Communications and Control Engineering

Transcription

1 Communications and Control Engineering Springer-Verlag London Ltd.

2 Published titles include: Stabilization ofnonlinear Uncertain Systems Miroslav Krstic and Hua Deng Passivity-based Control ofeuzer-lagrange Systems Romeo Ortega, Antonio Loria, Per [ohan Nicklasson and Hebertt Sira-Ramirez Stability and Stabilization ofinfinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics (znd edition) Bernard Brogliato Nonlinear Control Systems II Alberto Isidori Ly-Gain and Passivity Techniques in nonlinear Control Arjan van der Schaft Control oflinear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti Robust and Ho: Control BenM. Chen Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Dissipative Systems Analysis and Control Rogelio Lozano, Bernard Brogliato, Olav Egeland and Bernhard Maschke Control ofcomplex and Uncertain Systems StanislavV. Emelyanov and Sergey K. Korovin Robust Control Design Using H" Methods lan R.Petersen, Valery A. Ugrinovski and Andrey V.Savkin Model Reduction for Control System Design Goro Obinata and Brian D. O. Anderson Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan Positive ID and 2D Systems Tadeusz Kaczorek Identification and Control Using Volterra Models F.J. Doyle III, R.K. Pearson and B.A.Ogunnaike

3 Isabelle Fantoni and Rogelio Lozano Non-linear Control for Underactuated Mechanical Systems With 83 Figures Springer

4 Isabelle Fantoni, Doctor Rogelio Lozano, Professor Heudiasyc, UMR CNRS 6599, Universite de Technologie de Compiegne, BP 20529, Compiegne, France Series Editors E.D. Sontag M. Thoma British Library Cataloguing in Publication Data Fantoni,Isabelle Non-linear control for underactuated mechanical systems. (Communications and control engineering) I.Nonlinear control theory 2.Automatic control 3.Mechatronics 1.Title II.Lozano, R. (Rogelio), '36 ISBN Library ofcongress Cataloging-in-Publication Data Fantoni, Isabelle, Non-linear control for underactuated mechanical systems / Isabelle Fantoni and Rogelio Lozano. p. cm. -- (Communications and control engineering) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Automatic control. 2. Nonlinear theories. 1. Lozano, R. (Rogelio), II. Title. III. Series. TI213.L dc Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. Communications and Control Engineering Series ISSN ISBN Htt:// Springer-V erlag London 2002 Originally published by Springer-Verlag London Limited in 2002 Softcover reprint ofthe hardcover Ist edition 2002 The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by authors Printed and bound at the Cromwell Press, Trowbridge, Wiltshire 69/ Printed on acid-free paper SPIN

5 Preface The purpose of this book is to provide a detailed presentation of the control of underactuated non-linear mechanical systems. Control of underactuated systems is a very popular research field, since there exist many applications of underactuated systems in robotics, marine and aerospace vehicles. Modelling and contral of aseries of well-known examples of underactuated mechanical systems are presented in this book. The total energy of the system and its passivity properties have been extensively used in the contral design. The main goal is the stabilization of controlled dynamical systems by construction of Lyapunov functions. Simulations and real applications illustrate the performance of the algorithms on several experimental platforms. This book is expected to be used by students and researchers in the areas of non-linear contral systems, mechanical systems, rabotics and control of helicopters. The book originates from the Ph.D. thesis prepared by the first author at the University of Technology of Cornpiegne and supervised by the second author. It also contains four chapters of individual contributions on closely related subjects: Chapter 10 (Carlos Aguilar and Rogelio Lozano), Chapter 13 (Juan Carlos Avila-Vilchis, Bernard Brogliato and Rogelio Lozano), Chapter 14 (Robert Mahony and Rogelio Lozano) and Chapter 15 (Robert Mahony, Tarek Hamel, Alejandra Dzul and Rogelio Lozano). It would not have been possible to compile the book without the precious help and the contributions of the following persons that are gratefully acknowledged here: We are specially indebted to Mark W. Spong, with whom we have closely collaborated under an agreement between CNRS and the University of Illinois. He contributed to the material presented in Chapters 5 and 7. v

