Inertial Quasi-Velocity Based Controllers for a Class of Vehicles: With Simulation Applications for Underwater Vehicles, Hovercrafts, and Indoor Airships (Springer Tracts in Mechanical Engineering) 3030946460, 9783030946463

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Springer Tracts in Mechanical Engineering

Przemyslaw Herman

Inertial Quasi-Velocity Based Controllers for a Class of Vehicles With Simulation Applications for Underwater Vehicles, Hovercrafts, and Indoor Airships

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA Francisco Cavas-Martínez , Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers of Sfax, Sfax, Tunisia Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany Hamid Reza Karimi, Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

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Przemyslaw Herman

Inertial Quasi-Velocity Based Controllers for a Class of Vehicles With Simulation Applications for Underwater Vehicles, Hovercrafts, and Indoor Airships

Przemyslaw Herman Poznan University of Technology Poznan, Poland

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-3-030-94646-3 ISBN 978-3-030-94647-0 (eBook) https://doi.org/10.1007/978-3-030-94647-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to my Family

Foreword

The topic of the monograph fits into the multifaceted research problem of mobile robotics and autonomous systems. In particular, it is focused on modelling and motion control. In my opinion, it is important to emphasize that unlike many other books dealing with issues specific to wheeled vehicles, commonly regarded as kinematic nonholonomic systems, this monograph explores research areas concerning another class of vehicles for which the proper modelling of dynamics is particularly relevant. The author of this monograph, Prof. Przemysław Herman, whom I have personally known for 20 years, is a specialist in applications of the Inertial Quasi-Velocities (IQV) in robotics. He published many papers in well-recognized scientific journals in which he reflects his personal perspective combing analytical mechanics with nonlinear control theory in an original way. In this book, he draws on his many years of experience in modelling multibody serial mechanical structures, which are commonly used in industrial manipulators, and investigates the development of the application of the IQV for a new class of mechanical systems including vehicles interacting with a fluid environment such as robotized vessels, submarines, hovercrafts and airships. The monograph consists of 13 chapters and the appendix. The first chapters are devoted in detail to the issue of modelling kinematics and dynamics of selected systems in the framework of the IQV taking advantage of Generalized Velocity Component (GVC) and Normalized Generalized Velocity Component (NGVC) variants. The cases of planar and spatial motion are addressed separately. In the following chapters, the author deals with the design of feedback motion control algorithms for a selected class of vehicles. He takes into account the point stabilization and especially trajectory tracking control tasks. The considered motion controllers are designed with perfect and also imperfect knowledge of the system dynamics, and are presented in terms of GVC and NGVC using earth fixed and body fixed coordinate frames. In addition, adaptive techniques are employed to improve the robustness of these controllers to some class of bounded disturbances. The essential theoretical tools used in the book are based on well-established mathematical methods including, among others, nonlinear transformations of state variables to obtain an alternative description of motion dynamics and Lyapunov-based techniques to study the stability vii

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of the closed-loop system. In my opinion, an important element of the work is the presentation of simulation results obtained on many scenarios which give an insight into the performance of the investigated control algorithms. I think that these may be relevant for a preliminary assessment of the applicability of the proposed methods in practice. One of my favourite chapters concerns the author’s analysis focused on the possibility of studying the dynamics of vehicles based on the closed-loop system response evaluated in terms of the IQV. I believe that this book could be of interest to researchers and Ph.D. students working on motion control of mobile systems with non-trivial dynamics. Although much of the content of the book is theoretical, the author’s conclusions and the numerous simulation examples may also be of benefit to practitioners. Poznan, Poland

Dariusz Pazderski, Ph.D., D.Sc.

Preface

Autonomous vehicles have in the last years an increasing interest from research and industry. There are various kinds of such vehicles: ground vehicles, marine vehicles, aerial vehicles, and others. In this book, some of moving systems are considered. In this book, objects have been selected whose dynamics can be described by such equations of motion, which can then be used for model-based control purposes. For this reason, the following categories of vehicles are taken into account: • Autonomous Underwater Vehicles (AUV) which move in water and belong to marine vehicles; they can be described by 6 DOF nonlinear equations of motion; • autonomous unmanned hovercrafts which move in the air and belong to planar vehicles (similarly as ships or surface vessels); they can be described by 3 DOF nonlinear equations of motion arising from reduction of equations used for AUV; • autonomous unmanned airships which belong to Lighter Than Air Robots (LTAR); they can be described by 6 DOF nonlinear equations of motion. As a result of different constructions and works in different environments, the external generalized forces must be taken into account. Control of the vehicles is an active field both due to its theoretical challenges and practical applications such as transportation inspection and others. There exist many control strategies for unmanned systems. Here, the focus is on model-based control to show the relationship between the vehicle model and the control algorithms. However, other approaches used for the same purpose are mentioned in this book. The main goal of control is to move the robot vehicle according to the wishes of the operator. In order to achieve it, the model-based controller is in many cases not the optimal solution because different strategies may guarantee more effective motion of the vehicle. This book is limited only to the simulation verification of the presented control algorithms without their experimental validation. The most important reason is the fact that the considered control algorithms can be applied at the stage of checking the suitability of vehicle models for further research independently of the experiment or before the decision to conduct an experiment is made, which is always expensive and requires overcoming various difficulties at the stage of its preparation. ix

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In the beginning, the models usually applied for the robot motion description are shown. Next, the same models expressed in the Inertial Quasi Velocities (IQV) are given. Then different types of control algorithms containing IQV are presented and the theoretical considerations are supported by simulation results. Inspired by progress in the field of unmanned vehicles, this monograph deals with model-based control of moving robotic systems including underwater vehicles, hovercrafts, and airships. The book can be useful for senior and postgraduate students and researchers of marine technology, control and mechanical engineering, or mechatronics. Poznan, Poland May 2021

Przemyslaw Herman

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Book Subject Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Underwater Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vehicles Moving in Horizontal Plane . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Airships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Inertial Quasi-Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3 4 4 6

2

Models of Underwater Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Six Degrees of Freedom Equations of Motion . . . . . . . . . . . . . . . . 2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Body-Fixed Representation . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Earth-Fixed Representation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equations of Motion in Terms of IQV . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Equations of Motion Using GVC . . . . . . . . . . . . . . . . . . . . 2.2.2 Equations of Motion Using NGVC . . . . . . . . . . . . . . . . . . 2.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 11 16 19 21 21 25 28 28

3

Models of Hovercrafts and Vehicles in Horizontal Motion . . . . . . . . . 3.1 Three Degrees of Freedom Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Body-Fixed Representation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Earth-Fixed Representation . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Equations of Motion in Terms of IQV . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Equations of Motion Using GVC and NGVC in Body-Fixed Representation . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Equations of Motion Using GVC and NGVC in Earth-Fixed Representation . . . . . . . . . . . . . . . . . . . . . .

35 35 37 40 40 41 44

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3.2.3

Observation About Application of IQV Equations of Motion for Underactuated Vehicles in Horizontal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

Models of Airships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Equations of Motion in Body-Fixed Representation . . . . . . . . . . . 4.1.1 Equations in Terms of Body Velocity—Short Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Models of LTAR in Body-Fixed Representation . . . . . . . 4.1.3 Application of Models for Indoor and Outdoor Airships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Equations of Motion for Model Based Control . . . . . . . . 4.2 Equations of Motion in Earth-Fixed Representation . . . . . . . . . . . 4.2.1 LTAR Models in Earth-Fixed Representation . . . . . . . . . 4.3 Equations of Motion in Terms of IQV . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Equations of Motion Using GVC . . . . . . . . . . . . . . . . . . . . 4.3.2 Equations of Motion Using NGVC . . . . . . . . . . . . . . . . . . 4.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Model Free Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Some Model Free Control Strategies . . . . . . . . . . . . . . . . . 5.1.2 Model Free Control for Underwater Vehicles . . . . . . . . . 5.1.3 Model Free Control for Vehicles Moving Horizontally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Model Free Control for Airships . . . . . . . . . . . . . . . . . . . . 5.2 Control Strategies for Underwater Vehicles . . . . . . . . . . . . . . . . . . 5.2.1 Non-Model Based Approaches: Intelligent Control . . . . 5.2.2 Model Based Control Schemes in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Model Based Control Algorithms in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Control Strategies for Vehicles Moving Horizontally . . . . . . . . . . 5.3.1 Various Non-Model Based Strategies: Intelligent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Model Based Control Schemes in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Model Based Control Schemes in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Control Strategies for Airships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Various Non-Model Based Strategies: Intelligent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Model Based Control Schemes . . . . . . . . . . . . . . . . . . . . .

47 50 51 57 57 59 65 65 67 67 67 68 69 69 70 70 75 75 75 77 77 78 80 80 81 84 84 84 85 89 90 91 91

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5.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93

6

IQV Based PD Control in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . 6.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 PD-Type Controller Expressed in GVC . . . . . . . . . . . . . . 6.1.2 PD-Type Controller Expressed in NGVC . . . . . . . . . . . . . 6.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . . 6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 PD Control Using GVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 PD Control Using NGVC . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 111 116 119 120 120 126 129 130

7

IQV Position and Velocity Tracking Control in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Position and Velocity Tracking Controller Expressed in GVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Position and Velocity Tracking Controller Expressed in NGVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . . 7.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Trajectory Tracking Control Using GVC . . . . . . . . . . . . . 7.2.2 Trajectory Tracking Control Using NGVC . . . . . . . . . . . . 7.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 141 142 142 149 152 153

8

IQV Velocity Tracking Control in Body-Fixed Frame . . . . . . . . . . . . . 8.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Velocity Tracking Controller Expressed in GVC . . . . . . . 8.1.2 Velocity Tracking Controller Expressed in NGVC . . . . . 8.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . . 8.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Velocity Tracking Control Using GVC . . . . . . . . . . . . . . . 8.2.2 Velocity Tracking Control Using NGVC . . . . . . . . . . . . . 8.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 159 162 163 163 166 169 169

9

IQV Position and Velocity Tracking Control in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Position and Velocity Tracking Controller Expressed in GVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Position and Velocity Tracking Controller Expressed in NGVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . .

133 133 133

171 171 171 177 182

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9.2

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Trajectory Tracking Control Using GVC . . . . . . . . . . . . . 9.2.2 Trajectory Tracking Control Using NGVC . . . . . . . . . . . . 9.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 IQV Position and Velocity Tracking Control with Adaptive Term in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Position and Velocity Tracking Controller with Adaptive Term Expressed in GVC . . . . . . . . . . . . . . 10.1.2 Position and Velocity Tracking Controller with Adaptive Term Expressed in NGVC . . . . . . . . . . . . . 10.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . . 10.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Trajectory Tracking Control Using GVC . . . . . . . . . . . . . 10.2.2 Trajectory Tracking Control Using NGVC . . . . . . . . . . . . 10.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Velocity Tracking Controller with Adaptive Term Expressed in GVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Velocity Tracking Controller with Adaptive Term Expressed in NGVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . . 11.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Velocity Tracking Control Using GVC . . . . . . . . . . . . . . . 11.2.2 Velocity Tracking Control Using NGVC . . . . . . . . . . . . . 11.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 IQV Position and Velocity Tracking Control with Adaptive Term in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Control Algorithms Expressed in IQV . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Position and Velocity Tracking Controller Expressed in GVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Position and Velocity Tracking Controller Expressed in NGVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Simplified Forms of Controllers . . . . . . . . . . . . . . . . . . . . . 12.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Trajectory Tracking Control Using GVC . . . . . . . . . . . . . 12.2.2 Trajectory Tracking Control Using NGVC . . . . . . . . . . . .

183 183 187 189 189 191 191 191 196 200 201 201 204 207 208 209 209 209 214 218 218 218 219 224 224 227 227 227 232 237 237 237 246

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12.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 13 Vehicle Dynamics Study Based on Nonlinear Controllers Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Controllers and Gain Matrices Selection . . . . . . . . . . . . . . . . . . . . . 13.3 Vehicle Dynamics Study Using Procedure . . . . . . . . . . . . . . . . . . . 13.3.1 Analysis Based on GVC Controllers . . . . . . . . . . . . . . . . . 13.3.2 Analysis Based on NGVC Controllers . . . . . . . . . . . . . . . 13.4 Comparison of Vehicle Dynamics Using Indexes . . . . . . . . . . . . . . 13.4.1 Model Test Based on GVC Controller . . . . . . . . . . . . . . . 13.4.2 Model Test Based on NGVC Controller . . . . . . . . . . . . . . 13.5 Application of GVC Control Algorithms . . . . . . . . . . . . . . . . . . . . . 13.5.1 Position and Velocity Trajectory Tracking Controller in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . 13.5.2 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.3 Velocity Tracking Controller with Adaptive Term in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.4 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Application of NGVC Control Algorithms . . . . . . . . . . . . . . . . . . . 13.6.1 Position and Velocity Trajectory Tracking Controller in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . 13.6.2 Velocity Tracking Controller in Body-Fixed Frame . . . . 13.6.3 Position and Velocity Trajectory Tracking Controller in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . 13.6.4 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Body-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.5 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Earth-Fixed Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8.1 Summary of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8.2 Perspectives and Open Problems . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 253 255 256 256 259 263 263 266 269 269

274 289

298 305 307 317 321

326

353 358 359 359 360 360

Appendix: IQV Equations, Formulas, Vehicle Models . . . . . . . . . . . . . . . . . 363

Acronyms and Nomenclature

DOF IQV GVC NGVC TCM M Mη (η) ν, ν˙

η, η, ˙ η¨ C(ν) Cη (ν, η) D(ν) Dη (ν, η) g(η) gη (η)

Degree(s) of Freedom Inertial Quasi Velocities Generalized Velocity Components Normalized Generalized Velocity Components Thruster Control Matrix ∈ R 6×6 the inertia matrix in the Body-Fixed Representation that includes a rigid body mass matrix and an added mass matrix, ˙ =0 i.e. M = M R B + M A , and satisfies M = MT > 0 and M 6×6 ∈R the inertia matrix in the Earth-Fixed Representation ∈ R 6 the vector of Body-Fixed linear and angular velocity components, i.e. ν = [u, v, w, p, q, r ]T , which corresponds to the motion variable in surge, sway, heave, roll, pitch, and yaw, respectively, and its time derivative ∈ R 6 the vector of positions and the Euler angles, i.e. η = [x, y, z, φ, θ, ψ]T and its time derivatives ∈ R 6×6 the matrix of Coriolis and centripetal terms in the Body-Fixed Representation, i.e. C(ν) = C R B (ν) + C A (ν) that satisfies C(ν) = −CT (ν), ∀ν ∈ R 6 ∈ R 6×6 the matrix of Coriolis and centripetal terms in the EarthFixed Representation ∈ R 6×6 the matrix of hydrodynamic damping terms in the BodyFixed Representation (D(ν) > 0, ∀ν ∈ R 6 , ν = 0) ∈ R 6×6 the matrix of hydrodynamic damping terms in the EarthFixed Representation ∈ R 6 the vector of gravitational and buoyancy forces and moments in the Body-Fixed Representation ∈ R 6 the vector of gravitational and buoyancy forces and moments in the Earth-Fixed Representation

xvii

xviii

F˜ F˜ η τ τη J(η)

w wη N Nη (η) ϒ

Z(η) ξ, ξ˙ Cξ (ξ) Cξ (ξ, η) Dξ (ξ) Dξ (ξ, η) gξ (η) gηξ (η)

Acronyms and Nomenclature

∈ R 6 the vector which approximates internal and external disturbances forces and moments in the Body-Fixed Representation, i.e. F˜ = [ F˜ X , F˜Y , F˜ Z , F˜ K , F˜ M , F˜ N ]T ∈ R 6 the vector which approximates internal and external disturbances forces and moments in the Earth-Fixed Representation ∈ R 6 the forces and moments (control) vector in the Body-Fixed Representation, i.e. τ = [τ X , τY , τ Z , τ K , τ M , τ N ]T ∈ R 6 the forces and moments (control) vector in the Earth-Fixed Representation ∈ R 6×6 block diagonal transformation matrix between the Body-Fixed frame and the inertial reference frame (usually the earth) which depends on the Euler angle ∈ R 6 the lumped dynamics estimation error in the Body-Fixed Representation, i.e. w = [w X , wY , w Z , w K , w M , w N ]T ∈ R 6 the lumped dynamics estimation error in the Earth-Fixed Representation ∈ R 6×6 the diagonal inertia matrix in terms of the GVC arising from decomposition of the matrix M (it contains constant elements only for the considered class of vehicles) ∈ R 6×6 the diagonal inertia matrix in terms of the GVC arising from decomposition of the matrix Mη (η) ∈ R 6×6 the upper triangular, invertible matrix in terms of the GVC with constant elements arising from decomposition of the matrix M ∈ R 6×6 the upper triangular, invertible matrix in terms of the GVC arising from decomposition of the matrix Mη (η) ∈ R 6 the vector of velocities and its time derivative in terms of the GVC arising from the decomposition of the matrix M (or from the decomposition of the matrix Mη (η)) ∈ R 6×6 the Coriolis matrix in terms of the GVC arising from decomposition of the matrix M ∈ R 6×6 the Coriolis matrix in terms of the GVC arising from decomposition of the matrix Mη (η) ∈ R 6×6 the hydrodynamic damping matrix in terms of the GVC arising from decomposition of the matrix M ∈ R 6×6 the hydrodynamic damping matrix in terms of the GVC arising from decomposition of the matrix Mη (η) ∈ R 6 the gravitational and buoyancy forces and moments vector in terms of the GVC arising from decomposition of the matrix M ∈ R 6 the gravitational and buoyancy forces and moments vector in terms of the GVC arising from decomposition of the matrix Mη (η)

Acronyms and Nomenclature

F˜ ξ F˜ η ξ π πη wξ  (η) ζ, ζ˙ Cζ (ζ) Cζ (ζ, η) Dζ (ζ) Dζ (ζ, η) gζ (η) gηζ (η) F˜ ζ F˜ ηζ 

xix

∈ R 6 the vector which approximates internal and external disturbances forces and moments in terms of the GVC arising from decomposition of the matrix M ∈ R 6 the vector which approximates internal and external disturbances forces and moments in terms of the GVC arising from decomposition of the matrix Mη (η) ∈ R 6 the forces and moments (control) vector in terms of the GVC arising from decomposition of the matrix M ∈ R 6 the forces and moments (control) vector in terms of the GVC arising from decomposition of the matrix Mη (η) ∈ R 6 the lumped dynamics estimation error in terms of the GVC arising from decomposition of the matrix M or the GVC arising from decomposition of the matrix Mη (η) ∈ R 6×6 the upper triangular, invertible matrix in terms of the NGVC with constant elements arising from decomposition of the matrix M ∈ R 6×6 the upper triangular, invertible matrix in terms of the NGVC arising from decomposition of the matrix Mη (η) ∈ R 6 the vector of velocities and its time derivative in terms of the NGVC arising from the decomposition of the matrix M (or from the decomposition of the matrix Mη (η)) ∈ R 6×6 the Coriolis matrix in terms of the NGVC arising from decomposition of the matrix M ∈ R 6×6 the Coriolis matrix in terms of the NGVC arising from decomposition of the matrix Mη (η) ∈ R 6×6 the hydrodynamic damping matrix in terms of the NGVC arising from decomposition of the matrix M ∈ R 6×6 the hydrodynamic damping matrix in terms of the NGVC arising from decomposition of the matrix Mη (η) ∈ R 6 the gravitational and buoyancy forces and moments vector in terms of the NGVC arising from decomposition of the matrix M ∈ R 6 the gravitational and buoyancy forces and moments vector in terms of the NGVC arising from decomposition of the matrix Mη (η) ∈ R 6 the vector which approximates internal and external disturbances in terms of the NGVC arising from decomposition of the matrix M ∈ R 6 the vector which approximates internal and external disturbances forces and moments in terms of the NGVC arising from decomposition of the matrix Mη (η) ∈ R 6 the forces and moments (control) vector in terms of the NGVC arising from decomposition of the matrix M

xx

η wζ

I A−T B† = BT (BBT )−1 λmin {A}

Acronyms and Nomenclature

∈ R 6 the forces and moments (control) vector in terms of the NGVC arising from decomposition of the matrix Mη (η) ∈ R 6 the lumped dynamics estimation error in terms of the NGVC arising from decomposition of the matrix M or the NGVC arising from decomposition of the matrix Mη (η) the identity matrix the matrix inverse to matrix A after transposition, i.e. (A−1) )T ∈ R m×6 is the pseudo-inverse matrix of B matrix the minimal eigenvalue of the matrix A

Chapter 1

Introduction

Abstract In this chapter the subject of this book is presented. Based on the literature, some modeling aspects for underwater vehicles, surface vehicles, and airships that are relevant to the issues considered in this book are indicated here. Against this background, works in which different types of Inertial Quasi-Velocities (IQV) are discussed are pointed out. Finally, an overview of the issues discussed in the following chapters is given.

1.1 Book Subject Matter The subject of this book is presentation of nonlinear model based control methods which can be used for various kind of vehicles and their application to study the dynamics of vehicle models. The considered class of vehicles are limited only to underwater vehicles, hovercrafts, and indoor airships. This limitation arises from that the equations of motion are suitable for the model based control. It is assumed that the vehicles move at low speed and thus the equations can be simplified. In this chapter some general definitions related to vehicles and robots described in this book are introduced. Moreover, the motivation of the research work is also considered here. At the end of the chapter the structure of the book is shortly presented. Referring to the model based control methods the following important remarks are given. In some cases, it is difficult to classify whether the considered approach belongs to the group of model based control methods or not. Such situation occurs if, e.g. the vehicle model must be known and simultaneously soft computing based control method or a complex control strategy is applied. In this work it is assumed that only the control algorithms which use directly the mathematical model in the control loop (without intelligent control techniques or other method) are treated as the model based control methods. The presented control algorithms based on the IQV (Inertial Quasi-Velocities) are applied only to some models of vehicles (underwater vehicles, hovercrafts, indoor airships). They are suitable for simulation tests before real experiment or when it is necessary to decide whether an experiment has a chance of being carried out. The use of the algorithms for a real vehicle is still open problem. The approach is rather © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_1

1

2

1 Introduction

limited for some educational or research purposes. Other control methods used in robotic vehicles and known from the literature are only mentioned. The problem of modeling is out of scope of this book. For this reason the used models are taken from other references. This book is limited only to the simulation verification of the presented control algorithms without their experimental validation. An important value of the presented control schemes is that they can be applied at the stage of checking the suitability of vehicle models that are intended for further research. Thus, they can be used independently of the experiment or before deciding to conduct it because the experiment is expensive and requires not only overcoming various difficulties at the stage of its preparation but also additional knowledge concerning its preparation.

1.2 Underwater Vehicles Underwater vehicles, in particular autonomous vehicles have many applications not necessary commercial but also scientific. Autonomous Underwater Vehicles (AUV) operate independently of the ship or other vehicle and have no connecting cables. They belongs to a group of Unmanned Underwater Vehicles (UUV). Outside this group we distinguish Remotely Operated Vehicles (ROV) which denote some underwater vehicles physically linked, via the tether, to an operator on a submarine or on a surface ship. The AUV that navigate over the sea surface and can diving are very sophisticated marine vehicles. The control unit is only a part of much more complex system. For example the real control system consists of several subsystems: control unit, energy management, navigation, propulsion/immersion, safety, communication and data acquisition (Garcia-Valdovinos et al. 2014). Marine vehicles, as for example AUV, are useful for many operations. They can be applied for sea inspection, geological works, survey of undersea structures and other researches. Review of some possible application of marine robots can be found, e.g. in Yuh et al. (2011). In the robotic literature underwater marine vehicles are described with 6 DOF models (ABS Guide for Vessel Maneuverability 2017; Antonelli 2018; Do and Pan 2009; Fossen 1994; Lantos and Marton 2011; Roberts and Sutton 2008). This means that in order to describe the vehicle motion, we take into account three coordinates to define translations and three coordinates to define the orientation. Moreover, the coordinates are defined using the Earth-Fixed Reference Frame or the Body-Fixed Reference Frame. There exist many types of control strategies concerning AUV, e.g.: model free control methods, intelligent control methods, model based control methods or the hardware oriented methods based on complex architecture of the control system. In this book we refer to various control methods but only the model based algorithms are considered more particularly.

1.3 Vehicles Moving in Horizontal Plane

3

1.3 Vehicles Moving in Horizontal Plane Mathematical models of various surface vehicles or other vehicles moving in horizontal plane can be obtained by simplification of the full 6 DOF model. It is easy to observe that environmental forces are not the same for both kind of vehicles. Thus, it is possible to notice similarities between both system models but there exist also some differences which must be taken into account during modeling. Hovercrafts are vehicles which belongs to planar or surface vehicles. Despite of the fact that the ship motion should be described by 6 DOF equations (as in Do and Pan 2009; Perez 2005), the mathematical models of such robotic systems are described usually by reduced 3 DOF equations of motion (Do and Pan 2009; Fossen 1994; Lantos and Marton 2011). There are many references dealing with hovercrafts, e.g. Aguiar et al. (2003), Fantoni et al. (1999, 2000), Fu et al. (2017), Jeong and Chwa (2018), Kim et al. (2013), Munoz-Mansilla et al. (2012), Seguchi and Ohtsuka (2002, 2003), Serres and Ruffier (2015), Sira-Ramirez and Ibanez (2000a, b), Sira-Ramirez (2002). However, the inertial forces in dynamic models are often very simplified because the inertia matrix is assumed as a diagonal one. It simplifies control algorithms. In this book only fully actuated hovercrafts are considered. The model of marine surface vehicles are slightly more complex than hovercraft models, the environmental disturbance as well as the additional masses coming from water must be taken into consideration. Because some control methods given here need the symmetric inertia matrix then the hovercraft model seems to be more appropriate. For other surface vehicles it is necessary to make additional assumptions. The algorithms presented in this book they are not considered.

1.4 Airships The airships can be divided into two groups, namely airships controlled by a pilot and autonomous. The term Unmanned Aerial Vehicles (UAV) is used to describe flying systems without a pilot. From another point of view, it is possible to divide outdoor and indoor airships. One of categories of UAV are autonomous airships (Lighter Than Air Robots—LTAR). Other categories, i.e. rotary wings autonomous systems (helicopters) and fixed wing flying systems (airplanes) are out of scope of this book. On the one hand, comparing the structure, the mathematical model of an underwater vehicle and an airship is similar. Similarity is observable if the consider a rigid body only. For modeling of airships 6 DOF equations of motion are usually used, e.g Bestaoui Sebbane (2012), Khoury and Gillett (1999), Khoury (2012). On the other hand, it can be noted essential differences in environmental disturbances and elasticity of both vehicles which must be taken into account in the system models. The control strategies concerning LTAR arise from the above remarks. The most important is the efficiency of the control algorithm in the presence of existing disturbances. For this reason very often complex control units or non-model based

4

1 Introduction

controllers are used in order to ensure acceptable flight of the airship. Moreover, for indoor experiments quite different control approaches can be applied than for outdoor flight tests. It results from the fact that the environmental disturbances are different in both cases. Model based control algorithms seems to be more appropriate rather for the first situation.

1.5 Inertial Quasi-Velocities The idea of the Inertial Quasi-Velocities (IQV), namely some generalized velocities arising from decomposition of the inertia matrix in equations of motion is well known from robotics work. The IQV can be classified as quasi-velocities described in the literature, e.g. Kwatny and Blankenship (2000). However, decomposition of the inertia matrix leads to variables, which besides kinematic quantities, contains also dynamical and geometrical parameters of the system. Some kind of the IQV were described by: • • • •

Jain and Rodriguez (1995); Loduha and Ravani (1995); Junkins and Schaub (1997a, b); Hurtado (2004).

The Generalized Velocity Components (GVC) described by Loduha and Ravani (1995) have been modified to Normalized Generalized Velocity Components (NGVC) in Herman (2005, 2006). A review of the IQV can be found in Herman and Kozlowski (2006). Some of the IQV have been used for control purposes, e.g. Herman (2005), Kozlowski and Herman (2008), Schaub and Junkins (1997), Sinclair et al. (2006). It also turned out that IQV can be used to study the dynamics of manipulators, e.g. Herman (2008a) (GVC), Herman (2008b) (NGVC), Herman (2009) (IQV proposed by Hurtado Hurtado 2004).

1.6 Outline of the Book The rest of the book is organized in 12 chapters and appendix. A brief description of each chapter is given below. • Chapter 2 presents mathematical model which is usually used for underwater vehicles. The six degrees of freedom equations of motion both for the Body-Fixed Representation and the Earth-Fixed Representation are remained here. Next, the same equations of motion are expressed in terms of the Inertial Quasi Velocities (IQV).

1.6 Outline of the Book

5

• Chapter 3 shows analogous equations of motion for vehicles mowing in the horizontal plane, first of all, hovercrafts. In this case we obtain a model with three degrees of freedom (3 DOF). • Chapter 4 discusses the six degrees of freedom equations of motion used for airships. It is shown that the vehicle model depends on that if the outdoor or indoor airship is considered because different disturbance models are assumed. • Chapter 5 deals with various control strategies known from the literature. The control methods are considered depending on the type of vehicle, although some of them are suitable for various vehicles. • Chapter 6 concerns algorithms of PD Control expressed in terms of the Inertial Quasi-Velocities (IQV). The mathematical results are illustrated by simulations done for a 6 DOF model of an underwater vehicle and an indoor airship. • Chapter 7 presents non-adaptive trajectory (position and velocity) tracking controllers using the IQV in the Body-Fixed Frame and the corresponding simulation results. • Chapter 8 considers some non-adaptive velocity tracking controllers in terms of the IQV in the Body-Fixed Frame. The algorithms are supported by simulation tests. • Chapter 9 contains non-adaptive trajectory (position and velocity) tracking controllers in terms of the IQV in the Earth-Fixed Frame as well as their simulation validation. • Chapter 10 deals with trajectory (position and velocity control trajectory) problem taking into account an adaptive term. The control algorithms using the IQV are realized in the Body-Fixed Frame. This chapter contains also results of simulations. • Chapter 11 gives velocity tracking controllers based on the IQV also with an adaptive term and in the Body-Fixed Frame and the simulation verification. • Chapter 12 shows the adaptive version of the position and velocity tracking control algorithms using the IQV in the Earth-Fixed Frame and the results of simulation tests. • Chapter 13 considers the problem of vehicle dynamics study based on the proposed control algorithms. It is shown in which way the controllers can be applied to obtain information about the vehicle dynamics and the effect of existing couplings between variables describing the motion of the object. It also contains conclusions and perspectives arising from the book. • Appendix presents an additional information concerning the description of unmanned vehicles, namely derivation of equations of motion using IQV both for the Body-Fixed Representation and the Earth-Fixed Representation. It also includes parameters of the vehicle models used for the simulation tests.

6

1 Introduction

References ABS Guide for Vessel Maneuverability (2017) American bureau of shipping abs plaza 16855 Northchase Drive Houston, TX 77060 USA, (March 2006) (Updated February 2017) Aguiar AP, Cremean L, Hespanha JP (2003) Position tracking for a nonlinear underactuated hovercraft: controller design and experimental results. In: Proceedings of the 42nd IEEE conference on decision and control, HI, December, pp 3858–3863 Antonelli G (2018) Underwater robots. Springer International Publishing AG, part of Springer Nature Bestaoui Sebbane Y (2012) Lighter than air robots. Springer, Dordrecht Do KD, Pan J (2009) Control of ships and underwater vehicles. Springer, London Fantoni I, Lozano R, Mazenc F, Pettersen KY (1999) Stabilization of a nonlinear underactuated hovercraft. In: Proceedings of the 38th conference on decision and control, Phoenix, Arizona USA December, pp 2533–2538 Fantoni I, Lozano R, Mazenc F, Pettersen KY (2000) Stabilization of a nonlinear underactuated hovercraft. Int J Robust Nonlinear Control 10:645–654 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Fu M, Gao S, Wang C (2017) Safety-guaranteed trajectory tracking control for the underactuated hovercraft with state and input constraints. Hindawi Math Probl Eng 2017:Article ID 9452920, 12 Garcia-Valdovinos LG, Salgado-Jimenez T, Bandala-Sanchez M, Nava-Balanzar L, HernandezAlvarado R, Cruz-Ledesma JA (2014) Modelling, design and robust control of a remotely operated underwater vehicle. Int J Adv Robot Syst 11(1):1–16 Herman P (2005) Normalised-generalised-velocity-component-based controller for a rigid serial manipulator. IEE Proc-Control Theory & Appl 152:581–586 Herman P (2006) On using generalized velocity components for manipulator dynamics and control. Mech Res Commun 33:281–291 Herman P (2008a) Dynamical couplings reduction for rigid manipulators using generalized velocity components. Mech Res Commun 35:553–561 Herman P (2008b) Evaluation of the reduction of dynamical coupling for robot manipulators. Proc Inst Mech Eng: J Mech Eng Sci - Part C 222:339–347 Herman P (2009) Dynamical couplings analysis of rigid manipulators. Meccanica 44:61–70 Herman P, Kozlowski K (2006) A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators. Arch Appl Mech 76:579–614 Hurtado JE (2004) Hamel coefficients for the rotational motion of a rigid body. J Astron Sci 52(1 and 2):129–147 Jain A, Rodriguez G (1995) Diagonalized Lagrangian robot dynamics. IEEE Trans Robot Autom 11:571–584 Jeong S, Chwa D (2018) Coupled multiple sliding-mode control for robust trajectory tracking of hovercraft with external disturbances. IEEE Trans Ind Electron 65(2):4103–4113 Junkins JL, Schaub H (1997a) An instantaneous eigen-structure quasi-velocity formulation for nonlinear multibody dynamics. J Astron Sci 45:279–295 Junkins JL, Schaub H (1997b) Orthogonal square root eigenvector parametrization of mass matrices. J Guid Control Dyn 20:1118–1124 Khoury GA, Gillett JD (1999) Airship technology. Cambridge University Press, Cambridge Khoury GA (ed) (2012) Airship technology. Cambridge University Press, Cambridge Kim K, Lee YK, Oh S, Moroniti D, Mavris D, Vachtsevanos GJ, Papamarkos N, Georgoulas G (2013) Guidance, navigation, and control of an unmanned hovercraft. In: 2013 21st mediterranean conference on control & automation (MED) Platanias-Chania, Crete, Greece, June 25–28, pp 380–387 Kozlowski K, Herman P (2008) Control of robot manipulators in terms of quasi-velocities. J Intell Robot Syst 53:205–221

References

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Kwatny HG, Blankenship GL (2000) Nonlinear control and analytical mechanics. Birkhäuser, Boston Lantos B, Marton L (2011) Nonlinear control of vehicles and robots. Springer, London Loduha TA, Ravani B (1995) On first-order decoupling of equations of motion for constrained dynamical systems. Trans ASME J Appl Mech 62:216–222 Munoz-Mansilla R, Chaos D, Aranda J, Díaz JM (2012) Application of quantitative feedback theory techniques for the control of a non-holonomic underactuated hovercraft. IET Control Theory Appl 6(14):2188–2197 Perez T (2005) Ship motion control. Springer, London Roberts GN, Sutton R (eds): Advances in unmanned marine vehicles, pp 14–42. The Institution of Engineering and Technology, Stevenage Herts Schaub H, Junkins JL (1997) Feedback control law using the eigen-factor quasi-coordinate velocity vector. J Chin Soc Mech Eng 19:51–59 Seguchi H, Ohtsuka T (2002) Nonlinear receding horizon control of an RC hovercraft. In: Proceedings of the 2002 IEEE international conference on control applications, September 18–20, Glasgow, Scotland UK, pp 1076–1081 Seguchi H, Ohtsuka T (2003) Nonlinear receding horizon control of an underactuated hovercraft. Int J Robust Nonlinear Control 13:381–398 Serres JR, Ruffier F (2015) Biomimetic autopilot based on minimalistic motion vision for navigating along corridors comprising U-shaped and S-shaped turns. J Bionic Eng 12:47–60 Sinclair AJ, Hurtado JE, Junkins JL (2006) Linear feedback controls using quasi velocities. J Guid Control Dyn 29:1309–1314 Sira-Ramirez H, Ibanez CA (2000a) The control of the hovercraft system: a flatness based approach. In: Proceedings of the 2000 IEEE international conference on control applications Anchorage, Alaska, USA September 25–27, pp 692–697 Sira-Ramirez H, Ibanez CA (2000b) On the control of the hovercraft system. Dyn Control 10:151– 163 Sira-Ramirez H (2002) Dynamic second-order sliding mode control of the hovercraft vessel. IEEE Trans Control Syst Technol 10(6):860–865 Yuh J, Marani G, Blidberg DR (2011) Applications of marine robotic vehicles. Intell Serv Robot 4:221–231

Chapter 2

Models of Underwater Vehicles

Abstract In this chapter, a mathematical model of underwater vehicles known from the literature is recalled. In the references concerning ocean and marine vehicles deeper analyzes of specific problems were presented. The underwater vehicle models were presented earlier both in books and journal papers. For this reason, in this work, the known models are applied. Next, the transformed equations of motion in terms of so called the Inertial Quasi Velocities (IQV) are given (their derivation is presented in Appendix).

2.1 Six Degrees of Freedom Equations of Motion The problem of modeling of the vehicle is omitted here because this problem has been studied in the literature, e.g., ABS Guide for Vessel Maneuverability (2017), Antonelli (2018), Do and Pan (2009), Fjellstad and Fossen (1994), Fossen (1994), Lantos and Marton (2011), Roberts and Sutton (2008). The mathematical model (and modeling) of marine vehicles can be found also in many other references as journal papers (Antonelli 2007; Evans and Nahon 2004; Fjellstad and Fossen 1994). The structure of a real vehicle, e.g. Gomariz et al. (2015), Molnar et al. (2007) is however more complex than it can be taken into account in usually used mathematical models. In order to define the coordinate frames we refer to Fig. 2.1. The notation used for marine vehicles is shown in Table 2.1. The presented models come mainly from Antonelli (2018), Do and Pan (2009), Fossen (1994), Roberts and Sutton (2008). Moreover, the following quantities based on Table 2.1 are introduced: η1 = [η1 η2 ]T ,

η1 = [x y z]T ,

η2 = [φ θ ψ]T ,

(2.1)

ν = [v ω] ,

v = [u v w] ,

ω = [p q r] .

(2.2)

T

T

T

The control signals, in this book, are assumed as follows: τ = [τ f τ m ]T ,

τ f = [τ X τY τ Z ]T ,

τ m = [τ K τ M τ N ]T .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_2

(2.3)

9

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2 Models of Underwater Vehicles

Fig. 2.1 Motion variables in the Body-Fixed and the Earth-Fixed reference frames for an underwater vehicle

Table 2.1 Notation used for linear motion (1, 2, 3) and rotation (4, 5, 6) of marine vehicles DOF Forces and Linear and Position and moments angular velocities Euler angles 1 2 3 4 5 6

in x-direction (surge) in y-direction (sway) in z-direction (heave) about x-axis (roll) about y-axis (pitch) about z-axis (yaw)

τX τY τZ τK τM τN

u v w p q r

x y z φ θ ψ

The mathematical background given below can be found in many works, e.g. Do and Pan (2009), Fossen (1994), Lantos and Marton (2011), Roberts and Sutton (2008). The considerations regarding modeling and the vehicle models are based mainly on the cited references. In this book, various vehicles, namely marine vehicles, hovercrafts and indoor airships are considered. The models used for underwater vehicle are applied also for these objects under assumption of limited disturbances and vehicle movement at low speed.

2.1 Six Degrees of Freedom Equations of Motion

11

2.1.1 Kinematics In this work the attitude representation by Euler Angles is applied. In order to describe kinematics of an underwater vehicle the transformation between the vectors ν and η˙ is as follows:      v R(η2 ) 03×3 η˙ 1 (2.4) = η˙ = J(η)ν ⇔ η˙ 2 03×3 J(η2 ) ω ⎡ ⎤ cψcθ −sψcφ + cψsθ sφ sφsψ + cψcφsθ R(η2 ) = ⎣ sψcθ cψcφ + sφsθ sψ −cψsφ + sθ sψcφ ⎦ , (2.5) −sθ cθ sφ cθ cφ ⎡ ⎤ 1 sφtθ cφtθ J(η2 ) = ⎣ 0 cφ −sφ ⎦ , cθ = 0. (2.6) 0 sφ/cθ cφ/cθ where s· = sin(·), c· = cos(·), and t· = tan(·). Remark 2.1 Besides the presented kinematics description the attitude representation by quaternions is applied in the literature (e.g. in Do and Pan 2009; Fossen 1994).

2.1.2 Dynamics The nonlinear dynamic equations can be derived from the Newton-Euler formulation or Lagrangian mechanics (Antonelli 2018; Do and Pan 2009; Fjellstad and Fossen 1994; Fossen 1994; Lantos and Marton 2011). The models considered in this book are based on the approach given in these references. There exist also different dynamics descriptions in which the vehicle motion in currents is taken into account, e.g. Thomasson and Woolsey (2013). However, in the cited paper the equations of motion are expressed using the relative velocity. Recall the equations of motion known from the literature (Do and Pan 2009; Fossen 1994).

2.1.2.1

Rigid Body Equations of Motion

The full dynamic model in the Body-Fixed Frame containing the rigid-body dynamics and the added inertia, hydrodynamic damping, and restoring forces and moments yields can be written as follows: M R B ν˙ + C R B (ν)ν = τ R B ,

(2.7)

12

2 Models of Underwater Vehicles

where:    mI3×3 −mS(r gb ) M R B 11 M R B 12 = Ib M R B 21 M R B 22 mS(r gb ) ⎤ ⎡ m 0 0 0 mz g −myg ⎢ 0 m 0 −mz g 0 mx g ⎥ ⎥ ⎢ ⎢ 0 0 m myg −mx g 0 ⎥ ⎥. =⎢ ⎢ 0 −mz g myg Ix −Ix y −Ix z ⎥ ⎥ ⎢ ⎣ mz g 0 −mx g −I yx I y −I yz ⎦ −myg mx g 0 −Izx −Izy Iz 

MR B =

(2.8)

In the equation x g , yg , z g are elements of the vector of the center of gravity (CG), i.e. rbg = [x g , yg , z g ]T . Moreover, the appropriate matrices are defined as follows:   C R B 11 C R B 12 C R B (ν) = C R B 21 C R B 22   −mS(v) − mS(ω)S(rbg ) 03×3 = with (2.9) −S(Ib ω) −mS(v) + mS(rbg )S(ω) ⎡ ⎤ m(yg q + z g r ) −m(x g q − w) −m(x g r + v) C R B 12 = ⎣ −m(yg p + w) m(z g r + x g p) −m(yg r − u) ⎦ , −m(z g p − v) −m(z g q + u) m(x g p + yg q) ⎡ ⎤ −m(yg q + z g r ) m(yg p + w) m(z g p − v) C R B 21 = ⎣ m(x g q − w) −m(z g r + x g p) m(z g q + u) ⎦ , m(yg r − u) −m(x g p + yg q) m(x g r + v) ⎡ ⎤ 0 −I yz q − Ix z p + Iz r I yz r + Ix y p − I y q 0 −Ix z r − Ix y q + Ix p ⎦ . C R B 22 = ⎣ I yz q + Ix z p − Iz r 0 −I yz r − Ix y p + I y q Ix z r + Ix y q − Ix p 2.1.2.2

Hydrodynamic Forces and Moments

The hydrodynamic force and moment vector τ H can be described as follows: τ H = −M A ν˙ − C A (ν)ν − D(ν)ν − g(η),

(2.10)

where: ⎡

 MA =

A11 A21

X u˙ ⎢ Yu˙ ⎢  ⎢ Z u˙ A12 = −⎢ ⎢ K u˙ A22 ⎢ ⎣ Mu˙ Nu˙

X v˙ Yv˙ Z v˙ K v˙ Mv˙ Nv˙

X w˙ Yw˙ Z w˙ K w˙ Mw˙ Nw˙

X p˙ Y p˙ Z p˙ K p˙ M p˙ N p˙

X q˙ Yq˙ Z q˙ K q˙ Mq˙ Nq˙

⎤ X r˙ Yr˙ ⎥ ⎥ Z r˙ ⎥ ⎥, K r˙ ⎥ ⎥ Mr˙ ⎦ Nr˙

(2.11)

2.1 Six Degrees of Freedom Equations of Motion

13



−S(A11 v + A12 ω) 03×3 −S(A11 v + A12 ω) −S(A21 v + A22 ω) ⎡ ⎤ 0 0 0 0 −a3 a2 ⎢ 0 0 0 a3 0 −a1 ⎥ ⎢ ⎥ ⎢ 0 0 0 −a2 a1 0 ⎥ ⎥ =⎢ ⎢ 0 −a3 a2 0 −b3 b2 ⎥ , ⎢ ⎥ ⎣ a3 0 −a1 b3 0 −b1 ⎦ −a2 a1 0 −b2 b1 0



C A (ν) =

(2.12)

where: a1 = X u˙ u + X v˙ v + X w˙ w + X p˙ p + X q˙ q + X r˙ r, a2 = Yu˙ u + Yv˙ v + Yw˙ w + Y p˙ p + Yq˙ q + Yr˙ r, a3 = Z u˙ u + Z v˙ v + Z w˙ w + Z p˙ p + Z q˙ q + Z r˙ r, b1 = K u˙ u + K v˙ v + K w˙ w + K p˙ p + K q˙ q + K r˙ r, b2 = Mu˙ u + Mv˙ v + Mw˙ w + M p˙ p + Mq˙ q + Mr˙ r, b3 = Nu˙ u + Nv˙ v + Nw˙ w + N p˙ p + Nq˙ q + Nr˙ r.

2.1.2.3

Hydrodynamic Forces and Moments

In general, hydrodynamic damping for ocean vessels results from potential damping, skin friction, wave drift damping, and damping due to vortex shedding. From this fact it arises that it is difficult to give a general expression of the hydrodynamic damping matrix D(ν). However, the hydrodynamic damping matrix is assumed often as (Berge and Fossen 2000; Do and Pan 2009; Roberts and Sutton 2008): D(ν) = D + Dn (ν),

(2.13)

where the linear damping matrix D is described by: ⎡

 D=

D11 D21

Xu ⎢ Yu ⎢  ⎢ Zu D12 = −⎢ ⎢ Ku D22 ⎢ ⎣ Mu Nu

Xv Yv Zv Kv Mv Nv

Xw Yw Zw Kw Mw Nw

Xp Yp Zp Kp Mp Np

Xq Yq Zq Kq Mq Nq

⎤ Xr Yr ⎥ ⎥ Zr ⎥ ⎥, Kr ⎥ ⎥ Mr ⎦ Nr

(2.14)

and the nonlinear damping matrix Dn (ν) is usually modeled by using a third-order Taylor series expansion or quadratic drag. Moreover, it is assumed that D = DT . In reference Fossen (1994) it can be found that the total hydrodynamic damping matrix consists of:

14

2 Models of Underwater Vehicles

D(ν) = D P (ν) + D S (ν) + DW (ν) + D M (ν),

(2.15)

where D P (ν) is potential damping arising from forced body oscillation, D S (ν) is linear skin friction (arising from laminar boundary layers) and quadratic skin friction (arising from turbulent boundary layers), DW (ν) is wave drift damping, D M (ν) is damping arising from vortex shedding (Morison’s equation), respectively. Because, in general (at high speed), the damping is highly nonlinear and coupled, a rough approximation for underwater vehicles (for completely submerged body) is sometimes made (Antonelli 2018; Fossen 1994): 

D(ν) = −diag X u , Yv , Z w , K p , Mq , Nr

 −diag X |u|u |u|, Y|v|v |r |, Z |w|w |w|, K | p| p | p|, M|q|q |q|, N|r |r |r | . (2.16)

2.1.2.4

Restoring Forces and Moments

At present, a model for the term g(η) is described. Let rbb = [xb , yb , z b ]T denote the vector of the center of buoyancy (CB) (the center of gravity (CG) vector was define earlier). Moreover, the symbol ∇ denotes the volume of the fluid displaced by the underwater vehicle, g the acceleration of gravity (positive downwards), and ρ the water density. The submerged weight of the body W and buoyancy force B are given by: W = mg,

B = ρg∇.

(2.17)

The gravity and buoyancy forces are described by: ⎡

⎤ 0 τ g = R T (η2 ) ⎣ 0 ⎦ , W

⎡

⎤ 0 τ b = −R T (η2 ) ⎣ 0 ⎦ , B

(2.18)

and the restoring forces and moments are given in the vector:  τg + τb . g(η) = − b r g × τ g + rbb × τ b 

(2.19)

The restoring force and moment vector g(η) taking into account gravity and buoyancy forces is represented by: ⎤ (W − B)sθ ⎥ ⎢ −(W − B)cθ sφ ⎥ ⎢ ⎥ ⎢ −(W − B)cθ cφ ⎥ g(η) = ⎢ ⎢ −(yg W − yb B)cθ cφ + (z g W − z b B)cθ sφ ⎥ . ⎥ ⎢ ⎣ (z g W − z b B)sθ + (x g W − xb B)cθ cφ ⎦ −(x g W − xb B)cθ sφ − (yg W − yb B)sθ ⎡

(2.20)

2.1 Six Degrees of Freedom Equations of Motion

2.1.2.5

15

Environmental Disturbances

For control purposes it is necessary to take into account the effects of the ocean currents. The forces and moments induced by environmental disturbances including wa wi ocean currents τ cu E , waves (wind generated) τ E and wind τ E (Do and Pan 2009), can be written as follows: wa wi (2.21) τ E = τ cu E + τE + τE . The detailed analysis of these effects is presented in the cited references.

2.1.2.6

Propulsion Forces and Moments

The vector τ representing the propulsion forces and moments depends on a specific configuration of actuators of the vehicle. Thus, for a particular vessel we must consider propellers, rudders, and water jets. For real underwater vehicle the relationship between the force/moment τ and the control input of the thrusters uv is highly nonlinear. However, as it is known (Antonelli 2018) a common simplification is to consider a linear relationship between τ and uv ∈ Rs : (2.22) τ = Bv u v , where Bv ∈ R 6×s is a known constant matrix known as the Thruster Control Matrix (TCM). If the matrix Bv will be square or low rectangular, i.e., s ≥ 6 we obtain full control of force/moments of the vehicle. In other case, the vehicle is underactuated. Fully Actuated and Underactuated Vehicles If all propulsion forces and torques are available then the vehicle is fully actuated. In the literature we can find many examples of models which are limited to fully actuation. Some underwater vehicles which belong to this group are described, e.g. in: Antonelli et al. (2003, 2001), Caccia and Veruggio (2000), Chin and Lin (2018), Encarnacao and Pascoal (2000), Eslami et al. (2019), Fischer et al. (2011, 2014), Fjellstad and Fossen (1994), Garcia-Valdovinos et al. (2014), Guerrero et al. (2019, 2020), He et al. (2013), Jia et al. (2020), Joe et al. (2014), Lin and Chin (2017), Lin et al. (2015), Liu et al. (2016), Martin and Whitcomb (2018), Nie et al. (2000), Paulsen et al. (1994), Qiao and Zhang (2019a, 2020, 2019b), Rangel et al. (2020), Sebastian and Sotelo (2007), Shen and Shi (2020), Sun and Cheach (2009), Wang and Lee (2003), Yan et al. (2020), Yoerger and Slotine (1985), Yuh (1990), Yuh et al. (1999), Zhang et al. (2018), Zhao and Yuh (2005). In many moving systems the problem of underactuation must be taken into account. It occurs when in one or more directions we are not able to give the input signal. Underactuated underwater vehicles were investigated in the literature many times, e.g. in Aguiar and Pascoal (2007), Alonge et al. (2001), Batista et al. (2014), Borhaug and Pettersen (2005), Cao et al. (2020), Do et al. (2004a, b), Lapierre et al. (2003), Lapierre and Soetanto (2007), Li et al. (2020), Peng et al. (2019), Refsnes

16

2 Models of Underwater Vehicles

et al. (2008), Repoulias and Papadopoulos (2007a, b), Yu et al. (2020), Zheng et al. (2019). The algorithms, considered in this book are based on the IQV and they can be applied for fully actuated vehicles.

2.1.3 Body-Fixed Representation 2.1.3.1

Model Without Disturbances

Control of marine vehicles should take into account effects of ocean or sea currents. For this reason time-varying disturbances or the current are present in a dynamic model. However, sometimes these effects are not given in the equations of motion. The obtained model is simplified and not close of reality but it facilities the use of the mathematical theory. In the book Fossen (1994) some models of underwater ocean vehicles do not contain external disturbances. In the model disturbances are absent for tracking and stabilization nonlinear control (applied for a spherical underactuated underwater vehicle moving in a horizontal plane) (Do et al. 2004a), trajectory tracking (Alonge et al. 2001; Fjellstad and Fossen 1994; Repoulias and Papadopoulos 2007a, b), sliding control (trajectory control) (Yoerger and Slotine 1985), cross-track control scheme (using tracking error) (Borhaug and Pettersen 2005), nonlinear control for path following (Lapierre et al. 2003; Lapierre and Soetanto 2007). Environmental disturbances are also omitted in the dynamic model if set-point control algorithms are used (Sun and Cheach 2009), or intelligent control is applied (Ven et al. 2005). The full dynamic model in the Body-Fixed Frame containing the rigid body dynamics and the added inertia, hydrodynamic damping, and the restoring forces and moments can be written as follows (Fjellstad and Fossen 1994; Fossen 1994; Roberts and Sutton 2008): Mν˙ + C(ν)ν + D(ν)ν + g(η) = τ , η˙ = J(η)ν,

(2.23) (2.24)

where:  M11 M12 , M21 M22 C(ν) = C R B (ν) + C A (ν)   −S(M11 v + M12 ω) 03×3 . = −S(M11 v + M12 ω) −S(M21 v + M22 ω) 

M = MR B + MA =

There are true the following properties (Do and Pan 2009; Fossen 1994):

(2.25)

(2.26)

2.1 Six Degrees of Freedom Equations of Motion

17

M = MT > 0,

(2.27)

C(ν) = −C (ν) ∀ν ∈ R , D(ν) > 0 ∀ν ∈ R 6 . T

6

(2.28) (2.29)

Moreover, the following relationship is valid (Fossen 1994): ˙ − 2C(ν)]s = 0 ∀s ∈ R 6 . s T [M

(2.30)

Application of Equations of Motion As seen in the literature various moving systems can be described using (2.23) and (2.24). The equations can be applied for ocean or marine vehicles as well as for airships during indoor flight. Some examples can be found: (1) for underwater vehicles in Alonge et al. (2001), Choi et al. (2014), Conte and Serrani (1999), Garcia-Valdovinos et al. (2014), Kim (2015), Lapierre and Soetanto (2007), Paulsen et al. (1994), Shen et al. (2018), (2) for surface vehicles or ships in Behal et al. (2002), Bertaska and von Ellenrieder (2019), Chwa (2011), Do et al. (2002, 2005), Fang et al. (2004), Jiang (2002), Lefeber et al. (2003), Zhang and Wu (2015) or hovercrafts in Aguiar et al. (2003), Fantoni et al. (2000), Hayashi et al. (1994), Morales et al. (2015), MunozMansilla et al. (2012), Sira-Ramirez and Ibanez (2000), (3) for indoor airships in Oh et al. (2006), Ohata et al. (2007), Yamada et al. (2007), Zufferey et al. (2006), Zhang and Ostrowski (1999). It should be mentioned that equations of this form are valid for vehicles moving at low velocity. This form of equations is applied under assumption that all kind of uncertainties and disturbances can be neglected. Such equations are allowed if the vehicle moves slowly, i.e. at low velocity and additionally its parameters are known. In spite of that in many cases this model is considered.

2.1.3.2

Model with Environmental Disturbances

For the moving vehicle forces and moments induced by environmental disturbances including ocean currents, waves (wind generated) and wind must be taken into consideration as it is explained in books concerning ocean or marine vessels, i.e. Antonelli (2018), Do and Pan (2009), Lantos and Marton (2011), Roberts and Sutton (2008). In majority works the term coming from environmental disturbances is added (Antonelli 2007; Antonelli et al. 2003, 2001; Campos et al. 2017; Do et al. 2004b; Fischer et al. 2014; Healey and Lienard 1993; Li et al. 2020; Li and Lee 2005; Peng et al. 2019; Qiao and Zhang 2020; Refsnes et al. 2008; Soylu et al. 2008; Tee and Ge 2006; Vu et al. 2021; Wang and Lee 2003; Xie et al. 2020; Zhao and Yuh 2005). Taking into account the environmental disturbances we have τ R B = τ H + τ + τ E . As a result, instead of the dynamic equation in the form (2.23) it can be written: Do and Pan (2009), Lantos and Marton (2011):

18

2 Models of Underwater Vehicles

Mν˙ + C(ν)ν + D(ν)ν + g(η) = τ + τ E .

(2.31)

Sometimes another dynamic model is used (Roberts and Sutton 2008) if the effect of ocean currents is incorporated in the six equations of motion, namely: ν r = [u − u bc , v − vcb , w − wcb , p, q, r ]T ,

(2.32)

where u bc , vcb , wcb , are the current velocities in the Body-Fixed Frame, then we obtain the following dynamic equation: M R B ν˙ + C R B (ν)ν + M A ν˙ r + C A (ν r )ν r + D(ν r )ν r + g(η) = τ .

(2.33)

The concept of relative motion is used under the assumption that flow is not rotational. If the current velocity vector is slowly varying (what is a common practice), ν˙ r ≈ 0 such that ν˙ r ≈ 0 then we get the reduced equation: Mν˙ + C R B (ν)ν + C A (ν r )ν r + D(ν r )ν r + g(η) = τ .

(2.34)

In order to apply the Eq. (2.34) measurement of the ocean current must be available. Equations of Motion In the previous dynamic equation, i.e. (2.23) the disturbances were not taken into account. However, the equation in which parameters uncertainties as well external disturbances are present are also considered, namely: M∗ ν˙ + C∗ (ν)ν + D∗ (ν)ν + g∗ (η) + d(t) = τ ,

(2.35)

˜ C∗ (ν) = C(ν) + C(ν), ˜ ˜ D∗ (ν) = D(ν) + D(ν), g∗ (η) = g(η) where M∗ = M + M, + g˜ (η). Symbols M, C(ν), D(ν), g(η) denote matrices and vectors with nominal ˜ C(ν), ˜ ˜ values, while symbols M, D(ν), g˜ (η) denote matrices and vectors containing unknown parameters and uncertainties. Moreover, d(t) means the vector of all other disturbances depending on the time. Thus, the equations of motion (dynamic together with kinematic) can be rewritten as follows:  = τ, Mν˙ + C(ν)ν + D(ν)ν + g(η) + F η˙ = J(η)ν,

(2.36) (2.37)

where the disturbance vector is given in the form: ˜ ν˙ + C(ν)ν ˜ ˜ =M + D(ν)ν + g˜ (η) + d(t). F

(2.38)

2.1 Six Degrees of Freedom Equations of Motion

2.1.3.3

19

Form of Inertia Matrix

In the simplest models the inertia matrix is diagonal. Recall the following references in which the inertia matrix M is assumed as a diagonal one (Aguiar and Pascoal 2007; Batista et al. 2014; Caccia and Veruggio 2000; Campos et al. 2017; Cao et al. 2020; Do et al. 2004a, b; Kim et al. 2016; Lamraoui and Qidan 2019; Lapierre et al. 2003; Lapierre and Soetanto 2007; Li et al. 2020; Repoulias and Papadopoulos 2007b; Sebastian and Sotelo 2007; Shen et al. 2018; Smallwood and Whitcomb 2004; Yan and Yu 2018; Yan et al. 2020, 2019; Yoerger and Slotine 1985; Zheng et al. 2019; Zhou et al. 2020). However, when the inertial couplings are important the models contain a symmetric inertia matrix. In the literature it can be found references in which the inertia matrix M is assumed as a symmetric one (Aguiar and Hespanha 2007; Antonelli 2007; Antonelli et al. 2003, 2001; Batista et al. 2014; Borhaug and Pettersen 2005; Fjellstad and Fossen 1994; Martin and Whitcomb 2018; Paulsen et al. 1994; Qiao and Zhang 2019a; Refsnes et al. 2008; Sebastian and Sotelo 2007; Sun and Cheach 2009; Vu et al. 2021). The control problems considered in this work concern the case with a symmetric inertia matrix.

2.1.4 Earth-Fixed Representation The dynamic equations of motion (2.23) can be formulated in the Earth-Fixed Frame by utilizing the following relationships: ˙ ν = J−1 (η)η, −1 −1 ˙ ˙ ν˙ = J (η)[¨η − J(η)J (η)η],

(2.39) (2.40)

assuming that θ = ±π/2 (the Jacobian matrix J(η) is then invertible). According to Fossen (1994) the Earth-Fixed vector equations can be written as follows: Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) = τη ,

(2.41)

where: Mη (η) = J−T (η)MJ−1 (η),

(2.42)

−1 ˙ Cη (ν, η) = J−T (η)[C(ν) − MJ−1 (η)J(η)]J (η), −T −1 Dη (ν) = J (η)D(ν)J (η),

(2.43) (2.44)

gη (η) = J−T (η)g(η), τ η = J−T (η)τ .

(2.45) (2.46)

20

2 Models of Underwater Vehicles

In the Earth-Fixed Representation the following properties are true (Do and Pan 2009; Fossen 1994): Mη (η) = MηT (η) > 0 ∀η ∈ R 6 , ˙ η (η) − 2Cη (ν, η)]s = 0 ∀s ∈ R 6 , ν ∈ R 6 , η ∈ R 6 , s T [M

(2.47)

Dη (ν, η) > 0 ∀ν ∈ R , η ∈ R .

(2.49)

6

6

(2.48)

Taking into account the external disturbance the equation is valid (Do and Pan 2009): Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) = J−T (η)(τ + τ E ).

(2.50)

If the model inaccuracies and external disturbances are considered, the general dynamic equation can be written in the following form (referring to (2.36)):

where:

2.1.4.1

η = τ η , Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) + F

(2.51)

 η = J−T (η)F. F

(2.52)

Models Without Environmental Disturbances

In older publications the term coming from environmental disturbances is omitted (Fjellstad and Fossen 1994; Paulsen et al. 1994; Zhang et al. 2015).

2.1.4.2

Models with Environmental Disturbances

In many references the term coming from environmental disturbances is taken into account, e.g. Fischer et al. (2011, 2014), Guerrero et al. (2020), Liu et al. (2016), Munoz-Vazquez et al. (2017), Qiao and Zhang (2019b), Soylu et al. (2008), Wang and Lee (2003), Yuh et al. (1999), Zhao and Yuh (2005).

2.1.4.3

Form of Inertia Matrix

A symmetric inertia matrix for an underwater vehicle model was assumed, e.g. in Antonelli (2007), Eslami et al. (2019), Fischer et al. (2011, 2014), Fjellstad and Fossen (1994), Guerrero et al. (2020), Liu et al. (2016), Paulsen et al. (1994), Qiao and Zhang (2019b), Rangel et al. (2020), Soylu et al. (2008), Zhang et al. (2015).

2.2 Equations of Motion in Terms of IQV

21

2.2 Equations of Motion in Terms of IQV In this section the equations in terms of the IQV are presented. The first kind of the IQV is called the Generalized Velocity Components (GVC) and the second one the Normalized Generalized Velocity Components (NGVC). Although there are different types of IQV as reported in Chap. 1, in this book the GVC originally introduced in Loduha and Ravani (1995) and their modification from Herman (2005), Herman (2006) are applied. The described bellow approach, which is based on the decomposition of the inertia matrices M and Mη (η), is possible only if it is assumed that they are symmetric, positive definite and their elements are known. As it is known from the literature (Fossen 1994) for a class of marine vehicle models such approximation is reasonable. Moreover, as it was mentioned earlier models based on this assumption are widely used. The equations expressed in GVC in the Body-Fixed Representation for the class of vehicles under consideration can be found in Herman (2019, 2020a), Herman and Adamski (2017b, c), while in the Earth-Fixed Representation in Herman (2021a), Herman and Adamski (2019, 2020). However, equations described by NGVC in the Body-Fixed Representation were shown in Herman (2021b), Herman and Adamski (2017a, d) and in the Earth-Fixed Representation in Herman (2020b), Herman and Adamski (2017e), respectively. The properties of the equations using IQV were discussed in Herman (2020b, 2021b), Herman and Adamski (2017c, d, e). But, this chapter systematizes the knowledge of these properties while pointing out some differences that occur depending on the type of the IQV and the representation considered. Therefore, the information provided here represents some complete knowledge regarding this issue.

2.2.1 Equations of Motion Using GVC Here the 6 DOF equations of motion based on the GVC, for both representations, are given.

2.2.1.1

GVC Body-Fixed Representation

Introducing now a transformation of rates in the form: ν = ϒξ ,

(2.53)

where ϒ is an upper triangular, invertible matrix with constant elements. It is possible to decompose the matrix M into three matrices, i.e.:

22

2 Models of Underwater Vehicles

M = ϒ −T Nϒ −1 .

(2.54)

The determined matrix N is diagonal and contains constant elements. Because the matrix ϒ contains the constant elements only, then the time derivative of ν is ν˙ = ϒ ξ˙ . The terms of the transformed equations of motion (derived in Appendix) can be written in the following form (without disturbances): Nξ˙ + Cξ (ξ )ξ + Dξ (ξ )ξ + gξ (η) = π , η˙ = J(η)ϒξ ,

(2.55) (2.56)

where the appropriate matrices and the vectors are given as follows: N = ϒ T Mϒ,

(2.57)

Cξ (ξ ) = ϒ C(ν)ϒ, Dξ (ξ ) = ϒ T D(ν)ϒ,

(2.58) (2.59)

gξ (η) = ϒ T g(η), π = ϒT τ .

(2.60) (2.61)

T

Equations (2.55) and (2.53) together with (2.56) describe the motion of a marine vehicle, where N is a new diagonal inertia matrix. However, in the presence of disturbances we obtain instead of (2.55) the following dynamic equation:

where:

ξ = π , Nξ˙ + Cξ (ξ )ξ + Dξ (ξ )ξ + gξ (η) + F

(2.62)

 ξ = ϒ T F. F

(2.63)

The kinematic relationship is given by Eq. (2.56). Properties of GVC The discussed properties arise from the form of equations of motion expressed in terms of the IQV. The benefits which are observable in the equations are also noticeable if the control algorithms are applied. 1. Decoupled equations. The Eqs. (2.55) or (2.62) in terms of the GVC are decoupled because the matrix N is diagonal. Consequently, each element of the vector ξ˙ represents a decomposed inertial quasi-acceleration whereas each element of N is the corresponding inertia. In this sense the forces can be considered separately, what means that the element Nii ξi (i=1,…,6) represents an independent inertial quasi-force. 2. Properties of the vector ξ . This vector depends on the dynamic parameters (i.e. masses, inertia, geometrical parameters). In fact it includes the dynamical couplings existing in the vehicle.

2.2 Equations of Motion in Terms of IQV

23

3. Properties of the vector π. The input vector π contains the vehicle parameters which represents the couplings. It changes according to this part of the vehicle dynamics which reflects the couplings. 4. Properties of the matrix N. Each components of the matrix reflects exactly that part of the inertia which is related to ξ˙i . If it is included in the control gain then the controller works faster and always depends on the inertial parameters of the vehicle. 5. Kinetic energy calculation. The component of the diagonal inertia matrix N gives information about the inertia related to each quasi-acceleration. Moreover, each quasi-velocity ξi is independent from other quasi-velocities and it is possible to determine the kinetic energy reduced by the variable ξi , namely: 1 T 1 1 ν Mν = ξ T Nξ = Nii ξi2 . 2 2 2 i=1 6

KE =

(2.64)

In the control algorithms using the GVC the energy is reduced taking into account each ξi and often fast which leads to rapid convergence of errors.

2.2.1.2

GVC Earth-Fixed Representation

In order to transform the equations of motion expressed in the Earth-Fixed Representation we introduce the velocity transformation of the vector η˙ and calculate its time derivative as follows: η˙ = Z(η)ξ , ˙ η¨ = Z(η)ξ + Z(η)ξ˙ ,

(2.65) (2.66)

where Z(η) is an upper triangular, invertible matrix with elements arising from decomposition of the matrix Mη (η) into three matrices, what means that: Mη (η) = Z−T (η)Nη (η)Z−1 (η).

(2.67)

The determined matrix Nη (η) is a diagonal one. The transformed equations of motion, using inversion of (2.65), can be given in the form: Nη (η)ξ˙ + Cξ (ξ , η)ξ + Dξ (ξ , η)ξ + gη ξ (η) = π η , ˙ ξ = Z−1 (η)η, where the matrices and vectors are given as follows:

(2.68) (2.69)

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2 Models of Underwater Vehicles

Nη (η) = ZT (η)Mη (η)Z(η), ˙ Cξ (ξ , η) = ZT (η)[Mη (η)Z(η) + Cη (ν, η)Z(η)],

(2.70)

Dξ (ξ , η) = Z (η)Dη (ν, η)Z(η), gη ξ (η) = ZT (η)g(η),

(2.71) (2.72) (2.73)

π η = ZT (η)τ η .

(2.74)

T

Equations (2.68) and (2.69) describe the motion of a 6 DOF vehicle, where Nη (η) is a diagonal matrix. When the vehicle model is inaccurate and there are environmental disturbances the dynamic equation is as follows: η ξ = π η , Nη (η)ξ˙ + Cξ (ξ , η)ξ + Dξ (ξ , η)ξ + gη ξ (η) + F

(2.75)

η , η ξ = ZT (η)F F

(2.76)

where:

and the kinematic equation is given by (2.69). Properties of GVC Some properties concerning the GVC equations of motion in the Earth-Fixed Frame can be given. 1. Decoupled equations. The Eqs. (2.68) or (2.75) are decoupled because Nη (η) is a diagonal matrix in the Earth-Fixed Frame. Moreover, the properties given for the Body-Fixed Frame are valid. 2. Properties of the vector ξ . This vector also includes besides the velocity also the dynamical couplings but they are expressed in the Earth-Fixed Frame. The changes are observable in this frame. 3. Properties of the vector π . The vector depends both on the dynamics of the vehicle and the dynamical couplings. 4. Properties of the matrix Nη (η) are observable in the Earth-Fixed Frame. In allows one to detect changes of this matrix during the motion. 5. Kinetic energy calculation. The kinetic energy expression is as follows: 1 T 1 1 η˙ Mη (η)η˙ = ξ T Nη (η)ξ = Nη ii (η)ξi2 = K Ei . 2 2 2 i=1 i=1 6

KE =

6

(2.77)

Using (2.77) it can be determined which part of dynamical couplings is related to each component of the vector ξ . Moreover, each element of the diagonal matrix Nη (η) represents a decoupled inertia corresponding to ξi (i = 1, 2, . . . , 6).

2.2 Equations of Motion in Terms of IQV

25

2.2.2 Equations of Motion Using NGVC In this subsection the 6 DOF equations of motion based on the NGVC, for both representations, are shown.

2.2.2.1

NGVC Body-Fixed Representation

The control strategies considered in this book are based on transformed equations of motion with the identity inertia matrix. In order to obtain such equation the inertia matrix M is decomposed into two matrices: M = T .

(2.78)

ζ = ν,

(2.79)

Next, some new rates are defined:

in which the new equations will be expressed. The relationship between the matrix and the matrix ϒ from (2.53) is as follows (Herman 2005, 2010): = N 2 ϒ −1 . 1

(2.80)

Consequently, the matrix is upper triangular, invertible and it has constant elements only. In such case the time derivative of ζ is ζ˙ = ˙ν . The transformed equations of motion are as follows (derived in Appendix): ζ˙ + Cζ (ζ )ζ + Dζ (ζ )ζ + gζ (η) =  , η˙ = J(η) −1 ζ ,

(2.81) (2.82)

where the matrices and the vectors are: Cζ (ζ ) = −T C(ν) −1 , −T

−1

(2.83)

Dζ (ζ ) = D(ν) , gζ (η) = −T g(η),

(2.84) (2.85)

 = −T τ .

(2.86)

Equations (2.81) and (2.79) together with (2.82) describe the motion of the vehicle in terms of the transformed rates. Assuming a model of disturbances the dynamic equation has the form: ζ =  , ζ˙ + Cζ (ζ )ζ + Dζ (ζ )ζ + gζ (η) + F

(2.87)

26

2 Models of Underwater Vehicles

where the matrices and the vectors are:  ζ = −T F, F

(2.88)

and the kinematic equation is defined by (2.82). Properties of NGVC For these equations of motion the following properties can be specified. 1. Decoupled equations. The Eqs. (2.81) or (2.87) are decoupled in the sense that each quasi-acceleration ζ˙i is independent on other variables. However, the inertial forces are hidden partially in the vector ζ˙ . 2. Properties of the vector ζ . This vector besides the velocities includes also the dynamics of the vehicle (not only the dynamical couplings). 3. Properties of the vector  . The input vector  is related to the vehicle dynamics and includes the couplings which are hidden because of the matrix . 4. Properties of the matrix . In the matrix a part of dynamics of the vehicle is present because the full dynamics is described by T ). From the relationship (2.80) it follows that the dynamical couplings (included in ϒ) are mixed with the 1 inertia present in N 2 . If the matrix is applied in the NGVC based algorithm, then the dynamic of the vehicle gives an additional gain always corresponding to the dynamics. 5. Kinetic energy calculation. The elements of the vector ζ contain the dynamic parameters (i.e. masses, moments of inertia) and the geometrical dimensions of the system. The kinetic energy of the vehicle K E expressed in terms of the vector ζ is determined as follows: 1 T T 1 1 2 ν ν = ζ T ζ = ζ = K Ei . 2 2 2 i=1 i i=1 6

KE =

6

(2.89)

This energy is reduced quickly if the control algorithm is expressed in terms of the vector ζ . Consequently, the errors convergence (both in terms of ζ and ν) is guaranteed. It may be reduced faster than for an algorithm without the matrix because the system dynamics is included into the control process.

2.2.2.2

NGVC Earth-Fixed Representation

In order to obtain the equations of motion the inertia matrix is decomposed into two matrices as follows: (2.90) Mη (η) = T (η) (η). Based on this decomposition the transformation of rates is defined and its time derivative is calculated, i.e.:

2.2 Equations of Motion in Terms of IQV

27

˙ ζ = (η)η, ˙ ˙ζ = (η) η˙ + (η)¨η,

(2.91) (2.92)

where (η) is an upper triangular, invertible matrix determined from (2.90). The following mathematical formula is also true: 1

(η) = Nη2 (η)Z−1 (η).

(2.93)

The new kinematic relationship can be determined from (2.91) as follows: η˙ = −1 (η)ζ .

(2.94)

Thus, the transformed equations of motion have the form: ζ˙ + Cζ (ζ , η)ζ + Dζ (ζ , η)ζ + gζ (η) =  η ,

(2.95)

where the matrices and the vectors are are described by: −1 ˙ (η), Cζ (ζ , η) = [ −T (η)Cη (ν, η) − (η)] −T −1 Dζ (ζ , η) = (η)Dη (ν, η) (η),

(2.96) (2.97)

gζ (η) = −T (η)gη (η),  η = −T (η)τ η .

