fig. 8.14 Computer simulation of the MINCH robot motion on a plane surface

Exercise 8.1. Obtain the equation of motion for the model of the micro-robot, see Fig. 8.13, using the basic principles of dynamics, i.e. the principles of linear and angular momentum. Mass points A and B have the mass M, and mass point D the mass −→ −→ −→ m. The geometric parameters of the robot are |EA| = |EB| = l, |ED| = L and the −→ −→ angle between the segments EA and EB is 2 α . The normal forces on the points A and B vary according to the rules FNA = Aˆ sin ω t and FNB = Bˆ sin(ω t + ϕ0 ); the driving force is FD = Dˆ sin Ω t. Contact with ground at point D is assumed to be frictionless. Hint: Determine the position of the center of mass C first, i.e., find the length −→ a = |EC|. To compare the theoretical with the experimental results, a high-speed camera in connection with the automatic motion tracking software WINanalyze is used to determine the trajectories of the legs endpoint. The results of image processing confirm the theory and the induced oscillations of the piezo actuator leads to spatial bending vibrations in the legs. Therefore, the endpoints of the legs exhibit elliptical trajectories. They are different between the right and left leg because of the asymmetry. As mentioned above, this fact is important for the control of the direction of motion.

8.5 Miniaturized Vibration-Driven Robot with a Piezo Actuator

273 z[mm]

0,2 0,15 0,1 0,05

x [mm]

0 -0,6

-0,4

-0,2

0

-0,05

0,2

0,4

0,6

-0,1 -0,15 -0,2

Fig. 8.15 Trajectory of the left “leg” endpoint in the x − z-plane at the excitation frequency f = 1.85 kHZ

0,4

z [mm]

0,3 0,2 0,1

x [mm] 0 -0,6

-0,4

-0,2

-0,1

0

0,2

0,4

0,6

-0,2 -0,3 -0,4

Fig. 8.16 Trajectory of the right “leg” endpoint in the x − z-plane at the excitation frequency f = 1.85 kHZ

8.5 Miniaturized Vibration-Driven Robot with a Piezo Actuator In the future, miniaturized non-pedal locomotion systems are needed for many practical applications. New methods arising in microsystem technology open the door for new technical realizations in the field of micro-robots. The WLLS presented here is a mobile robot consisting of two small plates with spikes, which create asymmetric friction, see Fig. 8.17. The spikes are special ones, made of fiber-optic materials. In connection with the selected arrangement shown in the figure below, this results in a very large difference between the friction coefficients μ− and μ+ . The plates

274


are connected by a compliant mechanism to amplify the amplitudes of the piezo element by a factor of 10 [154].

Fig. 8.17 Miniaturized prototype of a WLLS with a piezo crystal (view from above – left) and a bristle structure (view from below – right)

The mechanics of this vibration-driven robot was described in Section 6.4.1. It exhibits steady-state motion with constant velocity V , determined from equation (6.25) or in dimensional variables from equation (6.26). For the presented prototype aΩ π F− −F+ . The measured using the formula V = sin · we calculate V = 4.3 cm s 2 2 F− +F+ . value of velocity is V = 4.5 cm s

8.6 Worm-like Locomotion System Based on Smart Materials Worm-like locomotion systems will prove to be an efficient form of locomotion in applications such as the inspection of pipes and sewers. Applications in medical technology are also envisioned. Applications like that create new demands on systems size, drive mechanisms, and operating principles. One of the key criteria is that this type of robot needs to work as autonomously as possible. In order to work efficiently in pipelines or areas that have been hit by an earthquake, the communication and power supply both need to be wireless. The use of smart materials, whose properties can be controlled using electric or magnetic fields, temperature or pH-value-gradients, and the like, present a new solution to this problem. Magnetic fluids, also known as ferrofluids and ferroelastomers, belong to this class of materials. Ferrofluids are suspensions of ferromagnetic particles, typically about 10 nanometers in diameter, in a carrier fluid. Commercial ferrofluids contain magnetite (Fe3 O4 ) particles suspended in fluids such as water, kerosene, or oil. Theoretical and experimental studies have shown that the peristaltic motion (contraction and extension) of a viscous fluid brought about by a wave on the material interface sets up a flow in that liquid. Artificial worms create such waveform surface deformations using ferrofluids in a magnetic field. From a technical point of view, the ferrofluid needs to be encapsulated in an extremely thin membrane in order prevent the membrane from reducing the waveform deformation of the surface and the

8.6 Worm-like Locomotion System Based on Smart Materials

275

transferable forces too much. Therefore, initial applications are based on the use of magnetic elastomers, which are plastics that are solid but elastically deformable. Cylindrically shaped bodies located in a cylindrical channel were used in the experiments. The channel diameter d exceeds the diameter dW of the body. The length of the worm is lW . The magnetic field is created by coils, with the axes of the coils being in the horizontal plane. Here, L is the distance between the axes of the coils, and I is the current in the coils, see Fig. 8.18. The coils are placed at the left and right sides of the channel. The magnetic field is created by three coils simultaneously (for example, coils 6–8 in Fig. 8.18); the axis of the middle coil is the symmetry axis of the magnetic field.

