Distributed Attitude Consensus of Multiple Flexible Spacecraft 9811942579, 9789811942570

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Ti Chen Jinjun Shan Hao Wen

Distributed Attitude Consensus of Multiple Flexible Spacecraft

Distributed Attitude Consensus of Multiple Flexible Spacecraft

Ti Chen · Jinjun Shan · Hao Wen

Distributed Attitude Consensus of Multiple Flexible Spacecraft

Ti Chen State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

Jinjun Shan York University Toronto, ON, Canada

Hao Wen State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

ISBN 978-981-19-4257-0 ISBN 978-981-19-4258-7 (eBook) https://doi.org/10.1007/978-981-19-4258-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To Piao and Rosie Ti Chen To Haiyan, Rory and Roger Jinjun Shan To Yonglian and Yuzhi Hao Wen

Preface

This book studies the attitude consensus of multiple spacecraft interconnected by a communication topology and focuses particularly on the consensus of flexible spacecraft. A spacecraft is a vehicle or machine designed to fly in outer space. Spacecraft formation flying is the coordination of multiple spacecraft to complete certain space missions, such as Earth-observation, on-orbit service, space telescope construction, high-resolution interferometry, and gravitational-wave observation. Networked small satellites have many advantages over a large single spacecraft including higher robustness, lower cost, shorter development cycles, and more flexible replacements. In spacecraft formation flying missions, a key problem is the attitude consensus. Since the rotation configuration space of a rigid body lies on the special Lie group S O(3) and the attitude dynamics is strongly nonlinear, the distributed attitude control is quite challenging compared with the classical linear multi-agent systems. Also, different from the attitude control for a monolithic spacecraft, the coupling between spacecraft due to information communication makes the controller design more difficult. A fundamental problem in spacecraft attitude consensus is the design of distributed protocols to guarantee the attitude consensus or synchronization with the feedback of the agent’s own state and local information exchange. The past decade has witnessed rapidly-growing research interest in cooperative attitude control due to controller design difficulty in highly dynamic space environments and with external constraints such as blinding celestial objects that may damage onboard sensitive instruments. For example, in Web of ScienceTM , with “spacecraft attitude consensus” as the search keyword, there have been 341 items published in the aerospace, control, dynamics, and robotics journals since 2012 (data gathered on Feb. 22nd, 2022). Some spacecraft may be large-scale and lightweight and carry flexible appendages, such as antennae, manipulators, and solar panels. In such a situation, the spacecraft should be treated as a flexible system rather than a rigid one. Due to the low damping of a space structure, it will take a long time for the flexible vibration to die out. Hence, it is necessary to consider the rigid-flexible coupling during the cooperative attitude motion. Significant efforts have been made towards

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the dynamics and control of a single flexible spacecraft. However, it is not straightforward to extend the control laws for a single flexible spacecraft or the consensus algorithm of the networked rigid spacecraft to multiple flexible spacecraft cases. Recently, the cooperative control of multiple flexible spacecraft has attracted the attention of some investigators. For example, Du and Li studied the distributed attitude control of the networked flexible spacecraft based on the backstepping technique and the neighbor-based design rule with or without communication delays. Huang et al. considered the attitude consensus control for multiple flexible spacecraft under actuator failures and saturation constraints based on the combination of tools from the Lyapunov stability theory and graph theory. However, most works on the attitude consensus of multiple spacecraft are based on the Modified Rodrigues Parameters or unit quaternions. As is well known, three-dimensional attitude representations, such as Euler angles and Modified Rodrigues Parameters, cannot describe the rigid body rotation globally. Even though unit quaternions can be used to remove the singularity, they double-cover the rotation configuration, i.e., two antipodal quaternions correspond to a single attitude. Hence, unit quaternions cannot represent the attitude uniquely. Under continuous control laws, undesirable unwinding phenomena may occur. To avoid the singularity and ambiguity of other attitude parameterizations, the rotation matrix or direction cosine matrix has been used in spacecraft attitude control because it can represent the attitude globally and uniquely. The main conclusion for the controller on S O(3) is that the continuous time-invariant feedback controllers cannot stabilize the rigid body rotation globally due to the topological structures of S O(3). The main reason is that there are at least four critical points where the gradient of any smooth potential function on S O(3) vanishes. The manifolds of the undesired points are of Lebesgue measure zero. Hence, the best stability result for the time-invariant continuous controllers based on smooth potential functions for the attitude error on S O(3) is almost global stable. In fact, the unstable equilibria may not be a big deal because the ubiquitous environmental disturbance can drive the system to leave these undesirable manifolds. The intention of this book is to summarize the authors’ recent works on the cooperative control of multiple spacecraft. The organization of this book is as follows. Chapter 1 introduces the notations in the book and some basic knowledge about the graph theory, attitude representations, and the dynamics models of the rigid and flexible spacecraft. Chapter 2 reviews the studies on spacecraft attitude consensus under various communication graphs, various attitude representations, and complicated conditions. The centralized, decentralized, and distributed cases are included. The works under Euler angles, Modified Rodrigues Parameters, unit quaternions, rotation matrix, and other attitude parameterizations are discussed. Finally, attitude consensus problems with state constraints and actuator nonlinearity, and some experimental results are surveyed. Chapter 3 focuses on the leader-follower attitude consensus of the networked flexible spacecraft based on Modified Rodrigues Parameters with the feedback of attitude

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information only. The passivity-based control algorithms are developed under undirected and directed graphs to achieve the attitude consensus and vibration suppression simultaneously. From Chaps. 4 to 9, the spacecraft attitude consensus problem is discussed directly on S O(3). In Chap. 4, the leader-follower attitude consensus of multiple flexible spacecraft is studied under an undirected tree graph considering the partial loss of actuator effectiveness. Chapter 5 presents the adaptive fault-tolerant control of the distributed attitude tracking problem for multiple flexible spacecraft under an undirected tree graph in the absence of the feedback of the modal variables. The observer-based control architecture is adopted in this work. Chapter 6 pays attention to the distributed attitude tracking and synchronization on S O(3) of a group of flexible spacecraft under a directed graph. A finite-time observer is embedded in each spacecraft to estimate the leader’s information. An adaptive control strategy with synchronization terms is proposed. Chapter 7 studies both the centralized and distributed attitude regulation of the networked flexible spacecraft subject to attitude constraints without the measurements of angular velocities and modal coordinates. A continuous controller based on a repulsive potential function is designed. Both the fixed and switching communication topologies are discussed. For the leaderless consensus of multiple rigid spacecraft under a connected undirected graph, Chap. 8 gives a solution to the design of an adaptive controller based on a distributed observer under the undirected tree graph generated from the original graph. Both cases with zero and nonzero final angular velocities are studied. Chapter 9 presents attitude control methods for a single satellite and the networked spacecraft based on the Koopman operator theory. The main concept is to find a set of observables to describe the original nonlinear attitude dynamics on S O(3) using a finite-dimensional linear system approximately. So, the powerful linear control theories can be used to realize attitude control. Both simulation and experimental results are presented to verify the developed control methods. The studies in this book would not have been possible without the efforts and the help of our colleagues, collaborators, and students. In particular, the first and third authors are indebted to their Ph.D. supervisor, Professor Haiyan Hu, a member of the Chinese Academy of Sciences, for leading us to the field of dynamics and control. We are also grateful to Professors Guanrong Chen, Dongping Jin, Weidong Chen, Hugh. H.T. Liu, Zaihua Wang, and Bin Jiang for their constant support and inspiration. We also acknowledge Dr. Shiyuan Jia, Dr. Marc Savoie, Hassan Alkomy, Samira Eshghi, former and current students of the second author, for their assistance in reviewing parts of the manuscript. We also thank our editors Padmavathi Jagadeishkumar and Wayne Hu for their interest and efforts in publishing this book. In addition, we acknowledge Elsevier, Springer Nature, and AIAA for permitting us to reuse materials from our publications copyrighted by these publishers in this book. We wish to thank our families for their support, patience, and endless love. We acknowledge the financial support by the National Natural Science Foundation of China under Grant Nos. 11832005 and 12102174 and the Research Fund of

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State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant No. MCMS-I-0122K01). This work was also supported partly by the Discovery Grant from the Natural Sciences and Engineering Research Council of Canada and the TalentEdge Fellowship Program from the Ontario Centres of Excellence. Nanjing, China Toronto, Canada Nanjing, China

Ti Chen Jinjun Shan Hao Wen

Contents

Part I

Preliminaries and Literature Review

1 Graph Theory and Attitude Representations . . . . . . . . . . . . . . . . . . . . . . 1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Graph Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Leader-Follower Communication Graph . . . . . . . . . . . . . . . . . 1.3 Attitude Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Rotation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Modified Rodrigues Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Kinematic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dynamics Equation of Rigid Spacecraft . . . . . . . . . . . . . . . . . . . . . . . 1.5 Dynamics Equation of Flexible Spacecraft . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 5 5 7 8 8 9 10 11 12 13 13 14

2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Attitude Consensus Under Various Communication Graphs . . . . . . . 2.2.1 Centralized Attitude Consensus . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Decentralized Attitude Consensus . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Distributed Attitude Consensus . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Attitude Consensus with Various Attitude Representations . . . . . . . . 2.3.1 Euler-Angles-Based Attitude Consensus . . . . . . . . . . . . . . . . . 2.3.2 MRPs-Based Attitude Consensus . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Quanternions-Based Attitude Consensus . . . . . . . . . . . . . . . . 2.3.4 Rotation-Matrix-Based Attitude Consensus . . . . . . . . . . . . . . 2.3.5 Attitude Consensus Based on Other Attitude Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Attitude Consensus Under Complicated Conditions . . . . . . . . . . . . .

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2.4.1 Attitude Consensus with Input Saturation . . . . . . . . . . . . . . . . 2.4.2 Attitude Consensus with Actuator Faults . . . . . . . . . . . . . . . . 2.4.3 Attitude Consensus with State Constraints . . . . . . . . . . . . . . . 2.4.4 Experimental Results on Attitude Consensus . . . . . . . . . . . . . 2.5 Attitude Consensus of Multiple Flexible Spacecraft . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

27 28 29 30 31 33 34

Leader-Follower Attitude Consensus of Networked Flexible Spacecraft

3 Distributed Passivity-Based Control with Attitude-Only Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Dynamic Equation of Undamped Flexible Spacecraft . . . . . . 3.2.2 Properties of Networked Flexible Spacecraft . . . . . . . . . . . . . 3.3 Distributed Passivity-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Passivity-Based Control Methodology . . . . . . . . . . . . . . . . . . 3.3.3 Distributed Controller Under Undirected Graph . . . . . . . . . . 3.3.4 Distributed Controller Under Directed Graph . . . . . . . . . . . . . 3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Case I: Leader-Follower Tracking Under Undirected Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Case II: Leader-Follower Tracking Under Directed Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Rotation-Matrix-Based Attitude Tracking Under an Undirected Tree Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dynamics of Flexible Spacecraft Based on Rotation Matrix . . . . . . . 4.3 Coordinated Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Case I: Fault-Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Case II: Fault-Tolerant Control from M1 with Initial Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Case III: Response Without Structural Damping Under Controller (4.24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Adaptive Fault-tolerant Attitude Tracking on SO(3) Under an Undirected Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Kinematics and Dynamics of Flexible Spacecraft . . . . . . . . . 5.2.2 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Distributed Attitude Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Finite-time Distributed Observer . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Modal Variable Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 82 83 83 83 86 87 93 95 99

6 Distributed Attitude Tracking and Synchronization on SO(3) Under Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Dynamics Equation of Flexible Spacecraft . . . . . . . . . . . . . . . 6.2.2 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Distributed Leader Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Distributed Tracking and Synchronization with Full State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Distributed Tracking and Synchronization with Partial State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Continuous Constrained Attitude Regulation on SO(3) . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Kinematics and Dynamics of Flexible Spacecraft . . . . . . . . . 7.2.2 Attitude Constraints Based on Rotation Matrix . . . . . . . . . . . 7.2.3 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Centralized Velocity-Free Attitude Regulation . . . . . . . . . . . . . . . . . . 7.3.1 Repulsive Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Centralized Velocity-Free Control . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Undesired Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Finite Control Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 103 103 103 104 104 105 115 117 124 126 126 129 129 130 130 131 132 132 132 133 135 136

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7.4 Distributed Attitude Regulation Control . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Finite-Time Distributed Observer . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Distributed Velocity-Free Regulation Controller . . . . . . . . . . 7.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Centralized Attitude Regulation . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Distributed Attitude Regulation . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 138 140 140 149 151 152

Part III Leaderless Attitude Consensus of Networked Rigid Spacecraft 8 Continuous Leaderless Synchronization Control of Multiple Rigid Spacecraft on SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Spacecraft Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Leaderless Consensus with Nonzero Final Angular Velocity . . . . . . 8.3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Generating an Undirected Tree Graph . . . . . . . . . . . . . . . . . . . 8.3.3 Distributed Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Leaderless Consensus with Zero Final Angular Velocity . . . . . . . . . . 8.4.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Distributed Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 159 159 159 160 160 164 168 168 168 169 170 175 175

9 Koopman-Operator-Based Attitude Dynamics and Control on SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Koopman Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Spacecraft Dynamics on SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Koopman Operator for Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . 9.3.1 Reduced Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Simulation Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Controller Design Based on Koopman Operator . . . . . . . . . . . . . . . . . 9.4.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Simulation Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Comparison with Traditional Optimal Control . . . . . . . . . . . . 9.4.5 Application to Leaderless Synchronization . . . . . . . . . . . . . . . 9.4.6 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 179 179 180 180 184 186 186 188 191 194 197 200

Contents

9.5 Attitude Control with Large Angular Velocities . . . . . . . . . . . . . . . . . 9.5.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Simulation Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

203 203 204 207 208 208

Part I

Preliminaries and Literature Review

This part aims to present some preliminaries and the literature review. In Chap. 1, we give the the notations used in this book, a brief introduction to graph theory and attitude representations and the dynamics models of rigid and flexible spacecraft. In Chap. 2, a survey of recent studies on the the spacecraft attitude coordination published in major aerospace, dynamics, automation and robotics journals is presented.

Chapter 1

Graph Theory and Attitude Representations

Abstract The communication between the networked spacecraft can be described by algebraic graphs. This chapter presents some basic knowledge of graph theory. Furthermore, the configuration space of the rigid-body rotation is the special Lie group S O(3). Attitude parameterizations are important to solve the attitude consensus problem. Hence, various attitude representation methods are introduced in this chapter.

1.1 Notations 0n and 1n represent vectors in Rn with all elements being 0 and 1, respectively. 0m×n denotes the zero matrix of m × n. I n is an identity matrix of n × n. For a vector x = [x1 x2 x3 ]T , |x| denotes [|x1 | |x2 | |x3 |]T and x × or s(x) is a skew-symmetric matrix defined as ⎡ ⎤ 0 −x3 x2 x × = s(x) = ⎣ x3 0 −x1 ⎦ (1.1) −x2 x1 0 S O(3) = {R ∈ R3×3 |det(R) = 1, R T R = R R T = I 3 } is a special orthogonal group in R3×3 with determinant of 1, where I 3 is the 3 × 3 identity matrix. Let ¯  ¯ ∈ so(3) be the group of skew-symmetric matrices in R3×3 ; that is, so(3) = {| T 3×3 ¯ ¯ R ,  = − }. so(3) is referred to as the Lie algebra of the Lie group S O(3). The mapping (·)× : R3 → so(3) is defined such that a× b = a × b for vectors a ∈ R3 and b ∈ R3 . The wedge mapping (·)∨ : so(3) → R3 denotes the inverse mapping of a× ; that is, (a× )∨ = a. For a ∈ R3 and R ∈ S O(3), we have the following equations [1]: tr(Ra× ) = −a T (R − R T )∨  (a× R + R T a× )∨ = tr(R)I 3 − R a (Ra)× = Ra× R T

(1.2)

For two matrices of arbitrary size, the Kronecker product ⊗ is defined as

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_1

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1 Graph Theory and Attitude Representations

⎡

⎤ α11 β · · · α1s β α ⊗ β = ⎣ · · · · · · · · · ⎦ ∈ Rr p×sq αr 1 β · · · αr s β

(1.3)

where α ∈ Rr ×s and β ∈ R p×q . Some properties of Kronecker product are [2] α ⊗ (β + γ ) = α ⊗ β + α ⊗ γ α ⊗ (β ⊗ γ ) = (α ⊗ β) ⊗ γ (α ⊗ β)(γ ⊗ ζ ) = (αγ ) ⊗ (βζ ) (α ⊗ β)−1 = α −1 ⊗ β −1 Suppose that α and β are two square matrices of size r and s. The eigenvalues of α ⊗ β are μi ξ j , i = 1, . . . , r, j = 1, . . . , s, where μ1 , . . . , μr and ξ1 , . . . , ξs are the eigenvalues of α and β, respectively. For z ∈ R, let sigα (z) = sgn(z)|z|α , where α > 0 and sgn(·) and | · | are the ¯ is defined standard sign function and absolute value function, respectively. sigα ( A) α m× ¯ n¯ m× ¯ n¯ ¯ with A = [a¯ i j ] ∈ R .  · 1 ,  · 2 and  · ∞ represent the as [sig (a¯ i j )] ∈ R 1-norm, 2-norm and infinity norm of a vector or matrix, respectively. For any matrix A = [ai j ] ∈ Rm×n , sign( A) is defined as [sgn(ai j )] ∈ Rm×n , i.e., sign( A) represents a matrix with the same size as A, where each element of sgn( A) is the sign of the corresponding element of A. By letting n = 1, one can get the definition of the sign function for a vector. For a symmetric matrix J = [Ji j ] ∈ R3×3 , J  is used to represent a 6-dimensional vector defined as J  = [J11 , J12 , J13 , J22 , J23 , J33 ]T .

1.2 Graph Theory In this book, we are concerned with the attitude consensus of multiple flexible spacecraft linked to each other. A communication graph can be used to represent the information flow between the agents. The agents in Multi-Agent Systems (MAS) are modeled as nodes in the graph and the communication edges correspond to the allowed flow of information between agents. Hence, some basic concepts of graph theory are given in this section. A detailed explanations of graph theory can be found in Ref. [3].

1.2 Graph Theory

5

1.2.1 Basic Definitions A graph for N agents can be represented by G = (V, E), where V = {1, 2, . . . , N } is the node set and E ⊂ V × V is the edge set. Elements of E can be expressed as (i, j), which means the node j can receive information from the ith node. Usually, only simple graphs are considered in the study of MAS, i.e., neither self-loops nor multiple edges between the same node pair are included in the graph. If (i, j) ∈ E, node i is a neighbor of the jth node, agent i is called the parent node and node j is the child node. The in-degree of the ith node is the number of edges having node i as a head. Similarly, the number of the edges with node i as a tail is called the out-degree of the node i. The neighbor set of the ith node is denoted by Ni = { j|( j, i) ∈ E}, which means the set of nodes with edges incoming to node i. It should be noted that the node number in Ni equals the in-degree of the ith node. The graph is said to be balanced if the in-degree equals the out-degree for all nodes in the graph. For any two nodes in the graph, if (i, j) ∈ E ⇒ ( j, i) ∈ E holds, the graph is said to be bidirectional, otherwise it is directed. A weight ai j can be defined for the edge ( j, i) ∈ E. If the graph is bidirectional and ai j = a ji holds, the graph is said to be undirected. It is clear that undirected graphs are special balanced graphs. In this book, the edge weights are assumed to be identical unless otherwise specified. A directed path is defined as a sequence of node i 1 , i 2 , . . . , il such that (i k , i k+1 ) ∈ E for k = 1, 2, . . . , l − 1. The ith node is connected to the node j if there exists a directed path from node i to node j. The graph is said to be strongly connected if any two distinct nodes are connected. A directed graph is weakly connected if the graph is not strongly connected, but the underlying undirected graph (i.e., considering all edges as undirected) is connected. Note that the qualifier ‘strongly’ is usually omitted for the strongly connected undirected graphs. A directed tree is a directed graph with every node having exactly one parent except the one called the root node. A spanning tree of a directed graph is a directed tree connecting all nodes of the graph. If a subset of the communication edges forms a directed tree, i.e., all nodes in the graph are reachable from the root node, the graph is said to have a spanning tree. Note that the graph may have multiple tree graphs. For a connected undirected graph, if two vertices are connected by exactly one path, the graph is called an undirected tree graph. Some graph examples are given in Fig. 1.1 to illustrate the different graph concepts. The edges in red in Fig. 1.2 represent a spanning tree for the graph in Fig. 1.1e with node 5 as the root.

1.2.2 Graph Matrices The adjacency matrix of the graph is defined as A = [ai j ], where ai j > 0 if ( j, i) ∈ E and ai j = 0 otherwise. Note that aii = 0 holds for simple graphs. In this book, the weight ai j is defined as 1. For the graph in Fig. 1.1e, its adjacency matrix is

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1 Graph Theory and Attitude Representations

Fig. 1.1 Different graphs with five nodes. a A connected undirected graph. b A strongly connected directed graph. c A balanced and strongly connected graph. d A weakly connected graph. e A directed graph having a spanning tree. f An undirected tree graph Fig. 1.2 A spanning tree for the graph in Fig. 1.1e with node 5 as the root

1.2 Graph Theory

7

⎡

0 ⎢0 ⎢ A=⎢ ⎢1 ⎣0 0

1 0 0 0 0

0 0 0 1 0

0 1 0 0 0

⎤ 0 1⎥ ⎥ 0⎥ ⎥ 1⎦ 0

(1.4)

The in-degree of node i is denoted by d

i , which equals to the summation of the ith row of the adjacency matrix A, i.e., di = Nj=1 ai j . The out-degree of the ith node is

defined as dio = Nj=1 a ji . D = diag{di } is called the diagonal in-degree matrix. The graph Laplacian matrix is defined as L = D − A. Matrices D and L of the graph in Fig. 1.1e are D = diag{1, 2, 1, 2, 0} (1.5) ⎡

1 ⎢0 ⎢ L=⎢ ⎢−1 ⎣0 0

−1 2 0 0 0

0 0 0 −1 1 0 −1 2 0 0

⎤ 0 −1⎥ ⎥ 0⎥ ⎥ −1⎦ 0

(1.6)

Similarly, for the undirected graph in Fig. 1.1f, L is as follows. ⎡

1 ⎢−1 ⎢ L=⎢ ⎢0 ⎣0 0

−1 2 0 −1 0

0 0 0 −1 1 −1 −1 3 0 −1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ −1⎦ 1

(1.7)

Some important properties of the graph Laplacian matrix are as follows. (1) All row sums of L are equal to zero. (2) For any undirected graph, L = L T holds and its eigenvalues can be ordered as 0 = λ1 ≤ λ2 ≤ · · · ≤ λ N . (3) L has the rank of N − 1 and λ1 = 0 is the nonrepeated eigenvalue if and only if the graph has a spanning tree.

1.2.3 Leader-Follower Communication Graph In the attitude consensus problem of concern, the desired attitude and angular velocity ¯ as the graph consisting may be determined by one (virtual) leader (node 0). Denote G of N follower agents and one leader. The graph of the N followers is represented ¯ is connected, the leader is the root node. For the graph G, ¯ by G. If the graph G denote H = L + diag{ai0 }, where ai0 > 0 if the ith node can receive the leader’s information, and ai0 = 0 otherwise.

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1 Graph Theory and Attitude Representations

Lemma 1.1 [4] In the connected undirected graph G, if there is at least one follower having access to the information of the leader, H is positive definite. Lemma 1.2 [5] The matrix H is positive stable if the leader is globally reachable.

1.3 Attitude Representations The configuration space of the rigid-body rotation is the set of 3 × 3 orthogonal matrices with determinant of one. Note that this configuration space is a Lie group, not a Euclidean space. So various attitude parameterizations are proposed in attitude dynamics and control. These parameterizations can be Euclidean, such as Euler angles, or non-Euclidean, such as the unit quaternions. This section aims to review the common attitude representation methods briefly.

1.3.1 Rotation Matrix ¯ describing the For the two right-hand frames in Fig. 1.3, the rotation matrix R orientation of the frame B relative to A reads ⎤ ⎡ b1 · a1 b1 · a2 b1 · a3 ¯ = ⎣ b2 · a1 b2 · a2 b2 · a3 ⎦ (1.8) R b3 · a1 b3 · a2 b3 · a3 The rotation matrix is also called the direction cosine matrix or coordinate ¯ in the above equation can be called the coordinate transformation transformation matrix. R matrix to B from A. Two important properties of the rotation matrix are that ¯R ¯ T = I and | R| ¯ = 1. Denote the set of rotation matrices as S O(3). RT R = R Denote the coordinate transformation matrix to A from B or the direction cosine matrix of A relative to B as R. One has ⎡ ⎤ a1 · b1 a1 · b2 a1 · b3 R = ⎣ a2 · b1 a2 · b2 a2 · b3 ⎦ (1.9) a3 · b1 a3 · b2 a3 · b3

Fig. 1.3 Two reference frames A and B

1.3 Attitude Representations

9

¯ hold. Consequently, the following intimate relationships between R and R ¯ R−1 = R T = R

(1.10)

¯T = R ¯ −1 = R R

(1.11)

In particular, C j (θi ) denotes the direction cosine matrix R of an elementary rotation about the jth axis of a frame with an angle θi . For example, ⎡

⎤ cos θ1 − sin θ1 0 C 3 (θ1 ) = ⎣ sin θ1 cos θ1 0 ⎦ 0 0 1

(1.12)

According to Euler’s formula [6], the rotation matrix R can be formulated as R = I 3 + (1 − cos θ R )n×R n×R + sin θ R n×R

(1.13)

where θ R and n R are rotation angle and rotation axis associated with the rotation described by R. Before moving on, some useful results on the rotation matrix, the maps × and ∨ are shown as follows. Lemma 1.3 [1] The following equations hold for x ∈ R3 and R ∈ S O(3). tr(Rx × ) = −x T (R − R T )∨  (x × R + R T x × )∨ = tr(R)I 3 − R x (Rx)× = Rx × R T

(1.14)

Lemma 1.4 [7] For the following attitude kinematics: ˙ = Rω× R

(1.15)

where R ∈ S O(3) and ω ∈ R3 , if R(0) ∈ S O(3), then R(t) ∈ S O(3) for ∀t ≥ 0.

1.3.2 Euler Angles The three-dimensional Euler angles can be obtained based on three successive bodyaxis rotations describing the orientation of the frame B relative to the frame A. The first rotation is about any axis. The second rotation is about either of the axes that are not used in the first rotation. The axes not used for the second rotation can be adopted as the axis of the third rotation. Hence, there are 12 sets of Euler angles available with various choices of rotation axes. Let’s consider the rotation with 3– 2–1 sequence shown in Fig. 1.4 as an example. The first rotation is about the axis b3

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1 Graph Theory and Attitude Representations

Fig. 1.4 Euler angles for the 3–2–1 sequence

with angle ψ. The second rotation is about the axis b 2 through angle θ and the last motion is the rotation of φ about b

1 . The rotation matrix R of such a rotation can be written as R =⎡ C 3 (ψ)C 2 (θ)C 1 (φ)⎤ ⎡ ⎤⎡ ⎤ cos ψ − sin ψ 0 cos θ 0 sin θ 1 0 0 1 0 ⎦ ⎣ 0 cos φ − sin φ ⎦ = ⎣ sin ψ cos ψ 0 ⎦ ⎣ 0 0 0 1 − sin θ 0 cos θ 0 sin φ cos φ ⎤ ⎡ cos ψ cos θ − sin ψ cos φ + cos ψ sin θ sin φ sin ψ sin φ + cos ψ sin θ cos φ ⎥ ⎢ ⎥ =⎢ ⎣ sin ψ cos θ cos ψ cos φ + sin ψ sin θ sin φ − cos ψ sin φ + sin ψ sin θ cos φ ⎦ − sin θ cos θ sin φ cos θ cos φ

(1.16) In general, compared with a rotation matrix with nine elements, three Euler angles can determine a unique orientation. However, it should be noted that for a given orientation, there is no unique set of Euler angles [8]. Also, Euler angles are kinematically singular since the transformation between their time derivative of angular velocity is not defined globally.

1.3.3 Modified Rodrigues Parameters In geometry, Euler’s rotation theorem states that, any rotation of a rigid body in three-dimensional space with a fixed point on the rigid body is equivalent to a single rotation about some axis that runs through the fixed point. That is, any rotation can be expressed as a single rotation about some axis. Such rotation axis is called the Euler axis. According to the Rodriguez formula, the rotation matrix R can be written as

1.3 Attitude Representations

11

R = I + v × sin θ + 2(v × )2 sin2

θ 2

(1.17)

where v is the vector along the rotation axis and θ is the rotation angle. The vector of Rodriguez parameters (sometimes called the Gibbs vector) is defined as γ = v tan

θ 2

(1.18)

Due to the introduction of tan θ2 , Rodriguez parameters have a singularity at the rotations of ±180 ◦ . Modified Rodrigues parameters (MRPs) can be expressed in terms of Euler axis and rotation angle by [9] σ = v tan

θ = [σi−1 , σi−2 , σi−3 ]T 4

(1.19)

which has singularities at the rotations of ±360 ◦ . Therefore, the Rodriguez parameters and MRPs are geometrically singular and the continuous attitude control based on the Rodriguez parameters and MRPs can only be used for local attitude maneuvers. Essentially, the three-dimensional attitude representations, including Euler angles, Rodriguez parameters and MRPs, are embedded subsets of R3 .

1.3.4 Quaternions In a four-dimensional vector space, Euler parameters or quaternions have been proven as a very useful method in representing rotations without any singular points. Four elements of quaternions are defined as follows [10]. θ θ θ θ q1 = v1 sin , q2 = v2 sin , q3 = v3 sin , q4 = cos 2 2 2 2

(1.20)

It is clear that quaternion parametrization satisfies the constraint q12 + q22 + q32 + = 1, i.e., quaternions lie in a non-Euclidean unit sphere in four-dimensional space. ¯ T ∈ S3 , where qˆ = [q1 , q2 , q3 ]T and Mathematically, q = [q1 , q2 , q3 , q4 ]T = [qˆ T , q] q¯ = q4 . With quaternions, two individual rotations can be combined using a quaternion product. Table 1.1 presents the key properties of various attitude representations. As shown in Table 1.1, three-dimensional attitude parametrizations cannot describe the rotation in 3-dimensional space globally. The mapping from the space S3 of quaternions to S O(3) is not unique. In particular, each attitude in S O(3) is represented by a pair of antipodal quaternions. If the quanternion-based controllers are not carefully designed, the unwinding phenomenon may happen [11]. Only a rotation matrix can describe the rotation in 3-dimensional space globally and uniquely, but it has nine elements with the constraints R T R = R R T = I and |R| = 1. q42

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1.3.5 Kinematic Differential Equations This section aims to present the angular velocity in terms of different attitude representations. Since this book focuses on networked spacecraft, the subscript “i” will be added to the symbols to show that they are associated with the ith spacecraft. Based on the rotation matrix, the kinematics of the ith spacecraft can be described as [1] ˙ i = Ri ωi× (1.21) R where Ri ∈ S O(3) is the rotation matrix from the body frame to the inertial frame and ωi = [ωi−1 ωi−2 ωi−3 ]T ∈ R3 is the angular velocity. The kinematics equation of a rigid body based on MRPs is written as σ˙ i = G i (σ i )ωi where G i (σ i ) = 21 ( 1−σ2i

T

σi

1+σ iT σ i

(1.22)

I 3 + s(σ i ) + σ i σ iT ) ∈ R3×3 . The matrix G i (σ i ) satisfies

G i (σ i )G i (σ i )T = ( 4 )2 I 3 . Based on the Euler angles for the 3–2–1 sequence shown in Sect. 1.3.2, the angular velocity can be expressed as ωi = Aiω i (1.23) where i = [φi , θi , ψi ]T is the vector of the Euler angles and ⎡

⎤ 1 0 − sin θi Aiω = ⎣ 0 cos φi sin φ cos θ ⎦ 0 − sin φ cos φ cos θ

(1.24)

Based on the quaternion q i = [qˆ iT , q¯i ]T , the kinematics equation of the ith spacecraft reads q˙ˆ i = 21 (qˆ i× + q¯i I 3 )ωi (1.25) q˙¯ i = − 21 qˆ iT ωi

Table 1.1 Key properties of popular attitude parametrizations [11] Attitude representation Global? Unique? Euler angles Rodriguez parameters MRPs Quaternions Rotation matrix

No No No Yes Yes

No No No No Yes

1.5 Dynamics Equation of Flexible Spacecraft

13

1.4 Dynamics Equation of Rigid Spacecraft In the body frame of the ith spacecraft, the angular momentum of the body reads H 0 = Jω

(1.26)

where J∈ R3×3 is the inertia matrix. The relationship between the inertial time derivative of H 0 and the external torque u about the center of mass is ˙ 0 = J ω˙ + ω × Jω = u H

(1.27)

The above equation is known as Euler’s equation. It describes the rotational dynamics of the rigid body. Therefore, for the ith rigid spacecraft, its dynamics equation reads J ω˙ i + ωi × Jωi = ui (1.28)

1.5 Dynamics Equation of Flexible Spacecraft According to Refs. [12, 13], the dynamic equations of the i-th flexible spacecraft can be expressed as 

J i ω˙ i + δ iT η¨ i = −s(ωi )( J i ωi + δ iT η˙ i ) + ui η¨ i + C ni η˙ i + K ni ηi = −δ i ω˙ i

(1.29)

where J i ∈ R3×3 is the total inertia matrix, ηi ∈ Rn is the modal coordinate vector, ui == [u i−1 , u i−2 , u i−3 ]T ∈ R3 is the control torque, δ i ∈ Rn×3 is the coupling matrix between the attitude motion and the flexible vibration, C ni = diag{2ζ j ωn j }∈ Rn×n is the damping matrix and K ni = diag{ωn2 j }∈ Rn×n is the stiffness matrix. Here, ωn j and ζ j is the j-th order natural frequency and damping ratio, respectively. In this study, without loss of generality, only the first n elastic modes are considered, i.e., ηi = [ηi−1 , ηi−2 , · · · ηi−n ]T . From Eq. (1.29), one can find that the attitude movement and flexible vibration are coupled, i.e., the attitude maneuver with non-zero acceleration will cause the vibration of flexible appendages and the flexible vibration will result in some disturbances acting on the attitude motion. Hence, in the following consensus law design for the networked spacecraft, both the attitude consensus and vibration suppression should be considered.

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References 1. Lee, T.: Exponential stability of an attitude tracking control system on S O(3) for large-angle rotational maneuvers. Syst. Control Lett. 61(1), 231–237 (2012) 2. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (2012) 3. Lewis, F.L., Zhang, H., Hengster-Movric, K., Das, A.: Cooperative Control of Multi-agent Systems: Optimal and Adaptive Design Approaches. Springer Science & Business Media (2013) 4. Du, H., Chen, M.Z., Wen, G.: Leader-following attitude consensus for spacecraft formation with rigid and flexible spacecraft. J. Guid. Control. Dyn. 39(4), 944–951 (2016) 5. Hu, J., Hong, Y.: Leader-following coordination of multi-agent systems with coupling time delays. Phys. A 374(2), 853–863 (2007) 6. Shuster, M.D., et al.: A survey of attitude representations. Navigation 8(9), 439–517 (1993) 7. Zou, Y., Meng, Z.: Velocity-free leader-follower cooperative attitude tracking of multiple rigid bodies on so(3). IEEE Trans. Cybern. 49(12), 4078–4089 (2019) 8. Wie, B.: Space vehicle dynamics and control. AIAA (1998) 9. Ye, D., Zou, A.M., Sun, Z.: Predefined-time predefined-bounded attitude tracking control for rigid spacecraft. IEEE Trans. Aerosp. Electron. Syst. (2022) (in press) 10. Xiao, Y., De Ruiter, A., Ye, D., Sun, Z.: Adaptive fault-tolerant attitude tracking control for flexible spacecraft with guaranteed performance bounds. IEEE Trans. Aerosp. Electron. Syst. (2022) (in press) 11. Chaturvedi, N.A., Sanyal, A.K., McClamroch, N.H.: Rigid-body attitude control. IEEE Control Syst. Mag. 31(3), 30–51 (2011) 12. Di Gennaro, S.: Passive attitude control of flexible spacecraft from quaternion measurements. J. Optim. Theory Appl. 116(1), 41–60 (2003) 13. Hu, Q., Xiao, B.: Intelligent proportional-derivative control for flexible spacecraft attitude stabilization with unknown input saturation. Aerosp. Sci. Technol. 23(1), 63–74 (2012)

Chapter 2

Literature Review

Abstract In the past two decades, spacecraft attitude coordination has gained significant developments. This chapter aims to present a survey of recent research on the spacecraft attitude consensus problem, paying particular attention to the papers published in major aerospace, dynamics, automation and robotics journals.

2.1 Background Multi-Agent Systems (MAS) are complex systems composed of multiple interacting autonomous agents with only local knowledge and limited abilities to achieve a global objective. An agent in MAS means an intelligent system capable of autonomous action. Compared with a single monolithic machine to perform some complex missions, a group of agents can complete the same or even more complicated missions in a cooperative manner with lower cost and higher robustness. Some examples of swarms in the natural world are given in Fig. 2.1. MAS control theory has gained significant theoretical and practical developments with plenty of research papers in this area. Survey papers by Cao et al. [1], Wang et al. [2], Ding et al. [3], Yu et al. [4], Abdulghafor et al. [5] and Zuo et al. [6] provided excellent overviews of multi-agent systems from different perspectives. As shown in Fig. 2.1, in space engineering, the collaboration of multiple spacecraft has many potential applications, such as high-resolution SAR interferometry [7] and distributed Earth observation missions [8], terrestrial planet finders [9] and on-orbit assembly [10]. As mentioned in Refs. [11, 12], multiple small satellites can deliver a comparable or greater mission capability than a monolithic satellite and significant attention has been attracted by spacecraft formation flying. Bandyopadhyay et al. reviewed 39 multisatellite missions from the classification of their mission type and status, the number of satellites, the lead institution, and funding source [11]. They concluded that constellation missions based on small satellites have matured and four or more small satellites may be used in the future formation flying missions. For spacecraft formation flying missions, a key challenge is the dynamics analysis and control algorithms design. The early works on spacecraft formation flying mainly focused on the modeling approaches and controller design of the relative © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_2

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Fig. 2.1 Swarms in natural world and space engineering: a flocks of birds, b a fish swarm in the form of a ball, c interferometry, d terrestrial planet finder, e GPS satellite network and f on-orbit assembly

translational motion between spacecraft. Sullivan et al. surveyed the relative motion dynamics models and assessed the choice of relative motion state parameterization, derivation methodology, fundamental assumptions on reference orbit eccentricity and interspacecraft separation, and the ability to incorporate perturbations into analytical solutions [13]. Kristiansen and Nicklasson summarized the modeling and control methods for spacecraft formation flying missions and presented five leader-follower tracking control approaches [12]. Essentially, the dynamics model of translational

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motion of a spacecraft is a linear system subject to complicated environmental forces. However, the configuration space of 3-D rotation is the special Lie group S O(3), which implies algebraic subtraction cannot be used to define the attitude tracking error on S O(3). Usually, some rotation parameterizations, such as Euler angles, Modified Rodrigues parameters (MRPs) and unit quaternions, are used to represent spacecraft attitude. The resulting attitude kinetics and dynamics are strongly nonlinear and the linear control method for MAS cannot be applied directly. Hence, the attitude consensus control of multiple spacecraft is a fairly challenging problem, which has attracted a great deal of interest in the past fifteen years. For the distributed attitude consensus problem, Ma et al. presented a survey with different system models, different communication topologies, different control goals and different techniques in 2015 [14]. Hence, this chapter aims to present an overview of the recent works on attitude consensus including the centralized, decentralized and distributed cases.

2.2 Attitude Consensus Under Various Communication Graphs It is important to find a control protocol to drive the attitude and angular velocity to the same value to achieve the attitude consensus. Similar to the classical linear multiagent systems, the main difference between networked spacecraft and a single one lies in the communication among its members. The structure and properties of the communication graph will have a great influence on the system’s performance. With different communication typologies, the attitude consensus mission can be classified into centralized, decentralized and distributed ones.

2.2.1 Centralized Attitude Consensus As shown in Fig. 2.2a, there is a central node in the centralized case, which implies that the control action of each node is taken based on the information stream from the central node. Such centralized control strategies are easier to install and implement. Note that the central node can be a practical powerful spacecraft or a ground station. Essentially, the centralized case can be considered as the attitude tracking of multiple individual rigid bodies, i.e., the controller for each follower node can be designed separately. Hence, the research community in the multi-agent field did not pay too much attention to this case because it is quite straightforward to use attitude control theory developed for a spacecraft to solve such a centralized attitude consensus problem. To solve the formation control of multiple spacecraft, Ren and Beard studied the centralized scheme of virtual structures [15]. In Ref. [16], Chen and Shan investigated the centralized attitude regulation of multiple flexible spacecraft with only the feedback of spacecraft attitude considering attitude constraints.

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2.2.2 Decentralized Attitude Consensus Since each member of the team will communicate to a central node in the centralized case, the fault or failure of the central satellite and the instability of communication edges may cause the consensus mission to fail. Hence, to improve the flexibility, reliability, and robustness of the closed-loop system, decentralized and distributed communication graphs shown in Fig. 2.2 can be adopted. In the decentralized case, each node knows the desired information via communication with the (virtual) leader and the members of the team can exchange the attitude tracking error information with their neighbors [17, 18]. This case is reasonable for the mission that drive the networked spacecraft to the pre-set target attitude synchronously, i.e., the spacecraft swarm is required to track the desired attitude while maintaining relative attitudes. Hence, some scholars have focused on this topic. For example, Zhou et al. addressed the cooperative attitude tracking problem in the presence of time-varying communication delays. They designed an attitude synchronization tracking controller and analyzed the sufficient delay-dependent stability condition based on a LyapunovKrasovskii function [17]. Zhang and Song designed consensus algorithms for the decentralized coordinated control problem with local information exchange in three cases: (i)with full state feedback; (ii)without velocity measurements; (iii)and with external disturbances and parametric uncertainty [19]. Xu et al. studied the decentralized finite-time adaptive control for the attitude consensus of multiple spacecraft under a directed graph with communication bandwidth constraints [20]. In the proposed control scheme, the attitude information exchange is activated only when the specified event is triggered.

2.2.3 Distributed Attitude Consensus However, in the distributed leader-follower consensus case in Fig. 2.2c, the desired attitude can be determined by the (virtual) leader whose information can only flow to partial nodes in the graph. The leaderless communication topology shown in Fig. 2.2d can also be adopted in the distributed consensus case, where the final consensus attitude may be determined by the communication graph, the initial conditions and the designed control strategies. In these distributed cases, the powerful central station is not necessary and all control commands are made based on a distributed protocol. In recent years, the distributed consensus control of multiple spacecraft has attracted almost all attention in the field of attitude consensus because of its wide practical applications and advantages, such as less system requirement and higher robustness. Note that in some works, the meanings of “distributed” and “decentralized” are mixed up. However, in this book, the word “decentralized” is used to represent the cases where each node can access the (virtual) leader’s information and exchange the error information between its own state and the desired state with its neighbors. If only partial nodes have access to the leader or if there is no leader at all in the graph, i.e.,

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Fig. 2.2 Attitude consensus strategies

some nodes have to try to reach the desired state only based on the communication with neighboring nodes, such cases are called “distributed” ones. The studies on attitude consensus of a group of spacecraft mainly focus on the consensus algorithm design, convergence analysis and its application prospects. The controller convergence heavily depends on the communication graph structure. VanDyke and Hall considered the attitude consensus in a spacecraft formation flying problem. They designed a class of distributed coordinated attitude consensus algorithms using behavior-based control theory based on relative attitude variables and presented convergence proof on the basis of a corollary of Barbalat’s Lemma [21]. Ren studied the distributed attitude consensus control of spacecraft swarm in the following cases: (i) in the presence of zero final angular velocities under an undirected communication graph; (ii) with nonzero final angular velocities under an undirected communication graph; (iii) under a directed communication topology [22]. Sarlette et al. addressed the autonomous attitude synchronization of multiple spacecraft on S O(3) under tree-like or all-to-all inter-satellite communication graphs [23]. Ren proposed a controller for the formation keeping and attitude alignment of multiple spacecraft with

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of the proposed approach [24]. Hayakawa and Mohanarajah proposed an asymptotic control law for spacecraft attitude consensus to drive the networked spacecraft to rotate around a fixed common rotational axis [25]. Septanto et al. designed two attitude alignment control methods based on a consensus algorithm and presented simulation results without theoretical proof [26]. Zhang and Demetriou considered the temporal adjustment of the consensus weights in the attitude synchronization via coupling terms from communication graphs. They assumed that each spacecraft in the team has identical dynamics but with unknown external disturbances and unknown inertia parameters and introduced the adaptation of the consensus weights to improve spacecraft synchronization results. The proposed adaptive control method will update both the parameter uncertainties and consensus penalty terms [27]. Mehrabian and Khorasani proposed two distributed control schemes for the attitude synchronization and tracking of multiple heterogeneous spacecraft. They considered both cases with and without angular velocity feedback [28]. Thunberg et al. presented several control laws for the consensus problem on S E(3) with directed and switching communication graphs. The first one is designed on a kinematic level. The second controller is developed on the dynamic level. They showed that the same type of control laws can be designed for many popular local attitude representations of S O(3) [29]. Latifu et al. paid attention to the distributed finite-time attitude synchronization and tracking problem of a group of spacecraft. A full order sliding mode controller is constructed with a fast non-singular terminal sliding mode surface as the inner layer and terminal sliding mode surface as the outer layer [30]. For the attitude consensus of multiple spacecraft under undirected communication typologies, Zou and Fan designed a distributed fixed-time controller based on the backstepping technique, a distributed fixed-time observer, and a power integrator. The developed controller is nonsingular and can finish the attitude consensus task within a fixed time [31]. Butcher and Maadani developed nonlinear consensus control algorithms for N heterogeneous rigid bodies in the tangent bundles T S O(3) and T S E(3) associated with Lie groups S O(3) and S E(3) to stabilize the relative pose under a static communication graph [32]. For the distributed attitude tracking of multiple spacecraft, Maadani et al. considered the rotation-matrix-based attitude consensus control of multiple rigid bodies. First, they designed a controller on the kinematic level. Next, the controller is designed on the dynamic level in the framework of the tangent bundle T S O(3) associated with S O(3). Note that the control torque is designed based on the relative attitudes and angular velocities [33]. Liu et al. proposed two adaptive distributed control laws for the attitude synchronization and tracking problems of multiple spacecraft with uncertain inertial parameters under a switching communication network [34]. Liu et al. studied the finite-time attitude tracking control of multiple spacecraft. The networked spacecraft is controlled based on the relative information acquired by active measurement units. They synthesized a non-singular control torque based on a finite-time sliding mode surface [35]. Hong and Anderson considered the fixed-time distributed attitude tracking problem under a directed communication graph. They developed a fixed-time tracking protocol for each follower

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spacecraft based on a distributed fixed-time observer with heterogeneous coefficients [36]. Long et al. considered the distributed adaptive attitude synchronization of a group of spacecraft with event-triggered information exchange. By introducing a reference system for each spacecraft and a pseudo-velocity filter, they designed an event-based adaptive attitude synchronization strategy under a directed graph [37]. Crnki´c et al. studied the attitude coordination and consensus on S O(3) and S 3 . Distributed algorithms are designed using gradient descent methods. They discussed the main differences in the convergence of controllers on the two groups [38]. Roughly, the distributed attitude consensus problem can be classified into two types, i.e., leader-follower tracking and leaderless synchronization.

2.2.3.1

Leaderless Attitude Consensus

For leaderless cases, there are no leaders in the graph. The networked spacecraft are expected to reach agreement with inter-communication only. The final consensus value mainly depends on the graph structure and initial conditions. Abdessameud et al. focused on both the leaderless and leader-follower attitude consensus problem with communication delays in the absence of angular velocity measurements. First, they presented a solution to the leaderless and leader-follower attitude consensus under time-varying communication delays and undirected communication topologies. Then, the leaderless attitude consensus problem under directed graphs is solved based on an attitude synchronization scheme with constant communication delays [39]. For the leaderless consensus of multiple spacecraft on S O(3), Chen presented an algorithm to block some communication edges of a connected undirected graph to get an undirected tree graph. Based on the distributed observer designed under the undirected tree graph, an adaptive attitude synchronization controller working under a general connected undirected graph is developed [40]. Li and Duan developed two distributed adaptive controllers for the distributed attitude consensus problems with unknown inertia matrices. Both the leaderless attitude synchronization under the communication topologies having a directed spanning tree and the distributed tracking problem with a virtual leader having a directed path to all other spacecraft are discussed [41]. Rezaee and Abdollahi focused on the leaderless attitude consensus problem with stochastic communication links. They designed a magnitudeconstrained consensus control strategy guaranteeing almost sure attitude consensus of multiple spacecraft with stochastically failed communication links [42]. To solve the leaderless attitude synchronization, Chen and Shan discovered a set of observables to describe the nonlinear attitude dynamics on S O(3) using a finite-dimensional linear system approximately with the assumption of low angular velocities. Based on the well-studied linear multiagent system theory, the leaderless attitude synchronization problem was solved. Both simulations and experiments were presented to verify the effectiveness of the control theory [43].

