Control of ships and underwater vehicles: design for underactuated and nonlinear marine systems [1 ed.] 9781848827301, 184882730X

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Advances in Industrial Control

Other titles published in this series: Digital Controller Implementation and Fragility Robert S.H. Istepanian and James F. Whidborne (Eds.)

Modelling and Control of Mini-Flying Machines Pedro Castillo, Rogelio Lozano and Alejandro Dzul

Optimisation of Industrial Processes at Supervisory Level Doris Sáez, Aldo Cipriano and Andrzej W. Ordys

Ship Motion Control Tristan Perez

Robust Control of Diesel Ship Propulsion Nikolaos Xiros

Hard Disk Drive Servo Systems (2nd Ed.) Ben M. Chen, Tong H. Lee, Kemao Peng and Venkatakrishnan Venkataramanan

Hydraulic Servo-systems Mohieddine Jelali and Andreas Kroll

Measurement, Control, and Communication Using IEEE 1588 John C. Eidson

Model-based Fault Diagnosis in Dynamic Systems Using Identification Techniques Silvio Simani, Cesare Fantuzzi and Ron J. Patton

Piezoelectric Transducers for Vibration Control and Damping S.O. Reza Moheimani and Andrew J. Fleming

Strategies for Feedback Linearisation Freddy Garces, Victor M. Becerra, Chandrasekhar Kambhampati and Kevin Warwick

Manufacturing Systems Control Design Stjepan Bogdan, Frank L. Lewis, Zdenko Kovaìiè and José Mireles Jr.

Robust Autonomous Guidance Alberto Isidori, Lorenzo Marconi and Andrea Serrani Dynamic Modelling of Gas Turbines Gennady G. Kulikov and Haydn A. Thompson (Eds.) Control of Fuel Cell Power Systems Jay T. Pukrushpan, Anna G. Stefanopoulou and Huei Peng Fuzzy Logic, Identification and Predictive Control Jairo Espinosa, Joos Vandewalle and Vincent Wertz Optimal Real-time Control of Sewer Networks Magdalene Marinaki and Markos Papageorgiou Process Modelling for Control Benoît Codrons Computational Intelligence in Time Series Forecasting Ajoy K. Palit and Dobrivoje Popovic

Windup in Control Peter Hippe Nonlinear H2/Hũ Constrained Feedback Control Murad Abu-Khalaf, Jie Huang and Frank L. Lewis Practical Grey-box Process Identification Torsten Bohlin Control of Traffic Systems in Buildings Sandor Markon, Hajime Kita, Hiroshi Kise and Thomas Bartz-Beielstein Wind Turbine Control Systems Fernando D. Bianchi, Hernán De Battista and Ricardo J. Mantz Advanced Fuzzy Logic Technologies in Industrial Applications Ying Bai, Hanqi Zhuang and Dali Wang (Eds.) Practical PID Control Antonio Visioli (continued after Index)

Khac Duc Do • Jie Pan

Control of Ships and Underwater Vehicles Design for Underactuated and Nonlinear Marine Systems

123

Khac Duc Do, PhD Jie Pan, PhD School of Mechanical Engineering The University of Western Australia 35 Stirling Highway Crawley, WA 6009 Australia [email protected] [email protected]

ISSN 1430-9491 ISBN 978-1-84882-729-5 e-ISBN 978-1-84882-730-1 DOI 10.1007/978-1-84882-730-1 Springer Dordrecht Heidelberg London New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009931136 © Springer-Verlag London Limited 2009 FUTABA® is a registered trademark of Futaba Corporation. Futaba Corporation, 629 Oshiba, Mobara, Chiba Prefecture 297-8588, Japan, http://en.futaba.co.jp Maple™ is a product provided by Maplesoft™ a trademark of Waterloo Maple Inc., Waterloo, Ontario, Canada, http://www.maplesoft.com “NI” refers to National Instruments and all of its subsidiaries, business units, and divisions worldwide. LabWindows™ is a trademark of National Instruments. National Instruments Corporation, 11500 N Mopac Expwy, Austin, TX 78759-3504, U.S.A, http://www.ni.com 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. 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. Cover design: eStudioCalamar, Figueres/Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Advances in Industrial Control Series Editors Professor Michael J. Grimble, Professor of Industrial Systems and Director Professor Michael A. Johnson, Professor (Emeritus) of Control Systems and Deputy Director Industrial Control Centre Department of Electronic and Electrical Engineering University of Strathclyde Graham Hills Building 50 George Street Glasgow G1 1QE United Kingdom Series Advisory Board Professor E.F. Camacho Escuela Superior de Ingenieros Universidad de Sevilla Camino de los Descubrimientos s/n 41092 Sevilla Spain Professor S. Engell Lehrstuhl für Anlagensteuerungstechnik Fachbereich Chemietechnik Universität Dortmund 44221 Dortmund Germany Professor G. Goodwin Department of Electrical and Computer Engineering The University of Newcastle Callaghan NSW 2308 Australia Professor T.J. Harris Department of Chemical Engineering Queen’s University Kingston, Ontario K7L 3N6 Canada Professor T.H. Lee Department of Electrical and Computer Engineering National University of Singapore 4 Engineering Drive 3 Singapore 117576

Professor (Emeritus) O.P. Malik Department of Electrical and Computer Engineering University of Calgary 2500, University Drive, NW Calgary, Alberta T2N 1N4 Canada Professor K.-F. Man Electronic Engineering Department City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong Professor G. Olsson Department of Industrial Electrical Engineering and Automation Lund Institute of Technology Box 118 S-221 00 Lund Sweden Professor A. Ray Department of Mechanical Engineering Pennsylvania State University 0329 Reber Building University Park PA 16802 USA Professor D.E. Seborg Chemical Engineering 3335 Engineering II University of California Santa Barbara Santa Barbara CA 93106 USA Doctor K.K. Tan Department of Electrical and Computer Engineering National University of Singapore 4 Engineering Drive 3 Singapore 117576 Professor I. Yamamoto Department of Mechanical Systems and Environmental Engineering The University of Kitakyushu Faculty of Environmental Engineering 1-1, Hibikino,Wakamatsu-ku, Kitakyushu, Fukuoka, 808-0135 Japan

The first author dedicates this book to his parents, Do Thi Duyen and Do Khac Yen, and his little daughter, Do Thu Trang. The second author dedicates this book to his parents, Liyou Pan and Hungxiu Diao.

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. New technology, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies..., new challenges. Much of this development work resides in industrial reports, feasibility study papers, and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. When advances are made in industrial control technology, for example, sensors, actuators, controllers, communications, and computing power, there are at least three possible consequences for system control. The control engineering may decide to use the new hardware to make an existing control system perform better. Another possibility is that the new hardware advances make a proposed but previously impractical control method feasible. Alternatively, it may be necessary to devise a completely new control technique in order to exploit the new hardware. However, in many cases, it is often economically impossible to advance an aspect of the system’s control hardware until there is a major upgrade of the system and the industrial control engineer has to grapple with the limitations of the system as it exists. (This is where control engineering science becomes an art!) Underactuated marine vessels are a case in point. In most configurations, a vessel’s main actuators are propellers and rudders, yet a marine vessel has six degrees of freedom in its motion, and the marine control engineer simply has to work with the control surfaces and sensors available. One area that may advance control performance is the use of better control designs, and recently control engineers have become more interested in what nonlinear control might have to offer. Researchers Khac Duc Do and Jie Pan have published a sequence of journal and conference papers on new control algorithms for underactuated systems. Most of this work has used models of marine systems (surface ships and underwater vehicles), but some of the work has been with other mechanical system models (wheeled mobile robots, vertical take-off and landing (VTOL) aircraft). Now they have taken the opportunity to capture this research and development work in a monograph en-

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titled Control of Ships and Underwater Vehicles: Design for Underactuated and Nonlinear Marine Systems for the Advances in Industrial Control series. This will enable industrial control engineers and control researchers to read and study a systematic presentation of their ideas and work. The book opens with chapters that introduce the appropriate nonlinear control theory and marine vessel models used in the research, proceeds to controller derivation and development, and finally details simulation results for a series of nonlinear control schemes devised for marine vessel control problems like point-to-point navigation and path-following. Also presented are some field results of a laboratory-scale vessel on a local river. In one chapter other applications fields are considered and the nonlinear control results for a simple wheeled robots and a simple VTOL aircraft model are given. Each of the applications chapters contains illustrative simulation studies and control results. The monograph will interest control researchers, graduate students, and industrial control engineers alike, particularly those involved in marine and wheeled robotic motion control problems. The Series Editors, being based in Glasgow, Scotland, have always had an interest in marine control problems and have endeavoured to ensure that the Advances in Industrial Control series has useful volumes from this field of control application. Past volumes have included: Ship Motion Control by Tristan Perez (ISBN 978-1-85233-959-3, 2005), Compressor Surge and Rotating Stall by Jan Tommy Gravdahl and Olav Egeland (ISBN 978-1-85233-067-5, 1998), and Robust Control of Diesel Ship Propulsion by Nikolaos Xiros (ISBN 9781-85233-543-4, 2002), and we are pleased to welcome this new volume, Control of Ships and Underwater Vehicles into the series. Industrial Control Center Glasgow Scotland, UK 2009

M.J. Grimble M.A. Johnson

Preface

Control of ocean vessels including ships and underwater vehicles is an active field due to its theoretical challenges and important applications such as passenger and goods transportation, environmental surveying, undersea cable inspection, offshore oil installations, and many others. Most ocean vessels are underactuated meaning that they have more degrees of freedom to be controlled than the number of independent control inputs. Ships do not usually have an independent sway actuator while for underwater vehicles there are often no independent sway and heave actuators. As a result, motion control of underactuated ocean vessels opened a new territory in applied nonlinear control, and attracted special attention from both marine technology and control engineering communities. If classical motion control systems designed for fully or overactuated vessels are directly used on underactuated vessels, the resulting performance of controlled systems is very poor or control objectives cannot be achieved. For example, the traditional approach, in which a combination of a conventional autopilot and a line-ofsight algorithm is used to steer an underactuated ship from one point to another on a straight line, does not impose on minimizing the lateral distance. Consequently, the shortest traveling distance is not achieved. Another example is that underactuated ocean vessels cannot be stabilized by any time-invariant continuous state feedback controllers although they are open loop controllable. This fact resulted from a direct application of the Brockett necessary condition to feedback stabilization of underactuated ocean vessels. Inspired by progress in the field, we present this monograph on control of underactuated ocean vessels including ships and underwater vehicles to senior and postgraduate students, researchers and practitioners of marine technology, control engineering, mechanical engineering, electrical engineering, and mechatronics. This is the first book in the literature that offers various solutions to advanced feedback control topics of practical importance including stabilization, trajectory-tracking, path-tracking, and path-following for underactuated ocean vessels. In the control development and stability analysis of the controlled systems, practical motivations as well as nontrivial techniques are carefully detailed. The techniques presented in

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the book can be readily applied to other underactuated mechanical systems such as aircraft, spacecraft, mobile robots, and robot arms.

Acknowledgements The first author would like to thank the Rectoral Board and his colleagues at Thai Nguyen University of Technology and the University of Western Australia for providing him with a friendly and efficient working environment during his time of writing this book. He is grateful to H.L. Nguyen for her support and encouragement in different ways. He also thanks Z.P. Jiang for inviting him to Polytechnic University for a period of three months from October 2001 to January 2002. Both authors thank the anonymous reviewers and editorial staff of various journals and conferences for their helpful comments on the authors’ research papers, which are the main source of research and results for this book. The authors’ special thanks go to M.A. Johnson, Joint Editor of the Advances in Industrial Control series, O. Jackson, Editor (Engineering), and A. Bunning, Editorial Assistant (Engineering) of Springer, UK, for their support and constructive comments in the process of publishing the book. The writing of this book was supported in part by the Australian Research Council under grants DP0453294, LP0219249, DP0774645, and DP0988424. Perth, Australia Thai Nguyen, Vietnam Perth, Australia May, 2009

Khac Duc Do Jie Pan

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview of Nonlinear Control Developments . . . . . . . . . . . . . . . . . . 1.2 Difficulties in Control of Underactuated Ocean Vessels . . . . . . . . . . . 1.3 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5

Part I Mathematical Tools 2

Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Lemmas and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Stability of Cascade Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Input-to-state Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Control Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Stabilization Under Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Barbalat-like Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Controllability and Observability of Linear Time-invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Controllability and Observability of Linear Time-varying Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Controllability and Observability of Nonlinear Systems . . . . 2.7.4 Brockett’s Theorem on Feedback Stabilization . . . . . . . . . . . . 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 13 16 18 20 21 23 25 27 27 29 31 34 34

Part II Modeling and Control Properties of Ocean Vessels 3

Modeling of Ocean Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Basic Motion Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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3.3 Modeling of Ocean Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Standard Models for Ocean Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Three Degrees of Freedom Horizontal Model . . . . . . . . . . . . . 3.4.2 Six Degrees of Freedom Model . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 44 45 54 54 57 60

Control Properties and Previous Work on Control of Ocean Vessels . . 4.1 Controllability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Acceleration Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Kinematic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Controllability at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Controllability About a Trajectory . . . . . . . . . . . . . . . . . . . . . . 4.2 Previous Work on Control of Underactuated Ocean Vessels . . . . . . . 4.2.1 Control of Nonholonomic Systems . . . . . . . . . . . . . . . . . . . . . 4.2.2 Control of Underactuated Ships and Underwater Vehicles . . 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 61 62 64 66 68 68 69 86

Part III Control of Underactuated Ships 5

Trajectory-tracking Control of Underactuated Ships . . . . . . . . . . . . . . . 89 5.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6

Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Selection of Design Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7

Partial-state and Output Feedback Trajectory-tracking Control of Underactuated Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Partial-state Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

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7.2.5 Selection of Design Constants . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.3.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.4 Robustness Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8

Path-tracking Control of Underactuated Ships . . . . . . . . . . . . . . . . . . . . 165 8.1 Full-State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.1.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.1.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.1.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.1.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.1.5 Dealing with Environmental Disturbances . . . . . . . . . . . . . . . 176 8.1.6 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.2 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.2.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.2.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.2.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.2.4 Control Design with Integral Action . . . . . . . . . . . . . . . . . . . . 190 8.2.5 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.2.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9

Way-point Tracking Control of Underactuated Ships . . . . . . . . . . . . . . . 213 9.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.2 Full-state Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.2.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 9.3 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 9.3.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 9.3.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.4.1 State Feedback Simulation Results . . . . . . . . . . . . . . . . . . . . . . 237 9.4.2 Output Feedback Simulation Results . . . . . . . . . . . . . . . . . . . . 238 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

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Path-following of Underactuated Ships Using Serret–Frenet Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.2 State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 10.2.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 10.2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 10.3 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.3.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.3.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 10.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 10.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.4.1 State Feedback Simulation Results . . . . . . . . . . . . . . . . . . . . . . 264 10.4.2 Output Feedback Simulation Results . . . . . . . . . . . . . . . . . . . . 265 10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

11

Path-following of Underactuated Ships Using Polar Coordinates . . . . . 271 11.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 11.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.2.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.2.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 11.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 11.4 Discussion of the Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 11.5 Parking and Point-to-point Navigation . . . . . . . . . . . . . . . . . . . . . . . . . 284 11.5.1 Parking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11.5.2 Point-to-point Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.6 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.6.1 Path-following Simulation Results . . . . . . . . . . . . . . . . . . . . . . 287 11.6.2 Point-to-point Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 287 11.6.3 Parking Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Part IV Control of Underactuated Underwater Vehicles 12

Trajectory-tracking Control of Underactuated Underwater Vehicles . 295 12.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 12.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 12.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 12.3.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 12.3.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 12.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 12.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Contents

13

xvii

Path-following of Underactuated Underwater Vehicles . . . . . . . . . . . . . 313 13.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 13.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 13.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 13.5 Discussion of the Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.6 Parking and Point-to-point Navigation . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.6.1 Parking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.6.2 Point-to-point Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.7 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 13.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Part V Control of Other Underactuated Mechanical Systems 14

Control of Other Underactuated Mechanical Systems . . . . . . . . . . . . . . 341 14.1 Mobile Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 14.1.1 Basic Motion Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 14.1.2 Modeling and Control Properties . . . . . . . . . . . . . . . . . . . . . . . 342 14.1.3 Output Feedback Simultaneous Stabilization and Trajectory-tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 14.1.4 Output Feedback Path-following . . . . . . . . . . . . . . . . . . . . . . . 360 14.1.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 14.2 Vertical Take-off and Landing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . 368 14.2.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 14.2.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 14.2.3 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 14.2.4 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 14.2.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 14.2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 14.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

15

Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 15.1 Summary of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 15.2 Perspectives and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 15.2.1 Further Issues on Control of Single Underactuated Ocean Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 15.2.2 Coordination Control of Multiple Underactuated Ocean Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Chapter 1

Introduction

The subject of this book is the application of nonlinear control theory to control underactuated ocean vessels and to analyze stability of their controlled systems. A vessel is said to be underactuated if it has more degrees of freedom to be controlled than the number of independent control inputs. Many ocean ships are equipped with propellers and rudders for surge and yaw motions only, but without any actuators for direct control of their sway motion. The control of underactuated ocean vessels has become an active area of research in recent years not only because it poses many challenging questions in applied nonlinear control theory, but also because of its practical importance. This chapter starts with a brief review of the development in nonlinear control theory and its applications. Next, difficulties in control of underactuated ocean vessels are discussed. The chapter ends with a description of the organization of the book.

1.1 Overview of Nonlinear Control Developments Over the last two decades nonlinear control theory has received considerable research effort and has undergone a period of significant progress. This is motivated from the fact that real world problems are often inherently nonlinear. Conventionally, control engineers studied a nonlinear system by using a linearized model around selected operating points, and suitable techniques for linear control systems founded on the basis of the superposition principle. A fundamental limitation of this linearization approach is that stability and performance can only be guaranteed in the neighborhood of the selected operating points. Consequently, studying nonlinear systems based on linear models is severely limited since real world systems are usually required to operate over a large number of operating points. In addition, many phenomena such as the existence of multiple equilibria or operating points, periodic variation of state variables or limit cycles, finite escape time, and bifurcations cannot be described or predicted by using linear models. The above-mentioned

1

2

1 Introduction

limitations led control researchers to directly study nonlinear systems. Moving from linear to nonlinear systems, we are faced with a much more difficult situation. The superposition principle no longer holds and analysis tools involve more advanced mathematics. In addition, the separation principle does not hold for nonlinear systems in general. Currently, the techniques that are available for nonlinear control system design include linearization and gain-scheduling techniques, feedback linearization, circle Popov criteria, small-gain theorems, passivity, averaging, singular perturbation, sliding mode control, Lyapunov design and redesign, backstepping and forwarding, adaptive control, input-to-state stability, integral input-to-state stability, and differential geometric approaches. For details of the above-mentioned approaches, the reader is referred to [1–10]. Nonlinear control methods are increasingly implemented in practice with great success such as electrical motors, diesel engines, ocean vessels, jet engine compressors and electromechanical systems. For some of these applications, the reader is referred to [3, 11–15].

1.2 Difficulties in Control of Underactuated Ocean Vessels The main difficulty in the control of an underactuated system is that the system has more degrees of freedom (outputs) to be controlled than the number of independent actuators (inputs). A direct application of the Brockett necessary condition indicates that an underactuated system cannot be stabilized by any time-invariant continuous state feedback controllers although it is open loop controllable. This topic is detailed in Section 2.7.4. Moreover, if classical motion control systems designed for fully or overactuated systems are directly used on the underactuated ones, the resulting performance of the controlled systems is very poor or the control objectives cannot be achieved. The above observations are also true for underactuated ocean vessels. Below we illustrate the aforementioned difficulties through a simple example. In Chapter 3, the equations of motion of an underactuated ocean vessel without environmetal disturbances are given by (see (3.31)): P D J ./v; M vP D C .v/v  D.v/v  g./ C ;

(1.1)

 D Œx; y; z; ; ; T ; v D Œu; v; w; p; q; rT

(1.2)

where

are the vectors of position/Euler angles and velocities, respectively. The matrices M , C .v/, and D.v/ denote inertia, Coriolis, and damping, respectively, g./ is the vector of gravitational/buoyancy forces,  2 R6 is the vector of the forces and moments provided by the actuators. Depending on the vessel’s configuration, several

1.2 Difficulties in Control of Underactuated Ocean Vessels

3

elements of  are zero. For example, for the case where an underwater vehicle does not have independent actuators in the heave and sway axes, the second and third elements of  are zero. This gives an underactuated situation. We will show that an underactuation causes serious problems for classical control design techniques. We now assume that we want to design the control input vector  to force the vector of position/Euler angles  to track a reference vector of position/Euler angles d with bounded P d and R d . As a standard application of the control design techniques in [3, 6] for such a strict feedback system like (1.1), we define the tracking error vector e as (1.3)  e D   d and differentiate e twice along the solutions of the system (1.1) to obtain   R e D JP ./v C J ./M 1  C .v/v  D.v/v  g./ C   R d :

(1.4)

Letting   JP ./v C J ./M 1  C .v/v  D.v/v  g./ C   R d D K1 e  K2 P e ; (1.5) where K1 2 R66 and K2 2 R66 are positive definite symmetric control gain matrices, results in R e D K1 e  K2 P e ;

(1.6)

where P e D J ./v  P d , which is globally exponentially stable (GES) at the origin. We now need to solve (1.5) for  for implementation. From (1.5), we have
  D MJ 1 ./  K1 e  K2 P e  JP ./v C R d CC .v/v C D.v/v C g./ WD  .d ; P d ; R d ; ; v/: (1.7) Let us assume that the actuators available are independent and assume that they number mc , where mc < 6. Consequently we can propose a decomposition of  as: H1. a 2 R6 , where a has zeros in the element positions identifying that there are no actuator inputs and where a contains the elements of  associated with actuator locations. H2. na 2 R6 . This vector complements a by having zeros in the locations associated with actuator inputs and contains the elements of  in the locations associated with no actuator inputs. H3. Similarly, the vector  .
/ can also be decomposed into vectors a .
/ and na .
/ according to the rules of schemes (H1) and (H2). Thus, the computed control vector  can be decomposed as  D a C na D a .
/ C na .
/ 2 R6

(1.8)

4

1 Introduction

and a D a .
/ 2 R6 ; na D na .
/ 2 R6 :

(1.9)

In practice the actuator configuration only allows a to be implemented, and the response of the vessel (system) will result from this input alone. If the vessel response is given by , where  D H ./;

(1.10)

the full response would have been full , where full D H ./ D H .a C na /;

(1.11)

but the actual response is partial , where partial D H .a / D Hs .ra /

(1.12)

and a 2 R6 , partial 2 R6 , H .
/ W R6 ! R6 but ra is the mc vector of nonzero actuator inputs, ra 2 Rmc , and Hs .
/ is the reduced functional representation with Hs W Rmc ! R6 . Therefore the fully actuated vessel is represented by H .
/ and the underactuated vessel is represented by Hs .
/. These two representations may have very different properties. The implications of not being able to implement the control action na , and having a vessel whose representation is Hs .
/ are potentially: 1. Degraded performance achieved by partial when compared to full . 2. Total loss of performance and inability to meet the control objectives in any useful way. 3. Loss of guaranteed properties like closed loop stability with possible catastrophic system failure. Ways to overcome these problems include: 1. Reduction of performance specification as a control objective that can be achieved by partial . 2. Modification of the control design procedure to accommodate the reduced degrees of freedom available for control. 3. Procedures to use the available actuators to minimize the deleterious effects caused by not being able to implement the na component. 4. The use of advanced control procedures like those described in this book. 5. The installation of additional actuators able to create access to effect and control the missing degrees of freedom.

1.3 Organization of the Book

5

1.3 Organization of the Book The rest of the book is organized in 14 chapters, see Table 1.1. A brief summary of each chapter is given below. Chapter 2 presents mathematical tools like Lyapunov stability theory, Barbalatlike lemmas, and the backstepping technique to be used in control design and stability analysis in the book. Chapter 3 sets out the basic material that will be used in the subsequent chapters. Basic motion control tasks for ocean vessels and modeling of vessels are given. Chapter 4 presents control properties of ocean vessels. Then, the existing literature on the control of underactuated ocean vessels is reviewed. Through the review of the previous work in the area of stabilization, trajectory-tracking, path-following, and output feedback control of underactuated ocean vessels, challenging questions are raised. These questions motivate contributions of the book to new solutions for the motion control of underactuated ships and underwater vehicles. Chapter 5 addresses the problem of trajectory-tracking control of underactuated ships. These ships do not have independent actuators in the sway axis. The reference trajectory is generated by a suitable virtual ship. The control development is based on an elegant coordinate transformation, Lyapunov’s direct method, and the backstepping technique, and utilizes passivity properties of the ship dynamics and their interconnected structure. Chapter 6 examines the problem of designing a single controller that achieves stabilization and trajectory-tracking simultaneously for underactuated ships. In comparison with the preceding chapter, a path approaching the origin and a set-point can also be included in the reference trajectory, i.e., stabilization/regulation is also considered. The control development is based on several special coordinate transformations plus the techniques in the preceding chapter. Chapter 7 presents global partial-state feedback and output feedback control schemes for trajectory-tracking control of underactuated ships. For the case of partial-state feedback, measurements of the ship sway and surge velocities are not needed, while for the case of output feedback, no ship velocities are required for feedback. Global nonlinear coordinate changes are introduced to transform the ship dynamics to a system affine in ship velocities to design observers to globally exponentially estimate unmeasured velocities. These observers plus the techniques in Chapters 5 and 6 facilitate the controllers’ development. Chapter 8 deals with the problem of path-tracking control of underactuated ships. In comparison with Chapters 5, 6, and 7, the requirement of the reference trajectory generated by a suitable virtual ship is relaxed. Both full state feedback and output feedback cases are considered. The control development is based on a series of nontrivial coordinate transformations plus the techniques in the previous chapters. Chapter 9 addresses the problem of way-point tracking control of underactuated ships. Both full state feedback and output feedback controllers are designed. The controllers in this chapter can be considered as an advanced version of the conventional course-keeping controllers in the sense that in addition to maintaining the

6

1 Introduction

desired heading, both the sway displacement (lateral distance) and sway velocity are controlled. Chapter 10 develops full state feedback and output feedback controllers that force underactuated ships to follow a predefined path. The control development is motivated by an observation that it is practical to steer a vessel such that the vessel is on the reference path and its total velocity is tangent to the reference path, and that the vessel’s forward speed is controlled separately by the main thruster control system. The techniques in the previous chapters plus the use of the Serret–Frenet frame facilitate the results. Chapter 11 presents a different approach from Chapter 10 to solve a pathfollowing problem for underactuated ships. Unlike in Chapter 10, here the control development is based on the method of generating reference paths by the helmsman. The path-following errors are first interpreted in polar coordinates, the techniques used in the previous chapters are then used to design path-following controllers. Interestingly, some situations of practical importance such as parking and pointto-point navigation are covered in this chapter as a by-product of the developed path-following system. Chapter 12 addresses the problem of trajectory-tracking control of underactuated underwater vehicles. These vehicles do not have independent actuators in the sway and heave axes. The control development is built on the techniques developed for underactuated ships in Chapters 5, 6, and 7. Due to complex dynamics of underwater vehicles in comparison with that of the surface ships, the control design and stability analysis require more complicated coordinate transformations and techniques than those developed for underactuated ships in Chapters 5, 6, and 7. Chapter 13 extends the results of Chapter 11 to the design of a path-following control system for underactuated underwater vehicles. A series of path-following strategies for the vehicles is first discussed. A practical approach is then chosen to facilitate the control development. We also address the parking and point-to-point navigation problems of the vehicles in this chapter. Chapter 14 illustrates several applications of the observer and control design techniques developed in the previous chapters to control of other underactuated mechanical systems, which are common in practice. These systems include mobile robots and VTOL aircraft. For mobile robots, a global exponential observer is first designed based on the observer design for underactuated ships in Chapter 7. Output feedback simultaneous stabilization and trajectory-tracking and path-following controllers are then developed using the control design techniques proposed for underactuated ships in Chapters 5 and 10. For VTOL aircraft, the observer and control design strategies used for underactuated ships in Chapters 5 and 6 are utilized to design a global output feedback trajectory-tracking controller. Finally, Chapter 15 concludes the book by briefly summarizing the main results presented in the previous chapters and presenting related open problems for further investigation.

1.3 Organization of the Book Table 1.1 Organization of the book 1. Introduction 2. Mathematical Preliminaries 3. Modeling of Ocean Vessels Part II: Modeling and Control 4. Control Properties and Previous Work on Control of Properties of Ocean Vessels Ocean Vessels 5. Trajectory-tracking Control of Underactuated Ships 6. Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships Part III: Control of 7. Partial-state and Output Feedback Trajectory-tracking Underactuated Ships Control of Underactuated Ships 8. Path-tracking Control of Underactuated Ships 9. Way-point Tracking Control of Underactuated Ships 10. Path-following of Underactuated Ships Using Serret– Frenet Coordinates 11. Path-following of Underactuated Ships Using Polar Coordinates Part IV: Control of 12. Trajectory-tracking Control of Underactuated UnderUnderactuated Underwater water Vehicles Vehicles 13. Path-following of Underactuated Underwater Vehicles Part V: Control of Other 14. Control of Other Underactuated Mechanical Systems Underactuated Mechanical Systems 15. Conclusions and Perspectives Part I: Mathematical Tools

7

Part I

Mathematical Tools

Chapter 2

Mathematical Preliminaries

This chapter presents mathematical tools, which will be used in control design and stability analysis in the subsequent chapters. Some standard theorems, lemmas and corollaries, which are available in references, are sometimes given without a proof.

2.1 Lyapunov Stability Stability theory is important in system theory and engineering. For a controlled system to be usable in practice, the least requirement is that the system is stable under unknown disturbances or noise. There are various types of stability problems that arise in the study of dynamical systems such as input–output stability, Lyapunov stability, absolute stability and stability of periodic solutions. These stability concepts have been studied extensively over the last 100 years. For control design and stability analysis in this book, Lyapunov stability plays an important role for the following reasons. First, Lyapunov’s direct method uses an energy-like function called the Lyapunov function to study the behaviors of dynamical systems analytically. This function reflects physical properties of the systems under study. Second, Lyapunov’s second method is applicable to both linear and nonlinear systems. Third, many results in input–output stability can be obtained directly using Lyapunov stability theory. Lyapunov stability theory generally includes Lyapunov’s first and second methods. The first method and the center manifold theory developed later are techniques based on the lowest-order approximation around a given point or a nominal trajectory. The stability results obtained using these two approximation methods are inherently local, and the stability regions are hard to estimate and often quite small. Since the objective of this book is to study nonlinear dynamics of ocean vessels, we omit the aforementioned two approximation techniques. Our primary interest is in stability based on Lyapunov’s second method for systems described by ordinary differential equations. This section is concerned with stability of equilibrium points in the sense of Lyapunov. An equilibrium point is stable if all solutions starting at nearby points

11

12

2 Mathematical Preliminaries

stay nearby; otherwise it is unstable. It is asymptotically stable if in addition, all solutions tend to the equilibrium point as time approaches infinity. These notions will be made mathematically rigorous in this section. Consider the following nonautonomous system: xP D f .t; x/;

(2.1)

where f W Π0; 1/  D ! Rn is piecewise continuous in t and locally Lipschitz in x on Π0; 1/  D and D  Rn is a domain that contains the origin x D 0.

2.1.1 Definitions Definition 2.1. The origin x D 0 is the equilibrium point of (2.1) if f .t; 0/ D 0; 8 t  0:

(2.2)

Definition 2.2. A continuous function ˛ W Œ 0; a/ ! RC is said to belong to class K if it is strictly increasing and ˛.0/ D 0. It is said to belong to class K1 if a D 1 and ˛.r/ ! 1 as r ! 1. Definition 2.3. A continuous function ˇ W Œ 0; a/  RC ! RC is said to belong to class KL if, for each fixed s, the mapping ˇ.r; s/ belongs to class K with respect to r and, for each fixed r, the mapping ˇ.r; s/ is decreasing with respect to s and ˇ.r; s/ ! 0 as s ! 1. It is said to belong to class KL1 if, in addition, for each fixed s the mapping ˇ.r; s/ belongs to class K1 with respect to r. Definition 2.4. The equilibrium point x D 0 of (2.1) is 1. stable if, for each " > 0, there is ı D ı."; t0 / > 0 such that kx.t0 /k < ı ) kx.t /k < "; 8t  t0  0;

(2.3)

2. uniformly stable if, for each " > 0, there is ı D ı."/ > 0 independent of t0 such that (2.3) is satisfied, 3. unstable if it is not stable, 4. asymptotically stable if it is stable and there is a positive constant c D c.t0 / such that x.t / ! 0 as t ! 1, for all kx.t0 /k < c; 5. uniformly asymptotically stable if it is uniformly stable and there is a positive constant c, independent of t0 , such that for all kx.t0 /k < c, x.t / ! 0 as t ! 1, uniformly in t0 ; that is, for each  > 0, there is T D T ./ > 0 such that kx.t /k < ; 8 t  t0 C T ./; 8 kx.t0 /k < c;

(2.4)

6. globally uniformly asymptotically stable (GUAS) if it is uniformly stable, ı."/ can be chosen to satisfy lim"!1 ı."/ D 1, and, for each pair of positive numbers  and c, there is T D T .; c/ > 0 such that

2.1 Lyapunov Stability

13

kx.t /k < ; 8 t  t0 C T .; c/; 8 kx.t0 /k < c:

(2.5)

Definition 2.5. The equilibrium point x D 0 of (2.1) is exponentially stable if there exist positive constants c, k, and  such that kx.t /k  k kx.t0 /k e .tt0 / ; 8 t  t0  0; 8 kx.t0 /k < c

(2.6)

and GES if (2.6) is satisfied for any initial state x.t0 /. Definition 2.6. The equilibrium point x D 0 of (2.1) is K-exponentially stable if there exist positive constants c and  and a class K function ˛ such that kx.t /k  ˛ .kx.t0 /k/ e .t t0 / ; 8 t  t0  0; 8 kx.t0 /k < c

(2.7)

and globally K-exponentially stable if (2.7) is satisfied for any initial state x.t0 /: Definition 2.7. The solutions of (2.1) are as follows: 1. Uniformly bounded if there exists a positive constant c, independent of t0  0, and for every a 2 .0; c/, there is ˇ D ˇ.a/ > 0, independent of t0 , such that kx.t0 /k  a ) kx.t /k  ˇ; 8t  t0 :

(2.8)

2. Globally uniformly bounded if (2.8) holds for an arbitrarily large a. 3. Uniformly ultimately bounded with ultimate bound b if there exist positive constants b and c, independent of t0  0, and for every a 2 .0; c/ there is T D T .a; b/  0, independent of t0 , such that kx.t0 /k  a ) kx.t /k  b; 8t  t0 C T:

(2.9)

4. Globally uniformly ultimately bounded if (2.9) holds for an arbitrarily large a.

2.1.2 Lemmas and Theorems The following lemma provides equivalent definitions of uniform stability and uniform asymptotic stability by using class K and class KL functions. Lemma 2.1. The equilibrium point x D 0 of (2.1) is 1. uniformly stable if and only if there exist a class K function ˛ and a positive constant c, independent of t0 , such that ˇ (2.10) kx.t /k  ˛ .kx.t0 /k/ ; 8 t  t0  0; 8 x.t0 /ˇ kx.t0 /k < c; 2. uniformly asymptotically stable if and only if there exist a class KL function ˇ and a positive constant c, independent of t0 , such that ˇ (2.11) kx.t /k  ˇ .kx.t0 /k ; t  t0 / ; 8 t  t0  0; 8 x.t0 /ˇ kx.t0 /k < c;

14

2 Mathematical Preliminaries

3. GUAS if and only if inequality (2.11) is satisfied with ˇ 2 KL1 for any initial state x.t0 /. Proof. See [6].  Lemma 2.2. Assume that d W Rn ! Rn satisfies P



  T @d @d C P  0; 8 x 2 Rn ; @x @x

(2.12)

where P D P T > 0. Then .x  y/T P .d.x/  d.y//  0; 8 x; y 2 Rn :

(2.13)

Proof. See [16].  Lemma 2.3. The following nonlinear interconnected system: xP 1 D f1 .t; x1 ; x2 / C g1 .t; x1 ; x2 /u; xP 2 D f2 .t; x1 ; x2 / C g2 .t; x1 ; x2 /u;

(2.14)

where xi 2 R; i D 1; 2; fi .t; x1 ; x2 / are locally Lipschitz in xi and piecewise continuous in t; u 2 R is the control input, and g2 .t; x1 ; x2 / ¤ 0; 8 t  0; xi 2 R, can be transformed to the following system zP 1 D 1 .t; z1 ; x2 /; xP 2 D 2 .t; z1 ; x2 / C '2 .t; z1 ; x2 /u:

(2.15)

Proof. Define z1 D x1 C .t; x1 ; x2 /;

(2.16)

where .t; x1 ; x2 / is to be determined. Differentiating both sides of (2.16) along the solutions of (2.14) yields   @.t; x1 ; x2 / zP1 D C 1 f1 .t; x1 ; x2 / C @x1 @.t; x1 ; x2 / @.t; x1 ; x2 / f2 .t; x1 ; x2 / C C (2.17) @x2 @t    @.t; x1 ; x2 / @.t; x1 ; x2 / C 1 g1 .t; x1 ; x2 / C g2 .t; x1 ; x2 / u : @x1 @x2 Now choosing the function .t; x1 ; x2 / such that   @.t; x1 ; x2 / @.t; x1 ; x2 / C 1 g1 .t; x1 ; x2 / C g2 .t; x1 ; x2 / D 0 @x1 @x2

(2.18)

results in (2.15), where the functions 1 .t; z1 ; x2 /, 2 .t; z1 ; x2 /, and '2 .t; z1 ; x2 / are defined as

2.1 Lyapunov Stability

15



 @.t; x1 ; x2 /

1 .t; z1 ; x2 / WD C 1 f1 .t; x1 ; x2 / C @x1 @.t; x1 ; x2 / @.t; x1 ; x2 / f2 .t; x1 ; x2 / C ; @x2 @t

2 .t; z1 ; x2 / WD f2 .t; x1 ; x2 /; '2 .t; z1 ; x2 / WD g2 .t; x1 ; x2 /;

(2.19)

with x1 being solved from (2.16) and substituted in.  Remark 2.1. 1. The success of the above lemma depends on whether a solution to the partial differential equation (2.18) can be found. Solving this partial differential equation is difficult in general but might be possible in some specific cases such as the vehicle systems in this book. 2. In some cases, designing a control input u for the transformed system (2.15) is simpler than that for the original system (2.14). The main Lyapunov stability theorem, which has a number of applications in studying stability of (2.1), is given below. ˇ Theorem 2.1. Let D D fx 2 Rn ˇkxk < rg and x D 0 be an equilibrium point of (2.1). Let V W D  Rn ! RC be a continuously differentiable function such that 8 t  0; 8 x 2 D,

1 .kxk/  V .x; t /  2 .kxk/ ; @V @V C f .t; x/   3 .kxk/ : @t @x

(2.20)

Then the equilibrium point x D 0 is 1. uniformly stable, if 1 and 2 are class K functions on Œ 0; r/ and 3  0 on Œ 0; r/, 2. uniformly asymptotically stable, if 1 ; 2 and 3 are class K functions on Œ 0; r/, 3. exponentially stable if i ./ D ki ˛ on Œ 0; r/ ; ki > 0; ˛ > 0; i D 1; 2; 3, 4. globally uniformly stable if D D Rn , 1 and 2 are class K1 functions, and

3  0 on RC , 5. GUAS if D D Rn , 1 and 2 are class K1 functions, and 3 is a class K function on RC , 6. GES, if D D Rn , i ./ D ki ˛ on RC ; ki > 0; ˛ > 0; i D 1; 2; 3. Proof. See [3].  Theorem 2.2. Let x D 0 be an equilibrium point of (2.1) and suppose that f is locally Lipschitz in x and uniformly continuous in t . Let V W Rn  RC ! RC be a continuously differentiable function such that

1 .kxk/  V .x; t /  2 .kxk/ ; @V @V VP D C f .t; x/  W .x/  0; @t @x

(2.21)

16

2 Mathematical Preliminaries

for all t  0 and x 2 Rn , where 1 and 2 are class K1 functions, and W is a continuous function. Then all solutions of (2.1) are globally uniformly bounded and satisfy lim W .x.t // D 0: (2.22) t!1

In addition, if W .x/ is positive definite, then the equilibrium point x D 0 is GUAS. Proof. See [3].  The following Lyapunov-like theorem is useful for showing uniform boundedness and ultimate boundedness. Theorem 2.3. Let V W Œ 0; 1/D ! R be a continuously differentiable function and D  Rn be a domain that contains the origin such that ˛1 .kxk/  V .x; t /  ˛2 .kxk/ ; @V @V C f .t; x/  W .x/ ; 8 kxk   > 0 @t @x

(2.23)

for all t  0 and x 2 D where ˛1 and ˛2 are class K functions, and W is a continuous positive definite function. Take r > 0 such that Br  D and suppose that  < ˛21 .˛1 .r//:

(2.24)

Then, there exists a class KL function ˇ and for every initial state x.t0 /, satisfying kx.t0 /k < ˛21 .˛1 .r//, there is T  0 (dependent on x.t0 / and / such that the solutions of (2.1) satisfies kx.t /k  ˇ .kx.t0 /k ; t  t0 / ; 8 t0  t  t0 C T; kx.t /k < ˛11 .˛2 .//; 8 t  t0 C T:

(2.25)

Moreover, if D D Rn and ˛1 belongs to class K1 , then (2.25) holds for any initial state x.t0 / with no restriction on how large  is. Proof. See [6]. 

2.1.3 Stability of Cascade Systems Consider the following cascade system: zP 1 D f1 .t; z1 / C g.t; z1 ; z2 /z2 ; zP 2 D f2 .t; z2 /;

(2.26)

where z1 2 Rn ; z2 2 Rm ; f1 .t; z1 / is continuously differentiable in .t; z1 /, and f2 .t; z2 / and g.t; z1 ; z2 / are continuous and locally Lipschitz in z2 and .z1 ; z2 /, respectively.

2.1 Lyapunov Stability

17

If we set z2 D 0, then the first equation of (2.26) becomes zP1 D f1 .t; z1 /. Therefore we can view the first equation of (2.26) as the system ˝1 W zP 1 D f1 .t; z1 /;

(2.27)

which is perturbed by the output of the system ˝2 W zP 2 D f2 .t; z2 /:

(2.28)

Now assume that the systems ˝1 and ˝2 are asymptotically stable at the origin, i.e., (2.27) and (2.28) yield lim t!1 z1 .t / D 0 and lim t!1 z2 .t / D 0, respectively. Based on these assumptions, it is plausible to conclude that the system (2.26) is asymptotically stable at the origin in general. In many cases, the solution z1 .t / of the system (2.26) goes to infinity in finite time as can be seen from the simple cascade system: zP 1 D k1 z1 C z12 z2 ; (2.29) zP 2 D k2 z2 ; where k1 and k2 are strictly positive constants. It is obvious that the subsystems zP 1 D k1 z1 and zP 2 D k2 z2 are GES at the origin. It is straightforward to show that the solution of (2.29) is z1 .t / D

z1 .t0 /.k1 C k2 / ; z1 .t0 /z2 .t0 /e k2 .tt0 / C .k1 C k2  z1 .t0 /z2 .t0 //e k1 .tt0 /

z2 .t / D z2 .t0 /e k2 .tt0 / :

(2.30)

It can be seen from (2.30) that if z1 .t0 /z2 .t0 / < k1 Ck2 , then both z1 .t / and z2 .t / are bounded and converge exponentially to zero, respectively. If z1 .t0 /z2 .t0 / D k1 C k2 , then z2 .t / is still bounded and converges exponentially to zero but z1 .t / tends to infinity exponentially fast. If z1 .t0 /z2 .t0 / > k1 C k2 , the situation is catastrophic, that is z1 .t / tends to infinity when t ! t0 C tf with   z1 .t0 /z2 .t0 / 1 : (2.31) ln tf D k1 C k2 z1 .t0 /z2 .t0 /  .k1 C k2 / The following theorem gives sufficient conditions of stability of the cascade system (2.26) based on stability of (2.27) and (2.28), and the connected term g.t; z1 ; z2 /. Theorem 2.4. Consider the following assumptions: 1. The systems (2.27) and (2.28) are both GUAS and we know explicitly a C 1 Lyapunov function V .t; z1 /, two class-K1 functions ˛1 and ˛2 , a class-K function ˛4 , and a positive semidefinite function W .z1 / such that ˛1 .kz1 k/  V .t; z1 /  ˛2 .kz1 k/ ;

@V @V @V

C f1 .t; z1 /  W .z1 /;

 ˛4 .kz1 k/ : @t @z1 @z1

(2.32)

18

2 Mathematical Preliminaries

2. For each fixed z2 , there exists a continuous function  W RC ! RC such that lim .s/ D 0; s!1



@V

@z g.t; z1 ; z2 /   .kz1 k/ W .z1 /: 1

(2.33)

kg.t; z1 ; z2 /k   .kz2 k/ ˛5 .kz1 k/

(2.34)

3. There exist continuous functions  W RC ! RC and ˛5 W RC ! RC such that

and a continuous nondecreasing function ˛6 W RC ! RC , and a nonnegative constant a such that 

 ˛6 .s/  ˛4 ˛11 .s/ ˛5 ˛11 .s/ ; Z1 ds D 1: (2.35) ˛6 .s/ a

4. For each r > 0, there exist constants  > 0 and  > 0 such that for all t  0 and all kz2 k < r

@V

(2.36) g.t; z ; z / 1 2  W .z1 /; 8 kz1 k  :

@z 1

5. There exists a class K function  such that the solution z2 .t / of (2.28) satisfies Z1 kz2 .t /k dt   .kz2 .t0 /k/ :

(2.37)

t0

Then we can conclude that if  Assumptions 1 and 2, or  Assumptions 1, 3 and 4, or  Assumptions 1, 3 and 5 hold then the cascade system (2.26) is GUAS. Proof. See [17]. 

2.2 Input-to-state Stability Definition 2.8. The system xP D f .t; x; u/;

(2.38)

where f is piecewise continuous in t and locally Lipschitz in x and u, is said to be input-to-state stable (ISS) if there exist a class KL function ˇ and a class K function

2.2 Input-to-state Stability

19

, such that, for any x.t0 / and for any input u.
/ continuous and bounded on Œ 0; 1/, the solution exists for all t  t0  0 and satisfies   (2.39) kx.t /k  ˇ .kx.t0 /k ; t  t0 / C sup ku. /k : t0 t

The following theorem establishes the equivalence between the existence of a Lyapunov-like function and the input-to-state stability. Theorem 2.5. Suppose that for the system (2.38) there exists a C 1 function V W RC  Rn ! RC such that for all x 2 Rn and u 2 Rm ,

1 .kxk/  V .t; x/  2 .kxk/; @V @V C f .t; x; u/   3 .kxk/; kxk  .kuk/ ) @t @x

(2.40)

where 1 ; 2 , and  are class K1 functions and 3 is a class-K function. Then the system (2.38) is ISS with D 11 ı 2 ı . Proof. If x.t0 / is in the set ˇ  ˚
 R t0 D x 2 Rn ˇkxk   sup ku. /kt0 ;

then x.t / remains within the set ˇ ˚ 
 S t0 D x 2 Rn ˇkxk  11 ı 2 ı  sup ku. /kt0 ;

(2.41)

(2.42)

for all t  t0 . Define B D Œt0 ; T / as the time interval before x.t / enters R t0 for the first time. In view of the definition of R t0 we have VP   3 ı 21 .V /; 8t 2 B:

(2.43)

Then, there exists a class-KL function ˇv such that V .t /  ˇv .V .t0 /; t  t0 /; 8t 2 B, which implies kx.t /k  11 .ˇv . 2 .kx.t0 /k; t  t0 // WD ˇ.kx.t0 /k; t  t0 /; 8t 2 B:

(2.44)

On the other hand, by (2.42), we conclude that kx.t /k  11 ı 2 ı .sup ku. /kt0 / WD .sup ku. /kt0 /;

(2.45)

for all t 2 Œt0 ; 1nB. Then by (2.44) and (2.45), kx.t /k  ˇ.kx.t0 /k; t  t0 / C .sup ku. /kt0 /; 8t  t0  0:

(2.46)

By causality, we have kx.t /k  ˇ.kx.t0 /k; t  t0 / C .sup ku. /k t0 t /; 8t  t0  0:

(2.47)

20

2 Mathematical Preliminaries

A function V satisfying conditions (2.40) is called an ISS Lyapunov function. 

2.3 Control Lyapunov Functions This section presents an extension of the Lyapunov function concept, which is a useful tool to design an adaptive controller for nonlinear systems. Assuming that the problem is to design a feedback control law ˛.x/ for the time-invariant system: xP D f .x; u/; x 2 Rn ; u 2 R; f .0; 0/ D 0;

(2.48)

such that the equilibrium x D 0 of the closed loop system: xP D f .x; ˛.x//

(2.49)

is globally asymptotically stable (GAS). We can take a function V .x/ as a Lyapunov candidate function, and require that its derivative along the solutions of (2.49) satisfy VP .x/  W .x/, where W .x/ is a positive definite function. We therefore need to find ˛.x/ to guarantee that for all x 2 Rn such that @V .x/ f .x; ˛.x//  W .x/: @x

(2.50)

This is a difficult problem. A stabilizing control law for (2.48) may exist but we may fail to satisfy (2.50) because of a poor choice of V .x/ and W .x/. A system for which a good choice of V .x/ and W .x/ exists is said to possess a control Lyapunov function (CLF). For systems affine in the control: xP D f .x/ C g.x/u; f .0/ D 0;

(2.51)

the CLF inequality (2.50) becomes @V @V f .x/ C g.x/˛.x/  W .x/: @x @x

(2.52)

If V .x/ is a CLF for (2.51), then a particular stabilizing control law ˛.x/, smooth for all x ¤ 0, is given by s 8 2  4 ˆ @V @V @V ˆ ˆ C f .x/ C f .x/ g.x/ ˆ ˆ ˆ @x @x @V < @x  ; g.x/ ¤ 0; u D ˛.x/ D @V @x ˆ g.x/ ˆ ˆ @x ˆ ˆ ˆ @V : 0; g.x/ D 0: @x (2.53)

2.4 Backstepping

21

It should be noted that (2.52) can be satisfied only if @V @V g.x/ D 0 ) f .x/ < 0; @x @x

8x ¤ 0

and that in this case (2.53) gives s  2  4 @V @V W .x/ D C > 0; f g @x @x

8 x ¤ 0:

(2.54)

(2.55)

The main drawback of the CLF concept as a design tool is that for most nonlinear systems a CLF is not known. The task of finding an appropriate CLF may be as complex as that of designing a stabilizing feedback law. The backstepping technique [3] presented in the next section gives a constructive way to construct an appropriate CLF for various systems.

2.4 Backstepping Assumption 2.1. Consider the system xP D f .x/ C g.x/u; f .0/ D 0;

(2.56)

where x 2 Rn is the state and u 2 R is the control input. There exist a continuously differentiable feedback control law u D ˛.x/; ˛.0/ D 0

(2.57)

and a smooth, positive definite, radially unbounded function V W Rn ! R such that @V Œf .x/ C g.x/˛.x/  W .x/  0; 8x 2 Rn ; @x

(2.58)

where W W Rn ! R is positive semidefinite. It should be noted that under this assumption, the control law (2.57) applied to the system (2.56) guarantees global boundedness of x.t ), and the regulation of W .x/ W lim t!1 W .x/ D 0. If W .x/ is positive definite, the control law (2.57) renders the global asymptotic stable equilibrium of (2.56). Theorem 2.6. Let the system (2.56) be augmented by an integrator: xP D f .x/ C g.x/; P D u;

(2.59)

and suppose that the first equation of (2.59) satisfies Assumption 2.1 with  as its control.

22

2 Mathematical Preliminaries

1. If W .x/ is positive definite then 1 Va D V .x/ C Œ  ˛.x/2 2

(2.60)

is a CLF for the system (2.59), that is, there exists a feedback control law u D ˛a .x; /, which renders x D 0;  D 0 the GAS equilibrium of (2.59). One such control law choice is u D c.  ˛.x// C

@V @˛ Œf .x/ C g.x/  g.x/; c > 0: @x @x

(2.61)

2. If W .x/ is only positive semidefinite, then there exists a feedback control law that renders VPa  Wa .x; /  0, such that Wa .x; / > 0 whenever W .x/ > 0 or  ¤ ˛.x/. This guarantees global boundedness and convergence of Œx.t / .t /T to the largest invariant set Ma contained in the set    x 2 RnC1 jW .x/ D 0;  D ˛.x/ : Ea D  Proof. We only prove the first part of the theorem. Proof of the second part is trivial. Introducing an error variable z D   ˛.x/ (2.62) and differentiating with respect to time, (2.59) can be written as xP D f .x/ C g.x/Œ˛.x/ C z; @˛ Œf .x/ C g.x/.˛.x/ C z/ : zP D u  @x

(2.63)

Using (2.58), the first time derivative of (2.60) along the solutions of (2.63) satisfies   @V @˛ .f .x/ C g.x/.˛.x/ C z// C VPa  W .x/ C z u  g.x/ : (2.64) @x @x Any control law, such as (2.61), which renders VPa  Wa .x; /  W .x/, with Wa positive definite in z, guarantees global boundedness of x and z, and regulation of W .x/ and z.t /.  Corollary 2.1. Let the system (2.56) satisfy Assumption 2.1 with ˛.x/ D ˛0 .x/ being augmented by a chain of k integrators so that u is replaced by 1 , the state of the last integrator in the chain is xP D f .x/ C g.x/1 ; P1 D 2 ; :: : Pk1 D k ; Pk D u:

(2.65)

2.5 Stabilization Under Uncertainties

23

For this system, repeated application of Theorem 2.6 with 1 ; : : : ; k as virtual controls, results in the Lyapunov function k

Va .x; 1 ; : : : ; k / D V .x/ C

1X Œi  ˛i1 .x; 1 ; : : : ; i1 /2 : 2

(2.66)

iD1

Any choice of feedback control which renders VPa  Wa .x; 1 ; : : : ; k /  0, with Wa .x; 1 ; : : : ; k / D 0 only if W .x/ D 0 and i ¤ ˛i1 .x; 1 ; : : : ; i1 /, i D 1; : : : ; k, T  guarantees that x T .t /; 1 .t /; : : : ; k .t / is globally bounded and converges to the largest invariant set Ma contained in the set h  ˇ iT nCk ˇ T Ea D x ; 1 ; : : : ;  k 2 R ˇW .x/ D 0; i D ˛i1 .x; 1 ; : : : ; i1 / for all i D 1; : : : ; k. Furthermore, if W .x/ is positive definite, that is, if x D 0 can be rendered GAS through 1 , then (2.66) is a CLF for (2.65) and the equilibrium x D 0; 1 D 0; : : : ; k D 0 can be rendered GAS through u. Proof. See [3]. 

2.5 Stabilization Under Uncertainties The power of adaptive control is exhibited in the presence of uncertain nonlinearities and unknown parameters. Such uncertainties in linear systems make the control design procedure difficult and become more serious in control of nonlinear systems. For nonlinear systems, the states can easily escape to infinity in a finite time. The following theorem introduces the use of a term in the control law called nonlinear damping to stabilize the system (2.56) in the presence of disturbance. Theorem 2.7. Consider the system (2.56) satisfying Assumption 2.1 which is perturbed as i h (2.67) xP D f .x/ C g.x/ u C '.x/T
.x; u; t / ;

where '.x/ is a .p  1/ vector of known smooth nonlinear functions, and
.x; u; t / is a .p  1/ vector of uncertain nonlinearities, which are uniformly bounded for all values of x, u, and t . If Assumption 2.1 satisfies with W .x/ being positive definite and radially unbounded, then the control u D ˛.x/  k

@V .x/g.x/ k'.x/k2 ; k > 0; @x

(2.68)

when applied to the system (2.67) renders the closed loop system ISS with respect to the disturbance input
.x; u; t / and hence guarantees global uniform boundedness of x.t / and convergence to the residual set

24

2 Mathematical Preliminaries

(

R D x W kxk 

11 ı 2 ı 31

k
k21 4k

!)

;

(2.69)

where 1 ; 2 ; 3 are class K1 functions such that

1 .kxk/  V .x/  2 .kxk/;

3 .kxk/  W .x/:

(2.70)

Proof. By using (2.68) and (2.70), the first time derivative of V .x/ is @V @V VP D Œf C gu C g' T
@x @x  2 @V @V @V Œf C g˛  k g k'k2 C g' T
D @x @x @x 2  @V @V g k'k2 C g' T
(2.71)  W .x/  k @x @x

2 

@V @V k
k21 2

 W .x/  k  W .x/ C g k'k C g : k'k k
k 1 @x @x 4k

From (2.71), it follows that VP is negative whenever W .x/  with the second equation of (2.70), we conclude that ! 2 1 k
k1 kx.t /k > 3 ) VP < 0: 4k This means that if kx.0/k  31



k
k2 1 4k

V .x.t // 

 , then

2 ı 31

which implies that kx.t /k 

11 ı 2 ı 31

If, on the other hand, kx.0/k  31



k
k2 1 4k



k
k2 1 . 4k

Combining this

! k
k21 ; 4k ! k
k21 : 4k

(2.72)

(2.73)

(2.74)

, then V .x/  V .x.0//, which implies

kx.t /k  11 ı 2 .kx.0/k/: Combining (2.74) and (2.75) leads to global uniform boundedness of x.t /: ( ! ) 2 1 1 k
k1 1 ; 1 ı 2 .kx.0/k/ ; kxk1  max 1 ı 2 ı 3 4k

(2.75)

(2.76)

2.6 Barbalat-like Lemmas

25

while (2.72) and the first equation of (2.70) prove the convergence of x.t / to the residual set defined in (2.69).  Combining the above theorem with Theorem 2.6 leads to the following corollary. Corollary 2.2. Consider the following system xP D f .x/ C g.x/u C F .x/
1 .x; u; t /;

(2.77)

where x 2 Rn ; u 2 R; F .x/ is an .n  q/ matrix of known smooth nonlinear functions, and
1 .x; u; t / is a .q  1/ vector of uncertain nonlinearities, which is uniformly bounded for all values of x, u and t. Suppose that there exists a feedback control law u D ˛.x/ that renders x.t / globally uniformly bounded, and that this is established via positive definite and radially unbounded functions V .x/, W .x/, and a constant b such that @V Œf .x/ C g.x/˛.x/ C F .x/
1 .x; u; t /  W .x/ C b: @x

(2.78)

Now consider the augmented system xP D f .x/ C g.x/ C F .x/
1 .x; u; t /; P D u C ' T .x; /
2 .x; ; u; t /;

(2.79)

where '.x; / is a .p 1/ vector of known smooth nonlinear functions,
2 .x; u; ; t / is a .p  1/ vector of uncertain nonlinearities, which are uniformly bounded for all values of x, u,  and t . For this system, the feedback control law u D c.  ˛.x// C

@˛ Œf .x/ C g.x/  @x (

2 )

@˛ @V 2

g.x/  k.  ˛.x// k'.x; /k C F .x/

@x @x

(2.80)

guarantees global uniform boundedness of x.t / and .t/ with any c > 0 and k > 0. Proof. See [3]. 

2.6 Barbalat-like Lemmas This section presents lemmas that are useful in investigating the convergence of time-varying systems. Lemma 2.4. (Barbalat’s lemma) Consider the function  W RC ! R. If  is uniRt formly continuous and lim t!1 . /d exists and is finite, then 0

26

2 Mathematical Preliminaries

lim .t / D 0:

(2.81)

t!1

Proof. See [6].  Lemma 2.5. Assume that a nonnegative scalar differentiable function f .t / enjoys the following conditions ˇ ˇ ˇ ˇd ˇ 1: ˇ f .t /ˇˇ  k1 f .t /; dt Z 1 f .t /dt  k2 (2.82) 2: 0

for all t  0, where k1 and k2 are positive constants, then lim t!1 f .t / D 0. Proof. Integrating both sides of 1) in (2.82) gives Z t f .s/ds  f .0/ C k1 k2 ; f .t /  f .0/ C k1 0 Z t f .t /  f .0/  k1 f .s/ds  f .0/  k1 k2 :

(2.83)

0

These inequalities imply that f .t / is a uniform bounded function. From (2.83) and the second condition in (2.82), we have that f .t / is alsoˇbounded ˇ on the half axis Œ0; 1, i.e., f .t /  k3 with k3 a positive constant. Hence ˇ dtd f .t /ˇ  k1 k3 . Now assume that lim t!1 f .t / ¤ 0. Then there exists a sequence of points ti and a positive constant  such that f .ti /  ; ti ! 1; i ! 1; jti  ti1 j > 2=.k1 k3 / and moreover f .s/  =2; s 2 Li D Œti  =.2k1 k3 /; ti C =.2k1 k3 /. Since the segments Li and Lj do not intersect for any i and j with i ¤ j , we have Z

1

f .t /dt  0

Z

0

T

f .t /dt 

XZ

ti T

f .t /dt 

Li

  M.T / 2 k1 k3

(2.84)

where M.T /Ris the number of points ti not exceeding T. Since limT !1 M.T / D 1, 1 the integral 0 f .t /dt is divergent. This contradicts Condition 2 in (2.82). This contradiction proves the lemma.  Remark 2.2. Lemma 2.5 is different from Barbalat’s lemma 2.4. While Barbalat’s lemma assumes that f .t / is uniformly continuous, Lemma 2.5 assumes that j dtd f .t /j is bounded by k1 f .t /. Lemma 2.6. Consider a scalar system xP D cx C p.t /;

(2.85)

where c > 0 and p.t / is a bounded and uniformly continuous function. If, for any initial time t0  0 and any initial condition x.t0 /, the solution x.t / is bounded and converges to 0 as t ! 1, then

2.7 Controllability and Observability

27

lim p.t / D 0:

t!1

(2.86)

Proof. See [18].  Lemma 2.7. Consider a first-order differential equation of the form xP D .a.t / C f1 ..t ///x C f2 ..t //;

(2.87)

where f1 and f2 are continuous functions, and  W Π0; 1/ ! Rm is a time-varying vector-valued signal that exponentially converges to zero and, for all t  t0  0, satisfies (2.88) jfi ..t //j  i .k.t0 /k/ e i .tt0 / ; where i > 0; i D 1; 2 and i are class-K functions. If a.t / enjoys the property that there is a constant 3 such that Zt2

a. /d  3 .t2  t1 /; 8t2  t1  0;

(2.89)

t1

then there exist a class-K function and a constant  > 0 such that jx.t /j  .k.x.t0 /; .t0 //k/ e .tt0 / :

(2.90)

Proof. See [19]. 

2.7 Controllability and Observability 2.7.1 Controllability and Observability of Linear Time-invariant Systems This section deals with the controllability and observability properties of systems described by linear time-invariant state-space representations. In particular, consider a linear and time-invariant system defined by the state–space representation: x.t P / D Ax.t / C Bu.t /; y.t / D C x.t / C Du.t /;

(2.91)

where A has n distinct eigenvalues 1 ; 2 ; : : : ; n , which define the poles and the corresponding modes e i t .

28

2 Mathematical Preliminaries

2.7.1.1 Controllability Conditions for controllability of the system (2.91) are given in the following lemmas. Lemma 2.8. The system (2.91) is controllable if and only if any one of the following (equivalent) conditions holds: 1. The controllability matrix Cc WD ŒB; AB; : : : ; A n1 B

(2.92)

has full rank n. 2. The n rows of e At B, where e At represents the unique state transition matrix of the system, are linearly independent over the real field R for all t . 3. The controllability Grammian Z tf T e A BB T e A  d (2.93) Gc .t0 ; tf / WD t0

is nonsingular for all tf > t0 . 4. The n  .n C m/ matrix ŒI  A; B has rank n at all eigenvalues i of A or equivalently, I  A and B are left comprime polynomial matrices. Proof. See [20], Chapter 8.  Since the solution to the system (2.91) is given by Z t x.t / D e A.tt0 / x.t0 / C e A.t/ Bu. /d;

(2.94)

t0

it follows that the controllability Grammian-based control input u.t / D B T e A

T

t

  Gc 1 .t0 ; tf / e Atf x.tf /  e At0 x.t0 /

(2.95)

transfers any initial state x.t0 / to any arbitrarily chosen final state x.tf / at any arbitrary tf > 0. This observation is consistent with the more traditional definition of controllability.

2.7.1.2 Observability Conditions for observability of the system (2.91) are given in the following lemmas. Lemma 2.9. The system (2.91) is observable if and only if any one of the following (equivalent) conditions holds: 1. The observability matrix

2.7 Controllability and Observability

29

3

2

C 6 CA 7 7 6 Co WD 6 7 :: 4 5 : CA n1

(2.96)

has full rank n. 2. The n columns of C e At , where e At represents the unique state transition matrix of the system, are linearly independent over the real field R for all t . 3. The observability Grammian Z tf T e A  C T C e A d (2.97) Go .t0 ; tf / WD t0

is nonsingular for all tf > t0 .  I  A has rank n at all eigenvalues i of A or 4. The .n C p/  n matrix C equivalently I  A and C are right comprime polynomial matrices. Proof. See [20], Chapter 8.  If the system (2.91) is observable, it then follows that the system initial state can be determined by Z tf T T x.t0 / D e At0 Go 1 .t0 ; tf /e A t0 e A .tt0 / C T f .t /dt; (2.98) t0

where f .t / WD y.t /  C

Z

t

e A.t/ Bu. /d  Du.t /:

(2.99)

t0

This observation is consistent with the more traditional definition of observability.

2.7.2 Controllability and Observability of Linear Time-varying Systems This section deals with controllability and observability properties of the following n-dimensional linear time-varying system x.t P / D A.t /x.t / C B.t /u.t /; y.t / D C .t /x.t / C D.t /u.t /:

(2.100)

We assume that the matrices A.t /, B.t /, C .t /, and D.t / are at least continuous functions of t .

30

2 Mathematical Preliminaries

2.7.2.1 Controllability The system (2.100) is said to be controllable on the interval Œt0 ; tf , where tf > t0  0, if for any states x0 and xf , a continuous input u.t / exists that drives the system to the state x.tf / D xf at time t D tf starting from the state x.t0 / D x0 at time t D t0 . Conditions for controllability of the system (2.100) are given in the following lemma: Lemma 2.10. Assuming that the matrix A.t / is n  2 times differentiable and B.t / is n  1 times differentiable, a sufficient condition for the system (2.100) to be controllable on the interval Œt0 ; tf  is that the following matrix has rank n for at least one t 2 Œt0 ; tf : (2.101) K .t / D ŒK0 .t / K1 .t / : : : Kn1 .t /; where Ki , i D 1; : : : ; n  1 are defined by K0 .t / D B.t /; Ki .t / D A.t /Ki1 .t / C KP i1 .t /:

(2.102)

Moreover, if the matrix K .t / has rank n for all t 2 Œt0 ; tf , then the system (2.100) is uniformly controllable on the interval Œt0 ; tf . Proof. See [20], Chapter 25. 

2.7.2.2 Observability Now suppose that the control input is zero, i.e., the system (2.100) is given by x.t P / D A.t /x.t /; y.t / D C .t /x.t /:

(2.103)

The system (2.103) is said to be observable on the interval Œt0 ; tf  if any initial state x.t0 / D x0 can be determined from the output y.t / for t 2 Œt0 ; tf . Conditions for observability of the system (2.103) are given in the following lemma. Lemma 2.11. Assuming that the matrix A.t / is n  2 times differentiable and B.t / is n  1 times differentiable, a sufficient condition for the system (2.100) to be observable on the interval Œt0 ; tf  is that the following matrix has rank n for at least one t 2 Œt0 ; tf  2 3 L0 .t / 6 L1 .t / 7 6 7 L.t / D 6 (2.104) 7; :: 4 5 : Ln1 .t /

where Li , i D 1; : : : ; n  1 are defined by

2.7 Controllability and Observability

31

L0 .t / D C .t /; P i1 .t /: Li .t / D Li1 .t /A.t / C L

(2.105)

Moreover, if the matrix L.t / has rank n for all t 2 Œt0 ; tf , then the system (2.100) is uniformly observable on the interval Œt0 ; tf . Proof. See [20], Chapter 25. 

2.7.3 Controllability and Observability of Nonlinear Systems 2.7.3.1 Controllability This subsection deals with controllability of the following nonlinear system xP D f .x/ C

m X

gi .x/ui ;

x 2 ˝x  Rn ;

(2.106)

iD1

where u D Œu1 ; u2 ; :::; um T 2 ˝u  Rm is the input vector. The system (2.106) is controllable if there exists an admissible input vector u.t / such that the state x.t / can travel from an initial point x.t0 / D x0 2 ˝x to x.tf / 2 ˝x within a finite time interval T D tf  t0 . The controllability reveals whether the control system has a set of “healthy” input channels through which the input can excite the states effectively to reach the destination xf . Based on this interpretation, the controllability of (2.106) should clearly depend on the function forms of all f .x/ and gi .x/. The controllability of the nonlinear system (2.106) is based on a useful mathematical concept called Lie algebra, which is defined as follows. Definition 2.9. A Lie algebra over the real field R or the complex field C is a vector space G for which a bilinear map .X; Y / ! ŒX; Y  is defined from G  G ! G such that 1. ŒX; Y  D ŒY; X ; X; Y 2 G; 2. ŒX; ŒY; Z C ŒY; ŒZ; X  C ŒZ; ŒX; Y  D 0 for X; Y; Z 2 G: According to this definition, a Lie algebra is a vector space where an operator Œ:; : is installed. Such a general operator, conventionally called a Lie bracket, can be defined arbitrarily as long as it satisfies the preceding two specified conditions simultaneously. The first condition is often called a skew symmetric relation and obviously implies that ŒX; X  D 0. The second condition is called the Jacobi identity, which reveals a closed loop cyclic relation among any three elements in a Lie algebra. Let us now define a special Lie algebra E that collects all n-dimensional differentiable vector fields in Rn along with a commutative derivative relation: For any two vector fields f and g 2 Rn , which are functions of x 2 Rn ,

32

2 Mathematical Preliminaries

Œf; g D

@f @g f g: @x @x

(2.107)

It can be immediately shown that this Lie bracket satisfies the two conditions of a Lie algebra. To extend the above Lie bracket between two vector fields to higher order derivatives, a more compact notation may be defined based on an adjoint operator, that is, Œf; g D adf g. This new notation treats the Lie bracket Œf; g as vector field g operated on by an adjoint operator adf D Œf; :. Therefore, for an n-order Lie bracket .n > 1/, one can simply write Œf; „ƒ‚… ::: Œf; g „ƒ‚… :::  D adfn g: n

n

For a general control system given by (2.106), we now specifically define a control Lie algebra Ec , which is spanned by all up to order .n  1/ Lie brackets among f and g1 through gm as Ec D spanfg1 ; :::; gm ; adf g1 ; :::; adf gm ; :::; adfn1 g1 ; :::; adfn1 gm g:

(2.108)

With the control Lie algebra concept, we can show that the following theorem is true and is also a general effective testing criterion for system controllability. Theorem 2.8. The control system (2.106) is controllable if and only if dim.Ec / D dim.˝x / D n. Proof. See [2].  Note that because each element in Ec is a function of x, the dimension of Ec may be different from one point to another. Thus, if the preceding condition of dimension is valid only in a neighborhood of a point in ˝x  Rn , we say that the system (2.106) is locally controllable. On the other hand, if the condition of dimension can cover all of region ˝x , then it is globally controllable. Moreover, it is not hard to show that Theorem 2.8 covers the controllability condition for a linear time-invariant system (2.91).