Fig. 8.18 Arrangement of the coils and geometrical parameters

Periodically, the left coil is switched off, and the next coil is switched on, with n as the number of the coil switches per second (i.e., the frequency), making T = 1n the period between change-over of the coils. Currents flowing through the coils are unidirectional. An electromagnetic system such as this forms a traveling magnetic which is a function of x, y, z,t (x is the coordinate along the channel, z field H, is parallel to axis of the coils and y is orthogonal to x and z). It has been shown experimentally that in such a periodic magnetic field the cylindrical, magnetizable body (“worm”) moves along the channel. The direction of the worm locomotion is opposite of the direction of the travelling magnetic field. In the experiment I = 4.6 A, L = 10 mm, d = 11 mm, and the parameters of the “worm” are: YOUNG’s modulus E = 49950 Pa, length lW = 48 mm, diameter dW = 4 mm. The frequency n changes from 5 Hz to 1000 Hz in the experiment. A cycle of body deformation caused by the traveling magnetic field is the process starts and ends when the body is not deformed The cycle covers the time when the travelling magnetic field covers the body, see Fig. 8.19.

276


Fig. 8.19 Magnetizable elastic body (worm) in a travelling magnetic field [175]

The worm velocity depends on the geometrical shape of the deformed body and that of the channel. If n is small enough and the body inertia does not affect the body velocity, the following formula is valid: lW ls − L . (8.1) , tc = (ks + 3)T , ks = v = ks tc ls Here, ls is the segment length (a segment is a part of the deformed body between two neighboring coils), and ks is the number of the segments. The symbol [·] denotes the integral part of a number; tc is the cycle time. The length of the segment may be determined under assumption about its form. The segment form is determined by the elastic and magnetic properties of the body material and the value of the magnetic field. Let us assume that the form of the body segment between two coils is determined by the model of an unextended elastic beam (the bending moment is due to the magnetic forces, assuming that magnetic forces are acting on the ends of the segment). In this case the equation of the central line of the segment is: y = ax3 + bx2 +

dW , 2

(8.2)

W W with a = −2 d−d and b = 3 d−d . For this assumption and for the parameters given L3 L2 above, the length of the segment is equal to 12.5 mm, ks = 4, and the analytically determined body velocity is v = 1.43 · n mm. For n < 100 Hz the theoretical result matches the experimental result. The maximum worm velocity measured in the experiment is v = 7.8 cm s for n = 100 Hz. For n > 700 Hz the prototype does not move because of the large inertial forces.

8.6 Worm-like Locomotion System Based on Smart Materials

277

Exercise 8.2. Let us assume that the segment of the body between two coils has a sinusoidal 1 form. In this case the equation of the central line of the segment is y = 2 (d − πx dW ) sin L . Calculate the worm velocity and compare the result with the model of the elastic beam. For simplification the term “worm” is used for the locomotion system presented in this section. However, it should be noted that the type of locomotion realized with the magnetic elastomer and shown in some sequences in Fig. 8.19 is a snake-like motion called concertina motion.