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Leader-Follower Attitude Consensus

In the leader-follower tracking cases, the final consensus values are determined by the leaders. Note that in the team there may exist more than one leader. There are mainly two methods to solve leader-follower attitude consensus. One is to design the controller with the feedback of the attitude and angular velocity errors between the agent and its neighbors. The other one is based on the distributed observer embedded in each follower spacecraft to estimate the leader’s information. Some examples using the first method are as follows. Yang and Liu designed a distributed robust attitude tracking controller for multiple spacecraft with external disturbances and unmodelled dynamics based on the relative state measurements under an undirected communication graph with the assumption that at least one follower can access the leader’s information [44]. Zhu et al. focused on the attitude synchronization problem with time-varying communication delays under a directed graph having a spanning tree with the leader as the root. They proposed a distributed attitude consensus controller based on the second-order consensus algorithm and graph theory [45]. Zhu et al. focused on the distributed attitude regulation problem of multiple spacecraft with unknown time-varying uncertainties and state-dependent disturbances under communication graphs having a spanning tree with the leader as the root. They introduced a self-adjustment matrix to reduce the overshoot of the system [46]. Wang and Huang focused on the leader-follower attitude tracking problem of multiple rigid bodies under jointly connected switching networks. They designed a distributed attitude feedback control law and used the generalized Barbalat’s lemma to overcome the difficulty in the proof of the theorem [47]. Cui et al. focused on the distributed finite-time attitude tracking control problem of multiple spacecraft without the feedback of the angular velocity. Two finite-time adaptive neural network observers are used to estimate the unavailable angular velocity of each follower spacecraft and the angular velocity of the dynamic leader in a distributed manner. A continuous finitetime control strategy is designed with these two observers and the power integrator technique [48]. Wang and Huang focused on the leader-follower attitude consensus of multiple spacecraft subject to disturbances under jointly connected switching graphs. They integrated the adaptive distributed observer technique and the adaptive learning control technique to design a controller that can handle bounded disturbances and the case under jointly connected switching graphs [49]. Some works based on a distributed estimator of leader’s states are as follows. Zhou et al. developed a nonlinear controller using the finite-time control technique for the distributed attitude synchronization and stabilization problem based on a finite time sliding-mode estimator. Under the designed controller, the networked spacecraft will converge to the attitude of the leader with zero final angular velocity in a finite time [50]. Ma et al. introduced a sliding-mode estimator for each follower spacecraft to avoid the necessity of the angular acceleration information and developed a distributed finite-time attitude tracking control protocol for the networked spacecraft with a dynamic leader under directed communication typologies [51]. Zou et al. developed a distributed finite-time velocity-free attitude controller for multiple

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spacecraft under undirected communication typologies. Two finite-time observers are introduced for each spacecraft to estimate the spacecraft’s angular velocity and the angular acceleration of the virtual leader. Based on the homogeneous method, a continuously distributed attitude control law is designed and its semi-global finitetime stability is shown using a rigorous proof [52]. Gui and de Ruiter extended the quaternion-based hybrid homogeneous controllers designed for a single spacecraft to the multi-agent systems based on a distributed finite-time observer of the leader’s information. They proved that a global finite-time attitude consensus can be achieved with the feedback of full state or attitude only [53]. Wang et al. designed a distributed attitude tracking controller with the compensation of parametric uncertainties and unknown disturbances based on an improved distributed observer to estimate the leader’s information for multiple spacecraft with time-varying communication delays. The Lyapunov–Krasovskii functional approach was used to show the controller convergence [54]. Peng and Geng developed three types of attitude consensus control laws on S O(3) with full state, biased angular velocity, and external disturbance combined with biased angular velocity, respectively, based on an intrinsically distributed observer to estimate the leader’s trajectory for each follower [55]. Huang and Meng addressed the leader-follower attitude tracking problem of multiple rigid bodies. They designed a quaternion-based nonlinear hybrid distributed observer that always evolves on the 3-D unit sphere. A distributed attitude tracking control law combining adaptive technique and hybrid theory is developed based on such an observer [56]. Zhu et al. considered the distributed global attitude consensus for multiple spacecraft with disturbance based on a fixed-time observer to estimate the leader’s states and a filter system to achieve velocity-free control [57].

2.3 Attitude Consensus with Various Attitude Representations To solve the attitude consensus problem, various attitude representations have been adopted. Different attitude parameterization methods usually result in different attitude errors between neighbors, which implied different difficulties in the consensus controller design. This section presents an overview of the studies on this topic according to classifications of the attitude representations.

2.3.1 Euler-Angles-Based Attitude Consensus The algebraic subtraction between the Euler angles of two rigid bodies can only be used to describe the attitude error under small rotation cases. Since the scholars in the field of the attitude consensus of multiple spacecraft usually concern themselves with a large rotation of spacecraft, there are not too many results available for the attitude

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consensus based on Euler angles. Chen et al. considered the flexible spacecraft as an underactuated Lagrange system and designed a distributed adaptive control strategy under a leader-follower architecture with an extended state observer to estimate the generalized accelerations [58]. Zhang et al. studied the attitude cooperative eventtriggered control problem of multiple spacecraft under undirected communication. In the designed controller, the event triggered conditions are irrelevant to the system states. They also proved that there exists a lower bound for the update interval to avoid the Zeno phenomenon [59]. In [18], even the decentralized adaptive sliding-mode controller was developed using unit quaternions, they used Euler angles converted from unit quaternions in simulation examples to present the results intuitively. However, in the field of formation flying of multiple aircraft, the Euler angles are widely used, especially for works based on the linearized model [60–63]. For example, Wang et al. used four linear subsystems to describe the quadrotor dynamics and designed distributed leadless consensus laws under fixed and switching topologies based on the position errors and attitude errors described using Euler angles [61].

2.3.2 MRPs-Based Attitude Consensus With MRPs, the Lie group S O(3) is projected to R3 . Since MRPs can express the attitude of a rigid body using three elements only, many works have tried to solve the attitude consensus problem based on MRPs. For example, Dimarogonas et al. assumed that only a certain subgroup of rigid bodies can access the leaders’ orientations and designed a feedback control law based on the Laplacian matrix of the communication graph. They showed that the system convergence depends on the connectivity of the communication graph [64]. Meng et al. proposed a modelindependent algorithm to drive the followers to the stationary convex hull determined by the stationary leaders and designed a non-singular finite-time sliding controller based on a distributed sliding-mode estimator for the case with multiple dynamic leaders [65]. Bhatia and Subbarao solved attitude synchronization and tracking problems for a group of rigid bodies by introducing coupled attitude dynamics based on the controller under a prior communication topology and imposing additional constraint functions to achieve specific consensus and formation objectives [66]. Du et al. investigated the distributed finite-time attitude control for a group of spacecraft under a leader-follower architecture with a rigorous proof [67]. Ma et al. focused on the distributed leader-follower attitude tracking problem of multiple spacecraft under jointly connected switching communication topologies. They considered both the cases with a static and dynamic leader. Based on the distributed observer for each follower, they designed controllers to drive the follower to track the leader’s attitude [68]. Cheng et al. paid attention to the finite-time attitude regulation of multiple spacecraft under a leader-follower communication graph. They developed a bounded finite-time attitude controller based on a distributed finite-time observer designed for each spacecraft to estimate the leader’s attitude [69]. To solve the distributed prescribed-time attitude tracking for multiple spacecraft with bounded external

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disturbances, Xu et al. designed a distributed prescribed-time observer to estimate the leader’s information, based on which a prescribed-time attitude cooperative controller is developed [70]. Cong et al. presented a time-varying sliding mode control algorithm together with a behavioral consensus algorithm for attitude synchronization considering both parametric uncertainty and external disturbance. Three cases are simulated to illustrate the effectiveness of the proposed strategy [71]. Nair and Subbarao used a time-domain approach and a frequency domain approach to analyze the delay-dependent stability criteria for the attitude consensus of multiple spacecraft with bounded, time-varying communication delays under a strongly connected topology [72]. Since MRPs have singularities for the rotations of ±360◦ , the attitude consensus controllers based on MRPs work well for the missions with rotations within ±360◦ . To achieve the attitude consensus with larger rotations, the following quaternionsbased control methods are developed.

2.3.3 Quanternions-Based Attitude Consensus Quaternions can represent the spacecraft’s attitude using four elements without singular points. To avoid the problems caused by the singularity of three-dimensional attitude representation, lots of attentions have been attracted to the quaternion-based attitude consensus. For example, Abdessameud and Tayebi introduced an auxiliary dynamical system aiming to estimate angular velocity and designed the coordinated attitude controller without the requirement of the absolute angular velocities and relative angular velocities for the distributed attitude tracking and distributed attitude synchronization problems [73]. Due to the nonlinearity of the dynamics and kinematics, difficulty in guaranteed stability and avoiding unwinding phenomenon for quaternion-based attitude dynamics, Septanto et al. designed a continuous quaternion-based controller for the attitude consensus of multiple rigid bodies [74]. Cai and Huang considered a distributed unit quaternion-based attitude control for the leader-follower attitude consensus problem of multiple rigid bodies. A distributed observer was designed to estimate the leader’s information. An auxiliary system was introduced to compensate for the unavailability of the angular velocity. The controller is developed based on the certainty equivalence principle [75]. They also designed an asymptotic reference tracking controller by combining the distributed observer approach with an internal model and adaptive control techniques to solve the leader-follower attitude consensus problem subject to sinusoidal disturbances with arbitrarily unknown amplitudes, initial phases, frequencies, and constant biases [76]. Zong et al. developed two finite-time attitude synchronization controllers based on continuous adaptive technology and an adaptive disturbance observer for spacecraft formation subject to inertia uncertainties and external disturbances without the feedback of angular velocity [77]. Lu and Liu addressed the leader-follower attitude consensus of multiple rigid spacecraft under a directed and switching communication network. The robustness and reliability of the attitude consensus can be

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greatly improved since the designed controller works under a switching communication graphs [78]. Zhang et al. investigated the attitude consensus problem of multiple spacecraft under undirected communication graphs with attitude and angular velocity measurement errors. They considered the measurement bias of attitude and angular velocity as mismatched and matched disturbances, respectively. The backstepping control method and adaptive control technique were used to solve such consensus problem [79]. Wang et al. addressed the attitude synchronization of multiple rigid bodies using an event-based approach. To estimate the leader’s information without continuous information exchange, they proposed a distributed event-triggered observer and synthesized a distributed attitude synchronization controller based on intermittent communication [80].

2.3.4 Rotation-Matrix-Based Attitude Consensus The minimal attitude representations cannot describe the rotation of a rigid body globally. Even quaternions can avoid singularities, they double-cover the rotation space of a rigid body, which implies undesirable unwinding phenomena may occur under continuous control laws [40, 81]. Hence, to avoid the singularities associated with Euler angles and MRPs and the ambiguity associated with quaternions, various control strategies have been developed directly on S O(3) for a single or networked spacecraft. For example, Wang et al. represented the orientation kinematics on S O(3) and designed a rigid-body orientation controller with the angular velocity as a class of piecewise sinusoidal signals [82]. Berkane et al. presented two global exponential attitude tracking controllers with the feedback of only attitude information on S O(3). The first controller is designed based on a combined hybrid observer-hybrid controller structure. The second approach is the attitude tracking controller with global exponential stability based on an auxiliary dynamical system [83]. For the spacecraft attitude tracking of arbitrary continuously differentiable trajectories, Sanyal et al. proposed a inertia-free nonlinear control law. They showed that such a controller is almost globally stable and the unwinding phenomenon associated with continuous controllers based on quaternions can be avoided [84]. Similarly, for the networked spacecraft on S O(3), many studies are available, especially within the past five years. For example, Tan et al. focused on the robust finite-time attitude tracking control for multiple spacecraft on S O(3) subject to external disturbances. Based on the terminal sliding mode (TSM) technology and adaptive laws, a distributed finite-time attitude controller is designed. The singularity and unwinding phenomenon can be avoided due to the usage of the rotation matrix [85]. Zou and Meng used the Lie group S O(3) to represent the attitude of a rigid body and developed a leader-follower attitude tracking controller for a group of rigid bodies without the feedback of the absolute and relative angular velocity information. A distributed-observer-based controller design method is adopted [86]. For the attitude formation control of multiple spacecraft on the special orthogonal group S O(3) under undirected tree graphs, Shi et al. designed a model-free distributed controller with an adaptive term to enhance the disturbance rejection ability and an actor-critic neural

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network to compensate for the system uncertainties [87]. Sharma and Kar focused on the distributed attitude consensus algorithm for multiple rigid bodies on the tangent bundle T S O(3) N under directed fixed communication topologies. They designed an almost globally asymptotic controller with a disturbance observer to compensate for constant or slow-varying external disturbances [88]. For small satellites with internal rotating-mass actuators that cannot drive the satellite to achieve complete rotations, Chavan et al. developed two attitude-feedback consensus controllers that are continuous but only piecewise-continuously differentiable sinusoids [89]. Maadani and Butcher focused on the full pose consensus control of multiple heterogeneous rigid bodies in Banach space associated with the tangent bundle T S E(3) N under an undirected communication graph. They assumed that the spacecraft communicated with homogenous or heterogeneous delays. Morse-Lyapunov-Krasovskii approach is used to prove the controller stability [90].

2.3.5 Attitude Consensus Based on Other Attitude Representations Except for the popular attitude representations, some works on attitude consensus are conducted based on other attitude representations, such as exponential coordinates. Note that the exponential coordinates for attitude are obtained using the logarithmic map, i.e., × = log(R) ∈ so(3). Thunberg et al. considered the attitude synchronization problem under various conditions. They studied both cases with absolute attitude measurements and with relative rotation information. Two intuitive controllers are proposed based on axis-angle attitude representations [91]. Nazari et al. considered the distributed consensus control of multiple spacecraft on the Lie group S E(3) with constant communication time delay. They converted the consensus problem to a local stabilization problem of the error dynamics associated with the Lie algebra se(3). The control convergence is proven using infinite-dimensional Floquet theory [92]. Essentially, exponential coordinates are a three-dimensional representation. Hence, they cannot be used to represent the rotation in 3D space globally, i.e., the attitude controllers based on exponential coordinates suffer from the singularity problem.

2.4 Attitude Consensus Under Complicated Conditions 2.4.1 Attitude Consensus with Input Saturation In practice, the actuator may not output the desired controller command due to physical constraints. Input saturation is a common class of nonlinear system constraints. For some systems, if the saturation nonlinearity is ignored, the system performance

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may degenerate, such as the increasing overshoot and longer settling time. Hence, the application of the input-saturated system to the networked spacecraft has gained more and more attention in recent years. Zou and Kumar studied the attitude coordination control problem in the case that only partial members can access the reference attitude and designed a distributed quaternion-based attitude tracking scheme considering control input saturation based on the sliding-mode observer, Chebyshev neural networks, smooth projection algorithm, and robust control technique. They show that the developed controller can finish the distributed tracking task using graph theory and a Lyapunov-based approach [93]. Zong and Shao considered the attitude synchronization of multiple spacecraft with external disturbances, inertia uncertainties and actuator saturation. First, they show that there is always a nonsingular terminal sliding mode controller that converges faster than a fast terminal sliding mode controller for the attitude control problem of concern. Then, they proposed a terminal sliding mode disturbance observer with some adaptive parameters. Some simulation results with or without actuator saturation are presented to testify the effectiveness of the designed decentralized attitude synchronization control laws [94]. Wang et al. investigated the quaternion-based attitude consensus problem of multiple rigid bodies under undirected graphs in the absence of relative angular measurements with actuator saturation. They considered the leaderless attitude synchronization problems with zero and non-zero final angular velocities [95]. Zhu and Guo designed a robust adaptive finite-time controller for the attitude tracking and synchronization of multiple spacecraft under leader-follower architecture with actuator saturation, unknown time-varying inertia and disturbance based on finite-time control, consensus algorithm and graph theory [96]. For the attitude consensus of multiple spacecraft with actuator saturation, inertia uncertainties and external disturbances, Gao et al. proposed a fast nonsingular terminal sliding mode controller with an adaptive update law and the saturation function based on the distributed finite-time observer [97]. Wang et al. designed a distributed attitude tracking controller with input saturation in the absence of angular velocity measurements with the help of the introduction of an auxiliary dynamical systems [98].

2.4.2 Attitude Consensus with Actuator Faults In the spacecraft attitude system, the general unpredictable actuator faults can be termed as: partial loss of effectiveness, lock-in-place and bias. Modern spacecraft controllers can be designed to handle some faults autonomously. The fundamentals of spacecraft fault-tolerant attitude control systems can be found in Ref. [99]. Some distributed fault-tolerant attitude controllers have been developed. Zhang et al. proposed a distributed attitude synchronization and tracking controller for multiple spacecraft based on an adaptive nonsingular fast terminal sliding mode surface. They showed that the proposed controller provided high precision finite-time convergence, robust to time-varying disturbances, uncertainties and accommodate actuator faults, limited inputs [100]. For the fault-tolerant attitude consensus problem, Zhu and Guo

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proposed an adaptive approach to reject the influence of external disturbances and inertia uncertainties to regulate all spacecraft to a common time-varying reference state [101]. Liu et al. investigated the attitude synchronization problem of multiple satellites with parametric perturbations, external disturbances and actuator faults. They designed a robust fault-tolerant attitude consensus controller based on the linear quadratic regulation method and the robust compensating theory [102]. Yu et al. considered the finite-time resilient control of the attitude consensus of multiple spacecraft. In order to estimate the leader’s information, they designed a distributed finite-time observer, whose resilience to communication link faults is proven by using the homogeneity property [103].

2.4.3 Attitude Consensus with State Constraints As stated in [104, 105], in some missions, some onboard instruments may be sensitive to certain celestial objects (e.g., the sun), or should point to some objects within a certain angle range to do some communication and observation. That is, there may be some attitude-forbidden zones during the attitude maneuver, such as, as shown in Fig. 2.3, the angle between the boresight vector of an on-board instrument and the vector pointing to a celestial object should be greater than a certain value. Okoloko and Kim used the quadratically constrained attitude control method to generate an optimal collision-free attitude trajectory for the attitude synchronization of multiple spacecraft with cone avoidance constraints [106]. Liu et al. focused on the attitude consensus problem of multiple spacecraft with external disturbance subject to convex attitude constraints. They adopted an adaptive law to estimate the disturbance and the attitude is forced to stay in the permission set by introducing an auxiliary system based on a logarithmic barrier potential function. Lyapunov approach and Barbalat’s Lemma are used to prove the stability of the closed-loop networked spacecraft system [107]. Li et al. introduced a distributed finite-time sliding-mode estimator to reconstruct the leader’s information accurately and solved the distributed attitude consensus problem of multiple spacecraft subject to attitude constraints using

Fig. 2.3 Attitude constraints [16]

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the integral barrier Lyapunov function technique and output feedback [108]. Also, in some cases, the angular velocity is required to be limited by a fixed value considering the measurement limitations of the inertial measurement unit. Zheng and Shen considered the attitude consensus of multiple spacecraft with actuator and angular velocity constraints based on the vector measurements. They proposed a vectormeasurement-based attitude synchronization controller using the barrier Lyapunov function and analyzed the undesired equilibria [109].

2.4.4 Experimental Results on Attitude Consensus In almost all studies on attitude consensus, only numerical simulation results are presented. Since attitude consensus is a topic inspired by practical missions, it is quite important to perform space flights to verify the designed control strategies. In the formation flying and proximity operation of networked spacecraft, some classical attitude consensus algorithms are verified [11, 12, 110, 111]. However, due to the expensive costs and time-consuming nature of the on-orbit experimental tests, ground experimental approaches are more feasible for the plenty of developed attitude consensus controllers. How to simulate the microgravity environment is the main difficulty for the ground experiments. The main methods to simulate microgravity conditions are drop tower [112, 113], parabolic flight [114], buoyancy tank testbed [115], air floating suspension [116] and planar air-bearing facilities [117] and spherical air-bearing systems [118]. Air-bearing techniques are the most popular and widely used for attitude control. For example, Chen and his colleagues developed the spacecraft simulators on the planar air bearing testbed shown in Fig. 2.4 and verified the proposed distributed PD controller for state consensus and the autonomous assembly controllers of multiple flexible spacecraft [119, 120]. Jung et al. adopted the State-Dependent Riccati Equation (SDRE) technique to solve the attitude synchronization problems and validated the proposed controller in real time based on a spherical-air-bearing-based Hardware-In-the-Loop simulator in Fig. 2.5 [121]. Based on the facilities in Fig. 2.5, Jung and his colleagues also verified the nonlinear control schemes based on the Lyapunov stability theorem under the situations: full-state feedback, full-state feedback with unknown inertia parameters, and output feedback without angular velocity measurements for spacecraft attitude synchronization [122]. Essentially, the spacecraft attitude motion in space can be considered as the attitude motion of a rigid body in a micro-gravity environment. Hence, except for the works based on classical test rigs for micro-gravity environments, some scholars have tried to demonstrate the effectiveness of the designed attitude consensus laws based on some rigid bodies with compensation for unbalanced gravity. For example, Chen and Shan used four AERO systems from Quanser Inc. under a helicopter configuration shown in Fig. 2.6 to verify the attitude consensus controller proposed in Ref. [123]. They also used the networked Quanser AERO systems to verify the Koopman-operator-based attitude synchronization controller developed in Ref. [43].

2.5 Attitude Consensus of Multiple Flexible Spacecraft

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Fig. 2.4 The spacecraft simulators in Ref. [120]

Fig. 2.5 The Hardware-In-the-Loop simulator in Ref. [121]

2.5 Attitude Consensus of Multiple Flexible Spacecraft In some missions, each or partial spacecraft may be equipped with some flexible appendages, such as solar panels and antennae. The attitude maneuver of rigid parts may cause the dynamic deformation of flexible appendages. In the space environment, structural damping is usually quite small. It may take a long time to damp out the excited flexible vibration. Hence, it is necessary to consider the attitude consensus problem of networked flexible spacecraft. Some works have been done to solve such a problem. For example, based on the backstepping technique and the neighbor-based controller design method, Du and Li developed a distributed attitude control scheme for the attitude synchronization of multiple flexible spacecraft. Cas-

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Fig. 2.6 The Quanser AERO systems adopted in Ref. [123]

caded system theory and graph theory are used to prove the controller convergence [124]. They also considered the attitude synchronization problem of multiple flexible spacecraft with communication delays. They designed a distributed asymptotic attitude control algorithm using the backstepping technique and the finite-time control method [125]. Du et al. focused on the leader-follower attitude consensus for a group of spacecraft consisting of rigid and flexible spacecraft. They assumed that the desired attitude is available only for a subset of group members. A distributed attitude controller is designed based on the backstepping technique and graph theory. Both the attitude consensus of the networked spacecraft and the vibration suppression of flexible spacecraft were achieved [126]. For the attitude synchronization and tracking problem of a group of flexible spacecraft under the leader-follower architecture, Wang et al. designed an adaptive sliding-mode controller without the measurement of the modal variable [127]. Chen and Chen considered the distributed leader-follower tracking problem of multiple planar flexible spacecraft under an undirected communication graph with uncertain parameters subject to various actuator and measurement limitations. They considered the following three scenarios: (i) the networked spacecraft are fully measured and driven, (ii) the rigid body is controlled with full state feedback and (iii) only the rigid part is measured and controllable [128]. Huang et al. developed three distributed consensus control laws based on Lyapunov’s stability and graph theories for the attitude consensus of flexible spacecraft formation with actuator failures and saturation constraints. The designed controllers did not depend on the modal variables of the flexible appendages [129]. Chen et al. investigated the distributed passivity-based control problem for the leader-follower attitude consensus of multiple under-actuated flexible spacecraft without structural damping with the attitude measurements only. They constructed two auxiliary EulerLagrange systems under undirected or directed communication graphs to synthesize the distributed controllers to finish the attitude tracking and vibration suppression simultaneously [130]. Wang et al. paid attention to the cooperative attitude tracking and vibration reduction of networked flexible spacecraft without modal variable measurement. They designed a distributed attitude synchronization controller with

2.6 Conclusions

33

a disturbance observer and a feedforward compensation scheme and the control law is further modified by integrating a disturbance observer and a feedforward compensating scheme [131]. Chen and Shan designed a finite-time observer for the follower spacecraft to estimate the leader’s information and developed a distributed adaptive fault-tolerant controller on S O(3) without modal variable measurement. For the stability of the closed-loop system, they adopted the separation principle between the finite-time observer and the proposed controller [132]. Chen and Shan focused on the distributed tracking problem of multiple flexible spacecraft on S O(3) under an undirected tree graph. Based on the assumption that partial loss of actuator effectiveness exists for each spacecraft, they developed a distributed fault-tolerant controller with the feedback of rigid bodies only [133]. Chen et al. used Partial Differential Equations (PDEs) to describe the dynamics of the spacecraft with an appendage and presented two distributed control methods for the attitude tracking of multiple flexible spacecraft without or with a PDE observer to estimate the deformation of the flexible appendage [58]. Chen and Shan solved both the centralized and distributed attitude regulation of a group of flexible spacecraft on S O(3) subject to attitude constraints. A continuously distributed attitude regulation controller is designed in the absence of the measurements of the angular velocities and modal coordinates and a repulsive potential function is introduced to deal with attitude constraints [16]. He et al. designed attitude consensus law for multiple heterogeneous flexible spacecraft. They used high-gain-based controller design technology and an internal model approach to compensate for the uncertainties and external disturbances, respectively [134]. Chen and Shan designed distributed adaptive controllers for the attitude tracking and synchronization on S O(3) of a group of flexible spacecraft under a directed graph. They introduced a distributed observer to estimate the leader’s information in a finite time. An adaptive controller is proposed to achieve the attitude tracking and synchronization and the suppression of flexible vibrations [123].

2.6 Conclusions Attitude coordination is one key and challenging problem for spacecraft formation flying missions. Since the configuration of 3D rotation is described by the special Lie group S O(3) and the attitude dynamics is strongly nonlinear, it is nontrivial to extend the control algorithms for classical linear multi-agent systems to the attitude consensus problem. Hence, significant attention has been attracted to the spacecraft attitude consensus in the past fifteen years, especially for the distributed cases. The attitude consensus of simple and classical spacecraft has been well studied. Some practical issues, such as the actuator dynamics and the fuel sloshing, may degrade the controller performance. Furthermore, in some future space missions, some spacecraft may carry some long flexible appendages. Some studies have considered this topic with a simplified dynamics model of flexible spacecraft, which is built using the assumed modes method. Such models can only describe flexible spacecraft with low

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velocity and small deformation. In the future, it is necessary to address the attitude consensus based on more accurate dynamics models.

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111. McCamish, S.B., Romano, M., Nolet, S., Edwards, C.M., Miller, D.W.: Flight testing of multiple-spacecraft control on spheres during close-proximity operations. J. Spacecr. Rocket. 46(6), 1202–1213 (2009) 112. Liu, T., Wu, Q., Sun, B., Han, F.: Microgravity level measurement of the Beijing drop tower using a sensitive accelerometer. Sci. Rep. 6(1), 1–9 (2016) 113. Selig, H., Dittus, H., Lämmerzahl, C.: Drop tower microgravity improvement towards the nano-g level for the microscope payload tests. Microgravity Sci. Technol. 22(4), 539–549 (2010) 114. Sawada, H., Ui, K., Mori, M., Yamamoto, H., Hayashi, R., Matunaga, S., Ohkami, Y.: Microgravity experiment of a space robotic arm using parabolic flight. Adv. Robot. 18(3), 247–267 (2004) 115. Sun, C., Chen, S., Yuan, J., Zhu, Z.: A six-DOF buoyancy tank microgravity test bed with active drag compensation. Microgravity Sci. Technol. 29(5), 391–402 (2017) 116. Sun, Z., Sun, X., Zhou, Q., Zou, B., Liu, G., Dong, H.: An air floating suspension microgravity simulator for multi specification of spaceborne SAR. In: 2021 2nd China International SAR Symposium (CISS), pp. 1–7. IEEE (2021) 117. Rybus, T., Seweryn, K.: Planar air-bearing microgravity simulators: review of applications, existing solutions and design parameters. Acta Astronaut. 120, 239–259 (2016) 118. Ustrzycki, T., Lee, R., Chesser, H.: Spherical air bearing attitude control simulator for nanosatellites. In: AIAA Modeling and Simulation Technologies Conference, p. 6272 (2011) 119. Chen, T.: State consensus and autonomous assembly for a team of flexible spacecraft. Ph.D. thesis, Nanjing University of Aeronautics and Astronautics (2017) 120. Hu, H., Chen, T., Wen, H., Jin, D.: Autonomous assembly of a team of flexible spacecraft. In: Proceedings of the 9th European Nonlinear Dynamics Conference, pp. 25–30 (2017) 121. Jung, J., Park, S.Y., Kim, S.W., Eun, Y., Chang, Y.K.: Hardware-in-the-loop simulations of spacecraft attitude synchronization using the state-dependent Riccati equation technique. Adv. Space Res. 51(3), 434–449 (2013) 122. Jung, J., Park, S.Y., Eun, Y., Kim, S.W., Park, C.: Hardware simulations of spacecraft attitude synchronization using Lyapunov-based controllers. Int. J. Aeronaut. Space Sci. 19(1), 120– 138 (2018) 123. Chen, T., Shan, J.: Distributed spacecraft attitude tracking and synchronization under directed graphs. Aerosp. Sci. Technol. 109, 106432 (2021) 124. Du, H., Li, S.: Attitude synchronization control for a group of flexible spacecraft. Automatica 50(2), 646–651 (2014) 125. Du, H., Li, S.: Attitude synchronization for flexible spacecraft with communication delays. IEEE Trans. Autom. Control 61(11), 3625–3630 (2016) 126. Du, H., Chen, M.Z., Wen, G.: Leader-following attitude consensus for spacecraft formation with rigid and flexible spacecraft. J. Guid. Control. Dyn. 39(4), 944–951 (2016) 127. Wang, Q., Duan, Z., Lv, Y.: Distributed attitude synchronization control for multiple flexible spacecraft without modal variable measurement. Int. J. Robust Nonlinear Control 28(10), 3435–3453 (2018) 128. Chen, T., Chen, G.: Distributed adaptive tracking control of multiple flexible spacecraft under various actuator and measurement limitations. Nonlinear Dyn. 91(3), 1571–1586 (2018) 129. Huang, D., Wang, Q., Duan, Z.: Distributed attitude control for multiple flexible spacecraft under actuator failures and saturation. Nonlinear Dyn. 88(1), 529–546 (2017) 130. Chen, T., Shan, J., Wen, H.: Distributed passivity-based control for multiple flexible spacecraft with attitude-only measurements. Aerosp. Sci. Technol. 94, 105408 (2019) 131. Wang, L., Song, Z., Su, H., Zou, Y., Tang, X.: Distributed cooperative attitude tracking for multiple flexible spacecraft under a directed graph. In: 2019 Chinese Control Conference (CCC), pp. 6082–6087. IEEE (2019) 132. Chen, T., Shan, J.: Distributed adaptive fault-tolerant attitude tracking of multiple flexible spacecraft on S O(3). Nonlinear Dyn. 95(3), 1827–1839 (2019) 133. Chen, T., Shan, J.: Rotation-matrix-based attitude tracking for multiple flexible spacecraft with actuator faults. J. Guid. Control. Dyn. 42(1), 181–188 (2019)

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134. He, J., Wang, J., Li, Q., Liu, C.: Attitude consensus of heterogeneous uncertain flexible spacecrafts with disturbance compensation. In: 2020 12th International Conference on Intelligent Human-Machine Systems and Cybernetics (IHMSC), vol. 2, pp. 126–129. IEEE (2020)

Part II

Leader-Follower Attitude Consensus of Networked Flexible Spacecraft

Cooperative attitude tracking of multiple spacecraft has attracted significant attention due to its wide applications such as interferometry, Earth monitoring and on-orbit service.According to the communication topology structure, the coordinated attitude control can be classified into three types, namely, centralized, decentralized and distributed coordination. In the centralized and decentralized cases it is required that each follower spacecraft knows the desired attitude. The loss of the leader can cause mission failure. Hence, to improve robustness, some missions prefer to determine control actions in a distributed way. The distributed attitude control has advantages such as less communication requirement and greater efficiency. Consequently, many investigators have designed distributed attitude controllers for multiple spacecraft. In some missions, spacecraft tend towards large-scale and light-weight and may carry some flexible appendages, such as manipulators, antennae and solar panels. The attitude motion will cause the vibration of flexible appendages and the flexible vibration may degrade the attitude control accuracy. Hence, it is necessary to consider the spacecraft as rigid-flexible one, instead of rigid one. The dynamics and control problem is much more difficult and challenging than that of rigid ones due to rigid flexible dynamics. Hence, in the following five chapters, we will present some results on the attitude consensus of multiple flexible spacecraft.

Chapter 3

Distributed Passivity-Based Control with Attitude-Only Measurements

Abstract This chapter focuses on the leader-follower attitude consensus of multiple under-actuated flexible spacecraft without system damping based on attitude measurements only. By constructing two different kinds of auxiliary Euler-Lagrange systems, passivity-based controllers are designed under undirected or directed communication graphs to achieve the distributed attitude tracking and vibration suppression simultaneously. For the leader-follower consensus under directed graphs, only the controller’s generalized coordinates and the spacecraft attitudes are exchanged among the followers with the help of an extended state observer. The Lyapunov theory is adopted for the stability analysis of the overall system. Finally, numerical simulations are conducted to verify the effectiveness of the proposed controllers.

3.1 Introduction The aim of distributed leader-follower attitude consensus is to drive multiple spacecraft to track a leader synchronously based on the information exchange with neighbors. In the previous studies on a single flexible spacecraft [1–3], the system damping plays a crucial role in the proof of the controller convergence. For example, in [1], the existence of solution for Lyapunov equation relies on the non-zero damping term, i.e., the Lyapunov equation will have no solution in the extreme situation where the damping ratio is zero. However, for a flexible spacecraft, the structural and environmental damping is generally very low [4, 5], i.e., it will take a long time to passively attenuate the vibration of the flexible appendages. In [1, 6–10], the controllers’ function in vibration suppression is indeterminate, i.e., it is uncertain whether the vibration can be eliminated by the controllers. An extreme example is that the flexible vibration will not be damped out under the controller in [10] if the system damping is set as zero. Furthermore, some spacecraft may not be equipped with the instruments to measure angular velocity to lower the cost or these instruments may fail. Hence, Reproduced from Ti Chen, Jinjun Shan and Hao Wen. Distributed passivity-based control for multiple flexible spacecraft with attitude-only measurements. Aerospace Science and Technology 2019; c 94: 105408. Copyright 2019 Elsevier Masson SAS. All rights reserved. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_3

43

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3 Distributed Passivity-Based Control with Attitude-Only Measurements

it is necessary to focus on the controller design with attitude-only measurements. In this chapter, the passivity-based control technique is adopted to design the distributed consensus controllers without the measurement of angular velocities and modal variables under undirected or directed communication graphs.

3.2 Problem Formulation 3.2.1 Dynamic Equation of Undamped Flexible Spacecraft In this chapter, the kinematics of a rigid body is described by the MRPs. According to Eqs. (1.22) and (1.29), the dynamic equations of the i-th flexible spacecraft without damping term can be written as 

J i ω˙ i + δ iT η¨ i = −s(ωi )( J i ωi + δ iT η˙ i ) + ui η¨ i + K ni ηi = −δ i ω˙ i

(3.1)

Obviously, the system in Eq. (3.1) is under-actuated because ηi cannot be controlled directly, and the flexible vibration can be suppressed based on the rigid-flexible dynamics only. The kinematics and dynamics of the networked flexible spacecraft can be described by σ˙ s = G s (σ s )ωs (3.2) and



J s ω˙ s + δ sT η¨ s = −S(ωs )( J s ωs + δ sT η˙ s ) + us η¨ s + K s ηs = −δ s ω˙ s

(3.3)

where σ s = [σ 1T , · · · , σ TN ]T , G s = diag{G i }, ωs = [ω1T , · · · , ω TN ]T , J s = diag{ J i }, S(ωs ) = diag{s(ωi )}, δ s = diag{δ i }, ηs = [η1T , · · · , η TN ]T , us = [u1T , · · · , u TN ]T and K s = diag{K ni } for i = 1, . . . , N . Note that N represents the number of the spacecraft of concern.

3.2.2 Properties of Networked Flexible Spacecraft Before analyzing the properties of the networked control plant, some basic knowledge of passive systems will be recalled in this section. Consider the system governed by the following equation  x˙ = f (x, u) : (3.4) y = h(x, u)

3.3 Distributed Passivity-Based Control

45

where a causal dynamic operator  : u → y [11] is defined. According to [12], the system (3.4) is passive if there is a continuously differentiable semidefinite function E such that (3.5) E˙ ≤ u T y, ∀(x, u) where E is called the storage function. Moreover, it is said to be lossless if E˙ = u T y. For the system in Eq. (3.3) with output ys = ωs , the storage function is defined as      1 1 ωs T J s δ sT ωs + ηsT K s ηs Es = (3.6) η˙ s δs I 2 η˙ s 2 where the first and second terms in E s are the kinetic energy and potential energy for the networked flexible spacecraft, respectively. Obviously, E˙ s = usT ys holds. Hence, the system in Eq. (3.3) is lossless. Remark 3.1 If the damping term is included in the dynamic equations of the networked flexible spacecraft, i.e., the system is governed by Eq. (1.29), the time derivative of the storage function reads E˙ s = usT ys − η˙ sT C s η˙ s ≤ usT ys

(3.7)

where C s = diag{C ni } for i = 1, . . . , N . Hence, the networked flexible spacecraft with damping term are passive. Remark 3.2 The dynamic equations of the networked flexible spacecraft is not a standard Lagrange formulation. According to [13], the first line in Eq. (3.3) can be rewritten as a standard Lagrange formulation by replacing ωs with σ˙ s .

3.3 Distributed Passivity-Based Control The distributed passivity-based controllers are designed to solve the leader-follower consensus problem with attitude-only measurements in this section. First, the control objective is given. Second, the passivity-based control methodology is briefly reviewed. Then the passivity-based control technique is used to solve the distributed tracking of multiple flexible spacecraft under undirected or directed graphs.

3.3.1 Control Objective The controllers are designed to drive the N flexible spacecraft to track a static leader and suppress the flexible vibration simultaneously with the assumption that only the MRPs of each spacecraft are available. The position of the static leader is represented

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3 Distributed Passivity-Based Control with Attitude-Only Measurements

Fig. 3.1 Interconnection between the EL controller and the control plant

by σ¯ d . The leader-follower consensus and vibration suppression problems are said to be achieved if lim ||ωi || = 0, lim ||σ i || = σ¯ d , lim ||ηi || = 0 and lim ||η˙ i || = 0 t→∞ t→∞ t→∞ t→∞ hold for i = 1, 2, . . . , N . Remark 3.3 In fact, the distributed tracking task and vibration suppression can be achieved by the control inputs only because the system in Eq. (3.3) is lossless. However, as stated in Remark 3.1, if the damping term is involved, the controller needs to drive the flexible spacecraft to track the leader only because the vibration can be attenuated by the system damping. Hence, the distributed control problem for the system in Eq. (3.3) is more difficult than that for multiple flexible spacecraft with damping. Furthermore, in general, the introduction of the damping term will accelerate the convergence of the distributed tracking controller designed based on the undamped system.

3.3.2 Passivity-Based Control Methodology The aim of passivity-based control methodology is to render the closed-loop system passive [11, 14]. Essentially, as shown in Fig. 3.1, the standard passivity-based control is achieved by interconnecting another Euler-Lagrange (EL) system with the control plant. The resulting controller can be called EL controller [15]. In this way, additional energies and damping can be included in the closed-loop system. If the total energy of the controller dynamics is designed as E c ( p, θ , θ˙ ) = E ck (θ, θ˙ ) + E cp ( p, θ ), where p and θ are the generalized coordinate vectors of the plant and the controller, respectively, the passivity-based controller is given as −∇ p E cp ( p, θ ).

3.3.3 Distributed Controller Under Undirected Graph The aim of this subsection is to design a distributed controller for the leader-follower tracking of a group of flexible spacecraft under undirected communication graph utilizing passivity-based control technique. Before moving on, several useful assumptions are presented. Assumption 3.1 The communication graph is undirected and connected. There exists at least one follower that can receive the information of the leader.

3.3 Distributed Passivity-Based Control

47

Assumption 3.2 For the plant governed by Eqs. (3.2) and (3.3), only the spacecraft MRP vector σ s is measurable. Under Assumption 3.1, the matrix H is positive definite from Lemma 1.1. In some attitude tracking mission, the facilities to measure the angular velocity may not be equipped or may fail. Consequently, it is necessary to study the control problem based on the attitude feedback only. From Assumption 3.2, ωs , ηs and η˙ s cannot be used as feedback variables in the following controller design. Inspired by [15], the controller dynamics is designed as follows M c θ¨ + C c θ˙ + K c (θ − σ s ) + (H ⊗ I 3 )(θ − σ d ) = 0

(3.8)

where M c = diag{M ci } ∈ R3N ×3N , C c = diag{C ci } ∈ R3N ×3N and K c = diag {K ci } ∈ R3N ×3N are constant positive definite matrices. For i = 1, 2, . . . , N , M ci ∈ R3×3 , C ci ∈ R3×3 and K ci ∈ R3×3 are positive definite matrices as well. θ ∈ R3N is generalized coordinate vector. σ d represents 1 N ⊗ σ¯ d , where 1 N is a vector with all entries equal to 1. Hence, the controller dynamics corresponding to the i-th spacecraft is M ci θ¨ i + C ci θ˙ i + K ci (θ i − σ i )  (3.9) + Nj=1 ai j (θ i − θ j ) + bi (θ i − σ¯ d ) = 0 where θ i represents the entries of θ from 3(i − 1) + 1 to 3i. Obviously, the controller dynamics is distributed because Eq. (3.9) only depends on its own and neighbors’ information. It should be noted that the information of the controller’s variables rather than the MRPs of each spacecraft will flow in the edges of the graph. The initial value of θ is chosen as the value of σ s at t = 0. The energies of the controller dynamics (3.8) can be expressed as

where

and

E c (σ s , θ , θ˙ ) = E ck (θ , θ˙ ) + E cp (σ s , θ )

(3.10)

1 T E ck (θ , θ˙ ) = θ˙ M c θ˙ 2

(3.11)

1 (θ − σ s )T K c (θ − σ s ) 2 1 + (θ − σ d )T (H ⊗ I 4 )(θ − σ d ) 2

(3.12)

E cp (σ s , θ ) =

Hence, the controller for the group of flexible spacecraft is us = G s (σ s )T K c (θ − σ s )

(3.13)

where the reason of the introduction of G s (σ s )T is that the dynamic equation in Eq. (3.3) is not a standard EL formulation. The definition of G s (σ s ) can be found in Section 3.2.1.

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3 Distributed Passivity-Based Control with Attitude-Only Measurements

Theorem 3.1 For the networked flexible spacecraft governed by Eqs. (3.2) and (3.3) with the controller dynamics (3.8), the controller in (3.13) can drive the group of flexible spacecraft to track the leader and suppress the flexible vibration asymptotically under Assumptions 3.1 and 3.2. Proof Consider the following Lyapunov function V1 = E s + E c (σ s , θ , θ˙ )

(3.14)

where E s is defined in Eq. (3.6). It should be noted that the solution to V1 = 0 is the unique equilibrium point of the closed-loop system. The time derivative of V1 is T V˙1 = ωs T us − θ˙ C c θ˙ − σ˙ sT K c (θ − σ s ) T = ωs T G s (σ s )T K c (θ − σ s ) − θ˙ C c θ˙ − σ˙ sT K c (θ − σ s ) T = −θ˙ C c θ˙ ≤0

(3.15)

According to LaSalle’s invariance principle, the closed-loop system will asymptotically converge to the largest invariant set B p = {σ s , ωs , ηs , η˙ s , θ , θ˙ |V˙1 = 0}. Due to the positive definiteness of C c , V˙1 = 0 implies θ˙ = 0. In the case of θ˙ ≡ 0, θ keeps constant and from Eq. (3.8), one has K c (θ − σ s ) + (H ⊗ I 3 )(θ − σ d ) = 0

(3.16)

Hence, σ s will keep constant because both K c and H ⊗ I 3 are positive definite, i.e., σ˙ s = 0. Equation (3.2) can be rewritten as

Note that

G s (σ s )T σ˙ s = G s (σ s )T G s (σ s )ωs

(3.17)

G s (σ s )T G s (σ s ) = diag{G i (σ i )T G i (σ i )} = diag{G i (−σ i )G i (−σ i )T } 1 + σ iT σ i 2 ) I 3} = diag{( 4 >0

(3.18)

Therefore, from Eq. (3.17), σ˙ s = 0 implies ωs = 0. In such a case, Eq. (3.3) can be recast as  T δ s η¨ s = G s (σ s )T K c (θ − σ s ) = constant (3.19) η¨ s + K s ηs = 0 or



δ iT η¨ i = constant i = 1, 2, . . . , N η¨ i + K i ηi = 0

(3.20)

3.3 Distributed Passivity-Based Control

49

Two possible solutions to the second equation in Eq. (3.20) are ηi ≡ 0 or simple harmonic vibration. In the case of simple harmonic vibration, each entry of ηi will oscillate with different frequencies due to the expression of K i , hence, the first line in Eq. (3.20) is false. Therefore, the unique solution to Eq. (3.20) is ηi ≡ 0. In conclusion, the solution to V˙1 = 0 is {ηs ≡ 0, θ ≡ σ s ≡ σ d }. Therefore, according to LaSalle’s invariance principle, the controller can complete the distributed tracking  and vibration suppression for the group of flexible spacecraft. Next, the validity of Theorem 3.1 will be presented based on the passivity-based control theory. Remark 3.4 In fact, the networked flexible spacecraft is an under-actuated EL system of high dimensions. The introduction of the EL controller only modifies the potential energy and dissipation properties of the under-actuated EL plant, hence, the designed controller preserves the EL structure. Consequently, the controller can also be named as a distributed EL controller. For the proposed controller, we have the following statements: (a) The total potential energy is 21 ηsT K s ηs + E cp (σ s , θ ), which have a global and unique minimum at the desired state, i.e., ηs = 0 and θ = σ s = σ d . (b) The damping injection is introduced by the term C c θ˙ in the controller dynamT ˙ 2 holds. ics. Due to the positive definiteness of C c , θ˙ C c θ˙ ≥ λmin (C c )||θ|| ∂ E cp (c) In the case of θ ≡ constant and ∂θ ≡ 0, based on the similar derivation of the largest invariant set in the proof of Theorem 3.1, one has σ s ≡ σ d and ηs = 0. Therefore, according to Proposition 3.6 in [11], the closed-loop system is asymptotically stable.

3.3.4 Distributed Controller Under Directed Graph The directed communication graph in the following assumption will be considered in this section. Assumption 3.3 The leader in the directed graph is globally reachable. Under the above assumption, there exists at least one path from the leader to each node in the graph, that is, the desired attitude can be propagated to all flexible spacecraft. According to Lemma 1.2, the matrix H of such a graph is positive stable, i.e., all eigenvalues of H have positive real part. Usually, the matrix H is asymmetric. The loss of symmetry makes the controller designed in Sect. 3.3.3 no longer effective. Hence, the controller dynamics can be redesigned as (H ⊗ I 3 )θ¨ + α(H ⊗ I 3 )θ˙ +β(H ⊗ I 3 )(θ − σ d ) + γ (H ⊗ I 3 )(θ − σ s ) = 0

(3.21)

where α, β and γ are three positive constants. The controller dynamics corresponding to σ i is

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3 Distributed Passivity-Based Control with Attitude-Only Measurements

N

 ai j (θ¨ i − θ¨ j ) + bi θ¨ i + α Nj=1 ai j (θ˙ i − θ˙ j )  +αbi θ˙ i + β Nj=1 ai j (θ i − θ j ) + βbi (θ i − σ¯ d ) N +γ j=1 ai j (θ i − θ j ) + γ bi θ i  −γ Nj=1 ai j (σ i − σ j ) − γ bi σ i = 0 j=1

(3.22)

It is clear that the response of θ i depends on the generalized coordinates, the generalized velocities, the generalized accelerations of the controller and the MRPs of itself and the adjacent spacecraft. Obviously, the controller dynamics is distributed. The matrix H⊗I 3 is invertible because all its eigenvalues have positive real parts. Hence, Eq. (3.21) can be rewritten as θ¨ + α θ˙ + β(θ − σ d ) + γ (θ − σ s ) = 0

(3.23)

Equation (3.23) describes a standard EL system. Hence, the distributed EL controller for the networked flexible spacecraft can be designed as us = G s (σ s )T γ (θ − σ s )

(3.24)

Clearly, the closed-loop system with controller in Eq. (3.24) has a unique equilibrium point at θ = σ s = σ d , θ˙ = 0, ωs = 0, ηs = 0 and η˙ s = 0. Theorem 3.2 Under Assumptions 3.2 and 3.3, the controller in Eq. (3.24) can achieve the attitude tracking and vibration suppression of multiple flexible spacecraft gradually. Proof The Lyapunov function is chosen as 1 T 1 V2 = E s + θ˙ θ˙ + (θ − σ d )T β(θ − σ d ) 2 2 1 + (θ − σ s )T γ (θ − σ s ) 2

(3.25)

and its time derivative is T V˙2 = ωs T us − α θ˙ θ˙ − σ˙s T γ (θ − σ s ) T = −α θ˙ θ˙ ≤0

(3.26)

Similar to the proof of Theorem 1, it can be concluded that θ˙ ≡ 0 implies ηs ≡ 0 and θ ≡ σ s ≡ σ d . From LaSalle’s invariance principle, θ → σ d , σ s → σ d , ηs → 0,  θ˙ → 0, ωs → 0 and η˙ s → 0 as time goes to infinity. Remark 3.5 Similar to the statements in Remark 3.4, the controller dynamics (3.21) maintains the EL structure of the overall system. Actually, the total potential energy can be expressed as 21 ηsT K s ηs + 21 (θ − σ d )T β(θ − σ d ) + 21 (θ − σ s )T γ (θ − σ s ), the global unique minimum of which is located at the desired configuration with

3.3 Distributed Passivity-Based Control

51

zero vibration. The system oscillation will be damped out due to the damping term α(H ⊗ I 3 )θ˙ in Eq. (3.21). The desired state is detectable in the case of θ ≡ constant. Hence, the controller in Eq. (3.24) can complete the distributed tracking and vibration suppression asymptotically. Remark 3.6 In [7], Zou et al. proposed a control algorithm with attitude-only measurements based on a nonlinear observer to estimate the unmeasurable angular velocities and modal variables under undirected graph. It is nontrivial to extend their controller to the case under directed graphs. However, the controller in Eq. (3.24) can solve the attitude consensus only with the feedback of the attitude under directed graphs. Remark 3.7 In Eq. (3.22), only MRPs in the feedback signals need to be measured. The generalized coordinates, generalized velocities and generalized accelerations of the controller can be calculated numerically. Hence, such a continuous controller is easy to be implemented in practice. However, the information flowing in the communication edges among the followers contains the controller’s generalized coordinates, generalized velocities, generalized accelerations and the spacecraft MRPs. To reduce the communication burden, an Extended State Observer (ESO) is used to estimate the generalized velocities and accelerations of the controller of the adjacent followers based on the available information of the generalized coordinates. Denote i = Nj=1 ai j θ j . A linear ESO is designed as ⎧ ¯ i − z i1 ) ⎨ z˙ i1 = z i2 + 3ω( z˙ i2 = z i3 + 3ω¯ 2 (i − z i1 ) ⎩ z˙ i3 = ω¯ 3 (i − z i1 )

(3.27)

where ω¯ is the positive observer bandwidth, z i1 , z i2 and z i3 are the estimations of ˙ i and  ¨ i , respectively. The initial value of z i1 is set as i (0). Based on the i ,  estimation vectors, the controller dynamics corresponding to the i-th spacecraft can be designed as (di + bi )θ¨ i − z i3 + α(di + bi )θ˙ i − αz i2 +(β + γ )(di + bi )θ i − (β + γ )i − βbi σ¯ d (3.28)  −γ Nj=1 ai j (σ i − σ j ) − γ bi σ i = 0 In the case that the controller dynamics is governed by Eqs. (3.27) and (3.28) and the distributed EL controller is designed as Eq. (3.24), the information transmitted among the agents only contains θ i and σ i . Eqs. (3.27) and (3.28) can be rewritten in the following format ⎧ ¯ − z1 ) ⎨ z˙ 1 = z 2 + 3ω( z˙ 2 = z 3 + 3ω¯ 2 ( − z 1 ) (3.29) ⎩ z˙ 3 = ω¯ 3 ( − z 1 ) and

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3 Distributed Passivity-Based Control with Attitude-Only Measurements

(H ⊗ I 3 )θ¨ + α(H ⊗ I 3 )θ˙ + β(H ⊗ I 3 )(θ − σ d ) ¨ + α(z 2 − ) ˙ +γ (H ⊗ I 3 )(θ − σ s ) = (z 3 − )

(3.30)

T T where  = [1T · · · TN ]T , z 1 = [z 11 · · · z TN 1 ]T , z 2 = [z 12 · · · z TN 2 ]T and z 3 = T T T [z 13 · · · z N 3 ] . ... If it is assumed that  is bounded by a constant h, the convergence of the controller in Eq. (3.24) governed by Eqs. (3.29) and (3.30) can be indicated by the following theorem.