(2.98) (2.99)

The pair of Eqs. (2.95) and (2.91) describe the motion of the vehicle in terms of the vector of rates ζ . If the vehicle with not exact model is considered and additionally external disturbances occur then the appropriate dynamic equation is given as follows: η ζ =  η , ζ˙ + Cζ (ζ , η)ζ + Dζ (ζ , η)ζ + gζ (η) + F

(2.100)

η , η ζ = −T (η)F F

(2.101)

where:

and the kinematic equations is described by (2.94). Properties of NGVC Several properties can be given if the vehicle is described using the NGV in the Earth-Fixed Frame. 1. Decoupled equations. The Eqs. (2.95) or (2.100) are also decoupled in the sense that each quasi-acceleration ζ˙i is independent on other variables in the Earth-Fixed Frame. 2. Properties of the vector ζ . This vector includes the vehicle dynamics and it is related to the Earth-Fixed Frame.

28

2 Models of Underwater Vehicles

3. Properties of the vector  . The quasi-forces vector  is related to the EarthFixed Frame and to the vehicle dynamics, and it includes the couplings which are hidden in the matrix (η). 4. Properties of the matrix (η). The matrix plays similar role as in the Body-Fixed Frame, however it is expressed in the Earth-Fixed Frame. 5. Kinetic energy calculation. Decomposition of the matrix Mη (η) leads to obtaining the new quasi-variables. Taking into consideration the kinetic energy is given by the formula: 1 T 1 1 η˙ Mη (η)η˙ = η˙ T T (η) (η)η˙ = ζ T ζ 2 2 2 6 6 1 2 = ζ = K Ei . 2 i=1 i i=1

KE =

(2.102)

From (2.102) it is possible to determine the part of the energy which is related to each quasi-velocity ζi .

2.3 Closing Remarks The models used for description of motion of underwater vehicles were presented it this chapter. It was shown that, in the literature, both models without disturbances and with disturbances are applied. However, if it is assumed that disturbances (internal and external) can be neglected, then accurate model knowledge and low vehicle speed must also be assumed. Moreover, it was point at the fact that in many works the symmetric inertia matrix is considered. The equations of motion expressed in terms of the IQV related to the classical equations were also reminded in this chapter. The IQV based equations of motion are derived using decomposition of the inertia matrix. They can be applied in the Body-Fixed Representation as well as in the Earth-Fixed Representation. Moreover, the form of the equations for underwater vehicles was considered here. The presented equations are the basis for building of control algorithms in terms of the IQV.

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Sira-Ramirez H, Ibanez CA (2000) On the control of the hovercraft system. Dyn Control 10:151–163 Smallwood DA, Whitcomb LL (2004) Model-based dynamic positioning of underwater robotic vehicles: theory and experiment. IEEE J Ocean Eng 29(1):169–186 Soylu S, Buckham BJ, Podhorodeski RP (2008) A chattering-free sliding-mode controller for underwater vehicles with fault-tolerant infinity-norm thrust allocation. Ocean Eng 35:1647–1659 Sun YC, Cheach CC (2009) Adaptive control schemes for autonomous underwater vehicle. Robotica 27:119–129 Tee KP, Ge SS (2006) Control of fully actuated ocean surface vessels using a class of feedforward approximators. IEEE Trans Control Syst Technol 14(4):750–756 Thomasson PG, Woolsey CA (2013) Vehicle motion in currents. IEEE J Oceanic Eng 38(2):226–242 Ven, van de P.W.J., Flanagan, C., Toal, D. (2005) Neural network control of underwater vehicles. Eng Appl Artif Intell 18:533–547 Vu MT, Le TH, Thanh HLNN, Huynh TT, Van M, Hoang QD, Do TD (2021) Robust position control of an over-actuated underwater vehicle under model uncertainties and ocean current effects using dynamic sliding mode surface and optimal allocation control. Sensors 21(747):1–25 Wang JS, Lee CSG (2003) Self-adaptive recurrent neuro-fuzzy control of an autonomous underwater vehicle. IEEE Trans Robot Autom 19(2):283–295 Xie T, Li Y, Jiang Y, An L, Wu H (2020) Backstepping active disturbance rejection control for trajectory tracking of underactuated autonomous underwater vehicles with position error constraint. Int J Adv Robot Syst May-April 1–12. https://doi.org/10.1177/1729881420909633 Yamada M, Taki Y, Katayama A, Funahashi Y (2007) Robust global stabilization and disturbance rejection of an underactuated nonholonomic airship. In: Proceedings of 16th IEEE international conference on control applications, part of IEEE multi-conference on systems and control, Singapore, 1–3 October, pp 886–891 Yan Y, Yu S (2018) Sliding mode tracking control of autonomous underwater vehicles with the effect of quantization. Ocean Eng 151:322–328 Yan Z, Gong P, Zhang W, Wu W (2020) Model predictive control of autonomous underwater vehicles for trajectory tracking with external disturbances. Ocean Eng 217:107884 Yan Z, Wang M, Xu J (2019) Global adaptive neural network control of underactuated autonomous underwater vehicles with parametric modeling uncertainty. Asian J Control 21(4):1–13 Yoerger DR, Slotine JJE (1985) Robust trajectory control of underwater vehicles. IEEE J Oceanic Eng 10(4):462–470 Yu H, Guo C, Shen Z, Yan Z (2020) Output feedback spatial trajectory tracking control of underactuated unmanned undersea vehicles. IEEE Access 8:42924–42936 Yuh J (1990) A neural net controller for underwater robotic vehicles. IEEE J Oceanic Eng 15(3):161– 166 Yuh J, Nie J, Lee CSG (1999) Experimental study on adaptive control of underwater robots. In: Proceedings of the 1999 IEEE international conference on robotics & automation, Detroit, Michigan, May, pp 393–398 Zhang H, Ostrowski JP (1999) Visual servoing with dynamics: control of an unmanned blimp. In: Proceedings of the 1999 IEEE international conference on robotics & automation, Detroit, Michigan, May, pp 618–623 Zhang M, Liu X, Yin B, Liu W (2015) Adaptive terminal sliding mode based thruster fault tolerant control for underwater vehicle in time-varying ocean currents. J Franklin Inst 352:4935–4961 Zhang W, Teng Y, Wei S, Xiong H, Ren H (2018) The robust H-infinity control of UUV with Riccati equation solution interpolation. Ocean Eng 156:252–262 Zhang Z, Wu Y (2015) Further results on global stabilisation and tracking control for underactuated surface vessels with non-diagonal inertia and damping matrices. Int J Control 88(9):1679–1692 Zhao S, Yuh J (2005) Experimental study on advanced underwater robot control. IEEE Trans Rob 21(4):695–703 Zheng Z, Ruan L, Zhu M (2019) Output-constrained tracking control of an underactuated autonomous underwater vehicle with uncertainties. Ocean Eng 175:241–250

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Zhou H, Wei Z, Zeng Z, Yu C, Tao B, Lian L (2020) Adaptive robust sliding mode control of autonomous underwater glider with input constraints for persistent virtual mooring. Appl Ocean Res 95:102027 Zufferey JC, Guanella A, Beyeler A, Floreano D (2006) Flying over the reality gap: from simulated to real indoor airships. Auton Robot 21:243–254

Chapter 3

Models of Hovercrafts and Vehicles in Horizontal Motion

Abstract The full surface vehicle model which is closer to real system should have 6 DOF. However, common practice in modeling is to reduce this model to 3 DOF (surge, sway, and yaw) model taking into account the horizontal motion of the vehicle. The model is highly nonlinear and as a result the used equations of motion are complicated. In order to simplify the control methods many researchers apply linearization of the dynamic model. The obtained controller is easier to design but usefulness of such approach is limited in comparison with a strategy based on more complicated nonlinear models. This chapter gives the equations of motion for a vehicle in horizontal motion and then shows the same equations expressed in terms of IQV.

3.1 Three Degrees of Freedom Model The modeling problems and derivation of equation of motion can be found, e.g. in Berge and Fossen (2000), Do and Pan (2009), Lantos and Marton (2011), Perez (2005). It can be mentioned that sometimes the ship model taking into consideration also roll motion is used (Perez and Blanke 2012). Sometimes models with 4 DOF are also used to describe hovercraft motion (Ding et al. 2021; Fu et al. 2021). However, here the only 3 DOF model of horizontally moving vehicles is considered. The main reason of this assumption arises from the fact that such model is most often used for description of surface vehicles (including hovercrafts). Consider a fully actuated hovercraft model (Fig. 3.1) which represents a horizontally moving vehicle. The horizontal motion of surface vehicles or underwater vehicles is described by 3 DOF under assumption that the roll, pitch, and heave motion is ignored. In other words it is assumed that the roll, pitch, and heave motions are small. It means that only the motion components in surge, sway, and yaw are taken into consideration. This implies that the Earth-Fixed Frame position vector and the Body-Fixed velocity are defined as follows: η = [x y ψ]T ,

v = [u v r ]T .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_3

(3.1) 35

36

3 Models of Hovercrafts and Vehicles in Horizontal Motion

Fig. 3.1 Planar model of a fully actuated hovercraft or planar vehicle and two coordinate frames

One of the important question is the model of the vehicle. In some references fully actuated systems are considered, e.g.: (1) hovercrafts in Hayashi et al. (1994), Roubieu et al. (2014), Ruffier et al. (2008), Serres and Ruffier (2015), Serres et al. (2008, 2006a, b); (2) surface vehicles or ships in Alfaro-Cid et al. (2006, 2005), Bertaska and von Ellenrieder (2019), Chen et al. (2020), Fahimi and Van Kleeck (2013), Godhavn et al. (1998), Qin et al. (2020), Skjetne et al. (2005), Tee and Ge (2006), Unar and Murray-Smith (1999), Wang et al. (2019a, 2016, 2018), Wondergem et al. (2011), Yang et al. (2020), Zhu and Du (2020). Surface vehicles are very often underactuated what make their control more difficult. Selected examples can be found, e.g. in: (1) for hovercrafts (Aguiar et al. 2003; Cabecinhas et al. 2018; Fantoni 1999; Fantoni et al. 2000; Fu et al. 2017, 2019; Jeong and Chwa 2018; Kim et al. 2013; MunozMansilla et al. 2012; Rigatos and Raffo 2014; Sira-Ramirez and Ibanez 2000a, b; Sira-Ramirez 2002; Wang et al. 2019b; Xie et al. 2019); (2) for surface vessels, ships or marine vehicle (Ashrafiuon et al. 2008, 2017; Ashrafiuon 2010; Behal 2001; Chwa 2011; Dai et al. 2019; Do 2010; Do et al. 2002, 2003, 2004, 2005; Do and Pan 2006; Harmouche et al. 2014; Jiang 2002; Lefeber et al. 2003; Li et al. 2009; Mahini et al. 2013; McNinch et al. 2010; McNinch and Ashrafiuon 2011; Panagou and Kyriakopoulos 2011, 2013, 2014; Pettersen and Fossen 2000; Pettersen and Nijmeijer 1998, 2001; Qiu et al. 2019; Reyhanoglu 1996, 1997; Serrano et al. 2014; Soltan et al. 2011; Svec et al. 2014; Xie and Ma 2015; Zhang et al. 2020a, b). In this book, fully actuated vehicles are considered only because their models are suitable for dynamics investigation.

3.1 Three Degrees of Freedom Model

37

3.1.1 Body-Fixed Representation 3.1.1.1

General Three Degrees of Freedom Model

In general the motion of a surface vehicle can be described as (Do and Pan 2009): Mν˙ + C(ν)ν + D(ν)ν = τ , η˙ = J(η)ν,

(3.2) (3.3)

where the matrices have the following form (slightly modified in comparison with Do and Pan 2009 because we assume that yg = 0): ⎡

⎤ m − X u˙ 0 −myg − X r˙ 0 m − Yv˙ mx g − Yr˙ ⎦ , M=⎣ −myg − Nu˙ mx g − Nv˙ Iz − Nr˙ C(ν) = C R B (ν) + C A (ν), ⎡ ⎤ 0 0 −m(x g r + v) 0 0 −m(yg r − u) ⎦ , C R B (ν) = ⎣ 0 m(x g r + v) m(yg r − u) ⎡ ⎤ 0 0 Yv˙ v + Yr˙ r 0 0 −X u˙ u − X r˙ r ⎦ , C A (ν) = ⎣ 0 −Yv˙ v − Yr˙ r X u˙ u + X r˙ r ⎡ ⎤ Xu 0 Xr D(ν) = D + Dn (ν), D = − ⎣ 0 Yv Yr ⎦ , N u N v Nr ⎤ ⎡ 0 X |r |r |r | + X |u|r |u| X |u|u |u| + X |r |u |r | ⎢ 0 Y|v|v |v| + Y|r |v |r | Y|v|r |v| + Y|r |r |r | ⎥ ⎥ Dn (ν) = − ⎢ ⎣ N|u|u |u| + N|r |u |r | N|v|v |v| + N|r |v |r | N|u|r |u| + N|v|r |v| ⎦ , +N|r |r |r | ⎡ ⎤ cψ −sψ 0 J(η) = ⎣ sψ cψ 0 ⎦ , (3.4) 0 0 1 and the propulsion force and moment vector is given τ = [τ X τY τ N ]T , and the environmental disturbance vector τ e = [τ X e τY e τ N e ]T if it is added to Eq. (3.2).

3.1.1.2

Simplified Three Degrees of Freedom Models

The simplification of the previously given model takes place if the off-diagonal terms of the matrices M and D, and all elements of the nonlinear damping matrix Dn (ν) are ignored. In such case we obtain (Do and Pan 2009):

38

3 Models of Hovercrafts and Vehicles in Horizontal Motion

Mν˙ + C(ν)ν + Dν = τ ,

(3.5)

where the matrices have the following form: ⎡

⎤ ⎡ ⎤ m 11 0 0 0 −m 22 v 0 0 m 11 u ⎦ , M = ⎣ 0 m 22 0 ⎦ , C(ν) = ⎣ 0 0 0 m 33 m 22 v −m 11 u 0 ⎡ ⎤ d11 0 0 D = ⎣ 0 d22 0 ⎦ > 0, 0 0 d33 m 11 = m − X u˙ , m 22 = m − Yv˙ , m 33 = Iz − Nr˙ ,

(3.6)

d11 = −X u , d22 = −Yv , d33 = −Nr . The matrix J(η) in the kinematic relationship η˙ = J(η)ν is defined by (3.4). Moreover, the vector τ = [τ X τY τ N ]T , and the environmental disturbance vector τ e = [τ X e τY e τ N e ]T are defined once the latter is added to Eq. (3.5).

3.1.1.3

Environmental Disturbances Issue

Taking into account environment of the vehicle it is important to decide if the disturbances are strong or if they can be omitted. In some works the term coming from environmental disturbances is absent, e.g.: (1) for hovercrafts in Aguiar et al. (2003), Cabecinhas et al. (2018), Fantoni (1999); Fantoni et al. (2000), Munoz-Mansilla et al. (2012), Sira-Ramirez (2002), Serres and Ruffier (2015), Sira-Ramirez and Ibanez (2000a, b), Tanaka et al. (2001); (2) for ships or surface vessels in Ashrafiuon et al. (2008), Ashrafiuon (2010), Behal et al. (2002), Behal (2001), Bertaska and von Ellenrieder (2019), Chwa (2011), Do et al. (2002, 2005), Fahimi and Van Kleeck (2013), Harmouche et al. (2014), Jiang (2002), Lefeber et al. (2003), Li et al. (2009), Mahini et al. (2013), McNinch et al. (2010), McNinch and Ashrafiuon (2011), Oh et al. (2010), Panagou and Kyriakopoulos (2011), Pettersen and Egeland (1996), Pettersen and Fossen (2000), Pettersen and Nijmeijer (1998, 2001), Reyhanoglu (1997), Serrano et al. (2014), Siramdasu and Fahimi (2013), Soltan et al. (2011), Svec et al. (2014), Unar and Murray-Smith (1999). It can be found other works containing equations of motion with environmental disturbances: (1) for hovercrafts in Fu et al. (2017, 2019), Jeong and Chwa (2018), Kim et al. (2013); (2) for ships or marine vehicles in Berge and Fossen (2000), Do (2010), Do et al. (2004), Do and Pan (2006), Godhavn et al. (1998), Loria et al. (2000), Panagou and Kyriakopoulos (2013, 2014), Qiu et al. (2020), Song et al. (2021), Serrano et al. (2014), Skjetne et al. (2005), Sorensen et al. (1996), Tee and Ge (2006),

3.1 Three Degrees of Freedom Model

39

Velagic et al. (2003), Wang and Er (2015), Wang et al. (2015), Yang et al. (2014), Zhang (2018). In this book equations of motion are considered both with and without disturbances model.

3.1.1.4

Form of Inertia Matrix

In the following references the inertia matrix M is assumed as a diagonal one: (1) for hovercrafts (Aguiar et al. 2003; Ashrafiuon et al. 2017; Fantoni 1999; Fantoni et al. 2000; Fu et al. 2017; Jeong and Chwa 2018; Kim et al. 2013; MunozMansilla et al. 2012; Rigatos and Raffo 2014; Serres and Ruffier 2015; SiraRamirez and Ibanez 2000a, b; Sira-Ramirez 2002); (2) for ships, surface vessels or other marine vehicles (Ashrafiuon et al. 2008; Behal et al. 2002; Behal 2001; Burns 1995; Chwa 2011; Do et al. 2002, 2005; Fahimi and Van Kleeck 2013; Harmouche et al. 2014; Huang et al. 2015; Jiang 2002; Lefeber et al. 2003; Li et al. 2009; Mahini et al. 2013; McNinch et al. 2010; McNinch and Ashrafiuon 2011; Menoyo Larrazabal and Santos Penas 2016; Oh et al. 2010; Panagou and Kyriakopoulos 2011, 2013; Pettersen and Egeland 1996; Pettersen and Fossen 2000; Pettersen and Nijmeijer 1998, 2001; Qiu et al. 2020; Reyhanoglu 1996, 1997; Siramdasu and Fahimi 2013; Soltan et al. 2011; Svec et al. 2014; Unar and Murray-Smith 1999; Xie and Ma 2015; Zhang et al. 2020b; Zhang 2018). This assumption allows to design control algorithms which have simpler form than controllers based on dynamical models with a symmetric inertia matrix. Despite the above it can be found references in which the inertia matrix M for surface vessels is assumed as a symmetric one (Dai et al. 2019; Do 2010; Qin et al. 2020; Serrano et al. 2014; Skjetne et al. 2004; Wang and Er 2015; Wang et al. 2015, 2016, 2018, 2019a; Yang et al. 2020; Zhang and Wu 2015; Zhu and Du 2020). A symmetrical inertia matrix occurs for hovercraft exceptionally, e.g. in Herman and Kowalczyk (2015). The reason for this may be that controllers are more difficult to design than using a diagonal matrix of inertia. However, sometimes such matrix can be found if planar motion of underwater vehicles is considered, e.g. in Herman and Kowalczyk (2016a, b). In this book models with a symmetric matrix M are taken into account only because the control algorithms are based on decomposition of this matrix.

40

3 Models of Hovercrafts and Vehicles in Horizontal Motion

3.1.2 Earth-Fixed Representation 3.1.2.1

General Three Degrees of Freedom Model

For a planar vehicle the dynamic equation in the Earth-Fixed Representation can be written as follows: Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ = τη ,

(3.7)

whereas the kinematic relationship is given by (3.3). This description is used less frequently than the description in the Body-Fixed representation.

3.1.2.2

Environmental Disturbances Issue

In many publications the term coming from environmental disturbances is omitted as, e.g. in Cremean et al. (2002), Fang et al. (2004), Jin et al. (2004), Seguchi and Ohtsuka (2002, 2003), Wondergem et al. (2011). However, in some works the term coming from environmental disturbances is taken into account as, e.g. in Wang et al. (2018).

3.1.2.3

Form of Inertia Matrix

A diagonal inertia matrix is applied for various hovercraft models, e.g. in Cremean et al. (2002), Jin et al. (2004), Seguchi and Ohtsuka (2002, 2003), Tanaka et al. (2001). Sometimes a symmetric inertia matrix in the model describing vehicle in planar motion is assumed (Chen et al. 2020; Fang et al. 2004; Herman 2017; Wondergem et al. 2011) but rather for other vehicles than hovercraft.

3.2 Equations of Motion in Terms of IQV In this section the equations in terms of the IQV are presented. The first kind of the IQV is called the Generalized Velocity Components (GVC) (Loduha and Ravani 1995) and the second one the Normalized Generalized Velocity Components (NGVC) (Herman 2005). For the class of vehicles considered here, the 3 DOF equations of motion in terms of the IQV are a reduced form of the equations expressed by 6 DOF and presented in Chap. 2. The described bellow approach, which is based on the decomposition of the inertia matrices M and Mη (η), is possible only if it is assumed that they are symmetric,

3.2 Equations of Motion in Terms of IQV

41

positive definite and their elements are known. It is assumed that such approximation is allowable at least for simulation tests.

3.2.1 Equations of Motion Using GVC and NGVC in Body-Fixed Representation In this subsection the 3 DOF equations of motion based on the GVC, for both representations, are given.

3.2.1.1

Equations in Terms of GVC

For a 3 DOF vehicle the equations of motion in terms of the GVC can be written as follows (in general with disturbances): ξ = π , Nξ˙ + Cξ (ξ )ξ + Dξ (ξ )ξ + F −1

ξ = ϒ ν, η˙ = J(η)ϒξ .

3.2.1.2

(3.8) (3.9) (3.10)

Application to Vehicle in Horizontal Motion and Hovercraft

Vehicle in Horizontal Motion. The main assumption which allows one to use the IQV based controller without simplifications arises from the fact that the inertia matrix M in not a diagonal matrix (in other case the appropriate controllers are also applicable but we loss properties and information about the vehicle dynamics). Physically a nondiagonal matrix means that there exist some dynamical couplings between linear and angular accelerations νi . For that reason if the model of the system has a diagonal matrix M then it is no need to use the proposed approach. Let the Earth-Fixed position {x, y} and heading ψ of the vessel relative to the Earth-Fixed Frame X E Y E Z E be expressed in vector form by η = [x, y, ψ]T , and let the velocities decomposed in a body-fixed reference frame be represented by the state vector ν = [u, v, r ]T . The vector elements are referred to the surge, sway and yaw of the planar vehicle. Note that it is assumed that, for the model of the vehicle, the center of the body frame is not the same as the center of gravity. This case is more realistic because it is very difficult to avoid dynamical couplings in a real vehicle. In order to show more clearly usefulness of the control algorithm it is assumed the vector ν s = [u, r, v]T instead of ν = [u, v, r ]T , and ηs = [x, ψ, y]T instead of η = [x, y, ψ]T . Referring to Eqs. (3.5), (2.55), and (2.61) in the reduced nonlinear equations of motion in the Body-Fixed Frame at low speed one gets:

42

3 Models of Hovercrafts and Vehicles in Horizontal Motion

Ms ν˙ s + C(ν s )ν s + D(ν s )ν s = τ s ,

(3.11)

⎡

⎤ ⎡ ⎤ m − X u˙ −myg − X r˙ m 11 m 12 0 0 Ms = ⎣ −myg − X r˙ Jz − Nr˙ mx g − Nv˙ ⎦ = ⎣ m 12 m 22 m 23 ⎦ , (3.12) 0 mx g − Nv˙ m − Yv˙ 0 m 23 m 33 ⎡ ⎤ 1 ϒ12 ϒ13 ϒ = ⎣ 0 1 ϒ23 ⎦ , (3.13) 0 0 1 where: ϒ12 = −

m 12 , m 11

ϒ13 = −

m 12 m 23 m 11 m 23 , ϒ23 = 2 . − m 11 m 22 m 12 − m 11 m 22

m 212

(3.14)

Moreover, τ s = [τ X , τ N , τY ]T . The matrices C(ν s ) and D(ν s ) have the form which results from the transformation of the dynamic equation. Thus, after transformation from (2.61), i.e. in terms of the GVC π = ϒ T τ s one obtains: π1 = τ X , π2 = ϒ12 τ X + τ N ,

(3.15) (3.16)

π3 = ϒ13 τ X + ϒ23 τ N + τY ,

(3.17)

which means that the vector π = [π1 , π2 , π3 ]T is full rank and the proposed control algorithm can be applied. However, as can be seen from the results above, only the element π1 is equal to τ X , whereas in π2 both τ X and τ N are included. The last variable π3 contains all forces and torques acting on the vehicle. On the other hand if any control algorithm is applied, then one has the vector τ ∗s = ϒ −T π ∗ where: τ X∗ = π1∗ , τ N∗ = ϒ¯ 12 π1∗ + π2∗ , τY∗ = ϒ¯ 13 π1∗ + ϒ¯ 23 π2∗ + π3∗ .

(3.18) (3.19) (3.20)

The symbols ϒ¯ 12 , ϒ¯ 13 , and ϒ¯ 23 are defined in Appendix. From the above relationships it can be concluded that the signal τ X∗ is dependent on the quantity π1∗ received from the controller. But, τ N∗ and τY∗ depend on previous signals coming from the controller output. For the considered vehicle, it is ϒ¯ 13 = 0 and consequently τY∗ = ϒ¯ 23 π2∗ + π3∗ . Therefore, the transfer of dynamical couplings in the moving system plays an important role. Note that if even is lack of one of the output signal π2∗ or π3∗ then the vehicle input signals τ N∗ and τY∗ will be nonzero. This is possible due to the dynamical couplings in the system (contained in ϒ¯ 12 and ϒ¯ 23 ). Such property can be useful in case dynamics investigation. However, when x g = 0 and yg = 0 the algorithm is reduced

3.2 Equations of Motion in Terms of IQV

43

to a special case for which ϒ = I (the identity matrix in the velocity transformation). Hovercraft. Consider a simplified model that is suitable for describing of a hovercraft motion. Taking into account the rearranged vectors we obtain the kinematic relationship η˙ s = J(ηs )ν s which can be written in the matrix-vector form as follows: ⎤ ⎡ ⎤⎡ ⎤ x˙ cos ψ 0 − sin ψ u ⎣ ψ˙ ⎦ = ⎣ 0 1 0 ⎦⎣r ⎦. sin ψ 0 cos ψ v y˙ ⎡

(3.21)

The corresponding reduced nonlinear equations of motion are, at low speed, described by (3.11), where: ⎡

⎤ m −myg 0 Ms = ⎣ −myg Jz mx g ⎦ , 0 mx g m ⎡ ⎤ 0 −m(x g r + v) 0 0 m(yg r − u) ⎦ , C(ν s ) = ⎣ m(x g r + v) 0 0 −m(yg r − u) ⎡ ⎤ Xu Xr 0 D(ν s ) = − ⎣ Nu Nr Nv ⎦ , 0 Yr Yv

(3.22)

and the vector τ s = [τ X , τ N , τY ]T . In (2.61) the vector  depends on the matrix 1 −T . Recall that (Herman 2005), the matrix = N 2 ϒ −1 where the matrices N and ϒ are determined using the method presented in Loduha and Ravani (1995) and next implemented in Herman (2005). As a result, the matrix −T in the form 1 −T = N− 2 ϒ T is obtained. The equation of motion is described by (3.11) whereas the kinematic equation by (3.21). Thus, according to the above mentioned method, the calculated matrices are as follows: ⎡ ⎤ ⎡ ⎤ N11 0 0 1 Y12 Y13 N = ⎣ 0 N22 0 ⎦ , ϒ = ⎣ 0 1 Y23 ⎦ , (3.23) 0 0 1 0 0 N33 where:  N11 = m,

N22 = Jz −

Y12 = yg ,

Y13 =

myg2 ,

mx g yg , myg2 − Jz

N33 = m 1 + Y23 =

The resulting matrix −T is given in the form:

mx g2 myg2 − Jz

mx g . myg2 − Jz

,

(3.24) (3.25)

44

3 Models of Hovercrafts and Vehicles in Horizontal Motion

⎤ ∗11 0 0 = ⎣ ∗21 ∗22 0 ⎦ , ∗31 ∗32 ∗33 ⎡

−T

(3.26)

and it has the following elements: −1

−1

−1

∗11 = N11 2 , ∗21 = N22 2 ϒ12 , ∗22 = N22 2 , −1

∗31 = N33 2 ϒ13 ,

−1

∗32 = N33 2 ϒ23 ,

−1

∗33 = N33 2 .

(3.27)

The components of the transformed control vector  = −T τ s are: 1 = ∗11 τ X , 2 = ∗21 τ X + ∗22 τ N ,

(3.28) (3.29)

3 = ∗31 τ X + ∗32 τ N + ∗33 τY ,

(3.30)

which shows that the vector  = [1 , 2 , 3 ]T is of full range. There exist also the relationships between 1 and the input signal τ X , among 2 and τ X , τ N , and among τ X , τ N , τY . Consequently, the outputs are related as follows: τ X depends on 1 , τ N depends on 1 and 2 , and τY depends on 1 , 2 , and 3 . This means that the signals obtained from a controller are distributed among the drives as it is mentioned above. Moreover, if the signals 2 or 3 are lost then the input signals τ X , τ N , τY still acting due to dynamical couplings in the vehicle.

3.2.2 Equations of Motion Using GVC and NGVC in Earth-Fixed Representation 3.2.2.1

Equations in Terms of GVC

The set of equations of motion in terms of the GVC in the Earth-Fixed Representation is as follows (in general with disturbances): η ξ = π η , Nη (η)ξ˙ + Cξ (ξ , η)ξ + Dξ (ξ , η)ξ + F −1 ˙ ξ = Z (η)η, η˙ = J(η)ν.

3.2.2.2

(3.31) (3.32) (3.33)

Application to Planar Vehicle

The matrices Ms and J(ηs ) where ηs = [x, ψ, y]T in Eqs. (3.2)–(3.3) (used instead of M and J(η)), have the following form:

3.2 Equations of Motion in Terms of IQV

45

⎤ μ11 μ12 0 Ms = ⎣ μ12 μ22 μ23 ⎦ , 0 μ23 μ33 ⎡

where μ11 = m − X u˙ , μ12 μ33 = m − Yv˙ . Consequently, are as follows: ⎡ m 11 Mη (η) = ⎣ m 12 m 13

⎡

⎤ cos ψ 0 − sin ψ 0 ⎦, J(ηs ) = ⎣ 0 1 sin ψ 0 cos ψ

(3.34)

= −myg − X r˙ , μ23 = mx g − Yr˙ , , μ22 = Jz − Nr˙ , the matrix Mη (η) and the appropriate matrix Z(η) ⎤ m 12 m 13 m 22 m 23 ⎦ , m 23 m 33

⎡

⎤ 1 Z 12 Z 13 Z(η) = ⎣ 0 1 Z 23 ⎦ , 0 0 1

(3.35)

where m 11 = μ11 cos2 ψ + μ33 sin2 ψ, m 12 = μ13 cos ψ − μ23 sin ψ, m 13 = (μ11 − μ33 ) cos ψ sin ψ, m 22 = μ22 , m 23 = μ12 sin ψ + μ23 cos ψ, m 33 = μ11 sin2 ψ + μ33 cos2 ψ, and Z 12 , Z 13 , Z 23 are calculated according to (A.24). Next, the diagonal matrix N(η) from Eq. (2.70) is determined, i.e.: ⎡

⎤ N11 0 0 N(η) = ⎣ 0 N22 0 ⎦ , 0 0 N33

(3.36)

with the elements: N11 = m 11 , N22 = m 22 − (m 212 /m 11 ),

 N33 = m 33 − (2m 12 m 13 m 23 − m 11 m 223 − m 22 m 213 )/(m 212 − m 11 m 22 ) . Based on (2.61) we define π = [π1 , π2 , π3 ]T and from (A.24) we determine Z¯ 12 , Z¯ 13 , Z¯ 23 in order to obtain: ⎡

τη = Z

−T

⎤ π1 ⎦. Z¯ 12 π1 + π2 (η)π = ⎣ ¯ ¯ Z 13 π1 + Z 23 π2 + π3

(3.37)

Taking into account that τ s = [τ X , τ N , τY ]T and making use of (2.46), i.e. calculating τ s = JT (ηs )τ η one gets the relationship: ⎡

⎤ ⎡ ⎤ cos ψ π1 + sin ψ ( Z¯ 13 π1 + Z¯ 23 π2 + π3 ) τX ⎣ τN ⎦ = ⎣ ⎦. Z¯ 12 π1 + π2 τY − sin ψ π1 + cos ψ ( Z¯ 13 π1 + Z¯ 23 π2 + π3 )

(3.38)

For a hovercraft the equations are simplified because of its simpler model in the Body-Fixed Frame.

46

3 Models of Hovercrafts and Vehicles in Horizontal Motion

3.2.2.3

Equations in Terms of NGVC

Similarly as previously also here the 3 DOF equations of motion based on the NGVC, for both representations, are presented. The set of equations of motion in terms of the NGVC in the Earth-Fixed Representation is given by (in general with disturbances): η ζ =  η , ζ˙ + Cζ (ζ , η)ζ + Dζ (ζ , η)ζ + F ˙ ζ = (η)η, η˙ = J(η)ν.

(3.39) (3.40) (3.41)

The transformation matrix is determined, using (2.93), as follows: ⎡

⎤ 11 12 13 (η) = Nη (η)Z−1 (η) = ⎣ 0 22 23 ⎦ , 0 0 33 1 2

1

1

1

1

(3.42)

1

where 11 = N12 , 12 = N12 Z¯ 12 , 13 = N12 Z¯ 13 , 22 = N22 , 23 = N22 Z¯ 23 , and 1

33 = N32 . Based on (2.99) and defining  η = [1 , 2 , 3 ]T it can be calculated: ⎡

⎤ 11 1 ⎦. 12 1 + 22 2 τ η = T (η) η = ⎣ 13 1 + 23 2 + 33 3

(3.43)

Taking into consideration that τ s = [τ X , τ N , τY ]T and making use of (2.46), i.e. calculating τ s = JT (ηs )τ η one gets the following relationship: ⎡

⎤ ⎡ ⎤ τX cos ψ 11 1 + sin ψ (13 1 + 23 2 + 33 3 ) ⎣ τN ⎦ = ⎣ ⎦. 12 1 + 22 2 τY − sin ψ 11 1 + cos ψ (13 1 + 23 2 + 33 3 )

(3.44)

This result means that: (1) the dynamical parameters including couplings existing in the vehicle are present in the control input signal; (2) the signals 1 , 2 , and 3 from the controller are included in τ X , τY , τ N whereas 3 is not present in τ N .

3.2 Equations of Motion in Terms of IQV

47

3.2.3 Observation About Application of IQV Equations of Motion for Underactuated Vehicles in Horizontal Motion 3.2.3.1

GVC Equations of Motion

Consider an underactuated hovercraft with 3 DOF. In this case one obtains, according to (3.5), the inertia matrix M and the corresponding matrix ϒ in the following form: ⎡

⎤ m 0 −myg m mx g ⎦ , M=⎣ 0 −myg mx g Jz

⎡

⎤ 1 0 yg ϒ = ⎣ 0 1 −x g ⎦ . 00 1

(3.45)

Consequently, from (2.61) and τ = [τ X , 0, τ N ]T it arises the relationship: π1 = τ X , π2 = 0,

π3 = yg τ X + τ N ,

(3.46)

which means that one input signal is not available. Putting the actuated quantities in the first components of the velocity vector, i.e. assuming ν s = [u, r, v]T the Eq. (3.5) can be rewritten as follows: Ms ν˙ s + C(ν s )ν s + Ds ν s = τ su ,

(3.47)

where: ⎡

⎤ m −myg 0 Ms = ⎣ −myg Jz mx g ⎦ , 0 mx g m ⎡ ⎤ 0 −m(x g r + v) 0 0 m(yg r − u) ⎦ , C(ν s ) = ⎣ m(x g r + v) 0 0 −m(yg r − u) ⎡ ⎤ Xu Xr 0 Ds = − ⎣ N u N r X v ⎦ , 0 Yr Yv

(3.48)

and the vector τ su = [τ X , τ N , 0]T . Moreover, the surge control force τ X = f 1 + f 2 and the yaw control moment τ N = 21 ( f 2 − f 1 )L are expressed in terms of propeller forces. The kinematic equation, taking into account ηs = [x, ψ, y]T , has the form: η˙ s = J(ηs )ν s , with the velocity transformation relationship:

(3.49)

48

3 Models of Hovercrafts and Vehicles in Horizontal Motion

⎡

⎤ cos ψ 0 − sin ψ 0 ⎦. J(ηs ) = ⎣ 0 1 sin ψ 0 cos ψ

(3.50)

The corresponding matrix ϒ s is: ⎡

⎤ 1 ϒ12 ϒ13 ϒ s = ⎣ 0 1 ϒ23 ⎦ , 0 0 1

(3.51)

where: ϒ12 = yg , ϒ13 =

mx g yg mx g , ϒ23 = . 2 myg − Jz myg2 − Jz

(3.52)

In terms of the GVC it is π = ϒ sT τ s which means that: π1 = τ X ,

(3.53)

π2 = ϒ12 τ X + τ N , π3 = ϒ13 τ X + ϒ23 τ N ,

(3.54) (3.55)

and the vector π = [π1 , π2 , π3 ]T has all elements. However, the vehicle is still underactuated because of lack of one signal in the Body-Fixed Frame. The signals obtained from the GVC controller are distributed to the vector τ ∗s = ∗ [τ X , τ N∗ ]T (τ X∗ and τ N∗ mean the new control signals) by: τ ∗s = A+ π ,

(3.56)

where the pseudoinverse matrix A+ is calculated for: ⎡

⎤ 1 0 A = ⎣ ϒ12 1 ⎦ , ϒ13 ϒ23

(3.57)

and the pseudoinverse matrix A+ is determined from the relationship A+ = (AT A)−1 AT . For the matrix (3.57) one gets: (AT A) =

2 2 + ϒ13 ϒ12 + ϒ13 ϒ23 1 + ϒ12 , 2 ϒ12 + ϒ13 ϒ23 1 + ϒ23

and after some calculations one obtains finally:

(3.58)

3.2 Equations of Motion in Terms of IQV

1 0 0 + + + , a22 a23 a21 −(ϒ12 + ϒ13 ϒ23 ) 1 + = , a22 = , 2 2 1 + ϒ13 1 + ϒ23 A+ =

+ a21

49



(3.59) + a23 =

ϒ23 . 2 1 + ϒ23

Consequently, the vector τ ∗s is as follows:

τ X∗ τ N∗

=

π1 , + + + a21 π1 + a22 π2 + a23 π3

(3.60)

in which all the quantities π1 , π2 , π3 are taken into account. Finally, the quantities f 1 and f 2 are determined as f 1 = 21 τ X∗ − L1 τ N∗ and f 1 = 21 τ X∗ + L1 τ N∗ . Comment. The analysis do not help to solve the control problem because of lack of formal proof. Nevertheless it can be treat as an observation arising from the use of the GVC. Moreover, using control algorithms based on the above results it is possible to investigate dynamics of the vehicle in the absence of an input signal.

3.2.3.2

NGVC Equations of Motion

Consider a 3 DOF planar vehicle described by equations of the form given in (3.47)– (3.48). As previously, the vector τ su = [τ X , τ N , 0]T is applied. The matrix N is described by (3.23)–(3.24), whereas the matrix ϒ s by (3.51)–(3.52). The signals from the controller expressed in terms of  are distributed to the vector τ ∗c = [τ X∗ , τ N∗ ]T where the vector  = [1 , 2 , 3 ]T results from the use of the control algorithm. The following procedure can be proposed. At the beginning it can be written: τ c = A+ n .

(3.61)

The pseudoinverse matrix A+ n is calculated for: ⎡

⎤ ∗11 0 An = ⎣ ∗21 ∗22 ⎦ , ∗31 ∗32 −1

−1

−1

(3.62)

−1

−1

where ∗11 = N11 2 , ∗21 = N22 2 ϒ12 , ∗22 = N22 2 , ∗31 = N33 2 ϒ13 , ∗32 = N33 2 ϒ23 . + T −1 The pseudoinverse matrix A+ n is determined from the relationship An = (An An ) T An . For the matrix (3.62) after some calculations one gets:

50

3 Models of Hovercrafts and Vehicles in Horizontal Motion

A+ n =



0 A+ 11 0 , + + A+ 21 A22 A23

(3.63) 1

A+ 11 A+ 22

1 2

A+ 21

= N11 ,

=

−1 −1 ϒ12 + N33 ϒ13 ϒ23 ) −N112 (N22 −1 −1 2 N22 + N33 ϒ13

−1

=

N22 2

−1 −1 2 N22 + N33 ϒ23

,

A+ 23

,

−1

=

N33 2 ϒ23

−1 −1 2 N22 + N33 ϒ23

.

Consequently, the vector τ c is as follows:

τ X∗ τ N∗

=

A+ 11 1 . + + A+ 21 1 + A22 2 + A23 3

(3.64)

It easy to observe that signals all quantities 1 , 2 , 3 are taken into account in the controller. Therefore, in terms of the NGVC we have full vector of quasi-forces. Moreover, because of τ X∗ = A+ 11 1 in the x direction the quasi-force signal is multi1

plied by the constant N112 . However, τ N∗ takes into account all quantities of the vector  what means that, thanks the mechanical couplings, vehicle motion in the y direction is ensured. Effectiveness of the controller depends on the vehicle parameters set + + because A+ 21 , A22 , and A23 play the role of gains. The motion task in the y direction is performed by the signal τ ∗ . If there are no couplings then the task realization is impossible. This analysis can be treated as observation only because no formal proof concerning the control problem solution was given.

3.3 Closing Remarks For vehicles moving in the horizontal plane the 3 DOF reduced model is applied. The vehicles can be divided into fully actuated and underactuated. Consequently, slightly different equations describe both systems. The same vehicle can be described in the Body-Frame Representation and in the Earth-Frame Representation. Moreover, reduction of the full 3 DOF model leads to simplified models which are also useful for motion investigation. Another simplification of the model relies on omitting environmental disturbances. It was shown equations of motion, expressed in terms of the IQV, for planar vehicle models. Some observations concerning underactuation for 3 DOF models have also been discussed. It appears that the vector of forces and torques used for an underactuated vehicle in the classical equations of motion, after transformation into the IQV space can be considered as the full rank vector in spite of that the model still represents an underactuated vehicle. The observations given in this chapter were not supported by any formal proof. Therefore, the presented results do not guarantee stable work of a controller if the vehicle is underactuated. The problem of application of models which describe underactuated vehicle in terms of the IQV is unsolved and it needs further research.

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Chapter 4

Models of Airships

Abstract In this chapter some models of airship called also Lighter-Than-Air Robot (LTAR) or Lighter-Than-Air (LTA) robotic system (unmanned robotic system) are considered. Modeling of LTAR is nontrivial problem because the object has a small mass in comparison with its geometric dimensions. Moreover, environment, i.e. air causes additional troubles arising from the earlier mentioned airship property. Various models of airships that are known from the literature as well as their applications are shown. For the selected model, the equations of motion are expressed in IQV.

4.1 Equations of Motion in Body-Fixed Representation The modeling approach for the nonlinear dynamics simulation of airships was presented, e.g. in Li and Nahon (2007). A literature review concerning airship dynamics modeling is contained in Li et al. (2011). In the paper the authors categorized the references according to the major topics in this area as follows: aerodynamics, flight dynamics, incorporation of structural flexibility, incorporation of atmospheric turbulence, and effects of ballonets. Moreover, the relevant analytical, numerical, and semi-empirical techniques as well as differences between LTA and HTA aircraft were discussed. Another review of conventional non-rigid, semi-rigid, and rigid airships and of unconventional airship-type air vehicles can be found in Liao and Pasternak (2009). Similar survey is presented also in Stockbridge et al. (2012). For modeling purposes the real flying vehicle must be taken into account. Classical airships equipped with two thrusters and fins were considered, e.g. in Acanfora (2011), Khoury and Gillett (1999), Khoury (2012), Kornienko (2006), LaGloria (2008). However, LTAR sometimes are different as Mk II presented in Liesk (2012), Peddiraju (2010), Peddiraju et al. (2009). There are many references about modeling of airships, e.g. Acanfora (2011), Bestaoui (2012), Gomes (1990), Hygounenc et al. (2004), Khoury and Gillett (1999), Khoury (2012), Kornienko (2006), LaGloria (2008), Li (2008), Liesk (2012), Moutinho (2007), Peddiraju (2010). The dynamics of the airship should be studied in detail before real experiment. Such approach is given in Gomes (1990) where many simulation tests for a model © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_4

57

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4 Models of Airships

Fig. 4.1 Motion variables in Body-Fixed and Earth-Fixed reference frames for an airship

of the large airship YEZ-2A were conducted. In other work, i.e. Kornienko (2006) the system identification approach for determining flight dynamical characteristics was given in detail. The model of AURORA airship as an evolution of the Airspeed Airships’ AS800 was described in Moutinho (2007). Usually, for modeling purposes, the airship is considered as rigid body. Sometimes it can be treated as flexible body, as for example in Li (2008). Consider an Lighter Than Air Robot (LTAR) or airship presented in Fig. 4.1. The airship models are usually expressed in the Body-Fixed Representation Khoury and Gillett (1999); Khoury (2012). For building the LTAR two approaches are considered: (1) equations based on the body velocity vector; (2) equations using the relative velocity vector in which the wind velocity is taken into account. In this chapter the two kind of equations are discussed. Moreover, the airships can be divided into two groups (depending on application): (1) indoor airships, e.g. Adamski et al. (2020), Bermudez et al. (2007), Frye et al. (2007), Furukawa and Shimada (2014), Gonzalez et al. (2016), Mueller (2013), Oh et al. (2006), Rottmann et al. (2007), Wang et al. (2020), Zhang and Ostrowski (1999), Zufferey et al. (2006); (2) outdoor airships, e.g. Azinheira et al. (2009), Hong et al. (2009), Liesk et al. (2012), Liesk et al. (2013), Liu et al. (2020), Liu et al. (2021), Moutinho et al. (2016), Nguyen et al. (2012), Yuan et al. (2020), Yuan et al. (2021), Zheng et al. (2018). The used LTAR models are based sometimes on equations of motion coming from marine underwater vehicle, e.g. Fossen (1994), Lewis et al. (1984). Other references

4.1 Equations of Motion in Body-Fixed Representation

59

takes into consideration equations which are appropriate for submarines as well as for airships, e.g. Thomasson (1995, 1996, 2000); Thomasson and Woolsey (2013). In spite of that there exist many works related directly to airships Bestaoui (2012), Cook (1987), Gomes (1990), Gomes and Ramos (1998), Khoury and Gillett (1999), Khoury (2012), Kornienko (2006), Li (2008), Peddiraju (2010), Peddiraju et al. (2009).

4.1.1 Equations in Terms of Body Velocity—Short Review In this subsection various dynamic equations for an airship are presented. The kinematic equation is assumed the same as for underwater vehicles. Many known airship models are based on the form of equations of motion introduced for the dynamics of underwater vehicles given by Lewis et al. (1984) (cited, e.g. in Thomasson 1995, 1996, 2000), namely: Mν˙ = τ d (ν) + τ A + τ f + τ ,

(4.1)

where M denotes the mass and inertia matrix with added-mass effects, ν = [u, v, w, p, q, r ]T is the vector of state variables (linear and angular velocity vector) (or ν = [v T , ω T ]T ), τ d (ν) the dynamics vector (vector of body axis forces and moments due to rotating axes), τ A is the vector of the fluid dynamics forces and moments due to relative velocity, τ f vector of body axis forces and moments due to the fluid inertial motion, τ vector of other external forces and moments. The added mass is also present in the vector τ d (ν). This type of model was also developed in Thomasson and Woolsey (2013).

4.1.1.1

Model Application for Airship

First Model The dynamic model (4.1) was successfully applied for an airship by Cook (1987) in slightly different form: Mν˙ = τ d (ν) + τ a (ν) + τ g + τ c + τ p + τ ad ,

(4.2)

where M = M R B + M A is the matrix composed of the mass and inertia matrix (for the rigid body) M R B and the matrix M A representing the added-mass effects, τ d (ν) is the dynamics vector containing also virtual mass effects, τ g includes gravitational and buoyancy induced forces and moments, τ c is the vector of aerodynamic forces and moments resulting from the control surfaces (rudder and elevator), τ p propulsion forces and moments generated by engine thrusts, τ ad is the vector of atmospheric disturbances. Here the vector τ a consists of aerodynamic forces and moments arising from the flow of air around the hull of the blimp.

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This model (without the term τ ad ) is well known from the airship literature Khoury and Gillett (1999), Khoury (2012) and it was used also in Frye et al. (2007), Jia et al. (2009), Nguyen et al. (2012), Park et al. (2003). The equations known from Lewis et al. (1984) were basis for airship model described by Gomes (1990) in the following form: Mν˙ = τ d (ν) + τ a (ν) + τ g + τ p ,

(4.3)

in which the control surface effects are contained in the aerodynamics vector τ a (ν). In other words the vector τ a (ν) includes the aerodynamic terms of the model, arising from the aerodynamics of the hull of the airship as well as the control surfaces. Moreover, in the vector τ d (ν) virtual masses are partially included. This airship model can be also found in Gomes and Ramos (1998). It was considered in many later works, e.g. in Azinheira et al. (2001), Azinheira et al. (2000), Elfes et al. (2003), Li et al. (2011), Liu et al. (2021), Paiva et al. (1999), Paiva et al. (2006), Trevino et al. (2007). Similar model as the given by Cook and Gomes are applied in other works Kim et al. (2003), Kim and Ostrowski (2003). Second Model This model was proposed by Thomasson (1995, 1996) for a vehicle moving in unsteady fluid and developed next in Thomasson (2000). Recall, a general form of the equation which can be deduced from Thomasson (2000): ¯ R ) + τ f (˙ν f , M A , M ¯ R) Mν˙ = τ d (ν, M R , M ¯ R) + τ , +τ p f (ν, ν r , c , M A , M

(4.4)

¯ R) where M means the inertia matrix with the added mass effects, τ d (ν, M R , M is the dynamics vector containing terms arising from body fixed rotating axes, ¯ R ) is a function which contains fluid motion terms depending only τ f (˙ν f , M A , M on the fluid inertial acceleration but which does not depend on the fluid inertial ¯ R ) is vector of the perfect fluid forces and moments velocity, τ p f (ν, ν r , c , M A , M (a function of the relative velocity and the velocity gradients), τ is vector of external forces and moments. Moreover, the used symbols means: ν r - relative velocity vector of the vehicle and fluid (ν r = ν − ν f − ν c ), ν f - vehicle or region velocity vector (including body axis components of the inertial velocity of the multiply connected region of fluid), ν˙ f - inertial acceleration of the fluid, ν c - circulating velocity vector, ¯ R - the displaced mass matrix, c - velocity gradient matrix related to body axis M components of steady circulating velocity relative to multiply connected region. Third Model The airship equations of motion can be derived for a body moving in vacuum as in Fossen (1994) (cf. Li and Nahon 2007): Mν˙ = τ d (ν) + τ g + τ pc ,

(4.5)

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61

where M = M R B + M A , the vector τ d (ν) = τ d R B (ν) + τ d A (ν) with the terms τ d R B (ν) and τ d A (ν) describing the rigid inertia and the added mass inertia, and τ pc is the control and propulsion (including the thruster, the deflection of control surfaces) vector, respectively. The aerodynamic forces and moments effects are included in M A ν˙ and τ d A (ν) . However, in the modeling procedure concerning real vehicle (even in simulation) many aspects must be taken into account Li (2008), Li and Nahon (2007). In spite of that the model is appropriate for underwater vehicle Fossen (1994) it can be applied for modeling of indoor airships as it shown in Oh et al. (2006), Ohata et al. (2007), Zufferey (2005), Zufferey et al. (2006), Zwaan et al. (2000). The indoor airship dynamic model due to simplification arising from the environment can be described as follows Zufferey (2005), Zufferey et al. (2006): Mν˙ = τ d (ν) + τ D + τ g + τ p ,

(4.6)

where τ D includes damping forces due to air friction and τ p means propelling forces (the tested indoor airship does not have control surfaces), respectively. Hence, the symbol τ pc is replaced by τ p . This model is very close to the model used for underwater vehicles Fossen (1994). Fourth Model The equations of motion for outdoor LTAR can be expressed according to Bestaoui (2012) as follows: Mν˙ = τ d R (ν) + τ D + τ a + τ g + τ p + τ A dist ,

(4.7)

with τ d R B (ν) = C R B (ν)ν the term related to Coriolis and centrifugal forces and moments, τ D = D(ν)ν the damping term containing the viscous and lumped effects, τ a aerodynamic forces and moments, and τ A dist the atmospheric disturbances vector. Fifth Model In the next airship model wind is considered as the atmospheric disturbance as it was shown in Azinheira et al. (2002) (corrected in Azinheira et al. 2008): Ma ν˙ = τ k + τ a + τ g + τ p + τ wind .

(4.8)

In this equation instead of the matrix M is replaced by the matrix Ma called the generalized apparent mass matrix (which is a symmetric matrix containing virtual mass and inertia) is applied. The vector τ k understood here as kinematics force and moment (including virtual mass effects) in other references is known as the dynamics vector τ (ν) Cook (1987), Thomasson (2000). Moreover, τ a means the aerodynamic forces and moments, τ wind the forces and torques vector induced by wind. The symbols τ g , τ p denote, as previously gravity and buoyancy, and propulsion vectors, respectively.

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Sixth Model Another airship model was developed at Stuttgart University. It can be given as follows Kaempf (2004) (compare also the later work Kornienko 2006): Ma ν˙ = τ d R B (ν) + τ a (ν A , ω, ν˙ w ) + τ g + τ c + τ p ,

(4.9)

where Ma is the apparent mass (a symmetric matrix containing the added mass and inertia), τ d R B (ν) is the dynamics vector which contains only rigid body components, ν A = ν − ν w (ν A means the airstream velocity and ν w is wind velocity and ν˙ w is it time derivative). The aerodynamics vector τ a (ν A , ω, ν˙ w ) is analyzed in detail in Kaempf (2004) as well as in Kornienko (2006) in which the vector τ c was contained in the thrust (propulsion) term τ p (for airship “Lotte”). The model Kornienko (2006) was also used in Jelenciak et al. (2013). Seventh Model Different airship model was developed in LAAS/CNRS project. For this model estimation of the aerodynamic parameters plays crucial role. Recall this model in the form presented in Hygounenc et al. (2004): M R B ν˙ + C R B (ν)(ν) + τ a (ν A ) + g(η) = τ p ,

(4.10)

where M R B is the symmetric mass and inertia matrix with respect to the nose of the hull, C R B (ν)(ν) is the torque of centrifugal and Coriolis term for rigid airship body and the torque of aerodynamic forces and moment is defined as: τ a (ν) = M A ν˙ A + D1 (ω)ν A + τ sta (va2 ),

(4.11)

where ν A = [vaT , ω T ]T , with va = v − vw , whereas M A means a symmetric added mass and inertia matrix at the nose of the hull, D1 (ω)ν A represents the torque of added centrifugal, Coriolis and damping terms of the fluid, D1 (ω) is a matrix which is only dependent on the rotational velocity rotation ω, τ sta (va2 ) is the torque of stationary forces and moments at the nose of the hull (proportional to the square of the aerodynamic velocity). The component includes the forces and moments produced by the control surface. This type of model were applied in Solaque et al. (2007) but related to the center of gravity. In the nominal case, when the external wind is weak or null, the following approximation holds: va = v, that is ν A = ν. Thus, the model of dynamics is as follows Hygounenc et al. (2004): Mν˙ + C R B (ν)(ν) + τ a (ν) + g(η) = τ p ,

(4.12)

where M is the matrix of inertia of the rigid airship body and the added mass of air at the nose of the hull.

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63

Comment. Similar model was derived using the Newton-Euler approach in Bestaoui (2012). However, the obtained dynamic equation was slightly different, namely: Mν˙ = C R B (ν)(ν) + D1 (ω)ν A + +τ sta (va2 ) + g(η) +M R B ν˙ w + Md ν + τ p ,

(4.13)

because the aerodynamic torques were described as τ a (ν) = M R B ν˙ A + D1 (ω)ν A + τ sta (va2 ). Moreover, g(η) in the motion equation means the static tensor due to the gravity and lift forces, ν˙ w the vector of acceleration of wind, Md the matrix containing derivatives of mass and inertia. Eighth Model The next model was derived for an unmanned fin-less airship in Peddiraju et al. (2009) and also in Liesk (2012), Peddiraju (2010). It was applied in Liesk et al. (2012); Mazhar (2012) (in this model effect of wind is given in the relative velocity).This model is described as follows: Ma ν˙ = τ d (ν) + τ aw (ν, vw , v˙ w ) + τ v + τ g + τ p ,

(4.14)

where Ma means the apparent mass matrix inertia, i.e. a symmetric matrix containing rigid-body inertia and added mass terms, τ aw is the inertial force and moment term which describes coupling effects between wind and the added mass, and the Munk moment. The vector τ v contains the force and moment terms arising from viscous drag, τ g is the gravity and buoyancy term, τ p is the propulsion (thrusters ) term. Ninth Model A different airship model was considered in Moutinho et al. (2016), Moutinho et al. (2007) (for fully actuated airship). It corresponds to reference Thomasson (2000), namely: Mν˙ = −6 Mν + Eg Sa g + τ 1 + τ 2 ,

(4.15)

where ν = [vaT , ω T ]T (va = v − vw where va , vw mean the air velocity and the wind velocity vectors in the local frame, respectively), 6 = diag{3 , 3 } ∈ R 6×6 (3 = ω× ∈ R 3×3 and C3 = c× ∈ R 3×3 is the cross product of the coordinates of the center of gravity expressed inthe local  frame.). The gravity vector in the inertial frame can m w I3 . In the sum of terms τ 1 + τ 2 = τ p + τ a the vector be expressed as Eg = mC3 τ 1 is understood as the state only depending part vector whereas τ 2 is the actuation or control force input vector which are equal the sum of the aerodynamic part vector and the propulsion part vector (a g is the gravity vector in the inertial frame, m is a scalar mass of the airship, m w is the airship’s weighting mass). If it is assumed that the wind is constant in the earth frame (with the linear velocity vector w and without

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angular velocity vector), i.e. v = va + vw = va + Sw, where S is a transformation matrix. This type of dynamic equation was also applied in Azinheira et al. (2009) but with the kinematic equations expressed in terms of quaternions. The airship position vector is defined as η = [pT , χ T ]T where p is Cartesian coordinates vector, χ is the angular Euler attitude. Next, denoting R as a coefficients matrix relating to Euler angles, the position derivative related to the airship air velocity can be expressed as follows: η˙ =

    T  va + Sw p˙ S 0 . = 0 R ω χ˙

(4.16)

Tenth Model The last considered, in this review, airship model was derived in Bestaoui (2012) an it was based on the Lagrange Approach. The dynamic equations of motion for LTAR can be written in the following form: Mν˙ + C R B (ν)(ν) + D(ν)(ν) + g(η) = τ dist + τ a + τ p ,

(4.17)

where M means the inertia matrix together with the aerodynamic virtual inertia (added mass), C R B (ν)(ν) includes the nonlinear forces and moments due to centrifugal and Coriolis forces, and D(ν)(ν) is the vehicle damping matrix containing the potential damping and the viscous effects, g(η) is a vector of the restoring terms (namely buoyancy and gravitational terms), τ dist denotes the atmospheric disturbance vector, τ a and τ p are tensors containing the aerodynamic and propulsion forces and moments, respectively. Comment. This review does not give all models of airships but only selected ones. Sometimes other models are used (or in slightly different form), e.g. Atmeh and Subbarao (2016), Beji and Abichou (2005), Sun and Zheng (2015), Yuan et al. (2021), Zheng et al. (2018), Zheng and Sun (2018).

4.1.1.2

Fully Actuated and Underacuated LTAR

The design of an airship decide if the model can be described as fully actuated or as underactuated system. Usually the airships, because of their design, must be considered as underactuated vehicles. Examples of underactuated airships are given (if only propulsion forces and moments are applied), e.g. in Frye et al. (2007), Furukawa and Shimada (2014), Gomes and Ramos (1998), Kahale et al. (2013), Kim et al. (2003), Kim and Ostrowski (2003), Liesk et al. (2012), Liesk et al. (2013), Nguyen et al. (2012), Shan (2009), Yamada et al. (2007), Yamada et al. (2005). However, if besides the propulsion term also the control forces and moments are used then we are able to avoid underactuation, e.g. in Gomes and Ramos (1998), Kim

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65

et al. (2003), Kim and Ostrowski (2003). Also in other works, namely Azinheira and Moutinho (2008), Azinheira et al. (2009), Han et al. (2016a), Han et al. (2016b), Moutinho (2007), Moutinho et al. (2016), Moutinho et al. (2007), Solaque et al. (2007), Yuan et al. (2020), Zhang and Ostrowski (1999), Zheng and Sun (2018) fully actuated (taking into account propulsion and control forces and moments) airships are considered.

4.1.2 Models of LTAR in Body-Fixed Representation 4.1.2.1

Model Without Disturbances

Taking into account the presented above review of dynamic equations and some similarities to equations describing underwater vehicles (2.23) and (2.24), the assumed, in this book, 6 DOF equations for an airship (without external disturbances, e.g. for an indoor LTAR) can be written as follows: Mν˙ + C(ν)ν + D(ν)ν + g(η) = τ , η˙ = J(η)ν.

(4.18) (4.19)

The Eqs. (4.18) and (4.19) are limited to indoor airships flying at low speed only. They are presented here because they are used to the considered, in this book, airship model.

4.1.2.2

Model with Disturbances

In similar way the 6 DOF model with external disturbances is given as:  = τ, Mν˙ + C(ν)ν + D(ν)ν + g(η) + F η˙ = J(η)ν.

(4.20) (4.21)

The two equations can be also applied to an indoor airship under assumption that the disturbance functions are limited.

4.1.3 Application of Models for Indoor and Outdoor Airships In general the inertia matrix in equations of motion describing LTAR is non-symmetric. However, making simplifying assumptions it is possible to use models which are more convenient from the mathematical point of view. Models of an airship are applied depending on its use, namely for indoor airships and for outdoor airship.

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For various model based control algorithms it is convenient to use a symmetric or even a diagonal inertia matrix in the dynamic equation of motion. If we consider indoor airships then assumptions about symmetric or diagonal inertia matrix is reasonable because of limited environmental effects. In analytical models the symmetric inertia matrix can be assumed Han et al. (2016a), Han et al. (2016b), Mo et al. (2003), Yamasaki and Goto (2003), Zheng et al. (2018). Recall thus some works in which such assumption was made.

4.1.3.1

Outdoor Airships

The symmetric matrix is usually applied for airship models contained a generalized apparent mass matrix of airship together with the inertia and masses. This condition is fulfilled for the Fifth Model, Sixth Model, Seventh Model, and Eighth Model. In many references the inertia matrix M is assumed as a symmetric one, e.g.: (1) “Lotte” airship Kaempf (2004), Kornienko (2006); (2) airship AS800 (AURORA project) Azinheira and Moutinho (2008), Azinheira et al. (2002), Moutinho et al. (2016); (3) ALTAV Quanser MkII Liesk et al. (2012), Liesk et al. (2013), Mazhar (2012), Peddiraju (2010); (4) Karma airship LAAS-CNRS project Hygounenc et al. (2004); (5) Karma and UrAn airships test Solaque et al. (2007); (6) the airship model considered in the review works Solaque et al. (2008) and Stockbridge et al. (2012); (7) the airship model taking into consideration in Li et al. (2011); (8) DIVA airship (Portuguese project) Moutinho et al. (2007) (the model was based on Azinheira et al. (2002)). Motion of outdoor airships can be also investigated in a hangar or a hall and in such cases they are treated as indoor airships.

4.1.3.2

Indoor Airships

Models with a symmetric inertia matrix are also considered for control of indoor airships. Such models were tested for vehicles described, e.g. for: (1) (2) (3) (4) (5)

indoor airship described in Zufferey (2005), Zufferey et al. (2006); indoor airship considered in Zwaan et al. (2000); Korean-Japanese indoor blimp Oh et al. (2006); York University indoor airship Shan (2009); Tri-Turbofan Remote-Controlled Airship Frye et al. (2007).

Sometimes models with a diagonal inertia matrix are taken into account. In the following works the inertia matrix M is assumed as a diagonal one:

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67

(1) Unmanned Aerial Vehicle (UAV) for GRASP project Zhang and Ostrowski (1999); (2) radio-controlled indoor blimp tested in Yamada et al. (2007), Yamada et al. (2005); (3) test platform described in Adamski et al. (2020).

4.1.4 Equations of Motion for Model Based Control In order to use, in the further part of the book, the airship model various assumptions are necessary. The vehicle modeling is based on the following hypotheses (compare, e.g. Acanfora 2011; Khoury and Gillett 1999): (1) the airship mass and its volume are considered as constant (this strong assumption neglects the variation of mass induced by the inflation of air ballonets, inside the hull, which change with temperature or pressure); (2) the use of the mechanical theory of a rigid body is allowed - the hull is considered as a solid, aero-elastic phenomena are neglected, and no phenomenon of inertial added fluid due to motion of helium inside the hull is taken into account; (3) the aerodynamic effects due to gravity (which are modeled by the Froude number) are decoupled from the dynamics; (4) the airship is symmetric respect to the longitudinal plan, to which belong the airship centers of gravity and buoyancy (in our models this condition is not necessary); (5) the airship has control surfaces and two independent thrust vectored propellers; (6) there are no turbulence effects and the steady air model is assumed; (7) the center of buoyancy is assumed to be the same as the hull’s center of volume and the position of the center of buoyancy cannot be modify significantly; (8) the velocity for a small airship is generally low (low Mach number), thus the couplings between dynamics and thermal phenomena are neglected and the density of air is not locally modified by the motion of the system; (9) the Earth is assumed as flat over the flight area.

4.2 Equations of Motion in Earth-Fixed Representation 4.2.1 LTAR Models in Earth-Fixed Representation Despite of the fact that in practical applications, such models are of minor importance, they may be successfully used for dynamics test of the vehicle. For this reason they are discussed below.

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4.2.1.1

4 Models of Airships

Model Without Disturbances

It is assumed that the airship equations of motion without disturbances have the form: Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) = τη , ˙ ν = J−1 (η)η.

(4.22) (4.23)

This kind of model is appropriate for indoor LTAR assuming that internal as well as external disturbances can be neglected.

4.2.1.2

Model with Disturbances

The model taking into account disturbances is assumed in the following form: η = τ η , Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) + F −1 ˙ ν = J (η)η.

(4.24) (4.25)

An example of such model can be found in Yang and Yan (2016). In this book the equation of motion are applied for indoor LTAR.

4.2.1.3

Form of Inertia Matrix

Equations of motion expressed in the Earth-Fixed Representation are used very rarely, indeed exceptionally. Some models with symmetric inertia matrix were considered, e.g. in Yang (2018), Yang and Yan (2015), Yang and Yan (2016), Yang et al. (2014). A diagonal inertia matrix in an indoor airship model was assumed in Furukawa and Shimada (2014).

4.3 Equations of Motion in Terms of IQV In this section the equations in terms of the IQV are presented. For the class of vehicles considered here, the same equations of motion apply as in Chap. 2, i.e. the GVC and the NGVC introduced in Sect. 2.2. The considered approach, which is based on the decomposition of the inertia matrices M and Mη (η), is possible only if it is assumed that they are symmetric, positive definite and their elements are known.

4.3 Equations of Motion in Terms of IQV

69

4.3.1 Equations of Motion Using GVC In this subsection the 6 DOF equations of motion based on the GVC, for both representations, are given.

4.3.1.1

GVC Body-Fixed Representation

The model taking into account disturbances has the following form: ξ = π , Nξ˙ + Cξ (ξ )ξ + Dξ (ξ )ξ + gξ (η) + F η˙ = J(η)ϒξ , ν = ϒξ .

(4.26) (4.27) (4.28)

ξ in Eq. (4.26) does not appear. If the disturbances are omitted then the term F 4.3.1.2

GVC Earth-Fixed Representation

For the model with disturbances the following equations set can be written: η ξ = π η , Nη (η)ξ˙ + Cξ (ξ , η)ξ + Dξ (ξ , η)ξ + gξ (η) + F ˙ ξ = Z−1 (η)η, η˙ = J(η)ν.

(4.29) (4.30) (4.31)

η ξ in Eq. (4.29) does not appear. If the disturbances are omitted then the term F

4.3.2 Equations of Motion Using NGVC In this subsection the full 6 DOF models of indoor airship based on the NGVC, for both representations, are presented. 4.3.2.1

NGVC Body-Fixed Representation

In this representation one has the following equations of motion: ζ = , ζ˙ + Cζ (ζ )ζ + Dζ (ζ )ζ + gζ (η) + F ζ = ν, η˙ = J(η)−1 ζ . ζ is omitted. If disturbances are not taken into account the term F

(4.32) (4.33) (4.34)

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4.3.2.2

4 Models of Airships

NGVC Earth-Fixed Representation

In this representation the equations of motion are as follows: η ζ = η , ζ˙ + Cζ (ζ , η)ζ + Dζ (ζ , η)ζ + gζ (η) + F ˙ ζ = (η)η, η˙ = J(η)ν.

(4.35) (4.36) (4.37)

η ζ does not If any disturbances are not taken into consideration the component F exist.

4.4 Closing Remarks There are many equations of motion representing model of an airship. It results from the fact that the problem of modeling is nontrivial. First of all, the airship model strictly depends on its application. Models for indoor airships and outdoor airships are quite different because in both cases environment is not the same. For outdoor airships the environmental disturbances play crucial role in the model. Moreover, various phenomena are taken into account depending on use of the airship. Some of them are omitted in the model of an indoor airship. In practical applications usually the Body-Fixed Representation is assumed. In this chapter various models expressed in this representation, known form the literature, were shown. Next, the equations of motion which will be considered in this book were presented. The equations expressed in IQV, namely GVC and NGVC, were also given.

References Acanfora M (2011) New approach and results on the stability and control of airship. PhD Thesis, University of Naples “Federico II”, Naples, Italy Adamski W, Pazderski D, Herman P (2020) Robust 3D tracking control of an underactuated autonomous airship. IEEE Robot Autom Lett 5(3):4281–4288 Atmeh G, Subbarao K (2016) Guidance, navigation and control of unmanned airships under timevarying wind for extended surveillance. Aerospace 3(1),8:1–25 Azinheira JR, Moutinho A (2008) Hover control of an UAV with backstepping design including input saturations. IEEE Trans Control Syst Technol 16(3):517–526 Azinheira JR, Moutinho A, de Paiva EC (2008) Influence of wind speed on airship dynamics (Erratum). J Guid Control Dyn 31(2):443–444 Azinheira JR, Moutinho A, de Paiva EC (2009) A backstepping controller for path-tracking of an underactuated autonomous airship. Int J Robust Nonlinear Control 19:418–441 Azinheira JR, de Paiva EC, Bueno SS (2002) Influence of wind speed on airship dynamics. J Guid Control Dyn 25(6):1116–1124

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Kim J, Keller J, Kumar V (2003) Design and verification of controllers for airships. In: Proceedings of the 2003 IEEE/RSJ international conference on intelligent robots and systems, Las Vegas, Nevada, October, pp 54–60 Kim J, Ostrowski JP (2003) Motion planning of aerial robot using rapidly-exploring random trees with dynamic constraints. In: Proceedings of the 2003 IEEE international conference on robotics & automation, Taipei, Taiwan, September 14-19, pp 2200–2205 Khoury GA, Gillett JD (1999) Airship technology. Cambridge University Press, Cambridge, United Kingdom Khoury GA (ed) (2012) Airship technology. Cambridge University Press, Cambridge, United Kingdom Kornienko A (2006) System identification approach for determining flight dynamical characteristics of an airship from flight data. PhD Thesis, University of Stuttgart, Stuttgart, Germany La Gloria N (2008) Simultaneous localization and mapping applied to an airship with inertial navigation system and camera sensor fusion. PhD Thesis, Universita’ degli Studi di Padova, Italy Lewis DJG, Lipscombe JM, Thomasson PG (1984) The simulation of remotely operated underwater vehicles. In: Proceedings of ROV ’84 conference and exposition, The Marine Technology Society, San Diego, CA, pp 245–251 Li Y (2008) Dynamics modeling and simulation of flexible airships. PhD Thesis, McGill University, Montreal, Canada Li Y, Nahon M (2007) Modeling and simulation of airship dynamics. J Guid Control Dyn 30(6):1691–1700 Li Y, Nahon M, Sharf I (2011) Airship dynamics modeling: a literature review. Prog Aerosp Sci 47:217–239 Liao L, Pasternak I (2009) A review of airship structural research and development. Prog Aerosp Sci 45:83–96 Liesk T (2012) Control design and validation for an unmanned, finless airship. PhD Thesis, McGill University, Montreal, Canada Liesk T, Nahon M, Boulet B (2012) Design and experimental validation of a controller suite for an autonomous, finless airship. In: Proceedings of 2012 American control conference Fairmont Queen Elizabeth, Montreal, Canada, June 27-29, pp 2491–2496 Liesk T, Nahon M, Boulet B (2013) Design and experimental validation of a nonlinear low-level controller for an unmanned fin-less airship. IEEE Trans Control Syst Technol 21(1):149–161 Liu SQ, Sang YJ, Whidborne JF (2020) Adaptive sliding-mode-backstepping trajectory tracking control of underactuated airships. Aerosp Sci Technol 97:105610 Liu SQ, Whidborne JF, He L (2021) Backstepping sliding-mode control of stratospheric airships using disturbance-observer. Adv Space Res 67:1174–1187 Mazhar H (2015) Dynamics, control and flight testing of an unmanned, finless airship. Master Thesis, McGill University, Montreal, Canada Mo YH, Kawashima M, Goto N (2003) Implementation of robust stability augmentation systems for a blimp. Trans Jpn Soc Aeronaut Space Sci 46(153):155–162 Moutinho AB (2007) Modeling and nonlinear control for airship autonomous flight. PhD Thesis, Instituto Superior Tecnico, Technical University of Lisbon, Portugal Moutinho A, Azinheira J (2005) Stability and robustness analysis of the AURORA airship control system using dynamic inversion. In: Proceedings of the 2005 IEEE international conference on robotics and automation Barcelona, Spain, April, pp 2265–2270 Moutinho A, Azinheira JR, de Paiva EC, Bueno SS (2016) Airship robust path-tracking: a tutorial on airship modelling and gain-scheduling control design. Control Eng Pract 50:22–36 Moutinho A, Mirisola L, Azinheira J, Dias J (2007) Project DIVA: guidance and vision surveillance techniques for an autonomous airship. In: XP Guo (Ed), Robotics research trends. Nova Science Publishers, Inc., New York, pp 77–120 Mueller M (2013) Autonomous navigation for miniature indoor airships. PhD Thesis, Faculty of Engineering, University of Freiburg, Germany

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Nguyen TA, Lee S, Park JS (2012) Design and implementation of embedded hardware and software architecture in an unmanned airship. In: Proceedings of 2012 IEEE 14th international conference on high performance computing and communications, Liverpool, United Kingdom, June 25-27, pp 1730–1735 Oh S, Kang S, Lee K, Ahn S, Kim E (2006) Flying display: autonomous blimp with real-time visual tracking and image projection. In: Proceedings of the 2006 IEEE/RSJ international conference on intelligent robots and systems, Beijing, China, October 9 - 15, pp 131–136 Ohata Y, Ushijima S, Nenchev DN (2007) Development of an indoor blimp robot with internetbased teleoperation capability. In: Proceedings of the 13th IASTED international conference on robotics and applications, Wurzburg, Germany, August 29-31, pp 186–191 Paiva de EC, Bueno SS, Gomes SBV, Ramos JJG, Bergerman M (1999) A control system: development environment for AURORA’s semi-autonomous robotic airship. In: Proceedings of the 1999 IEEE international conference on robotics & automation, Detroit, Michigan, May, pp 2328–2335 de Paiva EC, Azinheira JR, Ramos JG Jr, Moutinho A, Bueno SS (2006) Project AURORA: infrastructure and flight control experiments for a robotic airship. J Field Robot 23(3/4):201–222 Park CS, Lee H, Tahk MJ, Bang H (2003) Airship control using neural network augmented model inversion. In: Proceedings of 2003 IEEE conference on control applications, CCA 2003, Istanbul, Turkey, June 23-25, pp 558–563 Peddiraju P (2010) Development and validation of a dynamics model for an unmanned finless airship. Master Thesis, McGill University, Montreal, Canada (2010) Peddiraju P, Liesk T, Nahon M (2009) Dynamics modeling for an unmanned, unstable, fin-less airship. In: Proceedings of 18th AIAA lighter-than-air systems technology conference, 4-7 May, Seattle, Washington, AIAA 2009-2862 pp 1–18 Rottmann A, Plagemann, C., Hilgers, P., Burgard, W.: Autonomous blimp control using modelfree reinforcement learning in a continuous state and action space. In: Proceedings of the 2007 IEEE/RSJ international conference on intelligent robots and systems, San Diego, CA, USA, Oct 29-Nov 2, pp 1895–1900 Shan J (2009) Dynamic modeling and vision-based control for indoor airship. In: Proceedings of the 2009 IEEE international conference on mechatronics and automation August 9 - 12, Changchun, China, pp 2934–2939 Solaque L, Lacroix S (2007) Airship control. In: Ollero A, Maza I (Eds) Multiple heterogeneous unmanned aerial vehicles, STAR 37, pp 147–188 Solaque L, Pinzon Z, Duque M (2008) Nonlinear control of the airship cruise flight phase with dynamical decoupling. In: Proceedings of electronics, robotics and automotive mechanics conference, CERMA’08, September 30-October 3, Morelos, Mexico, pp 472–477 Stockbridge C, Ceruti C, Marzocca P (2012) Airship research and development in the areas of design, structures, dynamics and energy systems. Int J Aeronaut Space Sci 13(2):170–187 Sun L, Zheng Z (2015) Nonlinear adaptive trajectory tracking control for a stratospheric airship with parametric uncertainty. Nonlinear Dyn 82:1419–1430 Thomasson PG (1995) Motion of a rigid body in an unsteady non-uniform heavy fluid. College of Aeronautics, report No.9501, Cranfield University, Cranfield Bedford, England Thomasson PG (1996) Motion of a rigid body in an unsteady non-uniform heavy fluid, an extension. College of Aeronautics, report No.9610, Cranfield University, Cranfield Bedford, England Thomasson PG (2000) Equations of motion of a vehicle in a moving fluid. J Aircr 37(4):630–639 Thomasson PG, Woolsey CA (2013) Vehicle motion in currents. IEEE J Ocean Eng 38(2):226–242 Trevino R, Frye M, Franz JA, Qian Ch (2007) Robust receding horizon control of a tri-turbofan airship. In: Proceedings of 2007 IEEE international conference on control and automation, Guangzhou, China, May 30-June 1, pp 671–676 Wang Y, Zheng G, Efimov D, Perruquetti W (2020) Disturbance compensation based controller for an indoor blimp robot. Robot Auton Syst 124:103402 Yamada M, Taki Y, Katayama A, Funahashi Y (2007) Robust global stabilization and disturbance rejection of an underactuated nonholonomic airship. In: Proceedings of 16th IEEE international

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Chapter 5

Various Control Strategies

Abstract This chapter presents some control algorithms used for underwater vehicles, surface vehicles, and airships. At the beginning, a short review of the used techniques is given. Next, control schemes that are based on the underwater vehicle model are discussed. In some cases, if for example the vehicle model must be known and simultaneously soft computing based control methods are applied, it may be difficult to classify such approach. However, in this work, only the control algorithms which use directly the mathematical model in the control loop (without intelligent control techniques) are treated as the model based control methods.