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162. Witte, H., C. Lesch, H. Preuschoft and C. Loitsch (1995): Die Gangarten der Pferde: Sind Schwingungsmechanismen entscheidend? Teil II: Federschwingungen bestimmen den Trab und den Galopp; Pferdeheilkunde 11(4), pp. 265–272. 163. Witte, H., H. Hoffmann, R. Hackert, C. Schilling, M.S. Fischer and H. Preuschoft (2004): Biomimetic robotics should be based on functional morphology; J. Anat. 204, pp. 331–342. 164. Ye, X. (1999): Universal λ -tracking for nonlinearly-perturbed systems without restrictions on the relative degree; Automatica 35, pp. 109–119. 165. Zabczyk, J. (1992): Mathematical Control Theory: An Introduction; Birkhäuser, Boston. 166. Zajac, F.E. (1989): Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control; Crit. Rev. Biomed. Eng. 17(4), pp. 359–411. 167. Zeidis, I. and K. Zimmermann (2000): Ein mathematisches Modell für peristaltische Bewegung als Grundlage für das Design wurmartiger Mikroroboter; Technische Mechanik, 20(1), pp. 73–80. 168. Zimmermann, K. and I. Zeidis (2007): Worm-like locomotion as a problem of nonlinear dynamics; Journal of Theoretical and Applied Mechanics 45, pp. 179–187. 169. Zimmermann, K. and I. Zeidis (2008): Mobile robots based on vibrational systems and dry friction; Proceedings of the 17th CISM-IFToMM Symposium (RoManSy 17), Tokyo, pp. 51–59. 170. Zimmermann, K., I. Zeidis and M. Pivovarov (2005): Dynamics of a nonlinear oscillator in consideration of non-symmetric coulomb dry friction; 51th EUROMECH Nonlinear Dynamics Conference, Eindhoven, pp. 2333–2340. 171. Zimmermann, K., I. Zeidis and J. Steigenberger (2003): Mathematical model of a worm-like motion systems with finite and infinite degree of freedom; Proceedings of the 14th CISMIFToMM Symposium (RoManSy 14), pp. 507–515. 172. Zimmermann, K., I. Zeidis and J. Steigenberger (2003): On artificial worms as chain of mass points; Proceedings of the 6th Int. Conference of Climbing and Walking Robots (CLAWAR 2003), Catania, Italy, pp. 11-18. 173. Zimmermann, K., I. Zeidis, E. Gerlach and A. Tröbs (2007): Ein Beitrag zur Schwingungsanalyse an Nanopositionier- und Nanomessmaschinen unter der Bercksichtigung von nichtlinearnen Reibungsmodellen - Nichtsymmetrisches Stribeck-Modell; Technische Mechanik 27(2), pp. 81–93. 174. Zimmermann, K., I. Zeidis, M. Pivovarov and K. Abaza (2007): Forced nonlinear oscillator with nonsymmetric dry friction; Archiv Applied Mechanic 77, pp. 353–362. 175. Zimmermann, K., V.A. Naletova, I. Zeidis, V. Böhm and E. Kolev (2006): Modeling of locomotion systems using deformable magnetizable media; Journal of Physics: Condensed Matter 18, pp. 2973–2983. 176. Zimmermann, K., V.A. Naletova, I. Zeidis, V.A. Turkov, E. Kolev and S.A. Kalmykov (2008): Calculation of a magnetizable worm deformation in a magnetic field; Magnetohydrodynamics 44(2), pp. 143–148. 177. Zimmermann, K., V.A. Naletova, I. Zeidis, V.A. Turkov, D.A. Pelevina, V. Böhm and J. Popp (2008): Surface of a magnetic fluid containing magnetizable bodies in an applied uniform magnetic field; Magnetohydrodynamics 44(2), pp. 175–182. 178. Zimmermann, K., V.A. Naletova, I. Zeidis, V.A. Turkov, E. Kolev, M.V. Lukashevitch and G.V. Stepanov (2007): A deformable magnetziable worm in a magnetic field - A prototype of a mobile crawling robot; Journal of Magnetism and Magnetic Materials 311, pp. 450–453. 179. Zimmermann, K., I. Zeidis, J. Steigenberger, C. Behn, V. Böhm, J. Popp, E. Kolev and V.A. Naletova (2007): Worm-like locomotion systems (WLLS) - Theory, control and prototypes; Climbing and Walking Robots - Towards New Applications, Editor: H. Zhang, I-Tech Education & Publishing, Vienna, ISBN 978-3-902613-16-5, pp. 429–456.

Index

λ -tracking, 252 λ -tube, 253 adaptation law, 253 control objective, 252 controller, 255 acceleration, 10, 77, 81, 104, 112 acceleration energy, 79, 82, 83, 111, 114 actual displacement, 19 adaptive control, 251 analytical method, 15 angular acceleration, 82 angular velocity, 10, 82, 96, 103 APPELL’s equations, 29, 84, 111, 115 asymptotic solution, 193 asymptotically stable motion, 51, 205 autonomous system, 50 averaging method, 48, 172, 178, 193, 216, 228, 236 body-fixed coordinate system, 9, 201 bristle structure, 265 center of mass, 15, 76, 92, 108, 112 CHAPLYGIN’s equations, 27, 87 characteristic polynomial, 52, 183, 187 closed-loop control, 63, 103, 104 constrained mechanical system, 8 constraints, 96 differential, 13, 163, 267 geometric, 11, 96 holonomic, 13, 23, 163 kinematic, 191, 192, 213, 227, 239 non-holonomic, 13, 23, 98, 100, 109, 120 rheonomous, 12 scleronomous, 12 unilateral, 215

control input, 62 coordinates cylindrical, 11 generalized, 12, 79, 87, 98, 108, 119, 147, 151 spherical, 11 CORIOLIS acceleration, 80, 81 D’ALEMBERT’s principle, 19 degrees of freedom (DOF), 12, 97 dimensionless variable, 55, 192, 203, 215, 227, 233, 235, 243 disturbed motion, 51 double pendulum, 136 EINSTEIN’s summation convention, 9 embodiment, 156 energy dissipation, 132 error, 104, 253 fast variable, 49, 186, 218, 237 feedback high-gain output, 65 output, 64 state, 64 ferrofluid, 274 finite escape time, 254 force centrifugal, 159 COULOMB friction, 21, 36, 174, 192, 214 dissipative, 21 friction, 34 generalized, 20, 100, 121, 149, 150 gravity, 29, 158 harmonic, 235 NEWTON friction, 54 non-conservative, 143

