Spacecraft Dynamics and Control

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Space Science and Technologies Series Editor: Peijian Ye

Yongchun Xie · Yongjun Lei · Jianxin Guo · Bin Meng

Spacecraft Dynamics and Control

Space Science and Technologies Series Editor Peijian Ye, China Academy of Space Technology, Beijing, China

Space Science and Technologies publishes a host of recent advances and achievements in the field – quickly and informally. It covers a wide range of disciplines and specialties, with a focus on three main aspects: key theories, basic implementation methods, and practical engineering applications. It includes, but is not limited to, theoretical and applied overall system design, subsystem design, major space-vehicle supporting technologies, and the management of related engineering implementations. Within the scopes of the series are monographs, professional books or graduate textbooks, edited volumes, and reference works purposely devoted to support education in related areas at the graduate and post-graduate levels.

More information about this series at http://www.springer.com/series/16385

Yongchun Xie · Yongjun Lei · Jianxin Guo · Bin Meng

Spacecraft Dynamics and Control

Yongchun Xie Beijing Institute of Control Engineering Beijing, China

Yongjun Lei Beijing Institute of Control Engineering Beijing, China

Jianxin Guo Beijing Institute of Control Engineering Beijing, China

Bin Meng Beijing Institute of Control Engineering Beijing, China

ISSN 2730-6410 ISSN 2730-6429 (electronic) Space Science and Technologies ISBN 978-981-33-6447-9 ISBN 978-981-33-6448-6 (eBook) https://doi.org/10.1007/978-981-33-6448-6 Jointly published with Beijing Institute of Technology Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Beijing Institute of Technology Press. © Beijing Institute of Technology Press 2022 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain 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

Preface

Spacecraft Dynamics and Control is one of the 23 volumes of the Space Science and Technologies series. It is “a set of authoritative works of both theoretical and practical significance for front-line researchers in the space field, researchers in related fields, and teachers and students in universities.” In accordance with the above objectives, this book focuses on spacecraft dynamics and control. Based on years of research and practical experience, the book systematically introduces the laws of orbital and attitude motion of spacecraft, as well as the basic theory and engineering design of spacecraft control, in order to meet educational and practical needs in spacecraft engineering. A spacecraft’s trajectories and attitude have a significant role in spacecraft design. They are the primary factors that determine the function and performance of the spacecraft and directly affect the effectiveness of loads on the spacecraft. The orbit and attitude control of spacecraft is another core component of spacecraft flight control and management, which is related to spacecraft mission implementation and the lifetime of spacecraft in orbit. This book is a product of the union of theory and practice. It summarizes many years of research and practical experience in control theories of many engineers and technical personnel at the Beijing Institute of Control Engineering. The control schemes and systems designed based on their research accomplishments have been successfully applied to many types of spacecraft, including remote sensing satellites, communication satellites, navigation satellites, technology experiment satellites, Shenzhou manned spacecraft, Tianzhou cargo spacecraft, Tiangong-1/Tiangong-2 space laboratories, and Chang’e lunar exploration satellites for various missions. Furthermore, the schemes and systems developed are of great significance for future engineering development of manned lunar landings, deep-space exploration, on-orbit servicing, and other missions. The book consists of seven chapters: Chapter 1 is the introduction, which reviews concepts related to types of spacecraft and spacecraft control. Chapter 2 introduces the orbit and orbital dynamics of spacecraft. Chapter 3 presents methods for orbit control. Chapter 4 reviews spacecraft attitude kinematics and dynamics. Chapter 5 elaborates on spacecraft attitude determination methods. Chapter 6 presents methods for spacecraft attitude control. Chapter 7 introduces technologies of autonomous navigation, guidance, and control of spacecraft. v

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Preface

The leading authors and editors of this book are Yongchun Xie, Yongjun Lei, Jianxin Guo, and Bin Meng, who supervised the compilation and review of the whole book. Specifically, Yongchun Xie was responsible for the compilation of Chaps. 1 and 7; Jianxin Guo oversaw the compilation of Chaps. 2 and 3; and Yongjun Lei was responsible for the compilation of Chaps. 4–6. Many experts from the Beijing Institute of Control Engineering participated in the writing of this book. The authors of each chapter are as follows. Chapter 1 was written by Yongchun Xie; Chaps. 2 and 3 were written by Jianxin Guo, Shaochun Hu, Liang Zhou, and Tao Liu; Chap. 4 was written by Yongjun Lei, Ya Deng, and Zeguo Wang; Chap. 5 was written by Yongjun Lei; Chap. 6 was written by Yongjun Lei, Shuping Tan, Fei Li, Yujia Sun, Ya Deng, and Zeguo Wang; and Chap. 7 was written by Yongchun Xie, Tao Liu, Kai Xiong, Wenqiang Dong, Ji Li, Maodeng Li, Pengji Wang, Xue Ma, and Xinxin Yu. Furthermore, Meng Hu, Juan Wang, and Jingya Geng were involved in the compilation and review of the book, and data compilation was completed by Yipeng Li, Jingya Geng, and others. In a book that covers a wide range of topics, there will inevitably be omissions and mistakes, despite the best efforts of the authors. We warmly welcome criticisms, corrections, and suggestions from readers and experts. Beijing, China November 2020

Yongchun Xie Yongjun Lei Jianxin Guo Bin Meng

Acknowledgements

Many thanks to the colleagues who helped to prepare the material of this book: Shaochun Hu, Liang Zhou, Tao Liu, Ya Deng, Zeguo Wang, Shuping Tan, Fei Li, Yujia Sun, Kai Xiong, Wenqiang Dong, Ji Li, Maodeng Li, Pengji Wang, Xue Ma, Xinxin Yu, Yipeng Li, Meng Hu, Juan Wang, and Jingya Geng. We would especially like to express our deepest appreciation to Professors including Guo Li, Hongxin Wu, Liangdong Liu, Nanhua Wang, Jun Hu, and Jun Yuan. The chief reviewer of this book is Guo Li. The first draft of the book was carefully reviewed by Academician Hongxin Wu and four Professors: Liangdong Liu, Nanhua Wang, Jun Hu, and Jun Yuan. The compilation of this book is also attributed to the guidance and support received from the leaders of China Academy of Space Technology and Beijing Institute of Control Engineering. Special thanks should be given to Academician Peijian Ye and Professor Li Yuan for detailed guidance and dedicated support. The authors express their heartfelt gratitude to the National Defense Science and Technology Book Publishing Fund, the National Key Basic Research and Development Program of China (grant number: 2013CB733100), and National Natural Science Foundation (grant number: U20B2054) for financial support. Finally, we also would like to express our thanks to editors from Beijing Institute of Technology Press for their meticulous proofreading and valuable assistance.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Types of Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Low-Earth-Orbit Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Lunar and Deep-Space Probes . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Manned Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Near-Space Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Connotation of Spacecraft Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Orbital Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Attitude Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Guidance, Navigation, and Control . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 4 5 7 8 8 9 10 11 11 12 13 16

2 Spacecraft Orbits and Orbital Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Time Systems and Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Transformation Between Coordinate Systems . . . . . . . . . . . . 2.3 Two-Body Problem and Three-Body Problem . . . . . . . . . . . . . . . . . . 2.3.1 Overview of Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Constants of Two-Body Orbital Motion . . . . . . . . . . . . . . . . . 2.3.3 Geometric Equation of Two-Body Orbits . . . . . . . . . . . . . . . . 2.3.4 Geometric Properties of Two-Body Orbits . . . . . . . . . . . . . . . 2.3.5 Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . 2.3.6 Libration Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Orbital Properties of Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Orbital Parameters and Transformations . . . . . . . . . . . . . . . . . 2.4.2 Satellite Ground Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 17 20 22 25 25 26 28 30 32 35 36 36 39 ix

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2.4.3 Launch Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Geosynchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Sun-Synchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Critical Inclination Orbits and Frozen Orbits . . . . . . . . . . . . . 2.4.7 Repeat Ground Track Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.8 Reentry Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.9 Libration Point Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Orbital Perturbation Equations and Their Solutions . . . . . . . . . . . . . . 2.5.1 Osculating Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Lagrange Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Gauss Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Numerical Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Sources of Orbital Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Earth’s Non-sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Atmospheric Drag Near Earth . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Gravitational Forces of the Sun and Moon . . . . . . . . . . . . . . . 2.6.4 Solar Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 The Moon’s Non-sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Mars’ Non-sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.7 Atmospheric Drag Near Mars . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Relative Motion of Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Definitions of Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Equations of Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 43 44 46 46 47 50 58 60 60 61 62 63 66 66 69 69 71 71 72 73 73 73 74 77

3 Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basics of Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Governing Equation of Orbital Maneuvering . . . . . . . . . . . . . 3.2.2 Impulsive Thrust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Finite Thrust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Optimal Orbit Control Problem . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Orbit Control for Typical Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Perturbation Analysis and Stationkeeping of LEO Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Perturbation Analysis and Stationkeeping of HEO Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Orbit Transfer of HEO Spacecraft . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Return Orbit Control for Lunar Exploration . . . . . . . . . . . . . . 3.3.5 Orbit Dynamic Models for Lunar Exploration . . . . . . . . . . . . 3.3.6 Design of Cislunar Return Trajectory . . . . . . . . . . . . . . . . . . . 3.3.7 Precise Design of Cislunar Return Trajectory . . . . . . . . . . . . 3.3.8 Impulsive Thrust Orbit Control for Cislunar Transfer . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 80 82 84 86 87 87 98 119 124 124 128 146 154 158

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4 Spacecraft Attitude Kinematics and Dynamics . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Attitude and Attitude Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Attitude Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Attitude Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Attitude Dynamics of Rigid-Body Spacecraft . . . . . . . . . . . . 4.3.2 Attitude Dynamics of Flexible Spacecraft . . . . . . . . . . . . . . . 4.3.3 Attitude Dynamics of Liquid-Filled Spacecraft . . . . . . . . . . . 4.3.4 Attitude Dynamics of Multi-Body Spacecraft . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 159 159 163 167 167 169 172 180 199

5 Spacecraft Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Modeling of Attitude Sensor Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Modeling of Random Errors of Gyroscopes . . . . . . . . . . . . . . 5.2.2 Modeling of Star-Sensor Measurement Errors . . . . . . . . . . . . 5.3 Three-Axis Attitude Determination Based on State Estimation . . . . 5.4 Calibration of Relative Error of Attitude Sensor . . . . . . . . . . . . . . . . . 5.4.1 Calibration of Relative References for Star Sensors . . . . . . . 5.4.2 Calibration of Gyro Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Ground-Based Post-Event High-Precision Attitude Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Determination of Spin Angular Velocity with Abnormal Attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Principle of Determining Spin Angular Velocity of Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Strategies for Reducing the Determination Error of Spin Angular Rate and Improving the Determination Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Simulation Verification and Application . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 201 202 211 213 221 221 231

6 Spacecraft Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Attitude Control Based on Angular-Momentum Management Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Spacecraft Attitude Stabilization Control . . . . . . . . . . . . . . . . 6.2.2 Attitude Maneuver Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Steering Strategies for Angular-Momentum Management Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Flywheel Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 CMG Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Steering Strategies for Hybrid Actuators . . . . . . . . . . . . . . . . . 6.4 Liquid-Filled Spacecraft Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Control Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 248 248

253 256 260 263 263 263 264 270 299 299 314 338 347 347

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6.4.2 Design of Attitude Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Multi-Body Spacecraft Attitude Control . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Hybrid Attitude Control with Moving Antenna . . . . . . . . . . . 6.5.2 Hybrid Attitude Control of Combined Body . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

348 352 352 367 369

7 Autonomous Guidance, Navigation, and Control of Spacecraft . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Absolute Autonomous Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 SINS-GPS Integrated Navigation . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Attitude-Sensor-Based Autonomous Navigation . . . . . . . . . . 7.3 Relative Autonomous Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Relative State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Autonomous Orbit Determination and Relative State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Guidance and Control for Rendezvous and Docking . . . . . . . . . . . . . 7.4.1 Flight Phases and Mission Requirements . . . . . . . . . . . . . . . . 7.4.2 Rendezvous and Docking Guidance . . . . . . . . . . . . . . . . . . . . 7.4.3 Rendezvous and Docking Control . . . . . . . . . . . . . . . . . . . . . . 7.5 Guidance and Control for Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Ballistic Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Semi-ballistic Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Lifting Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Reentry Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Guidance and Control for Soft Landing on Extraterrestrial Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Soft Landing on the Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Soft Landing on Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Soft Landing on Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371 371 372 372 375 378 379 385 388 388 390 394 398 398 399 401 401 406 406 414 417 421

About the Authors

Yongchun Xie Professor Yongchun Xie is the Director of Science and Technology Committee of Beijing Institute of Control Engineering. She is currently the corresponding member of IAA (International Academy of Astronautics) and has been engaging in the field of guidance, navigation, and control (GNC) of spacecraft rendezvous and docking (RVD) for nearly 20 years. She has been participating in China’s manned space project as a leading engineer, who is responsible for the GNC systems design, and the GNC schemes have been successfully applied to the RVD tasks of China’s spacecraft. Her research interests are in autonomous guidance and control of rendezvous and intelligent control of spacecraft. She has published more than 100 articles in technical journals and conferences, authored 3 books, and won five ministry-level prizes and three national-level awards. Yongjun Lei Professor Yongjun Lei currently works as a Senior Chief Designer of spacecraft control system in Beijing Institute of Control Engineering. He has been engaged in the control scheme design for highperformance low-orbit satellites. His current research interest is in the area of modeling and control of complicated systems, including flexible structures and multibody spacecraft. He won six ministerial-level prizes (first class once, second class for three times, and third class twice).

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About the Authors

Jianxin Guo Professor Jianxin Guo currently works as a Director Designer of medium and high orbit satellite control system in Beijing Institute of Control Engineering. He has been engaged in the control scheme design for the BEIDOU navigation satellite system and new generation communication satellites of China. His current research interests are in spacecraft orbit control and multi-body spacecraft complex control. He won four ministerial-level prizes (first class for three times and second class once).

Bin Meng Professor Bin Meng currently works in Beijing Institute of Control Engineering. Her major is in spacecraft guidance and control, and adaptive control theory and application. As the Principle Investigator or Co-investigator, she has led or been involved in more than 10 research projects. She has published more than 30 research articles, 5 in-book chapters, and has been granted 10 patents. She won the second prize of the 2013 National Technological Invention Award and some other awards. She has served as a Member of IFAC, Member of the Chinese Association of Automation, and Member of editorial boards of Control Theory and its Application and Journal of System Science and Mathematical Science.

Chapter 1

Introduction

1.1 Types of Spacecraft A spacecraft is a vehicle designed to fly in outer space and can perform a variety of specific tasks, such as exploring, developing, or utilizing resources in space or celestial bodies. Spacecraft can be classified into different types depending on their tasks. Given that a spacecraft may perform different space missions at a time, it is difficult to classify spacecraft based on a single criterion alone. As shown in Fig. 1.1, spacecraft can be categorized into four groups according to the orbit and the ability to transport human beings: low-Earth-orbit (LEO) satellites, lunar and deep-space probes, manned spacecraft, and near-space vehicles (NSVs). It should be noted that NSVs may not be regarded as spacecraft in a strict sense.

1.1.1 Low-Earth-Orbit Satellites LEO satellites, known as artificial Earth satellites, mainly refer to unmanned spacecraft that revolve around the Earth in low earth orbit. More LEO satellites have seen launched than any other type of spacecraft, and they provide a wide range of services. In fact, the number of LEO satellites launched accounts for more than 90% of the total number of spacecraft launched. According to specific missions and the launch purpose, LEO satellites can be classified into application satellites, scientific experiment satellites, and technology experiment satellites. 1.