2.7.3.2 Observability We address the observability for the following nonlinear system xP D f .x/; y D h.x/;

(2.109)

where y 2 ˝y  Rp is the output vector. This system is said to be observable if for each pair of distinct states x1 and x2 , the corresponding outputs y1 and y2 are also distinguishable. Clearly, the observability can be interpreted as a testing criterion to check whether the entire system has sufficient output channels to measure (or

2.7 Controllability and Observability

33

observe) each internal state change. Intuitively, the observability should depend on the function forms of both f .x/ and h.x/. We now introduce a Lie derivative, which is virtually a directional derivative for a scalar field .x/, with x 2 Rn along the direction of an n-dimensional vector field f .x/. The mathematical expression is given as Lf .x/ D

@.x/ f .x/: @x

(2.110)

is a 1  n gradient vector of the scalar .x/ and the norm of a gradient Since @.x/ @x vector represents the maximum rate of function value changes, the product of the gradient and the vector field f .x/ in (2.109) becomes the directional derivative of .x/ along f .x/. Therefore, the Lie derivative of a scalar field defined by (2.110) is also a scalar field. If each component of a vector field h.x/ 2 Rp is considered to take a Lie derivative along f .x/ 2 Rn , then all components can be acted on concurrently and the result is a vector field that has the same dimension as h.x/; its i th element is the Lie derivative of the i th component of h.x/. Namely, if h.x/ D Œh1 .x/; :::; hp .x/T and each component hi .x/; i D 1; :::; p is a scalar field, then the Lie derivative of the vector field h.x/ is defined as 2 3 Lf h1 .x/ 7 6 :: Lf h.x/ D 4 (2.111) 5: : Lf hp .x/

With the Lie derivative concept, we now define an observation space ˝o over Rn as ˝o D spanfh.x/; Lf h.x/; :::; Lfn1 h.x/g:

(2.112)

In other words, this space is spanned by all up to order .n  1/ Lie derivatives of the output function h.x/. Then, we further define an observability distribution, denoted by d ˝o , which collects the “gradient” vector of every component in ˝o . Namely,   @ ˇˇ (2.113) d ˝o D span ˇ 2 ˝o : @x

With these definitions, we can present the following theorem for testing the observability. Theorem 2.9. The system (2.109) is observable if and only if dim.d ˝o / D n. Proof. See [2].  Similarly to the controllability case, this testing criterion also has locally observable and globally observable cases, depending on whether the condition of dimension in the theorem is valid only in a neighborhood of a point or over the entire state region.

34

2 Mathematical Preliminaries

2.7.4 Brockett’s Theorem on Feedback Stabilization The following theorem, which is due to Brockett [21], gives a necessary condition for the existence of a stabilizing control law for the system xP D f .x; u/

(2.114)

at an equilibrium point x0 with x being the state and u being the control input. Theorem 2.10. Let the system (2.114) be given with f .x0 ; 0/ D 0 and f .
;
/ continuously differentiable in a neighborhood of .x0 ; 0/. A necessary condition for the existence of a continuously differentiable control law that makes .x0 ; 0/ asymptotically stable is that 1. the linearized system should have no uncontrollable modes associated with eigenvalues whose real part is positive, 2. there exists a neighborhood ˝ of .x0 ; 0/ such that for each  2 ˝ there exists a control u .
/ defined on Π0; 1/ such that this control steers the solution of xP D f .x; u / from x D  at t D 0 to x D x0 at t D 1, 3. the mapping

W A  Rm ! Rn defined by W .x; u/ ! f .x; u/ should be onto an open set containing 0. Proof. See [21].  Remark 2.3. If the system (2.114) is of the form xP D f .x/ C

m X

ui gi .x/I

x.t / 2 ˝  Rn ;

(2.115)

iD1

then condition 3 of Theorem 2.10 implies that the stabilization problem cannot have a solution if there is a smooth distribution D containing f .
/ and g1 .
/; : : : ; gm .
/ with dim D < n. One further special case: If the system (2.114) is of the form xP D

m X

ui gi .x/I

x.t / 2 ˝  Rn

(2.116)

iD1

with the vectors gi .x/ being linearly independent at x0 , then there exists a solution to the stabilization problem if and only if m D n. In this case, we must have as many control parameters as we have dimensions of x.

2.8 Conclusions This chapter briefly provides the fundamental tools that will be used for control design and stability analysis of underactuated vehicles in the coming chapters. The

2.8 Conclusions

35

interested reader is referred to [3–6, 22–24] for a detailed and comprehensive coverage of the topics discussed in this chapter.

Part II

Modeling and Control Properties of Ocean Vessels

Chapter 3

Modeling of Ocean Vessels

In this chapter, we classify the basic motion tasks for ocean vessels and their mathematical models, which will be used for the design of various control systems in the subsequent chapters.

3.1 Introduction In automatic control, feedback improves system performance by allowing the successful completion of a task even in the presence of external disturbances and initial errors, and inaccuracy of the system parameters. To this end, real-time sensor measurements are used to reconstruct the vehicle state. Throughout this study, the latter is assumed to be available at every instant, as provided by local/global position and orientation measurement sensors. In some cases, we also assume that the vehicle velocities are measurable or constructible from position measurements. We will concentrate on the case of a vessel workspace free of obstacles. In fact, we implicitly consider the vessel controller to be embedded in a hierarchical architecture in which a higher-level planner solves the obstacle avoidance problem and provides a series of motion goals to the lower control layer. In this perspective, the controller deals with the basic issue of converting ideal plans into actual motion execution. The nonholonomic nature of the ocean vessels is related to the fact that the vessel does not usually have independent actuators in the sway and heave axes. This implies the presence of a nonintegrable set of second-order differential constraints on the configuration variables. While these nonholonomic constraints reduce the instantaneous motions that the vessel can perform, they still allow almost global controllability in the configuration space. This feature leads to some challenging problems in the synthesis of feedback controllers, which parallel the new research issues arising in nonholonomic motion planning. Indeed, the ocean vessel application has triggered the search for innovative types of feedback controllers that can be used also for more general nonlinear systems that describe the motion of

39

40

3 Modeling of Ocean Vessels

more complicated vessel systems such as ocean vessels and air vehicles working in a group.

3.2 Basic Motion Tasks In order to derive the most suitable feedback controllers for each case, it is convenient to classify the possible motion tasks as follows:  Point-to-point motion: The vessel must reach a desired goal configuration starting from a given initial configuration, see Figure 3.1a.  Path-following: The vessel must reach and follow a geometric reference path in the Cartesian space starting from a given initial configuration (on or off the path), see Figure 3.1b.  Trajectory-tracking and path-tracking: The vessel must reach and follow a reference trajectory/path in the Cartesian space (i.e., a geometric path with an associated timing law) starting from a given initial configuration (on or off the trajectory/path), see Figure 3.1c. Trajectory-tracking is referred to as the case where the reference trajectory is generated by a suitable virtual vessel whereas the reference path is not required to be generated by a virtual vessel for the pathtracking. The above tasks for an ocean vessel are sketched in Figure 3.1. Execution of these tasks can be achieved using either feedforward commands, or feedback control, or a combination of the two. Indeed, feedback solutions exhibit an intrinsic degree of robustness. Using a more control-oriented terminology, the point-to-point motion task is a stabilization problem for a (equilibrium) point in the vessel state space. When using a feedback strategy, the point-to-point motion task leads to a state regulation control problem for a point in the vessel state space. Posture stabilization is another frequently used term. Without loss of generality, the goal can be taken as the origin of the n-dimensional vessel configuration space. Contrary to the usual situation, trajectory-tracking, path-tracking, and path-following are easier than regulation for a nonholonomic vessel. An intuitive explanation of this can be given in terms of a comparison between the number of controlled variables (outputs) and the number of control inputs. For the ship or underwater vehicle moving in a horizontal plane, two input commands are available while three variables (position and orientation) are needed to determine its configuration. Thus, regulation of the surface ship or the underwater vehicle in a horizontal position to a desired configuration implies zeroing three independent configuration errors. In the path-following task, the controller is given a geometric description of the assigned Cartesian path. This information is usually available in a parameterized form expressing the desired motion in terms of a path parameter, which may be in particular the arc length along the path. For this task, time dependence is not relevant because one is concerned only with the geometric displacement between the vessel

3.2 Basic Motion Tasks

41

a Goal Start

b Start

eter Param s

Reference path

Start

c Reference trajectory/path

t

Time

Figure 3.1 Basic motion tasks for an ocean vessel

and the path. In this context, the time evolution of the path parameter is usually free and, accordingly, the command inputs can be arbitrarily scaled with respect to time without changing the resulting vessel path. It is then customary to set the vessel forward velocity (one of the inputs) to an arbitrary constant or time-varying value, leaving the other input variables for control. The path-following problem is thus rephrased as the stabilization to zero of a suitable scalar path error function using only the rest of the control inputs. In the trajectory-tracking and path-tracking tasks, the vessel must follow the desired Cartesian path with a specified timing law. Although the reference trajectory/path can be split into a parameterized geometric path and a timing law for the parameter, such separation is not strictly necessary. Often, it is simpler to specify the workspace trajectory as the desired time evolution for the position of some representative point of the vessel. The trajectory-tracking and path-tracking problems consist then in the stabilization to zero of the Cartesian errors using all the available control inputs. The point stabilization problem can be formulated in a local or in a global sense, the latter meaning that we allow for initial configurations that are arbitrarily far from the destination. The same is true also for path-following, trajectory-tracking, and path-tracking, although locality has two different meanings in these tasks. For

42

3 Modeling of Ocean Vessels

path-following, a local solution means that the controller works properly, provided that we start sufficiently close to the path; for trajectory-tracking and path-tracking, closeness should be evaluated with respect to the current position of the reference vessel and a current reference point, which moves on the reference path with a specified time law, on the reference path, respectively. The amount of information that should be provided by a high-level motion planner varies for each control task. In point-to-point motion, information is reduced to a minimum (i.e., the goal configuration only) when a globally stabilizing feedback control solution is available. However, if the initial error is large, such a control may produce erratic behavior and/or large control effort, which are unacceptable in practice. On the other hand, a local feedback solution requires the definition of intermediate subgoals at the task planning level in order to get closer to the final desired configuration. For the other motion tasks, the planner should provide a path that is kinematically feasible (namely, that complies with the nonholonomic constraints of the specific vessel), so as to allow its perfect execution in nominal conditions. While for a fully or overactuated vessel in which any path is feasible, some degree of geometric smoothness is in general required for nonholonomic vessels. Nevertheless, the intrinsic feedback structure of the driving commands enables it to recover transient errors due to isolated path discontinuities. Note also that the infeasibility arising from a lack of continuity in some higher-order derivative of the path may be overcome by appropriate motion timing. For example, paths with discontinuous curvature (like the Reeds and Shepp optimal paths under maximum curvature constraint) can be executed by choosing an appropriate point on the vessel provided that the vessel is allowed to stop, whereas paths with discontinuous tangent are not feasible. In this analysis, the selection of the vessel representative point for path/trajectory planning is critical. The timing profile is the additional item needed in trajectory-tracking and path-tracking control tasks. This information is seldom provided by current motion planners, also because the actual dynamics of the specific vessel are typically neglected at this level. The above example suggests that it may be reasonable at the planning stage to enforce requirements such as “move slower where the path curvature is higher”.

3.3 Modeling of Ocean Vessels Modeling of the ocean vessels is usually based on mechanics, principles of statics and dynamics. Statics is concerned with the equilibrium of bodies at rest or moving with a constant velocity. Dynamics deals with bodies having accelerated motion resulting from disturbances or/and control forces. Since we are interested in a mathematical model of the ocean vessels for the purpose of designing the control systems, this section focuses on dynamics of the vessels rather than statics. The following briefly presents the ocean vessel equations of motion based on the results in [11]. The resulting nonlinear model presented in this section is mainly intended for designing control systems in the next chapters. For a detailed and comprehensive derivation of the model, the reader is referred to [11,12,25]. The physical and control

3.3 Modeling of Ocean Vessels

43

properties of the model are also presented for control design and stability analysis. In this section, we use the notation, see Table 3.1 and Figure 3.2, that complies with the Society of Naval Architects and Marine Engineers (SNAME) [26].

( y, v) Sway

x

Ob Earth-fixed frame

XE

Yb (T , q) Pitch

( x, u ) Surge

Xb

(I , p Roll )

Body-fixed frame

OE ZE

YE

Zb

x CG (\ , r ) Yaw

( z , w) Heave

Figure 3.2 Motion variables for an ocean vessel

For an ocean vessel moving in six degrees of freedom, six independent coordinates are required to determine its position and orientation. The first three coordinates .x; y; z/ and their first time derivatives correspond to the position and translational motion along the x-, y- and z-axes, while the last three coordinates .; ; / and their first time derivatives describe orientation and rotational motion. Table 3.1 SNAME Notation for ocean vessels Degree of freedom Force and Linear and Position and moment angular velocity Euler angles 1 Surge X u x 2 Sway Y v y 3 Heave Z w z 4 Roll K p  5 Pitch M q  6 Yaw N r

According to SNAME, the six different motion components are defined as surge, sway, heave, roll, pitch, and yaw. To determine the equations of motion, two reference frames are considered: the inertial or fixed to earth frame OE XE YE ZE that may be taken to coincide with the vessel fixed coordinates in some initial condition and the body-fixed frame Ob Xb Yb Zb see Figure 3.2. Since the motion of the Earth hardly affects ocean vessels (different from air vehicles), the earth-fixed frame OE XE YE ZE can be considered to be inertial. For ocean vessels in general, the

44

3 Modeling of Ocean Vessels

most commonly adopted position for the body-fixed frame is such that it gives hull symmetry about the Ob Xb Zb -plane and approximate symmetry about the Ob Yb Zb plane. In this sense, the body axes Ob Xb , Ob Yb , and Ob Zb coincide with the principal axes of inertia and are usually defined as follows: Ob Xb is the longitudinal axis (directed from aft to fore); Ob Yb is the transverse axis (directed to starboard); and Ob Zb is normal axis (directed from top to bottom. Based on the notion in Table 3.1, the general motion of an ocean vessel can be described by the following vectors:  D Œ1 2 T ;

1 D Œx y zT ;

2 D Π

T ;

v D Œv1 v2 T ;  D Œ1 2 T ;

v1 D Œu v wT ; 1 D ŒX Y ZT ;

v2 D Œp q rT ; 2 D ŒK M N T ;

where  denotes the position and orientation vector with coordinates in the earthfixed frame, v denotes the linear and angular velocity vector with coordinates in the body-fixed frame, and  denotes the forces and moments acting on the vessel in the body-fixed frame. In deriving equations of motion of the ocean vessels, we divide the study of vessel dynamics into two parts kinematics, which treats only geometrical aspects of motion, and kinetics, which is the analysis of the forces resulting in the motion.

3.3.1 Kinematics The first time derivative of the position vector 1 is related to the linear velocity vector v1 via the following transformation: P 1 D J1 .2 /v1 ;

(3.1)

where J1 .2 / is a transformation matrix, which is related through the functions of the Euler angles: roll ./, pitch ./, and yaw . /. This matrix is given by 2

cos. / cos./  sin. / cos./ C sin./ sin./ cos. / J1 .2 / D 4 sin. / cos./ cos. / cos./ C sin./ sin./ sin. /  sin./ sin./ cos./ 3 sin. / sin./ C sin./ cos. / cos./  cos. / sin./ C sin./ sin. / cos./ 5 : cos./ cos./

(3.2)

It is noted that the matrix J1 .2 / is globally invertible since J11 .2 / D J1T .2 /. On the other hand, the first time derivative of the Euler angle vector 2 is related to the body-fixed velocity vector v2 through the following transformation: P 2 D J2 .2 /v2 ;

(3.3)

3.3 Modeling of Ocean Vessels

where the transformation matrix J2 .2 / is given by 3 2 1 sin./ tan./ cos./ tan./ cos./  sin./ 5 : J2 .2 / D 4 0 0 sin./= cos./ cos./= cos./

45

(3.4)

Note that the transformation matrix J2 .2 / is singular at  D ˙ 2 . However, during practical operations ocean vessels are not likely to enter the neighborhood of  D ˙ 2 because of the metacentric restoring forces. For the case where it is essential to consider a region containing  D ˙ 2 , a four-parameter description based on Euler parameters can be used instead. The interested reader is referred to [12] for more details. Combining (3.1) and (3.3) results in the kinematics of the ocean vessels:      J1 .2 / 033 v1 P 1 D , P D J ./v: (3.5) 033 J2 .2 / P 2 v2

3.3.2 Kinetics 3.3.2.1 Rigid Body Equations of Motion Let us define the following vectors: fOb D ŒX Y ZT : force decomposed in the body-fixed frame. mOb D ŒK M N T : moment decomposed in the body-fixed frame. vOb D Œu v wT : linear velocity decomposed in the body-fixed frame. !E D Œp q rT : angular velocity of the body-fixed frame relative to the earthOb fixed frame.  rOb D Œxg yg zg T : vector from Ob to C G (center of gravity of the vessel) decomposed in the body-fixed frame.

   

By the Newton–Euler formulation for a rigid body with a mass of m, we have the following balancing forces and moments: E E E PE mŒvP Ob C ! Ob  rOb C !Ob  vOb C !Ob  .!Ob  rOb / D fOb ; E E P Ob C !E PE Io ! Ob C !Ob  Io !Ob C mrOb  .v Ob  vOb / D mOb ;

where Io is the inertia matrix about Ob defined by 3 2 Ix Ixy Ixz Io D 4 Iyx Iy Iyz 5 : Izx Izy Iz

(3.6)

(3.7)

Here Ix , Iy , and Iz are the moments of inertia about the Ob Xb , Ob Yb , and Ob Zb axes, and Ixy D Iyx , Ixz D Izx , and Iyz D Izy are the products of inertia. These quantities are defined as

46

3 Modeling of Ocean Vessels

Ix D

Z

.y 2 C z 2 /m dV; Ixy D

Z

.x 2 C z 2 /m dV; Ixz D

Z

.x 2 C y 2 /m dV; Izy D

V

Iy D

V

Iz D

V

Z

ZV ZV

xym dV; xzm dV;

(3.8)

zym dV;

V

where m and V are, respectively, the mass density and the volume of the rigid body. , and rOb into (3.6) results in Substituting the definitions of fOb , mOb , vOb , !E Ob the following equations of motion of a rigid body: MRB vP C CRB .v/v D RB ;

(3.9)

where v D Œu v w p q rT is the generalized velocity vector decomposed in the body-fixed frame, RB D ŒX Y Z K M N T is the generalized vector of external forces and moments, the rigid body system inertia matrix MRB is given by 2 3 m 0 0 0 mzg myg 6 0 m 0 mzg 0 mxg 7 6 7 6 0 0 m my mx 0 7 g g 6 7; (3.10) MRB D 6 Ix Ixy Ixz 7 6 0 mzg myg 7 4 mzg 0 mxg Iyx Iy Iyz 5 myg mxg 0 Izx Izy Iz and the rigid body Coriolis and centripetal matrix CRB .v/ is given by 2 0 0 0 6 0 0 0 6 6 0 0 0 CRB .v/ D 6 6 m.yg q C zg r/ m.yg p C w/ m.z p  v/ g 6 4 m.xg q  w/ m.zg r C xg p/ m.zg q C u/ m.xg r C v/ m.yg r  u/ m.xg p C yg q/

(3.11) 3 m.yg q C zg r/ m.xg q  w/ m.xg r C v/ 7 m.yg p C w/ m.zg r C xg p/ m.yg r  u/ 7 m.zg p  v/ m.zg q C u/ m.xg p C yg q/ 7 7: 0 Iyz q  Ixz p C Iz r Iyz r C Ixy p  Iy q 7 7 Iyz q C Ixz p  Iz r 0 Ixz r  Ixy q C Ix p 5 Iyz r  Ixy p C Iy q Ixz r C Ixy q  Ix p 0

The generalized external force and moment vector, RB , is a sum of hydrodynamic force and moment vector H , external disturbance force and moment vector E , and propulsion force and moment vector . Each of these vectors is detailed in the following sections.

3.3 Modeling of Ocean Vessels

47

3.3.2.2 Hydrodynamic Forces and Moments In hydrodynamics, it is usually assumed that the hydrodynamic forces and moments on a rigid body can be linearly superimposed, see [27]. The hydrodynamic forces and moments are forces and moments on the body when the body is forced to oscillate with the wave excitation frequency and there are no incident waves. These forces and moments can be identified as the sum of three components: (1) added mass due to the inertia of the surrounding fluid, (2) radiation-induced potential damping due to the energy carried away by generated surface waves, and (3) restoring forces due to Archimedian forces (weight and buoyancy). The hydrodynamic force and moment vector H is given by H D MA vP  CA .v/v  D.v/v  g./;

(3.12)

where MA is the added mass matrix, CA .v/ is the hydrodynamic Coriolis and centripetal matrix, D.v/ is the damping matrix, and g./ is the position and orientation depending vector of restoring forces and moments. The added mass matrix MA is given by 2 3 XuP XvP XwP XpP XqP XrP 6 YuP YvP YwP YpP YqP YrP 7 6 7 6 ZuP ZvP ZwP ZpP ZqP ZrP 7 6 7; (3.13) MA D  6 7 6 KuP KvP KwP KpP KqP KrP 7 4 MuP MvP MwP MpP MqP MrP 5 NuP NvP NwP NpP NqP NrP where the SNAME notation has been used. For example, the hydrodynamic added mass force Y along the y-axis due to an acceleration uP in the x-direction is written as @Y Y D YuP u; : (3.14) P YuP WD @uP The hydrodynamic Coriolis and centripetal matrix is given by 3 2 0 0 0 0 a3 a2 6 0 0 0 a3 0 a1 7 7 6 6 0 0 0 a2 a1 0 7 7 (3.15) CA .v/ D 6 6 0 a3 a2 0 b3 b2 7 ; 7 6 4 a3 0 a1 b3 0 b1 5 a2 a1 0 b2 b1 0 where a1 D XuP u C XvP v C XwP w C XpP p C XqP q C XrP r; a2 D YuP u C YvP v C YwP w C YpP p C YqP q C YrP r; a3 D ZuP u C ZvP v C ZwP w C ZpP p C ZqP q C ZrP r; b1 D KuP u C KvP v C KwP w C KpP p C KqP q C KrP r;

48

3 Modeling of Ocean Vessels

b2 D MuP u C MvP v C MwP w C MpP p C MqP q C MrP r; b3 D NuP u C NvP v C NwP w C NpP p C NqP q C NrP r:

(3.16)

In general, hydrodynamic damping for ocean vessels is mainly caused by potential damping, skin friction, wave drift damping, and damping due to vortex shedding. It is difficult to give a general expression of the hydrodynamic damping matrix D.v/. However, it is common to write the hydrodynamic damping matrix D.v/ as D.v/ D D C Dn .v/: Here the linear damping matrix D is given by 2 Xu Xv Xw Xp 6 Yu Yv Yw Yp 6 6 Zu Zv Zw Zp D D 6 6 Ku Kv Kw Kp 6 4 Mu Mv Mw Mp Nu Nv Nw Np

Xq Yq Zq Kq Mq Nq

(3.17)

3 Xr Yr 7 7 Zr 7 7: Kr 7 7 Mr 5 Nr

(3.18)

The nonlinear damping matrix Dn .v/ is usually modeled by using a third-order Taylor series expansion or modulus functions (quadratic drag). If the xz-plane is a plane of symmetry (starboard/port symmetry) an odd Taylor series expansion containing first-order and third-order terms in velocity can be sufficient to describe most manoeuvres. An approximate expression of each of this matrices will be given in the next section when specific vessels are considered.

3.3.2.3 Restoring Forces and Moments In this section, a model for g./ is described. Let r be the volume of fluid displaced by the vessel, g the acceleration of gravity (positive downwards), and  the water density. The submerged weight of the body and buoyancy force are defined as W D mg; B D gr:

(3.19)

With the above definition, the restoring force and moment vector g./ is due to gravity and buoyancy forces, and is given by 3 2 .W  B/ sin./ 7 6 .W  B/ cos./ sin./ 7 6 7 6 .W  B/ cos./ cos./ 7 g./ D 6 6 .yg W  yb B/ cos./ cos./ C .zg W  zb B/ cos./ sin./ 7 ; (3.20) 7 6 5 4 .zg W  zb B/ sin./ C .xg W  xb B/ cos./ cos./ .xg W  xb B/ cos./ sin./  .yg W  yb B/ sin./

3.3 Modeling of Ocean Vessels

49

where .xb ; yb ; zb / denote coordinates of the center of buoyancy.

3.3.2.4 Environmental Disturbances In this section, we detail the vector, E , of forces and moments induced by environmental disturbances including ocean currents, waves (wind generated) and wind, i.e., we can write cu wa wi E D E C E C E ; (3.21) cu wa wi where E , E , and E are vectors of forces and moments induced by ocean currents, waves and wind, respectively.

Current-induced Forces and Moments cu of the current-induced forces and moments is given by The vector E cu E D .MRB C MA /vP c C C .vr /vr  C .v/v C D.vr /vr  D.v/v;

(3.22)

where vr D v  vc and vc D Œuc ; vc ; wc ; 0; 0; 0T is a vector of irrotational bodyfixed current velocities. Let the earth-fixed current velocity vector be denoted by E E T T ŒuE c ; vc ; wc  . Then, the body-fixed components Œuc ; vc ; wc  can be computed as 2 3 2 E3 uc uc 4 vc 5 D J1T .2 / 4 vcE 5 : (3.23) wc wcE Wave-induced Forces and Moments wa The vector E of the wave-induced forces and moments is given by

PN 3 iD1 gBLT cos.ˇ/si .t / P N 6 7 iD1 gBLT sin.ˇ/si .t / 6 7 6 7 0 6 7; D6 7 0 6 7 4 5 0 PN 1 2 2 2 iD1 24 gBL.L  B / sin.2ˇ/si .t / 2

wi E

(3.24)

where ˇ is the vessel’s heading (encounter) angle, see Figure 3.3,  is the water density, L is the length of the vessel, B is the breadth of the vessel, and T is the draft of the vessel. Ignoring the higher-order terms of the wave amplitude, the wave slope si .t / for the wave component i is defined as:

50

3 Modeling of Ocean Vessels

si .t / D Ai

2 sin.!ei t C i /; i

(3.25)

where Ai is the wave amplitude, i is the wave length, !ei is the encounter frequency, and i is a random phase uniformly distributed and constant with time in Œ0 2/ corresponding to the wave component i . Beam sea

E

60o

E 120o

Quartering sea

E

30o

Bow sea

E 150o Following sea

E

0o

Head sea

Figure 3.3 Definition of a vessel’s heading (encounter) angle

Wind-induced Forces and Moments wi For the case where the vessel is at rest (zero speed), the vector E of the windinduced forces and moments is given by 2 3 CX . w /AF w 6 CY . w /ALw 7 6 7 6 7 1 C . /A Z w Fw wi 26 7; (3.26) E D a Vw 6 7 C . /A H 2 6 K w Lw Lw 7 4 CM . w /AF w HF w 5 CN . w /ALw Loa

where Vw is the wind speed, a is the air density, AF w is the frontal projected area, ALw is the lateral projected area, HF w is the centroid of AF w above the water line, HLw is the centroid of ALw above the water line, Loa is the over all length of the vessel, w is the angle of relative wind of the vessel bow, see Figure 3.4, and is given by (3.27)

w D  ˇw  ;

3.3 Modeling of Ocean Vessels

51

with ˇw being the wind direction. All the wind coefficients (look-up tables) CX . w /, CY . w /, CZ . w /, CK . w /, CM . w /, and CN . w / are computed numerically or by experiments in a wind tunnel, see [28].

Ew

Xb

Yb Vw

Jw

x

Ob

\ Zb Figure 3.4 Definition of wind speed and direction wi For the case where the vessel is moving, the vector E is given by 3 2 CX . rw /AF w 7 6 CY . rw /ALw 7 6 6 1 CZ . rw /AF w 7 wi 2 6 7 E D a Vrw 6 CK . rw /ALw HLw 7 2 7 6 4 CM . rw /AF w HF w 5 CN . rw /ALw Loa

(3.28)

where Vrw D

q 2 ; u2rw C vrw

rw D arctan2.vrw ; urw /;

(3.29)

with urw D u  Vw cos.ˇw  /; vrw D v  Vw sin.ˇw  /:

(3.30)

52

3 Modeling of Ocean Vessels

3.3.2.5 Propulsion Forces and Moments The vector, , of propulsion forces and moments depends on a specific configuration of actuators such as propellers, rudders, and water jets on a particular vessel. In the next section where  is specified, we consider some classes of the ocean vessels that are common in practice. In this book, we neglect the dynamics of the actuators that provide the propulsion forces and moments since the response of the actuators such as hydraulic systems and electrical motors is much faster than the response of the vessel.

3.3.2.6 Model Summary and its Properties Body-fixed Representation Now substituting RB D H C E C  into (3.9) and combining it with (3.5) results in the equations of motion of an ocean vessel in six degrees of freedom as follows: P D J ./v; M vP D C .v/v  D.v/v  g./ C  C E ;

(3.31)

M D MRB C MA ; C .v/ D CRB .v/ C CA .v/:

(3.32)

where

Under the assumption that the body is at rest (or at most is moving at low speed) in ideal fluid, the matrix M is always symmetric positive definite, i.e., M D M T > 0:

(3.33)

For a rigid body moving in fluid, the Coriolis and centripetal matrix C .v/ can always be parameterized such that it is skew-symmetric, i.e., C .v/ D C T .v/; 8 v 2 R6 :

(3.34)

For a rigid body moving in an ideal fluid, the hydrodynamic damping matrix D.v/ is real, non-symmetric and strictly positive, i.e., D.v/ > 0; 8 v 2 R6 :

(3.35)

Earth-fixed Representation The mathematical model (3.31) can also be written using a representation of the earth-fixed coordinates by applying the following kinematic transformations (with

3.3 Modeling of Ocean Vessels

53

the assumption that J 1 ./ exists, i.e.,  ¤ ˙ 2 ): P v D J 1 ./; h i 1 vP D J ./ R  JP ./J 1 ./P :

(3.36)

Now substituting (3.36) into the second equation of (3.31) results in M ? ./R D C ? .v; /P  D ? .v; /P  g ? ./ C J T ./. C E /;

(3.37)

where M ? ./ D J T ./MJ 1 ./;   C ? .v; / D J T ./ C .v/  MJ 1 ./JP ./ J 1 ./;

D ? .v; / D J T ./D.v/J 1 ./; g ? ./ D J T ./g./:

(3.38)

Under the same assumptions used in the body-fixed representation, the model (3.37) using the earth-fixed representation has the following properties: M ? ./ D M ? ./T ; 8  2 R6 ;   ? P ./  2C ? .v; / s D 0; 8  2 R6 ; v 2 R6 ; s 2 R6 ; sT M D ? .v; / > 0;

8  2 R6 ; v 2 R6 :

Yb (Sway) YE X b (Surge)

y

OE

Ob

\ (Yaw) x

x

Figure 3.5 Motion variables for an ocean vessel moving in a horizontal plane

XE

(3.39)

54

3 Modeling of Ocean Vessels

3.4 Standard Models for Ocean Vessels In this section, the main results of the previous section are simplified to give a set of standard models for surface ships and underwater vehicles. These standard models will be extensively used for the control design in the coming chapters.

3.4.1 Three Degrees of Freedom Horizontal Model 3.4.1.1 Standard Three Degrees of Freedom Horizontal Model The horizontal motion of a surface ship or an underwater vehicle moving in a horizontal plane is often described by the motion components in surge, sway, and yaw. Figure 3.5 illustrates the motion variables in this case. Therefore, we choose  D Œx y T and v D Œu v rT . This model is obtained from the general model (3.31) under the following assumption. Assumption 3.1. 1. The motion in roll, pitch, and heave is ignored. This means that we ignore the dynamics associated with the motion in heave, roll, and pitch, i.e., z D 0, w D 0,  D 0, p D 0,  D 0, and q D 0. 2. The vessel has homogeneous mass distribution and xz-plane of symmetry so that Ixy D Iyz D 0:

(3.40)

3. The center of gravity C G and the center of buoyancy, CB, are located vertically on the z-axis. With Assumption 3.1, the dynamics of a surface ship or an underwater vehicle moving in a horizontal plane is simplified from the general model (3.31) as follows: P D J ./v; M vP D C .v/v  .D C Dn .v//v C  C E ;

(3.41)

where the matrices J ./, M ; C .v/; D, and Dn .v/ are given by 2 3 3 2 m  XuP cos. /  sin. / 0 0 0 m  YvP mxg  YrP 5 ; J ./ D 4 sin. / cos. / 0 5 ; M D 4 0 0 mxg  YrP Iz  NrP 0 0 1

3.4 Standard Models for Ocean Vessels

2

C .v/ D 4

55

0 0 0 0 m.xg r C v/  YvP v  YrP r mu C XuP u 3 m.xg r C v/ C YvP v C YrP r 5; mu  XuP u 0

3 Xu 0 0 D D  4 0 Yv Yr 5 ; 0 Nv Nr 3 2 Xjuju juj 0 0 5: Yjvjr jvj 0 Yjvjv jvj C Yjrjv jrj Dn .v/ D  4 0 Njvjv jvj C Njrjv jrj Njvjr jvj C Njrjr jrj 2

The propulsion force and moment vector  is given by 2 3 u  D 4 0 5: r

(3.42)

(3.43)

The above propulsion force and moment vector  implies that we are considering a surface vessel, which does not have an independent actuator in the sway, i.e., an underactuated vessel is under consideration. Such a vessel can be one equipped with a pair of water jets or a pair of propellers. The environmental disturbance vector E is given by 3 2 uE (3.44) E D 4 vE 5 ; rE where uE and vE are disturbance forces acting in surge and sway respectively, and rE is the disturbance moment acting in yaw.

3.4.1.2 Simplified Three Degrees of Freedom Horizontal Model In some cases, in addition to Assumption 3.1 we ignore the off-diagonal terms of the matrices M and D, all elements of the nonlinear damping matrix Dn .v/. These assumptions hold when the vessel has three planes of symmetry, for which the axes of the body-fixed reference frame are chosen to be parallel to the principal axis of the displaced fluid, which are equal to the principal axis of the vessel. Most ships have port/starboard symmetry, and moreover, bottom/top symmetry is not required for horizontal motion. Ship fore/aft nonsymmetry implies that the off-diagonal terms of the inertia and damping matrices are nonzero. However, these terms are small compared to the main diagonal terms. Furthermore, disturbances induced by waves, wind, and ocean currents are ignored. Under the just-mentioned assumptions, the

56

3 Modeling of Ocean Vessels

dynamics of a surface ship or an underwater vehicle moving in a horizontal plane is simplified from the three degrees of freedom model (3.41) as follows: P D J ./v; M vP D C .v/v  Dv C ; where the matrices J ./, M ; C .v/ and D are given by 3 2 3 2 m11 0 0 cos. /  sin. / 0 J ./ D 4 sin. / cos. / 0 5 ; M D 4 0 m22 0 5 ; 0 0 1 0 0 m33 2

(3.45)

(3.46)

3 3 2 0 m22 v d11 0 0 0 m11 u 5 ; D D 4 0 d22 0 5 ; C .v/ D 4 0 0 0 d33 m22 v m11 u 0 0

with m11 D m  XuP ; m22 D m  YvP ; m33 D Iz  NrP ; d11 D Xu ; d22 D Yv ; d33 D Nr :

(3.47)

The propulsion force and moment vector  is still given by (3.43), i.e.,  D Œu 0 r T .

3.4.1.3 Spherical Three Degrees of Freedom Horizontal Model In addition to the assumptions made in Subsection 3.4.1.2, we assume that the vessel has bottom/top symmetry. An example of this type of vessels is an ODIN moving in a horizontal plane, see Figure 3.6. In this case, the model is further simplified to P D J ./v; M vP D C .v/v  Dv C 

(3.48)

where the matrices J ./, M ; C .v/ and D are given by 3 2 3 2 0 cos. /  sin. / 0 mxy 0 J ./ D 4 sin. / cos. / 0 5 ; M D 4 0 mxy 0 5 ; 0 0 m33 0 0 1 3 2 3 0 mxy v dxy 0 0 0 mxy u 5 ; D D 4 0 dxy 0 5 ; C .v/ D 4 0 0 0 d33 mxy v mxy u 2

with

0 0

(3.49)

3.4 Standard Models for Ocean Vessels

mxy D m  XuP D m  YvP ; m33 D Iz  NrP ; dxy D Xu D Yv ; d33 D Nr :

57

(3.50)

The propulsion force and moment vector  is still given by (3.43), i.e.  D Œu 0 r T .

Figure 3.6 An omnidirectional intelligent navigator (ODIN). Courtesy http://www.math.hawaii.edu/ryan/STOMP/Photos 20ODIN.html

3.4.2 Six Degrees of Freedom Model 3.4.2.1 Standard Model In addition to the assumptions made in Section 3.3, we assume that the center of gravity and the center of buoyancy are located vertically on the Ob Zb -axis, and that there are no couplings (off-diagonal terms) in the matrices M , D, and Dn .v/. In this case, the model presented in Section 3.3 is simplified to P 1 D J1 .2 /v1 ; M1 vP 1 D C1 .v1 /v2  D1 v1  Dn1 .v1 /v1 C 1 C 1E ; P 2 D J2 .2 /v2 ; M2 vP 2 D C1 .v1 /v1  C2 .v2 /v2  D2 v2  Dn2 .v2 /v2  g2 .2 / C 2 C 2E ;

(3.51)

58

3 Modeling of Ocean Vessels

where J1 .2 / and J2 .2 / are given in (3.2) and (3.4). The matrices M1 and M2 are 3 2 m11 0 0 M1 D 4 0 m22 0 5 ; 0 0 m33 3 2 m44 0 0 M2 D 4 0 m55 0 5 ; (3.52) 0 0 m66 where m11 D m  XuP ; m22 D m  YvP ; m33 D m  ZwP ; m44 D Ix  KpP ; m55 D Iy  MqP ; m66 D Iz  NrP : The matrices C1 .v1 / and C2 .v2 / are 2 3 0 m33 w m22 v 0 m11 u 5 ; C1 .v1 / D 4 m33 w m22 v m11 u 0 3 2 0 m66 r m55 q 0 m44 p 5 : C2 .v2 / D 4 m66 r m55 q m44 p 0 The linear damping matrices D1 and D2 are 3 2 d11 0 0 D1 D 4 0 d22 0 5 ; 0 0 d33 3 2 d44 0 0 D2 D 4 0 d55 0 5 ; 0 0 d66 where d11 D Xu ; d22 D Yv ; d33 D Zw ; d44 D Kp ; d55 D Mq ; d66 D Nr :

(3.53)

(3.54)

3.4 Standard Models for Ocean Vessels

59

The nonlinear damping matrices Dn1 .v1 / and Dn2 .v2 / are 2 P3

iD2 dui juj

Dn1 .v1 / D 4

2 P3

i1

0 0

iD2 dpi jpj

Dn2 .v2 / D 4

0 0

3 0 0 i1 5; 0 iD2 dvi jvj P3 i1 0 iD2 dwi jwj 3 0 0 P3 i1 5 ; (3.55) 0 iD2 dqi jqj P3 i1 0 iD2 dri jrj

P3 i1

where dui , dvi , dwi , dpi , dqi , and dri with i D 2; 3 are the nonlinear hydrodynamic damping coefficients. The restoring force and moment vector g2 .2 / is given by 2

6 g2 .2 / D 4

grGMT sin./ cos./ grGML sin./ 0

3

7 5;

(3.56)

where , g, r, GMT and GML are the water density, gravity acceleration, displaced volume of water, transverse metacentric height and longitudinal metacentric height, respectively. The propulsion force and moment vectors 1 and 2 are 2 3 2 3 p u 1 D 4 0 5 ; 2 D 4 q 5 ; (3.57) 0 r which imply that the vessel under consideration does not have independent actuators in the sway and heave. The environmental disturbance vectors 1E and 2E are given by 3 2 3 2 Ep E u (3.58) 1E D 4 E v 5 ; 2E D 4 E q 5 ; E r E w where E u , E v , E w , Ep , E q , and E r are the environmental disturbance forces or moments acting on the surge, sway, heave, roll, pitch, and yaw axes, respectively.

3.4.2.2 Ignoring Nonlinear Damping Terms and Roll Model In addition to the assumptions made in Section 3.4.2.1, it is sometimes reasonable to ignore nonlinear hydrodynamic damping terms and roll, and environmental disturbances. This holds when the vessel is operating at low speed and is equipped with

60

3 Modeling of Ocean Vessels

independent internal/external roll actuators. As such, the model (3.51) is simplified to: 1. Kinematics xP D cos. / cos./u  sin. /v C sin./ cos. /w; yP D sin. / cos./u C cos. /v C sin./ sin. /w; zP D  sin./u C cos./w; P D q; P D r : cos./

(3.59)

2. Kinetics m33 d11 1 m22 vr  wq  uC u ; m11 m11 m11 m11 m11 d22 vP D  ur  v; m22 m22 d33 m11 uq  w; wP D m33 m33 d55 grGML sin./ 1 m33  m11 uw  q C q ; qP D m55 m55 m55 m55 m11  m22 d66 1 rP D uv  rC r : m66 m66 m66

uP D

(3.60)

3.5 Conclusions This chapter sets out material about the basic motion tasks and mathematical models of the ocean vessels that will be used in the subsequent chapters. More details on deriving the mathematical models of the ocean vessels are given in [11, 12, 25]. It has also been pointed out that regulation/stabilization is much more difficult than trajectory-tracking, path-tracking, and path-following for underactuated ocean vessels.

Chapter 4

Control Properties and Previous Work on Control of Ocean Vessels

In this chapter, first control properties of ocean vessels are presented. Then, the existing literature on the control of underactuated ocean vessels is reviewed. Through the review of the previous work in the areas of stabilization, trajectory-tracking, path-following, and output feedback control of underactuated ocean vessels, challenging questions are raised. Illustration of the background and process of solutions of those questions, as well as an explanation of the solutions in terms of their physical insights and practical applications are then presented in subsequent chapters.

4.1 Controllability Properties 4.1.1 Acceleration Constraints The number .mc / of independent control inputs (the number of nonzero elements of the propulsion force and moment vector ) is smaller than the number .nc / of degrees of freedom to be controlled for a standard model of the ocean vessels. As such, we remove all zero elements of  and denote the resulting vector by a . Thus, if a 2 Rmc and  2 Rnc , then mc < nc . For example, for the case of the vessels with six degrees of freedom to be controlled we have mc < 6, for the case of the vessels with five degrees of freedom to be controlled mc < 5, and mc < 3 for the case of the vessels with three degrees of freedom to be controlled. For clarity, we ignore the environmental disturbance forces and moments to investigate acceleration constraints on the aforementioned ocean vessels. Let Mu , Cu .v/, Du .v/, and gu ./ denote the rows of M , C .v/, D.v/, and g./ that correspond to those rows without propulsion forces or moments, i.e., Mu vP C Cu .v/v C Du .v/v C gu ./ D 0:

(4.1)

61

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The above equation describes the acceleration constraints, i.e., second-order constraints. The following results give the conditions whether the constraints given in (4.1) are partially integrable or totally integrable. Lemma 4.1. The constraints (4.1) are partially integrable if and only if the following conditions hold: 1. gu ./ is a constant vector. 2. .Cu .v/ C Du .v// is a constant matrix. 3. The distribution ˝ ? ./ D ker ..Cu .v/ C Du .v//J 1 .// is completely integrable. Proof. See [29].  Lemma 4.2. The constraints (4.1) are totally integrable if and only if the following conditions hold: 1. The constraints are partially integrable. 2. .Cu .v/ C Du .v//=0. 3. The distribution
./ D ker .Mu J 1 .// is completely integrable. Proof. See [29].  The following lemma gives a result on the stabilizability of an underactuated ocean vessel. Lemma 4.3. Consider the system (3.31) with E D 0nc 1 . Assume that the elements of the restoring force and moment vector g./ corresponding to the unactuated dynamics are zero, i.e., the vector g./ can be written in the form of " # ga ./ (4.2) g./ D .n m /1 ; 0 c c where ga ./ 2 Rmc is the restoring force and moment vector corresponding to the actuated dynamics. Let .; v/ D .e ; 0nc mc / be an equilibrium. There is no C 1 state feedback law ˛.; v/ W Rnc  Rnc ! Rmc that makes the equilibrium .e ; 0nc mc / asymptotically stable. Proof. See [29]. 

4.1.2 Kinematic Constraints In this section, we address controllability properties of ocean vessels. Since the vessel under consideration has a number of degrees of freedom to be controlled greater than control inputs (e.g., underwater vehicles do not have independent actuators in the heave and sway axes, see Section 3.4.2 and surface ships do not have an independent actuator in the sway axis, see Section 3.4.1), we can address the controllability

4.1 Controllability Properties

63

issue of the vessel kinematic by considering the kinematics with the linear velocity vector 2 3 u (4.3) v1 D 4 0 5 : 0

We analyze controllability properties of six degrees of freedom vessels. The case of three degrees of freedom vessels can be obtained directly from the results for the six degrees of freedom vessels. With (4.3), we now write the kinematics of the vessel as follows: P D 1 ./u C 2 ./p C 3 ./q C 4 ./r m P D  ./u;

(4.4)

where 3 cos./ cos. / 6 cos./ sin. / 7 7 6 6  sin./ 7 7;

1 ./ D 6 7 6 0 7 6 5 4 0 0 2

3 0 7 6 0 7 6 7 6 0 7; 6

3 ./ D 6 7 sin./ tan./ 7 6 4 cos./ 5 sin./ sec./ 2

and

2 3 0 607 6 7 607 7

2 ./ D 6 6 1 7; 6 7 405 0

3 0 7 6 0 7 6 7 6 0 7; 6

4 ./ D 6 7 cos./ tan./ 7 6 4  sin./ 5 cos./ sec./ 2

   ./ D 1 ./ 2 ./ 3 ./ 4 ./ ;

u D Œu p q rT :

(4.5)

(4.6)

From (4.4) and (4.5), a calculation shows that the following nonhonolomic (nonintegrable) constraints are satisfied: 
cos. / sin./ sin./  sin. / cos./ xP C 
sin. / sin./ sin./ C cos. / cos./ yP C cos./ sin./zP D 0; 
sin. / sin./ cos./  sin. / sin./ xP C (4.7) 
sin. / sin./ cos./  cos. / sin./ yP C cos./ cos./zP D 0:

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4 Control Properties and Previous Work on Control of Ocean Vessels

Based on (4.4), we will address the following controllability issues: Controllability about a point (i.e., stabilization) and controllability about a trajectory (i.e., trajectory-tracking).

4.1.3 Controllability at a Point We will first consider a linear approximation of the system (4.4) at an equilibrium point e . Let the error associated with the equilibrium point e be as follows: Q D   e :

(4.8)

With (4.8), we can write the tangent linearization of (4.4) at the equilibrium point e as PQ D  .e /u; (4.9) which is not controllable because the rank of the matrix  .e / is 4. This implies that a linear controller will never achieve posture stabilization, not even in a local sense. In order to study the controllability of the vessel in question, we need to use some tools (the Lie algebra rank condition and nilpotent concepts) from nonlinear control theory [4]. Given a set of generators or basis vector fields 1 ; 2 ; :::; mc , the length of a Lie product recursively defined as `f i g D 1;

i D 1; 2; :::; mc

`.ŒA; B/ D `ŒA C `ŒB;

(4.10)

where A and B are themselves Lie products. Alternatively, `ŒA is the number of generators in the expansion for A. A Lie algebra or basis is nilpotent if there exists an integer k such that all Lie products of length greater than k are zero. The integer k is called the order of nilpotency. The use of the nilpotent basis eliminates the need for cumbersome computations as we see that all higher order Lie brackets above some particular order are zero. The above concepts and conditions imply that Lie algebra Lf 1 ; 2 ; 3 ; 4 g is nilpotent algebra of order k D 2, i.e., the vector fields 1 ; 2 ; 3 , and 4 are the nilpotent basis. Thus all Lie brackets of order greater than two are zero. The only independent Lie brackets computed from the four basis vector fields are Œ 1 ; 3  and Œ 1 ; 4 . Therefore, for our system the Lie algebra rank condition becomes rankŒCc  D 6 , rankŒ 1 ; 2 ; 3 ; 4 ; Œ 1 ; 3 ; Œ 1 ; 4  D 6;

(4.11)

where Œ 1 ; 3  and Œ 2 ; 4  are the two independent Lie brackets computed from the four vector fields . 1 ; 2 ; 3 ; 4 / and Cc is called the controllability matrix. For two vector fields g.x/ and h.x/, a Lie bracket is computed based on the following formula:

4.1 Controllability Properties

65

@g @h g  h: (4.12) @x @x Using the definition (4.12), Lie brackets Œ 1 ; 3  and Œ 1 ; 4  are given by 3 2 cos. / sin./ cos./ C sin. / sin./ 6 sin. / sin./ cos./  cos. / sin./ 7 7 6 7 6 @ 1 @ 3 cos./ cos./ 7;

1 

3 D 6 Œ 1 ; 3  D 7 6 0 @ @ 7 6 5 4 0 0 Œg; h.x/ D

3  cos. / sin./ sin./ C sin. / cos./ 6  sin. / sin./ sin./  cos. / cos./ 7 7 6 7 6 @ 4 @ 1  cos./ sin./ 7: Œ 1 ; 4  D

1 

4 D 6 7 6 0 @ @ 7 6 5 4 0 0 (4.13) 2

Therefore, the controllability matrix Cc is given by 2 cos. / cos./ 0 0 0 6 sin. / cos./ 0 0 0 6 6  sin./ 0 0 0 Cc D 6 6 0 1 sin./ tan./ cos./ tan./ 6 4 0 0 cos./  sin./ 0 0 sin./ sec./ cos./ sec./ cos. / sin./ cos./ C sin. / sin./ sin. / sin./ cos./  cos. / sin./ cos./ cos./ (4.14) 0 0 0 3  cos. / sin./ sin./ C sin. / cos./  sin. / sin./ sin./  cos. / cos./ 7 7 7  cos./ sin./ 7: 7 0 7 5 0 0

It can be seen that the above matrix Cc has one nonzero minor of order 6. Therefore, this matrix is full rank provided that  ¤ 2 . This implies that the vessel is locally controllable and also globally controllable as long as the singular condition  ¤ 2 is avoided. As for the stabilizability of system (4.4) to a point, the failure of the previous linear analysis indicates that exponential stability cannot be achieved by

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4 Control Properties and Previous Work on Control of Ocean Vessels

smooth feedback [21]. Things turn out to be even worse: If smooth (in fact, even continuous) time-invariant feedback laws are used, Lyapunov stability cannot be used directly. This negative result is established on the basis of a necessary condition due to Brockett [21], see Section 2.7.4: Smooth stabilizability of a driftless regular system (i.e., such that the input vector fields are well defined and linearly independent at e ) requires that the number of inputs be equal to the number of states. The above difficulty has a deep impact on the control design. In fact, to obtain a posture stabilizing controller it is either necessary to give up the continuity requirement and/or to resort to time-varying control laws.

4.1.4 Controllability About a Trajectory For the system (4.4), let the reference trajectory d and the reference trajectory input ud be 2 3 xd .t / 3 2 6 yd .t / 7 ud .t / 6 7 6 pd .t / 7 6 zd .t / 7 7 6 7 (4.15) d D 6 6 d .t / 7 ; ud D 4 qd .t / 5 : 7 6 4 d .t / 5 rd .t / d .t / Indeed, the reference trajectory d and the reference trajectory input ud should satisfy the nonhonolomic constraints (4.7), i.e., 
cos. d / sin.d / sin.d /  sin. d / cos.d / xP d C 
sin. d / sin.d / sin.d / C cos. d / cos.d / yPd C cos.d / sin.d /zP d D 0; 
sin. d / sin.d / cos.d /  sin. d / sin.d / xP d C 
sin. d / sin.d / cos.d /  cos. d / sin.d / yPd C cos.d / cos.d /zP d D 0: (4.16)

Let the errors associated with the reference trajectory and the reference input trajectory be e D   d ; ue D u  ud :

(4.17)

Using (4.17), we can write (4.4) as
 P D  .d C e / ud C ue :

(4.18)

The Taylor series expansion of  .d C e / about the nominal solution d is given by

4.1 Controllability Properties

67

ˇ    @ ./ ˇˇ  .t / C HOT u .t / C u .t / : P D  .d ; t / C e d e @ ˇDd

(4.19)

Since the reference trajectory  and the reference input trajectory ud satisfy the nonholonomic constraints (4.16), we have P d D  .d ; t /ud .t /: Subtracting (4.19) by (4.20) and ignoring the high-order terms (HOT) gives ˇ   @ ./ ˇˇ P e D  .t / ud .t / C  .d ; t /ue .t / e @ ˇDd WD A.t /e .t / C B.t /ue .t /;

(4.20)

(4.21)

where 

   033 A1 .t / Jd1 .t / 033 A.t / D ; B.t / D ; 033 A2 .t / 031 Jd2 .t /

(4.22)

with A1 and A2 given by 2 3 0  cos. d / sin.d /ud  sin. d / cos.d /ud A1 .t / D 4 0  sin. d / sin.d /ud cos. d / cos.d /ud 5 ; 0  cos.d /ud 0 2

A2 .t / D 4

cos.d / tan.d /qd  sin.d / tan.d /rd  sin.d /qd  cos.d /rd cos.d / sec.d /qd  sin.d / sec.d /rd 3(4.23) sin.d / sec2 .d /qd C cos.d / sec2 .d /rd 0 0 0 5; sin.d / sec.d / tan.d /qd C cos.d / sec.d / tan.d /rd 0

and Jd1 .t / and Jd2 .t / given by 2 3 cos.d / cos. d / Jd1 .t / D 4 cos.d / sin. d / 5 ;  sin.d / 2 1 sin.d / tan.d / cos.d / Jd2 .t / D 4 0 0 sin.d / sec.d /

3 cos.d / tan.d /  sin.d / 5 : cos.d / sec.d /

(4.24)

In (4.23) and (4.24), the argument t of d , d , d , ud , pd , and rd is omitted for simplicity. The system (4.21) is linear time-varying. The controllability condition becomes

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4 Control Properties and Previous Work on Control of Ocean Vessels

rankfB; AB; A 2 B; A 3 B; A 4 B; A 5 Bg D 6:

(4.25)

A calculation shows that the above matrix has a nonzero minor of order 6 provided that (ud ¤ 0, pd ¤ 0, qd ¤ 0, rd ¤ 0) and (d ¤ 2 ). Therefore, we conclude that the kinematic system (4.21) can be locally stabilized by linear feedback about trajectories consisting of linear or circular or helix paths, which do not collapse to a point.

4.2 Previous Work on Control of Underactuated Ocean Vessels This section starts with a brief review on the control of nonholonomic systems, due to their relevance to the control of underactuated ocean vessels. Next, the existing methods on control of underactuated ocean vessels are reviewed. Limitations of the existing methods are then pointed out and hence motivate the contributions of the book.

4.2.1 Control of Nonholonomic Systems The term “nonholonomic system” originates from classical mechanics and has its widely accepted meaning as a “Lagrange system with linear constraints being nonintegrable”. A mechanical system is said to be nonholonomic if its generalized velocity satisfies an equality condition that cannot be written as an equivalent condition on the generalized position, see [30]. Control of nonholonomic dynamic systems has formed an active area in the control community – see surveys by Kolmanovsky and McClamroch in [31], Canudas de Wit et al. in [15], Murray and Sastry in [32], and references therein for an overview and interesting introductory examples in this expanding area. Nonholonomic systems have inherent difficulties in feedback stabilization at the origin or at a given equilibrium point since the tangent linearization of these systems is uncontrollable. In fact, a direct application of Brockett’s necessary condition, see Section 2.7.4 for more details, for feedback stabilization implies that nonholonomic systems cannot be stabilized by any stationary continuous state feedback although they are open loop controllable. As a consequence, the classical smooth control theory cannot be applied. This motivates researchers to seek novel approaches. These approaches can be roughly classified into discontinuous feedback, see for example [33–45] and time-varying feedback, see for example, [15, 32, 46–48]. The discontinuous feedback approach often uses the state scaling originated from the  process [49] and a switching control strategy to overcome the difficulty due to the loss of controllability. This approach results in a fast transient response and usually an exponential convergence can be achieved. The drawback is discontinuity in the control input. On the other hand, the time-varying feedback approach provides

4.2 Previous Work on Control of Underactuated Ocean Vessels

69

a smooth/continuous controller, i.e., no switching is required, however the price is slow convergence. The stability analysis is often based on linear time-varying system theory and Barbalat’s lemma. The backstepping technique [3] is usually used for high-order chained form systems in both discontinuous and time-varying approaches. Those aforementioned systems are either driftless or have weak nonlinear drifts. When nonholonomic systems are perturbed by drifts with uncertainties, robust and adaptive control approaches are often applied. The robust control design schemes are based on the size domination concept [50]. The control is conservative when a priori knowledge of uncertainties is poor. A class of nonholonomic systems with strong nonlinear uncertainties was recently considered in [51]. Discontinuous state feedback and output feedback controllers were designed to achieve global exponential stability. However the x0 -subsystem is required to be Lipschitz since a constant control input u0 is used to get around the difficulty due to the loss of controllability. The adaptive approach [38, 40, 46] provides less conservative control input but increases the dynamics of the closed loop system. The systems studied in these papers do not allow drifts in the x0 -subsystem. A difficulty in designing adaptive stabilization controllers for chained systems with drifts is that the state of the x0 -subsystem can have several zero crossings due to transient behavior of the unknown parameter estimate. This phenomenon causes difficulties in applying the state scaling. For a solution of the stabilization of nonholonomic systems in a chained form with strong nonlinear drifts and unknown parameters, the reader is referred to [52].

4.2.2 Control of Underactuated Ships and Underwater Vehicles Control of underactuated ships and autonomous underwater vehicles (AUVs) is an active field due to its important applications such as passenger and goods transportation, environmental surveying, undersea cable inspection, and offshore oil installations. Based on its practical requirement, motion control of underactuated ocean vessels has been divided into three areas: Stabilization, trajectory-tracking, and pathfollowing. These control problems are challenging due to the fact that the motion of underactuated surface ships and AUVs possesses more degrees of freedom to be controlled than the number of the independent controls under some nonintegrable second-order nonholonomic constraints [29, 53, 54]. In particular, underactuated ships do not usually have an actuator in the sway axis while in the case of AUVs there are no actuators in the sway and heave directions. This configuration is by far the most common among marine vessels. Therefore, Brockett’s condition indicates that any continuous time-invariant feedback control law does not make a null solution of the underactuated surface ship and AUV dynamics asymptotically stable in the sense of Lyapunov. Furthermore as observed in [22, 54], the underactuated ship and AUV system is not transformable into a standard chain system. Consequently, existing control schemes [15, 32–48] developed for chained systems

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4 Control Properties and Previous Work on Control of Ocean Vessels

cannot be applied directly. Nevertheless, in the past decade, stabilization, trajectorytracking control, and path-following of underactuated ocean vessels have been studied separately from different viewpoints.

4.2.2.1 Stabilization An underactuated ocean vessel belongs to a class of underactuated mechanical systems subject to some nonintegrable second-order nonholonomic constraints, see [29,53,54]. Therefore, design of a feedback stabilizer using linear and classical nonlinear control theories is not possible. There are two main approaches to deal with stabilization of an underactuated ocean vessel. They are (time-invariant and timevarying) discontinuous feedback and time-varying continuous/smooth feedback. We here mention some typical results of both approaches.

Time-invariant and Time-varying Discontinuous Approach A discontinuous state feedback control law was proposed in [55] using the  -process to exponentially stabilize an underactuated ship at the origin where the ship model is discontinuously transformed to an extended chained form system. The dynamics of an underactuated ship is considered in [55], see also Section 3.4.1: P D J ./v; M vP D C .v/v  Dv C ;  D Œx; y;

T ; v D Œu; v; rT ;  D Œu ; 0; r T ;

(4.26)

where .x; y/ denotes the earth-fixed position of the center of mass of the ship, denotes the orientation angle, .u; v/ and r are the linear and angular velocities in the body-fixed frame, and .u ; r / are the surge force and yaw moment. The matrices J ./, M ; C .v/, and D are given by 3 2 3 2 m11 0 0 cos. /  sin. / 0 J ./ D 4 sin. / cos. / 0 5 ; M D 4 0 m22 0 5 ; 0 0 1 0 0 m33 3 2 3 0 m22 v d11 0 0 0 m11 u 5 ; D D 4 0 d22 0 5 ; C .v/ D 4 0 0 d33 m22 v m11 u 0 2

0 0

(4.27)

where m11 , m22 , and m33 denote the ship inertia including added mass, and d11 , d22 , and d33 are hydrodynamic damping constants, see Chapter 3 for more details. The control objective is to design the control inputs u and r to stabilize (4.26) asymptotically at the origin. In [55], the coordinate transformation

4.2 Previous Work on Control of Underactuated Ocean Vessels

2

3

71

3

2

x1 6 x2 7 6 x cos. / C y sin. / 7 7 6 7 6 6 x3 7 6 x sin. / C y cos. / 7 7 6 7D6 7 6 x4 7 6 v 7 6 7 6 5 4 x5 5 4 r u x6

(4.28)

is used to transform the ship model (4.26) to the following system xP 1 D x5 ; xP 2 D x6 C x3 x5 ; xP 3 D x4  x2 x5 ; xP 4 D ˛x4  ˇx5 x6 ; xP 5 D ˝1 ;

(4.29)

xP 6 D ˝2 ; where ˛ D d22 =m22 , ˇ D m11 =m22 , and ˝1 D

r  d33 r C .m11  m22 /uv u C m22 vr  d11 u ; ˝2 D : m33 m11

(4.30)

It can be seen that the system (4.29) consists of two subsystems, namely .x1 ; x2 ; x3 ; x4 / and .x5 ; x6 /, connected to each other in a strict feedback form [3]. The control design can be simply carried out in two steps as follows.

Step 1 In this step, the author of [55] considers the first four equations of (4.29), and .x5 ; x6 / as controls .v1 ; v2 /. With the assumption of x1 ¤ 0, the coordinate transformation ( -process) y D x1 ; z 1 D x 2 ; z 2 D

x3 x4 ; z3 D x1 x1

(4.31)

results in yP D v1 ; zP 1 D v2 C yz2 v1 ; z1 C z2 zP 2 D z3  v1 ; y z3 C ˇv2 zP 3 D ˛z3  v1 : y The feedback control law is designed as

(4.32)

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4 Control Properties and Previous Work on Control of Ocean Vessels

v1 D k1 y; v2 D k21 z1  k22 z2  k23 z3 ;

(4.33)

where k1 , k21 , k22 , and k23 are the control gains chosen such that the matrix 3 2 k21 k22 k23 7 6 (4.34) A1 D 4 k1 k1 1 5 k1 ˇk21 k1 ˇk22 k1  ˛  k1 ˇk23 is Hurwitz.

Step 2 At this step, the last two equations of (4.29) are considered. Using the standard backstepping technique results in the following control law ˝1 D k3 .x5  v1 /  k1 x5 ; ˝2 D k4 .x6  v2 /  k21 .x6 C x3 x5 / (4.35) x4  x2 x5 x4 x5 ˛x4 C ˇx5 x6 x3 x5 k22 C k23 C k22 2 C k23 2 ; x1 x1 x1 x1 where k3 > k1 and k4 are positive constants. The actual controls u and r can be found from (4.35) and (4.30). In [55] it is shown that if the initial conditions x1 .t0 / ¤ 0 and x1 .t0 /.x5 .t0 / Ck1 x1 .t0 //  0 then .x1 .t /, x2 .t /, x3 .t /, x4 .t /, x5 .t /, x6 .t // is bounded for all t  t0  0, and exponentially converges to zero. If the above conditions do not hold, the controls ˝1 D jx1  ja sign.x1  /  jx5 jb sign.x5 /; ˝2 D 0;

(4.36)

with  ¤ 0, b 2 .0; 1/, and a > b=.2  b/ being constants, can be used to make the above conditions hold in finite time. For more details, the reader is referred to [55]. Remark 4.1. The aforementioned discontinuous stabilizer provides a fast convergence of the stabilizing errors to zero. However, the control inputs u and r are discontinuous. Moreover, under arbitrarily small nonvanishing environmental disturbances induced by waves, wind, and ocean currents, the closed loop system consisting of (4.35) and (4.29) can be unstable in the sense that the states .x1 .t /, x2 .t /, x3 .t /, x4 .t /, x5 .t /, x6 .t // can go to infinity exponentially fast. The work mentioned in [22, 53, 54, 56–58] can also be grouped in the discontinuous approach. The authors of [22] developed a discontinuous time-varying feedback stabilizer for a nonholonomic system and applied it to underactuated ships. Some local exponential stabilization results were reported in [53, 54] based on the time-varying homogeneous control approach. An application of averaging and backstepping tech-

4.2 Previous Work on Control of Underactuated Ocean Vessels

73

niques was proposed in [59] to design a global practical controller for stabilization and tracking control of surface ships. Experimental results on dynamic positioning of underactuated ships were reported in [56]. By transforming the underactuated ship kinematics and dynamics into the so-called skew form, some dynamic feedback results on stabilization were given in [57]. In [58], the authors proposed a discontinuous solution to the problem of steering an underactuated AUV to a point with desired orientation using the polar coordinate transformation motivated from the work in [60].

Time-varying Continuous/Smooth Approach A typical result on stabilization of the underactuated ship (4.26) in the time-varying continuous/smooth approach is given in [61]. In [61], the coordinate transformations (similar to the ones given in (4.30) and (4.28)) z1 D cos. /x C sin. /y; z2 D  sin. /x C cos. /y; z3 D ; r  d33 r C .m11  m22 /uv ˝1 D ; m33 u C m22 vr  d11 u ˝2 D m11

(4.37)

are first used to transform the ship system (4.26) to zP 1 D u C z2 r; zP 2 D v  z1 r; zP 3 D r; uP D ˝2 ; vP D cur  dv;

(4.38)

rP D ˝1 ; where c D m11 =m22 and d D d22 =m22 . Then the following nontrivial coordinate transformations v Z2 D z2 C ; d d d u D  z1  ; c c d d v c ˝2 D z1 C   Z2 r C r  ˝2 (4.39) c c d d are applied to (4.38) to obtain the system

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4 Control Properties and Previous Work on Control of Ocean Vessels

d d v zP 1 D  z1   C Z2 r  r; c c d ZP 2 D r; zP 3 D r; vP D dv C d.z1 C /r;

(4.40)

P D ˝2 ; rP D ˝1 : Let k2 , k3 , k , and kr be strictly positive constants such that 1  k2  k3 . The controls ˝1 and ˝2 are designed in [61] as
 ˝1 D kr .r  rf / C rPf   Z2 f C 2Z3 C 2Z2 k2 cos.t /f ; 
˝2 D k .  f / C P f   Z2 C 2Z3 k2 cos.t / r; (4.41)

where

k3 k3 sin.2t / 2V1 C V12  ; 3 6 .1 C V1 /2 Z3 D z3 C k2 cos.t /Z2 ;

 D 2C

V1 D Z22 C 2Z32 ; sin.t /Z22 ; 2.0:001 C Z22 / k3 Z3 C k2 sin.t /Z2 rf D : 1 C k2 cos.t /f f D 

(4.42)

In [61], it is proven that the closed loop system consisting of (4.41), (4.39), (4.37), and (4.26) is GAS at the origin. Remark 4.2. The design of the feedback given in (4.41) is nontrivial. Overall, convergence of the stabilizing errors to zero is slow. This is a well-known phenomenon of the continuous/smooth time-varying approach applying not to only underactuated ships but also to mobile robots. Moreover, since the stabilizer design mentioned above is nontrivial, it is difficult to extend the control design scheme to solve a trajectory-tracking problem, see the next section. In addition, the physical meaning of the feedback is not clear. In addition to the aforementioned results on stabilization of underactuated vessels, the following results are also related to the topic under discussion. In [62], several control configurations were considered, and a technique for synthesizing open loop controls was given. The first control scheme with the dynamic AUV model taken into account was proposed in [63]. A kinematic drift free model of the underwater vehicles with four control inputs was used to design a regulation controller in [64]. The authors of [65] proposed a controller that is able to stabilize an AUV to some equilibria based on the interconnection and damping assignment passivitybased control approach, which has been successfully applied to many other mechan-

4.2 Previous Work on Control of Underactuated Ocean Vessels

75

ical systems [66]. See also [67] for stabilization results of underactuated mechanical systems on Riemannian manifolds.

4.2.2.2 Trajectory-tracking Trajectory-tracking is here defined as a control problem of forcing an underactuated surface ship or AUV to track a reference trajectory generated by a suitable vessel model, i.e., the vessel model that has the same parameters as the real one. There are two main approaches to solve the trajectory-tracking control problems. The first approach is based on linear time-varying control system theory while the second approach relies on the Lyapunov direct method. We here briefly describe typical results of the two approaches.