287

288 nonlinear spring, 185 reaction, 89 spring, 21, 30, 174 static friction, 37 STOKES friction, 21 friction anisotropic, 35, 39, 246 asymmetric, 36, 161 coefficient of kinetic, 37 cone, 37 COULOMB, 36, 171, 174, 192, 200, 201, 214, 247 extended COULOMB, 39 force, 34 graph, 45 isotropic, 34, 246 KARNOPP model, 42 NEWTON, 54 smooth approximation, 43 static, 37 stiction, 37 viscous, 34, 200, 226, 243 FROUDE number, 158 gait, 259 kinematic, 165 HELMHOLTZ-Identities, 20 human walking, 157 HURWITZ criterion, 187 HURWITZ determinant, 53 inertial coordinate system, 9, 11, 96, 107 instantaneous axis of rotation, 10 invariant zeros, 69 inverted double pendulum, 142 inverted pendulum, 139 kinematic tree structure, 8 kinetic energy, 22, 85, 87, 99, 120, 137, 141, 142, 145, 148 knife-edge condition, 106 LAGRANGE’s equations of the 2nd kind, 20, 22, 137, 141, 143, 146, 147, 151, 155 LAGRANGE’s equations with multipliers, 86, 98 Lagrangian multiplier, 24, 100 LAPLACE transformation, 130 locomotion concertina, 277 non-pedal, 265 undulatory, 162 wheeled, 75

Index mass moment of inertia, 17, 76, 99, 112, 136, 139 mathematical pendulum, 134 method of section, 78 minimum-phase condition, 69 multibody system (MBS), 7 definition, 7 natural frequency, 132 non-autonomous system, 50 non-inertial frame, 81, 96 numerical methods, 59 omnidirectional wheel, 4, 74 open-loop control, 63, 103 open-loop force control, 104 oscillation harmonic, 207, 208 period of, 204 passive walker, 30, 143 pelvis, 128 pendulum, 30, 32 perturbation method, 48 phase portrait, 191 physical pendulum, 140 PI-controller, 104 piezo actuator, 270, 272 position vector, 9 post-resonant, 216 potential energy, 20, 137, 141, 143, 145 pre-resonant, 216 principle of angular momentum, 17, 89, 129, 140 principle of linear momentum, 16, 89, 129 product of inertia, 18 reduced length of pendulum, 136 relative acceleration, 80 relative degree, 65, 255 strict, 66 resonance, 132 RoboCup, 4, 95 small-size league, 102 rolling condition, 96, 109 rotation, 10 rotation matrix, 9 self-stabilization, 156 shank, 136, 138, 150 slackness condition, 167 slow motion, 49 slow variable, 49, 177, 186 smart materials, 274

Index spikes, 41, 163 stability condition, 183, 187 stable motion, 51, 199 stance phase, 133, 157 stationary regime, 179, 182, 184, 187 stationary solution, 195, 205 stationary velocity, 173, 184, 190, 198 steady-state motion, 172, 187, 205, 212, 216, 222, 230, 233 STEINER’s theorem, 18 stick-slip effect, 41, 175 stiffness, 30, 157 swing phase, 133, 157 synthetic method, 15, 88 thigh, 136, 138, 150 thorax, 128 transient behavior, 50, 165 translation, 10 trimaran, 118 trunk, 128 two-wheel vehicle, 76

289 unbalance exciter, 208, 213 unbalance rotor, 267, 268 uncertain system, 250 undisturbed motion, 51 unstable motion, 51 velocity, 9, 77, 97, 109, 121 velocity control loop, 104 vibration exciter, 216 virtual displacement, 19, 23, 84, 143, 154 virtual work, 19, 84, 87, 100, 121, 145, 149 VORONETS’ equations, 27, 88, 118, 121 walking machines, 125 wheelchair, 75 worm-like locomotion system (WLLS), 161 λ -tracking, 256 control input, 250, 266 crawling, 226 dynamic control system, 250 internal control input, 249 prototype, 265 vibration-driven, 200, 207, 210, 267, 274

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