Application satellites

Application satellites are satellites used for serving the needs of the national economy or performing specific tasks. In general, application satellites include remote-sensing (RS) satellites, reconnaissance satellites, communications satellites, navigation satellites, and satellites for on-orbit servicing. © Beijing Institute of Technology Press 2022 Y. Xie et al., Spacecraft Dynamics and Control, Space Science and Technologies, https://doi.org/10.1007/978-981-33-6448-6_1

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1 Introduction Low-dynamic vehicles Near-space vehicles High-dynamic vehicles Application satelites Low-Earth-orbit satellites

Scientific experiment satellites Technical experiment satellites

Spacecraft Lunar and deep-space probes

Lunar probes Deep-space probes Space station Manned spaceship

Manned spacecraft

Cargo spacecraft Space shuttles Space tourist flights

Fig. 1.1 Types of spacecraft

RS satellites are artificial satellites used for remote sensing platforms in outer space, and their orbits are designed according to mission requirements. RS satellites can cover the entire Earth or a designated area within a certain period of time, and they have been mainly used for the observation of ocean and land, environmental protection, applications in agriculture and forestry, and meteorological monitoring. Common RS satellites include land satellites, ocean satellites, and meteorological satellites. Reconnaissance satellites are mainly used to collect different types of information from countries or regions of interest, including information on military or nonmilitary facilities and activities; the distribution, transportation, and use of natural resources; or meteorological, marine, and hydrological data. A reconnaissance satellite typically takes photographs of the ground and targets with visible-light and infrared cameras, respectively. It can also intercept and capture radio signals in various frequency bands. A communications satellite transmits and amplifies radio communication signals via a transponder, and it acts as a repeater between a transmitter and a receiver on Earth. In general, a communications satellite consists of a communications payload (transponder, antenna, and information-exchange system), an attitude and orbit control system, a power subsystem, and a command and control system. Communications satellites can be used in the fields of television, telephone, radio, networks, and military. According to the type of communications service, communications satellites can be categorized into those for fixed satellite communications, mobile satellite communications, television broadcasting, maritime communications, and

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tracking and data relay satellites (TDRS). Further, according to the orbit, communications satellites can be grouped into four types: communications satellites in a geostationary orbit (GEO), high elliptical orbit (HEO), medium-Earth orbit (MEO), and LEO. A navigation satellite is equipped with special radio navigation equipment, which can provide all-day location and time information for data terminal equipment on the ground by continuously sending radio signals to a receiver on Earth. Its main functions include precision timing, navigation, and positioning. Well-known navigation satellite systems in the world include the Global Positioning System (GPS) owned by the U.S. government, Russia’s Global Navigation Satellite System (GLONASS), the Chinese Bei Dou Navigation Satellite System (BDS), and the European Union’s Global Satellite Navigation System (GNSS) called Galileo. Satellites for on-orbit servicing are designed to operate on orbit to help achieve a series of tasks such as extending the life of spacecraft and improving the capability of spacecraft to perform tasks with the assistance from human beings and space robots. The main tasks of on-orbit operations include on-orbit object detection, orbit transfers, on-orbit refueling, on-orbit fault repair, on-orbit assembly of components, and on-orbit target attack and defense. 2.

Scientific experiment satellites

Scientific experiment satellites are mainly satellites used for scientific detection and research. Typical examples are satellites for near-Earth geophysical exploration, astronomical satellites, and satellites for physical testing. Satellites for near-Earth geophysical exploration are mainly used to study the near-Earth space environment, including the upper atmosphere, ionosphere, Earth’s magnetosphere, Earth’s gravitational field, and solar radiation. Astronomical satellites are used for the observation and study of the Sun and other celestial bodies in the solar system. Satellites for physical testing are mainly used to test physical objects by utilizing the unique features of the space environment. On August 16, 2016, China launched the first quantum space satellite “Mozi,” also known as Quantum Experiments at Space Scale (QUESS), which achieved quantum communication between a satellite and the ground for the first time. In addition, the mission verified a previous prediction that pairs of entangled photons would remain coupled to one another even at a separation of hundreds of thousands of meters. 3.

Technology experiment satellites

A technology experiment satellite (TES) is designed to verify new technology or conduct tests for application satellites. Before launching a satellite, a large number of ground tests are required to verify the reliability of new technology. Considering the complexity of satellite technologies and high costs of application satellites, it is preferable to verify complex technologies by first testing them on less expensive TESs. Therefore, new technology is usually applied in a TES for verification before use. China’s satellite series “SJ” (Shi Jian, “practice”) is mainly used for scientific detection and technical tests. On March 3, 1971, China successfully launched its

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first scientific experiment satellite “SJ-1,” which was developed based on the Dong Fang Hong-1 (DFH-1) satellite. While in orbit, the satellite measured environmental parameters, such as the high-altitude magnetic field, X-rays, and cosmic rays, and tested long-life satellite technologies such as a power supply system based on silicon solar cells as well as an active and passive thermal control system. Following the success of SJ-1, China has launched over ten satellites for scientific experiments and technical tests. Thus far, 20 satellites have been launched for SJ missions.

1.1.2 Lunar and Deep-Space Probes Lunar and deep-space probes (including lunar probes and deep-space probes) typically refer to unmanned spacecraft that fly by, orbit, and land on the Moon or other celestial bodies in space. 1.

Lunar probes

Between 1958 and 1976, the United States and the Soviet Union launched 47 and 61 lunar probes, respectively. After that, the two countries stopped their lunar exploration programs. In 1990, Japan launched a lunar probe. Lunar probes generally explore the Moon with various approaches, such as lunar flyby, hitting the Moon (hard landing), soft landing, orbiting the of Moon, sampling of lunar soil, or returning to the Earth after sampling. China’s lunar exploration project has three-step objectives: “orbiting, landing, and sample returning.” First, the lunar exploration satellites Chang’e 1 and Chang’e 2 were launched to detect the surface environment and geomorphology of the Moon. Second, Chang’e 3 was designed to make a soft landing on the lunar surface to investigate the composition of rocks and minerals in the landing site by using patrol cars and robots. Third, Chang’e 5 will be launched for lunar roving and returning samples. 2.

Deep-space probes

Deep-space probes are generally sent into interplanetary space by using a launch vehicle to explore the space with different equipped instruments. The deep space networks on the ground offer support for the measurement and control of probes and receives the scientific detection data sent back by the detector via probes. Based on the capability of technical support and the arrangement of space missions, different approaches can be adopted to explore the Moon, planets, or other celestial bodies. Specifically, probes can fly by planets or other celestial bodies, during which they can perform close detection and photographic observation. A probe can also enter the orbit of a celestial body and become an artificial satellite for long-term repeated exploration. It can also perform a hard landing on a celestial body and detect the target during the fall before the landing. A soft landing on the surface of a celestial

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body can also be adopted for on-site investigation of the celestial body. The probe can return to the Earth with samples after a soft landing for detailed studies. The Japanese spacecraft Hayabusa (Japanese for falcon) explored asteroid 25,143 Itokawa and sent the collected samples back to the Earth. It was originally planned to return to the Earth in June 2007, but the return was delayed by 3 years to June 13, 2010 owing to a fuel leak from the probe. Hayabusa had a seven-year journey of six billion kilometers (four billion miles), achieving the first detection of an asteroid that threatens the Earth and returning material samples. Looking back at the history of deep-space probes, Voyager 1 made the longest journey ever with a travel distance of 21.2 billion kilometers (13.2 billion miles). Voyager 1 was an unmanned probe with a mass of 815 kg launched by the National Aeronautics and Space Administration (NASA) on September 5, 1977 for detection in the outer solar system. The main missions of Voyager 1 were to fly by Jupiter and Saturn and return with detailed images of these planets. On February 10, 2012, Voyager 1 entered the heliosheath (a terminal shock region between the solar system and interstellar matter), which is approximately 17.91 billion kilometers from the Sun. On September 22, 2013, NASA confirmed and announced that Voyager 1 was out of the area dominated by solar wind after 36 years of travel.

1.1.3 Manned Spacecraft A manned spacecraft is a manned aircraft that orbits the Earth or space in a certain trajectory. According to the operating style, manned spacecraft are divided into five major types: manned spaceships, cargo spacecraft, space stations, space shuttles, and space tourist flights. 1.

Manned spaceship

A manned spaceship is a spacecraft that can support astronauts to live and work in space to perform space missions, and to allow the astronaut cockpit to have a ballistic return and vertical landing. A manned spaceship is actually a type of space shuttle, and it is the smallest amongst manned spacecraft. Manned spaceships developed and launched by the Soviet Union/Russia include the Vostok (“East”), Voskhod (“Sunrise”), and Soyuz (“Union”). The United States has developed and launched the Mercury spacecraft, Gemini spacecraft, and Apollo spacecraft. The Apollo was a spacecraft that landed human beings on the Moon. The Shenzhou (“Divine land”) spacecraft, a manned spacecraft independently developed by China, has reached or surpassed third-generation manned spacecraft worldwide. 2.

Cargo spacecraft

A cargo spacecraft is a spaceship that transports cargo to space. Its main task is to transport food, air, water, fuel, and equipment to a space station. It is an important means of transportation for providing supplies to a space station as well as a ground

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system for supporting the space station. At present, cargo spacecraft serving the International Space Station (ISS) include the Japanese cargo spacecraft H-II Transfer Vehicle (HTV), Russia’s Progress cargo ship, Europe’s Autonomous Transfer Vehicle (ATV) cargo ship, and the American spacecraft Dragon. China’s Tianzhou (“heavenly vessel”) cargo spacecraft, launched on April 20, 2017, is an unmanned cargo spacecraft used for resupply services at China’s Space Station. The Tianzhou cargo spacecraft is 10.6 m in length and has a maximum diameter of 3.35 m. It has a launch mass of 12.91 t with a cargo capacity of approximately 6.5 t. Its main task is to supply propellant, air, drinking water, and food for astronauts, as well as replacement equipment for the maintenance of the space station for the Tiangong-2 (“heavenly place”) space laboratory and subsequent Chinese space stations in order to extend the operating life of space stations and ship waste from the space station back to the Earth. At present, most cargo spacecraft are not reusable. After separation from a space station, cargo spacecraft are normally burned in the Earth’s atmosphere on return to the Earth. 3.

Space station

A space station is a spacecraft that can operate on orbit for a long time under certain conditions for tests or production and can satisfy the needs of astronauts to live and work. There are two types of space stations: monolithic and modular. Monolithic stations are launched into orbit by a rocket in a single launch, while modular stations consist of units or modules that are launched into orbit separately. Modular space stations have two configurations: building blocks and the integrated truss structure (ITS). Space stations in the structure of building blocks are docked by multiple bay sections on orbit, such as the Mir (Russian for “peace”) space station. ITS space stations are based on assembled or deployable trusses up to several meters or hundreds of meters in length, which carry several bay sections and equipment, such as the ISS. 4.

Space shuttles

A space shuttle is a spacecraft that uses a booster rocket and its own rocket engine to lift off. It can use its own wings to glide and land on an airport runway, similar to an airplane, and can be reused many times. Space shuttles can be divided into two types according to their relationship with the carrier. The first type of spacecraft has no power during launch. It is only used as the payload of a launch vehicle and is sent into orbit by a rocket. The second type of spacecraft uses its own rocket engine power to work alongside the launch vehicle during the launch and ascent process. This type consists of a space shuttle that can re-enter the atmosphere and is equipped with three liquid-propellant rocket engines, two solid rocket boosters, and an external fuel tank. Apart from space shuttles that carry human beings and cargo, some space shuttles can launch upper-stage rockets on orbit to send the satellite into another orbit. 5.

Space tourist flights

Space tourism refers to activities in which tourists enter space to experience space life and partake in sightseeing in a spaceship. Owing to the extremely high cost of space

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launches and spaceships, Russia has opened up private funding. In exchange, sponsors can enter space in the spacecraft. At present, space tourism is mainly operated in Russia. The idea of self-funded space tourism originated from a proposal by an engineer named Dennis Anthony Tito who sponsored the Russian Mir space station in exchange for travelling to space. Dennis Tito took the spacecraft Soyuz TM-32 to the ISS on April 28, 2001, becoming the first space tourist. The second space tourist was a South African entrepreneur named Mark Shuttleworth, who entered the ISS on April 25, 2002 in the spacecraft Soyuz TM-34. The third space tourist was Gregory Olsen, an American businessman. An Iranian–American engineer, Anousheh Ansari, became the fourth space tourist and the first female space tourist. As space technologies continue to develop, space tourism is expected to become a part of the lives of ordinary people.

1.1.4 Near-Space Vehicles Near-space vehicles (NSVs) are aircraft that can only fly continuously for a long time in or near space at an altitude of 20–100 km. NSVs are not spacecraft in a strict sense. Hypersonic or suborbital vehicles flying in near space benefit from advantages that aeronautic and aerospace vehicles do not have. They have shown promise for application in communications security, intelligence gathering, electronic suppression, early warning, and civilian use. In recent years, a wide range of studies related to NSVs have been conducted at home and abroad. As a popular research subject, NSVs can be roughly divided into two types based on flight speed: low-dynamic vehicles (Mach number < 1) and high-dynamic vehicles (Mach number >1). 1.

Low-dynamic vehicles

Low-dynamic vehicles mainly include stratosphere airships, high-altitude balloons, and solar drones. They have a long suspension time, a large load capacity, a high flying altitude, strong survivability, and other features. In addition, they can carry visible-light, infrared, multispectral, and hyperspectral radars and other payloads for acquiring data, which can be used as a means of obtaining regional information to improve battlefield awareness and support combat operations. Moreover, they can carry a variety of payloads for electronic countermeasures to achieve electromagnetic suppression and electromagnetic strikes on the battlefield, as well as communications and other energy-relay payloads for field emergency communications, communication relay, and energy relay services. 2.

High-dynamic vehicles

High-dynamic vehicles mainly include hypersonic vehicles, suborbital vehicles, and space shuttles. They have the characteristics of fast speed, long travel distance, high maneuverability, strong survivability, and a high variety of suitable payloads. The advantages of this type of vehicles include long-range and fast arrival, high-speed

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and precise attack on targets, reusability, and fast long-range delivery of materials. Given their advantages, they are highly valuable for military applications and have potential business value. Hypersonic vehicles can be mainly grouped into two categories based on the driving mechanism: unpowered gliding hypersonic vehicles and airbreathing/combined-cycle hypersonic vehicles. An unpowered gliding hypersonic vehicle does not have a propulsion system; it relies on a launch vehicle to push it to a sub-orbit with an altitude of approximately 100 km and on its own lift to maneuver in the atmosphere. Air-breathing/combined-cycle hypersonic vehicles can fly at hypersonic speed on its own power. Specifically, an air-breathing aerospace vehicle relies on a carrier to accelerate it to the required speed for a supersonic-combustion ramjet (scramjet) to operate, following which it is powered by the scramjet to fly at hypersonic speed. A combined-cycle aerospace vehicle can complete the entire process from takeoff to hypersonic travel by itself. These vehicles combine aerospace engines with different working principles that work separately in different flight phases to give full play to their respective advantages. A reusable launch vehicle (RLV) is a type of vehicle that falls between a space shuttle and an ordinary aircraft. It can take off as an ordinary aircraft and fly at a supersonic speed in the atmosphere at an altitude of 20–100 km. In addition, it can directly accelerate and enter an LEO to release satellites and launch space probes. After completing tasks such as satellite recovery from space and space maintenance, it can return to the atmosphere and perform a level landing. As RLVs are reusable, they are economical and inexpensive carriers that can greatly reduce manufacturing costs and transportation cycles.

1.2 Connotation of Spacecraft Control The motions of a spacecraft include orbital and attitude motions. In order to control the orbit and attitude of a spacecraft, it is vital to first determine its orbit and attitude. Early spacecraft control mainly refers to orbit control and attitude control. Considering the increasing requirements for spacecraft functions and the development of space technologies, spacecraft control has spawned an important branch of control engineering: spacecraft guidance, navigation, and control (GNC). As technology continues to evolve, the form of spacecraft control has also developed from telemetry and telecontrol as well as manual control to automatic control and even autonomous control.

1.2.1 Orbital Motions The orbit is the trajectory of the center of mass of the spacecraft (as a spatially moving body) when it moves in space. The main factors affecting the orbital characteristics of

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a spacecraft include the initial motion characteristics of the spacecraft, the mechanical environment of the space in which it is located, the active control force it is under, and its geometric characteristics during the movement. The orbital dynamics of the spacecraft and the technology for orbit control are important aspects of spacecraft engineering. German astronomer Johannes Kepler proposed his first law of planetary motion (the Ellipse Law) and the second law (the Law of Areas) in 1609, and the third law (the Harmonic Law) proposed in 1619, providing a basis for the orbital dynamics of spacecraft. Sir Isaac Newton, a British physicist, proposed the law of gravity and the three laws of motion in 1687, which laid the foundation for spacecraft engineering. Ideally, the orbital motion of spacecraft can be regarded as a two-body motion around a central body, which means that the spacecraft is affected by the gravity of only the central body. The physical variables describing the orbital motion of spacecraft include the position of the center of mass, speed, acceleration, and time. It is generally assumed that the spacecraft mass is much less than that of the central body. The spacecraft moves in a plane of space, and its orbit can be described by six orbital elements: the semi-major axis of orbit, eccentricity, argument of perigee, right ascension of ascending node, orbit inclination, and time of perigee passage. While the orbital motion of a spacecraft is mainly affected by the gravity of the central body, any external force will have an influence on the orbit. Such an influence can lead to changes in the six orbital elements, resulting in a deviation from the designed orbit. Such a deviation is called an orbital perturbation. For LEO spacecraft, including LEO satellites and LEO manned spacecraft, the forces that can cause orbit perturbations include the gravity of the non-ideal spherical Earth, atmospheric drag of the LEO, gravitational attraction of the Sun and Moon, solar radiation pressure and Earth-reflected radiation pressure, control force of the spacecraft’s own execution module, magnetic force, effects of Earth tides, internal motions of the Earth, and atmospheric disturbances.