6 VI We thank Carlos Aguilar, Juan Carlos Avila-Vilchis, Bernard Brogliato, Alejandro Dzul, Tarek Hamel and Robert Mahony for their contributions in Chapters 10 and We are grateful to Anuradha M. Annaswamy, Joaquin Collado, Frederic Mazenc and Kristin Y. Pettersen for fruitful collaboration on the domain of underactuated mechanical systems. We also want to thank D. J. Block from the University of Illinois. The experimental results contained in Chapter 3 and in Chapter 5 wouldn't have been possible without his help. Isabelle Fantoni Rogelio Lozano

7 Contents 1 Introduction Motivation Outline of the book Energy-based control approaches for several underactuated mechanical systems The hovercraft model, the PVTOL aircraft and the helicopter. 8 2 Theoretical preliminaries Lyapunov stability Lyapunov direct method Passivity and dissipativity Stabilization Non-holonomic systems Underactuated systems Homoclinic orbit 19 3 The cart-pole system Introduction Model derivation System model using Newton's second law Euler-Lagrange's equations Passivity of the inverted pendulum Controllability of the linearized model Stabilizing control law The homoclinic orbit Stabilization around the homoclinic orbit Domain of attraction St ability analysis Simulation results 37 VB

8 Vlll CONTENTS 3.8 Experimental results 3.9 Conclusions A convey-crane system Introduction Model Passivity of the system Damping oscillations control law Asymptotic stability analysis Simulation results Concluding remarks 51 5 The pendubot system Introduction System dynamics Equations of motion via Euler-Lagrange formulation Passivity of the pendubot Linearization of the system Control law for the top position The homoclinic orbit Stabilization around the homoclinic orbit Stability analysis Simulation results Experimental results Conclusions The Furuta pendulum Introduction Modeling of the system Energy of the system Euler-Lagrange dynamic equations Passivity properties of the Furuta pendulum Controllability of th e linearized model Stabilization algorithm Stability analysis Simulation results Conclusions

9 CONTENTS IX 7 The reaction wheel pendulum 7.1 Introduction. 7.2 The reaction wheel pendulum Equations of motion Passivity properties of the system Linearization of the system Feedback linearization First energy-based control design 7.4 Second energy-based controller 7.5 Simulation results. 7.6 Conclusions. 7.7 Generalization for Euler-Lagrange systems. 8 The planar flexible-joint robot 8.1 Introduction. 8.2 The two-link planar robot Equations of motion Linearization of the system Passivity of the system Controllaw for the two-iink manipulator Equivalent closed-loop interconnection 8.4 Stability analysis Simulation results. 8.6 The three-link planar robot. 8.7 Controllaw for the three-link robot. 8.8 Stability analysis. 8.9 Simulation results 8.10 Conclusions The PPR planar manipulator Introduction System dynamics Equations of motion via Euler-Lagrange formulation, Passivity properties of the planar PPR manipulator Energy-based stabilizing controllaw Equivalent closed-loop interconnection Convergence and st ability analysis Simulation results Conclusions

10 x CONTENTS 10 The ball and beam acting on the ball Introduction Dynamical model Mechanical properties The controllaw Stabilityanalysis Simulation results Conclusions The hovercraft model Introduction The hovercraft model System model using Newton's second law Euler-Lagrange's equations Controllability of the linearized system Stabilizing controllaw for the velocity Stabilization of the position First approach Second approach Third approach Simulation results Conclusions The PVTOL aircraft Introduction The PVTOL aircraft model Input-output linearization of the system Second stabilization approach Third stabilization algorithm Forwarding control law First step: a Lyapunov function for the altitudeangle (y,b)-subsystem Boundedness of B(t) Second step: forwarding design Third step: last change of coordinates Simulation results Conclusions

11 CONTENTS Xl 13 Helicopter on a platform Introduction General considerations Flight modes Aerodynamic forces and torques Inertia moments and products The general model The helicopter-platforrn model Dissipativity properties of the 3-DOF model Control design Passivity-based control of the rotational part Take-off Altitude control Simulation results Simulation Simulation Conclusions Lagrangian helicopter model Introduction Helicopter model Energy-based control design Analysis and simulations Conclusions Newtonian helicopter model Introduction Modeling a helicopter using Newton's laws New dynamic model for control design Lyapunov-based tracking control design Analysis Simulations Conclusions 278 Bibliography 281 Index 293