Theorem 3.3 Consider the networked flexible spacecraft governed by Eqs. (3.2) and (3.3) under the communication graph in Assumption 3.2 with the controller in Eq. (3.24), where the dynamics of the control variable is governed by Eqs. (3.27) and (3.28). The closed-loop system will converge to a close neighborhood of the equilibrium point if ω¯ is set large enough. (ω¯ +ωα) ¯ Proof Denote c1 = q p ω¯ p+1 − 4αλ for p ∈ N and q > 0. Obviously, for any T min (H H) given ω¯ > 0, α > 0 and H, there must exist an integer p and a positive q such that c1 > 0 holds. A candidate Lyapunov function is chosen as 2

2

1 T 1 V3 = E s + θ˙ θ˙ + (θ − σ d )T β(θ − σ d ) 2 2 1 + (θ − σ s )T γ (θ − σ s ) + q p ω¯ p eT P e 2

(3.31)

T T T ˙ 1T − z 12 ¨ 1T − z 13 ˙ TN − z TN 2 )/ where e = [1T − z 11 ( )/ω¯ ( )/ω¯ 2 · · · TN − z TN 1 ( ¨ TN − z TN 3 )/ω¯ 2 ]T , and p ∈ N and q > 0 are chosen properly such that c1 > 0 ω¯ ( holds. The positive definite matrix P is the solution of the following Lyapunov equation AeT P + P Ae = −ω¯ I (3.32)

in which Ae = diag{ Aei ⊗ I 3 } for i = 1, . . . , N . The definition of Aei is ⎡

⎤ −3ω¯ ω¯ 0 Aei = ⎣ −3ω¯ 0 ω¯ ⎦ −ω¯ 0 0

(3.33)

Ae is a stable matrix because all its eigenvalues are −ω. ¯ P is independent of ω. ¯ The estimation error vector e is governed by e˙ = Ae e + h

(3.34)

... T ... T where h = [0T 0T 1 /ω¯ 2 · · · 0T 0T  N /ω¯ 2 ]T . Hence, h is bounded by h/ω¯ 2 . The time derivative of V3 is

3.3 Distributed Passivity-Based Control

53

Fig. 3.2 Undirected communication graph

Fig. 3.3 Attitudes of the four flexible spacecraft in Case I

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1000 400

500

Time (s) T T ¯ 2) V˙3 = −α θ˙ θ˙ − θ˙ (H ⊗ I 3 )−1 (ω¯ 2 e3 + ωαe p p+1 T e e + 2q p ω¯ p eT P h −q ω¯ ˙ ≤ −α||θ˙ ||2 + √ 1 T (ω¯ 2 + ωα)|| ¯ θ||||e|| λmin (H H)

−q p ω¯ p+1 ||e||2 + 2q p hλmax ( P)ω¯ p−2 ||e|| 2

√ ˙ (ω¯ 2 +ωα) ¯ α||θ|| − √ √ ||e|| =− T 2 α p

=

λmin (H H) −c1 ||e||2 + 2q hλmax ( P)ω¯ p−2 ||e||

2 √ ˙ (ω¯ 2 +ωα) ¯ − α||θ|| − √ √ ||e|| 2 α λmin (H T H)

−c1 ||e||(||e|| − c)

(3.35)

54

3 Distributed Passivity-Based Control with Attitude-Only Measurements

Fig. 3.4 Modal variables of the four flexible spacecraft in Case I

0.2

i-1

i=1

0

-0.2 0

100

i=2

0.005 0 -0.005 800 200 300

i=3

i=4

900 400

1000 500

900 400

1000 500

900 400

1000 500

900 400

1000 500

Time (s)

i-2

0.2

0

-0.2 0

100

0.005 0 -0.005 800 200 300

Time (s)

i-3

0.2

0

-0.2 0

100

0.001 0 -0.001 800 200 300

Time (s)

i-4

0.1

0

-0.1 0

100

0.001 0 -0.001 800 200 300

Time (s) Fig. 3.5 Directed communication graph

3.3 Distributed Passivity-Based Control Fig. 3.6 Attitudes of the four flexible spacecraft in Case II

55

2

i-1

i=1

0

-2 0

100

i=2

0.01 0 -0.01 300 200

i=3

i=4

400 300

400

500 500

400

500 500

400

500 500

Time (s)

i-2

3

0

-3 0

100

400 300

Time (s)

3

i-3

0.01 0 -0.01 300 200

0

-3 0

100

0.01 0 -0.01 300 200

400 300

Time (s)

where c = 2q hλmaxc1( P)ω¯ . ¨ + V˙3 is negative in the case of ||e|| > c. If ||e|| ≤ c, the right-hand terms (z 3 − ) 2 ˙ ¯ which decreases as ω¯ increases. If α(z 2 − ) in Eq. (3.30) is bounded by cω¯ + αcω, ¨ + α(z 2 − ) ˙ can be considered as small disturbances ω¯ is set large enough, (z 3 − ) acting on the unperturbed system governed by Eqs. (3.2), (3.3), (3.23) and (3.24). From Theorem 3.2, the unperturbed system is uniformly asymptotically stable, hence, the equilibrium point at θ = σ s = σ d , θ˙ = 0, ωs = 0, ηs = 0, η˙ s = 0 is totally stable [16]. Therefore, the closed-loop system will converge to a close neighborhood of the  equilibrium point if ω¯ is set large enough. p

p−2

Remark 3.8 In Theorem 3.3, the closed-loop system will not converge to the equilibrium point of the unperturbed system strictly. However, the system is driven to a close neighborhood of the equilibrium point and remains within it, which is acceptable for practical applications. Hence, the attitude consensus and vibration suppression can also be said to be achieved under the controller in Theorem 3.3.

56

3 Distributed Passivity-Based Control with Attitude-Only Measurements

Fig. 3.7 Modal variables of the four flexible spacecraft in Case II

0.05

i-1

i=1

0

-0.05 0

100

i=2

0.001 0 -0.001 300 200 300

i=3

i=4

400 400

500 500

400 400

500 500

400 400

500 500

400 400

500 500

Time (s)

i-2

0.05

0

-0.05 0

100

0.001 0 -0.001 300 200 300

Time (s)

i-3

0.05

0

-0.05 0

100

0.001 0 -0.001 300 200 300

Time (s)

i-4

0.05

0

-0.05 0

100

0.001 0 -0.001 300 200 300

Time (s)

3.4 Numerical Simulations In this section, four flexible spacecraft are taken as example to verify the effectiveness of the proposed controllers in Sect. 3.3. The system parameters J i , δ i and K ni are chosen as the same as those in [17]. At t = 0, the four agents are rest at σ 1 (0) = [0.5 1 − 2.5]T , σ 2 (0) = [−2 − 1.2 2]T , σ 3 (0) = [−1 − 3 0.8]T and σ 4 (0) = [1.5 3 − 0.5]T . Both the initial values of ηs and η˙ s are equal to zero, i.e., the flexible appendages have no initial deformation. The desired attitude is σ¯ d = [0 0 0]T .

3.4 Numerical Simulations

57

Fig. 3.8 Attitudes of the four flexible spacecraft in Case II with ESO

2

i-1

i=1

0

-2 0

100

i=2

0.01 0 -0.01 300 200

i=3

i=4

400 300

400

500 500

400

500 500

400

500 500

Time (s)

i-2

3

0

-3 0

100

400 300

Time (s)

3

i-3

0.01 0 -0.01 300 200

0

-3 0

100

0.01 0 -0.01 300 200

400 300

Time (s)

3.4.1 Case I: Leader-Follower Tracking Under Undirected Graph The controller in Eq. (3.13) is validated through this case. The undirected communication graph is shown in Fig. 3.2. The parameters in controller dynamics (3.9) are chosen as M ci = I 3 , C ci = 50 I 3 and K ci = 100 I 3 , respectively. The simulation results are shown in Figs. 3.3 and 3.4. Obviously, it can be observed that the designed distributed EL controller in Eq. (3.13) can control the spacecraft to arrive at the desired attitude without the measurements of angular velocities and modal coordinates, as well as suppress the vibration of flexible appendages gradually based on the rigid-flexible dynamics. Since the initial deformations of the flexible appendages are zero and the structural damping is ignored, the vibrations shown in Fig. 3.4 are excited by the spacecraft’s attitude maneuver and suppressed by the control input governed by Eq. (3.13). Hence, the proposed controller is able to play a positive role in vibration suppression.

58

3 Distributed Passivity-Based Control with Attitude-Only Measurements

Fig. 3.9 Modal variables of the four flexible spacecraft in Case II with ESO

0.05

i-1

i=1

0

-0.05 0

100

i=2

0.002 0 -0.002 300 200 300

i=3

i=4

400 400

500 500

400 400

500 500

400 400

500 500

400 400

500 500

Time (s)

i-2

0.05

0

-0.05 0

100

0.002 0 -0.002 300 200 300

Time (s)

i-3

0.05

0

-0.05 0

100

0.001 0 -0.001 300 200 300

Time (s)

i-4

0.05

0

-0.05 0

100

0.001 0 -0.001 300 200 300

Time (s)

3.4.2 Case II: Leader-Follower Tracking Under Directed Graph The performance of the distributed controller in Eq. (3.24) under directed graph is tested through this case. The controller parameters α, β and γ are set as 50, 1 and 100, respectively. The communication graph in Fig. 3.5 is adopted in this simulation. Clearly, the leader is globally reachable in such a graph. The responses of the four spacecraft are depicted in Figs. 3.6 and 3.7. It can be found that the four agents converge to the desired attitude and the vibrations of flexible appendages are damped out asymptotically. As shown in Figs 3.4 and 3.7, the excited dynamic deformations of flexible appendages in this case are smaller than those in Case I. The main reason is that the curves in Fig. 3.6 are more smooth than the same ones in Fig. 3.3.

3.4 Numerical Simulations Fig. 3.10 Estimation errors ˙ i − z i2 of ESO for the four  spacecraft

59 0.1

0

-0.1 0

100

200

300

400

500

Time (s) 0.1 1st element

2nd element

3rd element

0

-0.1 0

100

200

300

400

500

400

500

400

500

Time (s) 0.1

0

-0.1 0

100

200

300

Time (s) 0.1

0

-0.1 0

100

200

300

Time (s)

Next, the passivity-based controller with ESO in Theorem 3.3 will be illustrated via a numerical simulation. The ESO parameter ω¯ is set as 5 rad/s. As shown in Figs. 3.8 and 3.9, the leader-follower consensus and vibration suppression can be completed by the controller in Eq. (3.24) governed by Eqs. (3.27) and (3.28). The responses of the attitude and modal variables of each spacecraft are almost the same as those in Figs. 3.6 and 3.7. The main reason is that, as shown in Fig. 3.10, the ESO can quickly estimate the generalized velocities and accelerations of the controller dynamics of its neighbors for each spacecraft.

60

3 Distributed Passivity-Based Control with Attitude-Only Measurements

3.5 Conclusions In this study, the leader-follower consensus of multiple under-actuated flexible spacecraft without system damping is considered with the attitude measurements only under undirected or directed graphs. In both the undirected and directed cases, auxiliary dynamic systems are derived for the design of the distributed Euler-Lagrange controllers in the absence of the measurements of angular velocities and modal variables. For the leader-follower consensus under undirected graphs, only the controller’s generalized coordinates are exchanged among the followers. However, under directed graphs, the information to be exchanged among the followers contains the spacecraft attitudes, the generalized coordinates, velocities and accelerations of the controller. Consequently, an extended state observer is introduced to remove the requirement of the controller’s generalized velocities and accelerations of neighbors. Theoretical proofs and numerical simulations show that the proposed distributed controllers can drive the group of undamped flexible spacecraft to track the leader and suppress the flexible vibrations simultaneously.

References 1. Di Gennaro, S.: Output attitude tracking for flexible spacecraft. Automatica 38(10), 1719–1726 (2002) 2. Xiao, B., Yin, S., Kaynak, O.: Attitude stabilization control of flexible satellites with high accuracy: an estimator-based approach. IEEE/ASME Trans. Mechatron. 22(1), 349–358 (2017) 3. Di Gennaro, S.: Passive attitude control of flexible spacecraft from quaternion measurements. J. Optim. Theory Appl. 116(1), 41–60 (2003) 4. Hu, Q., Ma, G.: Adaptive variable structure controller for spacecraft vibration reduction. IEEE Trans. Aerosp. Electron. Syst. 44(3), 861–876 (2008) 5. Hu, Q., Jia, Y., Xu, S.: Adaptive suppression of linear structural vibration using control moment gyroscopes. J. Guidance Control Dyn. 37(3), 990–996 (2014) 6. Chen, T., Chen, G.: Distributed adaptive tracking control of multiple flexible spacecraft under various actuator and measurement limitations. Nonlinear Dyn. 91(3), 1571–1586 (2018) 7. Zou, A.M., de Ruiter, A.H., Kumar, K.D.: Distributed finite-time velocity-free attitude coordination control for spacecraft formations. Automatica 67, 46–53 (2016) 8. Du, H., Li, S.: Attitude synchronization for flexible spacecraft with communication delays. IEEE Trans. Autom. Control 61(11), 3625–3630 (2016) 9. Du, H., Chen, M.Z., Wen, G.: Leader-following attitude consensus for spacecraft formation with rigid and flexible spacecraft. J. Guidance Control Dyn. 39(4), 944–951 (2016) 10. Chen, T., Wen, H.: Autonomous assembly with collision avoidance of a fleet of flexible spacecraft based on disturbance observer. Acta Astronautica 147, 86–96 (2018) 11. Ortega, R., Perez, J.A.L., Nicklasson, P.J., Sira-Ramirez, H.J.: Passivity-Based Control of EulerLagrange Systems: Mechanical, Electrical and Electromechanical Applications. Springer, London (1998) 12. Khalil, H.: Nonlinear Syst. Prentice Hall, New Jersey (2002) 13. Lyu, J., Gao, D.: Attitude synchronization for multiple spacecraft with input constraints. Chin. J. Aeronaut. 27(2), 321–327 (2014) 14. Wen, H., Chen, T., Jin, D., Hu, H.: Passivity-based control with collision avoidance for a hub-beam spacecraft. Adv. Space Res. 59(1), 425–433 (2017)

References

61

15. Nuño, E., Ortega, R.: Achieving consensus of Euler-Lagrange agents with interconnecting delays and without velocity measurements via passivity-based control. IEEE Trans. Control Syst. Technol. 26(1), 222–232 (2018) 16. Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991) 17. Du, H., Li, S.: Attitude synchronization control for a group of flexible spacecraft. Automatica 50(2), 646–651 (2014)

Chapter 4

Rotation-Matrix-Based Attitude Tracking Under an Undirected Tree Graph

Abstract This chapter studies the distributed leader-follower consensus for multiple flexible spacecraft with actuator faults under an undirected tree graph. The attitudes of the networked flexible spacecraft are represented on the special orthogonal group S O(3) globally and uniquely. A fault-tolerant tracking controller is proposed to protect against the partial loss of actuator effectiveness based on sign function. Then the saturation function with an adaptive parameter is introduced to replace the sign function. Rigorous Lyapunov analyses are presented to prove the stability of the overall system. Finally, numerical simulations are conducted to verify the effectiveness of the proposed controllers.

4.1 Introduction Due to the strong nonlinearity of the attitude dynamics of spacecraft, it is challenging to design a distributed attitude controller. Various control algorithms have been proposed to achieve the attitude synchronization of multiple rigid spacecraft based on algebraic graph theory [1–4]. In these works, the Modified Rodriguez Parameters (MRPs) or quaternions are used to describe the attitude of rigid bodies. However, as stated in [5–7], the minimal representations, such as the Euler angles and MRPs, have singularities, and quaternions are not unique in representing an attitude, i.e., two antipodal unit quaternions are corresponding to the same attitude. The continuous control methods may cause the unstable unwinding phenomenon [7] with this ambiguity. One global and unique attitude representation is based on the rotation matrix [5]. Consequently, some researchers have tried to design attitude controllers via rotation matrix [6–10]. For instance, Lee defined a novel attitude error function on S O(3) to obtain a controller which guarantees a desirable performance uniformly for attitude tracking problem [6]. Song et al. developed a class of distributed control algorithms by solving the gradient of a potential function based on relative attitude information on S O(3) [10]. Recent years have also witnessed some studies on Reproduced from Ti Chen and Jinjun Shan. Rotation-matrix-based attitude tracking for multiple flexible spacecraft with actuator faults. Journal of Guidance, Control, and Dynamics 2019; 42(1): 181–188. Copyright © 2018 by Ti Chen and Jinjun Shan. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_4

63

64

4 Rotation-Matrix-Based Attitude Tracking …

distributed control for multiple flexible spacecraft, which is more complicated and challenging due to the rigid-flexible dynamics [11–14]. Furthermore, some unknown actuator faults may occur on a spacecraft. The actuator faults present a challenge in the control system design of flexible spacecraft. Some results are available on the fault-tolerant control for a single or multiple flexible spacecraft [15–18]. For instance, in [15], a fault-tolerant controller is proposed for a single flexible spacecraft based on the sign of the angular velocity. Sakthivel et al. focused on the fault-tolerant sampled-data control for flexible spacecraft with partial actuator failures, external disturbances and probabilistic time delays [17]. Huang et al. designed distributed controllers for networked flexible spacecraft with actuator failures and saturation constraints [18]. However, in the above-mentioned studies on a single or multiple flexible spacecraft and the work in last chapter, MRPs or unit quaternions are used to represent the attitude. Hence, this chapter tries to design almost globally convergent controllers for multiple flexible spacecraft with partial actuator failures based on rotation matrix. The integral term is put in a hyperbolic tangent function to avoid the integral windup. Furthermore, a continuous saturation function with adaptive parameter is introduced to remove the discontinuity of the presented controller.

4.2 Dynamics of Flexible Spacecraft Based on Rotation Matrix The dynamic equations for the i-th flexible spacecraft with the partial loss of actuator effectiveness can be expressed as [16] 

J i ω˙ i + δ iT η¨ i = −ωi× ( J i ωi + δ iT η˙ i ) + F i (ui ) η¨ i + C ni η˙ i + K ni ηi = −δ i ω˙ i

(4.1)

where J i ∈ R3×3 is the total inertia matrix of the flexible spacecraft, δ i ∈ Rn×3 is the coupling matrix between the rigid and flexible dynamics, ηi = [ηi−1 , . . . , ηi−n ]T ∈ Rn is the vibration modal coordinate vector, K ni = diag{21 , . . . , 2n } ∈ Rn×n is the stiffness matrix and C ni = diag{2ξ1 1 , . . . , 2ξn n } ∈ Rn×n is the structural damping matrix, in which  j and ξ j are the j-th order natural frequency and damping ratio, respectively. It should be noted that only the first n elastic modes are considered in this study. The control input vector with faults can be expressed as F i (ui ) = ρ i (t)ui , in which ρ i (t) = diag{ρi−1 (t), ρi−2 (t), ρi−3 (t)} is the effectiveness factor of spacecraft actuators and ui is the control command. It is assumed that 0 < μ0 ≤ ρi− j (t) ≤ 1 holds for i = 1, . . . , N and j = 1, 2, 3, where μ0 is a positive constant.

4.3 Coordinated Tracking Control

65

4.3 Coordinated Tracking Control 4.3.1 Control Objective This study aims to control N flexible spacecraft to track a static leader and suppress the flexible vibration simultaneously under the communication graph in the following Assumption 4.1. The control objective is said to be achieved if Ri → Rd , ωi → 0, ηi → 0 and η˙ i → 0 as time goes to infinity for i = 1, . . . , N , where the constant rotation matrix Rd ∈ S O(3) represents the attitude of the leader. Assumption 4.1 The undirected communication graph among the N flexible spacecraft forms a tree network. Only one node can receive the information of the static leader.

4.3.2 Controller Design Some auxiliary variables will be defined firstly. Denote Rid = RdT Ri and Ri j = R Tj Ri . The kinematic equations of Rid and Ri j are ˙ id = Rid ωi× R

(4.2)

˙ i j = Ri j ωi×j R

(4.3)

where ωi j = ωi − RiTj ω j and Ri j has the following properties [19] Ri j = R Tji

(4.4)

(Ri j − RiTj )∨ = −R ji (R ji − R Tji )∨

(4.5)

An auxiliary variable ωei is defined as ωei =

N 

T ai j (RiTj − Ri j )∨ + bi (Rid − Rid )∨

j=1

Clearly, ω˙ ei can be expressed as

(4.6)

66

4 Rotation-Matrix-Based Attitude Tracking …

ω˙ ei =

N 

T ai j (−ωi×j RiTj − Ri j ωi×j )∨ + bi (−ωi× Rid − Rid ωi× )∨

j=1

=−

N 

(4.7) ai j [tr(Ri j )I 3 −

RiTj ]ωi j

− bi [tr(Rid )I 3 −

T Rid ]ωi

j=1

Obviously, ωei only depends on its own and neighbors’ attitudes of rigid bodies. A lemma about ωei is given as follows. T − Rid )∨ = 0 hold Lemma 4.1 If ωei = 0 is satisfied, (RiTj − Ri j )∨ = 0 and (Rid under the communication graph in Assumption 4.1.

Proof For i = 1, . . . , N , j = 1, . . . , N and i = j, if i > j, i j is defined as (RiTj − Ri j )∨ ; otherwise, i j = Ri j (RiTj − Ri j )∨ . Clearly, i j = − ji holds. The leader can be considered as the (N + 1)-th node in the graph. Hence, one has T i(N +1) = (Rid − Rid )∨ . A directed spanning tree G = {V, E} with the leader as the root can be obtained by designating a proper direction to each undirected graph for the communication graph in Assumption 4.1. Clearly, the graph G has no cycles and contains N + 1 nodes and N directed edges. Hence, ωei = 0 can be rewritten as C = 0

(4.8)

where  is the column stack vector composed of i j for i = 1, . . . , N and j = 1, . . . , N + 1, and C = Cˆ ⊗ I 3 ∈ R3N ×3N , in which ⊗ denotes the Kronecker product. Cˆ is obtained by removing the last row of the incidence matrix of the graph G. Note that the first N − 1 columns of Cˆ contains exactly two non-zero entries, namely one 1 and one −1. From Property 2.1 in [20], the first N − 1 columns are linearly independent. Furthermore, since the last column only contains one non-zero entry, it is independent of the first N − 1 columns. Hence, Eq. (4.8) only has zero solution. Due to the orthogonality of Ri j , it can be concluded that (RiTj − Ri j )∨ = 0 T and (Rid − Rid )∨ = 0 hold in the case of ωei = 0.  Remark 4.1 Lemma 4.1 is derived under the graph consisting of a leader and an undirected tree network. In fact, this lemma is the extension of the results in [19, 21] under undirected tree graph. Lemma 4.2 [21] (RiTj − Ri j )∨ = 0 has two solutions, i.e., Ri = R j and tr(Ri j ) = −1. Denote q i = η˙ i + δ i ωi . The second equation in Eq. (4.1) can be rewritten as ξ˙ i = Ai ξ i + g i ωi

(4.9)

     ηi 0 In −δ i , Ai = , gi = where ξ i = . It is clear that all eigenqi −K ni −C ni C ni δ i values of Ai have negative real parts. Based on q i , the first equation in Eq. (4.1) can be recast as 

4.3 Coordinated Tracking Control

67

J 0i ω˙ i − δ iT K ni ηi − δ iT C ni (q i − δ i ωi ) = −ωi× ( J 0i ωi + δ iT q i ) + F i (ui ) (4.10) where J 0i = J i − δ iT δ i . For the i-th spacecraft, the following fault-tolerant controller is proposed to achieve the control objective defined in Sect. 4.3.1.  t  ui = −k2 ωi + k1 ωei + k3 tanh ωei (τ )dτ 0  t    1 − μ0 − Sign(ωi ) k1 |ωei | + k3 tanh ωei (τ )dτ μ0 0

(4.11)

where k1 , k2 and k3 are positive control gains and Sign(ωi ) = diag{sign(ωi−1 ), sign(ωi−2 ), sign(ωi−3 )}

(4.12)

in which sign(·) is the sign function. The hyperbolic tangent function is introduced to avoid the integral windup. From Eq. (4.11), it is clear that the proposed controller for the i-th spacecraft only depends on its own angular velocity and attitude and its neighbors’ attitude. The information of the vibration modal coordinates is not needed in the feedback loop. The convergence of the controller in Eq. (4.11) can be stated as follows. Theorem 4.1 Consider the networked flexible spacecraft governed by Eqs. (1.21) and (4.1). The fault-tolerant controller (4.11) can drive the spacecraft to the desired attitude and suppress the flexible vibration almost globally asymptotically. Proof The Lyapunov function is chosen as V = V1 + V2 where V1 =

(4.13)

N N N  k1   ai j i j + k1 bi id 2 i=1 j=1 i=1

1 T 1 T V2 = ωi J 0i ωi + ξ Pi ξi 2 i=1 2 i=1 i N

N

(4.14)

in which i j and id are the attitude error function between the i-th and j-th spacecraft and the attitude tracking error function of the i-th spacecraft, respectively. The positive definite matrix P i is defined as P i = diag{K ni , I n }. The detailed expressions of i j and id are i j = tr(I 3 − Ri j ) (4.15) id = tr(I 3 − Rid )

68

4 Rotation-Matrix-Based Attitude Tracking …

It should be noted that both Ri j and Rid are orthogonal matrices. Hence, according to Lemma 2 in [19], i j ∈ [0, 4] and id ∈ [0, 4] hold. Furthermore, both i j = 0 and id = 0 are true only if R j = Ri = Rd . Obviously, the positive definite matrix P i is the solution to the following equation T

P i Ai + Ai P i = −2 Q i

(4.16)

in which Q i = diag{0, −C ni }. The time derivatives of V1 and V2 are N N N  k1   ai j tr(−Ri j ωi×j ) + k1 bi tr(−Rid ωi× ) V˙1 = 2 i=1 j=1 i=1

=

N N N N k1   k1   ai j ωiT (Ri j − RiTj )∨ − ai j ω Tj Ri j (Ri j − RiTj )∨ 2 i=1 j=1 2 i=1 j=1 N 

+ k1

ωiT (−ωei +

N 

i=1

= −k1

N 

(4.17)

ai j (RiTj − Ri j )∨ )

j=1

ωiT ωei

i=1

V˙2 =

N 

ωiT {δ iT K ni ηi + δ iT C ni (q i − δ i ωi ) − ωi× ( J 0i ωi + δ iT q i ) + F i (ui )}

i=1

−

N 

ξ iT Q i ξ i +

i=1

=

N 

N 

ξ iT P i g i ωi

i=1

ωiT δ iT {[K ni C ni ]ξ i − C ni δ i ωi } +

i=1

−

N 

ξ iT Q i ξ i +

T N   δ i ωi i=1

ωiT F i (ui )

i=1

i=1

=−

N 

ξi

N 

ωiT δ iT [−K ni C ni ]ξ i

i=1



δω  i i ξi

 +

N 

ωiT F i (ui )

i=1

(4.18) C ni  where  = with  = [0 − C ni ]. Since C ni is positive definite and the T  Qi Schur complement of C ni in  is positive semidefinite,  is a positive semidefinite matrix. q i − δ i ωi = 0, i.e., η˙ i = 0, is the unique solution to the following equation



4.3 Coordinated Tracking Control

69



T

δ i ωi ξi



δω  i i ξi

 =0

(4.19)

Hence, V˙ can be written as V˙ = −k1

N 

ωiT ωei

i=1

−

T N   δ i ωi ξi

i=1

δω  i i ξi



 1 − μ0 + − k2 ωi + k1 ωei − k1 Sign(ωi )|ωei | μ0 i=1  t   t    N  T + ωi ρ i (t) k3 tanh ωei (τ )dτ − Sign(ωi )k3 tanh ωei (τ )dτ N 





ωiT ρ i (t)

0

i=1

≤−

N   i=1

− k1

δ i ωi ξi

N 

≤−

i=1

 

δ i ωi ξi

0

 −

N 

δ i ωi ξi

T

 

δ i ωi ξi

k2 ωiT ωi

i=1

ωiT (I 3 − ρ i (t))ωei −

i=1

N  

T

N  (1 − μ0 )k1 |ωi ||ωei | i=1

 −

N 

k2 ωiT ωi ≤ 0

i=1

(4.20) Hence, V is bounded, i.e. Ri , ηi , ωi and η˙ i are bounded. From the dynamic equations of the networked flexible spacecraft, both ω˙ i and η¨ i are bounded, i.e., V¨ is finite considering the boundedness of ρ i (t). According to Barbalat’s lemma, as t → ∞. Inthe case of ωi → 0 and η˙ i → 0, one has ηi → 0 and k1 ωei + V˙ → 0 t

ωei (τ )dτ → 0 from the dynamic equations. For the system governed  t  ωei (τ )dτ = 0, one has by k1 ωei + k3 tanh k3 tanh

0

0

ω˙ ei− j = −ktanh(ωei− j )

(4.21) 

t

where k = k3 /k1 and ωei− j is the j-th entry of 0

ωei (τ )dτ . By choosing

ln(cosh(ωei− j )) as Lyapunov function, it can be concluded that the system in Eq. (4.21) is asymptotically stable. Hence, ωei will converge to zero as t → ∞. From T − Rid )∨ → 0. According to Lemma 4.2, Lemma 4.1, (RiTj − Ri j )∨ → 0 and (Rid T ∨ the equation (Ri j − Ri j ) = 0 has two solutions, i.e., Ri = R j and tr(Ri j ) = −1. As shown in [21], these two solutions correspond to the minimum and maximum values of V1 , hence, they are the stable and unstable equilibriums of the closed-loop  system, respectively. Therefore, the conclusion in this theorem holds.

70

4 Rotation-Matrix-Based Attitude Tracking …

Remark 4.2 The proof of Theorem 4.1 depends on the structural damping term C ni . However, the structural damping of a flexible spacecraft is very low [22], i.e., it will take a long time to passively attenuate the flexible vibration. Hence, it is necessary to reveal the role of the controller (4.11) in the vibration control. To do this, the damping term is omitted in theory. In such a case, V˙ → 0 implies ωi → 0. If ωi ≡ 0 holds, the dynamic equations of the i-th flexible spacecraft can be rewritten as

  t δ iT η¨ i = ρ i (t) k1 ωei + k3 tanh 0 ωei (τ )dτ = ρ i (t) η¨ i + K ni ηi = 0

(4.22)

  t where = k1 ωei + k3 tanh 0 ωei (τ )dτ is a constant vector. Clearly, the solutions to η¨ i + K ni ηi = 0 are ηi ≡ 0 or simple harmonic oscillation. In the case of simple harmonic oscillation, each entry of ηi will vibrate with different frequency, hence, the first line of Eq. (4.22) cannot hold for the stochastic diagonal matrix ρ i (t) if = 0. Therefore, the unique solution to Eq. (4.22) is ηi = 0 and = 0. Hence, similar to the proof of Theorem 4.1, Ri → Rd , ηi → 0, ωi → 0 and η˙ i → 0 as time goes to infinity, i.e., the proposed controller is able to play a positive role in vibration suppression. Essentially, the controller can suppress the flexible vibration due to the rigid-flexible dynamics. However, if the corresponding elements of ρ i (t) and δ iT η¨ i have the same oscillation frequency, the controller probably does not have the ability to suppress the flexible vibration. Remark 4.3 If the node that can receive the information of the leader is numbered as k, denote the set M = {(R1 , . . . , R N , ω1 , . . . , ω N , η1 , . . . , η N , η˙ 1 , . . . , η˙ N )|(RiTj − T Ri j )∨ = 0, (Rkd − Rkd )∨ = 0, ωi = 0, ηi = 0, η˙ i = 0, i = 1, . . . , N , j ∈ Ni } and M0 = {(R1 , . . . , R N , ω1 , . . . , ω N , η1 , . . . , η N , η˙ 1 , . . . , η˙ N )| Ri = Rd , ωi = 0, ηi = 0, η˙ i = 0, i = 1, . . . , N }. The complement of M0 in M is represented by M1 = M\M0 . However, as indicated in the proof of Theorem 4.1, the states in M1 are unstable equilibriums of the closed-loop system. That is, if the system starts from one element in M1 , it will stay there. However, in practical applications, the ubiquitous disturbances may make the system deviate from M1 , hence, with the help of the ubiquitous disturbances, the networked flexible spacecraft will be driven to the desired attitude even from the states in M1 . Remark 4.4 As stated in [23], an important issue in vibration control is the control spillover. From Theorem 4.1, the proposed controller can achieve the control aim for the flexible spacecraft with the first n modes, where n can be chosen large enough such that the residual modes to the overall system becomes insignificant [24]. Hence, the proposed controller will not cause the spillover effects. In [15], a similar controller is presented for the attitude maneuvering of a flexible spacecraft. However, in this study, the distributed tracking controller is designed for the networked flexible spacecraft on S O(3) and the integral term is put in a hyperbolic tangent function to avoid the integral windup. Furthermore, the control

4.3 Coordinated Tracking Control

71

strategy in Eq. (4.11) is discontinuous because the sign function is involved. The discontinuity may cause high-frequency control chattering [25]. In order to eliminate the discontinuity, the function sign(x) can be replaced by the following continuous saturation function sat(x, α) = x/(|x| + α 2 ) (4.23) where is a small positive constant and α(t) > 0 is a time-varying variable to be defined. It is clear that xsat(x, α) can be rewritten as xsign(x) − |x|αg(x, α), where g(x, α) = α /(|x| + α 2 ) > 0. Based on the saturation function sat(x, α), the controller in Eq. (4.11) can be recast as  t  ui = −k2 ωi + k1 ωei + k3 tanh ωei (τ )dτ 0 (4.24)  t    1 − μ0 k1 |ωei | + k3 tanh ωei (τ )dτ − Sat(ωi , α i ) μ0 0 where the functions Sat(x, c) is defined as diag{sat(x1 , c1 ), sat(x2 , c2 ), sat(x3 , c3 )} for x = [x1 x2 x3 ]T and c = [c1 c2 c3 ]T . In the case where the j-th entries of ui and α i are denoted by u i− j and αi− j , j = 1, 2, 3, ωi− j ρi− j u i− j can be expressed as ωi− j ρi− j u i− j = ωi− j ρi− j u i− j  t    1 − μ0 + k1 ωei− j + k3 tanh ωei− j (τ )dτ μ0 0   · ωi− j ρi− j αi− j g ωi− j , αi− j

(4.25)

where u i− j is the j-th element of ui in Eq. (4.11). If αi− j is updated by the following law  t        α˙ i− j = −γ 1 − μ0 k1 ωei− j + k3 tanh ωei− j (τ )dτ ωi− j g ωi− j , αi− j 0

(4.26) where γ is a positive constant and αi− j (0) > 0, one has the following theorem. Theorem 4.2 Consider multiple flexible spacecraft governed by Eqs. (1.21) and (4.1) under the controller in Eq. (4.24) with the updating law in Eq. (4.26). The attitude tracking errors and flexible vibration will converge to zero almost globally asymptotically. Proof A candidate Lyapunov function Vs is defined as Vs = V +

3 N   1 2 α 2γ i− j i=1 j=1

(4.27)

72

4 Rotation-Matrix-Based Attitude Tracking …

where V is defined in Eq. (4.13). The time derivative of Vs satisfies the following inequality V˙s ≤ −

T N   δ i ωi ξi

i=1

+



δω  i i ξi

N  3   1 − μ0

μ0

i=1 j=1

 −

N 

k2 ωiT ωi

i=1

k1 |ωei− j |

 t   + k3 tanh ωei− j (τ )dτ |ωi− j |ρi− j αi− j g(ωi− j , αi− j ) 0  3 N   − αi− j (1 − μ0 )k1 |ωei− j | i=1 j=1

 t   + k3 tanh ωei− j (τ )dτ |ωi− j |g(ωi− j , αi− j ) 0  N N  δ i ωi T  δ i ωi   ≤−  k2 ωiT ωi − ξi ξi i=1

i=1

 t  3 N   − (1 − ρi− j )αi− j k3 tanh ωei− j (τ )dτ |ωi− j |g(ωi− j , αi− j ) 0

i=1 j=1

≤−

N   i=1

δ i ωi ξi

T

 

δ i ωi ξi

 −

N 

k2 ωiT ωi

i=1

 t N  3  2 2 − (1 − ρi− j )k3 ωei− j (τ )dτ |ωi− j |αi− j /(|ωi− j | + αi− j ) 0

i=1 j=1

≤−

T N   δ i ωi i=1

ξi

 

δ i ωi ξi

 −

N 

k2 ωiT ωi ≤ 0

i=1

(4.28) Similar to the proof of Theorem 4.1, it can be concluded that the conclusion of  this theorem holds. Remark 4.5 As shown in the proof of Theorem 4.2, the positive semi-definiteness of V˙s does not depend on αi− j , hence, the asymptotic stability of the closed-loop networked flexible spacecraft is established independently of the behavior of αi− j . Upon the updating law in Eq. (4.26), lim ωi− j = 0 implies lim α˙ i− j = 0, which t→∞ t→∞ means that αi− j will keep constant as time goes to infinity. A potential problem is that αi− j might go to zero before ωi− j arrives at zero. Such a situation is undesirable because in the case of αi− j = 0, the controller (4.24) becomes (4.11), which will cause the chattering problem. Based on the results in Theorem 4.2, there exists a constant β > 0 such that the following inequality holds

4.4 Numerical Simulations

73

 t        1 − μ0 k1 ωei− j + k3 tanh ωei− j (τ )dτ ωi− j g ωi− j , αi− j ≤ βαi− j

γ

0

(4.29)

Hence, Eq. (4.26) can be rewritten as α˙ i− j ≥ −βαi− j

(4.30)

With αi− j (0) > 0, integrating the preceding inequality yields αi− j (t) ≥ αi− j (0)exp(−βt)

(4.31)

From Eq. (4.31), it can be concluded that αi− j (t) has a positive lower bound for a finite time. In other words, in finite time, αi− j (0) > 0 holds and αi− j (t) cannot cross zero. Only at infinity, αi− j (t) may tend to zero.

4.4 Numerical Simulations This section takes four flexible spacecraft as example to verify the effectiveness of the proposed controller. The physical parameters of the four flexible spacecraft, the same as those in [15], are given in Table 4.1. The communication graph among the

Table 4.1 Parameters of flexible spacecraft Parameter Value ⎡ ⎤ 350 3 4 ⎢ ⎥ Ji ⎣ 3 270 10 ⎦ 4 10 190 ⎡ ⎤ 6.45637 1.27814 2.15629 ⎢ ⎥ δi ⎣ −1.25819 0.91756 −1.67264 ⎦ 1.11687 2.48901 −0.83674 1 2 3 ξ1 ξ2 ξ3 ρi− j

0.7681 1.1038 1.8733 0.0056 0.0086 0.013 1, t < 100 s 0.8 + 0.15sin(10t), t ≥ 100 s

Unit kg m2

kg1/2 m/s2 rad/s rad/s rad/s – – – –

74

4 Rotation-Matrix-Based Attitude Tracking …

Fig. 4.1 Communication graph

four flexible spacecraft is illustrated in Fig. 4.1. The desired attitude, i.e., the attitude of the leader, is Rd = I 3 . The time history of the function V1 defined in Eq. (4.14) will be presented in the following simulation results to reveal the tracking errors because, as shown in the proof of Theorem 4.1, the distributed attitude tracking is achieved when V1 = 0 is satisfied.

4.4.1 Case I: Fault-Tolerant Control The controller in Eq. (4.24) is verified through this simulation case. The parameters k1 , k2 , k3 , μ0 , γ and are chosen as 5, 10, 1, 0.5, 10 and 0.0001, respectively. At t = 0 s, these four spacecraft are rest at the following attitude without any flexible deformations. ⎡ ⎤ 0.7071 0.5 −0.5 R1 (0) = ⎣ 0 0.7071 0.7071 ⎦ , R2 (0) = I 3 ⎡ 0.7071 −0.5 0.5 ⎤ ⎡ ⎤ (4.32) 0.3062 0.9186 −0.25 −0.866 0 −0.5 R3 (0) = ⎣ −0.3536 0.3536 0.866 ⎦ , R4 (0) = ⎣ −0.5 0 0.866 ⎦ 0.8839 −0.1768 0.433 0 1 0 The initial value of ρi− j is set as 1 for i = 1, 2, 3, 4 and j = 1, 2, 3. The time histories of V1 , angular velocities, modal coordinates and control inputs are shown in Figs. 4.2, 4.3, 4.4 and 4.5. From Figs. 4.2 and 4.3, it can be observed that the proposed controller can compensate for the partial loss of actuator effectiveness and drive the four spacecraft to track the leader gradually. As indicated in Fig. 4.4, the flexible vibration converges to zero at about 250 s.

4.4 Numerical Simulations

75

25 4

20 15

× 10-4

V1

2

10

0 800

5

850

900

950

1000

0 0

100

200 300 Time (s)

400

500

Fig. 4.2 The time history of V1 (Case I)

ω 1 (rad/s)

0.2 0 ω 1-1

ω 1-2

ω 1-3

-0.2 0

100

200 300 Time (s)

400

500

ω 2 (rad/s)

0.2 0 ω

2-1

ω

ω

2-2

2-3

-0.2 0

100

200 300 Time (s)

400

500

ω 3 (rad/s)

0.2 0 ω 3-1

ω 3-2

ω 3-3

-0.2 0

100

200 300 Time (s)

400

500

ω 4 (rad/s)

0.25 0 ω 4-1

ω 4-2

ω 4-3

-0.25 0

100

200 300 Time (s)

Fig. 4.3 The time history of angular velocity in Case I

400

500

76

4 Rotation-Matrix-Based Attitude Tracking …

η1

Fig. 4.4 The time history of modal coordinate in Case I

0.8 0.4 0 -0.4 -0.8

η1-1

0

100

η1-2

200 300 Time (s)

η1-3

400

500

η2

0.5 0 η

-0.5 0

100

2-1

η

η

2-2

200 300 Time (s)

2-3

400

500

η3

0.5 0 η3-1

η3-2

η3-3

-0.5 0

100

200 300 Time (s)

400

500

η4

0.5 0 η

4-1

η

η

4-2

4-3

-0.5 0

100

200 300 Time (s)

400

500

4.4.2 Case II: Fault-Tolerant Control from M1 with Initial Disturbance In this case, to verify the statements in Remark 4.3, the initial conditions are set as follows R1 (0) = R2 (0) = R3 (0) = R4 (0) = diag{1, −1, −1}, ω1 (0) = ω2 (0) = ω3 (0) = ω4 (0) = 0, (4.33) η1 (0) = η2 (0) = η3 (0) = η4 (0) = 0, η˙ 1 (0) = η˙ 2 (0) = η˙ 3 (0) = η˙ 4 (0) = 0 Furthermore, it is assumed that there exists a small initial disturbance acting on the first order modal coordinate of the fourth flexible spacecraft. That is, in this case, the group of flexible spacecraft will start from M1 , but the system is subject to a small disturbance at t = 0. The control parameters k1 , k2 , k3 , μ0 , γ and are chosen as 5, 1, 5, 10 and 0.0001, respectively. The time history of V1 is illustrated in Fig. 4.6.

4.4 Numerical Simulations

77

Fig. 4.5 Control command in Case I

Fig. 4.6 The time history of V1 in Case II

V1 will converge to zero, but at the beginning, it is nearly constant. This is because the control input is very small in the close neighborhood of M1 . In summary, the small initial disturbance can make the system deviate from M1 and then the controller is able to drive the networked flexible spacecraft to the desired attitude.

78

4 Rotation-Matrix-Based Attitude Tracking …

4.4.3 Case III: Response Without Structural Damping Under Controller (4.24) In Case I, the role of the controller in vibration control is uncertain. Hence, as stated in Remark 4.2, the structural damping can be omitted in this case such that the vibration cannot be suppressed passively. All control parameters are chosen the same as those in Case I. The responses of the modal coordinates are given in Fig. 4.7. It can be found that the flexible vibrations can be damped out even without structural damping, but the settling time is a little longer than that in Case I. The main reason is that the structural damping is able to keep consuming system energy and then accelerate the vibration suppression. In conclusion, since each element of ρ i (t) has a single frequency only after t = 100 s with the selected parameters, the corresponding entries of δ iT η¨ i and ρ i (t) have different oscillation frequencies, hence, the proposed controller plays a role in vibration suppression.

Fig. 4.7 The time history of modal coordinate in Case II

References

79

4.5 Conclusions In this study, a fault-tolerant controller is proposed to solve the distributed tracking problem of a group of flexible spacecraft on S O(3) under an undirected communication graph. The attitude of each spacecraft is represented by a rotation matrix and it is assumed that the spacecraft is subject to partial loss of actuator effectiveness. A distributed fault-tolerant controller is proposed with the feedback of the information of rigid bodies only based on the sign function. The sign function is then replaced by a saturation function with an adaptive parameter to eliminate the discontinuity. Theoretical proofs and numerical simulations reveal that the distributed tracking and vibration suppression can be achieved almost globally considering the actuator faults.