Linear Time-varying Approach A typical work belonging to this approach is given in [68] on a global K-exponential tracking result for the underactuated ship (4.26). In [68], the authors consider a problem of designing the control u and r to force the position .x; y/ and orientation of the ship (4.26) to track the reference position .xd ; yd / and orientation d generated by the reference ship model Pd D J .d /vd ; M vP d D C .vd /vd  Dvd C d ; 2 3 2 3 3 2 ud ud xd d D 4 yd 5 ; vd D 4 vd 5 ; d D 4 0 5 : rd rd d

(4.43)

In [68], the coordinate transformations 8 8 < z1d D cos. d /xd C sin. d /yd ; < z1 D cos. /x C sin. /y; z D  sin. d /xd C cos. d /yd ; (4.44) z2 D  sin. /x C cos. /y; : : 2d z3d D d ; z3 D ; and the tracking errors

ue D u  ud ; ve D v  vd ; re D r  rd ; z1e D z1  z1d ; z2e D z2  z2d ; z3e D z3  z3d are used to obtain the tracking error dynamics of a chained form uP e D


d11 m22  1 ue C .u  ud /; ve re C ve rd C vd re  m11 m11 m11

(4.45)

76

4 Control Properties and Previous Work on Control of Ocean Vessels


d22 m11  ue re C ue rd C ud re  ve ; m22 m22
d33 1 m11  m22  ue ve C ue vd C ud ve  re C .r  rd /; rPe D m33 m33 m33 zP 1e D ue C z2e re C z2e rd C z2d re ; zP 2e D ve  z1e re  z1e rd  z1d re ; vP e D 

(4.46)

zP 3e D re : Assuming that ud , vd , z1d , and z2d are bounded, and that rd .t / is persistently exciting, the controls u D ud  k1 ue C k2 rd ve  k3 z1e C k4 rd z2e ; r D rd  .m11  m22 /.ue ve C vd ue C ud ve /  k5 re  k6 z3e ;

(4.47)

where the control gains ki , i D 1; : : : ; 6 satisfy k1 > d22  d11 ; m22 k4 .k4 C k1 C d11  d22 / k2 D ; d22 k4 C m11 k3 d22 ; 0 < k3 < .k1 C d11  d22 / m11 k4 > 0;

(4.48)

k5 > d33 ; k6 > 0; make the closed loop system consisting of (4.47), (4.43), and (4.26), that is, 2 3 3 3 2 k1 C d11 2 k3 k2 C m22 k4  rd .t /  rd .t / 6 ue 7 uP 7 76 m m m m 6 e7 6 11 11 11 11 76 7 6 7 6 m d 6 6 7 7 11 22 6 vP e 7 6   0 0 7 6 ve 7 6 7 6 m rd .t / 6 7C 7 m 22 22 6 7D6 76 7 6 7 6 6 7 7 6 zP 1e 7 6 1 0 0 rd .t / 7 6 z1e 7 4 5 4 7 56 4 5 zP 2e 0 1 rd .t / 0 z2e 2 m 3 22 .ve C vd / 0 6 m11 7 6 m11 7" # 6 7 6 m .ue C ud / 0 7 re ; 6 7 22 6 7 z3e 6 z2e C z2d 07 4 5 .z1e C z1d / 0

4.2 Previous Work on Control of Underactuated Ocean Vessels

2

3

2

6 rPe 7 6  7D6 6 5 4 4 zP 3e

3

d33 C k5 k6 " #  7 re m33 m33 7 ; 5 z 3e 1 0

77

(4.49)

globally K-exponentially stable at the origin. Proof of K-exponential stability of the above closed loop system is straightforward using the results on stability of cascade systems in [17] and [69], and those of linear time-varying system theory in [6], see [68] for details. It should be noted that the persistently exciting condition on the yaw reference velocity, rd , is required to prove K-exponential stability of the closed loop system (4.49). Remark 4.3. The persistent exciting condition on the yaw reference velocity rd excludes a straight-line reference trajectory. In comparison with the direct Lyapunov approach summarized below, it is difficult to deal with any external disturbances and/or actuator dynamics using the linear time-varying approach.

Direct Lyapunov Approach The direct Lyapunov method has been widely used in designing controllers for underactuated ocean vessels. However, the use of the Lyapunov direct method for designing control systems for underactuated ocean vessels is not straightforward due to the underactuated nature of ocean vessels. We here describe typical results of trajectory-tracking control based on the Lyapunov direct method. Local Trajectory-tracking Results An application of the recursive technique proposed in [70] for the standard chainform systems yields a high-gain based local tracking result in [59] for surface ships. The experimental results of this proposed controller were reported in [71]. The control design in [59,71] starts from (4.46) as follows. First of all, the following restrictive assumption is made on the yaw reference velocity: 0 < rd min < jrd .t /j < rd max ;

(4.50)

where rd min and rd max are strictly positive constants. Motivated by the work in [70], the authors define new error variables as !1 D z1e  z2 z3e ; !2 D z2e C z1 z3e ; y1 D ve C cuz3e C k2 !2 ; k2 .d  k2 / y2 D ue C k1 !1  !2 ; crd y3 D z3e ;

(4.51)

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4 Control Properties and Previous Work on Control of Ocean Vessels

where c D m11 =m22 , d D d22 =m22 , and k2 is a parameter to be determined later. It should be stressed that the condition (4.50) on the yaw reference velocity, rd is required so that the error transformations (4.51) are valid. With (4.51), the tracking error system (4.46) is written in a triangular-like structure as follows: k2 .d  k2 / !2 C !2 rd  .v  z1 re /y3 ; crd !P 2 D y1  k2 !2  !1 rd C ..1  c/u C z2 re /y3 ; yP1 D cy2 rd C .ck1  k2 /rd !1  .d  k2 /y1 C .c˝2 C cdu C !P 1 D y2  k1 !1 C

k2 ..1  c/u C z2 re //y3 ; yP2 D ˝2  ˝2d C k1 y2  k12 !1 C

1 k1 k2 .d  k2 /!2 C k1 rd !2 C crd

1 rPd k2 .d  k2 /.y1  k2 !2  rd !1 /  k .d  k2 /!2  2 2 crd crd 1 k2 .d  k2 /..1  c/u C z2 re //y3 ; .k1 .v  z1 re / C crd yP3 D re ; rPe D ˝1  ˝1d ;

(4.52)

where ˝1 and ˝2 are given in (4.30), and rd  d33 rd C .m11  m22 /ud vd ; m33 ud C m22 vd rd  d11 ud : D m11

˝1d D ˝2d

(4.53)

The triangular structure (4.52) allows us to use the backstepping technique [3] to design the controls ˝1 and ˝2 . In [71], the the controls ˝1 and ˝2 are designed as ˝1 D a3 .r  ˛r / C ˛P r  y3 ; 

˝2 D a1 y2  !1 C card y1   ˝2d C k1 y2  k12 !1 C

where

1 1 rPd k1 k2 .d  k2 /!2 C k1 rd !2 C 2 k2 .d  k2 /!2   crd crd crd
k2 .d  k2 /.y1  k2 !2  rd !1 / ; (4.54)

1  1 ˛r D  C .!1 z1 C !2 z2 / C ak2 y1 z2 C k1 y2 z1   k2 .d  k2 /y2 z2 crd   a2 y3 C !1 v  !2 .1  c/u  ay1 .c.˝2 C du/ C k2 .1  c/u/C

4.2 Previous Work on Control of Underactuated Ocean Vessels

k1 y 2 v C



79

1 k2 .d  k2 /.1  c/uy2 C rd ; crd

 D  C !1 z1 C !2 z2 C ak2 y1 z2 C k1 y2 z1 

1 k2 .d  k2 /y2 z2 : (4.55) crd

In (4.54) and (4.55), the control parameters k1 , k2 , a, a1 , a2 , a3 , and  are positive constants, and are chosen such that k2 .d  k2 /2 ; c 2 rd2 min a k1 .d  k2 / 1 < < : k2 .d  k2 / .ck1  k2 /2 rd2 max

k2 < d; k1 >

(4.56)

It is noted that the virtual control ˛r given in (4.55) is solvable if and only if   1 k2 .d  k2 /y2 z2 : (4.57)  >  !1 z1 C !2 z2 C ak2 y1 z2 C k1 y2 z1  crd Proof of local exponential stability of the closed loop system consisting of (4.54), (4.46), and (4.30) can be carried out by using the Lyapunov function 1 1 1 1  1 V D !12 C !22 C ay12 C y22 C y32 C .r  ˛r /2 : 2 2 2 2 2 2

(4.58)

Remark 4.4. There are two limitations of the aforementioned tracking controllers. These limitations are described in conditions (4.50) and (4.57). The condition (4.50) implies that the reference yaw velocity rd cannot be zero at any time. This restrictive condition excludes a straight-line reference trajectory. The condition (4.57) implies that the aforementioned trajectory-tracking result is inherently local. One can argue that by the control parameters k1 , k2 , a, a1 , a2 , a3 , and  may increase the size of the attraction region. However, it is very hard to ensure this property since the control parameters must satisfy various conditions specified in (4.56). In fact, this is true, as said in [71]. Global Trajectory-tracking Results Based on Lyapunov’s direct method and the passivity approach [72], two restricted tracking solutions of an underactuated surface ship were proposed in [19]. We here briefly summarize the result based on the passivity approach in [19]. The result based on the standard backstepping technique [3] is discussed later. In [19], the starting point is the tracking error system (4.46). The passivity based method consists of two steps as follows.

Step 1 Design of the surge force u : This force is designed based on the Lyapunov function

80

4 Control Properties and Previous Work on Control of Ocean Vessels

V1 D


2 1 2 1 0 1 z1e  1 z2e rd C z2e C ve2 C uN 2e ; 2 2 2 2

(4.59)

where uN e D ue  ˛0 with

˛0 D 2 .z1e  1 z2e rd /:

(4.60)

In (4.59) and (4.60), the control parameters 0 , 1 and 2 are chosen such that     d22 m22 ; 2 ; 2c1  c  ; c.t / D min 2.2  1 rd2 .t //; 2 1 rd2 .t /  .1  /0 d22 m22 (4.61) where c1 is a positive constant, 0 <  < 1, and c  is strictly positive. In (4.59) and (4.60), ˛0 is understood as a virtual control of ue . From the first time derivative of the Lyapunov function V1 given in (4.59) along the solutions of (4.46), a choice of the surge force u  m22 d11 u D ud C m11  .vr  vd rd / C ue  c1 .ue C 2 .z1e  1 z2e rd //  m11 m11   0 m11 .z1e  1 z2e rd /  rd ve  2 .ue C z2e rd C z2 re / C m22  (4.62) 1 2 rPd z2e C 1 2 rd .ve  z1e rd  z1 re / gives   0 m11 P V1  c.t /V1 C .z1e  1 z2e rd /.z2 C 1 rd z1 /  z2e z1  ve u re ; (4.63) m22 where c.t / is given in (4.61).

Step 2 Design of the yaw moment r : This moment is designed based on the Lyapunov function 1 1 2 V2 D V1 C z3e C rNe2 ; 2 2

(4.64)

where rNe D re  ˛1 and   0 m11 ve u C z3e ; (4.65) ˛1 D c2 .z1e  1 z2e rd /.z2 C 1 rd z1 /  z2e z1  m22 with c2 > 0. From the first time derivative of the Lyapunov function V2 given in (4.64) along the solutions of (4.46), a choice of the yaw moment

4.2 Previous Work on Control of Underactuated Ocean Vessels

81



m11  m22 d33 r D rd C m33  .uv  ud vd / C re  c3 rNe C ˛P 1  m33 m33   0 m11 ve u C z3e (4.66) .z1e  1 z2e rd /.z2 C 1 rd z1 /  z2e z1  m22 where c3 > 0 results in  P V2  c.t /V1  c2 .z1e  1 z2e rd /.z2 C 1 rd z1 /  z2e z1  0 m11 ve u C z3e m22

2

 c3 rNe2 :

(4.67)

This implies global asymptotic stability of the closed loop system at the origin as long as the control parameters 0 , 1 , and 2 are chosen such that (4.61) holds. Note that this condition is feasible only when the reference yaw velocity rd satisfies the following restrictive condition 0 < r?  jrd .t /j  r ? ;

(4.68)

where r? and r ? are positive constants. In [19], the result based on the standard backstepping technique also consists of two steps. The first step is to design the surge force u . This step is the same as Step 1 mentioned above. The second step is to design the yaw moment r . This step is slightly different from Step 2. In this step, a simple controller to stabilize the .z3e ; re /-subsystem, that is the third and last equations of (4.46), is designed as   m11  m22 d33 r D rd C m33  .uv  ud vd / C re  k1 z3e  k2 re : (4.69) m33 m33 With the surge force u and the yaw moment r designed as in (4.62) and (4.70), it is proven in [19] that the tracking errors .z1e , z2e , z3e , ue , ve , re / exponentially converge (not exponential stability of the closed loop system) to zero as long as the following restrictive condition on the reference yaw velocity rd holds Z t (4.70) rd2 . /d  r .t  t0 /; 80  t0  t < 1; t0

where r is a strictly positive constant. Remark 4.5. In comparison with the trajectory-tracking results in [71], we see that the control design in [19] is much simpler and gives global solutions. However, the restrictive conditions on the yaw reference velocity cannot be relaxed, see (4.68) and (4.70). Moreover, it is only possible to find the control parameters such that 22 , i.e., the c.t / given in (4.61) is strictly positive for vessels with a small ratio m d22 vessels with large damping in the sway axis.

82

4 Control Properties and Previous Work on Control of Ocean Vessels

Remark 4.6. A common restriction of the above results on trajectory-tracking control of underactuated ships is that the reference yaw velocity has to satisfy various kinds of persistently exciting conditions. This implies that the reference trajectory must be curved, and indeed excludes a straight-line reference trajectory, hence, it substantially limits the practical use of the aforementioned control systems. A curious question is why all the above controllers suffer from the must-be-curved reference trajectory restriction. An answer is that the design of the above controllers starts from the chained form (4.46). The reader will find that this book provides various solutions for trajectory-tracking control of underactuated ships without imposing a persistent exciting condition on the yaw reference velocity. As such, we will not use the chained form (4.46) but will project the tracking errors, x  xd , y  yd , and  d , on the body-fixed frame. Apart from the aforementioned results on trajectory-tracking control of underactuated ships there are a few more results that are worth reviewing. Using sliding mode control, output redefinition and results on tracking of nonlinear nonminimum phase system [73], a path controller for surface ships was proposed in [74]. However, the convergence of the combined output does not guarantee convergence of its components. A continuous time-invariant state feedback controller was developed in [75] to achieve global exponential position tracking under the assumption that the reference surge velocity is always positive. Unfortunately, the orientation of the ship was not controlled. In [76, 77], (see also [78]), the authors developed a highgain dynamic feedback control law to achieve global ultimate regulation and tracking of underactuated ships. The dynamics of the closed loop system is increased due to the controller designed to make the state of the transformed system track the auxiliary signals generated by some oscillator. The same approach was extended to the case of adaptive tracking control in [77]. It is worth mentioning that in [47], a time-varying velocity feedback controller was proposed to achieve both stabilization and tracking of unicycle mobile robots at the kinematics level motivated by the work in [18]. However this controller cannot be extended directly to the case of underactuated ships or AUVs due to the nonintegrable second-order constraint. Some related independent work includes [79,80] on local H1 tracking control and output redefinition, and the trajectory planning approach, see [81, 82].

4.2.2.3 Path-following Path-following is here defined as a control problem of forcing an underactuated ship or AUV to follow a specified path at a desired forward speed. Due to the high dependence on the reference model and complicated control laws of the trajectorytracking approach, several researchers have studied the path-following problem, which is more suitable for practical implementation. The problem of path-following for air and underwater vehicles was introduced in [83] where some local results were obtained using linearization techniques. In [84], a feedforward cancelation of simplified vessel dynamics scheme followed by a linear quadratic regulator design was proposed to obtain local results on “track-keeping”. A fourth-order ship model

4.2 Previous Work on Control of Underactuated Ocean Vessels

83

in Serret–Frenet frame was used in [85] to develop a control strategy to track both a straight line and a circumference under constant ocean current disturbance. The ocean-current direction was assumed to be known. A path-following controller was proposed in [86] by using a kinematic model written in polar coordinates, which is inspired by the solution for mobile robots in [60]. However, the controller was designed at the kinematic level with an assumption of constant ocean current and its direction known to be to achieve an adjustable boundedness of the path-following error. Since ocean vessels do not have direct control over velocities, a static mapping implementation might result in an unstable closed loop system due to nonvanishing environmental disturbances. Recently, a path-following controller based on a transformation of the ship kinematics to the Serret–Frenet frame, which was used for mobile robot control [44], on the path was proposed in [87], where an acceleration feedback and linearization of ship dynamics were used. It is worth mentioning that in [88, 89], a simple control scheme was proposed to make mobile robots follow a specified path using a polar coordinate transformation. Since underactuated surface ships have fewer numbers of actuators than the to-be-controlled degrees of freedom and are subject to nonintegrable acceleration constraints, their dynamic models are not transformable into a system without drifts. Therefore, the above control scheme is not directly applicable. In [54], a continuous, periodic time-varying feedback control law was proposed to locally exponentially stabilize an underactuated underwater vehicle at the origin. When the hydrodynamic restoring force in roll is large enough, this controller can be used without a roll control torque. However the closed loop system exhibits undesired oscillatory motions. In [90], a linearization technique with an assumption of reference trajectories of underwater vehicles, which are helices parameterized by the vehicles’ linear speed, yaw rate, and path angle, was introduced to develop the so-called time-invariant generalized vehicle error dynamics and kinematics. Various controllers were then designed based on the gain-scheduling technique to yield some local stability result about the trimming trajectories.

4.2.2.4 Output Feedback Output feedback control of an underactuated ocean vessel is here defined as a control problem of forcing the vessel to achieve the aforementioned tasks (stabilization, trajectory-tracking, and path-following) without using measurements of the vessel’s velocities for feedback. For ocean vessels, output feedback control usually consists of two stages. The first stage is to design an observer to reconstruct unmeasured states. Using the reconstructed states, a controller is designed to achieve control objectives in the second stage. In the literature, there are two main approaches to designing an observer for ocean vessels. The first approach is based on the output-injection method applied directly to the vessel’s equations of motion. This approach is simple and usually results in a semiglobal observer due to the quadratic terms of the vessel’s velocities. Belonging to this approach are the results presented in [11, 14, 91–94] on output feedback

84

4 Control Properties and Previous Work on Control of Ocean Vessels

control of fully actuated ocean vessels or Lagrange systems. In addition, the reader is referred to [93] for an exponential observer and output feedback controller for a special class of multi-degree of freedom Lagrange systems without cross terms of quadratic velocities, and to [14, 95–98] for output feedback control of robot manipulators and rigid body without measurements of angular velocities. Another method to design an observer is the use of contraction theory, see for example, [99, 100]. This method has been applied to Lagrange systems with monotonic velocity terms but without any quadratic velocity terms. Below, we summarize the aforementioned results on an observer design. We will show that a standard observer design cannot be used to obtain a global exponential/asymptotical observer for the ocean vessel system (1.1). Assume that the vessel velocity vector v is not measurable for feedback. We would then design an output injection observer to estimate v as follows: O PO D J ./vO C K01 .  /; P O vO  D.v/ O vO  g./ C  C K02 .  /; O M vO D C .v/

(4.71)

where O and vO are estimates of  and v, respectively, and the positive definite symmetric matrices K01 2 R66 and K02 2 R66 are the observer gain matrices. It is noted that in some of the aforementioned work, the observer gain matrices K01 and K02 depend on the measurable state . Letting the observer errors be O Q D   ; vQ D v  vO

(4.72)

and differentiating (4.72) along the solutions of (1.1) and (4.71) results in Q PQ D K01 Q C J ./v;     O vO  D.v/v  D.v/ O vO : M vPQ D K02 Q  C .v/v  C .v/

(4.73)


O vO does not cause a problem if the damping matrix D.v/ The term D.v/v  D.v/ 
T 
O vO is nonnegative for all v 2 R6 and is monotonic, i.e., v  vO D.v/v  D.v/ vO 2 R6 . However, we can see a serious
problem with (4.73) because of the Coriolis  O vO is not nonnegative for all v 2 R6 and vO 2 R6 . O T C .v/v  C .v/ matrix, i.e., v  v/ Therefore, only a local or semiglobal observer can be obtained. The second approach involves a nontrivial coordinate transformation to transform the vessel’s equations of motion to a new set of differential equations that are linear in unmeasured states. Then the output-injection method is used to design an observer. This approach usually results in a global observer if the nontrivial coordinate transformation can be found. Unfortunately, this coordinate transformation depends heavily on a solution of a set of partial differential equations, which in general are hard to solve. The main idea of this approach is to find a coordinate transformation X D Q./v;

(4.74)

4.2 Previous Work on Control of Underactuated Ocean Vessels

85

where Q./ is an invertible. This matrix is to be determined later. Substituting (4.74) into (4.26) results in P D J ./Q1 ./X ; h i P XP D Q./v  Q./M 1 C .v/v  Q./M 1 DQ1 X C Q./M 1 :

(4.75)

The goal is to determine the matrix Q./ such that P Q./v  Q./M 1 C .v/v D 0;

(4.76)

for all  2 R3 and v 2 R3 . With (4.76), we can write (4.75) as P D J ./Q1 ./X ; XP D Q./M 1 DQ1 X C Q./M 1 :

(4.77)

It is seen that the transformed system (4.77) is linear in the unmeasured state X . This allows us to design an exponential/asymptotical observer to estimate X . After O of v can be found from (4.74), i.e., that an estimate, v, vO D Q1 ./XO ;

(4.78)

where XO denotes an estimate of X . It is noted that combining the first equation of (4.26) and (4.76) results in a set of partial differential equations. Finding a solution to this set of partial differential equations is a hard task. A simple application of the above idea gives the results in [101–104] for some single degree of freedom Lagrange systems. It is noted that the method of solving the set of partial differential equations in [101–104] is not applicable for systems of more than one degree of freedom. For more complicated Lagrange systems, it is hard to find a result in this approach. However, the reader is referred to [105] where an output eedback control solution for simultaneous stabilization and tracking control of an underactuated ODIN is given. Remark 4.7. The main difficulty in designing an observer-based output feedback for surface ships and Lagrange systems in general is because of the Coriolis matrix, which results in cross terms of unmeasured velocities. In addition, the underactuation of surface ships makes the output feedback problem much more challenging. For example, many solutions proposed for robot control, see [14] and references therein, cannot directly be applied. The reader will find that a set of special coordinate transformations is derived in this book to transform the ship dynamics to a system that is linear in unmeasured velocities, and another set of coordinate transformations that makes it possible to design global output feedback control controllers for underactuated ships.

86

4 Control Properties and Previous Work on Control of Ocean Vessels

4.3 Conclusions This chapter presented the main control properties of ocean vessels. The literature on the control of underactuated ocean vessels including ships and underwater vehicles was then reviewed. Through this review, several challenging questions were raised. These questions motivate contributions of the coming chapters of this book.

Part III

Control of Underactuated Ships

Chapter 5

Trajectory-tracking Control of Underactuated Ships

This chapter addresses the problem of trajectory-tracking control of underactuated surface ships. In particular, we present a method to design a controller for underactuated surface ships with only surge force and yaw moment available to globally asymptotically track a reference trajectory generated by a suitable virtual ship. The reference yaw velocity does not have to satisfy a persistently exciting condition as was often required in previous literature. Hence, the reference trajectory is allowed to be a curve including a straight line and a circle. In addition, a new solution to global K-exponential tracking as given in previous work is obtained. The control development is based on Lyapunov’s direct method and the backstepping technique, and utilizes passive properties of ship dynamics and their interconnected structure.

5.1 Control Objective We consider an underactuated ship with simplified dynamics meaning that all offdiagonal terms of the linear and nonlinear damping matrices, and environmental disturbances induced by waves, wind and ocean currents are ignored. For the reader’s convenience the mathematical model of an underactuated ship moving in surge, sway, and yaw directions, see Section 3.4.1.2, is rewritten as xP D u cos. /  v sin. /;

yP D u sin. / C v cos. /; P D r;

m22 d11 1 vr  uC u ; m11 m11 m11 d22 m11 ur  v; vP D  m22 m22 m11  m22 d33 1 rP D uv  rC r ; m33 m33 m33 uP D

(5.1)

89

90

5 Trajectory-tracking Control of Underactuated Ships

where all symbols in (5.1) are defined as in Section 3.4.1.2. The available control inputs are the surge force u and the yaw moment r . Since the sway control force is not available, the ship model (5.1) is underactuated. The tracking control problem is to force the underactuated ship to track a reference trajectory generated by a virtual ship as xP d D ud cos.

yPd D ud sin. P d D rd ;

d /  vd

d / C vd

sin.

d /;

cos.

d /;

m22 d11 1 vd rd  ud C ud ; m11 m11 m11 d22 m11 ud rd  vd ; vP d D  m22 m22 .m11  m22 / d33 1 rPd D ud vd  rd C rd ; m33 m33 m33

uP d D

(5.2)

where all variables have similar meanings as in system (5.1) for the virtual reference ship. In this chapter, we propose a novel method to design a controller such that it forces the ship model (5.1) to globally asymptotically track a reference trajectory generated by a virtual ship described by (5.2) under the following assumption. Assumption 5.1. 1. The reference signals ud , rd , uP d , uR d and rPd are bounded. 2. One of the following conditions holds: (1) There exists a positive constant r such that, for any pair of .t0 ; t /; 0  t0  t < 1, Zt (5.3) rd2 . /d  r .t  t0 /: t0

(2) There exist constants umin > 0; 1  0 and 2 > 0 such that umin  jud .t /j ; jrd .t /j  1 e 2 .tt0 / ; 8 t  t0  0:

(5.4)

Remark 5.1. A persistently exciting condition similar to (5.3) is also required in [19, 71, 106], and (5.4) implies that the sign of ud .t / remains unchanged. However, we only require either (5.3) or (5.4) to be satisfied. Assumption 5.1 is quite realistic from a practical point of view. Roughly speaking, either rd or ud is allowed to approach zero. When both rd and ud are equal to zero, the tracking problem becomes one of stabilizing (5.1) and cannot be solved by any time-invariant smooth feedback. It is also noted that the exponential vanishing of rd in (5.4) can be easily replaced R1 by jrd .t /j dt  #; 0  # < 1. 0

In [19, 71, 106], a coordinate transformation of the form

5.1 Control Objective

91

z1 D cos. /x C sin. /y; z2 D  sin. /x C cos. /y;

(5.5)

z3 D

was used for both (5.1) and (5.2), then different controllers were designed for the resulting tracking error system. Consequently, the reference yaw velocity rd must satisfy persistently exciting conditions of various kinds. Hence, the way-point tracking is excluded from consideration. In addition, the physical meaning of the tracking errors is less clear. To overcome the above-mentioned drawbacks, we introduce the position and orientation errors x  xd ; y  yd ; 

(5.6)

d

in a frame attached to the ship body, see Figure 5.1. In this figure, OE XE YE is the earth-fixed frame, Ob Xb Yb is the body-fixed frame, and C G is the center of gravity of the ship. With the above definition in mind, we have the error coordinates Virtual ship

YE

Real ship

Yb

Yd vd

y

v Ob x

\

x

CG ye

ud X d xe

\e Od

yd

x

OE

xd

Xb

u

\d

x

XE

Figure 5.1 Definition of tracking errors

32 3 3 2 x  xd cos. / sin. / 0 xe 4 ye 5 D 4  sin. / cos. / 0 5 4 y  yd 5 : 0 0 1  d e 2

(5.7)

92

5 Trajectory-tracking Control of Underactuated Ships

Indeed, convergence of Œxe ye e T to zero implies that of Œx  xd y  yd since the matrix 3 2 cos. / sin. / 0 4  sin. / cos. / 0 5 0 0 1

is nonsingular for all



T d

2 R. We also define the velocity tracking errors as ue D u  ud ;

ve D v  vd ; re D r  rd :

(5.8)

Then, differentiating both sides of (5.7) and (5.8) along the solutions of (5.1) and (5.2) results in xP e D ue  ud .cos. yPe D ve  vd .cos. P e D re ;

e /  1/  vd

sin.

e /  1/ C ud sin.

e / C re ye

m22 d11 1 vr  uC u  uP d ; m11 m11 m11 m11 d22 m11 ue rd  .ue C ud /re  ve ; vP e D  m22 m22 m22 .m11  m22 / d33 1 rPe D uv  rC r  rPd : m33 m33 m33

uP e D

C rd ye ;

e /  re xe  rd xe ;

(5.9)

It is now clear that the tracking problem of underactuated surface ships becomes one of stabilizing (5.9) at the origin since, as has already been mentioned convergence of xe , ye , and e to zero implies that of x  xd , y  yd , and  d . In particular, we will design explicit expressions for u and r such that kXe .t /k 

.kXe .t0 /k/ e .tt0 / , t0  0, Xe .t0 / 2 R6 , with Xe D Œxe ; ye ; e ; ue ; ve ; re T , being a class-K function, and  being a nonnegative continuous decreasing function of k.Xe .t0 /k. When Assumption 5.1 satisfies (5.3), it can be shown that  is a positive constant, i.e., global K-exponential tracking is achieved.

5.2 Control Design In this section, a procedure to design a global asymptotic stabilizer for the tracking error system (5.9) is presented. The triangular structure of (5.9) suggests that we design the actual controls u and r in two stages. First, we design the virtual velocity controls ue and re to globally asymptotically stabilize xe ; ye ; e , and ve at the origin. Based on the backstepping technique, the controls u and r will then be designed.

5.2 Control Design

93

By looking at (5.9), we can directly see that xe and e can be stabilized at the origin by ue and re . There are several options to stabilize ye at the origin. We can either use re , ve or e . If re is directly used, the control design will be extremely complicated since re enters all of the three equations of (5.9). In addition, re couples with xe and ye in the first and second equations of (5.9), respectively. This causes difficulties in obtaining a global solution. On the other hand, the use of ve to stabilize ye at the origin will result in an undesired feature of the vehicle control practice, namely the vehicle will slide in the sway direction. Hence we will choose e to stabilize the sway error ye at the origin. This also coincides with the ship control practice as described in [107]: A good helmsman will use the ship course angle rather than use the ship sway velocity to steer the ship. Toward this end, we define the following coordinate ! k ud ye ; (5.10) ze D e C arcsin p 1 C xe2 C ye2 where k is a positive constant satisfying

kumax d  k

(5.11)

for some positive constant k < 1 to be chosen later in the stability analysis. umax d denotes the maximum value of jud j. It is seen that (5.10) is well defined and convergence of xe ; ze , and ye to zero implies that of e . Using (5.10) instead of ze D e C kye , we avoid the ship whirling around for large ye . This equation is also different from the one in [108] and [18], which resulted in a local tracking result for mobile robots. With (5.10), the ship error dynamics (5.9) are rewritten as xP e D ue C ud .1  $e / C kud vd ye $e1 C re ye C rd ye C px ; yPe D ve C vd .1  $e /  ku2d ye $e1  re xe  rd xe C py ;

vP e D ˛ ve  ˇ ue rd  ˇ .ue C ud /re ;
 1 2 1 1 .pz C pu /  kud xe ye $e2 $e1 uQ e ; zP e D 1  kud $e2 xe re C $e2 m22 d11 1 uP e D vr  .ue C ud / C u  uP d ; m11 m11 m11 1 d33 m11  m22 .re C rd / C rPe D uv  r  rPd ; m33 m33 m33 where, for notational simplicity, we have defined m11 d22 ;ˇD ; m22 m22 px D  ..cos.ze /  1/ud C sin.ze /vd / $e  .sin.ze /ud  .cos.ze /  1/vd / kud ye $e1 ; ˛D

py D  ..cos.ze /  1/vd  sin.ze /ud / $e  .sin.ze /vd C .cos.ze /  1/ud / kud ye $e1 ;

(5.12)

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5 Trajectory-tracking Control of Underactuated Ships


2 ; pz D kud py  kud ye xe px C ye py $e1

pu D kud ve C kud vd .1  $e /  k 2 u3d ye $e1  kud rd xe C k uP d ye   2 kud $e1 ye xe ude C xe ud .1  $e / C kud vd $e1 xe ye C ye ve C
ye vd .1  $e /  ku2d $e1 ye2 ; s 1 ; $e1 D 1 C xe2 C ye2 q $e2 D 1 C xe2 C .1  .kud /2 /ye2 ; $e D $e1 $e2 ;

(5.13)

with ude and uQ e being defined in (5.15). Before designing the control laws, u and r , we note from (5.11) that 0 < 1  k  1 

xe kud : $e2

(5.14)

This allows us to design global control laws u and r to asymptotically stabilize (5.12) at the origin. The control design consists of two steps as follows.

Step 1 We define the following virtual control errors uQ e D ue  ude ; rQe D re  red ;

(5.15)

where ude and red are the virtual velocity controls of ue and re , respectively. The virtual surge and yaw velocity controls are chosen as ude D k1 xe C k2 rd ye ; d d red D r1e C r2e

where d D r1e d r2e D

1 pu ; $e2  kud xe

1 .k3 $e2 ze C pz / ; $e2  kud xe

(5.16)

(5.17) (5.18)

and ki ; i D 1; 2; 3, are positive constants to be selected later. We have written red as d d and r2e to simplify notation in the stability analysis later. a sum of r1e Remark 5.2. Unlike the standard application of the backstepping technique, in order to reduce complexity of the controller expressions, we have chosen a simple virtual control law ude without canceling the known terms. The first term in ude is used to

5.2 Control Design

95

stabilize the xe -dynamics while the second term plays the role of stabilizing the d is Lipsye -dynamics when (5.3) holds. From (5.17) and (5.13), we observe that r1e d chitz in .xe ; ye ; ve / and with (5.18), r2e exponentially vanishes when ze does. This observation plays a crucial role in the stability analysis of the closed loop system.

Step 2 Differentiating (5.15) along the solutions of (5.12) and (5.16) yields m22 d11 1 vr  uC u  uP d  uP de ; uPQ e D m11 m11 m11 .m11  m22 / d33 1 rQPe D uv  rC r  rPd  rPed ; m33 m33 m33

(5.19)

where uP de and rPed are the first time derivatives of ude and red along the solutions of (5.12). From (5.19) we choose the actual controls u and r without canceling the useful damping terms as   m22 d11 d u D m11 ı1 uQ e  vr C .ue C ud / C uP d C uP de ; m11 m11   d33 d .m11  m22 / r D m33 ı2 rQe  (5.20) uv C .re C rd / C rPd C rPed ; m33 m33 where ı1 and ı2 are positive constants to be chosen later. Substituting (5.16) and (5.20) into (5.12) yields the closed loop system XP 1e D f1 .t; X1e / C g1 .t; X1e ; X2e /; XP 2e D f2 .t; X1e ; X2e /;

(5.21)

where X1e D Œxe ye ve T , X2e D Œze uQ e rQe T , 3 2 d k1 xe C k2 rd ye C ud .1  $e / C .kud vd $e1 C r1e C rd /ye d 5; f1 .t; X1e / D 4 ve C vd .1  $e /  ku2d $e1 ye  r1e xe  rd xe d d d ˛ ve  ˇ ue rd  ˇ .ue C ud /r1e (5.22) 3 2 d r2e ye C rQe ye C px C uQ e d 5 g1 .t; X1e ; X2e / D 4 r2e
;  xe  rQe xe C dpy d d d ˇ uQ e rd C uQ e .r1e C r2e C rQe / C .ue C ud /.r2e C rQe / (5.23) 3 2 2 k3 ze C .1  kud xe =$e2 / rQe  kud xe ye $e1 uQ e =$e2 5: f2 .t; X1e ; X2e / D 4  .ı1 C d11 =m11 / uQ e  .ı2 C d33 =m33 / rQe (5.24)

96

5 Trajectory-tracking Control of Underactuated Ships

The time dependence in f1 .t; X1e /, g1 .t; X1e ; X2e / and f2 .t; X1e ; X2e / results from the time-varying reference velocities. We now state our main result, the proof of which is given in the next section. Theorem 5.1. Assume that the reference signals .xd ; yd ; d ; vd / are generated by the virtual ship model (5.2) and that the reference velocities .ud ; rd / satisfy Assumption 5.1. If the state feedback control law (5.20) is applied to the ship system (5.1) then the tracking errors x.t /  xd .t /, y.t /  yd .t /, .t /  d .t /, and v.t /  vd .t / converge to zero asymptotically from arbitrary initial values with an appropriate choice of the design constants ki ; 1  i  3 and ıj ; j D 1; 2, i.e., the closed loop system (5.21) is GAS at the origin. Furthermore if Assumption 5.1 holds with (5.3), the closed loop system (5.21) is globally K-exponentially stable at the origin.

5.3 Stability Analysis To prove the global asymptotic stability at the origin of the closed loop system (5.21), we note that (5.21) consists of two subsystems, X1e and X2e , in an interconnected structure. If X1e does not appear in the second equation of (5.21), we can use the stability analysis approaches for a cascaded system in [17]. One might claim that if the system XP 1e D f1 .t; X1e /; XP 2e D f2 .t; X1e ; X2e /

(5.25)

is uniformly globally asymptotically stable (UGAS), and the term g1 .t; X1e ; X2e / somehow is of linear growth in X1e , then the closed loop system (5.21) is UGAS. However, we are interested in proving stronger stability properties of (5.21). Hence we present the following lemma. Lemma 5.1. Consider the following nonlinear system: xP D f .t; x/ C g.t; x; .t //

(5.26)

where x 2 Rn ; .t / 2 Rm , f .t; x/ is piecewise continuous in t and locally Lipschitz in x. Assume the following: C1. There exists a proper function V .t; x/ satisfying: c1 kxk2  V .t; x/  c2 kxk2 ;

@V

.t; x/  c3 kxk;

@x @V @V kxk2 C f .t; x/  l1 .t / kxk2  l2 .t / q ; @t @x 1 C kxk2

(5.27)

5.3 Stability Analysis

97

where ci > 0; i D 1; 2; 3, and the pair l1 .t / and l2 .t / satisfy one of the conditions: Rt 1. t0 l1 . /d  1 .t  t0 /; 1 > 0 and l2 .t /  0; 8t  t0  0, 2. 0 < 2  l2 .t / < 1; and jl1 .t /j  1 e 2 .tt0 / ; 1  0; 2 > 0; 8t  t0  0. C2. g.t; x; .t // grows linearly in x: kg.t; x; .t //k  .1 C 2 kxk/ k.t/k ; i  0; i D 1; 2: C3. .t/ vanishes exponentially: k.t/k  0 .k.t0 /k/ e 0 .tt0 / ; 0 > 0; 8t  t0  0; with 0 being a class-K function. Then, the solution x.t / of (5.26) is GAS in the sense that kx.t /k  .k.x.t0 /; .t0 //k/ e .1 C2 /.tt0 / ; 8t  t0  0; 1  0;

(5.28)

where is a class-K function, 2 is a nonnegative continuous decreasing function of k.x.t0 /; .t0 //k, and 1 > 0 if condition C1 holds with 1. It can be understood that .t/ is generated from a globally K-exponentially stable dynamic system. Proof. We first consider the case of condition C1 satisfying 1. From conditions C1, C2, and C3, we have   l1 .t / 1 p 2 VP   (5.29) V C c3 p V C V 0 .k.t0 /k/ e 0 .tt0 / : c2 c1 c1 p By defining .t / D V .t /, we can rewrite (5.29) as   c 3 1 l1 .t / c3 2 .t P / 

0 .k .t0 /k/ e 0 .tt0 / .t / C p  2c2 2c1 2 c1

0 .k .t0 /k/ e 0 .tt0 / :

(5.30)

Solving the differential inequality (5.30) readily yields (5.28). It is of interest to note that in this case, the system (5.26) is GES due to independence of 1 in (5.28) on k.x.t0 /; .t0 //k. We now move to the case of condition C1 satisfying 2. We first show that x.t / is bounded for all t  t0  0. From conditions C1, C2, and C3, we have 2

kxk VP  1 e 2 .tt0 / kxk2  l2 .t / q C 1 C kxk2

c3 kxk .1 C 2 kxk/ 0 .k.t0 /k/ e 0 .tt0 /   1 p 2 1  c3 p V C V 0 .k.t0 /k/ e 0 .tt0 / C e 2 .tt0 / V: c1 c1 c1

(5.31)

From (5.31), it is not hard to show that the solution x.t / is bounded by kx.t /k  b .k.x.t0 /; .t0 //k/ with b being a class-K function. By utilizing this bound and

98

5 Trajectory-tracking Control of Underactuated Ships

condition C1, we can rewrite the first line in (5.31) as   2 1 p 2 1 P V  p V C V 0 .k.t0 /k/ e 0 .tt0 / C  V C c3 p 2 c1 c1 c1 c2 1 C b

(5.32) e 2 .tt0 / V: p By defining .t / D V .t /, we rewrite (5.32) as follows:   1 2 .tt0 / c 3 2 2

0 .k .t0 /k/ e 0 .tt0 /  e  .t P /  p 2c1 2c1 2c2 1 C b 2 c3 1 .t / C p 0 .k .t0 /k/ e 0 .tt0 / : (5.33) 2 c1 Again, solving the differential inequality (5.33) readily yields (5.28).  We now view our closed loop system (5.21) as a form of the system studied in Lemma 5.1 with X1e .t / as x.t /, and X2e .t / as .t/. In the following, we will show that the closed loop system (5.21) satisfies all the conditions C1, C2, and C3. Verifying Condition C1. To verify condition C1 of Lemma 5.1 for the system XP 1e D f1 .t; X1e /;

(5.34)

the first time derivative of the following Lyapunov function 1 k4 1 V1 D xe2 C ye2 C .ve C k5 ye /2 ; 2 2 2

(5.35)

where k4 and k5 are positive constants, along the solutions of (5.34) satisfies VP1 D k1 xe2  ˇk2 k4 k5 rd2 ye2  ku2d .1 C k4 k52 /$e1 ye2  k4 .˛  k5 /ve2 C M; (5.36) where M D k2 rd xe ye C .1  $e / .ud xe C k4 k5 vd .ve C k5 ye / C vd ye / C kud vd $e1 

d ; xe ye  k2 k4 ˇrd2 ye ve  k4 k5 ku2d $e1 ye ve  ˇk4 .ve C k5 ye /.k2 rd ye C ud /r1e (5.37)

and we have chosen ˇ k1  k5 D 0; 1  k4 .˛  k5 /k5 D 0

(5.38)

to cancel some common terms. In addition, we note that .kud /2 ye2 p  : (5.39) 1  $e D p p 1 C xe2 C ye2 1 C xe2 C ye2 C 1 C xe2 C .1  .kud /2 /ye2

5.3 Stability Analysis

We now write M D M1 C

99

P4

iD1 Ni ,

where

M1 D k2 rd xe ye C .1  $e / .ud xe C k4 k5 vd .ve C k5 ye / C vd ye / C kud vd $e1 xe ye  k2 k4 ˇrd2 ye ve  k4 k5 ku2d $e1 ye ve ; d d ; N2 D ˇk4 ud ve r1e ; N1 D ˇk2 k4 rd ye ve r1e

d d N3 D ˇk2 k4 k5 rd ye2 r1e ; N4 D ˇk4 k5 rd ye r1e :

(5.40)

Applying (5.39) and the completing square to (5.40) yields
1  k2 C ku2d C kvd2 xe2 C .k2 "1 C ˇk2 k4 "2 / rd2 ye2 C 4"1 u2 y 2 .2k"1 C 2kk4 k5 "3 / p d e C k.1 C k4 k52 / jud vd j  1 C xe2 C ye2 ! kk4 k5 vd2 ˇk2 k4 rd2 kk4 k5 u2d ye2 C C C ve2 ; p 4"3 4"2 4"3 1 C xe2 C ye2

M1 

(5.41)

ˇ ˇ
1  2  2 jud rd j ud vd C rd2 C 2k ˇu3d ˇ C C jud rd j C p 2 4" 1  k 2 !!! uP 2d ˇ k 2 k4 2 2 2 4 2 2 ve2 C C .k1 C k2 rd /ud C k ud C k vd C jud vd j (5.42) p 1  k 1  k2 ! ! ˇ ˇ 1  k"2 3 C k1 C k2 C 2k ˇu3d ˇ C p C jud vd j C k2 "2 rd2 ye2 ; 1  k2

N1 

ˇ k2 k4 k 1  k

 1  2 2 2 ˇ k4 k 2u2d C u2d jrd j "1 C k u v C 2ku4d C uP 2d C k1 u2d C N2  1  k 4"3  d d

ˇkk4 u2d jrd j 2 x C k2 u2d rd2 C k2 u4d C ku4d vd2 C k2 u2d vd2 ve2 C (5.43) .1! k / 4"1 e u2 y 2 ˇ k4 "3 k 1 ; C k1 C k2 C k p d e 3k2 C 2ku2d C p 1  k 1  k2 1 C xe2 C ye2 !

ˇ k2 k 4 k 5 k  C jud j C k2 jud j rd2 ye2 C p 1  k 1! k2 ˇ k4 k5 k2 k 2k jud rd j juP d j C k1 jud j jrd j ye2 C C k 2 jrd vd j C (5.44) p p 2 1  k 1  k 1  k2
u2 y 2 ˇ k2 k4 k5 ku2d ve2 C ; k2 C k jrd vd j C k 2 jud rd vd j p d e p 1 C xe2 C ye2 .1  k /2"2 1  k2

ˇ k 2 k4 k5 k N3  1  k

2"2

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5 Trajectory-tracking Control of Underactuated Ships

ˇ k4 k5 ku2d 2 ˇk4 k5 ku2d 1 2 ˇ k4 k5 ku2d "1 2 2 v C x C rd ye C 1  k 2"3 e 1  k 4"1 e 1  k 2 ˇ k4 k5 k juP d j "3 ˇ k 4 k5 k jud j ye C C 2k2 jvd j C (5.45) p p p 2 2 2 1  k .1  k / 1  k 1 C xe C ye 1  k2
u2 y 2 : 2ku2d C k1 C k2 jrd j C k2 jud j C k jud vd j C 1 p d e 1 C xe2 C ye2

N4 

Substituting (5.37) with (5.41)–(5.45) into (5.36) yields 2

where

22 .t /ye VP1  1 .t /xe2  21 .t /ye2  p  3 .t /ve2 ; 1 C xe2 C ye2 ˇk4 ku2d jrd j ˇk4 k5 ku2d 1 C k2 C ku2d C kvd2 C 1 .t / D k1  4"1 1  k 1  k

(5.46)

!

;

21 .t / D 211 .t /rd2  212 .t / jrd j ;  ˇ k2 k4 .k"2 .3 C k1 C k2 C 211 .t / D ˇ k2 k4 .k5  "2 /  k2 "1  1  k ! ! ˇ 3ˇ 1 ˇ k2 k4 k5 2 2k ˇud ˇ C p  C jud vd j C k "2  2 1  k 1  k ! # ˇ k4 k5 ku2d "1 2k"2 C k jud j C kk2 jud j  ; p 1  k 1  k2 ! k juP d j ˇ k 2 k 4 k5 212 .t / D C kk1 jud j ; p 1  k 1  k2 22 .t / D 221 .t /u2d  222 .t / jud j ;

ˇ k4 "3 k  1  k  ˇk k k k  1 4 5 2  C k1 C k2 C k  3k2 C 2ku2d C p 2 1  k 1  k ! 2k jud rd j 2 2 2 C k jrd vd j C k C k jrd vd j C k jud rd vd j  p 1  k2

221 .t / D k.1 C k4 k52 /  2k"1  2k4 k5 k"3 

ˇ k4 k 5 k 1  k

"3 C 2k2 jvd j C 2ku2d C k1 C k2 jrd j C p 1  k2
k2 jud j C k jud vd j C 1 ; ˇ k4 k5 k juP d j 222 .t / D k.1 C k4 k52 / jvd j C ; p .1  k / 1  k2

5.3 Stability Analysis

101

kk4 k5 vd2 ˇk2 k4 rd2 kk4 k5 u2d    4"3 4"2 4"3 ˇ k2 k4 k5 ku2d ˇ k4 k5 ku2d ˇ k2 k4 k    p 1  k .1  k /2"2 1  k2 1  k 2"3

3 .t / D k4 .˛  k5 / 

ˇ ˇ 1  2 2 jud rd j C jud rd j C ud .vd C rd2 / C 2k ˇu3d ˇ C p 2 4"2 1  k !! uP 2d 2 2 2 4 2 2  C .k1 C k2 rd /ud C k ud C k vd C jud vd j p 1  k2  ˇ k4 k 1  2 2 2 2u2d C u2d jrd j "1 k u v C 2ku4d C uP 2d C 1  k 4"3  d d

k1 u2d C k2 u2d rd2 C k2 u4d C ku4d vd2 C k2 u2d vd2 (5.47)

for some positive constants "i ; i D 1; 2; 3. The time dependence of 1 , 21 , 22 , and 3 is due to the time-varying reference velocities. Hence condition C1 is satisfied if there exist positive constants k4 and k5 , and the design constants k, k1 , and k2 such that the following conditions are satisfied: 1. kumax  k < 1, ˇ k1  k5 D 0; 1  k4 .˛  k5 /k5 D 0, 1 .t /  1 ; 3 .t /  d  3 ; 8t  0, max max r m  2. 221 .t /  0; 211  m 212 rd  222 ud  21 , if (5.3) holds, see Assumption 5.1, 3. 22 .t /  22 , if (5.4) holds, see Assumption 5.1,

m where 1 , 21 , 22 , and 3 are some positive constants, m 212 and 222 are the values of 211 .t / and 212 .t / with the maximum values of jud .t /j, juP d .t /j and jrd .t /j substituted in, and rdmax denotes the maximum value of jrd j. We now need to show that the above constants always exist. From (5.38), we have s 1 ˛ k5 ˛2  : (5.48) k1 D ; k5 D  ˇ 2 4 k4

It is noted that we take k5 D

˛ 2



q

˛2 4

 k14 instead of k5 D

˛ 2

C

q

˛2 4

 k14 since we

want to have a small k5 for the condition 3 .t /  3 > 0. We choose k4  4=˛ 2 to make k5 real and positive, and set k2 D ık with ı being a small positive constant, say ı  max.umax ; uP max ; rdmax /, with uP max being the maximum of juP d j. By observing d d d from all functions i .t /; i D 1; 21; 22; 3 that their negative parts have k as a factor, and that the mass including added mass in the sway dynamics, m22 , is always larger than that in the surge dynamics, m11 , for surface ships, i.e., ˇ < 1, therefore we can always find a positive constant k or even k D 0 such that the conditions 1 and 2 hold for some small "i > 0; i D 1; 2; 3. For simplicity, one can take "i D ˇ. In fact, k D 0 results in the controllers proposed in [19, 106] (with nonzero k2 selected). However if k D 0 the condition 3 cannot be satisfied. Hence we should choose k to be a small positive constant. Note that a small k automatically implies a small k . The value

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5 Trajectory-tracking Control of Underactuated Ships

of this constant should be reduced if the reference yaw, rd , and surge, ud , velocities and surge acceleration, uP d , are large. The physical meaning of this interpretation is that the distance from the ship to the point it aims to track should be increased if the velocities are large, otherwise the ship will miss that point due to high velocities. Furthermore if ˛ is small, k5 is automatically small. This physically means that if the damping in the sway is small, the control gain in the surge dynamics should also be small, otherwise the ship will slide in the sway direction. This can be mathematically seen from the expression of 3 .t /. Due to complicated expressions of i .t /, we give a simple procedure to choose the design constants k1 and k2 rather than present their complex explicit expressions. ; rdmax ; uP max /, ı such that 1. Pick a small positive constant k, say k 1:5; ı1  0:5; ı2  1:

(5.55)

Hence condition C3 follows from (5.54). We have thus verified all the conditions of Lemma 5.1. Therefore the closed loop system (5.21) is GAS at the origin. Remark 5.4. From the proof of Theorem 5.1 and Lemma 5.1, it is seen that when the yaw reference velocity rd satisfies the persistently exciting condition as stated in Assumption 5.1, the tracking error system is GES. This is the case reported in [19,71,106] based on different approaches. Therefore the results in those papers are a special case of the result in Theorem 5.1.

5.4 Simulations This section validates the control laws (5.20) by simulating them on a monohull ship with the length of 32 m, mass of 118  103 kg, and other parameters are calculated by using MARINTEK Ship Motion program version 3.18, a program for calculating

104

5 Trajectory-tracking Control of Underactuated Ships

the added mass and damping matrices of the ship, as m11 D 120  103 kg; m22 D 177:9  103 kg; m33 D 636  105 kgm2 ; d11 D 215  102 kgs-1 ; du2 D 43  102 kgm1 ; du3 D 21:5  102 kgsm2 ; d22 D 117  103 kgs-1 ; dv2 D 23:4  103 kgm1 ; dv3 D 11:7  103 kgsm2 ; d33 D 802  104 kgm2 s 1 ; dr2 D 160:4  104 kgm2 ; dr3 D 80:2  104 kgm2 s; dui D 0; dvi D 0; dri D 0; 8i > 3: This ship has a minimum turning circle with the radius of 150 m, a maximum surge force of 5:2  109 N, and a maximum yaw moment of 8:5  108 Nm. From the ship parameters, we have ˛ D 0:54 and ˇ D 0:55. The reference trajectory is generated by the virtual ship (5.2) where the reference surge force ud is taken as ud D d11  m22 vd rd , and the reference yaw moment rd is taken as rd D .m11  m22 /ud rd for the first 800 seconds and rd D .m11  m22 /ud rd C 0:01d33 for the last 1000 seconds. This choice means that the reference surge velocity ud is 1ms1 over the entire simulation time, and that the reference trajectory is a straight line for the first 800 seconds and a circle with a radius of 200 m for the last 1000 seconds. We first pick k D 0:02 and k4 D 13:9, then increase these constants until conditions a, b, and c hold. We have k D 0:1; k1 D 0:34; k2 D 0:1; k3 D 2; k4 D 15:26; k5 D 0:19; ı1 D 5; ı2 D 5: The initial conditions are chosen as Œx.0/;   y.0/; .0/; u.0/; v.0/; r.0/ D 15m; 50m; 0:7rad; 0ms1 ; 0:5ms1 ; 0rads1 :

The reference trajectory is generated by a virtual ship with the initial conditions of Œxd.0/; yd .0/; d .0/; ud .0/; vd .0/; rd .0/  D 0:5m; 0:5m; 0:2rad; 1ms1 ; 0ms1 ; 0rads1 :

The tracking errors are plotted in Figure 5.2a and control inputs are plotted in Figure 5.2b. In addition, the real and reference trajectories in the .x; y/-plane are plotted in Figure 5.3. It can be seen from Figure 5.2a that the tracking errors asymptotically converge to zero as proven in Theorem 5.1. From Figure 5.2b, we can see that the control inputs u and r are quite large in the first few seconds because of a short transient response time. Indeed, we can reduce the control effort by tuning the control design gains such as reducing the control gains ı1 and ı2 . However, this will increase the transient response time. For a comparison, we also simulate the controller proposed in [19] with the same initial conditions and reference trajectory in Figures 5.4 and 5.5. It is clearly seen that the controller proposed in [19] cannot track the straight line as discussed before. It should be noted that the controllers proposed in [71, 106] also result in similar nonzero errors although they were designed based on different approaches.

5.5 Conclusions

105 a

Tracking errors

60 40 20 0 −20 0

200

400

600

5

Control inputs

5

800 1000 Time [s]

1200

1400

1600

1800

1200

1400

1600

1800

b

x 10

0

−5 0

200

400

600

800 1000 Time [s]

Figure 5.2 Simulation results of the controller proposed in this chapter: a. Convergence of tracking errors (x  xd [m]: solid line, y  yd [m]: dashed line,  d [rad]: dash-dotted line); and b. Control inputs (u [N]: solid line, r [Nm]: dashed line)

5.5 Conclusions The key to control development is the coordinate change (5.10) to transform the tracking error system, in which the tracking errors are interpreted in the frame attached to the ship’s body, to a triangular form, to which the popular backstepping technique can be applied. This coordinate change made it possible to relax the severe restrictions in [19,71,106] on the reference trajectories generated by the virtual ships in the sense that the developed controllers can force the vehicles to track a straight line (way-point tracking), a curve and a combination of both. Several assumptions on the ship dynamics have been made such as all off-diagonal terms of the damping matrix, nonlinear damping matrix, and environmental disturbances induced by waves, wind, and ocean currents being ignored. For a justification, the reader is referred to Section 3.4.1.2. It is seen from Assumption 5.1 that the surge and yaw reference velocities cannot be zero, i.e., the problem of stabilization and/or parking is not considered in this chapter. In the next chapter, this limitation will be removed when a simultaneous stabilization and tracking control is addressed. This chapter is based on [109–111].

106

5 Trajectory-tracking Control of Underactuated Ships

600 500

y [m]

400 300 200 100 0

100

200

300

400

500 x [m]

600

700

800

900

Figure 5.3 Simulation results of the controller proposed in this chapter: Tracking trajectory in the .x; y/-plane (.x; y/: solid line, .xd ; yd /: dashed line) a

Tracking errors

60 40 20 0 −20 0

200

400

600

5

Control inputs

5

800 1000 Time [s]

1200

1400

1600

1800

1200

1400

1600

1800

b

x 10

0

−5 0

200

400

600

800 1000 Time [s]

Figure 5.4 Simulation results of the controller proposed in [19]: a. Convergence of tracking errors (x  xd [m]: solid line, y  yd [m]: dashed line,  d [rad]: dash-dotted line); b. Control inputs (u [N]: solid line, r [Nm]: dashed line)

5.5 Conclusions

107

600 500

y [m]

400 300 200 100 0

100

200

300

400

500 x [m]

600

700

800

900

Figure 5.5 Simulation results of the controller proposed in [19]: Tracking trajectory in the .x; y/plane (.x; y/: solid line, .xd ; yd /: dashed line)

Chapter 6

Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

This chapter examines the problem of designing a single controller that achieves both stabilization and tracking simultaneously for underactuated surface ships without an independent sway actuator and with simplified dynamics. The proposed controller guarantees that stabilization and tracking errors converge to zero asymptotically from any initial values. In comparison with the preceding chapter, a path approaching the origin and a set-point can also be included in the reference trajectory, i.e., stabilization/regulation is also considered. The control development is based on some special coordinate transformations, Lyapunov’s direct method and the backstepping technique, and utilizes passive properties of ship dynamics and their interconnected structure.

6.1 Control Objective For the reader’s convenience, the mathematical model of the underactuated ship moving in surge, sway, and yaw, see Section 3.4.1.2, is recaptured as xP D u cos. /  v sin. /;

yP D u sin. / C v cos. /; P D r;

m22 d11 1 vr  uC u ; m11 m11 m11 m11 d22 vP D  ur  v; m22 m22 d33 1 .m11  m22 / uv  rC r ; rP D m33 m33 m33 uP D

(6.1)

where all symbols in (6.1) are defined as in Section 3.4.1.2. The available controls are the surge force u and the yaw moment r . Since the sway control force is not

109

110

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

available, the ship model (6.1) is underactuated. Similar to the preceding chapter, it is assumed that the reference trajectory is generated by a virtual ship as xP d D ud cos. yPd D ud sin. P d D rd ;

d /  vd

sin.

d /;

d / C vd cos.

d /;

m22 d11 1 vd rd  ud C ud ; m11 m11 m11 m11 d22 vP d D  ud rd  vd ; m22 m22 d33 1 .m11  m22 / rPd D ud vd  rd C rd ; m33 m33 m33

uP d D

(6.2)

where all variables have similar meanings as in the system (6.1) for the virtual reference ship. In this chapter, a single controller that simultaneously solves stabilization and tracking problems of underactuated surface ships is proposed under the following assumptions. Assumption 6.1. The reference velocities, ud and rd , are bounded and differentiable with bounded derivatives uP d , uR d and rPd . Assumption 6.2. One of the following conditions holds: C1. ud D rd D 0. Rt C2. rd2 . /d  r .t  t0 /; r > 0; 8 0  t0  t < 1. t0

C3. jud .t /j  u > 0 and

C4.

R1 0

R1 0

jrd .t /jdt  1 ; 0  1 < 1.

.jrd .t /j C jud .t /j C juP d .t /j/ dt  2 , 0  2 < 1.

Remark 6.1. The problems of regulation, stabilization, or dynamic positioning are included in condition C1. Tracking a circular path belongs to the case when condition C2 holds. Condition C3 covers the case of straight-line or way-point tracking. By Barbalat’s lemma together with Assumption 6.2, Condition C4 implies that lim t!1 rd .t / D lim t!1 ud .t / D 0. Hence the parking problem is captured by condition C4. Indeed, one can see that Condition C4 covers Condition C1. However we study these two conditions separately because Condition C1 has been studied in the literature but Condition C4 has not. Remark 6.2. The work in [19, 53–56, 71, 106] can deal with either Condition C1 or Condition C2. In the preceding chapter of this book, we allowed for either Condition C2 or Condition C3. Similar to the preceding chapter, we introduce position and orientation errors x  xd , y  yd , and  d in a frame attached to the ship body. This results in the error coordinates as

6.2 Control Design

111

2

3

2

32

3

cos. / sin. / 0 x  xd xe 4 ye 5 D 4  sin. / cos. / 0 5 4 y  yd 5 : 0 0 1  d e

(6.3)

One can see that the convergence of xe ; ye , and e to the origin implies that of x  xd , y  yd and  d . We also define the velocity tracking errors as ue D u  ud ; ve D v  vd ; re D r  rd :

(6.4)

Differentiating both sides of (6.3) along the solutions of (6.1) and (6.2) yields: xP e D ue  ud .cos. e /  1/  vd sin. yPe D ve  vd .cos. e /  1/ C ud sin. P e D re ;

e / C re ye

C rd ye ; e /  re xe  rd xe ;

m22 d11 1 vr  uC u  uP d ; m11 m11 m11 m11 m11 d22 vP e D  ue rd  .ue C ud /re  ve ; m22 m22 m22 d33 1 .m11  m22 / uv  rC r  rPd : rPe D m33 m33 m33

uP e D

(6.5)

The tracking and regulation problem of underactuated surface ships is therefore equivalent to stabilizing system (6.5) at the origin. In particular, we are interested in designing explicit expressions for u and r such that lim t!1 kXe .t /k D 0, for all t  t0  0 and Xe .t0 / 2 R6 with Xe D Œxe ; ye ; e ; ue ; ve ; re T . In case of Condition C2, it will be shown that kXe .t /k  .kXe .t0 /k/ e .tt0 / with being a class-K function, and  a positive constant, i.e., global K-exponential tracking is achieved.

6.2 Control Design Similar to the preceding chapter, we observe from (6.5) that xe and e can be stabilized by ue and re . On the other hand, e should be chosen as a virtual control to stabilize the sway error ye . However when the surge reference velocity is zero or approaches zero, i.e., the case of stabilization and parking, we cannot use e to stabilize ye since ud sin. e / is zero or approaches zero. In this case, we need some persistently exciting signal in re to stabilize ye . As such, we choose the coordinate transformation ! k.t /ye ze D e C arcsin p ; (6.6) 1 C xe2 C ye2 with

112

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

k.t / D 1 .ud C 2 cos.3 t // ; where the constants i ; 1  i  3 are such that sup jk.t /j < 1:

(6.7)

t0

They will be specified in the stability analysis later. Notice that (6.6) is well defined and that convergence of ze and ye to the origin implies that of e . Remark 6.3. Using the nonlinear coordinate transformation (6.6) instead of ze D ship moving left and e C k.t /ye , we avoid the   right largely when ye is large. If p one uses ze D e C arcsin k.t /ye = 1 C ye2 , the problem of the ship whirling around is avoided. Indeed using these coordinate transformations will result in a much simpler tracking error system than using (6.6). However, it is readily shown that these coordinate transformations result in a local control solution because of the term re xe in the ye -dynamics. The term 1 ud plays the role of stabilizing ye when ud is not zero and does not approach to zero while the term 1 2 cos.3 t / is used to introduce the persistently exciting signal in re to stabilize ye when ud is zero or approaches zero. Remark 6.4. The function arcsin in (6.6) can be replaced by several other smooth functions such as arctan and tanh, or the identity function. We have chosen the function arcsin because of its simplicity. Using the new coordinate (6.6), the ship error dynamics (6.5) are rewritten as .$2  $1 / k.t /vd ye C C re ye C rd ye C px ; $2 $2 .$2  $1 / k.t /ud ye   re xe  rd xe C py ; D ve C vd $2 $   2   .$2  $1 / 1 k.t /xe P  D 1 re C k.t /ye C k.t / ve C vd $1 $1 $2  k.t /ud ye k.t /ye .xe ue C ye ve C .xe ud C ye vd /   rd xe  $2 $22  k.t /ye .$2  $1 / C pz ; C .xe vd  ye ud / $2 $2 m22 d11 1 D vr  .ue C ud / C u  uP d ; m11 m11 m11 m11 m11 d22 ve  ue rd  .ue C ud /re ; D m22 m22 m22 1 d33 m11  m22 .re C rd / C uv  r  rPd ; (6.8) D m33 m33 m33

xP e D ue C ud yPe zP e

uP e vP e rPe

where, for notational simplicity, we have used k.t / instead of using its explicit expression given in (6.6), and defined as

6.2 Control Design

$1 px

py

pz

113

q q D 1 C xe2 C .1  k 2 .t //ye2 ; $2 D 1 C xe2 C ye2 ;   k.t /ye $1  C sin.ze / D ud .cos.ze /  1/ $2 $2   $1 k.t /ye vd sin.ze / ;  .cos.ze /  1/ $2 $2   $1 1 D vd .cos.ze /  1/ C sin.ze /k.t /$2 ye C $2   $1 k.t /ye ud sin.ze /  .cos.ze /  1/ ; $2 $2  
k.t /ye  1 D k.t /py  xe px C ye py : $1 $2

(6.9)

The effort that we have made so far is to put the tracking error dynamics in the triangular form of (6.8), and to have the term k.t /ud ye =$2 in the ye -dynamics. This term plays an important role in stabilizing the ye -dynamics. In this section, a procedure to design a universal controller for the tracking error system (6.8) is presented in detail. The triangular structure of (6.8) suggests that we design the actual controls u and r in two stages. First, we design virtual velocity controls ue and re to globally asymptotically stabilize xe , ye , ze and ve at the origin. Based on the backstepping technique, the controls u and r are then designed to force the error between the virtual velocity controls and their actual values exponentially to zero. Before designing the control laws, u and r , we note if the constants i ; 1  i  3 are chosen such that sup t0 jk.t /j < 1, then (6.6) implies that 0 < 1  sup jk.t /j  1  t0

k.t /xe < 2: $1

(6.10)

This allows us to design control laws for u and r to globally asymptotically stabilize (6.8). The control design consists of two steps as follows.

Step 1 In this step, we define the following virtual control errors uQ e D ue  ude ;

rQe D re  red ;

(6.11)

where ude and rde are the virtual velocity controls of ue and re . Unlike the standard application of backstepping, in order to reduce complexity of the controller expressions, we will choose a simpler virtual control law ude without canceling the known terms. The virtual surge and yaw velocity controls are chosen as

114

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

ude D k1 xe C k2 rd ye ; d d red D r1e C r2e

(6.12)

where d r1e

  1 1 $2  $1 P D  k.t /ye C k.t / ve C vd .1  k.t /xe =$1 / $1 $2  k.t /ud ye k.t /ye .xe .k1 xe C k2 rd ye / C ye ve C  rd xe  $2 $22  k.t /ye $2  $1 .xe ud C ye vd / C .xe vd  ye ud / ; (6.13) $2 $2

1  d .k3 ze C pz / ; D  .1  k.t /xe =$1 / r2e

(6.14)

and ki ; i D 1; 2; 3, are positive constants to be selected later. Step 2 Differentiating (6.11) along the solutions of (6.8) and (6.12) yields m22 d11 1 vr  uC u  uP d  uP de ; uPQ e D m11 m11 m11 d33 1 .m11  m22 / uv  rC r  rPd  rPed ; rPQe D m33 m33 m33

(6.15)

where uP de and rPed are the first time derivatives of ude and red along the solutions of (6.8). From (6.15) we choose the actual controls u and r as   m22 d11 d k.t /xe ye ze d u D m11 c1 uQ e  vr C .u C ud / C uP d C uP e C ; m11 m11 e $1 $22  .m11  m22 / d33 d r D m33 c2 rQe  uv C .r C rd /C m33 m33 e    k.t /xe (6.16) ze ; rPd C rPed  1  $1 where c1 and c2 are positive constants. Substituting (6.16) and (6.12) into (6.8) yields the closed loop system xP e D k1 xe C k2 rd ye C ud .$2  $1 /$21 C k.t /vd $21 ye C d d r1e ye C rd ye C r2e ye C uQ e C rQe ye C px ;

d yPe D ve C vd .$2  $1 /$21  k.t /ud $21 ye  r1e xe  rd xe  d r2e xe  rQe xe C py ;

6.3 Stability Analysis

115

d22 m11 d m11 m11 d d ve  ue rd  .ue C ud /r1e  .rd C r1e /uQ e  m22 m22 m22 m22 m11 d .uQ e C ude C ud /.r2e C rQe /; m22   k.t /xe ye uQ e k.t /xe ; rQe  zP e D k3 ze C 1  $1 $1 $22   d11 k.t /xe ye ze uPQ e D  c1 C ; uQ e C m11 $1 $22     k.t /xe d33 rQe  1  ze : (6.17) rPQe D  c2 C m33 $1 vP e D 

We now state our main result, the proof of which is given in the next section. Theorem 6.1. Assume that the reference signals .xd ; yd ; d ; vd / are generated by the virtual ship model (6.2) and that the reference velocities .ud ; rd / satisfy Assumptions 6.1 and 6.2. If the universal state feedback control law (6.16) is applied to the ship system (6.1) then the tracking errors .x.t /  xd .t /; y.t /  yd .t /; .t /  d .t /, and v.t /  vd .t // converge to zero asymptotically from any initial values, i.e., the closed loop system (6.17) is GAS at the origin, with an appropriate choice of the design constants i and ki ; 1  i  3. Furthermore, if Assumption 6.2 holds with C2, the closed loop system (6.17) is globally K-exponentially stable at the origin.