5.1 Model Free Approach Due to the variety of control methods and their combinations used, it seems that a strict division into model free and non-model based methods is almost impossible, or even impossible. The non-model based methods used for vehicle control can be divided into the model free control methods and the intelligent control methods. This is the division adopted in this book although, of course, it can sometimes be questioned. The model free method do not require exact knowledge of the system or, in other words, detailed mathematical description of the system. This remark also applies to control strategies for horizontally moving vehicles and airships unless additional methods are used. Although this book is about model based control methods, it is worth mentioning model free control methods at the beginning of this chapter. Some examples of intelligent control methods are mentioned later.

5.1.1 Some Model Free Control Strategies In the classical approach to control of mechanical systems, proportional integral (PI), proportional derivative (PD), proportional integral derivative (PID) controllers can be treated as methods without knowledge of the model. Nevertheless, as © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_5

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control theory and practice has developed, new approaches have emerged that require only minimal knowledge of the model and for this reason are treated as model free control methods. Under the name of model free control, various control methods requiring minimal knowledge of the system can be included. Therefore, a strict division is impossible and the considerations in this section are very limited. Only two approaches representing this class of methods are mentioned here: Active Disturbances Rejected Control (ADRC) and so-called intelligent PID controllers. Selected intelligent control methods are presented later in the chapter.

5.1.1.1

Active Disturbances Rejected Control (ADRC)

The ADRC type control method is currently being very effectively developed and the works that are based on this method include various theoretical as well as experimental achievements, although most of them are the results of simulation studies. The history of this approach dates back to the previous century as can be learned from the literature. However, since this control method has become very popular, it is impossible to list even all the most important works on this topic. It seems worth mentioning the important works in which the ideas of this control method were explained. The Han paper Han (2009) certainly belongs to this group, but also worth mentioning are other publications, e.g., co-authored by Gao Gao (2006), Gao et al. (2001). A methodology and theoretical analysis of ADRC was presented in Huang and Xue (2014). The practical applications of the ADRC method were highlighted many years ago, namely in Qing and Gao (2010), Zheng et al. (2009). It can also be noted that the ADRC approach lends itself very well to research and scientific applications, as indicated for example by papers Ramirez-Neria et al. (2014), Yang et al. (2016) or Mado´nski and Herman (2015) (the latter work is a survey which shows various methods of improving the overall estimation quality of extended state observers). The model free control method discussed here is still being analyzed and developed, as indicated by publications in recent years, e.g. Feng and Guo (2017), Jin et al. (2020), Wu et al. (2020). Considering the vast amount of literature on ADRC and, in particular, the application of this control in systems of various types, it can be noted that the class of vehicles considered in this book is devoted to a not very large number of works.

5.1.1.2

Intelligent PID Controllers

This model free control approach is far less practical despite significant theoretical advances. The origins of the development of this method can be traced to the older work of Fliess and Join Fliess and Join (2008), Fliess (2013).

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Nevertheless, a recently published papers by these authors Fliess and Join (2021), Fliess (2021) contain the results of a laboratory experiment (on a half-quadrotor, manufactured by Quanser company) which gives hope for practical applications of this method.

5.1.2 Model Free Control for Underwater Vehicles It is worthy to mention that for underwater vehicle control purposes some important approaches are omitted in this work. Consideration of selected methods as model free is only approximate and does not constitute a strict division into model free methods. For example such approaches are based on (they cannot be classified, in the sense of this book, as the model based methods): (1) proportional derivative (PD) controller Koh et al. (2006), Mazumdar and Asada (2014); (2) adaptive PD controllers Hoang and Kreuzer (2007), Sun and Cheach (2009); (3) adaptive PD controller for AUV which require only model of gravity and buoyancy regressor matrix (only partial knowledge about the AUV parameters is needed) Sun and Cheach (2009); (4) nonlinear PD and PD+ controller Campos et al. (2017); (5) nonlinear control design for a fully actuated autonomous underwater vehicle (AUV) using robust integral of the sign of the error (RISE) Fischer et al. (2014); (6) model free second order sliding mode control (taking into account ocean currents as disturbances and thruster dynamics) Garcia-Valdovinos et al. (2014); (7) a method which does not require knowledge about dynamic parameters which guarantees prescribed transient and steady state performance despite the presence of external disturbances acting on the vehicle Karras et al. (2014); (8) decoupled, distributed control architecture and SLAM Williams et al. (2001); (9) the method based on stochastic means to handle noisy measurements consisting of path planner and motion controller Noguchi and Maki (2021); (10) active disturbances rejected control (ADRC) for path following Lamraoui and Qidan (2019); (11) model free adaptive control combining an extended state observer (ESO) Li et al. (2020) or model free sliding mode control Zhu et al. (2021); (12) improved active disturbance rejection heading control (based on Active Disturbance Rejection Control) Ye et al. (2021).

5.1.3 Model Free Control for Vehicles Moving Horizontally For the purposes of this book, this group of methods also includes those that do not require knowledge of a nonlinear model, so a linearized model that is only

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approximate is sufficient. It should be noted that also for this class of vehicles for control purposes some important strategies which are out of scope of this work. For example such methods can be cited: (1) velocity and position stabilization controller for hovercraft based on a simplified vehicle model Fantoni et al. (1999), Fantoni et al. (2000); (2) feedback tracking controller using integrator backstepping for simplified surface vessel model Pettersen and Nijmeijer (1998); (3) simple model free local controller Jin et al. (2004); (4) nonlinear robust adaptive controller (based on coordinate transformations and backstepping technique) for steering of underactuated ships with unknown parameters along a given path (environmental disturbances induced by wave, wind and ocean current are taken into account) Do et al. (2004); (5) dynamic positioning (using PD controller) for ships Berge and Fossen (2000); (6) LQR controller Cremean et al. (2002); (7) biomimetic vision based control strategy for hovercraft Roubieu et al. (2014); (8) trajectory tracking technique computing the desired speed for each segment of the trajectory (using PID controller) Svec et al. (2014); (9) adaptive global output feedback controller for ships (without velocity measurements or knowledge of the ship parameters) Fang et al. (2004); (10) controller for hovercrafts which uses fly-like correlators Fuller and Murray (2011); (11) model free adaptive control based on adaptive Kalman filter Li et al. (2018).

5.1.4 Model Free Control for Airships Control systems designed for airships are often very complicated and the types of controllers are difficult to classify. Here, this group of methods also includes those that do not require a nonlinear model, so a linearized model that is only approximate is sufficient. Due to the purpose of the book, the following solutions can be included in the model free control group: (1) classical controllers or their combination: (a) guidance system (composed of path tracking controller PI plus heading controller PD) together with additional roll controller which uses aileron input for outdoor airship AURORA Azinheira et al. (2001); (b) guidance algorithm plus PID controller Mazhar (2012) for airship Mk-II ALTAV; (c) PI control based guidance strategy used for following of the trajectory path for an outdoor airship Ramos et al. (2001); (d) PID control for an outdoor airship Peddiraju (2010); Solaque and Lacroix (2007);

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(e) combination of PD (height and heading control) and PID (velocity) controllers Paiva de et al. (1999); (f) PI controller described in Frye et al. (2007); (g) PD controller based on wind disturbance observer Furukawa and Shimada (2014); (h) PD controller proposed in Kuhle et al. (2005); (2) remote control system for outdoor airship LOTTE Kungl et al. (2004); (3) autonomous control of unmanned outdoor airships by the means of SLAM Solaque et al. (2007); (4) flight control software structure for airship VIA-50 described in Lee et al. (2004); (5) Kharitonov’s theory based robust controller and H∞ controller Mo et al. (2003); (6) image based tracking controller for an indoor airship Fukao et al. (2003a), Fukao et al. (2003b), Fukao et al. (2007); (7) flight trajectory following (H∞ controller and path tracking controller (PI)) for the airship AURORA Azinheira et al. (2000), Elfes et al. (2003); (8) feedback controller for waypoint to waypoint navigation for an outdoor airship Kim et al. (2003); (9) visual servo control scheme for an outdoor underactuaded robotic airship Azinheira et al. (2002), Silveira et al. (2002); (10) decoupled lateral and longitudinal controllers for the lateral and longitudinal motions (linear based control) for airship AURORA de Paiva et al. (2006); (11) station keeping and docking based on visual feedback (image based controller) where longitudinal and lateral models were applied for an indoor airship Van der Zwaan et al. (2000); (12) feedback control algorithms and LQR for optimization for airship AIUX15 Acanfora (2011); (13) LQR gain scheduling control for airship AURORA Moutinho (2007); Moutinho et al. (2016); (14) control strategies for various flight phases (based on decoupling of the lateral and longitudinal dynamics) Hygounenc et al. (2004); (15) controller based on the real-time visual tracking system with image projection for an indoor airship Oh et al. (2006); (16) combined control method consisting of long straight trajectory tracking control, turning control in a horizontal plane, and longitudinal control to keep the altitude and pitching motion in the presence of wind for an outdoor airship Fukao et al. (2008); (17) combined automatic embedded hardware and software system for an outdoor unmanned airship Nguyen et al. (2012), Nguyen et al. (2012); (18) gain-scheduled feedback controller based on linear quadratic (LQ) methods Atmeh and Subbarao (2016); (19) path following controller based on wind coordinate and a path regeneration algorithm in wind coordinate for an outdoor airship Saiki et al. (2011);

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(20) flight control system based on the airship parameters identification for an indoor airship Yamasaki and Goto (2003); (21) a parallel-embedded mission control framework (embedded computer cluster) described in Ribeiro et al. (2017); (22) active disturbance rejection control (ADRC) based control strategy Zhu et al. (2014); (23) active disturbance rejection control (ADRC) applied to the attitude control of an airship Le et al. (2021). Some review of the methods used earlier for airship control is given in Liu et al. (2009).

5.2 Control Strategies for Underwater Vehicles Because the most important criterion is usefulness of the algorithm hence there are many strategies applied and techniques for vehicles control. Here it is assumed the classification of methods taking into account the directly use of the underwater vehicle model for control purposes. Therefore, at the beginning some non-model based control approaches are presented and next the model based methods are given. There are many control techniques applied to underwater vehicles. Recall, that the literature concerning ocean and marine vehicles is rich. Many problems concerning modeling and control strategies are described, e.g. in ABS Guide (2006), Acanfora (2011), Antonelli (2018), Do and Pan (2009), Fossen (1994), Lantos and Marton (2011).

5.2.1 Non-Model Based Approaches: Intelligent Control Intelligent control can be understood as a group of control strategies which are different than classical approach and which uses soft computing techniques. To this group belong, e.g. genetic algorithms, neural network, fuzzy logic approaches. The methods are qualified here even if the equations of motion including the vehicle dynamic parameters are given or used. Referring to underwater vehicles the control strategies apply: (1) genetic algorithms, e.g. Guo (2009), Naeem et al. (2004), Naeem et al. (2005); (2) fuzzy logic together with genetic algorithms, e.g. Loebis et al. (2007); (3) fuzzy control algorithm combined with the cascade PID control algorithm Kong et al. (2020); (4) a sliding mode control (SMC) with a direction-based genetic algorithm (GA) and fuzzy inference mechanism Chin and Lin (2018); (5) fuzzy logic, e.g. Akkizidis et al. (2003), Bessa et al. (2010), Guo et al. (2003), Lea et al. (1999), Sebastian and Sotelo (2007), Teo et al. (2012), Yu et al. (2020),

5.2 Control Strategies for Underwater Vehicles

(6) (7)

(8) (9)

(10) (11) (12) (13)

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Zhang et al. (2021) (a survey of fuzzy logic based methods can be found in Xiang et al. (2018)); hybrid control (using artificial intelligence, fuzzy logic) Molnar et al. (2007); neural networks, e.g. Carreras et al. (2005), Cui et al. (2017), Fischer et al. (2011), Ishii et al. (1995), Ishii and Ura (2000), Kodogiannis et al. (1996), Li and Lee (2005), Soylu et al. (2008), Ven et al. (2005), Venugopal et al. (1992), Yan et al. (2019), Yuh (1990a), Yuh (1994), Zheng et al. (2019); combination of neural network and fuzzy logic approach, e.g. Kim and Yuh (2001), Liu et al. (2016), Wang and Lee (2003); combination of neural network, receding horizon optimization and disturbance estimation Peng et al. (2019) or neural network, LOS guidance scheme and a backstepping technique Peng et al. (2019); combination of image based visual servoing and neural network Gao et al. (2016); combination of barrier Lyapunov function and backstepping algorithm and neural network Zheng et al. (2019); indirect adaptive controllers based on hybrid neuro-fuzzy network Hassanein et al. (2016); adaptive control methods without knowledge about the system model, e.g. Choi and Yuh (1996), Nie et al. (2000), Yuh (1990b), Yuh et al. (1999), Zhao and Yuh (2005).

5.2.2 Model Based Control Schemes in Body-Fixed Frame Taking into account the Body-Fixed Representation there are many control approaches which can be understood as model based. Some of them belong to linear control methods and the others to nonlinear control methods. In this chapter the algorithms are divided into two groups: realized in the Body-Fixed Frame and in the Earth-Fixed Frame. The algorithms can be divided into two categories taking into account the used model of the vehicle. The first group includes the methods based on simplified dynamic model (first of all with a diagonal inertia matrix). If the model of the vehicle contains non-diagonal inertia matrix then the method belongs to the second group.

5.2.2.1

Algorithms for Simplified Dynamic Vehicle Model

Recall some control algorithms which are applicable for simplified underwater vehicle models (i.e. with a diagonal inertia matrix): (1) hierarchical guidance and control architecture for unmanned underwater vehicles Caccia and Veruggio (2000); (2) backstepping techniques based approaches:

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(a) adaptive controller that is used for underactuated underwater vehicle with unknown parameters along a given path and taking into account environmental disturbances Do et al. (2004); (b) the method for solving simultaneous tracking and stabilization problem in horizontal plane based on backstepping technique and coordinate transformation Do et al. (2004); (c) tracking controller for underactuated marine vehicles Repoulias and Papadopoulos (2007), Repoulias and Papadopoulos (2007); (d) path-following controllers Encarnacao and Pascoal (2000), Lapierre et al. (2003), Lapierre and Soetanto (2007); (e) two steps control approach (integration guidance and control) to solve the docking problem of AUV Batista et al. (2014); (f) nonlinear controller based on coordinate transformation and backstepping techniques for dynamic positioning and way-point tracking of underactuated autonomous underwater vehicles Aguiar and Pascoal (2007); (g) combination of backstepping with active disturbance rejection control (ADRC) and barrier Lyapunov function for trajectory tracking Xie et al. (2020); (h) in combination with sliding mode control method Liu et al. (2017); (i) together with model predictive control Shen et al. (2018); (3) nonlinear dynamics model based controller for trajectory tracking with exact linearization and without linearization Smallwood and Whitcomb (2004); (4) hierarchical guidance and control architecture for UUV (unmanned underwater vehicles) Caccia and Veruggio (2000); (5) sliding mode control (SMC) approach Elmokadem et al. (2016), Yan and Yu (2018), Yoerger and Slotine (1985), Zhou et al. (2020), second-order sliding mode control Joe et al. (2014), integral sliding mode control (ISMC) Kim et al. (2016), Kim et al. (2015), integral time-delay sliding mode control Zhou et al. (2020), adaptive nonsingular terminal sliding mode control (TSMC) Cao et al. (2020), finite-time control scheme developed using the nonsingular fast terminal sliding mode control method Yang et al. (2021); (6) combination of an adaptive sliding mode observer with output feedback control strategy Yu et al. (2020); (7) an adaptive trajectory tracking controller for underactuated autonomous underwater vehicles under asymmetric time-varying line-of-sight (LOS) range and angle constraints Li et al. (2020); (8) model predictive control (MPC) schemes for trajectory tracking Shen and Shi (2020), Yan et al. (2020). The simplified inertia matrices are used to facilitate the control algorithms. Moreover, it can be noticed that for this type of models methods based on the backstepping techniques and various SMC approaches are often applied.

5.2 Control Strategies for Underwater Vehicles

5.2.2.2

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Algorithms for More Complex Dynamic Vehicle Model

For more general models the following approaches are offered: (1) H∞ control strategy Petrich (2009), Zhang et al. (2018); (2) position trajectory tracking and path-following controller for underactuated vehicles in the presence of some parametric uncertainties (based on backstepping) Aguiar and Hespanha (2007); (3) sliding mode autopilot based on state feedback (using a linearized model) Healey and Lienard (1993); (4) Vehicle–Fixed Frame model based controller tested in Antonelli (2007); (5) nonlinear Luenberger observers together with the controller (based on the observer backstepping technique) Refsnes et al. (2008); (6) cascade control strategy for trajectory tracking of UUV Alonge et al. (2001); (7) position and attitude tracking controllers based on quaternion feedback approach Fjellstad and Fossen (1994); (8) robust controller for tracking in spite of limited knowledge about the vehicle Conte and Serrani (1999); (9) cross-track controller for underactuated autonomous vehicles (which guarantees stability to straight-line trajectories in three dimensional space) Borhaug and Pettersen (2005); (10) a passivity-based controller with integral action port-Hamiltonian systems Donaire et al. (2017), energy based control for port-Hamiltonian systems Valentinis et al. (2015) or adaptive tracking control using port-Hamiltonian theory Jia et al. (2020); (11) adaptive non-singular integral terminal sliding mode tracking control Qiao and Zhang (2017), adaptive double-loop integral terminal sliding mode control Qiao and Zhang (2019a) or fast nonsingular integral terminal sliding mode control Qiao and Zhang (2020), the dynamic sliding mode control (DSMC) Vu et al. (2021); (12) command governor adaptive control (CGAC) Makavita et al. (2019); (13) the finite-time control technique and the finite-time state feedback controller Li et al. (2015); (14) feedback linearization technique using linear matrix inequality (LMI) Kim (2015); (15) cascaded adaptive control scheme described in Fossen and Fjellstad (1993); (16) nonlinear controllers based on Lyapunov theory for the underwater vehicle MARES Ferreira et al. (2010), Ferreira et al. (2009); (17) model predictive control (MPC) algorithm Lin and Chin (2017). Experimental comparisons between selected model free and model based control algorithms were presented in Martin and Whitcomb (2016), Martin and Whitcomb (2018).

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5.2.3 Model Based Control Algorithms in Earth-Fixed Frame Such algorithms are proposed less frequently than those mentioned previously. For more general underwater vehicle models, the following control schemes apply: (1) the adaptive controllers using quaternions representation described in Antonelli et al. (2003, 2001) (containing two sets of parameters, one related to the EarthFixed Frame and the other related to Vehicle-Fixed Frame); (2) cascaded adaptive control scheme described in Fossen and Fjellstad (1993); (3) position tracking feedback controller with wave filter Paulsen et al. (1994); (4) Earth-Fixed Frame model based controllers compared in Antonelli (2007); (5) sliding mode control (SMC) Eslami et al. (2019), Soylu et al. (2016), fractional sliding mode control Munoz-Vazquez et al. (2017), integral sliding mode controller Cui et al. (2017), decoupled adaptive high order sliding mode control Guerrero et al. (2019), adaptive terminal sliding mode based thruster fault tolerant control Zhang et al. (2015) or non-singular terminal sliding mode control Qiao and Zhang (2019b), Rangel et al. (2020); (6) L 1 type adaptive controller Maalouf et al. (2015); (7) application of Extended State Observer (ESO) technique and high order sliding mode control to improve PD controller and backstepping controller Guerrero et al. (2020).

5.3 Control Strategies for Vehicles Moving Horizontally There are many control techniques applied to surface vessels or ships. Recall, for example, the books Do and Pan (2009), Fossen (1994), Lantos and Marton (2011).

5.3.1 Various Non-Model Based Strategies: Intelligent Control Since selected of the model free methods have been discussed in Sect. 5.1 here only selected solutions based on intelligent control methods are listed. Some control strategies are based on intelligent control approach. For example the following methods belong to this category: (1) genetic algorithms for ships Alfaro-Cid et al. (2006), Alfaro-Cid et al. (2005), McGookin et al. (2000); (2) fuzzy logic control strategy for ships Rigatos and Tzafestas (2006), Velagic et al. (2003), Zhang et al. (1995) or hovercrafts Tanaka et al. (2001), an adaptive universe-based fuzzy control algorithm with retractable fuzzy partitioning for marine vehicles Wang et al. (2018), an adaptive robust online constructive

5.3 Control Strategies for Vehicles Moving Horizontally

(3)

(4) (5) (6) (7)

85

fuzzy control algorithm for surface vehicles Wang et al. (2016), Nussbaum-based adaptive fuzzy tracking controller for surface vehicles Wang et al. (2015), robust fuzzy neural control scheme proposed in Wang and Er (2015), use of fuzzy strategy to predictor LOS based trajectory linearization control for path following Qiu et al. (2020) or an event-triggered adaptive fuzzy fixed-time trajectory tracking controller for unmanned surface vehicle with unknown dynamics Song et al. (2021); neural networks for ships Burns (1995), Tee and Ge (2006), Unar and MurraySmith (1999); neural network based adaptive dynamic inversion controller used for a hovercraft model Kim et al. (2013); radical basis function neural networks used to stabilize complex hovercraft nonlinear dynamic system Fu et al. (2017); combination of neural network and SMC Qiu et al. (2019); a gain scheduling PID controller tuned by genetic algorithms and fuzzy logic controller given in Menoyo Larrazabal and Santos Penas (2016); an adaptive controller for trajectory tracking based on backstepping method, neural network, and low-frequency learning techniques Zhang et al. (2020); an adaptive robust control algorithm based on combination of the backstepping method, neural network, dynamic surface control, and the sliding mode control method Zhang et al. (2020).

5.3.2 Model Based Control Schemes in Body-Fixed Frame In this book the model based control algorithms are considered. The algorithms can be divided in different ways but here they are grouped as follows: (1) algorithms realized in the Body-Fixed Frame, and (2) algorithms realized in the Earth-Fixed Frame. Hovercraft models can be derived from the model used for surface vehicles or ships taking into account different environment. Therefore, similar controllers as the used for these vehicles can be applied. In the literature there exist not many control methods applicable strictly to hovercrafts. Contrary, one can found many references containing algorithms for control of other planar vehicles than hovercrafts.

5.3.2.1

Algorithms for Simplified Dynamic Vehicle Model

Some control algorithms are applicable for simplified surface vehicle models (with a diagonal inertia matrix): (1) continuous periodic time-varying feedback stabilization controller based on coordinate transformation for underactuated surface vessels Pettersen and Egeland (1996);

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(2) time-invariant discontinuous feedback controller based on coordinate transformation and Lie algebra for underactuated surface vessels Reyhanoglu (1996, 1997); (3) algorithms based on backstepping method: (a) nonlinear tracking controller for underactuated surface vessels Godhavn (1996); (b) robust nonlinear path following controller for surface vessels (which relies on feedback dominance, instead of feedback linearization) Li et al. (2009); (c) position tracking controller for underactuated hovercraft Aguiar et al. (2003); (d) state tracking feedback controller for an underactuated ships (using a coordinate transformation and integrator backstepping) Pettersen and Nijmeijer (2001); (e) global tracking state feedback controller for underactuated ships (based on coordinate transformations and backstepping technique) Jiang (2002); (f) tracking controllers (for global partial-state feedback and output-feedback) used for underactuated ship (based on coordinate transformations and backstepping technique) Do et al. (2005); (g) controllers which enable stabilization and tracking of underactuated ships (based on coordinate transformations and backstepping technique) Do et al. (2002); (h) global tracking control problem of underactuated ship subject to input saturation Huang et al. (2015); (i) control scheme to achieve global uniform asymptotic full-state stabilization of underactuated vessels if the modeling parameters are unknown Xie and Ma (2015); (j) a trajectory tracking robust controller based on a vectorial backstepping approach Yang et al. (2014); (k) the control design based on combination of results developed for stability and control of stochastic systems, potential projection functions, and backstepping and Lyapunov’s direct methods Do (2015); (l) a controller for trajectory tracking and path following for a hovercraft vehicle in the presence of uncertain parameters and unknown external based on backstepping technique Xie et al. (2019); (4) various variants of the sliding mode control approach: (a) robust sliding mode control algorithm for unmanned surface vessels trajectory tracking Fahimi and Van Kleeck (2013); (b) sliding mode controller together with dynamic model identification procedure for surface vessel Siramdasu and Fahimi (2013); (c) second-order sliding mode controller for hovercraft Sira-Ramirez (2002); (d) sliding mode based controllers for underactuated surface vessels Ashrafiuon et al. (2008), Mahini et al. (2013), McNinch et al. (2010), McNinch and Ashrafiuon (2011), Soltan et al. (2011);

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(e) sliding mode control method described in Ashrafiuon et al. (2017); (f) higher-order sliding mode control algorithm applied to a hovercraft model Liu and Han (2014); (5) linear and angular velocity controllers (based on PI control and taking into account the vehicle inertia) for surface vehicle Caccia et al. (2008); (6) tracking controller for an underactuated surface vessel Behal et al. (2002), Behal et al. (2001); (7) global tracking controllers for underactuated ships (under assumption bounded feedback) Harmouche et al. (2014); (8) tracking controller (in the sense of cascade control) for underactuated ships Lefeber et al. (2003); (9) feedback controller (for stabilization of position and orientation) including integral action for ships Pettersen and Fossen (2000); (10) trajectory tracking controller (of course and velocity) for hovercraft based on the quantitative feedback theory Munoz-Mansilla et al. (2012); (11) nonlinear receding horizon controller applied for hovercraft Seguchi and Ohtsuka (2002), Seguchi and Ohtsuka (2003); (12) flatness based approach dynamic feedback controller Sira-Ramirez and Ibanez (2000), Sira-Ramirez and Ibanez (2000); (13) nonlinear state feedback controller for hovercraft Hayashi et al. (1994); (14) global tracking controller using the dynamic surface control (DSC) method for underactuated ships taking into account input and velocity constraints Chwa (2011); (15) a barrier Lyapunov function based adaptive path following control scheme Fu et al. (2019); (16) the motion tracking controller based on system identification for ships Oh et al. (2010); (17) control scheme for underactuated systems (including surface vehicles) based on vector fields and viability theory Panagou and Kyriakopoulos (2011), Panagou and Kyriakopoulos (2013); (18) switching controller which guarantees practical stabilization of an underactuated marine vehicle in the presence of non-vanishing, perturbations induced by current (using vector fields approach) Panagou and Kyriakopoulos (2014); (19) nonlinear bounded-gain-forgetting with a line-of-sight guidance law path following algorithm Wang et al. (2019); (20) control approach based on differential flatness theory and using a nonlinear state vector and disturbances estimation method under the name of derivativefree nonlinear Kalman Filter for an underactuated hovercraft model Rigatos and Raffo (2014); (21) bioinspired autopilot for performing corridor-following tasks developed for a hovercraft Serres and Ruffier (2015); (22) a trajectory tracking controller for a nonholonomic hovercraft using an integrated parameter estimator Cabecinhas et al. (2018).

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A review of available tracking and set-point controllers for surface underactuated vessels (marine vehicles) can be found in Ashrafiuon et al. (2010). Very often the algorithms are designed to solve the problem of control of underactuated vehicles. Despite of that the simplified vehicle model was considered in many cases, the proposed controllers had very complicated form. Moreover, frequently only simulation results are delivered. Moreover, for this type of models methods using the backstepping techniques are often used. In addition, strategies based on sliding mode control were frequently encountered.

5.3.2.2

Algorithms for More Complex Dynamic Vehicle Model

Methods of control suitable for more general vehicle models (with non-diagonal inertia matrix) are used. This could include, for example: (1) control algorithms based on backstepping approach: (a) controllers for ships using adaptive backstepping techniques Godhavn et al. (1998); (b) robust path-following global controller based on backstepping technique and coordinate transformations Do and Pan (2006); (c) global smooth controllers for practical stabilization of arbitrary chosen reference trajectories for ships based on backstepping technique Do (2010); (d) position tracking controller for underactuated hovercraft model based on backstepping technique with coordinate transformation Aguiar and Hespanha (2003), Aguiar and Hespanha (2007) (applied however for the model with diagonal inertia matrix); (e) trajectory tracking controller schemes: a cascade of proportional derivative controllers and a nonlinear controller obtained through backstepping for surface vehicles Sonnenburg and Woolsey (2013); (f) two global stable robust adaptive trajectory tracking controllers using the adaptive vector-backstepping design method Zhu and Du (2020); (g) a dynamic sliding mode controller with disturbance observer Zhang (2018); (h) a backstepping based controller with compensation control effort for surface ships Qu et al. (2018); (i) stabilization and tracking control schemes for underactuated surface ships using backstepping method given in Zhang and Wu (2015); (j) controller for stabilization underactuated ships on a linear course which takes into account environmental disturbances induced by wave, ocean current and wind (based on backstepping technique) Do et al. (2003); (k) robust adaptive trajectory tracking control control algorithms for surface ships based on a virtual parameter, the adaptive vector-backstepping design method, and taking into account unknown parameters, time-varying disturbances, as well as input saturation Zhu and Du (2020);

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(2) nonlinear tracking control algorithm (feedback linearizing controller) for ships Berge and Fossen (2000); (3) adaptive maneuvering procedure for ships Skjetne et al. (2005); (4) dynamic positioning system with model based controller (consisting of a modified LQG feedback controller and a model reference feedforward controller) Sorensen et al. (1996); (5) stabilizing controller which enable regulation and dynamic positioning of ships and uses only position measurements (based on separation principle and a nonlinear observer) Loria et al. (2000); (6) path following control scheme (PID heading controller and speed controller using state feedback linearization) for marine surface vessels Moreira et al. (2007); (7) a method of incorporating multiobjective controller selection into a closed-loop control system Bertaska and von Ellenrieder (2019); (8) a nonlinear receding horizon control algorithm for an underactuated hovercraft Seguchi and Ohtsuka (2002), Seguchi and Ohtsuka (2003); (9) an adaptive robust finite-time tracking controller for marine surface vehicle Wang et al. (2016); (10) trajectory tracking controller for underactuated surface vessels based on Linear Algebra approach Serrano et al. (2014); (11) sliding mode control method for underactuated vessels described in Ashrafiuon et al. (2017); (12) adaptive trajectory tracking algorithm using barrier Lyapunov function Qin et al. (2020); (13) nonsingular fast terminal sliding mode (NFTSM) controller with disturbance observer Wang et al. (2019); (14) the control strategy which uses of transverse function, the integration of backstepping procedure, barrier function, and Lyapunov synthesis for making stable feedback control with prescribed performance, and additionally NN approximation and disturbance estimates techniques for the compensation of uncertain hydrodynamic damping and external disturbances Dai et al. (2019); (15) robust nonlinear model predictive control (NMPC) strategy for the dynamic positioning ship which serves also for trajectory tracking based on a nonlinear disturbance observer Yang et al. (2020). A review of various ship roll motion control systems can be found in Perez and Blanke (2012). In this book these methods are out of scope. As before, backstepping techniques are popular for these models.

5.3.3 Model Based Control Schemes in Earth-Fixed Frame Control algorithms designed in the Earth-Fixed Frame are rarely designed. However, some approaches can be found, e.g.:

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(1) weather optimal positioning control (WOPC) strategy for ships and free floating rigs Fossen and Strand (2001); (2) trajectory tracking observer based control scheme for ships Wondergem et al. (2011); (3) robust coupled multiple sliding-mode control (CMSMC) method for hovercraft Jeong and Chwa (2018); (4) nonlinear optimal trajectory tracking control law Chen et al. (2020).

5.4 Control Strategies for Airships Airships are systems which work in specific environment, namely in the atmosphere. There exist many problems concerning the systems, namely Bestaoui Sebbane (2012): (1) mission planning; (2) trajectory design; (3) control. The problem of motion planning was considered, e.g. in Kim and Ostrowski (2003). Some conceptual and implementation aspects of an internet based software architecture for the AURORA unmanned autonomous airship were described in Ramos et al. (2003). Control of Lighter Than Air Robots (LTAR) is only one of many problems concerning these flying vehicles. In order to obtain satisfactory flight results for airships, their systems must be tested in practice, e.g. Recoskie et al. (2013). For indoor airships moving at low velocity a part of questions can be omitted what makes the test much easier. The control strategies are used first of all from the practical point of view. However, besides practical applications theoretical solutions of the airship control problem is considered in the literature. It seems sensible to consider the problem of control strategies from two points of view: (1) methods for outdoor and indoor blimp (airship); (2) simulation and experimental validation of the control approach. In the given below review various kinds of methods are taken into account. There are many control techniques applied to unmanned airships. Recall, only the selected books, i.e. Bestaoui Sebbane (2012), Khoury and Gillett (1999), Khoury (2012). However, the problem of airship control was considered in some theses Acanfora (2011), Gomes (1990), Kornienko (2006), Li (2008), Liesk (2012), Mazhar (2012), Moutinho (2007), Peddiraju (2010). The control challenges based on simulations of the YEZ-2A airship model were pointed out in Gomes and Ramos (1998).

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5.4.1 Various Non-Model Based Strategies: Intelligent Control Some model free methods are discussed in Sect. 5.1. So here only a selection of solutions based on intelligent control methods are listed. The strategies belonging to this approach are also considered in the literature. They are based, for example, on: (1) fuzzy logic for an indoor Gonzalez et al. (2009); (2) vision-based fuzzy logic control strategy for an indoor airship Shan (2009); (3) combination fuzzy approach and sliding mode control Yang (2019), Yang et al. (2014); (4) neural networks, e.g. Hong et al. (2009), Kawamura et al. (2009), Park et al. (2003); (5) neuronal control model—inspired by biological organisms (indoor) Bermudez et al. (2007); (6) neural network with evolutionary computation technique Jia et al. (2009); (7) neural network and evolutionary technique for an indoor airship Zufferey (2005), Zufferey et al. (2006); (8) neural network combined with sliding mode control Yang and Yan (2015), Yang and Yan (2016); (9) neural network combined with backstepping Zheng and Zou (2016); (10) Gaussian processes and reinforcement learning Ko et al. (2007), Rottmann et al. (2007).

5.4.2 Model Based Control Schemes In practical applications mostly control algorithms are designed in the Body-Fixed Frame. Model based strategies are less frequently used to control the airship. Recall some known solutions.

5.4.2.1

Algorithms for Simplified Dynamic Vehicle Model

There exist control algorithms applicable to simplified airship models (with diagonal inertia matrix or other simplification): (1) robust feedback control algorithm for global stabilization of the position and orientation for an underactuated indoor airship Yamada and Tomizuka (2005); (2) visual servoing taking into account the blimp dynamics (vision based control) Zhang and Ostrowski (1999); (3) Model Predictive Control (MPC) for an indoor airship Fukushima et al. (2006);

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(4) nonlinear control strategy based on robust control Lyapunov function for the airship stabilization (taking into account unknown wind gusts) Kahale et al. (2013); (5) Generalized Predictive Control (GPC) for an outdoor airship Solaque and Lacroix (2007); (6) robust feedback controller for both stabilization of an underactuated airship and rejection against wind disturbances based on the coordinate transformation and the integral control for an indoor airship Yamada et al. (2010), Yamada et al. (2007); (7) nonlinear control by extended linearization (ELC) Solaque and Lacroix (2007), Solaque et al. (2008); (8) path following using the line-of-sight (LOS) guidance law and backstepping control Wang et al. (2019).

5.4.2.2

Algorithms for More Complex Dynamic Vehicle Model

For more complex airship model the following algorithms were proposed (they can be treated as some examples of known control strategies): (1) model based controller using decoupled controllers for an indoor airship Kuhle et al. (2005); (2) dynamic inversion control strategy Moutinho (2007), Moutinho and Azinheira (2005), de Paiva et al. (2006); (3) backstepping control approach Azinheira and Moutinho (2008), Azinheira et al. (2009), Han et al. (2016), Moutinho (2007), Moutinho et al. (2007), de Paiva et al. (2006); (4) backstepping sliding mode control Liu et al. (2020), Liu et al. (2021), Vieira et al. (2020), Zhou et al. (2019); (5) combined integrator backstepping approach and Lyapunov theory Beji and Abichou (2005); (6) nonlinear controller using Lyapunov and backstepping techniques Liesk (2012), Liesk et al. (2012), Liesk et al. (2013) for the outdoor airship Mk-II ALTAV which is based on the model described in Peddiraju et al. (2009); (7) disturbance observer together with a Lyapunov barrier function-based finitetime controller Yuan et al. (2021); (8) controller based on a non-certainty equivalence adaptive method Sun and Zheng (2015); (9) control system for compensation of oscillation about axes x and y of the airship (not given in explicit form) based on mathematical model for an outdoor airship Jelenciak et al. (2013); (10) Receding Horizon Control based approach using Linear Matrix Inequalities for optimization purpose Trevino et al. (2007);

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(11)

model predictive control (MPC) presented in Liu et al. (2018), Yuan et al. (2020); (12) sliding mode approach based controller Adamski et al. (2020). Because algorithms realized in the Earth-Fixed Frame are used very rarely, they are considered here as a part of group algorithms for more complex dynamic vehicle model. The following controllers can be mentioned: (1) (2) (3) (4)

adaptive backstepping control scheme Han et al. (2016); terminal sliding mode controller Yang (2018); adaptive fixed-time trajectory tracking controller Zheng et al. (2018) adaptive sliding mode trajectory control scheme Zheng and Sun (2018).

5.5 Closing Remarks There are many control strategies used for underwater vehicles, surface vehicles and airships. Significant part of them represents approaches that do not require knowledge of the system model. In this chapter various control methods, coming from the literature, were mentioned. However, the model free controllers or intelligent controllers are out of scope of this book. From the presented control methods review it results that many solutions are dedicated for underwater vehicles, surface and horizontally moving vehicles, indoor or outdoor airships. Hence, it can be concluded that the control problem that this type of flying vehicles is still actual. The algorithms considered in this book are applicable for a class of vehicle models, i.e. underwater vehicles, some horizontally moving vehicles and indoor airships. Moreover, the given, in next chapters, control algorithms are related to the case of knowledge of the vehicle. An advantage is that they are universal in the sense that they can be used for various vehicles mentioned above. However, the main purpose of the algorithms proposed in the later part of the work is to test the dynamics and more precisely the assumed vehicle model.

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Aguiar AP, Hespanha JP (2007) Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty. IEEE Trans Autom Control 52(8):1362–1379 Aguiar AP, Pascoal AM (2007) Dynamic positioning and way-point tracking of underactuated AUVs in the presence of ocean currents. Int J Control 80(7):1092–1108 Akkizidis IS, Roberts GN, Ridao P, Batlle J (2003) Designing a Fuzzy-like PD controller for an underwater robot. Control Eng Pract 11:471–480 Alfaro-Cid E, McGookin EW, Murray-Smith DJ (2006) GA-optimised PID and pole placement real and simulated performance when controlling the dynamics of a supply ship. IEE Proc-Control Theory Appl 153(2):228–236 Alfaro-Cid E, McGookin EW, Murray-Smith DJ, Fossen TI (2005) Genetic algorithms optimisation of decoupled Sliding Mode controllers: simulated and real results. Control Eng Pract 13:739–748 Alonge F, D’Ippolito F, Raimondi FM (2001) Trajectory tracking of underactuated underwater vehicles. In: Proceedings of the 44th IEEE conference on decision and control, Orlando, Florida, December, pp 4421–4426 Antonelli G (2018) Underwater robots. Springer International Publishing AG, part of Springer Nature Antonelli G (2007) On the use of adaptive/integral actions for six-degrees-of-freedom control of autonomous underwater vehicles. IEEE J Oceanic Eng 32(2):300–312 Antonelli G, Caccavale F, Chiaverini S, Fusco G (2003) A novel adaptive control law for underwater vehicles. IEEE Trans Control Syst Technol 11(2):221–232 Antonelli G, Chiaverini S, Sarkar N, West M (2001) Adaptive control of an autonomous underwater vehicle: experimental results on ODIN. IEEE Trans Control Syst Technol 9(5):756–765 Ashrafiuon H, Muske KR, McNinch LC, Soltan RA (2008) Sliding-mode tracking control of surface vessels. IEEE Trans Ind Electron 55(11):4004–4012 Ashrafiuon H, Muske KR, McNinch LC (2010) Review of nonlinear tracking and setpoint control approaches for autonomous underactuated marine vehicles. In: Proceedings of 2010 american control conference marriott waterfront, Baltimore, MD, USA June 30-July 02, pp 5203–5211 Ashrafiuon H, Nersesov S, Clayton G (2017) Trajectory tracking control of planar underactuated vehicles. IEEE Trans Autom Control 62(4):1959–1965 Atmeh G, Subbarao K (2016) Guidance, navigation and control of unmanned airships under timevarying wind for extended surveillance. Aerospace 3(1):8 1–25 Azinheira JR, Moutinho A (2008) Hover control of an UAV With backstepping design including input saturations. IEEE Trans Control Syst Technol 16(3):517–526 Azinheira JR, Moutinho A, de Paiva EC (2009) A backstepping controller for path-tracking of an underactuated autonomous airship. Int J Robust Nonlinear Control 19:418–441 Azinheira JR, de Paiva EC, Bueno SS (2002) Influence of wind speed on airship dynamics. J Guid Control Dyn 25(6):1116–1124 Azinheira JR, de Paiva EC, Carvalho JRH, Ramos JJG, Bueno SS, Bergerman M, Ferreira PAV (2001) Lateral/directional control for an autonomous, unmanned airship. Aircr Eng Aerosp Technol 73(5):453–458 Azinheira JR, de Paiva EC, Ramos J Jr.G, Bueno SS (2000) Mission path following for an autonomous unmanned airship. In: Proceedings of the 2000 IEEE international conference on robotics & automation, San Francisco, CA, April, pp 1269–1275 Azinheira JR, Rives P, Carvalho JRH, Silveira GF, de Paiva EC, Bueno SS (2002) Visual servo control for the hovering of an outdoor robotic airship. In: Proceedings of the 2002 IEEE international conference on robotics & automation, Washington, DC, pp 2787–2792 Batista P, Silvestre C, Oliveira P (2014) A two-step control approach for docking of autonomous underwater vehicles. Int J Robust Nonlinear Control 25(10):1528–1547 Behal A, Dawson DM, Dixon WE, Fang Y (2002) Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics. IEEE Trans Autom Control 47(3):495–500

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Chapter 6

IQV Based PD Control in Body-Fixed Frame

Abstract In this chapter PD controllers with gravity compensation expressed in terms of the IQV are studied. The PD control algorithms in terms of the IQV considered here differ from the classical PD controllers in that they include the vehicle dynamics in the control gains. These control algorithms are able to satisfy the position control objective globally for 6 DOF vehicles moving at low velocity. Both proposed controllers are realized in the Body-Fixed Frame. Moreover, the problem of robustness of the algorithms is considered. Simplified forms of algorithms are also discussed. Finally, some examples of simulation results for a 6 DOF model of an underwater vehicle and an airship are shown.

6.1 Control Algorithms Expressed in IQV The PD control schemes are well-known from the literature, e.g. for underwater vehicles (Fossen 1994; Koh et al. 2006; Mazumdar and Asada 2014), for ships (Berge and Fossen 2000), or for airships (Frye et al. 2007; Kuhle et al. 2005). Two types of the PD controller in terms of the IQV are described in this section, namely the algorithm using the GVC and the NGVC. The control algorithms expressed in terms of the IQV differ from the classical PD controllers in that they include the vehicle dynamics in the control gains.

6.1.1 PD-Type Controller Expressed in GVC The set-point control algorithm for underwater vehicles described by equations of motion using GVC was presented in Herman (2009b). The control proposition for a 6 DOF vehicle can be summarized in the following theorem.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_6

111

112

6 IQV Based PD Control in Body-Fixed Frame

Theorem 6.1 Consider the vehicle system model with dynamic equations given by (2.55), kinematic equations (2.53), and (2.56) for which after transformation of the vector π into τ it is: τ = Bu, (6.1) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Then the nonlinear controller expressed in terms of the GVC is proposed: π = −N k D ξ + ϒ T JT (η) N k P e + gξ (η), τ =ϒ

−T

π,

u = B τ, †

(6.2) (6.3)

where e = ηd − η is the position error, k D = JT (η)K DC J(η) (with k DC > 0 which denotes a symmetric matrix with constant elements only, and assuming that θ = ±90 deg), k P are symmetric positive definite matrices, and N is a diagonal strictly positive matrix. The system under the controller (6.2) is locally asymptotically stable and both limt→∞ η(t) = ηd and limt→∞ ξ (t) = 0 are achieved.

6.1.1.1

Remark 6.1: Thruster Force Allocation

The first problem is to obtain the appropriate forces and moments by means of the vehicle’s thrusters. It is known from the literature, e.g. Antonelli (2018), Choi et al. (2014), Liu et al. (2016), Qiao and Zhang (2019a, b, 2020), Sarkar et al. (2002), Vu et al. (2021). In Eq. (6.1) the matrix B ∈ R 6×m contains the control force coefficients (m ≥ 6 which means full control over the forces and moments acting on the vehicle). The force to be produced by each vehicle thruster can therefore be determined by u ∈ R m (6.3) where B† = BT (BBT )−1 ∈ R m×6 is the pseudo-inverse matrix of B matrix.

6.1.1.2

Remark 6.2: Input Saturation

The second problem concerns the saturation of actuators. In real vehicle the maximal forces and torques are limited. For each marine vehicle, there is an inherent certain value of actuator saturation. Taking into account the above, it is assumed in the literature, that the control signals do not exceed the limit values. Formally, this condition can be described as follows: τ min ≤ τ ≤ τ max ,

(6.4)

where τ min , τ max ∈ R 6 are vectors of minimal and maximal constant values of forces and moments. A more detailed description may also be used, i.e.:

6.1 Control Algorithms Expressed in IQV

τi min ≤ τi ≤ τi max or |τi | ≤ τi max

113

(6.5)

where i = X, Y, Z , K , M, N . The issue of actuator saturation is particularly relevant when there are nonlinearities and model inaccuracies of external disturbances. Since in algorithms expressed in IQV the vehicle dynamics is included in the gain matrices of the controller therefore the effect resulting from this fact can be treated as an additional disturbance. For this reason this problem should also be discussed. Paper Qiao and Zhang (2019a) contains the remark (based on the results obtained in Zhang et al. 2015) that when the saturation effect of the thrust is not serious, then it is reasonable to assume that the control input τ is bounded. This means that simple trajectories must be chosen which require only that the thrust forces occasionally violate the thrust saturation limits. Omission of the effect of actuator saturation dynamics in the equations of motion can be found in papers on underwater vehicles, e.g. Peng et al. (2019a, b), Xie et al. (2020), Yan et al. (2019), Yu et al. (2020), Zhou et al. (2020b), surface vehicles, e.g. Yang et al. (2020) and airships (Fukushima et al. 2006; Oh et al. 2006). However, it must be taken into account that there are control algorithms in which the dynamics of the saturation of the actuators is taken into account, e.g. for underwater vehicles (Campos et al. 2017; Kim 2015; Zhou et al. 2020), surface vehicles (Qin et al. 2020; Qiu et al. 2020, 2019), or airships (Azinheira and Moutinho 2008; Yuan et al. 2020). The algorithms considered in this book represent the approach mentioned in Qiao and Zhang (2019a) because the saturation effects of the actuators do not last long and occur mainly at the beginning of the vehicle movement and the trajectories implemented do not have a very complex shape. Proof Can be found in Herman (2009b). The short version of the stability proof is given below. As a Lyapunov function candidate it is assumed: L(e, ξ ) =

1 1 T ξ Nξ + eT Nk P e. 2 2

(6.6)

Calculating the time derivative of the function L (6.6) and making use of the equations of motion in terms of the GVC, i.e. (2.53), (2.55)–(2.56) one gets: dL = ξ T Nξ˙ − η˙ T Nk P e dt = ξ T [π − ϒ T C(ν)ν − ϒ T D(ν)ν − ϒ T g(η) − ϒ T JT (η)Nk P e].

(6.7)

Using e˙ = −η˙ = −J(η)ν = −J(η)ϒξ , ν T C(ν)ν = 0 for all ν ∈ R n , and inserting π (6.2) one obtains: dL = −ξ T [Nk D ξ + ϒ T D(ν)ν] = −ξ T [Nk D + ϒ T D(ϒξ )ϒ]ξ ≤ 0, dt

(6.8)

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6 IQV Based PD Control in Body-Fixed Frame

where the damping matrix D(ϒξ ) is a non-symmetric and strictly positive definite matrix. The function candidate decreases as long as ξ = 0 for all system trajectories. ≡ 0 only if ξ ≡ 0. Therefore, the closed-loop system is as follows: Note that dL dt d dt

    −J(η)ϒξ e   . = ξ −k D ξ + N−1 ϒ T JT (η)k P e − Cξ (ξ )ξ − Dξ (ξ )ξ

(6.9)

Because the closed-loop Eq. (6.9) represents an autonomous system, the LaSalle’s invariance principle can be applied to demonstrate the local asymptotic stability. Let ∗ be the set in which L˙ = 0 (where L˙ = dL ). The set ∗ is defined as: dt ˙ ξ ) = 0} = {e ∈ R n , ξ ∈ R n : ξ = 0}. ∗ = {e ∈ R n , ξ ∈ R n : L(e,

(6.10)

The largest invariant set in ∗ is the origin of the state space [eT , ξ T ]T . From (2.53), (2.55), (2.58), and (2.59) it follows that for ξ = 0 it is Cξ (ξ )ξ = 0 and Dξ (ξ )ξ = 0 in (6.9). It implies that also ξ˙ = N−1 ϒ T JT (η)k P e. This result leads us to contradiction with L˙ because ξ˙ suggests that the vector ξ = 0 is time dependent. Moreover, one obtains L˙ ≡ 0 only if N−1 ϒ T JT (η)k P e is equal to zero. At the “equilibrium point” one has N−1 ϒ T JT (η)k P e = 0 and also e = 0 if the matrix N−1 ϒ T JT (η) is nonsingular (i.e. full rank). Thus, the system with the controller (6.2) converges to the desired state which ends the proof.  Vehicle input signals. Using the relationship (2.53) between the vectors ξ and ν and (2.61) (note that τ = ϒ −T π ) the input forces vector τ can be rewritten as follows: τ = −ϒ −T Nk D ϒ −1 ν + JT (η)Nk P e + g(η).

(6.11)

Therefore, the control signal is gained by Nk D and Nk P . Consequently, the input signal τ is regulated in accordance with the dynamics of the vehicle. It means that for a heavier vehicle the control coefficients are greater than for a light vehicle. Recall that for well-known PD controller (Fossen 1994) one has: τ = −k D ν + JT (η)k P e + g(η).

(6.12)

The main difference between Eqs. (6.11) and (6.12) consists in the presence of the matrices N and ϒ −1 which means that for the proposed controller the system parameters are known. However, even if they are not exactly known, thanks to both matrices, the velocity and the position error tend to zero making use of the dynamical parameters of the system. The benefit of using the GVC controller is that the gains are strictly related to the dynamics of the vehicle.

6.1 Control Algorithms Expressed in IQV

6.1.1.3

115

Sensitivity Analysis

ˆ and This analysis is based on Herman (2009b). In order to decompose matrix M ˆ it is necessary to assume that it is symmetrical. Let define to obtain the matrix N, ˆ where M ˆ is the known part of the matrix the symmetric matrix M = M A − M, ˆ + M A . Therefore, each element of the matrix M A can be M A . Hence, M A = M denoted as m i j = mˆ i j + m i j (i = 1, . . . , 6, j = 1. . . . , 6). After decomposition of M A the elements mˆ i j and m i j are included in the obtained matrix N but they are not separated what results from the used decomposition method. The same statement can be made regarding the transformation matrix ϒ. It means that both the diagonal matrix N and the matrix ϒ depend on mˆ i j + m i j , i.e. N(mˆ i j + m i j ) and ϒ(mˆ i j + m i j ). ˆ = ϒ(mˆ i j + m i j ) the controller (6.2) can be ˆ = N(mˆ i j + m i j ) and ϒ Denoting N given in the following form: T

T

ˆ JT (η)Nk ˆ g(η). ˆ Pe + ϒ ˆ Dξ + ϒ π = −Nk

(6.13)

The Lyapunov function candidate is defined by: L(e, ξˆ ) = ˆ where ξˆ = ϒ

−1

1 ˆT ˆ ˆ 1 T ˆ ξ Nξ + e Nk P e, 2 2

(6.14)

ν (instead of ξ = ϒ −1 ν). The first time derivative is as follows: T dL ˆ T D(ϒ ˆ ξˆ )ϒ] ˆ ξˆ . ˆ D +ϒ = −ξˆ [Nk dt

(6.15)

ˆ N, ˆ ξˆ ) are positive definite ˆ and D(ϒ The condition L˙ ≤ 0 is fulfilled as long as ϒ, ˆ ˆ matrices. After decomposition it is ϒ > 0 because ϒ is always an upper triangular ˆ is N ˆ > matrix containing ones on the diagonal. The condition regarding the matrix N 0 which implies that each diagonal element of the matrix must be strictly positive, ˆ + M > 0 and i.e. Nˆ ii > 0. For the model of a real system the inertia matrix M ˆ one gets also Nii > 0. Taking into account the velocity transformation between the ˆ ξˆ ) is equal to D(ν). Because vector ξˆ and the vector ν it can be observed that D(ϒ the matrix D(ν) is strictly positive (Fossen 1994) therefore it can be concluded that the condition L˙ ≤ 0 for (6.15) is satisfied. Using the relationships between ξˆ and ν, (6.15) can be rewritten as follows: dL ˆ −1 + D(ν)]ν. ˆ −T Nk ˆ Dϒ = −ν T [ϒ dt

(6.16)

Assuming that k D = c I (where c is a constant and I is the identity matrix) one has ˆ −T Nk ˆ −1 = c (M ˆ Dϒ ˆ + M). Because M ˆ > 0 and M ˆ + M > 0 then also: ϒ dL ˆ + M) + D(ν)]ν ≤ 0. = −ν T [c (M dt

(6.17)

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6 IQV Based PD Control in Body-Fixed Frame

The next part of the stability analysis can be carried out using the same arguments as for the system with exactly known parameters. The set-point controller expressed in terms of the GVC, namely (6.2) is suitable both for underwater vehicle and indoor airship models testing. It can be also applied for some vehicles moving horizontally.

6.1.2 PD-Type Controller Expressed in NGVC The set-point controller in terms of the NGVC was proposed in Herman (2010). The theoretical result can be given in the form of the following theorem. Theorem 6.2 Consider the vehicle system model with dynamic equations given by (2.81), kinematic equations (2.79), and (2.82) for which after transformation of the vector  into τ it is: τ = Bu, (6.18) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Then the nonlinear controller expressed in terms of the NGVC has the following form: 1  = −k D ζ + −T JT (η)( k P + k P T ) e + gζ (η), 2 τ = T  , u = B† τ ,

(6.19) (6.20)

where e = ηd − η is the position error, k D = JT (η)K DC J(η) (where k DC > 0 denotes a symmetric matrix with constant elements only, with θ = ±90 deg) and k P are symmetric positive definite matrices. The system under the controller (6.19) locally asymptotically stable and both limt→∞ η(t) = ηd and limt→∞ ζ (t) = 0 are achieved. Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid. Proof Together with robustness analysis can be found in Herman (2010). Recall, the stability proof. As the Lyapunov function candidate the following expression is proposed: L(e, ζ ) =

1 1 T ζ ζ + eT k P e. 2 2

The time derivative of the above function, namely L˙ =

dL dt

1 L˙ = ζ T ζ˙ + e˙ T ( k P + k P T )e. 2

(6.21) has the form: (6.22)

6.1 Control Algorithms Expressed in IQV

117

Determining e˙ = −η˙ = −J(η)ν = −J(η) −1 ζ , inserting (6.19) into (6.22), using (2.79), (2.83)–(2.84) and taking into account the property ν T C(ν)ν = 0 for all ν ∈ R n (Fossen 1994) one gets: 1 L˙ = ζ T [ − Cζ (ζ )ζ − Dζ (ζ )ζ − gζ (η)] − ζ T −T JT (η)( k P + k P T )e 2 = −ζ T [k D + −T D( −1 ζ ) −1 ]ζ ≤ 0. (6.23) Recalling that (2.79) from Fossen (1994) it can be concluded that the damping matrix D( −1 ζ ) is a non-symmetric and strictly positive definite matrix. Then the power is dissipated passively by the matrix −T D( −1 ζ ) −1 and actively by the virtual matrix k D . The function L decreases as long as ζ = 0 for all system trajectories. ≡ 0 only if ζ ≡ 0. The closed-loop system can be written in the folMoreover, dL dt lowing form:     −J(η) −1 ζ e . = ζ −k D ζ + 21 −T JT (η)( k P + k P T )e − Cζ (ζ )ζ − Dζ (ζ )ζ (6.24) Because the closed-loop (6.24) represents an autonomous system, thus the LaSalle’s invariant theorem can be applied to demonstrate local asymptotic stability. One has L˙ = 0 only if ζ = 0. Let ∗ be the set in which L˙ = 0. Then, the set ∗ is defined as: d dt

˙ ζ ) = 0} = {e ∈ R N , ζ ∈ R N : ζ = 0}. ∗ = {e ∈ R N , ζ ∈ R N : L(e,

(6.25)

The largest invariant set in ∗ is the origin of the state space [eT , ζ T ]T . From (2.79), (2.81) and (2.83)–(2.84) it follows that for ζ = 0 also Cζ (ζ )ζ = 0 and Dζ (ζ )ζ = 0. In this case one obtains ζ˙ = 21 −T JT (η)( k P + k P T )e. This result, however, leads to contradiction with L˙ because ζ˙ suggests that ζ = 0 and it is time dependent. Moreover, L˙ ≡ 0 only if 21 −T JT (η)( k P + k P T )e is zero. At the “equilibrium point” it is 21 −T JT (η)( k P + k P T )e = 0 and also e = 0 if the matrix 1 −T T

J (η)( k P + k P T ) is full rank. On this basis, it can be concluded that the 2 system (2.81) with the controller (6.19) converges to the desired state which completes the proof of Theorem 6.2.  Vehicle input signals. In order to obtain the input signal it is necessary to transform the quasi-moment  . Because of the relationship (2.86) one has τ = T  (the matrix T is invertible) and next: 1 τ = − T k D ζ + JT (η)( k P + k P T )e + g(η) 2 1 T = − k D ν + JT (η)( k P + k P T )e + g(η). 2

(6.26)

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6 IQV Based PD Control in Body-Fixed Frame

Note that for k D = λI the first term of (6.26) is equal −λMν. Thus, it can be concluded that the full inertia matrix is used in the velocity control gain of the proposed controller. Recall that the PD controller used for a underwater vehicle can be written as follows (Fossen 1994): τ = −k D ν + JT (η)k P e + g(η).

(6.27)

Comparing (6.27) and (6.26) it can be seen that for the same matrices k D , the matrix

and its transformation give an additional gain. Consequently, the steady-state using the NGVC controller possibility of the system response shaping is greater than if the controller (6.27) is applied.

6.1.2.1

Sensitivity Analysis

The analysis is based on Herman (2010). In the algorithm (6.19) it is assumed that the parameters of the system are exactly known. If this condition is not fulfilled then the following reasoning can be proposed. The known part of the symmetric matrix ˆ and the matrix containing uncertainty of parameters is given M A is defined by M ˆ ˆ + M where each by M = M A − M (also symmetric matrix). Then, M A = M element of the matrix M A is m i j = mˆ i j + m i j (i = 1, . . . , 6, j = 1, . . . , 6). After decomposition the matrix depends on mˆ i j + m i j , i.e. (mˆ i j + m i j ). Assuming ˆ = (mˆ i j + m i j ) the controller (6.19) is expressed as follows:

1 ˆ −T T ˆ P + kP

ˆ −T g(η). ˆ T )e +

 = −k D ζˆ +

J (η)( k 2

(6.28)

The following Lyapunov function candidate is proposed: L(e, ζˆ ) =

1 T 1 ˆ ζˆ ζˆ + eT k P e, 2 2

(6.29)

ˆ (instead of ζ = ν). The first time derivative has the form: where ζˆ = ν dL T ˆ −T D(

ˆ −1 ζˆ )

ˆ −1 ]ζˆ . = −ζˆ [k D +

dt

(6.30)

ˆ and D(

ˆ −1 ζˆ ) are positive definite matrices the condition L˙ ≤ 0 is As long as

ˆ fulfilled. The applied decomposition method leads to an upper triangular matrix . ˆ ˆ Hence, it is > 0. For the model of a real system the inertia matrix M + M > 0 ˆ ii > 0. Using the velocity transformation between and one gets diagonal elements  −1 ˆ ζˆ ) is equal to D(ν). From Fossen (1994) it follows ζˆ and ν it can be noted that D(

that the matrix D(ν) is strictly positive. Then, the condition L˙ ≤ 0 for (6.30) is satisfied.

6.1 Control Algorithms Expressed in IQV

119

Taking into consideration the relationships between ζˆ and ν, the Eq. (6.30) is as follows: dL ˆ T kD

ˆ + D(ν)]ν. = −ν T [

dt

(6.31)

For simplicity, assuming that k D = c I (where c is a constant and I is the identity ˆ + M). Because M > 0, then also M ˆ + M > 0. ˆ T kD

ˆ = c (M matrix) one has

So it can be written that: dL ˆ + M) + D(ν)]ν ≤ 0. = −ν T [c (M dt

(6.32)

Using the same reasoning as for the system with exactly known parameters the stability analysis can be made. The controller (6.19) is applicable also to indoor airships. An example of implementation of the control algorithm (6.19) together with dynamics investigation can be found in Herman and Adamski (2014).

6.1.3 Simplified Forms of Controllers The controllers using the IQV can be simplified. Two forms are shown below. GVC-PD. Consider the case M = N what means that the inertia matrix M is a diagonal one because the matrix ϒ = I. Recalling Eq. (6.11) one obtains: τ = −Nk D ν + JT (η)Nk P e + g(η).

(6.33)

Comparing this equation with (6.12) it is noticeable that (6.33) describes more general controller in which the vehicle dynamics is taken into account. Contrary, in the classical control algorithm (6.12) the dynamical parameters are absent. In fact in the classical algorithm one has N = I what represents only ones on the diagonal in the matrix of inertia. NGVC-PD. Recall Eq. (6.26) and assume for simplification that M = N. Then, one 1 gets ϒ = I, = N 2 and also: 1 1 1 1 1 τ = −N 2 k D N 2 ν + JT (η)(N 2 k P + k P N 2 )e + g(η), 2

(6.34)

which means that the vehicle dynamics is present in realization of the control task. The classical PD controller is a particular case of the controller (6.34) for N = I (lack of the dynamical parameters set).

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6 IQV Based PD Control in Body-Fixed Frame

Algorithms for vehicle model with 3 DOF moving horizontally. The algorithms presented for a 6 DOF model of underwater vehicle can be also used for the reduced model, i.e. a 3 DOF model of the vehicle moving horizontally. In such case the appropriate variables are collected in the vector ν = [u, v, r ]T . However, in order to show better the couplings between variables the order of vector components is changed, namely ν s = [u, r, v]T .

6.2 Simulation Results This section illustrates performance of the set-point control algorithms. The results are obtained by simulation on a 6 DOF fully actuated underwater vehicle and an indoor airship. Simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

6.2.1 PD Control Using GVC 6.2.1.1

Controller for 6 DOF Underwater Vehicle

Some benefits arising from the use of the equations of motion in terms of the GVC were discussed in Herman (2009a). In reference Herman (2009b) a set-point algorithm in terms of the GVC was proposed and used to control of an underwater vehicle. In this subsection the results of the same algorithm for different initial point are presented. Objective: to show performance of the algorithm, its some properties, and to investigate robustness; comparison with performance of the CL control algorithm (without determination of the full dynamics effect). The desired positions were chosen as follows: [xd , yd , z d ]T =[0.4, −0.2, −0.3]T π m, and [φd , θd , ψd ]T = [ π4 , − 18 , π9 ]T rad. Remark There are two methods for the GVC gains selection. In the first, the dynamics of the vehicle is included in the matrix N. The gain matrices k∗D and k P are assumed as small as possible to ensure the error convergence using k D = Nk∗D and k P . Using this method advantages which result from including the vehicle dynamics can be shown. In the second approach the trial and error method is applied. This approach is useful for couplings investigation in the vehicle. Here, the first approach is offered. The calculated decoupled inertia and masses corresponding to each quasiacceleration (ξ˙1 − ξ˙6 ), i.e. the elements of the matrix: N = diag {N11 , N22 , N33 , N44 , N55 , N66 } ,

(6.35)

6.2 Simulation Results

121

are as follows: N11 = 300.0 kg, N22 = 500.0 kg, N33 = 500.0 kg, N44 = 40.0 kgm2 , N55 = 175.5 kgm2 , and N66 = 141.1 kgm2 . In the control algorithms the resulting gains set is applied together with the dynamics vehicle parameters. For this reason the below given gains (for the GVC controller) are selected in order to guarantee acceptable errors convergence: k∗D = diag{5, 5, 5, 5, 5, 5},

k P = diag{10, 10, 10, 10, 10, 10}, (6.36)

where the applied velocity gain matrix is k D = Nk∗D . For the classical controller (CL) the aim is the same but the control gains do not contain the dynamical parameters. Therefore, their values were chosen 10 times larger using the trial and error method, i.e.: k∗D = diag{50, 50, 50, 50, 50, 50}, k P = diag{100, 100, 100, 100, 100, 100},

(6.37)

where k D = k∗D . The gain matrices have no relation to the dynamics of the vehicle. Obviously it is possible to select different control gains set increasing their values to obtain different results. In Fig. 6.1a–b the linear and angular position errors obtained using the GVC controller are shown. The linear errors are close to zero after about 4 s, whereas the angular one 2 s later. Overshoot of these signals is not observed. From Fig. 6.2a– b it is seen that, in spite of used greater control gains the errors for the classical PD controller are close to zero after longer time than previously. Moreover, for the linear position errors oscillations are observed. Comparing the forces and moments from Figs. 6.1c–d and 6.2c–d it is noticeable that they have similar values for both algorithms with exception the first phase of motion. In Fig. 6.2e comparison between the reduced kinetic energy is given (in logarithmic scale). For the GVC algorithm this energy is reduced much faster then for the CL controller and almost proportionally. The results shown explain why using the GVC controller the error convergence is faster than if the CL algorithm is applied. All the above effects arise from the use of dynamics in control gains or its absence. Robustness investigation. At present robustness of the GVC controller will be tested using the same set of gains as for the vehicle with nominal parameters. The dynamical parameters of the vehicle and environmental parameters (damping terms and added masses) differ by 20% from nominal (they are bigger). It means that elements of M and D are bigger by 20%. As it arises from Fig. 6.3 the signals do not differ significantly from the signals obtained with nominal parameters (with exception of τY and τ Z which have smaller values in the first motion phase). Hence, it can be concluded that, in this case, the proposed algorithm is robust to these parameter changes.

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6 IQV Based PD Control in Body-Fixed Frame

Fig. 6.1 Simulation results for GVC controller and underwater vehicle (nominal parameters): a linear position errors; b angular position errors; c forces; d moments

6.2.1.2

Controller for 6 DOF Indoor Airship

Objective: to show performance of the algorithm, its some properties, and to investigate robustness. The following desired positions were chosen as: [xd , yd , z d ]T =[1.2, 0.5, −1.0]T π π T m, and [φd , θd , ψd ]T = [ 12 , − π9 , − 12 ] rad. The calculated inertia and masses corresponding to each quasi-acceleration (ξ˙1 , . . . , ξ˙6 ), i.e. the elements of the matrix: N = diag {N11 , N22 , N33 , N44 , N55 , N66 } ,

(6.38)

for the airship are as follows: N11 = 12.84 kg, N22 = 10.92 kg, N33 = 11.04 kg, N44 = 8.11 kgm2 , N55 = 9.40 kgm2 , and N66 = 0.54 kgm2 . The airship’s dynamics and work conditions are quite different than for the underwater vehicle. The set of control parameters for the GVC controller was tuned to ensure acceptable position and orientation convergence errors, namely: k∗D = diag{3, 3, 3, 3, 3, 15},

k P = diag{4, 4, 4, 4, 4, 45},

(6.39)

6.2 Simulation Results

123

Fig. 6.2 Simulation results for CL controller and underwater vehicle (nominal parameters): a linear position errors; b angular position errors; c forces; d moments; e comparison between the kinetic energy reduction for both controllers (GVC and CL)

124

6 IQV Based PD Control in Body-Fixed Frame

Fig. 6.3 Simulation results for GVC controller and underwater vehicle (robustness—20% increasing of parameters): a linear position errors; b angular position errors; c forces; d moments

where the velocity gain matrix is k D = Nk∗D . The gains related to ψ are significant bigger than for others errors. It means that the orientation error ψ converges to zero more slowly than others if the control gain is the same as other gains. By tuning this variable (increasing its value), the time can be shortened. From Fig. 6.4a–b it results that the position errors are stabilized within 5 s whereas the orientation errors within 10 s. Thus, stabilization of orientation variables is more difficult than the position ones. Moreover, it is observed slight over-regulation for x which indicates the effect of couplings on this variable. The couplings effect is also visible for θ and ψ in the first phase of motion. From Fig. 6.4c–d it can be seen that the forces and moments values are stabilized very fast. The end values of moments are constant and are not close to zero. Some useful information we obtain calculating the kinetic energy reduced by each quasi-velocity. This partial kinetic energy contains the energy related to each velocity together with the energy related to its couplings in the vehicle. It is observable (Fig. 6.4e–f) that bigger part of energy must be reduced by x (K N1 ) and z (K N3 ). Note that the initial points are further from the end point in these directions. Contrary, the kinetic energy related to orientation is small. From Fig. 6.4i it is shown that the kinetic energy is reduced proportionally in logarithmic scale similarly as for the underwater vehicle.