References 1. Gui, H., Vukovich, G.: Distributed almost global finite-time attitude consensus of multiple spacecraft without velocity measurements. Aerosp. Sci. Technol. 75, 284–296 (2018) 2. Abdessameud, A., Tayebi, A.: Attitude synchronization of a group of spacecraft without velocity measurements. IEEE Trans. Autom. Control 54(11), 2642–2648 (2009) 3. Zou, A.M., Kumar, K.D.: Quaternion-based distributed output feedback attitude coordination control for spacecraft formation flying. J. Guid. Control. Dyn. 36(2), 548–556 (2013) 4. Meng, Z., Ren, W., You, Z.: Decentralised cooperative attitude tracking using modified Rodriguez parameters based on relative attitude information. Int. J. Control 83(12), 2427– 2439 (2010) 5. Chaturvedi, N.A., Sanyal, A.K., McClamroch, N.H.: Rigid-body attitude control. IEEE Control Syst. Mag. 31(3), 30–51 (2011) 6. Lee, T.: Exponential stability of an attitude tracking control system on S O(3) for large-angle rotational maneuvers. Syst. Control Lett. 61(1), 231–237 (2012) 7. Bohn, J., Sanyal, A.K.: Almost global finite-time stabilization of rigid body attitude dynamics using rotation matrices. Int. J. Robust Nonlinear Control 26(9), 2008–2022 (2016) 8. Thakur, D., Akella, M.R.: Gyro-free rigid-body attitude stabilization using only vector measurements. J. Guid. Control. Dyn. 38(4), 811–818 (2014) 9. Mayhew, C.G., Teel, A.R.: Synergistic hybrid feedback for global rigid-body attitude tracking on S O(3). IEEE Trans. Autom. Control 58(11), 2730–2742 (2013) 10. Song, W., Tang, Y., Hong, Y., Hu, X.: Relative attitude formation control of multi-agent systems. Int. J. Robust Nonlinear Control 27(18), 4457–4477 (2017) 11. Du, H., Li, S.: Attitude synchronization control for a group of flexible spacecraft. Automatica 50(2), 646–651 (2014) 12. Du, H., Li, S.: Attitude synchronization for flexible spacecraft with communication delays. IEEE Trans. Autom. Control 61(11), 3625–3630 (2016) 13. Zou, A.M., de Ruiter, A.H., Kumar, K.D.: Distributed finite-time velocity-free attitude coordination control for spacecraft formations. Automatica 67, 46–53 (2016) 14. Du, H., Chen, M.Z., Wen, G.: Leader-following attitude consensus for spacecraft formation with rigid and flexible spacecraft. J. Guid. Control. Dyn. 39(4), 944–951 (2016) 15. Xiao, B., Hu, Q., Zhang, Y.: Fault-tolerant attitude control for flexible spacecraft without angular velocity magnitude measurement. J. Guid. Control. Dyn. 34(5), 1556–1561 (2011) 16. Hu, Q., Xiao, B.: Fault-tolerant sliding mode attitude control for flexible spacecraft under loss of actuator effectiveness. Nonlinear Dyn. 64(1–2), 13–23 (2011) 17. Sakthivel, R., Selvi, S., Mathiyalagan, K.: Fault-tolerant sampled-data control of flexible spacecraft with probabilistic time delays. Nonlinear Dyn. 79(3), 1835–1846 (2015)

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18. Huang, D., Wang, Q., Duan, Z.: Distributed attitude control for multiple flexible spacecraft under actuator failures and saturation. Nonlinear Dyn. 88(1), 529–546 (2017) 19. Zou, Y., Meng, Z., Zuo, Z.: Rotation-matrix-based attitude synchronization of multiple spacecraft without velocity measurements. In: 2017 11th Asian Control Conference (ASCC), pp. 96–101 (2017) 20. Abdessameud, A., Tayebi, A.: Motion Coordination for VTOL Unmanned Aerial Vehicles: Attitude Synchronisation and Formation Control. Springer Science & Business Media, London (2013) 21. Zheng, Z., Song, S.: Autonomous attitude coordinated control for spacecraft formation with input constraint, model uncertainties, and external disturbances. Chin. J. Aeronaut. 27(3), 602– 612 (2014) 22. Chen, T., Shan, J., Wen, H.: Distributed adaptive attitude control for networked underactuated flexible spacecraft. IEEE Trans. Aerosp. Electron. Syst. 55(1), 215–225 (2019) 23. Jia, S., Shan, J.: Optimal actuator placement for constrained gyroelastic beam considering control spillover. J. Guid. Control. Dyn. 41(9), 2073–2081 (2018) 24. Meirovitch, L., Baruh, H.: Robustness of the independent modal-space control method. J. Guid. Control. Dyn. 6(1), 20–25 (1983) 25. Xu, H., Mirmirani, M.D., Ioannou, P.A.: Adaptive sliding mode control design for a hypersonic flight vehicle. J. Guid. Control. Dyn. 27(5), 829–838 (2004)

Chapter 5

Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

Abstract This chapter presents the distributed adaptive fault-tolerant control for the attitude tracking of multiple flexible spacecraft on S O(3) without modal variable measurement. Assume that the communication graph among the followers are undirected and connected and there exists at least one follower linked to the leader. To deal with the distributed tracking on S O(3), a finite-time observer is designed to estimate the leader’s information for the followers. A distributed adaptive fault-tolerant controller is proposed to achieve the attitude tracking based on the estimation of the unmeasurable modal variables. The separation principle between the finite-time observer and the proposed controller is adopted to prove the controller convergence. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed control protocols.

5.1 Introduction Cooperative attitude tracking of multiple spacecraft has attracted significant attention due to its wide applications such as interferometry, Earth monitoring and on-orbit service [1]. For the attitude consensus of multiple flexible spacecraft, Du and Li addressed the attitude synchronization of multiple flexible spacecraft [2], which is more difficult and challenging than that of rigid ones due to rigid-flexible dynamics. The follow-up studies focused on such topic and designed several control schemes based on adaptive control [3], backstepping technique [4, 5] and sliding-mode control [6]. However, in most studies, MRPs or unit quaternions are adopted to describe the attitude kinematics. The minimal representations, such as the Euler angles and MRPs, have kinematic singularities. The quaternion representation is ambiguous, i.e., there are two unit quaternion vectors corresponding to the same attitude. This Reproduced from Ti Chen and Jinjun Shan. Distributed adaptive fault-tolerant attitude tracking of multiple flexible spacecraft on S O(3). Nonlinear Dynamics 2019; 95: 1827–1839. Copyright c Springer Nature B.V. 2018. All rights reserved. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_5

81

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5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

ambiguity may cause the unstable unwinding phenomenon for the continuous quaternion feedback control [7]. Hence, attention has been drawn to the attitude tracking based on rotation matrix [7–10], which is a unique and global attitude representation. Furthermore, the severe operating conditions increase the possibility of actuator faults [11], which can unlikely be fixed on orbit. This motivates the fault-tolerant control of single or multiple spacecraft [12–16]. For the attitude control of a flexible spacecraft with actuator faults, various control strategies have been proposed [17–20]. For example, Hu and Xiao considered the fault-tolerant attitude control for a flexible spacecraft with actuator faults and uncertain inertia parameters based on the sliding mode control technique [19]. Cao et al. developed a terminal sliding mode and an adaptive sliding mode control scheme for the attitude tracking of a flexible spacecraft subject to external disturbance, inertia uncertainties, external disturbance,wheel torque saturation, and configuration misalignment [21]. Recently, Huang et al. considered the distributed fault-tolerant control for a group of flexible spacecraft [22]. However, all these studies on the control of flexible spacecraft with actuator faults are based on MPRs or unit quaternions. Hence, to avoid the singularities and ambiguities of attitude representations, this chapter addresses the distributed adaptive fault-tolerant control for the attitude tracking and vibration suppression of multiple flexible spacecraft subject to external disturbances and without modal variable measurement. Essentially, this chapter extends the work in last chapter to the case under connected undirected graphs.

5.2 Problem Formulation 5.2.1 Kinematics and Dynamics of Flexible Spacecraft The attitude kinematics of the i-th spacecraft can be described on S O(3) by Eq. (1.21), where Ri ∈ S O(3) is the rotation matrix from the body frame to the inertial frame and ωi ∈ R3 is the angular velocity. The dynamic equations of the i-th flexible spacecraft with actuator faults and external disturbances can be expressed as [19, 23] J i ω˙ i + δ iT η¨ i = −ωi× ( J i ωi + δ iT η˙ i ) + F i (ui ) + d¯ i (5.1) η¨ i + C ni η˙ i + K ni ηi = −δ i ω˙ i where J i ∈ R3×3 is the total inertia matrix of the flexible spacecraft, δ i ∈ Rn×3 is the rigid-flexible matrix, ηi =∈ Rn is the modal coordinate vector, d¯ i ∈ R3 is the external disturbance vector, K ni = diag{21 , . . . , 2n } ∈ Rn×n is the stiffness matrix and C ni = diag{2ξ1 1 , . . . , 2ξn n } ∈ Rn×n is the damping matrix, in which  j and ξ j are the j-th order natural frequency and damping ratio, respectively. In this study, two types of actuator faults are taken into account, namely partial loss of effectiveness and additive bias fault. The control input vector with faults can be expressed as

5.3 Distributed Attitude Tracking Control

83

F i (ui ) = ρ i (t)ui + f i (t)

(5.2)

in which ui = [u i−1 u i−2 u i−3 ]T ∈ R3 , ρ i (t) = diag{ρi−1 (t), ρi−2 (t), ρi−3 (t)} ∈ R3×3 and f i (t) = [ f i−1 (t) f i−2 (t) f i−3 (t)]T ∈ R3 are the desired control torque, the effectiveness factor of spacecraft actuators and the additive bias fault, respectively. It is assumed that 0 < μ0 ≤ ρi− j (t) ≤ 1 and | f i− j (t)| ≤ μ1 hold for i = 1, . . . , N and j = 1, 2, 3, where μ0 and μ1 are positive constants. Let q i = η˙ i + δ i ωi . The kinematic and dynamic equations of the i-th flexible spacecraft can be expressed as Eq. (1.21) and η˙ i = q i − δ i ωi J 0i ω˙ i =

−ωi× ( J 0i ωi

+

δ iT q i )

+

δ iT (C ni (q i

(5.3) − δ i ωi ) + K ni ηi ) + F i (ui ) + d¯ i (5.4)

q˙ i + C ni (q i − δ i ωi ) + K ni ηi = 0

(5.5)

where J 0i = J i − δ iT δ i .

5.2.2 Control Objective This work aims to design a distributed fault-tolerant controller for multiple flexible spacecraft to track a time-varying leader with the following assumptions. Assumption 5.1 The undirected communication graph among the N flexible spacecraft is connected. There exists at least one node that can receive the information of the leader. Assumption 5.2 It is assumed that the leader’s angular velocity and its first two derivatives are bounded, i.e., there exist three positive constants such that ωd ∞ ≤ γ1 , ω˙ d ∞ ≤ γ2 and ω¨ d ∞ ≤ γ3 . Assumption 5.3 The external disturbance d¯ i is bounded, i.e.,  d¯ i 2 ≤ μ2 . Assumption 5.4 The modal variable vector ηi and the modal velocity vector η˙ i of each flexible spacecraft cannot be measured.

5.3 Distributed Attitude Tracking Control 5.3.1 Finite-time Distributed Observer As stated in [24], the finite-time observer is a simple way to realize the separation principle, known as a principle of separation of estimation and control. It is assumed

84

5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

˙ d = R d ω× that the states of the leader are governed by R d . To estimate the leader’s information for each spacecraft in the tracking task, inspired by [25, 26], the finitetime distributed observer is designed as follows   ˙ˆ = −r sigα1  N a ( R ˆ ˆ ˆ − R ) + b ( R − R ) R i 1 i j i j i i d j=1   N ˆ ˆ ˆ a ( R − R ) + b ( R − R ) −r2 sign i j i j i i d j=1   N α2 g˙ i = −r3 sig j=1 ai j (g i − g j ) + bi (g i − ωd )   N −r4 sign j=1 ai j (g i − g j ) + bi (g i − ωd )   N ˙ a (h − h ) + b (h − ω ) h˙ i = −r5 sigα3 i j i j i i d j=1   N ˙ a (h − h ) + b (h − ω ) −r6 sign i j i i d j=1 i j

(5.6)

√ ˆ i ∈ R3×3 , g i ∈ R3 , hi ∈ R3 , r1 > 0, r2 > 2 3γ1 , r3 > 0, r4 > γ2 , r5 > 0, where R ˆ i , g i and r6 > γ3 , 0 < α1 < 1, 0 < α2 < 1 and 0 < α3 < 1. The initial values of R hi are chosen as Ri (0), ωi (0) and ω˙ i (0), respectively. Basically, in each spacecraft, a finite-time observer is embedded to estimate the attitude, angular velocity and ˆ i , g i and hi are the estimations of Rd , ωd angular acceleration of the leader, i.e., R ˆ i ) ∈ R9 , which is obtained ¯ ri = reshape( R and ω˙ d for the i-th spacecraft. Define R by reshaping a matrix of 3 × 3 into a 9-dimensional vector. Hence, Eq. (5.6) can be rewritten as     r r ˙˜ r + R ˙¯ r = −r sigα1 (H ⊗ I ) R ˜ ˜ − r sign (H ⊗ I ) R R 1 9 2 9 d     ˙g˜ + ω˙¯ d = −r3 sigα2 (H ⊗ I 3 ) g˜ − r4 sign (H ⊗ I 3 ) g˜ (5.7)     h˙˜ + ω¨¯ d = −r5 sigα3 (H ⊗ I 3 ) h˜ − r6 sign (H ⊗ I 3 ) h˜ ¯ r1T . . . R ¯ rNT ]T − R ¯ rd , ω¯ d = (I 3 ⊗ ωd ), ¯ rd = (1 N ⊗ reshape(Rd )), R ˜ r = [R where R g˜ = [g 1T . . . g TN ]T − ω¯ d and h˜ = [h1T . . . h TN ]T − ω˙¯ d . √ Lemma 5.1 If r1 > 0, r2 > 2 3γ1 , r3 > 0, r4 > γ2 , r5 > 0, r6 > γ3 , 0 < α1 < 1, 0 < α2 < 1 and 0 < α3 < 1 are satisfied, with Assumptions 5.1 and 5.2, the disˆ i → Rd , tributed observer in Eq. (5.6) is globally finite-time convergent, i.e., R g i → ωd and hi → ω˙ d in finite time. ˆ i , construct the following Lyapunov function Proof: For R VoR =

1 ˜ rT ˜r R (H ⊗ I 9 ) R 2

Clearly, VoR is positive definite and satisfies VoR ≤ the time derivative of VoR is

(5.8)

λmax (H) ˜ 2  R2 . Along Eq. (5.7), 2

5.3 Distributed Attitude Tracking Control

85

˙¯ r ˜ H ) − r2  R ˜ H 1 − R ˜ TH R ˜ TH sigα1 ( R V˙oR = −r1 R d

(5.9)

˙¯ r , one has ˜ r . For R ˜ H = (H ⊗ I 9 ) R where R d ˙¯ r  = reshape( R ˙ d )∞ ≤  R ˙ R d ∞ √d ∞ √ × ≤ Rd ∞ ωd ∞ ≤ 2 3Rd 2 ωd ∞ = 2 3ωd ∞

(5.10)

Similar to the proof of Proposition 1 in [26], it follows that V˙oR

√ 1 ˜ H 1+α ˜ H 1 ≤ −r1 (9N )−α1  R − (r2 − 2√3ωd ∞ ) R 1 1 ˜ H 1+α ˜ H 2 ≤ −r1 (9N )−α1  R − (r − 2 3ω  ) R 2 2 (1+α1d)/2∞ 2VoR ≤ −r1 (9N )−α1 λmin (H)1+α1 λmax (H) 1/2  √ 2VoR −(r2 − 2 3γ1 )λmin (H) λmax (H)

(5.11)

ˆ i → Rd in finite time and the settling time TR satisfies the following Hence, R inequality

√ 2λmax (H)VoR (0) r1 (9N )α1 λmax (H)(1+α1 )/2 [2VoR (0)](1−α1 )/2 TR ≤ min , √ (1 − α1 )λmin (H)1+α1 (r2 − 2 3γ1 )λmin (H) (5.12) Similarly, by choosing the Lyapunov function as Vog =

1 T g˜ (H ⊗ I 3 ) g˜ 2

(5.13)

Voh =

1 ˜T h (H ⊗ I 3 ) h˜ 2

(5.14)

it can be proved that g i → ωd and hi → ω˙ d in finite time Tg and Th satisfying

 2λmax (H)Vog (0) r3 (3N )α2 λmax (H)(1+α2 )/2 [2Vog (0)](1−α2 )/2 Tg ≤ min , (1 − α2 )λmin (H)1+α2 (r4 − γ2 )λmin (H) (5.15)

√ r5 (3N )α3 λmax (H)(1+α3 )/2 [2Voh (0)](1−α3 )/2 2λmax (H)Voh (0) Th ≤ min , (1 − α3 )λmin (H)1+α3 (r6 − γ3 )λmin (H) (5.16)  Remark 5.1 The observer in Eq. (5.6) enables all spacecraft to obtain the information of the leader. As shown in Lemma 5.1, the distributed observer can track the attitude, angular velocity and angular acceleration of the leader in a finite time

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5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

and the convergence time can be arbitrarily short by properly selecting the observer parameters. Remark 5.2 In [25], a similar finite-time observer is used to solve the quaternionbased attitude consensus control of multiple rigid bodies. However, the observer in Eq. (5.6) has higher dimensions due to the attitude representation on S O(3) and the convergence conditions are different from those in [25]. Remark 5.3 Based on the positive semi-definiteness of V˙oR , V˙oh and V˙og and the ˆ i , g i and hi are bounded. boundedness of Rd , ωd and ω˙ d , it can be concluded that R

5.3.2 Modal Variable Observer According to Assumption 5.4, ηi and q i in Eq. (5.4) cannot be used during the controller design. Hence, similar to [2], a modal variable observer for the i-th spacecraft is designed as follows η˙ˆ i = qˆ i − δ i ωi q˙ˆ i + C ni (qˆ i − δ i ωi ) + K ni ηˆ i = 0

(5.17)

Lemma 5.2 ηˆ i and qˆ i in the observer (5.17) will converge to ηi and q i globally asymptotically. Proof: Denote eηi = ηˆ i − ηi and eqi = qˆ i − q i . From Eqs. (5.3), (5.5) and (5.17), one has e˙ ηi = eqi (5.18) e˙ qi + C ni eqi + K ni eηi = 0 A Lyapunov function candidate is Vom =

1 T 1 T eqi e K ni eηi + eqi 2 ηi 2

(5.19)

Differentiating Vom along Eq. (5.18) yields T C ni eqi ≤ 0 V˙om = −eqi

(5.20)

In the case of V˙om = 0, eqi = 0 and eηi = 0 hold from Eq. (5.18). Hence, according to LaSalle’s invariant principle, eqi and eηi will converge to zero globally asymptot ically. Remark 5.4 Due to the positive semi-definiteness of V˙om , both eηi and eqi are bounded, i.e., there exist two positive constants φ1 and φ2 such that eηi 2 ≤ φ1 and eqi 2 ≤ φ2 hold.

5.3 Distributed Attitude Tracking Control

87

5.3.3 Controller Design To facilitate the convergence proof of the distributed tracking controller, this section will firstly address the centralized tracking problem, where each spacecraft knows the desired attitude and angular velocity. The attitude and angular velocity tracking errors are defined as Rei = (RdT Ri − RiT Rd )∨ ∈ R3 (5.21) ωei = ωi − RiT Rd ωd ∈ R3 ˙ ei and ω˙ ei can be expressed as R × T T T ∨ ˙ ei = (RdT Ri ω× R ei + ωei R i R d ) = [(tr(R i R d )I 3 − R i R d )ωei ] × T T ω˙ ei = ω˙ i − Ri Rd ω˙ d + ωi Ri Rd ωd

(5.22)

Let the sliding variable as si = ωei + β Rei . Hence, s˙ i = ω˙ i + i , where i = ˙ ei . Due to the orthogonality of Ri and Rd , the terms −RiT Rd ω˙ d + ωi× RiT Rd ωd + β R in i have the following properties √ RiT Rd ω˙ d 2 ≤ ω˙ d 2 ≤ 3γ2 √ √ ωi× RiT Rd ωd 2 ≤ ωi× 2 ω i ∞ . . . 3γ1 ≤ 6γ1 ωi 2 √d 2 ≤ T2 3ω × T ˙ ei 2 ≤ 3β R ˙ ei ∞ ≤ 3 3βRd Ri ω× β R i R d 2 ei + ωei R√ √ × ≤ 6 3βωei 2 ≤ 36βωei 2 ≤ 36βωi 2 + 36 3βγ1

(5.23)

From Eq. (5.4), one has J 0i s˙ i =  i − β Rei + ρ i (t)ui

(5.24)

where  i = J 0i i − ωi× ( J 0i ωi + δ iT q i ) + δ iT (C ni (q i − δ i ωi ) + K ni ηi ) +β Rei + f i (t) + d¯ i

(5.25)

√ ωi× J 0i ωi 2 ≤ 2√ 3λmax ( J 0i )ωi 22 ωi× δ iT q i 2 ≤ 2 3δ i 2 ωi 2 q i 2 δ iT C ni (q i − δ i ωi )2 ≤ δ i 2 λmax (C ni )q i 2 + δ i 22 λmax (C ni )ωi 2 δ iT K ni ηi 2 ≤ δ i 2 λmax (K ni )ηi 2 √ βRei ∞ ≤ 2βRdT Ri − RiT Rd ∞ ≤ 4 3β

(5.26)

and

Considering the boundedness of f i and d¯ i , there always exist positive constants c, ¯ c0 , c1 , c2 , c3 , c4 and c5 such that

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5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

 i 2 ≤ c0 + c1 ωi 2 + c2 ωi 22 + c3 ωi 2 q i 2 + c4 q i 2 + c5 ηi 2 ≤ c ¯ i (5.27) where i = 1 + ωi 2 + ωi 22 + ωi 2 q i 2 + q i 2 + ηi 2 . Replacing ηi and q i with ηˆ i and qˆ i yields ˆ i = 1 + ωi 2 + ωi 22 + ωi 2 qˆ i 2 + qˆ i 2 + ηˆ i 2 

(5.28)

˜i ≤ where ηˆ i and qˆ i are governed by Eq. (5.17). According to Remark 5.4, one has  ˜ i = i −  ˆ i . Hence, (1 + ωi 2 )eqi 2 + eηi 2 ≤ φ2 ωi 2 + φ1 + φ2 , where  there must exist a positive constant Ci such that ˆi ˆ i + c(φ  i 2 ≤ c ¯ i ≤ c¯ ¯ 2 ωi 2 + φ1 + φ2 ) ≤ Ci 

(5.29)

The control command is then designed as u¯ i = −k0 si − where k0 > 0, =

μ ˆi 1+

ˆ i si Cˆ i  si 2 +

(5.30)

, μ > 0 and Cˆ i is updating by the following law ˆi si 22  C˙ˆ i = −σ1 Cˆ i + σ2 si 2 +

(5.31)

in which σ1 > 0, σ2 > 0 and Cˆ i (0) > 0. Theorem 5.1 The centralized tracking errors and flexible vibrations of the networked flexible spacecraft governed by Eqs. (1.21) and (5.1) under the controller in Eq. (5.30), the updating law in Eq. (5.31) and the observer in Eq. (5.17) are bounded with Assumptions 5.2 and 5.4. Proof: Construct the following Lyapunov function for the i-th spacecraft Vi =

1 T 1 si J 0i si + (Ci − μ0 Cˆ i )2 + βtr(I 3 − RdT Ri ) 2 2σ2 μ0

The time derivative of Vi is

(5.32)

5.3 Distributed Attitude Tracking Control

89

1 V˙i = siT J 0i s˙ i + (μ0 Cˆ i − Ci )C˙ˆ i + βωeiT Rei σ2

ˆ ˆ  s C 1 i i i = siT  i + siT ρ i −k0 si − − β 2 ReiT Rei + (μ0 Cˆ i − Ci )C˙ˆ i si 2 + σ2 ˆ μ s 2 Cˆ  1 ˆ i si 2 − μ0 k0 si 22 − i i 0 i 2 − β 2 Rei 22 + (μ0 Cˆ i − Ci )C˙ˆ i ≤ Ci  si 2 + σ2 ˆ i si 2 Ci  σ 1 2 2 2 = −μ0 k0 si 2 − β Rei 2 + + (Ci − μ0 Cˆ i ) Cˆ i si 2 + σ2 σ1 ≤ −μ0 k0 si 22 − β 2 Rei 22 + Ci μ + (Ci − μ0 Cˆ i ) Cˆ i σ2 (5.33) It should be noted that 1 1 (Ci − μ0 Cˆ i )Cˆ i = − (μ20 Cˆ i2 − μ0 Ci Cˆ i ) = − μ0 μ0

  C2 Ci 2 ˆ μ0 Ci − + i (5.34) 2 4μ0

Hence, V˙i satisfies the following inequality V˙i ≤ −μ0 k0 si 22 − β 2 Rei 22 + 0

(5.35)

σ C2

where 0 = Ci μ + 4μ1 0 σi 2 , which has positive correlation with μ and σσ21 . (s , Rei ) are outside of the set B1 = that V˙i < 0 holds when  It is clear √  i  0 0  (si , Rei ) si 2 ≤ μ0 k0 , Rei 2 ≤ β . Once the system moves in B1 , (si , Rei ) cannot get out. Thus, si and Rei are bounded globally. From the definition of si and the boundedness of ωd , it can be concluded that ωei and ωi are bounded. The further analysis of the attitude tracking and vibration suppression is divided into two steps. Step 1: Rei will converge to a close neighborhood of the origin. In fact, the possible solutions to Rei = 0 are Ri = Rd and tr(Rei ) = −1. That is, the equilibriums of the closed-loop attitude dynamics of the networked flexible spacecraft without actuator faults and external disturbances are 1 = {RdT Ri = I, ωi = 0, ηi = 0, η˙ i = 0} 2 = {tr(Rei ) = −1, ωi = 0, ηi = 0, η˙ i = 0}

(5.36)

Note that Rei defined in Eq. (5.21) is the gradient of the potential function tr(I 3 − RdT Ri ) ∈ [0, 4]. Obviously, RdT Ri = I, i.e., Ri = Rd , is located at the minimum of tr(I 3 − RdT Ri ) and tr(Rei ) = −1 is corresponding to the maximum of the potential function. Hence, 1 is the unique stable equilibrium of the overall system and 2 is unstable one. Theoretically, if the system starts from 2 , it will stay there. However, the time-varying desired attitude, the external disturbances and the additive actuator bias faults of concern will push the system out of the unstable equilibriums. Therefore, it can be concluded that the networked flexible spacecraft governed by Eqs. (1.21) and (5.1) with the controller u¯ i will converge to the set B1 .

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5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

Step 2: Consider the modal dynamics governed by Eqs. (5.3) and (5.5), which can be rewritten as ξ˙ i = i ξ i + g i ωi (5.37) 

where ξi =

     ηi 0 In −δ i , i = , gi = qi −K ni −C ni C ni δ i

(5.38)

Obviously, all the eigenvalues of i have negative real parts, hence, ξ˙ i = i ξ i is uniformly asymptotically stable. Therefore, with the bounded ωi , the responses of ηi and q i are bounded.  However, for the distributed tracking problem, only some spacecraft are able to receive the leader’s information. Based on the observer in Eq. (5.6), the sliding ˆ i , g i and hi , which are variable sˆ i can be calculated by replacing R d , ωd and ω˙ d with R bounded according to Remark 5.3. Hence, Eqs. (5.27) and (5.29) still hold. Therefore, for all spacecraft in the tracking task, as shown in Fig. 5.1, the distributed control command ui can be obtained with sˆ i and the updating law in Eq. (5.31). It should be ˆ i , g i and hi are used to design a controller even for the noted that the estimation R followers linked to the leader. Theorem 5.2 Under the control command ui with the updating law in Eq. (5.31), the tracking errors and the flexible vibrations of the networked flexible spacecraft subject to the actuator faults and external disturbance with observers in Eqs. (5.6) and (5.17) are bounded with Assumptions 5.1−5.4. Proof: From Lemma 5.1, it follows that ui = u¯ i holds for t ≥ Tc = max{TR , Tg , Th }. That is, ui will become the controller in Eqs. (5.30) and (5.31) when t ≥ Tc . Consider the following total energy of a flexible spacecraft      1 1 ωi T J i δ iT ωi + ηiT K ni ηi η˙ i δ In 2 η˙ i 2 1 T 1 T 1 T = ωi J 0i ωi + q i q i + ηi K ni ηi 2 2 2

Ei =

(5.39)

and its time derivative is T ¯ E˙ i = ωiT (ρ i (t)u  i + f i (t) + d i ) − η˙ i C ni η˙ i  ˆ i (si + i ) Cˆ i  T = ωi ρ i (t) −k0 (si + i ) − + ωiT ( f i (t) + d¯ i ) − η˙ iT C ni η˙ i si + i 2 + (5.40) where i , the difference between si and sˆ i , is bounded. That is, i 2 ≤ Di1 holds, where Di1 is a positive constant. From the definition of si , one has si = ωi + s¯ i , where s¯ i = −RiT Rd ωd + (RdT Ri − RiT Rd )∨ . Note that s¯ i is bounded, i.e., there exists a positive constant Di2 such that ¯si 2 ≤ Di2 . Hence, E˙ i satisfies the following inequality

5.3 Distributed Attitude Tracking Control

91

Fig. 5.1 The overall scheme of the proposed distributed attitude controller in this chapter

E˙ i ≤ −k0 μ0 ωi 22 + (Di1 + Di2 )k0 ωi 2 − μ0 ωi 22

ˆi Cˆ i  si + i 2 +

ˆi Cˆ i  + (μ1 + μ2 )ωi 2 si + i 2 + ˆi Cˆ i  ≤ (Di1 + Di2 )k0 ωi 2 + (Di1 + Di2 )ωi 2 si + i 2 + +(μ1 + μ2 )ωi 2 +(Di1 + Di2 )ωi 2

(5.41)

If the system response can escape within finite time, both ωi 2 and ηi 2 will go to infinity before t = Tc due to the rigid-flexible dynamics. In the situation where ωi 2 and ηi 2 are large enough, there exist positive constants p1 and p2 such that E˙ i < p1 E i + p2

(5.42)

Integrating both sides of the above equation yields E i (t) < − p2 / p1 + exp( p1 t)(E i (0) + p2 / p1 )

(5.43)

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5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

Therefore, within finite time, the total energy is bounded. That is, the networked flexible spacecraft cannot escape in finite time. Therefore, the separation principle [25, 27] can be adopted to show that for t ≥ Tc , ui equals to u¯ i and the conclusions in Theorem 5.1 still holds for ui when t ≥ Tc .  Remark 5.5 As shown in Step A in the proof of Lemma 5.1, the definition of Rei leads to the unstable equilibriums of the overall system. Consequently, some investigators have tried to define some complicated potential function [8, 28] to avoid this drawback. However, in this study, the traditional attitude tracking error Rei in Eq. (5.21) is used for two reasons. One is that for the controller designed for the ˆ i , which varies from distributed tracking task, the desired attitude is determined by R Ri (0) to Rd . Hence, the system will not start from 2 , 3 or 4 (this is the reason why the estimated states are used instead of the leader’s states for the nodes having access to the leader). The other reason is that the time-varying desired attitude, the external disturbances and the additive actuator bias faults may prevent the system from staying at the unstable equilibriums. ˆ i from the distributed observer (5.6) is likely to violate Remark 5.6 Before TR , R ˆi ∈ / S O(3) holds for the orthogonality of rotation matrix, i.e., it is possible that R t < TR . The violation does not have significant effect on the derivations of the sliding variable si , the controller ui and the updating law (5.31) because the boundedness in Eqs. (5.23) and (5.26) still holds from Remark 5.3. However, Ri can only slide on ˆi ∈ S O(3) due to the attitude kinematics (1.21), hence, it is not able to track R / S O(3). ˆi Fortunately, due to the finite-time convergence property, the observer (5.6) enable R to arrive at Rd ∈ S O(3) for t > TR . Furthermore, for an attitude tracking task, the responses of the distributed finite-time observer are numerically calculated because no local measurements are included in the observer after the initial time. Hence, the convergence precision can be very high, i.e., the violation of the orthogonality of the ˆ i can be omitted. Consequently, the designed controller can convergence value of R achieve the specified control objective eventually. Remark 5.7 From Theorem 5.2, the modal variables are bounded only, i.e., there exist residual vibrations. The main reason is that the term −δ i ω˙ i in Eq. (5.1) can be considered as the external excitation for the flexible vibration. However, in the case that the desired angular acceleration is zero and the angular velocity tracking errors are nearly zero, the flexible vibrations will be almost damped out as time goes to infinity. Remark 5.8 Since the controller in Eq. (5.30) is developed on S O(3) directly, based on the statements in [7–9], it can be concluded that the proposed controllers can avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions.

5.4 Numerical Simulations

93

Fig. 5.2 Communication topology

5.4 Numerical Simulations Four identical flexible spacecraft are taken as an example to verify the effectiveness of the proposed controller. In the following simulations, e Ri = tr(I 3 − RdT Ri ) ˆ i ) will be used to indicate the tracking error and the estiand e Riˆ = tr(I 3 − RdT R ˆ i = Rd , mation error. Note that e Ri = 0 and e Riˆ = 0 imply that Ri = Rd and R respectively. The angular velocity tracking errors will be illustrated using ωei = [ωei−1 ωei−2 ωei−3 ]T defined in Eq. (5.21). The communication graph among the four followers and the leader is shown in Fig. 5.2. The parameters of the four spacecraft, as shown in Table 5.1, are chosen the same as those in [29]. The external disturbance d¯ i , actuator effectiveness factor ρ i and additive bias fault f i are set as 0.1 sin(5t)13 , [0.5 + 0.2 sin(3t)]I 3 and sin(2t)13 , respectively. The initial attitude and the angular velocity of the leader are I 3 and [0.03 cos(0.1t) 0.03 sin(0.2t) 0.03 sin(0.3t)]T . For the distributed finite-time √ observer, the parameters r1 , r2 , r3 , r4 , r5 , r6 , α1 , α2 and α3 are chosen as 0.01, 3/5, 0.01, 0.1, 0.01, 0.1, 0.5, 0.5 and 0.5. The control parameters k0 , β, μ, σ1 and σ2 are set as 100, 0.1, 0.0001, 0.001 and 1, respectively. The initial value of Cˆ i , ηˆ i and qˆ i are 0.1, 0 and 0 for i = 1, 2, 3, 4. At t = 0, it is assumed that the attitude, angular velocity, modal variable and modal velocity are as follows

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5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

Table 5.1 Parameters of flexible spacecraft Parameter Value ⎡ ⎤ 350 3 4 ⎢ ⎥ J 0i ⎣ 3 270 10 ⎦ 4 10 190 ⎡ ⎤ 6.45637 1.27814 2.15629 ⎢ ⎥ δi ⎣ −1.25819 0.91756 −1.67264 ⎦ 1.11687 2.48901 −0.83674 1 0.7681 2 1.1038 3 1.8733 ξ1 0.0056 ξ2 0.0086 ξ3 0.013

Unit kg·m2

kg·m2 rad/s rad/s rad/s – – –

⎡

⎤ 0.7071 0.5 −0.5 R1 (0) = ⎣ 0 0.7071 0.7071 ⎦ , ω1 (0) = 0, η1 = 0.113 , η˙ 1 = 0 0.7071 −0.5 0.5

(5.44)

R2 (0) = diag{1, −1, −1}, ω2 (0) = 0, η2 = 0.113 , η˙ 2 = 0 (5.45) ⎡ ⎤ 0.3062 0.9186 −0.25 R3 (0) = ⎣ −0.3536 0.3536 0.866 ⎦ , ω3 (0) = 0, η3 = 0.113 , η˙ 3 = 0 (5.46) 0.8839 −0.1768 0.433 ⎡ ⎤ −0.866 0 −0.5 R4 (0) = ⎣ −0.5 0 0.866 ⎦ , ω4 (0) = 0, η4 = 0.113 , η˙ 4 = 0 (5.47) 0 1 0 The simulation results are shown in Figs. 5.3−5.7. As indicated in Figs. 5.3 and 5.4, the four flexible spacecraft and the finite-time observer can reach the desired attitude at about t = 150 s and t = 5.8 s, respectively. However, the estimation errors of the first and third spacecraft increase at the first 1.3 s. This is because the initial value of the estimation of the leader’s attitude is chosen as the initial attitude of each spacecraft. The estimation errors of these two spacecraft will be influenced by their neighbors, hence, they will not monotonously decreasing. Furthermore, as shown in Fig. 5.3, the non-monotonicity leads to the increase of the attitude tacking errors of these two spacecraft in the initial stage. From Fig. 5.6, it can be seen that the flexible vibrations are uniformly bounded and the first and third spacecraft are subject to larger deformations in the first 20 s due to the non-monotonicity of the attitude tracking errors, i.e., the larger angular accelerations.

5.5 Conclusions

95

Fig. 5.3 Distributed tracking errors

Fig. 5.4 Distributed estimation errors

5.5 Conclusions In this study, a distributed adaptive controller is proposed for the attitude tracking of multiple flexible spacecraft on S O(3) subject to actuator faults and external disturbances without modal variable measurement. Since only some spacecraft can obtain the leader’s information, a distributed finite-time observer is designed to estimate the attitude and the angular velocity of the leader. Based on the finite-time observer and the modal variable observer, a distributed fault-tolerant controller is designed to solve the distributed attitude tracking. The convergence of the proposed controller is proved via the separation principle between the observer and controller. Because the controller is designed on S O(3), the singularity and ambiguity in representing an attitude can be avoided. The estimation of the leader’s states is regarded as the desired attitude and angular velocity for all spacecraft in the attitude tracking task in order to avoid the unstable equilibriums caused by the definition of the attitude tracking error in the feedback loop. Numerical simulations demonstrate that the proposed controller can recover from the actuator faults and reject the external disturbances.

96

5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

Fig. 5.5 Angular velocity tracking errors

5.5 Conclusions Fig. 5.6 Modal variables

97

98

5 Adaptive Fault-tolerant Attitude Tracking on S O(3) Under an Undirected Graph

Fig. 5.7 Control commands

References

99

References 1. VanDyke, M.C., Hall, C.D.: Decentralized coordinated attitude control within a formation of spacecraft. J. Guidance Control Dyn. 29(5), 1101–1109 (2006) 2. Du, H., Li, S.: Attitude synchronization control for a group of flexible spacecraft. Automatica 50(2), 646–651 (2014) 3. Chen, T., Shan, J., Wen, H.: Distributed adaptive attitude control for networked underactuated flexible spacecraft. IEEE Trans. Aerosp. Electron. Syst. 55(1), 215–225 (2019) 4. Du, H., Li, S.: Attitude synchronization for flexible spacecraft with communication delays. IEEE Trans. Autom. Control 61(11), 3625–3630 (2016) 5. Zou, A.M., de Ruiter, A.H., Kumar, K.D.: Distributed finite-time velocity-free attitude coordination control for spacecraft formations. Automatica 67, 46–53 (2016) 6. Wang, Q., Duan, Z., Lv, Y.: Distributed attitude synchronization control for multiple flexible spacecraft without modal variable measurement. Int. J. Robust Nonlinear Control 28(10), 3435– 3453 (2018) 7. Bohn, J., Sanyal, A.K.: Almost global finite-time stabilization of rigid body attitude dynamics using rotation matrices. Int. J. Robust Nonlinear Control 26(9), 2008–2022 (2016) 8. Lee, T.: Exponential stability of an attitude tracking control system on S O(3) for large-angle rotational maneuvers. Syst. Control Lett. 61(1), 231–237 (2012) 9. Mayhew, C.G., Teel, A.R.: Synergistic hybrid feedback for global rigid-body attitude tracking on S O(3). IEEE Trans. Autom. Control 58(11), 2730–2742 (2013) 10. Zou, Y., Meng, Z., Zuo, Z.: Rotation-matrix-based attitude synchronization of multiple spacecraft without velocity measurements. In: 2017 11th Asian Control Conference (ASCC), pp. 96–101 (2017). doi: https://doi.org/10.1109/ASCC.2017.8287149 11. Bounemeur, A., Chemachema, M., Essounbouli, N.: Indirect adaptive fuzzy fault-tolerant tracking control for mimo nonlinear systems with actuator and sensor failures. ISA Transact. 79, 45–61 (2018) 12. Yin, S., Xiao, B., Ding, S.X., Zhou, D.: A review on recent development of spacecraft attitude fault tolerant control system. IEEE Trans. Industr. Electron. 63(5), 3311–3320 (2016) 13. Shen, Q., Wang, D., Zhu, S., Poh, E.K.: Robust control allocation for spacecraft attitude tracking under actuator faults. IEEE Trans. Control Syst. Technol. 25(3), 1068–1075 (2017) 14. Xiao, B., Huo, M., Yang, X., Zhang, Y.: Fault-tolerant attitude stabilization for satellites without rate sensor. IEEE Trans. Industr. Electron. 62(11), 7191–7202 (2015) 15. Bustan, D., Sani, S.H., Pariz, N.: Adaptive fault-tolerant spacecraft attitude control design with transient response control. IEEE/ASME Trans. Mechatron. 19(4), 1404–1411 (2014) 16. Gui, H., Vukovich, G.: Adaptive fault-tolerant spacecraft attitude control using a novel integral terminal sliding mode. Int. J. Robust Nonlinear Control 27(16), 3174–3196 (2017) 17. Sakthivel, R., Selvi, S., Mathiyalagan, K.: Fault-tolerant sampled-data control of flexible spacecraft with probabilistic time delays. Nonlinear Dyn. 79(3), 1835–1846 (2015) 18. Ma, Y., Jiang, B., Tao, G., Cheng, Y.: Uncertainty decomposition-based fault-tolerant adaptive control of flexible spacecraft. IEEE Trans. Aerosp. Electron. Syst. 51(2), 1053–1068 (2015) 19. Hu, Q., Xiao, B.: Fault-tolerant sliding mode attitude control for flexible spacecraft under loss of actuator effectiveness. Nonlinear Dyn. 64(1–2), 13–23 (2011) 20. Sun, L., Zheng, Z.: Finite-time sliding mode trajectory tracking control of uncertain mechanical systems. Asian J. Control 19(1), 399–404 (2017) 21. Cao, X., Yue, C., Liu, M.: Fault-tolerant sliding mode attitude tracking control for flexible spacecraft with disturbance and modeling uncertainty. Adv. Mech. Eng. 9(3), 1–9 (2017) 22. Huang, D., Wang, Q., Duan, Z.: Distributed attitude control for multiple flexible spacecraft under actuator failures and saturation. Nonlinear Dyn. 88(1), 529–546 (2017) 23. Di Gennaro, S.: Passive attitude control of flexible spacecraft from quaternion measurements. J. Optim. Theory Appl. 116(1), 41–60 (2003) 24. Lopez-Ramirez, F., Polyakov, A., Efimov, D., Perruquetti, W.: Finite-time and fixed-time observer design: Implicit Lyapunov function approach. Automatica 87, 52–60 (2018)

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25. Gui, H., Vukovich, G.: Distributed almost global finite-time attitude consensus of multiple spacecraft without velocity measurements. Aerosp. Sci. Technol. 75, 284–296 (2018) 26. Li, S., Wang, X.: Finite-time consensus and collision avoidance control algorithms for multiple AUVs. Automatica 49(11), 3359–3367 (2013) 27. Atassi, A.N., Khalil, H.K.: A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Autom. Control 44(9), 1672–1687 (1999) 28. Berkane, S., Abdessameud, A., Tayebi, A.: Hybrid global exponential stabilization on S O(3). Automatica 81, 279–285 (2017) 29. Cao, X., Yue, C., Liu, M.: Flexible satellite attitude maneuver via constrained torque distribution and active vibration suppression. Aerosp. Sci. Technol. 67, 387–397 (2017)

Chapter 6

Distributed Attitude Tracking and Synchronization on S O(3) Under Directed Graphs

Abstract Distributed adaptive controllers are developed for attitude tracking and synchronization on S O(3) of multiple flexible spacecraft under a directed graph. Since not all spacecraft can receive the virtual leader’s information, a finite-time observer is designed for each follower spacecraft to estimate the moving virtual leader’s information. An adaptive controller is then proposed to realize the distributed attitude tracking synchronously and achieve the bounded flexible vibration. A rigorous theoretical proof is presented based on the separation principle. Furthermore, an adaptive controller with a modal variable observer is designed for the case without the measurements of the modal variables. Finally, numerical and experimental verifications are presented.

6.1 Introduction A group of spacecraft is expected to align their attitudes synchronously in some potential applications, such as interferometry and synthetic aperture radar. Coordinated attitude control is a significant, but challenging problem in spacecraft formation flying because, essentially, the configuration space of 3-D rotation is the special Lie group S O(3) and the attitude tracking error on S O(3) cannot be defined based on the algebraic subtraction. Some studies have paid attention to the distributed attitude control of networked spacecraft [1]. Moreover, in some formation flying missions, the spacecraft are usually equipped with flexible appendages, such as antennas [2, 3]. As a consequence, the necessity for studying the attitude coordination of a group of flexible spacecraft arises. Several papers have considered such a topic under various operating conditions [4–7]. The rotation matrix or direction cosine matrix has significant advantages in spacecraft attitude control [8, 9] because it can represent the attitude globally and uniquely. As shown in [9], continuous time-invariant feedback controllers cannot stabilize the Reproduced from Ti Chen and Jinjun Shan. Distributed spacecraft attitude tracking and synchronization under directed graphs. Aerospace Science and Technology 2021; 109: 106432. Copyright ©2021 Elsevier Masson SAS. All rights reserved. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_6

101

102

6 Distributed Attitude Tracking and Synchronization …

spacecraft rotation globally due to the topological structures of S O(3). The main reason is that there are at least four critical points where the gradient of any smooth potential function on S O(3) vanishes. The manifolds of the undesired points are of Lebesgue measure zero. Hence, the stabilities of the time-invariant continuous controllers are almost global, which is the best result based on smooth potential functions [10]. Another critical problem is that such smooth potential functions can only provide very small control torque in the small neighborhood of the unstable equilibria, which will result in a long settling time. To accelerate the convergence, some attempts have been made to design global control algorithms based on discontinuous or switching potential functions [10–12]. Global control algorithms based on discontinuous schemes have reached a good level of maturity. However, the commonly used attitude control actuators in space missions, like control moment gyroscopes and reaction wheels, can only provide continuous torques and the physical realization of the discontinuous control systems may give rise to several complications, such as the chattering phenomenon. On the other hand, if the discontinuous control torque is acceptable, the attitude control algorithms in terms of the quaternion parameterization is adequate to achieve the attitude maneuver globally and without unwinding, such as the controllers in [13, 14]. In last two chapters, continuous attitude tracking controllers are proposed on S O(3) directly for the networked flexible spacecraft under undirected communication graphs. The main goal here is to extend the results of the distributed attitude tracking of multiple flexible spacecraft on S O(3) to more general directed graphs to further reduce the communication burden. In the attitude tracking problem, all spacecraft are required to track a prescribed trajectory determined by a virtual leader. To improve the efficiency and robustness of the control system, the spacecraft in the team are expected to rotate synchronously. Hence, in this chapter, the attitude tracking and synchronization are considered simultaneously. Furthermore, due to fuel consumption, the inertia matrix of the spacecraft may change slowly. Such uncertainty may deteriorate the controller performance. Therefore, some investigators have focused on the adaptive attitude control on S O(3) [11, 15, 16]. However, the adaptation laws in [11, 15, 16] do not take full advantage of the boundedness of the inertia matrix. Hence, the bound of the inertia matrix will be used in the development of adaptation laws in this study to get a better estimate of uncertain parameters. Since only some follower spacecraft know the desired trajectory, a distributed finite-time observer is introduced to estimate the virtual leader’s information for each follower. An adaptive controller with synchronization term is designed based on a smooth attitude error function to achieve the control aim with uncertain parameters and bounded disturbance. Furthermore, the situation in the absence of modal variable measurement is also considered.

6.2 Problem Formulation

103

6.2 Problem Formulation 6.2.1 Dynamics Equation of Flexible Spacecraft Letting Ri ∈ S O(3) represent the rotation matrix from the body frame of the ith spacecraft to the inertial reference frame, the attitude kinematics of this spacecraft is as shown in Eq. (1.21). As shown in Sect. 1.5, the dynamics equations of the ith flexible spacecraft subject to external disturbances are J i ω˙ i + δ iT η¨ i = −ωi× ( J i ωi + δ iT η˙ i ) + ui + d¯ i η¨ i + C ni η˙ i + K ni ηi = −δ i ω˙ i

(6.1)

where J i , δ i , ηi , ui and d¯ i are the total inertia matrix, the rigid-flexible matrix, the modal coordinate vector, the control input vector and the external disturbance vector, respectively. C ni = diag{2ξ1 1 , . . . , 2ξn n } is the damping matrix and K ni = diag{21 , . . . , 2n } is the stiffness matrix with ξ j and  j as the jth order damping ratio and natural frequency, respectively. Let q i = η˙ i + δ i ωi and J 0i = J i − δ iT δ i . Equation (6.1) becomes η˙ i = q i − δ i ωi J 0i ω˙ i = −ωi× ( J 0i ωi + δ iT q i ) +δ iT (C ni (q i − δ i ωi ) + K ni ηi ) + ui + d¯ i q˙ i + C ni (q i − δ i ωi ) + K ni ηi = 0

(6.2)

6.2.2 Control Objective In this study, a distributed controller will be developed to realize the synchronous tracking of a moving virtual leader for N identical flexible spacecraft based on full state or partial state feedback under Assumptions 6.1–6.3. Suppose that the attitude and angular velocity of the virtual leader are denoted by Rd and ωd . Assumption 6.1 There exist four positive constants μ0 , μ1 , μ2 and μ3 such that  d¯ i 2 ≤ μ0 , ωd 2 ≤ μ1 , ω˙ d 2 ≤ μ2 and ω¨ d 2 ≤ μ3 hold. Assumption 6.2 Considering that J i will vary slowly due to fuel consumption, J 0i is assumed to be uncertain. Assumption 6.3 The virtual leader is globally reachable, i.e., there exist directed path(s) from the virtual leader to all follower spacecraft. Assumption 6.3 implies the virtual leader’s information can be spread to all nodes only based on the local communication, i.e., there are no isolated nodes in the com-

104

6 Distributed Attitude Tracking and Synchronization …

munication graph. If some isolated nodes exist in the graph, obviously, these nodes cannot be controlled to track the virtual leader’s states because they cannot receive any command at all. Hence, Assumption 6.3 is reasonable in a formation flying mission.