1.2.2 Orbit Determination The key task for orbit determination is to study how to determine the position and velocity of a spacecraft in space at a certain moment. The orbit determination of spacecraft is a prerequisite for satellite applications. Observation data for a satellite orbit are obtained from parameters related to the motion of the satellite at several moments or in several time zones, rather than by directly measuring orbital elements. Orbital elements are then calculated using specific algorithms based on the principle of orbital dynamics. The determination of the satellite orbit involves three elementary processes: data acquisition, initial orbit determination, and improvement of orbit determination. After a certain amount of orbital observation data have been collected, the initial orbit of the spacecraft can be determined. Subsequently, more observational data can be used to obtain a more accurate orbit of the spacecraft through the improvement of orbit determination based on the initial orbit. In general, the longer

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the orbital arc covered by the observation data, the higher will be the accuracy of orbit determination.

1.2.3 Orbit Control Orbit control entails the exertion of a control force on the center of mass of a spacecraft at appropriate moments according to its current position and speed in order to ensure that the spacecraft reaches the desired position and speed at the desired time. Based on orbital elements, orbit control can be divided into in-plane control and out-of-plane control. In-plane control aims to control the semi-major axis, eccentricity, argument of perigee, and recurrent point, while out-of-plane control mainly involves the control of inclination and the right ascension of the ascending node. In most cases, the in-plane and out-of-plane control of the elements can be integrated separately. For example, the in-plane control of the semi-major axis and eccentricity vector, including the eccentricity and argument of perigee, can be jointly performed. For LEO satellites, different orbits require different control targets. For instance, the Sun-synchronous orbit (SSO) mainly requires the control of the semi-major axis and inclination; frozen and critical orbits mainly require the control of the semi-major axis, eccentricity, and argument of perigee; and recurrent orbits and recoverable spacecraft require the control of the semi-major axis and recurrent points across the equatorial plane. According to the work stages after the spacecraft enters orbit, orbit control can be generally divided into orbit maneuver, orbit establishment, orbit maintenance, orbit rendezvous, and re-entry control. (1)

(2)

(3)

Orbital maneuver refers to control for transferring a spacecraft from one freeflight-segment orbit to another. For example, after a GEO satellite is launched into orbit by a launch vehicle, it first enters a large elliptical transfer orbit. Subsequently, to enter the GEO, orbital maneuvering is required at a distant place in its transfer orbit. Orbit establishment refers to control for transferring a spacecraft from its launch orbit to its mission orbit. Orbit establishment usually requires a series of orbital maneuvers. For instance, after a GEO satellite is launched into orbit by a launch vehicle, it needs to undergo a series of orbital maneuvers to enter the GEO and achieve a fixed point. Orbit maintenance refers to control for overcoming the influence of orbit perturbation to keep certain parameters of a spacecraft’s orbit unchanged. For example, geosynchronous-orbit satellites regularly perform orbital corrections to accurately maintain their fixed-point positions, SSO and recurrent-orbit satellites impose control to maintain their inclination and period, and some control on LEO satellites was introduced to overcome atmospheric drag and extend the orbital lifetime.

1.2 Connotation of Spacecraft Control

(4)

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Orbital rendezvous refers to the control process implemented on two different spacecraft so that they simultaneously arrive at almost the same position with the same speed. Re-entry control refers to control for making a spacecraft depart from its original orbit and re-enter the Earth’s atmosphere.

1.2.4 Attitude Motion The attitude of a spacecraft quantifies the rotation of the spacecraft body about its center of mass. The attitude is used to describe the relationship between the fixed coordinate system of the spacecraft and the reference coordinate system. Attitude parameters include the directional cosine matrix, Euler angles, and quaternions. The equations describing attitude motion include attitude kinematics equations and attitude dynamics equations. Attitude kinematics equations describe the relationship between various attitude parameters and the attitude angular velocity. Attitude dynamics equations are used to describe the pattern of changes in the attitude angular velocity of the spacecraft after an external moment of force is introduced. The attitude dynamics equations are based on the theorem of angular momentum. For rigid-body spacecraft, the forms of attitude motion include spin around the maximum-inertia axis, precession, nutation, Earth-to-Earth three-axis stabilization, and inertial three-axis stabilization. Spacecraft can have very different dynamic characteristics depending on the structural characteristics. According to the structural characteristics, spacecraft can be divided into simple rigid-body spacecraft; large, complex liquid-filled spacecraft; and flexible spacecraft. The dynamic characteristics of a spacecraft have a great impact on its attitude motions. Furthermore, the moments of active control and space environmental disturbance, such as the gravity gradient torque, aerodynamic moment, solar radiation moment, and geomagnetic moments, are important factors that affect the spacecraft attitude.

1.2.5 Attitude Determination Attitude determination entails the determination of the attitude parameters of a spacecraft relative to a reference coordinate system, which can be an inertial coordinate system or an orbital coordinate system. The attitude of a spacecraft is generally determined based on measurement data obtained from an attitude sensor installed on the spacecraft. The attitude parameters of the proprio-coordinate system relative to the reference coordinate system can be obtained using suitable data processing methods. The accuracy of attitude determination depends on the selected data processing method and the accuracy of data obtained from the sensor. Sensors commonly used for

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spacecraft attitude determination include solar sensors, infrared Earth sensors, gyroscopes, star sensors, and magnetometers. Data processing methods mainly include the double-vector attitude determination algorithm and filters for attitude determination.

1.2.6 Attitude Control Attitude control is the process by which a spacecraft achieves or maintains the desired attitude. The two forms of attitude control are attitude stabilization and attitude maneuvering. Attitude stabilization refers to the process of maintaining the attitude of a spacecraft at the desired attitude through control, while attitude maneuvering refers to the process of switching the attitude of a spacecraft from one to another. Methods for attitude control include the control of reaction jets, reaction wheels, control moment gyro (CMG), magnetron, and gravity-gradient stabilization. The design of control laws commonly uses proportional–integral–derivative (PID) control and PID control with signal filtering, as well as phase-plane control, optimal control, self-adaptive control, and H∞ (i.e., “H-infinity”) robust control. The choice of the control law is strongly dependent on the dynamic characteristics of the object and requirements of performance standards. Attitude control is a prerequisite for the normal operation of a spacecraft. For example, when a satellite communicates with or observes the Earth, its antenna or remote sensor should point at the target on the ground; when the satellite is in orbit control, the engine must point at the required direction of thrust; and when the satellite re-enters the Earth’s atmosphere, heat shields must face the forward direction of airflow. All of the above would be required for a satellite to establish and maintain a certain attitude with respect to the celestial body. Terms related to attitude control include orientation, capture, search, and tracking. (1)

(2)

Orientation refers to the process of determining the relative position of a single axis or the three axes of the spacecraft body or appendages, such as solar panels, observation cameras, and antennas, with respect to a specified reference direction with certain accuracy. The reference direction can be stable in the inertial space. For example, astronomical observations are stable toward the Sun and are called orientation to the Sun. It can also be stable in the orbital coordinate system. Similarly, if the observation of the Earth is stable to the Earth, then it is called orientation to the Earth. Attitude orientation needs to overcome various space disturbances to keep the attitude of the spacecraft body or appendages in the reference direction. Therefore, the directions need to be maintained via control. Capture, also called initial alignment, refers to the maneuvering process of a spacecraft from an unknown and uncertain attitude to a known and desired attitude. For example, after a spacecraft enters an orbit and a satellite is separated from the rocket, the spacecraft is said to move from an uncertain attitude to an

1.2 Connotation of Spacecraft Control

(3) (4)

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attitude oriented towards the Sun or to the Earth. Another example is the orientation or re-orientation of a spacecraft towards the Sun after the attitude of the spacecraft becomes unstable because of errors during travel. To facilitate the design of the control system, attitude capture generally has two phases: coarse alignment and fine alignment. Coarse alignment refers to the process from the initial maneuver to the attainment of the desired attitude with a wide range of uncertain initial attitudes. Typically, a large control torque is used to shorten the maneuvering time, but it does not require high orientation accuracy. Fine alignment refers to the attitude correction that is required because of a lack of accuracy of coarse alignment. It is performed to ensure the requirements for orientation accuracy. Fine alignment generally requires a small control torque. Search refers to the capture of a moving target by the spacecraft body or appendages; an example is the search for targets by data relay satellites. Tracking refers to the continuous orientation of the spacecraft body or appendages to a moving target; an example is the tracking of targets by data relay satellites.

Attitude stabilization is the main method to achieve the stable long-term operation of a spacecraft. Based on attitude kinematics, the attitude stabilization of spacecraft can be broadly divided into two forms: spin stabilization and three-axis stabilization. (1)

(2)

In spin stabilization, a spacecraft such as a satellite rotates around its spin axis while keeping the spin-axis direction fixed in the inertial space based on the angular momentum. Spin stabilization is often associated with active attitude control to correct the errors in the spin-axis direction. A dual-spin satellite is composed of a spinning body and a despinning body, which are connected to each other by a despinning bearing. The spinning body rotates around the bearing axis (spin axis) to obtain the orientation of the spin axis, while the despinning body is controlled by the despinning motor on the bearing in addition to the orientation of the spin axis, thereby being three-axis stabilized. The payload is typically carried in the despinning body. Three-axis stabilization relies on active attitude control or the use of environmental torque to maintain the orientation of the three axes of the spacecraft in a certain reference coordinate system.

1.2.7 Guidance, Navigation, and Control LEO satellites work on mission orbits, and their attitude should satisfy the requirements for load orientation and tracking. In this regard, the control of LEO satellites mainly includes attitude control and orbit control. As the requirements for the functions and performance of satellites continue to increase, the requirements for the control accuracy and maneuverability of satellites, especially the attitude control accuracy, stability, and agility, have progressively increased. For manned spacecraft, lunar and deep-space probes, and even NSVs, new requirements have been proposed

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for the control of spacecraft trajectories in addition to conventional attitude and orbit control. In light of this, an important branch has been introduced to spacecraft control, i.e. Guidance, Navigation, and Control (GNC). GNC has a significant role to modern spacecraft. Guidance refers to the process of producing a series of maneuver time as well as the size and direction of speed increment on orbit by using certain algorithms and orbital dynamics based on the current position and speed of the spacecraft. A spacecraft requires guidance to reach a desired position and speed at a predetermined time or to fly along a desired trajectory. After orbital maneuvers are completed, the initial position and speed of the spacecraft can be guided to the desired position and speed, or the spacecraft can follow the desired trajectory. Guidance can be divided into different types according to the guidance algorithms or guidance laws. Standard ballistic guidance and forecasting guidance are usually suitable for the reentry of spacecraft; Clohessy–Wiltshire (CW) terminal guidance, line-of-sight (LOS) guidance, and optimal multiple-pulse guidance can be applied for rendezvous and docking; and optimal variable-thrust guidance, nominal orbit guidance, explicit guidance, and gravity-turn guidance are usually applied for a soft landing on the Moon. The design of guidance laws mainly considers time constraints, fuel consumption or thrust limitations, attitude requirements, autonomy, and the real-time performance of algorithms. Navigation refers to the determination of information about the spacecraft motion, such as attitude and orbit, by using data processing methods based on the measurement outputs of sensors installed on the spacecraft. It provides information on attitude and orbit for spacecraft guidance and control. The main data processing methods used for navigation are filtering methods. Spacecraft navigation includes absolute navigation, in which the motion of a single spacecraft is estimated, and relative navigation, in which the relative motion of two or more spacecraft is estimated. According to the measurement sensors used, absolute navigation can be further categorized into astronomical navigation, inertial navigation, radio navigation, satellite navigation, geomagnetic navigation, pulsar navigation, land (beacon) navigation, and integrated navigation. Relative navigation includes navigation based on satellite navigation devices, light detection and ranging (LiDAR), microwave radars, and imaging-based rendezvous and docking sensors. Control refers to the control of attitude or orbit of a spacecraft through engines intended for attitude control and orbital transfer, and it is performed based on the requirements for guidance, the current motion of the spacecraft, and the requirements for attitude control to achieve the tasks for guidance and attitude control. In a broad sense, the attitude control, orbit control, and GNC of spacecraft can be collectively referred to as spacecraft control. Methods for spacecraft control mainly include telemetry and telecontrol, manual control (including teleoperation control), automatic control, and autonomous control.

1.2 Connotation of Spacecraft Control

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Telemetry and telecontrol

Telemetry and telecontrol refer to control that relies on ground intervention and are jointly implemented using spacecraft instruments and equipment on the ground. In this method, the ground station sends instructions to a spacecraft to adjust its attitude or orbit. Generally, this method requires assistance from stations worldwide or relay satellites. Though telemetry and telecontrol do not have high requirements for spacecraft instruments, they involve long time delays in the communication between the spacecraft and ground. Therefore, this approach is only suitable for simple attitude maneuver and orbit control. For example, telemetry and telecontrol have been used for the attitude maneuver and orbit control of most single- and dual-spin satellites in early times. 2.

Manual control

Manual control (including teleoperation control) refers to the direct or indirect control by a human being. Relatively delicate operations and fine control can be achieved with this approach. Nevertheless, it requires professional training, and the time delay between the Earth and space has a great impact on teleoperation control. The design of a manual control system needs to meet the requirements for ergonomics and human factors. For example, it is important to consider ergonomic requirements in the manual attitude control of a spacecraft by an astronaut, manual control of close rendezvous and docking, manual control of returning and re-entry, teleoperation control by an astronaut in a space station for rendezvous and docking with a visiting cargo spacecraft, and teleoperation control of space robots. 3.

Automatic control

Automatic control refers to the control of the motion of a spacecraft by using equipment or devices on the spacecraft in accordance with a set program without relying on the ground and manual intervention. For example, space-borne control circuits or computers can obtain measurement data from space-borne sensors and calculate the amount of control to perform the automatic control of the attitude and orbit of the satellite. Complex attitude and orbit control tasks can be achieved using automatic control, which is suitable for systems with high requirements for time response. However, this approach requires high-performance computing devices for spacecraft control, and it still needs support from ground stations when errors occur in the control system. At present, automatic control is adopted for the attitude control of most satellites. It has also been widely used in GNC systems for tasks such as rendezvous and docking, return and re-entry, and soft landing on the Moon. 4.

Autonomous control

In autonomous control, the motion and environmental characteristics of a spacecraft are completely detected by the instruments and devices on the spacecraft without any ground or human intervention. This control system can make decisions and

16

1 Introduction

perform a series of motion control to achieve predetermined objectives. Both 180-day autonomous control of geostationary satellites and autonomous control of deep-space probes are examples of autonomous control.

Bibliography 1. Zhang R (1998) Attitude dynamics and control of satellite orbit. Beihang University Press, Beijing 2. Hu Q, Yang F (2010) An introduction to astronautics. China Science and Technology Press, Beijing 3. Xu F (2004) Satellite engineering. China Astronautic Publishing House, Beijing 4. Zhou J (2001) Principles of spacecraft control. Northwestern Polytechnical University Press, Xi’an 5. Yang B (2011) Spacecraft guidance, navigation and control. China Science and Technology Press, Beijing 6. Wu H, Tan S (2012) The status and future of spacecraft control. Aerosp Control Appl 38(5):1–7 7. Wu H, Jun Hu, Xie Y (2016) Spacecraft intelligent autonomous control: past, present and future. Aerosp Control Appl 42(1):1–6 8. Huang W, Cao D, Han Z (2012) Research progress and prospect of spacecraft dynamics and control. Adv Mech 42(4):367–394 9. Cui P, Rui Xu, Zhu S et al (2014) State of the art and development trends of on-board autonomy technology for deep space explorer. Acta Aeronaut ET Astronaut Sinica 35(1):13–28

Chapter 2

Spacecraft Orbits and Orbital Dynamics

2.1 Introduction The trajectory of a spacecraft’s center of mass in space is called its orbit, which entails physical features such as position, velocity, acceleration, and time. The orbital characteristics of a spacecraft are mainly affected by different factors including its initial kinematic properties, the mechanical environment of space, active control forces, and the geometric features of motion. Orbital dynamics and the associated control technologies are critical components of spacecraft engineering.