6.3 Stability Analysis We only need to show that the transformed tracking errors, .xe ; ye ; ze ; ve /, converge to zero from any initial values. Under Assumption 6.1, boundedness of the controls u and r follows readily. We note that the closed loop system (6.17) consists of two subsystems, .ze ; uQ e ; rQe / and .xe ; ye ; ve /, in an interconnected structure. To prove Theorem 6.1, we take the following Lyapunov function: V1 .X1e / D for the .ze ; uQ e ; rQe /-subsystem, and


1 2 z C uQ 2e C rQe2 2 e

1 V2 .X2e / D .xe2 C ye2 C k4 .ye C k5 ve /2 / 2

(6.18)

(6.19)

for the .xe ; ye ; ve /-subsystem, where k4 and k5 are positive constants to be selected later and 2 3 2 3 xe zQ e X1e D 4 uQ e 5 ; X2e D 4 ye 5 : ve rQe

116

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

From (6.18) and (6.19), if we consider V D V1 C V2 as a Lyapunov function for the closed loop system (6.17) it can then be seen that this function has a connection with the energy of the ship system as follows. We write the function V as V D X T AX;

(6.20)

 T where X D xe ; ye ; ze ; u  .ud C ude /; v  vd ; r  .rd C red / and A is a diagonal positive definite matrix. It is now seen that the function V consists of two positive definite parts. The first part related to .xe ; ye ; ze / can be regarded as the
“potential” energy. The second part related to u  .ud C ude /; v  vd ; r  .rd C red / is referred to as the “kinetic” energy with respect to a coordinate
 of the ship, which is defined that moves at a speed of .ud C ude /; vd ; .rd C red / . Hence we wish to bring the energy function V to zero. However considering this function together with the closed loop (6.17) is difficult, we consider the subsystems .ze ; uQ e ; rQe / and .xe ; ye ; ve / separately. .ze ; uQ e ; rQe /-subsystem. From the last three equations of (6.17), it can be seen that the .ze ; uQ e ; rQe /-subsystem is GES at the origin by considering the Lyapunov function V1 .X1e /, see (6.18), whose time derivative along the solutions of the last three equations of (6.17) is     d11 d33 uQ 2e  c2 C rQ 2 : (6.21) VP1 D k3 ze2  c1 C m11 m33 e Therefore we have kX1e .t /k  kX1e .t0 /k e 1 .tt0 / ;

(6.22)

where 1 D min fk3 ; .c1 C d11 =m11 / ; .c2 C d33 =m33 /g. .xe ; ye ; ve /-subsystem. To investigate stability of this subsystem, we take the Lyapunov function V2 .X2e /, see (6.19), whose first time derivative, after some manipulation, along the solutions of the first three equations of (6.17) is 2

k.t /ud .1 C k4 /ye VP2 D k1 xe2  p  ˇk2 k4 k5 rd2 ye2  1 C xe2 C ye2 k4 k5 .˛k5  1/ve2 C M C ˝ C ˚;

(6.23)

where for simplicity, we have defined ˛ D d22 =m22 ; ˇ D m11 =m22 , $2  $1 C $2 k.t /ye .vd xe  k4 k5 ud ve / $2  $1 C k4 k5 vd ve  $2 $2 d ; ˇk2 k4 k52 rd2 ye ve  ˇk4 k5 .ye C k5 ve / .k2 rd ye C ud / r1e

M D k2 rd xe ye C .ud xe C vd .k4 C 1/ye /

(6.24)

6.3 Stability Analysis

117

 ˝ D xe .px C uQ e / C ye py C k4 .ye C k5 ve / py  rQe xe    d d d /uQ e C .uQ e C ude C ud /.rQe C r2e / ; xe  ˇk5 .rd C r1e r2e

(6.25)

d C rd /xe C .1  k4 .˛k5  1//ye ve : (6.26) ˚ D k4 .ye C k5 ve /.1 C ˇk1 k5 /.r1e

We first choose the design constants such that 1  k4 .˛k5  1/ D 0;

ˇk1 k5  1 D 0

to cancel the common term ˚. Noticing that ˇ ˇ ˇ ˇ ˇ ˇ sin.ze / ˇ ˇ ˇ ˇ  1; ˇ cos.ze /  1 ˇ  1; ˇ ˇ z ˇ ˇ ze e $2  $1 k 2 .t /ye2 D ; $2 $2 .$2 C $1 / 1 1  p $1 $2 1  k 2 .t /

(6.27)

(6.28)

after a lengthy but simple calculation of upper bounds of M and ˝ by completing the squares, we arrive at p23 .t /ye2 p22 .t /ye2 VP2  p1 .t /xe2  p21 .t /ye2  p Cp  1 C xe2 C ye2 1 C xe2 C ye2 p3 .t /ve2 C .1 V2 C 2 / e 1 .tt0 / ;

(6.29)

where  1 ˇk4 k5 jk.t /j jud j jrd j p1 .t / D k1  k2 C jk.t /j jvd j C C 41 1  jk.t /j  ˇk1 k2 k4 k5 jk.t /j ; 1  jk.t /j

(6.30)

p21 .t / D p211 .t /rd2  p212 .t / jrd j ;

ˇk2 k4 k52 ˇk2 k4 k5  42 1  jk.t /j

jk.t /j C p 22 1  k 2 .t / ˇ 0 ˇˇ P /ˇˇ C jk.t /j k 2 .t / 2 k.t ˇ ˇk2 k4 k5 @ .k2 C 1 C k1 1 / jk.t /j/  C p 42 .1  jk.t /j/ 1  k 2 .t /  2 k .t / jud j C .1 C k1 C k2 / jk.t /j C jk.t /j k 2 .t / jud j C

p211 .t / D ˇk2 k4 k5  k2 1 

118

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships


ˇk4 k52 jk.t /j jud j ; jk.t /j k 2 .t / jvd j C k 2 .t / jud j C k 2 .t / jvd j  42 .1  jk.t /j/ ˇ ˇ ˇP ˇ /ˇ ˇk2 k4 k5 ˇk.t ; (6.31) p212 .t / D p .1  jk.t /j/ 1  k 2 .t / p22 .t / D .1 C k4 /k.t /ud  p221 .t /;

! ˇk2 k4 k5 1  p221 .t / D k4 k5 3 jk.t /j jud j C jk.t /j C 1 C p 1  jk.t /j 1  k 2 .t / ˇ 0 ˇˇ P /ˇˇ jud j k.t ˇ ˇk4 k5 @ jk.t /j jud j k 2 .t / jud j jrd j C C C k 2 .t / p p 2 1  jk.t /j 1  k .t / 42 1  k 2 .t /


1 jk.t /j jud j jrd j  C k1 jk.t /j jud j C k2 jk.t /j  jk.t /j jud j jvd j C u2d C p 1  k 2 .t / jk.t /j jud j C jk.t /j k 2 .t /u2d C jk.t /j k 2 .t / jud j jvd j C p jud j jrd j C 2 42 1  k .t / ˇ 0 ˇˇ P /ˇˇ jud j C jk 3 .t /j jud vd j 2 k.t ˇ
ˇk4 k5 @ C k 2 .t /.u2d C jud j jvd j/ C p 42 .1  jk.t /j/ 1  k 2 .t /

k 2 .t /u2d C .k1 C k2 jrd j/ jk.t /j jud j C jk.t /j k 2 .t /u2d C k 2 .t / .jk.t /j C 1/ 
(6.32) jud j jvd j C k 2 .t /u2d C k 2 .t /jud j; p23 .t / D


 2 ˇk4 k2 k5  k .t /.1 C k4 / C 1 jk.t /j C 3 k 2 .t /k4 k5 jvd j C 1  jk.t /j ! jk.t /j k 2 .t / jvd j jrd j 2 C k .t / .jk.t /j C 1/ jvd j jrd j ; (6.33) p 1  k 2 .t / k4 k5 jk.t /j jud j k4 k5 k 2 .t / jvd j C C 43 43
22 ˇk2 k4 k5 jk.t /j  2 ˇk2 k4 k52 rd2  p .1  jk.t /j/ 1  k 2 .t / ˇ 0 0 ˇˇ P /ˇˇ 2 k.t ˇ 2ˇ2 k4 k5 jk.t /j jud j ˇk2 k4 k5 @ @ 2 p  C 1  jk.t /j 1  jk.t /j 1  k 2 .t / 
jk.t /j k 2 .t /vd2 C k 2 .t / jud j C jk.t /j rd2 C k1 C k2 rd2  !
2 /j j jk.t jr d  jk.t /j C .jk.t /j C 1/ k 2 .t / .jvd j C jud j/ C p 1  k 2 .t /

p3 .t / D k4 k5 .k5 ˛  1/ 



6.3 Stability Analysis

119

ˇ ˇk4 k52  ˇˇ P ˇ 2 ˇk.t /ud ˇ C jk.t /ud j .jvd j C 3/ C .k1 C k2 jrd j/  1  jk.t /j

(6.34) jk.t /ud j C .jk.t /j C 2/ k 2 .t /u2d C k 2 .t / jud vd j .jk.t /j C 1/ ;

where 1 ; 2 are nondecreasing functions of kX1e .t0 /k, 1 is given in (6.22), and i > 0; i D 1; 2; 3. We are now in a position to select the design parameters, i , 1  i  3 and kj ; j D 1; 2; 4; 5 so that the closed loop system (6.17) is GAS at the origin. We proceed to choose these parameters by investigating all the cases of Assumption 6.2. For each case, we choose a subset of i and kj . As discussed before, we firstly choose 0 < 1 < 1=.umax d C 2 /; 2 > 0; 3 > 0;

(6.35)

so that sup t0 jk.t /j  k < 1, where umax is the maximum value of jud .t /j. Note d that this primary choice guarantees sup t0 jk.t /j < 1. Secondly, we choose i , 1  i  3 and kj such that p1 .t /  p1 > 0; p3 .t /  p3 > 0:

(6.36)

Before going further to choose i and kj , let us discuss each case of Assumption 6.2. Case C1. From the fourth equation of (6.2), we have vd .t / D vd .t0 /e ˛1 .tt0 / ; 8 t  t0  0; ˛1 > 0: Therefore there exists a nondecreasing function 1 of jvd .t0 /j such that p23 .t /  1 e ˛1 .tt0 / :

(6.37)

p21 .t / D p22 .t / D 0:

(6.38)

It is noted that in this case

From (6.35)–(6.38), we can write (6.29) as VP2  p1 xe2  p3 ve2 C 1 e ˛1 .tt0 / p

ye2 1 C xe2 C ye2

C .1 V2 C 2 / e 1 .tt0 / ; (6.39)

which yields

VP2  p1 xe2  p3 ve2 C .1 V2 C 2 / e 2 .tt0 / ; 2 D min.˛1 ; 1 /

(6.40)

for some nondecreasing functions 1 and 2 of k.X1e .t0 /; vd .t0 //k. We now present a technical lemma to investigate stability of the .xe ; ye ; ve /-subsystem, in this case based on (6.40). Lemma 6.1. Consider the following first-order scalar differential equation

120

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

xP D .ax C b/ e c.tt0 / C ı.t /x;

(6.41)

where a  0; b  0 and c > 0 are constants. If ı.t / enjoys the property that there exists a constant   0 such that Z1 jı.t /j dt   < 1

(6.42)

0

then the solution of (6.41) satisfies  b  jx.t /j  jx.t0 /j e  e a=c C e  e a=c  1 WD  .jx.t0 /j/ : a

(6.43)

Proof. Proof of this lemma is followed directly by solving the differential equation (6.41).  By applying Lemma 6.1 and the comparison principle found in [6] to (6.40), we have  2  1 =1 V2 .t /  V2 .t0 /e 1 =1 C e  1 WD 3 .k.X1e .t0 /; X2e .t0 //k/ : (6.44) 1 Therefore we now rewrite (6.40) as VP2  W2 .xe ; ve / C .1 3 C 2 / e 2 .tt0 /

(6.45)

where W2 .xe ; ve / D p1 xe2 C p3 ve2 . Integrating both sides of (6.45) yields Zt

t0

W2 .xe . /; ve . //d  V2 .t0 / C

Zt

t0

.1 3 C 2 / e 2 .t0 / d:

(6.46)

It is seen that the right-hand side of (6.46) exists and is bounded. On the other hand, W2 .xe .t /; ve .t // is uniformly continuous because its time derivative is bounded. Hence from Barbalat’s lemma, we have W2 .xe .t /; ve .t // ! 0 as t ! 1. Therefore .xe .t /; ve .t // ! 0 as t ! 1. To prove that ye .t / ! 0 as t ! 1, applying Lemma 2.6 to the xe -dynamic equation of (6.17) yields  $2  $1 k.t /vd ye d C C r1e ye C rd ye C lim k2 rd ye C ud t!1 $2 $2  d r2e ye C uQ e C rQe ye C px D 0: (6.47) Since ud D 0 and .X1e .t /; xe .t /; ve .t /; vd .t // ! 0 as t ! 1, (6.47) is equivalent to   d lim r1e (6.48) ye D 0: t!1

6.3 Stability Analysis

121

d From the expression of r1e , it is directly shown that (6.48) is equivalent to ! ye2 P lim p k.t / D 0: t!1 1 C .1  k 2 .t //ye2

(6.49)

On the other hand from (6.45), we have

which implies that

 d  V2 C 21 .1 3 C 2 / e 2 .tt0 /  0 dt V2 C 21 .1 3 C 2 / e 2 .tt0 /

is nonincreasing. Since V2 is bounded from below by zero, V2 tends to a finite nonnegative constant depending on kXe .t0 /k. This implies that the limit of jye .t /j exists and is finite, say lye . If lye were not zero, there would exist a sequence of increasing time instants fti g1 iD1 with ti ! 1, such that both of the limits of P i /y 2 .ti / were not zero, which is impossible because of (6.49). Hence P i / and k.t k.t e lye must be zero. Therefore we conclude from (6.49) that ye .t / ! 0 as t ! 1. Hence we have proven lim t!1 X2e .t / D 0 for this case. We define the subset of the design parameters that satisfy the conditions (6.27), (6.35), and (6.36) as 1k . Case C2. In this case, we choose the design parameters i and kj such that  > 0: p211 .t /  p211

(6.50)

Under Assumption 6.1, there exists a choice of i and kj such that (6.27), (6.35), (6.36), and (6.50) hold. Substituting (6.35), (6.36), and (6.50) into (6.29) yields
  2 VP2  p1 xe2  p211 rd  jp22 .t /j  p212 .t / jrd j  p23 .t / ye2  p3 ve2 C .1 V2 C 2 / e 1 .tt0 / :

(6.51)

On the other hand, it is noted that V2 


1 2 x C .1 C k4 C k4 k5 /ye2 C k4 k5 .1 C k5 /ve2 : 2 e

(6.52)

From (6.52) and (6.51), we have VP2  2 min

p1 ;

!  p211 rd2  jp22 .t /j  p212 .t /jrd j  p23 .t / p3 ;  1 C k4 C k4 k5 k4 k5 .1 C k5 /

V2 C .1 V2 C 2 / e 1 .tt0 / :

(6.53)

By applying Lemma 2.6 to (6.53), there exist a positive constant 3 independent of initial conditions and a nondecreasing function 3 of k.X1e .t0 /; X2e .t0 //k such that kX2e .t /k  3 .k.X1e .t0 /; X2e .t0 //k/ e 3 .tt0 / ;

(6.54)

122

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

as long as Z t
  2  p211 rd . /  p212 . / jrd . /j  jp22 . /j  p23 . / d  p21 .t  t0 /;

(6.55)

t0

 where p21 > 0. For simplicity of calculation, we can replace the last three terms in the left-hand side of (6.55) by their maximum values. Note that (6.54) implies that X2e .t / is globally K-exponentially stable at the origin but not GES in the sense of Lyapunov, i.e., 3 in (6.54) must be a linear function of k.X1e .t0 /; X2e .t0 //k. However, it can be shown that the local exponential stability (in the sense of Lyapunov) of the closed loop system (6.17) is guaranteed. We define the subset of i and kj satisfying the conditions (6.27), (6.35), (6.36), (6.50), and (6.55) as 2k .

Case C3. Under Assumption 6.2 and

R1 0

jrd .t /jdt  1 < 1, there exists

0  p21 < 1 such that

Z1 jp21 .t /jdt  p21 :

(6.56)

0

In this case, we choose i and kj such that  p22 .t /  p23 .t /  p22 > 0:

(6.57)

With (6.36) and (6.57), we rewrite (6.29) as ye2  VP2  p1 xe2  p22  p3 ve2 C p 1 C xe2 C ye2 jp21 .t /j ye2 C .1 V2 C 2 / e 1 .tt0 / :

(6.58)

Again by applying Lemma 6.1 to (6.58), we have  2 2p21  1 =1 1 e e 1 WD 3 .k.X1e .t0 /; X2e .t0 //k/ :

V2 .t /  V2 .t0 /e 2p21 e 1 =1 C

(6.59)

Substituting (6.59) and (6.52) into (6.58) yields VP2  4 V2 C 2 jp21 .t /j V2 C .1 V2 C 2 / e 1 .tt0 / ;

(6.60)

where  4 D 2 min p1 ; p

  p22 p3 : ; 1 C 23 .1 C k4 C k4 k5 / k4 k5 .1 C k5 /

(6.61)

6.3 Stability Analysis

123

From (6.52) and (6.60), it can be shown that kX2e .t /k  4 .k.X1e .t0 /; X2e .t0 //k/ e 4 .tt0 / ;

(6.62)

where if 4 ¤ 1 then 4 D 0:5 min .4 ; 1 / ; r 4 ./ D

with

  2p21 e 1 =1 V .t / C  j   j1 ; 41 4 2 0 2 1 4 e

if 4 D 1 then 4 D 0:5 .1  d4 / ; q 2p21 e 1 =1 .V .t / C   /; 4 ./ D 41 2 0 2 4 4 e

(6.63)

  q
2  4 D min 2; 1 C k4 C k4 k52  1 C k4  k4 k52 C 4.k4 k5 /2 ; 0 < d4 < 1 ;

4  .t  t0 /e d4 .tt0 / : Note that 4 is finite for an arbitrarily small d4 . It can be seen from (6.59), (6.61) and (6.63) that the rate 4 > 0 depends on the initial conditions k.X1e .t0 /; X2e .t0 //k. Hence the closed loop system (6.17) is GAS but not exponentially stable at the origin. In other words, global asymptotic tracking is achieved in this case. We define the subset of i and kj satisfying the conditions (6.27), (6.35), (6.36), and (6.57) as 3k . Case C4. We first show that V2 is bounded. Substituting (6.35) and (6.36) into (6.29) yields (6.64) VP2  p1 xe2  p3 ve2 C p4 .t /ye2 C .1 V2 C 2 / e 1 .tt0 / ; where p4 .t / D jp21 .t /j C jp22 .t /j C p23 .t /: Note that in this case, there exists 0  p4 < 1 such that (6.19) and (6.65), we have

R1 0

(6.65) p4 .t /dt  p4 . From

VP2  2p4 .t /V2 C .1 V2 C 2 / e 1 .tt0 / : Applying Lemma 6.1 to (6.66) yields

(6.66)

124

6 Simultaneous Stabilization and Trajectory-tracking Control of Underactuated Ships

 2 2p4  1 =1 1 e e 1 WD 4 .k.X1e .t0 /; X2e .t0 //k/ :

V2 .t /  V2 .t0 /e 2p4 e 1 =1 C

(6.67)

Hence ye is also bounded. Substituting this upper bound into (6.64) yields VP2  p1 xe2  p3 ve2 C 24 p4 .t / C .1 4 C 2 / e 1 .tt0 / : From (6.68), we have   Z t d 1 .tt0 / 1  0; p4 . /d C 1 .1 4 C 2 / e V2  24 dt 0

(6.68)

(6.69)

which implies that .xe ; ve / ! 0 as t ! 1 and lim t!1 ye D 0. We define the subset of i and kj satisfying the conditions (6.27), (6.35), and (6.36) as 4k .

6.4 Selection of Design Constants In this section, we show that

\ lk ¤ ;. From the above section, it can be seen

1l4

that these design constants must satisfy (6.27), (6.35), (6.36), (6.50), (6.55), and (6.57). From (6.27), we have k1 D

1 1 : ; k4 D ˇk5 ˛k5  1

(6.70)

Hence we first choose k5 > 1=˛ and set k2 D ık with ı being a small positive constant, say ı  max.umax ; uP max ; rdmax /, with uP max being the maximum value of juP d j. d d d Next we replace k in all negative terms of pi .t /; i D 1; 21; 22; 23; 3 by k . It is now observed that the negative parts of pi .t / have k as a factor. The mass including added mass in the sway dynamics, m22 , is always larger than that in the surge dynamics, m11 , for surface ships, i.e., ˇ < 1. Therefore we can always find positive constants k and k5 such that the conditions (6.36), (6.50), (6.55), and (6.57) hold with some small i > 0; i D 1; 2; 3, and i being chosen based on (6.35). The value of constant k should be reduced if the reference yaw, rd , and surge, ud , velocities and surge acceleration, uP d , are large. This physically means that the distance from the ship to the point it aims to track should be increased if the velocities and surge acceleration are large, otherwise the ship will miss that point. In fact, setting i D 0 and picking k2 > 0 results in the tracking controllers proposed in [19,71,106] with a restrictive assumption of the yaw reference velocity satisfying a persistently exciting condition. It can also be seen that setting 2 D 0 yields the tracking controller in the preceding chapter, which does not require the yaw reference velocity to be persistently exciting. Furthermore, small ˛ automatically results in large k5 and small k1 , see (6.70). This can be physically interpreted as follows: If the damping in the sway dynamics is small, the control gain in the surge dynamics should be small, otherwise

6.5 Sensitivity Analysis

125

the ship will slide in the sway direction. Due to complicated expressions of pi .t /, we provide some guidelines to choose the design constants rather than present their extremely complex explicit expressions. , 1. Pick positive constants i such that the constant k is small, say k 0; av > 0; au > 0; ar > 0:

(7.78)

It is not hard to show that there always exist k, c1 , c2 and ki ; i D 1; : : : ; 4 such that (7.78) holds for arbitrarily small "1 . We will discuss more detail of (7.78) later. With the choice of (7.78), we can write (7.76) as VP1  14"1 .V1 C .a0 C 10 .
//=.14"1 // C 11 .
/ .V1 C .a0 C 10 .
//=.14"1 // e 0 .tt0 / ;

(7.79)

which, together with (7.75), yields (7.74). Part 2. We now prove that the closed loop system (7.70) is GAS at the origin by taking the Lyapunov function q
1 V2 D 1 C xe2 C ye2  1 C k3 ve2 C k4 .ze2 C uQ 2e C rQe2 / (7.80) 2

whose time derivative along the solutions of (7.70), after some lengthy but simple calculation by completing squares, and using (7.74), satisfies VP2  x $22 xe2  y .t /$22 ye2  z .t /ze2  v .t /ve2  u uQ 2e  r rQe2 C 21 .
/V2 e 2 .tt0 / C 20 .
/e 2 .tt0 / ;

(7.81)

where x D k1  61 ; y .t / D ku2d  .kud /2 .k1 C jud j C jvd j C 0:5/  jkud j .k1 C 1 / 

0:2511 .kud vd /2  101 ; z .t / D k2 k4  11 .0:5.k1 C jud j/2 C 5.1 C vd2 / C 0:25.k3 m11 m1 22 

.k1 C jud j//2 .k22 C .kud /2 .k1 C jud j C vd2 / C 4 C 4vd2 //; 2 1 v .t / D k3 d22 m1 22  71  0:251 .jkud j C 1/  0:5.kud /  1 1 k3 m11 m22 .k1 C jud j/ jkud j .8 C jkud j/  0:251 .k3 m11 m1 22  .k1 C jud j//2 ..k uP d /2 =.1  .kud /2 / C .kud /4 .u2d C k12 C .kud vd /2 C rd2 C 2/ C .kud /2 /;

u D k4 .c1 C d11 =m11 /  29=.41 /; r D k4 .c2 C d33 =m33 /;

(7.82)

where 2 D 0  1 , 1 is a positive constant, and 21 .
/ and 20 .
/ are nondecreasQ 0 /; XQ .t0 /; Xe .t0 //k. We now choose the constants k, c1 , c2 , ing functions of k..t and ki ; i D 1; : : : ; 4, such that

7.4 Robustness Discussion

159

x  x ; y .t /  y ;

z .t /  z ; v .t /  v ; u  u ;

(7.83)

r  r

for all t  0, where x , y , z , v , u , and r are positive constants. Substituting (7.83) into (7.81) yields VP2  x $22 xe2  y $22 ye2  z ze2  v ve2  u uQ 2e  r rQe2 C 21 .
/V2 e 2 .tt0 / C 20 .
/e 2 .tt0 / :

(7.84)

From (7.84) we have VP2  21 .
/V2 e 2 .tt0 / C 20 .
/e 2 .tt0 / , which implies that V2  22 .
/, with 22 .
/ being a nondecreasing function of k .t0 /k. With V2  22 .
/ in mind, one can show from (7.84) that there exists 3 > 0 depending on k .t0 /k such that kXe .t /k  2 .
/e 3 .tt0 / , where 2 .
/ is a nondecreasing function of k .t0 /k, which in turn implies that the closed loop system (7.70) is asymptotically stable at the origin. However one can straightforwardly show that (7.70) is also locally exponentially stable at the origin. By carrying out a similar arguments in Section 7.2.5, one can show that there always exist the design constants k, c1 , and c2 and ki , i D 1; 2; 3; 4, such that jkud .t /j < 1 and that the conditions (7.78) and (7.83) hold.

7.4 Robustness Discussion In this section, we discuss robustness of the output feedback tracking controller. A discussion for the partial-state feedback can be carried out in a similar way. The control law (7.68) has been designed under the assumption that there are no environmental disturbances. Indeed, this assumption is unrealistic in practice. The aim of this section is to discuss the robustness property of our proposed controller in relation to environmental disturbances. Under additive environmental disturbances, it is not hard to show that the observer (7.51) guarantees that the observer errors Q /; XQ .t // globally exponentially converge to a ball centered at the origin. More..t over, one can prove that the tracking error vector Xe .t / also globally asymptotically converges to a ball centered at the origin. The radius of this ball can be adjusted by changing the control gains if the environmental disturbances are not too large. When the environmental disturbances are large enough, the observer (7.51) cannot provide a sufficiently accurate estimate of unmeasured velocities, and the control law (7.68) cannot compensate for considerable environmental disturbances acting on the sway axis. These will result in an unstable closed loop system, especially at a low forward speed. This phenomenon should not be surprising since the vessel in

160

7 Partial-state and Output Feedback Trajectory-tracking Control of Underactuated Ships

question is not actuated in the sway axis and does not have velocities available for feedback. One can see this phenomenon by observing the simple example system (7.71) with some additive disturbance in the first equation. It is easy to show that when this additive disturbance has magnitude larger than 1, then the system (7.71) will be unstable. The robustness issue is still a challenging problem in control of underactuated ocean vehicles and underactuated systems in general.

7.5 Simulations This section illustrates the soundness of the control laws (7.68) by simulating them on the same monohull ship in the previous two chapters. Details of the ship parameters are listed in Section 5.4. The initial conditions of the reference trajectories are chosen as .xd .0/, yd .0/, 1 1 d .0/, vd .0// D .0 m; 0 m; 0 rad; 0 ms /. The reference velocities are ud D 4ms 1 1 1 and rd D 0rads for the first 300 seconds, and ud D 4 ms and rd D 0:02 rads for the rest of the simulation time. This choice means that the reference trajectory is a straight line for the first 300 seconds and then followed by a circle with a radius of 200 m. Indeed the above choice of reference velocities satisfies Assumption 7.1. All of the initial conditions of O and vO are chosen to be zero. We first choose k2 D 5, c1 D 1, c2 D 2, ı1 D ı2 D 0:1, K01 D 10 diag.1; 1; 1/, T P01 D P02 D 0:5 diag(1,1,1), and K02 D .J ./P 1 .// . The other design constants are chosen as: k D 0:05, k1 D 1, k3 D 5, and k4 D 100. Simulation results are plotted in Figures 7.1 and 7.2 with the initial conditions of the ship: x D 10 m; y D 10 m; D 0:1 rad; u D 0 ms1 ; v D 0 ms1 , and r D 0 rads1 . It is seen from Figures 7.1a and 7.1b that all of the tracking and observer errors converge to zero. The control inputs, u and r , are within their limits. As always, the magnitude of u and r can be reduced by adjusting the control gains such as c1 and c2 . However, the trade-off is a longer transient response time. For clarity, we only plot the tracking errors for the first 180 seconds, and the observer errors for the first 10 seconds. To illustrate robustness of our proposed controller, we also simulate with the same control gains and initial conditions chosen as above, and the environmental disturbance vector w .t / D 0:5M .Œsin.t /; cos.t /; sin.t /T C 1:5/, i.e., the last equation of (7.1) is in the form of M vP D C .v/v  Dv C  C w .t/. Simulation results are plotted in Figures 7.3 and 7.4. For clarity, we only plot the tracking errors for the first 300 seconds and the observer errors for the first 10 seconds. It can be seen that the environmental disturbances deteriorate the performance of the controlled loop system in the sense that the tracking errors do not converge to zero but to a ball centered at the origin. This shows an important property of robustness of the controlled system with respect to the environmental disturbances.

7.6 Conclusions

161

7.6 Conclusions The key to the control developments is an introduction of the global nonlinear coordinate transformations (7.6), (7.24), (7.43), (7.58), and (7.61) to obtain an exponential observer and to transform the tracking error dynamics to a suitable nonlinear system, to which Lyapunov’s direct method and the backstepping technique can be applied. The work presented in this chapter is based on [94, 114, 116, 117].

b

10

Observer errors

Tracking errors

a

0 −10 0

50

100 Time [s]

10

5

0 0

150

5 Time [s]

10

c

400 y [m]

300 200 100 0 0

200

400

600 800 x [m]

1000

1200

1400

Figure 7.1 Simulation results without disturbances: a. Tracking errors (x  xd [m]: solid line,  d [rad]]: dotted line); b. Observer errors q y  yd [m]: dashed line,  p 2 2 2 ( .x  x/ O C .y  y/ O C .  O / : solid line, .u  u/ O 2 C .v  v/ O 2 C .r  r/ O 2 : dotted line; c. Ship position and orientation in the .x; y/-plane

162

7 Partial-state and Output Feedback Trajectory-tracking Control of Underactuated Ships 6

a

x 10

u

τ [N]

5

0

−5 0

2

4

6

8

10

6

8

10

Time [s] 8

b

x 10

0

r

τ [Nm]

5

−5 0

2

4 Time [s]

Figure 7.2 Simulation results without disturbances (control inputs): a. Surge force; b. Yaw moment b

10

Observer errors

Tracking errors

a

0 −10 0

100 200 Time [s]

10

5

0 0

300

5 Time [s]

10

c

400 y [m]

300 200 100 0 0

200

400

600 800 x [m]

1000

1200

1400

Figure 7.3 Simulation results with disturbances: a. Tracking errors (x  xd [m]: solid line,  d [rad]]: dotted line); b. Observer errors q y  yd [m]: dashed line,  p 2 2 2 O O C .y  y/ O C .  / : solid line, .u  u/ O 2 C .v  v/ O 2 C .r  r/ O 2 : dotted ( .x  x/ line; c. Ship position and orientation in the .x; y/-plane

7.6 Conclusions

163 6

a

x 10

u

τ [N]

5

0

−5 0

2

4

6

8

10

6

8

10

Time [s] 8

b

x 10

0

r

τ [Nm]

5

−5 0

2

4 Time [s]

Figure 7.4 Simulation results with disturbances (control inputs): a. Surge force; b. Yaw moment

Chapter 8

Path-tracking Control of Underactuated Ships

This chapter deals with the problem of designing controllers to force an underactuated surface ship with standard dynamics to track a reference path. Both full-state feedback and output feedback cases are considered. In comparison with the preceding three chapters, the requirement that the reference trajectories be generated by virtual ships is relaxed, and the control design is simpler and more amenable for implementation in practice. The control development is based on a series of coordinate transformations, the backstepping technique, and utilizing the dynamic structure of the ship.

8.1 Full-State Feedback 8.1.1 Control Objective In addition to the assumptions made in Section 3.4.1.1, we ignore the nonlinear damping terms. The case with the nonlinear damping terms requires more effort, and is treated in Section 8.2 where an output feedback control design is addressed. As such, the resulting mathematical model of the underactuated ship moving in surge, sway, and yaw is rewritten as P D J . /v; M vP D C .v/v  Dv C ; with

(8.1)

2 3 2 3 3 u u x 4 4 5 4 5 v y ;  D 0 5; ; vD D r r 2

165

166

8 Path-tracking Control of Underactuated Ships

3 cos. /  sin. / 0 J . / D 4 sin. / cos. / 0 5 ; 0 0 1 2 3 2 3 d11 0 0 m11 0 0 M D 4 0 m22 m23 5 ; D D 4 0 d22 d23 5 ; 0 d32 d33 0 m32 m33 2

3 0 0 m22 v  21 .m23 C m32 /r 5; 0 0 m11 u C .v/ D 4 0 m22 v C 12 .m23 C m32 /r m11 u 2

where a definition of all symbols is given in Section 3.3. Moreover, in order to clearly illustrate our idea we did not include the disturbances in (8.1) but will address this issue in Section 8.1.5 by a simple modification of the control design. In this section, we consider the following control objective. Control Objective. Design the controls u and r such that the position .x; y/ of the ship (8.1) globally tracks a reference path ˝ parameterized by .xd .s/; yd .s// with s being the path parameter, the ship total linear velocity is tangential to the reference trajectory ˝, and the desired surge velocity can be adjusted on-line. Since the ship system (8.1) is underactuated, it is not expected to force the ship to track an arbitrary path ˝. We here impose the following sufficient conditions on the path ˝: Assumption 8.1. There exist strictly positive constants "i ; 1  i  4, such that 0

0

xd2 .s/ C yd2 .s/  "1 ; 8s 2 R; p 2 ˛ uN d .t /  j j  "2 ; 8 t  0; 2 ˇ uN d .t / C uNP d .t /  "3 ; 8 t  0;

(8.2) (8.3) (8.4)

and the solutions of the following differential equation .uN d cos.ı/.1  ˛/ C /ıP D .ˇ uN d C uPN d / sin.ı/  .˛ uN d cos.ı/  /rNd ; ı.t0 / D 0 (8.5) satisfy jı.t /j  0

0

  "4 ; 8t  t0  0; 4

where xd .s/; yd .s/; uN d and rNd are defined as

(8.6)

8.1 Full-State Feedback

167

@xd 0 @yd 0 ; yd .s/ D ; xd .s/ D @s q @s 0 0 uN d D xd2 .s/ C yd2 .s/Ps ; 0

rNd D

00

00

.xd .s/yd .s/  xd .s/yd .s// .xd2 .s/ C yd2 .s// 0

0

(8.7)

0

sP ;

and the constants ˛, ˇ, and are ˛D

d22 d22 m23 m11 ;ˇD ; D 2 : m22 m22 m22  d23 =m22

(8.8)

Remark 8.1. 1. Condition (8.2) implies that the path ˝ is regular with respect to the path parameter s. If the path is not regular, one can break it into different regular paths and consider each path separately. 2. Conditions (8.3) and (8.4) mean that the path ˝ does not contain or approach a set-point, i.e., we do not consider stabilization/regulation problems. These conditions also imply that uN d .t / should not vary too fast. For clarity we only consider the forward tracking case, i.e. the reference velocity uN d .t / is always larger than a positive constant. The backward tracking case (i.e. uN d .t / is always less than a negative constant) can be done similarly. 3. Conditions (8.5) and (8.6) imply that the desired surge velocity is always larger than the desired sway velocity. These conditions are reasonable since the ship in question is underactuated in the sway axis, one should not expect to force it to track a path with an arbitrarily large curvature. These conditions place certain restrictions on the path ˝ but allow many types of trajectories such as a straight line, an arc, and a sinusoidal path. For vessels with three planes of symmetry (like spherical underwater vehicles see Section 3.4.1.3, i.e., ˛ D 1; D 0/, the differential equation (8.5) reduces to an algebraic equation, and condition (8.6) holds if uN d .t / jrNd .t /j (8.9)  1  "5 ; 8 t  0; ˇ uN d .t / C uPN d .t / where "5 is an arbitrarily small positive constant. For other cases, an application of Lemma 9.3 in [6] to (8.5) readily shows that sufficient conditions such that (8.6) holds can be expressed as p 0:5 2 uN d .t /.1  ˛/  j j > "6 ; 8t  0; (8.10) p ˛ uN d .t / C j j 2 (8.11)  "7 ; 8 t  0; jrNd .t /j  P 2 ˇ uN d .t / C uN d .t /

where "6 and "7 are arbitrarily small positive constants. 4. The variable sP can be used to specify the desired forward speed. The main ideas to solve the control objective are as follows:

168

8 Path-tracking Control of Underactuated Ships

1. Introduce a coordinate to change the ship position .x; y/ such that the ship model (8.1) can be transformed to a diagonal form to overcome difficulties caused by nonzero off-diagonal terms in the system matrices. 2. Interpret tracking errors in a frame attached to the path ˝ such that the tracking error dynamics are of a triangular form to which the backstepping technique can be applied. 3. Use the orientation tracking error as a virtual control to stabilize the cross-track error.

8.1.2 Coordinate Transformations 8.1.2.1 Transform Ship Dynamics to a “Diagonal Form” Introduce the following coordinate transformation (changing the ship position, see Figure 8.1): xN D x C " cos. /; (8.12) yN D y C " sin. /; vN D v C "r;

where " D m23 =m22 . Using the above change of coordinates, the ship dynamics (8.1) can be written as xPN D u cos. /  vN sin. /; yPN D u sin. / C vN cos. /; P D r;

(8.13)

uP D Nu ; vPN D ˛ur  ˇ vN C r; rP D Nr ;

where we have chosen the primary controls u and r as: m23 C m32 2 r C d11 u; 2   m22 m33  m23 m32 1 r D .m11 m22  m222 /uv C m11 m32  Nr  m22 m22  m23 C m32 ur C .m32 d22  m22 d32 /v  .m22 d33  m32 d23 /r ; m22 2 (8.14) u D m11 Nu  m22 vr 

with Nu and Nr being considered as new controls to be designed later.

8.1 Full-State Feedback

169

Yb YE

v

Yd

Os Ob H x x CG

y y ye

yd

u

Od

x

Xb

\ Xd xe

\e \d

ud

:

OE

xd

x

x

XE

Figure 8.1 Interpretation of tracking errors

Remark 8.2. After the ship system (8.1) is transformed to the diagonal form (8.13), the methods mentioned in the preceding three chapters can be used to obtain controllers for specific tasks such as stabilization and model reference trajectorytracking if .x; N y/ N are considered as the ship position instead of .x; y/. 8.1.2.2 Transform Path-tracking Errors We now interpret the path-tracking errors in a frame attached to the reference path ˝ as follows (see Figure 8.1): 3 3 2 xN  xd xe 7 6 6 7 T 4 ye 5 D J . / 4 yN  yd 5 ; Ne  d 2 where

d

0

is the angle between the path and the X -axis defined by ! 0 yd .s/ ; d D arctan 0 xd .s/ 0

(8.15)

(8.16)

with xd .s/ and yd .s/ being defined in (8.7). In Figure 8.1, OE XE YE is the earth-fixed frame, Od Xd Yd is a frame attached to the path ˝ such that Od Xd and Od Yd are parallel to the surge and sway axes of the ship, uN d is tangential to the path, C G  Ob is the center of gravity, and Os is referred to as the center of oscillation of the ship. Therefore xe , ye , and N e can be

170

8 Path-tracking Control of Underactuated Ships

referred to as the tangential tracking error, cross-tracking error, and heading error, respectively. Differentiating (8.15) along the solutions of the first three equations of (8.13) results in the kinematic error dynamics: xP e D u  uN d cos. N e / C rye ; yPe D vN C uN d sin. N e /  rxe ; PN D r  rN ; e d

(8.17)

where uN d and rNd are given in (8.7). From (8.17), it can be seen that .xe ; ye ; N e / D .0; 0; 0/ is the equilibrium point only if the transformed sway velocity is zero, i.e., vN D 0, which means that the ship must move on a straight line at the steady state. Since we allow the reference path ˝ to be different from a straight line, but also include it, the transformed sway velocity is generally different from zero at the steady state, i.e., vN can be different from zero. To resolve this difficulty we introduce an angle ı to the orientation error N e by defining e

D N e C ı:

(8.18)

Substituting (8.18) into (8.17) yields xP e D u  ud cos. e /  vd sin. e / C rye ; yPe D ud sin. e / C ve  vd .cos. e /  1/  rxe ; P e D r  rd ;

(8.19)

ve D vN  vd ; ud D uN d cos.ı/; vd D uN d sin.ı/; P rd D rNd  ı:

(8.20)

where

We refer to ud ; vd , and rd as the desired surge, sway, and yaw reference velocities. It can now be seen that .xe ; ye ; e / D .0; 0; 0/ is the equilibrium point of (8.19) if ve D 0. At the steady state, the surge and yaw velocities .u; r/ should converge to their desired values .ud ; rd / with the help of a control to be designed later. Therefore, to determine the angle ı that guarantees ve D 0 at the steady state, from the fifth equation of (8.13), the desired sway velocity should be chosen as vP d D ˛ud rd  ˇvd C rd :

(8.21)

Substituting (8.20) into (8.21) results in (8.5). We make the following important observations: Remark 8.3. With jı.t /j < =4; 8t  t0  0, we have jvd .t /j < jud .t /j since ud D uN d cos.ı/ and vd D uN d sin.ı/. In addition, with the help of a control (to be designed

8.1 Full-State Feedback

171

later), convergence of .xe ; ye ; e / to zero implies that N e converges to ı.t /. Since jı.t /j < =4, the ship will not turn around. To prepare for the control design, we rewrite the path-tracking error dynamics as follows: xP e D u  ud cos.

yPe D ud sin. P e D r  rd ;

e /  vd

sin.

e / C rye ;

e / C ve  vd .cos.

e /  1/  rxe ;

vP e D ˛.ur  ud rd /  ˇve C .r  rd /; uP D Nu ; rP D Nr :

(8.22)

The control objective has been converted to one of stabilizing (8.22) at the origin. The tracking error dynamics (8.22) are similar to the ones in the preceding chapters. The techniques in these chapters can be applied to design the controls Nu and Nr but will result in an extremely complex procedure to determine the control gains. In this chapter we will propose a much simpler technique to design Nu and Nr to globally stabilize (8.22) at the origin.

8.1.3 Control Design We divide the control design into two stages. At the first stage, we consider the first four equations of (8.22), with u and r being viewed as controls. At the second stage, the last two equations of (8.22) are considered to design the actual controls Nu and Nr using the backstepping technique. 8.1.3.1 Kinematic Control Design Substep 1 (Stabilizing xe -dynamics) Define ue D u  ˛u ;

(8.23)

where ˛u is a virtual control of u. The first equation of (8.22) suggests that we choose ˛u D k1 xe C ud ; (8.24) where k1 is a positive constant to be selected later. Substituting (8.23) and (8.24) into the first equation of (8.22) yields xP e D k1 xe  ud .cos.

e /  1/  vd

sin.

e / C rye

C ue :

(8.25)

172

8 Path-tracking Control of Underactuated Ships

Note that we have chosen ˛u not to cancel ud .cos. simplify the control design.

e /  1/  vd

sin.

e / C rye

to

Substep 2 (Stabilizing .ye ; ve /-dynamics) Introduce a coordinate change z1 D ye C

ve : ˇ

(8.26)

Differentiating (8.26) along the solutions of the second and fourth equations of (8.22), and using (8.23) and (8.24) yields zP 1 D ud sin. e /  vd .cos. e /  1/  rxe  ˛

..ue  k1 xe C ud /r  ud rd / C .r  rd /: ˇ ˇ

(8.27)

For simplicity, we now choose ˇ (8.28) ˛ to cancel the term rxe on the right-hand side of (8.27) to have z1 -dynamics “independent” of xe . Substituting (8.28) into (8.27) yields k1 D

zP 1 D ud sin.

e /  vd .cos.

e /  1/ 

1 ˛ .˛ud  /.r  rd /  ue r: ˇ ˇ

(8.29)

It is now seen that the third equation of (8.22) and (8.29) are of a forward structure with r being considered as a control. One can apply forwarding control design techniques to design a control law for r. However, it is difficult to find a proper Lyapunov function to handle cross terms. The reader is referred to [7] for more details on control design techniques for forward systems. Here we use the backstepping technique by using e as a virtual control to stabilize z1 . Define z2 D

e

˛

e

;

(8.30)

where ˛ e is a virtual control of e . By noticing (8.3), (8.7) and jı.t /j < =4; 8t  t0  0, we have ud .t / > 0. Therefore, a simple control law for ˛ e can be chosen as   p k2 z 1 (8.31) ;
D 1 C .k2 z1 /2 ; ˛ e D  arcsin


where k2 is a positive constant to be selected later. It is of interest to note that (8.31) is well defined for all z1 2 R. Substep 3 (Stabilizing z2 -dynamics) Define

8.1 Full-State Feedback

173

re D r  rd  ˛r ;

(8.32)

where ˛r is a virtual control of r. Differentiating both sides of (8.30) along the solutions of the third equation of (8.22) and (8.29) yields   ˛ud  zP 2 D 1  k2 .re C ˛r / C ˇ
2 k2 k2 ˛ .ud sin. e /  vd .cos. e /  1//  ue r: (8.33)
2 ˇ
2 We choose the constant k2 such that 1  k2

˛ud  ˇ : > 0 ) 0 < k2  ˇ
2 ˛ud  j j

(8.34)

There always exists k2 satisfying (8.34) under condition (8.3) since jı.t /j < =4 and ud D uN d cos.ı/; 8t  t0  0. We choose the virtual control ˛r from (8.33) as   1 k2 c1 z 2   .u sin. /  v .cos. /  1// ; ˛r D q e e d d 2 k2 1  ˇ
2 .˛ud  / 1 C z22
(8.35)

where c1 is a positive constant. It is noted that ˛r is bounded by some constant for all .z1 ; z2 ; e / 2 R3 . 8.1.3.2 Kinetic Control Design By differentiating both sides of (8.23) and (8.32) along the solutions of the first, third, fifth, and sixth equations of (8.22), the control Nu is chosen as Nu D c2 ue C

@˛u k2 ˛ @˛u xP e C uP d C z2 r; @xe @ud ˇ
2

(8.36)

and the control Nr is chosen as Nr D

@˛r @˛r P @˛r @˛r @˛r zP 1 C zP 2 C uP d C vP d  eC @z1 @z2 @ e @ud @vd   k2 1 .˛u 

/ z2  c3 re C rPd ; d ˇ
2

(8.37)

where c2 and c3 are positive constants. Substituting (8.30), (8.31), (8.35), (8.36), and (8.37) into (8.29) and (8.33), derivatives of ue and re along the solutions of the last two equations of (8.22) gives the following closed loop system:

174

8 Path-tracking Control of Underactuated Ships

xP e D k1 xe  ud .cos. ˇ 1 ve / C ue ;

e /  1/  vd

sin.

e / C .rd

C ˛r C re /.z1 

vP e D ˛..ue C ˛u /.rd C ˛r C re /  ud rd /  ˇve C .˛r C re /; zP 1 D k2 ud z1 =
 vd .cos. e /  1/  ˇ 1 .˛ud  /.˛r C re / C ud .sin.z2 / cos.˛ e / C .cos.z2 /  1/ cos.˛ e //  ˛ˇ 1 ue .rd C

˛r C re /; q zP 2 D c1 z2 = 1 C z22 C .1  k2 .˛ud  /ˇ 1
2 /re  k2 ˛ˇ 1
2 ue r;

uP e D c2 ue C k2 ˛ˇ 1
2 z2 r; rPe D c3 re  .1  k2 .˛ud  /ˇ 1
2 /z2 :

(8.38)

We now present the first main result of this chapter, the proof of which is given in the next section. Theorem 8.1. Assume that the path ˝ satisfies all conditions specified in Assumption 8.1, then the controls u and r given by (8.14), (8.36), and (8.37) force the transformed tracking errors .xe ; ye ; e / to converge to zero globally asymptotically and locally exponentially with an appropriate choice of k2 . As a result, the actual position tracking errors .x  xd ; y  yd / and orientation tracking error .  d / globally asymptotically and locally exponentially converge to balls with radii of jm23 =m22 j and jı.t /j  arcsin..˛ uN d C j j/ jrNd j =.ˇ uN d C uPN d // < =4, respectively. Furthermore, the desired forward speed of the ship on the path can be adjusted by adjusting sP .

8.1.4 Stability Analysis To prove Theorem 8.1, we first show that the closed loop system (8.38) is forward complete, i.e., there is no finite escape in the closed loop system. We then consider the .z2 ; ue ; re /-, z1 - and .xe ; ve /-subsystems separately. We first consider the Lyapunov function V1 D xe2 C z12 C z22 C ve2 C u2e C re2 : (8.39) A simple calculation shows that the derivative of V1 along the solutions of (8.38) satisfies

)

VP1  11 V1 C 12

V1 .t /  .V1 .t0 / C 12 =11 /e 11 .tt0 / ;

(8.40)

where 11 and 12 are some positive constants, which implies that the closed loop system (8.38) is forward complete. To investigate stability of the .z2 ; ue ; re /-subsystem, we consider the Lyapunov function

8.1 Full-State Feedback

175

1 V21 D .z22 C u2e C re2 /; (8.41) 2 whose derivative along the solutions of the last three equations of (8.38) satisfies 2

z  c2 u2e  c3 re2  0: VP21 D c1 q 2 2 1 C z2

(8.42)

Since VP21  0, we have V21 .t /  21 .
/ where 21 .
/ is a class-K1 function of X1 .t0 / D k.z2 .t0 /; ue .t0 /; re .t0 //k. With this upper bound of V21 , we have z22 VP21 D c1 p  c2 u2e  c3 re2 ; 1 C 221 .
/

(8.43)

which implies that the .z2 ; ue ; re /-subsystem is globally asymptotically and locally exponentially stable at the origin, i.e. there exist a class-K function 21 .
/ and a constant 21 depending on X1 .t0 / such that k.z2 .t /; ue .t /; re .t //k  21 .
/e 21 .tt0 / :

(8.44)

For the z1 -subsystem, we take the Lyapunov function 1 V22 D z12 : 2

(8.45)

By substituting ˛r , see (8.35), into the derivative of V22 along the solutions of the third equation of (8.38), we have   ud  jvd j 2 j˛ud  j P V22  k2 1  k2 z1 C .ˇ  k2 j˛ud  j


21 .
/.V22 C 22 .
//e 21 .tt0 / ;

(8.46)

where 21 .
/ and 22 .
/ are class-K1 functions of X1 .t0 /. We now choose k2 such that k2 .˛ud  j j/ >0 ˇ  k2 .˛ud  j j/ ˇ ; 0 < k2 < 0:5 ˛ud  j j

1 )

(8.47)

then (8.46) together with conditions (8.3) and (8.10) in Assumption 8.1 imply that VP22  21 .
/.V22 C 22 .
//e 21 .tt0 / :

(8.48)

Solving the above differential inequality results in V22 .t /  23 .
/ with 23 .
/ being a class-K1 function of X2 .t0 / D k.z1 .t0 /; z2 .t0 /; ue .t0 /; re .t0 //k. Substituting this upper bound of V22 into (8.46) yields

176

8 Path-tracking Control of Underactuated Ships

VP22

  k2 1  k2

j˛ud  j ˇ  k2 j˛ud  j

 24 .
/e 21 .tt0 / ;



ud  jvd j 2 z1 C 24 .
/e 21 .tt0 /
(8.49)

where 24 .
/ D 21 .
/. 23 .
/ C 22 .
//. The second inequality of (8.49) implies that there exists a class-K1 function 25 .
/ of X2 .t0 / such that jz1 .t /j  25 .
/. Substituting this upper bound into the expression of
and using the first inequality of (8.49) yields global asymptotic and local exponential stability of the z1 -subsystem, i.e., there exist a class-K1 function of X2 .t0 / and a constant 22 > 0 depending on X2 .t0 / such that jz1 .t /j  26 .
/e 22 .tt0 / . Note that condition (8.47) covers the condition (8.34). For the .xe ; ve /-subsystem, we consider the Lyapunov function 1 V3 D .xe2 C ve2 /: 2

(8.50)

By taking the derivative of this function along the solutions of the first two equations of (8.38), it can readily be shown that there exist class-K1 functions 31 .
/ and

32 .
/ of X2 .t0 / such that ˇ VP3   xe2  ˇve2 C 31 .
/.V3 C 32 .
//e 22 .tt0 / ˛  31 .
/.V3 C 32 .
//e 22 .tt0 / :

(8.51)

The second inequality of (8.51) means that V3 .t /  33 .
/ with 33 .
/ being a classK1 function of k.xe .t0 /; ve .t0 /z1 .t0 /; z2 .t0 /; ue .t0 /; re .t0 //k. Substituting this upper bound into the first line of (8.51) readily yields global asymptotic and local exponential stability of the .xe ; ve /-subsystem. We have so far proven that the transformed tracking errors .xe ; z1 ; z2 ; ve ; ue ; re / globally asymptotically and locally exponentially converge to zero. By (8.26), convergence of z1 and ve to zero implies that of ye . Convergence of the actual position tracking error .x  xd ; y  yd / and orientation tracking error .  ˇ d / to balls with radii of jm23 =m22 j and ˇ jı.t /j  .˛ uN d C j j/ jrNd j = ˇˇ uN d C uPN d ˇ < =4 follows from (8.15) and (8.18). Fiq 0 0 nally, since vd D uN d sin.ı/, ud D uN d cos.ı/ and uN d D xd2 C yd2 sP , we can see that the total linear velocity of the ship is tangential to the path and the desired forward speed ud can be adjusted by adjusting sP .

8.1.5 Dealing with Environmental Disturbances To clearly illustrate our idea, we have not included constant (or at least slowly timevarying) environmental disturbances induced by waves, wind, and ocean currents in the control design. To use the proposed controller in practice, the environmental disturbances should essentially be taken into account in the control design. The aim

8.1 Full-State Feedback

177

of this section is to discuss a simple way to modify the proposed control design in the previous section to handle these disturbances. When the constant disturbances are present, the last equation of (8.1) is of the form M vP D C .v/v  Dv C  C E ;

(8.52)

where E D ŒuE vE rE T with uE ; vE and rE being the environmental disturbances acting on the surge, sway, and yaw axes. Processing the same coordinate transformations as in Section 8.1.2, the last three equations of (8.13) are written as uP D Nu C NuE ; vPN D ˛ur  ˇ vN C r C NvE ;

(8.53)

1 uE ; m11 1 NvE D vE ; m22 .m23 vE C m22 rE / : NrE D .m22 m33  m23 m32 /

(8.54)

rP D Nr C NrE ;

with NuE D

We first deal with NvE . The idea to handle NvE is to introduce an angle to the yaw angle. This angle together with the ship forward speed will compensate for NvE . We design an observer to estimate NvE as ON  ˛ur  ˇ vON C r C ONvE ; vPON D k01 .vN  v/ PON D  proj.vN  v; ON ONvE /; vE 01

(8.55)

where k01 and 01 are positive constants, ONvE is an estimate of NvE , the operator proj represents the Lipschitz projection algorithm [118] as proj($; !) O D $ if  .!/ O  0; proj($; !) O D $ if  .!/ O  0 and !O .!/$ O  0; O > 0; proj($; !) O D .1   .!//$ O if  .!/ O > 0 and !O .!/$ 2 where  .!/ O D .!O 2  !M /=. 2 C 2!M /; !O .!/ O D @ .!/=@ O !, O  is an arbitrarily small positive constant, !O is an estimate of !, and j!j  !M . The projection algorithm is such that if !PO D proj($, !/ O and !.t O 0 /  !M , then

1. 2. 3. 4.

!.t O /  !M C ; 8 0  t0  t < 1; proj($; !) O is Lipschitz continuous, O  j$j ; jproj($; !)j !proj($; Q !) O  !$ Q with !Q D !  !. O

178

8 Path-tracking Control of Underactuated Ships

From (8.55), the second equation of (8.53), and property 4 of the proj operator, it is readily shown that lim t!1 QNvE .t / D 0 with QNvE .t / D NvE  ONvE .t /. Now, by comparing the transformed sway dynamics with disturbances (the second equation of (8.53)) and the one without disturbances (the fifth equation of (8.13)), the desired sway velocity vd dynamics should now be modified from (8.21) as vP d D ˛ud rd  ˇvd C rd C ONvE :

(8.56)

With this choice, we have vP e D ˛.ur  ud rd /  ˇve C .r  rd / C QNvE :

(8.57)

Notice that (8.56) is equivalent to .uN d cos.ı/.1  ˛/ C /ıP D .ˇ uN d C uPN d / sin.ı/ .˛ uN d cos.ı/  /rNd C ONvE ; ı.t0 / D 0:

(8.58)

Therefore the tracking objective is solvable if condition (8.11) is replaced by p .˛ uN d C j j/ jrNd j C max.jNvE j/ C 01 2  (8.59)  "7 ; 2 ˇ uN d C uPN d ˇ ˇ where property 1 of the projection algorithm that guarantees ˇONvE ˇ  max.jNvE j/ C 01 has been utilized, with 01 an arbitrarily small positive constant. Hence, if the disturbances are not too large and the path ˝ is feasible, i.e., Assumption 8.1 with condition (8.11) being replaced by (8.59) holds, then the control objective is solvable under constant disturbances. Now processing the same coordinate changes as in Section 8.1.2 results in the same tracking error dynamics as (8.22) but the last three equations of (8.22) are replaced by vP e D ˛.ur  ud rd /  ˇve C .r  rd / C QNvE ; uP D Nu C NuE ; rP D Nr C NrE :

(8.60)

Since lim t!1 NQvE .t / D 0, and NuE and NrE satisfy a matching condition, the control design is much the same as in Section 8.1.3 and is briefly described in the following. Kinematic Control Design. The same as in Section 8.1.3. Kinetic Control Design. The controls Nu and Nr are almost the same as in Section 8.1.3 but we cannot directly use zP 1 and zP 2 since they contain an unmeasured term QNvE .t / and we need to compensate NuE and NrE . These controls are designed as follows:

8.1 Full-State Feedback

179

@˛u @˛u k2 ˛ xP e C uP d C z2 r  ONuE ; @xe @ud ˇ
2 @˛r .ud sin. e /  vd .cos. e /  1/  D c3 re C rPd C @z1 0

Nu D c2 ue C Nr

1 ˛ @˛r B c1 z 2 C .˛ud  /.r  rd /  ue r C @ q ˇ ˇ @z2 1 C z22    k2 ˛ @˛r P @˛r k2 .˛ud  /  u r C uP d C r 1 e e eC 2 2 ˇ
ˇ
@ e @ud @˛r .1  k2 .˛ud  / vP d  z2  ONrE ; @vd ˇ
2

(8.61)

where ONuE and ONrE are estimates of NuE and NrE updated by PONuE D  02 proj.ue ; ONuE /; OPN D  proj.r ; ON /; rE

03

e

rE

(8.62)

where 02 and 03 are positive constants. From the above control design, we have a closed loop system as xP e D k1 xe  ud .cos. ˇ 1 ve / C ue ;

e /  1/  vd

sin.

e / C .rd

C ˛r C re /.z1 

vP e D ˛..ue C ˛u /.rd C ˛r C re /  ud rd /  ˇve C .˛r C re / C QNvE ; zP 1 D k2 ud z1 =
 vd .cos. e /  1/  ˇ 1 .˛ud  /.˛r C re / C

ud .sin.z2 / cos.˛ e / C .cos.z2 /  1/ cos.˛ e //  ˛ˇ 1 ue .rd C ˛r C re / C ˇ 1 QNvE ; q zP 2 D c1 z2 = 1 C z22 C .1  k2 .˛ud  /ˇ 1
2 /re  k2 ˛ˇ 1
2 ue r C

k2 ˇ 1
2 QNvE ; uP e D c2 ue C k2 ˛ˇ 1
2 z2 r C QNuE ; rPe D c3 re  .1 

QNvE @˛r k2 NQvE @˛r k2 .˛ud  //z2   C QNrE ; 2 ˇ
ˇ @z1 ˇ
2 @z2 (8.63)

where QNuE D NuE  ONuE and QNrE D NrE  ONrE . We now present a result for the case with constant disturbances. Theorem 8.2. Assume that the path ˝ satisfies all conditions specified in Assumption 8.1 except for (8.5) being replaced by (8.58) and that the disturbances are not too large, i.e., (8.59) holds, the controls u and r given by (8.14) and (8.61) force the transformed tracking errors .xe ; ye ; e / to globally asymptotically converge to zero with an appropriate choice of k2 . As a result, the ac-

180

8 Path-tracking Control of Underactuated Ships

tual position tracking errors .x  xd ; y  yd / and the orientation tracking error .  d / globally asymptotically converge to balls with radii of jm23 =m22 j and jı.t /j  arcsin..˛ uN d C j j/ jrNd j C max.jNvE j/ C 01 /=.ˇ uN d C uPN d // < =4, respectively. Note that convergence of the tracking errors in the case with disturbances is asymptotic (not locally exponential due to adaptation), and the magnitude of ı.t / is generally larger than the one in the case without disturbances. Since lim .QNuE .t /; QNuE .t /; QNrE .t // D 0;

t!1

@˛r =@z1 and @˛r =@z2 are bounded by some constants, the structure of (8.63) is very similar to (8.38), and the proof of Theorem 8.2 follows the same lines as for Theorem 8.1 plus an application of Barbalat’s lemma, we omit proof of Theorem 8.2.

8.1.6 Numerical Simulations We perform some numerical simulations to illustrate the effectiveness of our proposed control laws given by (8.14), (8.36), and (8.37) on a supply vessel with the length of L D 76:2 m and a mass of m D 226  103 kg. This vessel has a minimum turning circle with the radius of 250m, a maximum surge force of 6:5  1010 N, and a maximum yaw moment of 3:2  1012 Nm. The Bis-scale parameters of the vessel taken from [11], pp. 446, are as follows: m11 D 1:1274; m22 D 1:8902; m23 D 0:0744; m33 D 0:1278; d11 D 0:0358; d22 D 0:1183; d23 D 0:0124; d32 D 0:0041; d33 D 0:0308: From these values, we have ˛ D 0:5964; ˇ D 0:0626, and D 0:0041. The reference path p ˝ is chosen as xd .s/ D s; yd .s/ D 100 sin.0:01s/ and sP D 1. Therefore, uN d D 2 1 C cos.0:01s/2 and rNd D 0:02 sin.0:01s/=.1 C cos.0:01s/2 /. A simple calculation shows that all of conditions of Assumption 8.1 hold. In the simulations, the initial conditions are chosen as x.0/ D 20, y.0/ D 50, .0/ D 0:5, u.0/ D 0; v.0/ D 0:1, and r.0/ D 0. The control gains are chosen as k1 D ˇ=˛ D 0:1049, k2 D 0:025, and c1 D c2 D c3 D 2. A simple calculation shows that k2 satisfies the condition specified in (8.47). Simulation results without disturbances are plotted in Figures 8.2 and 8.3. In these figures, all variables are converted back to their original values using the following table with g being the gravitational acceleration: The tracking errors are plotted in Figure 8.2a, and the angle ı is plotted in Figure 8.2b. It is seen that the proposed controller nicely forces the ship to track the given reference path in the sense that the tracking errors asymptotically converge to zero. From (8.20) and the fact that the surge and sway velocities .u; v/ converge to their of the surge desired values .ud ; vd /, the change of the angle ı reflects the change p and sway velocities of the ship with the total ship velocity uN D u2 C v 2 as time

8.1 Full-State Feedback

181

Table 8.1 Normalization variables used for the Bis system Variable Bis-system p Time L=g Position L Angle p1 Linear velocity p Lg Angular velocity g=L Force mg Moment mgL

evolutes. The ship and reference trajectories in the .x; y/-plane, and the control inputs are plotted in Figure 8.3. To illustrate how our controller in Section 8.1.5 can compensate for the disturbances, we also simulate with the disturbances as E D Œ0:15; 0:08; 0:05T in the Bis system. This choice of E implies that the disturbance forces on the surge and sway are 0:15 and 0:08, and the disturbance moment on the yaw is 0:05. The adaptation gains are chosen as 0i D 1; k0i D 1, and 0i D 0:05; i D 1; 2; 3. With these disturbances, one can verify that condition (8.59) holds, i.e., ..˛ uN d C j j/ jrNd j C max.jNvE j/ C 01 /=.ˇ uN d C uPN d //  0:71. Simulation results are plotted in Figures 8.4 and 8.5. It is seen that our controller forces the heading angle to a value ı.t / to compensate for the disturbances.

a

Tracking errors

400 x−xd

200

y−y

d

ψ−ψd

0 −200 −400 0

500

1000 Time [s]

1500

2000

1500

2000

b

0.2

δ [rad]

0.1 0 −0.1 −0.2 0

500

1000 Time [s]

Figure 8.2 Simulation results without disturbances: a. Tracking errors ..x xd /; .y yd /; .  d //; b. Variation of angle ı

182

8 Path-tracking Control of Underactuated Ships

a

y [m]

5000 0 −5000 −1

0

1

6

u

τ [N]

4

4

5

6

7 4

x 10

2

500 8

1

r

3 x [m] b

x 10

0 0

τ [Nm]

2

1000 Time [s]

1500

2000

1500

2000

c

x 10

0 −1 −2 0

500

1000 Time [s]

Figure 8.3 Simulation results without disturbances: a. Ship position and orientation in the .x; y/plane; b. Control input u ; c. Control input r

a

400

Tracking errors

x−x

d

y−yd

200

ψ−ψd 0 −200 0

500

1000 Time [s]

1500

2000

1500

2000

b

0.8

δ [rad]

0.6 0.4 0.2 0 0

500

1000 Time [s]

Figure 8.4 Simulation results with disturbances: a. Tracking errors ..x  xd /; .y  yd /; . d //; b. Variation of angle ı



8.2 Output Feedback

183 a

y [m]

5000 0 −5000 −1

0

1

6

τu [N]

10

3 x [m]

4

5

6

7 4

x 10

b

x 10

5 0 −5 0

500 8

5

τr [Nm]

2

1000 Time [s]

1500

2000

1500

2000

c

x 10

0 −5 −10 0

500

1000 Time [s]

Figure 8.5 Simulation results with disturbances: a. Ship position and orientation in the .x; y/ plane; b. Control input u ; c. Control input r

8.2 Output Feedback 8.2.1 Control Objective For the reader’s convenience the mathematical model of an underactuated ship moving in a horizontal plane is rewritten here, see Section 3.4.1.1: P D J . /v; M vP D C .v/v  .D C Dn .v//v C ; where

with

2 3 2 3 3 u u x  D 4 y 5; v D 4 v 5;  D 4 0 5; r r 2 3 2 3 m11 0 0 cos. /  sin. / 0 J . / D 4 sin. / cos. / 0 5 ; M D 4 0 m22 m23 5 ; 0 0 1 0 m32 m33 2 3 3 2 D11 0 0 0 C13 0 C .v/ D 4 0 0 C23 5 ; D C Dn .v/ D 4 0 D22 D23 5 ; 0 D32 D33 C31 C32 0 2

(8.64)

184

8 Path-tracking Control of Underactuated Ships

m11 D m  XuP ; m22 D m  YvP ; m23 D mxg  YrP ; m32 D mxg  NvP ; m33 D Iz  NrP ;

C13 D C31 D .m  YvP /v  .mxg  0:5.YrP C NvP //r; C23 D C32 D .m  XuP /u; D11 D .Xu C Xjuju juj/; D22 D .Yv C Yjvjv jvj/; D23 D Yr ; D32 D Nv ; D33 D .Nr C Njrjr jrj/:

All the symbols used here are defined in Section 3.3. It is noted that we did not include environmental disturbances in the ship dynamics (8.64). In the control design, we will add integrators to the path-tracking error outputs to compensate for the disturbances. In this section, we consider the following control objective. Control Objective. Assume that the ship velocities, u, v, and r are not available and that Assumption 8.2 holds, design the controls u and r to force the ship (8.64) to follow a prescribed path ˝ parameterized by .xd .s/; yd .s// with s being the path parameter in the sense that the position of the ship (8.64) globally tracks the path ˝, the ship total linear velocity is tangential to the path ˝, and let the desired surge speed, ud 0 .t /, be adjusted on-line. The desired surge speed, ud 0 .t /, is assumed to be bounded and twice differentiable. Assumption 8.2. There exist strictly positive constants "i ; 0  i  5 such that 0

0

0

"0  xd2 .s/ C yd2 .s/  "1 ; 8s 2 R;

(8.65)

ud 0 .t /  "2 ; 8 t  0;

(8.66)

NvP D YrP ;

(8.67)

1  .˛.ud 0 .t / C "3 / C j j/=.ud 0 .t /  "4 /  "5 ; 8 t  0;

(8.68)

0

where xd .s/; yd .s/; ˛ and are defined as @xd 0 @yd ; yd .s/ D ; @s @s m11 Yr ˛D ; D : m22 m22 0

xd .s/ D

(8.69)

Remark 8.4. The following observations are made on Assumption 8.2. 1. Condition (8.65) implies that the path ˝ is regular with respect to the path parameter s. If the path is not regular, one can break it into different regular paths and consider each path separately. 2. Condition (8.66) means that the path ˝ does not contain or approach a set-point, i.e., we do not consider stabilization/regulation problems. For clarity we only consider forward tracking. Backward tracking can be done similarly. 3. Condition (8.67) implies that the added mass matrix is symmetric. This condition is needed for designing an observer. If this condition does not hold, we can use

8.2 Output Feedback

185

an acceleration feedback in the yaw dynamics [12] to reshape the added mass matrix. 4. Condition (8.68) is needed to guarantee stability of the sway dynamics under the proposed controller (to be designed later). Since ˛ is strictly smaller than 1 (added mass, XuP , in surge is smaller than that, YvP , in sway) and j j is small for most ships, Condition (8.68) is satisfied under (8.66) with "2 being strictly larger than j j. The main ideas in solving the above control objective are as follows: 1. Choose an appropriate body-fixed frame origin to avoid the yaw moment control r acting directly on the sway dynamics. This choice will overcome the difficulty caused by off-diagonal terms in the mass matrix. 2. Transform the ship dynamics to a system, which is linear and monotonic in unmeasured velocities, so that an observer can be easily designed. 3. Interpret path-tracking errors in a frame attached to the path ˝ such that the error dynamics are of a triangular form to which the backstepping technique can be applied. 4. Use the orientation error as a virtual control to stabilize the cross-track error where a filter of the sway velocity is designed and used in the virtual control. This filter and the derivative of the path parameter used as an additional control allow a global controller and desired path with arbitrary curvature. 5. Add projection integrators to the path-tracking error dynamics’ output to compensate for constant bias of the disturbances.

Yb v

YE

Yd

u

Ob

y x

CG ye x

\

x

xg

Xd xe

\e Od

yd

Xb

\ d ud

: OE

xd

Figure 8.6 Interpretation of path-tracking errors

x

XE

186

8 Path-tracking Control of Underactuated Ships

8.2.2 Coordinate Transformations 8.2.2.1 Choosing the Body-fixed Frame Origin To avoid the yaw moment control r acting directly on the sway dynamics, we choose the body-fixed frame origin such that it is on the center line of the ship (see Figure 8.6), i.e., yg D 0 and that: xg D

YrP : m

(8.70)

With this choice, the ship model (8.64) can be written as: P D J . /v; N vP D CN .v/v  .DN C DN n .v//v C ; M

(8.71)

where 3 2 3 dn1 juj 0 m11 0 0 0 N D 4 0 m22 0 5 ; DN n .v/ D 4 0 dn2 jvj 0 5 ; M 0 0 dn3 jrj 0 0 m33 2

3 2 3 0 0 m22 v d11 0 0 7 6 DN D 4 0 d22 d23 5 ; CN .v/ D 4 0 0 m11 u 5 ; 0 d32 d33 m22 v m11 u 0 2

with

(8.72)

d11 D Xu ; d22 D Yv ; d23 D Yr ; d32 D Nv ; d33 D Nr ; dn1 D Xjuju ; dn2 D Yjvjv ; dn3 D Njrjr :

It is noted that the ship position .x; y/ is the coordinates of the body-fixed frame origin (not the ship’s center of gravity) with respect to the earth-fixed frame.