6.2 Simulation Results

125

Fig. 6.4 Simulation results for GVC controller (airship nominal parameters): a linear position errors; b angular position errors; c forces; d moments; e comparison between the kinetic energy for airship K A and kinetic energy related to N1 , N2 , N3 ; f comparison between the kinetic energy related to N4 , N5 , N6 ; g kinetic energy in logarithmic scale

126

6 IQV Based PD Control in Body-Fixed Frame

6.2.2 PD Control Using NGVC 6.2.2.1

Controller for 6 DOF Underwater Vehicle

In reference Herman (2010) a set-point algorithm in terms of the NGVC was proposed and used to control of an underwater vehicle. In this subsection the results of the same algorithm as in Herman (2010) but using different initial point and vehicle model are presented. Objective: to show performance of the algorithm, its some properties, and to investigate robustness. The desired positions were chosen as follows: [xd , yd , z d ]T =[0.4, −0.2, −0.3]T π m, and [φd , θd , ψd ]T = [ π4 , − 18 , π9 ]T rad. The calculated decoupled inertia and masses corresponding to each quasiacceleration (ξ˙1 − ξ˙6 ), i.e. the elements of the matrix N are given by (6.35). The velocity control gains set consists of dynamics of the vehicle and the matrix k D . However, the dynamics is included in different way as in the GVC algorithm. The gains for the NGVC controller were selected to ensure the errors convergence in comparable time. For this reason for each variable they do not have the same values,

Fig. 6.5 Simulation results for NGVC controller and underwater vehicle (nominal parameters): a linear position errors; b angular position errors; c forces; d moments

6.2 Simulation Results

127

Fig. 6.6 Simulation results for NGVC controller and underwater vehicle (robustness—20% decreasing of parameters): a linear position errors; b angular position errors; c forces; d moments

namely: k D = diag{0.95, 0.95, 0.95, 0.05, 0.05, 0.05}, k P = diag{10, 10, 10, 20, 10, 10}.

(6.40)

From Fig. 6.5 it results that the time of error convergence is longer than for the PD GVC algorithm as it is shown in Fig. 6.1a–b. However, signals overshoot is absent as it is observed for the CL controller (Fig. 6.2a). The forces and moments have smaller maximal values what arises from that smaller values of k D were used. Robustness investigation. All parameters of the matrix M and D are 20% less. The simulation results are given in Fig. 6.6. Comparing Figs. 6.5 and 6.6 it is observed that the position errors convergence is similar whereas the angular errors tend to zero faster if the parameters are decreased. However, the τ Z is bigger in the first phase of motion and τ M —smaller. The differences can be explained by the presence of couplings in the vehicle. It can be also concluded that, in this case, the algorithm is robust to the parameters changes because the control task is realized correctly.

128

6 IQV Based PD Control in Body-Fixed Frame

Fig. 6.7 Simulation results for NGVC controller and indoor airship (nominal parameters): a linear position errors; b angular position errors; c forces; d moments; e comparison between the kinetic energy for airship K A and kinetic energy related to ζ1 , ζ2 , ζ3 ; f comparison between the kinetic energy related to ζ4 , ζ5 , ζ6

6.2 Simulation Results

6.2.2.2

129

Controller for 6 DOF Indoor Airship

One of possible applications of the algorithm to an indoor airship model was presented in Herman and Adamski (2014) where a set-point control algorithm in terms of the NGVC for dynamics investigation was given. In this subsection the results of the same algorithm as in Herman and Adamski (2014) but using different airship model, working conditions, and objective are presented. Objective: to show performance of the algorithm and its some properties. Control parameters for NGVC controller (the same work conditions as for the GVC algorithm) were selected to ensure the position and errors convergence in a comparable time, namely: k D = diag{1.0, 1.3, 1.0, 15.0, 4.0, 11.0}, k P = diag{1.0, 1.0, 1.0, 5.9, 6.4, 3.7}.

(6.41)

Note, that their values depend on the position and orientation variable. This algorithm also uses the vehicle dynamics as the GVC one but here the dynamics is included in the quasi-velocity vector ζ . However, they are included in different way into the gains matrix because of different decomposition method. Moreover, comparing (6.41) with (6.39) we observe that the control gains for the position errors are smaller for the NGVC algorithm. From Fig. 6.7a and b it results that the position orientation errors converge to zero in longer time than using the GVC control algorithm (smaller control gains). Note however, that the control effort measured by the input forces is here much smaller than for the GVC algorithm (Figs. 6.7c and 6.4c). On the contrary, the moments are comparable in both cases (Figs. 6.7d and 6.4d). Using the NGVC controller also less energy related to linear position must be reduced (Figs. 6.7e and 6.4e), whereas the energy related to the angular position is comparable (Figs. 6.7f and 6.4f). It is effect of the use of smaller applied forces and smaller gain values.

6.3 Closing Remarks The PD controller expressed in terms of the IQV guarantees the achievement of the constant end point. Two kind of controllers, namely using the GVC and the NGVC, were discussed in this chapter. There exists one significant difference between these controllers and the classical set-point control algorithms. The IQV based controllers use the set of vehicle geometrical and dynamical parameters in the velocity control gains. This feature allows one to shape the system response including the parameters set. Moreover, the selected gains always correspond to the dynamics of the vehicle. Robustness issue (sensitivity to parameter changes) of the given controllers was considered too. Next, two reduced forms of the PD algorithms were shown. Finally, simulation results obtained for both PD controllers in terms of the GVC and NGVC and for a full 6 DOF underwater vehicle as well as for indoor airship model were presented.

130

6 IQV Based PD Control in Body-Fixed Frame

References Antonelli G (2018) Underwater robots. Springer International Publishing AG, part of Springer Nature Azinheira JR, Moutinho A (2008) Hover control of an UAV with backstepping design including input saturations. IEEE Trans Control Syst Technol 16(3):517–526 Berge SP, Fossen TI (2000) On the properties of the nonlinear ship equations of motion. Math Comput Model Dyn Syst 6(4):365–381 Choi JK, Kondo H, Shimizu E (2014) Thruster fault-tolerant control of a hovering AUV with four horizontal and two vertical thrusters. Adv Robot 28(4):245–256 Campos E, Chemori A, Creuze V, Torres J, Lozano R (2017) Saturation based nonlinear depth and yaw control of underwater vehicles with stability analysis and real-time experiments. Mechatronics 45:49–59 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Frye MT, Gammon SM, Qian C (2007) The 6-DOF dynamic model and simulation of the triturbofan remote-controlled airship. In: Proceedings of the 2007 American control conference marriott marquis hotel at times square, New York City, USA, July 11–13, pp 816–821 Fukushima H, Saito R, Matsuno F, Hada Y, Kawabata K, Asama H (2006) Model predictive control of an autonomous blimp with input and output constraints. In: Proceedings of the 2006 IEEE international conference on control applications, Munich, Germany, October 4–6, pp 2184–2189 Herman P (2009a) Transformed equations of motion for underwater vehicles. Ocean Eng 36:306– 312 Herman P (2009b) Decoupled PD set-point controller for underwater vehicles. Ocean Eng 36:529– 534 Herman P (2010) Modified set-point controller for underwater vehicles. Math Comput Simul 80:2317–2328 Herman P, Adamski W (2014) Investigation of control algorithm for airship before indoor experiment. In: Proceedings of 2014 22nd mediterranean conference on control and automation (MED) University of Palermo, June 16–19, 2014, Palermo, Italy, pp 1196–1201 Kim DW (2015) Tracking of REMUS autonomous underwater vehicles with actuator saturations. Automatica 58:15–21 Koh TH, Lau MWS, Seet G, Low E (2006) A control module scheme for an underactuated underwater robotic vehicle. J Intell Rob Syst 46:43–58 Kuhle J, Roth H, Klein C (2005) Robot airship for education and research-modelling and control. In: Proceedings of 16th triennial world congress, Prague, Czech Republic, pp 1378–1378 Liu YC, Liu SY, Wang N (2016) Fully-tuned fuzzy neural network based robust adaptive tracking control of unmanned underwater vehicle with thruster dynamics. Neurocomputing 196:1–13 Mazumdar A, Asada HH (2014) Control-configured design of spheroidal, appendage-free, underwater vehicles. IEEE Trans Robot 30(2):448–460 Oh S, Kang S, Lee K, Ahn S, Kim E (2006) Flying display: autonomous blimp with real-time visual tracking and image projection. In: Proceedings of the 2006 IEEE/RSJ international conference on intelligent robots and systems, Beijing, China, October 9–15, pp 131–136 Peng Z, Wang J, Han QL (2019a) Path-following control of autonomous underwater vehicles subject to velocity and input constraints via neurodynamic optimization. IEEE Trans Industr Electron 66(11):8724–8732 Peng Z, Wang J, Wang J (2019b) Constrained control of autonomous underwater vehicles based on command optimization and disturbance estimation. IEEE Trans Industr Electron 66(5):3627– 3635 Qiao L, Zhang W (2019a) Double-loop integral terminal sliding mode tracking control for UUVs with adaptive dynamic compensation of uncertainties and disturbances. IEEE J Oceanic Eng 44(1):29–53 Qiao L, Zhang W (2019b) Adaptive Second-Order Fast Nonsingular Terminal Sliding Mode Control for Fully Actuated Autonomous Underwater Vehicles. IEEE J Oceanic Eng 44(2):363–385

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Qiao L, Zhang W (2020) Trajectory Tracking Control of AUVs via Adaptive Fast Nonsingular Integral Terminal Sliding Mode Control. IEEE Trans Industr Inf 16(2):1248–1258 Qin H, Li C, Sun Y, Wang N (2020) Adaptive trajectory tracking algorithm of unmanned surface vessel based on anti-windup compensator with full-state constraints. Ocean Eng 200:106906 Qiu B, Wang G, Fan Y (2020) Predictor LOS-based trajectory linearization control for path following of underactuated unmanned surface vehicle with input saturation. Ocean Eng 214:107874 Qiu B, Wang G, Fan Y, Mu D, Sun X (2019) Adaptive sliding mode trajectory tracking control for unmanned surface vehicle with modeling uncertainties and input saturation. Appl Sci 9(1240):1– 18 Sarkar N, Podder TK, Antonelli G (2002) Fault-accommodating thruster force allocation of an AUV considering thruster redundancy and saturation. IEEE Trans Robot Autom 18(2):223–233 Vu MT, Le TH, Thanh HLNN, Huynh TT, Van M, Hoang QD, Do TD (2021) Robust position control of an over-actuated underwater vehicle under model uncertainties and ocean current effects using dynamic sliding mode surface and optimal allocation control. Sensors 21(747):1–25 Xie T, Li Y, Jiang Y, An L, Wu H (2020) Backstepping active disturbance rejection control for trajectory tracking of underactuated autonomous underwater vehicles with position error constraint. Int J Adv Robot Syst May–April 2020, 1–12. https://doi.org/10.1177/1729881420909633 Yan Z, Wang M, Xu J (2019) Global adaptive neural network control of underactuated autonomous underwater vehicles with parametric modeling uncertainty. Asian J Control 21(4):1–13 Yang H, Deng F, He Y, Jiao D, Han Z (2020) Robust nonlinear model predictive control for reference tracking of dynamic positioning ships based on nonlinear disturbance observer. Ocean Eng 215:107885 Yu H, Guo C, Shen Z, Yan Z (2020) Output feedback spatial trajectory tracking control of underactuated unmanned undersea vehicles. IEEE Access 8:42924–42936 Yuan J, Zhu M, Guo X, Lou W (2020) Trajectory tracking control for a stratospheric airship subject to constraints and unknown disturbances. IEEE Access 8:31453–31470 Zhang M, Liu X, Yin B, Liu W (2015) Adaptive terminal sliding mode based thruster fault tolerant control for underwater vehicle in time-varying ocean currents. J Franklin Inst 352:4935–4961 Zhou H, Wei Z, Zeng Z, Yu C, Tao B, Lian L (2020a) Adaptive robust sliding mode control of autonomous underwater glider with input constraints for persistent virtual mooring. Appl Ocean Res 95:102027 Zhou J, Zhao X, Feng Z, Wu D (2020b) Trajectory tracking sliding mode control for underactuated autonomous underwater vehicles with time delays. Int J Adv Robot Syst May–June 2020, 1–12. https://doi.org/10.1177/1729881420916276

Chapter 7

IQV Position and Velocity Tracking Control in Body-Fixed Frame

Abstract This chapter focuses on the non-adaptive nonlinear control methods which serve for the position and velocity tracking of vehicles described in terms of the IQV. The presented algorithms guarantee the position (linear and angular) and velocity error convergence in the finite time. The discussed here control algorithms are realized using the Body-Fixed Representation. There are many controllers known which serve for achieving the same goal. The strategies proposed here can be included in the SMC group without adaptation. The robustness problem to vehicle parameters changes, the real input control signals, and the simplified controllers, among others for 3 DOF horizontal models of surface vehicle are also considered. Moreover, some simulation results for 6 DOF model of an underwater vehicle and an airship are delivered. It is also shown which effects are observable if the controllers are applied for 3 DOF underactuated horizontally moving vehicles. The chapter concludes with a summary of the results presented.

7.1 Control Algorithms Expressed in IQV Some examples model based algorithms suitable for underwater vehicles, surface vehicles and airships are given in Chap. 5. Many of them use backstepping techniques and sliding mode control (SMC) approach. However, the position and velocity tracking control task can be realized using the GVC and the NGVC schemes. In this section both controllers are considered. Moreover, the selected simplified form are discussed in the last subsection.

7.1.1 Position and Velocity Tracking Controller Expressed in GVC This type of algorithm for a 3 DOF model of a hovercraft is presented in reference Herman and Kowalczyk (2014), and for a full 6 DOF vehicle model in Herman and Adamski (2017b). In the latter reference also the robustness problem was taken into © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_7

133

134

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

account. Considerations contained in this subsection are based on work (Herman and Adamski 2017b). The general result is given in the theorem. Theorem 7.1 Consider the vehicle system model with dynamic equations given by (2.55), kinematic equations (2.53), and (2.56) for which after transformation of the vector π into τ one has: τ = Bu, (7.1) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Then the nonlinear controller expressed in terms of the GVC: ˜ π = Nξ˙ r + Cξ (ξ )ξ r + Dξ (ξ )ξ r + gξ (η) + k D sξ + JξT (η)k P η, τ =ϒ

−T

π,

u = B τ, †

(7.2) (7.3)

in which: ˙ d + η), ˜ ξ r = J−1 ξ (η)(η sξ = ξ r − ξ =

˙˜ J−1 ξ (η)(η −1 J˙ ξ (η)(η˜˙

s˙ξ = ξ˙ r − ξ˙ = Jξ (η) = J(η)ϒ,

(7.4) ˜ + η),

¨˜ + η), ˙˜ ˜ + J−1 + η) ξ (η)(η θ = ±90 deg,

(7.5) (7.6) (7.7)

where η˜ = ηd − η and η˙˜ = η˙ d − η˙ are the position error vector and the velocity vector, respectively, k D = k TD > 0, k P = k TP > 0,  = T > 0, and N is a diagonal strictly positive matrix, results in an exponentially stable equilibrium point [sξT , η˜ T ]T = 0. Consequently, η˜ → 0 and η˙˜ → 0 as t → ∞. Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) apply to this controller. Proof The closed-loop system consisting of (2.55), (2.56) together with the control algorithm (7.2) can be given in the form: N˙sξ + [Cξ (ξ ) + Dξ (ξ ) + k D ]sξ + JξT (η)k P η˜ = 0.

(7.8)

The following Lyapunov function candidate is assumed: ˜ = L(sξ , η)

1 T 1 ˜ s Nsξ + η˜ T k P η. 2 ξ 2

(7.9)

Next, the time derivative of L (7.9) is calculated: 1 ˙ ˙˜ T k P η. ˙ ξ , η) ˜ ˜ = sξT N˙sξ + sξT Ns L(s ξ +η 2

(7.10)

7.1 Control Algorithms Expressed in IQV

135

˙ = The matrices M and ϒ have constant elements only. Therefore, the matrix N d T (ϒ Mϒ) = 0. Taking into account (7.8) one has: dt T ˙ ξ , η) ˜ = sξT [−Cξ (ξ )sξ − Dξ (ξ )sξ − k D sξ − JξT (η)k P η] ˜ + η˙˜ k P η. ˜ L(s

(7.11)

Recalling (2.58) one gets sξT Cξ (ξ )sξ = (ϒsξ )T C(ν)(ϒsξ ) = sT C(ν)s = 0 because it is sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). Using Eq. (7.5) one obtains: ˙ ξ , η) ˜ ˜ = −sξT [Dξ (ξ ) + k D ]sξ − η˜ T T k P η. L(s

(7.12)

Defining the vector χ = [sξT , η˜ T ]T the Eq. (7.12) using (2.59) can be written in the form: ˙ ξ , η) ˜ = −χ T Aχ, L(s

(7.13)

 ϒ T D(ν)ϒ + k D 0 . 0 T k P

(7.14)

where:  A=

From (2.29) it results that the matrix A is positive definite. Thus assuming that λmin {A} > 0 it is possible to find an upper bound of the time derivative, namely: ˙ χ ) ≤ −λmin {A} χ2 , L(t,

(7.15)

for all t ≥ 0 and χ ∈ R 2N . Applying now the Lyapunov direct method (Slotine and Li 1991), the conclusion that the state space origin of the system (2.53), (2.55) together with the controller (7.2):  lim

t→∞

 sξ (t) = 0, ˜ η(t)

(7.16) 

is exponentially convergent can be made.

Vehicle input signals. Define the reference velocity ν r , its time derivative ν˙ r , and the vector s as follows: ˜ ν r = J−1 (η)(η˙ d + η), −1

(7.17)

˙˜ ˜ + J (η)(¨ηd + η), ν˙ r = J˙ (η)(η˙ d + η) −1 ˜ s = J (η)(η˜˙ + η). −1

(7.18) (7.19)

Taking into account Eqs. (2.61) and (7.2) one can determine the input forces vector from τ = ϒ −T π , i.e.:

136

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

˜ τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + ϒ −T k D ϒ −1 s + JT (η)k P η.

(7.20)

By comparison of Eqs. (7.2) and (7.20), the following relationships can be noticed: s = ϒsξ ,

s = ν r − ν,

s˙ = ν˙ r − ν˙ .

(7.21)

It means that in the control gain of the algorithm (7.20) instead of the matrix k D contains the matrix ϒ −T k D ϒ −1 and the dynamical couplings in the vehicle are included.

7.1.1.1

Robustness Analysis

The symbols M∗ = M∗ (ν), C∗ = C∗ (ν), D∗ = D∗ (ν), and g∗ = g∗ (η) are introduced. If the robustness issue is considered then the Lyapunov function candidate, using (7.21) (however, here the matrix M∗ is decomposed in order to obtain N∗ ), is given as follows: L=

1 1 1 1 T ∗ ˜ sξ N sξ + η˜ T k P η˜ = sT M∗ s + η˜ T k P η. 2 2 2 2

(7.22)

It is assumed that the matrix M∗ is symmetric and it has only constant elements. ˙ ∗ = 0 and the time derivative of L is: Therefore, M T T ˜ L˙ = sT M∗ s˙ + η˙˜ k P η˜ = sT (M∗ ν˙ r − M∗ ν˙ ) + η˙˜ k P η.

(7.23)

Making use of (7.21) one gets: M∗ ν˙ = τ − C∗ ν − D∗ ν − g∗ = τ − C∗ (ν r − s) − D∗ (ν r − s) − g∗ .

(7.24)

Applying the formula sT C∗ s = 0 (Fossen 1994) the Eq. (7.23) can be rewritten in the form: T ˜ L˙ = sT (M∗ ν˙ r + C∗ ν r + D∗ ν r + g∗ − D∗ s − τ ) + η˙˜ k P η.

(7.25)

The nominal control input is assumed as (7.20), where M, C, D, g, ϒ −T , and ϒ −1 contain known elements. Inserting (7.20) into (7.25), grouping the terms, and denot˜ = C∗ − C, D ˜ = D∗ − D, g˜ = g∗ − g one obtains: ˜ = M∗ − M, C ing M ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ − sT (D + ϒ −T k D ϒ −1 )s L˙ = sT (M T ˜ − sT JT (η)k P η˜ + η˙˜ k P η.

(7.26)

T Converting the expression J(η)s = η˙˜ + η˜ to −sT JT (η) + η˙˜ = −η˜ T T and applying (7.19) one has:

7.1 Control Algorithms Expressed in IQV

137

˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ − sT (D + ϒ −T k D ϒ −1 )s L˙ = sT (M ˜ − η˜ T T k P η.

(7.27)

Because for the nominal controller it is s = ϒsξ (7.21), then one gets: ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds)] ˜ L˙ = sξT [ϒ T (M − sξT (ϒ T D ϒ + k D )sξ ˜ − η˜ T T k P η.

(7.28)

Based on Slotine and Li (1991) the strictly positive constants βi where i = 1, . . . , 6 can be found to ensure convergence of the tracking error to zero. Selecting a constant which are greater than elements of the lumped uncertainty vector (they should be   T ˜ ˜ r + Dν ˜ r +˜g− norm-bounded by sufficiently large constants βi )βi ≥ [ϒ (Mν˙ r + Cν  ˜ i  and assuming that k D and T k P as symmetric or diagonal matrices one Ds)] obtains: 6  ˜ L˙ ≤ − (7.29) βi |sξ i | − sξT (ϒ T D ϒ + k D )sξ − η˜ T T k P η. i=1

The constant βi must guarantee that the derivative of the Lyapunov function is seminegative. If the nominal parameters of the system and the assumed reference trajectories are known then for a given value of uncertainty one can evaluate this parameter βi . In other case we choose its value in conservative manner, i.e. as high as possible but to ensure stability of the system under the controller. The second  solution relies on that in (7.28) the  matrix k D is selected to ensure  ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ . λmin (k D ) ≥ ϒ T (M From expression (7.29) it can be concluded that the tracking error convergence to a neighborhood of zero is guaranteed for t → ∞ if the parameters of the vehicle are not exactly known.

7.1.2 Position and Velocity Tracking Controller Expressed in NGVC The position and velocity tracking controller using the NGVC for a 3 DOF vehicle model was presented in Herman and Kowalczyk (2015), whereas for a 6 DOF model of a vehicle in Herman and Adamski (2017a). In this subsection the general case of the control scheme is discussed in the given below theorem. Theorem 7.2 Consider the vehicle system model with dynamic equations given by (2.81), kinematic equations (2.79), and (2.82) for which after transformation of the vector into τ it is: τ = Bu, (7.30)

138

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Then the nonlinear controller expressed in terms of the NGVC: ˜

= ζ˙ r + Cζ (ζ )ζ r + Dζ (ζ )ζ r + gζ (η) + k D sζ + JζT (η)k P η,

(7.31)

τ = ,

(7.32)

T

u = B τ, †

in which: ˙ d + η), ˜ ζ r = J−1 ζ (η)(η −1 J˙ ζ (η)(η˙ d

˙˜ ˜ + J−1 + η) ηd + η), ζ (η)(¨ ˜ sζ = ζ r − ζ = J−1 ˜˙ + η), ζ (η)(η s˙ζ = ζ˙ r − ζ˙ ,

ζ˙ r =

−1

Jζ (η) = J(η) ,

θ = ±90 deg,

(7.33) (7.34) (7.35) (7.36) (7.37)

where η˜ = ηd − η and η˜˙ = η˙ d − η˙ are the position error vector and the velocity vector, respectively, k D = k TD > 0, k P = k TP > 0,  = T > 0 leads in the exponentially stable equilibrium point [sζT , η˜ T ]T = 0. Consequently, η˜ → 0 and η˙˜ → 0 as t → ∞. Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) apply to this controller. Proof The closed-loop system composed of (2.81), (2.82), and the controller (7.31) may be described in the following form: s˙ζ + [Cζ (ζ ) + Dζ (ζ ) + k D ]sζ + JζT (η)k P η˜ = 0.

(7.38)

The proposed Lyapunov function candidate is as follows: ˜ = L(sζ , η)

1 T 1 ˜ sζ sζ + η˜ T k P η. 2 2

(7.39)

Calculating the time derivative of the function L (7.39) one gets: T ˙ ζ , η) ˜ ˜ = sζT s˙ζ + η˙˜ k P η. L(s

(7.40)

Making use of (7.38) it can be written: T ˙ ζ , η) ˜ = sζT [−Cζ (ζ )sζ − Dζ (ζ )sζ − k D sζ − JζT (η)k P η] ˜ + η˙˜ k P η. ˜ L(s

(7.41)

7.1 Control Algorithms Expressed in IQV

139

Recalling Eq. (2.83), one has sζT Cζ (ζ )sζ = ( −1 sζ )T C(ν)( −1 sζ ) = sT C(ν)s = 0 because sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). Next, applying (7.35) one obtains: ˙ ζ , η) ˜ ˜ = −sζT [Dζ (ζ ) + k D ]sζ − η˜ T T k P η. L(s

(7.42)

Defining the vector χ = [sζT , η˜ T ]T and using (2.84), Eq. (7.41) can be given in the following form: ˙ ζ , η) ˜ = −χ T Aχ , L(s

(7.43)

where:  −T D(ν) −1 + k D 0 . A= 0 T k P 

(7.44)

Taking into account (2.29) and based on Fossen (1994) it can be concluded that the matrix A is positive definite. Assuming now λmin {A} > 0 an upper bound of the time derivative can be found. Therefore, it can be written that: ˙ χ ) ≤ −λmin {A} χ2 , L(t,

(7.45)

for all t ≥ 0 and χ ∈ R 2N . Applying the Lyapunov direct method (e.g. Slotine and Li 1991), the conclusion that the state space origin of the system (2.79), (2.81) together with the control algorithm (7.31):  lim

t→∞

 sζ (t) = 0, ˜ η(t)

is exponentially convergent can be made.

(7.46) 

Vehicle input signals. Transforming (2.86) into the form τ = T , using next (7.31) and (7.17)–(7.19) it is possible to determine the input forces vector τ . Consequently, one gets: ˜ τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + T k D s + JT (η)k P η.

(7.47)

Comparing (7.31) and (7.47) it can be noted that ν r and its time derivative ν˙ r are the same as in (7.17)–(7.19) whereas: sζ = s,

s = ν r − ν, s˙ = ν˙ r − ν˙ .

(7.48)

140

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

This result means that in the control gain of (7.47) instead of the matrix k D the matrix T k D is present which includes the full vehicle dynamic (i.e. together with the dynamical couplings) which are contained in the matrix ϒ.

7.1.2.1

Robustness Analysis

The symbols M∗ = M∗ (ν), C∗ = C∗ (ν), D∗ = D∗ (ν), and g∗ = g∗ (η) are introduced. Moreover, it is assumed that the matrix M∗ is symmetric and it has only ˙ ∗ = 0 and the following Lyapunov function candidate is constant elements. Then, M considered: 1 1 1 1 1 1 T ˜ s sζ + η˜ T k P η˜ = sT ∗T ∗ s + η˜ T k P η˜ = sT M∗ s + η˜ T k P η, 2 ζ 2 2 2 2 2 (7.49) where the relationship (7.48) (the matrix N∗ results from decomposition of the matrix M∗ ) is used. The time derivative of L is defined by (7.23). Therefore, also (7.24) is true, and L˙ is given by (7.25). Moreover, the nominal control input vector is defined by (7.47), where the elements in M, C, D, g, T , and are known. Inserting (7.47) into (7.25) one gets: L=

L˙ = sT [(M∗ − M)˙ν r + (C∗ − C)ν r + (D∗ − D)ν r + g∗ − g T ˜ + η˙˜ k P η. ˜ − D∗ s − T k D s − JT (η)k P η]

(7.50)

˜ = M∗ − M, C ˜ = C∗ − C, D ˜ = D∗ − D, g˜ = g∗ − g one has: Denoting now M ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ − sT (D + T k D )s L˙ =sT (M ˜ − sT JT (η)k P η˜ + η˙˜ k P η. T

(7.51)

T From Eq. (7.19) and next J(η)s = η˙˜ + η˜ it follows that −sT JT (η) + η˙˜ = −η˜ T T . Consequently, one obtains:

˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ − sT (D + T k D )s) L˙ =sT (M ˜ − η˜ T T k P η.

(7.52)

Using the expression s = −1 sζ (7.48) for the nominal controller, after some manipulation it can be written: ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds)] ˜ L˙ = sζT [ −T (M − sζT ( −T D −1 + k D )sζ ˜ − η˜ T T k P η.

(7.53)

Based on Slotine and Li (1991) it is possible to find the strictly positive constants βi where i = 1, . . . , 6 to ensure convergence of the tracking error to zero. Thus, choos-

7.1 Control Algorithms Expressed in IQV

141

   ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds)] ˜ i , i.e. constants which are greater ing βi ≥ [ −T (M than elements of the lumped uncertainty vector (they should be norm-bounded by sufficiently large constants βi ) and assuming k D and T k P as symmetric or diagonal matrices it can be obtained: L˙ ≤ −

6 

˜ βi |sζ i | − sζT ( −T D −1 + k D )sζ − η˜ T T k P η.

(7.54)

i=1

The constant βi must guarantee that the derivative of the Lyapunov function is seminegative. If the nominal parameters of the system and the assumed reference trajectories are known then for a given value of uncertainty we can evaluate this parameter βi . In other case it is possible to choose this value in conservative manner, i.e. as high as possible but to ensure stability of the system under the controller. The second  solution relies on that in (7.28) the  matrix k D is selected to ensure  −T ˜  ˜ ˜ ˜ λmin (k D ) ≥  (Mν˙ r + Cν r + Dν r + g˜ − Ds). From the condition (7.54) it can be concluded that the tracking error convergence is guaranteed for t → ∞ if the vehicle parameters are not known exactly.

7.1.3 Simplified Forms of Controllers GVC-PVTCBF. Some particular cases of the controller (7.2) or (7.20) can be deduced. 1. Consider the case of a diagonal matrix M. Then it will be N = M because the matrix is diagonal and no transformation is needed. From (2.53) it results that ϒ = I and ξ = ν, and from (2.61) π = τ . Consequently, one obtains: ˜ τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + k D s + JT (η)k P η.

(7.55)

This algorithm is similar to the algorithm proposed in Fossen (1994) but slightly different. It is rather like the control scheme given in the work (Slotine and Li 1987). Here, the algorithm is changed in order to show better usefulness of the equations of motion it terms of the GVC. 2. For a symmetric vehicle in the x y-plane one gets yg = 0. As a result, the controller is simplified and reduced. However, the impact of dynamic couplings effect is reduced, too. NGVC-PVTCBF. Some simplified control algorithms can be deduced from the general controller (7.31) or (7.47). 1. If the matrix M is a diagonal matrix (as a result of the assumption that x g = 0 and yg = 0) then one has the simplified controller of the form:

142

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

˜ τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + T k D s + JT (η)k P η,

(7.56)

1

where = M 2 . However, if k D = δ I then also T k D = δ M (but the last term is a diagonal matrix). In such case all inertial elements are taken into account in the control process. This fact implies that the controller directly depends on the vehicle dynamics. 2. Sometimes it is assumed that the vehicle is symmetric in the x y-plane, and one gets yg = 0. If such model can be accepted then the controller is simplified and simultaneously reduced. Consequently, the dynamic and couplings effects are observable but weakened. Algorithms for Vehicle Model with 3 DOF Moving Horizontally. The considered algorithm can be also applied to a reduced 3 DOF vehicle model. The remark given in Chap. 6 is also useful for these controllers.

7.2 Simulation Results In this section some examples of the use of the discussed in this chapter algorithms are presented. Simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

7.2.1 Trajectory Tracking Control Using GVC 7.2.1.1

Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: to show performance of the GVC position and velocity trajectory tracking controller, to investigate robustness; fast error convergence without overshoot in position errors. For tracking it is assumed the following desired helical trajectory for linear position with constant desired angular position, and the appropriate velocity profile, respectively:  π π T π , − , − , (7.57) ηd = 5 cos(0.4 t) − 4, 5 sin(0.4 t) − 2, 0.3 t + 0.5, 6 10 6 η˙ d = [−2 sin(0.4 t), 2 cos(0.4 t), 0.3, 0, 0, 0]T . (7.58) The gain coefficients of the controller were selected due to the comparable time of convergence of position and velocity errors as follows: k D = diag{1000, 1000, 1000, 1000, 1000, 1000}, k P = diag{200, 200, 200, 200, 200, 200},  = diag{1, 1, 3, 1, 1, 1}.

(7.59)

7.2 Simulation Results

143

The third element of  is larger to obtain the errors convergence in comparable time. For this set of control gains and nominal parameters the results are shown in Fig. 7.1. Because  acts on the position and velocity, so to ensure fast convergence its third value has been increased. The control gains set is not related to the full dynamics of the vehicle (only to dynamical couplings) and they are selected using the trial and error method. The matrices k D and k P have great values due to considering only dynamic couplings instead of full dynamics. From Fig. 7.1a it can be seen that the trajectory tracking task is realized correctly. The position linear and angular errors (Fig. 7.1b and c) tend to zero very fast, i.e. in about 5 s. The overshoot effect is also avoided. Faster convergence is observed for z (increased control gain). From Fig. 7.1d–e result that all velocity errors are close to zero after about 5 s. However, overshoot is observed if the vehicle starts (it means that the coupling effect is observable) in velocities time history and some velocities increase. Because the helical trajectory is tracked in three dimensions then values of the the forces are also great (Fig. 7.1f). But also moments τ M and τ N , as it is noticed from Fig. 7.1g, have significant values (in spite of that they should achieve constant values only). High initial values of forces and moments arise from dynamical coupling among variables. Robustness investigation. For the same set of control gains the robustness test with 10% decreasing (reduced) of parameters was performed and the results are given in Fig. 7.2. The control task in 3 dimensional (3D) space is realized as it is shown in Fig. 7.2a. However, the position and velocity trajectory tracking is no longer as accurate as when we know the dynamic parameters as it is observed from Fig. 7.2b–e. From Fig. 7.2f–g it arises that the control effort measured by forces and moments are similar as in case of nominal parameters. The results obtained indicate that the GVC control algorithm is robust to small changes in parameters.

7.2.1.2

Controller for 6 DOF Fully Actuated Airship

In reference Herman and Adamski (2017b) the GVC position and orientation tracking algorithm was given for airship and its performance was shown for a model of indoor airship including robustness test. Objective: to show performance of the GVC position trajectory tracking controller for different airship model as in Herman and Adamski (2017b) and to obtain fast error convergence without overshoot in position errors and estimation of dynamical couplings. For tracking it is assumed the following desired 3D linear position trajectory with constant angular position and the appropriate velocity profile, respectively:  π π T π , − , , ηd = 0.5 t − 2.5, 0.3 t, 0.3 t, 12 5 12 η˙ d = [0.5, 0.3, 0.3, 0, 0, 0]T .

(7.60) (7.61)

144

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

Fig. 7.1 Simulation results for GVC controller and underwater vehicle (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

7.2 Simulation Results

145

Fig. 7.2 Simulation results for GVC controller and underwater vehicle (robustness—10% decreasing of parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

146

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

The gain coefficients of the controller were selected, to ensure acceptable performance of the controller as follows: k D = diag{200, 50, 50, 50, 50, 50}, k P = diag{20, 20, 20, 20, 20, 20},  = diag{0.3, 0.3, 0.3, 0.3, 0.3, 0.3}.

(7.62)

The gains should guarantee the position errors convergence in comparable time. They are not related to the dynamics of the airship. For this reason, the first coefficient of the matrix k D is higher. For this set of control gains and nominal parameters the results are shown in Fig. 7.3. The control task (Fig. 7.3a) is realized correctly. From Fig. 7.3b–c it is noticeable that after about 10 s all position errors are close to zero. The coupling effect occurs for z (Fig. 7.3b) what means that other variables have an impact on this variable. All velocity errors converge to zero after about 10 s as it is observed in Fig. 7.3d–e. However, the linear velocity errors d x/dt, dz/dt and the angular velocity errors dφ/dt, dψ/dt strongly depend on the dynamical couplings (overshoot is noticeable). From Fig. 7.3f it is seen that significant initial differences of the position cause high values of forces on the corresponding variable (namely τ X ). Values of moments (Fig. 7.3g) arise from the dynamical parameters of the airship. In particular τ M values are large because the airship’s model parameters.

7.2.1.3

Observation—Application to 3 DOF Underactuated Vehicle

In reference Herman and Kowalczyk (2014) the algorithm for a fully actuated hovercraft was shown. Based on the same idea similar controller was used for position tracking. For control of underactuated vehicle (lack of one input signal) no mathematical proof was given. For this reason the presented below results can be treated as some observations only. Objective: to show performance of the GVC position and velocity trajectory tracking controller for a hovercraft model and using nominal parameters. Underactuated Hovercraft. The presented simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s). The velocity transformation vector is as follows: ηd = [xd , ψd , yd ]T with a initial point η0 = [1.4, 0, 0.1]T . Moreover, the following trajectories for tracking were assumed: ηd = [0.8 cos(0.02 t), 0.02 t + 1.5708, 0.8 sin(0.02 t)]T , η˙ d = [−0.016 sin(0.02 t), 0.02, 0.016 cos(0.02 t)]T .

(7.63) (7.64)

The control gain matrices (to ensure acceptable position errors convergence) were selected as follows: k D = diag{14.0, 14.0, 14.0}, k P = diag{2.4, 2.4, 2.4},  = diag{0.4, 0.4, 0.4}.

(7.65)

7.2 Simulation Results

147

Fig. 7.3 Simulation results for GVC controller and indoor airship (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

148

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

Fig. 7.4 Simulation results for GVC controller and underactuated hovercraft (nominal parameters): a linear position desired and realized trajectory; b angular position desired and realized trajectory; c linear and angular position errors; d linear and angular velocity errors; e quasi-forces; f force and moment

The set of parameters should guarantee the trajectory tracking control. However, for an underactuated vehicle it is difficult to select gains because inappropriate selection of parameters prevents the task from being completed. Moreover, the obtained results can be treated only as a particular case because no mathematical proof of stability of system together with the GVC controller was given. From Fig. 7.4a it is seen that the desired trajectory is tracked after some time. If the vehicle starts then it tends to the desired trajectory indirectly. After about

7.2 Simulation Results

149

30 s the angle ψ is close to the desired value ψd (Fig. 7.4b). As it is observed from Fig. 7.4c–d the positions and velocities are tracked but tracking of the velocity in y direction (underactuated variable) is guaranteed only to a constant value. Comparing Fig. 7.4e–f it is noticeable that the quasi-forces π1 , π2 , and π3 have non-zero values in spite of that only τ X∗ and τ N∗ are available. The results offer some information concerning the coupling effect in the vehicle. For example it is possible to show how the position y and its time derivative is excited if the desired trajectory is tracked. Based on the time history of the signals π1 , π2 , and π3 it is observed changes of π2 (it exists only if the input signals are transformed into the quasi-velocity space).

7.2.2 Trajectory Tracking Control Using NGVC 7.2.2.1

Controller for 3 DOF Fully Actuated Hovercraft

Performance of the control algorithm applied to a 3 DOF hovercraft model can be found in reference Herman and Kowalczyk (2015).

7.2.2.2

Controller for 6 DOF Fully Actuated Airship

The NGVC based algorithm for a 6 DOF indoor airship model was tested in Herman and Adamski (2017a). Moreover, some further results will be given in Chap. 13.

7.2.2.3

Observation—Application to 3 DOF Underactuated Vehicle

The presented test can be useful for dynamics study but for trajectory tracking it is inappropriate because of lack of the closed-loop system stability proof. Underactuated Underwater Vehicle—circular desired trajectory. The velocity transformation vector is as follows: ηd = [xd , ψd , yd ]T with initial point η0 = [0.4, 0, −1.5]T . ηd = [0.6 cos(0.2 t), 0.2 t + 1.5708, 0.6 sin(0.2 t)]T , η˙ d = [−0.12 sin(0.2 t), 0.2, 0.12 cos(0.2 t)]T .

(7.66) (7.67)

The gain coefficients of the controller (to ensure acceptable convergence of position errors) were selected as follows: k D = diag{42.0, 42.0, 42.0}, k P = diag{70.0, 70.0, 70.0},  = diag{0.7, 0.7, 0.7}.

(7.68)

150

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

Fig. 7.5 Simulation results for NGVC controller and underactuated underwater vehicle (nominal parameters and circular trajectory): a position desired and realized trajectory; b orientation desired and realized trajectory; c linear and angular position errors; d linear and angular velocity errors; e quasi-forces; f force and moment

From Fig. 7.5a it is seen that the vehicle tracks the desired trajectory in spite of that the initial point is far from the goal. The orientation is achieved after about 7 s (Fig. 7.5b). However, the position errors x and y tend to some end value different than zero as it is shown in Fig. 7.5c. Similarly, from Fig. 7.5d it is noticeable that the velocity errors d x/dt and dψ/dt do not converge to zero. The quasi-forces

7.2 Simulation Results

151

are presented in Fig. 7.5e, and the available force and the moment Fig. 7.5f. All of them achieve a constant value. Based on this test it can be said that under assumed vehicle model, conditions, and the circular trajectory the errors x, y, d x/dt, and dψ/dt are sensitive to lack of the signal in y direction.

Fig. 7.6 Simulation results for NGVC controller and underactuated underwater vehicle (nominal parameters and linear trajectory): a position desired and realized trajectory; b orientation desired and realized trajectory; c linear and angular position errors; d linear and angular velocity errors; e quasi-forces; f force and moment

152

7 IQV Position and Velocity Tracking Control in Body-Fixed Frame

Underactuated Underwater Vehicle—linear desired trajectory. The velocity transformation vector was: ηd = [xd , ψd , yd ]T with initial point ηd0 = [−2, 0, 5]T . Saturation limits for forces and moments were assumed as follows: |τ X | ≤ 20 N, |τ N | ≤ 20 Nm. ηd = [t, 0.02 t + 1.5708, −0.3 t + 2]T , η˙ d = [1, 0.02, −0.3]T .

(7.69) (7.70)

The gain coefficients of the controller were selected to ensure linear position convergence. Their values were: k D = diag{70.0, 70.0, 70.0}, k P = diag{1.0, 1.0, 1.0},  = diag{0.5, 0.5, 0.5}.

(7.71)

From Fig. 7.6a it follows that the linear trajectory is tracked correctly in spite of lack of signal y. Taking into account Fig. 7.6b–f it turns out that the information was misleading. Moreover, the motion of the vehicle is strongly disturbed. Unfortunately, in this case, any analysis is impossible. It can be concluded that the dynamics test can only be performed for certain vehicle models and under special conditions. Using this algorithm it can be shown the time history of variables (position, velocity, and quasi-forces) during vehicle motion when the side input signal is not available. However, usefulness of this controller is very limited because real application is impossible. It has a value in simulation studies whose purpose is to estimate the dynamics of the vehicle.

7.3 Closing Remarks Two non-adaptive controllers in terms of the IQV ensuring the position and velocity error convergence were presented in this chapter. The schemes in which the crucial role plays introducing of the vehicle parameters in the control gains, are realized in the Body-Fixed Representation. In the tracking controllers in terms of the GVC and the NGVC, the dynamical and geometrical parameters affect the linear and angular position as well as the velocity errors. The algorithms guarantee tracking of the trajectory, i.e. the errors convergence similarly as the classical algorithms (algorithm realized without the IQV in control gains). The important difference relies on quite different idea of control. In the presented controllers the advantage is that the dynamical couplings or even the full (or partial) dynamics in the vehicle are used in order to ensure the tracking error convergence. It means that for control purposes both control gains and elements of the inertia matrix which are characteristic for different vehicles are applied. The idea of control can be given in two steps. First, the inertia matrix is decomposed. Next, the parameters set is included into the control

7.3 Closing Remarks

153

gain matrix which causes that the error convergence time strictly depends on the vehicle dynamics. Moreover, the robustness issue was discussed for both control algorithms expressed in terms of the IQV. It was shown that the presented algorithms are robust to parameters change. They can be applied both for a 6 DOF full vehicle model and a 3 DOF reduced model. Some simplified forms of the algorithms were given. Finally, the simulation results which can be found in the cited references as well as the presented here examples confirmed effectiveness of the control approach.

References Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Herman P, Adamski W (2017a) Nonlinear tracking control for some marine vehicles and airships. In: Proceedings of the 11th international workshop on robot motion and control, Wasowo Palace, Poland, July 3–5, 2017, pp 257–362 Herman P, Adamski W (2017b) Nonlinear trajectory tracking controller for a class of robotic vehicles. J Frankl Inst 354:5145–5161 Herman P, Kowalczyk W (2014) A nonlinear controller for trajectory tracking of hovercraft robot. In: Proceedings of 2014 22nd Mediterranean conference on control and automation (MED) University of Palermo, June 16–19, 2014, Palermo, Italy, pp 1311–1315 Herman P, Kowalczyk W (2015) Position tracking controller based on transformed equations of horizontal motion for a class of vehicles. In: Proceedings of 2015 23rd mediterranean conference on control and automation (MED), June 16–19, Torremolinos, Spain, pp 1148–1153 Slotine JJ, Li W (1987) On the adaptive control of robot manipulators. Int J Robot Res 6:49–59 Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Hoboken

Chapter 8

IQV Velocity Tracking Control in Body-Fixed Frame

Abstract The problem of velocity control is important in context of a guided motion control. The velocity controllers are often a part of the more complicated vehicle control system. This chapter presents two non-adaptive velocity tracking controllers using the IQV. It is shown that based on nonlinear techniques some algorithms which ensure velocity error convergence in the finite time can be designed. The controllers are expressed in the Body-Fixed Representation. The issue of algorithm robustness to vehicle parameters and the problem of input control signals is also discussed. Finally, the possibility of simplified form of the controllers (including 3 DOF horizontally moving vehicle) is also considered. Application of the approach is verified by simulations on a 3 DOF model of planar vehicle and on a 6 DOF model of an underwater vehicle as well as an indoor airship. At the end, some remarks are delivered.

8.1 Control Algorithms Expressed in IQV Some discussions concerning the velocity control issue can be found e.g. in Breivik (2010), Breivik et al. (2008), Li et al. (2018), Yoon et al. (2012). Velocity controllers for marine vehicles are known from the literature, e.g. Ferreira et al. (2010), Ferreira et al. (2009), Fossen and Fjellstad (1993), Fossen (1994), Yoon et al. (2012). In this chapter two non-adaptive control algorithms expressed in terms of the IQV which guarantee the velocity error convergence are described in this section. At the end their simplified forms are given. In the literature (Herman 2009) a simple velocity control algorithm in terms of the GVC can be found. However, it only provides vehicle velocity control.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_8

155

156

8 IQV Velocity Tracking Control in Body-Fixed Frame

8.1.1 Velocity Tracking Controller Expressed in GVC The algorithm for a 3 DOF underwater vehicle model was shown in Herman and Kowalczyk (2016b) and for 6 DOF full models, i.e. an indoor airship and an underwater vehicle in Herman and Adamski (2017b). The results can be summarized in the following theorem. Theorem 8.1 Consider the vehicle system model with dynamic equations given by (2.55), kinematic equations (2.53), and (2.56) for which after transformation of the vector π into τ it is: τ = Bu, (8.1) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Then the nonlinear controller expressed in terms of the GVC is defined as follows: π = Nξ˙ r + Cξ (ξ )ξ r + Dξ (ξ )ξ r + gξ (η) + k D sξ + ϒ T k I z, τ =ϒ

−T

π,

u = B τ, †

(8.2) (8.3)

in which:  z=

t

ν˜ (σ ) dσ,

(8.4)

0

ξ r = ϒ −1 (ν d + z), sξ = ξ r − ξ = ϒ −1 (˜ν + z), s˙ξ = ξ˙ r − ξ˙ = ϒ −1 (ν˙˜ + ˜ν ),

(8.5) (8.6) (8.7)

where ν˜ = ν d − ν is the velocity error vector between the desired velocity vector and the actual velocity vector, k D = k TD > 0, k I = k TI > 0,  = T > 0, and N is a diagonal strictly positive matrix, results in an exponentially stable equilibrium point [sξT , zT ]T = 0. Consequently, ν˜ → 0 and z → 0 as t → ∞. Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid for this controller. Proof The closed-loop system consisting of (2.55), (2.56) and including the controller (8.2) can be given in the following form: N˙sξ + [Cξ (ξ ) + Dξ (ξ ) + k D ]sξ + ϒ T k I z = 0.

(8.8)

The Lyapunov function candidate is proposed as follows: L(sξ , z) =

1 T 1 sξ Nsξ + zT k I z. 2 2

The time derivative of the function L (8.9) is:

(8.9)

8.1 Control Algorithms Expressed in IQV

157

˙ ξ , z) = sξT N˙sξ + 1 sξT Ns ˙ ξ + ν˜ T k I z. L(s 2

(8.10)

The matrices M and ϒ contain only constant elements. Consequently, ˙ = d (ϒ T Mϒ) = 0. Taking into account (8.8) one has: N dt ˙ ξ , z) = sξT [−Cξ (ξ )sξ − Dξ (ξ )sξ − k D sξ − ϒ T k I z] + ν˜ T k I z. L(s

(8.11)

From (2.58) and (7.21) it follows that sξT Cξ (ξ )sξ = (ϒsξ )T C(ν)(ϒsξ ) = sT C(ν)s = 0 because sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). Making use of (8.6) one obtains: ˙ ξ , z) = −sξT [Dξ (ξ ) + k D ]sξ − zT T k I z. L(s

(8.12)

Defining now the vector χ = [sξT , zT ]T and recalling (2.59) the expression (8.12) can be written given as follows: ˙ ξ , z) = −χ T Aχ, L(s

(8.13)

where:  A=

ϒ T D(ν)ϒ + k D 0 0 T k I

 .

(8.14)

From (2.29) and reference (Fossen 1994) it follows that the matrix A is positive definite. Thus, assuming λmin {A} > 0 it is possible to find an upper bound of the time derivative (8.13), namely: ˙ χ ) ≤ −λmin {A} χ2 , L(t,

(8.15)

for all t ≥ 0 and χ ∈ R 2N . Making use of the Lyapunov direct method (Slotine and Li 1991), the conclusion that the state space origin of the system (2.53), (2.55) together with the controller (8.2):  lim

t→∞

 sξ (t) = 0, z(t)

is exponentially convergent can be made.

(8.16) 

Vehicle input signals. The reference velocity ν r , its time derivative ν˙ r , and the vector s are defined in the following form: ν r = ν d + z, ν˙ r = ν˙ d + ˜ν , s = ν˜ + z.

(8.17) (8.18) (8.19)

158

8 IQV Velocity Tracking Control in Body-Fixed Frame

Based on (2.61), (7.21) and (8.2) it is possible to obtain the input forces vector (from τ = ϒ −T π ), i.e.: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + ϒ −T k D ϒ −1 s + k I z.

(8.20)

It means that in the control gain instead of the matrix k D the matrix ϒ −T k D ϒ −1 which includes the dynamical couplings in the vehicle is present.

8.1.1.1

Robustness analysis

The robustness issue was discussed in Herman and Adamski (2017b). The problem can be solved also in slightly different way. The following symbols M∗ = M∗ (ν), C∗ = C∗ (ν), D∗ = D∗ (ν), and g∗ = g∗ (η) are introduced. Next, the Lyapunov function candidate is proposed in the form: L=

1 1 1 1 T ∗ sξ N sξ + zT k I z = sT M∗ s + zT k I z. 2 2 2 2

(8.21)

It is assumed that the matrix M∗ is symmetric and it has only constant elements. ˙ ∗ = 0. The time derivative of (8.21) is: Therefore, M L˙ = sT M∗ s˙ + ν˜ T k I z = sT (M∗ ν˙ r − M∗ ν˙ ) + ν˜ T k I z.

(8.22)

Making use of (7.21) the equation in the same form as (7.24) is determined. Inserting now (7.24) into (8.22) and recalling that sT C∗ s = 0 (Fossen 1994) after some manipulations one gets: L˙ = sT (M∗ ν˙ r + C∗ ν r + D∗ ν r + g∗ − D∗ s − τ ) + ν˜ T k I z.

(8.23)

The control input τ is defined by (8.20), where the symbols M, C, D, g, ϒ −T , and ϒ −1 mean the known parameters. Substituting (8.20) into (8.23) one has: L˙ = sT [(M∗ − M)˙ν r + (C∗ − C)ν r + (D∗ − D)ν r + g∗ − g − D∗ s − ϒ −T k D ϒ −1 s − k I z] + ν˜ T k I z.

(8.24)

˜ = M∗ − M, C ˜ = C∗ − C, D ˜ = D∗ − D, g˜ = g∗ − g, and Introducing the symbols M applying next the formula s = ϒsξ (7.21) (ϒ represents the nominal parameters set) one obtains: ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ L˙ = sξT ϒ T (M − sξT (ϒ T Dϒ + k D )sξ − zT T k I z.

(8.25)

8.1 Control Algorithms Expressed in IQV

159

Based on reference (Slotine and Li 1991) it is possible to find the strictly positive constants βi where i = 1, . . . , 6 to ensure convergence of the tracking error to zero.  ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds)] ˜ i , assuming k D and T k I Selecting now βi ≥ [ϒ T (M as symmetric or diagonal matrices one gets: L˙ ≤ −

6 

βi |sξ i | − sξT (ϒ T Dϒ + k D )sξ − zT T k I z.

(8.26)

i=1

The constant βi must guarantee that the derivative of the Lyapunov function is seminegative. If the nominal parameters of the system and the assumed reference trajectories are known then for a given value of uncertainty one can evaluate this parameter βi . In other case its value is selected in conservative manner, i.e. as high as possible but to ensure stability of the system under the controller. The second  solution relies on that in (8.25) the  matrix k D is selected to ensure  ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ . λmin (k D ) ≥ ϒ T (M From the condition (8.26) it can be concluded that the tracking error convergence is guaranteed for t → ∞ if the vehicle parameters are not known exactly.

8.1.2 Velocity Tracking Controller Expressed in NGVC This type of velocity tracking control algorithm for a 3 DOF vehicle model can be found in Herman and Kowalczyk (2016a) whereas for a 6 DOF airship model in Herman and Adamski (2017a). The results are summarized in the presented below theorem. Theorem 8.2 Consider the vehicle system model with dynamic equations given by (2.81), kinematic equations (2.79), and (2.82) for which after transformation of the vector into τ it is: τ = Bu, (8.27) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Then the nonlinear controller expressed in terms of the NGVC is defined as:

= ζ˙ r + Cζ (ζ )ζ r + Dζ (ζ )ζ r + gζ (η) + k D sζ + −T k I z,

(8.28)

τ = ,

(8.29)

T

in which:

u = B τ, †

160

8 IQV Velocity Tracking Control in Body-Fixed Frame



t

z=

ν˜ (σ ) dσ,

(8.30)

0

ζ r = (ν d + z), sζ = ζ r − ζ = (˜ν + z), s˙ζ = ζ˙ r − ζ˙ = (ν˙˜ + ˜ν ),

(8.31) (8.32) (8.33)

where ν˜ = ν d − ν is the velocity error vector, k D = k TD > 0, k I = k TI > 0, and  = T > 0 results in an exponentially stable equilibrium point [sζT , zT ]T = 0. Consequently, ν˜ → 0 and z → 0 as t → ∞. Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid for this controller. Proof The closed-loop system composed of (2.79), (2.81), and the controller (8.28) can be written in the form: s˙ζ + [Cζ (ζ ) + Dζ (ζ ) + k D ]sζ + −T k I z = 0.

(8.34)

The Lyapunov function candidate is assumed as follows: L(sζ , z) =

1 T 1 sζ sζ + zT k I z. 2 2

(8.35)

The time derivative of the function L (8.35) is: ˙ ζ , z) = sζT s˙ζ + ν˜ T k I z. L(s

(8.36)

Inserting (8.34) into (8.36) one obtains: ˙ ζ , z) = sζT [−Cζ (ζ )sζ − Dζ (ζ )sζ − k D sζ − −T k I z] + ν˜ T k I z. L(s

(8.37)

Taking into account (2.83) one gets sζT Cζ (ζ )sζ = ( −1 sζ )T C(ν)( −1 sζ ) = sT C(ν)s = 0 because sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). Consequently, using (8.32) one has: ˙ ζ , z) = −sζT [Dζ (ζ ) + k D ]sζ − zT T k I z. L(s

(8.38)

Defining the vector χ = [sζT , zT ]T and applying Eqs. (2.84) (8.38) can be given in the form: ˙ ζ , z) = −χ T Aχ , L(s

(8.39)

where: 

−T D(ν) −1 + k D 0 A= 0 T k I

 .

(8.40)

8.1 Control Algorithms Expressed in IQV

161

Recalling (2.29) and the results shown in Fossen (1994) it can be concluded that the matrix A is positive definite. Assuming λmin {A} > 0 an upper bound of the time derivative can be determined. Finally, it can be written that: ˙ χ ) ≤ −λmin {A} χ2 , L(t,

(8.41)

for all t ≥ 0 and χ ∈ R 2N . Using the Lyapunov direct method (Slotine and Li 1991), the conclusion that the state space origin of the system (2.79), (2.81) with the controller (8.28):  lim

t→∞

 sζ (t) = 0, z(t)

is exponentially convergent can be made.

(8.42) 

Vehicle input signals. Using the reference velocity ν r , its time derivative ν˙ r , and the vector s according to (8.17)–(8.19), and inversion of (2.86) (because it is τ = T ) the input forces vector τ can be rewritten as follows: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + T k D s + k I z.

(8.43)

This result means that in the control gain instead of the matrix k D the matrix T k D includes the full vehicle dynamic (together with the dynamical couplings which are represented by the matrix ϒ).

8.1.2.1

Robustness analysis

The symbols M∗ = M∗ (ν), C∗ = C∗ (ν), D∗ = D∗ (ν), and g∗ = g∗ (η) are introduced. It is also assumed that the matrix M∗ is symmetric and it has only constant ˙ ∗ = 0. The Lyapunov function candidate is proposed in the folelements. Then, M lowing form: L=

1 1 1 1 1 1 T sζ sζ + zT k I z = sT ∗T ∗ s + zT k I z = sT M∗ s + zT k I z. (8.44) 2 2 2 2 2 2

It can be noted that the function L (8.44) is the same as for (8.21). Thus, its time derivative L˙ is described by (8.22) and also by (8.23). The nominal control input is defined by (8.43), where M, C, D, g, T , and are known parameters. Inserting (8.43) into (8.23) one gets: L˙ = sT [(M∗ − M)˙ν r + (C∗ − C)ν r + (D∗ − D)ν r + g∗ − g − D∗ s − T k D s − k I z] + ν˜ T k I z.

(8.45)

162

8 IQV Velocity Tracking Control in Body-Fixed Frame

˜ = M∗ − M, C ˜ = C∗ − C, D ˜ = D∗ − D, g˜ = g − g, and using the forDenoting M −1 mula s = sζ (7.48) (for the nominal parameter set) one has: ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ − (˜ν T + zT T )k I z L˙ = sT (M ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − Ds) ˜ − sT (D + T k D )s + ν˜ T k I z = sT (M − sT (D + T k D )s − zT T k I z,

(8.46)

and next: ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − D∗ s)] L˙ = sζT [ −T (M − sζT ( −T D −1 + k D )sζ − zT T k I z.

(8.47)

Based on reference Slotine and Li (1991) a strictly positive constants βi where i = 1, . . . , 6 can be found in order  to ensure convergence of the tracking error to  ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − D∗ s)]i  one can zero. Thus choosing a value βi ≥ [ −T (M receive (assuming k D and T k I as symmetric or diagonal matrices): L˙ ≤ −

6 

βi |sζ i | − sζT ( −T D −1 + k D )sζ − zT T k I z.

(8.48)

i=1

The constant βi must guarantee that the derivative of the Lyapunov function is seminegative. If the nominal parameters of the system and the assumed reference trajectories are known then for a given value of uncertainty one can evaluate this parameter βi . In other case its value in is selected in conservative manner, i.e. as high as possible but to ensure stability of the system under the controller. The second  solution relies on that in (8.47) the matrix k D is selected to ensure  ˜ ν˙ r + Cν ˜ r + Dν ˜ r + g˜ − D∗ s). λmin (k D ) ≥  −T (M From the condition (8.48) it can be concluded that the tracking error convergence is guaranteed for t → ∞ if the vehicle parameters are not known exactly.

8.1.3 Simplified Forms of Controllers GVC-VTCBF. Some reduced forms of the controller (8.2), which after transformation is described by (8.20) can be deduced. Two cases are indicated. 1. For a symmetric vehicle in the x y-plane one gets yg = 0; as a results the controller is simplified and reduced. Consequently, the impact of dynamic couplings effect is also reduced. 2. The matrix M is a diagonal one. It such case the simplified form is as follows:

8.1 Control Algorithms Expressed in IQV

τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + k D s + k I z,

163

(8.49)

because one obtains the identity matrix, i.e. ϒ = I. NGVC-VTCBF. Two particular cases of the controller (8.28) given in the form (8.43) can be shown. 1. For a symmetric vehicle in the x y-plane we get yg = 0; as a results the controller is simplified. Consequently, the impact of dynamic couplings effect is also reduced. 2. The matrix M is a diagonal one. It such case the simplified controller form is as follows: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + T k D s + k I z,

(8.50)

1

because one obtains = M 2 . Note that, if k D = δ I then also T k D = δ M. It can be observed that all inertial elements are taken into account in the control process. Consequently, the velocity controller gain matrix directly depends on the vehicle dynamics. Algorithms for Vehicle Model with 3 DOF Moving Horizontally. The algorithms can be also applied to a reduced planar model of the vehicle. The remark given in Chap. 6 is valid for these controllers.

8.2 Simulation Results At present some selected examples which illustrates the use of the considered in this section control algorithms are given. Simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

8.2.1 Velocity Tracking Control Using GVC 8.2.1.1

Controller for 3 DOF Fully Actuated Hovercraft

In Herman and Kowalczyk (2016b) a velocity tracking control algorithm for planar motion of an underwater vehicles was presented and its performance was shown. The results of the algorithm’s robustness test were also given. Objective: to show performance of the controller for a fully actuated hovercraft. In order to better show benefits of the velocity transformation the vectors were assumed as follows: νd = [u d , rd , vd ]T . The elements of the matrix N = diag {N11 , N22 , N33 } are: N11 = 10.0 kg, N22 = 0.5 kgm2 , N33 = 8.0 kg. The gains were selected to ensure acceptable velocity errors convergence, i.e.:

164

8 IQV Velocity Tracking Control in Body-Fixed Frame

k D = diag{5.0, 5.0, 5.0}, k I = diag{4.0, 4.0, 4.0},  = diag{7.0, 7.0, 7.0}.

(8.51)

Assuming equal control coefficient it is possible to determine effect of couplings for the velocity errors convergence. It results from that each variable tend to zero individually and with different overshoot. Based on the time response the effect of dynamical couplings in the system (i.e. from including of the matrix ϒ) can be shown. In Fig. 8.1a the desired velocity profiles are shown. It is assumed that only the rotation velocity is described by a function whereas the other velocities are constant. From Fig. 8.1b it is observed that for the surge velocity error converges to zero the longest. It is effect of weak dynamical couplings between u and r . The forces and the moment tend to small value very fast as it is shown in Fig. 8.1c. Robustness test. In the test the vehicle mass was equal m = 8.0 kg (it is reduced about 20 % because earlier was m = 10 kg). The elements of the matrix ϒ were the same as for the case of nominal parameters (they are calculated for the nominal set

Fig. 8.1 Simulation results for GVC controller and hovercraft (nominal parameters): a desired velocity profiles for u d , vd , and rd ; b velocity error convergence u = u d − u, v = vd − v in x and y direction, respectively, and orientation error convergence r = rd − r ; c control signals, i.e. forces τx , τ y , and moment τ N

8.2 Simulation Results

165

Fig. 8.2 Simulation results for GVC controller and hovercraft (robustness test): a velocity error convergence u = u d − u, v = vd − v in x and y direction, respectively, and orientation error convergence r = rd − r ; b control signals, i.e. forces τx , τ y , and moment τ N

of parameters because they are used in the control algorithm). Also the same set of control gains, namely (8.51) was applied. Comparing Fig. 8.2a–b with Fig. 8.1b–c it can be observed that both the velocity errors and the applied forces and the moment have similar values as previously. Hence, it can be concluded that the algorithm is robust to the mass change under the assumed working conditions.

8.2.1.2

Controller for 6 DOF Fully Actuated Underwater Vehicle

In Herman and Adamski (2017b) the GVC algorithm was proposed and used for control of an underwater vehicle. Objective: to show performance and the couplings effects based on the GVC velocity tracking controller for an underwater vehicle. For tracking the following desired velocity profile was assumed: π  π T  π t + 1, 0, 0.2 cos t + 0.2, 0, 0, 0.1 sin t . ν d = 0.5 sin 20 30 18 (8.52) The gain coefficients of the controller were selected to ensure realization of the control task: k D = diag{10, 10, 10, 10, 10, 10}, k I = diag{10, 10, 10, 10, 10, 10},  = diag{1.5, 1.5, 1.5, 1.5, 1.5, 1.5}.

(8.53)

Assuming the same gain values for each variable it is possible to determine the effect of dynamical couplings for velocity errors and detect which velocities are more sensitive to the couplings than others. Obviously the desired velocity profile is

166

8 IQV Velocity Tracking Control in Body-Fixed Frame

Fig. 8.3 Velocity profiles for GVC controller and underwater vehicle: a desired linear velocity profiles u d , vd , wd ; b desired angular velocity profiles pd , qd , rd

also important because there exist relationship between the vehicle dynamics and the realized trajectory (the couplings in the vehicle dynamics are included in the control gains). In Fig. 8.3 the desired velocity profiles are shown. Based on the velocity errors, the velocity integrals, and the input signals (forces and moments) the couplings effects in the vehicle can be determined. From Fig. 8.4a it is observed that significant changes concern the errors u and w. However, in Fig. 8.4b one can note that q is the most excited what results from the couplings in the vehicle. The observation is confirmed in Fig. 8.4c–d where the time history of artificial quantities z is given. It is because z u results from u and z q from q (they have maximal values for linear and angular variables, respectively). Figure 8.4e–f show that the applied forces τ X , τ Z and the moment τ M (related to u, w, and q, respectively) have high values.

8.2.1.3

Controller for 6 DOF Fully Actuated Airship

In Herman and Adamski (2017b) the GVC algorithm was also applied for an airship. Moreover, the robustness test was performed.

8.2.2 Velocity Tracking Control Using NGVC 8.2.2.1

Controller for 3 DOF Fully Actuated Hovercraft

The velocity tracking algorithm in terms of the NGVC for an underwater vehicle moving horizontally was shown and tested in Herman and Kowalczyk (2016a). Objective: to show performance of the controller for a hovercraft model.

8.2 Simulation Results

167

Fig. 8.4 Simulation results for GVC controller and underwater vehicle: a linear velocity errors; b angular velocity errors; c errors of z for linear velocities; d errors of z for angular velocities; e forces; f moments

168

8 IQV Velocity Tracking Control in Body-Fixed Frame

In this control algorithm the full dynamics of the vehicle is included in the velocity control gains. It is the main difference between the NGVC and the GVC algorithms because in the GVC controller the dynamical couplings are present in the gains only. The same conditions of work as for the GVC algorithm were assumed. In order to better show benefits of the velocity transformation we assume the vectors as follows: ν d = [u d , rd , vd ]T . The elements of the matrix N = diag {N11 , N22 , N33 } are as follows: N11 = 10.0 kg, N22 = 0.5 kgm2 , N33 = 8.0 kg. For realization velocity control the following gains set (ensuring acceptable errors convergence) was applied: k D = diag{3.0, 3.0, 3.0}, k I = diag{3.0, 3.0, 3.0},  = diag{5.0, 5.0, 5.0}.

(8.54)

The control gains matrices arise from that in the NGVC algorithm the full dynamics of the vehicle is present and the control gains in (8.54) serve rather for tuning. The desired velocity profiles for u d , vd , and rd are presented in Fig. 8.1a. From Fig. 8.5a it can be seen that the velocity errors converge to zero quickly. Figure 8.5b show that the forces and moment achieve the end values in a short period of time. Comparing the figures with Fig. 8.1b–c it is noticeable that the velocity errors convergence time has been shortened. Moreover, effects of vehicle dynamics are observed in the u and r time history. Robustness test. In the test we assumed the vehicle mass is equal m = 8.0 kg (it is reduced by about 20 % compared to m = 10 kg). The control gains were the same, namely (8.54). Comparing Fig. 8.5a–b with Fig. 8.6a–b one can see that the changes in time history are not great. This fact leads to conclusion that for the 3 DOF tested vehicle the NGVC velocity tracking algorithm is robust (to some extent) to dynamical changes.

Fig. 8.5 Simulation results for NGVC controller and hovercraft (nominal parameters): a velocity error convergence u = u d − u, v = vd − v in x and y direction, respectively, and orientation error convergence r = rd − r ; b control signals, i.e. forces τx , τ y , and moment τ N

8.2 Simulation Results

169

Fig. 8.6 Simulation results for NGVC controller and hovercraft (robustness test): a velocity error convergence u = u d − u, v = vd − v in x and y direction, respectively, and orientation error convergence r = rd − r ; b control signals, i.e. forces τx , τ y , and moment τ N

8.2.2.2

Controller for 6 DOF Fully Actuated Airship

The NGVC control algorithm was proposed and its performance for a fully actuated indoor airship together with a robustness test were presented in Herman and Adamski (2017a).

8.3 Closing Remarks The velocity tracking controllers presented in this chapter ensure the velocity error convergence to zero in the finite time. They can be applied to various vehicles, namely underwater vehicles, hovercraft or indoor airships. In this part of book only non-adaptive version of the velocity tracking algorithms in terms of the IQV were considered what means that disturbances arising from environment were not taken into account. It was shown that the algorithms are robust (to a limited extent) to parameters changes. The conducted on a 3 DOF reduced model as well as on a 6 DOF full model simulations (including the published results not presented here) confirmed satisfactory performance of the velocity controllers in terms of the IQV.

References Breivik M (2010) Topics in guided motion control of marine vehicles. PhD Thesis Norwegian University of science and technology faculty of information technology, mathematics and electrical engineering department of engineering cybernetics, Trondheim, June

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Breivik M, Hovstein VE, Fossen TI (2008) Straight-line target tracking for unmanned surface vehicles. Model Identif Control 29(4):131–149 Ferreira B, Matos A, Cruz N, Pinto M (2010) Modeling and control of the MARES autonomous underwater vehicle. Mar Technol Soc J 44(2):19–36 Ferreira B, Pinto M, Matos A, Cruz N (2009) Control of the MARES autonomous underwater vehicle. In: Proceedings of OCEANS 2009, Biloxi, MS, USA 26–29 October, pp 1–10 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Fossen TI, Fjellstad OE (1993) Cascaded adaptive control of ocean vehicles with significant actuator dynamics. In: Proceedings of the IFAC world congress, Sydney, Australia, 18–23 July, pp 1123– 1128 Li Y, Wang L, Liao Y, Jiang Q, Pan K (2018) Heading MFA control for unmanned surface vehicle with angular velocity guidance. App Ocean Res 80:57–65 Herman P (2009) Transformed equations of motion for underwater vehicles. Ocean Eng 36:306–312 Herman P, Adamski W (2017a) Velocity controller for a class of vehicles. Found Comput Decis Sci 42(1):43–58 Herman P, Adamski W (2017b) Non-adaptive velocity tracking controller for a class of vehicles. Bull Pol Acad Sci Techn Sci 65(4):459–468 Herman P, Kowalczyk W (2016a) Velocity tracking control of AUVs in horizontal motion. In: Proceedings of the 2016 3rd conference on control and fault-tolerant systems (SysTol), Barcelona, Spain, Sept 7–9, pp 105–110 Herman P, Kowalczyk W (2016b) Velocity tracking controller for planar motion of underwater vehicles. In: Proceedings of the 2016 3rd conference on control and fault-tolerant systems (SysTol), Barcelona, Spain, Sept 7–9, pp 139–144 Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs Yoon SM, Hong S, Park SJ, Choi JS, Kim HW, Yeu TK (2012) Track velocity control of crawler type underwater mining robot through shallow-water test. J Mech Sci Technol 26(10):3291–3298

Chapter 9

IQV Position and Velocity Tracking Control in Earth-Fixed Frame

Abstract This chapter focuses on the non-adaptive nonlinear control methods which serve for the position and velocity tracking of vehicles described in terms of the IQV. The presented algorithms guarantee the position and velocity error convergence in the finite time. In contrast to previously discussed algorithms the controllers are realized in the Earth-Fixed Representation. The robustness analysis to vehicle parameters changes, the input control signals, and some examples of the simplified controllers (including a 3 DOF horizontally moving vehicle model) are also shown. Validation of the controllers is done on a 6 DOF model of fully actuated underwater vehicle (GVC based algorithm) and of fully actuated indoor airship (NGVC based algorithm).

9.1 Control Algorithms Expressed in IQV Two types of algorithms using the IQV, namely the GVC based control algorithm and the NGVC based algorithm are considered here. Their simplified forms are discussed at the end of the section.

9.1.1 Position and Velocity Tracking Controller Expressed in GVC An algorithm of this type for a 6 DOF airship model was considered in Herman and Adamski (2019, 2020). Such a trajectory tracking algorithm using GVC in general form expressed in the Earth-Fixed Frame can be formulated in the theorem. Theorem 9.1 Consider the vehicle model fulfilled equations of motion (2.68), (2.69), and (2.24) together with the following controller: π η = Nη (η)ξ˙ r + Cξ (ξ , η)ξ r + Dξ (ξ , η)ξ r + gη ξ (η) ˜ +k D sξ + ZT (η)k P η, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_9

(9.1) 171

172

9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

in which: ˜ ξ r = Z−1 (η)(η˙ d + η), −1 ˙˜ ˜ + Z−1 (η)(¨ηd + η), ξ˙ r = Z˙ (η)(η˙ d + η) −1 ˜ sξ = ξ r − ξ = Z (η)(η˙˜ + η), −1 ˙˜ ˜ + Z−1 (η)(η¨˜ + η), s˙ξ = ξ˙ r − ξ˙ = Z˙ (η)(η˙˜ + η)

(9.2) (9.3) (9.4) (9.5)

˙ and η¨˜ = η¨ d − η¨ are the position error vector, the where η˜ = ηd − η, η˙˜ = η˙ d − η, velocity error vector, and the acceleration error vector, respectively, k D = k TD > 0, k P = k TP > 0,  = T > 0, and Nη (η) is a diagonal strictly positive matrix. Moreover, θ = ±90 deg. Then the equilibrium point [sξT , η˜ T ]T = 0 is exponentially stable. Consequently, η˜ → 0 and η˙˜ → 0 as t → ∞. Comment on Theorem 9.1. In the Body-Fixed Frame the formula applies τ = Bu, where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Using Eqs. (2.46) and (2.74) the relationship between the input signal in the Earth-Fixed Frame and in the BodyFixed Frame can be calculated as follows: τ η = Z−T (η)π η ,

τ = JT (η)Z−T (η)π η ,

u = B† τ .