2.2 Time Systems and Reference Frames 2.2.1 Time Systems Spacecraft kinematics focuses on a spacecraft’s state of motion, e.g., spatial position and velocity, at any instant of time. Time systems are established primarily based on the Earth’s rotation. Owing to the nonuniformity of the Earth’s rotation, time systems can become complicated under the impact of precession–nutation and polar motion [1]. Therefore, other time systems based on the Earth’s revolution around the Sun and atomic oscillation have been developed. Space research involves multiple types of time systems, with each having two essential elements: hour and unit time length. 1.

Sidereal time

Sidereal time is a time measurement system determined by the vernal equinox (alternatively, March equinox or spring equinox) and the diurnal motion based on the Earth’s rotation. The interval of time between two successive returns of the vernal equinox to an observer’s meridian is defined as one sidereal day. Clearly, the vernal

© Beijing Institute of Technology Press 2022 Y. Xie et al., Spacecraft Dynamics and Control, Space Science and Technologies, https://doi.org/10.1007/978-981-33-6448-6_2

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2 Spacecraft Orbits and Orbital Dynamics

equinox is unavailable for observation; therefore, the concept of hour angle is introduced to define the hours of sidereal time. The angle between an observer’s meridian on Earth and a fixed star or reference point on the geocentric celestial sphere is called the hour angle of this star or reference point for the current observation point, which is expressed as the positive west of the meridian plane. The hour angle of the vernal equinox with respect to the local meridian of an observation point is the local sidereal time of this point. During a sidereal day, the sidereal time travels from 0 to 24 h, where 0 h is the time when the vernal equinox is at the local meridian. The sidereal time is a basic time system for precisely studying the variation of the Earth’s rotation, which, in this sense, cannot be replaced by any other time system. The true equinox is an equinox moving with precession and nutation. The sidereal time with respect to the true equinox is the apparent sidereal time. An equinox that moves with only precession is defined as a mean equinox. The sidereal time with respect to the mean equinox is the mean sidereal time. 2.

Apparent solar time and mean solar time

In daily life, solar time systems based on the relative position of the Sun are convenient. The true solar time is also based on the Earth’s rotation, and it is measured by the hour angle of the true Sun. The interval between two successive returns of the Sun’s center to the same meridian is one apparent (or true) solar day. An apparent solar day can be divided into 24 apparent solar hours of 60 apparent solar minutes, each of which consists of 60 apparent solar seconds. The apparent solar time starts at the apparent noon, which is numerically equal to the sum of the corresponding hour angle of the apparent solar time and 12 h. If the hour angle is larger than 12 h, then 24 h should be subtracted from the apparent solar time. Owing to the inclination of the Earth’s axis, the interval of time in which the Earth orbits the Sun once varies over time. It is not appropriate to simply define this interval as one solar day. To solve this problem, the concept of mean solar time is introduced. The apparent solar day, as a variable, should not be adopted as the unit of time measurement. Hence, the mean solar time system, which is a uniform system similar to the apparent solar time, has been developed [2]. In order to establish the mean solar time system, we first introduce an auxiliary point that conducts uniform motion along the ecliptic; it has the same speed as the mean velocity of the Sun’s apparent motion and passes through the perihelion and aphelion simultaneously with the Sun. Subsequently, we introduce a second auxiliary point that moves uniformly along the equator at a speed equal to that of the first auxiliary point. These two auxiliary points pass through the vernal equinox simultaneously. The second auxiliary point is named the sun on the mean equator or mean sun for short. The mean sun is an imaginary sun that moves at a fixed angular velocity along its orbit from the perspective of the Earth, with a period equal to that of the actual Sun. The instant of time when the local mean sun is at the upper culmination is called the mean noon, and that when it is at the lower culmination is called the mean midnight. The time interval between two successive mean midnights is defined as a

2.2 Time Systems and Reference Frames

19

mean solar day, with the mean midnight being its starting point. The time defined with respect to the imaginary mean sun, similar to the definition of sidereal time, is called the mean solar time or mean time for short. The period of one rotation of the Earth with respect to the mean sun is a mean solar day. A mean solar day is divided into 24 mean solar hours of 60 mean solar minutes, each of which consists of 60 mean solar seconds. The mean solar hour is also location dependent and numerically equal to the mean solar hour angle plus 12 h. Owing to the Earth’s revolution around the Sun, a mean solar day is slightly longer than a sidereal day. During one period of the Earth’s orbit around the Sun, the Earth rotates 365.2422 times with respect to the Sun, i.e., the period is equal to 365.2422 mean solar days, which corresponds to 366.2422 rotations with respect to a certain distant star. 1 mean solar day = 24 h, 03 min, and 56.5554 s in sidereal time. 1 sidereal day = 23 h, 56 min, and 04.0905 s in mean solar time. The hour-angle difference between the apparent solar and mean solar time is referred to as the equation of time, which can be up to 17 min within a year. 3.

Universal time (UT)

For the convenience of everyday life and work, the Greenwich mean time (GMT) or universal time (UT) has been defined based on the mean solar time. GMT = Greenwich hour angle ± 12 h UT and the sidereal time are not two independent time systems. Since 1956, three versions of UT have been used: (1) UT0, directly measured at an observatory; (2) UT1, with the correction for meridian shifting caused by polar motion; and (3) UT2, a smoothed version with a correction for the Earth’s rotation-speed variation using an empirical formula extrapolation. 4.

Julian day (JD) and modified Julian day (MJD)

The Julian day (JD) is a continuous count of days starting at 12:00 UT (noon) on January 1, 4713 BC. Since the JD number can be quite large and it starts at noon, with a 12 h difference from the commonly used starting time 00:00 (midnight), a modified Julian day (MJD) is defined as follows: MJD = JD − 2400000.5 5.

(2.1)

International atomic time (TAI) and terrestrial dynamical time (TDT)

For an atom of some elements, the electromagnetic signal emitted with the transition of electrons between two known energy levels has a fixed oscillation frequency. A time standard established based on such oscillation frequencies is called an atomic time standard. The SI unit of time, second, was officially defined at the 13th General

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2 Spacecraft Orbits and Orbital Dynamics

Conference on Weights and Measures in 1967 as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of CS133 . The time standard established based on this time unit is called the international atomic time (TAI, from its French name temps atomique international). The dynamical time was adopted by the International Astronomical Union at its XVI General Assembly in 1976 for activities related to theoretical research on celestial mechanics and the compilation of celestial ephemerides starting from 1984. The unit of this time is the SI second. The time scale used for ephemerides from an observer on the surface of the Earth is called terrestrial dynamical time (TDT), which was later changed to terrestrial time (TT) in 1991 [1]. The conversion between TDT and TAI is TDT = TAI + 32S · 184

(2.2)

The difference between TDT and UT 1 is T = TDT − UT 1 = 32S · 184 + TAI − UT 1 6.

(2.3)

Coordinated universal time (UTC)

In the long term, UT increasingly lags behind TAI. To overcome this lack of synchronicity, the coordinated universal time (UTC) was adopted in broadcasting time signals in 1972. Its unit is the SI second, and it inserts leap seconds in the middle or end of the year. Every step adjustment is one SI second. The goal of insertion is to keep UTC within 0.9 s of UT.

2.2.2 Coordinate Systems The first step to describe an orbit is to find an appropriate reference frame, including the position of the origin, the fundamental plane (i.e., X–Y plane), the primary direction (i.e., direction of the X-axis), and the direction of the Z-axis. Different frames result in different forms and complexities of the orbit, which directly affect how intuitive the orbital parameters are and how difficult it is to obtain a solution. For orbits around the Sun, such as those of planets, asteroids, comets, and deep-space probes, the geocentric equation frame is suitable because the definition of this type of frame is closely related to the rotation of the Earth. Hence, this section focuses on coordinate systems that have the Earth’s center as their origins, as well as frames that are fixed on spacecraft bodies. Other types of frames will be introduced in subsequent sections. A full problem of the Earth’s rotation is composed of polar motion, precession, nutation, and variation in the Earth’s rotation speed. The variation in the Earth’s axis of rotation in space is analogous to the precession and nutation of a spinning gyroscope caused by external forces. Owing to the processional motion, the

2.2 Time Systems and Reference Frames

21

vernal equinox slowly moves westward. Consequently, the time when the Sun passes through the vernal equinox every year is earlier than its return to the same spot referred to as a certain distant star. That is, the tropical year is shorter than the sidereal year, which is a phenomenon called general precession. General precession consists of two components, namely, the lunisolar precession caused by the precession of the equatorial mean pole around the ecliptic pole and the nutation caused by the cyclic motion of the equatorial apparent pole relative to its mean pole. The polar motion of the Earth, or simply polar motion, refers to the motion of the Earth’s axis of rotation relative to its crust. Polar motion leads to local variations in latitudes and longitudes on the Earth’s surface. 1.

2.

3.

4.

5.

J2000 geocentric equatorial inertial coordinates OE XI YI ZI This coordinate system is also called the epoch mean equatorial inertial system or geocentric (1st) equatorial system, as denoted by SI . Its origin is at the Earth’s center, its OE XI axis passes through the J2000.0 mean equinox, and its OE ZI axis is normal to the J2000.0 epoch mean equatorial plane, which coincides with the angular velocity vector of the Earth’s rotation. The OE YI axis (on the equatorial plane), OE XI , and OE ZI form a right-handed system. The SI system is commonly used in calculations of spacecraft orbits and solar azimuth angles, as well as in navigation during the reentry phase. Instantaneous mean equatorial system OE XI _MOD YI _MOD ZI _MOD The origin of this system is the Earth’s center, its OE XI _MOD axis passes through the instantaneous mean equinox, and its OE ZI _MOD axis is normal to the instantaneous mean equatorial plane. OE YI _MOD , OE XI _MOD , and OE ZI _MOD satisfy the right-hand rule. Instantaneous apparent equatorial system OE XI _TOD YI _TOD ZI _TOD The origin of this system is the Earth’s center, its OE XI _TOD passes through the instantaneous apparent equinox, and its OE ZI _TOD axis is normal to the instantaneous apparent equatorial plane. OE YI _TOD , OE XI _TOD , and OE ZI _TOD satisfy the right-hand rule. Orbital frame OXo Yo Zo This frame is denoted by So . Its origin O is the center of mass of the spacecraft, its OZo axis passes through the Earth’s center, and its OYo axis is along the negative direction of the orbital angular velocity. OXo (on the orbital plane), OYo , and OZo satisfy the right-hand rule. So is used as the reference frame for the in-orbit attitude determination of spacecraft. Body-fixed frame OXb Yb Zb This frame is denoted by Sb . Its origin O is the center of mass of the spacecraft. OXb , OYb , and OZb are generally along the three principal inertial axes that are perpendicular to each other, forming a right-handed coordinate system. Sb is the installation frame for attitude sensors and actuators (engines) of a control system. During in-orbit operation of the spacecraft, the difference between the body frame Sb and orbital frame So is called the attitude error.

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6.

Launch coordinate system o0 x y z  This system is denoted by SL . Its origin is on the launch pad o0 . The o0 x axis is on the horizontal plane of the launch pad and in the launch direction. The angle between o0 x and the meridian plane of the launch pad is the launch azimuth, A0 , which is expressed as the positive east of the meridian plane. The o0 y axis is along the plumb line of the launch pad, with its positive direction pointing upwards. The angle between o0 y and the equatorial plane, B0 , is the geographic latitude of the launch pad. The o0 z  axis is determined based on the right-hand rule. SL is generally used for initial alignment before launch and emergency rescue during launch. WGS 84 system OE xW yW zW The origin of this system is the Earth’s center of mass. The OE zW axis passes through the Earth’s conventional terrestrial pole (CTP), where CTP is defined by the International Time Bureau (BIH) based on the coordinates of their observatory. The OE xW axis passes through the intersection between the WGS 84 fundamental meridian plane (i.e., the prime meridian defined by BIH based on the coordinates of their observatory) and the equatorial plane determined by CTP. OE yW , OE zW , and OE xW form a right-handed coordinate system.

7.

2.2.3 Transformation Between Coordinate Systems 1.

J2000 geocentric equatorial inertial coordinates OE XI YI ZI and orbital frame OXo Yo Zo A spacecraft position in OE XI YI ZI is expressed in terms of the right ascension of the ascending node , inclination i, and argument u. We have ⎡

⎤ ⎡ 0⎤ xo0 XI ⎣ y0 ⎦ = CoI ⎣ Y 0 ⎦ o I zo0 ZI0

(2.4)

where ⎡

⎤ − sin u cos  − cos u cos i sin  − sin u sin  + cos u cos i cos  cos u sin i ⎢ ⎥ CoI = ⎣ − sin i sin  sin i cos  − cos i ⎦ − cos u cos  + sin u cos i sin  − cos u sin  − sin u cos i cos  − sin u sin i

2.

Body-fixed frame OXb Yb Zb and orbital frame OXo Yo Zo During in-orbit operation of the spacecraft, the attitude error, or simply attitude, can be determined using the transformation between the two frames. The transformation matrix Cbo is also called the attitude matrix.

2.2 Time Systems and Reference Frames

23

⎡

⎤ ⎡ 0⎤ xb0 xo ⎣ y0 ⎦ = Cbo ⎣ y0 ⎦ o b zb0 zo0

(2.5)

An attitude matrix can be expressed in terms of direction cosines, Euler angles, or quaternions. The attitude of a space vehicle can be described more intuitively and conveniently using Euler angles. The commonly used attitude angles are based on the three ordered Euler angles between the body frame and orbital frame: the rotation about OXb yields the roll angle φ, that about OYb yields the pitch angle θ , and that about OZb yields the yaw angle ψ. When the Euler angles are used to describe attitude, the attitude matrix is dependent on the sequence of rotations. For instance, for a z–x–y sequence, the attitude matrix is ⎡ ⎢ Cbo = ⎣

⎤ cos θ cos ψ − sin θ sin ϕ sin ψ cos θ sin ψ + sin θ sin ϕ cos ψ − sin θ cos ϕ ⎥ − cos ϕ sin ψ coϕ cos ψ sin ϕ ⎦ sin θ cos ψ + cos θ sin ϕ sin ψ sin θ sin ψ − cos θ sin ϕ cos ψ cos θ cos ϕ

When these angles are small, the attitude matrix can be approximated as ⎡

Cbo

⎤ 1 ψ −θ = ⎣ −ψ 1 ϕ ⎦ θ −ϕ 1

Now, the attitude matrix no longer depends on the sequence of rotations. For large-angle attitude control, singularity can occur when solving the equations of motion with Euler angles. In this case, quaternions should be adopted. Thus, the attitude matrix is ⎡

Cb0

3.

⎤ q02 + q12 − q22 − q32 2(q1 q2 + q0 q3 ) 2(q1 q3 − q0 q2 ) = ⎣ 2(q1 q2 − q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 + q0 q1 ) ⎦ 2(q1 q3 + q0 q2 ) 2(q2 q3 − q0 q1 ) q02 − q12 − q22 + q32

The quaternions are defined as q = q0 +q1 i+q2 j+q3 k, with q02 +q12 +q22 +q32 = 1. The detailed operations can be found in Chap. 4. J2000 geocentric equatorial inertial coordinates OE XI YI ZI and instantaneous mean equatorial system OE XI _MOD YI _MOD ZI _MOD ⎡

⎤ ⎡ 0⎤ xI0_MOD xI ⎣ y0 ⎦ = CPR ⎣ y0 ⎦ I I _MOD zI0 zI0_MOD

(2.6)

The discrepancy between the two frames is attributed to axial precession. CPR is the precession matrix, which consists of three rotation matrices, i.e.,

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2 Spacecraft Orbits and Orbital Dynamics

1 = Rz (−zA )Ry (θA )Rz (−ζA )

(2.7)

where zA , θA , and ζA are equatorial precession angles. They can be calculated from the following equations: ⎧ ⎨ ζA = 2306 .2181T + 0 .30188T 2 + 0 .017998T 3 Z = 2306 .2181T + 1 .09468T 2 + 0 .018203T 3 ⎩ A θA = 2004 .3109T − 0 .43665T 2 + 0 .041833T 3

4.

where T represents the Julian century from epoch J2000.0 to the instant of observation t. Instantaneous mean equatorial system OE XI _MOD YI _MOD ZI _MOD and instantaneous apparent equatorial system OE XI _TOD YI _TOD ZI _TOD ⎡

⎤ ⎡ 0 ⎤ xI0_TOD xI _MOD ⎣ y0 ⎦ = CNR ⎣ y0 ⎦ I _TOD I _MOD 0 0 zI _TOD zI _MOD

(2.8)

The discrepancy between the two frames is attributed to nutation. CNR is the nutation, which consists of three rotation matrices, i.e., CNR = Rx (−(ε + ε))Rz (−ψ)Rx (ε)

(2.9)

where ε is the mean obliquity calculated from ε = 23◦ 26 21.448 − 46.8150 T − 0.00059 T 2

5.