8.2.2.2 Transformation of the Ship Dynamics to be Linear and Monotonic in Unmeasured State System The transformation here is very similar to that in Section 7.3.1. For the sake of completeness, we give the full derivation of the transformation. As such, we first remove the term CN .v/v in the right-hand side of the second equation of (8.71), which causes difficulties in the observer design, by introducing the following coordinate: X D P./v;

(8.73)

8.2 Output Feedback

187

where P./ 2 R33 is a global invertible matrix to be designed. With (8.73), the second equation of (8.71) is written as h i P N 1 CN .v/v  XP D P./v  P./M N 1 .DN C DN n .P 1 ./X //P 1 ./X C P./M N 1 : P./M

(8.74)

It can be seen that in order to remove the term CN .v/v, we need to find the matrix P./ such that h i P N 1 CN .v/v D 0; (8.75) P./v  P./M

for all .; v/ 2 R6 . Assuming that the elements of P./ are pij ./; i D 1; 2; 3; j D 1; 2; 3, (8.75) is expanded to: m22 m11 m11  m22 pi1 vr  pi2 ur C pi3 uv D 0; m11 m22 m33 i D 1; 2; 3; 8 .; u; v; r/ 2 R6 ; (8.76)

pPi1 u C pPi2 v C pPi3 r C

where for brevity, we omit the argument  of pij ./. With the first equation of (8.71), we expand (8.76) as     @pi1 @pi2 @pi2 @pi1 cos. / C sin. / u2 C  sin. / C cos. / v 2 C @x @y @x @y  @pi3 2 @pi1 @pi1 @pi2 @pi2 r C  sin. / C cos. / C cos. / C sin. /C @ @x @y @x @y    m11  m22 @pi1 @pi3 @pi3 m11 pi3 uv C C cos. / C sin. /  pi2 ur C m33 @ @x @y m22   @pi3 m22 @pi2 @pi3  sin. / C cos. / C pi1 vr D 0: (8.77) @ @x @y m11 Therefore (8.77) holds for all .x; y; ; u; v; r/ 2 R6 if @pi1 @pi1 cos. / C sin. / D 0; @x @y @pi2 @pi2  sin. / C cos. / D 0; @x @y @pi3 D 0; @     @pi2 @pi1 @pi1 @pi2 m11  m22 sin. / C cos. / C  C pi3 D 0; @y @x @y @x m33 @pi1 @pi3 @pi3 m11 C cos. / C sin. /  pi2 D 0; @ @x @y m22 @pi2 @pi3 @pi3 m22  sin. / C cos. / C pi1 D 0: (8.78) @ @x @y m11

188

8 Path-tracking Control of Underactuated Ships

A family of solutions of the above set of six partial differential equations is ..m11 Ci3 x C m33 Ci1 / sin. /  .m11 Ci3 y  m33 Ci2 / cos. // ; m33 m22 ..m11 Ci3 x C m33 Ci1 / cos. / C .m11 Ci3 y  m33 Ci2 / sin. // pi2 D ; m11 m33 pi3 D Ci3 ; (8.79)

pi1 D

where Ci1 ; Ci2 , and Ci3 are arbitrary constants. We now choose the constants Ci1 ; Ci2 , and Ci3 such that the matrix P./ is invertible for all .x; y; / 2 R3 . A trivial choice of C13 D C11 D 0, C12 D 1, C23 D C22 D 0, C21 D 1, C31 D C32 D 0, and C33 D 1 results in 3 2 m22 cos. /  sin. / 0 m11 6 7 7 6 m22 7: sin. / cos. / 0 P./ D 6 7 6 m11 5 4 m22 m11 .cos. /x C sin. /y/ .sin. /x  cos. /y/ 1 m33 m33 (8.80) Indeed this matrix is invertible for all .x; y; / 2 R3 . Substituting (8.80) into (8.74) gives N 1 : N 1 .DN C DN n .P 1 ./X //P 1 ./X C P./M XP D P./M

(8.81)

8.2.3 Observer Design It can be seen that system (8.81) is linear and monotonic in the unmeasured state vector X . We now design a simple observer to estimate X as: P N 1 .DN C DN n .P 1 ./XO //P 1 ./XO C P./M N 1 ; XO D P./M

(8.82)

where XO is an estimate of X . From (8.81) and (8.82), we have N 1 ./XQ  P./M N 1  N 1 DP XPQ D P./M ŒDNn .P 1 ./X //P 1 ./X  DNn .P 1 ./XO //P 1 ./XO ;

(8.83)

where XQ WD .xQ 1 ; xQ 2 ; xQ 3 /T D X  XO . We will use (8.83) in stability analysis of the closed loop system later. Define vO D Œu; O v; O r O T as being an estimate of the velocity vector v as: vO D P 1 ./XO : (8.84) O satisfies: Then the velocity estimate error vector, vQ WD Œu; Q v; Q r Q T D v  v,

8.2 Output Feedback

189

3

2

2 3 xQ 1 cos. / C xQ 2 sin. / uQ 7 6 7 6 6 6 7 6 m11 .xQ sin. / C xQ cos. // 7 7: 1 2 6 vQ 7 D 6 7 4 5 4 m22 5 m11 .xQ 1 y  xQ 2 x/ C xQ 3 rQ m33

(8.85)

Using (8.84), we rewrite the first equation of (8.71) and (8.82) as: xP D .uO C u/ Q cos. /  .v C v/ Q sin. /;

yP D .uO C u/ Q sin. / C .v C v/ Q cos. /; P D rO C r; Q m d11 dn1 m22 1 22 uPO D vO rO  uO  u C vO r; Q O uO C juj m11 m11 m11 m11 m11 m11 m11 d22 d23 dn2 O vO  vOP D  uO rO  vO  rO  uO r; Q jvj m22 m22 m22 m22 m22 d22 d33 dn3 m11  m22 O rO C uO vO  vO  rO  rPO D jrj m33 m33 m33 m33 1 m11 m22 r C uO vQ  vO u: Q m33 m33 m33

(8.86)

Transformation of Path-tracking Errors We now interpret the path-tracking errors in a frame attached to the reference path ˝ as follows (see Figure 8.6): 2

xe

3

2

x  xd

3

7 6 7 6 T 4 ye 5 D J . / 4 y  yd 5 ;  d e where

d

0

is the angle between the path and the X-axis defined by ! 0 yd .s/ ; d D arctan 0 xd .s/ 0

(8.87)

(8.88)

with xd .s/ and yd .s/ being defined in (8.69). In Figure 8.6, OE XE YE is the earth-fixed frame; Op Xp Yp is a frame attached to the path ˝ such that Op Xp and Op Yp are parallel to the surge and sway axes of the ship, respectively, ud is tangential to the path, C G is the center of gravity of the ship; and Ob Xb Yb is the body-fixed frame. Therefore xe , ye , and e can be referred to as tangential, cross and heading errors, respectively. Differentiating both sides of (8.87) along the solutions of the first three equations of (8.86) results in the kinematic path-tracking errors:

190

8 Path-tracking Control of Underactuated Ships

xP e D uO  ud cos. yPe D vO C ud sin. P e D rO  rd C r; Q

Q e e / C .rO C r/y

C u; Q Q e C v; Q e /  .rO C r/x

(8.89)

where ud and rd are given by: ud D rd D

q 0 0 xd2 .s/ C yd2 .s/Ps ; 0

00

00

0

xd .s/yd .s/  xd .s/yd .s/ xd2 .s/ C yd2 .s/ 0

0

sP :

(8.90)

8.2.4 Control Design with Integral Action There are often two methods to let a control system compensate for constant (or at least slowly-varying) environmental disturbances: (1) deriving an adaptive control law to estimate a constant bias (as the one in Section 8.1.5), and (2) adding extra integrators to the output of the path-tracking error system. Here we use the second method for simplicity since the first one will result in a complicated control law due to the observer developed in the previous section, see Section 8.2.6.1 for a discussion. We divide the control design procedure into two separate stages. At the first stage, we consider (8.89) with uO and rO being viewed as controls. At the second stage, the last three equations of (8.86) are considered to design the actual controls u and r using the backstepping technique.

8.2.4.1 Kinematic Control Design Substep 1 (Stabilizing .xe ; ye /-dynamics) First we add two projection integrators to the position path-tracking error dynamics as P xe D proj. 1 xe =
; xe /; P ye D proj. 2 ye =
; ye /; xP e D uO  ud cos. yPe D vO C ud sin.

Q e e / C .rO C r/y

C u; Q Q e C v; Q e /  .rO C r/x

(8.91)

p where
D 1 C xe2 C ye2 , 1 and 2 are positive constants and the operator proj represents the Lipschitz projection algorithm (repeated here for the reader’s convenience)

8.2 Output Feedback

191

proj($; !) O D $ if  .!/ O  0; proj($; !) O D $ if  .!/ O  0 and !O .!/$ O  0;

proj($; !) O D .1   .!//$ O if  .!/ O > 0 and !O .!/$ O > 0;

2 /=. 2 C 2!M /; !O .!/ O D @ .!/=@ O !, O  is an arbitrarily where  .!/ O D .!O 2  !M small positive constant and !M is the maximum value of the constant bias that !O will compensate for. The projection algorithm (repeated here for convenience of the reader) is such that if !PO D proj($, !/ O and !.t O 0 /  !M then:

1. 2. 3. 4.

!.t O /  !M C ; 8 0  t0  t < 1; proj($; !) O is Lipschitz continuous, O  j$j ; jproj($; !)j !proj($; Q !) O  !$ Q with !Q D !  !. O

It is noted that we have added the integrators via the projection algorithm to guarantee that xe and ye are always bounded by some predefined constants. Define ue D uO  ˛u ; Ne D e  ˛ e; where ˛u and ˛ e are virtual controls of uO and into the last two equations of (8.91) gives

e,

(8.92) respectively. Substituting (8.92)

xP e D ˛u C ue  ud cos.˛ e /  ud ..cos. N e /  1/ cos.˛ e /  sin. N e / sin.˛ e // C .rO C r/y Q e C u; Q yPe D vO C ud sin.˛ e / C ud .sin. N e / cos.˛ e / C .cos. N e /  1/  sin.˛

e

//  .rO C r/x Q e C v: Q

(8.93)

From the first equation of (8.91) and first equation of (8.93), we can easily design a control law for ˛u to stabilize the xe -dynamics as follows ˛u D '1  xe C ud cos.˛

e

/;

(8.94)

with '1 D k1

xe ;


where k1 is a positive constant to be specified later. It is noted that we have chosen ˛u not to cancel the known term,ud ..cos. N e /  1/ cos.˛ e /  sin. N e / sin.˛ e // C ry O e , to simplify the controller expression. However, stabilizing the second equation of (8.91) and second equation of (8.93) is more difficult. We cannot directly use ˛ e to cancel vO since the term ud sin.˛ e / cannot cancel vO without imposing a restriction on the initial conditions, i.e., no global result can be obtained. To resolve this difficulty, the derivative of the path parameter is used as an additional control and is designed as:

192

8 Path-tracking Control of Underactuated Ships

q 2 C .' C  C v /2 u2d 0 C xe 2 ye d sP D ; q 0 0 xd2 .s/ C yd2 .s/

(8.95)

with '2 D k2 y
e , where from now we drop the argument t of ud 0 and its derivatives, k2 is a positive constant to be specified later, and vd is a filtered value of vO to be designed later. From (8.95) and (8.90), we have: q 2 C .' C  C v /2 : ud D u2d 0 C xe (8.96) 2 ye d From (8.96), the second equation of (8.93), and (8.90), we design a control law for ˛ e as:   '2 C ye C vd ˛ e D  arctan ; (8.97)  ud 0

with ud 0 D

q

2 : u2d 0 C xe

(8.98)

Substituting (8.97), (8.95), and (8.94) into (8.93) results in xP e D k1 xe =
 xe C ue  ud ..cos. N e /  1/ cos.˛ sin. N e / sin.˛ e // C .rO C r/y Q e C u; Q yPe D k2 ye =
 ye C ve C ud .sin. N e / cos.˛ e / C .cos. N e /  1/ sin.˛

e

//  .rO C r/x Q e C v; Q

e

/

(8.99)

where ve D vO  vd . Substep 2 (Stabilizing N e dynamics) Define re D rO  ˛r ;

(8.100)

O Differentiating both sides of the second equation where ˛r is a virtual control of r. of (8.92) along the solutions of (8.97), and adding a projection integrator gives   Ne P N e D proj 3 q ; N e ; 1 C N e2 u .˘ C vP d C P ye / PN D b .r C ˛ C r/ Q  rd C d 0  e 1 e r u2d

(8.101)

Q C .1 C xe2 /.ve C v// Q .'2 C ye C vd /.uP d 0 ud 0 C P xe xe / k2 ud 0 .xe ye .ue C u/  ; u2d ud 0 u2d
3

8.2 Output Feedback

193

where 3 is a positive constant and  ud 0 x e ; u2d
3  vd C ud sin. ˘ D k2


b 1 D 1  k2

e/

ye .xe .˛u  ud cos.



e // C ye .vd
3

C ud sin.

e //



:

(8.102)

From the first equation of (8.102), we choose the design constant k2 such that: k2  min.ud 0 /  "6 ;

(8.103)

where "6 is a strictly positive constant. Then b1 is always larger than some strictly positive constant. Now the virtual control ˛r can be designed from (8.101) as 
'2 C ye C vd u  k3 N e 1 q ˛r D    N e C rd  2d 0 ˘ C vP d C P ye C b1 ud u2d ud 0 1 C N e2 .uP d 0 ud 0 C P xe xe // ;

(8.104)

where k3 is a positive constant. Substituting (8.104) into the second equation of (8.101) gives: N k2 ud 0 .1 C xe2 / k2  u d 0 x e y e PN D k q e .u C u/ Q C   Ne  e 3 e u2d
3 u2d
3 1 C N e2 .ve C v/ Q C b1 .re C r/: Q

(8.105)

To determine vd , we differentiate ve D vO  vd along the solutions of the fifth equation of (8.86) to obtain: m11 d22 d23 .˛u C ue /.˛r C re /  .ve C vd /  .˛r C re /  m22 m22 m22 m11 dn2 O .ve C vd /  uO rQ  vP d (8.106) jvj m22 m22

vP e D 

which suggests that we choose: vP d D 

Ne k2 ud 0 .1 C xe2 / d22 dn2 m11 ˛u C d23 ; ˛r  vd  q jvd j vd C 2 3 m22 m22 m22 ud
1 C N e2

(8.107)

where the last term on the right-hand side of (8.107) is added to take care of the second last term in the right-hand side of (8.105). Substituting (8.107) into (8.106) gives:

194

8 Path-tracking Control of Underactuated Ships

d22 dn2 m11 d23 m11 uO C d23 ve  re  ru O e re  O ve  jvj m22 m22 m22 m22 m22 Ne k2 ud 0 .1 C xe2 / m11 : (8.108) .ue C ˛u /rQ  q 2 3 m22 ud
1C N 2

vP e D 

e

Now notice that ˛r depends on vP d , see (8.104). Hence, we substitute (8.104) into (8.107) to obtain:   1 d22 vd C dn2 jvd j vd m11 ˛u C d23 k3 N e vP d D   N e C rd   q  b2 m22 b1 m22 1C N 2 e

  ud 0 .˘ C P ye / '2 C ye C vd ud 0 .1 C xe2 / N e ; C .uP d 0 ud 0 C P xe xe /  k2 q u2d u2d ud 0 u2
3 1 C N 2 e

d

(8.109)

where b2 D 1 

m11 ˛u C d23 ud 0 m22 u2d b1

)

b2  1 

m11 .ud 0 C k1 / C jd23 j : (8.110) m22 .ud 0  k2 /

From condition (8.68) in Assumption 8.2, if we pick k1 and k2 such that k1  "3 ; k2  "4 ;

(8.111)

then we have b2  "5 , i.e., there is no singularity in (8.109), see Assumption 8.2 for the constants "3 ; "4 and "5 .

8.2.4.2 Kinetic Control Design Before designing the actual controls u and r , we note that the virtual control ˛u is a smooth function of xe , ye , ud 0 , and xe , and the virtual control ˛r is a smooth function of xe , ye , e , xe , ye ,  N e , ud 0 , uP d 0 , vd , and s. By differentiating both sides of (8.100) and the first equation of (8.92), and noting (8.101) and (8.108), we choose the actual controls u and r as: O uC u D c1 m11 ue  m22 vO rO C .d11  dn1 juj/˛  @˛u @˛u m11 . uO  ud cos. e / C ry O e/ C .vO C ud sin. @xe @ye Ne

O e/ C e /  rx 1

k2  u d 0 x e y e m11 @˛u @˛u C C P xe C uP d 0 C rv O eA ; q @xe @ud 0 m u2d
3 22 1 C N e2

8.2 Output Feedback

195

O r C d32 vO  .m11  m22 /uO vO C r D c2 m33 re C .d33 C dn3 jrj/˛  @˛r @˛r m33 . uO  ud cos. e / C ry O e/ C .vO C ud sin. e /  rx O e/ C @xe @ye @˛r @˛r @˛r @˛r @˛r sP C .rO C rd / C P xe C P ye C P N C @s @ e @xe @ye @ N e e 1 N @˛r .m11 uO C d23 /ve C @˛r @˛r b1 e C uP d 0 C uR d 0 C vP d  q A; @ud 0 @uP d 0 @vd m22 1 C N e2

(8.112)

where c1 and c2 are positive constants. The closed loop system is:      

2 y e

3 N e

1 x e ; N e ; P xe D proj ; xe ; P ye D proj ; ye ; P N e D proj q

1C N 2 e

xe xP e D k1  xe C ue  ud ..cos. N e /  1/ cos.˛ e / 
sin. N e / sin.˛ e // C .rO C r/y Q e C u; Q N yPe D k2 ye =
 ye C ve C ud .sin. e / cos.˛ e / C .cos. N e /  1/ sin.˛ e //  .rO C r/x Q e C v; Q N k2  u d 0 x e y e PN D k q e   N e C b1 .re C r/ Q  .ue C u/ Q C e 3 u2d
3 1 C N e2 k2 ud 0 .1 C xe2 / .ve C v/; Q u2d
3 O d11 C dn1 juj m22 @˛u uP e D .c1 C /ue C vO rQ  .uQ C ye r/ Q  m11 m11 @xe Ne k2  u d 0 x e y e @˛u m11 C .vQ  xe r/ Q C rv O e; q 2 @ye ud
3 1 C N 2 m22 e

d22 dn2 m11 d23 m11 uO C d23 ve  re  ru O e re  O ve  jvj m22 m22 m22 m22 m22 Ne k2 ud 0 .1 C xe2 / m11 ; .ue C ˛u /rQ  q 2 m22 ud
3 1C N 2

vP e D 

e

d33 C dn3 jrj m11 m22 @˛r O /re C uO vQ  vO uQ  .uQ C ye r/ Q  m33 m33 m33 @xe @˛r @˛r .m11 uO C d23 /ve b1 N e ; .vQ  xe r/ Q  rQ C q @ye @ e m22 1C N 2

rPe D .c2 C

e

196

8 Path-tracking Control of Underactuated Ships

vP d D

0

0

1 B d22 vd C dn2 jvd j vd m11 ˛u C d23 B k3 N e   N eC  @ @ q b2 m22 b1 m22 1C N 2 e

! ud 0 .˘ C P ye / '2 C ye C vd rd  C .uP d 0 ud 0 C P xe xe /  u2d u2d ud 0 1 2 N u .1 C x / e C k2 d 0 q e A; u2
3 1 C N e2 d

xP d D

@xd ud ud @yd ; yPd D : q q 02 02 02 0 @s @s xd .s/ C yd .s/ xd .s/ C yd2 .s/

(8.113)

We now present the second main result of this chapter, the proof of which is given in the next subsection. Theorem 8.3. Assume that Assumption 8.2 holds, the controls u and r given by (8.112) solve the control objective with an appropriate choice of k1 and k2 such that (8.103) and (8.111) hold. Particularly, the transformed path-tracking errors .xe ; ye ; N e / globally asymptotically converge to zero. As a result, the actual position path-tracking errors .x  xd ; y  yd / and orientation path-tracking error .  d / globally asymptotically converge to zero and to a ball with a radius of smaller than 0:5, respectively. Furthermore, the desired forward speed of the ship on the path can be adjusted by adjusting ud 0 .t / and the total linear velocity of the ship is tangential to the path.

8.2.5 Stability Analysis To prove Theorem 8.3, we first show that the closed loop system (8.113) is forward complete (i.e. no finite escape in the closed loop system), and there exist an arbitrarily small positive constant 0 , and a class-K function 0 of k0 .t0 /k, with h 0 .t0 / WD xe .t0 /; ye .t0 /; e .t0 /; xd .t0 /; yd .t0 /; xe .t0 /; iT ye .t0 /;  N e .t0 /; ue .t0 /; ve .t0 /; re .t0 /; vd .t0 /

such that

k.xe .t /; ye .t /; xd .t /; xd .t //k  0 e 0 .tt0 / ; 8 t  t0 : The reason for doing this is that the term, r, Q contains .x; y/, see (8.85). We then consider the . N e ; N e ; ue ; ve ; re /-subsystem and prove that vd is bounded by some constant. Finally we consider the .xe ; ye ; xe ; ye /-subsystem.

8.2 Output Feedback

197

To prove that the closed loop system (8.113) is forward complete, we consider the Lyapunov function W D

1 1 log.1 C W0 / C K2 XQ T XQ ; 2 2

(8.114)

where 2 2 W0 D xe2 C ye2 C xd2 C yd2 C xe C ye C  2N e C K1 . N e2 C u2e C ve2 C re2 C vd2 /;

K1 and K2 are positive constants to be picked later. Differentiating both sides of (8.114) along the solutions of (8.113) and (8.83), and using properties of the projection algorithm, after some calculations we obtain: 1 1 C 2 kXQ k2 ; WP  0 C 2 1 C W0

(8.115)

with constants i ; i D 0; 1; 2 being defined as    1 d11 d22 d33 0 D 2 5"01 C 4"02 C 16"03 C  c12 C C C ; 2K1 "01 m11 m22 m33 2 2 N xe C N ye

9 2 2 2 2 2 C N 2N e /  C N ye C 3k22 / C .N xe C 3N ye .uN C N xe 4"01 "01 d 0   1 0:75 "04 jd23 j N N N C C C max.P xe ; P ye ; P N e / C K1 "03 b2 4"05 b 1 m22 !! m11 .k1 C uN d 0 / .2 C k2 C NP xe C NP ye C N xe C N ye / k3 C N N e C b 1 m22 ud 0

1 D

C

N C N 1 D/ 2 D K2 min .M A22 / C 8:25

2:25 1 1 C K1 . C .0:5k12 C 0:25.A21 C 2"01 "02 "03

m222 C .2k12 C 2k22 C 3.1 C k1 C uN d 0 /2 C 2A21 C m233

2.A22 C A23 //;

(8.116)

where c12 D c1 C c2 , "0i ; i D 1; :::; 5 are positive constants. The constants Ai ; i D 1; 2; 3 denote upper bounds of j@˛r =@xe j ; j@˛r =@ye j ; j@˛r =@ e j, respectively. These constants can be calculated by taking corresponding partial derivatives of ˛r , see (8.104). Inequality (8.115) implies that the solution of the closed loop system (8.113) exists. Now we pick the constants "0i ; i D 1; 2; 3 and K1 such that 0 is strictly less N 1 D/. N Then picking the constant K2 such that 2  0, we have than min .M 1 1 WP  0 C : 2 1 C W0

(8.117)

198

8 Path-tracking Control of Underactuated Ships

On the other hand, it is noted from (8.114) that WP 

WP 0 WP 0 N XQ k2   max .MN 1 D/k ; 2.1 C W0 / 2.1 C W0 /

(8.118)

N 1 D/ N > 0 is the maximum eigenvalue of M N 1 D. N From (8.117) and where max .M (8.118), we have (8.119) WP 0  0 W0 C 21 C 0 : From (8.119) and the expression for W0 , we have k.xe .t /; ye .t /; xd .t /; yd .t //k  0 e 0 .tt0 / ; 8 t  t0 : To show that XQ .t / globally exponentially converges to zero, we take the Lyapunov function 1 V0 D XQ T XQ ; 2 whose derivative along the solutions of (8.83) satisfies N 1 D/k N XQ k2 VP0  min .M

N 1 N

) kXQ .t /k  XQ .t0 / e min .M D/.tt0 / ;

8 t  t0  0;

(8.120)

where Lemma 2.2 has been used. To investigate stability of the . N e ; N e ; ue ; ve ; re /subsystem, we take the Lyapunov function q 1 1 2 V1 D 1 C N e2  1 C .u2e C ve2 C re2 / C  ; (8.121) 2 2 3 N e whose derivative along the solutions of the third, sixth, seventh, eighth, and ninth equations of (8.113) satisfied   N e2 d11 2  c C  " VP1  .k3  "06 / 1 06 ue  m11 1 C N e2     d33 d22 2  "06 ve  c2 C  "06 re2 C 1 e ı1 .tt0 / m22 m33  1 e ı1 .tt0 / ;

(8.122)

with 1 D A4 kXQ .t0 /k C A5 kXQ .t0 /k2 , where A4 D max.b 1 ; k2 u1 d 0 /;

1 1 2 2 A5 D 0:25"1 N d C vN d /2 C 06 max.k1 C b 1 ud 0 .k1 C k2 / .u 2 2 .m11 m1 N d /; 2k12 C .m11 m1 N u/ C 33 / .1 C v 33 / .1 C ˛ 1 2 2 b 1 N d C vN d /2 ; 2k12 C .m11 m1 22 /  1 ud 0 .k1 C k2 / .u

1 2 .1 C ˛N u / C b 1 N d C vN d /2 C k32 /; 1 ud 0 .k1 C k2 / .u

(8.123)

8.2 Output Feedback

199

N 1 D/ N  0 , "06 is a positive constant that is strictly smaller than and ı1 D min .M min.k3 ; c1 C d11 =m11 ; d22 =m22 ; c2 C d33 =m33 /. Since we have already proved that N 1 D/, N ı1 is a strictly positive constant. The second 0 is strictly less than min .M inequality of (8.122) implies that V1 .t /, i.e., . N e .t /; N e .t /; ue .t /; ve .t /; re .t //, is bounded. By integrating both sides of the first inequality of (8.122) and applying Barbalat’s lemma, we have lim t!1 . N e .t /; ue .t /; ve .t /; re .t // D 0. To show that vd is bounded, we take the Lyapunov function 1 V2 D vd2 ; 2 whose derivative along the solutions of the last equation of (8.113) yields
 2 2 VP2 D b21 dn2 m1 22 jvd j vd  B1 vd C B2 jvd j C B3 ;

(8.124)

where Bi ; 0  i  3 are given in (8.142). Since dn2 =m22 is a positive constant, an application of Theorem 4.18 in [6] shows that vd is bounded by some constant. It is further noted that if nonlinear damping terms are ignored, i.e., dni D 0; i D 1; 2; 3, then the path and desired surge velocity must satisfy an additional condition such that B1 is strictly positive otherwise the filtered sway velocity dynamics will be unstable. This will result in an unstable closed loop system. To investigate stability of the .xe ; ye ; xe ; ye /-dynamics, we take the Lyapunov function q 2  2 ye 1  xe V3 D 1 C xe2 C ye2  1 C ; C 2 1

2 whose derivative along the solutions of the first four equations of (8.113) satisfies .k1  3"07 /xe2 C .k2  3"07 /ye2 u2e C ve2 C 8u2d N e2 VP3   C C
2 4"07 2 min .MN 1 D/.tt N 0/ e "07 u2 C ve2 C 8u2d N e2 2 min .MN 1 D/.tt N 0/  e C e ; 4"07 "07

(8.125)

where "07 is a positive constant, 2 is a class-K function of k0 .t0 /k. We have picked this constant such that "07 < min.k1 ; k2 /=3. Since we have proved that . N e .t /; N e .t /; ue .t /; ve .t /; vd .t // is bounded and ud is given by (8.96), the second inequality of (8.125) implies that .xe .t /; ye .t /; xe .t /; ye .t // is bounded. Hence integrating both sides of the first inequality of (8.125) and applying Barbalat’s lemma yield lim t!1 .xe .t /; ye .t // D 0. It now ˇfollows from (8.97) and the second equation of (8.92) that j e .t /j conˇ verges to ˇarctan..'2 C ye C vd /=ud 0 /ˇ < 0:5, since we have already proven that lim t!1 .xe .t /; ye .t // D 0, vd .t / is bounded and ud 0 .t / is larger than some positive constant by assumption, see Assumption 8.2. Proof that the ship’s total linear velocity is tangential to the path follows readily, since e .t / converges to

200

8 Path-tracking Control of Underactuated Ships

arctan..'2 C ye C vd /=ud 0 /, and the proven fact that lim t!1 .v.t /  vd .t // D 0. Finally, we note that all constants "0i ; 1  i  7; Aj ; j D 1; 2; 3, K1 , and K2 are only used in the proof. They are not needed in controller implementation.

8.2.6 Discussion 8.2.6.1 Adding Integrator Approach Versus Adaptive Approach We have used the projection integral actions instead of an adaptive approach to compensate for the disturbances. The trade-off is that the integral action approach does not give deep insight into the ship dynamics with disturbances. However, it results in a simple controller that is suitable for practical implementation. An adaptive approach would be extremely complicated (even when nonlinear damping terms are ignored) if the controller design follows the methodology proposed in this chapter. In an adaptive approach, the disturbances, E , can be considered directly in the ship dynamics as: N vP D CN .v/v  Dv N C  C E ; M (8.126) where the nonlinear damping terms are ignored. Using the same coordinate transformation (8.73) with P./ given in (8.80) gives N 1 . C E /: N 1 DP N 1 ./X C P./M XP D P./M

(8.127)

Because of the unknown E , we would design a dynamic observer for X by defining XQ D X  XO  W E ;

(8.128)

where W 2 R33 is a matrix to be determined and XO is an estimate of X . Differentiating both sides of (8.128) and choosing

results in

P N 1 DP N 1 ./XO C P./M N 1 ; XO D P./M 1 1 N N 1 P N W D P./M DP ./W C P./M

(8.129)

N 1 ./XQ : N 1 DQ XPQ D Q./M

(8.130)

From (8.129) and the first equation of (8.71), a controller can be designed but this will be very complicated since some elements of P./ depend on .x; y/ and W must be generated by the second equation of (8.129).

8.2.6.2 Dealing with Parameter Uncertainties In the control design, it was assumed that all the system parameters are known. The ship’s mass and added mass and linear damping coefficients are determined quite

8.2 Output Feedback

201

accurately by using semi-empirical methods or hydrodynamic programs (such as MARINTEK). However, it is difficult to obtain nonlinear damping coefficients accurately. The purpose of this section is to discuss how the proposed observer and controller can be modified so that the inaccuracies of the nonlinear damping coefficients can be taken care of. Dealing with inaccuracies of the linear damping coefficients can be carried out similarly if these coefficients are not known accurately. The idea is to replace the real coefficients dn i ; i D 1; 2; 3 in the nonlinear damping matrix DN n .v/, see (8.72), by their estimates. These estimates will be used in the observer (8.82) instead of the real nonlinear damping coefficients. The estimates of the nonlinear damping coefficients are updated in such a way that the modified observer guarantees that the observer error vector XQ .t / globally asymptotically converges to zero. Then the proposed control design can be simply modified by replacing all the nonlinear damping coefficients dn i ; i D 1; 2; 3 by their estimates to take care of inaccuracies in the nonlinear damping coefficients. Observer Design Modifications Letting dOn i ; i D 1; 2; 3 be estimates of dn i and defining the estimate errors as dQn i D dn i  dOn i , we can write (8.81) as N 1   N 1 .DN C DON n .P 1 ./X //P 1 ./X C P./M XP D P./M N 1 DQN .P 1 ./X //P 1 ./X ; P./M (8.131) n

where 2

6 DON n .v/ D 4

dOn1 juj 0 0

0

0

2

3

7 6 dOn2 jvj 0 5 ; DNQ n .v/ D 4 0 dOn3 jrj

dQn1 juj 0 0

0

0

3

7 dQn2 jvj 0 5 ; 0 dQn3 jrj

with (8.73) having been used for a short notation. From (8.131), we design an observer to estimate X as P N 1 .DN C DON n .P 1 ./XO //P 1 ./XO C P./M N 1 : XO D P./M

(8.132)

From (8.131) and (8.132), we have the observer error dynamics N 1 ./XQ  P./M N 1 DP N 1  XPQ DP./M ŒDON .P 1 ./X /P 1 ./X DON .P 1 ./XO //P 1 ./XO   n

n

N 1 DQN n .P 1 ./X //P 1 ./X P./M N 1 ./XQ  Q./M N 1  N 1 DP DP./M Q (8.133) ŒDN n .P 1 ./X /P 1 ./X  DN n .P 1 ./XO //P 1 ./XO   ˚ ;

where XQ D X  XO and we have defined

202

8 Path-tracking Control of Underactuated Ships

3 2 3 dQn1 juj O uO 0 0 7 6 7 N 1 6 ˚ D P./M O vO 0 5 ; Q D 4 dQn2 5 : 4 0 jvj 0 0 jrj O rO dQn3 2

It is noted that we have used (8.84). To determine an update law for dOn i ; i D 1; 2; 3, we take the following Lyapunov function 1 1 L0 D XQ T XQ C Q T  2 2

1

Q ;

(8.134)

where  is a positive definite matrix. Using Lemma 2.2, the derivative of L0 along the solutions of (8.133) satisfies N 1 ./XQ  XQ T P./M N 1 DP N 1  LP 0 D XQ T P./M ŒDN n .P 1 ./X /P 1 ./X  DN n .P 1 ./XO //P 1 ./XO   Q XQ T ˚ ;

N 1 D/jj N XQ jj2  XQ T ˚ Q  PO T   min .M where

1

Q ;

(8.135)

T  O D dOn1 dOn2 dOn3 ;

N 1 D/ N > 0 is the minimum eigenvalue of M N 1 D. N From (8.135), choosand min .M ing an update law for O as P (8.136) O D  ˚ T .
/XQ results in N 1 D/jj N XQ jj2 ; P 0  min .M L

(8.137)

which means that lim t!10 XQ .t / D 0. Indeed, we can use the projection algorithm instead of (8.136). Now it is important to note that the update law (8.136) cannot be used since XQ contains the unknown vector X . We need to get around this problem. Substituting XQ D X  XO into the right-hand side of (8.136) and integrating both sides gives Zt O / D .t O 0/   ˚ T .X . /  XO . //d : (8.138) .t t0

On the other hand, from the first equation of (8.71) and (8.73), we have P X D .J . /P.//1 :

(8.139)

Substituting (8.139) into (8.138) gives O / D .t O 0/ C  .t

Zt

t0

˚ XO . /d   T

Z.t/

.t0 /

˚ T .J . /P. //1 d ;

(8.140)

8.2 Output Feedback

203

where we have abused a notation of J . / as J ./. It is seen that (8.140) is a linear Volterra equation, which can be numerically solved by using a number of methods available in the literature, see [119]. Noticing that the right-hand side of (8.140) conO / instead of (8.136). We tains only known terms, we can use (8.140) to calculate .t summarize this section as follows: The observer (8.82) is replaced by the adaptive observer consisting of (8.132) and (8.140) for the case where the nonlinear damping coefficients are not known accurately. Control Design Modifications The modified observer, see (8.132) and (8.140), guarantees that lim t!10 XQ .t / D 0. The control design is the same as in Section 8.2.4 but the nonlinear damping coefficients dn i ; i D 1; 2; 3 are replaced by their estimates dOn i ; i D 1; 2; 3. The result with the modified observer and controller as discussed above is almost the same as the one stated in Theorem 8.3. The only difference is that in the case where the nonlinear damping coefficients are known accurately, the observer error XQ .t / globally exponentially converges to zero but only globally asymptotically converges to zero in the case where the nonlinear damping coefficients are not known accurately due to the adaptation. Proof of the result under the modified observer and controller follows the same lines as that of Theorem 8.3, and is therefore excluded here.

8.2.6.3 Dealing with Actuator Saturation In Section 8.2.4, the controller was designed without imposing any limitations on the magnitude and rate of the ship actuators. Actuator saturation restricts the reference path, deteriorates the overall path-tracking performance, and can even destabilize the closed loop system. These problems are difficult to deal with but it is important that they are taken care of. In many cases, global stabilization and/or tracking results cannot be achieved when actuators are subject to saturation. For example, consider the scalar system xP D x 2 C u with x the state and u the control. For this system, no bounded control u can globally exponentially/asymptotically stabilize the system at the origin. In fact, any bounded control u can result in a finite escape time. On the other hand, only restrictive tracking can be achieved for many systems such as a simple system consisting of an integrator chain, see [120]. In this section, we will discuss how to use the controller proposed in Section 8.2.4 when the actuator saturation presents. We will focus on the magnitude saturation. A discussion on the rate saturation can be carried out similarly. The idea is to set ju j  umax and jr j  rmax where umax and rmax are the maximum values that the actuators can supply, then to find restrictions on the reference path, control gains, and initial conditions such that ju j  umax and jr j  rmax hold. From the control design in Subsection 8.2.4, we calculate the upper bounds of ˛u , ˛ e , and ˛r as

204

8 Path-tracking Control of Underactuated Ships

q 2 ; ˛ N ˛N u D k1 C N xe C uN 2d 0 C N xe

e

! k2 C N ye C vN d D arctan q ; (8.141) 2 uN 2d 0 C N xe

˛N r D b 1 N N e C rNd C u1 N d C uN d / C 1 .k3 C  d 0 ..k2 C 1/.v 2 N N N ˛N u C uN d C vP d C P ye / C u .uP d 0 uN d 0 C NP xe N xe //; d0

where uN d D

vN d D max

s

B3 ; B1

q s

2 C .k C  uN 2d 0 C N xe N ye C vN d /2 ; 2

! B2 m22 ; dn2

.m11 ˛N u C dN23 /˛N r C d22 vN d C dn2 vN d2 C k2 u1 d0 ; m22 0 00 00 0 jxd .s/yd .s/  xd .s/yd .s/j rNd D uN d ; q 0 0 .xd2 .s/ C yd2 .s//3

vNP d D

B1 D

B2 D

d22  2B0 ; m22

k2 .2 C k1 C k2 / C NP ye m11 .k1 C uN d 0 / C jd23 j k3 C N N e C C b 1 m22 ud 0 ! uN d 0NPud 0 C N xeNPxe k2 ; C 2 ud 0 ud 0

1 2 2 C 3k22 C 3N ye /; B0 .uN 2d 0 C N xe 2 ˇ ˇ 00 00 0 ˇ ˇ 0 .s/y .s/  x .s/y .s/ ˇ ˇx d d d m11 .k1 C uN d 0 / C dN23 d ; B0 D q 0 0 b 1 m22 .xd2 .s/ C yd2 .s//3

B3 D

(8.142)

and the notations N and  denote the upper and lower bounds of j  j, respectively. By substituting uO D ue C ˛u ; vO D ve C vd , and rO D re C ˛r into (8.112), we can calculate the upper bounds of u and r as ju j  #1 u2e C #2 ve2 C #3 re2 C #0 ; jr j  1 u2e C 2 ve2 C 3 re2 C 0 ; where #1 D 0:5.m11 .k1 C c1 / C dn1 /;

#2 D m22 C 0:5k2 C m11 m1 22 ;

(8.143)

8.2 Output Feedback

205

#3 D m22 C 0:5.k1 C k2 C m11 m1 22 /; 2 #0 D 0:5c1 m11 C m22 .0:5.˛N r C vN d2 C vN d ˛N r / C d11 ˛N u C 1:5dn1 ˛N u2 C m11  ..k1 C k2 /.1 C uN d C ˛N r / C 2NPxe C uNP d 0 C k2 u1 C 0:5m11 m1 ˛N 2 /; 1 D

2 D

r

22 d0 1 1 1 jm11  m22 j C b 1 ud 0 .k1 C k2 /.uN d C vN d / C m11 m22 ; 1 N d C vN d / C 0:5dN32 C jm11  m22 j C b 1 1 ud 0 m33 .k1 C k2 /.u 1 N m11 m33 m1 22 C 0:5m33 m22 d23 ;

1 3 D 0:5c2 m33 C 0:5dn3 C 21 N d C vN d / C 0:5b 1 1 ud 0 m33 .k1 C k2 /.u 1 k3 ; 2 2 0 D 0:5c2 m33 C 1:5dn3 ˛N C 0:5dN32 C dN23 vN d C jm11  m22 j.0:5vN C r

d

1 0:5˛N u2 C ˛N u vN d / C m33 .b 1 N d C vN d /.2 C ˛N u C 2uN d C 1 ud 0 .k1 C k2 /.u

2 2 2˛N r C vN d / C b 1 1 max.j.xd .s/yd .s/  xd .s/yd .s//.xd .s/ C yd .s//  0

00

00

0

0

0

00

0

0

000

00

0

0

00

2.xd .s/yd .s/  xd .s/yd .s//.xd .s/xd .s/  yd .s/yd .s///j 

0 0 1 N 2 .xd2 .s/ C yd2 .s//2 / C uN d C1 N r / C b 1 P xe C 1 .ud 0 . 1 k3 .rNd C 0:5 C ˛ 2 1 2 1 N (8.144) PN ye / C NP N C uNP C vNP d / C 0:5m11 m ˛N C 0:5m d23 /:

e

d0

22

u

22

On the other hand, integrating both sides of the second inequality of (8.122) results in (8.145) u2e .t / C ve2 .t / C re2 .t /  ˚.t0 /; 8t  t0  0; where q ˚.t0 / D 2 1 C N e2 .t0 /  1 C 0:5.u2e .t0 / C ve2 .t0 / C re2 .t0 // C  1

3  2N e .t0 / C ı11 A4 kXQ .t0 /k C ı11 A5 kXQ .t0 /k2 ; 2

(8.146)

with A4 ; A5 and ı1 given in (8.123). Now using (8.143) and (8.145), and setting ju j  uMax and jr j  rMax , we deduce that   umax  #0 rmax  0 ; (8.147) ˚.t0 /  min ; max.#1 ; #2 ; #3 / max.1 ; 2 ; 3 / which implies that under the actuator magnitude saturation, restrictions must be placed on the type of reference paths (both desired velocity of the vessel on the path and the path curvature cannot be arbitrarily large), control gains, and initial conditions such that (8.147) holds, i.e., the proposed controller under the actuator magnitude saturation is not only local but also restricts the commanded paths.

8.2.7 Experimental Results This section describes the experimental set-up and tested results performed on a model ship in the Swan River, Western Australia. The model ship has a length of

206

8 Path-tracking Control of Underactuated Ships

Figure 8.7 Equipment for experimental tests

1.2 m and a mass of 17.5 kg. The model ship is equipped with one differential global positioning system (DGPS) SF-2050G from NavCom, which can provide an accuracy of 25 cm, to obtain longitude, latitude and altitude, one compass TCM2-50 from PNI to measure yaw angle, two direct current motors Torpedo 850 with two remote speed controllers driving two propellers to provide surge force and yaw moment, three batteries to supply power for motors, DGPS and wireless communication equipment, and two sets of wireless transceivers to transmit and receive signals between the model ship and the host computer, see Figure 8.7. The parameters of the ship model are calculated by VERES: m11 D 25:8; m22 D 33:8; m33 D 2:76; m23 D m32 D 6:2; D11 D 12 C 2:5juj; D22 D 17 C 4:5jvj; D33 D 0:5 C 0:1jrj; D23 D 0:2; D32 D 0:5:

The global coordinates (longitude, latitude, altitude) are transformed to the “river coordinates” based on a three-point algorithm as follows, see Figure 8.8: 1. The DGPS is first used to measure the global coordinates of two fixed points M1 .h1 ; l1 ; 1 / and M2 .h2 ; l2 ; 2 / with hi ; li ; i ; i D 1; 2; being the altitude, longitude and latitude of the point Mi .hi ; li ; i /. These points are chosen along the river bank. 2. We calculate the distance, M1 M2 between these two fixed points by the following formula, see [11]:

8.2 Output Feedback

where

207

p M1 M2 D .xe1  xe2 /2 C .ye1  ye2 /2 C .ze1  ze2 /2 ;

(8.148)

xei D .Ni C hi / cos.i / cos.li /; yei D .Ni C hi / cos.i / sin.li /;   2 rp zei D N C h i i sin.i /; re2

re2 Ni D q ; i D 1; 2; re2 cos2 .i / C rp2 sin2 .i /

rp D 6356752;

re D 6378137:

(8.149)

In the experimental set-up, we adjusted the fixed points M1 .h1 ; l1 ; 1 / and M2 .h2 ; l2 ; 2 / such that M1 M2 =100 m. 3. The river coordinate system, OE XE YE , is formed such that the OE XE -axis coincides with the line connecting between M1 .h1 ; l1 ; 1 / and M2 .h2 ; l2 ; 2 /, and that the origin O coincides with M1 .h1 ; l1 ; 1 /. The OY axis is perpendicular to the OX axis. 4. From this coordinate system, the position coordinates .x; y/ of the point, say Mj .hj ; lj ; j /, of interest on the model ship are obtained by calculating the distances, Mj M1 and Mj M2 , from this point to the points M1 .h1 ; l1 ; 1 / and M2 .h2 ; l2 ; 2 /.

Model ship

YE

x

y

Ai (hi , li , Pi )

Swan river River bank

A1 (h1 , l1 , P1 ) OE

x Wireless communication

A2 (h2 , l2 , P2 )

x

x XE

Work station

Figure 8.8 Experimental set-up

The position coordinates .x; y/ and the yaw angle are sent to the host computer by the transceivers. The control signal is calculated in the host computer and is sent

208

8 Path-tracking Control of Underactuated Ships

back to the remote speed controllers by a Futabar transmitter connecting to the host computer via a data acquisition card. A program is written in LabWindowsTM /CVI, a product of National InstrumentsTM , to implement the control algorithm developed in this chapter. The reference path ˝ is chosen as follows: For the first 90 seconds, xd D s; yd D 60, and xd D 20 sin.0:1s/ C 60; yd D 20 cos.0:1s/ C 40 for the remainder of testing time. This choice implies that the reference path is a straight line for the first 90 seconds and is a circle centered at (60 m, 40 m) with a radius of 20 m. The desired forward speed is chosen as ud 0 D 0:5 .m/s/. Based on conditions specified in Theorem 8.3, the design constants and initial conditions of the observer and filters are chosen as c1 D c2 D 4, k1 D k2 D k3 D 1:5, i D 0:1, i D 0:1, i D 1; 2; 3, vd .0/ D 0, xe .0/ D 0, ye .0/ D 0,  N e .0/ D 0, XO .0/ D .0; 0; 0/T , N xe D N ye D 10, and N N e D 6:5:

Figure 8.9 A screen shot of the user interface with integral actions

The initial conditions of the model ship are x D 2, y D 40, D 0:5, and u D v D r D 0. Experimental results are given in Figures 8.9 and 8.10. Figure 8.9 is a screen shot of the user interface of the developed program. The user interface displays general information of the ship model such as global coordinates, forward speed, and heading angle. The history of the ship position and the reference path are also displayed in the user interface panel. Figure 8.10 plots the position and orientation tracking errors. It can be seen that these errors are actually not zero as proved in Theorem 8.3. These nonzero errors are due to inaccuracy of the DGPS and compass. A close look at this figure shows that the position and orientation errors are in the range of the DGPS and compass accuracy. To illustrate the role

4 2

10 0

−10

0

−2 0

Orientation error: ψe (rad)

209

Position error: ye (m)

Position error: xe (m)

8.2 Output Feedback

−20

100 Time [s]

200

−30 0

100 Time [s]

200

2 1 0

−1 0

50

100 Time [s]

150

200

Figure 8.10 Position and orientation errors with integral actions

of integral actions, we also tested the control algorithm without integral actions by setting xe D ye D  N e D 0. The tested results without integral actions are given in Figures 8.11 and 8.12. It can be seen that the integral actions play an important role in experimental tests. Poor performance in the test without integral actions can be understood as follows: The disturbances result in velocities, for which the controller does not compensate since the integral actions are switched off. The velocities induced by disturbances result in poor performance and even destroy the stability of the path-tracking error dynamics when these disturbing velocities are large enough (larger than min.ki /; i D 1; 2; 3 in magnitude). Theoretically, we can see this phenomenon by looking at the fourth, fifth, and sixth equations of the closed loop system (8.113):

210

8 Path-tracking Control of Underactuated Ships

xe  xe C ue  ud ..cos. N e /  1/ cos.˛ e / 
sin. N e / sin.˛ e // C .rO C r/y Q e C uQ C uE ; ye yPe D k2  ye C ve C ud .sin. N e / cos.˛ e / C
.cos. N e /  1/ sin.˛ e //  .rO C r/x Q e C vQ C vE ; Ne k2  u d 0 x e y e PN D k q .ue C u/ Q C   N e C b1 .re C r/ Q  e 3 u2d
3 1 C N e2 xP e D k1

k2 ud 0 .1 C xe2 / .ve C v/ Q C rE ; u2d
3

(8.150)

where uE , vE , and rE are velocities induced by the disturbances E . From (8.150), we can see that when uE , vE and rE are larger than k1 , k2 and k3 , respectively, in magnitude, the xe -, ye -, and N e -dynamics will be unstable, respectively. It is noted that xe , ye , and  N e do not compensate for uE , vE , and rE since they are switched off. The constants k1 and k2 cannot be arbitrarily large since they have to satisfy conditions (8.103) and (8.111).

Figure 8.11 A screen shot of the user interface without integral actions

211

Position error: ye (m)

5

e

Position error: x (m)

8.3 Conclusions

0

10 0

−10

−5

Orientation error: ψe (rad)

−10 0

−20

100 Time [s]

200

−30 0

100 Time [s]

200

2 1 0

−1 −2 0

50

100 Time [s]

150

200

Figure 8.12 Position and orientation errors without integral actions

8.3 Conclusions This chapter provides global path-tracking controllers for underactuated ships. Both state feedback and output feedback cases have been considered. Of particular note is the use of a “nontraditional” adaptive observer and a simple modification of the proposed controller to deal with inaccuracies of the nonlinear damping coefficients. Moreover, the proposed controllers in this chapter are much simpler than those in the preceding chapters. The simplicity of the proposed controllers in this chapter makes them more suitable for use in practice than those proposed in the previous chapters. The work presented in this chapter is based on [121–125].

Chapter 9

Way-point Tracking Control of Underactuated Ships

This chapter presents state feedback and output feedback controllers that force underactuated ships to globally ultimately track a straight line under environmental disturbances induced by waves, wind, and ocean currents. When there are no environmental disturbances, the controllers are able to drive the heading angle and cross-tracking error to zero asymptotically. Based on the backstepping technique and several technical lemmas introduced for a nonlinear system with nonvanishing disturbances, a full state feedback controller is first designed. An output feedback controller is then developed by using a nonlinear observer, which globally exponentially estimates the unmeasured sway and yaw velocities from the measured sway displacement and the measured yaw angle.

9.1 Control Objective In addition to the assumptions made in Section 3.4.1.1, we assume that the surge velocity is controlled by the main propulsion control system. As such, the resulting mathematical model of the underactuated ship moving in sway and yaw is rewritten as yP D u sin. / C cos. /v; P D r; X dvi d22 1 m11 u r v wv .t /; vP D  jvji1 v C m22 m22 m22 m22

(9.1)

i2

rP D

X dri d33 1 1 .m11  m22 /u v r r C wr .t /; jrji1 r C m33 m33 m33 m33 m33 i2

where y, v, , r, and u are sway displacement, sway velocity, yaw angle, yaw velocity, and forward speed controlled by the main thruster control system, respectively. Without loss of generality, we assume that the forward speed u is positive and if 213

214

9 Way-point Tracking Control of Underactuated Ships

time-varying, has a bounded derivative u.t P /, i.e., 0 < umin  u.t /  umax < 1 and P /j  M < 1; 8t  0. The positive constant terms mjj ; 1  j  3 denote the ju.t ship’s inertia including added mass. The positive constant terms d22 ; d33 ; dvi and dri ; i  2 represent the hydrodynamic damping in sway and yaw. The bounded time-varying terms, wv .t / and wr .t /, are the environmental disturbance moments induced by wave, wind, and ocean current with an assumption that jwv .t /j  wv max < 1 and jwr .t /j  wr max < 1. In this chapter, we study two control objectives. The first is full state feedback. In this case, we assume that all states y; v; , and r are available for feedback. In the design of an output feedback controller, only sway and yaw displacements are measurable. For both full state and output feedback cases, we design a control law, r , that forces the ship to track a linear course with ultimate boundedness, i.e. the tracking errors are globally ultimately bounded. When there are no environmental disturbances, the sway displacement and velocity, y and v, yaw angle and velocity, and r, asymptotically converge to zero.

9.2 Full-state Feedback 9.2.1 Control Design We define the following coordinate transformation z1 D

C arcsin p

ky 1 C .ky/2

!

;

(9.2)

where k is a positive constant to be selected later. Note that the convergence of z1 and y to zero implies that of . Upon application of the coordinate transformation (9.2), the ship dynamics (9.1) are rewritten as v u .sin.z1 /  .cos.z1 /  1/ky/ Cp C C p 2 1 C .ky/ 1 C .ky/2 v ..cos.z1 /  1/ C ky sin.z1 // ; p 1 C .ky/2 X dvi m11 u d22 1 vP D  r v wv .t /; jvji1 v C m22 m22 m22 m22

yP D  p

kuy

1 C .ky/2

i2

2

zP 1 D r 

k uy

.1 C .ky/2 /3=2

C

kv

.1 C .ky/2 /3=2

kv ..cos.z1 /  1/ C ky sin.z1 //

C

ku .sin.z1 /  .cos.z1 /  1/ky/ .1 C .ky/2 /3=2

; .1 C .ky/2 /3=2 X dri .m11  m22 /u d33 1 1 rP D v r r C wr .t /: jrji1 r C m33 m33 m33 m33 m33 i2

C

(9.3)

9.2 Full-state Feedback

215

Therefore the problem of stabilizing (9.1) at the origin becomes that of stabilizing (9.3) at the origin. The structure of the model (9.3) suggests that we design the control r in two stages by applying the popular backstepping technique. At the first step, we design an intermediate control rd for r and at the second step the actual control r will be designed to eliminate the error between rd and r:

Step 1 Define z2 D r  rd ;

(9.4)

where rd is an intermediate control designed as rd D k1 z1 C

k 2 uy

kv

 .1 C .ky/2 /3=2 ku .sin.z1 /  .cos.z1 /  1/ky/ kv ..cos.z1 /  1/ C ky sin.z1 //  ; .1 C .ky/2 /3=2 .1 C .ky/2 /3=2 .1 C .ky/2 /3=2



(9.5)

where k1 is a positive constant to be selected later.

Step 2 With (9.5), the time derivative of (9.4) along the solutions of the last equation of (9.3) is zP 2 D

X dri d33 1 1 .m11  m22 /u v r r C wr .t /  jrji1 r C m33 m33 m33 m33 m33 i2  @rd kuy @rd @rd v .k1 z1 C z2 /  Cp C p uP  @u @z1 @y 1 C .ky/2 1 C .ky/2  u .sin.z1 /  .cos.z1 /  1/ky/ v ..cos.z1 /  1/ C ky sin.z1 // C  p p 1 C .ky/2 1 C .ky/2 0 1 X dvi @rd @ m11 u d22 1 i1  (9.6) r v wv .t /A ; jvj v C @v m22 m22 m22 m22 i2

where

k2y @rd k .sin.z1 /  .cos.z1 /  1/ky/ ; D  3=2 2 @u .1 C .ky/ / .1 C .ky/2 /3=2 ku .cos.z1 / C ky sin.z1 // kv . sin.z1 / C ky cos.z1 // @rd D k1   ; @z1 .1 C .ky/2 /3=2 .1 C .ky/2 /3=2

216

9 Way-point Tracking Control of Underactuated Ships

@rd 3k 3 y .kuy  u .sin.z1 /  y.cos.z1 /  1///  D @y .1 C .ky/2 /5=2 v ..cos.z1 /  1/ C y sin.z1 //

C

k 2 .u cos.z1 /  v sin.z1 //

.1 C .ky/2 /5=2 .1 C .ky/2 /3=2 k @rd .cos.z1 / C sin.z1 /ky/ : D @v .1 C .ky/2 /3=2

; (9.7)

We now choose the actual control without canceling the useful damping terms as  X dri .m11  m22 /u d33 vC rd C r D m33 z1  k2 z2  jrji1 rd C m33 m33 m33 i2  @rd kuy @rd @rd v p uP C .k1 z1 C z2 / C Cp C 2 @u @z1 @y 1 C .ky/ 1 C .ky/2  u .sin.z1 /  .cos.z1 /  1/ky/ v ..cos.z1 /  1/ C ky sin.z1 // C C p p 1 C .ky/2 1 C .ky/2   X dvi @rd m11 u d22 i1  r v (9.8) jvj v  @v m22 m22 m22 i2     1 z2 1 @rd @rd z2 wr max tanh wv max  ; tanh m33 1 m22 @v @v 2 where k2 , 1 , and 2 are positive constants to be chosen later. Substituting (9.4), (9.5), and (9.8) into (9.3) results in the following closed loop system: yP D  p

kuy 1 C .ky/2

v u .sin.z1 /  .cos.z1 /  1/ky/ Cp C C p 2 1 C .ky/ 1 C .ky/2

v ..cos.z1 /  1/ C ky sin.z1 // ; p 1 C .ky/2 X dvi m11 u k 2 uy  kv d22 vP D   v jvji1 v  m22 m22 m22 .1 C .ky/2 /3=2 i2 ku m11 u  .sin.z1 /  .cos.z1 /  1/ky/  k1 z1 C z2  m22 .1 C .ky/2 /3=2  kv 1 ..cos.z /  1/ C ky sin.z // wv .t /; C 1 1 3=2 2 m 22 .1 C .ky/ /

zP 1 D k1 z1 C z2 ;

X dri d33 1 .wr .t /wr max  z2  jrji1 z2 C m33 m33 m33 i2      @rd @rd z2 1 @rd z2 wv .t /  wv max tanh C  : (9.9) tanh 1 m22 @v @v @v 2

zP 2 D z1  k2 z2 

9.2 Full-state Feedback

217

9.2.2 Stability Analysis The following two lemmas will be used extensively in stability analysis. Lemma 9.1. Consider the following nonlinear system: xP D f .t; x/ C g.t; x; .t //;

(9.10)

where x 2 Rn ; .t / 2 Rm , f .t; x/ is piecewise continuous in t and locally Lipschitz in x. If there exist positive constants ci ; 1  i  4, j ; 1  j  2, 0 , "0 , 0 , c0 , and a class-K function ˛0 such that the following conditions are satisfied: C1. There exists a proper function V .t; x/ satisfying: c kxk2  V .t; x/  c2 kxk2 ;

1

@V



@x .t; x/  c3 kxk ; @V @V C f .t; x/  c4 kxk2 C c0 : @t @x C2. The vector function g.t; x; .t // satisfies: kg.t; x; .t //k  .1 C 2 kxk/ k.t/k : C3. .t/ globally exponentially converges to a ball centered at the origin: k.t/k  ˛0 .k.t0 /k/ e 0 .tt0 / C "0 ; 8t  t0  0: C4. The following gain condition is satisfied: c4  2 c3 "0 

1 c3 "0 > 0: 40

Then the solution x.t / of (9.10) globally exponentially converges to a ball centered at the origin, i.e., kx.t /k  ˛ .k.x.t0 /; .t0 //k/ e .tt0 / C "; 8t  t0  0; p where " D a4 =c1 a1 and

if a1 D 0 then v u a2 .s/ u 0

  te ˛.s/ D c2 s 2 C a3 .s/ C a11 a2 .s/a4  c1  D 0:5.a1  d /I

(9.11)

218

9 Way-point Tracking Control of Underactuated Ships

if a1 ¤ 0 then v u a2 .s/   u 0 a1 a3 .s/ C a2 .s/a4 te 2 ˛.s/ D c2 s C c1 a1 ja1  0 j  D 0:5 min .a1 ; ja1  0 j/ I

with   1 1 c3 "0 c4  2 c3 "0  ; c2 40 c3 a2 .s/ D .1 C 2 /˛0 .s/ ; c1 1 c 3 ˛0 .s/ ; a3 .s/ D 4 a4 D c0 C 1 c3 "0 0 ; a1 D

0 < d < a1 ;

  .t  t0 /e d.tt0 / ;

8t  t0  0; s  0:

When c0 D 0 and "0 D 0, we have " D 0 and the system (9.10) is globally Kexponentially stable. Note that a finite value of the constant  exists for an arbitrarily small positive d . Proof. From conditions C1, C2, and C3, we have @V @V @V C f .t; x/ C g.t; x; .t // VP D @t @x @x  .c4  2 c3 "0  1 c3 "0 =40 / kxk2 C c3 kxk .1 C 2 kxk/˛0 .k.t0 /k/ e 0 .tt0 / C 1 c3 "0 0 C c0 :

Upon application of the completing square, (9.12) can be rewritten as   VP   a1  a2 e 0 .tt0 / V C a3 e 0 .tt0 / C a4 :

(9.12)

(9.13)

Solving the above differential inequality results in V .t /  V .t0 /e

a2 0

e

a1 .tt0 /

  a Zt 2 a4 a2 a4 0 a1 tC0 t0 e .a1 0 / d C ; C a3 C e e a1 a1 t0

(9.14) which yields (9.11) readily.  Lemma 9.2. Consider the following nonlinear system: xP D f .t; x/ C g.t; x; .t //;

(9.15)

9.2 Full-state Feedback

219

where x1 2 Rn1 , x2 2 Rn2 , x D Œx1 x2 T 2 Rn1 Cn2 , .t/ 2 Rm , f .t; x/ is piecewise continuous in t and locally Lipschitz in x. If there exist positive constants c0 , c1 , c2 , c31 , c32 , i , 0  i  2, 0 , "0 , c0 , and a class-K function ˛0 such that the following conditions are satisfied. C1. There exists a proper function V .t; x/ such that: c1 kxk2  V .t; x/  c2 kxk2 ;

@V c32 kx2 k2 @V C c0 ; C f .t; x/  c31 kx1 k2  q @t @x 1 C c4 kx2 k2 0 1

2

@V

B 2 kx2 k C 2

A k.t/k :

@x g.t; x; .t //  @0 C 1 kx1 k C q 1 C c4 kx2 k2

C2. .t/ globally exponentially converges to a ball centered at the origin: k.t/k  ˛0 .k.t0 /k/ e 0 .tt0 / C "0 ; 8t  t0  0: C3. The following gain conditions are satisfied: c31  1 "0 > 0 and

c32  2 "0 > 0:

C4. x2 .t / is bounded: kx2 .t /k  $; where $is a nondecreasing function of k.x.t0 /; .t0 //k, then the solution x.t / of (9.15) globally asymptotically converges to a ball centered at the origin, i.e., kx.t /k  ˛ .k.x.t0 /; .t0 //k/ e .k.x.t0 /;.t0 //k/.tt0 / C ".s/; 8t  t0  0; (9.16) q where ".s/ D c1 aa14.s/ and if a1 .s/ D 0 then q 

 ˛.s/ D c11 e a2 .s/=0 c2 s 2 C a3 .s/ C a11 .s/a2 .s/a4   .s/ D 0:5.a1 .s/  d /I

if a1 .s/ ¤ 0 then s ˛.s/ D

c11 e a2 .s/=0

  a1 .s/a3 .s/ C a2 .s/a4 2 c2 s C a1 .s/ ja1 .s/  0 j

 .s/ D 0:5 min .a1 .s/; ja1 .s/  0 j/ I

220

9 Way-point Tracking Control of Underactuated Ships

with ! c32  2 "0 1 ; a1 .s/ D min c31  1 "0 ; p c2 1 C c4 $ 2 .s/ 1 a2 .s/ D max.1 ; 2 /˛0 .s/ ; c1 a3 .s/ D 0 ˛0 .s/ ; a4 D c0 C 0 "0 ; 0 < d < a1 .s/;   .t  t0 /e d.tt0 / ;

8t  t0  0; s  0:

When c0 D 0 and "0 D 0, we have " D 0 and the system (9.15) is GAS. Note that a finite value of the constant  exists for an arbitrarily small positive d . Proof. The proof of this lemma is similar to that of Lemma 9.1.  Remark 9.1. It is important to note that the rate  > 0 in (9.16) and a1 depend on the initial conditions. In addition, around the origin, both  and a1 are bounded below from zero. We first need to show that the closed loop system (9.9) is forward complete. It is straightforward to show that the derivative of the function V0 D z12 C z22 C v 2 C y 2 along the solutions of the closed loop system (9.9) satisfies VP0  a0 V0 C b0 where a0 and b0 are nonnegative constants. The inequality VP0  a0 V0 C b0 implies that the closed loop system (9.9) is forward complete. We now apply Lemmas 9.1 and 9.2 to analyze the closed loop system (9.9). We view .z1 ; z2 / as .t/, v as x in Lemma 9.1, and .v; y/ as x in Lemma 9.2. Hence it is necessary to verify all the conditions of Lemmas 9.1 and 9.2.

.z1 ; z2 /-subsystem We take the following quadratic function: 1 V1 D .z12 C z22 /; 2

(9.17)

whose time derivative along the solutions of the last two equations of (9.9) satisfies z2 d33 2 X dri .wr .t /wr max  z2  VP1 D k1 z12  k2 z22  jrji1 z22 C m33 m33 m33 i2      @rd z2 z2 @rd @rd z2 tanh wv .t /  wv max tanh C  1 m22 @v @v @v 2

9.2 Full-state Feedback

221

wr max   k1 z12  k2 z22 C m33 ˇ ˇ      wv max ˇˇ @rd ˇˇ @rd z2 @rd z2  C tanh z z jz2 j  z2 tanh ˇ @v 2 ˇ @v 2 1 m22 @v 2   1 1  k1 z12  k2 z22 C 0:2785 (9.18) wr max 1 C wv max 2 ; m33 m22 where we have used jxj  x tanh.x=/  0:2785; 8x 2 R and  > 0. From (9.17) and (9.18), it can be shown that kz.t /k  kz.t0 /k e 0 .tt0 / C "0 8t  t0  0 ;

(9.19)

where z D Œz1 z2 T and 0 D min.k1 ; k2 /; s 0:2785 .wr max 1 =m33 C wv max 2 =m22 / "0 D : 0

(9.20)

Therefore the .z1 ; z2 /-subsystem is globally ultimately stable at the origin. Furthermore, (9.19) implies that .t/ WD .z1 ; z2 /T globally exponentially converges to a ball centered at the origin. The radius of this ball can be made arbitrarily small by increasing k1 and k2 and/or reducing 1 and 2 .

Boundedness of v To prove that v is bounded, we consider the second equation of (9.9). In order to apply Lemma 9.1, define x D v; .t / D Œz1 z2 T and consider y as a function of time t , f ./ D 

X dvi d22 v jvji1 v  m22 m22 i2

kv k 2 uy m11 u  m22 .1 C .ky/2 /3=2 .1 C .ky/2 /3=2 m11 u .k1 z1 C z2  g./ D  m22

!

C

1 wv .t /; m22

k .u .sin.z1 /  .cos.z1 /  1/ky/ C v ..cos.z1 /  1/ C ky sin.z1 /// .1 C .ky/2 /3=2

This abuse of notation is introduced for simplicity and is possible because: ˇ ˇ ˇ ˇ ky 1 ˇ ˇ < 1; 8 y 2 R; 0  1; 0  ˇ ˇ .1 C .ky/2 /3=2 .1 C .ky/2 /3=2

(9.21) ! :

222

9 Way-point Tracking Control of Underactuated Ships

and we have shown that the closed loop system is forward complete. We now verify all of the conditions of Lemma 9.1. Verifying Condition C1. We take the function V2 D 0:5v 2 whose time derivative along the solutions of the differential equation vP D f .t; v/, see (9.21), satisfies d22 2 X dvi v m11 uv k 2 uy  kv VP2 D  C v  wv .t / jvji1 v 2  3=2 m22 m22 m22 .1 C .ky/2 / m22 i2   2 m11 ku2max C wv 1 d22 m11 kumax m11 ku2max 1 max    : v2 C  m22 m22 m22 m22 41 m22 (9.22) Hence, the condition C1 is satisfied with
 1 2 m11 ku2max C wv max ; c1 D c2 D 0:5; c3 D 1; 41 m22 m11 ku2max m11 kumax 1 d22 c4 D  1   ; m22 m22 m22 m22 c0 D

(9.23)

where 1 > 0 and k > 0 are chosen such that c4 > 0. Verifying Condition C2. It is directly shown from (9.21) that jg.t; v; z.t //j  .1 C 2 jvj/ kz.t /k ;

(9.24)

where 1 D

2km11 umax m11 umax .1 C k1 C 2kumax /; 2 D : m22 m22

(9.25)

Verifying Condition C3. This condition follows directly from (9.19). Verifying Condition C4. It can be shown from (9.20), (9.23), and (9.25) that we can find positive constant k such that the condition C4 is satisfied, i.e., c4  2 c3 "0 

1 c3 "0 > 0: 40

(9.26)

All of the conditions of Lemma 9.1 have been verified, hence the sway velocity is bounded and satisfies jv.t /j  ˛1 .k.v.t0 /; z.t0 //k/ e 1 .tt0 / C "1 ;

(9.27)

where "1 , 1 , and ˛1 are calculated as in Lemma 9.1, and the constants ci , 1  i  4, j , 1  j  2, 0 , "0 , 0 , and c0 are given in (9.20), (9.23), and (9.26).

9.2 Full-state Feedback

223

.v; y/-subsystem In this section, we will apply Lemma 9.2 to prove global ultimate boundedness of the .v; y/-subsystem. It can be seen that the first two equations of (9.9) are in the form of the system in Lemma 9.2 with x1 D v, x2 D y, .t/ D z.t /, and 2

3 P dvi d22 1 m11 u k 2 uy  kv i1 6 C v wv .t / 7 jvj v  6 m22 7 m22 .1 C .ky/2 /3=2 m22 i2 m22 6 7; f ./ D 6 7 5 4 kuy v p Cp 1 C .ky/2 1 C .ky/2 2 !3 m11 u sin.z1 /.u C kvy/ C .cos.z1 /  1/.v  kuy/ k C k1 z1  z2 7 6 6 m22 7 .1 C .ky/2 /3=2 6 7: g./ D 6 7 4 u .sin.z1 /  .cos.z1 /  1/ky/ v ..cos.z1 /  1/ C sin.z1 /ky/ 5 C p p 1 C .ky/2 1 C .ky/2 (9.28) We now need to verify all of the conditions of Lemma 9.2. Verifying Condition C1. To verify this condition, we take the function V3 D 0:5.v 2 C y 2 /. It can be directly shown that this function satisfies condition C1 with jv.t /j  ˛1 .k.v.t0 /; z.t0 //k/ e 1 .tt0 / C "1 and 2 wv max ; c1 D c2 D 0:5; 43 m22 d22 m11 kumax m11 k 2 u2max 3   2  2  ; c31 D m22 m22 m22 m22   1 m11 k 2 u2max 1C ; c32 D kumin  42 m22 umax m11 umax .1 C k1 C 2kumax / C 0 D ; 44 m22 44 m11 umax 2km11 umax .2kumax C k1 C 1/ 4 C 1 D C 4 ; m22 m22 1 2 D C umax 4 C kumax C k.˛1 C "1 /; 44

c0 D

(9.29)

where k > 0 and 2 > 0 are chosen such that c31 > 0 and c32 > 0. Verifying Condition C2. This condition follows directly from (9.19). Verifying Condition C3. It can be shown that there exists a positive constant k such that the condition C3 satisfies, i.e.,

224

9 Way-point Tracking Control of Underactuated Ships

c31  1 "0 > 0; c32  2 "0 > 0:

(9.30)

Verifying Condition C4. From the boundedness of the sway velocity v.t / proven in the previous subsection and noting that "0 in (9.19) can be made arbitrarily small, it is shown that there exists a nondecreasing function $ of k..v.t0 /; y.t0 //; z.t0 //k such that jy.t /j  $ by applying Lemma 9.1 to the first equation of (9.9) with the Lyapunov function Vy D 0:5y 2 . All of the conditions of Lemma 9.2 have been verified. Therefore we have k.v.t /; y.t //k  ˛2 .k..v.t0 /; y.t0 //; z.t0 //k/ e 2 .tt0 / C "2 ; 8t  t0  0; (9.31) where "2 , 2 , and ˛2 are calculated as in Lemma 9.2, and all other constants given in (9.29). It can be seen that when there are no environmental disturbances, since "2 D 0, .v.t /; y.t // globally asymptotically converges to zero. We have thus proven the first main result of this chapter. Theorem 9.1. The full-state feedback control problem stated in Section 9.1 is solved by the control law (9.8) as long as the design constants k, k1 , and k2 are chosen such that (9.26) and (9.30) hold.