(9.6)

Moreover, Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid for this controller. In Qiao and Zhang (2019) (based on the results of Zhang et al. 2015) it has been noted that when the saturation effect of thrust is not serious, then it is reasonable to assume that the control input τ η is bounded. Such a statement implies that simple trajectories must be selected which require that the forces only occasionally violate the thrust saturation limits. Proof The closed-loop system (2.68), (2.69) together with the controller (9.1) can be given in the form: Nη (η)˙sξ + [Cξ (ξ , η) + Dξ (ξ , η) + k D ]sξ + ZT (η)k P η˜ = 0.

(9.7)

The following formula is proposed as a Lyapunov function candidate: ˜ = L(sξ , η)

1 T 1 ˜ sξ Nη (η)sξ + η˜ T k P η. 2 2

(9.8)

The time derivative of the function L (9.8) is: 1 ˙ ˙˜ T k P η. ˙ ξ , η) ˜ ˜ = sξT Nη (η)˙sξ + sξT N L(s η (η)sξ + η 2 By recalling (2.70) it can be calculated:

(9.9)

9.1 Control Algorithms Expressed in IQV

173

˙ η (η) = d (ZT (η)Mη (η)Z(η)) N dt T ˙ ˙ η (η)Z(η) + ZT (η)Mη (η)Z(η). ˙ = Z (η)Mη (η)Z(η) + ZT (η)M

(9.10)

Applying now (9.7) one obtains: 1 ˙ ˙ ξ , η) ˜ = sξT [−Cξ (ξ , η)sξ − Dξ (ξ , η)sξ − k D sξ ] + sξT N L(s η (η)sξ 2 T ˜ −sT ZT (η)k P η˜ + η˜˙ k P η. ξ

(9.11)

Substituting (2.71) and (9.10) to (9.11) one gets: T ˙ ξ , η) ˙ ˜ = sξT [−ZT (η)Mη (η)Z(η)s L(s ξ − Z (η)Cη (ν, η)Z(η)sξ − Dξ (ξ , η)sξ  1 ˙T ˙ η (η)Z(η)+ZT (η)Mη (η)Z(η) ˙ Z (η)Mη (η)Z(η)+ZT (η)M −k D sξ + sξ ] 2   T −sT ZT (η)k P η˜ + η˜˙ k P η˜ = −sT Dξ (ξ , η) + k D sξ ξ

ξ

1 1 T ˙ ˙ +sξT [ ZT (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η) + Z˙ (η)Mη (η)Z(η) 2 2   1 ˙ η (η) − 2Cη (ν, η) Z(η)]sξ − sξT ZT (η)k P η˜ + η˙˜ T k P η. ˜ + ZT (η) M 2

(9.12)

Using Eq. (2.48) the above expression is simplified, i.e:   T ˙ ξ , η) ˜ = −sξT Dξ (ξ , η) + k D sξ − sξT ZT (η)k P η˜ + η˙˜ k P η˜ L(s 1 T ˙ + sξT [Z˙ (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η)]s ξ. 2

(9.13)

˙ is a It can be proved that the matrix W = [Z˙ (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η)] skew-symmetric one, and as a result the last term of (9.13) vanishes.  T

˙ ˙ T (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η) is skewLemma 9.1 The matrix W = Z T symmetric, i.e. W = −W. Proof Calculating the matrix WT one gets: T T ˙ WT = (Z˙ (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η)) T ˙ = (Z˙ (η)Mη (η)Z(η))T − (ZT (η)Mη (η)Z(η)) T

˙ = ZT (η)MηT (η)Z(η) − Z˙ (η)MηT (η)Z(η). T

One has also:

(9.14)

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9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

˙ −W = −(Z˙ (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η)) T

T ˙ = −Z˙ (η)Mη (η)Z(η) + ZT (η)Mη (η)Z(η).

(9.15)

Recalling (2.47) and comparing (9.14) with (9.15) it can be concluded that: WT = −W,

(9.16)

˙ (η)Mη (η)Z(η) − ZT (η) which shows the skew-symmetry of the matrix W = [Z ˙  Mη (η)Z(η)]. T

˙ ξ , η) ˜ as follows: Therefore, taking into account (9.4) one gets L(s   T ˙ ξ , η) ˜ = −sξT Dξ (ξ , η) + k D sξ − sξT ZT (η)k P η˜ + η˙˜ k P η˜ L(s   ˜ = −sξT Dξ (ξ , η) + k D sξ − η˜ T T k P η.

(9.17)

Defining the vector χ e = [sξT , η˜ T ]T equation (9.17) using (2.72) can be written in the form: ˙ ξ , η) ˜ = −χ eT Aχ e , L(s

(9.18)

where:  ZT (η)Dη (ν, η)Z(η) + k D 0 . A= 0 T k P 

(9.19)

Recalling (2.49) it is noted that the matrix A is positive definite. Thus, assuming that λmin {A} > 0 it is possible to find an upper bound of the time derivative, i.e.: ˙ χ e ) ≤ −λmin {A} χ e 2 , L(t,

(9.20)

for all t ≥ 0 and χ e ∈ R 2N . At present, based on the Lyapunov direct method (Slotine and Li 1991), the conclusion that the state space origin of the system (2.68), (2.69) together with the controller (9.1):  lim

t→∞

 sξ (t) = 0, ˜ η(t)

(9.21)

is exponentially convergent can be made. Earth-Frame vehicle input signal. Define the reference velocity η˙ r , its first time derivative η¨ r , and the vectors s, and s˙, i.e.:

9.1 Control Algorithms Expressed in IQV

175

˜ η˙ r = η˙ d + η, ˙˜ η¨ r = η¨ d + η, ˜ s = η˙˜ + η, ˙˜ s˙ = η¨˜ + η.

(9.22) (9.23) (9.24) (9.25)

After transformation into the external space, namely after inversion of (2.74), one gets the following control input: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) ˜ +Z−T (η)k D Z−1 (η) s + k P η,

(9.26)

By comparison of Eqs. (9.1) and (9.26), and using (9.4) the relationships can be noticed: s = Z(η)sξ ,

˙ s = η˙ r − η,

s˙ = η¨ r − η¨ .

(9.27)

It means that in (9.26) the control gain instead of the matrix k D includes the matrix Z−T (η)k D Z−1 (η). Because the matrix Z(η) changes during the movement of the vehicle so one gets information on how the dynamical couplings change when the vehicle is moving.

9.1.1.1

Robustness Analysis

Consider the vehicle model with not exactly known parameters. Denote, for simplification, that Mη = Mη (η), Cη = Cη (ν, η), Dη = Dη (ν, η), gη = gη (η), and Zη = Z(η). The robustness problem can be considered in the following way. Equation (9.26) can be given in the form: −1 ˜ τ η = Mη η¨ r + Cη η˙ r + Dη η˙ r + gη + Z−T η k D Zη s + k P η.

(9.28)

Moreover, the symbols M∗η = M∗η (η), C∗η = C∗η (ν, η), D∗η = D∗η (ν, η), and g∗η = g∗η (η) are introduced. They mean the matrices and vectors including the known and unknown parameters. Then, the unknown parts of these matrices are denoted as: ˜ η = C∗η − Cη , D ˜ η = D∗η − Dη , g˜ η = g∗η − gη . It is assumed that ˜ η = M∗η − Mη , C M ∗ the matrix Mη is symmetric but its elements are not constant. Taking into account the relationships (9.27) (for Z∗η ) the Lyapunov function candidate using (2.67) can be proposed as follows: L=

1 1 1 1 T ∗ ˜ sξ Nη sξ + η˜ T k P η˜ = sT M∗η s + η˜ T k P η. 2 2 2 2

The time derivative of this function is:

(9.29)

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9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

1 ˙∗ ˙˜ T k P η˜ L˙ = sT M∗η s˙ + sT M ηs + η 2 1 ˙ T ∗ ˙˜ T k P η. ˜ = s (Mη η¨ r − M∗η η¨ ) + sT M ηs + η 2

(9.30)

From (9.27) it follows that η˙ = η˙ r − s. Remembering Eq. (2.41) one can write: M∗η η¨ = τ η − C∗η (η˙ r − s) − D∗η (η˙ r − s) − g∗η .

(9.31)

Inserting (9.31) into (9.30) one gets: L˙ = sT (M∗η η¨ r + C∗η η˙ r + D∗η η˙ r + g∗η − C∗η s − D∗η s − τ η ) 1 ˙∗ ˙˜ T k P η˜ = sT (M∗ η¨ r + C∗ η˙ r + D∗ η˙ r + g∗ − D∗ s − τ η ) + sT M η η η η η ηs + η 2 1 ˙ ∗η − 2C∗η )s + η˙˜ T k P η. ˜ + sT (M (9.32) 2 The nominal control input signal is defined by (9.28), where the parameters in Mη , −1 Cη , Dη , gη , Z−T η , and Zη are known. Using property (2.48) and inserting (9.28) into (9.32) one obtains: L˙ = sT (M∗η η¨ r + C∗η η˙ r + D∗η η˙ r + g∗η − D∗η s − Mη η¨ r − Cη η˙ r − Dη η˙ r − gη T −1 ˜ + η˙˜ k P η˜ = sT [(M∗η − Mη )¨ηr + (C∗η − Cη )η˙ r −Z−T η k D Zη s − k P η) T −1 ˜ + η˙˜ k P η. ˜ +(D∗η − Dη )η˙ r + g∗η − gη − D∗η s + Z−T η k D Zη s + k P η]

(9.33)

Next, grouping the terms one has: −1 ˜ η η¨ r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s) − sT (Dη + Z−T L˙ = sT (M η k D Zη )s

˜ −sT k P η˜ + η˙˜ k P η. T

(9.34)

T Recalling the formula (9.24) one determines sT = η˙˜ + η˜ T T . Taking now into consideration that for the nominal controller signal one has s = Zη sξ it can be written that:

˜ η η¨ r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s) L˙ = sT (M −1 ˙˜ T + η˜ T T )k P η˜ + η˙˜ T k P η˜ −sT (Dη + Z−T η k D Zη )s − (η

˜ η η¨ r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s)] = sξT [ZηT (M ˜ −sξT (ZηT Dη Zη + k D )sξ − η˜ T T k P η.

(9.35)

Based on Slotine and Li (1991) it can be concluded that it is possible to find a strictly positive constants βi where i = 1, . . . , 6 to ensure convergence of the tracking

9.1 Control Algorithms Expressed in IQV

177

   ˜ η η¨ r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s)]i  and error to zero. Thus, choosing βi ≥  [ZηT (M assuming that k D and T k P are symmetric or diagonal matrices, one obtains: L˙ ≤ −

6

˜ βi |sξ i | − sξT (ZηT Dη Zη + k D )sξ − η˜ T T k P η.

(9.36)

i=1

The constant βi must guarantee that the derivative of the Lyapunov function is seminegative. If the nominal parameters of the system and the assumed reference trajectories are known then for a given value of uncertainty one can evaluate this parameter βi . In other case its value can be chosen in conservative manner, i.e. as high as possible but to ensure stability of the system under the controller. The second k D is selected to ensure  solution relies on that in (9.35) the matrix   T ˜  ˜ ˜ ˜ λmin (k D ) ≥ Zη (Mη η¨ r + Cη η˙ r + Dη η˙ r + g˜ η − Dη s). From the condition (9.36) it can be concluded that the tracking error convergence is guaranteed for t → ∞ if the vehicle parameters are not known exactly.

9.1.2 Position and Velocity Tracking Controller Expressed in NGVC This type of algorithm for a 6 DOF vehicle model was shown and discussed first time in Herman and Adamski (2017). In the given below theorem the control problem in terms of the NGVC is considered. Next the extended proof is shown. Theorem 9.2 Consider the vehicle model fulfilled equations of motion (2.95), (2.94), and (2.24) together with the following controller: ˜ η = ζ˙ r + Cζ (ζ , η)ζ r + Dζ (ζ , η)ζ r + ge ζ (η) + k D sζ + −T (η)k P η,

(9.37)

in which: ˜ ζ r = (η)(η˙ d + η), ˙˜ ˙ ˙ζ r = (η)( ˜ + (η)(¨ηd + η), η˙ d + η) ˜ sζ = ζ r − ζ = (η)(η˙˜ + η), ˙ ˙˜ ˙ ˙ ˙ ˜ + (η)(η¨˜ + η), s˙ζ = ζ − ζ = (η)(η˜ + η) r

(9.38) (9.39) (9.40) (9.41)

˙ η¨˜ = η¨ d − η¨ are the position error vector, the velocwhere η˜ = ηd − η, η˙˜ = η˙ d − η, ity error vector, and the acceleration error vector, respectively, k D = k TD > 0, k P = k TP > 0,  = T > 0. Moreover, θ = ±90 deg. Then the equilibrium point [sζT , η˜ T ]T = 0 is exponentially stable. Consequently, η˜ → 0 and η˙˜ → 0 as t → ∞.

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9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

Comment on Theorem 9.2. In the Body-Fixed Frame the formula applies τ = Bu, where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0. Using Eqs. (2.46) and (2.99) the input signal in the Earth-Fixed Frame and in the Body-Fixed Frame can be calculated as follows: τ η = T (η) η ,

τ = JT (η)T (η) η ,

u = B† τ .

(9.42)

Moreover, Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid for this controller. Consider some relationships. Taking into account (9.37), (9.24), and (9.40) one obtains: sζ = (η)s,

˙ s = η˙ r − η,

s˙ = η¨ r − η¨ .

(9.43)

Proof The closed-loop system (2.94), (2.95) together with the controller (9.37) can be written in the form: s˙ζ + [Cζ (ζ , η) + Dζ (ζ , η) + k D ]sζ + −T (η)k P η˜ = 0.

(9.44)

The following expression as the Lyapunov function candidate is proposed: ˜ = L(sζ , η)

1 T 1 ˜ sζ sζ + η˜ T k P η. 2 2

(9.45)

Next, the time derivative of L (9.45) is calculated and then one gets: T ˙ ζ , η) ˜ ˜ = sζT s˙ζ + η˙˜ k P η. L(s

(9.46)

Using the formula (9.44) one has: ˙ ζ , η) ˜ = sζT [−Cζ (ζ , η)sζ − Dζ (ζ , η)sζ − k D sζ ] L(s ˜ −sζT −T (η)k P η˜ + η˙˜ k P η. T

(9.47)

Below it will be shown that the property sζT Cζ (ζ , η)sζ = 0 is true. sζT Cζ (ζ , η)sζ

Lemma 9.2 There is true the relationship T (η)Cζ (ζ , η)(η) is a skew-symmetric one.



= 0 because the matrix

Proof Using (2.96) and (9.43) one can write: −1 ˙ (η)(η)s sζT Cζ (ζ , η)sζ = sT T (η)[−T (η)Cη (ν, η) − (η)] T T ˙ = s [Cη (ν, η) −  (η)(η)]s.

The above expression is rewritten as follows:

(9.48)

9.1 Control Algorithms Expressed in IQV

179

1 ˙T ˙ sζT Cζ (ζ , η)sζ = sT [Cη (ν, η) − T (η)(η) +  (η)(η) 2 1 ˙T 1 ˙ −  (η)(η)]s = sT [Cη (ν, η) − T (η)(η) 2 2 1 1 ˙T 1 ˙T ˙ − T (η)(η) +  (η)(η) −  (η)(η)]s. 2 2 2

(9.49)

˙ η (η) =  ˙ T (η)(η) + T (η) Calculating the time derivative of (2.90) one gets M ˙ (η). Rearranging now (9.49) one has: 1 ˙ η (η) − 2Cη (ν, η)]s sζT Cζ (ζ , η)sζ = − sT [M 2 1 ˙ T (η)(η) − T (η)(η)]s. ˙ + s T [ 2

(9.50)

Recalling property (2.48) one can observe that the first term is equal zero. Thus: sζT Cζ (ζ , η)sζ =

1 T ˙T ˙ s [ (η)(η) − T (η)(η)]s. 2

(9.51)

˙ T (η)(η) − T (η)(η) ˙ It can be shown that the matrix W =  is skew-symmetric, T T i.e. W = −W. Calculating the matrix W one obtains: T ˙ (η)(η) − T (η)(η)) ˙ ˙ ˙ (η)(η). = T (η)(η) − WT = ( T

T

(9.52)

It is also: ˙ ˙ ˙ (η)(η). ˙ (η)(η) − T (η)(η)) = T (η)(η) − − W = −( T

T

(9.53)

Comparing (9.52) with (9.53) one gets: WT = −W,

(9.54)

˙ T (η)(η) − T (η)(η)]. ˙ which shows the skew-symmetry of the matrix [ Taking into account (9.51) it can be concluded that: sζT Cζ (ζ , η)sζ = 0, which ends the proof.

(9.55) 

Therefore, using (9.40) Eq. (9.47) can be written in the form:   ˙ ζ , η) ˜ = −sζT Dζ (ζ , η) + k D sζ L(s  T T ˜ ˜ − (η)(η˙˜ + η) −T (η)k P η˜ + η˙˜ k P η.

(9.56)

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9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

The formula (9.56) can be simplified as follows:   ˙ ζ , η) ˜ = −sζT Dζ (ζ , η) + k D sζ − η˜ T T k P η. ˜ L(s

(9.57)

Defining the vector χ e = [sζT , η˜ T ]T the Eq. (9.57) can be rewritten in the following form (using (2.97)): ˙ ζ , η) ˜ = −χ eT Aχ e , L(s

(9.58)

where:  −T (η)Dη (ν, η)−1 (η) + k D 0 . A= 0 T k P 

(9.59)

Taking into account (2.49) it is observed that the matrix A is positive definite. Assuming that λmin {A} > 0 one can find an upper bound of the time derivative, i.e.: ˙ χ e ) ≤ −λmin {A} χ e 2 , L(t,

(9.60)

for all t ≥ 0 and χ e ∈ R 2N . Using the Lyapunov direct method (Slotine and Li 1991), the conclusion that the state space origin of the system (2.94), (2.95) together with the controller (9.37):  lim

t→∞

 sζ (t) = 0, ˜ η(t)

(9.61)

is exponentially convergent can be made. Earth-Frame vehicle input signal. After transformation, i.e. after inversion of (2.99), and using (9.22)–(9.24) one gets the control input in the Earth-Fixed Frame: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) ˜ +T (η)k D (η)s + k P η.

(9.62)

The term T (η)k D (η)s contains the velocity gain matrix, namely K D = T (η) k D (η) which takes into account dynamics of the vehicle. For example when it is assumed that k D = δ I (where δ is a constant) then K D = δ T (η)(η) because T (η)(η) includes the inertia matrix. Consequently, the gain matrix is selected according to the dynamical parameters of the system and the input signal τ η is strictly related to the kinematics as well as the dynamics of the vehicle. The matrix k D serves rather for tuning the control gains. An additional benefit relies on that if the system parameters are not exactly known (or approximately known) then the term containing (η) causes that the tracking velocity error tends to zero quickly because in the regulation process the dynamics of the vehicle is included.

9.1 Control Algorithms Expressed in IQV

181

By comparison of Eqs. (9.37) and (9.62), and using (9.40) the following relationships can be observed: sζ = (η)s,

˙ s = η˙ r − η,

s˙ = η¨ r − η¨ ,

(9.63)

which is equivalent to (9.43).

9.1.2.1

Robustness Analysis

Consider the robustness issue, namely taking the vehicle model with not exactly known parameters. Denote, for simplification, that Mη = Mη (η), Cη = Cη (ν, η), Dη = Dη (ν, η), gη = gη (η), and Zη = Z(η). Recall also (9.62) and relationships (9.22)–(9.25). Consequently, Eq. (9.62) can be written in the following form: ˜ τ η = Mη η¨ r + Cη η˙ r + Dη η˙ r + gη + ηT k D η s + k P η.

(9.64)

The following symbols are applied: M∗η = M∗η (η), C∗η = C∗η (ν, η), D∗η = D∗η (ν, η), and g∗η = g∗η (η). They define the matrices and vectors including the known and ˜η= unknown parameters. The unknown parts of these matrices are denoted as: M ∗ ∗ ∗ ∗ ˜ ˜ Mη − Mη , Cη = Cη − Cη , Dη = Dη − Dη , g˜ η = gη − gη . It is assumed that the matrix M∗η is symmetric and its elements depend on η. Decomposition of this matrix leads to sζ = ∗η s instead of (9.63). The proposed Lyapunov function candidate, using (2.90), is as follows: L=

1 1 1 1 T ˜ sζ sζ + η˜ T k P η˜ = sT M∗η s + η˜ T k P η. 2 2 2 2

(9.65)

Note that the obtained function is the same as (9.29). Therefore, its time derivative is described by (9.30). The time derivative L˙ in the form (9.32) one obtains applying (9.31). The nominal control input in the following form is defined by (9.64), where the parameters in Mη , Cη , Dη , gη , ηT , and η are known. Making use of the property (2.48) and inserting (9.64) into L˙ (9.32) one gets: L˙ = sT (M∗η η¨ r + C∗η η˙ r + D∗η η˙ r + g∗η − D∗η s − Mη η¨ r − Cη η˙ r T ˜ + η˙˜ k P η˜ = sT [(M∗η − Mη )¨ηr −Dη η˙ r − gη − ηT k D η s − k P η)

+(C∗η − Cη )η˙ r + (D∗η − Dη )η˙ r + g∗η − gη − D∗η s + ηT k D η s T ˜ + η˙˜ k P η. ˜ +k P η]

Grouping the terms one can write:

(9.66)

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9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

˜ η η¨ r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s) L˙ = sT (M T T T ˜ −s (Dη +  k D η )s − s k P η˜ + η˜˙ T k P η. η

(9.67)

T Recalling (9.24) one obtains sT = η˙˜ + η˜ T T . Consequently, taking into account that for the nominal controller signal it is sζ = η s one has:

˜ η η¨ r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s) L˙ = sT (M T T −sT (Dη + ηT k D η )s − (η˙˜ + η˜ T T )k P η˜ + η˙˜ k P η˜

˜ ¨r + C ˜ η η˙ r + D ˜ η η˙ r + g˜ η − D ˜ η s)] = sζT [−T η (Mη η −1 ˜ ˜ T T k P η. −sζT (−T η Dη η + k D )sζ − η

(9.68)

As it can be concluded from Slotine and Li (1991) a strictly positive constants βi where i = 1, . . . , 6 can error to zero.   be found to ensure convergence of the tracking   −T ˜ ˜ ˜ ˜ Thus, choosing βi ≥  [η (Mη¨ r + Cη˙ r + Dη˙r + g˜ − Dη s)]i  one gets, assuming that k D and T k P are symmetric or diagonal matrices, the following inequality: L˙ ≤ −

6

−1 ˜ ˜ T T k P η. βi |sζ i | − sζT (−T η Dη η + k D )sζ − η

(9.69)

i=1

The constant βi must guarantee that the derivative of the Lyapunov function is seminegative. If the nominal parameters of the system and the assumed reference trajectories are known then for a given value of uncertainty we can evaluate this parameter βi . In other case we choose its value in conservative manner, i.e. as high as possible but to ensure stability of the system under the controller. The second  solution relies on that in (7.28) the matrix  k D is selected to ensure  −T ˜  ˜ ˜ ˜ λmin (k D ) ≥ η (Mη η¨ r + Cη η˙ r + Dη η˙ r + g˜ η − Dη s). From the condition (9.69) it can be concluded that the tracking error convergence is guaranteed for t → ∞ if the vehicle parameters are not known exactly.

9.1.3 Simplified Forms of Controllers GVC-PVTCEF. Because the matrix Mη (η) is usually a symmetric one (it is assumed that this condition is fulfilled), then it is possible to determine the diagonal matrix Nη (η). However, in particular cases (e.g. for hovercraft), one has a diagonal matrix Mη (η) that implies Mη (η) = Nη (η). Consequently, Z(η) = I and instead of (9.26) one obtains:

9.1 Control Algorithms Expressed in IQV

183

τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) ˜ ˜ + k P η, +k D (η˜˙ + η)

(9.70)

which means that the coupling effects are not present in the gain matrix. However, the controller can be applied too. NGVC-PVTCEF. The matrix Mη (η) may be a diagonal one instead of a symmetric one, then it is possible to determine the diagonal matrix Nη (η). In particular cases (e.g. for hovercraft), one has a diagonal matrix Mη (η) from which it follows that 1

Mη (η) = Nη (η). Consequently, Z(η) = I and (η) = Nη2 (η) and instead of (9.62) one gets: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) 1

1

˜ ˜ + k P η, +Nη2 (η)k D Nη2 (η)(η˙˜ + η)

(9.71)

which means that the coupling effects are not present in the gain matrix. In spite of that, the controller can be also used. Algorithms for Vehicle Model with 3 DOF Moving Horizontally. The algorithms may be applied to a reduced planar model of the vehicle. The remark given in Chap. 6 is valid also for these controllers.

9.2 Simulation Results In this section the selected results concerning both control algorithms are shown. Simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

9.2.1 Trajectory Tracking Control Using GVC 9.2.1.1

Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the GVC algorithm for control using dynamical couplings and for couplings investigation in the vehicle model. For tracking it is assumed the following desired linear 3D trajectory and angular constant values as well as the appropriate velocity profile, respectively:

π π T π , − , − , ηd = 0.3 t − 2.0, 0.1 t, 0.2 t, 12 5 12 η˙ d = [0.3, 0.1, 0.2, 0, 0, 0]T .

(9.72) (9.73)

184

9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

The gain coefficients of the controller were selected as follows (to ensure acceptable position and velocity errors convergence): k D = diag{1500, 1500, 1500, 1500, 1500, 1500}, k P = diag{2000, 2000, 2000, 2000, 2000, 2000},  = diag{0.05, 0.05, 0.05, 0.01, 0.01, 0.01}.

(9.74)

The matrices k D and k P have large values because in the algorithm only couplings (not full dynamics) are taken into account. In contrast, values of  are small what means that realization of the tracking task depends mainly on the position and velocity errors gains. Moreover, if the algorithm in the in Earth-Fixed Frame is applied then usually large gains coefficients are needed. Saturation values were: |τη | ≤ 1500 N or Nm. For this set of control gains and nominal parameters the results are shown in Fig. 9.1. It is noticeable that the first phase of the movement takes 4 s (quick reaction of the system). As shown in Fig. 9.1a, the algorithm works correctly. Taking into account Fig. 9.1b and c the effect of dynamical couplings is observed for y and z (excitation of variables) and φ (fast error convergence). From Fig. 9.1d it follows that because of couplings not only velocity d x/dt has big values but also linear velocities dy/dt and dz/dt are excited (they should be equal zero if the couplings are absent). Moreover, as it results from Fig. 9.1e that the largest changes concern the variable dθ/dt. However, the dynamic couplings also cause a sudden increase in variables dφ/dt and dψ/dt. As is observed in Fig. 9.1f in due to the large distance from the trajectory, the force τηX is also large when the vehicle starts. High values of the force τηZ are related to the movement in the vertical direction. Due to the increase in angular position errors and angular velocity errors in the initial phase of the movement, there is a large increase in the moments as shown in the Fig. 9.1g. Robustness investigation. In the next simulation robustness of the GVC control algorithm was tested using the same set of gains as for the vehicle with nominal parameters. The dynamical parameters of the vehicle and environmental parameters (damping terms and added masses) differ by 5% from nominal (they are greater). Elements of M and D are bigger by 5%. Despite the fact that Fig. 9.2a shows that the trajectory is tracked correctly, in Fig. 9.2b–c it is seen that not all linear and angular position errors tend to zero. It is effect of couplings which make worse the effectiveness of the controller. However, the linear and angular velocities as shown in the Fig. 9.2d–e are tracked with sufficient accuracy. The forces and moments (Fig. 9.2f–g) have similar values as for the vehicle when we know its dynamic parameters. It can be concluded that the algorithm allows velocity tracking if the parameters are not known exactly. Unfortunately, the position tracking is not accurate in this case.

9.2 Simulation Results

185

Fig. 9.1 Simulation results for GVC controller and underwater vehicle (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

186

9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

Fig. 9.2 Simulation results for GVC controller and underwater vehicle (changed parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

9.2 Simulation Results

187

9.2.2 Trajectory Tracking Control Using NGVC 9.2.2.1

Controller for 6 DOF Fully Actuated Airship

In reference Herman and Adamski (2017) the NGVC controller expressed in the Earth-Fixed Frame was proposed and tested both for nominal and changed parameters set. Objective: to show performance of the NGVC control algorithm for an indoor airship model. For tracking it is assumed the following desired linear 3D trajectory and the appropriate velocity profile, respectively, i.e. (9.72)–(9.73). The gain coefficients of the controller were selected as follows (to ensure acceptable errors convergence): k D = diag{40, 40, 40, 160, 160, 160}, k P = diag{70, 70, 70, 70, 70, 70},  = diag{0.2, 0.2, 0.2, 0.2, 0.2, 0.2}.

(9.75)

For this set of control gains and nominal parameters the results are shown in Fig. 9.3. Moreover saturation was assumed for |τη | ≤ 150 N or Nm. Realization of the tracking task for the airship model is done using smaller gain matrices values than for the earlier considered underwater vehicle. It results from the vehicle dynamics. Also the differences in the time history of variables arise from the same reason. From Fig. 9.3a it follows that the desired trajectory is tracked after some time. All linear position errors are close to zero after about 15 s, and y and z are only slightly excited (Fig. 9.3b). The angular position errors do not converge in the same time as shown in Fig. 9.3c. The angular error θ tends to zero more slowly than other errors. It means that the variable θ is less coupled with other variables. The linear velocity errors are dynamically coupled (dynamics guarantees fast error convergence as it is observed in Fig. 9.3d. The angular velocity errors dφ/dt and dψ/dt decrease quickly to the presence of dynamics in control gain matrices (Fig. 9.3e). Contrary, the error dθ/dt tends to zero slowly (weak couplings between the velocity θ˙ and other velocities). Note that the desired velocity is equal to zero. From Fig. 9.3f–g it is seen that the maximal values of forces at moments is in the initial phase of motion. Remark Using the same gains set the CL controller does not work at all.

188

9 IQV Position and Velocity Tracking Control in Earth-Fixed Frame

Fig. 9.3 Simulation results for NGVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

9.3 Closing Remarks

189

9.3 Closing Remarks Two non-adaptive controllers ensuring the position and velocity error convergence in the Earth-Fixed Representation were considered in this chapter. In the control algorithms the key feature is including of the vehicle parameters in the control gains. The algorithms allow the desired position and velocity trajectory tracking and guarantee the errors convergence similarly as the classical algorithms (without the dynamics in the control gains). The main difference is the idea of control. In the IQV type controllers the dynamical couplings (or the full dynamics) are applied in the control gains in order to shape the system response which is always dependent on the vehicle parameters. Moreover, the total gain is composed of the selected control gains and elements of the inertia matrix which are characteristic for the vehicle. The decomposition of the inertia matrix is made in the Earth-Fixed Frame. The algorithms designed in the Earth-Fixed Frame are more complicated than in the Body-Fixed Frame and consequently the obtained information is slightly different. However, it can be concluded that practical realization of these algorithms is more difficult than the algorithms in terms of the Body-Fixed Frame. It was shown that the IQV based algorithms are in some extent robust to parameters change. They can be applied, in general, for full 6 DOF models. Finally, the simulation results which are known from the literature and the results presented in this chapter confirmed effectiveness of the control approach.

References Herman P, Adamski W (2017) A trajectory tracking controller for vehicles moving at low speed. In: Proceedings of 2017 25th mediterranean conference on control and automation (MED) July 3–6, 2017, Valletta, Malta, pp 1183–1188 Herman P, Adamski W (2019) Model-based controller using quasi-velocities for some vehicles. In: Proceedings of 2019 24th international conference on methods and models in automation and robotics (MMAR), Miedzyzdroje, Poland, 26–29 Aug, pp 48–51 Herman P, Adamski W (2020) Trajectory tracking control algorithm in terms of quasi-velocities for a class of vehicles. Math Comput Simul 172:175–190 Qiao L, Zhang W (2019) Adaptive second-order fast nonsingular terminal sliding mode control for fully actuated autonomous underwater vehicles. IEEE J Ocean Eng 44(2):363–385 Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs Zhang M, Liu X, Yin B, Liu W (2015) Adaptive terminal sliding mode based thruster fault tolerant control for underwater vehicle in time-varying ocean currents. J Frankl Inst 352:4935–4961

Chapter 10

IQV Position and Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

Abstract This chapter deals with nonlinear trajectory tracking strategies with an adaptive term. The proposed controllers including the IQV are suitable for a class of vehicles and allow trajectory position and velocity tracking if disturbances can be described by a model. The control algorithms considered in this chapter guarantee the position and velocity error convergence in the finite time. They are realized in the Body-Fixed Representation. The input control signals, and the form of simplified controllers are also discussed. Additionally, simulation results for a 6 DOF model of underwater vehicle and indoor airship are presented.

10.1 Control Algorithms Expressed in IQV In this section two types of algorithms that is the GVC based control algorithm and the NGV based control algorithm are considered. At the end selected simplified forms of the controllers are given.

10.1.1 Position and Velocity Tracking Controller with Adaptive Term Expressed in GVC The controller with an adaptive term using the GVC and its application for a 6 DOF underwater vehicle dynamics testing was presented in Herman (2019). The control problem is formulated in the given below theorem. Theorem 10.1 Consider the vehicle system model with dynamic equations given by (2.62), kinematic equations (2.53), and (2.56) for which after transformation of the vector π into τ it is: τ = Bu,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_10

(10.1)

191

192

10 IQV Position and Velocity Tracking Control with Adaptive Term …

where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0 together with the nonlinear controller expressed in terms of the GVC: π = Nξ˙ r + Cξ (ξ )ξ r + Dξ (ξ )ξ r + gξ (η) + k D sξ + JξT (η)k P η˜ + f ξ , τ =ϒ

−T

π,

u = B τ, †

(10.2) (10.3)

in which: ˙ d + η), ˜ ξ r = J−1 ξ (η)(η

(10.4)

˙˜ + η), ˜ sξ = ξ r − ξ = J−1 ξ (η)(η

(10.5)

s˙ξ = ξ˙ r − ξ˙ =

−1 J˙ ξ (η)(η˙˜

Jξ (η) = J(η)ϒ, f˙ ξ = sξ ,

¨˜ + η), ˙˜ ˜ + J−1 + η) ξ (η)(η

θ = ±90 deg,

(10.6) (10.7) (10.8)

where η˜ = ηd − η, η˜˙ = η˙ d − η˙ are the position error vector and the velocity ξ is the lumped dynamics estimation error, error vector, respectively, wξ = f ξ − F ˙ ˙ ˙ ξ = f ξ − Fξ is the time derivative of the unknown lumped dynamics estimation w error, k D = k TD > 0, k P = k TP > 0,  = T > 0,  =  T > 0, and N is a diagonal strictly positive matrix. The system trajectories converge locally to a bounded by a constant value ρ neighborhood of the origin [sξT , η˜ T , wξT ]T = 0. Comment on Theorem 10.1. Both Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid for this controller. In this control algorithm a term representing the lumped uncertainty occurs. The lumped uncertainty vector, i.e. the term representing the unknown dynamics has been used many times in the literature, for example in Cui et al. (2017), Peng et al. (2019), Qiao and Zhang (2016, 2019), Qu et al. (2018), Soylu et al. (2008, 2016). In this work it is expressed ξ but it has the same meaning. in terms of the IQV, i.e. F In order to prove the stability of a system with the proposed controller, two assumptions are necessary. The assumptions given below are made taking into account references Qiao and Zhang (2016), Soylu et al. (2008, 2016) in which similar conditions concerning underwater vehicles control were applied. Assumption 10.1 (a) The desired trajectories ηd , η˙ d are differentiable in time and η˙ d , η¨ d are known and bounded.  ν, ν˙ , t) ξ (η, ν, ν˙ , t) = ϒ T F(η, (b) The nonlinear uncertainty vector defined by F and the time derivatives are bounded in the Euclidean norm, i.e.: ξ (η, ν, ν˙ , t)|| ≤ a1 < ∞, ||F ˙ ξ (η, ν, ν˙ , t)|| ≤ a2 < ∞, ||F ¨ ξ (η, ν, ν˙ , t)|| ≤ a3 < ∞. ||F

(10.9)

10.1 Control Algorithms Expressed in IQV

193

Comment on Assumption 10.1. The problem of boundedness of the lumped uncertainty vector was discussed in detail in Qiao and Zhang (2019). In that reference was shown that if Eqs. (2.36)–(2.38) are taken into account then the lumped uncertainty  ≡ F(η,  ν, ν˙ , t)): can be bounded by the following form (F  ≤ b0 + b1 ||ν|| + b2 ||ν||2 , ||F||

(10.10)

where bi (i = 0, 1, 2) are unknown positive constants. The lumped system uncertainty bound (10.10) can be applied if the required thrust forces only sometimes exceed the thruster saturation limit. Based on the first three assumptions made in Qiao and Zhang (2019) (or Qiao and Zhang 2017) and recalling that the matrix ϒ results from decomposition of the inertia matrix M, it can be concluded that also ξ and its time derivative will be bounded if the required thrust forces the vector F do not always exceed saturation limits of the thruster. Also, Assumption 10.1 can be deduced from Qiao and Zhang (2016) based on similar reasoning. Assumption 10.2 The following inequality holds: T ˙ ξ  −1 wξ | sξT [Dξ (ξ ) + k D ]sξ + η˜ T T k P η˜ ≥ |F T ˙ ξ  −1 wξ < 0. only when F

(10.11)

Remark 10.1 If elements of the matrices k D and  are selected then the dynamical couplings are taken into consideration in the controller (10.2). However, it is possible to apply matrices k D = Nk∗D and  = N ∗ in which k∗D and  ∗ are chosen whereas the matrix N is calculated from the inertia matrix M. In this case, the full dynamics of the vehicle is taken into account in the controller’s equation. ˙ ξ has small Remark 10.2 If the vehicle moves slowly then values of the vector F values. Moreover, values of the adaptive term wξ are decreasing when the vehicle is moving (apart from the initial phase of the movement in which the effect of dynamics is noticeable). Thus, the adaptive term are able to compensate the unknown error term. In such case the assumptions seem to be reasonable. Proof The closed-loop system (2.62), (2.56) together with the controller (10.2) can be written as: N˙sξ + [Cξ (ξ ) + Dξ (ξ ) + k D ]sξ + JξT (η)k P η˜ + wξ = 0.

(10.12)

As a Lyapunov function candidate the following expression is proposed: ˜ wξ ) = L(sξ , η,

1 1 T 1 sξ Nsξ + η˜ T k P η˜ + wξT  −1 wξ . 2 2 2

(10.13)

194

10 IQV Position and Velocity Tracking Control with Adaptive Term …

Calculating the time derivative of the function L (10.13) one obtains: 1 ˙ ˙˜ T k P η˜ + w ˙ ξ , η, ˜ wξ ) = sξT N˙sξ + sξT Ns ˙ ξT  −1 wξ . L(s ξ +η 2 ˙ = Because the matrices M, and ϒ have only constant elements, thus N 0. Using now the relationship (10.12) one gets:

(10.14) d (ϒ T Mϒ) dt

=

˙ ξ , η, ˜ wξ ) = sξT [−Cξ (ξ )sξ − Dξ (ξ )sξ − k D sξ − JξT (η)k P η˜ − wξ ] L(s T ˙ ˙ ξ )T  −1 wξ = −sξT [Dξ (ξ ) + k D ]sξ − sξT JξT (η)k P η˜ +η˙˜ k P η˜ + (f ξ − F T T ˙ ξ  −1 wξ . −sξT wξ + η˙˜ k P η˜ + sξT  T  −1 wξ − F

(10.15)

From (2.58) it results that sξT Cξ (ξ )sξ = (ϒsξ )T C(ν)(ϒsξ ) = sT C(ν)s = 0 because sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). Moreover, using (10.8), the relationships  =  T and  −1 = I, and taking into account (10.5) one has: T ˙ ξ , η, ˜ ˜ wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − (η˙˜ + η) ˜ T J−T L(s ξ (η)Jξ (η)k P η T T ˙ ξ  −1 wξ . +η˙˜ k P η˜ − F

(10.16)

Finally one obtains: T ˙ ξ , η, ˙ ξ  −1 wξ . ˜ wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − η˜ T T k P η˜ − F L(s

(10.17)

Consider the result given in (10.17). Two kind of argumentation are used in the literature to show stability of the closed-loop system in the presence of model uncertainties and disturbances. Method 1 was applied for manipulators (Zeinali and Notash 2010) and for an engine torque control (Fazeli et al. 2012). This strategy can, however after it modification, be successfully applied to marine vehicles. In Soylu et al. (2007) and next in Soylu et al. (2008) slow motion of underwater vehicles was taken into account and for this reason, the proposed approach is valid. For marine vehicles, similar considerations can be found, e.g. in Gan et al. (2018), Ismail et al. (2016), Lakhekar and Waghmare (2017). Two different assumptions are made to perform the stability analysis: slowly time-varying uncertainties and fast time-varying uncertainties. If the T ˙ ξ is zero or uncertainties are arbitrarily large and slowly varying with time, then F negligible. Consequently, one gets: ˙ ξ , η, ˜ wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − η˜ T T k P η˜ ≤ 0, L(s

(10.18)

which is negative or zero. Note that from (10.18) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired

10.1 Control Algorithms Expressed in IQV

195

state in the finite time. This problem is solved using Barbalat’s lemma Slotine and Li (1991) as it is shown in Soylu et al. (2007) where underwater vehicles under assumption analogous to Assumption 10.2 were considered. For slowly time-varying uncertainties an asymptotic stability is guaranteed (Zeinali and Notash 2010). If the uncertainties are assumed as arbitrarily large and fast time-varying but norm bounded (Zeinali and Notash 2010) then the sufficient condition for L˙ to be negative T ˙ ξ > 0 and wξ > 0 are met then ˙ ξ  −1 wξ ≥ 0. When the conditions F in (10.17) is F stabilization of the closed-loop system is ensured. For a well designed controller T ˙ ξ  −1 wξ < 0 one gets the it can be assumed that wξ → 0. In the worse case, i.e. F inequality: ˙ ξ , η, ˜ wξ ) ≤ −sξT [Dξ (ξ ) + k D ]sξ − η˜ T T k P η˜ + ρ, L(s

(10.19)

T ˙ ξ  −1 wξ means a positive scalar value (in the sense of maximal in which ρ = −F    ˙ T −1  ξ  wξ  in Eq. (10.19)). The boundedness and convalue of this term, i.e. ρ = F  vergence to a small vicinity of the state space origin is guaranteed and it is possible to reduce the tracking error using the design parameters k D , k P , and . Therefore, even in the worst case occurs, the system stability can also be ensured by increasing k D , k P , and . Method 2 is applied to underwater vehicles (Qiao and Zhang 2016). This approach results from Soylu et al. (2007, 2008) and it is more rigorously. The argumentation appropriate for our case is as follows. If the Assumptions 10.1 and 10.2 are valid then one obtains: T ˙ ξ , η, ˙ ξ  −1 wξ ≤ 0. ˜ wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − η˜ T T k P η˜ − F L(s

(10.20)

From Soylu et al. (2008) it is known that inequality of type as (10.20) does not imply that L˙ → 0 as t → ∞ what means that the system trajectories may not converge to the desired state in the finite time. It is possible to solve this problem applying Barbalat’s lemma, e.g. Slotine and Li (1991). Because L is lower bounded (L ≥ 0), and L˙ is negative semi-definite, i.e. L˙ ≤ 0, then one obtains that L must have limit (there exists a non-negative constant ρ) such that limt→∞ L(t) = ρ. Moreover, ˜ and wξ Eq. (10.20) leads to conclusion that L(t) ≤ L(0), which implies that sξ , η, are bounded. Using Assumption 10.1 and Eqs. (10.2)–(10.8) it may be verified that ˙ ξ , and F ¨ ξ are also bounded. Consequently, L¨ can be determined and it ˙ ξ, F s˙ξ , w t is bounded. One has limt→∞ 0 L(σ )dσ = limt→∞ L(t) − L(0) = ρ − L(0) < ∞. ˙ = 0, then L˙ → 0 as t → ∞. This From Barbalat’s lemma it results that limt→∞ L(t) in turn implies that sξ , η˜ → 0 as t → ∞.  Vehicle input signals. Because the controller (10.2) is expressed in terms of the vector ξ then the control signal is determined from the relationship (2.61) τ = ϒ −T π:

196

10 IQV Position and Velocity Tracking Control with Adaptive Term …

τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + ϒ −T k D ϒ −1 s +JT (η)k P η˜ + f,

(10.21)

t where f = ϒ −T ϒ −1 0 f s(t)dt and the symbols are defined in (7.17)–(7.19) and (7.21). Note that in (10.21) the velocity gain matrix is ϒ −T k D ϒ −1 . Thus, the gain coefficients in this matrix contain the dynamic parameters (i.e. masses, inertia) as well as the geometrical dimensions of the system. If a diagonal matrix k D is selected, it serves rather for precise tuning of the controller for the vehicle. The same remark concerns the matrix . This property may facilitate selection of control. Therefore, it is no need to assume the matrix k D as a symmetric one because thanks dynamical couplings in the system the matrix ϒ −T k D ϒ −1 is symmetric even if k D is a diagonal matrix only. In this way it is possible to evaluate effects of the dynamical couplings. However, if instead of selecting k D the matrix k∗D is selected and the matrix N is used, then k D = Nk∗D . In this case effects of the vehicle dynamics is investigated. t Consequently, f = ϒ −T N ∗ ϒ −1 0 f s(t)dt (the matrix  ∗ is a diagonal one).

10.1.2 Position and Velocity Tracking Controller with Adaptive Term Expressed in NGVC The control scheme with an adaptive term using the NGVC and its application for a 6 DOF underwater vehicle dynamics can be found in Herman (2021). The controller in terms of the NGVC is proposed in the below given theorem. Theorem 10.2 Consider the vehicle system model with dynamic equations given by (2.87), kinematic equations (2.79), and (2.82) for which after transformation of the vector into τ it is: τ = Bu, (10.22) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0 together with the nonlinear controller expressed in terms of the NGVC:

= ζ˙ r + Cζ (ζ )ζ r + Dζ (ζ )ζ r + gζ (η) + k D sζ + JζT (η)k P η˜ + f ζ ,

(10.23)

τ = ,

(10.24)

T

where:

u = B τ, †

10.1 Control Algorithms Expressed in IQV

197

˙ d + η), ˜ ζ r = J−1 ζ (η)(η

(10.25)

−1 J˙ ζ (η)(η˙ d

˙˜ ˜ + J−1 + η) ηd + η), ζ (η)(¨ ˜ sζ = ζ r − ζ = J−1 ˜˙ + η), ζ (η)(η

ζ˙ r =

s˙ζ = ζ˙ r − ζ˙

−1 = J˙ ζ (η)(η˙˜ −1

Jζ (η) = J(η) , f˙ ζ = sζ ,

¨˜ + η), ˙˜ ˜ + J−1 + η) ζ (η)(η

θ = ±90 deg,

(10.26) (10.27) (10.28) (10.29) (10.30)

and η˜ = ηd − η, η˙˜ = η˙ d − η˙ are the position error vector and the velocity error ζ is the lumped dynamics estimation error, w ˙ζ = vector, respectively, wζ = f ζ − F ˙ ˙ f ζ − Fζ is the time derivative of the unknown lumped dynamics estimation error, k D = k TD > 0, k P = k TP > 0,  = T > 0, and  =  T > 0. The system trajectories converges locally to a bounded neighborhood ρ (a constant value) of the origin [sζT , η˜ T , wζ ]T = 0. Comment on Theorem 10.2. Both Remark 6.1 (about thruster force allocation) and Remark 6.2 (about input saturation) remain valid for this controller. In this control algorithm a term representing the lumped uncertainty occurs which is expressed in ζ . terms of the IQV, i.e. F In order to prove the stability of a system with the proposed controller, two assumptions are necessary. The assumptions given below are made, as previously, taking into account references Qiao and Zhang (2016), Soylu et al. (2008, 2016) in which similar conditions concerning underwater vehicles control were applied. Assumption 10.3 (a) The desired trajectories ηd , η˙ d are differentiable in time and η˙ d , η¨ d are known and bounded.  ν, ν˙ , t) ζ (η, ν, ν˙ , t) = −T F(η, (b) The nonlinear uncertainty vector defined by F and the time derivatives are bounded in the Euclidean norm, i.e.: ζ (η, ν, ν˙ , t)|| ≤ a1 < ∞, ||F ˙ ζ (η, ν, ν˙ , t)|| ≤ a2 < ∞ ||F ¨ ζ (η, ν, ν˙ , t)|| ≤ a3 < ∞. ||F

(10.31)

Comment on Assumption 10.3 is analogous to Comment on Assumption 10.1 however, ζ . considering vector F Assumption 10.4 The following inequality holds: T ˙ ζ  −1 wζ | sζT [Dζ (ζ ) + k D ]sζ + η˜ T T k P η˜ ≥ |F T ˙ ζ  −1 wζ < 0. only when F

(10.32)

198

10 IQV Position and Velocity Tracking Control with Adaptive Term …

˙ ζ (and its time derivative) is ensured Remark 10.3 Boundedness of the vector F when the vehicle speed is low. Additionally, the adaptive vector wζ decreases during realization of control task when the vehicle moves (apart from the initial phase of the movement in which the effect of dynamics is noticeable). Thus, the adaptive term are able to compensate the unknown error term. Therefore, making the above assumptions appears to be justified. Proof The closed-loop system (2.87), (2.82) together with the controller (10.23) can be written as: s˙ζ + [Cζ (ζ ) + Dζ (ζ ) + k D ]sζ + JζT (η)k P η˜ + wζ = 0.

(10.33)

As a Lyapunov function candidate the following expression is proposed: ˜ wζ ) = L(sζ , η,

1 1 T 1 sζ sζ + η˜ T k P η˜ + wζT  −1 wζ . 2 2 2

(10.34)

Calculating the time derivative of the function L (10.34) one obtains: T ˙ ζ , η, ˜ wζ ) = sζT s˙ζ + η˙˜ k P η˜ + w ˙ ζT  −1 wζ . L(s

(10.35)

Using the relationship (10.33) one gets: ˙ ζ , η, ˜ wζ ) = sζT [−Cζ (ζ )sζ − Dζ (ζ )sζ − k D sζ − JζT (η)k P η˜ − wζ ] L(s T ˙ ˙ ζ )T  −1 wζ = −sζT Cζ (ζ )sζ − sζT Dζ (ζ )sζ − sζT k D sζ +η˜˙ k P η˜ + (f ζ − F T T ˙ ζ  −1 wζ . −sζT JζT (η)k P η˜ − sζT wζ + η˙˜ k P η˜ + sζT  T  −1 wζ − F

(10.36)

Recall from (2.58), that sζT Cζ (ζ )sζ = ( −1 sζ )T C(ν)( −1 sζ ) = sT C(ν)s = 0 because sT C(ν)s=0 for all s ∈ R n (Fossen 1994). Moreover,  =  T and  −1 = I. Therefore, it is: ˙ ζ , η, ˜ wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − sζT JζT (η)k P η˜ L(s T T ˙ ζ  −1 wζ , +η˙˜ k P η˜ − F

(10.37)

Taking into account (10.27) is finally obtained: T ˙ ζ , η, ˙ ζ  −1 wζ . ˜ wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − η˜ T T k P η˜ − F L(s

(10.38)

Argumentation concerning (10.38) to show stability of the closed-loop system taking into account the model uncertainties and disturbances is similar as previously. Method 1 was applied for manipulators (Zeinali and Notash 2010) and for an engine torque control (Fazeli et al. 2012). For the considered vehicle the approach

10.1 Control Algorithms Expressed in IQV

199

can be modify in the following manner. Two different assumptions are made to perform the stability analysis: slowly time-varying uncertainties and fast time-varying uncertainties. If the uncertainties are arbitrarily large and slowly varying with time, T ˙ ζ is zero or negligible. Consequently, one gets: then F ˙ ζ , η, ˜ wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − η˜ T T k P η˜ ≤ 0, L(s

(10.39)

which is negative or zero. Note that from (10.39) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. This problem is solved using Barbalat’s lemma (Slotine and Li 1991) as it is shown in Soylu et al. (2007) where underwater vehicles under assumption analogous to Assumption 10.4 were considered. For slowly time-varying uncertainties an asymptotic stability is guaranteed (Zeinali and Notash 2010). In Soylu et al. (2007) and next in Soylu et al. (2008) slow motion of underwater vehicles was taken into account and for this reason, the proposed approach is valid. Similar considerations can be found in other works, i.e. in Gan et al. (2018), Ismail et al. (2016), Lakhekar and Waghmare (2017). If the uncertainties are assumed as arbitrarily large and fast time-varying but norm bounded (Zeinali and Notash 2010) then the sufficient condition for L˙ to be negative T ˙ ζ > 0 and wζ > 0 are met then ˙ ζ  −1 wζ ≥ 0. When the conditions F in (10.38) is F stabilization of the closed-loop system is ensured. For a well designed controller T ˙ ζ  −1 wζ < 0 one gets the it can be assumed that wζ → 0. In the worse case, i.e. F inequality: ˙ ζ , η, ˜ wζ ) ≤ −sζT [Dζ (ζ ) + k D ]sζ − η˜ T T k P η˜ + ρ, L(s

(10.40)

T ˙ ζ  −1 wζ means a positive scalar value (in the sense of maximal in which ρ = −F    ˙ T −1    value of this term, i.e. ρ = Fζ  wζ  in Eq. (10.40)). The boundedness and con-

vergence to a small vicinity of the state space origin is guaranteed and it is possible to reduce the tracking error using the design parameters k D , k P , and . Therefore, even in the worst case occurs, the system stability can also be ensured by increasing k D , k P , and . Method 2 is applied to underwater vehicles (Qiao and Zhang 2016). Using it the results from Soylu et al. (2007, 2008) are more rigorously presented. The argumentation appropriate for our case is as follows. If the Assumptions 10.3 and 10.4 are valid then one obtains: T ˙ ζ , η, ˙ ζ  −1 wζ ≤ 0. ˜ wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − η˜ T T k P η˜ − F L(s

(10.41)

From Soylu et al. (2008) it is known that inequality of type as (10.41) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not

200

10 IQV Position and Velocity Tracking Control with Adaptive Term …

converge to the desired state in the finite time. It is possible to solve this problem applying Barbalat’s lemma, e.g. Slotine and Li (1991). Because L is lower bounded (L ≥ 0), andL˙ is negative semi-definite, i.e.L˙ ≤ 0, then one obtains that L must have limit (there exists a non-negative constant ρ) such that limt→∞ L(t) = ρ. Moreover, ˜ and Eq. (10.41) leads one to conclusion that L(t) ≤ L(0), which implies that sζ , η, wζ are bounded. Using Assumption 10.3 and Eqs. (10.23)–(10.30) it may be verified ˙ ζ , and F ¨ ζ are also bounded. Consequently, L¨ can be determined and it ˙ ζ, F that s˙ζ , w t is bounded. One obtains limt→∞ 0 L(σ )dσ = limt→∞ L(t) − L(0) = ρ − L(0) < ˙ = 0, then L˙ → 0 as t → ∞. ∞. From Barbalat’s lemma it results that limt→∞ L(t) This in turn implies that sζ , η˜ → 0 as t → ∞.  Vehicle input signals. Consider benefits of the proposed control algorithm. Because the controller (10.23) is expressed in terms of the vector ζ then the control signal is determined from the relationship τ = T (recall that (2.86) the matrix T is invertible): τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + T k D s +JT (η)k P η˜ + f,

(10.42)

t where f = T  0 f s(t)dt, and the symbols are defined in (7.17)–(7.19) and (7.48). Note that in (10.42) the full velocity gain matrix is K D = T k D . If it is assumed for example that k D = δ I then one gets T k D = δ M. Thus, the gain coefficients in the matrix K D contain the dynamic parameters (i.e. masses, inertia) as well as the geometrical dimensions of the system. If a diagonal matrix k D is selected, it serves rather for precise tuning of the controller. This property may facilitate selection of control. Therefore, it is no need to assume the matrix k D as a symmetric one because thanks dynamical couplings in the system the matrix T k D symmetric even if k D is a diagonal matrix only.

10.1.3 Simplified Forms of Controllers GVC-APVTCBF. The simplified controller of (10.21) is, e.g. if M is a diagonal matrix. In this case ϒ = I and consequently: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + k D s + JT (η)k P η˜ + f.

(10.43)

Now one has the following gain matrix K D = k D which means that the set of dynamical parameters is not taken into account. NGVC-APVTCBF. The simplified controller of (10.42) is, e.g. if M is a diagonal 1 matrix. In this case ϒ = I and consequently = N 2 . Thus: 1 1 τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + N 2 k D N 2 s + JT (η)k P η˜ + f.

(10.44)

10.1 Control Algorithms Expressed in IQV

201 1

1

Now one has the following gain matrix K D = N 2 k D N 2 which means that the set of dynamical parameters is still taken into consideration. Algorithms for Vehicle Model with 3 DOF Moving Horizontally. The algorithms can be also applied to a reduced planar model of the vehicle. The remark given in Chap. 6 is valid for these controllers.

10.2 Simulation Results The examples which present the use of the considered previously control algorithms are given in this section. Simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

10.2.1 Trajectory Tracking Control Using GVC 10.2.1.1

Controller for 6 DOF Fully Actuated Airship

The GVC control algorithm was presented in Herman (2019) and applied to dynamics test of an underwater vehicle. In this subsection its use for an indoor airship model is tested. Objective: to show performance of the GVC trajectory tracking controller for an airship model in the presence of disturbances model. Fast error convergence of position and velocity errors avoiding great overshoot is expected. For tracking it is assumed the following desired helical position trajectory together with exponential angular trajectories, and the appropriate velocity profile, respectively, namely: ηd = [5 cos(0.2 t) − 4, 5 sin(0.2 t) − 2, 0.3 t + 0.1, 0.5e−0.3t , 0.4e−0.3t , −0.5e−0.3t ]T , η˙ d = [− sin(0.2 t), cos(0.2 t), 0.3, −0.15e−0.3t , −0.12e−0.3t , 0.15e−0.3t ]T .

(10.45) (10.46)

The vector including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments: ⎤ 10 f 1 (t) e−1 + 20 ⎢ f 1 (t) e−1 + 2 ⎥ ⎥ ⎢ ⎢ 20 f 2 (t) e−1 − 20 ⎥ ⎥, ⎢  F=⎢ f 2 (t) e−1 − 1 ⎥ ⎥ ⎢ ⎣ 500 f 3 (t) e−3 + 40 ⎦ 50 f 3 (t) e−3 + 4 ⎡

(10.47)

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10 IQV Position and Velocity Tracking Control with Adaptive Term …

X , F

Y , F

Z , b disturbance moments F

K , Fig. 10.1 Disturbance functions: a disturbance forces F

M , F

N F

where the functions f 1 (t) = 6 cos(0.5 t) + sin(0.8 t), f 2 (t) = 0.4 sin(0.4 t)e−5 + π t) + π4 cos( π5 t). 0.4 ecos(−0.3t) , and f 3 (t) = π5 sin( 15 The control gains set was assumed as follows: k∗D = diag{20, 20, 20, 40, 40, 40}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.35, 0.35, 0.35, 0.45, 0.45, 0.45},  ∗ = diag{200, 200, 200, 600, 600, 600},

(10.48)

where k D = Nk∗D and  = N ∗ in order to include the vehicle dynamics in the gains (the matrix N is determined as a result of the decomposition of matrix M). Because of dynamical effects observed in the first phase of motion the input signals are limited (applied forces, moments, and disturbance estimated functions). Saturation values are: |τ | ≤ 150 N or Nm and | f e | ≤ 150 N or Nm. From Fig. 10.2a it can be noted that the desired trajectory is tracked correctly. As it is shown in Fig. 10.2b–c both linear and angular position errors reach the end values close to zero after about 15 s. At the same time (Fig. 10.2d–e) the linear and angular velocity errors reach values very close to zero. It is observed in Fig. 10.3a–b that all lumped dynamics estimation errors converge to limited final value very quickly (after 2 s), which is the result of taking into account the dynamics in the controller’s gain. From Fig. 10.3c–d it is seen that if the airship moves then the force τ X an the moment τ M have great values. Therefore, there exists a relationship between these signals and the disturbance functions from Fig. 10.1a–b.

10.2 Simulation Results

203

Fig. 10.2 Simulation results for GVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

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10 IQV Position and Velocity Tracking Control with Adaptive Term …

Fig. 10.3 Simulation results for GVC controller and indoor airship: a lumped dynamics estimation errors w related to linear velocities; b lumped dynamics estimation errors w related to angular velocities; c forces; d moments

10.2.2 Trajectory Tracking Control Using NGVC 10.2.2.1

Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: to show performance of the NGVC trajectory tracking controller for an underwater vehicle model in the presence of a disturbances model. Acceptable convergence of position and velocity errors must be guaranteed. For tracking it is assumed the following desired 3D linear position trajectory together with constant angular values, and the appropriate velocity profile, i.e. (9.72)– (9.73). The model including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments:

10.2 Simulation Results

205

X , F

Y , F

Z , b disturbance moments F

K , Fig. 10.4 Disturbance functions: a disturbance forces F

M , F

N F

⎤ 300 f 4 (t) e−2.5 + 60 ⎢ 50 f 4 (t) e−2.5 + 10 ⎥ ⎥ ⎢ ⎢ 100 f 5 (t) e−1 + 100 ⎥ ⎥, ⎢  F=⎢ −1 ⎥ ⎢ 2 f 5 (t) e −3+ 2 ⎥ ⎣ 150 f 6 (t) e + 12 ⎦ 50 f 6 (t) e−3 + 4 ⎡

(10.49)

where the functions f 4 (t) = 0.8 cos(0.6 t) + 0.3 sin(0.1 t), f 5 (t) = 0.2 sin(t)e−1 + 0.7 0.1 ecos(−t ) , and f 6 (t) = 0.6 sin(0.2 t) + 0.8 cos(0.6 t). The functions are shown in Fig. 10.4. The control gains set is as follows: k∗D = diag{5, 5, 5, 5, 5, 5}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.5, 0.5, 0.5, 0.5, 0.5, 0.5},  ∗ = diag{15, 15, 15, 25, 25, 25}. (10.50) The selected gains are not the same for all variables because the goal is to show the effectiveness of the controller without comparing with results obtained from another algorithm. In order to limit the dynamical effects observed in the first phase of motion the input signals are limited (forces, moments, and disturbance estimation function). Saturation values are: |τ | ≤ 1500 N or Nm and | f e | ≤ 1500 N or Nm. From Fig. 10.5a it can be seen that the realized trajectory tracks the desired trajectory correctly. It follows from Fig. 10.5b–c that both the linear position errors and angular position errors are close to zero after about 10 s. Similar observation concerns the linear and angular velocity errors shown in Fig. 10.5d–e. The effect of dynamics causes that velocity errors (for all variables) first increase and then decrease. The lumped dynamics estimation errors w related to linear and angular velocities reach limited values close to zero after about 3 s which indicates the quick reaction of the system with the controller (Fig. 10.6a–b). At the beginning values most of the forces and moments have great values (which results also from the dynamics effect) as

206

10 IQV Position and Velocity Tracking Control with Adaptive Term …

Fig. 10.5 Simulation results for NGVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

10.2 Simulation Results

207

Fig. 10.6 Simulation results for NGVC controller and underwater vehicle: a lumped dynamics estimation errors w related to linear velocities; b lumped dynamics estimation errors w related to angular velocities; c forces; d moments

shown in Fig. 10.6c–d. However, after a short time they tend to limited values resulting from dynamic parameters of the vehicle. Taking into consideration the presented results it can be concluded that the control algorithm works quickly and correctly.

10.3 Closing Remarks The control algorithms containing an adaptive term and using the IQV for the position and velocity trajectory tracking were presented in this chapter. The IQV based controllers are realized in the Body-Fixed Frame. They ensure the trajectory tracking in the presence of a disturbances model and they can be applied, in general, to a full 6 DOF vehicle. The main difference between the presented algorithms and the classical ones relies on that the dynamical and geometrical parameters are included in the control gain matrices. Consequently, the resultant gain matrix always depends directly on the dynamics of the vehicle. Moreover, it is possible to obtain fast system response what causes that the position and velocity tracking errors quickly tends to zero. In the algorithms the system dynamics plays the crucial role in the control

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10 IQV Position and Velocity Tracking Control with Adaptive Term …

process. If the IQV controllers are applied then a diagonal control gain matrix can be used because the resulting matrix containing the vehicle parameters is a symmetric matrix. The presented simulations and also the results known from the literature confirmed effectiveness of the proposed control algorithms.

References Cui R, Chen L, Yang Ch, Chen M (2017) Extended state observer-based integral sliding mode control for an underwater robot with unknown disturbances and uncertain nonlinearities. IEEE Trans Ind Electron 64(8):6785–6795 Fazeli A, Zeinali M, Khajepour A (2012) Application of adaptive sliding mode control for regenerative braking torque control. IEEE/ASME Trans Mechatron 17(4):745–755 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Gan W, Zhu D, Ji D (2018) QPSO-model predictive control-based approach to dynamic trajectory tracking control for unmanned underwater vehicles. Ocean Eng 158:208–220 Herman P (2019) Application of nonlinear controller for dynamics evaluation of underwater vehicles. Ocean Eng 179:59–66 Herman P (2021) Preliminary design of the control needed to achieve underwater vehicle trajectories. J Mar Sci Technol 26:986–998 Ismail ZH, Mokhar MBM, Putranti VWE, Dunnigan MW (2016) A robust dynamic region-based control scheme for an autonomous underwater vehicle. Ocean Eng 111:155–165 Lakhekar GV, Waghmare LM (2017) Robust maneuvering of autonomous underwater vehicle: an adaptive fuzzy PI sliding mode control. Intel Serv Robot 10:195–212 Peng Z, Wang J, Wang J (2019) Constrained control of autonomous underwater vehicles based on command optimization and disturbance estimation. IEEE Trans Ind Electron 66(5):3627–3635 Qiao L, Zhang W (2016) Double-loop chattering-free adaptive integral sliding mode control for underwater vehicles. In: Proceedings MTS/IEEE OCEANS conference Shanghai, China, 10–13 April 2016, pp 1–6 Qiao L, Zhang W (2017) Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles. IET Control Theory & Appl 11(8):1293–1306 Qiao L, Zhang W (2019) Double-loop integral terminal sliding mode tracking control for UUVs with adaptive dynamic compensation of uncertainties and disturbances. IEEE J Oceanic Eng 44(1):29–53 Qu Y, Xiao B, Fu Z, Yuan D (2018) Trajectory exponential tracking control of unmanned surface ships with external disturbance and system uncertainties. ISA Trans 78:47–55 Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs Soylu S, Buckham BJ, Podhorodeski RP (2007) Robust control of underwater vehicles with faulttolerant infinity-norm thruster force allocation. In: Proceedings of OCEANS 2007, 29 September– 4 October, Vancouver, BC, Canada, 2007, pp 1–10 Soylu S, Buckham BJ, Podhorodeski RP (2008) A chattering-free sliding-mode controller for underwater vehicles with fault-tolerant infinity-norm thrust allocation. Ocean Eng 35:1647–1659 Soylu S, Proctor AA, Podhorodeski RP, Bradley C, Buckham BJ (2016) Precise trajectory control for an inspection class ROV. Ocean Eng 111:508–523 Zeinali M, Notash L (2010) Adaptive sliding mode control with uncertainty estimator for robot manipulators. Mech Mach Theory 45:80–90

Chapter 11

IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

Abstract This chapter deals with the velocity tracking controllers containing an adaptive term in the equations expressed using the IQV. The control schemes can be applied to underwater vehicles, some horizontally moving vehicles, and indoor airships. They are realized in the Body-Fixed Frame and guarantee the velocity error convergence in the finite time if a disturbance model is assumed. The input control signals obtained after variables transformation, and the simplified forms of controllers are also considered. Moreover, the relationship between both kind of algorithms is indicated. Simulation results for a full 6 DOF model of an underwater vehicle and an indoor airship show performance of the control algorithms in terms of the GVC and the NGVC. Finally, some selected simulation results are delivered.

11.1 Control Algorithms Expressed in IQV In this section two kinds of controllers are discussed, i.e. the GVC based velocity controller and the NGVC based velocity controller. At the end some simplified forms of the algorithms are shown.

11.1.1 Velocity Tracking Controller with Adaptive Term Expressed in GVC A controller of this type and its application to a 6 DOF underwater vehicle is shown in Herman (2020). The general form of the algorithm is presented in the theorem given below.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_11

209

210

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

Theorem 11.1 Consider the vehicle system model with dynamic equations given by (2.62), kinematic equations (2.53), and (2.56) for which after transformation of the vector π into τ it is: τ = Bu, (11.1) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0 together with the nonlinear controller expressed in terms of the GVC: π = Nξ˙ r + Cξ (ξ )ξ r + Dξ (ξ )ξ r + gξ (η) + k D sξ + ϒ T k I z + f ξ , τ =ϒ

−T

π,

u = B τ, †

(11.2) (11.3)

where:  z=

t

ν˜ (σ ) dσ,

(11.4)

0

ξ r = ϒ −1 (ν d + z), sξ = ξ r − ξ = ϒ −1 (˜ν + z), s˙ξ = ξ˙ r − ξ˙ = ϒ −1 (ν˙˜ + ˜ν ), f˙ ξ = sξ ,

(11.5) (11.6) (11.7) (11.8)

ξ is the lumped dynamics and ν˜ = ν d − ν is the velocity error vector, wξ = f ξ − F ˙ ˙ ˙ ξ = f ξ − Fξ is the time derivative of the unknown lumped dynamestimation error, w ics estimation error, k D = k TD > 0, k I = k TI > 0,  = T > 0, = T > 0 and N is a diagonal strictly positive matrix. The system trajectories converges globally to a bounded neighborhood ρ (a constant value) of the origin [sξT , zT , wξT ]T = 0. Comment on Theorem 11.1. The Comment on Theorem 10.1 is also valid here. Two assumptions are necessary to prove the stability of a closed-loop system containing the control algorithm. Such assumptions can be found in Qiao and Zhang (2016), Soylu et al. (2008, 2016). Assumption 11.1 (a) The desired trajectories in the vector ν d are differentiable in time and ν d , ν˙ d are known and bounded. ξ (η, ν, ν˙ , z, t) = (b) The nonlinear uncertainty vector defined by the formula F T ϒ F(η, ν, ν˙ , z, t) and its time derivatives are bounded in the Euclidean norm, i.e.: ξ (η, ν, ν˙ , z, t)|| ≤ a1 < ∞, ||F ˙ ξ (η, ν, ν˙ , z, t)|| ≤ a2 < ∞ ||F ¨ ξ (η, ν, ν˙ , z, t)|| ≤ a3 < ∞. ||F

(11.9)

11.1 Control Algorithms Expressed in IQV

211

Comment on Assumption 11.1. The relevant argumentation is set out in Comment on Assumption 10.1. Assumption 11.2 The following inequality holds: T ˙ ξ −1 wξ | sξT [Dξ (ξ ) + k D ]sξ + zT T k I z ≥ |F T ˙ ξ −1 wξ < 0. only when F

(11.10)

Remark 11.1 If elements of the matrices k D and are selected then the dynamical couplings are taken into consideration in the controller (11.2) only. However, it is possible to apply matrices k D = Nk∗D and = N ∗ in which k∗D and ∗ are chosen whereas the matrix N is calculated from the inertia matrix M. In this case, the full dynamics of the vehicle is taken into account in the controller’s equation. Remark 11.2 The presented algorithm is suitable for vehicle moving at low veloc˙ ξ has small values. Moreover, values of the adaptive ities because then the vector F vector wξ decrease during the vehicle motion (with exception of the start phase). Taking the above into account the assumption concerning the bounded signals and our way of thinking seem reasonable. Proof The closed loop system (2.62) together with the controller (11.2) can be written as: N˙sξ + [Cξ (ξ ) + Dξ (ξ ) + k D ]sξ + ϒ T k I z + wξ = 0.