In Eq. (2.9), ψ and ε are the nutation in longitude and nutation in obliquity, respectively. They can be taken from the IAU1980 series. WGS 84 system OE xW yW zW and instantaneous apparent equatorial system OE XI _TOD YI _TOD ZI _TOD The WGS84 system rotates with the Earth’s rotation, deviating from the instantaneous apparent equatorial system by the Earth’s rotation angle. Let θ0 be the Greenwich sidereal time at instant t0 , and ωe be the speed of the Earth’s rotation. Then, at instant t, we have ⎡

⎤ ⎡ 0 xW cos(θ0 + ωe (t − t0 )) sin(θ0 + ωe (t − t0 )) ⎣ y0 ⎦ = ⎣ − sin(θ0 + ωe (t − t0 )) cos(θ0 + ωe (t − t0 )) W 0 zW 0 0

⎤ ⎤⎡ 0 xI _TOD 0 0 ⎦⎣ yI0_TOD ⎦ 1 zI0_TOD (2.10)

2.3 Two-Body Problem and Three-Body Problem

25

2.3 Two-Body Problem and Three-Body Problem 2.3.1 Overview of Two-Body Problem The general equation of relative motion of two objects that are subject to perturbations from other objects can be simplified into an equation of motion for only two objects. This is the two-body problem. First, we need two assumptions for simplification. 1. 2.

The objects are spherically symmetric such that their masses are concentrated at their spherical centers. There are no external or internal forces other than the gravitational force along the line joining the centers of mass of the two bodies.

Next, we need to determine an inertial frame (a frame without acceleration or rotation) so as to measure the state of motion of an object. Newton described the inertial frame as one fixed in absolute space, which in essence is independent of the outside world and remains at rest indefinitely. However, Newton did not clarify how to find such a frame of absolute rest. Researchers had to presume the existence of such a frame to conduct research on relative motion; in reality, we have no choice but to find an “approximately” inertial frame. Consider a system consisting of two objects that have masses of M and m. Their position vectors in the inertial frame are rM and rm , respectively. Furthermore, let r = rm − rM . Now, in an inertial frame, by using Newton’s law of gravitation, we write GMm r r2 r GMm r M r¨M = r2 r

m¨rm = −

(2.11)

That is, r¨m = −

GM Gm r r¨M = 3 r r3 r

(2.12)

Hence, r¨ = r¨m − r¨M = −

GM Gm r− 3 r 3 r r

(2.13)

Equation (2.3) is the vector differential equation of relative motion of a two-body problem. As we primarily study artificial satellites, manned spacecraft, and space probes orbiting a planet or the Sun in this book, it is fair to state that the mass of a spacecraft, m, is much smaller than that of a celestial body, M. Thus, G(M + m) ≈ GM .

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For the sake of convenience and generality, we call M the primary body. Further, we define a gravitational constant μ ≡ GM. Then, Eq. (2.13) becomes r¨ +

μ r=0 r3

(2.14)

which is the two-body orbital equation of motion. μ varies for different primary bodies. For the Earth, μ = 3.986005 × 105 km/s; for the Moon, μ = 4.902794 × 103 km/s; and for the Sun, μ = 1.32712438 × 1011 km/s.

2.3.2 Constants of Two-Body Orbital Motion In fact, Eq. (2.14) alone provides great insight into the motion of a spacecraft orbiting the Earth or other primary bodies, determining the spacecraft orbit and some fundamental properties of in-orbit motion. For example, the conservation of mechanical energy and angular momentum are two important laws that can be derived from this equation. 1.

Conservation of mechanical energy First, the dot products of r˙ and the two sides of Eq. (2.14) are calculated. r˙ · r¨ + r˙ ·

μ μ r = v · v˙ + 3 r˙ · r = 0 r3 r

(2.15)

which can be written as 1 d 2 μ (v ) + 3 r˙ r = 0 2 dt r

(2.16)

Equation (2.16) can be rewritten as  μ d v2 − =0 dt 2 r

(2.17)

 μ d v2 − +c =0 dt 2 r

(2.18)

Thus,

where c is any constant. Hence, the variable defined in the equation below will be constant as well. E=

μ v2 + c− 2 r

(2.19)

2.3 Two-Body Problem and Three-Body Problem

27

E is called the specific mechanical energy. Clearly, the first term on the righthand side (RHS) of the above equation is the kinetic energy per unit mass of the spacecraft, and the second term is the potential energy per unit mass, which is equal to the work done against the gravitational force from one point in space to another. The value of the constant c depends on the selection of the reference point of zero potential energy. If the surface of the primary body, e.g., the Earth, is set as the zero point, then c = μr , where r is the Earth’s radius. If c = 0, which implies that a point infinitely far away is used as the zero point, then the potential energy of the spacecraft will always be negative. Thus, it can be concluded that when a satellite moves along an orbit, its specific mechanical energy E remains constant. E is expressed as E= 2.

v2 μ  + − 2 r

(2.20)

Conservation of angular momentum By taking the cross products of r and the two sides of Eq. (2.14), we obtain r × r¨ + r ×

μ r=0 r3

(2.21)

Because r × r = 0, the second term on the left-hand side (LHS) of the above equation is 0, yielding r × r¨ = 0. Thus, d d (r × r˙) = 0, (r × v) = 0 dt dt

(2.22)

which shows that the vector r × v is a constant of motion, defined as the specific angular momentum and denoted by h. Therefore, we have proven that the orbital angular momentum of a spacecraft is a constant written as h=r×v

(2.23)

Clearly, h is perpendicular to the plane formed by r and v. This indicates that the motion of a spacecraft is confined to a fixed plane in space, i.e., the orbital plane. By examining the vectors r and v that lie in the orbital plane, as well as their relationships with the primary body, we find that at any location in space, the local plumb line is along the vector r. Hence, the local horizontal plane must be perpendicular to r. We define the angle between the velocity vector v and local plumb line as the zenith angle γ, and we define the angle between v and the local horizontal plane as the flight path elevation angle, or simply flight path angle. Based on the definition of the cross product, the magnitude of h is calculated as h = |h| = |r × v| = |r||v| sin γ

(2.24)

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2 Spacecraft Orbits and Orbital Dynamics

Fig. 2.1 Relationship between the local plumb line and vector r

It is more convenient to express h in terms of . Because γ and are complementary, this equation can be written as (Fig. 2.1). h = rv cos

(2.25)

2.3.3 Geometric Equation of Two-Body Orbits In the previous section, we presented the equation of motion, Eq. (2.14), for a spacecraft orbiting a primary body in a two-body problem. Although this equation has a simple form, it fully describes the shape and size of an orbit. By taking the cross products of h and the two sides of Eq. (2.14), we obtain h × r¨ = −

u h×r r3

(2.26)

By using the conservation of angular momentum and the property (a × b) × c = b(a · c) − a(b · c) for cross products of vectors, which can be written as d (h × r˙) = h × r¨ + h˙ × r˙ = h × r¨ dt μ μ dr d r μ μ v − r = μ h × r = (r × v) × r = r3 r3 r r 2 dt dt r

(2.27)

Thus, d r d (˙r × h) = μ dt dt r The integration of both sides yields

(2.28)

2.3 Two-Body Problem and Three-Body Problem

r r˙ × h = μ + B r

29

(2.29)

where B is a constant vector. Taking the dot products of r and both sides of the above equation r r · (˙r × h) = r · μ + r · B r

(2.30)

Further, we obtain the scalar equation h2 = μr + rB cos f

(2.31)

where f is the angle between the constant vector B and position vector r. The geometric equation of the orbit is obtained by solving for r: r=

h2 /μ 1 + (Bμ) cos f

(2.32)

Let p = h2 /μ and e = B/μ. Then, the above equation becomes r=

p 1 + e cos f

(2.33)

It is not difficult to conclude that the geometric equation of the orbit represents a conic section in a polar coordinate system. The center of mass of the primary body is the origin of the polar coordinates, located at a focus of the conic. The true anomaly f is the angle between r and the line joining the focus and its nearest point on the conic. The constant p is called the semi-latus rectum. The constant e is called eccentricity and determines the type of conic represented by Eq. (2.33), as shown in Fig. 2.2. This not only proves Kepler’s first law, but also extends it to the motion of any conic orbit (not just ellipses). Thus, the following points can be summarized 1. 2.

A conic section (e.g., circle, ellipse, parabola, and hyperbola) is the only possible shape of a spacecraft orbit in the two-body problem. The center of the primary body must be a focus of the conic orbit.

Fig. 2.2 General equation of a conic section in polar coordinates

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2 Spacecraft Orbits and Orbital Dynamics

3.

When a spacecraft moves along a conic orbit, its specific mechanical energy remains constant. However, conversion between kinetic and potential energies is possible. This suggests that when the altitude of the spacecraft increases (i.e., when r increases), its velocity certainly decreases; conversely, a decrease in r leads to a higher velocity. Consequently, E is always a constant. For a spacecraft orbiting the primary body, when its r and v change along the orbit, its specific angular momentum h remains constant. Orbital motion is always confined to a plane fixed in inertial space.

4. 5.

2.3.4 Geometric Properties of Two-Body Orbits 1.

Geometric parameters of conic orbits Conic orbits can be classified into four types: circular, elliptical, parabolic, and hyperbolic. Figure 2.3 shows some common geometric parameters and the relations among these different types of orbits.

Fig. 2.3 Common geometric parameters of conic sections

2.3 Two-Body Problem and Three-Body Problem

31

All conic sections have two foci, F and F’. The primary focus F represents the location of the primary body, whereas the secondary focus (or virtual focus) F’ has little physical meaning in orbital mechanics. The distance between the two foci is 2c. For a circle, the two foci coincide; therefore, 2c is zero. For a parabola, the virtual focus can be thought to be at infinity; therefore, 2c is infinitely large. For a hyperbola, 2c is negative. The chord passing through the foci is the major axis of the conic section, which is denoted by 2a, where a is called the semi-major axis or major radius. For a circle, 2a is exactly the diameter. For a parabola, 2a is infinitely large. For a hyperbola, 2a is negative. The width of the conic at a focus is a positive quantity called the latus rectum, which is denoted by 2p in Fig. 2.3. Except for parabolas, all the other conic sections have the following eccentricity: e=

c a

p = a(1 − e2 ) 2.

(2.34) (2.35)

Periapsis and apoapsis of the orbit The two endpoints of the major axis of the orbit are called apsides: the one near the primary focus is the periapsis, while the one far from the primary focus is the apoapsis. In orbital research, depending on the primary body, these apsides can be called perigee/apogee for the Earth, perihelion/aphelion for the Sun, and perilune/apolune for the Moon. It should be noted that these points are not fixed for circular orbits. Further, the apoapsis is not physically meaningful for open curves (parabolas and hyperbolas). To calculate the distance from the primary focus to the periapsis or apoapsis (if they exist), we only need to substitute f = 0◦ or f = 180◦ in the general conic-section equation in polar coordinates. Hence, for any conic section, rp =

p = a(1 − e) 1+e

(2.36)

ra =

p = a(1 + e) 1−e

(2.37)

Furthermore, at the periapsis or apoapsis (if they exist) of any conic orbit, the position vector and velocity vector are always perpendicular to each other. Therefore, as a special case of the above equation, h = rp vp = ra va

3.

(2.38)

where vp and va are the velocities at the two apsides, respectively. Shape of the orbit From Eqs. (2.20), (2.36), (2.37), and (2.38), as f = 0◦ or f = 180◦ , we get E = −μ/2a, and the energy E remains constant at any position in orbit. The

32

2 Spacecraft Orbits and Orbital Dynamics

simple relation that is valid for all conic orbits indicates that the semi-major axis of the orbit is only dependent on the specific mechanical energy of the spacecraft E. Circular and elliptical orbits: a > 0, E < 0. Parabolic orbits: a = ∞, E = 0. Hyperbolic orbits: a < 0, E > 0. Therefore, the sign of E alone is sufficient to determine the type of conic orbit a spacecraft is in. Further, because p = h2 /μ, for any conic orbit,  e=

1+

2Eh2 μ2

(2.39)

Considering that p is determined solely by h and a solely by E, we can state that h and E collectively specify e, i.e., the specific shape of the conic orbit. Moreover, because h = rvcos , r > 0, and v > 0 for a regular spacecraft, it is concluded that the value of the flight path angle (0 ≤ ≤ 180°) determines the sign of h. When = 90°, i.e., h = 0, the following are applicable. If E < 0, then e < 1; the orbit is elliptical or circular. If E = 0, then e = 1; the orbit is parabolic. If E > 0, then e > 1; the orbit is hyperbolic. When = 90°, i.e., h = 0, e is always unity irrespective of the value of E. In this case, the spacecraft orbit is a straight line joining the primary body’s center of mass and the current position of the spacecraft, which is also a degenerate conic. Because this is considered a singularity that rarely occurs in reality, we will not go into the details of this situation.

2.3.5 Circular Restricted Three-Body Problem As discussed above, the two-body problem concerns the motion of two interacting point masses in an inertial frame of reference. In contrast, the three-body problem studies the motion of three gravitational bodies. Among the three bodies M 1 , M 2 , and m, m is infinitely small, and its gravitational effects on M 1 and M 2 are negligible; only the integrations between M 1 and M 2 and their gravitational effects on m are considered. Therefore, this type of problem of three-body motion is referred to as a restricted three-body problem (RTBP). For a libration-point mission of the Sun–Earth system, the Sun, Earth, Moon, and satellite constitute a restricted four-body problem (RFBP). It is very difficult to handle an RFBP directly. Hence, we generally simplify it into an RTBP. There are two main approaches for such simplification. Because a satellite at the libration point in a transfer orbit has to avoid the sphere of influence of the Moon, the influence

2.3 Two-Body Problem and Three-Body Problem

33

of the Moon on the satellite can be assumed to be small treated as a perturbation. This is how the first approach converts the RFBP into an RTBP consisting of the Sun, Earth, and satellite. In the second approach, the Earth and Moon are treated as a single point mass at their joint center of mass, provided the satellite is far away from the Earth–Moon barycenter. In this manner, we have another RTBP consisting of the Sun, Earth + Moon, and satellite. In an RTBP, M 1 and M 2 are finite-mass bodies, whereas m is the infinitely small body. Assuming M 1 ≥ M 2 , M 1 and M 2 are called primary and secondary massive bodies, respectively. Because the gravitational effects of m on M 1 and M 2 are neglected, M 1 and M 2 only have the gravitational forces acting on each other. Thus, the motion of M 1 and M 2 can be described by a two-body problem, where their positions and velocities are known functions of time. In addition, the gravitational forces of M 1 and M 2 exerted on m are also known. Therefore, the RTBP essentially studies the motion of m under the gravitational attraction from M 1 and M 2 while M 1 and M 2 exhibit unperturbed motion in the conic orbit. Depending on the motion curves of M 1 and M 2 , there are hyperbolic, parabolic, and elliptical (circular) RTBPs. The circular RTBP, or CRTBP, was first proposed by Euler in 1772 for studying the Moon’s motion around the Earth under perturbation from the Sun in the Sun–Earth–Moon system. Assuming that the Sun and Earth move in a circular orbit around their joint center of mass, the system consisting of the Sun, Earth, and satellite is a CRTBP. To describe the orbital motion of the satellite, we adopt a barycentric rotating frame C-XYZ, as shown in Fig. 2.4, where M 1 and M 2 represent the Sun and Earth, respectively. For the convenience of analysis, the equations of motion are usually written in the dimensionless form. The mass unit [M], distance unit [L], and time unit [T ] are calculated as follows:

Fig. 2.4 Barycentric inertial coordinates C-X ei Y ei Z ei and barycentric rotating coordinates C-XYZ

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2 Spacecraft Orbits and Orbital Dynamics

⎧ ⎨ [M ] = M1 + M2 [L] = M 1 M2 ⎩ [T ] = [L]3 /G[M ] = 1/n

(2.40)