9.3 Output Feedback This section is devoted to the development of an output feedback controller to fulfill the output feedback control objective. A nonlinear observer is first designed so that it globally exponentially drives the observer error dynamics to a ball centered at the origin. When there are no environmental disturbances, the observer error dynamics are GES at the origin. A controller is then designed based on the approach in the preceding section and the proposed observer. Before designing an observer and output feedback controller, we impose the following assumption, see [12]. Assumption 9.1. For the ship model (9.1), the matrix 2

d22 m11 u 6  6 m22 m22 K2 D 6 4 .m  m /u d33 11 22  m33 m33

3 7 7 7 5

(9.32)

is Hurwitz. The above assumption implies that the ship (when the nonlinear damping terms P dvi P dri i1 i1 v and r are ignored) is dynamic stable in straight-line m33 jvj m33 jrj

i2

i2

motion. Straight-line stability physically implies that a new path of the ship will be a straight line after an action in yaw. The direction of the new path will usually be different from that of the initial path, as mentioned in [12]. On the other hand,

9.3 Output Feedback

225

unstable ships will go into a starboard or port turn without any rudder deflection. We impose Assumption 9.1 to make our observer design possible. Note that this assumption does not hold for several types of surface ships such as large tankers and high-speed crafts with sufficiently small ratios d22 =m22 and d33 =m33 , and the added mass in the sway axis sufficiently larger than the added mass in the surge axis. Consequently, for these ships the real part of at least one of the eigenvalues of the matrix K2 is positive.

9.3.1 Observer Design The ship dynamics (9.1) represent some difficulties for output feedback control deP dvi i1 v and sign. These difficulties are mainly due to the nonlinear terms m22 jvj i2 P dri i1 r, the nonlinear kinematic term cos. /, and the underactuated situm33 jrj

i2

ation. However we first observe that the nonlinear terms are monotonic, i.e., they satisfy 0 1 X X dvi d vi .v1  v2 / @ jv1 ji1 v1  jv2 ji1 v2 A  0; 8 v1 2 R; v2 2 R; m22 m22 i2 i2 0 1 X dri X d ri .r1  r2 / @ jr1 ji1 r1  jr2 ji1 r2 A  0; 8r1 2 R; r2 2 R: m33 m33 i2

i2

(9.33)

Based on the structure of the underatuated ship dynamics (9.1) and property (9.33), we propose the following nonlinear observer: yOP D u sin. / C cos. /vO C k11 .y  y/ O C k12 .  O /; OP D rO C k21 .y  y/ O C k22 .  O /; X dvi m11 u d22 O i1 vO C k31 .y  y/ vPO D  rO  vO  O C jvj m22 m22 m22

(9.34)

i2

O .k13 C cos. //.y  y/; X dri d33 1 .m11  m22 /u vO  rO  r C rPO D O i1 rO C jrj m33 m33 m33 m33 i2

k42 .  O / C .k24 C 1/.  O /;

where y; O O ; v, O and rO are the estimate of y; ; v and r respectively. All the constants k11 , k12 , k21 , k22 , k13 , k31 , and k42 will be chosen later.

226

9 Way-point Tracking Control of Underactuated Ships

By defining the observer errors as yQ D y  y, O Q D r  r, O the observer error dynamics can be rewritten as

 O , vQ D v  v, O and rQ D

Q yPQ D k11 yQ  k12 Q  k13 vQ C .k13 C cos. //v; QP D k21 yQ  k22 Q  k24 rQ C .k24 C 1/r; Q   m u X d d 22 vi 11 vPQ D k31 yQ  vQ  O i1 vO  rQ  jvji1 v  jvj m22 m22 m22 i2

1 wv .t /; m22 .m11  m22 /u d33 rPQ D k42 Q  .k24 C 1/ Q C vQ  rQ  m33 m33  X dri  i1 1 O i1 rO C wr .t /: jrj r  jrj m33 m33 .k13 C cos. //yQ C

(9.35)

i2

We now show that there exist suitable observer gains k11 , k12 , k13 , k21 , k22 , k24 ; k31 , and k42 such that the observer error dynamics (9.35) is globally ultimately stable. Consider the Lyapunov function 1 Vobs D xQ T xQ 2

(9.36)

T  where xQ D yQ Q vQ rQ . The time derivative of (9.36) along the solutions of (9.35) and property (9.33) results in Q 2 C q0 ; VPobs  p0 kxk where 

 1 wv max wr max p0 D max .A/  max ; ; m22 m33 40   wv max wr max 0 ; 0 > 0; q0 D max ; m22 m33 3 2 7 6 k11 k12 k13 0 7 6 6 7 7 6 6 7 6 k21 k22 0 k24 7 6 7 AD6 7: 6 m11 u 7 d22 6 7   6 k31 0 7 6 m22 m22 7 7 6 4 .m11  m22 /u d33 5 0 k42  m33 m33

The above matrix A is made negative definite by choosing

(9.37)

9.3 Output Feedback

227

k13 D k24 D k31 D k42 ;   k11 k12 < 0; K1 WD k21 k22

(9.38)

K2  K12 K11 K12 < 0; where K12 D

"

k13 0

0

k24

#

;

(9.39)

and K2 is defined in (9.32). Here are details of choosing the observer gains such that (9.38) holds. The condition (9.38) is expanded as " # k11 k12 < 0; k21 k22 2 3 2 k13 k22 d22 m11 u k13 k12 k24 6 7  C   6 m22 k11 k22  k12 k21 m22 k11 k22  k12 k21 7 6 7 < 0: (9.40) 4 .m  m /u 5 2 k24 k11 k24 k21 k13 d33 11 22   C m33 k11 k22  k12 k21 m33 k11 k22  k12 k21 From (9.40), it suffices that 2 k13 k22 d22  > 0; m22 k11 k22  k12 k21 2 k24 k11 d33  > 0; m33 k11 k22  k12 k21 k13 k12 k24 m11 u D ; m22 k11 k22  k12 k21 k13 k21 k24 .m11  m22 /u D ; m33 k11 k22  k12 k21 k11 > 0; k22 > 0;

(9.41)

k11 k22  k12 k21 > 0: For simplicity, we choose p k13 D k24 D  u;

k11 k22  k12 k21 D ;

where  > 0 is to be selected later. Substituting (9.42) into (9.41) yields 0 < k22
 2 :  m22 m33

k12 D 

(9.43)

Hence, under Assumption 9.1, we can always pick a suitable constant  > 0 such that (9.43) holds. In summary, the observer gains k11 , k12 , k13 , k21 , k22 , k24 , k31 , and k42 are chosen such that (9.42) and (9.43) hold. We choose A and 0 such that p0 > 0. Hence (9.36) and (9.37) yield Q /k  kx.t Q 0 /k e .tt0 / C 0 ; 8t  t0  0; kx.t

(9.44) p with 0 D q0 =p0 and  D p0 . When there are no environmental disturbances, we have 0 D 0. The observer error dynamics (9.35) is thus GES at the origin.

9.3.2 Control Design We use the coordinate transformation (9.2) to rewrite the ship dynamics (9.3) in conjunction with (9.34) as follows vO u .sin.z1 /  ky.cos.z1 /  1// Cp C C p 2 1 C .ky/ 1 C .ky/2 vO ..cos.z1 /  1/ C sin.z1 /ky/ vQ .cos.z1 / C sin.z1 /ky/ ; Cp p 2 1 C .ky/ 1 C .ky/2

yP D  p

kuy

1 C .ky/2

zP 1 D rO C rQ 

k 2 uy  k vO

.1 C .ky/2 /3=2

C

ku .sin.z1 /  ky.cos.z1 /  1//

k vO ..cos.z1 /  1/ C sin.z1 /ky/

.1 C .ky/2 /3=2 k vQ .cos.z1 / C sin.z1 /ky/

C

C ; .1 C .ky/2 /3=2 .1 C .ky/2 /3=2 X dvi m11 u d22 rO  vO  Q O i1 vO C .k31 C k13 C cos. //y; vPO D  jvj m22 m22 m22 i2

X dri .m11  m22 /u d33 1 rPO D vO  rO  r C .k42 C k24 C 1/ Q : O i1 rO C jrj m33 m33 m33 m33 i2

(9.45)

Similarly to the full state feedback case, we design the control law r in two steps.

Step 1 Define

9.3 Output Feedback

229

z2 D rO  rOd ;

(9.46)

where rOd is an intermediate control designed as rOd D k1 z1 

ku .sin.z1 /  ky cos.z1 // .1 C .ky/2 /3=2



k vO .cos.z1 / C sin.z1 /ky/ .1 C .ky/2 /3=2

;

(9.47)

with k1 being a positive constant to be selected later.

Step 2 The first time derivative of (9.46) along the solutions of the last equation of (9.45) together with (9.47) is zP 2 D

X dri .m11  m22 /u d33 1 vO  rO  r C O i1 rO C jrj m33 m33 m33 m33 i2

@rOd @rOd uP  .k1 z1 C z2 /  .k42 C k24 C 1/ Q  @u @z1 @rOd @y @rOd @vO @rOd @z1 @rOd @y where

! u .sin.z1 /  ky cos.z1 // vO .cos.z1 / C sin.z1 /ky/  C p p 1 C .ky/2 1 C .ky/2 1 0 X dvi @ m11 u rO  d22 vO  (9.48) O i1 vO A  jvj m22 m22 m22 i2 ! k .cos.z1 / C sin.z1 /ky/vQ C rQ  .1 C .ky/2 /3=2 @rOd 1 .cos.z1 / C sin.z1 /ky/ vQ  .k31 C k13 C cos. // y; Q p @vO 1 C .ky/2

k . cos.z1 /ky C sin.z1 // @rOd k .cos.z1 / C sin.z1 /ky/ @rOd D D ; ; 3=2 @u @vO .1 C .ky/2 / .1 C .ky/2 /3=2

@rOd ku .cos.z1 / C ky sin.z1 // k vO . sin.z1 / C ky cos.z1 //  ; D k1  @z1 .1 C .ky/2 /3=2 .1 C .ky/2 /3=2

3k 3 y .kuy  u .sin.z1 /  .y C v/.cos.z O O sin.z1 // @rOd 1 /  1//  vy D C 5=2 @y .1 C .ky/2 / k 2 .u cos.z1 /  vO sin.z1 // .1 C .ky/2 /3=2

:

(9.49)

We now choose the actual control without canceling the useful damping terms as

230

9 Way-point Tracking Control of Underactuated Ships

 .m11  m22 /u d33 r D m33 z1  k2 z2  vO C rOd C m33 m33 X dri @rOd @rOd uP C O i1 rOd C .k1 z1 C z2 / C jrj m33 @u @z1 i2

! u .sin.z1 /  ky cos.z1 // vO .cos.z1 / C sin.z1 /ky/ C C p p 1 C .ky/2 1 C .ky/2 11 0 X dvi @rOd @ m11 u d22 (9.50) rO  vO  O i1 vO AA ;  jvj @vO m22 m22 m22

@rOd @y

i2

where k2 is a positive constant to be chosen later. Substituting (9.46), (9.47), and (9.50) into (9.45) results in the following closed loop system: vO u .sin.z1 /  ky.cos.z1 /  1// kuy yP D  p Cp C C p 2 2 1 C .ky/ 1 C .ky/ 1 C .ky/2 vO ..cos.z1 /  1/ C sin.z1 /ky/ vQ .cos.z1 / C sin.z1 /ky/ C ; p p 1 C .ky/2 1 C .ky/2 X dvi d22 m11 u .k1 z1 C z2  vOP D  vO  O i1 vO  jvj m22 m22 m22 i2 ! ku .sin.z1 /  ky cos.z1 // k vO .cos.z1 / C sin.z1 /ky/ C  .1 C .ky/2 /3=2 .1 C .ky/2 /3=2 .k31 C k13 C cos. //y; Q k .cos.z1 / C sin.z1 /ky/ vQ C r; Q zP 1 D k1 z1 C z2 C .1 C .ky/2 /3=2 X dri d33 zP 2 D z1  k2 z2  z2  O i1 z2  jrj m33 m33 i2 ! @rOd k .cos.z1 / C sin.z1 /ky/ vQ C rQ C .k42 C k24 C 1/ Q  @z1 .1 C .ky/2 /3=2 @rOd 1 .cos.z1 / C sin.z1 /ky/ vQ @rOd .k31 C k13 C cos. //yQ  : (9.51) p @vO @y 1 C .ky/2

9.3.3 Stability Analysis It is not difficult to show that the closed loop system (9.51) is forward complete. We now use Lemmas 9.1 and 9.2 to prove that the closed loop (9.51) is globally ultimately stable. From (9.49), it can be seen that the closed loop (9.51) is different from (9.9) since vO enters the .z1 ; z2 /-subsystem. To remove this obstacle, we first

9.3 Output Feedback

231

prove that vO is bounded. We then prove the convergence of .z1 ; z2 / and finally vO and y.

Boundedness of vO To prove that vO is bounded, we view the last three equations of (9.51) as the system  T studied in Lemma 9.1 with x1 D vO z1 z2 as x, xQ as .t/ and xP 1 D f1 .t; x1 / C g1 .t; x1 ; x/; Q

(9.52)

where 3

2

7 6 ˝1 7 6 6 7 7 6 7; f1 .t; x1 / D 6 k z C z 1 1 2 7 6 6 ! 7 7 6 4 z  k C d33 C P dri rj 5 j O i1 z2 1 2 m33 i2 m33 3 2

7 6 .k31 C k13 C cos. //yQ 7 6 7 6 7 6 k vQ .cos.z1 / C sin.z1 /ky/ g1 .t; x1 ; x/ Q D6 7; C r Q 3=2 7 6 2 .1 C .ky/ / 7 6 5 4 ˝2

(9.53)

with ˝1 D 

X dvi d22 m11 u vO   O i1 vO  jvj m22 m22 m22 i2

k1 z1 C z2 

k .u .sin.z1 /  ky cos.z1 // C vO .cos.z1 / C sin.z1 /ky//

@rOd ˝2 D .k42 C k24 C 1/ Q  @z1

.1 C .ky/2 /3=2

k .cos.z1 / C sin.z1 /ky/ vQ

!

;

!

C rQ  .1 C .ky/2 /3=2 @rOd @rOd .cos.z1 / C sin.z1 /ky/ vQ ; .k31 C k13 C cos. //yQ  p @vO @y 1 C .ky/2

where again with abuse of notation, rO is considered as a function of time. We now need to verify all of the conditions of Lemma 9.1. Verifying Condition C1. We take the following Lyapunov function:

232

9 Way-point Tracking Control of Underactuated Ships

V1 D


1 2 vO C ı1 .z12 C z22 / ; 2

(9.54)

where ı1 is a positive constant. The first time derivative of (9.54) along the solutions of the differential equation xP 1 D f1 .t; x1 /, see (9.52) and (9.53), satisfies d22 2 vO  VP1 D  m22

X dvi m11 uvO .k1 z1 C z2  O i1 vO 2  jvj m22 m22 i2

k .u .sin.z1 /  ky cos.z1 // C vO .cos.z1 / C sin.z1 /ky//

!

 .1 C .ky/2 /3=2 X dri d33 2 z 2  ı1 O i1 z22 ı1 k1 z12  ı1 k2 z22  ı1 jrj m33 m33 i2   d22 m11 umax   .1 C k1 1 C 4k C 2k1 umx / vO 2  m22 m22     m11 k1 umax 2 m11 umax 2 5m11 ku2max ı1 k 1  z 1  ı 1 k2  z2 C : 41 m22 41 m22 2m22 (9.55) Hence the condition C1 of Lemma 9.1 is verified with 1 5m11 ku2max ; c1 D min.1; ı1 /; 2m22 2 1 c2 D max.1; ı1 /; c3 D max.1; ı1 /; 2   d22 m11 umax c4 D min  .1 C k1 1 C 4k C 2k1 umx / ; m22 m22     m11 umax m11 k1 umax ; ı1 k 2  ; ı1 k1  41 m22 41 m22 c0 D

(9.56)

where 1 > 0 and k > 0 are chosen such that c4 > 0. Verifying Condition C2. To verify this condition of Lemma 9.1, we note from (9.49) that ˇ ˇ ˇ @rOd ˇ ˇ ˇ O ; ˇ @z ˇ  k1 C 2k .umax C jvj/ 1 ˇ ˇ ˇ ˇ ˇ @rOd ˇ ˇ @rOd ˇ 2 ˇ ˇ ˇ ˇ (9.57) O : ˇ @vO ˇ  2k; ˇ @y ˇ  3k .umax C 3kumax C 3k jvj/ From (9.56) and (9.57), a simple calculation shows that the condition C2 of Lemma 9.1 is satisfied with

9.3 Output Feedback

233

1 D .2k C 1/.k31 C k13 C 1/ C k42 C k24 C 1 C .2k C 1/.k1 C 2kumax C 1/ C 6k 2 umax .3k C 1/; 2 D 2k.2k C 1/ C 18k 3 :

(9.58)

Verifying Condition C3. This condition follows directly from (9.44). Verifying Condition C4. It can be shown that there exists a positive constant k such that the condition C4 of Lemma 9.1 satisfies c4  2 c3 "0 

1 c3 "0 > 0; 40

(9.59)

where c4 , c3 , "0 , 1 , and 2 given in (9.44), (9.56) and (9.58). All of the conditions of Lemma 9.1 have been verified, therefore we have Q 0 //k/ e 1 .tt0 / C "1 ; 8t  t0  0; O  kx1 .t /k  ˛1 .k.x1 .t0 /; x.t jvj

(9.60)

where ˛1 , 1 , and "1 are in the form of ˛,  , and " in Lemma 9.1 with all constants given in (9.44), (9.56), and (9.58).

.z1 ; z2 /-subsystem Having proven that vO is bounded in the previous section, we now apply Lemma 9.1 to the .z1 ; z2 /-subsystem. It is clear that the last two equations of (9.51) are in the form of the system studied in Lemma 9.1 with z D Œz1 z2 T as x; xQ as .t/ and zP D fz .t; z/ C gz .t; y; z; x/; Q

(9.61)

where 2

6 fz .t; z/ D 6 4

3

k1 z1 C z2

! 7 7; P dri d33 5 i1 z1  k2 C O z2 C jrj m33 i2 m33 2

6 gz .t; y; z; x/ Q D4

k .cos.z1 / C sin.z1 /ky/ vQ .1 C .ky/2 /3=2 ˝2

(9.62)

3

C rQ 7 5:

Proceeding with the same steps as in the previous section, it is shown that all the conditions of Lemma 9.1 hold with the Lyapunov function V2 D 0:5.z12 C z22 / and c0 D 0; c1 D c2 D 0:5; c3 D 1; c4 D min.k1 ; k2 /; 1 D 2k .k13 C k31 C 1/ C .2k C 1/.2 C 2kumax / C k42 C

234

9 Way-point Tracking Control of Underactuated Ships

k24 C 1 C 6k 2 umax .1 C 3k/ C .18k 3 C 2k.2k C 1//.˛1 C "1 /; 2 D 0; (9.63) where ˛1 and "1 are given in (9.60). The condition C4 of Lemma 9.1 becomes c4 

1 c3 "0 > 0; 40

(9.64)

where c4 ; c3 ; "0 , and 2 are calculated from in (9.44) and (9.63). All of the conditions of Lemma 9.1 have been verified, therefore we have Q 0 //k/ e 2 .tt0 / C "2 ; 8t  t0  0 kz.t /k  ˛2 .k.z.t0 /; x.t

(9.65)

where ˛2 , 2 , and "2 are in the form of ˛,  , and " in Lemma 9.1 with all constants given in (9.63).

.y; v/-subsystem O It can be seen that the first two equations of (9.51) are in the form of the system  T  T studied in Lemma 9.2, i.e., x3 D vO y , xQ 3 D z1 z2 yQ vQ , and xP 3 D f3 .t; x3 / C g3 .t; x3 ; xQ 3 /;

(9.66)

where P dvi d22 m11 u k 2 uy  k vO O i1 vO  jvj 6  m vO  m22 .1 C .ky/2 /3=2 6 22 i2 m22 f3 .t; x3 / D 6 kuy vO 4 p Cp 2 1 C .ky/ 1 C .ky/2 2

g3 .t; x3 ; xQ 3 / D



3

7 7 7; 5

(9.67)



˝31 ; ˝32

with m11 u ku .sin.z1 /  ky.cos.z1 /  1//  k1 z1 C z2  m22 .1 C .ky/2 /3=2 ! k vO ..cos.z1 /  1/ C sin.z1 /ky/ Q C .k31 C k13 C cos. //y; .1 C .ky/2 /3=2 u .sin.z1 /  ky.cos.z1 /  1// vO ..cos.z1 /  1/ C sin.z1 /ky/ ˝32 D C C p p 1 C .ky/2 1 C .ky/2 .cos.z1 /  sin.z1 /ky/ vQ : p 1 C .ky/2

˝31 D 

9.3 Output Feedback

235

We now need to verify all conditions of Lemma 9.2 for the system (9.66). Verifying Condition C1. To verify this condition of Lemma 9.2, we take the following proper function
1 V3 D vO 2 C y 2 ; (9.68) 2 whose time derivative along (9.67) satisfies y2 VP3  c31 vO 2  c32 p ; 1 C c4 y 2

where

m11 k 2 u2max d22 m11 kumax   2  3 ; m22 m22 m22 1 m11 k 2 u2max 1 c32 D kumin   ; 42 43 m22

(9.69)

c31 D

(9.70)

with 2 > 0 and 3 > 0 chosen such that c31 > 0 and c32 > 0. From (9.67) and (9.68), it is easy to show that ! ˇ ˇ ˇ @V3 ˇ y2 2 ˇ ˇ kxQ 3 k ; ˇ @x g3 .t; x3 ; xQ 3 /ˇ  0 C 1 vO C 2 p 3 1 C c4 y 2

with

(9.71)



m11 umax .k1 C 1 C 2kumax / C .k31 C k13 C 1/C m22
umax C 1 C .˛1 C "1 /2 ; 2m11 kumax m11 umax 4 .k1 C 1 C 2kumax / C C 4 .k31 C k13 C 1/; 1 D m22 m22 (9.72) 2 D .k C 4 /umax C 24 C k .˛1 C "1 C ˛2 C "2 / ; 1 0 D 44

where 4 > 0, ˛1 , and "1 are given in (9.60), and ˛2 and "2 are given in (9.65). Verifying Condition C2. To verify this condition, we note that

   

z1 .t / v.t Q /

:

C kxQ 3 .t /k  Q / z2 .t / y.t

(9.73)

Therefore we can write (9.73) from (9.44) and (9.65) as

Q 0 //k/ e 3 .tt0 / C "3 ; kxQ 3 .t /k  ˛3 .k.z.t0 /; x.t where

(9.74)

236

9 Way-point Tracking Control of Underactuated Ships

˛3 .k.z.t0 /; x.t Q 0 //k C ˛2 .k.z.t0 /; x.t Q 0 //k/ D k.z.t0 /; x.t Q 0 //k/ ; 3 D min.; 2 /; "3 D 0 C "2 ;

(9.75)

with ˛2 and "2 given in (9.65), 0 and  given in (9.44). Verifying Condition C3. This condition is satisfied if c31  1 "3 > 0 and c32  2 "3 > 0;

(9.76)

where c31 , c32 , 1 , 2 , and "3 are given in (9.72) and (9.75). After some lengthy but simple calculation, it can be shown that, under the assumption of small enough environmental disturbances, the condition (9.76) holds for a suitable choice of the observer gains k11 , k12 , k13 , k21 , k22 , k24 , k31 , and k42 , and the control gains k, k1 , and k2 . Verifying Condition C4. From the boundedness of the sway velocity estimate, v.t O /, proven in the previous section and noting that "2 in (9.65) can be made arbitrarily small, it is directly shown that there exists a nondecreasing function $ of k..v.t0 /; y.t0 //; z.t0 //k such that jy.t /j  $ by applying Lemma 9.1 to the first equation of (9.51) with the Lyapunov function Vy D 0:5y 2 . All of the conditions of Lemma 9.2 have been verified, the closed loop (9.51) is globally ultimately stable, i.e., kx3 .t /k  ˛4 .k.x.t0 /; .t0 //k/ e 4 .tt0 / C "4 ; 8t  t0  0;

(9.77)

where ˛4 ; 4 , and "4 are calculated as in Lemma 9.2. It is noted that when there are no environmental disturbances, " D 0. Therefore the closed loop (9.51) is GAS. We note that the convergence of z1 and z2 implies the convergence of rO and . The convergence of v and r results from that of vO and rO due to the global exponential property of the observer. We have thus proven the second main result of this chapter. Theorem 9.2. Under Assumption 9.1, the output feedback control problem stated in Section 9.1 is solved by the observer (9.34) and the control law (9.50) as long as the observer gains k11 , k12 , k13 , k21 , k22 , k24 , k13 , k31 , and k42 , and the control gains k, k1 , and k2 are chosen such that (9.64), (9.70), (9.76), and (9.43) hold. Remark 9.2. Due to underactuaction and nonzero-mean environmental disturbances in the sway dynamic, our controller is only able to force the sway and its velocity to converge to a ball centered at the origin. The radius of this ball cannot be made arbitrarily small. This phenomenon should not be surprising since there is no control force in the sway direction. In addition, the yaw angle cannot be made arbitrarily small due to the effect of the sway. In fact, to guarantee the sway displacement bounded under nonzero-mean environmental disturbances acting on the sway dynamics, our controller forces the heading angle to a small value. This value together with the forward speed will prevent the sway from growing unbounded.

9.4 Simulations

237

Remark 9.3. The choice of k depends on the ship parameters and forward speed, which coincides with the steering practice of a helmsman. The helmsman uses the ship’s course angle to steer the ship toward the straight line rather than use the sway velocity, which will cause the ship to glide sideways. Furthermore, the design constant k is reduced when the ship forward speed is large, see (9.2), (9.23), (9.30), (9.56), and (9.59), otherwise the ship will miss the point on the straight line and slide in the sway direction. Remark 9.4. By setting the value of k equal to zero, our proposed controller reduces to a course-keeping controller. In this case, the heading angle can be made arbitrarily small. However the sway will grow linearly unbounded under nonzero-mean environmental disturbances, see Figures 9.3 and 9.6.

9.4 Simulations This section validates the control laws (9.8) and (9.50) for both cases of state and output feedback on a monohull ship with the parameters given in Section 5.4. The ship surge velocity is chosen as u D 10 C 0:5 sin.3t / ms-1 . The environmental disturbances wv .t / and wr .t / are taken as wv .t / D 105  0:5  .1 C rand.
// and wr .t / D 1:5  107  rand.
/, with rand.
/ being zero mean random noise with the uniform distribution on the interval Œ0:5 0:5. We run simulations for both state feedback and output feedback cases.

9.4.1 State Feedback Simulation Results The control design parameters are chosen as k D 0:05, k1 D 0:2, k2 D 0:5, and 1 D 2 D 0:05. It can be directly verified that this choice satisfies all the conditions stated in Theorem 9.1. The initial values are   Œy.0/; v.0/; .0/; r.0/ D 15 m; 0:2 ms-1 ; 0:5 rad; 0:1 rads-1 :

Simulation results are plotted in Figure 9.1 for the case without disturbances. In this case, it can be seen that all sway displacement, sway velocity, and yaw angle converge to zero as desired. The large control effort is due to the fact that we simulate our controllers on a real surface ship but it is within the limit of the maximum yaw moment. For the case with disturbances, simulation results are plotted in Figure 9.2. In this case, all the states converge to a ball centered at the origin as proven in Theorem 9.1. To illustrate Remark 9.4, we simulate our controller with the design constant k D 0. The simulation results for this case are given in Figure 9.3. The sway displacement y grows linearly unbounded due to nonvanishing environmental disturbances. It should be noted that all of the course-keeping controllers, see for

238

9 Way-point Tracking Control of Underactuated Ships

a

b

1

ψ [rad]

y [m]

20 0 −20 0

10

20

0 −1 0

30

10

c

0 −5 0

10

20

20

30

0 −2 0

30

8

10

e

x 10

0

r

τ [Nm]

30

2

r [rad/s]

v [m/s]

5

5

20 d

−5 0

5

10

15 Time [s]

20

25

30

Figure 9.1 State feedback control results without disturbances: a. Sway displacement y; b. Heading angle ; c. Sway velocity v; d. Yaw velocity r; e. Yaw moment r

example [12], which do not take the sway displacement into account, will result in similar unboundedness of the sway that was pointed out in Remark 9.4.

9.4.2 Output Feedback Simulation Results The control design parameters are chosen as k D 0:05, k1 D 0:2, k2 D 0:5, and m11 , 1 D 2 D 0:05. The observer gains are selected as k11 D k22 D 2, k12 D  m 22 p m11 m22 k21 D m33 , k31 D k13 D k24 D k42 D  u, and  D 0:015. A calculation shows that this choice satisfies all the conditions stated in Theorem 9.2. The initial values are   Œy.0/; v.0/; .0/; r.0/D 15 m; 0:2 ms-1 ; 0:5 rad; 0:1 rads-1 ;  O .0/; r.0/ y.0/; O v.0/; O O D 10 m; 0 ms-1 ; 0:2 rad; 0:2 rads-1 :

Simulation results are plotted in Figure 9.4 for the case without disturbances and in Figure 9.5 for the case with disturbances. From Figure 9.4, it is seen that all sway displacement, sway velocity, and yaw angle converge to zero asymptotically. It is also observed that the observer states (dotted lines) exponentially converge to

9.5 Conclusions

239

a

b

1

ψ [rad]

y [m]

20 0 −20 0

10

20

0 −1 0

30

10

c

0 −5 0

10

20

20

30

0 −2 0

30

8

10

e

x 10

0

r

τ [Nm]

30

2

r [rad/s]

v [m/s]

5

5

20 d

−5 0

5

10

15 Time [s]

20

25

30

Figure 9.2 State feedback control results with disturbances: a. Sway displacement y; b. Heading angle ; c. Sway velocity v; d. Yaw velocity r; e. Yaw moment r

their unknown estimated ones (solid lines). For the case with disturbances, all the states converge to a ball centered at the origin as proven in Theorem 9.2. The simulation results with the design constant k D 0 are plotted in Figure 9.6. Again, the sway displacement y grows linearly unbounded due to nonvanishing environmental disturbances as mentioned in Remark 9.4.

9.5 Conclusions The control design was based on the idea of an interaction between the ship behavior and the action of a helmsman on a linear course. Although our proposed state feedback controller has been designed by using precise knowledge of the ship parameters, we can easily change them to an adaptive version to take inaccurate knowledge of the system parameters into account, see (9.6). However, for the case of output feedback, an adaptive observer will be required, see (9.34) and (9.45). This chapter is based on [116, 126, 127].

240

9 Way-point Tracking Control of Underactuated Ships

a

b

1

ψ [rad]

y [m]

20 10 0 0

10

20

0 −1 0

30

10

c

1 0 0

10

20

20

30

0 −0.5 0

30

8

τ [Nm]

30

0.5

r [rad/s]

v [m/s]

2

1

20 d

10

e

x 10

r

0 −1 0

5

10

15 Time [s]

20

25

30

Figure 9.3 State feedback control results with disturbances and k D 0: a. Sway displacement y; b. Heading angle ; c. Sway velocity v; d. Yaw velocity r; e. Yaw moment r

a

b

hat

ψ, ψ [rad]

y, yhat [m]

10 0 −10 0

10

20

30

1 0 −1 0

10

2

hat

0 −2 0

10

20

30

8

30

20

30

2 0 −2 0

10

e

x 10

0

r

τ [Nm]

5

20 d

r, r [rad/s

v, vhat [m/s]

c

−5 0

5

10

15 Time [s]

20

25

30

Figure 9.4 Output feedback control results without disturbances: a. Sway displacement y; b. Heading angle ; c. Sway velocity v (solid line) and its estimate vO (dotted line); d. Yaw velocity r (solid line) and its estimate rO (dotted line); e. Yaw moment r

9.5 Conclusions

241

a

b

1

hat

ψ, ψ [rad]

y, yhat [m]

20 0 −20 0

10

20

30

0 −1 0

10

r, r [rad/s 10

30

20

30

2 0

hat

0 −2 0 8 x 10 4

20 d

2

τ [Nm]

v, vhat [m/s]

c

20

30

e

−2 0

10

2

r

0 −2 0

5

10

15 Time [s]

20

25

30

Figure 9.5 Output feedback control results with disturbances: a. Sway displacement y; b. Heading angle ; c. Sway velocity v (solid) and its estimate vO (dot); d. Yaw velocity r (solid) and its estimate rO (dot); e. Yaw moment r

a

b

ψ, ψhat[rad]

y, yhat [m]

20 10 0 0

10

20

30

1 0 −1 0

10

1

10

20

8

20

30

0

30

−0.5 0

10

e

x 10

0

r

τ [Nm]

1

30

0.5

hat

0.5 0 0

20 d

r, r [rad/s

v, vhat [m/s]

c

−1 0

5

10

15 Time [s]

20

25

30

Figure 9.6 Output feedback control results with disturbances and k D 0: a. Sway displacement y; b. Heading angle ; c. Sway velocity v (solid line) and its estimate vO (dotted line); d. Yaw velocity r (solid line) and its estimate rO (dotted line); e. Yaw moment r

Chapter 10

Path-following of Underactuated Ships Using Serret–Frenet Coordinates

This chapter presents state feedback and output feedback controllers that force an underactuated surface ship to follow a predefined path under the presence of environmental disturbances induced by waves, wind, and ocean currents. The control solutions originated an observation that it is reasonable in practice to steer a vessel such that it is on the reference path and its total velocity is tangent to the path, and that the vessel’s forward speed is independently controlled by the main propulsion control system. The proposed controllers are designed using Lyapunov’s direct method, the popular backstepping technique, and the Serret–Frenet frame. The unmeasured sway and yaw velocities are estimated by introducing a novel nonlinear passive observer.

10.1 Control Objective In addition to the assumptions made in Section 3.4.1.1, we assume that the surge velocity is a positive constant and is independently controlled by the main propulsion control system. The resulting mathematical model of the underactuated ship moving in surge, sway, and yaw is rewritten as xP D u cos. /  v sin. /; yP D u sin. / C v cos. /; P D r; vP D 

3

X dvi m11 d22 1 ur  v wv .t /; jvji1 v C m22 m22 m22 m22

(10.1)

iD2

3

rP D

X dri .m11  m22 / d33 1 1 uv  r r C wr .t /; jrji1 r C m33 m33 m33 m33 m33 iD2

243

244

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

where all symbols in (10.1) are defined in Section 3.4.1.2. The positive constant terms dv , dr , dvi , and dri ; i D 2; 3 represent the hydrodynamic damping in surge, sway and yaw. The bounded time-varying terms, wv .t / and wr .t /, are the environmental disturbances induced by wave, wind, and ocean current with jwv .t /j  wv max < 1 and jwr .t /j  wr max < 1. The available control is the yaw moment r . Since the sway control force is not available in the sway dynamics, the ship model (10.1) is again underactuated. The control objective of this chapter is to design the yaw moment r to force the underactuated ship (10.1) to follow a specified path ˝, see Figure 10.1, where M is the orthogonal projection of the ship point P on ˝, xn and x t are the normal and the tangent unit vectors to the path at M , respectively. We assume that this point is uniquely defined. This assumption holds if the interior of any circle tangenting ˝ at two or more points does not contain any point of the path and the distance between the ship and ˝ is not too large. Furthermore, it is assumed that the radius of any osculating circle of the path is larger than or equal to Rmin which is feasible for the ship to follow. Let the signed distance between M and P be ze . Also s is the signed distance along the path between some arbitrary fixed point on the path and M , d is the angle between x t and Xb , c.s/ is the curvature of the path at the point M . We assume that c.s/ is uniformly bounded and differentiable. Let the total velocity of the ship be u t and e D  d . The variables s, ze , and e form a new set of state coordinates for the ship. It can be seen that when the path ˝ coincides with the XE -axis, the above variables coincide with the ship variables x, y, and . By using the above parameterization, it is straightforward, see [128, 129], to transform the kinematics of (10.1) to zP e D u sin.

e / C v cos.

e /;

c.s/ .u cos. e /  v sin. 1  c.s/ze 1 .u cos. e /  v sin. e // : sP D 1  c.s/ze Pe D r 

e // ;

(10.2)

Note that the above transformation is singular when ze c.s/ D 1. We first assume that 1  ze c.s/  ı  > 0: (10.3) We then find the initial conditions after the controllers are designed such that this hypothesis holds. Ideally, we want the ship to be on the path and tangent to it, i.e., ze D e D 0. However when the curvature of the path is different from zero, the sway velocity v is also different from zero. Hence ze D e D 0 cannot be the equilibrium point of (10.2) with r as the control input. This feature distinguishes the ship from mobile robots, see [15, 130–133] for some work on controlling mobile robots. If one designs a controller as was proposed in [129], to achieve e D 0, then the error ze might be very large (depending on v/. This phenomenon can be seen by substituting e D 0 into the first equation of (10.2). Therefore, in this chapter, we formulate the

10.1 Control Objective

245

ut

v

I u \

YE

y

x

P

ze

xn

xt

x

M

s

\d

ȍ OE

x

XE

Figure 10.1 General framework of ship path-following

control problem such that .ze ; e / D .0; / is the equilibrium point of (10.2), with  being the angle between the surge velocity and the total velocity. This is desirable in practice since it guarantees that the ship is on the path and the ship’s total velocity is tangent to the path. Define  (10.4) e D e C ; where e is referred to as the modified heading error. With (10.4), the last four equations of the ship model (10.1) with (10.2) are transformed to zP e D u t sin. e /;  2 P e D r 1  m11 u  c.s/u t cos. m22 u2t 1  c.s/ze u u2t

 e /

! 3 X 1 dvi d22 i1 vC wv .t / ; jvj v  m22 m22 m22 iD2

3

vP D 

X dvi m11 d22 1 ur  v wv .t /; jvji1 v C m22 m22 m22 m22

(10.5)

iD2

3

rP D

X dri .m11  m22 / d33 1 1 uv  r r C wr .t /; jrji1 r C m33 m33 m33 m33 m33 iD2

where ut D

p u2 C v 2 :

(10.6)

246

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

As a result,
the path-following objective has been converted to a problem of stabilizing ze ; e in (10.5) at the origin. We study this stabilization problem under either Assumption 10.1 or Assumption 10.2 as follows. Assumption 10.1. The ship parameters satisfy the condition m11 < m22 . All of the ship states (position, orientation, and velocities) are measurable. Assumption 10.2. The ship parameters satisfy the condition m11 < m22 . The sway and yaw velocities are not available for feedback. Assumption 10.1 means a robust state feedback path-following problem, while Assumption 10.2 gives a robust output feedback path-following problem. It is noted that the condition m11 < m22 always holds for surface ships since the added mass in sway is larger than that in surge. The triangular structure of (10.5) suggests that we design the actual control r in two stages. First, we design the virtual velocity control r to stabilize ze and e at the origin. Based on the backstepping technique, the control r will then be designed to make the error between the virtual velocity control and its actual values exponentially tend to a small ball centered at the origin. A new nonlinear observer is introduced to estimate the sway and yaw velocities, and an output feedback controller is then designed.

10.2 State Feedback 10.2.1 Control Design Observing from the first equation of (10.5) that the ze -dynamics have to be stabilized by using the angle e , we introduce the following coordinate transformation ! kze   we D e C arcsin p ; (10.7) 1 C .kze /2

where k is a positive constant to be selected later.

Remark 10.1. The above coordinate transformation is well defined and convergence of we and ze implies that of e . The function arcsin is not unique. It can be replaced by some other smooth bounded functions such as tanh; arctan or an identity function. We here use the arcsin function due to its simplicity. Using (10.7) instead of a linear coordinate change such as we D e C kze , we avoid the ship whirling around when ze is large. In addition the nonlinear change of coordinate (10.7) will result in a “global design”. Upon an application of the coordinate transformation (10.7), the system (10.5) is rewritten as

10.2 State Feedback

247

u t sin.we /


ku t ze ku t ze zP e D  p Cp p cos.we /  1 ; 1 C .kze /2 1 C .kze /2 1 C .kze /2   2 ku t sin. e / c.s/u t cos. e / m11 u C wP e D r 1    m22 u2t 1 C .kze /2 1  c.s/ze ! 3 X u d22 dvi 1 i1 vC wv .t / ; jvj v  m22 m22 u2t m22 iD2

3

vP D 

X dvi d22 1 m11 ur  v wv .t /; jvji1 v C m22 m22 m22 m22

(10.8)

iD2

3

rP D

X dri .m11  m22 / d33 1 1 uv  r r C wr .t /; jrji1 r C m33 m33 m33 m33 m33 iD2

where we leave the variable e in the second equation of (10.8) for simplicity of presentation. As discussed in Remark 10.1, we will design the control r to stabilize (10.8) at the origin.

Step 1 Define b D 1

m11 u2 : m22 u2t

(10.9)

The first condition of Assumptions 10.1 implies that b  1

m11 WD b  > 0: m22

(10.10)

Introduce the yaw velocity control error rQ as rQ D r  rd ;

(10.11)

where rd is a virtual control of the yaw velocity r. From the second equation of (10.8), we choose the virtual control rd as k1  1 ku t sin. e / 1 c.s/u t cos. e / w  C C b e b 1 C .kze /2 b 1  c.s/ze ! 3 X 1 u d22 dvi i1 vC jvj v  b u2t m22 m22 iD2    1 u wv max we wv max tanh ; b u2t m22 "1 m22

rd D 

(10.12)

where k1 and "1 are positive design constants to be selected later. It is of interest to note that rd is a smooth function of s; ze ; v, and e . Substituting (10.12) and

248

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

(10.11) into the second equation of (10.8) yields wP e

D

k1 we C b rQ C

   u wv .t / u wv max we wv max  2 tanh : "1 m22 u2t m22 u t m22

(10.13)

Step 2 Differentiating both sides of (10.11) along the solutions of the last equation of (10.8) and (10.12) yields 3

X dri d33 1 1 .m11  m22 / uv  r r C wr .t /  rPQ D jrji1 r C m33 m33 m33 m33 m33 iD2

@rd 1 .u cos. @s 1  c.s/ze

e /  v sin.

e // 

@rd .u sin. @ze

e / C v cos.

! 3 X dvi d22 1 m11 i1 ur  v wv .t /   jvj v C m22 m22 m22 m22 iD2   @rd c.s/ .u cos. e /  v sin. e // : r @ e 1  c.s/ze

e // 

@rd @v

(10.14)

From (10.14), we choose the actual control r without canceling the useful damping terms as 3

r D m33 k2 rQ  bwe 

X dri d33 .m11  m22 / uv C rd C jrji1 rd C m33 m33 m33 iD2

@rd @rd 1 .u cos. e /  v sin. e // C .u sin. e / C v cos. e // C @s 1  c.s/ze @ze !  3 X d22 dvi m11 @rd ur  v  jvji1 v C @v m22 m22 m22 iD2   @rd c.s/ .u cos. e /  v sin. e //  r @ e 1  c.s/ze     rQ wr max @rd rQ wv max @rd wv max wr max  ; tanh tanh m33 "2 m33 @v m22 @v "3 m22 (10.15) where k2 ; "2 , and "3 are positive constants to be selected later. Substituting (10.15) into (10.14) yields   3 X 1 dri PrQ D  k2 C d33 rQ  bw   wr .t / jrji1 rQ C e m33 m33 m33 iD2

10.2 State Feedback

249





wr max rQ wr max @rd 1  tanh wv .t /  m33 "2 m33 @v m22   @rd rQ wv max @rd wv max tanh : @v m22 @v "3 m22

(10.16)

We now present the first main result of this chapter, the proof of which is given in the next section. Theorem 10.1. Under Assumption 10.1, if the state feedback control law (10.15) is applied to the ship system (10.1) then there exist feasible initial conditions such that the regulation errors .ze .t /; e .t // converge to a small ball centered at the origin with an appropriate choice of the design constants k, k1 , k2 and "i , 1  i  3. Furthermore if there are no environmental disturbances, the regulation errors converge to zero asymptotically. In addition, the sway velocity v.t / is always bounded.

10.2.2 Stability Analysis For the reader’s convenience we write the closed loop consisting of the first and third equations of (10.8), (10.13), and (10.16) as follows: 
ku t ze u t sin.we / ku t ze zP e D  p Cp p cos.we /  1 ; 2 2 2 1 C .kze / 1 C .kze / 1 C .kze /    u we wv max  .t /  u wv wv max ;  2 tanh wP e D k1 we C b rQ C 2 "1 m22 u t m22 u t m22 3 X m11 dvi d22 1 u.rQ C rd /  v wv .t /; jvji1 v C m22 m22 m22 m22 iD2   3 X d33 dri 1 rQ  bwe  rPQ D  k2 C wr .t /  jrji1 rQ C m33 m33 m33 iD2   wr max @rd 1 rQ wr max  tanh wv .t /  m33 "2 m33 @v m22   @rd wv max @rd rQ wv max ; tanh @v m22 @v "3 m22

vP D 

(10.17)

where rd is given in (10.12). A direct calculation of the Lyapunov function candidate V0 D ze2 C we2 C v 2 C rQ 2 along the solutions of the closed loop system (10.17) shows that there exist nonnegative constants a0 and b0 such that VP0  a0 V0 C b0 . This means that the closed loop system (10.17) is forward complete. Therefore, to prove Theorem 10.1, we can first consider the .we ; r/-dynamics, Q then move to the .ze ; v/-dynamics.

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10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

.we ; r/-dynamics Q Consider the following Lyapunov function: V1 D


1  2 w C rQ 2 : 2 e

(10.18)

It is not hard to show that the time derivative of (10.18) along the solutions of (10.13) and (10.16) satisfies   3 X d33 2 P V1  k1 we  k2 C rQ 2 C 0:2785 "i ; m33

(10.19)

iD1

which in turn implies that

 


w .t /; r.t Q /  ˛wr .
/e wr .tt0 / C wr ; e

(10.20)

where


˛wr .
/ D we .t0 /; r.t Q 0/ ;    d33 wr D min k1 ; k2 C ; m33 v u 3 u 0:2785 X "i : wr D t 1

(10.21)

iD1

From (10.21), we can see that wr can be made arbitrarily small by adjusting the design constants, k1 ; k2 and "i ; 1  i  3. This observation is important in stability analysis of the .ze ; v/-dynamics.

.ze ; v/-dynamics For the reader’s convenience, we rewrite the .ze ; v/-dynamics from the first and third equations of the closed loop system (10.17) with the virtual control rd substituted in the third equation of the closed loop system (10.17) from (10.12) as 
u t sin.we / ku t ze ku t ze cos.we /  1 ; zP e D  p Cp p 2 2 2 1 C .kze / 1 C .kze / 1 C .kze /     3 X dvi d22 m11 u2 m11 u2 vP D  1C 1 C v  jvji1 v C 2 m22 m22 bm22 u2t bm u 22 t iD2  1 m11 urQ m11 u ku t sin. e / k1 we  wv .t /   C m22 m22 m22 b 1 C .kze /2

10.2 State Feedback

251

c.s/u t cos. 1  c.s/ze

 e/



we



wv max u wv max : (10.22) tanh 2 m "1 m22 ut 22 
If we view that (10.22) and the we ; rQ -subsystem are in a cascade form, we might think that stability results developed for cascade systems, see Section 2.1.3 can be applied. However, stability results in that section are developed for the cascade systems without nonvanishing disturbances. In fact, nonvanishing disturbances may destroy stability of a cascade system that satisfies all conditions stated in Section 2.1.3. To illustrate this fact, we give the following simple example. 

xP 1 D  q

x1 x2 Cq ; 1 C x12 1 C x12 x1

xP 2 D x2 C d.t /:

(10.23)

When there is no disturbance d.t /, by applying stability results in Section 2.1.3, the cascade system (10.23) is GAS at the origin. However, whenever the magnitude of bounded disturbance d.t / is larger than 1, we have the fact that x1 grows unbounded although x2 is bounded. This fact results from the first equation being not globally ISS with respect to x2 as input, see Section 2.2. Therefore, to analyze stability of (10.22) we first present the following lemma. Lemma 10.1. Consider the following nonlinear system: xP D f .t; x/ C g.t; x; .t //

(10.24)

where x 2 Rn ; .t / 2 Rm , f .t; x/ is piecewise continuous in t and locally Lipschitz in x. If there exist 0 > 0, positive constants ci , 1  i  4, j , 1  j  2, c, "0 , mu0 , and c0 , and a class-K function ˛0 such that the following conditions hold: C1. There exists a proper function V .t; x/ such that: c kxk2  V .t; x/  c2 kxk2 ;

1

@V



@x .t; x/  c3 kxk ;

c0 @V c4 kxk2 @V Cq : C f .t; x/   q @t @x 2 2 1 C c kxk 1 C c kxk

C2. g.t; x; .t // satisfies:

1 .1 C 2 kxk/ k.t/k : kg.t; x; .t //k  q 1 C c kxk2

C3. .t/ globally asymptotically converges to a ball centered at the origin: k.t/k  ˛0 .k.t0 /k/ e 0 .tt0 / C "0 ; 8t  t0  0:

252

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

C4. The following gain condition is satisfied:   1 c4  c3 "0 2 C > 0: 40 Then the solution x.t / of (10.24) globally asymptotically converges to a ball centered at the origin, i.e., kx.t /k  ˛ .k.x.t0 /; .t0 //k/ e .tt0 / C " .k.x.t0 /; .t0 //k/ ; 8t  t0  0; (10.25) q p a4 2 where ".s/ D c1 a1 1 C cxm .s/,  and ˛ are given as follows: a1 if p D 0 then 2 1 C cxm v u a2     u 0 q a2 a4 te 2  ; c2 s 2 C a3 C ˛.s/ D 1 C cxm c1 a1 1  D .a1  d /I 2

with

a1 if p ¤ 0 then 2 1 C cxm v   1 p u a2 0 2 q u 0 1 C cx a a C a a 1 3 2 4 m ue @ 2 2 A; ˇ ˇ c2 s C ˛.s/ D t 1 C cxm p ˇ c1 2 ˇˇ a1 ˇa1  0 1 C cxm ˇ! ˇ ˇ ˇ a1 a1 1 ˇ ˇ  .s/ D min p ;ˇp  0 ˇ I 2 ˇ 1 C cx 2 ˇ 2 1 C cxm m   1 c3 "0 c3 1 c4  2 c3 "0  ; a2 D .1 C 2 /˛0 .s/ ; c2 40 c1 1 c 3 ˛0 .s/ ; a4 D c0 C 1 c3 "0 0 ; a3 D 4 0 < d < a1 ;   .t  t0 /e d.tt0 / ; s   1 a4 p C Wm .s/ ; xm .s/ D c1 a1   2.a Ca Ca a =a /   2 3 2 4 1 a4 0 e Wm .s/ D V .s/  C a1    2.a Ca Ca a =a /  2 3 2 4 1 a3 C a2 a4 =a1 0 1 : e 8.a2 .s/ C a3 .s/ C a2 .s/a4 =a1 /

a1 D

10.2 State Feedback

253

When c0 D 0 and "0 D 0, we have " D 0 and the system (10.24) is GAS. Note that a finite value of  exists for an arbitrarily small positive d and the convergence rate  depends on the initial conditions. Proof. We first prove that x is bounded. From conditions C1, C2, and C3, we have    c3 kxk 1 kxk2 P Cq  V   c4  c3 "0 2 C q 40 1 C c kxk2 1 C c kxk2 1 c3 "0 0 C c0 .1 C 2 kxk/˛0 .k.t0 /k/ e 0 .tt0 / C q : 2 1 C c kxk

(10.26)

Upon application of completing squares, (10.26) can be rewritten as

  a3 e 0 .tt0 / a4 V Cq Cq VP   a1  a2 e 0 .tt0 / q 1 C c kxk2 1 C c kxk2 1 C c kxk2  .V  a =a / .a C a a =a /e 0 .tt0 /  4 1 3 2 4 1   a1  a2 e 0 .tt0 / q C : q 2 1 C c kxk 1 C c kxk2

(10.27)

Now consider the differential equation   P D  a1  a2 e 0 .tt0 / q

 1 C c kxk2

Take the following Lyapunov function:

C

.a3 C a2 a4 =a1 /e 0 .tt0 / : q 1 C c kxk2

1 W D 2; 2

(10.28)

(10.29)

whose time derivative along the solutions of (10.28) with the use of condition C4 satisfies WP  a2 e 0 .tt0 / q

2 1 C c kxk2

C

.a3 C a2 a4 =a1 /e 0 .tt0 / q 1 C c kxk2

a2 a4 /  .tt / a1 0 0

.a3 C a2 a4 /W e 0 .tt0 / C  2.a2 C a3 C a1 4

e

:

(10.30)

Hence W .t /  Wm which implies from (10.30) and the comparison principle found p in [6] that V .t /  a4 =a1 C 2Wm or kx.t /k  xm . Substituting this inequality into (10.27) yields

254

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

! a1 0 .tt0 / P  a2 e V C a3 e 0 .tt0 / C a4 : V  p 2 1 C cxm

(10.31)

Solving the inequality (10.31) readily yields (10.25). 

To investigate stability of the .ze ; v/-dynamics, we apply Lemma 10.1.

ze -dynamics We will apply Lemma 10.1 to estimate convergence of the path-following error, ze , dynamics. Indeed, it is seen that ze - dynamics, the first equation of (10.22), is in the form of the system studied in Lemma 10.1, i.e., zP e D fze .
/ C gze .
/;

(10.32)

ku t ze ; fze .
/ D  p 1 C .kze /2 
u t sin.we / ku t ze gze .
/ D p p cos.we /  1 : 2 2 1 C .kze / 1 C .kze /

(10.33)

where

We now verify all conditions of Lemma 10.1.

Verifying Condition C1. By taking the proper function V2 D 0:5ze2 , it is not hard to show that this condition holds with c D k 2 ; c1 D c2 D 0:5; c3 D 1; c4 D ku t ; c0 D 0: Verifying Condition C2. By noting that ˇ ˇ ˇ u sin.w  / 
ˇˇ ku t ze ˇ t e  cos.we /  1 ˇ  p ˇp ˇ 1 C .kze /2 ˇ 1 C .kze /2 condition C2 holds with

1 u t .1 C k jze j/ jwe j ; p 1 C .kze /2 1 D u t ;

2 D ku t :

(10.34)

(10.35)

(10.36)

Verifying Condition C3. This condition holds directly from (10.20). Verifying Condition C4. From (10.34), (10.36), and (10.20), this condition becomes

10.2 State Feedback

255

    ku t k > 0 , k  wr 1 C > 0: ku t  wr u t C 40 40

(10.37)

Since 0 is an arbitrarily positive constant, see Lemma 10.1, and wr can be made arbitrarily small, we can always pick a positive constant k such that (10.37) holds. Therefore, we have (10.38) jze .t /j  ˛z .
/e z .tt0 / C z ; where ˛z .
/; z and z are calculated as in Lemma 10.1. From (10.38), the requirement of initial conditions such that 1  c.s/ze  ı  > 0 becomes ˛z .
/ 

1  ı  z : jc.s/j

(10.39)

Boundedness of v To show that v is bounded, we take the following Lyapunov function 1 V3 D v 2 ; 2

(10.40)

whose time derivative along the solutions of the second equation of (10.22), after some simple calculation, satisfies    m11 urQ d22 1 m11 u2 2 VP3   wv .t /   C v 1C v m22 m22 m22 bm22 u2t  m11 u ku t sin. e / c.s/u t cos. e / k1 we  C  m22 b 1 C .kze /2 1  c.s/ze     u wv max we wv max : (10.41) tanh "1 m22 u2t m22 From (10.41), (10.20), (10.38), and (10.7), it can be shown that (10.42) jv.t /j  ˛v .
/e v .tt0 / C v ;

 
where ˛v .
/ is a nondecreasing function of we .t0 /; r.t Q 0 /; ze .t0 / , and v and v are positive constants. The constant v depends on the environmental disturbances and the path curvature. Indeed, this constant is equal to zero when there are no environmental disturbances and c.s/ D 0. Remark 10.2. From (10.39), it is seen that the initial conditions are mainly limited by the path curvature c.s/. If the initial conditions do not satisfy (10.39), one can generate a smooth curve segment, ˝0 , such that it is tangent to the path ˝ and that (10.39) holds, see Figure 10.2.

256

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

YE

ȍ

ȍ0

\ (t0 )

y(t0 )

OE

x(t0 )

XE

Figure 10.2 Overcoming “bad” initial conditions

10.3 Output Feedback 10.3.1 Observer Design To design an observer that estimates the sway and yaw velocities from the ship position and orientation measurements, we rewrite (10.1) as follows xP D u cos. /  v sin. /; xP 1 D F1 .x1 / C J1 .x1 /x2 ; xP 2 D K2 x2  D2 .x2 /x2 C 2 C w2 .t /;

(10.43)

where 









 u sin. / x1 D ; x2 D ; F1 .x1 / D ; 0 2 3 2 3 2 3 1 wv .t / 7 0 6 cos. / 0 7 6 m22 7 ; J1 .x1 / D 4 5; 2 D 4 1 5 ; w2 .t / D 6 5 4 1 0 1 r wr .t / m33 m33 y

y

3 2 3 3 d P vi d22 m11 u i1 0 jvj 7 6   6 m 7 6 m22 m22 7 7: D2 .x2 / D 6 iD2 22 7 ; K2 D 6 4 3 P dri 4 d33 5 .m11  m22 /u i1 5  0 jrj m33 m33 iD2 m33 (10.44) 2

10.3 Output Feedback

257

If we use the observer designed in [116, 126] and Chapter 9, then the matrix K2 has to be Hurwitz, i.e., the ship has to possess straight-line stability. This limitation motivates us to seek a new observer in this section. The idea is to find a coordinate transformation to transform the system (10.1) to a form that does not require K2 to be a Hurwitz matrix to design an observer. To this end, we define z2 D x2  2 .x; x1 /;

(10.45)

where 2 .x; x1 / is a locally Lipschitz vector function, which will be determined later. Using (10.45), we write the last two equations of (10.43) as follows: xP 1 D F1 .x1 / C J1 .x1 /2 .x; x1 / C J1 .x1 /z2 ;   @2 .x; x1 / @2 .x; x1 / g2 .x1 /  J1 .x1 / z2  D2 .z2 C zP 2 D K2 C @x @x1 2 .x; x1 //.z2 C 2 .x; x1 // C ˚.x; x1 / C 2 C w2 .t /; (10.46) where ˚.x; x1 / D K2 2 .x; x1 / 

@2 .x; x1 / .u cos. /  g2 .x1 /2 .x; x1 //  @x

@2 .x; x1 / .F1 .x1 / C J1 .x1 /2 .x; x1 // ; @x1  T g2 .x1 / D sin. / 0 ;

(10.47)

D2 .z2 C 2 .x; x1 // D 3 2 3 d P vi i1 7 6 0 7 6 iD2 m22 jz21 C 21 .x; x1 /j 6 7; 7 6 3 4 5 P dri 0 jz22 C 21 .x; x1 /ji1 m iD2 33

where z2i and 2i .x; x1 /; i D 1; 2, are the first and second elements of z2 and 2 .x; x1 /, respectively. From (10.46), one can design a reduced order observer. However it is often noise-sensitive. We here propose the following observer: xPO 1 D F1 .x1 / C J1 .x1 /2 .x; x1 / C J1 .x1 /Oz2 C K01 .x1  xO 1 /;   @2 .x; x1 / @2 .x; x1 / zPO 2 D K2 C g2 .x1 /  J1 .x1 / zO 2  D2 .Oz2 C @x @x1 2 .x; x1 //.Oz2 C 2 .x; x1 // C ˚.x; x1 / C 2 C K02 .x1  xO 1 / (10.48) where xO 1 WD ŒyO O T and zO 2 are estimates of x1 and z2 , respectively, and K01 and K02 are the observer gains to be selected later. From (10.46) and (10.48), we have the observer error dynamics:

258

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

xPQ 1 D K01 xQ 1 C J1 .x1 /Qz2 ;   @2 .x; x1 / @2 .x; x1 / J1 .x1 / zQ 2  g2 .x1 /  zQP 2 D K02 xQ 1 C K2 C @x @x1  D2 .z2 C 2 .x; x1 //.z2 C 2 .x; x1 //  D2 .Oz2 C  2 .x; x1 //.Oz2 C 2 .x; x1 // C w2 .t /; (10.49)

where xQ 1 WD ŒyQ Q T D x1  xO 1 , and zQ 2 D z2  zO 2 . Now, considering the Lyapunov function (10.50) V0 D xQ 1T P01 xQ 1 C zQ T2 P02 zQ 2 ;

where P01 and P02 are positive definite matrices, whose time derivative along the solutions of (10.49) satisfies: T .t /P02 zQ 2 C zQ T2 P02 w2 .t /; VP0  xQ 1T Q01 xQ 1  zQ T2 Q02 zQ 2 C w2

(10.51)

where T Q01 D K01 P01 C P01 K01 ; T  @2 .x; x1 / @2 .x; x1 / Q02 D  K2 C J1 .x1 / P02  g2 .x1 /  @x @x1   @2 .x; x1 / @2 .x; x1 / P02 K2 C J1 .x1 / ; (10.52) g2 .x1 /  @x @x1

and we have chosen P02 K02  J1T .x1 /P01 D 0

(10.53)

and used the following inequality: .D2 .z2 C 2 .x; x1 //.z2 C 2 .x; x1 //  D2 .Oz2 C 2 .x; x1 // .Oz2 C 2 .x; x1 /// P02 zQ 2 C zQ T2 P02 .D2 .z2 C 2 .x; x1 //.z2 C 2 .x; x1 //  D2 .Oz2 C 2 .x; x1 //.Oz2 C 2 .x; x1 ///  0:

(10.54)

It is not hard to show that there always exist K01 , K02 and 2 .x; x1 / such that Q01 and Q02 are positive definite and (10.53) holds. For example, one can take T K01 D K01 > 0;

1 T K02 D P02 J1 .x1 /P01 ;  T m11 u 2 .x; x1 / D  0 : m22

(10.55)

Therefore (10.51) implies that kQx0 .t /k  '0 kQx0 .t0 /k e 0 .tt0 / C 0 ;

(10.56)

10.3 Output Feedback

259

where '0 , 0 , and 0 are some positive constants, and xQ 0 D ŒxQ 1 zQ 2 T . It is noted .Q02 / that 0 cannot be made arbitrarily small since the ratio min is maximized with max .P02 / the choice of Q02 D diag.1; 1/. By defining an estimate of the velocity vector as xO 2 D zO 2 C 2 .x; x1 /;

(10.57)

where xO 2 D ŒvO r Q T satisfy: O T , then the velocity observer errors xQ 2 D ŒvQ r Q /; r.t Q //k  '0 kxQ 0 .t0 /k e 0 .tt0 / C 0 : k.v.t

(10.58)

It is seen that when there are no environmental disturbances, the observer errors globally exponentially tend to zero.

10.3.2 Control Design The control design in this section is very similar to the one of the state feedback controller. We therefore present it briefly. Define   vO ; O D arctan u p uO t D u2 C vO 2 ; (10.59)  O e D e C : O O using the Taylor series expansion, it can be shown that By denoting Q D   , ˇ ˇ ˇ ˇ ˇ O v  vO ˇˇ 1 ˇ.t Q /ˇ D ˇ arctan.v=u/  arctan.v=u/  jv.t Q /j : (10.60) ˇ .v  v/=u O u ˇ u Similar to the state feedback control design, we apply the following coordinate transformation ! kz e   (10.61) wO e D O e C arcsin p 1 C .kze /2

to (10.48) in conjunction with (10.59). After some simple calculation, we have k uO t ze uO t sin.wO e / .cos.wO e /  1/k uO t ze zP e D  p Cp  C p 1 C .kze /2 1 C .kze /2 1 C .kze /2 O O vQ cos.wO e  / vQ sin.wO   /kz e C p e ; p 2 2 1 C .kze / 1 C .kze /   m11 u2 c.s/uO t cos. O e /  P wO e D rO 1    2 m22 uO t 1  c.s/ze

260

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

! 3 X d22 dvi k uO t sin. O e / i1 O vO C vO C C jvj m22 m22 1 C .kze /2 iD2   m11 u2 k vQ cos. e / c.s/vQ sin. e / u C ; cos. / y Q C r Q 1  C m22 uO 2t 1 C .kze /2 1  c.s/ze uO 2t u uO 2t

3

X dvi m11 u d22 m11 vPO D  O i1 vO C cos. /yQ  rO  vO  ur; Q jvj m22 m22 m22 m22 iD2

3

X dri .m11  m22 /u d33 1 rPO D vO  rO  r C Q O i1 rO C jrj m33 m33 m33 m33

(10.62)

iD2

where for simplicity, we have taken K02 D J1T .x1 /;  m11 u 2 .x; x1 / D  m22

T 0 :

(10.63)

The control design consists of two steps as follows:

Step 1 Introduce the virtual yaw velocity error as re D rO  rOd ;

(10.64)

where rOd is the virtual yaw velocity control which is chosen as  1 c.s/uO t k1 wO e C cos. O e /C rOd D m11 u2 1  c.s/ze 1  m22 uO 2 t ! ! 3 X u d22 k uO t sin. O e / d1 wO e dvi i1 vO C vO   O ; jvj m22 1 C .kze /2 .1  c.s/ze /2 uO 2t m22 iD2

(10.65)

where k1 and d1 are positive constants. The term multiplied by d1 is the nonlinear damping term to overcome the effect of the observer errors. It is seen that rOd is a smooth function of s; ze ; e , and v. O Substituting (10.64) and (10.65) into the second equation of (10.62) results in   m11 u2 d1 wO e O T .
/xQ  wPO e D k1 wO e C re 1  C f ; (10.66) 1 m22 uO 2t .1  c.s/ze /2 where

10.3 Output Feedback

261

u cos. / 6 uO 2t 6 6 6 0 6 fO1 .
/ D 6 c.s/ sin. e / k cos. e / 6 6 1  c.s/ze C 1 C .kze /2 6 4 m11 u2 1 m22 uO 2t 2

3

2 3 7 yQ 7 6 7 7 7 6 7 6 Q7 7 7 6 Q ; x D 7 6 7: 7 7 6 v Q 7 4 5 7 5 rQ

(10.67)

Step 2 Differentiating both sides of (10.64) yields 3

rPe D

X dri d33 1 .m11  m22 /u vO  rO  r  O i1 rO C jrj m33 m33 m33 m33 iD2

cos. O e /

! c.s/ cos. O e / rO   1  c.s/ze ! i1 Q O vO C fO2T .
/x; (10.68) jvj

@rOd @rOd @rOd  sin. O e /  @s 1  c.s/ze @ze @ e @rOd @vO

3



X dvi d22 m11 u rO  vO  m22 m22 m22 iD2

where 2



@rOd cos. / @vO 1

3

6 7 6 7 6 7 6 7 7 fO2 .
/ D 6 @ r O @ r O sin. c.s/ sin. / / @ r O e 7: e d d 6 d  cos. e /  6 7 @ e 1  c.s/ze 7 6 @s 1  c.s/ze @ze 4 5 @rOd @rOd m11  u  @ e @vO m22

(10.69)

From (10.68), the control r is designed without canceling the useful nonlinear damping terms as follows: 3

X dri d33 .m11  m22 /u vO C rOd C O i1 rOd C jrj m33 m33 m33 iD2 !  O e / O @rOd @ r O c.s/ cos. @rOd @rOd cos. e / d  C uO t sin. O e / C rO    @s 1  c.s/ze @ze @ e 1  c.s/ze @vO !   3 4 X X m11 u2 dvi d22 m11 u i1 ; d2 fO2i2 .
/re  wO e 1  rO C vO C vO  O jvj m22 m22 m22 m22 uO 2t

r D m33 k2 re 

iD2

iD1

(10.70)

262

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

where k2 and d2 are positive design constants, fO2i .
/ is the i th component of fO2 .
/. Substituting (10.70) into (10.68) yields   3 X dri d33 m11 u2 i1 O wO e C rPe D k2 re  re  re  1  jrj m33 m33 m22 uO 2t iD2

fO2T .
/xQ 

4 X iD1

d2 fO2i2 .
/re :

(10.71)

We now present the second main result of this chapter, the proof of which is given in the next section. Theorem 10.2. Under Assumption 10.2, if the output feedback control law (10.70) and the observer (10.48) are applied to the ship system (10.1) then there exist feasible initial conditions such that the regulation errors .ze .t /; e .t // converge to a ball centered at the origin with an appropriate choice of the design constants k, ki and di with i D 1; 2. Furthermore when there are no environmental disturbances, the regulation errors converge to zero asymptotically. In addition, the velocity v.t / is always bounded.

10.3.3 Stability Analysis To prove Theorem 10.2, we first consider the .wO e ; re /-dynamics then move to the .v; O ze / dynamics. .wO e ; re /-dynamics Consider the following Lyapunov function 1 1 V1 D wO e2 C re2 ; 2 2

(10.72)

whose time derivative along the solutions of (10.66) and (10.71) satisfies VP1 D k1 wO e2 C wO e fO1T .
/xQ 

d33 2 d1 wO e2  k2 re2  r  .1  c.s/ze /2 m33 e

4 3 X X dri i1 2 T 2 O Q O fO2i2 .
/ re C re f2 .
/x  d2 re jrj m33 iD2

Q 2;   .k1  11 / wO e2  k2 re2 C 12 kxk

iD1

(10.73)

where 11 and 12 are some positive constants. From (10.73) and (10.56), it is direct to show that

10.3 Output Feedback

263

 



wO .t /; re .t /  wO  .t0 /; re .t0 / e 1 .tt0 / C 1 ; e

e

(10.74)

for some positive constants 1 and 1 . It is noted that 1 can be made arbitrarily small by adjusting the design constants k1 , k2 , d1 , and d2 .