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs 6.3.1 Distributed Leader Observer Since only some followers know the virtual leader’s information, inspired by [17], a distributed observer is designed as   ˙ˆ = −γ sign  ˆ ˆ ˆ R di 1 j∈Ni ai j ( R di − R d j ) + bi ( R di − R d )   ˆ ˆ ˆ a ( ω − ω ) + b ( ω − ω ) ω˙ˆ di = −γ2 sign ij di dj i di d  j∈Ni  ˙hˆ = −γ sign  ˆ ˆ ˆ ˙ d) di 3 j∈Ni ai j ( hdi − hd j ) + bi ( hdi − ω

(6.3)

where the definitions of ai j and bi can be found in Sect. 1.2, γ1 > μ1 , γ2 > μ2 , ˆ di , ωˆ di and hˆ di are the estimates of the virtual γ3 > μ3 , i = 1, . . . , N . Essentially, R leader’s attitude, angular velocity and angular acceleration for spacecraft i. Note that the observers in Eq. (6.3) are embedded in follower spacecraft and aim to provide the reference signals for all followers. The reason why the followers linked to the virtual leader use the estimate of the leader’s information as the reference signal will be discussed in Remark 6.4. According to the definition of bi in Sect. 1.2, bi = 0 if the ith follower cannot receive the leader’s information, i.e., Rd , ωd and ω˙ d will vanish in the observers embedded in the spacecraft having no access to the leader. ˆ di , ωˆ di and hˆ di are chosen as Ri (0), ωi (0) and 03 . Denote The initial conditions of R ˆ ˆ di and Rd , respectively. The kth elements of Rdi,kr and Rd,kr as the (k, r ) entry of R ωˆ di , ωd , hˆ di and ω˙ d are represented by ωˆ di,k , ωd,k , hˆ di,k and ω˙ d,k , respectively. ˆ di = Lemma 6.1 If Assumption 6.3, γ1 > μ1 , γ2 > μ2 and γ3 > μ3 are satisfied, R ˆ Rd , ωˆ i = ωd and hdi = ω˙ d hold for t ≥ max{TR , Tω , Th }, where TR = ˆ (0)−ω˙ d,k (0)|} maxk,r,i {| Rˆ di,kr (0)−Rd,kr (0)|} (0)−ωd,k (0)|} , Tω = maxk,i {|ωˆ di,k and Th = maxk,i {|h di,k . γ1 −μ1 γ2 −μ2 γ3 −μ3 ˙ d 2 ≤ ˙ d satisfies | R˙ d,kr | ≤  R Proof The time derivative of each entry of R × Rd 2 ω×  ≤ ω  = ω  , where k = 1, 2, 3 and r = 1, 2, 3. It follows from d 2 d 2 d 2 ˆ di,kr (0)−Rd,kr (0)|} max {| R i Theorem 3.1 in [17] that Rˆ di,kr will equal to Rd,kr after t R,kr = . γ1 −μ1

ˆ di = Rd holds for t ≥ TR = maxk,r {t R,kr }. Similarly, the kth element of Hence, R (0)−ωd,k (0)|} ωˆ di will be equal to the corresponding element of ωd after tω,k = maxi {|ωˆ di,k . γ2 −μ2 Hence, ωˆ di will reach ωd after Tω = maxk {tω,k }. Based on the same proof, it can be

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs

105

concluded that the kth element of hˆ di will be equal to the corresponding element of ˆ (0)−ω˙ d,k (0)|} ˆ di = Rd , ωˆ di = ωd and hˆ di = ω˙ d ω˙ d after th,k = maxi {|h di,k . Therefore, R γ3 −μ3 hold for t ≥ max{TR , Tω , Th }, where Th = maxk {th,k }.  Remark 6.1 Essentially, the main role of the observers in Eq. (6.3) is to provide the ˆ di , ωˆ di and hˆ di will be used reference trajectory for each follower spacecraft, i.e., R as the reference attitude, angular velocity and angular acceleration in the following controller design. Remark 6.2 Usually, the attitude errors should be defined based on RiT R j . Such nonlinear definitions cause that the traditional consensus algorithms cannot be applied directly to the attitude consensus of multiple spacecraft under a general directed graph. Hence, the distributed-observer-based method is used in this chapter. ˆ di may not always stay in S O(3) due to the algebraic subtraction in Eq. However, R ˆ di will equal (6.3). Fortunately, as shown in Lemma 6.1, after TR each element of R ˆ di will be to the corresponding element of Rd , which belongs to S O(3). Hence, R one element of S O(3) finally even though it may not be a rotation matrix before the convergence of the distributed observer (6.3). Furthermore, as shown in Lemma 6.1, the upper bound of the convergence time of the observer (6.3) will decrease with the increase of γ1 . Hence, one can choose a large γ1 such that the observer (6.3) can converge to Rd within a short time. Furthermore, the effect of the nonorthogonality ˆ di before TR on the system performance is discussed in Remark 6.3. of R

6.3.2 Distributed Tracking and Synchronization with Full State Feedback Based on the distributed observers in Eq. (6.3), a sliding variable is defined as si = ωei + β Rei

(6.4)

T

ˆ di Ri − RiT R ˆ di )∨ and ωei = ωi − RiT R ˆ di ωˆ di . where β > 0 is a constant, Rei = ( R si can be rewritten as si = ωi + i (6.5) with

ˆ di ωˆ di + β Rei i = −RiT R

(6.6)

Based on the sliding variable si , the controller is designed as ui = − Jˆ 0i i◦ + ωi× ( Jˆ 0i ωi + δ iT q i ) − κ1 si ˆ i −δ iT (C ni (q i − δ i ωi ) + K ni ηi ) − s di s+α 2 i 2 i  −κ2 j∈Ni ai j [β1 (R Tj Ri − RiT R j )∨ + ωi − ω j ]

(6.7)

106

6 Distributed Attitude Tracking and Synchronization …

T T ˆ di ˆ di ˆ di ωˆ di − RiT R ˆ di hˆ di + β[−(tr( R where i◦ = ωi× RiT R Ri )I 3 − R Ri )ωˆ di + (tr ˆ di )I 3 − RiT R ˆ di )ωi ], Jˆ 0i is the estimate of J 0i , dˆi and αi are two adaptive (RiT R parameters and κ1 , κ2 , β1 are positive constants. In fact, i◦ is the time derivaˆ di = Rd , ωˆ di = ωd and hˆ di = ω˙ d . In (6.7), the terms tive of i in the case of R ˆ i × ˆ T ◦ ˆ − J 0i i +ωi ( J 0i ωi + δ i q i ) − δ iT (C ni (q i − δ i ωi ) + K ni ηi ) − s di s+α 2 − κ1 s i and i 2 i  T T ∨ −κ2 j∈Ni ai j [β1 (R j Ri − Ri R j ) + ωi − ω j ] are used to achieve the attitude tracking and synchronization, respectively. J 0i i◦ − ωi× J 0i ωi can be rewritten as Y (i◦ , ωi ) J 0i , where J 0i = [J0i,11 J0i,12 J0i,13 J0i,22 J0i,23 J0i,33 ]T ∈ R6 and Y (i◦ , ωi ) ∈ R3×6 is the regressor matrix. Note that J0i,kr represents the (k, r ) element of the symmetric matrix J 0i for k = 1, 2, 3 and r = 1, 2, 3. Therefore, inspired by [18], the adaptation law of Jˆ 0i ∈ R6 associated with Jˆ 0i can be defined as

  J˙ˆ 0i = proj1 0 Y (i◦ , ωi )T si , Jˆ 0i , Jmax

(6.8)

where 0 is a positive constant. Considering that J 0i is positive definite, the initial value of Jˆ 0i can be set as γ0 I 3 , i.e., Jˆ 0i (0) = γ0 [1 0 0 1 0 1]T . Jmax >  J 0i max is a positive constant, where  · max means the max norm. The mapping g = proj1 (Z, z, z max ) with two vectors Z = [Z 1 · · · Z 6 ]T , z = [z 1 · · · z 6 ]T and a positive constants z max is defined as

gk =

⎧ ⎪ ⎨ ⎪ ⎩

if |z k | ≤ z max

Zk

if |z k | > z max and Z k z k ≤ 0

Zk Z k (1 −

χ+1 zk

+

χ z k z max )

(6.9)

otherwise

where gk is the kth element of g for k = 1, . . . , 6 and χ is a positive even number. As shown in Eq. (6.9), in the case of z k  z max or z k −z max , gk z k ≤ 0 holds, which implies |z k | will not increase with time. Therefore, all elements of Jˆ 0i cannot go to infinity with the updating law in Eq. (6.8). Moreover, dˆi and αi are updated by d˙ˆi = proj2 ( 1 si 2 , dˆi ) ˆ 2 αi di si 2 α˙ i = − s  +α 2 i 2

(6.10)

i

with  proj2 ( 1 si 2 , dˆi ) =

1 si 2

if dˆi ≤ μ¯ 0

χ χ

1 si 2 (1 − dˆi 1 + μ¯ 0 1 ) otherwise

(6.11)

where χ1 > 0 is an even number and 1 , 2 and μ¯ 0 are positive constants. Similar to the properties of the mapping defined in Eq. (6.9), the introduction of proj2 ( 1 si 2 , dˆi ) in Eq. (6.11) aims to ensure the boundedness of dˆi .

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs

107

Theorem 6.1 If μ¯ 0 > κ2 (N − 1)(2β1 + 4β + 2μ1 ) + μ0 and Assumptions 6.1−6.3 hold, with the controller (6.7), the updating laws (6.8) and (6.10) and the finite-time observer (6.3), the distributed attitude tracking task of a moving virtual leader for N spacecraft can be completed almost globally and flexible vibration is bounded. Proof The proof will be presented according to the separation principle between the finite-time distributed observers in Eq. (6.3) and the adaptive controller in Eq. (6.7). Hence, in the first step, the centralized tracking problem will be considered ˆ di = Rd , ωˆ di = ωd and hˆ di = ω˙ d . Then, it will be proven that the response with R of the closed-loop system under the distributed controller in Eq. (6.7) will not go to infinity. ˆ di , ωˆ di and hˆ di are replaced by Rd , ωd and ω˙ d , respecStep A: In this step, R tively. Hence, si = ωi − RiT Rd ωd + β(RdT Ri − RiT Rd )∨ and i◦ equals to the time derivative of i = −RiT Rd ωd + β(RdT Ri − RiT Rd )∨ . Based on Rodrigues’ rotation formula, the matrix R Tj Ri ∈ S O(3) is written as R Tj Ri = I 3 + (1 − cos θ )n¯ × n¯ × + sin θ n¯ ×

(6.12)

where θ and n¯ are rotation angle and rotation axis associated with the rotation described by R Tj Ri . Hence, one has ¯ 2≤2 (R Tj Ri − RiT R j )∨ 2 = 2 sin θ n T T  2 =  − R

Rd ωd + β(RdT R − R

Rd )∨ 2 ≤ μ1 + 2β

(6.13)

(6.14)

 T T ∨ where represents i or j. Denote  d si = −κ2 j∈Ni ai j [β1 (R j Ri − Ri R j ) − i +  j ], which is bounded by κ2 j∈Ni ai j (2β1 + 4β + 2μ1 ) ≤ κ2 (N − 1)(2β1 + 4β + 2μ1 ). That is, d si can be treated as a bounded disturbance. Hence, the closedloop system of the ith spacecraft is ˙ i − ωi× J˜ 0i ωi − κ1 si J 0i s˙ i = J˜ 0i   ˆ i − s di s+α 2 − κ2 j∈Ni ai j (s i − s j ) + d i i 2

(6.15)

i

where J˜ 0i = J 0i − Jˆ 0i and d i = d¯ i + d si . Note that d i is bounded by κ2 (N − 1)(2β1 + 4β + 2μ1 ) + μ0 . Denote S = [s1T , . . . , s TN ]T , J 0 = diag{ J 0i } and f = ˙ i − ωi× J˜ 0i ωi − dˆi si 2 + d i . Equation (6.15) [ f 1T , . . . , f TN ]T , where f i = J˜ 0i  si 2 +αi can be rewritten as J 0 S˙ = f − (κ1 I 3N + κ2 L ⊗ I 3 )S (6.16) where ⊗ is the Kronecker product. A Lyapunov function candidate is

108

6 Distributed Attitude Tracking and Synchronization … N 1

V1 =

2

i=1

siT J 0i si +

1 ˜2 1 2 1 ˜ T ˜ d + J J α + 2 1 i 2 2 i 2 0 0i 0i

 (6.17)

where d˜i = μ¯ 0 − dˆi and J˜ 0i ∈ R9 is obtained by reshaping J˜ 0i into a 9-dimensional vector. V˙1 reads

V˙1 = −ST (κ1 I 3N + κ2 L ⊗ I 3 )S +

N 



˙ i , ωi ) J˜ 0i siT Y (

i=1 αi2 dˆi si 2 dˆi siT si 1 ˜ ˙˜ T − s  +α2 + si d i + 1 di di − s  +α2 i 2 i 2 i i  T − J˜ 0i proj1 ( 0 Y (i◦ , ωi )T si , Jˆ 0i , Jmax )/ 0

(6.18)

If dˆi ≤ μ¯ 0 , d˜i d˙˜i = −d˜i proj2 ( 1 si 2 , dˆi ) = −d˜i 1 si 2 holds; otherwise, d˜i d˙˜i = χ χ ˙ i , ωi ) J˜ 0i − J˜ T −d˜i 1 si 2 (1 − dˆi 1 + μ¯ 0 1 ) ≤ −d˜i 1 si 2 . Denote W J = siT Y ( 0i proj1 ( 0 Y (i◦ , ωi )T si , Jˆ 0i , Jmax )/ 0 . W J can be rewritten as ◦ T ˙ i , ωi ) J˜ 0i − J˜ T W J = siT Y ( 0i Y (i , ωi ) s i 6 χ T χ − Sign k [ J˜ 0i ]k [Y (i◦ , ωi )T si ]k [ Jˆ 0i ]k (Jmax − [ Jˆ 0i ]k )

(6.19)

k=1

where Sign k = 0 if the first two conditions in Eq. (6.9) are satisfied; otherwise, Sign k = 1. [·]k represents the kth element of a vector. Note that if both |[ Jˆ 0i ]k | > Jmax and [Y (i◦ , ωi )T si ]k [ Jˆ 0i ]k > 0 hold, one has

T

χ

χ Sign k [ J˜ 0i ]k [Y (i◦ , ωi )T si ]k [ Jˆ 0i ]k (Jmax − [ Jˆ 0i ]k ) > 0

(6.20)

Therefore, W J satisfies ◦ T ˙ i , ωi ) J˜ 0i − J˜ T W J ≤ siT Y ( 0i 0 Y (i , ωi ) s i = 0

(6.21)

V˙1 satisfies the following inequality N 

dˆi siT si + μ¯ 0 si 2 − d˜i si 2 si 2 + αi2 i=1 α 2 dˆ si 2  − ST (κ1 I 3N + κ2 L ⊗ I 3 )S − si i +α 2

V˙1 ≤

i 2

−

i

= −κ1 ST S − κ2 ST [(L + L T ) ⊗ I 3 ]S/2 κ2 λmin (L + L T ) T S S ≤ −κ1 ST S − 2 T ≤ −κ1 S S

(6.22)

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs

109

Due to κ1 > 0, V˙1 ≤ 0 holds. Hence, V1 is monotonically nonincreasing and 0 ≤ V1 (t) ≤ V1 (0) holds, where V1 (0) represents the initial value of V1 . From the definition of V1 in Eq. (6.17), si , d˜i , αi and J˜ 0i are bounded. In the centralized tracking case, si is si = ωi − RiT Rd ωd + β(RdT Ri − RiT Rd )∨

(6.23)

Hence, ωi will be bounded considering the boundedness of ωd and si , and the orthogonalities of Ri and Rd . Furthermore, due to the boundedness of d˜i and J˜ 0i , dˆi and Jˆ 0i are bounded. i with Rd and ωd can be expressed as i = −RiT Rd ωd + β(RdT Ri − RiT Rd )∨

(6.24)

˙ i = Ri ωi× , R ˙ d = R d ω× Since R d and Assumption 6.1 holds, one can conclude that ˙ i is also bounded. From Eq. (6.2), one has x˙ i = Ai x i + Bi ωi

(6.25)

     ηi 0 In −δ , Ai = , Bi = and ωi can be considered qi −K ni −C ni C ni δ as a bounded input. In order to analyze the stability of the autonomous system x˙ i = Ai x i , one can choose the Lyapunov function as 

where x i =

Vx,i = x iT P i x i

(6.26)

wherethe subscript i means that Lyapunov function is for the ith spacecraft and  K ni 0 P i = 0 I > 0 is the solution to the equation AiT P i + P i Ai = − Q i , in which n   0 0 Q i = 0 2C . The time derivative of Vx,i along x˙ i = Ai x i is ni

V˙x,i = −x iT Q i x i = −2q iT C ni q i ≤ 0

(6.27)

V˙x,i = 0 implies q i = 0. From the system equation x˙ i = Ai x i , one has ηi = 0. Hence, according to LaSalle’s invariant principle, x˙ i = Ai x i is the globally asymptotically stable. Therefore, the response of Eq. (6.25) is finite with the bounded ωi as the input. So far, it has been shown that all terms in the right hand side of Eq. (6.15) are bounded. Therefore, s˙ i is bounded. Hence, both si and −κ1 ST S are uniformly continuous. Since V1 is monotonically  t nonincreasing and bounded, V1 converges as t → ∞. Integrating V˙1 yields κ1 0 ST Sdτ ≤ V1 (0) − V1 (t). Therefore, t lim 0 ST Sdτ exists and is finite. It can be concluded from Lemma 8.2 in [19] that t→∞ T

S S → 0, i.e., si → 0, as t → ∞ for the ith spacecraft in the team. The definition of si can be rewritten as

110

6 Distributed Attitude Tracking and Synchronization …

ωi − RiT Rd ωd + β(RdT Ri − RiT Rd )∨ = si

(6.28)

Choose the Morse-Lyapunov function as Vs = tr(I 3 − RdT Ri ) ∈ [0, 4]

(6.29)

According to Lemma 2 in [20], the unique solution to Vs = 0 is Ri = Rd . V˙s is V˙s = ωdT (RiT Rd − RdT Ri )∨ + ωiT (RdT Ri − RiT Rd )∨ = ωdT (RiT Rd − RdT Ri )∨ +[RiT Rd ωd − β(RdT Ri − RiT Rd )∨ + si ]T (RdT Ri − RiT Rd )∨ ≤ −β[(RdT Ri − RiT Rd )∨ ]T (RdT Ri − RiT Rd )∨ + 2si 2

(6.30)

√ Hence, V˙s < 0 holds if (RdT Ri − RiT Rd )∨ 2 > 2si 2 /β, which goes to zero as t → ∞. The term (RdT Ri − RiT Rd )∨ 2 is ultimately bounded. According to [21], the solutions to (RdT Ri − RiT Rd )∨ = 0 are Ri = Rd and tr(RiT Rd ) = −1, at which Vs = 0 and Vs = 4 hold, respectively. Therefore, they are the stable and unstable equilibria, respectively. Note that tr(RiT Rd ) = −1 represents the set of all rotations of 180◦ and tr(RiT Rd ) = −1 implies tr(I 3 − RiT Rd ) = 4. From Proposition 2 in [22], the system governed by ωi − RiT Rd ωd + β(RdT Ri − RiT Rd )∨ = si is inputto-state stable with respect to Ri = Rd almost globally. Furthermore, as shown in the paragraph after Eq. (6.25), x i governed by Eq. (6.25) is bounded with a finite ωi . ˆ di = Rd , ωˆ di = ωd Hence, according to Theorem 2 in [23], Theorem 6.1 holds with R ˆ and hdi = ω˙ d . Step B: In the distributed case, the total energy of N flexible spacecraft is 1 E1 = 2 N



i=1

ωi η˙ i

T 

J i δ iT δi I n



ωi η˙ i



 + ηiT

K ni ηi

(6.31)

The time derivative of E 1 is E˙ 1 =

N  T ˆ di ωˆ di − RiT R ˆ di hˆ di ωi (− Jˆ 0i (ωi× RiT R i=1

T T ˆ di ˆ di −β(tr( R Ri )I 3 − R Ri )ωˆ di ˆ di )I 3 − RiT R ˆ di )ωi ) +β(tr(RiT R −δ iT (C ni η˙ i + K ni ηi )) − η˙ i C ni η˙ i   dˆi si +ωiT − − κ1 s i − κ2 ai j (si − s j ) + d i 2 si 2 + αi j∈N

(6.32)

i

ˆ di , ωˆ di and hˆ di , si satisfies si  ≤ c1 ωi  + c2 , where Due to the boundedness of R c1 and c2 are two large enough positive constants. It follows from the updating laws in (6.8) and (6.10) that the adaptive parameters Jˆ 0i , dˆi and αi are bounded. Since d¯ i , ˆ di , ωˆ di and Ri are bounded, d i is finite as well. Hence, there must exist positive conR

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs

stants cm , m = 3, . . . , 7, such that E˙ 1 ≤

N

c6 ωi ηi  + c7 η˙ i 2 ] holds. Considering

2 i=1 [c3 ωi 

N E 1 ≥ 21 i=1

111

+ c4 ωi  + c5 ωi η˙ i  + 

λmin



J i δ iT δ In



(ωi 2 + η˙ i 2 ) +

, one can find two positive constants c8 and c9 such that E˙ 1 ≤ c8 E 1 + c9 , integrating which yields

λmin (K ni )ηi 2

E 1 ≤ −c9 /c8 + exp(c8 t)(E 1 (0) + c9 /c8 )

(6.33)

Hence, E 1 cannot go to infinity in a finite time. After t = max{TR , Tω , Th }, the distributed attitude control problem becomes the centralized one. It can be concluded from the separation principle [1] between the finite-time distributed observer and the adaptive controller that, the conclusions in Step A still hold for the distributed tracking task, i.e., the proposed controller in Eq. (6.7) can achieve the distributed tracking almost globally asymptotically and keep the flexible vibration bounded.  ˆ di may not always stay in S O(3) Remark 6.3 As discussed in Remark 6.2, R before the convergence of the distributed observer (6.3). It should be noted that T ˆ di ˆ di is always an antisymmetric matrix whether R ˆ di belongs to S O(3) Ri − RiT R R T T ∨ ˆ di Ri − Ri R ˆ di ) can provide a 3-dimensional vector and Rei can or not. Hence, ( R be rewritten as Rei = (RdT Ri − RiT Rd )∨ + δ i (6.34) ˆ di − Rd )T Ri − RiT ( R ˆ di − Rd )]∨ . From the observer (6.3), it is where δ i = [( R straightforward to conclude that the bounded δ i equals to zero after the finite time TR . The controller proposed in Eq. (6.7) can be written as the controller in Step A in the proof of Theorem 6.1 with a disturbance term related to δ i . Such a disturbance term will go to zero after TR . Hence, essentially, the closed-loop system under the controller (6.7) can be considered as the one discussed in Step A in the proof of Theorem 6.1 with a disturbance going to zero with the convergence of the observer (6.3). Also, as shown in Step B of the proof of Theorem 6.1, such a disturbance will not cause the system’s instability in a finite time. Consequently, the proposed controller (6.7) is able to finish the distributed attitude tracking task. Remark 6.4 Note that the estimates of the virtual leader’s information are used in the designed controller (6.7) even for the spacecraft linked to the virtual leader. The main reason is that the estimate for each spacecraft is advantageous for the spacecraft to escape from the attraction of the unstable equilibria in some cases. For example, for a certain attitude tracking task with zero desired angular velocity, if the spacecraft linked to the virtual leader stays at the unstable equilibria statically at the initial time, the algorithm without synchronization term based on Rd can only control the spacecraft to leave the unstable equilibrium with the help of disturbances. As discussed in the last remark, the controller (6.7) for ith spacecraft can provide a disturbance before the convergence of the finite-time observer in Eq. (6.3). Hence, the introduction of the observer (6.3) can play a positive role in escaping from the unstable critical points.

112

6 Distributed Attitude Tracking and Synchronization …

Remark 6.5 As shown in Lemma 6.1, the distributed observers can estimate virtual leader’s information in a finite time. However, according to the discussion in Remark 6.3, the networked spacecraft may not be in the right direction to achieve the distributed tracking before the observer convergence because a disturbance term is added due to the introduction of the distributed observers. To shorten the time of such a stage, one can increase γ1 to get a smaller upper bound of the convergence time of the observer. If γ1 is chosen large enough such that the convergence time of the distributed observer is much shorter than the maneuver time of the networked spacecraft, the distributed tracking mission almost becomes a centralized one, i.e., the spacecraft can be controlled separately with the estimate of the virtual leader’s information as reference signals. Hence, to keep the advantages of the distributed system, such as the synchronous movement, the controller (6.7) introduces the syn chronization term −κ2 j∈Ni ai j [β1 (R Tj Ri − RiT R j )∨ + ωi − ω j ], which can help to finish the attitude tracking mission synchronously. In some cases, the synchronization term −κ2 j∈Ni ai j [β1 (R Tj Ri − RiT R j )∨ + ωi − ω j ] can help the spacecraft leave the attraction of the unstable critical points. For example, the finite-time observer can converge quickly and the leader’s attitude is a constant matrix ⎡ ⎤ I 3. 1 0 0 Assume that R1 (0) = diag{1, −1, −1}, R2 (0) = ⎣ 0 −0.99995 0.01 ⎦ and 0 −0.01 −0.99995 ⎡ ⎤ 10 0 R3 (0) = ⎣ 0 0 −1 ⎦. It is clear that s1 = 0, s2 = [−0.02, 0, 0]T and s3 = [2, 0, 0]T 01 0 because spacecraft 1 is located at an unstable equilibrium and the second spacecraft is in its close neighborhood. However, if the first two spacecraft have access to the attitude of the third spacecraft, (R3T R1 − R1T R3 )∨ 2 = 2  s1 2 = 0 and (R3T R2 − R2T R3 )∨ 2 = 1.9999  s2 2 = 0.02. That is, if the spacecraft i starts from an unstable equilibrium or its close neighborhood, the sliding variable si is zero or almost zero. It will take a long time for this spacecraft to leave the close neighborhood of the unstable equilibria without the synchronization term because the control command in Eq. (6.7) is small with κ2 = 0. If there exists one neighbor spacecraft j far away from the unstable equilibria, (R Tj Ri − RiT R j )∨ 2 will be much larger than si 2 . In this case, the synchronization term can help the spacecraft to leave the close neighborhood of the unstable critical points quickly. If, unfortunately, all neighbors of spacecraft i exactly stay at the critical points, the synchronization term will also equal to zero, hence, it will not contribute to escaping from the unstable equilibria. However, even in such a case, as shown in last remark, the observer (6.3) can still play a positive role. Remark 6.6 The controller (6.7) is proposed under a directed topology structure. In fact, it is easy to extend the result to the connected undirected graph. The stability of the closed-loop system with the controller (6.7) under undirected graph can be indicated by the following corollary. Corollary 6.1 Assume the communication graph among the followers is undirected and connected, and there exists at least one follower linked to the virtual leader. If

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs

113

Assumptions 6.1−6.2, κ1 > 0, κ2 > 0 and μ¯ 0 > κ2 (N − 1)(2β1 + 4β + 2μ1 ) + μ0 are satisfied and the controller is designed as Eq. (6.7) with the finite-time observer (6.3), and the adaptation laws (6.8) and (6.10), the distributed attitude tracking can be achieved almost globally asymptotically and the flexible vibration is bounded. The proof of Corollary 6.1 can also be divided into two steps. First, the finite-time observer in Eq. (6.3) still works under connected undirected graph because the virtual leader has path(s) to all followers. Second, the proof of this theorem is almost the same as that of Theorem 6.1. The only difference is the time derivative of V1 . For the connected undirected graph, the Laplacian matrix L is positive semi-definite and κ1 I N + κ2 L is positive definite [24]. Hence, Eq. (6.22) can be recast as V˙1 ≤ −ST (κ1 I 3N + κ2 L ⊗ I 3 )S ≤ 0

(6.35)

Then, based on a similar proof to that in Theorem 6.1, the conclusion in this corollary can be drawn. Remark 6.7 Different from [11, 25] where the unstable equilibria are avoided by defining a discontinuous attitude error function, the finite-time observer and synchronization term are introduced for each follower spacecraft to speed away from the attraction of the unstable equilibria. Essentially, this study presents a method to escape from the small neighborhood of the undesired critical points (or some manifolds from which the system will go to the unstable critical points) based on the distribution characteristic of formation flying mission, rather than eliminating the unstable equilibria. Furthermore, the proposed controller (6.7) is continuous. The continuity allows for easy physical realization. Remark 6.8 Inspired by [26–28], the distributed attitude tracking and synchronization controller can be designed as ui = − Jˆ 0i i◦ + ωi× ( Jˆ 0i ωi + δ iT q i ) −δ iT (C ni (q i − δ i ωi ) + K ni ηi )  ˆ i − s di s+α 2 − κ1 s i − κ2 j∈Ni ai j (s i − s j ) i 2

(6.36)

i

Note that the errors between the sliding variables are used to realize the attitude synchronization in such a controller. As shown in [28], the attitude synchronization can only be achieved locally. The main reason is that si = s j does not if Rd = imply ⎡Ri = R j . For example, ⎤ ⎡ I 3 and ωi = ω j =⎤ωd hold, assume that cos ψ − sin ψ 0 − cos ψ − sin ψ 0 Ri = ⎣ sin ψ cos ψ 0 ⎦ and R j = ⎣ sin ψ − cos ψ 0 ⎦, where 0 < ψ < π2 . 0 0 1 0 0 1 It is clear that Ri = R j . However, si equals to s j because (Ri − RiT )∨ = (R j − R Tj )∨ = [0, 0, 2 sin ψ]T in this example. Therefore, si = s j cannot ensure the attitude synchronization. For the synchronization term in Eq. (6.7), consider the system governed by β1 (R Tj Ri − RiT R j )∨ + ωi − ω j = 0. Construct the Lyapunov function

114

6 Distributed Attitude Tracking and Synchronization …

Vi j = tr(I 3 − R Tj Ri )

(6.37)

V˙i j = −β1 [(R Tj Ri − RiT R j )∨ ]T (R Tj Ri − RiT R j )∨ ≤ 0

(6.38)

whose time derivative is

V˙i j = 0 implies (R Tj Ri − RiT R j )∨ = 0. From [21], the solutions to (R Tj Ri − R j )∨ = 0 are Ri = R j and tr(RiT R j ) = −1, which correspond to the  minimum and maximum of Vi j , respectively. In conclusion, the proposed term −κ2 j∈Ni ai j [β1 (R Tj Ri − RiT R j )∨ + ωi − ω j ] can achieve attitude synchronization almost globally. RiT

Remark 6.9 The designed controller can be used to deal with the case for networked rigid spacecraft just by setting δ i = 0. It is straightforward to show that the conclusion of the attitude tracking in Theorem 6.1 still holds. Remark 6.10 Sliding-mode finite-time observer is introduced in Eq. (6.3). With sign function, the terms on the right hand of the observer are discontinuous and the chattering phenomenon may happen to the distributed observers. However, even with chattering, the responses of the observers are still continuous, hence, the chattering of the designed controller will not happen. That is, the sliding-mode finite-time observer will not cause the control chattering of the networked flexible spacecraft. Remark 6.11 In a practical attitude consensus mission, the communication edges may not be reliable all the time. For example, there may exist time delays and packet loss in communication channels due to the long distance between spacecraft. How to solve the case with a long-time delay and a fast-switching graph is still an open problem. However, if the time delay is short and the packet loss probability is low, the effect from the unstable communication can be considered as a small disturbance acting on the system, which can be compensated for by the designed controller. Remark 6.12 The communication graph in Fig. 6.1 will be taken as an example to explain how to implement the designed controller in Eq. (6.7) in a distributed attitude control mission. In this study, it is assumed that only partial spacecraft know the virtual leader’s information. For example, only the first spacecraft can receive information from the virtual leader in Fig. 6.1. To implement the designed controller in Eq. (6.7) , the onboard computer of each spacecraft will integrate the observer (6.3), the updating laws (6.8) and (6.10) in real time. The spacecraft in the team will send the variables of the observer (6.3) and its attitude and angular velocity to its ˆ d3 , ωˆ d3 , hˆ d3 , neighbors. For example, the fourth spacecraft in Fig. 6.1 will receive R R3 and ω3 . The control command for the fourth spacecraft will be calculated based on the received information, its own state and adaptation parameters computed based on the updating laws (6.8) and (6.10).

6.3 Distributed Attitude Tracking and Synchronization Under Directed Graphs

115

Fig. 6.1 Communication topology

Remark 6.13 This remark aims to explain the meanings of the control parameters used in the designed controller and give some fundamental rules to tune these parameters. γi represents the observer gain, where i = 1, 2, 3. A large γi will accelerate the convergence of the observer in Eq. (6.3). β and β1 are two gains to indicate the weights in the sliding variable si and the synchronization  of the attitudeT error signal T a [β (R R − R R )∨ + ωi − ω j ]. These two gains can be chosen term i j 1 i j j i j∈Ni based on the tuning methods of PD controller. κ1 and κ2 represent the weights of the attitude tracking and attitude synchronization, respectively. κ2 can be larger if the attitude synchronization is more important in the mission of concern. 0 , 1 and 2 are the updating gains of the adaptation laws in Eqs. (6.8) and (6.10). A large i will result in a fast updating rate of the adaptive parameters for i = 0, 1, 2.

6.3.3 Distributed Tracking and Synchronization with Partial State Feedback In some missions, it may not be easy to measure the modal variables directly. In such a situation, inspired by [29], the following observer can be introduced for the ith spacecraft η˙ˆ i = qˆ i − δ i ωi (6.39) q˙ˆ i + C ni (qˆ i − δ i ωi ) + K ni ηˆ i = 0 where ηˆ i and qˆ i are the estimates of ηi and q i , respectively. Denote eηi = ηi − ηˆ i and eqi = q i − qˆ i . Lemma 6.2 ([29]) The observer in Eq. (6.39) can estimate ηi and q i globally asymptotically. Then, the control algorithm can be redesigned as

116

6 Distributed Attitude Tracking and Synchronization …

ui = − Jˆ 0i i◦ + ωi× ( Jˆ 0i ωi + δ iT qˆ i ) ˆ i −δ iT (C ni (qˆ i − δ i ωi ) + K ni ηˆ i ) − s di s+α 2 − κ1 s i i 2 i  −κ2 j∈Ni ai j [β1 (R Tj Ri − RiT R j )∨ + ωi − ω j ]

(6.40)

Theorem 6.2 If μ¯ 0 > κ2 (N − 1)(2β1 + 4β + 2μ1 ) + μ0 and Assumptions 6.1−6.3 hold, the controller (6.40) with the finite-time observer (6.3), the modal variable observer (6.39) and the updating laws (6.8), (6.10) can drive the networked spacecraft to track the virtual leader almost globally and control the flexible vibration bounded. Proof With the controller (6.40), the closed-loop system is ˙ i − Jˆ 0i i◦ − ωi× [ J˜ 0i ωi + δ iT eqi ] J 0i s˙ i = J 0i  dˆi si +δ iT (C ni eqi + K ni eηi ) − − κ s − κ ai j (si − s j ) + d i 1 i 2 si 2 + αi2 j∈Ni (6.41) with Eqs. (6.39), (1.21), (6.8), (6.10) and (6.3). Essentially, this attitude control system is a cascade system. Similar to the proof of Theorem 6.1, the centralized tracking problem is considered first. Choose the Lyapunov function as V2 = V1 +

N

Vom,i

(6.42)

i=1 T T T where Vom,i = eiT P i ei , in which ei = [eηi , eqi ] and P i is defined the same as that in the paragraph after Eq. (6.25). It follows from the definition of si that ωi can be expressed as (6.43) ωi = si + RiT Rd ωd − β(RdT Ri − RiT Rd )∨

From Lemma 6.2, it can be concluded that both eηi  and eqi  are bounded. Due to the boundedness of Rd , ωd , Ri , C ni , K ni , δ i , eηi and eqi , siT [−ωi× δ iT eqi + δ iT (C ni eqi + K ni eηi )] = siT [−(RiT Rd ωd − β(RdT Ri − RiT Rd )∨ )× δ iT eqi + δ iT (C ni eqi + K ni eηi )]≤c¯i si 1 holds, where c¯i represents the upper bound of −(RiT Rd ωd − β(RdT Ri − RiT Rd )∨ )× δ iT eqi + δ iT (C ni eqi + K ni eηi ). Obviously, c¯i → 0 as eηi → 0 and eqi → 0. Moreover, from Lemma 6.2, c¯i → 0 as t → ∞. Denoting c¯m = maxi {c¯i }, one has κ2 λmin (L + L T ) T S S V˙2 ≤ −κ1 ST S −  N 2T +c¯m S1 − 2 i=1 eqi C ni eqi √ N T ≤ −κ1 S22 + 3N c¯m S2 − 2 i=1 eqi C ni eqi

(6.44)

√ ˙ It is √clear that V2 < 0 when S2 > c¯1 3N /κ1 . Furthermore, the set I = {S|S2 ≤ c¯1 3N /κ1 } → 03N as t → ∞. Therefore, S = 03N is asymptotically stable. Based

6.4 Numerical Simulations

117

on the same statements in Step A in the proof of Theorem 6.1, it can be concluded that the centralized attitude tracking can be achieved and the flexible vibration is bounded. In the second step, it is easy to show the boundedness of the total energy based on the similar proofs in Step B in the proof of Theorem 6.1. Finally, based on the  separation principle [1], one can draw the conclusions in Theorem 6.2.

6.4 Numerical Simulations In this section, a group consisting of one moving virtual leader and four follower flexible spacecraft is used as the example to verify the effectiveness of the proposed controller. Only the controller (6.40) will be demonstrated because the controller (6.40) will become that in Eq. (6.7) as the modal variable observer converges. Fig. 6.1 gives the communication graph among the networked spacecraft. Obviously, the virtual leader is globally reachable for all followers. The spacecraft parameters are indicated in Table 6.1 [30]. According to Lemma 2 in [20], for Rk ∈ S O(3) and Rr ∈ S O(3), the necessary and sufficient condition of tr(I 3 − RkT Rr ) = 0 is Rk = Rr . Hence, the attitude tracking error is indicated by e Ri = tr(I 3 − RdT Ri ). ˆ di may not belong to S O(3) before the However, as discussed in Remark 6.2, R ˆ di 2 is adopted converge of the observer in Eq. (6.3). Therefore, e Riˆ = Rd − R to showthe attitude estimation error of the observer in Eq. (6.3). Furthermore, N −1  N T Vsyn = i=1 tr(I − R R ) is used to represent the synchronization error. 3 j i j=i+1 ¯d i is set as 0.1 sin(5t)13 Nm. The finite-time observer’s parameters γ1 , γ2 and γ3 are √ chosen as 0.01 3, 0.005 and 0.005. The control parameters β, β1 , κ1 , κ2 , γ0 , 0 , 1 ,

2 , Jmax , χ , μ¯ 0 and χ1 are set as 0.01, 0.025, 12, 100, 1, 1000, 0.001, 0.001, 500, 2, 100 and 2, respectively. The initial value of dˆi , αi , ηˆ i and qˆ i are chosen as 0.1, 0.01, 03 and 03 for i = 1, 2, 3, 4. Clearly, μ¯ 0 > κ2 (N − 1)(2β1 + 4β + 2μ1 ) + μ0 is satisfied. The initial attitude and angular velocity of the virtual leader are I 3 and [0.003 cos(0.1t), 0.003 sin(0.2t), 0.003 sin(0.3t)]T , respectively. At t = 0, the

Table 6.1 Spacecraft parameters Parameter Value ⎤ ⎡ 350 3 4 ⎥ ⎢ J 0i ⎣ 3 270 10 ⎦ 4 10 190 ⎡ ⎤ 6.45637 1.27814 2.15629 ⎢ ⎥ δi ⎣ −1.25819 0.91756 −1.67264 ⎦ 1.11687 2.48901 −0.83674 1 , 2 ,3 0.7681, 1.1038, 1.8733 ξ1 , ξ2 , ξ3 0.0056, 0.0086, 0.013

Unit kg m2

– rad/s –

118

6 Distributed Attitude Tracking and Synchronization …

Fig. 6.2 Attitude estimate errors of the observer (6.3)

4

10-3

5 2 0 116

0 0

118

200

120 400

600

Time (s)

⎡

⎤ 0.707 0.5 −0.5 states of the four flexible spacecraft are as R1 (0) = ⎣ 0 0.707 0.707 ⎦, ω1 (0) = 0.707 −0.5 0.5 ˙2 = 0, η1 = 0.11⎡3 , η˙ 1 = 0, R2 (0) = diag{1, −1, −1}, ω 2 (0) = 0, η 2 = 0.113 , η ⎤ 0.306 0.92 −0.25 0, R3 (0) = ⎣ −0.354 0.354 0.866 ⎦, ω3 (0) = 0, η3 = 0.113 , η˙ 3 = 0, R4 (0) = 0.884 −0.177 0.433 ⎡ ⎤ −0.866 0 −0.5 ⎣ −0.5 0 0.866 ⎦, ω4 (0) = 0, η4 = 0.113 and η˙ 4 = 0. That is, at the beginning, 0 1 0 these four spacecraft rest at four different attitudes with non-zero initial vibration of flexible appendages. From Fig. 6.2, the virtual leader’s attitude can be estimated by the finite-time observer at about 118 s. Since the estimation error of the first spacecraft is influenced by the second and fourth spacecraft from the communication graph in Fig. 6.1, the solid blue curve in Fig. 6.2 is not monotonously decreasing. As indicated in Fig. 6.3, the initial value of e R2 is approximately equal to 4, which implies that the second spacecraft is located in the close neighborhood of an unstable equilibrium at the initial time according to statements about the unstable equilibria in the paragraph after Eq. (6.30). As shown in Figs. 6.3 and 6.4, the developed control algorithm can actuate the four spacecraft to track the virtual leader synchronously even though the second spacecraft starts from the close neighborhood of an unstable equilibrium. As indicated in Figs. 6.3 and 6.4, the attitude synchronization and tracking are achieved at 153 s and 363 s, respectively. The control torques and the angular velocities of

Fig. 6.3 Spacecraft attitude tracking errors

4

eR

4

10 -3

e e

2

2

0 350 400 450

R1 R2

e R3 e R4

0 0

100

200

300

Time (s)

400

500

600

6.4 Numerical Simulations

119

20

10 -3

Vsyn

4 2

10

0 150

0 0

100

200 200

300

250 400

500

600

Time (s) Fig. 6.4 Spacecraft attitude synchronization error

10 5 0 -5 -10 0

100

200

300

400

500

600

400

500

600

400

500

600

400

500

600

Time (s) 10 5 0 -5 -10 0

100

200

300

Time (s) 10 5 0 -5 -10 0

100

200

300

Time (s) 10 5 0 -5 -10 0

100

200

300

Time (s) Fig. 6.5 Control inputs of the four spacecraft

6 Distributed Attitude Tracking and Synchronization …

1

(rad/s)

120

0.2 0.1 0 -0.1 -0.2

1-1

0

100

200

1-2

300

400

1-3

500

600

2

(rad/s)

Time (s) 0.2 0.1 0 -0.1 -0.2

2-1

0

100

200

2-2

300

400

2-3

500

600

3

(rad/s)

Time (s) 0.2 0.1 0 -0.1 -0.2

3-1

0

100

200

3-2

300

400

3-3

500

600

4

(rad/s)

Time (s) 0.2 0.1 0 -0.1 -0.2

4-2

4-1

0

100

200

300

400

4-3

500

600

Time (s) Fig. 6.6 Angular velocities of the four spacecraft

the four spacecraft are given in Figs. 6.5 and 6.6. Figure 6.7 shows that the flexible vibrations of the four spacecraft are bounded. The time histories of the first element of Jˆ 0i , dˆi and αi are given in Figs. 6.8, 6.9 and 6.10. As shown in Fig. 6.8, Jˆ 0i does not converge to the real value of J 0i . The main reason is that the distributed attitude tracking can be achieved by many possible values of Jˆ 0i , besides J 0i . Therefore, it is not necessary for the parameter adaptation law (6.8) to find out the true parameter. To further verify the effectiveness of the proposed controller, a Monte Carlo simulation is conducted with stochastic initial attitudes. All other initial conditions, communication graph and control parameters are chosen the same as those in the simulation shown in Figs. 6.3, 6.4, 6.5, 6.6 and 6.7. The convergence times of the observer (6.3) and the controller are defined as the times after which e Riˆ = Rd − ˆ di 2 < 2 × 10−3 and e Ri = tr(I 3 − RdT Ri ) < 1 × 10−3 hold, respectively, for i = R 3  N T −3 is 1, 2, 3, 4. The time after which Vsyn = i=1 j=i+1 tr(I 3 − R j R i ) < 1 × 10 satisfied is called the time of achieving attitude synchronization. Table 6.2 gives the average convergence time of the observer (6.3), the average convergence time

6.4 Numerical Simulations

121

1

1 1-1

0.5

1-2

1-3

0 -0.5 0

100

200

300

400

500

600

Time (s) 2

1 2-2

2-1

0.5

2-3

0 -0.5 0

100

200

300

400

500

600

Time (s) 3

1 3-2

3-1

0.5

3-3

0 -0.5 0

100

200

300

400

500

600

Time (s) 4

1 4-2

4-1

0.5

4-3

0 -0.5 0

100

200

300

400

500

600

Time (s) Fig. 6.7 Modal variables of the four spacecraft

4 3 2 1 0

200

400

Time (s) Fig. 6.8 The first element of Jˆ 0i

600

122

6 Distributed Attitude Tracking and Synchronization …

0.11

0.105

0.1 0

200

400

600

400

600

Time (s) Fig. 6.9 Adaptive parameter dˆi

0.01

0.005

0 0

200

Time (s) Fig. 6.10 Adaptive parameter αi Table 6.2 Monte Carlo Results (ACTO: Average convergence time of the observer (6.3); ACTC: Average convergence time of the controller; ATAS: Average time of achieving attitude synchronization.) Spacecraft 1 Spacecraft 2 Spacecraft 3 Spacecraft 4 ACTO (s) ACTC (s) ATAS (s)

116.28 376.68

116.43 372.30

116.42 373.09 167.34

116.60 380.92

of the controller, and the average time of achieving attitude synchronization in 50 simulations. It is clear that the attitude synchronization is achieved earlier than the attitude tracking. As shown in Fig. 6.11, in all 50 simulations, the attitude tracking is achieved within 500 s. Furthermore, to illustrate the positive role of the estimates of virtual leader’s information and the synchronization term in controller (6.7) in escaping from the unstable equilibria, the following cases are compared: (a) centralized tracking without synchronization term (CT); (b) distributed tracking without synchronization term (DT); (c) centralized tracking with synchronization term (CTS) and (d) distributed tracking with synchronization term (DTS). In centralized cases, direct links to the virtual leader exist for all follower spacecraft, i.e., every spacecraft can receive the virtual leader’s information and the observer (6.3) is removed from the designed

6.4 Numerical Simulations

123 4

Attitude tracking error

Attitude tracking error

4 3 2 1 0

3 2 1 0

0

200

400

600

0

200

Time (s) (a) Spacecraft 1 4

600

400

600

4

Attitude tracking error

Attitude tracking error

400

Time (s) (b) Spacecraft 2

3 2 1 0

3 2 1 0

0

200

400

600

0

200

Time (s) (c) Spacecraft 3

Time (s) (d) Spacecraft 4

Fig. 6.11 Attitude tracking errors in 50 simulations with stochastic initial attitudes

e R2

4 CT DT CTS DTS

2

0 0

100

200

300

400

500

600

Time (s) Fig. 6.12 Attitude tracking errors of the 2nd spacecraft in four cases

control system. Figure 6.12 gives the attitude tracking errors of the second spacecraft in these four cases. By comparing the blue dashed and red chain curves, one can find that the distributed observer can help the second spacecraft leave the close neighborhood of the undesired critical point more quickly. As shown by the black dotted curve in Fig. 6.12, the controller with the synchronization term can drive the second spacecraft to escape from the small neighborhood of the unstable critical faster than that without the synchronization term. This is because the synchronization term can provide a large control torque with the given initial conditions. From the responses of CTS and DTS, one can find that the synchronization term plays the main role in escaping the close neighborhood of the unstable equilibrium. However, the settling time in CTS is a little bit less than that in case DTS. The possible reason is that the second spacecraft in case CTS exactly knows the reference attitude 118 s earlier than that in case DTS.

124

6 Distributed Attitude Tracking and Synchronization …

6.5 Experimental Verification In studies on the distributed attitude control of multiple flexible spacecraft [4–7, 29], only simulation results have been reported. The main reasons are the high cost of on-orbit missions and the difficulties in the development of a ground experimental system of several flexible spacecraft due to the gravity. Hence, in this section, an alternative method is presented based on four Quanser AERO systems with helicopter configurations in Fig. 6.13 to demonstrate the effectiveness of the proposed controller in attitude control, i.e., the controller in Eq. (6.7) with δ i = 0 is adopted. In other words, the experimental verification in this section is degenerated to the case discussed in Remark 6.9. The Quanser AERO system mainly contains two propellers attached to the motors, an integrated data acquisition device, QFLEX 2 interface and a built-in amplifier. By setting the two motors in the horizontal and vertical directions, respectively, the Quanser AERO system can be considered as a dual-rotor helicopter. Each helicopter can only rotate along pitch and yaw directions, hence, only the last two elements of the control signal from Eq. (6.7) are applied. Two optical encoders are equipped to measure pitch and yaw angles. Note that the rotation matrix representing the helicopter attitude is calculated based on the measured pitch and yaw angles and zero roll angle. The angular velocity of the helicopter is provided by the integrated Inertial Measurement Unit mounted within the AERO body. The real-time control is realized using QUARC software from Quanser Inc. with the sampling frequency of 500 Hz. The moments of inertia in pitch and yaw directions are 0.0219 kg kg·m2 and 0.0220 kg kg·m2 , respectively. The stiffness and damping of the system are compensated for such that the helicopter can rotate in pitch and yaw directions freely. Therefore, Eqs. (1.21) and (6.1) with δ i = 0 can describe the dynamics of the helicopters in Fig. 6.13. The communication topology among the four helicopters follows the graph in Fig. 6.1. Suppose that the virtual leader’s information is Rd (0) = diag{−1, −1, 1} and ωd = 0.01 × [0, sin(0.2t), sin(0.3t)]T . The initial conditions of the four heli-

Fig. 6.13 Experimental systems

6.5 Experimental Verification

125

Fig. 6.14 Estimate errors of the observer (6.3) in the experimental verification

2

10-3

1 0.5

1

0 1

0 0

2

3

20

40

60

Time (s) Fig. 6.15 Attitude tracking errors in experimental verification

4

eR

8

10-3

eR1 eR2

4

2

0 10

eR3

30

50

eR4

0 0

20

40

60

Time (s)

⎡

⎤ ⎡ ⎤ 0.7071 −0.7071 0 0 −1 0 copters are R1 (0) = I 3 , R2 (0) = ⎣ 0.7071 0.7071 0 ⎦, R3 (0) = ⎣ 1 0 0 ⎦, 0 0 1 0 0 1 ⎡ ⎤ −0.7071 −0.7071 0 R4 (0) = ⎣ 0.7071 −0.7071 0 ⎦, ω1 (0) = 0, ω2 (0) = 0, ω3 (0) = 0 and ω4 (0) = 0 0 1 0. The√parameters γ1 , γ2 , γ3 , β, β1 , κ1 , κ2 , 0 , 1 , 2 , Jmax , χ , μ¯ 0 and χ1 are chosen as 0.5 3, 0.5, 0.5, 1, 1, 0.05, 0.05, 0.001, 0.01, 0.001, 1, 2, 1 and 2, respectively. The initial value of Jˆ 0i , dˆi and αi are set as 0.01I 3 , 0.01 and 0.2 for i = 1, 2, 3, 4. d¯ i is set as 0.001 sin(5t)13 N·m. e Ri , e Riˆ and Vsyn defined in Sect. 6.4 are used to represent the attitude tracking errors, the estimation errors of the observer in Eq. (6.3) and the attitude synchronization errors, respectively. Figures 6.14, 6.15 and 6.16 give the time histories of e Riˆ , e Ri and Vsyn . As shown in Fig. 6.14, the observer in Eq. (6.3) converges to the leader’s attitude at 1.33 s. After ˆ di will be one element of S O(3). The attitude synchronization and tracking that, R are achieved at 9.11 s and 23.82 s, respectively. Furthermore, it should be noted that tr(I 3 − R1 (0)Rd (0)) = 4 holds for the first helicopter. As indicated in Fig. 6.15, the first helicopter can leave its initial attitude quickly, not stay in a small neighborhood of R1 (0) for a long time.