(11.11)

As a Lyapunov function candidate the following expression is considered: L(sξ , z, wξ ) =

1 T 1 1 s Nsξ + zT k I z + wξT −1 wξ . 2 ξ 2 2

(11.12)

Calculating the time derivative of the function L (11.12) one has: ˙ ξ , z, wξ ) = sξT N˙sξ + 1 sξT Ns ˙ ξ + ν˜ T k I z + w ˙ ξT −1 wξ . L(s 2

(11.13)

˙ = Because the matrices M and ϒ have only constant elements, therefore N d T (ϒ Mϒ) = 0. Using the relationship (11.8) and (11.11) one can write: dt ˙ ξ , z, wξ ) = sξT [−Cξ (ξ )sξ − Dξ (ξ )sξ − k D sξ − ϒ T k I z − wξ ] L(s ˙ ˙ ξ )T −1 wξ = −sξT Cξ (ξ )sξ − sξT Dξ (ξ )sξ − sξT k D sξ +˜ν T k I z + (f ξ − F T ˙ ξ −1 wξ . −sξT ϒ T k I z − sξT wξ + ν˜ T k I z + sξT T −1 wξ − F

(11.14)

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11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

From (2.58) it follows that sξT Cξ (ξ )sξ = (ϒsξ )T C(ν)(ϒsξ ) = sT C(ν)s = 0 because sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). Besides it is = T and −1 = I. Consequently, one gets: ˙ ξ , z, wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − sξT ϒ T k I z L(s T ˙ ξ −1 wξ . +˜ν T k I z − F

(11.15)

Applying (11.6) finally one obtains: T ˙ ξ , z, wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − zT T k I z − F ˙ ξ −1 wξ . L(s

(11.16)

Consider the result received in (11.16). Two kind of argumentation are used in the literature to show stability of the closed loop system in the presence of model uncertainties and disturbances. Method 1 was applied for manipulators (Zeinali and Notash 2010) and an engine torque control (Fazeli et al. 2012). For the considered vehicle the approach can be modify in the following manner. Two different assumptions are made to perform the stability analysis: slowly time-varying uncertainties and fast time-varying uncertainT ˙ ξ ties. If the uncertainties are arbitrarily large and slowly varying with time, then F is zero or negligible. Consequently, one gets: ˙ ξ , z, wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − zT T k I z ≤ 0, L(s

(11.17)

which is negative or zero. Note that from (11.17) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. This problem is solved using Barbalat’s lemma (Slotine and Li 1991) as it is shown in Soylu et al. (2007) where underwater vehicles under assumption analogous to Assumption 11.2 were considered. For slowly time-varying uncertainties an asymptotic stability is guaranteed (Zeinali and Notash 2010). In reference Soylu et al. (2007) and next in Soylu et al. (2008) slow motion of underwater vehicles was taken into account and for this reason, the proposed approach is valid. Similar considerations can be found in other works, i.e. Gan et al. (2018), Ismail et al. (2016), Lakhekar and Waghmare (2017). If the uncertainties are assumed as arbitrarily large and fast time-varying but norm bounded (Zeinali and Notash 2010) then the sufficient condition for L˙ to be negative T ˙ ξ > 0 and wξ > 0 are met then ˙ ξ −1 wξ ≥ 0. When the conditions F in (11.16) is F stabilization of the closed-loop system is ensured. For a well designed controller T ˙ ξ −1 wξ < 0 one gets the it can be assumed that wξ → 0. In the worse case, i.e. F inequality: ˙ ξ , z, wξ ) ≤ −sξT [Dξ (ξ ) + k D ]sξ − zT T k I z + ρ, L(s

(11.18)

11.1 Control Algorithms Expressed in IQV

213

T ˙ ξ −1 wξ means a positive scalar value (in the sense of maximal in which ρ = −F    ˙ T −1   ξ wξ  in Eq. (11.18)). The boundedness and convalue of this term, i.e. ρ = F  vergence to a small vicinity of the state space origin is guaranteed and it is possible to reduce the tracking error using the design parameters k D , k I , and . Therefore, even in the worst case occurs, the system stability can also be ensured by increasing k D , k I , and . Method 2 is applied to underwater vehicles (Qiao and Zhang 2016). Using it the results from Soylu et al. (2007, 2008) are more rigorously presented. The argumentation appropriate for our case is as follows. If the Assumptions 11.1 and 11.2 are valid then one obtains: T ˙ ξ , z, wξ ) = −sξT [Dξ (ξ ) + k D ]sξ − zT T k I z − F ˙ ξ −1 wξ ≤ 0. L(s

(11.19)

From the literature (Soylu et al. 2008) it is known that inequality of type as (11.19) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. It is possible to solve this problem applying Barbalat’s lemma, e.g. Slotine and Li (1991). Because L is lower bounded (L ≥ 0), and L˙ is negative semi-definite, i.e. L˙ ≤ 0, then one obtains that L must have limit (there exists a non-negative constant ρ) such that limt→∞ L(t) = ρ. Moreover, Eq. (11.19) leads to conclusion that L(t) ≤ L(0), which implies that sξ , z, and wξ are bounded. Using Assumption 11.1 and Eqs. (11.2)–(11.8) it may be verified ¨ can be determined and it ˙ ξ , and F ¨ ξ are also bounded. Consequently, L ˙ ξ, F that s˙ξ , w t is bounded. One obtains limt→∞ 0 L(σ )dσ = limt→∞ L(t) − L(0) = ρ − L(0) < ˙ = 0, then L˙ → ∞ as t → ∞. ∞. From Barbalat’s lemma it results that limt→∞ L(t) This in turn implies that sξ , z → 0 as t → ∞.  Vehicle input signals. Using inversion of the relationship (2.61) (note that τ = ϒ −T π ) and (8.17)–(8.19) the input forces vector τ can be rewritten as follows: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + ϒ −T k D ϒ −1 s + k I z + f,

(11.20)

t where f = ϒ −T ϒ −1 0 f s(t)dt. Therefore, it can be observed that the gain matrix ϒ −T k D ϒ −1 includes dynamical couplings of the system. Consequently, the input signal τ is strictly related not only to kinematics but also to the vehicle couplings. This means that the matrix k D is chosen according to dynamics of the controlled system. Because the couplings are present in the gain by the matrix ϒ, they may make it difficult to choose the matrix k D . In order to avoid it the matrix N is applied. If it is necessary to evaluate the effects of the vehicle dynamics (instead of the couplings effect only), then it should be assumed k D = Nk∗D and = N ∗ where the diagonal matrices k∗D and ∗ are selected.

214

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

11.1.2 Velocity Tracking Controller with Adaptive Term Expressed in NGVC An algorithm of this type was proposed in Herman (2019). The control problem can be summarized in the given below theorem. Theorem 11.2 Consider the vehicle system model with dynamic equations given by (2.87), kinematic equations (2.79), and (2.82) for which after transformation of the vector into τ it is: τ = Bu, (11.21) where τ is the control force vector, B is the thruster configuration matrix, u is the thruster force vector under assumption that BBT > 0 together with the nonlinear controller expressed in terms of the NGVC:

= ζ˙ r + Cζ (ζ )ζ r + Dζ (ζ )ζ r + gζ (η) + k D sζ + −T k I z + f ζ ,

(11.22)

τ = ,

(11.23)

T

u = B τ, †

where:  z=

t

ν˜ (σ ) dσ,

(11.24)

0

ζ r = (ν d + z), sζ = ζ r − ζ = (˜ν + z), ν), s˙ζ = ζ˙ r − ζ˙ = (ν˙˜ + Q

(11.27)

f˙ ζ = sζ ,

(11.28)

(11.25) (11.26)

ζ is the lumped dynamics and ν˜ = ν d − ν is the velocity error vector, wζ = f ζ − F ˙ ˙ ˙ ζ = f ζ − Fζ is the time derivative of the unknown lumped dynamestimation error, w ics estimation error, k D = k TD > 0, k I = k TI > 0,  = T > 0, = T > 0. The system trajectories converges globally to a bounded neighborhood ρ (a constant value) of the origin [sζT , zT , wζT ]T = 0. Comment on Theorem 11.2. Information contained in Comment on Theorem 10.2 remains valid. Two assumptions are necessary to prove the stability of a closed-loop system containing the control algorithm. Similar assumptions can be found in Qiao and Zhang (2016), Soylu et al. (2008, 2016). Some assumptions are made. Assumption 11.3 (a) The desired trajectories in the vector ν d are differentiable in time and ν d , ν˙ d are known and bounded. ζ (η, ν, ν˙ , z, t) = (b) The nonlinear uncertainty vector defined by the formula F −T  F(η, ν, ν˙ , z, t) and its time derivative are bounded in the Euclidean norm, i.e.:

11.1 Control Algorithms Expressed in IQV

ζ (η, ν, ν˙ , z, t)|| ≤ a1 < ∞, ||F ˙ ζ (η, ν, ν˙ , z, t)|| ≤ a2 < ∞ ||F ¨ ζ (η, ν, ν˙ , z, t)|| ≤ a3 < ∞. ||F

215

(11.29)

Comment on Assumption 11.3. The argumentation is set out in Comment on Assumption 10.3. Assumption 11.4 The following inequality holds: T ˙ ζ −1 wζ | sζT [Dζ (ζ ) + k D ]sζ + zT T k I z ≥ |F T ˙ ζ −1 wζ < 0. only when F

(11.30)

˙ ζ Remark 11.3 For a vehicle which moves at low velocity values of the vector F are also small. Moreover, values of the adaptive vector wζ are reduced fast because it included the system dynamics. Under such conditions assumptions (11.29) and (11.30) seem reasonable. Proof The closed loop system (2.87), (2.82) together with the controller (11.22) can be written in the form: s˙ζ + [Cζ (ζ ) + Dζ (ζ ) + k D ]sζ + −T k I z + wζ = 0.

(11.31)

As a Lyapunov function candidate the following expression is proposed: L(sζ , z, wζ ) =

1 T 1 1 sζ sζ + zT k I z + wζT −1 wζ . 2 2 2

(11.32)

Calculating the time derivative of the function L (11.32) one obtains: ˙ ζ , z, wζ ) = sζT s˙ζ + ν˜ T k I z + w ˙ ζT −1 wζ . L(s

(11.33)

Taking into account Eqs. (11.31) and (11.28) one gets: ˙ ζ , z, wζ ) = sζT [−Cζ (ζ )sζ − Dζ (ζ )sζ − k D sζ − −T k I z − wζ ] L(s ˙ ˙ ζ )T −1 wζ = −sζT Cζ (ζ )sζ − sζT Dζ (ζ )sζ − sζT k D sζ +˜ν T k I z + (f ζ − F T ˙ ζ −1 wζ . −sζT −T k I z − sζT wζ + ν˜ T k I z + sζT T −1 wζ − F

(11.34)

From (2.83) it follows that sζT Cζ (ζ )sζ = ( sζ )T C(ν)( sζ ) = sT C(ν)s = 0 because sT C(ν)s = 0 for all s ∈ R n (Fossen 1994). It is also = T and −1 = I. Therefore, one has:

216

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

˙ ζ , z, wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − sζT −T k I z L(s T ˙ ζ −1 wζ , +˜ν T k I z − F

(11.35)

Referring to Eq. (11.26) one gets finally: T ˙ ζ , z, wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − zT T k I z − F ˙ ζ −1 wζ . L(s

(11.36)

Argumentation concerning (11.36) to show stability of the closed loop system in the presence of uncertainties and disturbances model is similar as previously. Method 1 was applied for manipulators (Zeinali and Notash 2010) and an engine torque control (Fazeli et al. 2012). For the considered vehicle the approach can be modify in the following manner. Two different assumptions are made to perform the stability analysis: slowly time-varying uncertainties and fast time-varying uncertainT ˙ ζ ties. If the uncertainties are arbitrarily large and slowly varying with time, then F is zero or negligible. Consequently, one obtains: ˙ ζ , z, wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − zT T k I z ≤ 0, L(s

(11.37)

which is negative or zero. Note that from (11.37) does not imply that L˙ → 0 as t → ∞ what means that the system trajectories may not converge to the desired state in the finite time. This problem is solved using Barbalat’s lemma (Slotine and Li 1991) as it is shown in Soylu et al. (2007) where underwater vehicles under assumption analogous to Assumption 11.4 were considered. For slowly time-varying uncertainties an asymptotic stability is guaranteed (Zeinali and Notash 2010). In reference Soylu et al. (2007) and next in Soylu et al. (2008) slow motion of underwater vehicles was taken into account and for this reason, the proposed approach is valid. Similar considerations can be found in other works, i.e. Gan et al. (2018), Ismail et al. (2016), Lakhekar and Waghmare (2017). If the uncertainties are assumed as arbitrarily large and fast time-varying but norm bounded (Zeinali and Notash 2010) then the sufficient condition for L˙ to be negative T ˙ ζ > 0 and wζ > 0 are met then ˙ ζ −1 wζ ≥ 0. When the conditions F in (11.36) is F stabilization of the closed-loop system is ensured. For a well designed controller T ˙ ζ −1 wζ < 0 one gets the it can be assumed that wζ → 0. In the worse case, i.e. F inequality: ˙ ζ , z, wζ ) ≤ −sζT [Dζ (ζ ) + k D ]sζ − zT T k I z + ρ, L(s

(11.38)

T ˙ ζ −1 wζ means a positive scalar value (in the sense of maximal in which ρ = −F    ˙ T −1    value of this term, i.e. ρ = Fζ wζ  in Eq. (11.38)). The boundedness and con-

vergence to a small vicinity of the state space origin is guaranteed and it is possible

11.1 Control Algorithms Expressed in IQV

217

to reduce the tracking error using the design parameters k D , k I , and . Therefore, even in the worst case occurs, the system stability can also be ensured by increasing k D , k I , and . ˙ ζ If the assumptions Assumptions 11.3 and 11.4 are fulfilled then the vectors F and wζ are bounded. Thus, based on control theory one can conclude that the system trajectories converges globally to a bounded neighborhood ρ (small constant value) of the origin of the system (2.79), (2.87) together with the controller (11.22). Method 2 is applied to underwater vehicles (Qiao and Zhang 2016). Using it the results from Soylu et al. (2007, 2008) are more rigorously presented. The argumentation appropriate for our case is given below. If the Assumptions 11.3 and 11.4 are valid then one obtains: T ˙ ζ , η, ˙ ζ −1 wζ ≤ 0. ˜ wζ ) = −sζT [Dζ (ζ ) + k D ]sζ − zT T k I z − F L(s

(11.39)

From Soylu et al. (2008) it is known that inequality of type as (11.39) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. It is possible to solve this problem applying Barbalat’s lemma, e.g. Slotine and Li (1991). Because L is lower bounded ˙ is negative semi-definite, i.e. L ˙ ≤ 0, then one obtains that L must (L ≥ 0), and L have limit (there exists a non-negative constant ρ) such that limt→∞ L(t) = ρ. Moreover, Eq. (11.39) leads one to conclusion that L(t) ≤ L(0), which implies that sζ , z, and wζ are bounded. Using Assumptions 11.3 and Eqs. (11.22)–(11.28) it may ¨ can be deter˙ ζ , and F ¨ ζ are also bounded. Consequently, L ˙ ζ, F be verified that s˙ζ , w t mined and it is bounded. One obtains limt→∞ 0 L(σ )dσ = limt→∞ L(t) − L(0) = ˙ = 0, then L˙ → 0 ρ − L(0) < ∞. From Barbalat’s lemma it results that limt→∞ L(t) as t → ∞. This in turn implies that sζ , z → 0 as t → ∞.  Vehicle input signals. The controller (11.22) is expressed in terms of the vector ζ . The input signal is obtained from the relationship (2.86) calculating τ = T

because the matrix T is invertible, namely: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + T k D s + k I z + f,

(11.40)

t where f = T 0 f s(t)dt, and the symbols are defined in (8.17)–(8.19). Now it is easy observable that the velocity gain matrix (after transformation) is T k D . Assuming that k D = δ I one has the following gain matrix K D = T k D = δ M. First benefits resulting from the controller form is that the gain coefficients are strictly related to the dynamics of the vehicle hidden in the matrix K D . This is because they depend on the parameters set. Consequently, it is not necessary to choose a symmetric gain matrix. The diagonal matrix k D serves for rather for precise tuning of the controller. Moreover, the appropriate gains are selected for the considered system that allows one to avoid searching their values using experience of the researcher only. The controller can be useful for fast error convergence. This

218

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

property is ensured thanks to the dynamical parameters which are included in the matrix .

11.1.3 Simplified Forms of Controllers GVC-AVTCBF. In the simplified form of the (11.20) for a diagonal matrix M (ϒ = I) the input forces vector τ has the following form: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + k D s + k I z + f.

(11.41)

Therefore, it is observed that the gain matrix K D = k D does not include the dynamical parameters of the system (more precisely dynamical couplings). NGVC-AVTCBF. The simplified controller of (11.40) is, e.g. if M is a diagonal 1 matrix. In this case ϒ = I and consequently = N 2 . Thus: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + N 2 k D N 2 z + k I z + f. 1

1

1

1

(11.42)

Now one has the following gain matrix K D = N 2 k D N 2 which means that the set of dynamical parameters is still taken into consideration. Algorithms for Vehicle Model with 3 DOF Moving Horizontally. The algorithms can be also applied to a reduced planar model of the vehicle according to the remark mentioned in Chap. 6.

11.2 Simulation Results In this section selected results of application of both controllers are given. Simulations were performed using Matlab/Simulink (the fifth-order Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

11.2.1 Velocity Tracking Control Using GVC 11.2.1.1

Controller for 6 DOF Fully Actuated Underwater Vehicle

The GVC control algorithm for fully actuated underwater vehicle was considered in Herman (2020). Moreover, in the reference its application to the vehicle dynamics analysis was shown. Other example of use will be given in Chap. 13.

11.2 Simulation Results

219

11.2.2 Velocity Tracking Control Using NGVC 11.2.2.1

Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: verification of the NGVC algorithm performance taking into consideration the vehicle dynamics of an underwater vehicle model. For tracking it is assumed the following desired velocity profile (Fig. 11.1):  π  t + 0.4, 0, 0.1 cos t − 0.05, 30 20  π 0.2 sin t + 0.1, 0, 0.2e−t ]T . 10

ν d = [0.5 sin

π

(11.43)

The disturbances function including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments described by:

Fig. 11.1 Desired velocity and disturbances profiles: a desired linear velocity profiles u d , vd , wd ; b desired angular velocity profiles pd , qd , rd ; c disturbances forces profiles  F X , FY ,  F Z, d disturbances moments profiles  FK,  F M,  FN

220

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

⎤ 100 f 7 (t) e−2.5 + 20 ⎢ 20 f 7 t) e−2.5 + 4 ⎥ ⎥ ⎢ −1 ⎥ ⎢  = ⎢ 150 f 8 (t) e−1 + 150 ⎥ , F ⎢ 3 f 8 (t) e + 3 ⎥ ⎥ ⎢ ⎣ 150 f 9 (t) e−3 + 12 ⎦ 50 f 9 (t) e−3 + 4 ⎡

(11.44)

π where the functions f 7 (t) = π4 cos( π5 t) + 10 sin( π3 t), f 8 (t) = 0.2 sin(t)e−1 + 0.7 π π π π 0.1 ecos(−t ) , and f 9 (t) = 5 sin( 15 t) + 4 cos( 5 t). The gain coefficients for the NGVC controller were selected to ensure fast errors convergence as follows:

k D = diag{10, 10, 10, 10, 10, 10}, k I = diag{1, 1, 1, 1, 1, 1},

(11.45)

 = diag{0.1, 0.1, 0.1, 0.1, 0.1, 0.1}, = diag{40, 40, 40, 50, 50, 50}. (11.46) The used here gain matrices k D and (instead of k∗D and ∗ ) have slightly smaller values than for the GVC controller in order to show differences between answers from the NGVC controller. At it is observed in Fig. 11.2a–b the steady state for linear and angular velocities is provided after about 1 s. The errors v, q are close to zero in a short time. The velocities are tracked correctly. The velocity convergence results from that the quantities s also quickly achieve the steady-state as it is observed in Fig. 11.2c–d. The lumped dynamics estimation errors w are approaching the end value quickly (Fig. 11.2e–f). However, in the first phase of vehicle motion their values are great. It arises from that the full vehicle dynamics is included into the control. For the same reason the forces (especially τ X , τ Z ), and moments (especially τ M , τ N ) have large values after the start as it is noticeable in Fig. 11.2g–h.

11.2.2.2

Controller for 6 DOF Fully Actuated Airship

Objective: to show performance of the NGVC algorithm for a fully actuated indoor airship. For tracking it is assumed the following desired velocity profile: π   π T  π  t + 0.3, 0, −0.5 cos t + 0.5, 0, 0, 0.5 sin t ν d = 1.2 sin . 20 10 10

(11.47) In Fig. 11.3a–b the desired linear and angular velocity profiles are given. The used disturbance forces and moments are shown in Fig. 11.3c–d. In this case vd , pd , and qd are equal to zero. It is expected that these variables will be close to zero if the dynamics of the vehicle is not very important. The disturbances vector including the unknown dynamics as well as the external disturbances is expressed by the vector of forces and moments:

11.2 Simulation Results

221

Fig. 11.2 Simulation results for NGVC controller and underwater vehicle: a linear velocity errors; b angular velocity errors; c errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

222

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

Fig. 11.3 Desired velocity and disturbances profiles: a desired linear velocity profiles u d , vd , wd ; b desired angular velocity profiles pd , qd , rd ; c disturbances forces profiles  F X,  FY,  F Z, d disturbances moments profiles  FK,  F M,  FN

⎤ 10 f 10 (t)e−1 + 20 ⎢ f 10 (t)e−1 + 2 ⎥ ⎥ ⎢ ⎢ 20 f 11 (t) − 20 ⎥ ⎥, ⎢  F=⎢ ⎥ f 11 (t) − 1 ⎥ ⎢ ⎣ 500 f 12 (t)e−3 + 40 ⎦ 50 f 12 (t)e−3 + 4 ⎡

(11.48)

where the functions are defined as: f 10 (t) = 6 cos(0.5 t) + sin(0.8 t), f 11 (t) = π t) + π4 cos( π5 t). 0.4 sin(0.4 t) e−5 + 0.4 ecos(−0.3) t , and f 12 (t) = π5 sin( 15 The gain coefficients of the NGVC controller were selected as follows: k∗D = diag{25, 25, 25, 25, 25, 25}, k I = diag{1, 1, 1, 1, 1, 1},  = diag{0.1, 0.1, 0.1, 0.1, 0.1, 0.1} ∗ = diag{400, 400, 400, 400, 400, 400}.

(11.49)

For the NGVC controller it is assumed that k D = k∗D and = ∗ . From Fig. 11.4a–b it is noticeable that the linear and angular velocities tend to limited values quickly (after about 1 s). However, the error v increases rapidly in

11.2 Simulation Results

223

Fig. 11.4 Simulation results for NGVC controller and indoor airship: a linear velocity errors; b angular velocity errors; c errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

224

11 IQV Velocity Tracking Control with Adaptive Term in Body-Fixed Frame

the first phase of airship’s motion as can be seen in Fig. 11.4a. This is unexpected because it should be close to zero. This effect can be explained the presence of vehicle dynamics in the control gain matrix. It can be concluded that the velocity v is very sensitive to dynamical parameters changes. Similar conclusion can be made from Fig. 11.4b for the errors p and r (q is less sensitive). There is a relationship between the velocities and quantities si . For this reason if si tend to limited values quickly then they are reduced in a short time as shown in Fig. 11.4c–d. All lumped dynamics estimation errors w converge to their end value quickly as can be seen from Fig. 11.4e–f. However, in the first phase of motion their values are great. It is because the full vehicle dynamics is included in the control gains (the full dynamics in included). For the same reason the forces (τ X ), and torques (τ M , τ N ) have large values after the vehicle starts as is observed in Fig. 11.4g–h.

11.3 Closing Remarks Two control algorithms with an adaptive term for velocity tracking realized using the IQV were proposed in this chapter. The adaptation means that in the control algorithm the model of disturbances is described by an additional term. The functions in this term represent approximately all disturbances, namely internal and external as well as not exactly known parameters. The controllers can be applied both to full 6 DOF vehicle models and reduced 3 DOF planar models of vehicles. They are suitable for underwater vehicles, indoor airships, and some horizontally moving vehicles at low speed. From the controllers form some simplified algorithms were deduced. Simulation research conducted on two full 6 DOF models showed the usability of the control strategy.

References Fazeli A, Zeinali M, Khajepour A (2012) Application of adaptive sliding mode control for regenerative braking torque control. IEEE/ASME Trans Mechatron 17(4):745–755 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Gan W, Zhu D, Ji D (2018) QPSO-model predictive control-based approach to dynamic trajectory tracking control for unmanned underwater vehicles. Ocean Eng 158:208–220 Herman P (2019) Numerical test of underwater vehicle dynamics using velocity controller. In: Proceedings of 2019 12th international workshop on robot motion and control (RoMoCo), Poznan, Poland July 8–10, pp 26–31 Herman P (2020) Velocity tracking controller for simulation analysis of underwater vehicle model. J Mar Eng Technol 19(4):229–239 Ismail ZH, Mokhar MBM, Putranti VWE, Dunnigan MW (2016) A robust dynamic region-based control scheme for an autonomous underwater vehicle. Ocean Eng 111:155–165 Lakhekar GV, Waghmare LM (2017) Robust maneuvering of autonomous underwater vehicle: an adaptive fuzzy PI sliding mode control. Intell Serv Robot 10:195–212

References

225

Qiao L, Zhang W (2016) Double-loop chattering-free adaptive integral sliding mode control for underwater vehicles. In: Proceedings MTS/IEEE OCEANS conference Shanghai, China, 10–13 April 2016, pp 1–6 Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs Soylu S, Buckham BJ, Podhorodeski RP (2007) Robust control of underwater vehicles with faulttolerant infinity-norm thruster force allocation. In: Proceedings of OCEANS 2007, 29 September– 4 October, Vancouver, BC, Canada, 2007, pp 1–10 Soylu S, Buckham BJ, Podhorodeski RP (2008) A chattering-free sliding-mode controller for underwater vehicles with fault-tolerant infinity-norm thrust allocation. Ocean Eng 35:1647–1659 Soylu S, Proctor AA, Podhorodeski RP, Bradley C, Buckham BJ (2016) Precise trajectory control for an inspection class ROV. Ocean Eng 111:508–523 Zeinali M, Notash L (2010) Adaptive sliding mode control with uncertainty estimator for robot manipulators. Mech Mach Theory 45:80–90

Chapter 12

IQV Position and Velocity Tracking Control with Adaptive Term in Earth-Fixed Frame

Abstract This chapter is devoted to adaptive nonlinear controllers using the IQV, for a class of vehicles, which allow the trajectory position and velocity tracking in the Earth-Fixed Frame in the presence of disturbances described by a model. The proposed controllers guarantee the position and velocity error convergence in the finite time. The input control signals in the Earth-Fixed Frame, some simplified controllers and application for 3 DOF horizontally moving vehicles are discussed either. Additionally, simulation results for a 6 DOF model of an underwater vehicle and an indoor airship are given here.

12.1 Control Algorithms Expressed in IQV This section deals with two types of controllers, namely the GVC based controller and the NGVC based controller.

12.1.1 Position and Velocity Tracking Controller Expressed in GVC This type of control algorithm for a 3 DOF underwater vehicle moving horizontally was proposed in Herman (2017) and some attempt to extend it for a vehicle with 6 DOF was made in Herman (2021). The problem, in general form, can be presented in the below given theorem. Theorem 12.1 Consider the vehicle model described by equations of motion (2.75), (2.69), and (2.37) together with the following controller: π η = Nη (η)ξ˙ r + Cξ (ξ , η)ξ r + Dξ (ξ , η)ξ r + ge ξ (η) +k D sξ + ZT (η)k P η˜ + f ξ ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_12

(12.1)

227

228

12 IQV Position and Velocity Tracking Control with Adaptive …

where: ˜ ξ r = Z−1 (η)(η˙ d + η), −1 ˙˜ ˜ + Z−1 (η)(¨ηd + η), ξ˙ r = Z˙ (η)(η˙ d + η) −1 ˜ sξ = ξ r − ξ = Z (η)(η˙˜ + η), −1 ˙˜ ˜ + Z−1 (η)(η¨˜ + η), s˙ξ = ξ˙ r − ξ˙ = Z˙ (η)(η˙˜ + η) f˙ ξ = sξ , θ = ±90 deg,

(12.2) (12.3) (12.4) (12.5) (12.6)

˙ η¨˜ = η¨ d − η¨ are the position error vector, the velocin which η˜ = ηd − η, η˙˜ = η˙ d − η, η ξ is the lumped ity vector, and the acceleration vector, respectively, wξ = f ξ − F ˙ ˙ η ξ is the time derivative of the unknown ˙ ξ = f ξ − F dynamics estimation error, w lumped dynamics estimation error. In general f ξ = f ∗ξ + c (where c is a scalar vector). Moreover, k D = k TD > 0, k P = k TP > 0,  = T > 0,  =  T > 0 and Nη (η) is a diagonal strictly positive matrix. The system trajectories converges locally to a bounded neighborhood ρ (a constant value) of the origin [sξT , η˜ T , wξT ]T = 0. Comment on Theorem 12.1. The input signal in the Earth-Fixed Frame and in the Body-Fixed Frame can be calculated from Eq. (9.6). The Comment on Theorem 9.1 is also valid. Moreover, in the control scheme a vector representing the lumped uncertainty occurs. Such lumped uncertainty vector representing the unknown dynamics and external disturbances has been discussed in the literature, e.g. in Cui et al. (2017), Peng et al. (2019), Qiao and Zhang (2016, 2018), Qu et al. (2018), Soylu et al. (2008, η ξ because it is expressed 2016). In the proposed algorithm the vector is denoted by F in the GVC. Two assumptions were made to prove the stability of the system with the controller. These are based on information from Qiao and Zhang (2016), Soylu et al. (2008, 2016), which used similar underwater vehicle control algorithms. Assumption 12.1 (a) The desired trajectories ηd , η˙ d are differentiable in time and η˙ d , η¨ d are known and bounded. η ξ is defined by the following formula (b) The nonlinear uncertainty vector F T η (η, η, η ξ (η, η, ˙ η¨ , ν, t) = Z (η)F ˙ η¨ , ν, t) and the time derivatives are bounded F in the Euclidean norm, i.e.: η ξ (η, η, ˙ η¨ , ν, t)|| ≤ a1 < ∞, ||F ˙ ˙ η¨ , ν, t)|| ≤ a2 < ∞, ||Fη ξ (η, η, ¨ ˙ η¨ , ν, t)|| ≤ a3 < ∞. ||Fη ξ (η, η,

(12.7)

Comment on Assumption 12.1. The problem of the boundedness of the lumped uncertainty vector was discussed in detail in Qiao and Zhang (2019). It was shown there that if Eqs. (2.51)–(2.52) are taken into account then the lumped uncertainty can be η (η, η, η ≡ F ˙ η¨ , ν, t)): bounded by the expression (F

12.1 Control Algorithms Expressed in IQV

η || ≤ b0 + b1 ||η|| ˙ + b2 ||ν|| ||η|| ˙ + b3 ||η|| ˙ 2, ||F

229

(12.8)

where bi (i = 0, 1, 2) are unknown positive constants. The lumped system uncertainty bound (12.8) can be applied if the required thrust forces only sometimes exceed the thruster saturation limit. Moreover, in Qiao and Zhang (2019) it was stated that (taking into account results given in Zhang et al. 2015) when the effect of thrust saturation is not serious, it can be assumed that the control effort τ η is bounded. Based on the assumptions made in Qiao and Zhang (2019) and recalling that the matrix Z(η) results from decomposition of the inertia matrix Mη (η), it can be η ξ and its time derivative will be bounded if the concluded that also the vector F required thrust forces do not always exceed saturation limits of the thruster. Also, Assumption 12.1 can be deduced from Qiao and Zhang (2016) based on similar reasoning. Assumption 12.2 The following inequality holds: T ˙ η ξ  −1 wξ | sξT [Dξ (ξ , η) + k D ]sξ + η˜ T T k P η˜ ≥ |F T ˙ η ξ  −1 wξ < 0. only when F

(12.9)

˙ η ξ are small. MoreRemark 12.1 For slowly moving vehicles values of the vector F over, using the adaptive term values of the vector wξ decrease. Hence, assumptions and conclusions seem to be reasonable. Proof The closed loop system, (2.69), (2.75) together with the controller (12.1) can be given in the form: Nη (η)˙sξ + [Cξ (ξ , η) + Dξ (ξ , η) + k D ]sξ + ZT (η)k P η˜ + wξ = 0.

(12.10)

The following formula is proposed as a Lyapunov function candidate: ˜ wξ ) = L(sξ , η,

1 1 T 1 s Nη (η)sξ + η˜ T k P η˜ + wξT  −1 wξ . 2 ξ 2 2

(12.11)

Calculating the time derivative of L (12.11) one has: 1 ˙ ˙˜ T k P η˜ + w ˙ ξ , η) ˜ = sξT Nη (η)˙sξ + sξT N ˙ ξT  −1 wξ . L(s η (η)sξ + η 2

(12.12)

Taking into account (2.70) one determines: ˙ η (η) = d (ZT (η)Mη (η)Z(η)) N dt T ˙ ˙ η (η)Z(η) + ZT (η)Mη (η)Z(η). ˙ = Z (η)Mη (η)Z(η) + ZT (η)M

(12.13)

230

12 IQV Position and Velocity Tracking Control with Adaptive …

Making use of (12.10) one gets: ˙ ξ , η, ˜ wξ ) = sξT [−Cξ (ξ , η)sξ − Dξ (ξ , η)sξ − k D sξ − wξ ] L(s 1 ˙ T ˙ T T ˙ η ξ )T  −1 wξ . ˜ + η˙˜ k P η˜ + (f ξ − F + sξT N η (η)sξ − sξ Z (η)k P η 2

(12.14)

Applying next (2.71) and (12.13) one can write: T ˙ ξ , η, ˙ ˜ wξ ) = sξT [−ZT (η)Mη (η)Z(η)s L(s ξ − Z (η)Cη (ν, η)Z(η)sξ −Dξ (ξ , η)sξ − k D sξ − wξ ]  1  T ˙ η (η)Z(η) + ZT (η)Mη (η)Z(η) ˙ sξ + sξT Z˙ (η)Mη (η)Z(η) + ZT (η)M 2   T ˙ ˙ η ξ )T  −1 wξ = −sξT Dξ (ξ , η) + k D sξ −sξT ZT (η)k P η˜ + η˙˜ k P η˜ + (f ξ − F 1 1 ˙T ˙ ˙ +sξT [ ZT (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η) + Z (η)Mη (η)Z(η) 2 2   1 ˙ η (η) − 2Cη (ν, η) Z(η)]sξ − sξT ZT (η)k P η˜ + η˜˙ T k P η˜ + ZT (η) M 2 ˙ ˙ η ξ )T  −1 wξ . −sξT wξ + (f ξ − F (12.15)

Using the property (2.48) and (12.6) the above expression can be rewritten in the following form: ˙ ξ , η, ˜ wξ ) = −sξT [Dξ (ξ , η) + k D ]sξ L(s  1  T ˙ + sξT Z˙ (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η) sξ 2 T T ˙ η ξ  −1 wξ . −sξT ZT (η)k P η˜ + η˙˜ k P η˜ − sξT wξ + sξT  T  −1 wξ − F

(12.16)

  T ˙ is a skewHowever, the matrix W = Z˙ (η)Mη (η)Z(η) − ZT (η)Mη (η)Z(η) symmetric one (Lemma 9.1), and as a result, the second term of the above expression vanishes. Moreover, taking into account (12.4) and the relationships  =  T and  −1 = I one obtains: ˙ ξ , η, ˜ wξ ) = −sξT [Dξ (ξ , η) + k D ]sξ L(s T T ˙ η ξ  −1 wξ , ˜ T Z−T (η)ZT (η)k P η˜ + η˙˜ k P η˜ − F −(η˙˜ + η)

(12.17)

and finally: T ˙ ξ , η, ˙ η ξ  −1 wξ . ˜ wξ ) = −sξT [Dξ (ξ , η) + k D ]sξ − η˜ T T k P η˜ − F L(s

(12.18)

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231

Consider the result obtained in (12.18). Two kind of argumentation are used in the literature to show stability of the closed loop system in the presence of model uncertainties and disturbances. Method 1 was applied for manipulators (Zeinali and Notash 2010) and an engine torque control (Fazeli et al. 2012). For the considered class of vehicles the approach can be modify in the following manner. Two different assumptions are made to perform the stability analysis: slowly time-varying uncertainties and fast time-varying uncertainties. If the uncertainties are arbitrarily large and slowly varying with time, T ˙ η ξ is zero or negligible. Consequently, one gets: then F ˙ ξ , η, ˜ wξ ) = −sξT [Dξ (ξ , η) + k D ]sξ − η˜ T T k P η˜ ≤ 0, L(s

(12.19)

which is negative or zero. Note that from (12.19) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. This problem is solved using Barbalat’s lemma (Slotine and Li 1991) as is shown in Soylu et al. (2007) where underwater vehicles under assumption analogous to Assumption 12.2 were considered. For slowly time-varying uncertainties an asymptotic stability is guaranteed (Zeinali and Notash 2010). In Soylu et al. (2007) and next in Soylu et al. (2008) slow motion of underwater vehicles was taken into account and for this reason, the proposed approach is valid. Similar considerations can be found in other works, i.e. Gan et al. (2018), Ismail et al. (2016), Lakhekar and Waghmare (2017). If the uncertainties are assumed as arbitrarily large and fast time-varying but norm bounded (Zeinali and Notash 2010) then the sufficient condition for L˙ to be negative T T ˙ η ξ > 0 and wξ > 0 are met then ˙ η ξ  −1 wξ ≥ 0. When the conditions F in (12.18) is F stabilization of the closed-loop system is ensured. For a well designed controller it T ˙ η ξ  −1 wξ < 0 one gets the can be assumed that wξ → 0. In the worse case, i.e. F inequality: ˙ ξ , η, ˜ wξ ) ≤ −sξT [Dξ (ξ , η) + k D ]sξ − η˜ T T k P η˜ + ρ, L(s

(12.20)

T ˙ η ξ  −1 wξ means a positive scalar value (in the sense of maximal in which ρ = −F    ˙ T −1  η ξ  wξ  in Eq. (12.20)). The boundedness and convalue of this term, i.e. ρ = F  vergence to a small vicinity of the state space origin is guaranteed and it is possible to reduce the tracking error using the design parameters k D , k P , and . Therefore, even in the worst case occurs, the system stability can also be ensured by increasing k D , k P , and . Method 2 is applied to underwater vehicles (Qiao and Zhang 2016). Using it the results from Soylu et al. (2007, 2008) are more rigorously presented. The argumentation appropriate for our case is as follows. If the Assumptions 12.1 and 12.2 are valid then one obtains:

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T ˙ ξ , η, ˙ η ξ  −1 wξ ≤ 0. ˜ wξ ) = −sξT [Dξ (ξ , η) + k D ]sξ − η˜ T T k P η˜ − F L(s

(12.21)

From Soylu et al. (2008) it is known that inequality of type as (12.21) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. It is possible to solve this problem applying Barbalat’s lemma, e.g. Slotine and Li (1991). Because L is lower bounded (L ≥ 0), andL˙ is negative semi-definite, i.e.L˙ ≤ 0, then one obtains that L must have limit (there exists a non-negative constant ρ) such that limt→∞ L(t) = ρ. Moreover, ˜ and wξ are Eq. (12.21) leads to conclusion that L(t) ≤ L(0), which implies that sξ , η, ˙ ξ, bounded. Using Assumption 12.1 and Eqs. (12.1)–(12.6) it may be verified that s˙ξ , w ¨ ˙ ¨ Fη ξ , and Fη ξ are also bounded. Consequently,L can be determined and it is bounded. t One obtains limt→∞ 0 L(σ )dσ = limt→∞ L(t) − L(0) = ρ − L(0) < ∞. From ˙ = 0, then L˙ → 0 as t → ∞. This in Barbalat’s lemma it results that limt→∞ L(t) turn implies that sξ , η˜ → 0 as t → ∞.  Earth-Fixed Frame vehicle input signal. After transformation into the velocity space, i.e. after inversion of (2.74) and recalling (9.22)–(9.24), one gets the following control input: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) +Z−T (η)k D Z−1 (η)s + k P η˜ + f η ,

(12.22)

t where  f η = Z−T (η) 0 f Z−1 (η(t))s(t)dt. Note that, in the fifth term, i.e. Z−T (η)k D Z−1 (η) the resulting velocity gain matrix K D = Z−T (η)k D Z−1 (η) includes also dynamical couplings of the system which occur if the vehicle moves. Consequently, the input signal τ η is strictly related not only to kinematics but also to the dynamical couplings. This means that the matrix k D is selected taking into account the couplings. However, if instead of selecting k D the matrix k∗D is selected and the matrix N is used, then k D = Nk∗D (and also  D = N ∗D ). In this case, the matrix N arises from decomposition of the matrix M and effects of the vehicle ˙ η (η). dynamics can be evaluated. Note that this matrix is different than the matrix N However, even if the system parameters are not exactly known, due to the matrix Z(η), the velocity, and after appropriate transformation, the position and velocity errors tend to zero. However, for Eq. (12.22) the couplings may make it difficult to choose the appropriate k D matrix.

12.1.2 Position and Velocity Tracking Controller Expressed in NGVC An algorithm of this type for an underwater vehicle model was presented in Herman (2020). The general control scheme for a 6 DOF vehicle model expressed in the Earth-Fixed Frame is summarized in the below given theorem.

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Theorem 12.2 Consider the vehicle model described by equations of motion (2.100), (2.94), and (2.37) together with the following controller: η = ζ˙ r + Cζ (ζ , η)ζ r + Dζ (ζ , η)ζ r + ge ζ (η) + k D sζ +−T (η)k P η˜ + f ζ ,

(12.23)

where: ˜ ζ r = (η)(η˙ d + η), ˙˜ ˙ ˜ + (η)(¨ηd + η), η˙ d + η) ζ˙ r = (η)( ˜ sζ = ζ r − ζ = (η)(η˙˜ + η), ˙ ˙˜ ˙ ˙ ˙ ˜ + (η)(η¨˜ + η), s˙ζ = ζ − ζ = (η)(η˜ + η) r

f˙ ζ = sζ ,

θ = ±90 deg,

(12.24) (12.25) (12.26) (12.27) (12.28)

˙ η¨˜ = η¨ d − η¨ are the position error vector, the velocin which η˜ = ηd − η, η˙˜ = η˙ d − η, η ζ is the lumped ity vector, and the acceleration vector, respectively, wζ = f ζ − F ∗   dynamics estimation error. In general f ζ = f ζ + c (where c is a scalar vector). ˙ ˙ η ζ is the time derivative of the unknown lumped dynam˙ ζ = f ζ − F Moreover, w ics estimation error, k D = k TD > 0, k P = k TP > 0,  = T > 0,  =  T > 0. The system trajectories converge locally to a bounded neighborhood, i.e. constant value ρ of the origin [sζT , η˜ T , wζT ]T = 0. Comment on Theorem 12.2. The Comment on Theorem 12.1 is valid but here it refers ˙ η ζ and the input signal from Eq. (9.42). to vector F Two assumptions were made to prove the stability of the system with the controller. These are based taking into account (Qiao and Zhang 2016; Soylu et al. 2008, 2016), in which similar underwater vehicle control algorithms were used. Assumption 12.3 (a) The desired trajectories ηd , η˙ d are differentiable in time and η˙ d , η¨ d are known and bounded. η ζ is defined by the following formula (b) The nonlinear uncertainty vector F −T   ˙ η¨ , ν, t) =  (η)Fη (η, η, ˙ η¨ , ν, t) and its time derivatives are bounded in Fη ζ (η, η, the Euclidean norm, i.e.: η ζ (η, η, ˙ η¨ , ν, t)|| ≤ a1 < ∞, ||F ˙ ˙ η¨ , ν, t)|| ≤ a2 < ∞ ||Fη ζ (η, η, ¨ η ζ (η, η, ˙ η¨ , ν, t)|| ≤ a3 < ∞. ||F

(12.29)

Comment on Assumption 12.3. The Comment on Assumption 12.1 is valid. However here, the assumptions made in Qiao and Zhang (2019) shall be referred to the matrix (η) results from decomposition of the inertia matrix Mη (η), it can be concluded

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η ζ and its time derivative will be bounded if the required thrust that also the vector F forces do not always exceed saturation limits of the thruster. Also, Assumption 12.3 can be deduced from Qiao and Zhang (2016) based on similar reasoning. Assumption 12.4 The following inequality holds: T ˙ η ζ  −1 wζ | sζT [Dζ (ζ , η) + k D ]sζ + η˜ T T k P η˜ ≥ |F T ˙ η ζ  −1 wζ < 0. only when F

(12.30)

Remark 12.2 The control algorithm (12.23) like the algorithm (12.1) is suitable ˙ η ζ are small then and the for slowly moving vehicles because values of the vector F adaptive term values of the vector wζ decrease. Proof The closed loop system (2.100), (2.94) together with the controller (12.23) has the form: s˙ζ + [Cζ (ζ , η) + Dζ (ζ , η) + k D ]sζ + −T (η)k P η˜ + wζ = 0.

(12.31)

The following Lyapunov function candidate is proposed: ˜ wζ ) = L(sζ , η,

1 1 T 1 s sζ + η˜ T k P η˜ + wζT  −1 wζ . 2 ζ 2 2

(12.32)

Calculating the time derivative of L (12.32) one gets: T ˙ ζ , η, ˜ wζ ) = sζT s˙ζ + η˙˜ k P η˜ + w ˙ ζT  −1 wζ . L(s

(12.33)

Using Eqs. (12.31) and (12.28) one obtains: ˙ ζ , η, ˜ wζ ) = sζT [−Cζ (ζ , η)sζ − Dζ (ζ , η)sζ − k D sζ L(s T ˙ ˙ η ζ )T  −1 wζ −−T (η)k P η˜ − wζ ] + η˙˜ k P η˜ + (f ζ − F = −sζT Cζ (ζ , η)sζ − sζT Dζ (ζ , η)sζ − sζT k D sζ − sζT −T (η)k P η˜ T T ˙ η ζ  −1 wζ . −sζT wζ + η˙˜ k P η˜ + sζT  T  −1 wζ − F

(12.34)

However, as was shown in Lemma 9.2 sζT Cζ (ζ , η)sζ = 0. Recalling that,  =  T and  −1 = I and taking into consideration (12.26) one can write: ˙ ζ , η, ˜ wζ ) = −sζT [Dζ (ζ , η) + k D ]sζ L(s  T T T ˙ η ζ  −1 wζ . ˜ − (η)(η˙˜ + η) −T (η)k P η˜ + η˙˜ k P η˜ − F

(12.35)

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235

The above expression can be simplified as follows: T ˙ ζ , η, ˙ η ζ  −1 wζ . ˜ wζ ) = −sζT [Dζ (ζ , η) + k D ]sζ − η˜ T T k P η˜ − F L(s

(12.36)

Consider the result obtained in (12.36). Two kind of argumentation are used in the literature to show stability of the closed-loop system if the model of uncertainties and disturbances is taken into consideration. Method 1 was applied for manipulators (Zeinali and Notash 2010) and an engine torque control (Fazeli et al. 2012). For the considered class of vehicles the approach can be modify in the following manner. Two different assumptions are made to perform the stability analysis: slowly time-varying uncertainties and fast time-varying uncertainties. If the uncertainties are arbitrarily large and slowly varying with time, T ˙ η ζ is zero or negligible. Consequently, one gets: then F ˙ ζ , η, ˜ wζ ) = −sζT [Dζ (ζ , η) + k D ]sζ − η˜ T T k P η˜ ≤ 0, L(s

(12.37)

which is negative or zero. Note that from (12.37) does not imply that L˙ → 0 as t → ∞ what means that the system trajectories may not converge to the desired state in the finite time. This problem is solved using Barbalat’s lemma (Slotine and Li 1991) as it is shown in Soylu et al. (2007) where underwater vehicles under assumption analogous to Assumption 12.4 were considered. For slowly time-varying uncertainties an asymptotic stability is guaranteed (Zeinali and Notash 2010). In Soylu et al. (2007) and next in Soylu et al. (2008) slow motion of underwater vehicles was taken into account and for this reason, the proposed approach is valid. Similar considerations can be found in other works, i.e. Gan et al. (2018), Ismail et al. (2016), Lakhekar and Waghmare (2017). If the uncertainties are assumed as arbitrarily large and fast time-varying but norm bounded (Zeinali and Notash 2010) then the sufficient condition for L˙ to be negative T T ˙ η ζ > 0 and wζ > 0 are met then ˙ η ζ  −1 wζ ≥ 0. When the conditions F in (12.36) is F stabilization of the closed loop system is ensured. For a well designed controller it T ˙ η ζ  −1 wζ < 0 one gets the can be assumed that wζ → 0. In the worse case, i.e. F inequality: ˙ ζ , η, ˜ wζ ) ≤ −sζT [Dζ (ζ , η) + k D ]sζ − η˜ T T k P η˜ + ρ, L(s

(12.38)

T ˙ η ζ  −1 wζ means a positive scalar value (in the sense of maximal in which ρ = −F    ˙ T −1    value of this term, i.e. ρ = Fη ζ  wζ  in Eq. (12.38)). The boundedness and convergence to a small vicinity of the state space origin is guaranteed and it is possible to reduce the tracking error using the design parameters k D , k P , and . Therefore, even in the worst case occurs, the system stability can also be ensured by increasing k D , k P , and .

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Method 2 is applied to underwater vehicles (Qiao and Zhang 2016). Using it the results from Soylu et al. (2007, 2008) are more rigorously presented. The argumentation appropriate for our case is as follows. If the Assumptions 12.3 and 12.4 are valid then one obtains: T ˙ ζ , η, ˙ η ζ  −1 wζ ≤ 0. ˜ wζ ) = −sζT [Dζ (ζ , η) + k D ]sζ − η˜ T T k P η˜ − F L(s

(12.39)

From Soylu et al. (2008) it is known that inequality of type as (12.39) does not imply that L˙ → 0 as t → ∞ which means that the system trajectories may not converge to the desired state in the finite time. It is possible to solve this problem applying Barbalat’s lemma, e.g. Slotine and Li (1991). Because L is lower bounded (L ≥ 0), andL˙ is negative semi-definite, i.e.L˙ ≤ 0, then one obtains that L must have limit (there exists a non-negative constant ρ) such that limt→∞ L(t) = ρ. Moreover, ˜ and Eq. (12.39) leads one to conclusion that L(t) ≤ L(0), which implies that sζ , η, wζ are bounded. Using Assumption 12.3 and Eqs. (12.23)–(12.28) it may be verified ˙ η ζ , and F ¨ η ζ are also bounded. Consequently,L¨ can be determined and it ˙ ζ, F that s˙ζ , w t is bounded. One obtains limt→∞ 0 L(σ )dσ = limt→∞ L(t) − L(0) = ρ − L(0) < ˙ = 0, then L˙ → 0 as t → ∞. ∞. From Barbalat’s lemma it results that limt→∞ L(t) This in turn implies that sζ , η˜ → 0 as t → ∞.  Earth-Fixed Frame vehicle input signal. After transformation of variables, i.e. after inversion of (2.99), and recalling (9.22)–(9.24), one gets the control input in the Earth-Fixed Frame: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) +T (η)k D (η)s + k P η˜ + f η ,

(12.40)

t where f η = T (η) 0 f (η(t))s(t)dt. The term T (η)k D (η) causes that the resultant velocity gain matrix is equal to K D = T (η)k D (η) which takes into account dynamics of the vehicle. For example if is assumed k D = δ I (where δ is a constant) then K D = δ T (η)(η) because T (η)(η) includes the inertia matrix. Consequently, the gain matrix is selected according to the dynamical parameters of the system and the input signal τ η is strictly related to the kinematics as well as to the dynamics of the vehicle. The matrix k D serves rather for tuning the control gains. An additional benefit relies on that if the system parameters are not exactly known (or approximately known) then the term containing (η) causes that the tracking velocity error tends to zero quickly because in the regulation process the dynamics of the vehicle is included.

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237

12.1.3 Simplified Forms of Controllers GVC-APVTCEF. Because the matrix Mη (η) is usually a symmetric one (it is assumed that this condition is fulfilled), then it is possible to determine the diagonal matrix Nη (η). However, in particular cases (e.g. for hovercraft), one has a diagonal matrix Mη (η) that implies Mη (η) = Nη (η). Consequently, Z(η) = I and instead of (12.40) one obtains: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) +k D s + k P η˜ + f η ,

(12.41)

which means that the gain matrix does not depend on the dynamical effects. Thus, the controller has a reduced form. NGVC-APVTCEF. The matrix Mη (η) may be a diagonal one instead of a symmetric one, then the diagonal matrix Nη (η). In particular cases (e.g. for hovercraft), the system matrix is diagonal Mη (η) that implies Mη (η) = Nη (η). Consequently, Z(η) = 1

I and (η) = Nη2 (η) and instead of (12.40) one gets: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η) 1

1

+Nη2 (η)k D Nη2 (η)s + k P η˜ + f η

(12.42)

which means that the dynamical effects are partially present in the gain matrix. However, the controller can be applied too. Algorithms for Vehicle Model with 3 DOF Moving Horizontally. The considered controllers can be also applied for a reduced planar model of the vehicle. The remark from Chap. 6 is valid in this case.

12.2 Simulation Results Some selected examples of use of the discussed, in this chapter, control algorithms are presented below. Simulations were performed using Matlab/Simulink (the fifthorder Dormand–Prince formula ODE 5 with the fixed step size t = 0.01 s).

12.2.1 Trajectory Tracking Control Using GVC 12.2.1.1

Controller for 3 DOF Fully Actuated Underwater Vehicle

Recall an example from work (Herman 2017). In order to better show benefits of the velocity transformation the following vectors ν d = [u d , rd , vd ]T and

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ηd = [xd , ψd , yd ]T are assumed (changing the order of elements in vectors). The elements of the matrix N = diag {N11 , N22 , N33 } are: N11 = 10.0 kg, N22 = 0.5 kgm2 , N33 = 8.0 kg. For tracking the following desired circular 2D trajectory and the appropriate velocity profile were used:

T π ηd = cos(0.1 t), 0.1 t + , sin(0.1 t) , 36 η˙ d = [−0.1 sin(0.1 t), 0.1, 0.1 cos(0.1 t)]T .

(12.43) (12.44)

The disturbances vector including the unknown dynamics as well as the external disturbances are described by the following vector of forces and forces and moment (in external space): ⎡

⎤ 50 f 1η (t)e−2 + 2 η = ⎣ 50 f 2η (t)e−4 + 5 ⎦ , F −20 f 1η (t)e−2 − 0.8

(12.45)

where f 1η (t) = cos(0.05 t) + sin(0.2 t) and f 2η (t) = 2 sin(0.1 t) + cos(0.05 t). The above given velocity and disturbance functions profiles were used for both adaptive velocity controllers, namely GVC (with the matrix N) and CL. Moreover, saturation f η | ≤ 100 N or Nm. was assumed as |τη | ≤ 100 N or Nm and |  The gain matrices for the GVC controller were selected as follows (to ensure fast errors convergence in a comparable time): k∗D = diag{10, 10, 10}, k P = diag{0.1, 0.1, 0.1},  = diag{1.0, 1.0, 1.0},  ∗ = diag{0.2, 2.0, 0.2}.

(12.46)

For tracking the desired angular trajectory, larger value of the second element of the matrix  ∗ is needed. Besides, k D = Nk∗D ,  = N ∗ for the GVC algorithm and k D = k∗D ,  =  ∗ for the CL controller were applied. From Fig. 12.1a it follows that the desired circular trajectory is tracked correctly. The disturbance functions for the GVC and CL controllers are given in Fig. 12.1b. From Fig. 12.2a–b it is observed that both the position and velocity errors tend to zero after about 5 s. Also all lumped dynamics estimation errors are close to zero after a very short time (Fig. 12.2c). As shown in Fig. 12.2d, the steady state of both forces and the moment also disappears very quickly. However, in the first phase of motion values of wηX and τηX are very large. It is effect of the vehicle dynamics which is taken into account in the GVC controller. Comparing Figs. 12.2 and 12.3, it can be seen that there are more oscillations using the CL controller than the GVC controller. In addition, the values wηX and τηX are now smaller at the beginning of the vehicle’s movement. The lack of dynamics in control matrices is most evident when the vehicle moves forward. Concluding,

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239

Fig. 12.1 Position trajectories and disturbance functions: a desired and realized trajectories; b  ,  disturbances forces profiles F ηX F ηY , and disturbances moment profile F ηN

Fig. 12.2 Simulation results for GVC controller and underwater planar vehicle: a linear and angular position errors; b linear and angular velocity errors; c lumped dynamics estimation errors w related to linear and angular velocities in terms of wη ; d forces and moment in terms of τη

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Fig. 12.3 Simulation results for CL controller and underwater planar vehicle: a linear and angular position errors; b linear and angular velocity errors; c lumped dynamics estimation errors w related to linear and angular velocities in terms of wη ; d forces and moment in terms of τη

under the considered conditions, the x-direction is the most sensitive for dynamical parameters of the vehicle.

12.2.1.2

Controller for 3 DOF Fully Actuated Hovercraft

In order to better show benefits of the velocity transformation the order of the elements in the vectors has been changed, i.e. ν d = [u d , rd , vd ]T and ηd = [xd , ψd , yd ]T . The elements of the matrix N = diag {N11 , N22 , N33 } were calculated as: N11 = 10.0 kg, N22 = 0.5 kgm2 , N33 = 8.0 kg. For tracking it is assumed the following desired elliptical 2D trajectory and the appropriate velocity profile, respectively:

T π ηd = cos(0.2 t), 0.2 t + , 1.6 sin(0.2 t) , 36 η˙ d = [−0.2 sin(0.2 t), 0.2, 0.32 cos(0.2 t)]T .

(12.47) (12.48)

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241

The disturbances vector including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moment (in external space): ⎡

⎤ 100 f 3η (t)e−3 − 1 η = ⎣ 50 f 4η (t)e−3 + 1 ⎦ , F −30 f 3η (t)e−3 + 0.3

(12.49)

where f 3η (t) = cos(0.05 t) + sin(0.3 t) and f 4η (t) = sin(0.2 t) + cos(0.05 t). The above given velocity and disturbance functions profiles were used for both adaptive velocity controllers, namely GVC (with matrix N ) and CL. Saturation limit was f η | ≤ 50 N or Nm. assumed as |τη | ≤ 50 N or Nm and |  The gain matrices for the NGVC controller were selected, in order to ensure fast errors convergence in a comparable time, as follows: k∗D = diag{25, 25, 25}, k P = diag{1, 1, 1},  = diag{0.8, 0.8, 0.8},  ∗ = diag{0.25, 2.5, 0.25}.

(12.50)

In Fig. 12.4 the desired and realized trajectories profiles as well as the disturbances functions are presented. The tracking position trajectory task is realized correctly as it is shown in Fig. 12.4a. Based on the graphs from Figs. 12.5 and 12.2 some observations can be made. Despite the different working conditions and the desired trajectory, the nature of the results is similar. As can be seen from Figs. 12.5a–b and 12.2a–b the errors are close to zero in a comparable time. Some differences are observable in the first phase of motion between the signals presented in Figs. 12.5c–d and 12.2c–d. However, the GVC control algorithm works properly. In Fig. 12.6 the results obtained using the CL controller for hovercraft are presented. By comparing the relevant graphs in Figs. 12.5 and 12.6, some observations

Fig. 12.4 Position trajectories and disturbance functions: a desired and realized trajectories; b  ,  disturbances forces profiles F ηX F ηY , and disturbances moment profile F ηN

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12 IQV Position and Velocity Tracking Control with Adaptive …

Fig. 12.5 Simulation results for GVC controller and hovercraft: a linear and angular position errors; b linear and angular velocity errors; c lumped dynamics estimation errors w related to linear and angular velocities in terms of wη ; d forces and moment in terms of τη

can be made. Tracking of the desired position trajectory in not exactly for the CL controller as for the GVC algorithm (Figs. 12.5a and 12.6a). Velocities are tracked with oscillations if the vehicle starts as it is given in Figs. 12.5b and 12.6b. The same remark can be made for the graphs in Figs. 12.5c–d and 12.6c–d. Therefore, it can be concluded that the vehicle dynamics has a decisive influence on the control results. This means that taking into account dynamic vehicle parameters in the control gain matrices is important for the realized task in the considered case.

12.2.1.3

Controller for 6 DOF Fully Actuated Airship

Objective: to show performance of the GVC position and velocity trajectory tracking controller for an indoor airship model in the presence of a disturbances model. Fast position and velocity errors convergence in comparable time is expected. For tracking it is assumed the following desired helical trajectory profile together with exponential angular trajectory:

12.2 Simulation Results

243

Fig. 12.6 Simulation results for CL controller and hovercraft: a linear and angular position errors; b linear and angular velocity errors; c lumped dynamics estimation errors w related to linear and angular velocities in terms of wη ; d forces and moment in terms of τη

ηd = [4 cos(0.1 t) − 3, 4 sin(0.1 t) − 1, 0.2 t + 0.2, 0.4e−0.5t , −0.8e−0.3t , 0.2e−t ]T , η˙ d = [−0.4 sin(0.1 t), 0.4 cos(0.1 t), 0.2, −0.20e−0.5t , 0.24e−0.3t , −0.20e−t ]T .

(12.51) (12.52)

The disturbances function including the unknown dynamics as well as the external disturbances is expressed by the following vector of the forces and moments: ⎤ 10 f 5η (t)e−1 + 20 ⎢ f 5η (t)e−1 + 2 ⎥ ⎥ ⎢ ⎥ ⎢ η = ⎢ 20 f 6η (t) − 20 ⎥ , F ⎥ ⎢ f 6η (t) − 1 ⎥ ⎢ ⎣ 500 f 7η (t)e−3 + 40 ⎦ 50 f 7η (t)e−3 + 4 ⎡

(12.53)

where the functions are defined as follows: f 5η (t) = 6 cos(0.5 t) + sin(0.8 t), π t) + π4 cos( π5 t). f 6η (t) = 0.4 sin(0.4 t) e−5 + 0.4 ecos(−0.3) t , and f 7η (t) = π5 sin( 15

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12 IQV Position and Velocity Tracking Control with Adaptive …

Fig. 12.7 Disturbance functions in the Earth-Fixed Reference Frame: a disturbances forces profiles  F ηX ,  F ηY ,  F ηZ , b disturbances moments profiles  F ηK ,  F ηM ,  F ηN

The above given velocity and disturbance functions profiles were used for the GVC (with the matrix N) controller, assuming however k D = N k∗D ,  = N  ∗ . The model of disturbances functions in the Earth-Fixed Reference Frame is presented in Fig. 12.7. Note that the great values of disturbance functions are for surge (u) and pitch (q). The gain matrices were selected to ensure acceptable tracking error convergence without great overshoot and the lumped dynamics error convergence. This set of gains can worsen the results but allows to more clearly recognize the impact of dynamics on individual variables. The following set of parameters was assumed: k∗D = diag{20, 20, 20, 40, 40, 40}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{200, 200, 200, 600, 600, 600}. (12.54) Note that the set of parameters is different only for position and orientation tracking but not for each quantity. The forces and moments are limited by |τη | ≤ 150 N or Nm and | f η e | ≤ 150 N or Nm, whereas, the lumped dynamics estimation errors were shown assuming the constant values equal to zero. The results obtained using the GVC controller are presented in Figs. 12.8 and 12.9. From Fig. 12.8a it is seen that the position trajectory tracking is realized correctly. The linear and angular position errors convergence is acceptable as is it observed from Figs. 12.8b–c. Slightly worse results are for the linear and angular velocity errors (Fig. 12.8d–e). The effect of dynamics on velocity errors is particularly visible for d x/dt and dφ/dt. As shown in Fig. 12.9a–b the lumped dynamics estimation errors wη are reduced very quickly (after about 2 s). The dynamical effects are observable mainly for

12.2 Simulation Results

245

Fig. 12.8 Simulation results for GVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

wηX and wηM . But limited values of all these variables are achieved. Similar effect is noticeable if the forces and moments are considered (Fig. 12.9c–d). All forces and moments in the Earth-Fixed Reference Frame are sensitive to the presence of dynamics in the control gain matrices.

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12 IQV Position and Velocity Tracking Control with Adaptive …

Fig. 12.9 Simulation results for GVC controller and indoor airship: a lumped dynamics estimation errors wη related to linear velocities; b lumped dynamics estimation errors wη related to angular velocities; c forces τη ; d moments τη

12.2.2 Trajectory Tracking Control Using NGVC 12.2.2.1

Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: to show performance of the NGVC trajectory and velocity tracking controller for an underwater vehicle model in the presence of disturbances model. Fast error convergence without overshoot in position errors. The used disturbance forces and moments functions are shown in Fig. 12.10. For tracking it is assumed the desired helical trajectory profile together with exponential angular trajectory defined by (12.51)–(12.52). The disturbances function including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments:

12.2 Simulation Results

247

Fig. 12.10 Disturbance functions: a disturbance forces  F ηX ,  F ηY ,  F ηZ , b disturbance moments  F ηK ,  F ηM ,  F ηN (in the Earth-Frame Representation)

⎤ 2 30 f 8η (t) e−1 + 2 f 8η (t) e−2 + 50 ⎢ 9 f 8η (t) e−1 + 0.6 f 2 (t) e−2 + 15 ⎥ 8η ⎥ ⎢ ⎥ ⎢ f 9η (t) − 70 ⎥, η = ⎢ F ⎥ ⎢ 0.02 f 9η (t) − 1.4 ⎥ ⎢ −3 ⎦ ⎣ 150 f 10η (t) e + 12 −3 50 f 10η (t) e + 4 ⎡

(12.55)

where f 8η (t) = 8 cos(0.2 t) + sin(0.05 t), f 9η (t) = 28 sin(0.4 t) e−5 + 35 ecos(−0.3 t) , π t) + π4 cos( π5 t). The control gains set is selected as (it is and f 10η (t) = π5 sin( 15 ∗ assumed that k D = k D and  =  ∗ ): k∗D = diag{90, 90, 90, 90, 90, 90}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{80, 80, 80, 80, 80, 80}. (12.56) Note that the diagonal elements are the same for each gain matrix, i.e.: k∗D = 90 I, k P = I,  = 0.4 I, and  ∗ = 80 I. Because of dynamical effects observed in the first phase of motion the input signals are limited (forces and moments). Saturation values were: |τη | ≤ 1500 N or Nm and | f η e | ≤ 1500 N or Nm. Moreover, the lumped dynamics estimation errors were obtained assuming the constant values (for all cases): c X = −20, cY = 10, c Z = −20, c K = 10, c M = 20, and c N = 35. From Fig. 12.11a it is seen that the desired position trajectory is realized correctly. Observing Fig. 12.11b–e it is noticeable that all position errors and velocity errors are close to zero after about 15 s. Thus, the controller is working properly. Moreover, oscillations are not observed. The obtained errors convergence is strictly related to the quantities sξ which is presented in Fig. 12.12a–b. The signals tend to zero very quickly (after about 2 s) what guarantees also fast errors convergence. From Fig. 12.12c–d it

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12 IQV Position and Velocity Tracking Control with Adaptive …

Fig. 12.11 Simulation results for NGVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

follows that all lumped dynamics estimation errors have limited values. As shown in Fig. 12.12e–f the forces and moments have great values only if the vehicle starts. Oscillations are not observed. It results from presence of the system dynamics in the control gains.

12.3 Closing Remarks

249

Fig. 12.12 Simulation results for NGVC controller and underwater vehicle: a quantities sζ for linear variables; b quantities sζ for angular variables; c lumped dynamics estimation errors wη related to linear velocities; d lumped dynamics estimation errors wη related to angular velocities; e forces τη ; f moments τη

12.3 Closing Remarks Two control algorithms with and adaptive term using the IQV for the position and velocity tracking were presented in this chapter. The IQV based controllers are realized in the Earth-Fixed Frame. They ensure the trajectory tracking in the presence of

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12 IQV Position and Velocity Tracking Control with Adaptive …

disturbances described by a model and they can be applied both for full 6 DOF and reduced 3 DOF vehicle models. The main difference between the presented algorithms and the classical ones relies on that the dynamical and geometrical parameters of the system in the control matrices are included. Consequently, the matrices always depend directly on the dynamics. For properly selected gains the system response is fast and the position and velocity tracking errors quickly tend to end values. Even if the error convergence is not fast the results depend always on the vehicle model parameters. Moreover, in the IQV algorithms a diagonal control gain matrix may be used because the resulting matrix contains the vehicle parameters and it is a symmetric one. It means that the parameters set plays the crucial role for the vehicle control. The simulation results obtained for a 6 DOF indoor airship and an underwater vehicle model confirmed effectiveness of the proposed control approach.

References Cui R, Chen L, Yang Ch, Chen M (2017) Extended state observer-based integral sliding mode control for an underwater robot with unknown disturbances and uncertain nonlinearities. IEEE Trans Ind Electron 64(8):6785–6795 Fazeli A, Zeinali M, Khajepour A (2012) Application of adaptive sliding mode control for regenerative braking torque control. IEEE/ASME Trans Mechatron 17(4):745–755 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, Chichester Gan W, Zhu D, Ji D (2018) QPSO-model predictive control-based approach to dynamic trajectory tracking control for unmanned underwater vehicles. Ocean Eng 158:208–220 Herman P (2017) Adaptive trajectory tracking controller for planar vehicles. In: Proceedings of 2017 25th mediterranean conference on control and automation (MED) July 3–6, 2017, Valletta, Malta, pp 1170–1175 Herman P (2020) A method for numerical simulation for dynamics and control of underwater vehicles based on quasi-velocities. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734. 2020.1751197 Herman P (2021) Use of a nonlinear controller with dynamic couplings in gains for simulation test of an underwater vehicle model. Int J Adv Rob Syst May-June 2021, 1–18. https://doi.org/10. 1177/17298814211016174 Ismail ZH, Mokhar MBM, Putranti VWE, Dunnigan MW (2016) A robust dynamic region-based control scheme for an autonomous underwater vehicle. Ocean Eng 111:155–165 Lakhekar GV, Waghmare LM (2017) Robust maneuvering of autonomous underwater vehicle: an adaptive fuzzy PI sliding mode control. Intel Serv Robot 10:195–212 Peng Z, Wang J, Wang J (2019) Constrained control of autonomous underwater vehicles based on command optimization and disturbance estimation. IEEE Trans Ind Electron 66(5):3627–3635 Qiao L, Zhang W (2016) Double-loop chattering-free adaptive integral sliding mode control for underwater vehicles. In: Proceedings MTS/IEEE OCEANS conference shanghai, China, 10–13 April 2016, pp 1–6 Qiao L, Zhang W (2018) Double-loop integral terminal sliding mode tracking control for UUVs with adaptive dynamic compensation of uncertainties and disturbances. IEEE J Oceanic Eng 44(1):29–53 Qiao L, Zhang W (2019) Adaptive second-order fast nonsingular terminal sliding mode control for fully actuated autonomous underwater vehicles. IEEE J Ocean Eng 44(2):363–385 Qu Y, Xiao B, Fu Z, Yuan D (2018) Trajectory exponential tracking control of unmanned surface ships with external disturbance and system uncertainties. ISA Trans 78:47–55

References

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Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs Soylu S, Buckham BJ, Podhorodeski RP (2007) Robust control of underwater vehicles with faulttolerant infinity-norm thruster force allocation. In: Proceedings of OCEANS 2007, 29 September– 4 October, Vancouver, BC, Canada, 2007, pp 1–10 Soylu S, Buckham BJ, Podhorodeski RP (2008) A chattering-free sliding-mode controller for underwater vehicles with fault-tolerant infinity-norm thrust allocation. Ocean Eng 35:1647–1659 Soylu S, Proctor AA, Podhorodeski RP, Bradley C, Buckham BJ (2016) Precise trajectory control for an inspection class ROV. Ocean Eng 111:508–523 Zeinali M, Notash L (2010) Adaptive sliding mode control with uncertainty estimator for robot manipulators. Mech Mach Theory 45:80–90 Zhang M, Liu X, Yin B, Liu W (2015) Adaptive terminal sliding mode based thruster fault tolerant control for underwater vehicle in time-varying ocean currents. J Frankl Inst 352:4935–4961

Chapter 13

Vehicle Dynamics Study Based on Nonlinear Controllers Conclusions and Perspectives

Abstract The modeling problem for the class of vehicles was discussed in Chaps. 2–4. However, if the model it is assumed other problems arise, such as the identification of parameters or the manner of using this model, e.g. in order to design an appropriate controller. The presented earlier control schemes can be useful for studying vehicle dynamics. In this chapter application of the presented algorithms for dynamics investigation of the class of vehicles is considered. The presented in previous chapters controllers using the IQV, namely expressed in terms of the GVC and the NGVC can serve not only for realization of the control task. Some interesting feature is that they can be used for evaluation of the dynamics as well as for comparison of various vehicle structures. The last section concludes the issues considered in this book regarding control algorithms containing IQV and their use for control and analysis of dynamic models of a class of vehicles. The prospects for the use of such algorithms and some unsolved problems are also pointed out.

13.1 General Scheme If the model is already known in terms of equations of motion, other problems arise, such as identification of parameters or how to use the model in further research. Some ways to solve these problems for marine vehicles can be found for example in Chin and Lum (2011), Evans and Nahon (2004), Hassanein et al. (2016), Martin and Whitcomb (2014), Miskovic et al. (2011), Wang et al. (2015). This section is devoted to the general vehicle dynamics study using the control algorithms expressed in terms of the IQV. Some attempts to use control algorithms including the IQV have been made in Herman (2019), Herman (2020a), Herman (2020b), Herman (2021a) while some proposals for the analysis of vehicle dynamics based on indexes have been included in Herman (2019), Herman (2021b). However, this chapter shows a systematic approach to the problem of analyzing a vehicle dynamics model using the IQV and a set of indexes for various types of controllers containing the IQV. In addition, this chapter not only summarizes the theoretical developments that were known only in a partial way from earlier publications but includes some comprehensive insights into the issues of analyzing the dynamics of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-94647-0_13

253

254

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.1 Diagram explaining the proposed dynamics analysis strategy using position and velocity trajectory tracking controller

Fig. 13.2 Diagram explaining the proposed dynamics analysis strategy using velocity trajectory controller

discussed class of vehicles based on different types of control algorithms expressed by IQV. A study with established objectives was conducted and indicated for which dynamics studies these control schemes might be intended. Based on the simulation results in this chapter, a certain universality of the algorithms can be deduced (they are suitable both for control purposes and for gaining insight into vehicle dynamics). The proposed approach to the problem of dynamics study consists of two steps. First of all, the control scheme that is appropriate to the dynamics test is proposed. Then, the procedure for estimating the vehicle dynamics based on the analysis of selected variables and possibly a set of indexes to understand dynamic properties is indicated. Diagram which present the proposed method using position and velocity tracking controller is depicted in Fig. 13.1, whereas using velocity tracking controller only in Fig. 13.2. If the dynamics analysis concerns the airship model then in the given figures instead of ‘underwater vehicle model’ one can insert ‘indoor airship’ model.

13.2 Controllers and Gain Matrices Selection

255

13.2 Controllers and Gain Matrices Selection The following control algorithms are used for the dynamics testing of underwater vehicle models and indoor vehicle models. GVC Controllers/ NGVC Controllers: (1) position and velocity tracking control algorithm in the Body-Fixed Frame (7.20)/(7.47) and with the adaptive term (10.21)/(10.42), (2) velocity tracking control algorithm in the Body-Fixed Frame (8.20)/(8.43) and with the adaptive term (11.20)/(11.40), (3) position and velocity tracking control algorithm in the Earth-Fixed Frame (9.26)/(9.62) and with the adaptive term (12.22)/(12.40). Hints for gain matrices values selection. Some hints which may be useful for selecting of the gain matrices are given below. 1. Initially it can be assumed that each matrix is the identity matrix, namely: k∗D = I, k P = I, k I = I,  = I,  ∗ = I.

(13.1)

Using the gains for the GVC controller the couplings effect in the vehicle can be estimated whereas for the NGVC controller effect of full dynamics is evaluated. For algorithms realized in the the Body-Fixed Frame the same results for GVC and NGVC are obtained (if k∗D and  ∗ have the same diagonal elements): k D = Nk∗D ,  = N ∗ .

(13.2)

For the NGVC algorithm it is always k D = k∗D ,  =  ∗ . The same matrices are selected for the CL algorithms in order to evaluate the effects of dynamics. 2. The time system response is determined (velocity, position errors, and their time derivatives convergence, applied forces and torques) and analyzed. If the graphs have unacceptable time history then the gain matrices must be changed. a. The matrix k∗D is important and it causes that if its values increase then the dynamics in the control process plays a greater role. Their elements should be more than one (e.g. equal to about 10, 100, sometimes larger). It ensures also the response without significant overshoot. b. It is difficult to say if values of the matrix k I are essential for realization the control task. It is usually assumed as the identity matrix or ten times bigger. c. The matrix k P causes that the errors tend to zero faster (if it has large values) or more slowly in the opposite case. If its values are too large then overshoot effect is observable. d. Selection larger values for  make that the errors convergence is achieved in shorter time. In many cases it is sufficient to assume its values between 0.1 and 1. Overshoot appears when the values are to large. In such case we may reduce its values or assume larger values of k∗D .

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

e. High values of the matrix  ∗ cause that the lumped dynamics estimation error is reduced in a short time but its initial values are very large. f. Often the compromise in selection of the matrices k∗D , k P , , and  ∗ is needed. 3. For the dynamics estimation the gain matrices k∗D , k I , k P ,  ∗ and  are usually selected as diagonal and constant. In such way each gain value corresponds to appropriate variable. Three cases can be taking into account. a. The matrices are multiplied by the unit matrix, i.e. k∗D = α D I, k I = α I I, k P = α P I,  = α I and  ∗ = α I, where α D , α I , α P , α , α are constant coefficients, and I means the unit matrix. b. At least one of the diagonal matrices k∗D , k I , k P ,  ∗ , and  has the same coefficients for linear or angular variables, e.g. k∗D = [k DL k D A ]T , where k DL = β DL I, k D A = β D A I (β DL and β D A are constant coefficients). c. Elements of the matrix k∗D , k I , k P ,  ∗ , and  can have different values according to the assumed goal (taking into account the desired trajectory, disturbances model).