.v; O ze /-dynamics To obtain convergence of ze , we apply Lemma 10.1 to the first equation of (10.62) by verifying all conditions of this lemma. Indeed, the first equation of (10.62) is in the form of the system studied in Lemma 10.1 by writing this equation as zP e D fze .
/ C gze .
/;

(10.75)

where k uO t ze fze .
/ D  p ; 1 C .kze /2 uO t sin.wO e / k uO t ze gze .
/ D p  .cos.wO e /  1/ p C 1 C .kze /2 1 C .kze /2 O O vQ sin.wO   /kz vQ cos.wO e  / e C p e : p 1 C .kze /2 1 C .kze /2

(10.76)

Verifying Condition C1. By taking the Lyapunov function V2 D 0:5ze2 , it can be shown that this condition holds with c0 D 0; c1 D c2 D 0:5; c3 D 1; c D k 2 ; c4 D k uO t :

(10.77)

Verifying Condition C2. From (10.76), we have


1 .uO t C 1 C .uO t C 1/k jze j/ v; Q wO e ; jgze .
/j  p 1 C .kze /2

(10.78)

which implies that condition C2 holds with

1 D uO t C 1; 2 D k .uO t C 1/ :

(10.79)

Verifying Condition C3. This condition directly holds from (10.74) and (10.56). Verifying Condition C4. From (10.77), (10.74), and (10.56), this condition becomes   uO t C 1 k uO t  .1 C 0 / C k.uO t C 1/ > 0: (10.80) 40 It is noted that 1 can be made arbitrarily small but 0 cannot, see (10.56). Therefore Condition C4 is satisfied if the environmental disturbances are not too large and

264

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

the ship forward speed u is high enough such that (10.80) holds for some positive constant 0 . This can be explained as follows: When there are large environmental disturbances, the observer cannot estimate sufficiently accurate velocities. In addition, the low ship speed cannot compensate for the environmental disturbances. Consequently, the ship under large environmental disturbances in question will diverge from the path. All the conditions of Lemma 10.1 have been verified, we have jze .t /j  ˛z .
/ e z .tt0 / C z ;

(10.81)

where ˛z ; z and z are calculated as in Lemma 10.1. The feasibility of initial conditions such that 1  c.s/ze  ı  > 0 is similar to the case of the state feedback design, see (10.39). Similar to the state feedback control design, it is not hard to show that jv.t /j  ˛v .
/ e v .tt0 / C v

(10.82)

where ˛v ; v and v are calculated similarly.

10.4 Simulations This section validates the control laws (10.15) and (10.70) by simulating them on a monohull ship with the parameters given in Section 5.4. In the simulation, we assume that the disturbances are wv D 26  104 .1 C rand.
//, and wr D 950  105 .1 C rand.
//, where rand.
/ is random zero-mean noise with the uniform distribution on the interval Œ0:5 0:5. This choice results in nonzero-mean disturbances. It should be noted that only boundaries of the environmental disturbances are needed in our proposed controllers. We simulate both state feedback and output feedback cases. For both cases, the ship forward speed is 4 m/s.

10.4.1 State Feedback Simulation Results The initial conditions are chosen as Œx.0/; y.0/; .0/; v.0/; r.0/ D Œ 250; 50; 0:5; 0; 0 : The reference path is a circle centered at the origin with a radius of 200 m. The control parameters are taken as k D 0:5, k1 D k2 D 10, and "i D 0:2. Figure 10.3 plots the ship positions and orientation in the .x; y/-plane. The path-following errors are plotted in Figure 10.4. It can be seen from this figure that the orientation error  e converges close to zero but not e due to nonzero sway velocity, and that the position path-following error, ze , also converges closely to zero. The sway and yaw velocities and control torque r are drawn in Figure 10.5. The high control effort

10.5 Conclusions

265

is due to the fact that we simulate on data of a real ship and large environmental disturbances. However, the control magnitude is within the limit of the maximum yaw moment. It is clearly seen from Figure 10.3 that the controller forces the vessel to move from its initial position to the path in the direction perpendicular to the circle. When the vessel closes to the path, the controller is able to drive it along the path in the desired direction. These observations coincide with our control objective. All the simulation results plotted in Figures 10.3, 10.4, and 10.5 illustrate that our control goal is achieved as stated in Theorem 10.1. It is noted that for clarity, we only plot the path-following errors, sway, and yaw velocities, and control input for the first 20 seconds.

10.4.2 Output Feedback Simulation Results The initial conditions are Œx.0/; y.0/; .0/; v.0/; r.0/iD Œ 250; 50; 0:5; 0:2; 0:2 ; h O .0/; v.0/; x.0/; O y.0/; O O r.0/ O D Œ 220; 40; 0:2; 0:2; 0:1; 0:1 :

The observer gain matrix is matrix chosen as K01 D diag.1:5; 1:5/. The control gains are k D 0:5, k1 D k2 D 15, d1 D d2 D 0:5, and "i D 0:2. The environmental disturbances are the same as full state feedback case. Simulation results are plotted in Figures 10.6– 10.8. All the comments on the results are similar to the state feedback simulation results. However, due to the effect of the observer, the performance of the output feedback case is slightly worse than the performance of the state feedback case.

10.5 Conclusions Although the sway velocity v is controlled in some conventional course-keeping control systems, see for example [12, 134], using Nomoto’s second-order model, the sway displacement y is not controlled and can be unbounded. This phenomenon can be seen from the second equation of (10.1), i.e., yP D u sin. / C v cos. /, that boundedness of the sway velocity v does not imply that of the sway displacement y. The controllers proposed in this chapter did control the sway displacement y and kept the sway velocity v bounded, and cover the ones in [126] where the linear course stabilization of the underactuated surface ships is addressed. This can be seen from (10.2) by setting ze D y and the curvature of the path c.s/ equal zero. The work in this chapter is based on [135]. A combination of the results in this chapter with the results in [127] is straightforward to cover the case where the roll and pitch modes are not ignored. By setting the value of k equal zero, the proposed controllers in this chapter reduce to the conventional course-keeping controllers,

266

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

200 150 100

y [m]

50 0 −50 −100 −150

−200

−100

0

100

200

x [m]

Figure 10.3 State feedback, ship positions and orientation in the .x; y/-plane: Real path (solid line) and reference path (dash-dotted line)

see for example [11, 12, 134]. However, the sway displacement will grow linearly unbounded under nonzero-mean environmental disturbances. The main limitation of both state feedback and output feedback controllers designed in this chapter is that the ship must not be too far away from the reference path at the initial time, i.e., the condition (10.3) holds at the initial time t0 . This limitation will be removed in the next chapter where a different approach for solving the path-following problem is proposed.

10.5 Conclusions

267

a

e

Z [m]

100 50 0 0

5

10 Time [s]

15

20

15

20

15

20

0

e

ψ* [rad]

b

2

−2 0

5

10 Time [s]

0

e

ψ [rad]

c

2

−2 0

5

10 Time [s]

Figure 10.4 State feedback, path-following errors: a. Position error ze ; b. Modified heading error  e e ; c. True heading error

a

v [m/s]

2 0 −2 0

5

10 Time [s]

15

20

15

20

15

20

r [rad/s]

b

5 0 −5 0

5

c

x 10

0

r

τ [KNm]

5

5

10 Time [s]

−5 0

5

10 Time [s]

Figure 10.5 State feedback: a. Sway velocity v; b. Yaw velocity r; c. Control torque r

268

10 Path-following of Underactuated Ships Using Serret–Frenet Coordinates

200 150 100

y [m]

50 0 −50 −100 −150

−200

−100

0

100

200

x [m]

Figure 10.6 Output feedback, ship positions and orientation in the .x; y/-plane: Real path (solid line) and reference path (dash-dotted line)

a

e

Z [m]

100 50 0 0

5

10 Time [s]

15

20

15

20

15

20

0

* e

ψ [rad]

b

2

−2 0

5

10 Time [s]

−1

e

ψ [rad]

c

0

−2 0

5

10 Time [s]

Figure 10.7 Output feedback, path-following errors: a. Position error ze ; b. Modified heading error e ; c. True heading error e

10.5 Conclusions

269

a

v, vhat [m/s]

0.5 0 −0.5 0

5

10 time [s]

15

20

15

20

15

20

r, r

hat

[rad/s]

b

2 0 −2 0

5

c

x 10

0

r

τ [KNm]

5

5

10 Time [s]

−5 0

5

10 Time [s]

Figure 10.8 Output feedback results: a. Sway velocity v (solid line) and its estimate vO (dashdotted line); b. Yaw velocity r (solid line) and its estimate rO (dash-dotted line); c. Control torque r

Chapter 11

Path-following of Underactuated Ships Using Polar Coordinates

This chapter is devoted to a different path-following approach from the preceding chapter. The approach is motivated by the practical experience of steering a ship in the sense that when traveling in open sea the helmsman first looks at the weather map, then generates way-points to avoid the vessel moving into bad weather areas. A continuous reference path curve is then generated so that it goes via (almost) all of the way-points. It is then practical to steer a vessel so that it is in a tube of nonzero adjustable diameter centered on the reference path, and moves along the path with the desired speed. In comparison with the approach in the preceding chapter, the control system developed in this chapter allows the ship to be far away from the desired path at the initial time, and covers situations of practical importance such as parking and point-to-point navigation.

11.1 Control Objective For the reader’s convenience, the mathematical model of the underactuated ship moving in surge, sway, and yaw, see Sections 3.4.1.1 and 3.4.1.2, is rewritten as xP D u cos. /  v sin. /; yP D u sin. / C v cos. /; P D r; 3

uP D

X dui m22 d11 1 1 vr  u u C wu .t /; juji1 u C m11 m11 m11 m11 m11

(11.1)

iD2

3

vP D 

X dvi m11 d22 1 ur  v wv .t /; jvji1 v C m22 m22 m22 m22 iD2

3

rP D

X dri d33 1 1 .m11  m22 / uv  r r C wr .t /; jrji1 r C m33 m33 m11 m33 m33 iD2

271

272

11 Path-following of Underactuated Ships Using Polar Coordinates

where all of symbols have been defined in Section 3.3. The bounded time-varying terms, wu .t /; wv .t /; and wr .t /, are the environmental disturbances induced by waves, wind, and ocean currents with jwu .t /j  wu max < 1, jwv .t /j  wv max < 1, and jwr .t /j  wr max < 1. In this chapter, we consider a control objective of designing the surge force u and the yaw moment r to force the underactuated ship (11.1) to follow a specified path ˝, see Figure 11.1. In this figure, P is the ship’s center of mass and Pd is a point attached to the virtual ship, which moves along the path with speed of u0 . If we are able to steer the ship to closely follow a virtual ship that moves along the path with a desired speed u0 , then the control objective is fulfilled, i.e., the ship is in a tube of nonzero diameter centered on the reference path and moves along the specified path at the speed u0 . Roughly speaking, the approach is to steer the ship so that it heads toward the virtual ship and eliminates the distance between itself and the virtual ship.

Virtual ship

Real ship

YE

u0 Pd

ȍ

yd

x

ze

\e y

OE

Ts

\d \

P x

x

xd

XE

Figure 11.1 General framework of ship path-following

Define

where

xe D xd  x; ye D yd  x; e Dp  d ; ze D xe2 C ye2 ; d

D arcsin



 ye : ze

(11.2)

(11.3)

11.1 Control Objective

273

With (11.2), the above control objective is mathematically stated as follows. Control Objective. Under Assumption 11.1, design the surge force u and the yaw moment r to make the underactuated ship (11.1) follow the path ˝ given by xd D xd .s/; yd D yd .s/;

(11.4)

where s is the path parameter variable, such that lim ze .t /  ze ;

t!1

lim j

t!1

with ze and

e

e .t /j



e;

(11.5)

being arbitrarily small positive constants.

Assumption 11.1.  2  2 @yd d C  Rmax < 1. 1. The reference path is regular, i.e., 0 < Rmin  @x @s @s 2. The minimum radius of the osculating circle of the path is larger than or equal to the minimum possible turning radius of the ship. Remark 11.1. 1. Assumption 11.1 ensures that the path is feasible for the ship to follow. 2. The angle d is not defined at ze D 0 but limze !0 d D s with s being the orientation angle of the virtual ship on the path, see Figure 11.1. Therefore, the fulfillment of the control objective guarantees that the ship closely follows the path in terms of both position and orientation. If the reference path is not regular, then we can often split it into regular pieces and consider each of them separately. This is a case of point-to-point navigation. 3. The path parameter, s, is not the arclength of the path in general. For example, a circle with radius of R centered at the origin can be described as xd D R cos.s/ and yd D R sin.s/, see [136] for more details. If one differentiates both sides of e D  d to get the P e -dynamics, there will be discontinuity in the P e -dynamics on the ye -axis. This discontinuity will cause difficulties in applying the backstepping technique. To get around this problem, we compute the P e -dynamics based on xe sin. /  ye cos. / ; ze xe cos. / C ye sin. / cos. e / D : ze

sin.

e/

D

We now use (11.2) and (11.6) to transform (11.1) to   xe @xd ye @yd sP ; C zPe D  cos. e /u C sin. e /v C ze @s ze @s

(11.6)

274

11 Path-following of Underactuated Ships Using Polar Coordinates

  sP @xd sin. / xe sin. e /   cos. e / @s ze ze2   @yd cos. / ye sin. e / sin. e /u C cos. C C 2 @s ze ze ze

Pe D r C



e /v

;

3

uP D

X dui d11 1 1 m22 vr  u u C wu .t /; juji1 u C m11 m11 m11 m11 m11 iD2

3

vP D 

X dvi m11 d22 1 ur  v wv .t /; jvji1 v C m22 m22 m22 m22 iD2

3

rP D

X dri d33 1 1 .m11  m22 / uv  r r C wr .t /: jrji1 r C m33 m33 m11 m33 m33 iD2

(11.7)

It is noted that the P e -dynamic is not defined at e D ˙0:5. However, our controller will guarantee j e .t /j < 0:5; 8 0  t < 1 for feasible initial conditions. Therefore, we will design the surge force u and the yaw moment r for (11.7) to yield the control objective. A procedure to design a stabilizer for the path-following error system (11.7) will be presented in detail. The triangular structure of (11.7) suggests that we design the actual controls u and r in two stages. First, we design the virtual velocity controls for u, r and choose sP to ultimately stabilize ze and e at the origin. Based on the backstepping technique, controls u and r will then be designed to make the errors between the virtual velocity controls, and their actual values exponentially tend to a small ball centered at the origin. Since the ship parameters are unknown, an adaptation scheme is also introduced in this step to estimate their values used in the control laws. The nonzero lower bound of ze to guarantee the existence of d and the boundedness of the sway velocity, v, are analyzed.

11.2 Control Design 11.2.1 Step 1 The ze and e dynamics have three inputs that can be chosen to stabilize ze and e , namely sP ; u, and r. The input r should be designed to stabilize the e -dynamics at the origin. Therefore, two inputs, sP , u, can be used to ultimately stabilize ze at the origin. We can either choose the input u or sP and then design the remaining input. If we fix sP , then the virtual ship is allowed to move at a desired speed. The real ship will follow the virtual one on the path by the controller, and vice versa. In this chapter we choose to fix sP . This allows us to adjust the initial conditions in most cases without moving the ships, see Section 11.3. Since the transformed system (11.7) is not defined at ze D 0 and e D ˙0:5, we first assume in the following

11.2 Control Design

275

that ze .t /  ze > 0 and j e .t /j < 0:5; 8 0  t < 1. In Section 11.3, we will then show that there exist initial conditions such that this hypothesis holds. Define uQ D u  ud ; rQ D r  rd ;

(11.8)

where ud and rd are the virtual controls of u and r, respectively. As discussed above, we choose the virtual controls ud and rd , and sP as follows:   xe @xd ye @yd u0 .t; ze / ud D k1 .ze  ıe / C C r 2  2 C tan . e / v; ze @s ze @s @yd @xd C @s @s (11.9)

where 

   @xd sin. / xe sin. e / @yd cos. / ye sin.   C rd D  @s ze ze2 @s ze ze2 u0 .t; ze / sin. e /ud C cos. e /v  k2 e ; s 2  2  ze @xd @yd C @s @s and

u0 .t; ze / cos. e / sP D s    ; @yd 2 @xd 2 C @s @s

e/





(11.10)

(11.11)

where k1 , k2 , and ıe are positive constants to be selected later, u0 .t; ze / ¤ 0; 8t  t0  0, and ze 2 R, is the speed of the virtual ship on the path. Indeed, one can choose this speed to be a constant. However, the time-varying speed and position pathfollowing dependence of the virtual ship on the path is more desirable, especially when the ship starts to follow the path. For example, one might choose u0 .t; ze / D u0 .1  1 e 2 .tt0 / /e 3 ze ;

(11.12)

where u0 ¤ 0; i > 0; i D 1; 2; 3; 1 < 1. The choice of u0 .t; ze / in (11.12) has the following desired feature: When the path-following error, ze , is large, the virtual ship will wait for the real one; when ze is small, the virtual ship will move along the path at the speed closed to u0 and the real one follows it within the specified look-ahead distance. This feature is suitable in practice because it avoids using a high gain control for large signal ze . Remark 11.2. If the sway velocity is assumed to be bounded by the surge velocity as in [129], the term sin. e /v is not required to be canceled. This controller can be designed similarly to the one in this chapter. Lemma 10.1 can be directly applied to the stability analysis. It is noted that the sway velocity is not needed to be bounded by the surge velocity with a relatively small constant.

276

11 Path-following of Underactuated Ships Using Polar Coordinates

Substituting (11.8)–(11.11) into the first two equations of (11.7) results in zP e D k1 cos. P e D k2

 ıe /  cos. sin. e / uQ C r: Q eC ze e /.ze

Q e /u; (11.13)

11.2.2 Step 2 By noting that with ze .t /  ze > 0 and j e .t /j < 0:5; 8 0  t < 1, the virtual controls ud and rd are smooth functions of xe ; ye ; s; u0 ; and v, differentiating both sides of (11.8) with (11.9) and (11.10) yields 3

X dui d11 1 1 m22 vr  u u C wu .t /  uPQ D juji1 u C m11 m11 m11 m11 m11 iD2

@ud @ud @ud @ud P @ud @ud xP e  yPe  sP  uP 0    @xe @ye @s @ @u0 @v ! 3 X m11 d22 @ud 1 dvi i1  ur  v wv .t /; jvj v  m22 m22 m22 @v m22 iD2

3

X dri d33 1 .m11  m22 / uv  r r C rPQ D jrji1 r C m33 m33 m11 m33

(11.14)

iD2

1 @rd @rd @rd @rd P @rd wr .t /  xP e  yPe  uP 0  sP   m33 @xe @ye @s @ @u0 ! 3 X @rd d22 @rd 1 dvi m11 ur  v wv .t /;  jvji1 v  @v m22 m22 m22 @v m22 iD2

where for convenience of choosing u0 , the terms

@ud @xe

,

@ud @ye

,

@rd @xe

and

@rd @ye

0 0 include @u and @u , which are lumped into uP 0 . From (11.14), we choose @xe @ye controls u and r without canceling useful nonlinear damping terms as

sin. e / u D k3 uQ  O1T f1 .
/  ze ! O32 u Q  @u @u d d ; tanh O32 @v @v "2

uQ O31 O e  31 tanh "1

rQ O33 r D k4 rQ  O2T f2 .
/  e  O33 tanh "3 and the update laws as

!

do not

the actual

 (11.15)

!

! O34 @r @r r Q  d d  O34 ; tanh @v @v "4

11.2 Control Design

277

  P Q 1j .
/; O1j ; 1  j  9; O1j D 1j proj uf   P O2j D 2j proj rf Q 2j .
/; O2j ; 1  j  9;   P O31 D 31 proj juj Q ; O31 ; ˇ ˇ  ˇ P d ˇ O ; O32 D 32 proj ˇuQ @u ; @v ˇ 32   PO 33 D 33 proj jrj Q ; O33 ; ˇ ˇ  ˇ dˇ O P ;  O34 D 34 proj ˇrQ @r ˇ 34 @v

(11.16)

where k3 , k4 , "i , 1j , 2j , 3i , 1  i  4, and 1  j  9 are positive constants to be selected later, and f1j .
/ and f2j .
/ are the j th elements of f1 .
/ and f2 .
/, respectively, with "  f1 .
/ D vr ud  juj ud u2 ud  @ud xP e C @ud yPe C @ud sP C @xe @ye @s   @ud P @ud @ud @ud @ud @ud 2 T ur v uP 0 C jvj v jvj v ; @ @u0 @v @v @v @v "  f2 .
/ D uv rd  jrj rd r 2 rd  @rd xP e C @rd yPe C @rd sP C @xe @ye @s   @rd P @rd @rd @rd @rd @rd 2 T uP 0 ur v C jvj v jvj v ; (11.17) @ @u0 @v @v @v @v Oij ; 1  i  3 is the j th element of Oi , which is an estimate of i with 1 D

"

2 m22 d11 du2 du3 m11 m11 m22

d22 m11 dv2 m11 dv3 m11 m22 m22 m22

#T

; #T

"

2 D .m11  m22 / d33 dr2 dr3 m33 m11 m33 d22 m33 dv2 m33 dv3 m33 ; m22 m22 m22 m22 #T " : (11.18) 3 D wu max m11 wv max wr max m33 wv max m22 m22 The operator, proj, is the Lipschitz continuous projection algorithm (repeated here for the reader’s convenience) as follows: proj.$; !/ O D$

if  .!/ O  0;

proj.$; !/ O D$

if  .!/ O  0 and !O .!/ O $  0;

proj.$; !/ O D .1   .!// O $

if  .!/ O > 0 and !O .!/ O $ > 0;

(11.19)

278

11 Path-following of Underactuated Ships Using Polar Coordinates !O 2 ! 2

!/ O M O D @. where  .!/ O D 2 C2! ,  is an arbitrarily small positive con; !O .!/ @!O M stant, !O is an estimate of ! and j!j  !M . The projection algorithm is such that if !PO D proj($ , !/ O and !.t O 0 /  !M then

1. 2. 3. 4.

!.t O /  !M C ; 8 0  t0  t < 1; proj($; !) O is Lipschitz continuous, O  j$j ; jproj($; !)j !proj($; Q !) O  !$ Q with !Q D !  !. O

Substituting (11.15) into (11.14) yields ! 3 d11 X dui 1 sin. e / k3 i1 C C uQ  juj e m11 m11 m11 m11 ze iD2 ! 1 OT uQ O31 1 T 1 O C  f1 .
/ C 31 tanh  f1 .
/  m11 1 m11 1 m11 "1 ! @ud 1 @ud 1 @ud uQ O32 O  wu .t /  32 wv .t /; tanh m11 @v @v "2 @v m22 ! 3 X 1 OT dri 1 i1 PrQ D  k4 C d33  rQ   f2 .
/ C jrj e m33 m33 m11 m33 m33 2 iD2 ! 1 T rQ O33 1 1 O  f2 .
/  wr .t /  33 tanh C m33 2 m33 "3 m33 ! 1 O @rd @rd 1 @rd rQ O34 34  tanh wv .t /: (11.20) m33 @v @v "4 @v m22

uPQ D 

We now present the following result, the proof of which is given in the next section. Theorem 11.1. Assume that (a) the ship inertia, added mass and damping matrices are diagonal and (b) Assumption 11.1 is satisfied. If the adaptive state feedback control law (11.15) and adaptation law (11.16) are applied to the ship system (11.1) then there exist feasible initial conditions such that the path-following errors .ze .t /; e .t // converge to a small ball centered at the origin with an appropriate choice of the design constants ki , "i , 1j , 2j , and 3i for all 1  i  4; 1  j  9.

11.3 Stability Analysis To prove Theorem 11.1, we first consider the . ze - and v-dynamics.

Q r/-dynamics, Q e ; u;

then move to the

11.3 Stability Analysis

.

279

Q r/-dynamics Q e ; u;

Consider the following Lyapunov function V1 D

3

1 2

2 e C

m11 2 m33 2 1 X Q T 1 Q i i i ; uQ C rQ C 2 2 2

(11.21)

iD1


where Qi D i  Oi and i D diag ij . Differentiating both sides of (11.21) along (11.14), (11.15), and (11.16) yields VP1  k2

2 Q 2  .k4 C d33 / rQ 2 C 0:2785 e  .k3 C d11 /u

4 X

"i :

(11.22)

iD1

By subtracting and adding

1 2

3 P QiT i1 Qi to the right-hand side of (11.22), we ar-

iD1

rive at

VP1  ıV1 C ;

(11.23)

where   2.k3 C d11 / 2.k4 C d33 / ; ; ı D min 1; 2k2 ; m11 m33 4

3

D

X 1 X Q T 1 Q i i i C 0:2785 "i : 2

(11.24)

iD1

iD1

From (11.23), it is direct to show that  V1 .t /  V1 .t0 /e ı.tt0 / C ; 8t  t0  0; ı

(11.25)

which further yields   1 m11 m33 min ; ; kX1 .t /k2  2 2 2   1 m11 m33 1  1 1 2 ; ; ; ; ; e ı.tt0 / C ; kX1 .t0 /k max 2 2 2 2 1j 2 2j 2 3j ı (11.26) where X1 .t / D X1 .t0 / Therefore we have



D

e .t /



u.t Q / r.t Q /

e .t0 /

T

;

u.t Q 0 / r.t Q 0 / Q1T .t0 / Q2T .t0 / Q3T .t0 /

kX1 .t /k  ˛1 e 1 .tt0 / C 1 ; 8t  t0  0;

T

:

(11.27)

280

11 Path-following of Underactuated Ships Using Polar Coordinates

where v   u 1 m11 m33 1 1 1 u ; ; ; ; ; u max t 2 2 2 2 1j 2 2j 2 3j p1 D ; 0:5 min .1; m11 ; m33 / ˛1 D p1 kX1 .t0 /k ; ı 1 D ; 2 r  1 D : 0:5ı min .1; m11 ; m33 /

(11.28)

Remark 11.3. It is important to note that, due to the use of the projection algorithm, by adjusting k3 ; k4 ; "i ; 1j ; 2j ; 3i ; 1  i  4; 1  j  9, we can make 1 arbitrarily small. This observation plays a crucial role in the stability analysis of the ze -dynamics.

ze -dynamics Lower-bound of ze . It can be seen from (11.13) and (11.27) that   zPQ e  k1 zQ e  ˛1 e 1 .tt0 / C 1 ;

(11.29)

where zQe D ze  ıe , which , with 1 ¤ k1 , further yields

 p1 X1 .t0 /  1 .tt0 / k1 .tt0 / zQ e .t /  zQ e .t0 /e e  e k1 .tt0 /  C 1  k1  1  k1 .tt0 / : (11.30) 1e k1

Hence k1 .tt0 /

ze .t /  .ze .t0 / C ıe /e  1  1  e k1 .tt0 / C ıe : k1

 p1 X1 .t0 /  1 .tt0 / C e  e k1 .tt0 /  1  k1

(11.31)

Therefore, the condition ze .t /  ze > 0 holds when 1 1 > k1 ; ıe  ze C ; k1

p1 X1 .t0 / ze .t0 /   ıe : 1  k1

(11.32)

11.3 Stability Analysis

281

Upper Bound of ze . We rewrite the first equation of (11.13) as zPQ e D k1 zQ e  .cos.

Qe e /  1/k1 z

 cos.

Q e /u:

(11.33)

It can be seen that (11.33) is of the form of the system studied in Lemma 10.1. Therefore, we will apply Lemma 10.1 to investigate stability of (11.33). We need to verify all conditions C1–C4 of Lemma 10.1 to (11.33). Verifying Condition C1. Take the following Lyapunov function: 1 V2 D ze2 : 2

(11.34)

It is direct to show that C1 holds with c0 D 0; c1 D c2 D 0:5; c3 D 1; c4 D k1 :

(11.35)

Verifying Condition C2. By noting that j.cos.

Qe e /  1/k1 z

 cos.

Q e /uj

 .k1 jzQ e j C 1/ kX1 .t /k ;

we have 1 D 1; 2 D k1 :

(11.36)

Verifying Condition C3. This condition directly holds from (11.27). Verifying Condition C4. From (11.35), (11.36), and (11.27), condition C4 becomes k1  k1 1  0:251 =0 > 0:

(11.37)

From Remark 11.3 and noting that 0 is an arbitrarily positive constant, we can see that there always exists k1 such that (11.37) holds. All the conditions of Lemma 10.1 have been verified, and we therefore have

  (11.38) jze .t /j  ˛2 X1 .t0 /; ze .t0 / e 2 .tt0 / C 2 ; where ˛2 ; 2 , and 2 are calculated as in Lemma 10.1.

v-dynamics From (11.9) and (11.10) the sway velocity dynamics can be rewritten as vP D 1 .
/v C 2 .
/v 2  where

dv2 dv3 3 v C 3 .
/; jvj v  m22 m22

(11.39)

282

11 Path-following of Underactuated Ships Using Polar Coordinates

 d22 m11 1 tan. e /rQ  1 .
/ D   uC Q m22 m22 ze cos. e /    a1 sin. e / ; a2  a1 tan. e /  ze ze cos. e / m11 tan. e / 2 .
/ D  ; m22 ze cos. e /    m11 sin. e / 3 .
/ D  Q uQ rQ C a1 rQ C a2 C a1 uC m22 ze    1 sin. e / a2 C a1 a1 C wv .t /; ze m22

(11.40)

with 

xe @xd ye @yd C a1 D k1 .ze  ıe / C ze @s ze @s



u0 .t; ze /    ; @yd 2 @xd 2 C @s @s      @xd sin. / xe sin. e / @yd cos. / ye sin. e / a2 D   C   @s ze ze2 @s ze ze2 u0 .t; ze / s    k2 e :  @yd 2 @xd 2 C @s @s s

To show that v is bounded, we take the following Lyapunov function: 1 V3 D v 2 ; 2 whose derivative along the solutions of (11.39), for any  > 0, satisfies     2M C 3M dv3 M 4 M P  2  v C  1 C v 2 C 3M ; V3   m22 4

(11.41)

(11.42)

where iM ; 1  i  3, are the upper bounds of i .
/. It can be seen from (11.40) that iM exist and are finite since their arguments are bounded as shown above. Therefore, we can pick  > 0 such that dv3 =m22  2M  >  > 0; then, see [6], the inequality (11.42) guarantees a finite upper-bound of the sway velocity v.

11.4 Discussion of the Initial Condition We now discuss how to obtain the initial conditions such that j .t /j < 0:5 and the last inequality of (11.32) holds. Since 0:5 2 .t /  V1 .t /, from (11.25), the condition j .t /j < 0:5 can be rewritten as

11.4 Discussion of the Initial Condition

v u u t

283

2 Q 2 .t0 / C m33 rQ 2 .t0 / C e .t0 / C m11 u

3 X iD1

On the other hand, the term

1 QiT .t0 /i1 Qi .t0 / C =ı < : (11.43) 2

3 P QiT .t0 /i1 Qi .t0 / C ı can be made arbitrarily small.

iD1

Noticing that xe .t0 / D x.t0 /  xd .s.t0 // and ye .t0 / D y.t0 /  yd .s.t0 //, it can be seen that the initial value, s.t0 /, can be adjusted such that (11.43) holds if: 1. the ship heads toward “almost” of the half-plane containing the initial part of the path to be followed, 2. the path satisfies Assumption 11.1, see Figure 11.2, 3. the initial velocities u.t0 /; v.t0 / and r.t0 / are not too large, and 4. the design constants are chosen such that k1 is small and k2 is large. The angle ı0 (see Figure 11.2) should be increased if the initial velocities u.t0 /; v.t0 /, and r.t0 / are large. Otherwise the ship might cross the edge line of the plane in question, which might result in e D ˙0:5. Similarly the last inequality of (11.32) is rewritten as

q p1 X1 .t0 / 2 2 .x.t0 /  xd .s.t0 /// C .y.t0 /  yd .s.t0 ///   ıe : (11.44) 1  k1

Again s.t0 / can be adjusted such that (11.44) holds. It is noted that if the abovementioned conditions are not satisfied, one might generate an additional path segment such that it makes the above conditions hold and “smoothly” connects to the path to be followed.

G0

Forbidden area

ȍ

G0 Figure 11.2 Feasible initial conditions

284

11 Path-following of Underactuated Ships Using Polar Coordinates

11.5 Parking and Point-to-point Navigation 11.5.1 Parking Parking Objective. Design the surge force u and the yaw moment r to park the underactuated ship (11.1) from the initial position and orientation, .x.t 
0 /, y.t0 /, .t0 //, to the desired parking position and orientation of xp ; yp ; p under the following conditions: 1. q There exists a large enough positive constant $p such that 
2 
2 x.t0 /  xp C y.t0 /  yp  $p . 2. The ship heads toward “almost” of the half-plane of the desired parking orientation. 3. At the desired parking position and orientation, the environmental disturbances are negligible. The above conditions normally hold for parking practice. However, if the first two conditions do not hold, one can apply the strategy in the preceding section to move the ship until they hold. Having formulated the parking problem as above, one might claim that the path-following controller proposed in Section 11.2 can be applied by setting u0 equal to zero. However, this will result in a yaw angle that may be very different from the desired parking one, at the desired parking position, since our proposed path-following controller is designed to drive ze to a small ball, not to zero for reasons of robustness. To resolve this problem, we first generate a regular curve, ˝p .xd ; yd /, which goes via the parking position and its tangent angle at the parking position is equal to the desired parking yaw angle, see Figure 11.3. For simplicity of calculation, the curve can be taken as a straight line in almost all cases of the vessel’s initial conditions. Then the proposed path-following controller can be used to make the vessel follow ˝p .xd ; yd /. In this case, the velocity u0 should be chosen such that it is equal to zero or tends to zero when thepvirtual ship tends to the desired parking position, i.e., limzep !0 u0 D 0 with zep D .xd  xp /2 C .yd  yp /2 . A simple choice can be taken as u0 D u0 .1  e 1 zep /e 2 ze ;

(11.45)

where i > 0; i D 1; 2. Special care should be taken when choosing the initial values of .xd .t0 /; yd .t0 // and the sign of u0 such that they result in a short parking time. Remark 11.4. At the desired parking position and orientation, if there are large environmental disturbances, there will be an oscillatory behavior in the yaw dynamics and the ship might diverge from its desired position. This phenomenon is well known in dynamic positioning systems, [11]. However, for parking practice, we here assume that the environmental disturbances are negligible since the parking place is usually in a harbor.

11.6 Numerical Simulations

285

YE

\p

yp ȍp

\ (t0 )

y (t0 ) OE

x (t0 )

xp

XE

Figure 11.3 Ship parking problem

11.5.2 Point-to-point Navigation As seen in Section 11.1, the requirement of the reference path to be a regular curve might be too cumbersome in practice, since this curve has to go via desired points generated by the helmsman, and its derivatives are needed in the path-following controller. These restrictions motivate us to consider the point-to-point navigation problem as follows: Point-to-point Navigation Objective. Design the surge force u and the yaw moment r to make the underactuated ship (11.1) go from the initial position and orientation, .x.t0 /; y.t0 /; .t0 //, via desired points generated by a path planner. To solve this control objective, we first assume that the path planner generates desired points, which are feasible for the ship to be navigated through. We then apply the proposed path-following controller in Section 11.2 to each regular curve segment connecting desired points in sequence, see Figure 11.4. The regular curve segments can be straight line, arc, or known regular curves. A fundamental difference between point-to-point navigation and the proposed smooth path-following is that there are a finite number of “spikes”, equal to the number of points, in the errors ze and e . This phenomenon is due to the path being nonsmooth in the orientation at the points.

11.6 Numerical Simulations The same 32 m long monohull ship used in Section 5.4 is also used in this section. The ship parameters are again listed below for the convenience of the reader:

286

11 Path-following of Underactuated Ships Using Polar Coordinates

YE $

$

$

$ OE

XE

Figure 11.4 Point-to-point navigation

m11 D 120  103 kg; m22 D 177:9  103 kg; m33 D 636  105 kgm2 ; du D 215  102 kgs-1 ; du2 D 43  102 kgm1 ; du3 D 21:5  102 kgsm2 ; dv D 117  103 kgs-1 ; dv2 D 23:4  103 kgm1 ; dv3 D 11:7  103 kgsm2 ; dr D 802  104 kgm2 s 1 ; dr2 D 160:4  104 kgm2 ; dr3 D 80:2  104 kgm2 s; dui D 0; dvi D 0; dri D 0; 8i > 3: This ship has a minimum turning circle with a radius of 150 m, a maximum surge force of 5:2  109 N, and the maximum yaw moment of 8:5  108 Nm. The above ship parameters are assumed to be those of real ships and are estimated on-line by adaptation laws (11.16). We assume that these parameters fluctuate around the above values ˙15% to calculate the maximum and minimum values used in the choice of the design constants in (11.37). We assume that the environmental disturbances are wu D 11  104 .1 C rand.
//, wv D 26  104 .1 C rand.
// and wr D 950105 .1Crand.
//, where rand.
/ is random noise with the uniform distribution on the interval Œ0:5 0:5. This choice results in nonzero-mean disturbances. In practice, the environmental disturbances may be different. We take the above disturbances for simplicity of generation. It should be noted that only the boundaries of the environmental disturbances are needed in our proposed controller. In simulations, the control parameters are taken as k1 D 1:5, k2 D 7:5, k3 D 6  105 , k4 D 3  108 , 1j D 2j D 3j D 2  107 , "i D 0:2 and ıe D 0:95. The initial conditions are Œx.0/; y.0/; .0/; u.0/, v.0/; r.0/; s.0/ DŒ-200, -20,  0.4, 0, 0, 0, 0, and all initial values of parameter estimates are taken to be 70% of their assumed true ones.

11.6 Numerical Simulations

287 a

200

y [m]

100 0 −100 −200 −200

0

200 x [m]

400

600

b

c

300

0.2 0

200

ψe [rad]

ze [m]

800

−0.2

100 0 0

−0.4 100

200 Time [s]

300

−0.6 0

100

200 Time [s]

300

Figure 11.5 Path-following: a. Ship position and orientation in the .x; y/-plane; b. Ship position error ze ; c. Ship orientation error e

11.6.1 Path-following Simulation Results The reference is .xd D s; yd D 200 tanh.0:02s// for the first 120 seconds, then followed by a circle with a radius of 200 m. The virtual ship velocity on the path is taken as u0 .t; ze / D 5.1  0:8e 2t /e 0:5ze . The simulation results of the ship’s position and orientation, path-following error ze , and path-following orientation error e are plotted in Figure 11.5. The control inputs u and r are plotted in Figure 11.6. For simplicity of presentation, we plot some samples of parameter estimates in Figure 11.7. It can be seen that the projection algorithm clearly prevents instability in adaptation due to non-vanishing disturbances.

11.6.2 Point-to-point Simulation Results For the case of point-to-point navigation, we want the ship to go via points: ..0 m; 0 m/; .400 m; 200 m/; .1000 m; 200 m/; .1400 m; 0 m//, then follow a horizontal straight line. For simplicity, we use straight-line segments to connect the above desired points. The virtual ship velocity is taken to be the same as for the case of path-following. The simulation results of the ship position and orientation, path-

288

11 Path-following of Underactuated Ships Using Polar Coordinates 6

a

x 10

τu [N]

4 2 0 0

50

100

7

τr [Nm]

1

150 200 Time [s]

250

300

350

250

300

350

b

x 10

0

−1 0

50

100

150 200 Time [s]

Figure 11.6 Path-following: a. Surge force u [N]; b.Yaw moment r [Nm]

following error ze , and path-following orientation error e are plotted in Figures 11.8 and 11.9. The “spikes” in ze , e , u and r are due to the nonsmooth reference path. However, these spikes are moderate in magnitude.

11.6.3 Parking Simulation Results We assume that the parking position is at the origin and the parking orientation is p D =4 without environmental disturbances at the desired parking place. We choose u0 D 5.1  e 0:2zep /e 0:2ze and a straight-line segment starting at .20; 20/ with a slope of 1. The simulation results are plotted in Figures 11.10 and 11.11. In summary, it can be seen from Figures 11.5–11.11 that our proposed controller is able to force the underactuated ship in question to follow the predefined paths. The path-following position error ze converges to a small nonzero value specified by ıe and does not cross zero as expected in the control design. It can be seen from Figures 11.5 and 11.8 that, under the nonvanishing environmental disturbances, the proposed controller forces the yaw angle to a small value. This value together with the ship forward speed prevents the ship from drifting away from the path. The high magnitude of controls results from the fact that we simulate the proposed con-

11.7 Conclusions

289

5

[kg]

1.5122

a

x 10

1.5122 1.5122 1.5121 0

50

100

150

4

[kg]

−4.9

200

250

300

350

200

250

300

350

200 time [s]

250

300

350

b

x 10

−5 −5.1 0

50

100

150

5

[kg]

2

c

x 10

1.8 1.6 1.4 0

50

100

150

Figure 11.7 Path-following: a. Estimate O11 ; b. Estimate O21 ; c. Estimate O31

troller on a real ship and environmental disturbances with large magnitude. Indeed, the magnitude of the control inputs, can be reduced by adjusting the control and adaptation gains. However, this will result in a slow transient response. Finally, the transient response of e is much shorter than that of ze due to the control gains chosen such that k2 >> k1 to guarantee a nonzero lower bound of ze .

11.7 Conclusions The proposed results in this chapter can be readily combined with the observer designs presented in Chapter 7 to design an output feedback path-following system for an underactuated ship. It is seen from Section 11.2 that the ship should not be too close to the desired path at the initial time. However, the strategy in Section 11.4 can be used to overcome the mentioned limitation of the control system proposed in this chapter. The work presented in this chapter is based on [137, 138]. The approach in this chapter will be extended to the case of the underactuated underwater vehicle in six degrees of freedom in Chapter 13.

290

11 Path-following of Underactuated Ships Using Polar Coordinates a

400

y [m]

200 0 −200 0

500

1000

1500

x [m] b

c

0.2 0

ψ [rad]

200

−0.2

e

e

z [m]

300

100

−0.4

0 0

100

200 Time [s]

300

−0.6 0

100

200 Time [s]

300

Figure 11.8 Point-to-point navigation: a. Ship position and orientation in the .x; y/ plane; b. Ship position error ze ; c. Ship orientation error e 6

5

a

x 10

τu [N]

4 3 2 1 0 0

50

100

7

1

150 200 Time [s]

250

300

350

250

300

350

b

x 10

τr [Nm]

0.5 0 −0.5 −1 0

50

100

150 200 Time [s]

Figure 11.9 Point-to-point navigation: a. Surge force u [N]; b.Yaw moment r [Nm]

11.7 Conclusions

291 a

20

y [m]

0 −20 −40 −200

−150

−100 x [m]

−50 c

200

0.4

150

0.2

ψe [rad]

ze [m]

b

0

100 50

0

−0.2

0 0

100

200 Time [s]

300

−0.4 0

100

200 Time [s]

300

Figure 11.10 Parking: a. Ship position and orientation in the .x; y/ plane; b. Ship position error; c. Ship orientation error in the .x; y/ plane 6

5

a

x 10

3

u

τ [N]

4 2 1 0 0

50

100

7

250

300

350

250

300

350

b

x 10

0.5 0

r

τ [Nm]

1

150 200 Time [s]

−0.5 −1 0

50

100

150 200 Time [s]

Figure 11.11 Parking: a. Surge force u [N]; b.Yaw moment r [Nm]

Part IV

Control of Underactuated Underwater Vehicles

Chapter 12

Trajectory-tracking Control of Underactuated Underwater Vehicles

This chapter addresses the problem of trajectory-tracking control of underactuated underwater vehicles. These vehicles do not have independent actuators in the sway and heave axes. Based on the techniques developed for underactuated surface ships in Chapters 5, 6, and 7, we present a method to design a controller for an underactuated underwater vehicle to globally asymptotically track a reference trajectory generated by a suitable virtual underwater vehicle. The yaw and pitch reference velocities do not have to satisfy a persistently exciting condition as was often required in previous literature. Due to the complex dynamics of the underwater vehicles in comparison with that of the ships, the control design and stability analysis require more complicated coordinate transformations and techniques than those developed for underactuated ships in Chapters 5, 6, and 7.

12.1 Control Objective In this chapter, we consider the following mathematical model of an underactuated underwater vehicle when nonlinear hydrodynamic damping terms and roll motion are ignored, see Section 3.4.2.2: 1. Kinematics xP D cos. / cos./u  sin. /v C sin./ cos. /w; yP D sin. / cos./u C cos. /v C sin./ sin. /w; zP D  sin./u C cos./w; P D q; P D r : cos./

(12.1)

2. Kinetics

295

296

12 Trajectory-tracking Control of Underactuated Underwater Vehicles

m22 m33 d11 1 vr  wq  uC u ; m11 m11 m11 m11 d22 m11 ur  v; vP D  m22 m22 m11 d33 wP D uq  w; m33 m33 d55 grGML sin./ 1 m33  m11 uw  q C q ; qP D m55 m55 m55 m55 d66 1 m11  m22 uv  rC r : rP D m66 m66 m66

uP D

(12.2)

In (12.1) and (12.2) the symbols (see Figure 12.1)  , , q, and r denote the roll, pitch, and yaw angles and velocities while x, y, z, u, v, and w are the surge, sway, and heave displacements and velocities, respectively. The available control inputs are u , q , and r . Since the sway and heave control forces are not available in the sway and heave dynamics, the underwater vehicle in question is underactuated. Notice that (12.1) is not defined when the pitch angle is equal to ˙900 . However, during practical operations with underwater vehicles, this problem is unlikely to happen due to the metacentric restoring forces. One way to avoid the singularities is to use a four-parameter description known as the quaternion. Here, we use the above Euler parameters because of their physical representation and computational efficiency. We first assume that j.t0 /j < 0:5. Then we find feasible initial conditions such

(I , p Roll )

Xb

Yb

( y, v) Sway

(T , q) Pitch

Body-fixed frame

( x, u ) Surge

x

Ob

(\ , r ) Yaw

Earth-fixed frame

OE

YE

XE ZE

x CG

Zb

( z , w) Heave

Figure 12.1 Motion variables of an underwater vehicle

that our proposed controller guarantees j.t /j < 0:5; 8 t  t0  0. We consider a control objective of designing the control inputs u , q , and r to force the underactuated vehicle given in (12.1) and (12.2) to asymptotically track a

12.2 Coordinate Transformations

297

reference trajectory generated by the following virtual vehicle xP d D cos. yPd D sin.

d / cos.d /ud

 sin. / cos. /u C cos. d d d

d /vd

C sin.d / cos. /v d d C sin.d / sin.

d /wd ; d /wd ;

zP d D  sin.d /ud C cos.d /wd ; Pd D qd ; P d D rd ; cos.d / m22 m33 d11 1 uP d D vd rd  wd qd  ud C ud ; (12.3) m11 m11 m11 m11 d22 m11 ud rd  vd ; vP d D  m22 m22 m11 d33 wP d D ud qd  wd ; m33 m33 d55 grGML sin.d / 1 m33  m11 ud wd  qd  C qd ; qP d D m55 m55 m55 m55 d66 1 m11  m22 ud vd  rd C rd ; rPd D m66 m66 m66 where all of the symbols in (12.3) have the same meaning as the real vehicle. In this section, we impose the following assumption on the reference model (12.3): Assumption 12.1. 1. The reference signals ud , qd , rd , uP d , rPd , and qP d are bounded. There exists a strictly positive constant ud min , such that jud .t /j  ud min ; 8 t  0. The reference sway and heave velocities satisfy: jvd .t /j < jud .t /j, jwd .t /j < jud .t /j, 8 t  0. 2. The reference pitch angle satisfies jd .t /j < 21 , 8 t  0. Remark 12.1. The condition jud .t /j  ud min ; 8 t  0, covers both forward and backward tracking since the reference surge velocity is always nonzero but can be either positive or negative. Indeed, the condition jud .t /j  ud min ; 8 t  0, is much less restrictive than a persistently exciting condition on the yaw reference velocity. The condition jvd .t /j < jud .t /j and jwd .t /j < jud .t /j implies that the underactuated underwater vehicle cannot track a helix with an arbitrarily large curvature and twist due to the vehicle’s high inertia and underactuation in the sway and heave directions.

12.2 Coordinate Transformations Since designing the control inputs u , q , and r to achieve the control directly from (12.1) and (12.2) is difficult, we interpret the tracking errors in a frame attached to the vehicle body as follows:

298

12 Trajectory-tracking Control of Underactuated Underwater Vehicles

2

xe 6 6 ye 6 6 ze 6 6 4 e

e

3

2

cos. / cos./ sin. / cos./  sin./ 7 6 cos. / 0 7 6  sin. / 7 6 7 D 6 sin./ cos. / sin./ sin. / cos./ 7 6 7 6 0 0 0 5 4 0

0

0

32 00 76 0 0 76 76 6 0 07 76 76 1 0 54 01

x  xd

3

7 y  yd 7 7 z  zd 7 7 : (12.4) 7   d 5  d

Indeed, convergence of .xe ; ye ; ze ; e ; e / to the origin implies that of .x  xd ; y  yd ; z  zd ;   d ;  d / since the matrix 3 2 cos. / cos./ sin. / cos./  sin./ 0 0 7 6 6  sin. / cos. / 0 0 07 7 6 6 sin./ cos. / sin./ sin. / cos./ 0 0 7 7 6 7 6 0 0 0 1 0 5 4 0 0 0 01 is nonsingular for all . ; / 2 R2 . We also define the velocity tracking errors as ue D u  ud ; ve D v  vd ; we D w  w d ; qe D q  qd ;

(12.5)

re D r  rd : Differentiating (12.4) along the solutions of (12.1) and (12.3) yields: xP e D ue  .cos.e /  1 C cos./ cos.d /.cos. e /  1// ud  cos./ sin. e /vd C .sin.e /  cos./ sin.d /.cos. e /  1// wd C .rd C re /ye  .qd C qe /ze ; yPe D ve C cos.d / sin. e /ud  .cos. .xe C tan./ze / .rd C re /;

e /  1/vd

 sin.d / sin.

e /wd



zP e D we  .sin.e / C sin./ cos.d /.cos. e /  1// ud  sin./ sin. e /vd  .cos.e /  1 C sin./ sin.d /.cos. e /  1// wd C tan./.rd C re /ye C .qd C qe /xe ; P e D qe ; re rd Pe D .cos.d /.1  cos.e // C sin.d / sin.e // : C cos./ cos./ cos.d / (12.6) From (12.6), we can directly see that xe , e , and e can be stabilized by ue , qe , and re . There are several options to stabilize ye and ze . We can either use qe , re , ve , we , xe , e , or e . If qe and re are used, the control design will be extremely

12.2 Coordinate Transformations

299

complicated since qe and re enter all of the first three equations of (12.6). On the other hand, the use of ve and we to stabilize ye and ze will result in an undesired feature of marine vehicle control practice, namely the vessel will slide in the sway and heave directions. If we use xe and ze , the reference yaw and heave velocities must then satisfy persistently exciting conditions. Hence we will choose e and e to stabilize the sway error ye and the heave error ze , respectively. This choice also coincides with the ship control practice. Toward this end, we define the following coordinates ! k1 y e ; z1 D e C arcsin p 1 C xe2 C ye2 C ze2 ! k2 z e ; (12.7) z2 D e  arcsin p 1 C xe2 C ye2 C ze2 where the constants ki ; i D 1; 2, are such that jki j < 1. These constants will be specified later. It can be seen that (12.7) is well defined and that convergence of .z1 ; ye / and .z2 ; ze / implies that of e and e .

Remark 12.2. Using the nonlinear coordinate transformations (12.7) instead of z1 D y and e C k1 ye and z2 D e  k2 ze , we avoid the vessel whirling around  when p e  ze are large. If one uses the transformations z1 D e C arcsin k1 ye = 1 C ye2   p and z2 D e  arcsin k2 ze = 1 C ze2 , the problem of the vessel whirling around is avoided. Indeed using these coordinate transformations will result in a much simpler tracking error system than using (12.7). However, these coordinate transformations make the design of the control inputs u , q , and r difficult because the terms .xe C tan./ze /re and .tan./re ye C xe qe / appear in the ye - and ze -dynamics. Using the coordinate changes (12.7), the tracking error system in the .xe , ye , ze , z1 , z2 , ue , ve , we , qe , re / coordinates can be written as: xP e D ue  .$2  $ C cos./ cos.d /.$1  $//$ 1 ud C cos./k1 vd $ 1 ye C .k2 ze  cos./ sin.d /.$1  $//$ 1 wd C .rd C re /ye  .qd C qe /ze C px ; yPe D ve  cos.d /k1 ud $ 1 ye  .$1  $/$ 1 vd C sin.d /k1 wd $ 1 ye  .xe C tan./ze /.rd C re / C py ; zP e D we  .k2 ze C sin./ cos.d /.$1  $//$ 1 ud C sin./k1 vd $ 1 ye  .$2  $ C sin./ sin.d /.$1  $//$ 1 wd C tan./.rd C re /ye C .qd C qe /xe C pz ; 
zP 1 D 1  k1 $11 .cos./xe C sin./ze / cos./1 re  k1 $11 $ 2 xe ye ue C

fz1 C pz1 ;
 zP 2 D 1  k2 $21 xe qe  k2 tan./$21 ye re C k2 $21 $ 2 xe ze ue C fz2 C pz2 ;

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12 Trajectory-tracking Control of Underactuated Underwater Vehicles

uP e D vP e D wP e D qP e D rPe D

m22 m33 d11 1 vr  wq  uC u  uP d ; m11 m11 m11 m11 d22 m11 .ue re C ue rd C ud re /  ve ;  m22 m22 m11 d33 .ue qe C ue qd C ud qe /  we ; m33 m33 d55 grGML sin./ 1 m33  m11 uw  q C q  qP d ; m55 m55 m55 m55 d66 1 m11  m22 uv  rC r  rPd ; m66 m66 m66 (12.8)

where, for simplicity, we have defined the following notations: 1. The terms $, $1 and $2 : q $ D 1 C xe2 C ye2 C ze2 ; q $1 D 1 C xe2 C .1  k12 /ye2 C ze2 ; q $2 D 1 C xe2 C ye2 C .1  k22 /ze2 :

(12.9)

2. The terms exponentially tend to zero when z1 and z2 do: px D  ..cos.z2 /  1/$2  sin.z2 /k2 ze C cos./ cos.d /..cos.z1 / 1/$1 C sin.z1 /k1 ye // $ 1 ud C .sin.z2 /$2 C .cos.z2 /  1/ k2 ze  cos./ sin.d /..cos.z1 /  1/$1 C sin.z1 /k1 ye // $ 1  wd  cos./.sin.z1 /$1  .cos.z1 /  1/k1 ye /$ 1 vd ; py D cos.d / .sin.z1 /$1  .cos.z1 /  1/k1 ye / $ 1 ud  ..cos.z1 / 1/$1 C sin.z1 /k1 ye / $ 1 vd  sin.d /.sin.z1 /$1  .cos.z1 /  1/k1 ye /$ 1 wd ; pz D  .sin.z2 /$2 C .cos.z2 /  1/k2 ze C sin./ cos.d /..cos.z1 / 1/$1 C sin.z1 /k1 ye // $ 1 ud  ..cos.z2 /  1/$2  sin.z2 / k2 ze C sin./ sin.d / ..cos.z1 /  1/$1 C sin.z1 /k1 ye // $ 1  wd  sin./.sin.z1 /$1  .cos.z1 /  1/k1 ye /$ 1 vd ; 
pz1 D k1 $11 py  ye $ 2 .xe px C ye py C ze pz / C cos./1 $ 1 

..1  cos.z2 //$2 C sin.z2 /k2 ze C tan.d /.sin.z2 / C .cos.z2 /  1/k2 ze ;
 pz2 D k2 $21 pz  ze $ 2 .xe px C ye py C ze pz / :

3. The terms fz1 and fz2 :

(12.10)

12.3 Control Design

301

fz1 D .$  $2 C tan.d /k2 ze / cos./1 $ 1 C k1 $11 .ve  cos.d / k1 ud $ 1 ye  $1  $/$ 1 vd C sin.d /k1 wd $ 1 ye  .xe C tan./ze /rd  ye $ 2 .xe . .$2  $ C cos./ cos.d /.$1  $//  $ 1 ud C cos./k1 vd $ 1 ye C .k2 ze  cos./ sin.d /.$1  $// 
$ 1 wd C ye .ve  cos.d /k1 ud $ 1 ye  .$1  $/$ 1 vd C
sin.d /k1 wd $ 1 ye C ze .we  .k2 ze C sin./ cos.d /.$1  $//  fz2

$ 1 ud C sin./k1 vd $ 1 ye  .$2  $ C sin./ sin.d /.$1  $// 



$ 1 wd ;  1 D k2 $2 we  .k2 ze C sin./ cos.d /.$1  $// $ 1 ud C sin./  k1 vd $ 1 ye  .$2  $ C sin./ sin.d /.$1  $// $ 1 wd C xe qd C tan./ye rd  ze $ 2 .xe . .$2  $ C cos./ cos.d /.$1  $//  $ 1 ud C cos./k1 vd $ 1 ye C .k2 ze  cos./ sin.d /.$1  $// 
$ 1 wd C ye .ve  cos.d / k1 ud $ 1 ye  .$1  $/$ 1 vd C
sin.d /k1 wd $ 1 ye C ze .we  .k2 ze C sin./ cos.d /

.$1  $/$ 1 ud C sin./k1 vd $ 1 ye  .$2  $ C



: sin./ sin.d /.$1  $//$ 1 wd

(12.11)

It is now clear that the problem of forcing the underactuated underwater vehicle given in (12.1) and (12.2) to track the virtual ship (12.3) becomes one of stabilizing the system (12.8). The efforts we have made so far are to put the tracking error dynamics in the triangular form of (12.8), and to have the terms  cos.d /k1 ud $ 1 ye and k2 ud $ 1 ze in the ye and ze -dynamics, respectively. These terms play an important role in stabilizing the ye - and ze -dynamics. In the next section we will design the control inputs u , q , and r to asymptotically stabilize (12.8) at the origin.

12.3 Control Design The triangular structure of (12.8) suggests that we design the actual controls u , q , and r in two stages. First, we design the virtual velocity controls of ue , qe , and re to asymptotically stabilize xe , ye , ze , z1 , z2 , we , and ve at the origin. Based on the backstepping technique, the controls u , q , and r will then be designed to force the errors between the virtual velocity controls and their actual values exponentially to zero. Since ue enters the ve - and we -dynamics, we will design a bounded virtual control of ue to simplify the stability analysis. The virtual controls of qe and re are chosen to stabilize the z1 - and z2 -dynamics.

302

12 Trajectory-tracking Control of Underactuated Underwater Vehicles

12.3.1 Step 1 Define the virtual control errors as uQ e D ue  ude ; qQ e D qe  qed ; rQe D re  red ;

(12.12)

where ude , qed , and red are the virtual velocity controls of ue , qe , and re , respectively. The virtual controls ude , qed , and red are chosen as follows: ude D k0 $ 1 xe C .$2  $ C cos./ cos.d /.$1  $//$ 1 ud  cos./k1 vd $ 1 ye  .k2 ze  cos./ sin.d /.$1  $//$ 1 wd ; d d C r2e ; red D r1e d d qed D q1e C q2e ;

(12.13)

where   cos./ d 1 2 ;  f $ x y u k $ z1 e e 1 e 1 1  k1 $11 .cos./xe C sin./ze / cos./ d .c1 z1  pz1 / ; r2e D 1 1  k1 $1 .cos./xe C sin./ze /   1 1 d 1 2 d d q1e k tan./$ y r  k $ $ x z u  f D 2 e 1e 2 2 e e e z2 ; 2 1  k2 $21 xe   1 d 1 d q2e D z C k tan./$ y r  p (12.14) c 2 2 2 e 1e z2 ; 2 1  k2 $21 xe

d D r1e

with k0 being a positive design constant to be specified later, and c1 and c2 being positive constants. It is not hard to show that the virtual control ude is bounded as ˇ ˇ
 ˇ dˇ ˇue ˇ  k0 C .1 C k12 C k22 / jud j C jk1 vd j C jk2 j C k12 jwd j WD uem : (12.15)

Remark 12.3. Unlike the standard application of backstepping, in order to reduce complexity of the controller expressions, we have chosen a simple virtual control d d law ude without canceling some known terms. We have written red D r1e C r2e and d d d d d qe D q1e C q2e to simplify stability analysis, because q2e and r2e exponentially vanish when z1 and z2 do. From (12.14), we observe that red and qed are Lipschitz in .xe ; ye ; ze ; z1 ; z2 ; ve ; we /. This observation plays a crucial role in the stability analysis of the closed loop system.

12.3.2 Step 2 By differentiating (12.12) along the solutions of (12.14) and (12.6), the actual controls u , q , and r without canceling the useful damping terms are chosen as

12.3 Control Design

303



m22 m33 d11 d vr C wq C .u C ud / C uP d C m11 m11 m11 e  uP de C k1 $11 $ 2 xe ye z1  k2 $21 $ 2 xe ze z2 ;  d66 d m11  m22 r D m66 2 rQe  uv C .r C rd / C rPd C rPed  m66 m66 e  
z1 1  k1 $11 .cos./xe C sin./ze / C k2 tan./$21 ye z2 ; cos./  m33  m11 d55 d q D m55 3 qQ e  uw C .q C qd /C m55 m55 e !
 grGML sin./ 1 d C qP d C qP e  1  k2 $2 xe z2 ; (12.16) m55

u D m11 1 uQ e 

where i ; i D 1; 2; 3, are positive constants. Substituting (12.16), (12.13), and (12.14) into (12.8) yields the closed loop system xP e D k0 $ 1 xe  .qd C qe /ze C .rd C re /ye C px C uQ e ; yPe D ve  cos.d /k1 ud $ 1 ye  .$1  $/$ 1 vd C sin.d /k1 wd $ 1 ye  .xe C tan./ze /.rd C re / C py ; zP e D we  .k2 ze C sin./ cos.d /.$1  $//$ 1 ud C sin./k1 vd $ 1 ye  .$2  $ C sin./ sin.d /  .$1  $//$ 1 wd C tan./.rd C re /ye C .qd C qe /xe C pz ; d22 ve  m11 m1 22  m22 d ..ude C ud /.r2e C rQe / C uQ e .red C rd C rQe //; d33 d d d d we C m11 m1 wP e D m11 m1 33  33 .ue q1e C ue rd C ud q1e /  m33 d ..ude C ud /.q2e C qQ e / C uQ e .qed C qd C qQ e //; 
zP 1 D c1 z1 C 1  k1 $11 .cos./xe C sin./ze /  d d d d vP e D m11 m1 22 .ue r1e C ue rd C ud r1e / 

zP 2 D uPQ e D qPQ e D rPQe D

cos./1 rQe  k1 $11 $ 2 xe ye uQ e ;
 c2 z2 C 1  k2 $21 xe qQ e  k2 tan./$21 ye rQe C k2 $21 $ 2 xe ze uQ e ;
 Q e C k1 $11 $ 2 xe ye z1  k2 $21 $ 2 xe ze z2 ;  1 C d11 m1 11 u

 Q e  1  k2 $21 xe z2  2 C d55 m1 55 q 
1  3 C d66 m1 66 rQe C k2 tan./$2 ye z2  
1  k1 $11 .cos./xe C sin./ze / cos./1 z1 : (12.17)

We now state the main result of this section, the proof of which is given in the next section.

304

12 Trajectory-tracking Control of Underactuated Underwater Vehicles

Theorem 12.1. Assume that the reference signals .xd ; yd ; zd ; d ; d ; vd ; wd / generated by the virtual vessel model (12.3), and Assumption 12.1 holds. If the state feedback control law (12.16) is applied to the vessel system (12.1) and (12.2), then the tracking errors .x.t /  xd .t /, y.t /  yd .t /, z.t /  zd .t /, .t /  d .t /, .t /  d .t /, v.t /  vd .t /, and w.t /  wd .t // asymptotically converge to zero with an appropriate choice of the design constants k0 , k1 , and k2 , i.e., the closed loop system (12.17) is locally asymptotically stable at the origin.

12.4 Stability Analysis To prove Theorem 12.1, we just need to show that the closed loop system (12.17) is asymptotically stable at the origin. To simplify stability analysis of this closed loop system, we observe that (12.17) consists of two subsystems .xe ; ye ; ze ; ve ; we / and .z1 ; z2 ; uQ e ; qQ e ; rQe / in an interconnected structure. Therefore we first consider the .z1 ; z2 ; uQ e ; qQ e ; rQe /-subsystem then move to .xe ; ye ; ze ; ve ; we /-subsystem.

.z1 ; z2 ; uQ e ; qQ e ; rQe /-subsystem From the last five equations of (12.17), it is direct to show that this subsystem is exponentially stable at the origin by taking the following Lyapunov function V1 D


1 2 z1 C z22 C uQ 2e C qQ e2 C rQe2 ; 2

(12.18)

whose time derivative along the solutions of (12.17) satisfies 
2
2 
2  VP1 D c1 z12  c2 z22  1 C d11 m1 Q e  3 C d66 m1 Q e  2 C d55 m1 66 rQe ; 55 q 11 u (12.19) which in turn implies that k.z1 .t /; z2 .t /; uQ e .t /; qQ e .t /; rQe .t //k  k.z1 .t0 /; z2 .t0 /; uQ e .t0 /; qQ e .t0 /; rQe .t0 //k e 1 .tt0 / ;

(12.20)

where



   1 1 1 D min c1 ; c2 ; 1 C d11 m1 : 11 ; 2 C d55 m55 ; 3 C d66 m66 (xe ; ye ; ze ; ve ; we )-subsystem To analyze stability of this subsystem, we consider the following Lyapunov function:

12.4 Stability Analysis

305

q
1  (12.21) V2 D 1 C xe2 C ye2 C ye2  1 C k3 ve2 C we2 ; 2 where k3 is a positive constant to be specified later. The time derivative of (12.21) along the solutions of the first five equations of (12.17), after a lengthy but simple calculation using completed squares, satisfies VP2  x .t /$ 2 xe2  y .t /$ 2 ye2  z .t /$ 2 ze2  v .t /ve2  w .t /we2 C .1 .
/V2 C 2 .
// e 1 .tt0 / ;

(12.22)

where i .
/; i D 1; 2 are some nondecreasing functions of k.z1 .t0 /; z2 .t0 /; uQ e .t0 /, qQ e .t0 /; rQe .t0 //k,  k3 m11 jrd j "1 k3 m11 uem "1 jk1 rd j ˇˇ 3 ˇˇ C k1 ud C k12 jvd j C  x .t / D k0  m22 m22 .1  2 jk1 j/ 1  k12
k3 m11 jqd j "1 k3 m11 uem jk2 tan./j "1 .k12 C k22 / jk1 wd j    m33 m33 .1  jk2 j/ ˇ ˇ
 jk1 rd j C ˇk13 ud ˇ C k12 jvd j C .k12 C k22 / jk1 wd j C uem    k3 m11 uem "1 jk2 qd j ˇˇ 3 ˇˇ 2 C k k ; (12.23) u C u j jk 1 2 d 1 d m33 .1  jk2 j/ 1  k12 y .t / D k1 ud cos.d /  jk1 wd j  "3  k12 .jvd j C jud j C jwd j/ 
k3 m11 uem "1  2 k3 m11 uem "1 k1 .jud j C jwd j/ C jk1 vd j   m22 .1  jk2 j/ m22 .1  2 jk1 j/ ˇ ˇ ˇ ˇ  uem jk1 j C k12 .jvd j C 2 jud j C 2 jwd j/ C ˇk13 wd ˇ C 2 ˇk13 vd ˇ C
k3 m11 jqd j "1  2
.k12 C k22 / jk1 ud j  k1 .jud j C jwd j/ C jk1 vd j  m33 k3 m11 uem jk2 tan./j "1  uem jk1 j C k12 .jvd j C 2 jud j C 2 jwd j/ C m33 .1  jk2 j/ ˇ 3ˇ 
ˇk ˇ .2 jvd j C jwd j/ C .k 2 C k 2 / jk1 ud j  k3 m11 uem "1 k 2 jk2 j  1 1 2 m33 .1  jk2 j/ 1  jk1 vd j jk2 rd j .jud j C jwd j/ C jk1 k2 j .jvd j C jud j/ C  ; (12.24) 2 4"2 1  k2
k3 m11 jrd j "1  2 z .t / D k2 ud  "3  jk1 vd j  k22 jwd j  k2 jud j C jk2 wd j  m22   k3 m11 uem "1  2 jk1 rd j k3 m11 uem jk2 tan./j "1 k2 C C C m33 .1  jk2 j/ m22 .1  2 jk1 j/ 1  k12
k3 m11 jqd j "1  jk2 wd j C k22 jud j  jk2 j C jk1 k2 j .jud j C jwd j//  m33

306

12 Trajectory-tracking Control of Underactuated Underwater Vehicles


k3 m11 uem "1  ˇˇ 3 ˇˇ  2 2 k2 wd C k2 C k12 jk2 j .jud j C jvd j C jwd j/ C m33 .1  jk2 j/ (12.25) jk1 k2 j .jud j C jvd j C 2 jwd j/ ;/ ; k3 d22 1 k3 m11 jrd j  2 k0 C .k12 C k22 / jud j C jk1 vd j C   m22 4"3 m22 4"1
k3 m11 uem .jk1 j .uem C 2:5C jk2 wd j C k12 jwd j  m22 .1  2 jk1 j/4"1 jrd j .1 C tan2 ./// C k22 C k12 .jud j C 3 jvd j C 2 jwd j/ C .k12 C
ˇ ˇ k22 / jk1 j .jud j C jwd j/ C jk1 k2 j .jud j C jwd j/ C ˇk13 ˇ  .jud j C 2

v .t / D

jvd j C jwd j/ C jk2 j/ 

k3 k22 m11 uem k3 k22 m11 uem jtan./j  ; m33 .1  jk2 j/8"1 m33 .1  jk2 j/8"1 (12.26)

k3 d33 1 k3 m11 jqd j  2   k0 C .k12 C k22 / jud j C jk1 vd j C m33 4"3 m33 4"1
k3 m11 uem jk2 tan./j .jk1 j .uem C 2:5"1 C jk2 wd j C k12 jwd j  m33 .1  jk2 j/8"1

w .t / D

jrd j .1 C tan2 ./// C k22 C k12 .jud j C 3 jvd j C 2 jwd j/ C .k12 C ˇ ˇ k22 / jk1 j .jud j C jwd j/ C jk1 k2 j .jud j C jwd j/ C ˇk13 ˇ .jud j C jvd j C k3 jk2 j m11 uem  k0 C .k12 C k22 / jud j C jk1 vd j C jwd j/ C jk2 j/  m33 .1  jk2 j/4"1 

 k3 m11 uem jk2 j .2:5"1 C tan2 ./C jk2 j C k22 jwd j  m33 .1  jk2 j/4"1 jqd j/ C k12 jk2 j .3 jud j C 2 jvd j C 2 jwd j/ C k22 .jud j C 2 jwd j/ C ˇ 3ˇ
ˇk ˇ .jud j C 2 jwd j/ C 2 jk1 k2 j .jud j C jvd j C jwd j/ ; (12.27) 2

with "i ; i D 1; 2; 3 being some positive constants, and

k2 ze .t /

!