126

6 Distributed Attitude Tracking and Synchronization …

Fig. 6.16 Attitude synchronization error in experimental verification

10

10-3

V syn

5 5 0 5

0 0

15

25

20

35 40

45 60

Time (s)

6.6 Conclusions A distributed attitude tracking control strategy with an almost global synchronization term is proposed on S O(3) for a group of flexible spacecraft subject to parameter uncertainties with or without modal variable measurement under directed communication topologies. Based on the assumption that the virtual leader is globally reachable, a distributed observer is embedded in each flexible spacecraft to estimate the virtual leader’s information. The distributed controllers are designed based on local information exchange only. Rigorous proof, numerical simulations and experimental study verify the controller’s effectiveness. This work mainly provides a solution to the distributed attitude tracking on S O(3) in the missions accomplished by lots of spacecraft with limited communication channels. The finite-time observer and the synchronization term can help the spacecraft escape from the attraction of the unstable equilibria in some cases and are able to ensure the continuity of the proposed controller.

References 1. Gui, H., Vukovich, G.: Distributed almost global finite-time attitude consensus of multiple spacecraft without velocity measurements. Aerosp. Sci. Technol. 75, 284–296 (2018) 2. Miao, Y., Hwang, I., Liu, M., Wang, F.: Adaptive fast nonsingular terminal sliding mode control for attitude tracking of flexible spacecraft with rotating appendage. Aerosp. Sci. Technol. 93, 105312 (2019) 3. He, T., Wu, Z.: Multirate iterative learning disturbance observer with robustness to frequency deviation for high precision attitude stabilization of flexible spacecraft subject to complex disturbances. Aerosp. Sci. Technol. 96, 105583 (2020) 4. Du, H., Chen, M.Z., Wen, G.: Leader-following attitude consensus for spacecraft formation with rigid and flexible spacecraft. J. Guid. Control. Dyn. 39(4), 944–951 (2016) 5. Zou, A.M., de Ruiter, A.H., Kumar, K.D.: Distributed finite-time velocity-free attitude coordination control for spacecraft formations. Automatica 67, 46–53 (2016) 6. Huang, D., Wang, Q., Duan, Z.: Distributed attitude control for multiple flexible spacecraft under actuator failures and saturation. Nonlinear Dyn. 88(1), 529–546 (2017) 7. Wang, Q., Duan, Z., Lv, Y.: Distributed attitude synchronization control for multiple flexible spacecraft without modal variable measurement. Int. J. Robust Nonlinear Control 28(10), 3435– 3453 (2018)

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8. Nazari, M., Butcher, E.A., Sanyal, A.K.: Spacecraft attitude fractional feedback control using rotation matrices and exponential coordinates. J. Guid. Control. Dyn. 41(10), 2185–2198 (2018) 9. Bhat, S.P., Bernstein, D.S.: A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Syst. Control Lett. 39(1), 63–70 (2000) 10. Berkane, S., Tayebi, A., et al.: Construction of synergistic potential functions on S O(3) with application to velocity-free hybrid attitude stabilization. IEEE Trans. Autom. Control 62(1), 495–501 (2017) 11. Fadakar, I., Fidan, B., Huissoon, J.: Robust adaptive attitude synchronisation of rigid body networks on S O(3). IET Control Theory Appl. 9(1), 52–61 (2014) 12. Berkane, S., Abdessameud, A., Tayebi, A.: Hybrid global exponential stabilization on S O(3). Automatica 81, 279–285 (2017) 13. Thienel, J., Sanner, R.M.: A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise. IEEE Trans. Autom. Control 48(11), 2011–2015 (2003) 14. Mayhew, C.G., Sanfelice, R.G., Teel, A.R.: Quaternion-based hybrid control for robust global attitude tracking. IEEE Trans. Autom. Control 56(11), 2555–2566 (2011) 15. Lee, T.: Robust adaptive attitude tracking on S O(3) with an application to a quadrotor UAV. IEEE Trans. Control Syst. Technol. 21(5), 1924–1930 (2012) 16. Fernando, T., Chandiramani, J., Lee, T., Gutierrez, H.: Robust adaptive geometric tracking controls on S O(3) with an application to the attitude dynamics of a quadrotor UAV. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 7380– 7385. IEEE (2011) 17. Cao, Y., Ren, W., Meng, Z.: Decentralized finite-time sliding mode estimators and their applications in decentralized finite-time formation tracking. Syst. Control Lett. 59(9), 522–529 (2010) 18. Pomet, J.B., Praly, L., et al.: Adaptive nonlinear regulation: estimation from the Lyapunov equation. IEEE Trans. Autom. Control 37(6), 729–740 (1992) 19. Khalil, H.: Nonlinear Systems. Prentice Hall, New Jersey (2002) 20. Zou, Y., Meng, Z., Zuo, Z.: Rotation-matrix-based attitude synchronization of multiple spacecraft without velocity measurements. In: 2017 11th Asian Control Conference (ASCC), pp. 96–101 (2017) 21. Mahony, R., Hamel, T., Pflimlin, J.M.: Nonlinear complementary filters on the special orthogonal group. IEEE Trans. Autom. Control 53(5), 1203–1217 (2008) 22. Angeli, D., Praly, L.: Stability robustness in the presence of exponentially unstable isolated equilibria. IEEE Trans. Autom. Control 56(7), 1582–1592 (2011) 23. Angeli, D.: An almost global notion of input-to-state stability. IEEE Trans. Autom. Control 49(6), 866–874 (2004) 24. Hu, J., Hong, Y.: Leader-following coordination of multi-agent systems with coupling time delays. Phys. A 374(2), 853–863 (2007) 25. Lee, T.: Exponential stability of an attitude tracking control system on S O(3) for large-angle rotational maneuvers. Syst. Control Lett. 61(1), 231–237 (2012) 26. Zheng, Z., Xu, Y., Zhang, L., Song, S.: Decentralized attitude synchronization tracking control for multiple spacecraft under directed communication topology. Chin. J. Aeronaut. 29(4), 995– 1006 (2016) 27. Wu, B., Xu, C., Zhang, Y.: Decentralized adaptive control for attitude synchronization of multiple spacecraft via quantized information exchange. Acta Astronaut. 175, 57–65 (2020) 28. Chen, T., Shan, J.: Rotation-matrix-based attitude tracking and synchronization of multiple flexible spacecraft under directed graph. In: 2019 Chinese Control Conference (CCC), pp. 301–306 (2019) 29. Du, H., Li, S.: Attitude synchronization control for a group of flexible spacecraft. Automatica 50(2), 646–651 (2014) 30. Xiao, B., Hu, Q., Zhang, Y.: Fault-tolerant attitude control for flexible spacecraft without angular velocity magnitude measurement. J. Guid. Control. Dyn. 34(5), 1556–1561 (2011)

Chapter 7

Continuous Constrained Attitude Regulation on S O(3)

Abstract In this chapter, both the centralized and distributed attitude regulation of multiple flexible spacecraft are studied on S O(3) with attitude constraints in the absence of the measurements of the angular velocities and modal coordinates. For centralized regulation, each spacecraft knows the desired attitude. A repulsive potential function is constructed to handle attitude constraints. Then a continuous controller is designed without the measurements of angular velocities. With the assumption that only some spacecraft know the desired attitude, a finite-time observer is introduced to estimate the leader’s information under fixed or switching communication graphs. A continuous control law is proposed for the distributed attitude regulation with the estimate of the leader as the reference signal.

7.1 Introduction Control system plays a fundamental role in the fulfillment of spacecraft formation flying missions. Essentially, the configuration space of a rigid-body rotation is the special orthogonal group S O(3), which leads to strong nonlinear attitude dynamics and makes the cooperative attitude control challenging [1]. According to the communication topology structure, the coordinated attitude control can be classified into two types, namely, centralized and distributed coordination. In the centralized case, it is required that each follower spacecraft knows the desired attitude. The loss of the leader can cause mission failure. Hence, to improve robustness, some missions prefer to determine control actions in a distributed way. Therefore, the necessity for considering the distributed attitude control problem arises. Hence, the past few years have witnessed the growth in the number of studies on the attitude control based on the rotation matrix [2–4], which is a global and unique attitude representation. Also, the rotation-matrix-based attitude representation has been used to solve the attitude consensus [5–8]. For instance, Fadakar et al. developed a robust adaptive controller for the attitude synchronization based on a modified Reproduced from Ti Chen and Jinjun Shan. Continuous constrained attitude regulation of multiple spacecraft on S O(3). Aerospace Science and Technology 2020; 99: 105769. Copyright © 2020 Elsevier Masson SAS. All rights reserved. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_7

129

130

7 Continuous Constrained Attitude Regulation on S O(3)

attitude error on S O(3) [5]. Nazari et al. addressed the decentralized control for the formation flying of multiple rigid spacecraft with a constant communication time delay under an acyclic directed graph [6]. Zheng et al. focused on the decentralized rotation-matrix-based control for the attitude tracking of a group of spacecraft with the feedback of its own and neighbors’ sliding variables [7]. Sarlette et al. presented a locally asymptotically stable attitude synchronization law under a fixed connected undirected communication graph and an almost-global distributed control law for the attitude synchronization under directed graphs [8]. However, as shown in [7, 9, 10], time-invariant continuous feedback controllers can only stabilize the rotation of a rigid body almost globally on S O(3). The reason is that there exist some unstable critical points with any continuous attitude error function on S O(3). One way to overcome this drawback is to introduce a discontinuous attitude error function [5, 10], which will lead to a discontinuous control law. On the one hand, the commonly used attitude control actuators, like reaction wheels and control moment gyroscopes, can only provide continuous torques. On the other hand, if the discontinuous control command is acceptable, it is enough to obtain some globally convergent controllers based on the lower dimensional unit quaternions, such as the controllers in [11]. In fact, in practical missions, the spacecraft cannot stay at the unstable critical points due to the ubiquitous noise. Hence, the most important problem is not to eliminate the unstable critical points, but how to escape from the close neighborhoods of the unstable equilibria. Hence, it is desirable to design a continuous rotation-matrix-based controller that is able to drive the spacecraft to leave the close neighborhoods of the unstable critical points fast. Furthermore, as stated in [12, 13], the sensitive on-board instruments in some missions cannot be directly exposed to certain celestial objects (e.g., the sun), i.e., there exist some attitude-forbidden zones during the attitude maneuver. Hence, this study aims to design continuous controllers for the attitude regulation of the networked flexible spacecraft with attitude constraints in the absence of the measurements of angular velocities.

7.2 Problem Formulation 7.2.1 Kinematics and Dynamics of Flexible Spacecraft As a global and unique attitude representation, the rotation matrix Ri ∈ S O(3) from the body frame of the ith agent to the inertial reference frame is used in the following kinematics equation of the ith spacecraft. From Eq. (1.21), one has ˙ i = Ri ωi× R

(7.1)

According to Eq. (1.5), the dynamics equations of the ith flexible spacecraft can be expressed as

7.2 Problem Formulation

131

J i ω˙ i + δ iT η¨ i = −ωi× ( J i ωi + δ iT η˙ i ) + ui

(7.2)

η¨ i + C ni η˙ i + K ni ηi = −δ i ω˙ i

(7.3)

where J i ∈ R3×3 is the total inertia matrix of the flexible spacecraft, δ i ∈ Rn×3 is the rigid-flexible matrix, ηi ∈ Rn is the modal coordinate vector, C ni ∈ Rn×n is the damping matrix and K ni ∈ Rn×n is the stiffness matrix. ui represents the control torque applied to the agent i. It should be noted that only the first n vibration modes are taken into consideration in the dynamics equations. Denote q i = η˙ i + δ i ωi . Equations (7.2) and (7.3) can be recast as η˙ i = q i − δ i ωi J 0i ω˙ i = −ωi× ( J 0i ωi + δ iT q i ) + δ iT (C ni (q i − δ i ωi ) + K ni ηi ) + ui q˙ i + C ni (q i − δ i ωi ) + K ni ηi = 0

(7.4) (7.5) (7.6)

where J 0i = J i − δ iT δ i .

7.2.2 Attitude Constraints Based on Rotation Matrix As shown in Fig. 7.1, suppose that the angle between the normalized vector x pointing towards a celestial object and the normalized boresight vector y of an on-board instrument should be greater than θ . This constraint can be expressed as x · y < cos θ

(7.7)

where y = R y denotes the boresight vector expressed in the inertial coordinate with the rotation matrix R. Note that the vector x is represented in the inertial coordinate.

Fig. 7.1 Attitude constraints

132

7 Continuous Constrained Attitude Regulation on S O(3)

Suppose that there are m 1 sensitive on-board instruments and m 2 constraint objects. Hence, at most, m 1 m 2 attitude-forbidden zones exist [13]. For the p1 th sensitive instrument on the ith spacecraft, the inequality x Tp2 Ri y p1 < cos θi, p1 , p2 should be satisfied with respect to the p2 th unwanted object. Without loss of generality, it is assumed that θi, p1 , p2 ∈ (0, π ) holds for all i = 1, . . . , N , p1 = 1, . . . , m 1 and p2 = 1, . . . , m 2 . For the ith spacecraft, the set of the allowed attitudes is denoted by Ri = {Ri ∈ S O(3)|x p2 Ri y p1 < cos θi, p1 , p2 , p1 = 1, . . . , m 1 , p2 = 1, . . . , m 2 }.

7.2.3 Control Objective This works aims to develop continuous controllers for centralized or distributed attitude regulation and vibration suppression of N identical flexible spacecraft considering the attitude constraints shown in Sect. 7.2.2 with attitude-only measurements. In centralized regulation missions, every node knows the desired attitude. However, in distributed cases, the spacecraft can only calculate the control command based on its own and neighbors’ information. Furthermore, the following assumptions will be used in the controller design. Assumption 7.1 The desired attitude Rd is constant and Rd ∈ R1 ∩ R2 ∩ · · · ∩ R N . Assumption 7.2 Only the attitude of each spacecraft is measured.

7.3 Centralized Velocity-Free Attitude Regulation In this section, the centralized regulation with attitude constraints is considered with Assumptions 7.1 and 7.2. In fact, the work in this section can be considered as an extension of the study in [12, 14] to the reorientation control on S O(3).

7.3.1 Repulsive Potential Function For the ith spacecraft, the set of the allowed attitudes is denoted by Ri = {Ri ∈ S O(3)|x p2 Ri y p1 < cos θi, p1 , p2 , p1 = 1, . . . , m 1 , p2 = 1, . . . , m 2 }. To deal with the attitude constraints, inspired by Ref. [12], the following potential function V pi : Ri → R is introduced for the ith spacecraft V pi (Ri ) = Vei

m2 m1   p1 =1 p2

γ cos θ − x Tp2 Ri y p1 i, p , p 1 2 =1

(7.8)

7.3 Centralized Velocity-Free Attitude Regulation

133

where Vei = tr(I 3 − RdT Ri ) and γ is a positive constant. Note that Vei ∈ [0, 4] is the classical tracking potential function on S O(3) [4] and the unique solution to Vei = 0 is Ri = Rd . As x Tp2 Ri y p1 → cos θi, p1 , p2 , V pi (Ri ) goes to infinity. Lemma 7.1 Some properties of the potential function in Eq. (7.8) are: (i) (ii) (iii)

V pi (Rd ) = 0; V pi (Rd ) > 0 for all Ri ∈ Ri \Rd ; V˙ pi (Ri ) can be expressed as ωi T i (Ri , Rd ), where i (Ri , Rd ) =     γ (x T Ri y× )T Rei mp11=1 mp22=1 cos θi, p , p γ−x T Ri y − Vei mp11=1 mp22=1 (cos θi, p , pp2 −x Tp1Ri y 1

2

p2

p1

1

2

p2

2 p1 )

.

Proof (i) Since Ri = Rd is the unique solution to tr(I 3 − RdT Ri ) = 0, V pi (Rd ) = 0 holds. (ii) For any Ri = Rd , tr(I 3 − RdT Ri ) ∈ (0, 4] holds. Hence, V pi (Rd ) > 0 holds because cos θi, p , p γ−x T Ri y > 0 in the set Ri . p1 p2 1 2 (iii) Note that V˙ei = tr(−RdT Ri ωi× ) = tr(ωi× RiT Rd ) = −ωiT (RiT Rd − RdT Ri )∨ = ωiT Rei (7.9) where Rei = (RdT Ri − RiT Rd )∨ ∈ R3 . Hence, V˙ pi (Ri ) can be expressed as V˙ pi (Ri ) = ωiT Rei

m2 m1   p1 =1 p2

+Vei

m2 m1  

p1 =1 p2 =1

γ cos θi, p1 , p2 − x Tp2 Ri y p1 =1 −γ x Tp2 Ri y×p1 ωi

(cos θi, p1 , p2 −

x Tp2 Ri

(7.10) y p1

)2

= ωi i (Ri , Rd ) T

Note that Ri = Rd is one solution to i (Ri , Rd ) = 0, but maybe not the unique solution. That is, the potential function V p (Ri ) may have some local minimums, local maximums or saddle points. Fortunately, according to Lemma 7.1, only Ri = Rd is  corresponding to the global minimum of V pi (Ri ).

7.3.2 Centralized Velocity-Free Control To solve the velocity-free control, an auxiliary system is constructed as ˙ i = Q i × Q i

(7.11)

where i = e Qi with a positive constant . The method to choose the initial value of Q i (0) ∈ S O(3) will be given in the following. Define the error between Q i and Ri as e Qi = √ 1 T ( Q iT Ri − RiT Q i )∨ ∈ R3 . As shown in [4], such an attitude 2

1+tr(Ri Q i )

134

7 Continuous Constrained Attitude Regulation on S O(3)

error is not well defined at the points satisfying tr(RiT Q i ) = −1. Such a drawback can be overcome by choose the initial value of Q i far away from the points satisfying tr(RiT Q i ) = −1. Hence, a continuous controller ui can be designed as ui = −κ1 e Qi − χi2 (t)Rei − i (Ri , Rd )

(7.12)

where κ1 is a positive constant and χ i (t) is a time-varying parameter with positive initial value, whose second-order updating law is defined as χ¨ i (t) = −γχ Vei (χ χ˙ i (t) + χi (t))

(7.13)

where γχ and χ are two positive constants and χ˙ i (0) can be chosen as 0. Construct the following Lyapunov function   N  χ˙ i2 1 T 1 T 1 T 2 ω J 0i ωi + q i q i + ηi K ni ηi + κ1 VeQi + χ i Vei + V pi (Ri ) + Vc = 2 i 2 2 γχ i=1



(7.14)

where VeQi = 2 − 1 + Ri ) ∈ [0, 2]. It is clear that Vc ≥ 0 holds and the necessary and sufficient condition to Vc = 0 is {ωi = 0, q i = 0, ηi = 0, Q i = Ri = Rd , χ˙ i = 0}. Note that q i = 0 implies η˙ i = 0 with ωi = 0 and V˙eQi = √ 1 T [iT (RiT Q i − Q iT Ri )∨ + ωiT ( Q iT Ri − RiT Q i )∨ ] = (ωi − i )T e Qi . 2

tr( Q iT

1+tr(Ri Q i )

The time derivative of Vc can be expressed as V˙c =

N  

− η˙ iT C ni η˙ i + ωiT [−κ1 e Qi − χi2 Rei − i (Ri , Rd )] + χi2 ωiT Rei

i=1

+ κ1 (ωi − i )T e Qi + i (Ri , Rd )T ωi + 2χ¨i χ˙ i /γχ + 2χi χ˙ i Vei =

N  



 − η˙ iT C ni η˙ i − κ1 eTQi e Qi − 2χ Vei χ˙ i2 ≤ 0

i=1

(7.15) Since the possible solutions to V˙c = 0 are {η˙ i = 0, e Qi = 0, Vei = 0, χ˙ i = 0}, {η˙ i = 0, e Qi = 0, Vei = 0, χ˙ i = 0} or {η˙ i = 0, e Qi = 0, χ˙ i = 0, Vei = 0}, the invariant set I = {(ωi , η˙ i , ηi , Q i , Ri , χ˙ i )|V˙c = 0} can be discussed in the following two cases: Case (I) {η˙ i = 0, e Qi = 0, Vei = 0, χ˙ i = 0} or {η˙ i = 0, e Qi = 0, Vei = 0, χ˙ i = 0}: Vei = 0 implies Ri = Rd . It follows from the dynamics equations (1.21)–(7.3) that ωi = 0 with Ri = Q i = Rd and e Qi = 0, hence, ηi = 0 holds. Hence, the attitude regulation is achieved and the flexible vibration is suppressed in such a case. Case (II) {η˙ i = 0, e Qi = 0, χ˙ i = 0, Vei = 0}: As shown in [4], the only solution to e Qi = 0 is Q i = Ri . It follows from e Qi = 0 and the definition of i that i = 0 and Q i = Ri = Constant. Hence, ωi = 0 holds. It can be concluded from Eqs. (1.21)–(7.3) that ηi = 0 and ui = 0. That is, −χi2 (t)Rei − i (Ri , Rd ) = 0.

7.3 Centralized Velocity-Free Attitude Regulation

135

According to the updating law of χi (t), χi (t) = 0 holds with non-zero Vei . Then, one has i (Ri , Rd ) = 0. Note that Ri = Rd is one solution to i (Ri , Rd ) = 0, but it may not be the unique one. Hence, it can be concluded that the closed-loop system is just stable in the sense of Lyapunov, but not necessarily asymptotically stable. Note that in both cases, η˙ i and ωi will go to zero as time goes to infinity. Therefore, the flexible vibration will be damped out gradually. The discussions of the undesired critical points and the method to avoid these points will be given in the following subsection.

7.3.3 Undesired Critical Points There may exist some solutions to i (Ri , Rd ) = 0 except Ri = Rd and these solutions can be local maximums, local minimums and saddle points. These undesired critical points can be classified into the following two types. (i) Local maximums and saddle points: These critical points are unstable, i.e., the system will stay at these points forever only when the system starts from here or some specific initial conditions. Hence, it’s easy to escape these critical points by introducing nonzero initial control torque. At these points, if Rei = 0, with the help of the term −χi2 (t)Rei , the system will leave these unstable points and manifolds. However, the term Rei might also be equal to zero at these local maximums and saddle points. For example, assume that only one sensitive on-board instrument and one constraint celestial object. Let x = [−1, 0, 0]T , y = [1, 0, 0]T and Rd = I 3 . θ ∈ (0, π ) can be any value. At the attitude position Ri = diag{1, −1, −1}, both Rei and i (Ri , Rd ) are equal to 0. If Ri is expressed based on the Euler angles (φ¯ i , θ¯i , ψ¯ i ) associated with the sequence (1, 2, 3), θ¯i = 0, ψ¯ i = 0 and φ¯ i = π hold⎡at Ri = diag{1, −1, −1}. ⎤ Ignoring higher order terms, Ri can be lin1 ψ¯ i −θ¯i earized as ⎣ ψ¯ i −1 (π − φ¯ i ) ⎦. Hence, the gradient of i (Ri , Rd ) can ¯ ¯ −θi −(π − φi ) −1 be written as ∇ E123 i = diag{c1 , −2c1 /(cos(θ ) + 1), −2c1 /(cos(θ ) + 1)}, where E123 = [φ¯ i , θ¯i , ψ¯ i ]T and c1 = 2γ /(cos(θ ) + 1) > 0. Since ∇ E123 i is indefinite, Ri = diag{1, −1, −1} is a saddle point. At this unstable saddle point, the terms −χi2 (t)Rei − i (Ri , Rd ) in Eq. (7.12) are equal to zero. Hence, we can take full advantage of the term −κ1 e Qi . If the Q i is in the set Su = { Q i | Q i = Ri }, the term −κ1 e Qi will be equal to zero. Hence, the initial value of Q i can be chosen as one element in Ri a little far from the element of the set Su and the points where e Qi is not well defined, i.e., Q i (0) ∈ { Q i |( Q iT Ri (0) − Ri (0)T Q i )∨ 2 > 1 , Q i ∈ Ri }, where 1 is a positive constant. The aim of the condition Q i ∈ Ri is to avoid the situation that the initial control torque drives the spacecraft to the attitude-forbidden zones fast, which may cause large overshoot. (ii) Local minimums: The number of the local minimums may increase with the number of the attitude constraints and the local minimums will result in some local stable equilibriums. A potential method to escape from local minimums is injecting

136

7 Continuous Constrained Attitude Regulation on S O(3) Algorithm 7.1

Initialize: Nχi = 0, N i = 0 If i (Ri , Rd )2 < 0.01 && tr(I 3 − RdT Ri ) > 0.01 && Rei 2 > 0.01 then Modify Eq. (7.13) as χ¨ i (t) = −Vei (χ˙ i (t) + χi (t)) + κχ Nχi δ(t) and let Nχi = Nχi + 1 elseif i (Ri , Rd )2 < 0.01 && tr(I 3 − RdT Ri ) > 0.01 && Rei 2 < 0.01 then Modify i as i = e Qi + κ N i δ(t) and let N i = N i + 1 end if

some energies such that the system can leave the attraction domains of the local minimums. In the case that Rei is not equal to zero, the term −χi2 (t)Rei can be utilized to escape from the local minimums. However, according to the updating law in Eq. (7.13), χi2 (t) decreases over time. Hence, if the system is trapped in the close neighborhood of one local minimum, one heuristic method to drive the spacecraft to escape from such undesired equilibriums is to add new energy by resetting the value of χi (t) at some time. If Rei is zero at these local minimums, the term −κ1 e Qi can be used to add some energy to the system. To keep the controller continuous, a possible algorithm is to schedule the execution of Algorithm 7.1 except initialization multiple times. One easy way is to run Algorithm 7.1 every T0 s. In Algorithm 7.1, δ(t) represents Dirac delta function and κχ and κ are two positive constant. The introduction of the term κχ Nχi δ(t) will only make χ˙ i (t) change suddenly and χ i (t) will be continuous but not differentiable. Similarly, Q i is continuous with Algorithm 7.1 as well. Hence, the controller in Eq. (7.12) is still continuous. Note that Nχi + 1 and N i + 1 represent the attempts of injecting energy by the terms χi2 (t)Rei and −κ1 e Qi , respectively. The injected energy increases with the attempts. Furthermore, in fact, the conditions i (Ri , Rd )2 < 0.01 && tr(I 3 − RdT Ri ) > 0.01 && Rei 2 > 0.01 and i (Ri , Rd )2 < 0.01 && tr(I 3 − RdT Ri ) > 0.01 && Rei 2 < 0.01 may include some local maximums and saddle points, both of which are unstable critical points. For these points, Algorithm 7.1 does not need work because the method in the previous discussion is enough to control the system to leave the attraction of these points. Hence, both Nχi and N i are initialized to zero in Algorithm 7.1.

7.3.4 Finite Control Torque In the controller (7.12), the avoidance of the forbidden attitude cone is achieved by the following term i (Ri , Rd ) = Rei

m1  m2  p1 =1 p2

− Vei

γ T cos θ i, p1 , p2 − x p2 R i y p1 =1

m2 m1  

γ (x Tp2 Ri y×p1 )T

p1 =1 p2 =1

(cos θi, p1 , p2 − x Tp2 Ri y p1 )2

(7.16)

7.4 Distributed Attitude Regulation Control

137

which is defined in Ri . However, if the spacecraft approaches the edges of the forbidden zones, i (Ri , Rd ) goes to infinity, i.e., the control command will be infinite. Hence, to avoid the infinite control command, it is assumed that the initial atti¯ i = {Ri ∈ S O(3)|x p2 Ri y p < tude of the ith spacecraft belongs to a smaller set R 1 cos(θi, p1 , p2 + ), p1 = 1, . . . , m 1 , p2 = 1, . . . , m 2 }, where  is a small positive con¯ i , the term i (Ri , Rd ) will be finite with an upper bound stant. In the set R 2γ m 1 m 2 /[cos θi, p1 , p2 − cos(θi, p1 , p2 + )] + 4γ m 1 m 2 /[cos θi, p1 , p2 − cos(θi, p1 , p2 ¯ i , it may enter the +)]2 . Nevertheless, even if the spacecraft starts from the set R ¯ i since the control command is the synthesis of the control commands for the Ri \ R attitude regulation and the forbidden-zone avoidance, both of which might be mutually contradictory. Fortunately, the main term for attitude regulation Rei is bounded by 2, but the term cos θi, p , p γ−x T Ri y goes to infinity with x Tp2 Ri y p1 → cos θi, p1 , p2 . p1 p2 1 2 Hence, as long as  is chosen small enough, the control command from the repulsive potential function will play the main role, i.e., the controller will drive the spacecraft to keep away from the attitude-forbidden zone. In such a case, the control command governed by Eq. (7.12) will be bounded.

7.4 Distributed Attitude Regulation Control In this section, the distributed attitude regulation task will be studied with Assumptions 7.1 and 7.2. Both the cases under fixed and time-varying communication topologies will be discussed. The following assumptions are made for the fixed and timevarying graphs, respectively. Assumption 7.3 The communication graph is fixed and the leader has directed path(s) to all follower spacecraft. Assumption 7.4 For the time-varying topologies, assume that there exists time sequence tk , k ∈ {0, 1, . . . , ∞} such that the communication graph is time invariant during each interval t ∈ [tk , tk+1 ], k = 0, 1, 2, . . .. Suppose that the leader is globally reachable during each time interval.

7.4.1 Finite-Time Distributed Observer Since some spacecraft are not linked to the leader directly, inspired by [15], the following sliding mode observer is introduced for each spacecraft to estimate the leader’s attitude. ⎧ ⎫ ⎨ ⎬ ˙R ˆ i = −γ1 sign ˆi − R ˆ j ) + bi ( R ˆ i − Rd ) ai j ( R (7.17) ⎩ ⎭ j∈Ni

138

7 Continuous Constrained Attitude Regulation on S O(3)

where i = 1, . . . , N , γ1 > 0, sign(R) : R3×3 → R3×3 denotes the matrix with eleˆ i is the estimate of the ment being the sign of the corresponding element of R and R ˆ i is chosen as Ri (0). Essenleader’s attitude on S O(3). The initial conditions of R tially, the main role of the observer in Eq. (7.17) is to provide the reference trajectory for each follower spacecraft in the communication graph. Based on Theorem 3.1 in [15], one has the following lemma. Lemma 7.2 Consider the system in Eq. (7.17) with Assumption 7.3. If γ1 > 0 is satisfied, thenthe finite-time consensus is achieved with convergent time upper bound   maxi {| Rˆ i,k,r (0)−Rd,k,r (0)|} TR1 = maxk maxr , where Rˆ i,k,r and Rd,k,r are the (k, r ) γ1 ˆ i and Rd , respectively. entry of the matrix R Essentially, the observer in Eq. (7.17) aims to estimate leader’s attitude element ˆ i may not be able to represent by element. However, a potential problem is that R ˆi = a rotation in 3D space. Fortunately, after t = TR1 , Rˆ i,k,r equals to Rd,k,r , i.e., R ˆ Rd ∈ S O(3). Hence, after the convergence of the observer in Eq. (7.17), Ri will be an element of S O(3) and can be used to specify the desired attitude. Furthermore, as shown in Lemma 7.2, the upper bound of the convergence time TR1 is inversely proportional to γ1 in a specified task. Hence, one can choose a large γ1 such that the finite-time observer can converge fast.

7.4.2 Distributed Velocity-Free Regulation Controller First, the controller is designed for a fixed communication graph under Assumption 7.3. Since the angular velocity is immeasurable, the same auxiliary system of Q i as that in Sect. 7.3.2 is built. Based on the estimate of the leader’s attitude and the similar potential function, the distributed controller can be designed as ˆ ei −  ˆ i (Ri , R ˆ i) ui = −κ1 e Qi − χi2 (t) R where

ˆ iT Ri − R T R ˆ ei = ( R ˆ i )∨ R i

ˆ iT R ˆ i − I 3 2 ) R

m 1

p1 =1

and ˆ i (Ri , Rˆ i ) = Rˆ ei

m 2

m2 m1   p1 =1 p2 =1

γ (x Tp2 Ri y×p1 )T p2 =1 (cos θi, p1 , p2 −x Tp Ri y p )2 1 2

(7.18)

γ cos θi, p1 , p2 −x Tp2 Ri y p1

with

Vˆei =

−(|Vˆei | + T

ˆ i Ri ). tr(I 3 − R

ˆ i ) = i (Ri , Rd ) holds in the case of R ˆ i (Ri , R ˆ i = Rd . It should be noted that 

7.4 Distributed Attitude Regulation Control

139

With the assumption that the ith spacecraft start from R¯ i , the stability of the closedloop system under the controller (7.18) can be proven by the separation principle between the controller and the finite-time observer. As shown in Sect. 7.3, when ˆ i = Rd , the closed-loop system is stable in the sense of Lyapunov. Next, we should R prove that the system will not escape within a finite time. Consider the following total energy of the ith flexible spacecraft. Ei =

     1 1 ωi T J i δ iT ωi + ηiT K ni ηi ˙ ˙ η η δ In 2 2 i i 

From the above equation, E i ≥ 21 λmin (K ni )ηi 22 holds. The time derivative of E i is E˙ i =

N 

J i δ iT δ In

 (ωi 22 + η˙ i 22 ) + 21 λmin

[ωiT ui − η˙ i C ni η˙ i ]

i=1

=

N 

(7.19)

(7.20) [−κ1 ωiT e Qi

−

χi2 (t)ωiT

ˆ ei − R

ωiT i (Ri ,

ˆ i ) − η˙ i C ni η˙ i ] R

i=1

Since both Ri and Q i are elements in S O(3), e Qi 2 ≤ 2 holds. According to the updating law of χ i (t) in Eq. (7.13) and Vei ∈ [0, 4], χi is bounded from above by ˆ i is bounded during the the initial value and from below by zero. From Lemma 2, R ˆ distributed regulation mission. Hence, Rei is bounded as well. If the spacecraft starts ¯ i , it will stay in a proper subset of Ri , in which the term i (Ri , R ˆ i ) will be from R ˙ bounded. Hence, there must exist a positive constant c1 , such that E i satisfies the inequality E˙ i =

N  [ωiT ui − η˙ i C ni η˙ i ] i=1

=

N  ˆ i ) − η˙ i C ni η˙ i ] ˆ ei − ωiT i (Ri , R [−κ1 ωiT e Qi − χi2 (t)ωiT R

(7.21)

i=1

≤ c1 ωi 2 Hence, one has E˙ i ≤ c2 E i + c3

(7.22)

140

7 Continuous Constrained Attitude Regulation on S O(3)



 J i δ iT with a positive constant c¯2 and c3 = c12 /(4c¯2 ) > 0. δ In Integrating E˙ i ≤ c2 E i + c3 yields

where c2 = 2c¯2 /λmin

E i ≤ −c3 /c2 + exp(c2 t)(E 1 (0) + c3 /c2 )

(7.23)

Hence, in a finite time, E i cannot be equal to infinity. After the convergence of the finite-time observer (7.17), the distributed controller in Eq. (7.18) will become the centralized one. Therefore, based on the separation principle [16], the closed-loop system under the actuation of the distributed controller (7.18) is still stable in the sense of Lyapunov. In controller (7.18), the communication graph is fixed. However, in practice, due to limited communication ranges, environmental obstacles and possible packet dropout, some communication edges may be unreliable. Hence, this subsection will consider the distributed regulation under a communication graph with Assumption 7.4. The aim is still to design a continuous controller for the distributed attitude regulation in the absence of velocity measurements. Unfortunately, the controller directly based on neighbors’ information is probably discontinuous because the received signal of each spacecraft is likely not continuous due to the switching communication graph. Therefore, in this study, to develop a continuous controller, the observer in Eq. (7.17) is introduced as well. It can be concluded from Theorem 1 in [17] that the following lemma holds. ˆ i in the observer (7.17) will be Lemma 7.3 With Assumptions 7.4 and γ1 > 0, R  maxi {| Rˆ i,k,r (0)−Rd,k,r (0)|} . equal to Rd after TR2 = (N − 1) maxk maxr γ1 The observer (7.17) under the switching communication graph can provide a continuous reference signal for the attitude regulation mission even though the observer is designed based on the probably discontinuous signal that each spacecraft received from its neighbors. Essentially, the observer (7.17) can be considered as a nonlinear filter that can remove the discontinuity of the received signal. The distributed regulation controller can be designed the same as that in Eq. (7.18). Obviously, the ˆ i . The stability controller is also continuous with the continuous reference attitude R of the closed-loop system can be proven using the same method in the case under fixed graphs satisfying Assumption 7.3.

7.5 Numerical Simulations 7.5.1 Centralized Attitude Regulation 7.5.1.1

An Example of Centralized Attitude Regulation

Simulations of the centralized regulation of four spacecraft are conducted with the parameters shown in Table 7.1. The control parameters γ , , κ1 , γχ , χ ,

7.5 Numerical Simulations

141

Table 7.1 Parameters of flexible spacecraft Parameter Value ⎡ ⎤ 350 3 4 ⎢ ⎥ J 0i ⎣ 3 270 10 ⎦ 4 10 190 ⎡ ⎤ 6.45637 1.27814 2.15629 ⎢ ⎥ δi ⎣ −1.25819 0.91756 −1.67264 ⎦ 1.11687 2.48901 −0.83674 1 0.7681 2 1.1038 3 1.8733 ξ1 0.0056 ξ2 0.0086 ξ3 0.013

Unit kg m2

– rad/s rad/s rad/s – – –

κχ and κ are chosen as 0.1, 1, 10, 1, 1, 5 and 5, respectively. Suppose that the leader’s attitude is I 3 . Assume there are one sensitive on-board instrument and three constraint celestial objects, i.e., m 1 = 1 and m 2 = 3. The normalized boresight vector of the instrument y = [0, 0, −1]T . The normalized vectors of the three celestial objects are x 1 = [1, 0, 0]T , x 2 = [0.5, 0.5, 0.707]T and x 3 = [−0.5, 0.5, 0.707]T . The allowed maximum angle of the three constraints are θi,1,1 = π/10 rad, θi,1,2 = π/6 rad and θi,1,3 = π/8 rad,⎤respectively. The initial atti⎤ ⎡ ⎡ tudes of the four spacecraft are ⎣

0.5 −0.0453 −0.8648 0.7071 −0.6533 0.2706 0 0.9986 −0.0523 ⎦, ⎣ 0 −0.3827 −0.9239 ⎦, 0.8660 0.0262 0.4993 0.7071 0.6533 −0.2706

⎤ 0 0.9239 −0.3827 ⎣ 0 −0.3827 −0.9239 ⎦ and diag{1, −1, −1}. The initial value of ηi , ωi , η ˙ i , χi −1 0 0 ⎡

and χ˙ i

are 0, 0, 0, 1 and 0. In this⎤example, Q 1 (0), Q 2 (0),⎤Q⎡3 (0) and Q 4 (0) are chosen as ⎤ ⎡ ⎡

0.5220 0.8464 −0.1057 0.2098 0.4538 −0.8660 −0.0396 −0.9491 0.3125 ⎣ −0.4547 0.3810 0.8051 ⎦, ⎣ 0.9465 0.1280 0.2963 ⎦, ⎣ 0.4107 −0.3006 −0.8608 ⎦ and 0.7217 −0.3721 0.5837 0.2453 −0.8818 −0.4027 0.9109 0.0942 0.4017

⎡

⎤ 0.7995 0.4968 0.3375 ⎣ −0.2222 0.7667 −0.6023 ⎦. It is trivial to show that Q i (0) ∈ { Q i |( Q iT Ri (0) − −0.5580 0.4065 0.7234 Ri (0)T Q i )∨ 2 > 0.01, Q i ∈ Ri } holds. Since the necessary and sufficient condition of Vei = 0 is Ri = Rd , Vei defined in the paragraph after Eq. (7.8) is used to reveal the attitude regulation error. The simulation results are shown in Figs. 7.2 and 7.3. With the proposed control algorithm, the closed-loop system will converge to the desired attitude within 200 s. In practical missions, there exist measurement noises and environmental disturbances. All these factors can be considered as random disturbances applied to the spacecraft. Hence, one more example with random disturbances is simulated.

142

7 Continuous Constrained Attitude Regulation on S O(3) 4 3 2 1 0 0

200

400

600

Time (s) Fig. 7.2 Attitude regulation errors 5

5

Control torque (Nm)

Control torque (Nm)

10

0 -5 -10 -15

0

-5

-10

-20 0

200

400

0

600

400

600

(b) 2nd spacecraft

(a) 1st spacecraft 5

Control torque (Nm)

5

Control torque (Nm)

200

Time (s)

Time (s)

0

-5

0

-5

-10

-10 0

200

400

Time (s)

(c) 3rd spacecraft

Fig. 7.3 Control torques

600

0

200

400

Time (s)

(d) 4th spacecraft

600

7.5 Numerical Simulations

143

4 3 2 1 0 0

200

400

600

Time (s) Fig. 7.4 Attitude regulation errors with random disturbances

Suppose that the magnitudes of random disturbances are less than 1 Nm. The attitude errors and control torques are shown in Figs. 7.4 and 7.5. Compared with Fig. 7.2, the random disturbances drive the fourth spacecraft to leave the unstable equilibrium a little faster. The attitude regulation task is completed successfully; hence, the designed controller is robust to the disturbances.

7.5.1.2

Monte Carlo Simulations

As shown in the theoretical discussion in Sect. 7.3.2, it is not easy to show the global or almost global stability of the closed-loop system, hence, the Monte Carlo method is used to prove the effectiveness of the proposed control method numerically, i.e., lots of simulations will be conducted with stochastic initial attitudes on S O(3). All control parameters and attitude constraints are chosen the same as those in Sect. 7.5.1.1. Figure 7.6 gives 100 simulation results with Q i (0) ∈ { Q i |( Q iT Ri (0) − Ri (0)T Q i )∨ 2 > 0.01, Q i ∈ Ri }. Since the centralized regulation is considered, i.e., each spacecraft knows the desired attitude, essentially, the Monte Carlo simulations here can be considered as 400 simulations of the maneuver of the individual flexible spacecraft. As indicated in Fig. 7.6, no spacecraft are trapped at the undesired critical points and all regulation tasks can be fulfilled. In more than 90 simulations, the attitude maneuvers are finished within 200 s. However, in several simulation results, the settling time is longer than 200 s and some oscillations exist in the final stage. The main reason is that the control parameters identical in all 100 simulations may not be the best control gains for these several simulations. Also, a 400 s settling time is still acceptable. Next, more Monte Carlo simulations are conducted to verify the performance of the controller further with non-zero initial angular velocity. Suppose that the initial angular velocity is random and the norm of initial angular velocity is less than 0.01 rad/s. 100 simulations results are shown in Fig. 7.7. In all simulations, the attitude regulation is achieved within 400 s.

144

7 Continuous Constrained Attitude Regulation on S O(3) 5

5

Control torque (Nm)

Control torque (Nm)

10

0 -5 -10 -15

0

-5

-10

-20 0

200

400

0

600

(a) 1st spacecraft

400

600

(b) 2nd spacecraft 5

Control torque (Nm)

5

Control torque (Nm)

200

Time (s)

Time (s)

0

-5

0

-5

-10

-10 0

200

400

600

Time (s)

(c) 3rd spacecraft

0

200

400

600

Time (s)

(d) 4th spacecraft

Fig. 7.5 Control torques with random disturbances

7.5.1.3

Comparison

The proposed controller is compared with several approaches that can be obtained directly by extending the studies in [13, 14] to attitude regulation on S O(3). Inspired by the studies in [13, 14], two control laws can be designed as Controller A : ui = −κ1 e Qi − i (Ri , Rd )

(7.24)

Controller B : ui = −κ1 e Qi − κ2 Rei − i (Ri , Rd )

(7.25)

where κ1 and κ2 are two positive constants. The definitions of e Qi , Rei and i (Ri , Rd ) can be found in Sect. 7.3.2.

7.5 Numerical Simulations

145

(a) 1st spacecraft

(b) 2nd spacecraft

(c) 3rd spacecraft

(d) 4th spacecraft

Fig. 7.6 Simulation results of the centralized regulation with stochastic initial attitudes

Take one flexible spacecraft as an example. Assume that there exist one sensitive on-board instrument and nine constraint celestial objects, i.e., m 1 = 1 and m 2 = 9. The normalized boresight vector of the instrument y = [0, 0, −1]T . The normalized vectors of the nine celestial objects are x 1 = [1, 0, 0]T , x 2 = [0.8090, 0.5878, 0]T , x 3 = [0.3090, 0.9511, 0]T , x 4 = [−0.3090, 0.9511, 0]T , x 5 = [−0.8090, 0.5878, 0]T , x 6 = [−1, 0, 0]T , x 7 = [−0.8090, −0.5878, 0]T , x 2 = [−0.3090, −0.9511, 0]T and x 3 = [0.3090, −0.9511, 0]T . For j = 1, . . . , 9, θi,1, j = π/10 holds. As shown in Fig. 7.8, the end point of y must not enter the 9 interconnected cycles in red. Assume that the spacecraft starts from diag{1, −1, −1} and the desired attitude is I 3 . The end point of y is equal to [0, 0, 1]T at the initial time and this point should reach [0, 0, −1]T in the case that the spacecraft arrives at the desired attitude. Hence, the end point of y should cross the narrow range between the forbidden zones caused by the first and ninth celestial objects during the attitude maneuver. κ2 is chosen as χ i (0) = 1. The remaining control parameters γ , , κ1 , γχ , χ , κχ and κ are chosen as 0.01, 1, 10, 0.001, 0.1, 5 and 5, respectively. As shown in Figs. 7.8 and 7.9, only the proposed controller can drive the spacecraft to the desired attitude, and under Controllers A and B, the spacecraft will stay at a local minimum. The main reason is that the closed-loop system with Controller A has some local

146

7 Continuous Constrained Attitude Regulation on S O(3)

(a) 1st spacecraft

(b) 2nd spacecraft

(c) 3rd spacecraft

(d) 4th spacecraft

Fig. 7.7 Simulation results of the centralized regulation with stochastic initial attitudes and angular velocities

Fig. 7.8 Trajectory of the end point of y

7.5 Numerical Simulations Fig. 7.9 Attitude regulation errors

147 4 3 2 1

Proposed controller Controller B Controller A

0 0

200

400

600

800

1000 1200

Time (s)

minimums especially with a lot of attitude-forbidden zones. Controller B adds a term based on the feedback of the attitude regulation error with a constant gain. Essentially, as stated in [14], this method can change the location of the local minimum, i.e., some local minimums can be removed, but Controller B may introduce some new local minimums. This statement can be verified by the different steady states in the blue and green curves in Figs. 7.8 and 7.9. However, the time-varying gain in the proposed controller (7.12) leads to the time-varying local minimums, hence, the proposed controller is more likely to find a trajectory to the desired attitude. Furthermore, Algorithm 7.1 is able to keep the local minimums time-varying except when the closed-loop system arrive at the global minimum.

7.5.1.4

Effect of Threshold in Algorithm 7.1

To test the effect of the threshold in Algorithm 7.1, some simulations are performed by taking the ith spacecraft with parameters in Table 7.1 as example. Suppose that one attitude-forbidden zone with x 1 = [−1, 0, 0]T and y1 = [1, 0, 0]T exists. The desired attitude and the allowed minimum angle are I 3 and π/10. κ is chosen as 50. All the remaining control parameters are set the same as⎡those in Sect. 7.5.1.1. The ⎤ 1 0 0 initial value of spacecraft attitude and Q i are chosen as ⎣ 0 −0.99995 0.01 ⎦ 0 −0.01 −0.99995 ⎡ ⎤ 1 0 0 and ⎣ 0 −1 0 ⎦. At the initial time, the spacecraft angular velocity is zero. 0 0 −1 With such initial conditions, i (Ri , Rd )2 = 0.001, tr(I 3 − RdT Ri ) = 3.9999 and Rei 2 = 0.02 hold. In Cases a, b and c, the thresholds in Algorithm 7.1 are chosen as 0.1, 0.01 and 0.001, respectively. Hence, at t = T0 = 50 s, in Case a, the second judgment condition is satisfied and Cases b and c will redefine the updating law in Eq. (7.13). The simulation results in these three cases are shown in Fig. 7.10. With threshold 0.1, in the neighborhood of the desired attitude, at least one judgment

148 Fig. 7.10 Attitude regulation errors with various thresholds in Algorithm 7.1

7 Continuous Constrained Attitude Regulation on S O(3) 4 3 2 1 0 0

200

400

600

800

1000

Time (s) Fig. 7.11 The adaptive parameter χi in Cases b and c

4 Case b Case c

3 2 1 0 0

200

400

600

800

1000

Time (s)

condition in Algorithm 7.1 is satisfied, which results in that the attitude regulation cannot be achieved. As shown in Fig. 7.11, in Case b, after four trials to increase the adaptive parameter, the spacecraft leaves the close neighbor of the undesired equilibrium and arrives at the desired attitude finally. It can be found by comparing Cases b and c that the proposed control algorithm needs a longer settling time with a smaller threshold. Essentially, the judgment conditions in Algorithm 7.1 are used to check whether the system is trapped in the close neighborhoods of undesired equilibria. The threshold indicates the ranges of the close neighborhoods. A large threshold may cause that the judgment conditions are satisfied at too large neighborhoods of the undesired equilibria, which may deteriorate the control performance. If the threshold is small, the judgment conditions will only hold in the very small neighborhoods of the undesired equilibria, which may cause a longer settling time.