Thus, within this barycentric rotating frame, the dimensionless equations of motion of the satellite are ⎧ (1−ρ)(X +ρ) ¨ ˙ − ⎪ r13 ⎨ X − 2Y − X = − (1−ρ)Y ρY ¨ ˙ Y + 2X − Y = r 3 − r 3 1 2 ⎪ ⎩¨ ρZ − Z = (1−ρ)Z 3 3 r r 1

ρ(X −1+ρ) r23

(2.41)

2

 where ρ = M /(M + M = r ), (X + ρ)2 + Y 2 + Z 2 , and r2 = 2 1 2 1  2 2 2 (X − 1 + ρ) + Y + Z . We can observe that the equations of motion of the satellite in a CRTBP are a system of nonlinear differential equations. The above equations of motion can also be expressed using a pseudopotential function, U = U (X , Y , Z), as follows: ⎧ ∂U ⎨ X¨ − 2Y˙ = ∂X = UX = UY Y¨ + 2X˙ = ∂U ∂Y ⎩¨ = U Z = ∂U Z ∂Z

(2.42)

where U is defined as U =

1 2 1−ρ ρ (X + Y 2 ) + + 2 r1 r2

(2.43)

Thus, Jacobi’s integral is written as C = 2U − (X˙ 2 + Y˙ 2 + Z˙ 2 )

(2.44)

Thus far, this is the only integral we have found in CRTBP. The zero-velocity surface is the surface on which the velocity of the infinitely small body equals zero. 2U (X , Y , Z) = C

(2.45)

The Jacobi constant C is determined by initial conditions, i.e., C = 2U (X0 , Y0 , Z0 ) − (X˙ 02 + Y˙ 02 + Z˙ 02 )

(2.46)

The geometric structure of the zero-velocity surface varies with the Jacobi constant C:

2.3 Two-Body Problem and Three-Body Problem

C = (X 2 + Y 2 ) +

35

2(1 − ρ) 2ρ + r1 r2

(2.47)

2.3.6 Libration Points There are five dynamic equilibrium points in Eq. (2.41), which are known as libration points (L-points) or Lagrangian points. All of them are in the XY plane, as shown in Fig. 2.5. The three colinear points L1 , L2 , and L3 are unstable; a small perturbation can cause the objects in the vicinity of these points to leave and move further away. L4 and L5 are stable; small motions near these points remain in the vicinity. L1 and L2 lie on the line connecting the Sun and Earth, approximately 1,500,000 km away from Earth. L3 is at the opposite side of the Sun and one Earth–Sun distance from the Sun. The triangular libration points L4 and L5 lie at the third vertices of the two equilateral triangles that share a common base on the Sun–Earth line. The libration points are quite useful. For example, satellites orbiting the L1 point in the Sun–Earth system can conduct continuous observation of the Sun, providing early prediction of the solar cycle; continuous observation of the sunlit side of Earth is also possible. The L2 point in the Sun–Earth system can avoid the influence from the Sun and Earth, which is important for studying the cosmic-ray background and the existence of terrestrial planets or for the continuous observation of the dark side of the Earth. Another good example is the L2 point in the Earth–Moon system, the satellites near which can continuously observe the far side of the Moon or provide continuous communication with Earth. This is important in solving the communication problem on the far side of the Moon, as shown in Fig. 2.6. L4

Fig. 2.5 Schematic of the libration points in the Sun–Earth system

Earth’s orbit

L1

L3

L2

Moon’s orbit

L5

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2 Spacecraft Orbits and Orbital Dynamics

Fig. 2.6 Communication link with the far side of the Moon

2.4 Orbital Properties of Spacecraft 2.4.1 Orbital Parameters and Transformations 1.

Definitions of orbital parameters The shape and position of a spacecraft’s orbit can be described using six quantities, which are known as orbital elements or Keplerian elements. These quantities are independent and normally have unambiguous physical meanings. We now examine them in an elliptical orbit around earth. Orbital elements are important parameters used to describe and determine the orbital properties of a spacecraft, and they are defined as follows. (1)

(2)

Inclination i. The plane on which the spacecraft’s orbit lies is called the orbital plane, which passes through the Earth’s center. The angle between the orbital plane and the equatorial plane of Earth is defined as inclination. Right ascension of the ascending node . It is the longitude of the ascending node measured from the reference direction of the vernal equinox, expressed as positive in the direction of the Earth’s rotation (0 ≤  ≤ 2π). The intersection of the orbital plane and the equatorial plane yields two points on the celestial body. The point through which the spacecraft passes during its ascension from the Southern Hemisphere

2.4 Orbital Properties of Spacecraft

(3)

(4) (5) (6)

2.

37

to the Northern Hemisphere is called the ascending node, whereas the point through which the spacecraft passes during its descension from the Northern Hemisphere to the Southern Hemisphere is called the descending node. Argument of perigee ω. It is the angle made by the direction of spacecraft motion, from the ascending node to the perigee on the celestial sphere’s projected elliptical orbit, with respect to the Earth’s center. Semi-major axis √ of the orbit. Eccentricity e = a2 − b2 /a, where b is the semi-minor axis of elliptical orbit. Time of perigee passage t p . Of course, there are orbital parameters other than these six quantities. However, these independent elements are sufficient to determine the orbital properties; all the other elements can be calculated from the six elements introduced above.

Practical meanings of the orbital parameters (1)

Determination of the orientation of the spacecraft’s orbital plane: specified by the inclination i and right ascension of the ascending node . When i = 0°, the orbit is called an equatorial orbit. When i = 90°, the orbit is called a polar orbit. When 0° < i < 90°, the spacecraft’s motion is in the same direction as the Earth’s rotation; therefore, such orbits are called prograde orbits. When 90° < i < 180°, the spacecraft’s motion is in the direction opposite to the Earth’s rotation; therefore, such orbits are called retrograde orbits. When i = 180°, the spacecraft is an equatorial spacecraft that orbits in the direction opposite to the Earth’s rotation. Figure 2.7 schematically shows these orbits. To send spacecraft-like satellites into a prograde orbit, the launch vehicle should be launched in the eastern direction. A great benefit associated with this type of orbits is that the launch vehicle can take advantage of the

Fig. 2.7 Different types of orbits based on the direction of motion: a prograde orbit, b polar orbit, c retrograde orbit, and d equatorial orbit

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2 Spacecraft Orbits and Orbital Dynamics

(2) (3)

(4)

Earth’s rotation from west to east, reducing fuel consumption. Clearly, the Earth’s rotational speed is fully used if the rocket is launched due east on the equator; on the other hand, if the inclination is 90°, the Earth’s rotational speed cannot be taken advantage of. All satellites launched in the early days of space exploration worldwide adopted prograde orbits. Even the satellites launched later tended to use prograde orbits, as long as they had no special missions. Therefore, we can observe from the ground that most satellites moving in the sky orbit in the southwest-northeast or northwest-southeast directions. The orbital plane of a polar orbit passes through the North Pole and South Pole of the Earth. The advantage of this type of orbit is that the spacecraft can fly over any part of the Earth’s surface. To send a spacecraft into a retrograde orbit, the launch vehicle should be launched in the western direction. As opposed to a prograde orbit discussed above, a retrograde orbit cannot take advantage of the Earth’s rotational speed to save fuel; in fact, the launch vehicle has to consume additional fuel to overcome the influence of the Earth’s rotation. Hence, this type of orbit is generally not a good choice. However, satellites in a Sun-synchronous orbit (SSO), which will be introduced later, must use retrograde orbits. If only the inclination is known, the orientation of the orbital plane cannot be determined without knowing the right ascension of the ascending node, because numerous orbital planes can exist with the same inclination. Similarly, with the right ascension of the ascending node known but the inclination unknown, the orbital plane cannot be determined, because numerous planes exist with the same right ascension of the ascending node. Therefore, an orbital plane can be determined only when both parameters are given. Determination of the direction of major axis in the orbital plane: specified by the argument of perigee. Determination of the shape and size of the orbit: specified by the semimajor axis a and eccentricity e. Only five independent orbital elements are necessary to determine the size, shape, and orientation of an orbit. However, to accurately specify the spacecraft’s position in the orbit at a particular time, we need the sixth element. Determination of the orbital position of the spacecraft: time and space (position of the spacecraft in the orbit) are linked by the time of perigee passage t p . When the above-mentioned orbital parameters are known, we can determine the position and velocity of the spacecraft at any instant of time t to realize the observation and prediction of the spacecraft’s motion in the inertial frame.

2.4 Orbital Properties of Spacecraft

3.

39

Transformation of orbital parameters The orbital elements (a, e, i, , ω, M ) can be used to calculate the position and velocity vectors (r, r˙). This is known as the calculation of ephemerides. The principle and procedure of calculation are the same for both invariant and instantaneous ellipses, as shown below. First, with e and M, E, sin E, and cos E are obtained using Kepler’s equation, which can be solved for using iterative algorithms such as Newton’s method: E − e sin E = M Next, the two unit vectors (P, Q) are calculated: ⎡

⎤ cos  cos ω − sin  sin ω cos i P = ⎣ sin  cos ω + cos  sin ω cos i ⎦ sin ω sin i ⎤ ⎡ − cos  sin ω − sin  cos ω cos i Q = ⎣ − sin  sin ω + cos  cos ω cos i ⎦ cos ω sin i

(2.48)

(2.49)

In fact, Q = P(ω → ω + 90◦ ). To calculate P, we only need to substitute ω in the equation for Q with ω + 90◦ . Then, the position and velocity vectors (r, r˙) can be obtained as follows:  r = r(σ ) = a(cos E − e)P + a 1 − e2 sin EQ √ 

   μa (− sin E)P + (2.50) r˙ = r˙(σ ) = 1 − e2 cos E Q r

2.4.2 Satellite Ground Track In many practical problems, it is desirable to study a spacecraft’s motion relative to the Earth’s surface. For instance, to design the orbit of a reconnaissance satellite or observe a satellite from ground stations, it is necessary to know information at a certain instant of time, such as the location of the satellite and its height from a point on the Earth’s surface. The line joining the in-orbit satellite (S) and the Earth’s center (i.e., the radius vector) has a point of intersection (S ) on the Earth’s surface, which is the projection of the satellite on the Earth’s surface, called the subsatellite point. Following the motion of the satellite, the subsatellite points also form a trajectory on the Earth’s surface, which is known as the ground track. It reflects the satellite’s motion relative to the Earth’s surface. If the Earth’s rotation is neglected, the ground track can be

40

2 Spacecraft Orbits and Orbital Dynamics

Fig. 2.8 Satellite ground track

Satellite orbit

Equator

viewed as a large circle generated by the intersection of the orbital plane and the Earth’s surface, as shown in Fig. 2.8. Because the orbital plane has a fixed orientation, it is not affected by the Earth’s rotation. Therefore, the collective effects of the Earth’s rotation and the orbital plane’s fixed-orientation property lead to an expansion of the area on the Earth’s surface swept by the ground track. After one orbital period of the satellite, the subsatellite points move westward to different longitudes owing to the Earth’s rotation. A satellite with an orbital period of 120 min passes over a point every 24 h. The ground track is a collection of subsatellite points on the surface of the Earth, resulting from the combined effects of the satellite’s orbital motion and the Earth’s rotation. A subsatellite point represents the geocentric latitude and longitude of the intersection point of the satellite’s radius vector and the Earth’s surface. The satellite’s orbit is defined in the equatorial inertial frame, in which the right ascension α and declination δ can be obtained based on the position coordinates of the satellite (x, y, z): α = arctan

y

(2.51)

x



z δ = arcsin 2 2 (x + y + z 2 )1/2

 (2.52)

Alternatively, they can be calculated from orbital elements. Based on spherical trigonometry, we have α =  + arctan(tan u cos i) δ = arcsin(sin u sin i)

(2.53)

where u = ω + f is the argument of the ascending node and the true anomaly f is obtained by solving Kepler’s equation. The geocentric longitude of the satellite λ is equal to the difference between the satellite’s right ascension and the Greenwich hour angle, i.e., λ = α − [G0 + ωe (t − t0 )]

(2.54)

2.4 Orbital Properties of Spacecraft

41

where G0 is the Greenwich hour angle at the initial instant of time. If we adopt the Earth ellipsoid model (i.e., the cross section of the Earth along its meridian is an ellipse), the semi-major axis of the ellipse a should be the equatorial radius, whereas its semi-minor axis should be the polar radius of the Earth. The oblateness ε and eccentricity e of the ellipse are defined as ε= e2 =

a−b a

(2.55)

a2 − b2 a2

(2.56)

The basic constants are a = 6,378.145 km, b = a (a–ε) = 6,356.76 km, ε = 1/298.257, and e = 0.08182. The satellite’s geocentric latitude ϕ is equal to the declination δ, which has the following relationship with the geographic latitude ϕg : tan ϕ = (1 − f )2 tan ϕg

(2.57)

2.4.3 Launch Window Normally, carrier rockets do not implement lateral maneuver during launch. That is, the launch ballistic plane is not altered. When sending spacecraft to geostationary orbits, the carrier rockets are occasionally required to change the launch ballistic plane to reduce the inclination of the transfer orbit for reducing fuel consumption or extending the longevity of the spacecraft. This book mainly addresses the situation in which the carrier rockets do not need to change the inclination. Factors determining the orientation of the orbital plane of the launched spacecraft in space include the geocentric latitude ϕ and geocentric longitude λ of the launch site S, launch azimuth A, and launch time t. 1.

Determination of the launch azimuth The orientation of the orbital plane of a spacecraft is determined by the inclination i and the right ascension of the ascending node . The spacecraft can be launched either in an ascending manner with the launch azimuth Aa or in a descending manner with the launch azimuth Ad . Both result in the same orbital plane, i.e., the same inclination. The relationship between Aa and Ad is Ad = 180◦ − Aa

(2.58)

Based on spherical trigonometry, the relationship between the inclination and launch azimuth is

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2 Spacecraft Orbits and Orbital Dynamics

cos i = sin Aa cos ϕ

(2.59)

From Eqs. (2.4)–(2.11)–(2.4)–(2.12), the accuracy of Aa is directly related to i. Hence, the alignment of the rocket before launch is critical. When Aa = 90◦ , i equals the geocentric latitude of the launch site. When Aa = 90◦ , i is greater than the geocentric latitude of the launch site. If i = 90° is desired, Aa should be 0° or 180°. This can be achieved by either an ascending launch due north or a descending launch due south, but the latter is usually adopted in practice. 2.

Determination of the launch time In the above discussion, the launch azimuth is determined based on the requirement of orbital inclination, but neither the right ascension of the ascending node nor the launch time is determined. Let us examine these factors here. At the time of launch, the increment of the right ascension of the launch site meridian relative to the right ascension of the ascending node of the orbit, D , is given by  D = arcsin

tan ϕ tan i

(2.60)

To achieve a given right ascension of the ascending node , neglecting the Earth’s rotation for now, the sidereal hour angle of the site at launch time, αL , is obtained using αL =  + D

(2.61)

where αL is fixed in the inertial frame. Considering that the Earth rotates with an angular velocity of ωe , αL should be equal to the sum of the Greenwich hour angle at midnight αG , the geocentric longitude of the launch site from the prime meridian λ, and the UT hour angle at launch time ωe tG . In the analysis of launch time, the hour angle needs to be converted to a regular unit of time, e.g., h. We also need to consider the interval between the launch of the rocket and injection of the spacecraft into the orbit, which is denoted by tA . That is, the actual launch time should be earlier by tA . Taking ascending launch as an example, to achieve the target right ascension of the ascending node, the UT of the launch time, tG , is calculated using the following equation, which must include a term for tA :    tan ϕ 1 1  + arcsin − αG − λ − tA tG = 15 tan i 60

(2.62)

The unit in Eq. (2.62) has been converted to h. The unit of the term in square brackets is degrees, and 1 h corresponds to 15°. The unit of the interval of satellite injection, tA , is min; therefore, it is divided by 60 for conversion to h.