: .t / D d .t / C z2 .t / C arcsin p 1 C xe2 .t / C ye2 .t / C ze2 .t /

(12.28)

We now choose the design constants ki ; 0  i  3 such that x  x ; y .t /  y ; z .t /  z ; v .t /  v ; w .t / 

w ;

(12.29)

12.4 Stability Analysis

307

for all t  t0  0 for some positive constants x ; y ; z ; v ; w , and  being replaced by ! k2 ze .t0 / .t0 / D d .t0 / C z2 .t0 / C arcsin p : (12.30) 1 C xe2 .t0 / C ze2 .t0 / C ze2 .t0 /

Substituting (12.29) into (12.26) yields

VP2  x $ 2 xe2  y $ 2 ye2  z $ 2 ze2  v ve2  w we2 C .1 .
/V2 C 2 .
// e 1 .tt0 / :

(12.31)

From (12.31) we have VP2  .1 .
/V2 C 2 .
// e 1 .tt0 / , which implies that V2 .t /  3 .
/ with 3 .
/ being a class-K function of k.xe .t0 /, ye .t0 /, ze .t0 /, ve .t0 /, we .t0 /, z1 .t0 /, z2 .t0 /, uQ e .t0 /, qQ e .t0 /, rQe .t0 //k. Substituting V2 .t /  3 .
/ into (12.31) results in VP2  x $ 2 xe2  y $ 2 ye2  z $ 2 ze2  v ve2  w we2 C .1 .
/3 .
/ C 2 .
// e 1 .tt0 / :

(12.32)

From (12.21) and (12.32) it is not hard to show that there exists a nonnegative constant 2 and a class-K function 2 .
/ depending on the initial conditions such that k.xe .t /; ye .t /; ze .t /; ve .t /; we .t //k  2 .
/e 2 .tt0 / :

(12.33)

The dependence of 2 > 0 on the initial conditions implies that the closed loop system (12.17) is asymptotically stable at the origin. However one can straightforwardly show that (12.17) is also locally exponentially stable at the origin. To complete the proof of Theorem 12.1, we need to show that there exist the design constants ki ; 0  i  3 such that jk1 j < 1; j k2 j < 1, condition (12.29) holds, and that j.t /j < 0:5. Before discussing these conditions, we note the following observations. 1. Under Assumption 12.1, the reference surge velocity ud is always nonzero, and the magnitudes of the reference sway and heave velocities are always less than that of the reference surge velocity. 2. The mass, including added masses in the sway and heave dynamics, m22 and m33 , is often larger than that in the surge dynamics, m11 , for underwater vehicles, 1 i.e., m11 m1 22 < 1 and m11 m33 < 1. The condition of jk1 j < 1 and j k2 j < 1 can be satisfied easily by picking small enough k1 and k2 . A close look at (12.29) shows that it always holds if we pick k1 and k2 such that they are small and have the same sign with the surge reference velocity ud , and pick large enough k3 for some small positive constants "i ; i D 1; 2; 3. For example, one can choose "i D jk1 j D jk2 j. We can see from the above choice that the value of jk1 j and j k2 j should be decreased if jud j is large. This physically means that the distance from the vessel to the point it aims to track should

308

12 Trajectory-tracking Control of Underactuated Underwater Vehicles

be increased if the surge velocity is large, otherwise the vessel will miss that point. 1 and d33 m1 Furthermore when d22 m22 33 are small, jk1 j and j k2 j should also be decreased. Small jk1 j and j k2 j also imply a small value of k0 . This can be physically interpreted as follows: If the damping in the sway and heave dynamics is small, the control gain in the surge dynamics should also be small otherwise the vessel will slide in the sway and heave directions. The condition j.t /j < 0:5 can be written as j.t /j  jd .t /j C 2 .
/ C jarcsin .k2 2 .
//j < 0:5: Hence there always exist initial conditions such that this condition holds under Assumption 12.1. Due to complicated expressions of x .t /; y .t /; z .t /; v .t / and w .t /, we provide some general guidelines to choose the design constants rather than present their extremely complex explicit expressions. Select small values for jk1 j and j k2 j, set "i D jk1 j ; i D 1; 2; 3, large enough value for k3 . Then increase k3 and/or decrease jk1 j and j k2 j until (12.29) holds. Finally, we note that for the ease of choosing the design constants, one can replace the absolute values of ud ; vd ; wd and rd in all of the negative terms in x .t /; y .t /; z .t /; v .t / and w .t / by their maximum values. The trade-off is that the control gains ki ; 0  i  2 may be very small, which results in slow convergence of the tracking errors, if the surge reference velocity ud .t / varies largely, i.e. ud max >> ud min with ud max and ud min being the maximum and minimum values of ud .t /, respectively.

12.5 Simulations This section illustrates the effectiveness of the control law (12.16) by simulating it on an underwater vehicle with a length of 5.56 m, a mass of 1089:8 kg, and other parameters taken from [139] as follows: m11 D 1116 kg; m22 D 2133 kg; m33 D 2133 kg; m44 D 36:7 kgm2 ; m55 D 4061 kgm2 ; m66 D 4061 kgm2 ; d11 D 25:5 kgs1 ; d22 D 138 kgs1 ; d33 D 138 kgs1 ; d44 D 10 kgm2 s1 ; d55 D 490 kgm2 s1 ; d66 D 490 kgm2 s1 ; du2 D 0; du3 D 0; dp2 D 0; dp3 D 0; dq2 D 0; dq3 D 0; dr2 D 0; dr3 D 0; dv2 D 920:1 kgm2 s; dv3 D 750 kgm3 s2 ; dw2 D 920:1 kgm2 s; dw3 D 750 kgm3 s2 : This vehicle has a minimum turning circle with a radius of 75 m, a maximum surge force of 2  104 N, a maximum yaw moment of 1:5  104 Nm, a maximum pitch moment of 1:5  104 Nm, and the maximum roll moment of 120 Nm. The reference trajectory is generated by (12.3) with

12.6 Conclusions

309

ud D 5d11  .m22 vd rd  m33 wd qd / ; qd D .d C 0:2  Pd /m55  .m33  m11 /ud wd  d55 qd  grGML sin.d /; rd D .m11  m22 /ud vd

(12.34)

for the first 200 seconds, and ud D 5d11  .m22 vd rd  m33 wd qd / ; qd D .d C 0:2  Pd /m55  .m33  m11 /ud wd  d55 qd  grGML sin.d /; rd D .m11  m22 /ud vd C 0:02d66

(12.35)

for the rest of simulation time. This choice means that the reference trajectory is a straight line for the first 200 seconds followed by a helix with constant curvature and torsion. The initial conditions are picked as follows: .x.t0 /; y.t0 /; z.t0 /; .t0 /; .t0 /; u.t0 /; v.t0 /; w.t0 /; q.t0 /; r.t0 // D .50; 50; 0; 0; 0; 0; 0; 0; 0; 0/; .xd .t0 /; yd .t0 /; zd .t0 /; d .t0 /; d .t0 /; ud .t0 /; vd .t0 /; wd .t0 /; qd .t0 /; rd .t0 // D .0; 0; 20; 0; 0; 0; 0; 10; 0; 0; 0/:

(12.36)

Based on the proof of Theorem 12.1, the design constants are chosen as k0 D 0:8, k1 D k2 D 0:4, ci D 2, and i D 5 with i D 1; 2; 3. The reference and real trajectories in three dimensions are plotted in Figure 12.2. The vessel position and orientation are plotted in Figure 12.3a while the tracking errors are plotted in Figures 12.3b-c. The control inputs plotted in Figures 12.4a-c. As proven in Theorem 12.1, the tracking errors asymptotically converge to the origin. Moreover, the control inputs have not reached their limits. This means that we still can further shorten the transient time by increasing the control gains.

12.6 Conclusions The key to the control development is the coordinate transformations (12.7) to transform the tracking error system, in which the tracking errors are interpreted in the frame attached to the vehicle body, to a triangular form, to which the backstepping technique can be applied. The control development in this chapter is based on the approach developed for underactuated surface ships in Chapter 5.

310

12 Trajectory-tracking Control of Underactuated Underwater Vehicles

Reference trajectory

0

z [m]

−200

Real trajectory

−400 1500 −600 1000 −800 500

500 400

300

200

0 100

0

x [m]

−500

−100

y [m]

Figure 12.2 Reference and real trajectories in three dimensions a

y [m]

400 300 200 100 0 −200

0

200

400

50 0

−50

−100 0

800

1000

1200

1400

c

0.6 ψ−ψd, θ−θd ([rad])

x−xd, y−yd, z−zd ([m])

b

600 x [m]

200 400 Time [s]

600

0.4 0.2 0

−0.2 0

200 400 Time [s]

600

Figure 12.3 a. Position and orientation in the (x,y)-plane; b. Tracking errors: x  xd (solid line), y  yd (dash-dotted line), z  zd (dash); c. Tracking errors:  d (solid line),   d (dashdotted line)

12.6 Conclusions

311

a

u

τ [N]

2000 1000 0 0

100

200

300 Time [s]

400

500

600

400

500

600

400

500

600

b

0

q

τ [Nm]

5000

−5000 0

100

200

300 Time [s] c

0

r

τ [Nm]

5000

−5000 0

100

200

300 Time [s]

Figure 12.4 a. Control input: u ; b. Control input q ; c. Control input r

Chapter 13

Path-following of Underactuated Underwater Vehicles

This chapter extends the approach proposed for underactuated surface ships in Chapter 11 to design a path-following system for six degrees of freedom underactuated underwater vehicles. Although the control design is much more involved in comparison with that for underactuated surface ships in Chapter 11, it still guarantees that path-following errors asymptotically converge to a ball, with an adjustable radius, centered on a desired path, and covers both parking and point-to-point navigation problems.

13.1 Control Objective For the reader’s convenience, we rewrite the mathematical model of an underactuated underwater vehicle, which is described in detail in Section 3.4.2.1, moving in six degrees of freedom as follows: P 1 D J1 .2 /v1 ; M1 vP 1 D C1 .v1 /v2  D1 v1  Dn1 .v1 /v1 C 1 C 1E ; P 2 D J2 .2 /v2 ; M2 vP 2 D C1 .v1 /v1  C2 .v2 /v2  D2 v2  Dn2 .v2 /v2 

(13.1)

g2 .2 / C 2 C 2E ; where J1 .2 / and J2 .2 / are given by 2 cos. / cos./  sin. / cos./ C sin./ sin./ cos. / J1 .2 / D 4 sin. / cos./ cos. / cos./ C sin./ sin./ sin. /  sin./ sin./ cos./

3 sin. / sin./ C sin./ cos. / cos./  cos. / sin./ C sin./ sin. / cos./ 5 ; cos./ cos./ 313

314

13 Path-following of Underactuated Underwater Vehicles

3 1 sin./ tan./ cos./ tan./ cos./  sin./ 5 : J2 .2 / D 4 0 0 sin./= cos./ cos./= cos./ 2

The matrices M1 and M2 are 3 2 3 2 m44 0 0 m11 0 0 M1 D 4 0 m22 0 5 ; M2 D 4 0 m55 0 5 : 0 0 m66 0 0 m33

(13.2)

(13.3)

The matrices C1 .v1 / and C2 .v2 / are 2

3 0 m33 w m22 v 0 m11 u 5 ; C1 .v1 / D 4 m33 w m22 v m11 u 0 3 2 0 m66 r m55 q 0 m44 p 5 : C2 .v2 / D 4 m66 r 0 m55 q m44 p

(13.4)

The linear and nonlinear damping matrices D1 , D2 , Dn1 .v1 /, and Dn2 .v2 / are 3 2 3 2 d44 0 0 d11 0 0 D1 D 4 0 d22 0 5 ; D2 D 4 0 d55 0 5 ; 0 0 d66 0 0 d33 3 2 P3 i1 0 0 iD2 dui juj P3 i1 5; Dn1 .v1 / D 4 0 0 iD2 dvi jvj P3 i1 d jwj 0 0 iD2 wi 3 2 P3 i1 0 0 iD2 dpi jpj P3 i1 5 : (13.5) Dn2 .v2 / D 4 0 0 iD2 dqi jqj P3 i1 0 0 iD2 dri jrj The restoring force and moment vector g2 .2 / is given by 2 3 grGMT sin./ cos./ 6 7 7: g2 .2 / D 6 grGML sin./ 4 5 0 The propulsion force and moment vectors 1 and 2 are 2 3 2 3 p u 1 D 4 0 5 ; 2 D 4 q 5 ; 0 r

(13.6)

(13.7)

which imply that the vehicle under consideration does not have independent actuators in the sway and heave. The environmental disturbance vectors 1E and 2E

13.1 Control Objective

315

are given by 1E

3 2 3 Ep .t / E u .t / D 4 E v .t / 5 ; 2E D 4 E q .t / 5 ; E r .t / E w .t / 2

(13.8)

where E u .t /, E v .t /, E w .t /, Ep .t /, E q .t /, and E r .t / are the environmental disturbance forces or moments acting on the surge, sway, heave, roll, pitch, and yaw axes, respectively. We assume that these disturbances are bounded as follows: max max j .t /j  Emax u < 1; jE v .t /j  E v < 1; jE w .t /j  E w < 1; ˇ Eu ˇ ˇ ˇ ˇEp .t /ˇ   max < 1; ˇE q .t /ˇ   max < 1; jE r .t /j   max < 1: Ep Eq Er

(13.9)

Since the sway and heave control forces are not available in the sway and heave dynamics, the vehicle model (13.1) is underactuated. In this chapter, we consider a control objective of designing the control inputs 1 and 2 to force the underactuated vehicle (13.1) to follow a specified path ˝, see Figure 13.1. If we are able to drive the vehicle to follow closely a virtual vessel that moves along the path with a desired speed u0 , then the control objective is fulfilled, i.e., the vessel is in a tube of nonzero diameter centered on the reference path and moves along the specified path at the speed u0 . Roughly speaking, the approach is to steer the vessel such that it heads toward the virtual one and diminishes the distance between the real and the virtual vessels.

Virtual vehicle

Real vehicle

XE

OE A x YE

a1

J

D

E a3

de ZE

a2 ȍ x As

Figure 13.1 General framework of underwater vehicle path-following

u0

316

13 Path-following of Underactuated Underwater Vehicles

In Figure 13.1, A is the center of the real vehicle and As is a point on the reference path attached to the virtual vehicle. Define the following path-following errors xe D xd  x; ye D yd  y; ze D zd  z; q de D xe2 C ye2 C ze2 ;

(13.10)

where xd ; yd and zd are the coordinates of As . Then the terms ai ; 1  i  3 in Figure 13.1 are obtained from xe ; ye and ze by rotating the body frame around the earth-fixed frame OE XE YE ZE the roll, pitch, and yaw angles, i.e., 2 3 2 3 a1 xe 4 a2 5 D J1 .2 / 4 ye 5 : (13.11) a3 ze Expanding (13.11) yields

a1 D xe J111 .2 / C ye J121 .2 / C ze J131 .2 /; a2 D xe J112 .2 / C ye J122 .2 / C ze J132 .2 /;

(13.12)

a3 D xe J113 .2 / C ye J123 .2 / C ze J133 .2 /; where J1ij .2 / is the element of J1 .2 / at the i th row and j th column. Therefore the path-following orientation errors are defined by the angles ˛ and ˇ. It is noted that the angles ; ˛, and ˇ are not defined at de D 0 but with the aid of a desired controller, limk.de ;˛;ˇ /k!0 .; / D .s ; s / with s and s being the orientation angles of the virtual vessel. Hence in this chapter, we will design a controller such that it guarantees de  de with de being an arbitrarily small positive constant to avoid chattering caused by de D 0. With the above definitions, our control objective can be mathematically stated as follows: Path-following Objective. Under Assumption 13.1, design the control inputs 1 and 2 to force the underactuated vehicle (13.1) to follow the path ˝ given by xd D xd .s/; yd D yd .s/; zd D zd .s/; where s is the path parameter variable, such that lim de .t /  de ; lim j˛.t /j  ˛;

t!1

t!1

lim jˇ.t /j  ˇ; lim j.t /j  ;

t!1

t!1

(13.13)

13.2 Coordinate Transformations

317

with de , , ˛, and ˇ being arbitrarily small positive constants. Assumption 13.1. The reference path is regular, i.e. there exist strictly positive constants a3 min ; a3 max ; a2 min and a2 max such that s       @xd 2 @yd 2 @zd 2 a3 min  C C  a3 max ; @s @s @s s     @xd 2 @yd 2 C  a2 max : (13.14) a2 min  @s @s Remark 13.1. 1. We might refer to the above objective as a path-tracking one. However, we use the term “path-following” since our approach is to make the real vehicle follow the virtual one, see Figure 13.1. 2. Assumption 13.1 ensures that the path is feasible for the vessel to follow, see Section 13.3. The condition (13.14) implies that the reference trajectory cannot contain a vertical straight line to avoid singularity of J2 .2 / at the pitch angle  D ˙0:5. 3. If the reference path is not regular, then we can often split it into regular pieces and consider each of them separately. This is the case of point-to-point navigation, which will be addressed in Section 13.6. 4. The path parameter, s, is not the arc length of the path in general. For example, a circle with radius R centered at the origin can be described as xd D R cos.s/ and yd D R sin.s/.

13.2 Coordinate Transformations From (13.10) and (13.12), we have the position kinematic error dynamics as follows:   @xd 1 @yd @zd a1 a2 a3 dPe D xe sP  u  v  w: C ye C ze (13.15) de @s @s @s de de de For the path-following orientation errors, referring to Figure 13.1 and the control objective stated in the previous section, one can see that the following holds a1 D cos. / D cos.˛/ cos.ˇ/; de which in turn implies that ( lim ˛.t / D 0 t!1

lim ˇ.t / D 0

t!1

, lim .t / D 0 ) lim t!1

t!1



a1 .t / de .t /

(13.16)



D 1:

(13.17)

318

13 Path-following of Underactuated Underwater Vehicles

Hence we can either choose the angles ˛ and ˇ, or the angle , or the term a1 =de as the orientation coordinates for the control design. We now discuss the above options and then choose one that results in a simple control design and enhance feasible initial conditions. Using Angles ˛ and ˇ. In this case, the path-following orientation errors are defined as follows, see Figure 13.1: ˛ D e˛  2 n˛ .e˛ /; ˇ D eˇ  2 nˇ .eˇ /;

(13.18)

where 8
0; i D 1; 2; 3; 1 < 1. The choice of u0 .t; de / in (13.44) has the following desired feature: When the path-following error, de , is large, the virtual vessel will wait for the real one; when de is small the virtual vessel will move along the path at the speed closed to u0 and the real one follows it within the specified look ahead distance. This feature is suitable in practice because it avoids using a highgain control for large signal de . It is noted that ud is not defined at a1 D 0. Since the terms a2 =a1 and a3 =a1 can be written as .a2 =de /=.a1 =de / and .a3 =de /=.a1 =de /, and recalling that cos. / D a1 =de , the intermediate control ud is well defined if (13.40) holds and (13.45) j .t /j < 0:5; 8t  t0  0: We will come back to this issue in Section 13.4. Remark 13.2. 1. If we design the virtual control ud without canceling the terms a2 v and a3 w in the de -dynamics, then the condition (13.45) is not required for ud being well defined. However, an assumption of the sway and heave velocities being bounded is needed in advance in the stability analysis, i.e. assume stability to prove stability. 2. If the sway and heave velocities are assumed to be bounded by the surge velocity, the terms a2 v and a3 w are not required to be canceled either. This controller can be designed similarly to the one in this chapter. It is noted that the sway velocity does not require to be bounded by the surge velocity with a relatively small constant as in [129] for the case of path-following in the horizontal plane. Substituting (13.42) and (13.43) into the first equation of (13.37) results in a1 a1 Q dPe D k1 .de  ıe /  u: de de

(13.46)

By noticing that under Assumption 13.1, see (13.40) and (13.45), the intermediate control ud is a smooth function of xe ; ye ; ze ; s; u0 ; 2 ; v, and w, differentiating both sides of (13.41) with (13.42) and (13.43) yields 3

X dui m33 d11 1 m22 vr  wq  u u C uPQ D juji1 u C m11 m11 m11 m11 m11 iD2

1 @ud @ud @ud @ud @ud wu .t /  xP e  yPe  zP e  sP  uP 0  m11 @xe @ye @ze @s @u0 @ud @ud P 2  @2 @v

3

X dvi m11 d22 m33 wp  ur  v jvji1 vC m22 m22 m22 m22 iD2

13.3 Control Design

325



1 @ud wv .t /  m22 @w



m11 m22 d33 uq  vp  w m33 m33 m33 ! 3 X dwi 1 i1 ww .t / ; jwj w C m33 m33

(13.47)

iD2

where for convenience of choosing u0 , the terms @u0 @u0 ; @xe @ye

and control 1 or u D

@ud @xe

;

@ud @ye

, and

@ud @ze

do not include

@u0 , which are lumped into uP 0 . From (13.47), we choose @ze u without canceling useful nonlinear damping terms as

c1 uQ  O1T f1 .
/  O21 tanh

uQ O21 "21

!

the actual

@ud @ud uQ O22  O22 tanh @v @v "22

! O23 @u u Q  @u d d O23 ; tanh @w @w "23

!



(13.48)

and the update law as   P O1j D 1j proj uf Q 1j .
/; O1j ; 1  j  16;   P O21 D 21 proj juj Q ; O21 ; ˇ ˇ  ˇ @ud ˇ P ˇ ; O22 ; O22 D 22 proj ˇˇuQ @v ˇ ˇ ˇ  ˇ @ud ˇ P ˇ ; O23 ; O23 D 23 proj ˇˇuQ @w ˇ

(13.49)

where c1 ; "2i ; 1j ; 2i ; 1  i  3 and 1  j  16; are positive constants to be selected later, and f1j .
/ are the j th elements of f1 .
/, respectively, with   @ud @ud @ud f1 .
/ D vr; wq; ud ;  juj ud ; u2 ud ;  xP e C yPe C zP e C @xe @ye @ze  @ud @ud @ud @ud @ud @ud @ud uP 0 C P 2 ;  sP C wp; ur; v; jvj v; @s @u0 @2 @v @v @v @v  @ud @ud @ud @ud 3 T @ud 3 @ud ; (13.50) v ; uq; vp; w; w jwj w; @v @w @w @w @w @w Oij ; 1  i  2 is the j th element of Oi , which is an estimate of i with 1 D



m11 m33 m211 m11 d22 m11 dv2 ; ; ; ; m22 m22 m22 m22 T m11 dv3 m211 m11 m22 m11 d33 m11 dw2 m11 dw3 ; ; ; ; ; ; m22 m33 m33 m33 m33 m33 m22 ; m33 ; d11 ; du2 ; du3 ; m11 ;

326

13 Path-following of Underactuated Underwater Vehicles

T  max m11 max m11 max : ;   2 D wu ; m22 wv m33 ww

(13.51)

The operator, proj, is the Lipschitz continuous projection algorithm repeated here for the convenience of the reader as follows: proj.$; !/ O D$ if  .!/ O  0; O $  0; proj.$; !/ O D$ if  .!/ O  0 and !O .!/ proj.$; !/ O D .1   .!// O $ if  .!/ O > 0 and !O .!/ O $ > 0;

(13.52)

!O 2 ! 2

!/ O M ; !O .!/ ,  is an arbitrarily small positive conwhere  .!/ O D 2 C2! O D @. @!O M stant, !O is an estimate of ! and j!j  !M . The projection algorithm is such that if !PO D proj($, !/ O and !.t O 0 /  !M and then

1. 2. 3. 4.

!.t O /  !M C ; 8 0  t0  t < 1; proj($; !) O is Lipschitz continuous, O  j$j ; jproj($; !)j !proj($; Q !) O  !$ Q with !Q D !  !. O

Substituting (13.48) into (13.47) yields the error dynamics ! 3 X 1 T 1 OT 1 i1 uQ C c1 C d11 C  f1 .
/ C dui juj  f1 .
/  uQP D  m11 m11 1 m11 1 iD2 ! @ud 1 1 1 O 1 O @ud uQ O21  21 tanh 22 wu .t /  wv .t /   m11 m11 "21 @v m22 m11 @v ! ! @ud uQ O22 @ud 1 1 O @ud @ud uQ O23 tanh 23  : ww .t /  tanh @v "22 @w m33 m11 @w @w "23 (13.53) Define vQ 2 D v2  v2d ;

(13.54)

T

where v2d D Œpd ; qd ; rd  is the intermediate control of v2 . Recalling that our goal is to ultimately stabilize 2 D Œ; 1 ; 2 T at the origin, the second equation of (13.37) suggests that we choose this intermediate control as follows:
 v2d D J21 .2 / f2 .
/  K2 2 ; (13.55)

where K2 D diag.k21 ; k22 ; k23 / is a positive definite diagonal matrix. Substituting (13.54) and (13.55) into the second equation of (13.37) yields P 2 D K2 2 C J2 .2 /vQ 2 :

(13.56)

13.3 Control Design

327

To design the actual control 2 , we first note that under Assumption 13.1, (13.40) and (13.45), the intermediate control v2d is a smooth function of xe ; ye ; ze ; s; u0 ; 2 , and v1 . Differentiating both sides of (13.54) and multiplying by M2 , along the solutions of the last two equations of (13.37) results in M2 vPQ 2 D C2 .v2 /vQ 2  D2 .v2 /vQ 2 C F .
/3 C G .
/4 .t / C 2 ;

(13.57)

where F .
/3 D C1 .v1 /v1  C2 .v2 /v2d  D2 .v2 /v2d  g2 .2 /   @v2d @v2d @v2d @v2d @v2d M2 xP e C yPe C zP e C sP C uP 0 C @xe @ye @ze @s @u0  @v2d 1 @v2d P 2  M2 M .C1 .v1 /v2  D1 .v1 /v1 C 1 /; @2 @v1 1 @v2d 1 M w1 .t /; (13.58) G .
/4 .t / D w2 .t /  M2 @v1 1 with F .
/ 2 R3m3 and G .
/ 2 R3m4 being the regression matrices, 3 2 R3m3 and 4 .t / 2 Rm4 being the vectors of unknown vessel and environmental disturbance parameters. For the sake of simplicity, the regression matrices F .
/ and G .
/, and the vectors 3 and 4 .t / are not written down explicitly. From (13.56) and (13.57), we choose the actual control 2 and update laws as follows:  T 2 D K3 vQ 2  T2 J2 .2 /  F .
/O3  G .
/O4 ; ! 3 P PO O vQ 2j fj i ; 3i ; 1  i  m3 ; 3i D 3i proj j D1 ! 3 ˇ ˇ P PO ˇvQ 2j gj i ˇ; O4i ; 1  i  m4 ; 4i D 4i proj

(13.59)

(13.60)

j D1

where "4i > 0; 1  i  3, 3j > 0; 1  j  m3 , 4l > 0; 1  l  m4 , K3 D diag.k31 , k32 , k33 / is a positive definite diagonal matrix,O3i is an estimate of the i th element of 3 , and O4i is an estimate of the maximum value of the i th element of 4 .t /. For simplicity of notation, we have defined  3 3 2 m3 2 m4 m2 P P P 1 O O O f  g  tanh " v Q g  1i 4i 7 41 21 6 iD1 1i 3i 7 6 iD1 1i 4i iD1 7 6m 6m 7  m 3 4 2 7 6P 7 6P P 1 O O O 6 7 6 O O f2i 3i 7 ; G .
/4 WD 6 g2i 4i tanh "42 vQ 22 g2i 4i 7 F .
/3 WD 6 7; iD1 7 6 iD1 6 iD1  7 m m m 4 P3 5 4 P4 5 P2 f3i O3i Q 23 g3i O4i tanh "1 g3i O4i 43 v iD1

iD1

iD1

(13.61)

328

13 Path-following of Underactuated Underwater Vehicles

with fj i ; 1  j  3; 1  i  m3 , being the element in j th column and i th row of the regression matrix F .
/. Similarly gj i ; 1  j  3; 1  i  m4 , is the element in j th column and i th row of the regression matrix G .
/. Substituting (13.59) into (13.57) yields the error dynamics  T M2 vPQ 2 D C2 .v2 /vQ 2  .K3 C D2 .v2 //vQ 2  T2 J2 .2 / C F .
/3 C G .
/4 .t/  F .
/O3  G .
/O4 :

(13.62)

We now present the main result of this chapter, the proof of which is given in the next section. Theorem 13.1. Assume that 1. the vessel inertia, added mass and damping matrices are diagonal; 2. the environmental disturbances are bounded; 3. the vessel parameters are unknown but constant; 4. the reference path satisfies Assumption 13.1. If the state feedback control laws (13.48) and (13.59), and the update laws (13.49) and (13.60) are applied to the vessel system (13.1) then there exist feasible initial conditions such that the path-following errors .x.t /  xd .t /; y.t /  yd .t /; z.t /  zd .t /; 1 .t /; 2 .t // converge to a ball centered on the desired path ˝ asymptotically. Furthermore, the radius of this ball can be made arbitrarily small by adjusting the control gains.

13.4 Stability Analysis 
To prove Theorem 13.1, we first consider the 2 ; vQ 2 -subsystem and then the Q .de ; u/-subsystem.
 2 ; vQ 2 -subsystem

To investigate stability of this subsystem, we consider the following Lyapunov function 4 1 T 1 X Q T 1 Q 1 T V1 D 2 2 C vQ 2 M2 vQ 2 C  i  i i ; (13.63) 2 2 2 iD3 
where Qi D i  Oi and i D diag ij ; 1  j  m3 for i D 3; 1  j  m4 for i D 4. Differentiating both sides of (13.63) along (13.56), (13.60), and (13.62) yields VP1  T2 K2 2  vQ 2T .D20 C K3 / vQ 2 C 0:2785

3 X iD1

"4i ;

(13.64)

13.4 Stability Analysis

329

where D20 D diag.d44 ; d55 ; d66 / and we have used jxj  x tanh.x=/  0:2785 for all x 2 R and  > 0. From (13.64), we conclude that 2 and vQ 2 are ultimately asymptotically stable at the origin. To estimate the upper bound of 2 and vQ 2 , we 4 1P subtract and add Q T  1 Qi to the right-hand side of (13.64) to obtain 2 iD3 i i VP1  1 V1 C 1 ;

(13.65)

where   2min .D20 C K3 / 1 D min 1; 2min .K2 / ; ; max .M2 / 1 D

4

3

iD3

iD1

X 1 X Q T 1 Q "4i : i i i C 0:2785 2

(13.66)

From (13.65), it is direct to show that V1 .t /  V1 .t0 /e 1 .tt0 / C

1 ; 1

(13.67)

which further yields 1 1 r

p  .tt0 / .tt0 /  21

2 .t/  2V1 .t0 /e 2 WD ˛ .
/e 2 C C  ; s s1 1 1  .tt0 / 2V1 .t0 /  .tt0 / 21 C C v : e 2 WD ˛v .
/e 2 kvQ 2 .t/k  min .M2 / 1 min .M2 / (13.68)

.de ; u/-subsystem Q To analyze the stability of this subsystem more easily, we first consider the uQ dynamics and then the de -dynamics. u-dynamics. Q Consider the following Lyapunov function 2

V2 D

m11 2 1 X Q T 1 Q uQ C i i i ; 2 2

(13.69)

iD1


where Qi D i  Oi and i D diag ij , 1  j  16 for i D 1; 1  j  3 for i D 2. Differentiating both sides of (13.69) along (13.47), (13.48), and (13.49) yields VP2  .c1 C d11 /uQ 2 C 0:2785

3 X iD1

"2i :

(13.70)

330

13 Path-following of Underactuated Underwater Vehicles

Subtracting and adding

2 1P Q T  1 Qi to the right-hand side of (13.70) yields 2 iD1 i i

VP2  2 V2 C 2 ;

(13.71)

where   2 .c1 C d11 / ; 2 D min 1; mmin 11 2 D

2

3

iD1

iD1

X 1 X Q T 1 Q "2i ; i i i C 0:2785 2

(13.72)

with mmin 11 being the minimum value of m11 . From (13.71), it is direct to show that 2 ; 2

(13.73)

22 WD ˛u .
/e 2 =2.tt0 / C u : 2

(13.74)

V2 .t /  V2 .t0 /e 2 .tt0 / C which further yields p Q /j  2V2 .t0 /e 2 =2.tt0 / C ju.t

s

Remark 13.3. It is noted that, due to the use of the projection algorithm, by adjusting K2 , K3 , c1 , "2i , "4i , 1j , 2i , 3n , 4l , 1  i  3, 1  j  16, 1  n  m3 , and 1  l  m4 , we can make u ,  and v arbitrarily small. This observation plays an important role in the stability analysis of the de -dynamics. de -dynamics. We first calculate the lower-bound of de . We now show that there exist initial conditions such that de .t /  de > 0. From (13.46) and (13.16), we have   (13.75) dPQe  k1 dQe  ˛u e 2 =2.tt0 / C u ;

where dQe D de  ıe , which with 2 > 2k1 further yields

 ˛u .
/e k1 .tt0 /  dQe .t /  dQe .t0 /e k1 .tt0 / C 1 C e .2 =2k1 /.tt0 /  2 =2  k1  u  k1 .tt0 / 1e : (13.76) k1

Therefore, the condition de .t /  de > 0 holds when 2 > 2k1 ; ıe  de C

u ˛u .
/ ; de .t0 /  C ıe  u : k1 2 =2  k1

(13.77)

We will come back to this issue in the next section. We now calculate the upperbound of de . We rewrite (13.46) as

13.4 Stability Analysis

331

dPQe D k1 dQe  cos. /uQ  k1 .cos. 1 /  1/ C cos. /.cos. 2 /  1// dQe :

(13.78)

Since (13.78) and the .u; Q 1 ; 2 /-subsystem are in the cascade form, one might think that the stability results developed for cascade systems in [17] and [69] can be applied. However, the stability results in those papers were developed for cascade systems without nonvanishing disturbances. In fact, nonvanishing disturbances may destroy the stability of a cascade system that satisfies all conditions stated in the above papers. Therefore, we will use Lemma 10.1 to investigate the stability of the system (13.78) by verifying all conditions C1–C4. Verifying Condition C1. Take the Lyapunov function 1 V3 D de2 : 2

(13.79)

It is direct to show that C1 holds with c0 D 0; c1 D c2 D 0:5; c3 D 1; c4 D k1 : Verifying Condition C2. By noting that ˇ ˇ ˇ ˇ ˇcos. /uQ C k1 .cos. 1 /  1/ C cos. /.cos. 2 /  1// dQe ˇ  ˇ ˇ ˇ ˇ Q C k1 .j 1 j C j 2 j/ ˇdQe ˇ ; juj

(13.80)

(13.81)

we have

1 D 1; 2 D k1 :

(13.82)

Verifying Condition C3. This condition directly holds from (13.74) and (13.68). Verifying Condition C4. From (13.80) and (13.82), condition C4 becomes k1  max. ; u / .k1 C 0:250 / > 0:

(13.83)

From Remark 13.3 and noting that 0 is an arbitrarily positive constant, we can see that there always exists k1 such that (13.83) holds. All conditions of Lemma 10.1 have been verified, and we therefore have jde .t /j  ˛de .
/ e de .tt0 / C de ;

(13.84)

where ˛de ; de and de are calculated as in Lemma 10.1. .v; w/-dynamics. Expanding (13.55) gives 
qd D  J221 .2 /k21  C J222 .2 /k22 1 C J223 .2 /k23 2  

J222 .2 / f1s sP C f1u u  J223 .2 / f2s sP C f2u u   22

 J2 .2 /f1v C J223 .2 /f2v v  J222 .2 /f1w C J223 .2 /f2w w;

332

13 Path-following of Underactuated Underwater Vehicles


rd D  J231 .2 /k21  C J232 .2 /k22 1 C J233 .2 /k23 2  

J232 .2 / f1s sP C f1u u  J233 .2 / f2s sP C f2u u 

  32 J2 .2 /f1v C J233 .2 /f2v v  J233 .2 /f1w C J233 .2 /f2w w; (13.85)

where J2ij .2 / is the element at the i th row and j th of J21 .2 /. To show that the sway and heave velocities are bounded, we take the following quadratic function 1 1 V4 D m222 v 2 C m233 w 2 ; 2 2

(13.86)

whose derivative along (13.37), (13.54), and (13.85) satisfies 2 max 2 VP4  m22 dv3 v 4  m33 dw2 jwj w 2  m33 dw3 w 4 C Amax 3 v C A4 w C 1 max 1 max A C A ; (13.87) 4"5 1 4"6 2

where "i > 0; i D 1; 2 , Ajmax ; 1  j  4 is the maximum value of Aj with  A1 D m11 m22 u rQ  J231 .2 /k21   J232 .2 /k22 1  J233 .2 /k23 2  


J232 .2 / f1s sP C f1u u  J233 .2 / f2s sP C f2u u C m22 wv.t/ ;  A2 D m11 m33 u qQ  J221 .2 /k21   J222 .2 /k22 1  J223 .2 /k23 2  


J222 .2 / f1s sP C f1u u  J223 .2 / f2s sP C f2u u C m33 ww.t/ ; ˇ 
ˇ m11 m22 ˇuJ233 .2 / f1w C f2w ˇ C A3 D m22 d22  m22 dv2 jvj C "5 jA1 j C 2 ˇ
ˇ !  m11 m33 ˇuJ222 .2 / f1v C f2v ˇ ; 2 ˇ
ˇ  m11 m22 ˇuJ233 .2 / f1w C f2w ˇ C A4 D m33 d33  m33 dw2 jwj C "6 jA2 j C 2 ˇ 
ˇ ! m11 m33 ˇuJ222 .2 / f1v C f2v ˇ : (13.88) 2 exist and are finite since their arguments are It can be seen from (13.88) that Amax i bounded as shown above. Hence (13.87) and (13.86) guarantee a finite upper bound of the sway and heave velocities. Initial Conditions for j .t /j < 2 ; j i .t /j < 2 ; i D 1; 2; 8t  t0  0. Since j i .t /j 

2 .t / ; i D 1; 2; 8t  t0  0, from (13.68), it is direct to show that the condition j i .t /j < 0:5; i D 1; 2; 8t  t0  0 holds if the initial conditions are such that p p 2V1 .t0 / C 21 =1 < 0:5; (13.89)

13.4 Stability Analysis

333

which is further equivalent to v u 4

2 r 

u X  2

1 2 1 Q t  .t /

C max .M2 / kvQ 2 .t0 /k C i i .t0 / C 2 < ; (13.90) 2 0 1 2 iD1

for all t  t0  0. It is noted that the terms

4 P

iD1

2

i 1 Qi .t0 / and 1 can be made

arbitrarily small, see Remark 13.3. From (13.33) and (13.16), the condition j .t /j < 0:5; 8t  t0  0 holds if the initial conditions are such that cos. 1 .t // C

ae .t / cos..t //.cos. 2 .t /  1/ > 0: de .t /

(13.91)

Under the assumption that j.t /j < 0:5; 8t  t0  0, the above condition is equivalent to cos. 1 .t // C cos. 2 .t // > 1: (13.92) From (13.68), the condition (13.92) holds if the initial conditions are such that v u 4

2 u X

t  .t /

2 C max .M2 / kvQ 2 .t0 /k2 C i1 Qi .t0 / C 2 0 iD1

r 1 2 < arccos.0:5/; 8t  t0  0: 1

(13.93)

Since arccos.0:5/ < 0:5, the condition (13.93) covers the condition (13.90). Initial Conditions for ae .t /  ae > 0; 8t  t0  0. Since ae2 D de2  ze2 , we have


‚…„ƒ ae2 D 2 .de .k1 cos. /de C k1 cos. /ıe  cos. /u/ Q    @zd sP C sin./u  cos./ sin./v  cos./ cos./w ze @s

D 2k1 cos. /ae2 C 2k1 cos. /ze2 C 2k1 cos. /de ıe  2 cos. /de uQ    @zd 2ze sP C sin./u  cos./ sin./v  cos./ cos./w : (13.94) @s

From (13.94), it is not hard to see that under Assumption 13.1 and de .t /  de > 0, if there exists a strictly positive constant ae0 such that ae .t0 /  ae0 ;

(13.95)

then there exists a strictly positive constant ae such that the condition ae .t /  ae > 0; 8t  t0  0 holds.

334

13 Path-following of Underactuated Underwater Vehicles

In summary, the feasible initial conditions are such that the conditions (13.77), (13.93) and (13.95) hold. Roughly speaking, with the above initial conditions, due to the underactuated configuration in the sway and heave, the sway and heave velocities are not able to push the vehicle to the point ae D 0 and de D 0.

13.5 Discussion of the Initial Condition We now discuss how to obtain the initial conditions such that (13.77), (13.93), and (13.95) hold. A close look at these conditions shows that they are always satisfied by selecting the initial value, s.t0 /, if the vessel heads toward the conical space containing the initial path to be followed, see Figure 13.2. If the vessel does not, the surge control should be turned off and the yaw and pitch controls should make the vessel turn until (13.77), (13.93), and (13.95) hold before applying the proposed path-following controller. The angle ı0 (see Figure 13.2) should be increased if the initial velocities v1 .t0 / and v2 .t0 / are large. Otherwise the vessel might cross the edge-line of the subspace in question, which might result in i D ˙0:5 and/or

D ˙0:5.

13.6 Parking and Point-to-point Navigation 13.6.1 Parking Parking Objective. Design the controls 1 and 2 to park the underactuated underwater vehicle (13.1) from the initial position and orientation .x.t0 /, y.t0 /, z.t0 /, .t0 /, .t0 /, .t0 // to the desired parking position and orientation of .xp , yp , zp , p , p , p / under the following conditions: 1. There exists a large enough positive constant $p such that q
2
2 
2   x.t0 /  xp C y.t0 /  yp C z.t0 /  zp  $p :

2. The vessel heads toward the feasible cone containing the desired parking orientation, see Section 13.5. 3. At the desired parking position and orientation, the environmental disturbances are negligible. The above conditions normally hold for parking practice. However, if the first two conditions do not hold, one can apply the strategy in Section 13.5 to move the vessel until they do hold. Having formulated the parking problem as above, one might claim that the path-following controller proposed in Section 13.3 can be applied by setting u0 equal zero. However, this will result in an orientation that may be very different from the desired parking one, at the desired parking position, since

13.6 Parking and Point-to-point Navigation

335

Feasible cone of initial conditions Edge line

G0

ȍ

G0 Figure 13.2 Feasible initial conditions

our proposed path-following controller is designed to drive de to a small ball, not to zero for reasons of robustness. To resolve this problem, we first generate a regular curve, ˝p .xd ; yd ; zd /, which goes via the parking position and its orientation at the parking position is equal to the desired parking condition. For simplicity of calculation, the curve can be taken as a straight line in almost all cases of the vessel’s initial conditions. Then the proposed path-following controller can be used to make the vessel follow ˝p .xd ; yd ; zd /. In this case, the velocity u0 should be chosen such that it goes to zero when the virtual p vessel tends to the desired parking position, i.e., limdep !0 u0 D 0 with dep D .xd  xp /2 C .yd  yp /2 C .zd  zp /2 . A simple choice can be taken as u0 D u0 .1  e 1 dep /e 2 de ;

(13.96)

where i > 0; i D 1; 2. Special care should be taken to choose the initial values of .xd .t0 /; yd .t0 /; zd .t0 //, see Section 13.5, and the sign of u0 such that it results in a short parking time. Remark 13.4. Once at the desired parking position and orientation, if there are large environmental disturbances, there will be an oscillatory behavior in the yaw and pitch dynamics, and the vessel might diverge from its desired position. This phenomenon is well known in ship dynamic positioning.

13.6.2 Point-to-point Navigation As seen in Section 13.1, the requirement that the reference path be a regular curve might be too cumbersome in practice, since this curve has to go via desired points generated by the helmsman and its derivatives are needed in the path-following controller. These restrictions motivate us to consider the point-to-point navigation problem as follows.

336

13 Path-following of Underactuated Underwater Vehicles

Point-to-point Navigation Objective. Design the surge force u and the yaw moment r to force the underactuated underwater vehicle (13.1) from the initial position and orientation, .x.t0 /; y.t0 /; z.t0 /; .t0 /; .t0 /; .t0 //, to go via the desired points generated by a path planner. To achieve this control objective, we first assume that the path planner generates desired points, which are feasible for the vehicle to be navigated through. We then apply the path-following controller proposed in Section 13.3 to each regular curve segments connecting desired points in sequence. The regular curve segments can be straight line, arc, or known regular curve ones. It is, however, noted that a fundamental difference between point-to-point navigation and the proposed smooth path-following is that there are a finite number of “peaks”, equal to the number of points, in the orientation errors, 1 and 2 . This phenomenon is because the path is non-smooth in the orientation at the points.

13.7 Numerical Simulations This section validates the control laws (13.48) and (13.59) by simulating them on a 5.56 m long underwater vehicle whose parameters are given in Section 12.5. The values of the vehicle parameters are assumed to be of the real vessel and are estimated on-line by the adaptation laws (13.49) and (13.60). We assume that these parameters fluctuate around the above values ˙15%. This fluctuation is chosen here for the purpose of calculating the maximum and minimum values used in the choice of the design constants. Indeed, a different fluctuation of the vehicle parameters results in different maximum and minimum values used in the choice of the design constants. In the simulation, we assume that the environmental disturbances are wu D 0:2m11 d.t /; wv D 0:2m22 d.t /; ww D 0:2m33 d.t /; wp D 0:2m44 d.t /; wq D 0:2m55 d.t /; wr D 0:2m66 d.t /; where d.t / D 1 C 0:1 sin.0:2t /. This choice results in nonzero-mean disturbances. In practice, the environmental disturbances may be different. We take the above disturbances for an illustration of the robustness properties of our proposed controller. It should be noted that only upper bounds of the environmental disturbances are needed in our proposed controller. In the simulation, based on Section 13.3 the control parameters and initial conditions are taken as k1 D 0:5; c1 D 2; K2 D diag.0:05/; K3 D diag.2/; i D diag.10/; ıe D 0:2;  T T 1 .t0 /; 2 T .t0 /; v1 T .t0 /; v2 T .t0 /; s.t0 / D Œ145; 15; 5; 0; 0:2; 0:5; 0; 0; 0; 0; 0; 0; 0T ; and all initial values of parameter estimates are taken to be 70% of their assumed true ones. The virtual vessel velocity on the path is taken as u0 .t; de / D 5.1  0:8e 2t /e 0:5de . The reference path is given by .xd D 90 cos.s/, yd D

13.8 Conclusions

337

90 sin.s/; zd D 3s/, i.e., a helix with constant curvature and torsion. Figure 13.3 plots the trajectory of the vessel and the path (dotted line) to be followed in three dimensions. The trajectory of the vessel in the horizontal plane and the path following error are plotted in Figure 13.4. Figure 13.5 plots the control inputs, u , r , p , and q . With nonvanishing environmental disturbances, our proposed controller is able to force the vehicle to follow a predefined path as expected in the control design. As can be seen in Figure 13.5, de converges to a nonzero small value, i.e., the sway and heave velocities cannot push the vessel to the point where de D 0. From Figure 13.5, it can be seen that the control inputs are below their limits. Therefore, we can still further shorten the transient time by increasing the control gains.

13.8 Conclusions The control scheme developed for path-following of underactuated surface ships in Chapter 11 was extended to design a path-following system for six degrees of freedom underactuated underwater vehicles. The key to the development of the proposed path-following system is the proper selection of the coordinate transformations in Section 13.2. The work presented in this chapter is based on [140, 141].

Reference trajectory 0

z [m]

−5 −10

Real trajectory

−15 −20 −25 100 50 0 −50 y [m]

−100

−150

−100

−50

0

50

100

x [m]

Figure 13.3 Simulation results: Path-following real and reference trajectories in three-dimensional space

338

13 Path-following of Underactuated Underwater Vehicles a

y [m]

50 0 −50 −300

−200

−100

0

100

200

x [m] b

de [m]

60 40 20 0 0

50

100

150

Time [s]

Figure 13.4 Simulation results: a. Path-following trajectory in the horizontal plane; b. Pathfollowing error de b

8000

4000

6000

τ [Nm]

3000

4000

r

τu [N]

a

5000

2000 1000 0

2000 50 100 Time [s]

0 0

150

c

150

d

60

5000 4000

40

3000

q

τ [Nm]

τp [Nm]

50 100 Time [s]

20 0 0

2000 50 100 Time [s]

150

1000 0

50 100 Time [s]

150

Figure 13.5 Simulation results (control inputs): a. Surge force u [N]; b. Yaw moment r [Nm]; c. Pitch moment p [Nm]; d. Roll moment q [Nm]

Part V

Control of Other Underactuated Mechanical Systems

Chapter 14

Control of Other Underactuated Mechanical Systems

This chapter presents several applications of the observer and control design techniques developed in the previous chapters to the control of other underactuated mechanical systems including mobile robots and VTOL aircraft. For mobile robots, a global exponential observer is first designed based on the observer design for underactuated ships in Chapter 7. Output feedback simultaneous stabilization and trajectory-tracking, and path-following controllers are then developed using the control design techniques proposed for underactuated ships in Chapters 6 and 11. For VTOL aircraft, the observer and control design strategies used for underactuated ships in Chapters 5 and 6 are utilized to design a global output feedback trajectorytracking controller.

14.1 Mobile Robots 14.1.1 Basic Motion Tasks In order to derive the most suitable feedback controllers for each case, it is convenient to classify the possible motion tasks as follows:  Point-to-point motion: The robot must reach a desired goal configuration starting from a given initial configuration, see Figure 14.1a.  Path-following: The robot must reach and follow a geometric path in the Cartesian space starting from a given initial configuration (on or off the path), see Figure 14.1b.  Trajectory-tracking: The robot must reach and follow a trajectory in the Cartesian space (i.e., a geometric path with an associated timing law) starting from a given initial configuration (on or off the trajectory), see Figure 14.1c. The three tasks are sketched in Figure 14.1with reference to a three wheel car-like robot. Execution of these tasks can be achieved using either feedforward commands, or feedback control, or a combination of the two. Indeed, feedback solutions exhibit 341

342

14 Control of Other Underactuated Mechanical Systems

a

Goal Start

b Start

eter Param s

Path

c Start

Time t

Trajectory

Figure 14.1 Basic motion tasks for a mobile robot

an intrinsic degree of robustness. Explanations of the above motion tasks are similar to those of ocean vehicles, see Section 3.2.

14.1.2 Modeling and Control Properties 14.1.2.1 Modeling We consider a unicycle-type mobile robot, which under the assumption of no wheel slips has the following dynamics [142]:

14.1 Mobile Robots

343

P D J ./!; P D C ./! P  D! C ; M!

(14.1)

where  D Œx y T denotes the position .x; y/, the coordinates of the middle point, P0 , between the left and right driving wheels, and heading angle  of the robot coordinated in the earth-fixed frame OX Y , see Figure 14.2, ! D Œ!1 !2 T , with !1 and !2 being the angular velocities of the wheels of the robot,  D Œ1 2 T , with 1 and 2 being the control torques applied to the wheels of the robot. The P and damping matrix rotation matrix J ./, mass matrix M , Coriolis matrix C ./, D in (14.1) are given by 3 2   cos./ cos./ r4 m11 m12 5 sin./ sin./ ; M D J ./ D ; m12 m11 2 b 1 b 1     0 c P d11 0 P D ; D D C ./ ; (14.2) 0 d22 c P 0 with 1 2 1 1 r mc a; m11 D 2 r 2 .mb 2 C I / C Iw ; m12 D 2 r 2 .mb 2  I /; 2b 4b 4b m D mc C 2mw ; I D mc a2 C 2mw b 2 C Ic C 2Im ; (14.3) cD

where mc and mw are the masses of the body and wheel with a motor; Ic ; Iw , and Im are the moments of inertia of the body about the vertical axis through Pc (center of mass), the wheel with the rotor of a motor about the wheel axis, and the wheel with the rotor of a motor about the wheel diameter, respectively; r, a, and b are defined in Figure 14.2; the nonnegative constants d11 and d22 are the damping coefficients. If these damping coefficients are zero, we have an undamped case. On the other hand, if the damping coefficients are positive, we have a damped case. We take the physical parameters from [142]: b D 0:75 m, a D 0:3 m, r D 0:15 m, mc D 30 kg, mw D 1 kg, Ic D 15:625 kgm2 , Iw D 0:005 kgm2 , Im D 0:0025 kgm2 , d11 D d22 D 5 kgs1 for numerical simulations. For convenience, we convert the wheel angular velocities .!1 ; !2 / of the robot to its linear, v, and angular, w, velocities by:   1 1 b ; (14.4) $ D B 1 !; B D r 1 b where $ D Œv wT and B is invertible since det.B/ D 2b=r. With (14.4), we can write the robot dynamics (14.1) as follows: xP D v cos./; yP D v sin./; P D w;

P D C .w/$  D$ C B; M$

(14.5)

344

14 Control of Other Underactuated Mechanical Systems

Y

Sway axis

s axi rge u S I

2r

y

2b

P0

Pc

xa


2 (see Assumption 14.3) and that conditions (14.126) and (14.128) hold. The simulation results are plotted in Figures 14.9, 14.10, and 14.11. It is seen that the proposed controller forces the aircraft to track the reference trajectory very well. The oscillation in the control input u2 is due to the initial conditions and the above selection of the control and observer gains. By further tuning these gains, we can improve the transient response. In general, higher control and observer gains result in a shorter transient response time but more overshoot in the velocities and control inputs of the aircraft.

380

14 Control of Other Underactuated Mechanical Systems a

b

10

15

y1, y1r [m]

x1, x1r [m]

5 0 −5 −10 0

20 Time [s]

10 5 0 0

40

c

y , y [m/s]

10 5

2r

0

2r

−2

0

2

2

40

d

2

x , x [m/s]

20 Time [s]

−4 0

20 Time [s]

−5 0

40

20 Time [s]

40

Figure 14.9 Reference (dotted line) and real (solid line) position trajectories: a. Position in x direction; b. Position in y direction; c. Velocity in x direction; d. Velocity in y direction a

b

0.6

0.5

ω, ω [rad/s]

θ, θr [rad]

0.4

0 −0.2 0

0

r

0.2

20 Time [s]

−0.5 −1 0

40

c

2

30

0

u2

u1

40

d

40

20

−2

10 0 0

20 Time [s]

20 Time [s]

40

−4 0

20 Time [s]

40

Figure 14.10 Reference (dotted line) and real (solid line) angular trajectories, and control inputs: a. Angles; b. Angular velocities; c. Control input u1 ; d. Control input u2

14.2 Vertical Take-off and Landing Aircraft

381

a

b

2hat

2

, y −y

−2

2

1

1hat

0

x −x

0

1

, y −y

2

x −x

2hat

2

1hat

4

−2 0

20 Time [s]

−4 0

40

20 Time [s]

c

40

θ−θhat, ω−ωhat

0.3 0.2 0.1 0 −0.1 0

5

10

15

20 Time [s]

25

30

35

40

Figure 14.11 Observer errors: a. Errors x1  xO 1 (solid line) and y1  yO 1 (dash-dotted line); b. Errors x2  xO 2 (solid line) and y2  yO 2 (dash-dotted line); c. Errors   O (solid line) and !  !O (dash-dotted line)

14.2.6 Notes and References The main difficulty in controlling VTOL aircraft is that they are underactuated and nonminimum phase. An approximate input–output linearization approach was used in [1, 153, 157–159] to develop a controller for stabilization and output tracking/regulation of a VTOL aircraft. In these papers, the controller was initially designed by ignoring the coupling between roll moment and thrust. The controller parameters were then selected to take into account the effects of the coupling. In [154], by noting that the output at a fixed point with respect to the aircraft body (Huygens center of oscillation) can be used, an interesting approach was introduced to design an output tracking controller. However, the proposed controller was not defined in the whole space. A simple approach was developed in [160] to provide a global controller for the stabilization of a VTOL aircraft. An optimal controller was provided in [6] for robust hovering control of a VTOL aircraft. In [161], dynamic inversion and robust control techniques were used to deal with the nonminimum phase dynamics. However, this approach imposed restrictions on the desired reference trajectories. Recently, a dynamic high-gain approach was used in [162] to design a controller to force the VTOL aircraft to globally practically track a reference trajectory generated by a reference model. In all of the aforementioned papers, all of the VTOL aircraft states are required for feedback. The output feedback tracking controller for a VTOL

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aircraft in this section is based on [163]. Under this controller, the VTOL aircraft globally asymptotically tracks the reference trajectory generated by the reference model. Indeed, the tracking controller proposed in this section covered stabilization and output tracking/regulation problems studied in the above-mentioned papers.

14.3 Conclusions This chapter has illustrated that a careful investigation of the observer design and control design techniques developed for the ocean vessels in this book can result in various solutions of many control problems for other underactuated mechanical systems that are common in practice.

Chapter 15

Conclusions and Perspectives

15.1 Summary of the Book This monograph has concentrated on the control of underactuated ocean vessels including ships and underwater vehicles. These vessels have more degrees of freedom to be controlled than the number of independent control inputs. Ships do not have an independent sway actuator while for underwater vehicles there are no independent sway and heave actuators. The book started with a review of the necessary background on ocean vessel dynamics and nonlinear control theory. The authors then demonstrated a systematic approach based on various nontrivial coordinate transformations together with advanced nonlinear control design methodologies founded on the basis of Lyapunov’s direct method, backstepping, and parameter projection techniques for the development and analysis of a number of ocean vessel control systems to achieve advanced motion control tasks. These tasks include stabilization, trajectory-tracking, path-tracking, and path-following. The book has offered new knowledge regarding the nonlinear control of underactuated ocean vessels, efficient controllers for practical control of underactuated ocean vessels, and general methods and strategies to solve nonlinear control problems of other underactuated systems, including underactuated land and aerial vehicles. Numerical simulations and real-time implementations of the designed control systems on a scaled model ship have been included to illustrate the effectiveness and practical guidance of the control systems developed in the book. As an illustration of the constructive feature of the book, the observer design, control design, and stability analysis techniques developed for underactuated ocean vessels have been applied to control of other underactuated mechanical systems including mobile robots and VTOL aircraft. The book consists of 15 Chapters. Chapter 1 presented a brief review of the development in nonlinear control theory and its applications, and difficulties in the control of underactuated ocean vessels. The main contributions of the book are organized into five parts.

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In Part I (Chapter 2), various mathematical tools such as Barbalat-like lemmas, Lyapunov stability theory, and the backstepping technique were presented for control design and stability analysis of controlled systems designed in the book. Part II consists of Chapters 3 and 4. This part addressed modeling, motion control tasks, and control properties of ocean vessels. The existing literature on the control of underactuated ocean vessels was also reviewed in this part. This review motivated contributions of the book on new solutions for the motion control of underactuated ships and underwater vehicles. Part III consists of Chapters 5, 6, 7, 8, 9, 10, and 11. This part focused on underactuated surface ships by proposing a number of solutions for various control problems including stabilization, trajectory-tracking, path-tracking and path-following. Chapter 5 addressed the problem of trajectory-tracking control of underactuated ships. These ships do not have independent actuators in the sway axis. The reference trajectory was generated by a suitable virtual ship. The control development was based on an elegant coordinate transformation, Lyapunov’s direct method, and the backstepping technique, and utilized passivity properties of ship dynamics and their interconnected structure. Chapter 6 examined the problem of designing a single controller that achieved stabilization and trajectory-tracking simultaneously for underactuated ships. In comparison with the preceding chapter, a path approaching the origin and a set-point were included in the reference trajectory, that is, stabilization/regulation was also considered. The control development was based on several special coordinate transformations plus the techniques in the preceding chapter. Chapter 7 presented global partial-state feedback and output feedback control schemes for trajectory-tracking control of underactuated ships. For the case of partial-state feedback, measurements of the ship sway and surge velocities were not needed, while for the case of output feedback, no ship’s velocities were required for feedback. Global nonlinear coordinate changes were introduced to transform the ship dynamics to a system affine in the ship velocities. This affine form allowed us to design observers that globally exponentially estimate unmeasured velocities. These observers plus the techniques in Chapters 4 and 5 facilitated the development of the controllers. Chapter 8 dealt with the problem of path-tracking control of underactuated ships. In comparison with Chapters 5, 6, and 7, the requirement of the reference trajectory generated by a suitable virtual ship was relaxed. Both full state feedback and output feedback cases were considered. The control development was based on a series of nontrivial coordinate transformations plus the techniques in the previous chapters. Chapter 9 addressed the problem of way-point tracking control of underactuated ships. Both full state feedback and output feedback controllers were designed. The controllers in this chapter could be regarded as an advanced version of the conventional course-keeping controllers in the sense that in addition to maintaining the desired heading, both the sway displacement (lateral distance) and sway velocity were controlled. Chapter 10 developed full state feedback and output feedback controllers that forced underactuated ships to follow a predefined path. The control development was motivated by an observation that it is practical to steer a vessel such that the vessel is on the reference path and its total velocity is tangent to the reference path, and that the vessel forward speed is controlled separately by the

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main thruster control system. The techniques in the previous chapters plus the use of the Serret–Frenet frame facilitated the results. Chapter 11 presented a different approach from Chapter 10 to solve a path-following problem for underactuated ships. Unlike Chapter 10, the control development was based on the method of generating reference paths by the helmsman. The path-following errors were first interpreted in polar coordinates, then the techniques developed in the previous chapters were used to design path-following controllers. Interestingly, some situations of practical importance such as parking and point-to-point navigation were covered in this chapter as a by-product of the developed path-following system. Part IV consists of Chapters 12 and 13. This part focused on underactuated underwater vehicles. Chapter 12 addressed the problem of trajectory-tracking control of underactuated underwater vehicles. These vehicles do not have independent actuators in the sway and heave axes. The controller development was built on the techniques developed for underactuated ships in Chapters 5, 6, and 7. Due to complex dynamics of the underwater vehicles in comparison with that of the ships, the control design and stability analysis required more complicated and coordinate transformations and control design than those already developed for underactuated ships in Chapters 5, 6, and 7. Chapter 13 extended the results of Chapter 11 to the design of a path-following control system for underactuated underwater vehicles. A series of path-following strategies for the vehicles was first discussed. A practical approach was then chosen to facilitate the control development. We also addressed the parking and point-to-point navigation problems for the vehicles in this chapter. Part V (Chapter 14) presented several applications of the observer and control design techniques developed in the previous chapters to other underactuated mechanical systems including mobile robots, and VTOL aircraft. Finally this chapter concludes the book by briefly summarizing the main results presented in the previous chapters and presenting related open problems for further investigation.

15.2 Perspectives and Open Problems Control of underactuated ocean vessels in particular and of underactuated mechanical systems in general is far from being well completed. There are in fact numerous important problems, which are still open to further investigation. In the previous chapters, we provided a number of control objectives and their solutions. However, this book has not covered all aspects of dynamics and control of the ocean vessels. Here, we highlight some open problems with detailed discussion. The problems are mainly chosen based on the following two criteria. First, to our knowledge these problems are still open and have not been solved in the literature. Second, we think that these problems potentially have important impacts on the development of control theory for underactuated systems and significant applications in practice. These criteria are highly subjective and mainly reflect our personal experience and points

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of view. However, the discussions are usually made in an elementary and intuitive manner.

15.2.1 Further Issues on Control of Single Underactuated Ocean Vessels 15.2.1.1 Theoretical issues Robust and Adaptive Issues. Although the trajectory-tracking controllers in this book possess a certain robustness, more research is needed to design such a controller that explicitly takes the environmental disturbances into account. Disturbances entering the actuated dynamics (e.g., surge and yaw for ships) can be easily taken care of. However, disturbances that enter the unactuated dynamics (e.g., sway for ships) are more difficult to handle. A possible way to reject the constant part of these disturbances on the unactuated dynamics is to introduce a small angle to the heading error to compensate for the disturbances. Since the controllers in the book require exact knowledge of the system parameters, it is of interest to examine the problem of entirely unknown vessel parameters. As opposed to the approach based on time-varying linear system stability, the proposed control designs in this book generate explicit Lyapunov functions. This feature can be further exploited in conjunction with adaptive control algorithms in the literature to address the adaptive tracking problem. Output Feedback Issues. The global coordinate transformations in the book bring the ship and mobile robot systems to an affine form for which a global exponential nonlinear observer can be easily designed. It is worth investigating the possibility of extending this type of transformation to a certain class of Lagrange systems. Indeed, the most difficult task is to obtain a solution of a set of partial differential equations, which are in general not easy to solve. Furthermore, an adaptation rule should be included in the observers to estimate the constant components (bias) of the environmental disturbances.

15.2.1.2 Real Application Issues Although several controllers presented in this book have been successfully implemented on a scaled model ship, the following problems of technology transfer to real application studies should be considered: 1. assessment of benefits obtainable from implementing nonlinear controllers on real ocean-going vessels; 2. the issues of controller implementation in practice; 3. water-tank trials of controllers with scaled models;

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4. sea-going trials: a full set of application trials; 5. commercial implementation.

15.2.2 Coordination Control of Multiple Underactuated Ocean Vessels Coordination control involves controlling positions of a group of ocean vehicles such that they perform desired tasks such as optimizing objective functions from measurements taken by each vessel, and stabilization/tracking desired locations relative to reference point(s). Therefore from a reference trajectory/path point of view, we can divide coordination control of a network of vessels into two main classes: (1) Known reference trajectories in advance and (2) unknown reference trajectories in advance, for the agents in the network. Three popular approaches to the known reference trajectory in advance class are leader-following (e.g. [164, 165]), behavioral (e.g., [166, 167]), and use of virtual structures (e.g., [168, 169]). Most research work investigating formation control utilize one or more of these approaches in either a centralized or a decentralized manner. Centralized schemes ( [165, 170, 171]) use a single controller that generates collision free trajectories in the workspace. Although these guarantee a complete solution, centralized schemes require high computational power and are not robust. Decentralized schemes (e.g., [57, 172]) require less computational effort, and are relatively more scalable to the team size. However, it is very difficult to predict and control the critical points. For the unknown reference trajectory in advance class, coordination control often involves optimizing objective functions to give reference trajectories for the agents in the network to track. An application of formation control based on the potential field method [165] and Lyapunov’s direct method [173] to gradient climbing was addressed in [174]. Geometric formation based on Voronoi’s partition optimization is given in [175]. Some other work belonging to this class includes [176] on geometric formation, [177,178] on pattern formation, [57] on flocking, [179] on swarm aggregation, and [180] on deployment and task allocation. The main problem with this class is that the final arrangement of the agents cannot be foretold due to the fact that the aforementioned results use local optimization methods to optimize nonconvex objective functions. This means that only local results can be obtained. For both classes, the aforementioned works assume that the vehicles have very simple dynamics such as single or double integrators. Vehicle dynamics are usually complex in the sense of being nonlinear and are subject to disturbances. Therefore, motion coordination control systems and deployment algorithms for networked vehicles should consider these complex features of the vehicle dynamics. It is suggested to combine the control developments for the single underactuated ocean vessels in this book with the available coordination control algorithms to achieve the aforementioned coordination objectives. Indeed, this combination is not a trivial task. It requires a careful examination of the algorithms developed for single vessels and coordination objectives. The reader is referred to [181–185] for several coordination control results based

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on the control of single mobile robots in Chapter 14 of this book and the formation control results in [186].

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Index

absolute stability, 11 acceleration feedback, 83, 185 actuator saturation, 203 adaptive control, 2, 20, 23, 69, 190, 386 added mass, 47, 70, 101, 104, 124, 146, 184, 185, 200, 214, 246, 278, 328 adjoint operator, 32 affine, 20, 135, 384, 386 air and underwater vehicles, 82 angular velocities, 70, 84, 343, 349, 368 autonomous underwater vehicles, 69 backstepping technique, 21, 105, 109, 165, 243, 246, 362, 373 backward tracking, 136, 167, 184, 297 Barbalat’s lemma, 25, 26, 69, 110, 120, 180, 199, 356 body-fixed frame, 43–46, 70, 82, 91, 185, 186, 189 bottom/top symmetry, 55, 56 bounded control, 203, 373 Brockett’s condition, 69 cascade systems, 77, 145, 156, 251, 331, 367 center manifold theory, 11 chained form, 69, 70, 82, 348 comparison principle, 120, 253, 365 control Lyapunov function, 20 controllability, 27–33, 39, 62–65, 67, 345–347 coordinate transformations, 73, 75, 85, 109, 112, 155, 161, 165, 177, 299, 309, 383–386 Coriolis and centripetal matrix, 46, 52 course-keeping, 237, 265, 384 damping matrix, 47, 48, 52, 55, 84, 201, 343

differential global positioning system (DGPS), 206 dynamic positioning, 73, 110, 284, 335 earth-fixed frame, 43–45, 91, 169, 186, 189, 343 Euler angles, 2, 3, 43, 44 Euler parameters, 45, 296 exponential observer, 84, 139, 161, 341, 370 exponential stability, 65, 69, 77, 79, 81, 122, 176, 345, 358 feedback linearization, 2 gravitational/buoyancy forces, 2 gravity acceleration, 59 hydrodynamic Coriolis and centripetal matrix, 47 hydrodynamic forces and moments, 47 hydrodynamic restoring force, 83 inertia matrix, 45, 46 input–output stability, 11 input-to-state stability, 2, 19 integral input-to-state stability, 2 Lie algebra, 31, 32, 64 Lie bracket, 31, 32, 345 Lie derivative, 33 Lipschitz projection algorithm, 177, 190 longitudinal metacentric height, 59 Lyapunov stability, 11, 66, 345 Lyapunov’s direct method, 11, 79, 89, 109, 161, 243, 367, 383, 384, 387 Lyapunov-like theorem, 16 main thruster control system, 213, 243

399

400 marine vessels, 69 matching condition, 178 metacentric restoring forces, 45, 296 mobile robots, 74, 82, 83, 93, 244, 341, 348, 366, 367, 383 monotonic velocity, 84 nilpotent algebra, 64 nilpotent basis, 64 nonautonomous system, 12 nonholonomic constraints, 39, 42, 67, 69, 70, 366 robots, 348 systems, 68, 69 nonintegrable acceleration constraints, 83 second-order constraint, 82 nonlinear control methods, 2 control theory, 1, 64, 383 damping, 23, 48, 55, 59, 89, 105, 145, 154, 155, 165, 199–201, 203, 211, 260, 261, 276, 314, 325, 355, 363, 364, 367, 377 drifts, 69 interconnected system, 14 passive observer, 243 nonminimum phase, 369, 381 nonvanishing disturbances, 251, 331 observability, 27–30, 32, 33 obstacle avoidance, 39 ocean currents, 49, 55, 89, 105 off-diagonal terms, 55, 57, 89, 105, 168, 185 output feedback controllers, 69, 213, 243, 266, 384 tracking controller, 137, 159, 381 overactuated systems, 2 vessel, 42 partial differential equations, 84, 85, 148, 188, 351, 386 path-following of underactuated ships, 243, 271 underactuated underwater vehicles, 313 pitch

Index angle, 296, 297, 317 dynamics, 335 point-to-point motion, 40, 42, 341 navigation, 271, 273, 285, 287, 313, 317, 335, 336, 385 posture stabilization, 40, 64, 345 projection integral actions, 200 integrators, 185, 190 propellers, 1, 52, 55, 136, 206 propulsion force and moment vector, 46, 55–57, 59, 61, 136, 314 forces and moments, 52 radiation-induced potential damping, 47 restoring forces and moments, 47 restrictive tracking, 203 rudders, 1, 52 separation principle, 2, 154 simplified dynamics, 89, 109 simultaneous stabilization and tracking, 105, 129, 350 starboard symmetry, 55 strict feedback system, 3 trajectory-tracking control of mobile robots, 366 underactuated ships, 82, 89, 109, 135, 384 underactuated underwater vehicles, 295, 385 transverse metacentric height, 59 underactuated mechanical systems, 70, 75, 341, 382, 383, 385 ocean vessels, 1, 2, 60, 61, 68, 69, 77, 383, 385–387 omni directional intelligent navigator, 57 systems, 160, 383, 385 vertical take-off and landing aircraft (VTOL), ix, x, 6, 341, 368, 381, 383, 385 way-point tracking, 91, 110, 213, 384

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