7.5 Numerical Simulations

149

Fig. 7.12 Communication topology

7.5.2 Distributed Attitude Regulation 7.5.2.1

Monte Carlo Simulations Under Fixed Graphs

Monte Carlo simulations are conducted under the fixed communication graph shown in Fig. 7.12. The control parameters and the forbidden attitude zones are set the same as those in Sect. 7.5.1.2. It can be seen that the leader is globally reachable. The parameter γ1 in the observer (7.17) is chosen as 0.1. Note that the initial condition of Q i is a random matrix satisfying the condition ( Q i (0)T Ri (0) − Ri (0)T Q i (0))∨ 2 > 0.01 and Q i (0) ∈ Ri . 100 simulation results with random initial attitudes are given in Fig. 7.13. All spacecraft can track the virtual leader within 400 s with the developed control algorithm.

7.5.2.2

Monte Carlo Simulations Under Switching Graphs

In this subsection, the Monte Carlo simulation results are presented under switching communication topologies. The attitude constraints and control parameters are set the same as those in previous subsection. Assume that the possible communication graphs are shown in Fig. 7.14. Two random sequences are generated as [6.11; 5.75; 4.09; 0.69; 3.23; 3.02; 0.42; 2.44; 5.06; 4.94; 0.754; · · · ], [c; a; b; d; a; d; a; b; c; b; d; . . .]. The first sequence represents the dwell time of the corresponding graph in the second sequence. That is, for the first 6.11 s, the graph (c) ˆ i − Rd 2 is adopted and in the next 5.75 s, the first graph is active and so on. E oi =  R is used to indicate the estimation errors of the observer (7.17). 100 simulation results are shown in Fig. 7.15. Obviously, all distributed attitude regulation missions are completed within 400 s. Figure 7.16 gives the estimation errors of the observer (7.17) and the control input of the first spacecraft in the first simulation. It is obvious that ˆ i will be equal to the leader’s attitude at the estimation errors are continuous and R ˆ i as the reference attitude, the control torque shown in Fig. 7.16b about 25 s. With R is also continuous.

150

7 Continuous Constrained Attitude Regulation on S O(3)

(a) 1st spacecraft

(b) 2nd spacecraft

(c) 3rd spacecraft

(d) 4th spacecraft

Fig. 7.13 Simulation results of the distributed regulation under fixed communication graph

Fig. 7.14 Switching communication topology

7.6 Conclusions

151

(a) 1st spacecraft

(b) 2nd spacecraft

(c) 3rd spacecraft

(d) 4th spacecraft

Fig. 7.15 Simulation results of the distributed regulation under switching communication graph

7.6 Conclusions Continuous attitude regulation controllers are proposed on S O(3) for a group of flexible spacecraft with attitude-only measurements subject to attitude constraints. Both the centralized and distributed cases are considered. To avoid forbidden attitudes, a repulsive potential filed is constructed. However, the introduction of the repulsive potential function may lead to undesired critical points. It is shown that the closedloop system is stable in the sense of Lyapunov. Two methods are presented to drive the system to leave these undesired critical points, which can be local maximums, local minimums and saddle points. Monte Carlo simulations are conducted to verify the effectiveness of the proposed controllers numerically. In the distributed case, a finite-time observer is designed to estimate the leader’s attitude for each follower spacecraft element by element under fixed or switching communication topologies. Then a continuous controller is developed based on the estimate of the leader’s information and the repulsive potential filed. The singularities and ambiguities associated with other attitude representations can be avoided because the controller is designed

152

7 Continuous Constrained Attitude Regulation on S O(3)

(a) Estimation errors of the observer (7.17) under a switching communication graph

(b) Control torque of the first spacecraft

Fig. 7.16 Simulation results of the distributed regulation under switching communication graph

on S O(3) directly. The continuities of the proposed controllers have advantages for practical implementation.

References 1. Zheng, Z., Shen, M.: Inertial vector measurements based attitude synchronization control for multiple spacecraft formation. Aerosp. Sci. Technol. 93, 105309 (2019) 2. Walsh, A., Forbes, J.R.: Constrained attitude control on S O(3) via semidefinite programming. J. Guid. Control Dyn. 41(11), 2483–2488 (2018) 3. Berkane, S., Abdessameud, A., Tayebi, A.: Hybrid global exponential stabilization on S O(3). Automatica 81, 279–285 (2017) 4. Lee, T.: Exponential stability of an attitude tracking control system on S O(3) for large-angle rotational maneuvers. Syst. Control Lett. 61(1), 231–237 (2012) 5. Fadakar, I., Fidan, B., Huissoon, J.: Robust adaptive attitude synchronisation of rigid body networks on S O(3). IET Control Theory Appl. 9(1), 52–61 (2014) 6. Nazari, M., Butcher, E.A., Yucelen, T., Sanyal, A.K.: Decentralized consensus control of a rigid-body spacecraft formation with communication delay. J. Guid. Control. Dyn. 39(4), 838– 851 (2016)

References

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7. Zheng, Z., Xu, Y., Zhang, L., Song, S.: Decentralized attitude synchronization tracking control for multiple spacecraft under directed communication topology. Chin. J. Aeronaut. 29(4), 995– 1006 (2016) 8. Sarlette, A., Sepulchre, R., Leonard, N.E.: Autonomous rigid body attitude synchronization. Automatica 45(2), 572–577 (2009) 9. Bhat, S.P., Bernstein, D.S.: A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Syst. Control Lett. 39(1), 63–70 (2000) 10. Berkane, S., Tayebi, A., et al.: Construction of synergistic potential functions on S O(3) with application to velocity-free hybrid attitude stabilization. IEEE Trans. Autom. Control 62(1), 495–501 (2017) 11. Thienel, J., Sanner, R.M.: A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise. IEEE Trans. Autom. Control 48(11), 2011–2015 (2003) 12. Shen, Q., Yue, C., Goh, C.H.: Velocity-free attitude reorientation of a flexible spacecraft with attitude constraints. J. Guid. Control. Dyn. 40(5), 1293–1299 (2017) 13. Lee, U., Mesbahi, M.: Feedback control for spacecraft reorientation under attitude constraints via convex potentials. IEEE Trans. Aerosp. Electron. Syst. 50(4), 2578–2592 (2014) 14. Hu, Q., Chi, B., Akella, M.R.: Anti-unwinding attitude control of spacecraft with forbidden pointing constraints. J. Guid. Control. Dyn. 42(4), 822–835 (2019) 15. Cao, Y., Ren, W., Meng, Z.: Decentralized finite-time sliding mode estimators and their applications in decentralized finite-time formation tracking. Syst. Control Lett. 59(9), 522–529 (2010) 16. Gui, H., Vukovich, G.: Distributed almost global finite-time attitude consensus of multiple spacecraft without velocity measurements. Aerosp. Sci. Technol. 75, 284–296 (2018) 17. Li, C., Qu, Z.: Distributed finite-time consensus of nonlinear systems under switching topologies. Automatica 50(6), 1626–1631 (2014)

Part III

Leaderless Attitude Consensus of Networked Rigid Spacecraft

In last part, the leader-follower attitude control of a group of flexible spacecraft is considered. The designed control methods can achieve the distributed attitude tracking mission and vibration suppression at the same time. As for distributed control directly on S O(3), two main strategies are used. The first one is designed for a specific type of communication graph, such as the connected undirected tree graph with a globally reachable leader in Chap. 4. The other is based on an observer to estimate the leader’s information and provide a reference trajectory for each follower spacecraft, such as the controllers in Chaps. 5–7. However, the leader-follower architecture relies heavily on the leader. Leader error or the loss of the leader will cause mission failure. The leaderless consensus strategy can avoid this problem. Also, in practical space engineering, the leaderless attitude consensus has some potential applications, such as the inter-communication between satellites. Thus, it is necessary to consider the leaderless consensus of multiple spacecraft. This part will take the networked rigid spacecraft as an example to show several controller design methods to solve the leaderless attitude synchronization problem.

Chapter 8

Continuous Leaderless Synchronization Control of Multiple Rigid Spacecraft on S O(3)

Abstract This chapter presents a solution to the leaderless consensus of multiple spacecraft on S O(3) under a connected undirected graph. An algorithm is proposed to generate an undirected tree graph from a communication topology. A distributed observer is designed to estimate the desired attitude and angular velocity of each spacecraft under the generated tree graph. An adaptive controller with a general connected undirected graph is developed to complete the synchronization task. Cases with zero and nonzero final angular velocities are considered. Theoretical proofs and numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.

8.1 Introduction Attitude coordination is crucial to spacecraft formation flying activities such as interferometry and inter-spacecraft communication [1]. However, because the configuration space of the spacecraft attitude is the special orthogonal group S O(3), the strongly nonlinear attitude dynamics poses serious challenges for controller design. A continuous time-invariant control law cannot globally stabilize the attitude maneuvering, and almost global asymptotic stability is the best result, because S O(3) is not diffeomorphic to any Euclidean space [2]. Attempts have been made to improve the performance of the controller on S O(3). Existing methods can be roughly classified into three categories. First, Lee [3] proposed a discontinuous attitude error function and presented an exponential stable control law. The main advantage of this method is that the attitude error vector is proportional to the rotation about the Euler axis between the attitude and the desired angular position. However, the proposed attitude error is not well-defined globally. Second, a hybrid control scheme [4, 5] Reproduced from Ti Chen. Continuous leaderless synchronization control of multiple spacecraft on S O(3). Astrodynamics 2021; 5: 279-291. Copyright ©Springer Nature B.V. 2021. All rights reserved. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_8

157

158

8 Continuous Leaderless Synchronization Control . . .

was developed to achieve global asymptotic stability on S O(3). The key concept is switching among a family of smooth potential functions, which results in a discontinuous controller. Third, a continuous controller [6] based on a finite-time observer was developed to provide a reference trajectory starting from the follower’s initial condition and ending at the leader’s attitude. Although this method cannot ensure global stability theoretically, the introduction of the finite-time observer can make it easier for networked spacecraft to escape the attraction of unstable equilibria. Commonly used actuators such as reaction wheels and control moment gyroscopes can provide only continuous control torques. However, if discontinuous feedback schemes are acceptable, it is unnecessary to use a high-dimensional rotation matrix because a discontinuous controller [7] based on quaternions can also achieve global stability. Thus, a continuous controller is desired for practical applications. Although only almost global stability can be ensured by using continuous controllers, methods can be developed to reduce the negative effects of unstable equilibria. Furthermore, in attitude consensus control, as shown in last part, the leaderfollower case has been widely studied using various attitude representations [8–11]. As for distributed control directly on S O(3), two strategies are used. One uses a specific type of communication graph, such as the connected undirected tree graph with a globally reachable leader [12]. The other is based on an observer, which can estimate the leader’s information and provide a reference trajectory for each follower spacecraft [6, 13–15]. For instance, Zou and Meng addressed the distributed leader-follower attitude tracking of multiple rigid bodies on S O(3) using a novel estimator satisfying the rotation matrix dynamics [14]. Chen and Shan proposed an adaptive fault-tolerant controller for the distributed attitude tracking of multiple flexible spacecraft on S O(3) under a connected undirected communication graph with a finite-time observer to estimate the leader’s information for each follower [6]. Peng et al. considered leader-follower attitude synchronization based on exponential distributed observers designed using the geometric convexity on S O(3) [15]. However, the leader-follower architecture relies heavily on the leader. Leader error or the loss of the leader will cause mission failure. The leaderless consensus strategy can avoid this problem. Thus, it is necessary to consider the leaderless consensus of multiple spacecraft [16, 17]. Nazari et al. presented solutions to the decentralized consensus of multiple rigid spacecraft on S O(3) under certain directed graphs [16]. Zou et al. investigated the attitude synchronization of multiple spacecraft under an undirected tree graph without angular velocity measurements [18]. However, all published studies on leaderless attitude consensus are based on specified communication topologies or those with local stability. To solve the leaderless attitude synchronization problem, a straightforward approach is to use the relative error on S O(3) of the real state as feedback. However, it is quite difficult to demonstrate its effectiveness under a general graph theoretically. Available methods using a distributed observer to estimate the leader’s information do not work for leaderless consensus under a general undirected graph. For example, the observers in [14] and [15] were designed for a specified communication topology. It is almost impossible for the finite-time observer in [6] to converge to a matrix on S O(3) without the leader’s information; that is, the observer cannot provide a feasible reference trajectory.

8.3 Leaderless Consensus with Nonzero Final Angular Velocity

159

Thus, leaderless attitude synchronization on S O(3) under a general connected undirected graph is still unsolved. This work provides a solution to this open problem. Furthermore, in space engineering, cases with zero and nonzero final angular velocities both have potential applications. For example, if a group of spacecraft needs to tune the antenna orientation to realize communication among the spacecraft, leaderless consensus with nonzero final angular velocity is necessary. For Earth observation, the cameras on all the spacecraft may need to point to a certain target. Zero final angular velocity is preferred for such a mission. Therefore, cases with zero or nonzero final angular velocities are both considered in this study. The main contribution of this study is that the leaderless consensus of multiple spacecraft is solved directly on S O(3) under a general connected undirected graph.

8.2 Spacecraft Dynamics According to Sects. 1.3.5 and 1.4, the dynamics of the ith spacecraft on S O(3) can be described by the following equations: ˙ i = Ri ωi× R J i ω˙ i =

(8.1)

−ωi× J i ωi

+ ui + d¯ i

(8.2)

where Ri ∈ S O(3) is the rotation matrix from the body frame to the inertial reference frame, ωi ∈ R3 is the angular velocity, J i ∈ R3×3 is the inertia matrix of the spacecraft, ui is the control torque, and d¯ i is the external disturbance. Suppose that the inertia matrix J i is uncertain and d¯ i is bounded by a positive constant μ0 ; that is,  d¯ i 2 ≤ μ0 holds.

8.3 Leaderless Consensus with Nonzero Final Angular Velocity 8.3.1 Control Objective This section presents the design of a distributed controller to synchronize the attitude and angular velocity of multiple spacecraft under a connected undirected graph. That is, the control input ui is designed such that lim Ri = lim R j

(8.3)

lim ωi = lim ω j

(8.4)

t→∞

t→∞

t→∞ t→∞

160

8 Continuous Leaderless Synchronization Control . . .

hold for ∀i, j ∈ V and i = j. Note that the final synchronized angular velocity can be nonzero.

8.3.2 Generating an Undirected Tree Graph There are many algorithms [19, 20] for traversing or searching tree structures, such as breadth-first search. However, these algorithms typically require global information. Thus, a method is proposed to generate an undirected tree graph based only on local information to facilitate the design of a distributed controller (Algorithm 8.1). The generated undirected tree graph is referred to as G below. Algorithm 8.1 essentially blocks some communication edges to obtain an undirected tree graph. Note that the information that flows in the edge contains a layer number, a friend flag (0 or 1), and the spacecraft’s attitude. Figure 8.1 gives an example showing how to generate an undirected tree graph using the proposed algorithm. In Step A, the five nodes are divided into three layers. According to the rules in Step B, the communication indicated by red dotted lines should be blocked because the friend flags are equal to 0, and two possible tree graphs (Figs. 8.1iii and 8.1iv) can be obtained. Algorithm 8.1 Initialize: The first node is listed in Layer 1; that is, L y1 = 1. The layer numbers of the other nodes are −1; that is, L yi = −1, where i = 2, · · · , N . The friend flag Fk at each communication edge is initialized to 0. Algorithm Main(): Step A: Assign a layer number to each node. For node i, if L yi = −1, the layer number is set to 1 plus the minimum layer number that node i can receive. Step B: Extract an undirected tree graph. Rule 1: All nodes in layer j must choose only one neighbor (friend) from the previous layer for j ≥ 2 and change the friend flag of the relevant undirected edge to 1. Rule 2: All nodes drop the information received from the nodes in the same layer; that is, the communication edges between two nodes in the same layer have Fk = 0.

8.3.3 Distributed Observer A distributed observer based on the generated tree graph G in Sect. 8.3.2 is designed for each spacecraft in the group, as follows.

8.3 Leaderless Consensus with Nonzero Final Angular Velocity

161

Fig. 8.1 Example of tree graph generation using Algorithm 8.1

˙ˆ = R ˆ i ωˆ i∨ R i ω˙ˆ i = uˆ i

(8.5) (8.6)

ˆ i ∈ R3×3 and ωˆ i ∈ R3 are the estimated final convergence attitude and anguwhere R lar velocity, respectively. uˆ i ∈ R3 is the virtual control input of the designed observer and is defined as uˆ i = −κ1

N 

ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ − κ 2 ai j ( R

j=1

N 

ai j (ωˆ i − ωˆ j )

(8.7)

j=1

where ai j is the (i, j) element of the adjacency matrix of the communication graph ˆ i and ωˆ i are set to Ri (0) and ωi (0), respectively. G . Note that the initial values of R Lemma 8.1 Almost global asymptotic synchronization of N observers governed by Eqs. (8.5) and (8.6) can be achieved under the generated tree graph G . Furthermore, the acceleration ω˙ˆ i will converge to zero asymptotically. Proof Consider the following Lyapunov function: V1 =

N N κ1   ˆ T ˆ ˆ Tj R ˆ i) + 1 a tr(I 3 − R 2 i=1 j=1 i j 2

(8.8)

8 Continuous Leaderless Synchronization Control . . .

162

ˆ is defined as [ωˆ 1T , · · · , ωˆ TN ]T . where tr(·) represents the trace of a matrix, and  ˆ i and ωˆ i = ωˆ j = 0 hold for ∀i, j ∈ V. ˆj=R V1 = 0 holds if and only if R The time derivative of V1 along Eqs. (8.5) and (8.6) is as follows: κ1 V˙1 = 2 +

N N  

  ˆ iT R ˆ Tj R ˆ Tj R ˆ iT R ˆj−R ˆ i )∨ + ωˆ iT ( R ˆi − R ˆ j )∨ ai j ωˆ Tj ( R

i=1 j=1 N 

⎡

ωˆ iT

⎣−κ1

i=1

N 

ˆ Tj ai j ( R

ˆi − R

ˆ iT R

ˆ j )∨ − κ 2 R

j=1

N 

⎤ ai j (ωˆ i − ωˆ j )⎦

(8.9)

j=1

Because the graph G is undirected, ai j = a ji holds, and one has N N  

ˆ iT R ˆ Tj R ˆj−R ˆ i )∨ = ai j ωˆ Tj ( R

i=1 j=1

N N  

ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ a ji ωˆ iT ( R

j=1 i=1

=

N N  

ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ ai j ωˆ iT ( R

j=1 i=1

=

N  N 

ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ ai j ωˆ iT ( R

(8.10)

i=1 j=1

Equation (8.9) can be rewritten as V˙1 = κ1

N N  

  ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ ai j ωˆ iT ( R

i=1 j=1

+

N  i=1

⎡

ωˆ iT ⎣−κ1

N 

ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ − κ 2 ai j ( R

j=1

N 

⎤ ai j (ωˆ i − ωˆ j )⎦

j=1

T

ˆ (L ⊗ I 3 ) ˆ = − ˆ1− ˆ2− ˆ3 ˆ 1T L  ˆ 2T L  ˆ 3T L  = −

(8.11)

ˆ k ∈ R N is a column vector consisting of the kth element of ωˆ 1 , · · · , ωˆ N , where  and L is the Laplacian matrix of the graph G . The real symmetric matrix L can be decomposed as L = Q Q T (8.12) where  is a diagonal matrix whose entries are the eigenvalues of L , and Q is an orthogonal matrix whose columns are the eigenvectors of L . Because zero is a simple eigenvalue of L , only the (1,1) element of  is zero, and the first column √ ˆ 1 , the solution to  ˆ 1T L  ˆ1 = ˆ1 = ˆ 1T Q Q T  of Q is 1/ N . If we let Y 1 = Q T 

8.3 Leaderless Consensus with Nonzero Final Angular Velocity

163

ˆ 1 = α1 with a constant α. Similar Y 1T Y 1 = 0 is Y 1 = [α, 0, · · · , 0]T ∈ R N and  ˆ ˆ results can be obtained for 2 and 3 . Thus, the solution to V˙1 = 0 is ωˆ i = ωˆ j for ∀i, j ∈ V. For ωˆ i = ωˆ j , one has N  k=1

ˆ kT R ˆ iT R ˆ i1 − R ˆ k )∨ = ai 1 k ( R 1

N 

ˆ kT R ˆ iT R ˆ i2 − R ˆ k )∨ = ω˙ˆ 1 ai 2 k ( R 2

(8.13)

k=1

where (i 1 , i 2 ) is one edge in G . Note that Eq. (8.13) essentially contains N equations N N ˆ kT R ˆ iT R ˆ kT R ˆ iT R ˆ i1 − R ˆ k )∨ = k=1 ˆ i2 − R ˆ k )∨ holds for all because k=1 ai1 k ( R ai2 k ( R 1 2 N − 1 communication edges. ˆ kT R ˆ iT R ˆi − R ˆ k )∨ . Note that ik = −ki . Because the undiWe define ik = ( R ¯ with the first node as the root rected graph G is a tree, a directed spanning tree G can be obtained by assigning a proper direction to each undirected edge in G . Note ¯ has N agents and N − 1 directed communication edges. that the graph G Equation (8.13) can be expressed as ¯ =0 (C¯ ⊗ I 3 )

(8.14)

¯ = [T , ω˙ˆ T ]T .  is the column stack vector, which consists of ik and where  1 ¯ ¯ . Note C = [C − 1 N ] ∈ R N ×N , where C ∈ R N ×(N −1) is the incidence matrix of G that each column of C contains only two nonzero elements, that is, 1 or −1. Because ¯ is a directed spanning tree, C is full column rank; that is, the N − 1 columns of G C are linearly independent. It is easy to verify that the inner product of the vector −1 N and each column of C is zero; that is, −1 N is orthogonal to each column of C. Thus, the N columns of C¯ are linearly independent. The only solution to Eq. (8.14) ¯ = 0. As shown in [21], ( R ˆ kT R ˆ iT R ˆ i and ˆi − R ˆ k )∨ = 0 has two solutions, R ˆk = R is  T ˆk R ˆ i ) = −1, which correspond to the minimum and maximum of the potential tr( R

ˆ kT R ˆ i ), respectively. Because G is an undirected tree graph, only function tr(I 3 − R ˆ N is stable. According to LaSalle’s invariance principle, the almost ˆ 1 = ··· = R R global asymptotic synchronization of N observers can be achieved. ¯ = Furthermore, because the N observers will be driven to the equilibrium { 0, ωˆ 1 = · · · = ωˆ N }, ω˙ˆ i will converge to zero asymptotically according to Eq. (8.6). 

Remark 8.1 Note that the convergence of the observers in Eqs. (8.5) and (8.6) does not depend on the real-time state of the networked spacecraft, and its responses are continuous. Essentially, the observers (8.5) and (8.6) provide the reference attitude and angular velocity for each node in spacecraft formation flying activities. After the observer converges, the desired states for all spacecraft are identical. Once each spacecraft can track this desired state, leaderless synchronization can be achieved. ˆ i and ωˆ i start from the initial conditions of the ith spacecraft. Furthermore, both R

8 Continuous Leaderless Synchronization Control . . .

164

8.3.4 Controller Design This section describes the design of a synchronization controller under a connected undirected graph. On the basis of a distributed estimator governed by Eqs. (8.5) and (8.6), for the ith spacecraft, a sliding variable can be defined as si = ωi + i

(8.15)

ˆ iT Ri − RiT R ˆ i )∨ − RiT R ˆ i ωˆ i with a positive constant β1 . The time where i = β1 ( R derivative of i can be expressed as



 ˆ iT Ri )I 3 − R ˆ iT Ri ωˆ i ˙ i = β1 tr(RiT R ˆ i )I 3 − RiT R ˆ i ωi − β1 tr( R  ˆ i ωˆ i − RiT R ˆ i ω˙ˆ i +ωi× RiT R

(8.16)

ˆ i ω˙ˆ i in the above equation goes to zero as time goes to Note that the term −RiT R infinity according to Lemma 8.1. Denote ˙ i + RiT R ˆ i ω˙ˆ i i◦ = 

(8.17)

Thus, Eq. (8.2) can be rewritten as ˙ i − ωi× J i ωi + ui + d¯ i = J i (i◦ − RiT R ˆ i ω˙ˆ i ) − ωi× J i ωi + ui + d¯ i J i s˙ i = J i  (8.18) The controller can be designed as follows: ui = −k1 si + ωi× Jˆ i ωi − Jˆ i i◦ −

dˆi si si 2 + αi2

− k2



[β2 (R Tj Ri − RiT R j )∨ + si − s j ]

j∈Ni

(8.19) ˆ i , dˆi , and αi are adaptive parameters. where k1 , k2 , and β2 are positive constants, and J Note that the terms −k1 si and −k2 j∈Ni ai j [β2 (R Tj Ri − RiT R j )∨ + si − s j ] are used to track the observer state and achieve synchronization, respectively. Essentially, Jˆ i is the estimated positive definite matrix J i . Note that J i i◦ − ωi× J i ωi can be expressed as J i i◦ − ωi× J i ωi =  i J i (8.20) where  i ∈ R3×6 , and J i = [Ji,11 , Ji,12 , Ji,13 , Ji,22 , Ji,23 , Ji,33 ]T ∈ R6×1 , and Ji,kr  represents the (k, r ) element of J i . Similarly, Jˆ i can be obtained for Jˆ i ; that is,   Jˆ i = [ Jˆi,11 , Jˆi,12 , Jˆi,13 , Jˆi,22 , Jˆi,23 , Jˆi,33 ]T . The update law of Jˆ i can be defined as J˙ˆ i = 0  iT si where 0 is a positive constant.

(8.21)

8.3 Leaderless Consensus with Nonzero Final Angular Velocity

165

Because R Tj Ri is also an element of S O(3), according to Rodrigues’ rotation formula, one has R Tj Ri = I 3 + (1 − cos θ )n¯ × n¯ × + sin θ n¯ × , where θ and n¯ are the rotation angle and unit vector, respectively. Thus, (R Tj Ri − RiT R j )∨ satisfies the following inequality: ¯ 2≤2 (R Tj Ri − RiT R j )∨ 2 = 2 sin θ n

(8.22)

The closed-loop system can be expressed as  dˆi si ˆ i ω˙ˆ i − ω× J˜ i ωi − k1 si − J i s˙ i = J˜ i i◦ − J i RiT R − k2 (si − s j ) + d i i 2 si 2 + αi j∈N

(8.23)

i

where J˜ i = J i − Jˆ i and d i = d¯ i − k2 j∈Ni β2 (R Tj Ri − RiT R j )∨ . Note that d i is bounded by μ0 + 2(N − 1)β2 κ2 . dˆi is updated by the following law to estimate the bound of d i . 1 si 2 if dˆi ≤ μ¯ 0 ˙ (8.24) dˆi = χ χ 1 si 2 (1 − dˆi 1 + μ¯ 0 1 ) otherwise where χ1 > 0 is an even number, 1 > 0, and μ¯ 0 is chosen large enough that μ¯ 0 > μ0 + 2(N − 1)β2 κ2 holds. αi is introduced to avoid possible control chattering according to the following adaption law: α˙ i = −

2 αi dˆi si 2 si 2 + αi2

(8.25)

where 2 > 0. Theorem 8.1 For the observer in Eqs. (8.5) and (8.6) and the update laws (8.21), (8.24), and (8.25), if μ¯ 0 > μ0 + 2(N − 1)β2 κ2 holds, the controller (8.19) can complete the attitude synchronization almost globally asymptotically under a connected undirected graph. Proof The following Lyapunov function is chosen: V2 =

N  1 i=1

2

siT

1 ˜ T ˜  1 ˜2 1 2 J i si + α di + Ji Ji + 20 21 22 i

 (8.26)

where d˜i = μ¯ 0 − dˆi . The time derivative of V2 along Eqs. (8.5), (8.6), (8.23), (8.24), and (8.25) is

8 Continuous Leaderless Synchronization Control . . .

166 N 

V˙2 =

siT J i s˙ i +

i=1

N  

1 ˜ T ˙˜  1 ˜ ˙˜ 1 αi α˙ i J J + di di + 0 i i 1 2



dˆi si si 2 + αi2 i=1   1 ˜ ˙˜ 1 T −k2 (si − s j ) + d i ) − J˜ i iT si + αi α˙ i di di + 1 2 j∈Ni  N   dˆi si ˆ i ω˙ˆ i − k1 si − siT (− J i RiT R − k2 (si − s j ) + d i ) = 2 si 2 + αi i=1 j∈N i  1 1 ˜ ˙˜ αi α˙ i + di di + 1 2 (8.27) For dˆi ≤ μ¯ 0 , −siT d i + 11 d˜i d˙˜i = −siT d i − d˜i si 2 ≤ dˆi si 2 holds. Otherwise, one has =

ˆ i ω˙ˆ i − ωi× J˜ i ωi − k1 si − siT ( J˜ i i◦ − J i RiT R

1 ˜ ˙˜ dd 1 i i

χ χ = −siT d i − (μ¯ 0 − dˆi )si 2 (1 − dˆi 1 + μ¯ 0 1 ) χ χ ≤ μ¯ 0 si 2 − (μ¯ 0 − dˆi )si 2 (1 − dˆi 1 + μ¯ 0 1 ) χ χ χ χ 1 1 = μ¯ 0 si 2 − μ¯ 0 si 2 (1 − dˆi + μ¯ 0 ) + dˆi si 2 (1 − dˆi 1 + μ¯ 0 1 ) χ χ χ χ 1 1 1 1 = μ¯ 0 si 2 (−dˆi + μ¯ 0 ) + dˆi si 2 + dˆi si 2 (−dˆi + μ¯ 0 ) < dˆi si 2 (8.28) Thus, V˙2 satisfies the following inequality:

−siT d i +

V˙2 ≤

N  

ˆ i ω˙ˆ i − k1 si − siT (− J i RiT R

i=1

 dˆi si − k2 (si − s j )) 2 si 2 + αi j∈N

 αi2 dˆi si 2 ˆ +di si 2 − si 2 + αi2   N   T T ˆ ˙ si (− J i Ri Ri ωˆ i − k1 si − k2 = (si − s j )) i=1

≤

N  

ˆ i ω˙ˆ i − k1 si ) siT (− J i RiT R



i

(8.29)

j∈Ni

i=1

If si 2 >  J i 2 ω˙ˆ i 2 /k1 , V˙2 < 0 holds. According to Lemma 8.1, ω˙ˆ i 2 → 0 as time goes to infinity. Thus, si goes to zero as t → ∞. From the definition of si , ˆ iT Ri − RiT R ˆ i ωˆ i + β1 ( R ˆ i )∨ = si , where we have a system governed by ωi − RiT R si is considered to be a control input that decreases with time. For this system, ˆ iT Ri ) is chosen as the Morse–Lyapunov function; its time derivative Vsi = tr(I 3 − R is

8.3 Leaderless Consensus with Nonzero Final Angular Velocity

167

ˆ iT Ri − RiT R ˆ iT Ri )∨ ˆ i )∨ + ωˆ iT (RiT R ˆi − R V˙si = ωiT ( R (8.30) ˆ iT Ri − RiT R ˆ iT Ri − RiT R ˆ iT Ri − RiT R ˆ i )∨T ( R ˆ i )∨ + siT ( R ˆ i )∨ = −β1 ( R ˆ iT Ri − RiT R ˆ i )∨ 2 > si 2 /β1 , V˙si < 0 holds. Because si → 0 as t → For ( R T ˆ i Ri − RiT R ˆ iT Ri − ˆ i )∨ goes to zero as time goes to infinity. Note that ( R ∞, ( R ˆ i or tr(RiT R ˆ i )∨ = 0 indicates that Ri = R ˆ i ) = −1, where Vsi is equal to 0 or RiT R ˆ 4. Thus, only Ri = Ri is a stable equilibrium. Thus, considering the conclusions in Lemma 8.1, the almost global asymptotic  attitude synchronization of multiple spacecraft can be achieved. Remark 8.2 Note that the conclusion of Theorem 8.1 represents the attitude synchronization of multiple spacecraft; that is, Ri → R j and ωi → ω j as t → ∞ almost globally. The final values of Ri can be time-varying, and ωi can converge to a nonzero value. Remark 8.3 Different controllers will typically produce different final consensus values. One can easily design an attitude consensus law based directly on various attitude tracking errors between neighbors. One possible controller is ui = −κ3

N  j=1

ai j (R Tj

Ri −

RiT

∨

R j ) − κ4

N 

ai j (ωi − ω j )

(8.31)

j=1

where κ3 and κ4 are positive constants. It is not clear how to show the convergence of this controller theoretically. In some cases, it may be able to finish the leaderless consensus task. However, the final convergence attitude under the designed controller is determined by the distributed observer in Eqs. (8.6)–(8.8). Because the design of the distributed observer is based on a tree graph generated from the original undirected graph, the final consensus value will probably differ from that under the controller in Eq. (8.31). However, in this study, the leaderless consensus of multiple spacecraft just implies the attitude and angular velocity synchronization. Therefore, regardless of the final attitude value, the most important point is that the proposed control method can achieve leaderless attitude consensus, as demonstrated by a rigorous theoretical proof. Remark 8.4 Note that the designed controllers work under a connected undirected graph, not just tree graphs. In Sect. 8.3.2, an algorithm is proposed to generate an undirected tree graph from a connected undirected graph by blocking some communication edges. A distributed observer is designed to estimate the desired attitude and angular velocity of each spacecraft under the generated tree graph. An adaptive controller with a general connected undirected graph is developed in this section to complete the synchronization process.

8 Continuous Leaderless Synchronization Control . . .

168

8.4 Leaderless Consensus with Zero Final Angular Velocity The final attitude obtained as described in the previous section can be time-varying. That is, the networked spacecraft may rotate synchronously when the control objective is achieved. However, this type of response might be unacceptable for some activities, such as the observation of a specific celestial body. Thus, in this section, the leaderless consensus of multiple spacecraft with zero final angular velocity under a connected undirected graph is derived.

8.4.1 Control Objective A controller with a connected undirected graph is designed such that lim Ri = lim R j

(8.32)

lim ωi = lim ω j = 03

(8.33)

t→∞

t→∞

t→∞

t→∞

hold for ∀i, j ∈ V and i = j.

8.4.2 Distributed Observer The following distributed observer is proposed. ˙ˆ = R ˆ i ωˆ i∨ R i ω˙ˆ i = −κ1

N 

(8.34) ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ − κ2 ωˆ i ai j ( R

(8.35)

j=1

where ai j represents the (i, j) element of the adjacency matrix of the undirected tree ˆ i and ωˆ i are graph G described in Sect. 8.3.2. As for the observer in Sect. 8.3.3, R used to provide the reference attitude and angular velocity of each spacecraft. Their initial values are chosen to be Ri (0) and ωi (0), respectively. Lemma 8.2 For the observers in Eqs. (8.34) and (8.35), under the generated tree ˆ i can be achieved, and graph G , the almost global asymptotic synchronization of R ωˆ i goes to zero asymptotically. Proof The chosen Lyapunov function is the same as V1 . Its time derivative along Eqs. (8.34) and (8.35) is ˆ ˆ T (8.36) V˙1 = −

8.4 Leaderless Consensus with Zero Final Angular Velocity

169

ˆ = 0. From Eqs. (8.34) and (8.35), one has Thus, this equation for V˙1 implies that  ˙R ˆ i = 0 and Nj=1 ai j ( R ˆ Tj R ˆ iT R ˆi − R ˆ j )∨ = 0. Using steps similar to those in the proof ˆ kT R ˆ iT R ˆi − R ˆ k )∨ = 0, which has two solutions, of Lemma 8.1, it can be found that ( R

ˆ i and tr( R ˆ kT R ˆ kT R ˆk = R ˆ i ) = −1. The potential function tr(I 3 − R ˆ i ) is equal to R ˆ N is stable. ˆ 1 = ··· = R 0 and 4, respectively, for the two solutions. Thus, only R That is, the almost global asymptotic synchronization of the observers governed by Eqs. (8.34) and (8.35) can be achieved, and the final value of ωˆ i approaches zero. 

8.4.3 Controller Design By using the sliding mode variable defined in Eq. (8.15), the distributed controller can be designed as follows:  dˆi si ui = −k1 si + ωi× Jˆ i ωi − Jˆ i i∗ − − k2 [β2 (R Tj Ri − RiT R j )∨ + si − s j ] 2 si 2 + αi j∈Ni

(8.37)

where i∗ is defined as

 ˆ i )I 3 − RiT R ˆ i ωi i∗ = β1 tr(RiT R

(8.38)

 Note that J i i∗ − ωi× J i ωi can be expressed as  i∗ J i . Jˆ i is updated according to the following law: (8.39) J˙ˆ i = 0  i∗T si

The update laws of dˆi and αi are defined in the same way as those in Eqs. (8.24) and (8.25). ˆ i and ωˆ i governed by Eqs. (8.34) and (8.35), almost global Theorem 8.2 By using R asymptotic attitude synchronization under a connected undirected graph can be achieved under the controller (8.37) with the adaption laws (8.39), (8.24), and (8.25) for μ¯ 0 > μ0 + 2(N − 1)β2 κ2 . Furthermore, the angular velocity of each spacecraft converges to zero. Proof The closed-loop system can be expressed as J i s˙ i = J˜ i i∗ + J i



 ˆ iT Ri )I 3 − R ˆ iT Ri ωˆ i + ωi× RiT R ˆ i ωˆ i − RiT R ˆ i ω˙ˆ i − β1 tr( R

−ωi× J˜ i ωi − k1 si −



 dˆi si − k2 (si − s j ) + d i 2 si 2 + αi j∈N i

(8.40) A Lyapunov function can also be used as V2 in Eq. (8.26). Its time derivative along Eqs. (8.24), (8.25), (8.39), and (8.40) satisfies the following inequality.

8 Continuous Leaderless Synchronization Control . . .

170 V˙2 ≤

N  

 ˆ ˆ i − RT R ˆ ˙ˆ i ) − k1 s T si ˆ iT Ri ωˆ i + ω× R T R ˆ iT Ri )I 3 − R siT J i (−β1 tr( R i iω i iω i i



(8.41)

i=1

ˆ i and Ri are elements of S O(3), R ˆ iT Ri also belongs to S O(3). Because both R ˆ iT Ri = That is, there exist an Euler angle θ ∗ and Euler axis n¯ ∗ such that R

ˆ iT Ri )I 3 has the cos θ ∗ I 3 + (1 − cos θ ∗ )n¯ ∗ n¯ ∗T − sin θ ∗ n¯ ∗× holds. The term tr( R



ˆ iT Ri )I 3 2 = (1 + 2 cos θ ∗ )I 3 2 ≤ 3. Thus, −β1 tr( R ˆ iT Ri )I 3 − property  tr( R  ˆ iT Ri ωˆ i + ωi× RiT R ˆ i ωˆ i − RiT R ˆ i ω˙ˆ i satisfies the following inequality. R

 ˆ iT Ri )I 3 − R ˆ iT Ri ωˆ i + ωi× RiT R ˆ i ωˆ i − RiT R ˆ i ω˙ˆ i 2  − β1 tr( R ≤ (4β1 + ωi 2 )ωˆ i 2 + ω˙ˆ i 2

(8.42)

It follows from Eq. (8.41) that V˙2 ≤ 0 holds if si 2 >  J i 2 [(4β1 + ωi 2 )ωˆ i 2 + ω˙ˆ i 2 ]/k1 . From Lemma 8.2, both ω˙ˆ i and ωˆ i go to zero as time goes to infinity, and the convergence does not depend on the real-time attitudes and angular velocities of the networked spacecraft. Thus, as t → ∞,  J i 2 [(4β1 + ωi 2 )ωˆ i 2 + ω˙ˆ i 2 ]/k1 → 0, and thus si also goes to zero. By following the steps ˆ i is an almost globally used in the proof of Theorem 8.1, it can be found that Ri = R  stable equilibrium, and ωi → ωˆ i → 03 . Remark 8.5 As shown in Theorems 8.1 and 8.2, attitude synchronization can be achieved using the developed controllers. However, it is not easy to predict the final consensus state before the controller converges because both the attitude dynamics and the designed controllers are strongly nonlinear. If the final consensus state is highly important for a practical activity, it is preferable to adopt the leader-follower strategy, where the final consensus state can be specified by the leader’s state. Then the controllers in [6, 13–15] can be used to solve the distributed attitude tracking task.

8.5 Simulations In this section, two numerical simulations are presented to demonstrate the effectiveness of the proposed control schemes in Eqs. (8.19) and (8.37) using five rigid spacecraft under the communication graph in Fig. 8.1i. The tree graph in Fig. 8.1iii is used as the generated graph G . The inertia matrix J i is set to diag{25, 20, 8}. The external disturbance d¯ i is set to 0.1 sin(5t)I 3 Nm. The initial attitudes and angular velocities of the five spacecraft are as follows.

8.5 Simulations

171

⎡

⎤ 0.7474 0.5387 −0.3887 R1 (0) = ⎣ −0.5534 0.8287 0.0844 ⎦ , ω1 (0) = [0.0337, 0.0145, 0.0711]T 0.3676 0.1521 0.9175 (8.43) ⎡ ⎤ 0.3478 −0.7321 −0.5858 R2 (0) = ⎣ 0.5453 0.6662 −0.5088 ⎦ , ω2 (0) = [0.3455, 0.0671, 0.2494]T 0.7627 −0.1425 0.6309 (8.44)   −0.1476 0.346 −0.9265 0.7834 −0.5309 −0.3231 −0.6037 −0.7735 −0.1927

R3 (0) = ⎡

, ω3 (0) = [−0.1604, −0.1939, −0.0512]T ⎤

(8.45)

0.7255 −0.6582 −0.2011 R4 (0) = ⎣ 0.5594 0.7342 −0.3848 ⎦ , ω4 (0) = [−0.2, −0.208, −0.0381]T 0.4009 0.1667 0.9008 (8.46) ⎡ ⎤ 0.7822 −0.5668 −0.2586 R5 (0) = ⎣ −0.5456 −0.4229 −0.7235 ⎦ , ω5 (0) = [0.022, 0.0235, 0.0719]T 0.3007 0.707 −0.6401 (8.47) A necessary and sufficient condition of tr(I 3 − R Tj Ri ) = 0 is R j = Ri . Thus, Vsyn = tr(I 3 − R1T R2 ) + tr(I 3 − R2T R3 ) + tr(I 3 − R3T R4 ) + tr(I 3 − R4T R5 ) ∈ [0, 16] is used to indicate the synchronization error of the five spacecraft. It is straightforward that Vsyn = 0 holds if and only if the networked spacecraft ˆ 1T R ˆ 2T R ˆ 2 ) + tr(I 3 − R ˆ 3 ) + tr(I 3 − are synchronized. Similarly, Vsyn−ob = tr(I 3 − R

ˆ 3T R ˆ 4T R ˆ 4 ) + tr(I 3 − R ˆ 5 ) ∈ [0, 16] is adopted to represent the synchronization error R ˆ iT Ri ) ∈ [0, 4] denotes the of the attitudes of the distributed observer. Vt,i = tr(I 3 − R tracking error of the ith spacecraft when the attitudes of the distributed observers are used as the reference trajectory. Case I: In this case, the observer in Eqs. (8.5) and (8.6) is adopted. The observer parameters κ1 and κ2 are set to 0.1 and 0.5, respectively. The remaining control parameters, β1 , k1 , k2 , β2 , μ¯ 0 , 0 , 1 , and 2 are 0.1, 2, 1, 0.1, 1, 0.01, 0.1, and 0.01,  respectively. The initial values of the adaptive parameters Jˆ i , dˆi , and αi are set to 0, 0, and 0.1, respectively. The responses of the observers in Eqs. (8.5) and (8.6) and the closed-loop networked spacecraft are shown in Figs. 8.2–8.5. As indicated in Figs. 8.2 and 8.3, the observers in Eqs. (8.5) and (8.6) are synchronized at approximately 65 s, and ωˆ i converges to a nonzero value. For this observer, the controller finishes the leaderless consensus of the networked spacecraft at 70 s. Note that Fig. 8.5 gives the tracking error of each spacecraft. The potential function Vt,i is found to be less than 1 during the entire process; that is, all five spacecraft are far from the undesired equilibria, where Vt,i = 4 holds.

8 Continuous Leaderless Synchronization Control . . .

172

V

syn-ob

15 10

1

5 0

10-4

2

0 60 0

80

20

40

100

60

80

Time (s)

100

Fig. 8.2 Synchronization error Vsyn−ob in Case I

0.4 0.2 0 -0.2 -0.4

0

0.3 0.15 0 -0.15 -0.3 0.3 0.15 0 -0.15 -0.3

20

40

0

20

40

0

20

40

Time (s)

Time (s)

Time (s)

60

80

100

60

80

100

60

80

100

Fig. 8.3 ωˆ in observer (8.6) in Case I

V

syn

15 2

10

1

5 0

-4

10

0 60 0

20

80 40

60

Time (s)

Fig. 8.4 Synchronization error Vsyn in Case I

100 80

100

8.5 Simulations

173

1

V

t,i

Fig. 8.5 Tracking error Vt,i in Case I

i=1 i=1 i=3 i=4 i=5

0.5 0

0

20

15

Vsyn

Fig. 8.6 Synchronization error Vsyn under the controller in Eq. (8.31)

40

60

Time (s)

80

100

0.05

10 5 0

0 60 0

20

40

80 60

Time (s)

100 80

100

To demonstrate the advantages of the proposed controller in Eq. (8.19), the system responses of these five rigid spacecraft under the controller (8.31) are simulated for the initial conditions in Eqs. (8.43–8.47). The control parameters κ3 and κ4 in the above controller are set to 0.2 and 1, respectively. As shown in Fig. 8.6, the controller in Eq. (8.31) takes longer to achieve attitude synchronization than that in Fig. 8.4. The main reason may be that the terms that track the observer state and achieve synchronization are both included in the designed controller (8.19), but only the synchronization term appears in the controller (8.31). Case II: In this case, the effectiveness of the controller (8.37) with the distributed observers (8.34) and (8.35) is verified. The parameters κ1 , κ2 , β1 , k1 , k2 , β2 , μ¯ 0 , 0 , 1 , and 2 are set to 0.1, 0.1, 0.1, 2, 1, 0.1, 1, 0.01, 0.1, and 0.01, respectively.  The initial conditions of the adaptive parameters Jˆ i , dˆi , and αi are the same as those in Case I. The simulation results are presented in Figs. 8.7–8.10. As shown in Figs. 8.7 and 8.8, the distributed observers in Eqs. (8.34) and (8.35) are synchronized at approximately 23.5 s, and ωˆ i goes to zero. Figure 8.9 indicates that the leaderless consensus task of the five spacecraft is complete at approximately 33.3 s. During this task, each spacecraft is far from the unstable equilibria because, as shown in Fig. 8.10, the potential function Vt,i is much less than 4. The final angular velocity is the main difference from the simulation in Case I. The controllers in both cases can finish the synchronization task efficiently.

8 Continuous Leaderless Synchronization Control . . .

174

Vsyn-ob

15 2

10

1

5 0

10 -4

0 20 0

30

20

40

40 60

50 80

Time (s)

100

Fig. 8.7 Synchronization error Vsyn−ob in Case II

0.4 0.2 0 -0.2 -0.4

0

0.3 0.15 0 -0.15 -0.3 0.4 0.2 0 -0.2 -0.4

20

0

0

20

20

40

40

40

Time (s)

Time (s)

Time (s)

60

80

100

60

80

100

60

80

100

Fig. 8.8 ωˆ in observer (8.35) in Case II

15

V

syn

Fig. 8.9 Synchronization error Vsyn in Case II

2

10

1

5 0

-4

10

0 30 0

20

40 40

60

Time (s)

50 80

100

References

175

1.5

V

t,i

Fig. 8.10 Tracking error Vt,i in Case II

i=1 i=1 i=3 i=4 i=5

1 0.5 0

0

20

40

60

Time (s)

80

100

8.6 Conclusions The leaderless consensus of multiple rigid spacecraft is studied directly on S O(3) under connected undirected graphs. Distributed observers that provide a reference signal for each spacecraft were designed under an undirected tree graph. Two control algorithms were presented to achieve either zero or nonzero final angular velocity. The stability conditions were presented on the basis of rigorous theoretical analyses. Numerical results showed that the proposed controllers can complete leaderless synchronization activities efficiently. The controller continuity facilitates practical implementation. This work presents a solution to the leaderless consensus control of networked spacecraft on S O(3).