13.3 Vehicle Dynamics Study Using Procedure 13.3.1 Analysis Based on GVC Controllers Body-Fixed Frame Control. Position and Velocity Tracking Control Algorithm (PVTC-BF). Velocity Tracking Control Algorithm (VTC-BF). Both controllers can be used for study of underwater vehicle dynamics models. The procedure is given below. 1. At the beginning it is necessary to decide what quantities will be estimated. The equation of the GVC control algorithm is transformed to obtain the input signals, i.e. (7.20) or (10.21) for PVTC-BF and (8.20) or (11.20) for VTC-BF. Assuming k D = k∗D ,  =  ∗ only the effect of couplings can be evaluated (from the matrix ϒ). On the contrary, if k D = Nk∗D ,  = N ∗ are selected then one receives K F D = ϒ −T Nk∗D ϒ −1 and  F D = ϒ −T N ∗ ϒ −1 and the effect of full dynamics can be estimated. The matrix N accelerates the velocity error and the lumped dynamics estimation error convergence. 2. Some insight into the vehicle dynamics under the controllers (10.21) or (11.20) is possible if the performance are compared with performance of an algorithm in which the dynamical couplings are omitted (called CL controller), i.e.: τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + k D s + JT (η)k P η˜ + f, τ = Mν˙ r + C(ν)ν r + D(ν)ν r + g(η) + k D s + k I z + f,

(13.3) (13.4)

13.3 Vehicle Dynamics Study Using Procedure

257

t where f =  0 f s(t) dt (s is defined by (7.19) or (8.19)) for PVTC-BF and VTCBF, respectively. These algorithms do not contain, in the control gains matrices, any matrix resulting from the inertia matrix decomposition. 3. Based on the results obtained for the GVC and CL controllers the following time ˜ η˜˙ for PVTC-BF responses are compared: 1) the position and velocity errors η, or the velocity errors ν˜ for VTC-BF, 2) the input signals (the forces and torques) τ , and the lumped dynamics estimation errors w. The purpose of the analysis is to estimate the effect of dynamics on the work of the controller and to identify the most sensitive variables to change of the vehicle dynamics. This way it is possible to recognize the effect of dynamics on the behavior of the vehicle while performing the control task (dynamics is included in the control gains). 4. From the diagonal inertia matrix N information about the inertia related to each quasi-acceleration can be obtained. Because in the dynamics equation each quasi-velocity ξi is independent from other quasi-velocities then the corresponding kinetic energy is determined. The vector ξ includes the dynamic parameters (i.e. masses, inertia) and geometry of the vehicle. The kinetic energy of the vehicle K E in terms of the vector ξ is expressed as follows:  1 T 1 1 Nii ξi2 = K Ei . ν Mν = ξ T Nξ = 2 2 2 i=1 i=1 6

KE =

6

(13.5)

Each component ξi takes into account a part of the kinetic energy coming from other velocities which are coupled with the i-th variable. Consequently, the contribution of the kinetic energy associated with each variable to the total kinetic energy of the vehicle, i.e. δ K Ei = KKEiE · 100% can be calculated. If the vehicle dynamics in the controller is used then the kinetic energy in the vector ξ is reduced quickly which results in fast errors convergence. 5. The dynamic couplings are determined because each variable ξi leads to evaluation of the impact of other velocities at the i-th velocity. The formal equation is given as follows: j=6  ξi = νi + ϒi−1 (13.6) j νj. j=i+1

From (13.6) it results that in each quasi-velocity ξi the couplings between itself and the other velocities are contained. Consequently, using the GVC controller, the effect of couplings for dynamics of the vehicle can be observed and an insight into the control process is accessible. Moreover, it is: 

j=i−1

πi = τi +

ϒi j τ j .

(13.7)

j=1

The effect of the dynamical couplings is determined due to the presence of a matrix ϒi j . Two quantities are proposed as the effect measures:

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

ξi = ξi − νi ,

(13.8)

πi = πi − τi .

(13.9)

6. Finally, discussion of the obtained simulation results is carried out. Remark 13.1 Simplified analysis can be done without steps 4 and 5. Earth-Fixed Frame Control. Position and Velocity Tracking Control Algorithm (PVTC-EF). The model analysis using the controller is realized can be divided into several steps. 1. Initially the purpose of research is chosen. The signal from the GVC controller after transformation into τ η is defined by (9.26) or (12.22). In order to test the dynamical couplings it is assumed that k D = k∗D and  =  ∗ from which it follows that in the Earth-Fixed Frame K DC = Z−T (η)k D Z−1 (η) and  C = Z−T (η). However, to show effect of the full dynamics in the Body-Fixed Frame which is transferred to the Earth-Fixed Frame the matrices k D = N k∗D and  = N  ∗ are assumed. The matrix N is determined in the same way as Nη (η) but from the inertia matrix M instead of Mη (η). The matrices k D , k P ,  and  should ensure acceptable error convergence and eventually fast system response (k∗D and  ∗ are selected as diagonal and serve rather for tuning because the vehicle dynamics is included into N). Then it is obtained K F D = Z−T (η)Nk D Z−1 (η) and  F D = Z−T (η)N. 2. The results for the GVC controller are compared to the results for the CL algorithm which does not contain the dynamical couplings in the gain matrices, i.e.: τ η = Mη (η)¨ηr + Cη (ν, η)η˙ r + Dη (ν, η)η˙ r + gη (η)  tf +k D s + k P η˜ +  s(t)dt,

(13.10)

0

where s is defined by Eq. (9.24). 3. After the test of both controllers, i.e. GVC and CL the following time responses are ˙˜ (3) the control signals ˜ (2) the velocity errors η, analyzed: (1) the position errors η, (the forces and torques) τ η , and the lumped dynamics estimation errors wη . In this way, the influence of dynamics on vehicle control process and identification of variables which are the most sensitive to changes of control gains can be analyzed (because dynamics is included in the control gains). 4. The vector ξ contains the dynamic parameters (i.e. masses, inertia) and the geometry of the system. For this reason, the kinetic energy of the vehicle K E in terms of the vector ξ can be presented in the following form: 1 T −T 1 η˙ Z (η)Nη (η)Z−1 (η)η˙ = ξ T Nη (η)ξ 2 2 6 6   1 = Nη ii (η)ξi2 = K Ei . 2 i=1 i=1

KE =

(13.11)

13.3 Vehicle Dynamics Study Using Procedure

259

Each element ξi is related to the kinetic energy of all other velocities which are coupled with it. Therefore, it takes into account the kinetic energy arising from all other coupled velocities. Consequently, the contribution of the kinetic energy associated with each variable to the total kinetic energy of the vehicle, namely δ K Ei = KKEiE · 100% can be evaluated. 5. The dynamical coupling between variables are determined. Each speed ξi allows one to estimate the impact of other velocities on the i-th velocity. The corresponding formula is as follows: ξi = η˙ i +

j=6 

Z i−1 ˙ j. j (η)η

(13.12)

j=i+1

It means that the variable ξi includes coupling between itself and the other velocities. Therefore, the effects of couplings for the vehicle dynamics and an insight into the controller work are available. In addition, the following relationship can be given: j=i−1  πi = τηi + Z i j (η)τη j . (13.13) j=1

In the second component, the effects of dynamic couplings in the vehicle are estimated. The measures of couplings are defined by means of the formula:

ξi = ξi − η˙ i ,

(13.14)

πi = πi − τηi .

(13.15)

6. Finally, an analysis of the results obtained is carried out. Remark 13.2 Simplified analysis can be done without steps 4 and 5.

13.3.2 Analysis Based on NGVC Controllers Body-Fixed Frame Control. Position and Velocity Tracking Control Algorithm (PVTC-BF). Velocity Tracking Control Algorithm (VTC-BF). The approach consists in performing simulation tests and their analysis. It is possible to determine in which way the dynamics included in the control gains changes the system response. The test relies on comparing the results for the NGVC control algorithm and the algorithm in which the inertia matrix is not taken into account in the gains. Therefore, one can estimate the impact of dynamics on the behavior of the vehicle at the simulation stage (or determining dynamical couplings effect) before real experiment avoiding unnecessary costs. The scheme of the procedure is shown below.

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1. As in the case of the GVC algorithm, a decision must be made at the beginning which quantities will be estimated. The equation of the NGVC control algorithm is transformed to the input signals, i.e. (7.47) or (10.42) for PVTC-BF and (8.43) or (11.40) for VTC-BF. Selecting k D = k∗D ,  =  ∗ we have K F D = T k D and  F D = T  and the effect of full dynamics is evaluated. It is because in the NGVC controller the matrix arising from the M inertia matrix decomposition (and the obtained matrices from it) is used. The diagonal matrices k D and  serve rather for precise tuning of the control rule whereas the matrix shortens the time of convergence of the velocity error and the lumped dynamics estimation error convergence. 2. For comparison the corresponding CL algorithm without dynamics in control gains, namely (13.3) or (13.4), it is assumed to show some differences between the time response of the system. 3. Taking into consideration the results obtained for the NGVC and CL controllers the following time responses are analyzed: (1) the position and velocity errors ˜ η˙˜ for PVTC-BF or the velocity errors ν˜ for VTC-BF, (2) the input signals (the η, forces and torques) τ , and the lumped dynamics estimation errors w. Based on the analysis of the above signals the effect of dynamics and identification of the most sensitive variables can be carried out. 4. The elements of the vector ζ include both the dynamic parameters (i.e. masses, inertia) and the geometrical dimensions of the system. The kinetic energy of the vehicle K E expressed in terms of the vector ζ can be written as follows: 1 T T 1 1 2  ν ν = ζ T ζ = ζ = K Ei . 2 2 2 i=1 i i=1 6

KE =

6

(13.16)

Reducing the kinetic energy included in the vector ζ (in which the vehicle dynamics is present) it is possible to ensure fast velocity error convergence (both in terms of ζ and ν). The contribution of the kinetic energy associated with each variable to the total kinetic energy of the vehicle, can be calculated using δ K Ei = KKEiE · 100%. 5. After calculation the components ζi the impact of other velocities on the ith velocity is available. The corresponding formula is as follows: ζi = ii νi +

j=6 

i j ν j .

(13.17)

j=i+1

In the second component the dynamical couplings coming from other quasivelocities are reflected. Denoting  = −T the quasi-forces can be written as: 

j=i−1

i = ii τi +

j=1

i j τ j .

(13.18)

13.3 Vehicle Dynamics Study Using Procedure

261

Effect of the dynamical couplings is determined by i j . If the elements are zero then we obtain some other measure of signals deformation arising from the vehicle dynamics. The proposed coupling measures are as follows:

ζi = ζi − Iii νi ,

i = i − Iii τi ,

(13.19) (13.20)

where Iii mean elements of the identity matrix, or alternatively:

ζi = ζi − ii νi ,

i = i − ii τi .

(13.21) (13.22)

6. Finally, after the time response analysis of the selected quantities conclusions about the impact of dynamics on system behavior are drawn. Remark 13.3 Simplified analysis can be done without steps 4 and 5. Earth-Fixed Frame Control. Position and Velocity Tracking Control Algorithm (PVTC-EF). The NGVC controller performs two tasks. First, it is a tracking control algorithm. Second, it is suitable for testing the dynamics of the vehicle via simulation in order to redesign its model. Using the proposed approach, it is possible to study the vehicle dynamics before a real experiment. It is also possible to provide information on whether an experiment is needed at all. The steps of the procedure for the dynamics investigation are summarized below. 1. The aim of the dynamics study is to learn the interactions resulting from the couplings between velocities and their impact on the vehicle’s motion. The signal obtained from the NGVC controller after transformation into τ η is described by (9.62) or (12.40). In order to test the dynamical interactions it is assumed that k D = k∗D and  =  ∗ from which it follows that in the Earth-Fixed Frame K DC = T (η)k D (η) and  C = Z−T (η). The control gain matrices are selected taking into account the dynamical parameters of the system. The input signal τ η is strictly related to the kinematics and to the dynamics of the vehicle. The matrix k D serves rather for tuning of the control gains. Moreover, it is  f = T (η) t F (η(t))s(t)dt. The matrix (η) containing the parameters of 0 the system gives an additional control gain. 2. An insight into the vehicle dynamics under the controller is possible. It can be obtained if performance of the NGVC algorithm is compared with performance of the algorithm in which the dynamical couplings are omitted, namely (13.10). Using the results for the two controllers, the effect of dynamics on the time response of the system is evaluated. 3. Taking into account the results obtained for both controllers, i.e. NGVC and ˜ 2) the CL the following time responses are analyzed: 1) the position errors η,

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

˙˜ 3) the control signals (the forces and torques) τ η , and the lumped velocity errors η, dynamics estimation errors wη . In this way, the influence of dynamics on vehicle control process and identification of variables which are the most sensitive to changes of control gains can be analyzed. 4. The vector ζ includes the set of dynamic parameters (i.e. masses, inertia) and the geometry of the system. The kinetic energy of the vehicle K E using the vector ζ is described as: 1 T T 1 1 2  ˙ = ζTζ = η˙ (η) (η)eta ζ = K Ei . 2 2 2 i=1 i i=1 6

KE =

6

(13.23)

From Eq. (13.23) it arises that each variable ζi takes into consideration the kinetic energy from all other velocities which are coupled with the i-th variable. The kinetic energy is the matrix (η) and the vector ζ dependent and it contains the vehicle parameters (including couplings). The dynamics in the control gain matrices causes that, the kinetic energy is reduced quickly, and the components of the vector ζ tend to the final values in shorter time than if the dynamics is absent. This effect is transferred on each coupled variable η˙ i and fast velocity error convergence is ensured. The contribution of the kinetic energy associated with each variable to the total kinetic energy of the vehicle, namely δ K Ei = KKEiE · 100% can be evaluated. 5. Based on the components ζi time history it is possible to estimate the influence of other velocities on the i-th velocity. The following equation explains this statement: j=6  ζi = ii (η)η˙ i + i j (η)η˙ j . (13.24) j=i+1

From Eq. (13.24) it can be indirectly learn (by using ζi or η˙ i ) how the couplings deform speeds of the vehicle. It is observed that for a light vehicle (weight less than 1 kg or inertia less that 1 kgm2 ) it is ii (η) < 0 what means that kinetic energy reduction of the i-the quantity decreases. The kinetic energy arising from other velocities is expressed by the second component. For heavier vehicles opposite effect can be observed. Denoting (η) = −T (η) the following equation can be written: 

j=i−1

i = ii (η)τηi +

i j (η)τη j .

(13.25)

j=1

Effect of the dynamical couplings is determined by i j (η). But some information can be obtained calculating a measure without this matrix. The measures of couplings are proposed in the form:

13.3 Vehicle Dynamics Study Using Procedure

263

ζi = ζi − Iii η˙ i ,

(13.26)

i = i − Iii τηi ,

(13.27)

where Iii mean elements of the identity matrix, or alternatively:

ζi = ζi − ii (η)η˙ i ,

i = i − ii (η)τηi .

(13.28) (13.29)

6. Finally, the results are analyzed and conclusions are drawn. Remark 13.4 Simplified analysis can be done without steps 4 and 5.

13.4 Comparison of Vehicle Dynamics Using Indexes 13.4.1 Model Test Based on GVC Controller The procedures presented previously are suitable for evaluation of the vehicle dynamics model. In the case where different vehicle models should be compared, an index based method is proposed. This approach is applied in the Body-Fixed Frame because for the dynamics study it is sufficient test various models of the vehicle. It can be formulated as follows. At the beginning of the test, the initial model, called Case 1, is chosen in which the inertia matrix is symmetric. Next, after neglecting all off diagonal elements a diagonal matrix, called Case 0 (it has the subscript d in all indexes), is obtained. It is approximation of the model understood as Case 1. Then other models with changed parameters for simulation tests are selected, e.g. Case 2, Case 3, …. Comparison of the models is based on the proposed set of indexes. Four groups of such indexes are considered, namely: quality of control investigation, input signal effort and energy determination, velocity deformation evaluation, and dynamic couplings investigation. ˜ η˙˜ have acceptable Remark 13.5 It is assumed that the position and velocity errors η, values if the vehicle moves. Indexes for testing quality of control. The main task of the controller is to track the desired position and velocity trajectory. Two indexes which serve for evaluation of the control task are introduced. Using these two indexes, one can check whether the control task is being carried out correctly. Index 1 is related to the position errors testing whereas Index 2 to velocity errors. In all indexes the subscript s denotes the tested model, and d the model with the diagonal inertia matrix.

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

 δ η˜ s , δ η˜ s,d = mean |η˜ i |. δ η˜ d i=1

(13.30)

 δ η˙˜ s Index 2 I2 = , δ η˙˜ s,d = mean |η˙˜ i |. δ η˙˜ d

(13.31)

6

Index 1 I1 =

6

i=1

Indexes for input signal effort and energy determination. The two quantities are important because the same control task can be realized with different control system (forces and moments) and various kinetic energy consumption. Index 3 is related to energy of the input signal. Index 3 I3 =

δπs , δπd

δπs,d =

6  

mean(πi2 ).

(13.32)

i=1

For the diagonal matrix with subscript d we have πd = τd because the matrix ϒ is the identity one. Index 4 refers to the kinetic energy reduced while the vehicle is moving. It allows also to determine K i reduced by each variable, i.e.:  δ Ks , δ K s,d = mean (K ) = mean (K i ). δ Kd i=1 6

Index 4 I4 =

(13.33)

The index arises from that the kinetic energy of the vehicle K E is the sum of each quasi-variable multiplied by the corresponding inertia (13.5). Indexes for velocity deformation evaluation. In order to evaluate deformation of velocity as a result of couplings two indexes are introduced, namely Index 5 (the average value of velocity deformation) and Index 6 (the difference between their maximum and minimum value). Index 5 I5 =

 δz s , δz s,d = mean |ξi |. δz d i=1

(13.34)

Index 6 I6 =

 δξs , δξs,d = |max ξi − min ξi |. δξd i=1

(13.35)

6

6

Indexes for dynamic couplings evaluation. The indexes enable determination of couplings and the effect of dynamic couplings on vehicle movement. The couplings may deteriorate the control performance but sometimes recognition of couplings can be useful in vehicle design. ss Index 7 results from the matrix ϒ. Using the matrix

13.4 Comparison of Vehicle Dynamics Using Indexes

265

(ϒ −1 − I) off diagonal elements can be calculated because the matrix ϒ is upper triangular. The index represent dynamical couplings and it is useful for symmetric matrix M only.  −1  ϒ − I  δϒcs of f s Index 7 I7 = , δϒcs = 1 +  −1  , ϒ  δϒcd diag s

(13.36)

where δϒcd = 1 because the subscript d refers to a diagonal inertia matrix. Index 8 is deduced from that in the matrix (ϒ T − I) the effect of coupling affecting for input signals τ is reflected. For the last matrix is the zero matrix. Index 8 I8 =

δπcs πcs , δπcs = 1 + , δπcd δπs 6   where πcs = [mean(πi −τi )]2 ,

(13.37) (13.38)

i=1

and δπcd = 1 (a diagonal inertia matrix). Index 9 gives information about the kinetic energy of dynamical couplings, i.e.:  δ K cs , δ K cs = 1 + mean (K c ) = 1 + mean (K ci ), δ K cd i=1 6

Index 9 I9 =

where K ci =

1 Ni (ξi − νi )2 , 2

(13.39) (13.40)

and δ K cd = 1. Index 10 allows one to evaluate the effect of couplings for the velocity vector ν. The index represents couplings arising from the vehicle model and it is not deformation of velocity but difference between both speeds ξ and ν.  δξcs |(max (ξi − νi )| , = , δξc = 1 + δξcd i=1 6

Index 10 I10

(13.41)

where δξcd = 1. Each lower index s corresponds to the considered vehicle model, whereas d denotes the reference model (the approximated model with a diagonal matrix is marked as Md ). The initially chosen model with a symmetric inertia matrix has subscript s = 1, i.e. M1 . The reference model d is obtained by eliminating all off-diagonal elements the matrix M1 . For this reason the model d is understood as an approximate model. The complex index for each case (i.e. 0, 1, 2, . . . where subscript d means the case 0). The ten indexes are collected in a complex index ICs (for each considered case). The new index is defined using the equation:

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

ICs =

10 

aj Ij.

(13.42)

j=1

The set of indexes give information about the behavior of the tested vehicle during its movement (more precisely the vehicle model). The values of each index I j ( j = 0, 1, 2, . . . , 10) are equal to I j ≥ 1 (I j = 1 represents the reference model), and 0 ≤ a j ≤ 1. The weight coefficients a j depend on importance of each index for control purposes and estimation of dynamics. Finally, the results are analyzed and conclusions concerning the tested vehicle structures are drawn.

13.4.2 Model Test Based on NGVC Controller The NGVC controller is useful for various models comparison via simulation and to redesign the vehicle model by changing its parameters. The essential advantage of such approach is that before real experiment it is possible to study the vehicle dynamics using the proposed algorithm based on the simulation test. This method is applied in the Body-Fixed Frame. Taking into account the formulas several criteria for dynamics investigation are proposed. The control effort is expressed by τ and the kinetic energy which is reduced by K . It is possible to determine a quantity kinetic energy consumed by each ζi . This information is useful for trajectory profile design and for selection parameters for modification of the vehicle. Moreover, ζi reflects deformation of each velocity νi through dynamical couplings in the vehicle. The couplings are determined from the matrix . The variable i represents deformation of each input signal τi resulting from couplings. The variables ζi and i are normalized. For this reason the information about the vehicle movement is indirect. Based on proposed formulas several indexes are introduced. The initial part of the method is the same as for using the GVC control algorithm. However, the set of indexes is slightly different. Moreover, Remark 13.5 is valid. Indexes for testing quality of control. If the position and velocity tracking controller in terms of the NGVC is used, then several criteria for dynamic vehicle evaluation can be offered. The subscript in all indexes s denotes the tested model, and d the model with the diagonal inertia matrix. Index 1a is related to the position errors testing whereas Index 2a to test the velocity errors (as for the GVC controller test).  δ η˜ s , δ η˜ s,d = mean |η˜ i |. δ η˜ d i=1 6

Index 1a I1a =

(13.43)

13.4 Comparison of Vehicle Dynamics Using Indexes

267

 δ η˙˜ s , δ η˙˜ s,d = mean |η˙˜ i |. δ η˜˙ d 6

Index 2a I2a =

(13.44)

i=1

Indexes for input signal effort and energy determination. In Index 3a the control effort is obtained indirectly because the real signal is scaled by inertia (or mass). Index 3a I3a =

δs , δd

δs,d =

6  

mean(i2 ).

(13.45)

i=1

The formula means that Index 3a is related to the normalized control effort. The effort increases when the value of this Index is higher. Index 4a gives information about the kinetic energy consumption (its value shows the energy which must be reduced if the vehicle moves).  δ Ks , δ K s,d = mean (K ) = mean (K i ). δ Kd i=1 6

Index 4a I4a =

(13.46)

It results from the following relationship (13.16). Each element ζi takes into consideration the kinetic energy of all other velocities which are coupled with the i-th variable. Including the vehicle dynamics in the control gains leads to that the kinetic energy corresponding to the vector ζ is reduced quickly and fast velocity error convergence is ensured. Indexes for velocity deformation evaluation. In order to evaluate deformation of velocity as a result of couplings two indexes are introduced.  δz s , δz s,d = mean |ζi |. δz d i=1 6

Index 5a I5a =

 δζs = , δζs,d = |max ζi − min ζi |. δζd i=1

(13.47)

6

Index 6a I6a

(13.48)

Index 5a represents average value of quasi-velocities whereas Index 6a gives information about maximum changes between their values. Indexes for dynamic couplings evaluation. The below given index provides information on the proportions of elements beyond diagonal values to the value of elements on the diagonal, which allows for some detection of couplings. Index 7a

I7a =

o f f s δcs , δcs = 1 + , diag s δcd

(13.49)

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

where δcd = 1. Information concerns the normalized couplings in the vehicle. The greater the value of the index, the stronger are the couplings. Index 8a shows the effect of coupling affecting for the vector of input signals τ in the normalized form. If the inertia matrix is diagonal then the last matrix is the zero matrix. δcs cs , δcs = 1 + , δcd δs  6  −1 where cs = [mean(i −Nii 2 τi )]2 ,

I8a =

Index 8a

(13.50) (13.51)

i=1

and δcd = 1. Index 9a reflects the kinetic energy consumed due to the occurrence of couplings in the dynamic system as follows: Index 9a

I9a =

δ K cs , δ K cs = 1 + mean (K c ) δ K cd

=1+

6 

mean (K ci ), where δ K ci =

i=1

1 (ζi − ii νi )2 , (13.52) 2

and δ K cd = 1. The last Index 10 is defined in the form:  δζcs |(max (ζi − ii νi )| , , δζcs = 1 + δζcd i=1 6

Index 10a

I10a =

(13.53)

where δζcd = 1. Index 10a represents the effect of couplings between the quasivelocity and velocity in a scaled value. Each lower index s corresponds to the considered vehicle model, whereas d denotes the reference model (the approximated model with a diagonal matrix is marked as Md ). The initially chosen model with a symmetric inertia matrix has subscript s = 1, i.e. M1 . The reference model d is obtained by eliminating all off-diagonal elements the matrix M1 . For this reason the model d is understood as an approximate model. The complex index for each case (i.e. 0, 1, 2, . . . where subscript d means the case 0) is defined by: ICs =

10 

aj Ij.

(13.54)

j=1

The set of indexes give information about the behavior of the tested vehicle during its movement (more precisely the vehicle model). The values of each index I j ( j = 0, 1, 2, . . . , 10) are equal to I j ≥ 1 (I j = 1 represents the reference model), and 0 ≤ a j ≤ 1. The weight coefficients a j depend on importance of each index for control purposes and estimation of dynamics.

13.4 Comparison of Vehicle Dynamics Using Indexes

269

After calculating the indexes the obtained results are analyzed taking into account all selected vehicle models. Finally, some conclusions are made.

13.5 Application of GVC Control Algorithms The essential advantage of the considered algorithms is that they can be applied for dynamics study of the vehicle. In this section selected algorithms using the GVC are tested to get information on the dynamics of the vehicle. Simulations were performed using Matlab/Simulink and the fifth-order Dormand-Prince formula ODE 5 with the fixed step size t = 0.01 s.

13.5.1 Position and Velocity Trajectory Tracking Controller in Earth-Fixed Frame 13.5.1.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Airship

Objective: use the GVC algorithm to estimate the effect resulting from couplings in an indoor airship model. For tracking it was assumed the following desired helical trajectory together with exponential functions and the appropriate velocity profile: ηd = [5 cos 0.1 t − 4, 5 sin 0.1 t − 1, 0.3 t, 0.5 e−0.3t , 0.4 e−0.3t − 0.5 e−0.3t ]T , η˙ d = [−0.5 sin 0.1 t, 0.5 cos 0.1 t, 0.3, −0.15 e−0.3t , −0.12 e−0.3t 0.15 e−0.3t ]T .

(13.55) (13.56)

Note that all position variables are time dependent while the vehicle is moving. The gain matrices of the controller were selected as follows (to ensure acceptable position and velocity errors tracking): k D = diag{200, 200, 200, 150, 150, 150}, k P = diag{200, 200, 200, 700, 700, 700},  = diag{0.5, 0.5, 0.5, 0.05, 0.05, 0.05}.

(13.57)

Saturation values are: |τη | ≤ 150 N or Nm. For this set of control gains and nominal parameters the results are shown in Fig. 13.3. As can be observed in Fig. 13.3a the desired trajectory is tracked correctly. From Fig. 13.3b–c the effect of couplings is noticeable for the position errors z, φ, and ψ. Similarly, the effect

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.3 Simulation results for GVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f applied forces; g applied torques

13.5 Application of GVC Control Algorithms

271

of dynamic couplings can be seen in Fig. 13.3d–e for the velocity errors dy/dt,

dz/dt, dφ/dt, and dψ/dt. The reaction to the coupling in the vehicle is clear in Fig. 13.3f–g where the forces and moments are presented (great values occur only if the airship starts). Remark 13.6 The classical control algorithm using the same parameters gain set does not work. The comparison with performance of the CL algorithm is impossible. This example can serve only for couplings effect determination based on the GVC controller. It is impossible to estimate the couplings effect using the gain matrices set. Dynamics Test Based on Comparison the GVC Controller with the CL At present it is shown that using another gains set the CL controller works correctly and it is possible to test dynamics of the vehicle. This test can serve for investigation of couplings between variables. Reverse method of dynamics analysis. In order to show some features it is assumed the same work conditions and trajectories as previously. However, first the CL algorithm is examined. The gains selected for the CL controller are applied next for the GVC control algorithm. The gain matrices for the CL controller were selected to ensure acceptable position and velocity convergence. They are as follows: k D = diag{150, 150, 150, 50, 50, 50}, k P = diag{90, 90, 90, 700, 700, 700},  = diag{0.2, 0.2, 0.2, 0.1, 0.1, 0.1}.

(13.58)

This set of control gain matrices will be used next also for the GVC algorithm in order to determine the dynamical couplings in the airship. The simulation results for the control gains and nominal parameters are shown in Fig. 13.4. Saturation values are: |τη | ≤ 150 N or Nm. As is shown in Fig. 13.4a the CL control algorithm works properly. From Fig. 13.4b–c it follows that the linear position errors are close to zero after about 6 s (the algorithm works more slowly than the GVC previously). The angular position errors are achieved faster but with oscillations. It suggests that the gain coefficients in the matrix k D related to angular velocities ought to be increased. However, it should be reminded that for earlier parameter set (13.57) in which these coefficients were larger, the algorithm did not work at all. From Fig. 13.4d–e it is observed that the errors tend to zero but with overshoot in the first phase of motion. Oscillations of forces and moments if the vehicle starts are noticeable also in Fig. 13.4f–g. In order to investigate the effect of couplings in the vehicle the same set of control gains was assumed for the GVC algorithm. Saturation values are: |τη | ≤ 150 N or Nm. The results are shown in Fig. 13.5. Such comparison is useful if the task is to determine the effect of dynamic couplings on the airship’s motion because the GVC algorithm contains the couplings only. It can be seen from Fig. 13.5a that the tracking task is realized. Comparing Fig. 13.5b–c with Fig. 13.4b–c it is observed that really all position errors converge to the end value. But the coupling effect occurs for the linear position errors y, z

272

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.4 Simulation results for CL controller (couplings test) and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

13.5 Application of GVC Control Algorithms

273

Fig. 13.5 Simulation results for GVC controller (couplings test) and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

274

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

and all angular position errors (significant change of signals). From Fig. 13.5d–e with Fig. 13.4d–e it follows that the deterioration of performance is for all velocity errors (more oscillations and disturbances), especially for dy/dt, dz/dt, dφ/dt, and

dψ/dt. Essential changes are also visible to the forces τηY , τηZ and the moments τηK , τηN (comparing Fig. 13.5f-g and 13.4f-g). Conclusions: (1) couplings cause more overshoot and degrade the tracking task; (2) it is possible to identify the effect comparing CL and GVC control algorithms. The results of couplings investigation can be useful when designing a vehicle and for selection desired trajectories.

13.5.2 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Body-Fixed Frame 13.5.2.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: to show performance of the GVC controller for an underwater vehicle model in the presence of a disturbances model. Fast error convergence without overshoot in position errors is expected. Evaluation of dynamics effect in the vehicle comparing the GVC controller with the corresponding CL controller. For tracking it is assumed the following desired helical linear position trajectory profile together with exponential angular position: ηd = [4 cos(0.1 t) − 3, 4 sin(0.1 t) − 1, 0.2 t + 0.2, 0.4e−0.5t , −0.8e−0.3t , 0.2e−t ]T , η˙ d = [−0.4 sin(0.1 t), 0.4 cos(0.1 t), 0.2, −0.20e−0.5t , 0.24e−0.3t , −0.20e−t ]T .

(13.59) (13.60)

The disturbances vector including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments: ⎤ 2 30 f 13 (t) e−1 + 2 f 13 (t) e−2 + 50 ⎢ 9 f 13 (t) e−1 + 0.6 f 2 (t) e−2 + 15 ⎥ 13 ⎥ ⎢ ⎥ ⎢ f 14 (t) − 70 ⎥, ⎢  F=⎢ ⎥ (t) − 1.4 0.02 f 14 ⎥ ⎢ ⎦ ⎣ 150 f 15 (t) e−3 + 12 50 f 15 (t) e−3 + 4 ⎡

(13.61)

where f 13 (t) = 8 cos(0.2 t) + sin(0.05 t), f 14 (t)=28 sin(0.4 t) e−5 + 35 ecos(−0.3 t) , π t) + π4 cos( π5 t). and f 15 (t) = π5 sin( 15 The gain matrices were selected to ensure position tracking error convergence without great overshoot and the lumped uncertainty error convergence with a small

13.5 Application of GVC Control Algorithms

275

X , F Y , F Z , b disturbance moments F K , Fig. 13.6 Disturbance functions: a disturbance forces F M , F N F

limit. In order to better show effects of dynamics in the control gains it is assumed that all elements of each gain matrix have the same values. However, because in the matrices k D and  the matrix N is used to include the vehicle dynamics hence the initial matrices k∗D and  ∗ are applied. The algorithm is the GVC controller. Moreover, the position and velocity errors convergence should be guaranteed. As a results, the following set of control parameters set was selected: k∗D = diag{8, 8, 8, 8, 8, 8}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{30, 30, 30, 30, 30, 30},

(13.62)

which means that k∗D = 8 I, k P = I,  = 0.4 I, and  ∗ = 30 I. The above given gains were used for the GVC controller (with the matrix N, i.e. k D = N k∗D ,  = N  ∗ ). Because of dynamical effects observed in the first phase of motion the input signals are limited (forces, moments, and estimated disturbances functions). Saturation values are: |τ | ≤ 1500 N or Nm and | f e | ≤ 1500 N or Nm. The used disturbance forces and moments functions are shown in Fig. 13.6. The GVC algorithm containing the matrix N is still the same. The difference relies on selecting the matrices k D and  which are related to the vehicle dynamics included in the matrix N. From Fig. 13.7a it is seen that the desired position trajectory is realized correctly. Observing Fig. 13.7b it is noticeable that all linear position errors are close to zero after about 10 s. Thus, the controller gives acceptable position error convergence. Moreover, oscillations are avoided. Similarly, the angular position error convergence is ensured at the same time as it arises from Fig. 13.8c. As it shown in Fig. 13.8d–e the velocity error convergence is also guaranteed. It can be concluded that the gain parameters are selected properly. An advantage of the control gains selection is that, because of the use the matrix N , the time system response is strictly related to the vehicle dynamics.

276

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.7 Simulation results for GVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

13.5 Application of GVC Control Algorithms

277

Fig. 13.8 Simulation results for GVC controller and underwater vehicle: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

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As can be seen from Fig. 13.8a–b the quantities s for linear and angular variables (responsible for the position and velocity convergence) tend to a value close to zero very fast (after about 2 s). Based on the time history of s it can be concluded if the position and velocity errors converge to the end value with acceptable convergence. From Fig. 13.8c–d it follows that all lumped dynamics estimation errors reach limited values very quickly. The applied forces and moments are given in Fig. 13.8e–f. Their values are high only in the first phase of motion when the vehicle starts. This phenomenon is related to presence the vehicle dynamics in the control gains. Comparing Fig. 13.9a–b it can be noted that the largest part of the reduced kinetic energy concerns N1 , N2 , and to lesser extent N3 . It means that the most energy is related to the movement of linear positions, whereas energy regarding angular variables is negligible. From Fig. 13.9c–d it is seen that ξ2 , ξ3 , and ξ4 have noticeable values. Therefore, it can be concluded that the most deformed velocities are v, w, and p. However, their deformation is small. The values of quasi-forces indicate that changes in moments only concern the moments τ M and τ N . Dynamics Test Based on Comparison the GVC Controller with the CL Objective: evaluation of dynamics effect in the vehicle comparing the GVC controller with the corresponding CL controller. Case 1—the control parameters set (13.62), but using k D = k∗D ,  =  ∗ . In order to estimate the influence of dynamics on control quality, the results obtained from the GVC controller with the results obtained for the CL controller (which does not contain dynamic parameters in the control gain matrices) were compared. The results for the CL controller are given in Figs. 13.10 and 13.11. Comparing Fig. 13.7b–e with Fig. 13.10a–d it is observed that the CL control algorithm does not work correctly. The trajectory tracking task is not realized. However, the results are important for evaluation of the vehicle dynamics. Some insights are possible taking into account performance obtained from the GVC controller (with the matrix N) because the matrix N is strictly related to the system dynamics. Effect of dynamics is noticeable both for the linear position errors and the linear velocity errors with exception of φ (position) and dφ/dt. It can be also concluded that the control gain matrices were not selected appropriately for the CL control algorithm and the desired trajectories. There is a visible effect of absence the dynamics in these matrices. From Fig. 13.11a–b it follows that also the quantities s do not tend to a limit which is close to zero. For this reason, other results are not satisfactory (tracking of the desired position and velocity trajectory). The lumped dynamics estimation errors w related to linear velocities tend to lower limit value using the GVC control algorithm than using the CL controller (Figs. 13.8c and 13.11c). Moreover, the lumped dynamics estimation errors w related to angular velocities are limited by higher values (Figs. 13.8d and 13.11d). Taking into account Fig. 13.8e–f it is seen that the forces and moments have similar time history as when using the GVC controller with exception of the first phase of motion (Fig. 13.11e–f). From the results it can be concluded that all position and velocity errors are sensitive to changes in dynamics. However, the greatest effect of dynamics is observed for x and θ and their time derivatives.

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Fig. 13.9 Simulation results for GVC controller and underwater vehicle: a comparison between kinetic energy for underwater vehicle K U V and kinetic energy related to N1 , N2 , N3 ; b comparison between kinetic energy related to N4 , N5 , N6 ; c ξ time history related to linear velocities; d ξ time history related to angular velocities; e π time history related to forces; f ξ time history related to moments

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Fig. 13.10 Simulation results for CL controller and underwater vehicle—Case 1: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

Case 2—increased values of control parameters for the CL algorithm to improve the performance. In order to show possibility of obtaining acceptable results for the CL controller a different control gain matrices were proposed and another simulation test was carried out. For this reason the new gains set was selected. The matrices k∗D ,  ∗ have 80 times bigger values (the matrices are crucial for control purposes in this case), i.e.: k∗D = diag{640, 640, 640, 640, 640, 640}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{2400, 2400, 2400, 2400, 2400, 2400}, (13.63) where k∗D = 640 I, k P = I,  = 0.4 I, and  ∗ = 2400 I. It can therefore be said that the greater values of the gains must replace, in a way, lack of the vehicle dynamics. Using the control gains set the CL algorithm works correctly and leads to satisfactory response of the system as it is shown in Figs. 13.12 and 13.13. Moreover, comparing graphs from Figs. 13.12 and 13.13 with corresponding graphs from Figs. 13.7 and 13.8 it can be observed that the results are basically comparable (with exception if the vehicle starts). However, for the CL control algorithm to avoid overshoot values

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Fig. 13.11 Simulation results for CL controller and underwater vehicle—Case 1: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

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Fig. 13.12 Simulation results for CL controller and underwater vehicle—Case 2: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

of the matrices should be increased (k∗D ,  ∗ ). Based on the graphs obtained for the CL control algorithm, one can say anything about couplings or dynamics effect in the vehicle. Therefore, the main difference is that the control matrix values selected for the CL controller are not related in any way to the dynamics of the vehicle. Moreover, any similar comparison useful for dynamics effects investigation as using the GVC controller cannot be done.

13.5.2.2

Models Comparison Based on the GVC Controller and Indexes for 6 DOF Fully Actuated Underwater Vehicle

The idea the the method was presented in Herman (2019) but with slightly different conditions and vehicles parameters as well the set of indexes. Objective: to show performance of the GVC position and velocity trajectory tracking controller for basic dynamical model of an underwater vehicle and to present the proposed analysis of selected models based on the set of indexes. For tracking it is assumed the following desired 3D cycloid trajectory profile for linear variables together with the exponential trajectory for angular variables:

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Fig. 13.13 Simulation results for CL controller and underwater vehicle—Case 2: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

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ηd = [0.6 t − 0.3 sin(0.5 t) − 1.2, 0.6 − 0.3 cos(0.5 t), 0.3 t + 0.2, 0.8e−0.4t , −0.6e−0.7t , −0.4e−0.2t ]T , η˙ d = [0.6 − 0.15 cos(0.5 t), 0.15 sin(0.5 t), 0.3, −0.32e−0.4t , 0.42e−0.7t , 0.08e−0.2t ]T .

(13.64) (13.65)

The disturbances model including the unknown dynamics as well as the external disturbances is expressed by the following vector of applied forces and moments defined by (13.61). The gain matrices for GVC (with N) controller were selected to ensure the position and velocity tracking errors convergence without great overshoot and the lumped uncertainty error convergence. It order to better show effects of dynamics included in the control gains it is assumed that all elements of each gain matrix have the same values. Moreover, the position and velocity errors convergence should be guaranteed. After some number of attempts, the following set of control parameters was selected: k∗D = diag{10, 10, 10, 10, 10, 10}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{30, 30, 30, 30, 30, 30}, (13.66) which means that k∗D = 10 I, k P = I,  = 0.4 I, and  ∗ = 30 I. Moreover, the resultant gains are k D = Nk∗D and  = N ∗ . Because of dynamical effects observed in the first phase of motion the input signals are limited (forces, moments, and disturbance estimation functions). Saturation values are: |τ | ≤ 1500 N or Nm and | f e | ≤ 1500 N or Nm. First step—analysis of time responses for basic vehicle model. At the beginning the basic model is tested using the controller with the selected set of parameters. The parameters are given in Table 13.1 (Case 1). The obtained results are presented in Figs. 13.14 and 13.15. Table 13.1 Parameters of tested vehicles model Symbol Case 0 Case 1 Case 2 L b xG yG zG m Ix Iy Iz Jx y Jx z Jyz

2.0 0.5 0 0 0 250 20 150 140 0 0 0

2.0 0.5 −0.15 0 0.03 250 20 150 140 0 −30 0

2.0 0.5 0.2 0 0 270 30 170 100 10 −20 0

Case 3

Unit

2.0 0.5 −0.1 −0.05 0.05 200 10 120 120 −10 −10 10

m m m m m kg kgm2 kgm2 kgm2 kgm2 kgm2 kgm2

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Fig. 13.14 Simulation results for GVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

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Fig. 13.15 Simulation results for GVC controller and underwater vehicle: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

13.5 Application of GVC Control Algorithms Table 13.2 Values of quantities for indexes Case 0 Case 1  −1  ϒ − I  0.0000 0.7435 of f  −1  ϒ  2.4495 2.4495 diag δξ δz δξc δπ πc δK δ Kc δ η δ η˙

1.7623 0.9737 1.0000 1308.9 0.0000 75.2687 1.0000 0.2949 0.1176

1.8525 0.9781 1.0314 1773.3 106.4773 75.2893 1.0036 0.2950 0.1178

287

Case 2

Case 3

0.4349

0.4889

2.4495

2.4495

1.8207 0.9766 1.0104 1673.4 92.3639 79.5519 1.0033 0.2948 0.1175

3.0638 1.0819 1.0831 2737.6 247.7274 81.7516 1.0174 1.5338 0.4476

From Fig. 13.14a it is observed that the desired position trajectory is tracked correctly. As it results from Fig. 13.14b–c after about 15 s all position errors are close to zero. Similar observation can be made based on Fig. 13.14d–e where velocity errors are shown. However, if the vehicle starts some changes are seen. It arises from the fact that dynamics of the vehicle is included into the control gains. The quantities s given in Fig. 13.15a–b tend to limited values near zero very fast what ensures fast convergence the position and velocity errors presented earlier. The lumped dynamics estimation errors w related to linear and angular velocities shown in Fig. 13.15c– d also converge to the limited values in short time. Finally, in Fig. 13.15e–f the forces and moments can be seen. Summarizing, it can be concluded that the control algorithm works correctly using the selected gain matrices. Second step—comparing various vehicles models. After examining that the algorithm is working properly and provides satisfying performance, a comparison of different vehicle models with changed dynamic parameters takes place. The proposed models are shown in Table 13.1 and denoted as Case 2 and Case 3 (Case 0 means the simplified basic model described as Case 1, in which couplings are reduced by assuming the diagonal inertia matrix and symmetry). The purpose of the comparison is to investigate effect of various couplings (arising from dynamical and geometrical parameters). The results obtained for the selected models are summarized in Table 13.2. Analysis of δ η and δ η˙ leads to conclusion that Case 3 (despite the reduced vehicle weight) should be rejected in this test. It can be considered in another test with different goal. Other cases give comparable values of these quantities. Moreover, for Case 3 couplings are the strongest (δ K c , πc , δξc ) and the kinetic energy consumption δ K , the input signal effort δπ and velocity deformation δz, δξ are also the largest. Case 0 is the comparative basis for selected models and it is an ideal case. Taking into account Case 1 and Case 2 it is observed that the selected gain values causes that the

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Table 13.3 Values of indexes Index Case 0 I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 IC1 IC2 IC3 IC4

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.000 2.800 6.000 1.500

Case 1

Case 2

Case 3

1.000 1.002 1.355 1.000 1.005 1.051 1.304 1.060 1.004 1.031 2.179 2.999 6.455 1.501

1.000 0.999 1.279 1.057 1.003 1.033 1.178 1.055 1.003 1.010 2.168 2.957 6.282 1.528

5.358 3.806 2.092 1.086 1.111 1.739 1.200 1.090 1.017 1.083 12.342 6.983 7.240 5.125

controller works correctly even if the parameters set is changed. It can be said that to some extent the assumed set of control gains gives acceptable tracking results. In order to obtain more information some indexes and together with their weight coefficients are applied. Values of the indexes for the same cases are given in Table 13.3. According to the calculations of indexes, it was confirmed that the worst case is the Case 3 and it is unacceptable. It can be noted that the values of indexes I1 ÷ I10 are comparable for Case 1 and Case 2. Based on these values it is difficult to decide which structure Case 1 or Case 2 is better. The decision must be related to the goal of such comparison. This goal is to calculate complex indexes. To obtain more information four sets of weight coefficients are assumed. They are chosen taking into account quantities important for the designer. The final values for all cases are shown in the lower part of Table 13.3. Set of weight coefficients 1. The most important are tracking performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ), the kinetic energy consumption and the control effort w2 = 0.5 (this value is multiplied by the sum of I3 , I4 ). The index IC1 (calculated from (13.42)) is given in Table 13.3. For Case 2 (1.001 + 1.178) the index is slightly lesser than for Case 1 (1.000 + 1.168). For Case 3 its value is unacceptable. Set of weight coefficients 2. Important are: tracking performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ), the kinetic energy consumption and the control effort w2 = 0.3 (this value is multiplied by the sum of I3 , I4 ), and couplings w3 = 0.2 (this value is multiplied by the sum of I5 ÷ I10 ). The index IC2 indicates that Case 2 (1.000 + 0.701 + 1.256) is better than Case 1 (1.001 + 0.707 + 1.291), whereas for Case 3 its value is very great.

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Set of weight coefficients 3. It is assumed that they are equally important velocity deformation and dynamic couplings. Therefore, w1 = 1.0 (this value is multiplied by the sum of I5 ÷ I10 ). Other parameters can be omitted. The index IC3 indicates that the strongest deformation and dynamical couplings are for Case 3. Velocity deformation and couplings are bigger for Case 1 (2.056 + 4.399) than for Case 2 (2.036 + 4.246). Set of weight coefficients 4. It is assumed that they are equally important control performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ) and the kinetic energy consumption w2 = 0.5 (this value is multiplied by I4 ). Now Case 1 (1.001 + 0.500) is better than Case 2 (1.000 + 0.528). Discussion of results. The performed test showed that using the proposed approach it was possible to indicate among various models the model which not only guaranteed good performance of the controller but also which was better from the kinetic energy point of view or control effort and had the greater couplings. Based on the results it is possible to eliminate structures which do not realize the control task or to ensure properly work of the control algorithm but with high energy consumption. In the test the worst values were for Case 3 and this case should be eliminate because of unsatisfactory results. It is harder to choose between Case 1 and Case 2. However, even slightly differences can give us an answer for this question (Case 1 represents the best structure it the kinetic energy consumption is very important). If the kinetic energy consumption and the mass of the vehicle are not crucial Case 2 is better than Case 1. Small differences between indexes can be explained by the fact that in the control gains dynamics and dynamical couplings are taken into consideration (because of the matrix N). The procedure is also useful for determination of dynamical effects in the vehicle. If we want to analyze couplings without the full vehicle dynamics then the matrix N should be omitted as in Herman (2019).

13.5.3 Velocity Tracking Controller with Adaptive Term in Body-Fixed Frame 13.5.3.1

Velocity Tracking Controller Frame for 6 DOF Fully Actuated Underwater Vehicle

Algorithm of this type and its application to underwater vehicle dynamics evaluation was considered in Herman (2020a). Objective: showing performance of the GVC algorithm with the matrix N, dynamics study by comparison with the CL algorithm. For tracking it is assumed the desired velocity profile defined by (11.43). The unknown dynamics as well as the external disturbances is expressed by the vector of forces and moments (11.44). The gain matrices of the GVC controller, ensuring acceptable errors convergence, were selected as follows:

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k∗D = diag{12, 12, 12, 12, 12, 12}, k I = diag{1, 1, 1, 1, 1, 1},  = diag{0.1, 0.1, 0.1, 0.1, 0.1, 0.1},  ∗ = diag{50, 50, 50, 60, 60, 60}. (13.67) For the GVC controller the matrices k D and  are selected in such way that they are multiplied by the matrix N. Hence, it is assumed that k D = Nk∗D and  = N ∗ instead of k D = k∗D and  =  ∗ . In fact it is only another way to choose the matrices values. As a result, the gains depend on full dynamics of the vehicle what enables the dynamics evaluation. This algorithm is applied to show the effects of dynamics while the vehicle is moving. If the matrix is omitted then using the GVC controller evaluation of couplings effects is possible only. The desired linear and angular velocity profiles are presented in Fig. 11.1a–b, and the disturbances forces and moments profiles in Fig. 11.1c–d, respectively. Note that, vd and qd are equal to zero. This is important because these variables should be also close to zero if the effects of dynamics on the system’s response is of little importance. From Fig. 13.16a–b it follows that the steady state for linear and angular velocities occur after about 1 s. Moreover, the errors v, q are close to zero during the motion. The velocities are tracked in a very short time. There exist a relationship between the velocities and quantities s which are also quickly tracked as it is presented in Fig. 13.16c–d. All lumped dynamics estimation errors w converge to their end value quickly Fig. 13.16e–f. However, in the first phase of motion their values are great. It is because the full vehicle dynamics is included into the control gains (presence of the matrix N). For the same reason the forces (especially τ X , τ Z ), and moments (especially τ M , τ N ) have large values after the vehicle starts as it is shown in Fig. 13.16g–h. Dynamics Test Based on Comparison the GVC Controller with the CL In order to show benefits of using the GVC algorithm, two comparative tests with the CL control algorithm were performed. In Case 1 the same set of gains parameters are used whereas in Case 2 different gains which ensure the errors convergence for the CL controller are selected. Both cases are useful for the test of the vehicle dynamics. Case 1—control using (13.67). For the CL controller it is assumed that k D = k∗D , k I , , and  =  ∗ . The results are given in Fig. 13.17. As it is seen from Fig. 13.17a– b any acceptable results are achieved. It is observed that u, w, and q are very sensitive for lack of dynamic parameters in the control gains. This observation is confirmed in Fig. 13.17c–d where the quantities s are shown. From Fig. 13.17e–f it follows that lumped dynamics estimation errors have smaller values but the main task, namely reducing these variables is not guaranteed. It means that the control algorithm does not work correctly (especially w Z and w M are to large if the vehicle moves. This can be explained by the fact that the dynamics of the vehicle is not included in the controller’s gain matrix. Comparing Fig. 13.16g–h with Fig. 13.17g– h it is seen that the most sensitive (for lack of dynamical parameters) is the force τ X . Likewise, all moments are sensitive in this situation. Case 2—gain matrices which guarantee the errors convergence. The gain coefficients of the controller were selected, in order to ensure the velocity errors convergence, as

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Fig. 13.16 Simulation results for GVC and underwater vehicle: a linear velocity errors b angular velocity errors; c errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

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Fig. 13.17 Simulation results for CL controller and underwater vehicle—Case 1: a linear velocity errors; b angular velocity errors; c errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

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follows: k∗D = diag{120, 120, 120, 120, 120, 120}, k I = diag{1.0, 1.0, 1.0, 1.0, 1.0, 1.0},  = diag{0.5, 0.5, 0.5, 0.5, 0.5, 0.5},  ∗ = diag{1000, 1000, 1000, 1200, 1200, 1200}.

(13.68)

However, in this example the gains are not in any relation to the vehicle dynamics. The performance is improved because all velocity errors tend to zero. But oscillations occur and time of convergence is longer than using the GVC controller. In order to reduce this phenomenon increased elements of the matrix k D were assumed. Comparing Fig. 13.16a with Fig. 13.18a it is observable that x and w are excited with overshoot if the vehicle dynamics is not taken into account in the gains. Similarly, as it arises from Fig. 13.18b q and r the signals are over-regulated. Moreover, the oscillations of the variable q occur the longest. The effects are confirmed in Fig. 13.18c–d where the time history of quantities s are given. From Fig. 13.18e–f it is seen that the lumped dynamics estimation errors w are reduced to their end values. However, w X , w Z , w M , and w N converge slowly. There exists some relationship between these variable and the appropriate velocity errors x, w, q, and r . Comparing Fig. 13.16g–h with Fig. 13.18g–h it is noticeable that the applied forces and moments have now similar values with exception of the initial phase of motion. From the results, it can be observed that assuming control gains without dynamical parameters the variables sensitive to dynamics are recognized.

13.5.3.2

Velocity Tracking Controller Frame for 6 DOF Fully Actuated Airship

Objective: showing performance of the GVC algorithm with N matrix, comparing it with performance of the CL algorithm, studying the dynamics of the airship model. For tracking it is assumed the desired velocity profile defined by (11.47). The disturbances function including the unknown dynamics as well as the external disturbances is expressed by equation (11.48). The above given velocity and disturbance function profiles were used for all adaptive velocity controllers. The gain coefficients of the GVC (with N) controller were selected as follows to ensure the error convergence: k∗D = diag{25, 25, 25, 25, 25, 25}, k I = diag{1, 1, 1, 1, 1, 1},  = diag{0.1, 0.1, 0.1, 0.1, 0.1, 0.1},  ∗ = diag{400, 400, 400, 400, 400, 400},

(13.69)

which means that k∗D = 25 I, k I = I,  = 0.1 I, and  = 400 I. Moreover, for the GVC algorithm it is assumed k D = N k∗D ,  = N  ∗ to show differences which are

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Fig. 13.18 Simulation results for CL controller and underwater vehicle—Case 2: a linear velocity errors; b angular velocity errors; c errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

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observed if the vehicle dynamics is taken into account. In Fig. 11.3a–b the desired linear and angular velocity profiles are given. The used disturbance forces and moments are shown in Fig. 11.3c–d. From Fig. 13.19a–b it is observed that the steady state for linear and angular velocities is achieved very quickly (after about 1 s). However, the error v increases rapidly after the start of the airship (Fig. 13.19a). This is unexpected because it should be close to zero. This effect can be explained the presence of vehicle dynamics in the control gain matrix. We conclude that the velocity v is very sensitive to the dynamical parameters. Similar conclusion can be made from Fig. 13.19b) for the errors p ( q is less sensitive). There exist a relationship between the velocities and quantities sv which are also quickly tracked as it is presented in Fig. 13.19c–d. All lumped dynamics estimation errors w converge to their end value quickly as shown in Fig. 13.19e–f. However, in the first phase of motion their values are great. It is because the full vehicle dynamics is included into the control gains (presence of the matrix N ). For the same reason the forces (especially τ X , τ Z ), and moments (especially τ M , τ N ) have large values after the vehicle starts as it is shown in Fig. 13.19g–h. Dynamics Test Based on Comparison the GVC Controller with the CL In order to compare the obtained results the test was done for the CL control algorithm for the same set of parameters (13.69) where k D = k∗D and  =  ∗ . In this algorithm the matrix N is not present what enables to show effects of dynamics. Comparing Figs. 13.19a–b and 13.20a–b it is observed that errors u and q are the most changed due to the vehicle’s dynamics (note that qd is zero). Similar effect is for

w. The error v is excited too (vd = 0). Moreover, many oscillations are visible what suggests that elements of the matrix k∗D are too small. It can be concluded that the signals u and w are sensitive using the desired velocity profile if the dynamics of the vehicle is taken into account. Additionally, it turned out that the influence of dynamics on signals v and q is significantly large. The same conclusion can be made if we compare Figs. 13.19c–d and 13.20c–d where the signals of the vector s are shown. The reason for the long convergence time of errors with many oscillations is inadequate choice of control gains. It means that if the dynamics of the vehicle is not taken into account then one has to choose the gain matrices in a different way. From the comparison of Figs. 13.19e–f and 13.20e–f it follows that the lumped dynamics estimation errors for the CL controller go to the final values after a longer time than for the GVC algorithm. In addition, noticeable changes concern the signals w X , w Z , and w M , which indicates their dependence on the dynamics of the vehicle. Lack of dynamics in the gain matrices is also observable in the time history of forces and moments as it results from Figs. 13.19g–h and 13.20g–h. The symptoms are longer time of achieving the signals in steady state and over-regulation of signals for the CL control algorithm, grater value of τ X for the GVC controller when the vehicle starts. Conclusion: taking into account the vehicle dynamics in the control gain matrices makes it possible to reduce overshoot and fast velocity error convergence. Besides, it can be shown which variables are sensitive to the dynamical parameters (for the considered vehicle model, desired velocities and disturbance functions).

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Fig. 13.19 Simulation results for GVC and indoor airship: a linear velocity errors; b angular velocity errors; b errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

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Fig. 13.20 Simulation results for CL controller and indoor airship: a linear velocity errors; b angular velocity errors; c errors of s for linear velocities; d errors of s for angular velocities; e lumped dynamics estimation errors w related to linear velocities; f lumped dynamics estimation errors w related to angular velocities; g forces; h moments

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13.5.4 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Earth-Fixed Frame 13.5.4.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the GVC position and velocity trajectory tracking controller for underwater vehicle model in the presence of disturbances model. Fast error convergence without overshoot in position errors. Evaluation of dynamics effect in the vehicle comparing the GVC controller and the corresponding CL controller. For tracking it is assumed the following desired helical trajectory and exponential trajectory profiles defined by (12.51)–(12.52). The disturbances function including the unknown dynamics together with the external disturbances is expressed by the vector of forces and moments (12.55). The gain matrices for GVC controller (with the matrix N) were selected to ensure velocity tracking error convergence without great overshoot and the lumped uncertainty error convergence. It order to better show effects of dynamics in the control gains it is assumed that all elements of each gain matrix have the same values. Moreover, the position and velocity errors convergence should be guaranteed. As a results, the following set of control parameters was selected: k∗D = diag{100, 100, 100, 100, 100, 100}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{120, 120, 120, 120, 120, 120},

(13.70)

i.e. k∗D = 100 I, k P = I,  = 0.4 I, and  ∗ = 120 I. The above given gains were used for the GVC controller in the form k D = N k∗D ,  = N  ∗ . Because of dynamical effects observed in the first phase of motion the input signals are limited (applied forces and moments). Saturation values are: |τη | ≤ 1500 N or Nm and | f η e | ≤ 1500 N or Nm. Moreover, the lumped dynamics estimation errors were obtained assuming the constant values (for all cases): c X = −20, cY = 10, c Z = −20, c K = 10, c M = 20, and c N = 35. The used disturbance forces and moments functions are shown in Fig. 12.10. From Fig. 13.21a it is seen that the desired position trajectory is realized correctly. Observing Fig. 13.21b–e it is noticeable that all position errors and velocity errors are close to zero after about 15 s. Thus, the controller is working properly. Moreover, oscillations are not observed. The obtained errors convergence is strictly related to the quantities sξ which are shown in Fig. 13.22a–b. These signals tend to zero very quickly (after about 2 s) what guarantees also fast errors convergence. From Fig. 13.22c–d it follows that all lumped dynamics estimation errors are reaching limited values. As is shown in Fig. 13.22e–f the forces and moments only if the vehicle starts have great values. Oscillations are not observed. It results from presence of the system dynamics in the control gains. Fig. 13.23a–b correspond to equation (13.11). It can be seen that the dominant part of kinetic energy is caused by variables acting in

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Fig. 13.21 Simulation results for GVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

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Fig. 13.22 Simulation results for GVC controller and underwater vehicle: a quantities sξ for linear variables; b quantities sξ for angular variables; c lumped dynamics estimation errors wη related to linear velocities; d lumped dynamics estimation errors wη related to angular velocities; e forces τη ; f moments τη

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Fig. 13.23 Simulation results for GVC controller and underwater vehicle: a comparison between kinetic energy for underwater vehicle K U V and kinetic energy related to N1 , N2 , N3 ; b comparison between kinetic energy related to N4 , N5 , N6 ; c ξ time history related to linear velocities; d ξ time history related to angular velocities; e π time history related to forces; f π time history related to moments

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the x and y direction. Energy in the z direction after about 10 seconds is constant. The energy caused by the vehicle’s rotation is negligible (as it is observed in Fig. 13.23b). This information regarding the reduction of kinetic energy corresponding to each variable is not available when the equations of motion are not expressed in quasivariables. Figure 13.23c–d show the result of using Eq. (13.14). They show the velocity deformations caused by dynamic effects from other variables expressed in terms of the vector ξ . Their time history depends on dynamical couplings in the vehicle and the realized position and velocity trajectories. It can be seen that the speed differences due to dynamics are small. At the beginning of the movement, the vehicle dynamics increase linear velocities in the y and z directions the most, as well as the speed of rotation around the x (angular velocity p) axis and then in the x direction. Based on Eq. (13.15) it is possible to find the time history of forces and moments deformation. From Fig. 13.23e–f it results that the greatest deformations are forces τηY , τηZ and moment τηM . It means that the greatest forces are when the vehicle moves sideways and up and rotates around the y axis. Dynamics Test Based on Comparison the GVC Controller with the CL Objective: evaluation of dynamics effect in the vehicle comparing results of the GVC controller and the corresponding CL controller. Case 1—the same control parameters set, i.e. (13.70) and using k D = k∗D ,  =  ∗ . In order to estimate the influence of dynamics on control quality, the results obtained from the GVC controller with the results obtained for the CL controller (which does not contain dynamic parameters in the control gain matrices) were compared. The corresponding results for the CL controller are given in Figs. 13.24 and 13.25. Comparing Fig. 13.21b–e with Fig. 13.24a–d it is observed that in this case the CL control algorithm does not work correctly. The trajectory tracking task is not realized. In spite of that the presented results are important for estimating the vehicle dynamics. Drawing conclusions is possible if the test was previously performed for the GVC controller. Effect of dynamics is noticeable especially for the linear position errors and the linear velocity errors. Based on these results, it can be concluded that the control gain matrices were not selected appropriately for the control task. There is a visible effect of not taking into account the dynamics in these matrices. From Fig. 13.25a–b it follows that the quantities s do not tend to zero. For this reason the tracking of the desired position and velocity trajectory is incorrect. The lumped dynamics estimation errors wη related to linear velocities tend to a lower limit value using the GVC control algorithm than using the CL controller (Figs. 13.22c and 13.25c). The lumped dynamics estimation errors wη related to angular velocities are limited by similar values (Figs. 13.22d and 13.25d). Recalling Fig. 13.22e–f it is seen that the applied forces and moments have similar time history with exception of the first phase of motion (Fig. 13.25e–f). On the basis of the obtained results it can be concluded that all position and velocity errors are sensitive to changes in dynamics. However, the greatest effect of dynamics can be observed for x and θ and their time derivatives.

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Fig. 13.24 Simulation results for CL controller and underwater vehicle—Case 1: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

Case 2—increased values of control parameters for the CL algorithm. In order to show possibility of obtaining acceptable results for the CL controller a different control gain matrices were proposed. Thus, another study was carried out. For this reason the gains set was selected in which the matrices k∗D ,  ∗ have 25 times bigger values (the matrices are crucial for control purposes in this case), namely: k∗D = diag{2500, 2500, 2500, 2500, 2500, 2500}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{3000, 3000, 3000, 3000, 3000, 3000},

(13.71)

i.e. k∗D = 2500 I, k P = I,  = 0.4 I, and  ∗ = 3000 I. Using the control gains set the CL algorithm works correctly as it is shown in Figs. 13.26 and 13.27. Moreover, comparing graphs from Figs. 13.26 and 13.27 with corresponding graphs from Figs. 13.21 and 13.22 it is noticeable that the results are comparable. However, they do not say anything about couplings or dynamics effect in the vehicle. The main

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Fig. 13.25 Simulation results for CL controller and underwater vehicle—Case 1: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors wη related to linear velocities; d lumped dynamics estimation errors wη related to angular velocities; e forces τη ; f moments τη

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Fig. 13.26 Simulation results for CL controller and underwater vehicle—Case 2: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

difference is that the control matrix values selected for the CL controller have no relation to the dynamics of the vehicle. Additionally, any similar comparison useful for dynamics effects investigation as using the GVC controller cannot be done because the differences between the GVC and CL controllers are not observable distinctly.

13.6 Application of NGVC Control Algorithms In this section it is shown in which way the selected algorithms using NGVC can be applied to investigation of the vehicle dynamics.

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Fig. 13.27 Simulation results for CL controller and underwater vehicle—Case 2: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors wη related to linear velocities; d lumped dynamics estimation errors wη related to angular velocities; e forces τη ; f moments τη

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13.6.1 Position and Velocity Trajectory Tracking Controller in Body-Fixed Frame 13.6.1.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the control algorithm ensuring fast position errors convergence, comparison with the GVC algorithm, and dynamic study based on the NGVC controller. For tracking it is assumed the following desired helical trajectory for linear position together with constant desired angular position and the appropriate velocity profile described by (7.57) and (7.58). In this case the vehicle dynamics is included in the vector ζ (not only couplings as in the vector ξ for the GVC control algorithm. The gain coefficients of the controller were selected as follows (to ensure acceptable the position and velocity error convergence in comparable time): k D = diag{5, 3, 9, 15, 15, 5}, k P = diag{50, 50, 50, 50, 50, 50},  = diag{0.8, 0.8, 0.8, 0.6, 0.6, 0.6}. (13.72) Different elements of k D allow to ensure positions convergence in a comparable time. This is understandable because the full dynamics is included into the control gain matrices (not only couplings in the vehicle). The values of k D , k P , and  are smaller than for the GVC algorithm because here the vehicle dynamics gives an additional gain. The gain matrices are composed together with the vehicle dynamics (masses, inertia, and geometrical parameters are taken into account). The benefit is that the time response of the closed-loop system is strictly related to the vehicle dynamics. If the GVC controller is applied only effects arising from the dynamical couplings are observable. For this set of control gains (13.72) and nominal parameters the results for the NGVC control algorithm are shown in Fig. 13.28. From Fig. 13.28a it follows that the desired trajectory is tracked correctly. This fact is confirmed in Fig. 13.28b. The linear position error convergence time is about 6 s. Also the angular position errors are close to zero in comparable time as it is observed from Fig. 13.28c. From Fig. 13.28d–e it can be seen that the velocity errors tend to zero also in about 6 s. The coupling effect between velocities it is noticeable especially for the variables d x/dt and for all angular velocity errors. Values of the applied forces and moments is given in Fig. 13.28f–g are similar as the values obtained for the GVC controller. It can be concluded that both algorithms, namely the GVC and the NGVC realize the desired trajectory tracking task. However, in the NGVC controller the vehicle dynamics plays the crucial role whereas in the GVC algorithm only dynamical couplings are taken into account in the gain matrices. Consequently, the set of gain coefficients has smaller values if the NGVC controller is applied. Moreover, they are related to the system dynamics. Robustness investigation. For the same set of control gains the robustness test with 10% decreasing of parameters was done and the results are given in Fig. 13.29. From

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Fig. 13.28 Simulation results for NGVC controller and underwater vehicle (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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Fig. 13.29 Simulation results for NGVC controller and underwater vehicle (robustness test—10% decreasing of parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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Fig. 13.29a it is seen trajectory tracking is not accurate. As it is observable from results presented in Fig. 13.29b–e the position errors as well as the velocity errors are not very close to zero. It means that they are sensitive to the vehicle parameters changes. As it is noticeable from Figs. 13.29f–g and 7.2f–g the forces and moments have similar time history. It can be concluded that the NGVC controller is not very useful in case of unknown parameters, in spite of that some model inaccuracies are acceptable. For small parameters changes the controller works almost correctly. However, application of the NGVC controller can give some answer concerning sensitivity of the errors and input signals to changes of the control gain matrices. Usability of the NGVC algorithm is based on something else, namely the study of vehicle dynamics together with the dynamic couplings. Consider now its use for this purpose. Dynamics Test Based on Comparison the NGVC Controller with the CL Test 1. Assuming for the CL control algorithm (controller without matrices arising from the inertia matrix decomposition) the same set of gains as for the NGVC controller some dynamical properties of the system can be recognized. It means that it is possible to determine indirectly the sensitivity of variables to the couplings in the system. For the same trajectories (7.57)-(7.58) and set of control parameters (13.72) the results for the CL controller are given in Fig. 13.30. This system response should be compared with the results given in Fig. 13.28. It is observed that the CL control algorithm works also correctly using the same gains set as for the NGVC controller. However, the important benefits relies on additional information which is accessible from this comparison. From Fig. 13.30a it is seen that the desired trajectory is tracked after a longer time than for the NGVC controller. When comparing Fig. 13.28b–c with Fig. 13.30b–c, it is observed that position x, z and ψ are sensitive to dynamical couplings in the vehicle (errors x, z and ψ differ significantly for each of controllers). Large differences of variables d x/dt, dφ/dt, and dψ/dt indicate the effect of dynamics on the speed of the vehicle (from Figs. 13.28d–e and 13.30d–e). Additional information is accessible from Figs. 13.28f–g and 13.30f–g. Significant differences are noticeable at the beginning of the vehicle motion for signals τ X , τY , τ M , and τ N . This can be explained by the fact that in the NGVC controller the dynamics of the vehicle is taken into consideration. It is noticeable that the errors converge to zero in different time. If the dynamics is absent in the gains then the control performance are deteriorated. Increasing the gain values it is possible to improve the results but information about dynamical effects in the vehicle will be lost. Test 2. In order to show another benefits arising from application of the NGVC controller consider the next case. In order to obtain other information about the system the different set of control parameters was selected, i.e.: k D = diag{100, 100, 100, 100, 100, 100}, k P = diag{500, 500, 500, 500, 500, 500},  = diag{0.8, 0.8, 0.8, 0.8, 0.8, 0.8},

(13.73)

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Fig. 13.30 Simulation results for CL controller and underwater vehicle (nominal parameters— Test 1): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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which means that k D = 100 I, k P = 500 I, and k D = 0.8 I. The gain matrices arise from the use of the trial and error method. They are assumed taking into account some hints. Firstly, the matrix  should ensure fast time response of the system. Secondly, together with the matrix k P it can be guarantee satisfactory position and velocity errors convergence. Thirdly, larger values of the matrix k D serve for reducing of overshoot. In this case the control parameters were selected independently on the vehicle dynamics but ensuring the errors convergence. It can be observed that the gains have great values. The results from Fig. 13.31 should be compared with the signal presented in Fig. 13.28. The desired trajectory is tracked after longer time now (Figs. 13.28a and 13.31a). Comparing results from Figs. 13.28a–b and 13.31a–b it is noticeable that all position and angular errors are sensitive for parameters selection and overshoot is observed in the position and velocity errors. Such information is not satisfying because now it would be necessary to introduce some measure of sensitivity. As is apparent from Figs. 13.28d–e and 13.31d–e important differences concern practically all velocities which suggests that a measure ought to be introduced. From Fig. 13.28f–g and 13.31f–g it can be seen that in the first phase of motion the forces and moments have great values if any of two controllers is applied. However, for the CL algorithm the signals and control gains are not related to the vehicle dynamics. Therefore, the conclusion is that without additional measures to assess the impact of dynamics on control, it is better to examine the system response for both algorithms with the same controller gain coefficients. Dynamics investigation based on analysis of response obtained from the NGVC and the CL algorithm even using different control gains can give some useful information about the influence of dynamics on the trajectory tracking. It must be remembered that the analysis of vehicle dynamics is done for the assumed trajectory and with strictly defined operating conditions of the controller. From Fig. 13.31b–e it arises that values of the matrix k D should be larger to decrease the oscillation effect. In contrast, the gain matrix k D used together with the vehicle dynamics for the NGVC algorithm guarantees reduction of the phenomenon (the couplings are transformed from one variable to others). Consequently, taking into account the comparison between signals from the NGVC and the CL controller some information about the impact of vehicle dynamics on the regulation process for selected desired trajectory is accessible.

13.6.1.2

Position and Velocity Trajectory Tracking Controller in Body-Fixed Frame for 6 DOF Fully Actuated Airship

The NGVC control algorithm was presented and tested on an airship model both for nominal and changed parameters in Herman (2017). Its extended use is given here. Objective: showing performance of the NGVC controller for an indoor airship model, robustness test of the algorithm, and dynamic study based on the NGVC controller. For tracking it is assumed the desired 3D linear position trajectory together with constant angular position values and the appropriate velocity profile described by (7.60) and (7.61). The gain coefficients of the controller were assumed to ensure

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Fig. 13.31 Simulation results for CL controller and underwater vehicle (nominal parameters— Test 2): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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acceptable errors convergence (because of this reason values of each matrix are not the same): k D = diag{15, 5, 5, 5, 5, 5}, k P = diag{10, 5, 5, 6, 6, 6},  = diag{0.3, 0.3, 0.3, 0.3, 0.3, 0.3}.