2.4 Orbital Properties of Spacecraft

43

Because the geocentric longitude of the launch site is λ from the prime meridian, the local time of the launch site at launch is ts1 =

λ + tG 15

(2.63)

2.4.4 Geosynchronous Orbits A geosynchronous orbit (GSO) is one along which a spacecraft orbiting the Earth has the same period as the Earth’s rotation, i.e., one sidereal day (23 h, 56 min, and 4.1 s). Satellites in this type of orbit are called GSO satellites or 24 h synchronous satellites. A special case of GSO, the geostationary orbit, is the most commonly adopted orbit for communication and Earth-observation satellites, with a rotation period of approximately 24 h. In the case of a circular orbit, the radius of the orbit r ≈ 6.63Re, where Re is the Earth’s radius. For a geostationary orbit with a period identical to that of the Earth’s rotation, an inclination of 0°, and an eccentricity of 0, the ground track of the satellite is a fixed point. Thus, it has a wide application in areas such as Earth observation and communication. However, with various perturbating effects, the orbital parameters of the satellite continuously change, making it difficult to maintain an ideal geostationary orbit. Hence, geostationary orbits, in reality, are all quasi-stationary orbits with small eccentricities and inclinations. When the regular orbital elements are used for perturbation analysis, a singularity will be encountered when finding the analytical solution. Therefore, we often employ the six elements of synchronous orbits to describe the orbital characteristics and the orbit’s relationship with the Earth. ⎧ ex = e sin( + ω) ⎪ ⎪ ⎪ ⎪ ⎪ ey = e cos( + ω) ⎪ ⎪ ⎨ i = sin i cos  x i ⎪ y = sin i sin  ⎪ ⎪ ⎪ s) ⎪ D = − 3(a−a ⎪ 2as ⎪ ⎩ L=+ω+M

(2.64)

where as is the synchronous radius, D the orbital drift rate, and L the mean right ascension of the satellite. Elliptical GSO satellites have various forms of ground tracks. Particularly, with certain inclinations, a satellite can remain stationary relative to high-latitude regions for a fairly long period of time. If multiple satellites are configured in the same orbit, they will have the same orbital parameters, except for the time of perigee passage t p and the right ascension of the ascending node . They pass over a point at constant intervals (a fraction of the period T ), realizing continuous Earth observation and communication for high-latitude regions in a relay fashion.

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2 Spacecraft Orbits and Orbital Dynamics

2.4.5 Sun-Synchronous Orbits A Sun-synchronous orbit (SSO) is one in which the orbital plane of a spacecraft rotates with the same angular velocity Ω as that of the Earth’s orbit around the Sun, i.e., it has the same rotation orientation and period. Satellites in this type of orbit are called SSO satellites. The perturbing effects of the Earth’s non-spherical shape lead to a long-term change in the spacecraft’s right ascension of the ascending node with the following rate of change: Ω˙ = −

3nJ2 R2e cos i 2a2 (1 − e2 )2

(2.65)

For a nearly circular orbit, the average daily change in the right ascension of the ascending node is  3.5 Re cos i Ω = −9.97 a

(2.66)

Fig. 2.9 Relationship between the altitude and inclination of Sun-synchronous orbits

Inclination/(º )

By setting an appropriate semi-major axis a and inclination i, the rate of change of the right ascension of the ascending node can be made equal to 0.9856°/d such that the precession orientation and rate of orbit are identical to those of the Earth’s orbit around the Sun. That is, a full orbit of the Earth is completed in 365.24 mean solar days. Orbits designed in this manner are SSOs. Figure 2.9 shows the relationship between the altitude and inclination of nearly circular SSOs, where the maximum altitude is determined as 5,976 km. The time required for the Earth to orbit the Sun once is a sidereal year, with a movement of 0.9856° every day on average. On the other hand, oblateness perturbation leads to the precession of the orbital plane as well. For retrograde orbits in which the rotation of the orbital plane is in the same direction as the Earth’s orbit around the Sun, if orbital parameters are appropriately selected, the orbital plane of

Altitude of orbit /(km)

2.4 Orbital Properties of Spacecraft

45

Fig. 2.10 Motion of the orbital plane of a Sun-synchronous orbit

the spacecraft can be ensured to rotate once in a sideral year. Thus, when the Earth orbits the Sun, the angle between the orbital plane and the Sun–Earth line, i.e., the solar illumination angle, is kept constant, as can be seen from Fig. 2.10. SSO provides a constant solar azimuth angle. Consequently, the satellite passes certain latitude at the same local time during its ascending or descending phase. Hence, with the same amounts of time and illumination, the satellite can observe a region at certain latitudes multiple times, facilitating the processing and interpretation of observation data. By carefully choosing the launch time, certain regions will always be under good lighting conditions when the satellite flies over. Hence, the local times of the ascending and descending nodes are critical SSO parameters. This is another reason why an SSO is occasionally called “SSO of XX time.” For instance, as the orbital precession is synchronous with the motion of the Sun–Earth line, the shadow time of SSO exhibits tiny changes. The shadow time is the shortest when the local time of the ascending node is 6:00 AM/PM. SSOs have two general features: (1) (2)

They are all retrograde orbits with an inclination i higher than 90°. When e = 0, the altitude of the orbit is less than 6,000 km.

In practice, Earth remote-sensing satellites often adopt SSOs with an altitude usually less than 1,000 km. The inclinations i corresponding to different altitudes are shown in Fig. 2.9. Clearly, this type of low-to-medium SSOs all has an inclination of approximately 90°. All satellites passing through the Earth’s poles can be referred to as polar-orbit satellites. Of course, a true polar orbit satellite should have an inclination of exactly 90°. In this case, the oblateness of the Earth no longer affects the inclination or right ascension of the ascending node. In other words, the orbital plane is stable in the inertial plane; changes in the orbital plane are consequences of third-body gravitational perturbations.

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2 Spacecraft Orbits and Orbital Dynamics

An important feature of SSO is that the local times are fixed when the satellite passes through the ascending and descending nodes. Hence, SSO satellites are particularly suitable for low-altitude remote sensing, ensuring relatively consistent Earth observation times and simplified control on the solar panel.

2.4.6 Critical Inclination Orbits and Frozen Orbits The Earth has a bulge around the equator, where more mass is distributed than around the poles. Consequently, when a satellite passes the equatorial plane, the gravity acting on the satellite is greater than that in other sections of the orbit. Hence, when the non-spherical gravity perturbation is taken into consideration, there are apsidal and shape changes associated with the satellite’s orbit. Only considering the gravity perturbations due to the J2 and J3 terms, the rates of change of eccentricity and argument of perigee are  2  J3 Rg sin i − e cos2 i sin ω (5 sin i − 4) 1 + ω˙ = − 2 4a (1 − e2 )2 2aJ2 (1 − e2 ) sin i e 3 3nJ3 Rg sin i (5 sin2 i − 4) cos ω (2.67) e˙ = − 3 3a (1 − e2 )2 3nJ2 R2g



2

When the inclination is 63.43° or 116.57°, the argument of the periapsis and the eccentricity are unaffected by the perturbing effects of the non-spherical shape of the Earth. This type of orbit is called a critical inclination orbit. i ˙ = e˙ = 0. In this case, ω is constant at If ω = 90° or 270°, e = cos2 sin 2J2 g ; then, ω i sin i

∓J

3 Rg

90° or 270°, and e is also a constant. However, the semi-major axis and inclination can be freely chosen. This type of orbit is called a frozen orbit. Frozen orbits are often adopted by observation satellites that have fairly constant altitudes. Satellites with communication and surveillance missions for high-latitude regions can use 12 h critical inclination orbits.

2.4.7 Repeat Ground Track Orbits If the ground track of a satellite in an orbit repeats itself after a certain period of time, the resulting orbit is called a repeat ground track orbit (RGTO). In this case, the longitude change of the satellite on the equator between two successive periods is ˙ λ = TN (ωe − Ω)

(2.68)

2.4 Orbital Properties of Spacecraft

47

where T N is the orbital period taking perturbations into account, ωe the rotation rate of the Earth, and Ω˙ the average rate of change of the right ascension of the ascending node. 

a3 TN = 2π μ

 1/2   2J2 R2e 5 2 1− 3 − sin (i) 2a2 2

(2.69)

By selecting an appropriate semi-major axis and inclination, we obtain ˙ = N2π MTN (ωe − Ω)

(2.70)

where M and N are irreducible positive integers. This RGTO has a repetition cycle of N days or M orbits.

2.4.8 Reentry Orbits It is clearly not economical to decelerate a returning spacecraft using its propulsion system, and this method of deceleration is never directly used in engineering applications. A better method is to utilize aerodynamic drag from the atmosphere of the Earth to realize deceleration. The deceleration process is summarized as follows. The spacecraft first leaves the original orbit using active thrust toward the atmosphere. Subsequently, the rockets are shut down. Atmospheric drag is generated from the relative velocity between the spacecraft and the atmosphere, decelerating the spacecraft. The spacecraft altitude gradually reduces before the final landing on the Earth’s surface (Fig. 2.11). From leaving the original orbit around the Earth to landing on the surface of the Earth, the returning spacecraft roughly experiences four phases in succession, namely, a deorbiting phase, transitional phase, reentry phase, and descent phase. Correspondingly, the motion trajectories of the spacecraft’s center of mass are, respectively, called the deorbiting-phase orbit, transitional-phase orbit, reentry-phase orbit, and descent-phase orbit, which collectively constitute the reentry orbit of the spacecraft. 1. (1)

Design constraints of the reentry orbit Constraints on initial conditions

The constraints on the initial conditions are mainly related to information on the location in the atmosphere from which reentry begins, such as the state values of the entry point. The initial state values of the entry point are determined based on the specific mission. Normally, the probe enters the atmosphere in a stable state. At the time of entry, variables of the probe such as position, velocity, and attitude angle have

48

2 Spacecraft Orbits and Orbital Dynamics

Reentry orbit

Atmosphere Earth Fig. 2.11 Schematic of skip reentry

been already determined. The constraints on the initial conditions usually include those imposed on the initial velocity, initial altitude, and initial fight-path angle. (2)

Constraints during atmospheric entry

During atmospheric entry, effective maneuvering at low speed should be ensured to avoid ablation due to aerodynamic heating, and the heat flux and dynamic pressure of the probe must be maintained within allowable ranges to prevent mechanical damage. The dynamic pressure during atmospheric entry should be kept below a given maximum value, i.e., q = 0.5ρv2 < qmax

(2.71)

Further, to reduce aerodynamic heating, it should be kept below a given maximum value, i.e., Q=k

ρ 2 3 v < Qmax 0.66 m

(2.72)

where ρ is the atmospheric density, m the mass of the probe, v the velocity of the probe, and k = 0.00019027. During entry, the overload constraint must be carefully considered to ensure structural and equipment safety. In this phase, a large overload could be experienced by the lander in both the axial and normal directions. Hence, for safety, the total overload constraint is always adopted in engineering applications.

2.4 Orbital Properties of Spacecraft

(3)

49

Terminal state constraints

The entry phase terminates as a parachute is deployed. In a landing mission, the deployment of a deceleration parachute is decisive to the success of the mission. Therefore, the extent to which the terminal state satisfies parachute-deployment requirements is one of the optimization indicators for accurate landing. In general, the location at which the parachute is deployed has constraints on the altitude and speed of the lander. If the speed is too high, the parachute may be damaged. 2.

Characteristics of ballistic reentry orbits

In semi-ballistic reentry, on the basis of ballistic reentry, the entering spacecraft can generate a certain level of lift by offsetting its center of mass from its axis of symmetry. This type of spacecraft usually has a lift-to-drag ratio less than 0.5. The special design of the spacecraft in terms of mass, inertia, and shape ensures that its entry attitude is maintained around the trim angle of attack (AOA), and the total lift cannot be manually adjusted. During reentry, the probe has some capacity to maneuver by controlling the roll angle. Trajectory control is achieved by adjusting the bank angle and, thereby, the vertical component of lift. In this manner, landing errors are greatly reduced, the entry trajectory is made gentler, and the maximum overload and dynamic pressure are kept low. Therefore, semi-ballistic vehicles, while retaining the advantages of ballistic vehicles such as a simple structure and easy thermal protection, can overcome some of the shortcomings of ballistic vehicles to some extent by making use of the generated lift; in addition, semi-ballistic vehicles have the advantage of limited trajectory-maneuvering capabilities. However, the lift generated from semiballistic vehicles is not sufficient to enable smooth horizontal landing. Like ballistic vehicles, spacecraft in the semi-ballistic category is also vertical-landing vehicles. 3.

Characteristics of space-shuttle reentry orbits

A space shuttle usually has a lift-to-drag ratio greater than unity, suggesting that lift is higher than drag during entry. Such a large lift can no longer be realized by offsetting the center of mass. Space shuttles have gentle reentry trajectories; therefore, the ranges and times of the process are much longer than those in ballistic and semiballistic reentry. Although the peak heat flux and overload are fairly small, the total heating of a space shuttle is considerable. Horizontal landing is normally used, and this process requires infrastructure such as runways; therefore, it is usually not used for interstellar exploration missions. Horizontal landing avoids the two drawbacks of vertical landing, i.e., the impact overload that leads to potential damage to the vehicle and its payload and the difficulty in controlling the fall point. In this manner, non-destructive and accurate landing is made possible for reusable spacecraft.

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2 Spacecraft Orbits and Orbital Dynamics

2.4.9 Libration Point Orbits 1.

Equations of motion relative to collinear Libration points

To study the motion around the collinear Libration point L1 (the same applies to other collinear points), David Richardson [8] introduced a frame of reference by translating the origin C of the barycentric rotating coordinates C−XYZ to the collinear Libration point L1 . That is, the original frame C−XYZ is transformed to another rotating frame L1 − xyz with its origin located at L1 . The x and y axes of the new frame are still in the plane of motion of M1 and M2 . Figure 2.12 shows such a transformation by using L1 as an example. The equations of motion with respect to the collinear Libration point L1 are written as

 ⎧ ∂  x l ⎪ x ¨ − 2˙ y − (1 + 2c )x = c (ρ)q P 2 l l ⎪ ∂x q ⎪ ⎪ l≥3 ⎪

 ⎨ ∂  y¨ + 2˙x + (c2 − 1)y = ∂y cl (ρ)ql Pl qx (2.73) ⎪ l≥3

 ⎪ ⎪  ⎪ z¨ + c z = ∂ ⎪ cl (ρ)ql Pl qx ⎩ 2 ∂z l≥3

where Pl is a Legendre polynomial of degree l, with P0 (υ) = 1, P1 (υ) = υ, and  P2 (υ) = (3υ 2 − 1)/2; ρ is the distance from M1 to the barycenter; and q = x2 + y2 + z 2 . γ1 is calculated from γ15 − (3 − ρ)γ14 + (3 − 2ρ)γ13 − ργ12 + 2ργ1 − ρ = 0

(2.74)

  l+1 1 l (1 − ρ)γ1 cl (ρ) = 3 ρ + (−1) (1 − γ1 )l+1 γ1

(2.75)

Fig. 2.12 Transformation from barycentric rotating coordinates C−XYZ to Libration point rotating coordinates L1 − xyz

2.4 Orbital Properties of Spacecraft

2.

51

The linear situation

The homogeneous equations corresponding to Eq. (2.73) are the linear equations of the system, i.e., ⎧ ⎨ x¨ − 2˙y − (1 + 2c2 )x = 0 y¨ + 2˙x + (c2 − 1)y = 0 ⎩ z¨ + c2 z = 0

(2.76)

In the linear approximation, motions in the xy plane are coupled but independent of the motion in the z-direction. The eigenvalues of the system of linear equations in Eq. (2.76) are

a1 = −a2 =

   9c2 − 8c + c − 2  2 2 2

0, 2    9c2 − 8c − c − 2  2 2 2 , a3 = −a4 = i 2 √ a5 = −a6 = i c2

From the structures of the linear ordinary differential equations, we see that a collinear Libration point serves as both a saddle and a center. The existence of the real eigenvalue a1 > 0 suggests that the collinear Libration points are unstable, whereas the existence of the complex eigenvalues suggests that periodic and quasi-periodic orbits can be found around the collinear Libration points. The general solution to the system of linear equations in Eq. (2.76) is ⎧ ⎨ x = C1 ea1 t + C2 e−a1 t + C3 cos β1 t + C4 sin β1 t y = α1 C1 ea1 t − α1 C2 e−a1 t − α2 C3 sin β1 t + α2 C4 cos β1 t ⎩ z = C5 cos β2 t + C6 sin β2 t

(2.77)

where Ci (i = 1,…,6) is an integration constant determined by the initial conditions, β1 = Im(a3,4 ), β2 = Im(a5,6 ), α1 = (a12 − 2c2 − 1)/2a1 , and α2 = (β12 + 2c2 + 1)/2β1 . Equation (2.30), which contains hyperbolic and centric terms, is the firstorder analytical solution to the motion around a collinear Libration point. Because a1 > 0 an exponential divergence term exists in the solution. Hence, the linear solution is unstable. This suggests that the collinear Libration points are also unstable, featuring a “saddle × center × center” structure in terms of dynamics. That is, the motion around these points is both periodic and hyperbolic. Figure 2.13 shows the projection of the motion around a collinear Libration point on the xy plane, where different types of orbits are determined by the values of C1 and C2 .