References 1. Di Mauro, G., Lawn, M., Bevilacqua, R.: Survey on guidance navigation and control requirements for spacecraft formation-flying missions. J. Guidance Control Dyn. 1–22 (2017) 2. Bhat, S.P., Bernstein, D.S.: A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Syst. Control Lett. 39(1), 63–70 (2000) 3. Lee, T.: Exponential stability of an attitude tracking control system on S O(3) for large-angle rotational maneuvers. Syst. Control Lett. 61(1), 231–237 (2012) 4. Berkane, S., Abdessameud, A., Tayebi, A.: Hybrid global exponential stabilization on S O(3). Automatica 81, 279–285 (2017) 5. Mayhew, C.G., Teel, A.R.: Hybrid control of rigid-body attitude with synergistic potential functions. In: Proceedings of the 2011 American Control Conference, pp. 287–292 (2011) 6. Chen, T., Shan, J.: Distributed adaptive fault-tolerant attitude tracking of multiple flexible spacecraft on S O(3). Nonlinear Dyn. 95(3), 1827–1839 (2019) 7. Mayhew, C.G., Sanfelice, R.G., Teel, A.R.: Quaternion-based hybrid control for robust global attitude tracking. IEEE Trans. Autom. Control 56(11), 2555–2566 (2011) 8. Zou, A.M., Kumar, K.D.: Quaternion-based distributed output feedback attitude coordination control for spacecraft formation flying. J. Guidance Control Dyn. 36(2), 548–556 (2013) 9. Gui, H., Vukovich, G.: Distributed almost global finite-time attitude consensus of multiple spacecraft without velocity measurements. Aerosp. Sci. Technol. 75, 284–296 (2018) 10. Maadani, M., Butcher, E.A.: Attitude consensus control of a rigid body multi-vehicle system with heterogeneous communication delays. In: AIAA Scitech 2021 Forum, p. 971 (2021)

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11. Koru, A.T., Maadani, M., Sarsilmaz, S.B., Johnson, E.N., Butcher, E.A., Yucelen, T.: Adaptive consensus control of rigid body attitude motion based on rotation matrices. In: AIAA Scitech 2020 Forum, p. 1824 (2020) 12. Chen, T., Shan, J.: Rotation-matrix-based attitude tracking for multiple flexible spacecraft with actuator faults. J. Guidance Control Dyn. 42(1), 181–188 (2019) 13. Chen, T., Shan, J.: Distributed spacecraft attitude tracking and synchronization under directed graphs. Aerosp. Sci. Technol. 109, 106432 (2021) 14. Zou, Y., Meng, Z.: Velocity-free leader-follower cooperative attitude tracking of multiple rigid bodies on so(3). IEEE Trans. Cybern. 49(12), 4078–4089 (2019) 15. Peng, X., Geng, Z., Sun, J.: The specified finite-time distributed observers-based velocityfree attitude synchronization for rigid bodies on S O(3). IEEE Trans. Syst. Man Cybern. Syst. 50(4), 1610–1621 (2019) 16. Nazari, M., Butcher, E.A., Yucelen, T., Sanyal, A.K.: Decentralized consensus control of a rigid-body spacecraft formation with communication delay. J. Guidance Control Dyn. 39(4), 838–851 (2016) 17. Chen, T., Shan, J.: Koopman-operator-based attitude dynamics and control on so(3). J. Guidance Control Dyn. 43(11), 2112–2126 (2020) 18. Zou, Y., Meng, Z., Zuo, Z.: Rotation-matrix-based attitude synchronization of multiple spacecraft without velocity measurements. In: 2017 11th Asian Control Conference (ASCC), pp. 96–101 (2017) 19. Kapoor, S., Ramesh, H.: Algorithms for enumerating all spanning trees of undirected and weighted graphs. SIAM J. Comput. 24(2), 247–265 (1995) 20. Matsui, T.: A flexible algorithm for generating all the spanning trees in undirected graphs. Algorithmica 18(4), 530–543 (1997) 21. Zheng, Z., Song, S.: Autonomous attitude coordinated control for spacecraft formation with input constraint, model uncertainties, and external disturbances. Chinese J. Aeronaut. 27(3), 602–612 (2014)

Chapter 9

Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Abstract This chapter presents an attitude control method based on Koopman operator theory. A set of observables is discovered to represent the nonlinear attitude dynamics on S O(3) using an infinite-dimensional linear system. With the assumption of low angular velocities, a finite-dimensional linear system is obtained by removing the high-order terms. An attitude control synthesis method is developed on the basis of the linear optimal control algorithm for the reduced linear system. The proposed controller design method is compared with some classical nonlinear optimal controllers to show its advantages. Also, the leaderless attitude synchronization problem is solved with the help of the reduced linear system and the well-studied linear multi-agent system theory. Furthermore, a possible solution to the case with a high angular velocity is provided. Simulations and experiments are conducted to verify the effectiveness of the theory.

9.1 Introduction Attitude control system is a critical component of a spacecraft. The configuration space of spacecraft attitude motion is the Lie group S O(3), a set of special orthogonal matrices with determinant one. Since this configuration is not an Euclidean space and the attitude dynamics is strongly nonlinear, the attitude control on S O(3) still attracts significant attention. For a nonlinear system, the superposition principle for linear system does not hold any longer and the nonlinear systems will show some nonlinear phenomena, such as multiple isolated equilibria, limit cycles and chaos [1]. Because of the powerful linear system analysis tools, a straightforward way to analyze the nonlinear system is to linearize it. However, the traditional linearization is just an approximation in the vicinity of an operating point. Hence, such linearization can only predict the behavior of nonlinear system in the neighborhood of the point of concern. In 1931, Koopman proved that the behavior of a nonlinear system can be represented by an infinite-dimensional linear operator acting on the Reproduced from Ti Chen and Jinjun Shan. Koopman-operator-based attitude dynamics and control on S O(3). Journal of Guidance, Control, and Dynamics 2019; 43(11): 2112-2126. Copyright ©2020 by Ti Chen and Jinjun Shan. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Chen et al., Distributed Attitude Consensus of Multiple Flexible Spacecraft, https://doi.org/10.1007/978-981-19-4258-7_9

177

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

observables (or measurement functions) of system states [2]. Essentially, the Koopman operator describes the flow of the observables. Once the proper observables have been found, a linear system can be built to describe the nonlinear system in the full state space. Since the Koopman operator is usually infinite-dimensional, in practice, the infinite-dimensional system should be truncated to get a finite-dimensional linear system, which can be well analyzed using linear system analysis tools. The Koopman-operator-based method has been widely used to analyze the behavior of the nonlinear systems [3–8] with the development of computer technology. More recently, increasing researchers focus on the control of nonlinear system based on Koopman operator because of the well-developed linear control theory. For example, Huang et al. proposed a systematic approach to design feedback laws based on a linear representation for a nonlinear dynamical system [9]. Brunton et al. restricted the Koopman operator to an invariant subspace spanned by specified measurement functions to obtain the finite-dimensional linear equation representing the nonlinear systems and designed the controllers for such nonlinear dynamical systems based on linear optimal control theory [10]. Korda and Mezi´c used the Koopman operator to linearize controlled dynamical systems and developed optimal control method based on linear model predictive control technique [11]. As stated in Ref. [10], it is still challenging to select the proper nonlinear observables to generate the reduced linear models. As shown in [3, 10], a possible way is to numerically calculate the Koopman modes based on the collected data. However, the resulting linear model may not be able to predict some nonlinear characteristics of the original system if the artificially collected data do not cover some critical states. For example, for a simple pendulum, if the collected data only cover the range of small swinging angle, it is clear that the reduced linear model built based on Koopman operator cannot reveal the nonlinear phenomenon at a large angular displacement. In literature, the sets of analytical observables have been reported for some low-order nonlinear systems, such as the examples in [10, 12]. However, the attitude dynamics on S O(3) is described by 12 nonlinear equations. As a 12-dimensional nonlinear system usually only with 3 control inputs at most, the attitude dynamics on S O(3) is much more complicated than such simple systems. This work can be considered as the first trial to provide a set of analytical observables such that the nonlinear attitude dynamics can be represented by a finite-dimensional linear system approximately and the attitude control laws can be designed based on the linear optimal control theory. Furthermore, the application of the proposed method in attitude synchronization is investigated and the proposed method is extended to the case with large angular velocity.

9.2 Problem Formulation

179

9.2 Problem Formulation 9.2.1 Koopman Operator Koopman operator is an infinite-dimensional linear operator that is able to represent the full nonlinear dynamics in the Hilbert space of possible observables [13, 14]. Consider the following autonomous dynamical system x˙ = F(x)

(9.1)

where x ∈ Rn is the state vector, an element of the state space M ⊆ Rn , and F : M → Rn is a smooth vector field. The solution to Eq. (9.1) with the initial condition x 0 at time t is denoted by S(t, x 0 ). An observable means a function of the state. Let O represents the set of all observables f : M → C. The Koopman operator Kt : O → O is defined as (9.2) (Kt f )(x) = f (S(t, x)) It should be noted that Koopman operator maps the observables to observables, not states to states. It is straightforward to show that the Koopman operator is linear on the space of observables even though the dynamical system in Eq. (9.1) can be nonlinear. That is, [Kt (α1 f 1 + α2 f 2 )](x) = (α1 f 1 + α2 f 2 )(S(t, x)) = α1 f 1 (S(t, x)) + α2 f 2 (S(t, x)) = α1 (Kt f 1 )(x) + α2 (Kt f 2 )(x)

(9.3)

Essentially, the Koopman operator defines a new linear, but infinite-dimensional dynamic system on O. The infinite dimensional characteristic may cause problems for the applications of Koopman operator. Fortunately, in practice, the infinitedimensional system can be truncated without a great loss of accuracy, i.e., a linear and finite-dimensional system can be obtained to approximate the Koopman operator. The most appealing point of using a finite-dimensional linear system to approximate the nonlinear system is that the powerful system analysis and control design tools developed for linear systems can be used for such a linear system on O.

9.2.2 Spacecraft Dynamics on S O(3) The dynamics equations of a spacecraft on S O(3) can be written as ˙ = Rω× R

(9.4)

J ω˙ = −ω× Jω + u

(9.5)

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

where R ∈ S O(3) represents the rotation matrix from the body frame to the inertial reference frame, ω ∈ R3 is the angular velocity expressed in the body frame, J ∈ R3×3 is the inertia matrix of the spacecraft and u ∈ R3 is the control torque vector. Let u¯ = −ω× Jω + u. Equation (9.5) can be rewritten as ω˙ = J u¯

(9.6)

where J = J −1 is also a positive definite matrix.

9.3 Koopman Operator for Attitude Dynamics 9.3.1 Reduced Linear Model Koopman operator is an appealing way to approximate the evolution of a nonlinear system on a subspace spanned by a finite set of measurables. How to construct such a set of measurables is the most challenging step [3]. This section aims to present a possible set of measurement functions of states for the attitude dynamics governed by Eqs. (9.4) and (9.6). Let z 1 = Rω× ∈ R3×3 . Equation (9.4) can be rewritten as ˙ = z1 R

(9.7)

¯ × z˙ 1 = z 1 ω× + R(J u) × ¯ = z 2 + R(J u)

(9.8)

¯ × ω× + z 1 (J u) ¯ × z˙ 2 = z 2 ω× + R(J u) × × × ¯ ω + Rω (J u) ¯ × = z 3 + R(J u)

(9.9)

The time derivative of z 1 is

where z 2 = z 1 ω× ∈ R3×3 and

in which z 3 = z 2 ω× ∈ R3×3 . The time derivative of z 3 is ¯ × ω×2 + z 1 (J u) ¯ × ω× + z 2 (J u) ¯ × z˙ 3 = z 3 ω× + R(J u) × ×2 × × × ×2 ¯ ω + Rω (J u) ¯ ω + Rω (J u) ¯ × = z 4 + R(J u)

(9.10)

where z 4 = z 3 ω× . Consequently, for k = 2, 3, 4, . . ., the time derivative of z k = z k−1 ω× = Rω×k is

9.3 Koopman Operator for Attitude Dynamics

181

¯ × ω×(k−1) + Rω× (J u) ¯ × ω×(k−2) + · · · + Rω×(k−1) (J u) ¯ × z˙ k = z k+1 + R(J u) k ×(i−1) × ×(k−i) ¯ ω = z k+1 + R i=1 ω (J u) (9.11) For any matrix A = [ai j ] ∈ R3×3 , A = [a11 , a21 , a31 , a12 , a22 , a32 , a13 , a23 , a33 ]T ∈ R9 represents a 9-dimensional vector obtained by reshaping the matrix A. Reshaping the matrices R and z k yields ˙ =z (9.12) R 1 z˙ k = z k+1 + B k u¯

(9.13)

u¯ is the 9-dimensional vector generated by reshaping the matrix where k B k×(i−1) ¯ × ω×(k−i) . Note that B k is a matrix of 9 × 3. If the (i, j)th ω (J u) R i=1 elements of the rotation matrix R and J are denoted by ri j and Ji j , the detailed expression of B 1 is ⎡

−r13 J21 + r12 J31 ⎢ −r23 J21 + r22 J31 ⎢ ⎢ −r33 J21 + r32 J31 ⎢ ⎢ r13 J11 − r11 J31 ⎢ B1 = ⎢ ⎢ r23 J11 − r21 J31 ⎢ r33 J11 − r31 J31 ⎢ ⎢ −r12 J11 + r11 J21 ⎢ ⎣ −r22 J11 + r21 J21 −r32 J11 + r31 J21

−r13 J22 + r12 J32 −r23 J22 + r22 J32 −r33 J22 + r32 J32 r13 J12 − r11 J32 r23 J12 − r21 J32 r33 J12 − r31 J32 −r12 J12 + r11 J22 −r22 J12 + r21 J22 −r32 J12 + r31 J22

⎤ −r13 J23 + r12 J33 −r23 J23 + r22 J33 ⎥ ⎥ −r33 J23 + r32 J33 ⎥ ⎥ r13 J13 − r11 J33 ⎥ ⎥ r23 J13 − r21 J33 ⎥ ⎥ r33 J13 − r31 J33 ⎥ ⎥ −r12 J13 + r11 JJ 23 ⎥ ⎥ −r22 J13 + r21 J23 ⎦ −r32 J13 + r31 J23

An important property of B 1 is ⎡

B 1T

⎤ 2 + 2J 2 + 2J 2 2J11 2J23 J31 + 2J11 J21 + 2J22 J21 2J31 J11 + 2J23 J21 + 2J31 J33 21 31 2 2 2 ⎣ B 1 = 2J23 J31 + 2J11 J21 + 2J22 J21 2J21 + 2J22 + 2J23 2J31 J21 + 2J22 J23 + 2J23 J33 ⎦ 2 + 2J 2 + 2J 2 2J31 J11 + 2J23 J21 + 2J31 J33 2J31 J21 + 2J22 J23 + 2J23 J33 2J23 31 33

(9.14)

It is clear that B 1T B 1 is a constant matrix in R3×3 . The determi2 2 2 2 nant of B 1T B 1 is 8(J11 J22 J33 − J11 J23 − J21 J33 + 2J23 J31 J21 − J22 J31 ) = T 2 8Det(J ) . Since J is positive definite, the determinant of B 1 B 1 is positive. Therefore, B 1T B 1 is invertible. Hence, the attitude dynamics in Eqs. (9.4) and (9.6) can be described as X˙ = AX + B u¯

(9.15)

X = [R T , z 1T , z 2T , z 3T , . . .]T

(9.16)

where

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

⎡

0 ⎢0 ⎢ A=⎢ ⎢0 ⎣0 ···

I9 0 0 0 ···

0 I9 0 0 ···

0 0 I9 0 ···

⎤ ··· · · ·⎥ ⎥ · · ·⎥ ⎥ · · ·⎦ ···

(9.17)

⎡

⎤ 0 ⎢ B1⎥ ⎢ ⎥ ⎥ B=⎢ ⎢ B2⎥ ⎣ B3⎦ ···

(9.18)

It is trivial to show that any solution to Eqs. (9.4) and (9.6) satisfies Eq. (9.15). Suppose that the solution to Eq. (9.15) with u¯ = u¯ s at any t > 0 is represented by {R = Rs , z 1 = z 1s , z 2 = z 2s , . . . , z k = z ks , . . .}. First, RsT Rs = I 3 is shown in the control-free case since the attitude kinematics has no relationship with the control input. Based on the Taylor series of a real function, RsT Rs can be written as RsT Rs = R(0)T R(0) +

∞ 1 dn (RsT Rs )|t=0 t n n n! dt n=1

(9.19)

where R(0) ∈ S O(3) is the initial attitude. Note that d i R T d n−i Rs dn s T T (R R )| = Cni |t=0 = Cni z is z (n−i)s |t=0 s t=0 s n i n−i dt dt dt i=0 i=0 n

n

(9.20)

where z 0s = Rs . Considering that z is |t=0 = R(0)ω(0)×i holds, one has dn (RsT Rs )|t=0 = Cni (−1)i ω(0)×i R(0)T R(0)ω(0)×(n−i) = (−1)i Cni ω(0)×n = 0 n dt n

n

i=0

i=0

(9.21) Therefore, RsT Rs = I 3 holds and RsT z 1s is antisymmetric. Denote ωs = ˙ It is straightforward to show that z 1s = Rs ω× s = R s . The time derivative of ωs can be expressed as (RsT z 1s )∨ .

T ¯ × )]∨ ω˙ s = [z 1s z 1s + RsT (z 2s + Rs (J u) T T × ¯ × ]∨ = [z 1s z 1s + Rs z 1s ωs + (J u) T × T × × ¯ × ]∨ = [(Rs ωs ) Rs ωs + Rs Rs ω× s ωs + (J u) = J u¯

(9.22)

Therefore, Eqs. (9.4) and (9.6) hold at the solution to Eq. (9.15) and the angular velocity in the original attitude dynamics can be calculated according to

9.3 Koopman Operator for Attitude Dynamics

183

ωs = (RsT z 1s )∨ . It can be concluded that the linear system in Eq. (9.15) is equivalent to the original equations (9.4) and (9.6). According to the definitions of z k , {R, z 1 , z 2 , z 3 , . . .} is a set of measurement functions of the states of the original attitude dynamics governed by Eqs. (9.4) and (9.6). Essentially, Eq. (9.15) defines a linear system that governs the evolution of observables R, z 1 , z 2 , z 3 , . . .. However, Eq. (9.15) is infinite-dimensional, hence, it cannot be used for dynamics analysis and controller design directly. It should be noted that the angular velocity of spacecraft is usually slow. For example, a geostationary satellite completes one orbit per day and its angular velocity is 7.2 × 10−5 rad/s. The slow angular velocity implies that z k → 0 as k → ∞. Hence, a finite-dimensional linear system can be obtained by truncating the infinite-dimensional system in Eq. (9.15). Suppose that the first n orders are adopted, i.e., R, z 1 , . . ., z n−1 are the variables of the following reduced linear system. ˙ = AX + Bu¯ X

(9.23)

T X = [R T , z 1T , z 2T , . . . , z n−1 ]T ∈ R9n

(9.24)

where

⎡

0 ⎢0 A=⎢ ⎣· · · 0

I9 0 ··· 0

0 ··· I9 · · · ··· ··· 0 ···

⎤ 0 0⎥ ⎥ ∈ R9n×9n · · ·⎦ 0

(9.25)

⎡

⎤ 0 ⎢ B1 ⎥ ⎢ ⎥ 9n×3 ⎥ B=⎢ ⎢ B2 ⎥ ∈ R ⎣ ··· ⎦ B n−1

(9.26)

In this chapter, the system in Eq. (9.23) is named nth-order reduced system of the original nonlinear system. Note that an nth-order reduced system has 9n dimensions. Therefore, a 9n-dimensional linear system has been obtained to describe the nonlinear attitude dynamics approximately. Certainly, in Eq. (9.23), one cannot choose n as 1 ˙ = 0 will be because the control input will disappear and a poor reduced system R generated. Hence, in the implementation of such Koopman-operator-based reduced linear system, n ≥ 2 should be chosen. Equation (9.23) is obtained by ignoring the term z n in the first 9n rows of Eq. (9.15). As shown in the paragraph after Eq. (9.18), the infinite dimensional linear system governed by Eq. (9.15) is equivalent to the original nonlinear attitude dynamics. Hence, the error between Eq. (9.23) and nonlinear attitude dynamics is mainly determined by the term z n . According to the definition of z n , it can be

tassumed that z n 2 ≤ ωn2 . Hence, z n−1 in Eq. (9.23) satisfies z n−1 − Rω×(n−1) ≤ 0 ωn2 I 3 dt ≤ n I 3 t,

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

where  is an upper bound of ω. Hence, it can concluded that the approximation error is smaller with smaller angular velocity, higher order of Eq. (9.23), and shorter simulation time.

9.3.2 Simulation Verification In this section, some simulation examples will be presented to show the approximation accuracy of the reduced linear model with various conditions. First, the control-free case is considered with J = I 3 . The approximation error is indicated by the error between the first elements of R governed by Eq. (9.4) and Eq. (9.23). The following three cases are considered. Case (a). The initial rotation matrix and angular velocity are set as I 3 and [0.005, 0.005, 0.005]T , respectively. As shown in Fig. 9.1, the difference between the linear system and the original attitude dynamics becomes smaller with the increase of the linear system order. Case (b). In this case, 50 random initial conditions are used for the simulation. The norm of the angular velocity is limited to 0.01 rad/s and the reduced system order is set as 10. As shown in Fig. 9.2, the linear system can approximate the nonlinear attitude dynamics accurately. Case (c). An example with a longer simulation time is performed in this case for the 10th-order linear system. The initial rotation matrix and angular velocity are set the same as those in Case (a). As shown in Fig. 9.3, as the simulation time increases, the approximation error becomes larger. The main reason is that the integral of the truncated terms increases with time. Second, the case with control input is simulated. Suppose that J = I 3 and the control input u = 0.0001[sin(0.1t), sin(0.01t), sin(0.02t)]T . At the initial time, the 1

Approximation errors

Fig. 9.1 Approximation errors with different linear system orders

n=3 n=5 n=7 n=9 n=11 n=13 n=15

0.8 0.6 0.4 0.2 0

0

100

Time (s)

200

300

9.3 Koopman Operator for Attitude Dynamics

185

Fig. 9.2 Approximation errors with random initial conditions

Fig. 9.3 Approximation errors versus simulation time

spacecraft rests at I 3 . The simulation results are shown in Fig. 9.4. The approximation errors are smaller with higher-order linear system and shorter simulation time. As shown in these simulation results, the proposed Koopman-operator-based linear finite-dimensional system can approximately describe the nonlinear attitude dynamics. The approximation error is small in a short simulation time and tends to zero with higher order of linear reduced model. This conclusion agrees well with the analysis in the last paragraph of Sect. 9.3.1. In all following simulations, the 10th-order linear system will be used unless otherwise specified.

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Fig. 9.4 Approximation errors with control input and different truncation numbers

9.4 Controller Design Based on Koopman Operator 9.4.1 Controller Design Suppose that the desired attitude is represented by a smooth function Rd (t). Denote ... T ˙ dT , R ¨ dT , Rd , . . . , d(n−1) RdT /dt (n−1) ]T ∈ R9n , where d(n−1) RdT /dt (n−1) Xd = [RdT , R represents the (n − 1)th time derivative of Rd . It should be noted that the first 9(n − ... T ˙ d= [R ˙ dT , R ¨ dT , Rd , . . . , d(n−1) RdT /dt (n−1) , dn RdT /dt n ]T ∈ R9n and 1) elements of X ... T ˙ dT , R ¨ dT , Rd , . . . ,d(n−1) RdT /dt (n−1) , 01×9 ]T ∈ R9n are the same, but the last AXd = [ R ˙ d and AXd are dn R T /dt n and 0, respectively. Considering that usually 9 entries of X d the magnitude and frequency of the desired angular velocity of a spacecraft are low, dn RdT /dt n ≈ 0 holds with a large n. Therefore, in this study, it is reasonable to assume ˙ d equals to AXd . Hence, the tracking error can be defined as E = X − Xd , that X whose time derivative is E˙ = AX + Bu¯ − AXd = AE + Bu¯

(9.27)

From the above equation, it can be concluded that the attitude tracking error E is also governed by a linear system with a state-dependent input matrix, which implies that the controller should be designed at each state. To generate a linear time-invariant system, a measurement function of the states and the control inputs can be defined ¯ Hence, one has as u∗k = B k u. ¯ E˙ = AE + BU (9.28)  09×9(n−1) ∗T ∗T T 9(n−1) ¯ where U = [u∗T . The controllability B = , u , . . . , u ] ∈ R , 1 2 n−1 I 9(n−1) matrix of the above linear system is

9.4 Controller Design Based on Koopman Operator

187

 ¯ c = 09×9(n−1) I 9(n−1) A ˆ 1 · · · A(9n−3) A ˆ 1 AA ˆ1 M I 9(n−1) 09×9(n−1)

(9.29)

ˆ 1 represents the matrix obtained by removing the last 9 columns from where A ⎡ ⎡ ⎤ ⎤ 0 I9 0 · · · 0 0 0 I9 · · · 0 ⎢ ⎢ ⎥ ⎥ ˆ 1 = ⎢ 0 0 0 ··· 0 ⎥ ∈ ˆ 1 = ⎢ 0 0 I 9 · · · 0 ⎥ ∈ R9n×9(n−1) . AA A, i.e., A ⎣· · · · · · · · · · · · · · ·⎦ ⎣· · · · · · · · · · · · · · ·⎦ 0 0 0 ··· 0 0 0 0 ··· 0  I 0 R9n×9(n−1) . It should be noted that 9×9(n−1) 9(n−1) has 18(n − 1) linearly indeI 9(n−1) 09×9(n−1) j ˆ pendent column vectors. A A1 has 9(n − 2 − j) linearly independent  column vectors for j = 0, 1, . . . , n − 3. Hence, there are in total 18(n − 1) + 9 n−3 j=0 (n − 2 − ¯ c . Since n ≥ 2, 9(n+2)(n−1) ≥ 9n always j) = 9(n+2)(n−1) non-zero columns in M 2 2 holds. It should be noted that each column only has one non-zero element. Con¯ c has at least one non-zero element, one can find 9n linearly sidering every row of M ¯c non-zero columns. Hence, the rank of M independent unit vectors from 9(n+2)(n−1) 2 is equal to 9n. Hence, one can conclude that the system in Eq. (9.28) is controllable. Suppose the feedback controller is U = −KE

(9.30)

The infinite-horizon linear-quadratic regulator (LQR) technique can be used to get an optimal feedback gain. The cost function is defined as 1 J= 2

∞

[ET QE + U T RU]dt

(9.31)

0

where Q ∈ R9n×9n is positive-semidefinite and R ∈ R9(n−1)×9(n−1) is positive definite. The optimal feedback gain can be expressed as ¯ P K = R−1 B T

(9.32)

where P is the solution to the following algebraic Riccati equation ¯ P +Q=0 ¯ −1 B AT P + PA − P BR T

(9.33)

It should be noted that the control input vector U has 9(n − 1) dimensions. The next challenging question is how to realize such a high-dimensional control input ¯ One can minimize the following cost function to using the 3-dimensional input u. ¯ determine u. 1 ¯ T (Bu¯ − BU) ¯ (9.34) J1 = (Bu¯ − BU) 2

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Essentially, this is a least-square problem. The minimum of J1 can be found by setting the gradient to zero. The solution to this least-square problem is ¯ u¯ = B+ BU

(9.35)

where B+ represents the Moore–Penrose inverse of B, i.e., B+ = (BT B)−1 BT = −1  n−1 T n−1 T BT . In the case with small angular velocity, i=1 B i B i i=1 B i B i ≈ B 1T B 1 = constant holds according to Eq. (9.14). Also, B 1T B 1 is invertible. Therefore, u¯ can be rewritten as ¯ u¯ = (B 1T B 1 )−1 BT BU T = (B 1T B 1 )−1 [B 1T B 2T · · · B n−1 ]U T T T T −1 ]KE = −(B 1 B 1 ) [B 1 B 2 · · · B n−1

(9.36)

Note that BT B is a matrix of both spacecraft parameters and states, however, B 1 is a matrix of only the inertia parameters from Eq. (9.14). Hence, the computational burden can be reduced by replacing (BT B)−1 with (B 1T B 1 )−1 since (B 1T B 1 )−1 can be calculated in advance. Consequently, the control input for the system in Eq. (9.5) becomes B 1T

T ]KE + ω× Jω u = −(B 1T B 1 )−1 [B 1T B 2T · · · B n−1

(9.37)

It should be noted that the controller (9.37) is nonlinear since both T ]KE and ω× Jω are nonlinear terms with state(B 1T B 1 )−1 [B 1T B 2T · · · B n−1 dependent B j for j = 2, . . . , n − 1.

9.4.2 Simulation Verification The inertia matrix is chosen as in [15] ⎡

⎤ 10 0.25 −0.25 J = ⎣ 0.25 9.75 −0.15⎦ −0.25 −0.15 8.25

(9.38)

The matrices Q and R are chosen as diag{I 9 , 1000 I 9 , 0, . . . , 0} ∈ R9n×9n and 1000 I 9(n−1) , respectively. According to Rodrigues’ rotation formula, the rotation matrix RdT R can be expressed as I 3 + sin θ nˆ × + (1 − cos θ )nˆ ×2 , where nˆ is the unit axis vector and θ is the rotation angle associated with RdT R. Hence, V1 = tr(I 3 − RdT R) = 2(1 − cos θ ) ∈ [0, 4]. It is straightforward to show that the necessary and sufficient condition of V1 = 0 is θ = 0, i.e., R = Rd . Hence, V1 is used to indicate the attitude tracking error in the following results.

9.4 Controller Design Based on Koopman Operator

189

Fig. 9.5 Attitude tracking errors in 50 simulations

Case (a). Let Rd = I 3 and ωd = 0. 50 simulations are performed with random initial conditions. Suppose that the norm of the initial angular velocity is less than 0.01 rad/s. Figure 9.5 gives the simulation results. The time histories of the attitude tracking errors in 50 simulations are various with different initial conditions, but all curves converge to zero, i.e., the designed controller succeeds in driving the spacecraft to the desired attitude in all simulations. Case (b). The desired states are the same as those in Case (a). Suppose that the initial attitude is diag{1, −1, −1} and the initial angular velocity is zero. For the controller designed based on the classical attitude tracking error (RdT R − R T Rd )∨ , such an initial attitude is an unstable equilibrium [16]. As shown in Fig. 9.6, with the numerical simulation error in the first 900 s, the spacecraft can leave the initial attitude. That is, such an initial attitude is an unstable equilibrium. Hence, unstable equilibria still exist under the proposed control architecture. The main reason is that the reduced linear system is obtained by deleting some high order terms containing ω. For the spacecraft staying at unstable equilibria statically, the angular velocity is zero. Hence, at these points, the reduced linear system is exactly the same as the nonlinear attitude dynamics. Hence, the continuous time-invariant controller designed based on the reduced system should also suffer from unstable equilibria. This agrees with the statement that time-invariant smooth feedback laws can achieve almost global stability at most [17]. Furthermore, considering that the reduced linear system (9.28) ¯ has only a global equilibrium with the control input U, the error between Bu¯ and BU is the main reason of the existence of unstable equilibria under the controller (4.3). Fortunately, the ubiquitous disturbance can help the spacecraft escape from unstable equilibria. Hence, this study will not pay more attention to unstable equilibria. Case (c). The desired attitude Rd is chosen as I 3 . The desired angular velocity is 0.001 × [sin(0.03t), sin(0.01t), sin(0.02t)]T . 50 simulation results are presented in Fig. 9.7 with random initial attitude and angular velocity. In these simulations, the spacecraft can track the time-varying desired attitude successfully.

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9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Fig. 9.6 Attitude tracking error of spacecraft starting from an unstable equilibrium

Table 9.1 Q and R in seven examples Example number Q 1 2 3 4 5 6 7

diag{I 9 , 1000 I 9 , 0, . . . , 0} diag{I 9 , 100 I 9 , 0, . . . , 0} diag{I 9 , 10 I 9 , 0, . . . , 0} diag{I 9 , 1000 I 9 , 0, . . . , 0} diag{I 9 , 1000 I 9 , 0, . . . , 0} diag{I 9 , 100 I 9 , 0, . . . , 0} diag{I 9 , 100 I 9 , 0, . . . , 0}

R 1000 I 9(n−1) 1000 I 9(n−1) 1000 I 9(n−1) 100 I 9(n−1) 10 I 9(n−1) 100 I 9(n−1) 10 I 9(n−1)

Case (d). As shown in Table 9.1, seven examples with various Q and R are simu⎡ ⎤ −0.1476 0.346 −0.9265 lated. The initial conditions are R(0) = ⎣ 0.7834 −0.5309 −0.3231 ⎦ and ω = 0. −0.6037 −0.7735 −0.1927 The spacecraft is expected to stay at I 3 . Figures 9.8 and 9.9 show the attitude tracking errors and the first elements of the control input with various Q and R. As indicated in Fig. 9.9b, in Examples 1, 4 and 5, the control inputs are different in the first 2 s. As shown in Table 9.1, only the control weight matrices in these three examples are different. One can find that the larger weight of control input in cost function results in a smaller control effort. After the first 2 s, the control inputs of Examples 1, 4 and 5 are almost the same. The different control inputs in a short time may not change the system response a lot. Consequently, as shown in Fig. 9.8, the time histories of the attitude tracking errors almost coincide in Examples 1, 4 and 5. The similar phenomenon can be found by comparing Examples 6 and 7. In Examples 1, 2 and 3, only the weights of z 1 are different. From Fig. 9.8, one can find that a larger weight of z 1 leads to a higher damping.

9.4 Controller Design Based on Koopman Operator

191

Fig. 9.7 Attitude tracking error with time-varying desired attitude

Fig. 9.8 Attitude tracking errors with various cost functions

9.4.3 Experimental Verification In this section, a Quanser AERO system with the 2 DOF helicopter configuration is used to verify the proposed controller. As shown in Fig. 9.10, Quanser AERO is a fully integrated dual-motor experimental equipment that mainly consists of two propellers driven by DC motors and a built-in amplifier, a built-in data acquisition device, and an interchangeable QFLEX 2 interface panel in the compact base. Two optical encoders are equipped to measure the pitch and yaw angles. The gyroscope sensors in the Inertial Measurement Unit (IMU) board can provide the angular velocity of the helicopter. 360◦ and 124◦ rotations are allowed for pitch and yaw angles, respectively. The real-time control is achieved with QUARC software with 500 Hz sampling rate. The rotation matrix used to describe the attitude of the helicopter is calculated with zero roll angle and the pitch and yaw angles measured by the two encoders. By compensating for the stiffness and damping, the system can almost rotate freely with 2 DOFs, i.e., the attitude dynamics equation based on the rotation matrix with

192

9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Fig. 9.9 The first elements of control input with various cost functions

Fig. 9.10 Quanser AERO experimental system

9.4 Controller Design Based on Koopman Operator

193

Fig. 9.11 Experimental result of attitude tracking error

zero roll angle can describe such a system. The moments of inertia in the Y and Z directions are 0.0219 kgm2 and 0.0220 kgm2 , respectively. Since only the rotation along Y and Z axes can be controlled, the last two elements in the proposed controller in Eq. (4.3) will be applied to the helicopter. Suppose that the helicopter attitude is I 3 and the angular ⎡ velocity is zero ⎤at the 0.707 −0.707 0 initial time. The desired attitude and angular velocity are ⎣ 0.707 0.707 0 ⎦ and 0 0 1 T [0, 0.2 sin(0.2t), 0.1 sin(0.5t)] rad/s. A 4th-order reduced linear system is used to design controller with Q = diag{10 I 9 , I 9 , 0, . . . , 0} and R = 0.001I 9(n−1) . Figure 9.11 indicates the attitude tracking error V1 = tr(I 3 − RdT R) in the experimental test. The attitude tracking can be achieved at about 10.9 s and the tracking error is less than 0.005. Compared with the simulation results in previous section, the experimental result needs less settling time. The main reason is that the R is chosen much smaller than that in previous section, i.e., the larger control input U is allowed for the reduced linear system in the experimental test. However, it should be noted that a large U does not mean large control torques for the Quanser AERO system. This is because the moments of inertia of the helicopter system are much smaller than the norm of T ] the inertia matrix in previous section, i.e., the term (B 1T B 1 )−1 [B 1T B 2T · · · B n−1 in the experimental test is much smaller than that in the simulations in Sect. 9.4.2. Furthermore, due to the small moments of inertia of the helicopter system, the values of Q and R in Sect. 9.4.2 will cause small control commands for the motors that are less than the bounds of the dead zones of the motors. Hence, R is decreased significantly in the experimental test.

194

9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

9.4.4 Comparison with Traditional Optimal Control In Sect. 9.4.1, the optimal feedback gain is obtained via LQR technique. Equation (9.35) presents a way to approximate the desired optimal feedback control signal by ¯ minimizing the sum of the squares of the residuals between Bu¯ and BU. However, the three-dimensional control input may not be able to realize the optimal 9(n − 1)dimensional controller exactly. Hence, the proposed control method can only provide a near-optimal analytical feedback controller. To show the difference between the developed method and the existing optimal control, the controller in Eq. (4.3) will be compared with the controllers from the General Pseudospectral Optimization Software (GPOPS) in this section. A brief introduction to GPOPS is presented here. To find the details, please refer to [18, 19]. GPOPS is a software to minimize the following cost function J = (x(t0 ), t0 , x(t f ), t f ; p) subject to the dynamic constraints, the inequality path constraints and the boundary conditions. Note that t0 and t f are the initial and final times, and x and p are the state and parameter vectors. All constraints can be nonlinear. The Radau Pseudospectral Method (RPM) is used in GPOPS. All the collocation points are the Legendre-Gauss-Radau points. The cost function and the differential-algebraic equations can be discretized at these points. Now the continuous-time optimal control problems have become a nonlinear programming problem (NLP), which can be solved by SNOPT [20]. GPOPS 5.1 is used in this study with MATLAB R2018b. The inertia matrix is chosen the same as that ⎡ ⎤ −0.1476 0.346 −0.9265 in Eq. (9.38). The initial attitude is set as R(0) = ⎣ 0.7834 −0.5309 −0.3231 ⎦ −0.6037 −0.7735 −0.1927 and the angular velocity is zero at the initial time. The desired attitude and angular velocity are I 3 and 0, respectively. For the proposed control method, the weight matrices Q and R are chosen the same as those in the paragraph after Eq. (9.38) in Sect. 9.4.2. Also, V1 = tr(I 3 − RdT R) is adopted to indicate the tracking error. For the optimal controller calculated by GPOPS, the cost function is set as

1 t T ˙ T ]diag{I 9 , 1000 I 9 }[R T − RdT , R ˙ T ]T + 1000 u¯ T B 1T B 1 udt ¯ [R − RdT , R J∗ = 2 0 (9.39) T T ˙T T T ˙T T where the first term [R − Rd , R ]diag{I 9 , 1000 I 9 }[R − Rd , R ] is exactly the same as the first term in the cost function in Eq. (9.31) with the weight matrices defined in the paragraph after Eq. (9.38) in Sect. 9.4.2. Note that u¯ T B 1T B 1 u¯ ≈  n ¯ T B kT B k u¯ = U T U holds in the case with small angular velocity. Hence, a k=1 u cost function nearly identical to that in Eq. (9.31) with the weight matrices in the paragraph after Eq. (9.38) is applied to the nonlinear optimal control. No path, state and control input constraints are introduced. Essentially, the solution to such a nonlinear optimal control problem obtained from GPOPS is a set of control efforts at discrete time instants, i.e., the control input is independent on state feedback. However, the proposed controller in Eq. (4.3) is based on state feedback. Hence, to make a fair comparison, an optimal feedback controller with the format in the following equation is designed

9.4 Controller Design Based on Koopman Operator

u = −K p1 (R − Rd ) − K p2 (ω − ωd )

195

(9.40)

where the K p1 ∈ R3×9 and K p2 ∈ R3×3 are the control parameters to be optimized. It should be noted that the classical attitude and angular velocity tracking errors on S O(3) are defined as (RdT R − R T Rd )∨ and ω − R T Rd ωd [21]. Consequently, another possible controller format can be u = −K p3 (RdT R − R T Rd )∨ − K p4 (ω − R T Rd ωd )

(9.41)

where K p3 ∈ R3×3 and K p4 ∈ R3×3 are the feedback gains to be optimized. In the designed controller (4.3), the nonlinear term ω× Jω is included. To present a fairer comparison, the following two controller are also adopted. u = −K p5 (R − Rd ) − K p6 (ω − ωd ) + ω× Jω u = −K p7 (RdT R − R T Rd )∨ − K p8 (ω − R T Rd ωd ) + ω× Jω

(9.42) (9.43)

where K p5 ∈ R3×9 , K p6 ∈ R3×3 , K p7 ∈ R3×3 and K p8 ∈ R3×3 are the feedback gains to be optimized. Based on the controllers in Eqs. (9.40–9.43), the nonlinear optimal control problem becomes a parametric programming problem, which can also be solved by GPOPS. The cost functions in these four parametric programming problems are set as Eq. (9.39). A Lenovo Thinkpad 470p with Intel Core i7-7700HQ processor and 8G RAM is used to perform the calculation. “GPOPS”, “GPOPS-P1”, “GPOPS-P2”, “GPOPS-P3” and “GPOPS-P4” are used to represent the traditional nonlinear optimal control, the parametric programmings based on formats in Eqs. (9.40), (9.41), (9.42) and (9.43), respectively. The computational times of these six methods are given in Table 9.2. Note that the most time-consuming part of the implementation of the proposed controller is the calculation of optimal gain using lqr function in Matlab. Hence, the computational time for the proposed method represents the time for lqr function. The results of the proposed method, traditional nonlinear optimal control and the parametric programmings are shown in Fig. 9.12. With the control commands from GPOPS, GPOPS-P1, GPOPS-P2, GPOPS-P3 and GPOPS-P4, the spacecraft attitude can converge faster than that with the proposed controller. The proposed method needs about 35 s more to reach the same tracking error. Hence, the proposed method can only provide a near-optimal solution. However, as shown in Fig. 9.13, the proposed method needs smaller control torque than the remaining three methods. Furthermore, the control input for the nonlinear optimal control needs to be recalculated if the initial condition changes. To check the sensitivities of the remaining five controllers from different methods (Proposed, GPOPS-P1, GPOPSP2, GPOPS-P3 and GPOPS-P4) to initial conditions, a different initial condition is applied to check the performance of these five methods. Suppose the spacecraft stays

196

9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Fig. 9.12 Attitude tracking error with different methods

Fig. 9.13 First element of the control input vector with different methods

⎡

⎤ −0.1987 −0.2554 0.9462 at ⎣ −0.9151 −0.2974 −0.2724 ⎦ with zero angular velocity at the initial time. As 0.3509 −0.9200 −0.1746 indicated in Fig. 9.14, controllers in Eqs. (9.40), (9.42) and (9.43) with the optimal control parameters cannot finish the attitude tracking task in 200 s. The green dot-dashed line indicates that under the controller from GPOPS-P2, the optimality cannot be ensured with the new initial condition. Hence, all the controllers in Eqs. (9.40–9.43) with optimal gains are sensitive to initial conditions, i.e., the optimal gains should be recalculated if the initial conditions change. However, the proposed controller can still drive the spacecraft to the desired attitude efficiently. The detailed comparison among these methods is given in Table 9.2. In conclusion, the developed control design method needs less computational time and the near-optimality does not depend on the initial conditions because the controller is designed based on the reduced linear system.

9.4 Controller Design Based on Koopman Operator

197

Table 9.2 Comparison of different methods Proposed

GPOPS

GPOPS-P1

Control signal

Continuous

Discrete

Continuous

Continuous

Continuous

Continuous

State feedback

Yes

No

Yes

Yes

Yes

Yes

Computational time

1.5 s

135.7 s

161.8 s

43.8 s

138.7 s

45.8 s

Sensitivity to ini- No tial conditions

Yes

Yes

Yes

Yes

Yes

Optimality

Optimal

Optimal

Optimal

Optimal

Optimal

Nearoptimal

GPOPS-P2

GPOPS-P3

GPOPS-P4

Fig. 9.14 Attitude tracking error starting from a new initial condition

9.4.5 Application to Leaderless Synchronization The literature on the attitude consensus of multiple spacecraft on S O(3) can be classified into two groups: one is the leader-follower consensus; the other is the leaderless synchronization. The leader-follower cases have been solved based on a distributed observer to estimate the leader’s information [22, 23] or under some specified communication graphs [24, 25]. However, for the leaderless attitude synchronization, the distributed-observer-based method does not work any more and the available results only are applicable to some specified graphs [26]. Therefore, the leaderless synchronization of nonlinear attitude dynamics under a general graph on S O(3) is still an open problem. The fundamental challenge comes from the strong nonlinearity of the attitude dynamics. In this study, a reduced linear system is developed to approximate the nonlinear attitude dynamics and may provide a solution to this problem.

198

9.4.5.1

9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Controller Design

Suppose that the directed graph with N nodes of concern contains spanning tree(s). Hence, all the eigenvalues of the graph Laplacian matrix L have non-negative real parts and zero is a simple eigenvalue with 1 as the corresponding right eigenvector [27, 28]. That is, the eigenvalues of graph Laplacian matrix L can be represented as 0, λ2 , . . ., λ N , where Re(λ j ) > 0 for j = 2, . . . , N . Assume that the attitude dynamics of the ith spacecraft is ˙ i = Ri ωi× (9.44) R J ω˙ i = −ωi× Jωi + ui

(9.45)

where the subscript i is used to indicate that the variable is associated with the ith spacecraft. From Sect. 9.3, the reduced linear system for the ith spacecraft is ˙ i = AXi + BU ¯ i X

(9.46)

The following consensus protocol can be used Ui = K c



ai∗j (X j − Xi )

(9.47)

j∈Ni

where K c is the control gain and A∗ = [ai∗j ] is used to represent the adjacency matrix of a communication graph in this chapter. Note that the distributed protocol in Eq. (9.47) depends only on the state errors between the agent i and its neighbors. From Theorem 1 in [29], it can be concluded that the networked systems governed by Eq. ¯ is stabilizable and the communication graph (9.46) is consensusable because (A, B) has a spanning tree. That is, there exists a feedback gain K c such that lim X j − t→∞

Xi  = 0 for i, j = 1, 2, . . . , N . Based on the Kronecker product in Sect. 1.2, the closed-loop networked reduced linear systems can be expressed as ˙ T = (I N ⊗ A − L ⊗ BK ¯ c )XT X

(9.48)

where XT = [X1T , X2T , . . . , XTN ]T . The consensus controller design now is equivalent ¯ c are in the open left half to find a gain K c such that the eigenvalues of A − λi BK plane for i = 2, 3, . . . , N [29]. One possible way to choose the gain matrix is [30] ˆ −1 B ¯ T Pˆ K c = cR

(9.49)

1 ˆ ∈ R9(n−1)×9(n−1) is ,R where c is a positive constant not less than 2 mini=2,...,N Re(λi ) a positive definite matrix and Pˆ is the solution of the following algebraic Riccati equation ˆ − Pˆ B ¯R ˆ −1 B ¯ T Pˆ = 0 ˆ +Q AT Pˆ + PA (9.50)

9.4 Controller Design Based on Koopman Operator

199

ˆ ∈ R9n×9n is a positive definite matrix. Now the controller for the ith spacewhere Q craft can be written as −1

ˆ ui = c[B 1 (Ri )T B 1 (Ri )]−1 [B 1 (Ri )T B 2 (Ri , ωi )T · · · B n−1 (Ri , ωi )T ]R

¯ T Pˆ B



ai∗j (X j − Xi ) + ωi× Jωi

j∈Ni

(9.51)

9.4.5.2

Simulation Verification

Four spacecraft with the same parameters as those in Sect. 9.4.2 are used to verify the proposed synchronization algorithm in Eq. (9.51). Four possible communication graphs in Fig. 9.15 are adopted. Obviously, all graphs in Fig. 9.15 contain a 1 are 0.33, 0.5, 0.5 and 0.25, spanning tree. For these four graphs, 2 mini=2,...,N Re(λi ) ˆ and R ˆ are chosen as I 9n respectively. Hence, in the simulations, c is chosen as 2. Q and 1000 I 9(n−1) . 50 numerical simulations are performed for each graph in Fig. 9.15 with random initial conditions. Suppose that the norm of initial angular velocity is

Fig. 9.15 Communication graphs

200

9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Fig. 9.16 Attitude synchronization errors under the communication graphs in Fig. 9.15

less than 0.01. V2 = tr(I 3 − R1T R2 ) + tr(I 3 − R2T R3 ) + tr(I 3 − R3T R4 ) ∈ [0, 12] is used to indicate the attitude synchronization error. Note that the synchronization is achieved if and only if V2 = 0 holds. The attitude synchronization errors in these simulations for the graphs in Fig. 9.15 are shown in Fig. 9.16. In all 200 simulations, the synchronization is achieved within 200 s. Hence, the proposed control algorithm in Eq. (9.51) designed based on the reduced linear system can finish the leaderless attitude synchronization mission of multiple spacecraft under a directed graph having a spanning tree.

9.4.6 Experimental Verification In this section, four Quanser AERO systems are used to verify the synchronization controller in Eq. (9.51) under the communication graph in Fig. 9.15a. At the initial time, these four helicopter systems shown in Fig. 9.17 rest at the following attitudes.

9.4 Controller Design Based on Koopman Operator

201

Fig. 9.17 Four Quanser AERO systems

R1 (0) = ⎡ I 3 , R2 (0)⎤= diag{−1,⎡−1, 1} ⎤ 0 10 −0.7071 −0.707 0 R3 (0) = ⎣ −1 0 0 ⎦ , R4 (0) = ⎣ 0.707 −0.7071 0 ⎦ 0 01 0 0 1

(9.52)

ˆ and R ˆ are chosen as I 36 and 0.01I 27 , respecThe parameter c is chosen as 8. Q tively. The potential function V2 defined the same as that in last section is used to indicate the attitude synchronization error. Figure 9.18 indicates the synchronization error in this experimental test. The snapshots of this test are given in Fig. 9.19. It is clear that the attitude synchronization is achieved within 15 s. However, there exists a steady-state error after 15 s. The main reason may be some unknown system uncertainties or disturbances. Furthermore, as shown in Fig. 9.19, at 96 s, a disturbance is applied to the first helicopter artificially. As shown in Fig. 9.18, such a disturbance results in a large attitude synchronization error. However, the designed controller is still able to control the networked system back to the synchronization status.

Fig. 9.18 Experimental result of attitude synchronization error

202

9 Koopman-Operator-Based Attitude Dynamics and Control on S O(3)

Fig. 9.19 Snapshots of the experimental test using four Quanser AERO systems

9.5 Attitude Control with Large Angular Velocities

203

9.5 Attitude Control with Large Angular Velocities 9.5.1 Controller Design In the previous sections, it is assumed that the angular velocity is small. Theoretically, the methods in Sects. 9.3 and 9.4 are applicable in the case that the norm of the angular velocity is less than 1 because the terms with high order of the angular velocity are omitted. When the norm of the angular velocity approaches 1, a higher order reduced linear system is necessary to approximate the original attitude dynamics. For example, if a 10th order reduced linear model is used, ω×10 2