(13.74)

For the set of control gains and nominal parameters the results are shown in Fig. 13.32. Values of the coefficients were selected in relation to the dynamics of the vehicle. Because the initial point in x direction is far from the desired trajectory then the first coefficients in k D and k P are bigger than others to ensure faster error convergence. From Fig. 13.32a it is noticeable that the linear position trajectory tracking task is realized correctly. As it is shown in Fig. 13.32b–c after about 12 s all linear and angular position errors are close to zero. Similar observation can be made from Fig. 13.32d–e where velocity errors are presented. However, from Fig. 13.32e it is seen that the angular velocity errors are excited in spite of that the initial error values are zero. Moreover, the most sensitive is the error dθ/dt because overshoot may be observed. This phenomenon can be explained by dynamical relationship among velocities in the vehicle. From Fig. 13.32f–g it arises that the forces and moments tend to end values after about 7 s. The long distance between the error x causes that the initial value of τ X is also large. Other large value, i.e. τ M results from the airship dynamics. Comparing Figs. 7.3 and 13.32 it is noticeable that the variables are more excited when the NGVC algorithm is used. It results from that the full dynamics is included in the control gains instead of couplings only. This effect depends on the airship parameters and the assumed gains set. Note however, that the obtained from both controllers information is qualitatively similar (τ X is lesser for the NGVC controller). Robustness investigation. At present robustness of the NGVC control algorithm will be tested using the same set of gains as for the vehicle with nominal parameters. The dynamical parameters of the vehicle and environmental parameters (damping terms and added masses) differ by 3 % from nominal (they are greater). It means that elements of M and D are bigger by 3 %. From Fig. 13.33a it follows that the desired trajectory is tracked correctly. However, from Fig. 13.33b–c it is seen that not all linear and angular position errors tend to zero which indicates the sensitivity of the algorithm to changes in parameters. On the contrary, the velocity errors have acceptable end values (Fig. 13.33d–e). It can be concluded that the algorithm guarantees the convergence of velocity errors with small changes in dynamic parameters of the vehicle. Observing the time history of forces and moments in Fig. 13.33f–g, it is noticeable that the forces τY and τ Z have bigger end values as previously. Based on the results we can say that the control algorithm of this type is very sensitive when we do not know exactly the dynamics of the vehicle. It is worth emphasizing that the essential value of this algorithm is that it can be used to analyze vehicle dynamics.

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Fig. 13.32 Simulation results for NGVC controller and indoor airship (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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Fig. 13.33 Simulation results for NGVC controller and indoor airship (robustness test): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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Dynamics Test Based on Comparison the NGVC Controller with the CL In this test the classical controller using the same gains and trajectories is considered. Comparing Figs. 13.32a and 13.34a it can be seen that if the CL controller is used then the task is realized correctly but in other way than for the NGVC controller. Recalling Fig. 13.32b–c and looking at Fig. 13.34b–c one can observe that linear position variable x and all angular variables are sensitive to dynamical parameters. It is because the errors x, φ, θ , and ψ converge to the end value with overshoot. Moreover, the errors φ and ψ are significantly deformed in the first phase of motion. From Figs. 13.32d–e and 13.34d–e it is noticeable that all velocity errors (linear and angular) are disturbed if the dynamics of the airship is taken into account. The maximal value of d x/dt increases essentially whereas other linear position errors converge to zero with oscillations. In the same time large fluctuations in velocity errors dθ/dt and dψ/dt are observable. Large changes in velocity errors lead to disturbances in vehicle movement (there is a relationship between the position and velocity errors). The velocity changes are also related to deformation of the applied forces τ X , τY in Fig. 13.34f and the moments τ K , τ N in Fig. 13.34g. Making use of the comparison (with the same control gains set) sensitivity of each variable to dynamics of the airship can be determined. Introducing additionally some indexes quantitative information about this phenomenon is available (e.g. differences between the maximal and minimal values of the errors). Comparing the results obtained for the NGVC and the CL control algorithms it can be observed that for the CL controller absence of dynamics leads to deterioration of performance.

13.6.2 Velocity Tracking Controller in Body-Fixed Frame 13.6.2.1

Velocity Tracking Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the NGVC algorithm, dynamics investigation based on NGVC, i.e. determining the effects of including the full dynamics into the control gain matrices by comparing with the corresponding CL controller. For tracking it is assumed the desired velocity profile described by Eq. (8.52). The velocity trajectory profiles were given in Fig. 8.3. The gain coefficients of the controller were selected as follows to ensure the velocity errors convergence: k D = diag{2, 2, 2, 2, 2, 2}, k I = diag{10, 10, 10, 10, 10, 10},  = diag{0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, (13.75) which means that k D = 2 I, k I = 10 I, and  = 0.5 I. The values are less than for the GVC algorithm because here the full dynamics is included in the control gains. From Fig. 13.35 it is seen that both the velocity errors (a)–(b) and their integrals (c)–(d) tend to zero. Comparing appropriate time history of variables from this figure

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Fig. 13.34 Simulation results for CL controller and indoor airship (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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Fig. 13.35 Simulation results for NGVC controller and underwater vehicle: a linear velocity errors; b angular velocity errors; c errors of z for linear velocities; d errors of z for angular velocities; e forces; f moments

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Fig. 13.36 Simulation results for CL controller and underwater vehicle: a linear velocity errors; b angular velocity errors; c forces; d moments

with Fig. 8.4 some observation can be made. The linear errors tend to zero faster for the NGVC controller than for the GVC one. The angular velocity error q has smaller maximal values in Fig. 13.35b. Smaller values have also integrals zlin and z ang . The forces and moments values are comparable in Figs. 13.35e–f and 8.4e–f. Finally the coupling effect is recognizable using the GVC controller whereas the effect of full dynamics is observed if the NGVC algorithm is applied. The NGVC algorithm is useful when we want to shorten the time of convergence of velocity errors. Dynamics Test Based on Comparison the NGVC Controller with the CL Some information about the impact of vehicle dynamics on velocity tracking can be obtained by comparing the proposed algorithm with an algorithm that does not contain dynamic parameters in the velocity gain matrix. In this case, the same set of controller parameters, namely (13.75) should be used in both cases. The results for the CL controller (without dynamics in gains) are given in Fig. 13.36. Comparing Fig. 13.35a–b with Fig. 13.36a–b one can see that the velocity errors tend to zero faster if the NGVC controller is used. Moreover, for the NGVC algorithm the errors

u, w, and q were reduced which means that the couplings between each of velocity u, w, q, and others are strong. However, the presence of dynamics causes

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that in the first phase of motion the τ X and τ Z have bigger values. In the next phase forces and moments have comparable values for both controllers. Similar analysis of dynamics can be done for different desired velocity trajectories. It can be concluded that if the full vehicle dynamics is included in the control gain matrices then the time response of the system is faster and the effects of couplings (in the full dynamics) are observable but indirectly (for the NGVC controller not only couplings are present in the gain matrix but the vehicle dynamics model). For this reason k D and  have smaller values if the NGVC algorithm is applied.

13.6.3 Position and Velocity Trajectory Tracking Controller in Earth-Fixed Frame 13.6.3.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the NGVC controller for an underwater vehicle model, robustness test of the algorithm, and dynamic study based on the NGVC controller. For tracking it is assumed the desired linear 3D trajectory and the appropriate velocity profile defined by equations (9.72)–(9.73). The gain coefficients of the controller were selected, to ensure acceptable errors convergence, as follows: k D = diag{80, 80, 80, 220, 220, 220}, k P = diag{100, 100, 100, 100, 100, 100},  = diag{0.8, 0.8, 0.8, 0.8, 0.8, 0.8}.

(13.76)

The above given gains serve rather for tuning because the dynamics of the vehicle is included in the controller. For this reason values of k D and k P are not so great as for the GVC algorithm. In contrast, the matrix  has larger value. It means that if k D and k P have reduced values then to ensure fast error convergence it is necessary to increase vales of . For this set of control gains and nominal parameters the results are shown in Fig. 13.37. As it follows from Fig. 13.37a the desired trajectory is tracked correctly. From Fig. 13.37b–c it is noticeable that the position errors converge more slowly to the end value than using the GVC algorithm. But the errors y and z are close to zero all the time. The effects result from including the vehicle dynamics in the control gains. From Fig. 13.37d–e it is seen that all velocity errors tend to zero without overshoot. The system response is strictly related to its dynamics. As it is observed from Fig. 13.37f–g the characteristic feature of control algorithms including dynamics in the gains are high values of forces and moments in the initial phase of the movement, which must be limited in the real control system (including the couplings).

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Fig. 13.37 Simulation results for NGVC controller and underwater vehicle (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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In comparison to the GVC controller, the time of convergence of errors is longer but the overshoots are smaller. It arises from the fact that instead of couplings the full dynamics of the vehicle serve for control. Using the NGVC algorithm the vehicle movement is not fast as using the GVC controller, which makes the algorithm appear more practical because the indoor airship flies slowly. Robustness investigation. At present robustness of the NGVC control algorithm will be tested using the same set of gains as for the vehicle with nominal parameters. The dynamical parameters of the vehicle and environmental parameters (damping terms and added masses) differ by 10 % from nominal (they are greater). Elements of M and D are bigger by 10 %. Comparing Figs. 13.37 and 13.38 it can be seen that changes in time history of selected variables are almost imperceptible. The control task is still realized as expected. This fact suggests that the NGVC algorithm is more robust to parameters changes then the GVC controller because the previously considered algorithm delivered worse results (Fig. 9.2). Dynamics Test Based on Comparison the NGVC Controller with the CL In order to estimate the dynamics of the airship, the CL (classical algorithm) was applied using the same gains and trajectories. Comparing Figs. 13.37 and 13.39 several observations can be made. First, using the CL controller the linear position error

x has constant value which make the linear position trajectory tracking impossible (Figs. 13.37a and 13.39a). The error x is the most sensitive to inappropriate selection of the control gain matrices. Secondly, the errors y and z are excited as a result of the omitted dynamics (Figs. 13.37b and 13.39b). Thirdly, the angular position errors are comparable for both controllers (Figs. 13.37c and 13.39c). Fourthly, the linear velocity error d x/dt has small values for the CL algorithm, but dy/dt and dz/dt are not close to zero (Figs. 13.37d and 13.39d). Fifthly, angular velocities are lesser excited than for the NGVC algorithm but some overshoot is observable as it is shown in Figs. 13.37e and 13.39e. This means that these signals are coupled together. Sixthly, the forces and moments have comparable values (for both controllers) with exception of the initial phase of motion when use of the NGVC leads to great values (Figs. 13.37e–f and 13.39e–f). Concluding, the dynamics effect is observable even if the gain coefficients are incorrect selected for the CL algorithm. However, the gains used for the NGVC controller must enable the trajectory tracking task to be carried out correctly. In the considered case there exist strong couplings between the linear positions and linear velocities. Also angular velocities are coupled together. The NGVC algorithm seems to be useful for vehicle dynamics study (together with the test of the CL controller). However, for control application limitation of forces and moments is necessary because in the first phase of motion they have great values. It is a result of including the vehicle dynamics in the control gains.

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Fig. 13.38 Simulation results for NGVC controller and underwater vehicle (robustness test): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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Fig. 13.39 Simulation results for CL controller and underwater vehicle (nominal parameters): a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors; f forces; g moments

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13.6.4 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Body-Fixed Frame 13.6.4.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the NGVC trajectory tracking controller for underwater vehicle model in the presence of disturbances model. Ensuring fast error convergence without overshoot in position errors. Evaluation of dynamics effect in the vehicle comparing the NGVC controller and the corresponding CL controller. For tracking it is assumed the desired 3D cycloid trajectory profile for linear variables together with the exponential trajectory for angular variables defined by (13.64)–(13.65). The disturbances model including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments: ⎤ 225 f 16 (t) e−2 + 45 ⎢ −90 f 16 (t) e−2 − 18 ⎥ ⎥ ⎢ ⎢ 20 f 17 (t) e−2 + 20 ⎥ ⎥, ⎢  F=⎢ −2 ⎥ ⎢ 5 f 17 (t) e −3+ 5 ⎥ ⎣ 150 f 18 (t) e + 12 ⎦ 50 f 18 (t) e−3 + 4 ⎡

(13.77)

where f 16 (t) = 0.8 cos(0.4 t) + 0.2 sin(0.1 t), f 17 (t) = 0.5 sin(t) e−1 + ecos(−t ) , and f 18 (t) = 0.5 sin(0.1 t) + 0.9 cos(0.4 t). The gain matrices for NGVC controller were selected to ensure the position and velocity tracking errors convergence without great overshoot and the lumped uncertainty error convergence. It order to better show effects of dynamics included in the control gains it is assumed that all elements of each gain matrix have the same values. Moreover, the position and velocity errors convergence should be guaranteed. Taking the above into account, the following set of control parameters was selected: 0.5

k∗D = diag{10, 10, 10, 10, 10, 10}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{30, 30, 30, 30, 30, 30},

(13.78)

which means that k∗D = 10 I, k P = I,  = 0.4 I, and  ∗ = 30 I. The above given gains were used the NGVC assuming k D = k∗D ,  =  ∗ . Because of dynamical effects observed in the first phase of motion the input signals are limited (forces, moments, and disturbance estimation functions). Saturation values were: |τ | ≤ 1500 N or Nm and | f e | ≤ 1500 N or Nm. The used disturbance forces and moments functions are shown in Fig. 13.40. From Fig. 13.41a is noticeable that the desired position trajectory is realized correctly. Observing Fig. 13.41b–c it can be seen that all position errors and velocity

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X , F Y , F Z ; b disturbance moments F K , Fig. 13.40 Disturbance functions: a disturbance forces F M , F N F

errors are close to zero after about 15 s. Thus, it can be concluded that the trajectories (linear and angular) are tracked and the errors convergence is ensured. Moreover, the velocity trajectories are tracked too as it arises from Fig. 13.41d–e. The results obtained are closely related to the quantities s shown in Fig. 13.42a–b. Note that both the signals s corresponding to linear and angular variables tend to the end values close to zero after about 2 s. Consequently, the position and velocity errors convergence is guaranteed. The lumped dynamics estimation errors wY , w Z , w M , and w N shown in Fig. 13.42c–d which are related to τY , τ Z , τ M , and τ N are great. The presence of the vehicle dynamics causes this phenomenon. However, all lumped dynamics estimation errors are reaching limited values quickly (after about 2 s). From Fig. 13.42e–f it arises that if the vehicle starts then values of τY , τ Z , τ M , and τ N are great which is the effect of dynamics in the control gains. In Fig. 13.43a–b the total kinetic energy which must be reduced by the vehicle K U V as well as the kinetic energy corresponding to all quasi-velocities are presented. The values are calculated according to (13.16). In Fig. 13.43c–d the quasi-velocities error ζi , determined from (13.19), are given. It can be concluded that if the vehicle moves then the linear velocities u and v are sensitive to dynamics ( ζ1 and ζ2 changes during motion essentially). The angular velocities are sensitive to the motion changes if the vehicle starts only (if ζi tend to zero then also quasi-velocity tend to corresponding velocity νi ). The time history of i (Fig. 13.43e–f), calculated using (13.20) is similar to the history of forces and moments given in Fig. 13.42c–d. Thus, for the considered vehicle and position trajectory, more information is unavailable. More information can be obtained from Fig. 13.44a–b were ζi , determined from (13.21), are presented. Not very large changes concern ζ2 , ζ3 , and ζ4 which means that the sensitive for the dynamical couplings are velocities v, w, and p. Recalling Fig. 13.43c–d, it can be seen that the test results are ambiguous. However, the results shown in Fig. 13.44a–b seem to be more reliable because the velocities were scaled and therefore more closely related to the quasi-velocities. In order to get more accurate information about the effect of dynamics, it is worthwhile to use both measures of quasi-velocity errors. In Fig. 13.44c–d i obtained from (13.22) are

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Fig. 13.41 Simulation results for NGVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

given. Now it is obvious that the most sensitive to changes in dynamic parameters is the moment τ M . Dynamics Test Based on Comparison the NGVC Controller with the CL Objective: evaluation of dynamics effect in the vehicle comparing the NGVC controller and the corresponding CL controller.

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Fig. 13.42 Simulation results for NGVC controller and underwater vehicle: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

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Fig. 13.43 Simulation results for NGVC controller and underwater vehicle: a comparison between kinetic energy for underwater vehicle K U V and kinetic energy related to ζ1 , ζ2 , ζ3 ; b comparison between kinetic energy related to ζ4 , ζ5 , ζ6 ; c ζ time history related to linear velocities; d ζ time history related to angular velocities; e  time history related to forces; f  time history related to moments

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Fig. 13.44 Simulation results for NGVC controller and underwater vehicle: a ζ time history related to linear velocities; b ζ time history related to angular velocities; c  time history related to forces; d  time history related to moments

The same control parameters set, i.e. (13.78) and k D = k∗D ,  =  ∗ as for the NGVC control algorithm are used. In order to estimate the influence of dynamics on control quality, the results obtained from the NGVC controller with the results obtained for the CL controller (which does not contain dynamic parameters in the control gain matrices) were compared. The corresponding results for the CL controller are given in Figs. 13.45 and 13.46. Taking into account Fig. 13.45a–d with Fig. 13.41b–e it can be seen that the CL control algorithm does not track the position or velocity. It shows the influence of dynamics on the behavior of the vehicle. Therefore, comparison of results is important for evaluation of vehicle dynamics. Effect of dynamics is noticeable especially for the linear position errors and the linear velocity errors. Only φ and its time derivative give satisfactory results. In the CL control algorithm the gains do not include the vehicle dynamic. It can be concluded that the position x is the most sensitive to dynamical parameters (from x) whereas y, z slightly less (from y, z). From Figs. 13.46a–b it can be noticed that the quantities s are not close to zero. Consequently, the tracking position and velocity errors differ significantly from zero. The lumped dynamics estimation errors w related to linear velocities tend to a lower

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Fig. 13.45 Simulation results for CL controller and underwater vehicle: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

limit value using the NGVC controller than using the CL controller (Figs. 13.42c and 13.46c) which results from the influence of dynamic parameters. Observing the lumped dynamics estimation errors w related to angular velocities it can be seen that using the NGVC controller end values have also lower limit (Figs. 13.42d and 13.46d). Recalling Fig. 13.42e–f and comparing them with Fig. 13.46e–f it is seen that the forces and moments have similar time history with exception of the first phase of motion. Therefore, it can be concluded that all position and velocity errors are sensitive to changes in dynamics (especially linear positions). However, the greatest effect of dynamics can be observed for x and its time derivative.

13.6.4.2

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Airship

Objective: showing performance of the NGVC position and velocity trajectory tracking controller for indoor airship model in the presence of disturbances model. Evaluation of dynamics effect in the vehicle and comparing the NGVC controller with the corresponding CL controller.

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Fig. 13.46 Simulation results for CL controller and underwater vehicle: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

For tracking it is assumed the following desired linear position 3 D cycloid, desired angular position exponential trajectory profiles, and the appropriate desired velocities:

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ηd = [0.5 t − 0.3 sin(0.4 t) − 0.9, 0.5 − 0.3 cos(0.4 t), 0.3 t + 0.1, 0.8e−0.5t , −0.6e−0.5t , −0.4e−0.5t ]T , η˙ d = [0.5 − 0.12 cos(0.4 t), 0.12 sin(0.4 t), 0.3, −0.40e−0.5t , 0.30e−0.5t , 0.20e−0.5t ]T .

(13.79) (13.80)

The disturbances function including the unknown dynamics as well as the external disturbances is expressed by the following vector of forces and moments: ⎤ 40 f 19 (t) e−2 + 5 ⎢ 16 f 19 (t) e−2 + 2 ⎥ ⎥ ⎢ ⎥ ⎢ 50 f 20 (t) ⎥, ⎢  F=⎢ ⎥ (t) 20 f 20 ⎥ ⎢ ⎣ 120 f 21 (t) e−3 + 20 ⎦ −42 f 21 (t) e−3 − 7 ⎡

(13.81)

where f 19 (t) = cos(0.5 t) + 5 sin(0.2 t), f 20 (t) = 0.5 sin(0.5 t) e−3 + 0.5 e−0.1 t , and f 21 (t) = 0.7 sin(0.2 t) + 0.4 cos(0.5 t). The gain matrices for NGVC controller were selected to ensure velocity tracking error convergence without overshoot in position errors and the lumped uncertainty error convergence. It order to better show effects of dynamics in the control gains it is assumed that all elements of each gain matrix have the same values. Moreover, the position and velocity errors convergence should be guaranteed. In order to meet the indicated conditions the following set of control parameters was selected: k∗D = diag{20, 20, 20, 30, 30, 30}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.7, 0.7, 0.7, 0.7, 0.7, 0.7},  ∗ = diag{200, 200, 200, 400, 400, 400}.

(13.82)

The above given gains were assuming k D = k∗D ,  =  ∗ . Note also the gains are different only for linear and angular variables. Because of dynamical effects observed in the first phase of motion the input signals are limited (forces, moments and disturbance estimation function). Saturation values were: |τ | ≤ 150 N or Nm and | f e | ≤ 150 N or Nm. The disturbance forces and moments functions are given in Fig. 13.47. Observing Fig. 13.48a it is apparent that the desired position trajectory is tracked correctly. From Fig. 13.48b–e it is noticeable that the position and velocity errors are close to zero after about 8 s. Therefore, the controller performs its task. The vehicle velocities increase only if the vehicle starts. There exist some relationship between the obtained errors convergence and the quantities s which are shown in Fig. 13.49a–b. The variables s are close to zero very quickly (after about 1 s). Such result guarantees also fast errors convergence. From Fig. 13.49c–d it arises that all lumped dynamics estimation errors are reaching limited values after about 1 s. As is

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X , F Y , F Z ; b disturbance moments F K , Fig. 13.47 Disturbance functions: a disturbance forces F M , F N F

shown in Fig. 13.49e–f the forces and moments have great values only if the vehicle starts. It is the effect of including the system dynamics in the control gains. From Fig. 13.50a–b it is seen that most of the kinetic energy is must be reduced by ζ1 which means that the linear velocity u is the most important. The kinetic energy concerning this variable is the greatest (assuming the desired trajectory and disturbances model). Less kinetic energy is reduced by the linear velocity w. Other velocities generate a small amount of the kinetic energy. The values are calculated according to (13.16). In Fig. 13.50c–d the quasi-velocity errors ζi , determined from (13.19), are presented. The results suggest that the linear velocities u and w are sensitive to dynamical couplings (the time history of ζ1 and ζ3 ). The angular velocity q seems to be more sensitive to the motion changes than others (based on ζ5 ). However, at the beginning of motion p is also sensitive to the changes ( ζ4 graph). The time history of i shown in Fig. 13.50e–f) are calculated from Eq. (13.20). There exist similarity of the quasi-forces to the history of forces and moments from Fig. 13.49e–f. Consequently, more information is unavailable. For additional information the quantities ζi in Fig. 13.51a–b, calculated according to (13.21), are shown. Changes are noticeable for ζ2 , ζ3 , and ζ4 what suggests that velocities v, w, and p are sensitive to the dynamical couplings. It seems that the results shown in Fig. 13.51a–b are more reliable because the velocities were scaled and therefore more closely related to quasi-velocities. In order to get more accurate information about the effect of dynamics both measures of quasi-velocity errors may be used. Based on Fig. 13.51c–d, where i obtained from Eq. (13.22) it can be concluded that the forces are insensitive to changes in dynamics. However, the moments τ M and τ N are sensitive to changes in the dynamic parameters. Dynamics Test Based on Comparison the NGVC Controller with the CL Objective: evaluation of dynamics effect in the vehicle comparing the results from the NGVC controller with the results obtained for the CL controller. The control parameters set (13.82), namely k D = k∗D ,  =  ∗ was assumed.

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Fig. 13.48 Simulation results for NGVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

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Fig. 13.49 Simulation results for NGVC controller and indoor airship: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

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Fig. 13.50 Simulation results for NGVC controller and indoor airship: a comparison between kinetic energy for indoor airship K A and kinetic energy related to ζ1 , ζ2 , ζ3 ; b comparison between kinetic energy related to ζ4 , ζ5 , ζ6 ; c ζ time history related to linear velocities; d ζ time history related to angular velocities; e  time history related to forces; f  time history related to moments

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Fig. 13.51 Simulation results for NGVC controller and indoor airship: a ζ time history related to linear velocities; b ζ time history related to angular velocities; c  time history related to forces; d  time history related to moments

In order to estimate the effect of dynamics on control process, the results from the NGVC controller with the results from the CL controller (which does not contain dynamic parameters in the control gain matrices) were compared. The response of the system obtained for the CL controller are given in Figs. 13.52 and 13.53. Comparing Fig. 13.48b–e with Fig. 13.52a–d it can be seen that the CL controller does not work correctly. Although the errors of the angular positions are acceptable, but the linear positions are unfortunately not. Based on the results it is possible to conclude about dynamical effects during the airship motion. Effect of dynamics is noticeable especially for the linear position and velocity errors and much less for the angular position and velocity errors. The impact of vehicle dynamics on linear position errors is noticeable for x, z and for their time derivatives. Moreover, signals overshoot is observable. The effects of lacks of the dynamics is confirmed in Fig. 13.53a–b where the quantities s are shown. All signals are characterized by over-regulation and moreover the quantities s corresponding to linear variables are not close to zero. The lumped dynamics estimation errors w related to linear and velocities tend to the end limit value faster and with less overshoot using the NGVC control algorithm than using the CL controller as it is presented in Figs. 13.49c-d and 13.53c–d. Taking into consideration Figs. 13.49e–f and 13.53e–f it can be seen that

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Fig. 13.52 Simulation results for CL controller and indoor airship: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

the forces and moments have similar time history with exception of the first phase of motion. For the CL control algorithm the overshooting phenomenon is still present. On the basis of the obtained results it can be concluded that all position and velocity errors are sensitive to changes in dynamics. The greatest effect of dynamics can be observed for x and z, their time derivatives, and the corresponding variables. Therefore, dynamic parameters have the greatest impact on forward and upward movement. 13.6.4.3

Models Comparison Based on the NGVC Controller and Indexes for 6 DOF Fully Actuated Underwater Vehicle

Objective: showing performance of the NGVC position and velocity trajectory tracking controller for basic dynamical model of an underwater vehicle and presentation the proposed analysis of selected models using a set of indexes. For tracking it is assumed the desired helical trajectory profile for linear variables together with the exponential trajectory for angular variables described by (10.45)– (10.46). The disturbances model including the unknown dynamics as well as the external disturbances is expressed by the vector of forces and moments (13.77).

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Fig. 13.53 Simulation results for CL controller and indoor airship: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

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The gain matrices for the NGVC controller were selected to ensure the position and velocity tracking errors convergence without great overshoot and the lumped uncertainty error convergence. It order to better show effects of dynamics included in the control gains it is assumed that all elements of each gain matrix have the same values. Moreover, the position and velocity errors convergence should be guaranteed. Consequently, the following set of control parameters was selected: k∗D = diag{10, 10, 10, 10, 10, 10}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{20, 20, 20, 20, 20, 20}, (13.83) which means that k∗D = 10 I, k P = I,  = 0.4 I, and  ∗ = 20 I. Moreover, the resultant gains are k D = k∗D and  =  ∗ . Because of dynamical effects observed in the first phase of motion the input signals are limited (forces, moments, and disturbance estimation functions). The saturation values were assumed as: |τ | ≤ 1500 N or Nm and | f e | ≤ 1500 N or Nm. First step—analysis of time responses for basic vehicle model. At the beginning the basic model is tested using the controller with the selected set of parameters. The parameters are given in Table 13.1 (Case 1). The obtained results are presented in Figs. 13.54 and 13.55. From Fig. 13.54a it results that the desired position trajectory is tracked correctly. As is shown in Fig. 13.54b–c after about 15 s all position errors are close to zero. This is also observable in Fig. 13.54d–e where velocity errors are presented. When the vehicle starts some changes are noticeable (dynamics of the vehicle is included into the control gains). The quantities s given in Fig. 13.55a–b tend to limited values near zero very fast which ensures fast convergence the position and velocity errors presented earlier. The lumped dynamics estimation errors w which correspond to linear and angular velocities presented in Fig. 13.55c–d are fast approaching the limited value. In Fig. 13.55e–f the forces and moments are shown. Based on the results, it can be concluded that the controller works correctly using the selected gain matrices. Second step—comparing various vehicles models. After the initial examination concerning the algorithm performance, a comparison of different vehicle models with various dynamic parameters is made. The proposed models are given in Table 13.1 and denoted as Case 2 and Case 3 (Case 0 is the simplified with reduced couplings). The goal is to know the effects of couplings (as a result of dynamical and geometrical parameters changes). The results obtained for the selected models are presented in Table 13.4. Taking into account δ η and δ η˙ it can be noticed that Case 3 is unacceptable because of the quantities high values compared with other cases. Therefore, it should be rejected or tested in different research under different conditions. In two other cases studied values of these quantities are comparable. For Case 3 also couplings are the strongest (δ K c , c , δζc ). However, the kinetic energy consumption δ K and velocity deformation δz are not the greatest (but the input signal effort δ and δζ have the greatest values). As before Case 0 is only ideal. Taking into consideration Case 1 and Case 2

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Fig. 13.54 Simulation results for NGVC controller and underwater vehicle: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

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Fig. 13.55 Simulation results for NGVC controller and underwater vehicle: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

13.6 Application of NGVC Control Algorithms Table 13.4 Values of quantities for indexes Case 0 Case 1 o f f diag δζ δz δζc δ c δK δ Kc δ η δ η˙

0.0000 41.1096 87.4859 30.2326 1.0000 74.2492 0.0000 196.3129 1.0000 0.3783 0.1521

5.5169 40.7377 87.6302 30.2162 1.2478 106.8575 8.7018 196.3645 1.0009 0.3785 0.1523

345

Case 2

Case 3

4.0797 41.5133 88.6855 30.9538 1.0499 107.4153 7.1459 206.1212 1.0006 0.3784 0.1512

3.5180 38.3096 94.8083 27.4586 1.7319 172.7541 23.9708 176.9534 1.0106 2.0409 0.3354

it is difficult to answer the question, which variant is better because the results are not straightforward. It can be observed that using the selected gain values the controller works correctly and gives acceptable results. In order to obtain more information the indexes and together with their weight coefficients bare applied. Values of the indexes for the same cases are given in Table 13.5. Comparing indexes values I1 and I2 (the main task performance, i.e. tracking trajectory task), it was confirmed that the worst case is the Case 3 and it should be rejected. In spite of that other values of indexes give inconclusive results. It can be noted that the values of indexes I1 ÷ I10 are comparable for Case 1 and Case 2 but for Case 1 they are slightly worse. Based on these values it is still difficult to choose structure Case 1 or Case 2. The decision must be confirmed by a more accurate comparison. Thus, the complex indexes are determined. Four situations are taken into account in which four sets of weight coefficients are used. They are selected according to criteria important for the designer. The final values for all cases are shown in the lower part of Table 13.5. Set of weight coefficients 1. The most important are tracking performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ), the kinetic energy consumption and the control effort w2 = 0.5 (this value is multiplied by the sum of I3 , I4 ). The index IC1 (calculated from (13.54)) is given in Table 13.5. For Case 1 (1.001 + 1.220) the index is slightly lesser than for Case 2 (0.997 + 1.249). For Case 3 the values are unacceptable. Set of weight coefficients 2. Important are: tracking performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ), the kinetic energy consumption and the control effort w2 = 0.3 (this value is multiplied by the sum of I3 , I4 ), and couplings w3 = 0.2 (this value is multiplied by the sum of I5 ÷ I10 ). The index IC2 indicates that Case 2 (0.997 + 0.749 + 1.251) is better than Case 1 (1.001 + 0.732 + 1.293), whereas for Case 3 its value is the greatest.

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Table 13.5 Values of indexes Index Case 0 I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 IC1 IC2 IC3 IC4

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.000 2.800 6.000 1.500

Case 1

Case 2

Case 3

1.001 1.001 1.439 1.000 0.999 1.002 1.135 1.081 1.001 1.248 2.221 3.026 6.466 1.501

1.000 0.994 1.447 1.050 1.024 1.014 1.098 1.067 1.001 1.050 2.246 2.997 6.254 1.522

5.395 2.205 2.327 0.901 0.908 1.084 1.092 1.139 1.011 1.732 5.414 5.962 6.966 4.251

Set of weight coefficients 3. It is assumed that they are equally important velocity deformation and dynamic couplings. Therefore, w1 = 1.0 (this value is multiplied by the sum of I5 ÷ I10 ). Other parameters can be omitted. The index IC3 indicates that the strongest deformation and dynamical couplings are for Case 3 (1.992 + 4.974). Velocity deformation and couplings are bigger for Case 1 (2.001 + 4.465) than for Case 2 (2.038 + 4.216). Set of weight coefficients 4. It is assumed that they are equally important control performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ) and the kinetic energy consumption w2 = 0.5 (this value is multiplied by I4 ). Now Case 1 (1.001 + 0.500) is better than Case 2 (0.997 + 0.525). Discussion of results. The test showed that it is possible to indicate among various models the model which not only guarantees good performance of the control algorithm as well as which is better from kinetic energy point of view or control effort and has the greater couplings. Moreover, the structure which do not realize the control task was eliminated. In the test the worst values were for Case 3 and this case should be eliminate because of unsatisfactory results. It is harder to select between models called Case 1 and Case 2. As previously, even slightly differences can give us an answer for this question (Case 1 represents the best structure it the kinetic energy consumption is very important). If the kinetic energy consumption and the mass of the vehicle are not crucial Case 2 is better than Case 1. Small differences between indexes can be explained by the fact that in the control gains the full is taken into account. The procedure can be applied for investigation of dynamical effects in the vehicle. One fact can be observed, namely, in spite of slightly different set of gains and disturbances as well as desired trajectories, similar quality results were obtained for

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347

GVC and NGVC controllers (due to the same coefficients of the controller gain matrix we can not talk about different algorithms). For this reason, it can be concluded that indexes based dynamics testing is important and the results may be similar in quality and representative.

13.6.4.4

Models Comparison Based on the NGVC Controller and Indexes for 6 DOF Fully Actuated Indoor Airship

Objective: showing performance of the NGVC position and velocity trajectory tracking controller for basic dynamical model of an indoor airship and presentation of the proposed analysis of selected models based on the set of indexes. For tracking it is assumed the desired helical position trajectory together with exponential angular trajectories, and the appropriate velocity profile, respectively, namely: ηd = [5 cos(0.2 t) − 4, 5 sin(0.2 t) − 2, −0.3 t − 0.1, 0.5e−0.3t , 0.4e−0.3t , −0.5e−0.3t ]T , η˙ d = [− sin(0.2 t), cos(0.2 t), −0.3,

(13.84)

−0.15e−0.3t , −0.12e−0.3t , 0.15e−0.3t ]T .

(13.85)

The disturbances function including the unknown dynamics as well as the external disturbances is expressed by the vector of forces and moments (13.81). The gain matrices for the NGVC controller were selected to ensure the position and velocity tracking errors convergence without great overshoot and the lumped uncertainty error convergence. It order to better show effects of dynamics included in the control gains it is assumed that all elements of each gain matrix have the same values. Moreover, the position and velocity errors convergence should be guaranteed. Consequently, the following set of control parameters was selected: k∗D = diag{15, 15, 15, 30, 30, 30}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{150, 150, 150, 400, 400, 400},

(13.86)

which means that k∗D and  ∗ have different values for the linear and angular variables. This choice is related to the dynamics of the vehicle (the values depend on the dynamics). Because of dynamical effects observed in the first phase of motion the input signals are limited (forces, moments, and disturbance estimation functions). Saturation values were: |τ | ≤ 150 N or Nm and | f e | ≤ 150 N or Nm. First step—analysis of time responses for basic vehicle model. At the beginning the basic model is tested using the controller with the selected set of parameters. The parameters are given in Table 13.6 (Case 1). The obtained results are presented in

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Table 13.6 Parameters of tested vehicles model Symbol Case 0 Case 1 Case 2 L b xG yG zG m Ix Iy Iz Jx y Jx z Jyz

7.6 1.6 0 0 0 13 10 50 50 0 0 0

7.6 1.6 −1.0 0 0.15 13 10 50 50 0 −10 0

7.6 1.6 −0.5 0.2 −0.1 16 5 40 60 0 5 −5

Case 3

Unit

7.6 1.6 −0.6 0 0.2 10 15 40 50 −5 −5 0

m m m m m kg kgm2 kgm2 kgm2 kgm2 kgm2 kgm2

Figs. 13.56 and 13.57. As is shown in Fig. 13.56a the desired position trajectory is tracked correctly. From Fig. 13.56b–c it can be seen that after about 15 s all position errors are close to zero. As it is observed in Fig. 13.56d–e the velocity errors are near zero about 10–15 s. At the beginning of motion great values of velocity errors occur what results from including of the vehicle dynamics into the control gains. From Fig. 13.57a–b it is noticed that the quantities s values are quickly limited and approaching zero what ensures convergence of the position and velocity errors. It is observed that the lumped dynamics estimation errors w related to linear and angular velocities in Fig. 13.57c–d also converge to the limited values quickly. Finally, in Fig. 13.57e–f the forces and moments are shown. Therefore, it can be concluded that the controller works properly using selected gains. Second step—comparing various vehicles models. After the first test that showed that the algorithm is working properly and provides satisfying performance, a comparison of different vehicle models with various dynamic parameters is performed. The investigated models are given in Table 13.6 and denoted as Case 2 and Case 3 (Case 0 means the simplified basic model described as Case 1, in which couplings are reduced by assuming the diagonal inertia matrix and symmetry). The the goal is to recognize the dynamic effects in the vehicle. The results of the test for the selected models are presented in Table 13.7. For all cases the trajectory tracking performance is acceptable. It is observed that the best error values δ η and δ η˙ were obtained for Case 2 (despite the increased vehicle weight). However, in this case the greatest value of kinetic energy δ K must be reduced (the lowest consumption is for Case 3). The velocity deformation δz, δζ are also the largest for this model. Strong couplings (measured by δ K c , δc , δζc ) occur for Case 1, Case 2, and Case 3. The input signal effort δ is the largest for 1 and slightly smaller for Case 2. It can therefore be concluded that the results regarding the model variant are not conclusive if only data from Table 13.7 are taken into consideration.

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Fig. 13.56 Simulation results for NGVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

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Fig. 13.57 Simulation results for NGVC controller and indoor airship: a quantities s for linear variables; b quantities s for angular variables; c lumped dynamics estimation errors w related to linear velocities; d lumped dynamics estimation errors w related to angular velocities; e forces; f moments

13.6 Application of NGVC Control Algorithms Table 13.7 Values of quantities for indexes Case 0 Case 1 o f f diag δζ δz δζc δ c δK δ Kc δ η δ η˙

0.0000 8.3490 15.3647 5.1764 1.0000 19.8764 0.0000 5.8747 1.0000 0.3800 0.1519

Table 13.8 Values of indexes Index Case 0 I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 IC1 IC2 IC3 IC4

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.000 2.800 6.000 1.500

351

Case 2

Case 3

4.1055 7.2699 15.5117 5.1604 2.9852 61.8495 5.3779 5.8795 1.0050 0.3894 0.1573

4.3218 7.4181 17.5064 5.7985 1.6055 73.5303 16.6196 7.3642 1.0006 0.3843 0.1536

2.9534 6.8544 14.1271 4.4675 1.7075 98.7142 32.0874 4.3930 1.0025 0.3875 0.1568

Case 1

Case 2

Case 3

1.025 1.036 3.112 1.001 0.997 1.010 1.565 1.087 1.005 2.985 3.087 3.994 8.649 1.531

1.011 1.011 3.699 1.254 1.120 1.139 1.583 1.226 1.001 1.606 3.488 4.032 7.675 1.638

1.020 1.032 4.966 0.748 0.863 0.919 1.431 1.325 1.003 1.708 3.883 4.190 7.249 1.400

More information are accessible if indexes and together with their weight coefficients are applied. Values of the indexes for the same cases are given in Table 13.8. In spite of that the position and velocity error indexes are the lowest for Case 2 it is necessary to compare also other criteria because all results are acceptable. It can be noted that values of indexes I1 ÷ I1 are comparable for Case 1 and Case 3. Based on these values it is difficult to decide which structure is the best. The decision must be confirmed using a different comparison. Four situations are considered applying four sets of weight coefficients. The final values for all cases are shown in the lower part of Table 13.8.

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Set of weight coefficients 1. The most important are tracking performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ), the kinetic energy consumption and the control effort w2 = 0.5 (this value is multiplied by the sum of I3 , I4 ). The index IC1 (calculated from (13.54)) is given in Table 13.8. For Case 1 (1.030 + 2.057) the index is lesser than for Case 2 (1.011 + 2.477) or Case 3 (1.026 + 2.857). Set of weight coefficients 2. Important are: tracking performance w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ), the kinetic energy consumption and the control effort w2 = 0.3 (this value is multiplied by the sum of I3 , I4 ), and couplings w3 = 0.2 (this value is multiplied by the sum of I5 ÷ I10 ). The index IC2 also suggests that Case 1 (1.030 + 1.234 + 1.730) is the best because value for other case are greater, i.e. Case 2 (1.011 + 1.486 + 1.535) and Case 3 (1.026 + 1.714 + 1.450). Set of weight coefficients 3. It is assumed that the velocity deformation and dynamic couplings are equally important. Therefore, w1 = 1.0 (this value is multiplied by the sum of I5 ÷ I10 ). Other parameters can be omitted. The index IC3 indicates that the strongest deformation and dynamical couplings are for Case 1 (2.007 + 6.642). Velocity deformation and couplings are bigger for Case 2 (2.259 + 5.416) than for Case 3 (1.782 + 5.467). Set of weight coefficients 4. It is assumed that equally important control performance are w1 = 0.5 (this value is multiplied by the sum of I1 , I2 ) and the kinetic energy consumption w2 = 0.5 (this value is multiplied by I4 ). Now Case 3 (1.026 + 0.374) is better than Case 1 (1.030 + 0.501) and Case 2 (1.011 + 0.627). Discussion of results. The performed test showed that using the proposed approach the best model using selected index can be determined from various models which guarantee good performance of the controller and kinetic energy consumption as well as to point at the greater control effort or the greater couplings. All selected models ensure acceptable the position and velocity errors convergence. In the conducted test the worst values were obtained for Case 2. Better results give Case 1 and Case 3 . However, their usefulness strictly depend on the assumed criteria. Taking into account the task performance and the kinetic energy the best is Case 3 whereas if additionally the control effort is taken into account then Case 1 is the best. However, we must remember that the control effort is normalized what leads us to conclusion that for more particular investigation it is reasonable to add the test using the GVC control algorithm as for the underwater vehicle. In spite of that indication of a promising model or models is possible using this study.

13.6 Application of NGVC Control Algorithms

353

13.6.5 Position and Velocity Trajectory Tracking Controller with Adaptive Term in Earth-Fixed Frame 13.6.5.1

Position and Velocity Trajectory Tracking Controller for 6 DOF Fully Actuated Airship

Objective: showing performance of the NGVC trajectory and velocity tracking controller for indoor airship model in the presence of a disturbances model. Ensuring fast error convergence without overshoot in position errors. Evaluation of dynamics effect in the vehicle comparing results from the NGVC controller with results from the corresponding CL controller. For tracking it is assumed the desired helical trajectory profile with angular exponential trajectory profile, i.e. (12.51)-(12.52). The disturbances function including the unknown dynamics as well as the external disturbances is expressed by the vector of forces and moments (12.53). The gain matrices for NGVC controller were selected to ensure the position and velocity tracking error convergence without great overshoot and the lumped uncertainty error convergence. Consequently, the following set of parameters was chosen: k∗D = diag{50, 50, 50, 50, 50, 50}, k P = diag{1, 1, 1, 1, 1, 1},  = diag{0.4, 0.4, 0.4, 0.4, 0.4, 0.4},  ∗ = diag{400, 400, 400, 700, 700, 700},

(13.87)

where k D = k∗D ,  =  ∗ were used (both for the controllers NGVC and CL). The maximal values have been limited to: |τη | ≤ 150 N or Nm and | f η e | ≤ 150 N or Nm. Moreover, the lumped dynamics estimation errors were obtained assuming the constant values equal to zero. The model of disturbances functions in the Earth-Fixed Reference Frame is presented in Fig. 12.7. Note that the great values of disturbance functions are for surge (u) and pitch (q). As shown in Fig. 13.58a the realized position trajectory tracks the desired trajectory correctly. From Fig. 13.58b–c it follows that all position errors are convergent to the final value at similar time. The velocity errors have big values only if the airship starts (Fig. 13.58d–e). It is effect of the assumed velocity profiles and the vehicle dynamics included into the control gain matrices. Then each velocity error reaches the final value without overshoot. The lumped dynamics estimation errors also have great values at the beginning of the vehicle’s movement as it follows from Fig. 13.59a–b. Next, they achieve the final values quickly. In Fig. 13.59c–d the forces and moments are given. It is observed that in the first phase of motion their values are large. Moreover, if the airship moves then the value of the force τηX and the moment τηM are the greatest. High values of these variables can be explained by both the models of disturbances assumed for simulation and the influence of vehicle dynamics contained in the controller gain matrices.

354

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.58 Simulation results for NGVC controller and indoor airship: a desired and realized trajectory; b linear position errors; c angular position errors; d linear velocity errors; e angular velocity errors

13.6 Application of NGVC Control Algorithms

355

Fig. 13.59 Simulation for NGVC controller and indoor airship: a lumped dynamics estimation errors wη related to linear velocities; b lumped dynamics estimation errors wη related to angular velocities; c forces; d moments

Figure 13.60a–b correspond to equation (13.23). The dominant part of kinetic energy is caused by variables acting in the ζ1 and ζ2 . The energy value concerning ζ3 after about 10 s is constant. However, the kinetic energy which is reduced is small what results from the airship dynamical parameters. The energy caused by the vehicle’s rotation is negligible as it is shown in Fig. 13.60b. It can also be noted that the kinetic energy corresponds to the desired velocity trajectory. This information regarding the reduction of kinetic energy corresponding to each variable is only available when the equations of motion are not expressed in quasi-variables. z Figure 13.60c–d show result calculating from Eq. (13.26). They represent the quasi-velocity deformation caused by dynamic effects from other variables expressed in terms of ζi . The information is different than for the GVC because the quasi-velocities are not the same. Their time history depends on dynamics of the vehicle (including couplings) and on realized position and velocity trajectories but the information is indirect. It is observed that ζ1 and ζ2 are more deformed than ζ3 for linear velocities and ζ4 is the most deformed among angular quasi-velocities. Based on Eq. (13.27) the time history of quasi-forces and quasi-moments is available. From Fig. 13.60e–f it results that the obtained information is similar to earlier obtained from the forces and moments history presented in Fig. 13.59c–d.

356

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.60 Simulation results for NGVC controller and indoor airship: a comparison between kinetic energy for airship K A and kinetic energy related to ζ1 , ζ2 , ζ3 ; b comparison between kinetic energy related to ζ4 , ζ5 , ζ6 ; c ζ time history related to linear velocities; d ζ time history related to angular velocities; e  time history related to forces; f  time history related to moments

13.6 Application of NGVC Control Algorithms

357

Fig. 13.61 Simulation results for CL controller and indoor airship: a linear position errors; b angular position errors; c linear velocity errors; d angular velocity errors

Dynamics Test Based on Comparison the NGVC Controller with the CL Objective: evaluation of dynamics effect in the vehicle comparing the NGVC controller results with results from the corresponding CL controller. For the same conditions and control gains the results for CL controller are presented in Figs. 13.61 and 13.62. It is obvious that the gain matrices values are too small to ensure proper work of the controller. In spite of that the results obtained from the CL controller are useful for estimating the effect of dynamics on various variables. Comparing Fig. 13.61a–b with Fig. 13.58b–c it is noticeable that the errors x and

θ are changed more than others. Inappropriate selection of the corresponding gain coefficients just increases these errors. Similar observation can be made comparing Fig. 13.61c–d with Fig. 13.58d–e. The biggest changes concern the signals d x/dt and dθ/dt, but also the dz/dt and dψ/dt signals are essentially changed. There exist, for the airship model, dynamical couplings between these variables. The failure to account for the dynamics in the CL controller also causes significant changes in other signals, i.e. lumped dynamics estimation errors (Figs. 13.62a–b and 13.59a–b) or forces and moments (Figs. 13.62c–d and 13.59a–b) when the airship starts to move.

358

13 Vehicle Dynamics Study Based on Nonlinear Controllers …

Fig. 13.62 Simulation results for CL controller and indoor airship: a lumped dynamics estimation errors wη related to linear velocities; b lumped dynamics estimation errors wη related to angular velocities; c forces; d moments

Summarizing the presented test, it can be concluded that on the basis of a comparison of the results obtained from NGVC and CL controllers the effect of dynamics on the airship movement with the assumed vehicle model, position, velocity profiles and disturbances model can be estimated.

13.7 Closing Remarks The main purpose of this chapter was to show the potential applications of control algorithms considered in this book. Undoubtedly, such application is the study of vehicle dynamics and determination of dynamical couplings in the system as well as the influence of these couplings on the movement of the vehicle when the controller is started. In this chapter the selected control algorithms were applied for dynamics study of an underwater vehicle and an indoor airship. Both, the GVC and the NGVC type of algorithms were tested. At the beginning the procedure for testing of the vehicle models was presented in detail. Moreover, application of the control schemes was given. Next, a comparison of vehicle dynamics using indexes

13.7 Closing Remarks

359

method was considered. The simulation results were shown for the following control algorithms (in terms of the GVC and the NGVC): position and velocity trajectory tracking controllers in the Body-Fixed Frame and the Earth-Fixed Frame, velocity tracking controllers in the Body-Fixed Frame. Some of the algorithms included an adaptive term which allows one to evaluate dynamical effects if the disturbances can be described by deterministic models. In spite of that, many simplifications were done in the models and control schemes, it is possible to obtain some insight into the vehicle dynamics. Concluding, it can be said that control algorithms using IQV can be useful for analysis of vehicles dynamics (underwater vehicles or indoor airships) during motion.

13.8 Conclusions and Perspectives 13.8.1 Summary of the Book In this book some model based control algorithms which can be applied for a class of vehicles are collected and discussed. This class includes: (1) underwater vehicles, (2) some surface vehicles as hovercrafts, and (3) indoor airship moving at low velocity. The described models of this class of vehicles, without and with disturbances, are taken from the literature. The main goal of this book is to show that the presented control algorithm allow to study the vehicle dynamics before planning the experiment. As a result, some useful information which enables the correction of the dynamic vehicle model is obtained at the design stage. The equations of motion for underwater vehicles, hovercrafts (or more general vehicles moving in the horizontal plane), and airships were given in separate chapters. Based on these equations the proposed velocity transformation which leads to the differential equations ssexpressed in terms of the IQV was shown. Various control strategies concerning the considered vehicles have been reminded in the next chapter. As it arises from the cited references the model based control strategy is only one of the possible methods. The proposed control algorithms based on the IQV approach were considered in several chapters. The following propositions were discussed as follows: (1) (2) (3) (4) (5) (6)

PD controllers in the Body-Fixed Frame; non-adaptive position and velocity trajectory tracking controllers in the BodyFixed Frame; non-adaptive velocity tracking control algorithms in the Body-Fixed Frame; non-adaptive trajectory position and velocity tracking controllers in the EarthFixed Frame; position and velocity tracking controllers with an adaptive term in the BodyFixed Frame; velocity tracking controllers with an adaptive term in the Body-Fixed Frame;

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13 Vehicle Dynamics Study Based on Nonlinear Controllers …

(7)

position and velocity tracking controllers with an adaptive term in the EarthFixed Frame.

All algorithms were applied for various selected vehicles and verified via simulations. In this chapter it was shown that the presented algorithms can be used for dynamics study of various vehicles which belong to the considered class. A procedure is also proposed for comparing different vehicle models. The considered in the book algorithms of control based on the IQV can be useful at the stage of verification of the mathematical model of the vehicle, because they allow to some extent to predict the behavior of this vehicle at the designated task and in specific conditions. At this stage, the experiment is not foreseen, because using the simulation results one can decide both on the necessity to conduct the experiment and on the decision to change or correct the vehicle model. Experimentation on a real object is almost always expensive and requires overcoming various difficulties and additional knowledge about the system under study. On the other hand, the presented control schemes can be used independently from the experiment. Therefore, the IQV based controllers can be treated as some useful tool at the design stage of vehicle because they also avoid experiments that might not guarantee success.

13.8.2 Perspectives and Open Problems It seems that the main application of algorithms expressed in terms of the IQV is testing of vehicle dynamic models. They can be also applied if we want to control variables in accordance to the vehicle dynamics. Moreover, from the control point of view it is interesting to design control algorithms with different adaptive terms. Such schemes may be useful for comparison of models and their investigation. One of the open problem is still lack of control algorithms using the IQV which are suitable for underactuated vehicles at least with 3 DOF but also for 6 DOF models.

References Chin CS, Lum SH (2011) Rapid modeling and control systems prototyping of a marine robotic vehicle with model uncertainties using xPC Target system. Ocean Eng 38:2128–2141 Evans J, Nahon M (2004) Dynamics modeling and performance evaluation of an autonomous underwater vehicle. Ocean Eng 31:1835–1858 Hassanein O, Anavatti SG, Shim H, Ray R (2016) Model-based adaptive control system for autonomous underwater vehicles. Ocean Eng 127:58–69 Herman P (2019) Application of nonlinear controller for dynamics evaluation of underwater vehicles. Ocean Eng 179:59–66 Herman P (2019) Numerical test of underwater vehicle dynamics using velocity controller. In: Proceedings of 2019 12th international workshop on robot motion and control (RoMoCo), Poznan, Poland July 8-10, pp 26–31

References

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Herman P (2020) Velocity tracking controller for simulation analysis of underwater vehicle model. J Marine Eng Technol 19(4):229–239 Herman P (2020) A method for numerical simulation for dynamics and control of underwater vehicles based on quasi-velocities. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734. 2020.1751197 Herman P (2021) Use of a nonlinear controller with dynamic couplings in gains for simulation test of an underwater vehicle model. Int J Adv Robot Syst 2021:1–18. https://doi.org/10.1177/ 17298814211016174 Herman P (2021) Preliminary design of the control needed to achieve underwater vehicle trajectories. J Mar Sci Technol 26:986–998 Herman P, Adamski W (2017) Nonlinear tracking control for some marine vehicles and airships. In: Proceedings of the 11th international workshop on robot motion and control, Wasowo Palace, Poland, July 3-5, 2017, pp 257–362 Martin SC, Whitcomb LL (2014) Experimental identification of six-degree-of-freedom coupled dynamic plant models for underwater robot vehicles. IEEE J Ocean Eng 39(4):662–671 Miskovic N, Vukic Z, Bibuli M, Bruzzone G, Caccia M (2011) Fast in-field identification of unmanned marine vehicles. J Field Robot 28(1):101–120 Wang C, Zhang F, Schaefer D (2015) Dynamic modeling of an autonomous underwater vehicle. J Marine Sci Technol 20:199–212

Appendix

IQV Equations, Formulas, Vehicle Models

In this part of the book some important questions concerning the use of the IQV for underwater vehicles, hovercrafts (vehicles moving in the horizontal plane), and indoor airship models are explained.

Derivation of Equations of Motion in Terms of Inertial Quasi-velocities Below the derivation of the equations of motion using the IQV given in Chap. 2 are presented.

Equations of Motion in Terms of GVC Nonlinear 6 DOF Equations in Body-Fixed Representation Recall, Eqs. (2.23) and (2.24): Mν˙ + C(ν)ν + D(ν)ν + g(η) = τ , η˙ = J(η)ν.

(A.1) (A.2)

Note also that (as can be seen from Chap. 2) the time derivative of ν is ν˙ = ϒP ξ . Inserting this time derivative and the relationship (2.53) into (2.23), and pre-multiplying next both sides by ϒ T (Herman, 2009) the classical dynamic equation into its GVC form are transformed. It can be done as follows:

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Herman, Inertial Quasi-Velocity Based Controllers for a Class of Vehicles, Springer Tracts in Mechanical Engineering https://doi.org/10.1007/978-3-030-94647-0

363

364

Appendix: IQV Equations, Formulas, Vehicle Models

Mϒ ξ˙ + C(ν)ϒξ + D(ν)ϒξ + g(η) = τ , ϒ T Mϒ ξ˙ + ϒ T C(ν)ϒξ + ϒ T D(ν)ϒξ + ϒ T g(η) = ϒ T τ .

(A.3) (A.4)

After grouping the terms the transformed equations of motion can be written in the following form (without disturbances): Nξ˙ + Cξ (ξ )ξ + Dξ (ξ )ξ + gξ (η) = π , η˙ = J(η)ϒξ .

(A.5) (A.6)

The appropriate values of the components of (A.5) are given in Chap. 2 by (2.57)– (2.61). If the disturbances term is added then Eq. (2.36) is used.

Nonlinear 6 DOF Equations in Earth-Fixed Representation Now recall Eq. (2.41) (the kinematic equation is described by (A.2)): Mη (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) = τη .

(A.7)

Taking into account (2.66) and inserting (2.65) into (A.7), and pre-multiplying next both sides by ZT (η) one gets: ˙ + Z(η)ξ˙ ) + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) = τ η , Mη (η)(Z(η)ξ T ˙ + ZT (η)Mη (η)Z(η)ξ˙ + ZT (η)Cη (ν, η)η˙ Z (η)Mη (η)Z(η)ξ +ZT (η)Dη (ν, η)η˙ + ZT (η)gη (η) = ZT (η)τ η .

(A.8) (A.9)

Grouping the terms one obtains the transformed equations of motion in the form (using inversion of (2.65)): Nη (η)ξ˙ + Cξ (ξ , η)ξ + Dξ (ξ , η)ξ + gξ (η) = π, ˙ ξ = Z−1 (η)η,

(A.10) (A.11)

where the final matrices and vectors are given in Chap. 2 by (2.70)–(2.74). If the disturbances term is added then Eq. (2.51) is applied.

Equations of Motion in Terms of NGVC Nonlinear 6 DOF Equations in Body-Fixed Representation Refer to Eqs. (A.1), (A.2), (2.79) and the time derivative of ζ (Chap. 2), namely ζ˙ = ˙ν . Taking the above into account, inserting (2.78) into (A.1), and pre-multiplying

Appendix: IQV Equations, Formulas, Vehicle Models

365

both sides by −T Herman (2010) one can write: T ˙ν + C(ν) −1 ζ + D(ν) −1 ζ + g(η) = τ , ˙ν + −T C(ν) −1 ζ + −T D(ν) −1 ζ + −T g(η) = −T τ .

(A.12) (A.13)

After grouping the terms of this equation one get the transformed equations of motion in the following form: ζ˙ + Cζ (ζ )ζ + Dζ (ζ )ζ + gζ (η) = , η˙ = J(η) −1 ζ ,

(A.14) (A.15)

where the matrices and vectors are given in Chap. 2 by (2.83)–(2.86). If the disturbances term is added then Eq. (2.36) is true.

Nonlinear 6 DOF Equations in Earth-Fixed Representation Recall Eq. (A.7). Inserting (2.90) into (A.7), and next pre-multiplying both sides of the equation by −T (η) one obtains: T (η) (η)¨η + Cη (ν, η)η˙ + Dη (ν, η)η˙ + gη (η) = τ η , ˙ ˙ (η)¨η + (η) η˙ − (η) η˙ + −T (η)Cη (ν, η)η˙ + −T (η)Dη (ν, η)η˙ + −T (η)gη (η) = −T (η)τ η .

(A.16) (A.17)

Grouping the terms of the equation, using (2.92) and (2.94), one gets: −1 ˙ ζ˙ + [ −T (η)Cη (ν, η) − (η)] (η)ζ + −T (η)Dη (ν, η) −1 (η)ζ + −T (η)gη (η) = −T (η)τ η .

(A.18)

Thus, the transformed equations of motion have the form: ζ˙ + Cζ (ζ , η)ζ + Dζ (ζ , η)ζ + gζ (η) = , η˙ = −1 (η)ζ ,

(A.19) (A.20)

where the final matrices and vectors are given in Chap. 2 by (2.96)–(2.99). If the disturbances term is added then Eq. (2.100) is used.

Some General Formulas The formulas given are useful for transforming the equations of motion to be expressed in IQV.

366

Appendix: IQV Equations, Formulas, Vehicle Models

Decomposition of the Symmetric Matrix M(3, 3) Constant Inertia Matrix. The general formula for calculating the matrix ϒ(3, 3) from the symmetric matrix M(3, 3), which can be deduced from the method presented in Loduha and Ravani (1995), is as follows: ⎤ m 11 m 12 m 13 M = ⎣ m 12 m 22 m 23 ⎦ , m 13 m 23 m 33 ⎡

⎡

⎤ 1 ϒ12 ϒ13 ϒ = ⎣ 0 1 ϒ23 ⎦ , 0 0 1

(A.21)

where: m 12 m 12 m 23 − m 13 m 22 , ϒ13 = − , m 11 m 212 − m 11 m 22 m 12 m 13 − m 11 m 23 =− . m 212 − m 11 m 22

ϒ12 = − ϒ23

Obviously the conditions that m 11 = 0 and m 212 = m 11 m 22 must be fulfilled. The inverse matrix ϒ −1 can be determined according to the formula: ⎡

ϒ −1

1 = ⎣0 0

ϒ¯ 12 1 0

⎤ ϒ¯ 13 ϒ¯ 23 ⎦ , 1

(A.22)

where: ϒ¯ 12 =

m 12 m 13 m 12 m 13 − m 11 m 23 , ϒ¯ 13 = , ϒ¯ 23 = . m 11 m 11 m 212 − m 11 m 22

The matrix is calculated based on (2.80). From (2.57) the matrix N is obtained 1 taking into account the matrices M and ϒ (A.21). Next, N 2 ϒ −1 is determined where −1 ϒ comes from (A.22). Variable Dependent Inertia Matrix The general formula for calculating the matrix Z(η) from the three dimensional symmetric matrix Mη (η), which can be deduced from the method given in Loduha and Ravani (1995), is as follows (cf. Herman and Adamski 2020): ⎡

⎤ μ11 μ12 μ13 Mη (η) = ⎣ μ12 μ22 μ23 ⎦ , μ13 μ23 μ33 where:

⎡

⎤ 1 Z 12 Z 13 Z(η) = ⎣ 0 1 Z 23 ⎦ , 0 0 1

(A.23)

Appendix: IQV Equations, Formulas, Vehicle Models

Z 12 = −

μ12 , μ11

Z 13 = −

μ12 μ23 − μ13 μ22 , μ212 − μ11 μ22

367

Z 23 = −

μ12 μ13 − μ11 μ23 . μ212 − μ11 μ22

Obviously the conditions that μ11 = 0 and μ212 = μ11 μ22 must be satisfied. The inverse matrix Z−1 (η) can be determined according to: ⎡

1 Z−1 (η) = ⎣ 0 0

Z¯ 12 1 0

⎤ Z¯ 13 Z¯ 23 ⎦ , 1

(A.24)

where: μ12 , Z¯ 12 = μ11

μ13 Z¯ 13 = , μ11

μ12 μ13 − μ11 μ23 Z¯ 23 = . μ212 − μ11 μ22

The matrix (η) is calculated based on (2.93). From (2.70) the matrix N(η) is deter1 mined taking into account the matrices Mη (η) and Z(η) (A.23). Next, N 2 (η)Z−1 (η) is obtained where Z−1 (η) comes from (A.24).

Decomposition of the Symmetric Matrix M(6, 6) Constant Inertia Matrix This issue was discussed in paper Herman and Adamski (2017). However, the decomposition procedure can be expressed somewhat differently. Consider the inertia matrix (including the added mass) M for a general 6 DOF model according to (2.23). If the formulas are expressed in matrix-vector form then the successively decomposed matrices can be written as follows: P1 = ϒ 1T Mϒ 1 , P2 = ϒ 2T P1 ϒ 2 , P3 = ϒ 3T P2 ϒ 3 , P4 = ϒ 4T P3 ϒ 4 , N = ϒ 5T P4 ϒ 5 ϒ = ϒ 1ϒ 2ϒ 3ϒ 4ϒ 5 N = ϒ T Mϒ, where:

(A.25)

368

Appendix: IQV Equations, Formulas, Vehicle Models

⎡

m 11 ⎢ m 12 ⎢ ⎢ m 13 M=⎢ ⎢ m 14 ⎢ ⎣ m 15 m 16

m 12 m 22 m 23 m 24 m 25 m 26

m 13 m 23 m 33 m 34 m 35 m 36

m 14 m 24 m 34 m 44 m 45 m 46

m 15 m 25 m 35 m 45 m 55 m 56

⎤ m 16 m 26 ⎥ ⎥ m 36 ⎥ ⎥. m 46 ⎥ ⎥ m 56 ⎦ m 66

The intermediate matrices are calculated as follows: ⎡ 12 13 14 15 16 1 −m −m −m −m −m m 11 m 11 m 11 m 11 m 11 ⎢0 1 0 0 0 0 ⎢ ⎢0 0 1 0 0 0 ϒ1 = ⎢ ⎢0 0 0 1 0 0 ⎢ ⎣0 0 0 0 1 0 0 0 0 0 0 1 ⎡

1 ⎢0 ⎢ ⎢0 ϒ2 = ⎢ ⎢0 ⎢ ⎣0 0 ⎡

1 ⎢0 ⎢ ⎢0 ϒ3 = ⎢ ⎢0 ⎢ ⎣0 0 ⎡

1 ⎢0 ⎢ ⎢0 ϒ4 = ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 0 0 1 − PP11 2223 − PP11 2224 − PP11 2225 − PP11 2226 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

0 1 0 0 0 0

0 1 0 0 0 0

0 0 0 0 0 0 0 0 1 − PP22 3433 − PP22 35 − PP22 36 33 33 0 1 0 0 0 0 1 0 0 0 0 1

0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 − PP33 4445 − PP33 46 44 0 1 0 0 0 1

(A.26)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(A.27)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(A.28)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(A.29)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(A.30)

Appendix: IQV Equations, Formulas, Vehicle Models

⎡

1 ⎢0 ⎢ ⎢0 ϒ5 = ⎢ ⎢0 ⎢ ⎣0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 − PP44 56 55 0 1

369

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(A.31)

The final matrices have the form:: N = diag{N11 , N22 , N33 , N44 , N55 , N66 }, ⎤ ⎡ 1 ϒ12 ϒ13 ϒ14 ϒ15 ϒ16 ⎢ 0 1 ϒ23 ϒ24 ϒ25 ϒ26 ⎥ ⎥ ⎢ ⎢ 0 0 1 ϒ34 ϒ35 ϒ36 ⎥ ⎥. ⎢ ϒ=⎢ ⎥ ⎢ 0 0 0 1 ϒ45 ϒ46 ⎥ ⎣ 0 0 0 0 1 ϒ56 ⎦ 0 0 0 0 0 1

(A.32)

(A.33)

Taking into consideration the above results the Generalized Velocity Components vector ξ = [ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 ]T for 6 DOF is defined as follows: ξ1 = u − ϒ12 v + μ7 w + μ8 p + μ9 q + μ10 r,

(A.34)

ξ2 = v − ϒ23 w + μ4 p + μ5 q + μ6r, ξ3 = w − ϒ34 p + μ2 q + μ3r,

(A.35) (A.36)

ξ4 = p − ϒ45 q + μ1 r ξ5 = q − ϒ56r, ξ6 = r,

(A.37) (A.38) (A.39)

where: μ1 = ϒ45 ϒ56 − ϒ46 , μ2 = ϒ34 ϒ45 − ϒ35 , μ3 = ϒ35 ϒ56 − ϒ34 μ1 − ϒ36 , μ4 = ϒ23 ϒ34 − ϒ24 , μ5 = ϒ24 ϒ45 − ϒ23 μ2 − ϒ25 , μ6 = ϒ25 ϒ56 − ϒ23 μ3 − ϒ24 μ1 − ϒ26 , μ7 = ϒ12 ϒ23 − ϒ13 , μ8 = ϒ13 ϒ34 − ϒ12 μ4 − ϒ14 , μ9 = ϒ14 ϒ45 − ϒ12 μ5 − ϒ13 μ2 − ϒ15 , μ10 = ϒ15 ϒ56 − ϒ12 μ6 − ϒ13 μ3 − ϒ14 μ1 − ϒ16 . The matrix is calculated based on (2.80). From (2.57) the matrix N (A.32) is 1 determined taking into account the matrices M (A.26) and ϒ (A.33). Next, N 2 ϒ −1 is derived where ϒ −1 is the inverse matrix to the matrix ϒ (A.33). Variable Dependent Inertia Matrix. The matrix (η) is calculated based on (2.93). At the beginning the matrix Mη (η) according to (2.42) is computed. For this matrix, using the same decomposition

370

Appendix: IQV Equations, Formulas, Vehicle Models

Table A.1 Parameters of underwater vehicle Symbol Value L b m Ix Iy Iz Jx y Jx z Jyz

2.0 0.5 250 20 150 140 0 -30 0

Unit m m kg kgm2 kgm2 kgm2 kgm2 kgm2 kgm2

method, the matrix Z(η) is determined (instead of the matrix ϒ). From (2.70) the matrix N(η) is obtained taking into account the matrices Mη (η) and Z(η). Finally, 1 N 2 (η)Z−1 (η) is determined where Z−1 (η) is the inverse matrix to the matrix Z(η).

Vehicle Models Underwater Vehicle Parameters (6 DOF) The model of the underwater vehicle considered in this book was tested earlier, e.g. in Herman (2009, 2020). Vehicles with lower masses are, for example AUV model given in Antonelli (2007), Cyclops underwater vehicle Joe et al. (2014); Kim et al. (2016, 2015), Tethra vehicle Molnar et al. (2007) or NEROV vehicle Spangelo and Egeland (1994) while the C-Ranger platform He et al. (2013) has a similar weight. The underwater vehicle tested here has a similar shape to the AUV BA-1 Choi et al. (2014) but is much lighter than it. In Table A.1 L is length and b is breadth of the vehicle. The submerged weight of the body is defined as W = mg, and the buoyancy force as B = ρg∇ with ρ = 1000 kg/m3 , g = 9.81 m/s2 where ∇ = 0.36 m3 . The center of gravity is r G = [x G , yG , z G ] with x G = −0.15 m, yG = 0 m, z G = 0.03 m and the center of buoyancy is r B = [x B , y B , z B ] with x B = y B = z B = 0 m (Table A.2). The hydrodynamic damping matrix is defined as:  D(ν) = − diag X u , Yv , Z w , K p , Mq , Nr  − diag X |u|u |u|, Y|v|v |v|, Z |w|w |w|, K | p| p | p|, M|q|q |q|, N|r |r |r | . (A.40) This type of model based on Antonelli (2018); Fossen (1994) is suitable for fully submerged body. It was used, e.g. in Hassanein et al. (2016). From the latter work,

Appendix: IQV Equations, Formulas, Vehicle Models

371

Table A.2 Hydrodynamic derivatives and damping terms Symbol Value Unit Symbol X u˙ Yv˙ = Z w˙ X q˙ = Yr˙ = Mu˙ = Nv˙ Y p˙ = Z q˙ = K v˙ = Mw˙ Xu Yv Zw Kp Mq Nr

−50 −250 10 −10 −50 −100 −100 −200 −300 −300

kg kg kgm kgm kg/s kg/s kg/s kgm2 /s kgm2 /s kgm2 /s

Table A.3 Parameters of hovercraft Symbol Value m Jzz xg yg

10 0.4 0.1 0.1

Value

Unit

K p˙ Mq˙ = Nr˙ K r˙ = N p˙

−20 −30 1

kgm2 kgm2 kgm2

X |u|u X |v|v Z |w|w K | p| p M|q|q N|r |r

−50 −100 −100 −200 −300 −300

kg/m kg/m kg/m kgm2 kgm2 kgm2

Unit kg kgm2 m m

which contains experimental results, arises that if the off-diagonal elements are small then they can be negligible.

Hovercraft Parameters (3 DOF) The model of the hovercraft used in this book was tested in Herman and Kowalczyk (2014). Parameters of the model are shown in Table A.3. The inertia Jz of the vehicle is determined according to the formula Jz = Jzz + m(x g2 + yg2 ). The damping terms were X u = −1.0 kg/s, Yv = −1.0 kg/s, Nr = −1.0 kgm2 /s, and X r = Yr = Nu = Nv = 0. The model can be treated as a modification of the hovercraft models investigated in Aguiar et al. (2003); Cremean (2002); Kim et al. (2013).

372

Appendix: IQV Equations, Formulas, Vehicle Models

Table A.4 Parameters of the AUV Symbol Value m Jzz xg yg

7 0.2 0.1 0.1

Unit kg kgm2 m m

Underwater Vehicle Parameters (3 DOF) The model of the underwater vehicle moving horizontally was tested, e.g. in Herman and Kowalczyk (2016). Parameters of the model are shown in Table A.4. The vehicle inertia Jz = 0.34 kgm2 is calculated according to the formula Jz = Jzz + m(x g2 + yg2 ). Moreover, X u˙ = −3 kg, X r˙ = 0.3 kg„ Yv˙ = −3 kg, Yr˙ = −0.3 kg, Nr˙ = −0.26 kgm2 . The damping terms were X u = −10.0 kg/s, Yv = −10.0 kg/s, Nr = −10.0 kgm2 /s, and X r = Yr = Nu = Nv = 0. The test vehicle has a lower mass than, e.g. Intelligence Ocean I (IO-I) Kong et al. (2020) but is assumed to have a similar shape to it.

Airship Parameters (6 DOF) The airship model used in this book is a modification of the AS500 airship model described in Bestaoui (2007) subsequently used in Herman and Adamski (2017). It may be also treated as changed version of the airship model AS200 Beji and Abichou (2005). Parameters of the model are shown in Table A.5 whereas elements of added mass and damping terms in Table A.6. The symbols mean: L is length and b is breadth of the vehicle. The body weight is W = mg, and the buoyancy force B = ρg∇ with ρ = 1.225 kg/m3 , g = 9.81 m/s2 where ∇ = 10.6 m3 . The center of gravity is r G = [x G , yG , z G ] with x G = −1.0 m, yG = 0 m, z G = 0.15 m and the center of buoyancy is r B = [x B , y B , z B ] with x B = y B = z B = 0 m. Model of damping terms of the type (A.40) was considered, e.g. in Beji and Abichou (2005).

Appendix: IQV Equations, Formulas, Vehicle Models Table A.5 Parameters of indoor airship Symbol Value L b m Ix Iy Iz Jx y Jx z Jyz

Unit

7.6 1.6 13.0 10.0 50.0 50.0 0 −10.0 0

m m kg kgm2 kgm2 kgm2 kgm2 kgm2 kgm2

Table A.6 Elements of added mass and damping terms Symbol Value Unit Symbol a11 a15 a22 a24 a26 a33 a35 a44 a46 a55 a66

0.13ρ 0 1.7ρ −0.12ρ −5.5ρ 1.6ρ 5.5ρ 1.3ρ 0.3ρ 30.0ρ 30.0ρ

kg kg kg kg kg kg kg kgm2 kgm2 kgm2 kgm2

373

Xu X |u|u Yv = Z w Y|v|v = Z |w|w K p = Nr K | p| p Mq M|q|q = N|r |r

Value

Unit

−0.2 −0.2 −0.8 −0.8 −0.1 −0.01 −0.3 −0.1

kg/s kg/m kg/s kg/m kgm2 /s kgm2 kgm2 /s kgm2

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