2 Spacecraft Orbits and Orbital Dynamics

Fig. 2.13 Projection of the motion around a collinear Lagrangian point on the xy plane

Forbidden region

Non-transit orbit

Transit orbit

Asymptotic orbit

L2

Periodic orbit

Forbidden region

y (nondimensional units, rotating frame)

52

x (nondimensional units, rotating frame)

(1)

Asymptotic orbits

When C1 · C2 = 0 and the two constants are not simultaneously zero, the unstable component comes into play, corresponding to asymptotically stable orbits (C1 = 0, C2 = 0) and asymptotically unstable orbits (C1 = 0, C2 = 0). The asymptotically stable orbits asymptotically approach the periodic or quasi-periodic orbits, whereas the asymptotically unstable orbits asymptotically depart the periodic or quasi-periodic orbits. (2)

Transit orbits and non-transit orbits

When C1 · C2 < 0, the orbits are called transit orbits. When C1 · C2 > 0, the orbits are called non-transit orbits. (3)

Periodic orbits and quasi-periodic orbits

When C1 = C2 = 0, the unstable component is suppressed. Therefore, the general solution of Eq. (2.77) can be written as ⎧ ⎨ x(t) = −Ax cos(λt + φ) y(t) = κAx sin(λt + φ) ⎩ z(t) = Az sin(vt + ψ)

(2.78)

where Ax is amplitude in the xy plane, Az the out-of-plane amplitude, κ = α2 , λ = β1 , v = β2 , φ and ψ the initial orbital phases, and λ and v the motion frequency within and out of the xy plane, respectively. The motion represented by Eq. (2.78) is usually quasi-periodic.

2.4 Orbital Properties of Spacecraft

a.

53

Lissajous trajectories

Because λ and v in Eq. (2.78) are incommensurable, the motion is characterized by a non-closed quasi-periodic orbit in the rotating frame at L1 , named the Lissajous trajectory, as shown in Fig. 2.14. This type of orbit is named after the French physicist Jules Antoine Lissajous because the shapes of the projections of this orbit on different planes are highly similar to some of the research results of Lissajous published in 1857. b.

Halo orbits

When the amplitudes of the orbit, Ax and Az , increase because of the influence of the nonlinear terms, the planar motion frequency λ and vertical motion frequency v also vary. When the amplitudes are sufficiently large, some of their combinations might lead to λ = v, producing a periodic orbit called a halo orbit. Clearly, halo orbits exist only when the higher-order terms on the RHS of Eq. (2.73) are considered. Figure 2.15 shows halo orbits of different sizes about the Sun–Earth L 1 Libration

8

x 10

4

1

z[km]

0.5

2 y[km]

0 L1

-0.5

-1.5 1

L1

-4

0.5

4 -6

2

0

6

0

-0.5

x 10

-2 -1

y[km]

-4

5

-8 -3

x[km]

x 10

5

1.5

1

0.5

0.5

z[km]

1

0

L1

-1

-1

-4

-2

0 y[km]

0 x[km]

1

2

3 x 10

5

x 10

5

L1 -0.5

-6

-1

0

-0.5

-1.5 -8

-2

(b) Projection on xy plane

(a) 3D view

z[km]

0 -2

-1

1.5

5

6

1.5

x 10

x 10

5

2

4

6

8 x 10

(c) Projection on yz plane

-1.5 -3

-2

-1

5

0 x[km]

1

(d) Projection on xz plane

Fig. 2.14 Lissajous trajectory about the Sun–Earth L 1 Libration point

2

3 x 10

5

54

2 Spacecraft Orbits and Orbital Dynamics 1

x 10

5

8

0.6

6

0.4

4

6

0.2

y[km]

z[km]

x 10

0.8

2 L1 0

0 L1 -0.2

-2

-0.4

-4 -0.6

1

-6 -2

0.5 0 2 x 10

-0.8

0 x 10

-0.5

4

5

6

-1

6

-1 -2

-1

0

5

4

3

2

1 x[km]

y[km]

x 10

x[km]

(b) Projection on xy plane

(a) 3D view 8

x 10

5

5

8

6

4

4

2

2

5

z[km]

z[km]

6

x 10

0

0

L1

-2

-2

-4

-4

-6 -1

-0.5

0 y[km]

0.5

1 x 10

6

(c) Projection on yz plane

-6 -2

L1

-1

0

1

2

3

x[km]

4

5 x 10

5

(d) Projection on xz plane

Fig. 2.15 Halo orbits of different sizes about the Sun–Earth L 1 Libration point

point (in rotating coordinates centered at L1 ). Compared to conventional low-Earth orbits, halo orbits and Lissajous trajectories have the ability to providing continuous observations in the target mission. Because halo orbits are symmetric about the xz plane, their projections on this plane show the special shape shown in Fig. 2.15d. c.

Planar and vertical Lyapunov orbits

In addition to periodic halo orbits and quasi-periodic Lissajous trajectories, there are two types of periodic orbits around the collinear libration points: planar Lyapunov orbits and vertical Lyapunov orbits. A planar Lyapunov orbit is restrained within the xy plane, representing the nonlinear state of a small vibration in the xy plane. Although a vertical Lyapunov orbit is a 3D periodic orbit, it in fact represents the nonlinear state of a small vibration in the z-direction. Halo orbits are caused by the bifurcation of planar Lyapunov orbits. As the motion amplitude of a planar Lyapunov orbit grows and eventually approaches the critical value, bifurcation occurs. By setting initial conditions, the amplitude Az in Eq. (2.78) can equal zero. That is, the periodic motion around the libration point is only limited to the motion plane

2.4 Orbital Properties of Spacecraft Fig. 2.16 Planar Lyapunov orbits of different sizes about the Sun–Earth L 1 libration point

55

1.5

x 10

6

1

y[km]

0.5

0

L1

-0.5

-1

-1.5 -3

-2

-1

0

2

1 x[km]

3

4

6

5 x 10

5

of the two primaries, whereas there is no motion perpendicular to the orbital plane. In this case, we arrive at a planar Lyapunov orbit in L1 rotating coordinates, as shown in Fig. 2.16. A higher amplitude caused by nonlinear effects leads to a more distorted orbit. On the other hand, if the initial conditions are set such that the amplitude Ax in Eq. (2.78) equals zero (that is, the periodic motion around a Libration point is restricted to the z-axis), then we arrive at a vertical Lyapunov orbit in L1 rotating coordinates. With bounded amplitudes, vertical Lyapunov orbits have x and y components but only one point of intersection with the xy plane, as shown in Fig. 2.17. 3.

The nonlinear situation

The above discussion was made in the context of linear models. We can still draw similar conclusions in the nonlinear situation, i.e., taking higher-order terms into account. When the orbital energy is close to the energy corresponding to the collinear Libration points, the motion around these points can be classified into the same four groups: periodic or quasi-periodic orbits, asymptotic orbits, transit orbits, and non-transit orbits. When higher-order terms are added to the linear equations in Eq. (2.73), i.e., the original nonlinear form in Eq. (2.73), periodic or quasi-periodic orbits can still exist in the system. The Lindstedt–Poincaré method can be employed to find the periodic or quasi-periodic solution as follows. First, the periodic or quasi-periodic solution is expressed in a trigonometric series. Second, the series is substituted in the equations of motion with the considered higher-order terms. Third, the coefficients of each term in the trigonometric series are determined. In this manner, higher-order halo orbits and Lissajous trajectories can be obtained. Considering the higher-order terms on the RHS of Eq. (2.73), under certain initial conditions, solution to the halo orbits can be written as power series of the amplitudes Ax and Az .

56

2 Spacecraft Orbits and Orbital Dynamics 1.5 x 10

x 10

5

6

1.5

1

1

0.5 y[km]

z[km]

0.5 L1

0

0 L1

-0.5

-0.5 -1 -1.5 0

-1

1 2

0

4 x 10

6

5

-1

0.5

-0.5

x 10

5

y[km]

x[km]

-1.5 -1

x 10 1.5

1

1

0.5

0.5

z[km]

z[km]

4

6

5 x 10

5

(b) Projection on xy plane

6

1.5

0 L1

L1 -0.5

-1

-1

-1.5

6

0

-0.5

-1

3

2 x[km]

(a) 3D view x 10

1

0

-1.5 -0.5

0 y[km]

1

0.5 x 10

-1

0

1

5

(c) Projection on yz plane

3

2

4

x[km]

6

5 x 10

5

(d) Projection on xz plane

Fig. 2.17 Vertical Lyapunov orbits near the Sun–Earth L 1 libration point

⎧   ∞ ⎪   ⎪ j ⎪ x(t) = xljk cos(kθ0 ) Alx Az ⎪ ⎪ ⎪ |k|≤l+j l,j=1 ⎪ ⎪   ⎪ ⎨ ∞   j y(t) = yljk sin(kθ0 ) Alx Az ⎪ l,j=1 |k|≤l+j ⎪ ⎪   ⎪ ⎪ ∞ ⎪   ⎪ j ⎪ zljk cos(kθ0 ) Alx Az ⎪ ⎩ z(t) = l,j=1

(2.79)

|k|≤l+j

where θ0 = ωm t + φ and n0 = l + j is the order of the solution. By considering the terms on the RHS of Eq. (2.73) up to the third order, Richardson [9] obtained the third-order analytical solution to the halo orbits by using the Lindstedt–Poincaré method. In a practical mission analysis, this can be used as an initial guess, and high-accuracy initial values of the halo orbits can then be acquired using the differential correction method. Considering the higher-order terms on the RHS of Eq. (2.73), under certain initial conditions, the solution to the Lissajous trajectories can also be written as power

2.4 Orbital Properties of Spacecraft

57

series of amplitudes Ax and Az : ⎧   ∞ ⎪   ⎪ j ⎪ x(t) = xljku cos(kθ1 + uθ2 ) Alx Az ⎪ ⎪ ⎪ l,j=1 |k|≤l,|u|≤j ⎪ ⎪   ⎪ ⎨ ∞   j y(t) = yljku sin(kθ1 + uθ2 ) Alx Az ⎪ |k|≤l,|u|≤j l,j=1 ⎪ ⎪   ⎪ ⎪ ∞ ⎪   ⎪ j ⎪ zljku cos(kθ1 + uθ2 ) Alx Az ⎪ ⎩ z(t) = l,j=1

(2.80)

|k|≤l,|u|≤j

where θ1 = ωm t + φ, θ2 = vm t + ψ, and n0 = l + j is the order of the solution. In practice, the two-level differential corrector proposed by Howell and Pernicka is often adopted when calculating Lissajous trajectories. It is noted that halo orbits are periodic orbits in CRTBP. Actual force models have no strict periodic halo orbits; rather, they have quasi-periodic orbits. Nevertheless, by properly selecting the in-plane and out-of-plane amplitudes of the quasiperiodic Lissajous trajectories, we can have quasi-periodic Lissajous trajectories that are almost periodic. In the literature, they are still loosely referred to as halo orbits. 4.

Invariant manifolds

In the study of dynamical systems, a manifold is a mathematical term referring to the surface formed by trajectories or orbits. Based on the properties of motion around the collinear Libration points, we know that near periodic and quasi-periodic orbits, there exist asymptotic orbits that asymptotically approach or depart these orbits. The collection of these orbits has a tubular structure and is, therefore, called a manifold or tube. The collection of asymptotically stable orbits is called the stable manifold, whereas the collection of asymptotically unstable orbits is called the unstable manifold. The collection of periodic and quasi-periodic orbits near the collinear Libration points are referred to as the center manifold. These manifolds have the property of invariance. That is, orbits starting from this manifold will remain on the manifold throughout the course of their dynamical evolution. Hence, stable, unstable, and center manifolds are all invariant manifolds, which are dynamical structures closely related to the periodic and quasi-periodic orbits near the collinear Libration points. Transit orbits are inside the invariant manifold tubes, whereas non-transit orbits are outside the tubes, as shown in Fig. 2.13. Figures 2.18, 2.19, 2.20, 2.21 show stable and unstable manifolds of halo orbits near the Sun–Earth L 1 and L 2 Libration points. By using invariant manifolds, it is possible to achieve low-energy transfer from massive celestial bodies such as the Earth and Moon to Libration point orbits, as well as transfer between Libration point orbits.

58

2 Spacecraft Orbits and Orbital Dynamics

Fig. 2.18 Stable manifold of a halo orbit about the Sun–Earth L 1 Libration point

10

x 10

5

8 6

Y[km]

4 2 0 L1

E

-2 -4 -6 -8 1.475

Fig. 2.19 Unstable manifold of a halo orbit about the Sun–Earth L 1 Libration point

8

x 10

1.485

1.48

1.49 X[km]

1.495

1.5

1.505 x 10

8

5

6 4 2

Y[km]

E 0 L1 -2 -4 -6 -8 -10 1.475

1.48

1.485

1.49 X[km]

1.495

1.5

1.505 x 10

8

2.5 Orbital Perturbation Equations and Their Solutions The gravity of the primary body acting on a spacecraft is a key factor determining the orbital characteristics of the spacecraft. Therefore, many orbital mechanical problems can be described by two-body problems. However, any external force could affect the orbit. In most cases, these external forces are significantly smaller than the force of gravity due to the primary body. Hence, their effects on the orbit can be regarded as perturbations in the two-body problem. For satellites orbiting the Earth, perturbations affecting the orbit mainly include the gravitational force due to the Earth’s non-spherical shape, the Sun’s gravitational force, the Moon’s gravitational force, atmospheric drag, and radiation pressure. Their influences on the orbit are

2.5 Orbital Perturbation Equations and Their Solutions Fig. 2.20 Invariant manifolds of a halo orbit about the Sun–Earth L 1 Libration point

59

Unstable manifold Stable manifold

Unstable Earth manifold Stable manifold

Fig. 2.21 Invariant manifolds of halo orbits about the Sun–Earth L 1 and L 2 Libration points

Unstable Stable manifold Earth manifold Stable Unstable manifold manifold

L1 Unstable Stable manifold manifold L2 Stable Unstable manifold manifold

mostly dependent on the properties of the orbit itself and simultaneously related to the attitude dynamics of the spacecraft. With the existence of these factors, in reality, the spacecraft does not move along a Keplerian orbit. The orbital parameters of the spacecraft vary continuously, leading to deviations from the orbit that is determined by Kepler’s laws. These deviations are referred to as perturbations. To simplify the problem, the Keplerian orbits can be viewed as approximate orbits for satellites and other spacecraft. The Keplerian orbit obtained in the ideal situation is also called the unperturbed orbit. The goal of the perturbation study is to examine how unperturbed orbits of celestial bodies (including artificial celestial bodies) vary under the effects of various perturbations. The actual orbits of spacecraft including satellites

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2 Spacecraft Orbits and Orbital Dynamics

are referred to as perturbed Keplerian orbits. Solving for perturbed Keplerian orbits is the most fundamental topic in the research on orbital mechanics.

2.5.1 Osculating Orbit Owing to perturbating effects, the orbital elements of a satellite are not always constant and instead vary with time. We can imagine that at any instant of time, the satellite is in an elliptical orbit. This orbit and the actual orbit of the satellite osculate; therefore, it is called an osculating orbit, and its orbital elements are called osculating elements.

2.5.2 Lagrange Perturbation Equations Lagrange planetary equations are a well-known set of equations in celestial mechanics. In this section, we will introduce them in a slightly different form that is convenient for the description of a satellite’s motion. The motion of an Earth satellite can be described by r¨ = ∇U

(2.81)

where r is the position vector of the satellite and U is the potential function. ∇U T  , ∂U , ∂U , where x, y, and z are the three is a vector expressed as ∇U = ∂U ∂x ∂y ∂z components of r. For an Earth satellite, U can be written as U =−

μ + R(r) r

(2.82)

where μ is the Earth constant, μr the potential function of the central force field, and R(r) the potential function of perturbation. If R is neglected, this becomes a two-body problem, the exact solution to which is r = r(a, e, M , , i, ω) r˙ = r˙(a, e, M , , i, ω)

(2.83)

Moreover, r˙ = n

∂r ∂M

(2.84)

2.5 Orbital Perturbation Equations and Their Solutions

61

 where n = aμ3 represents average motion. We will not present the detailed orbit derivation here, which can be found in astronomy textbooks. Instead, we write the equations of perturbed motion with respect to the six orbital elements: ⎧ da 2 ∂R = na ⎪ dt ∂M ⎪ √ ⎪ ⎪ de = 1−e2 ∂R − 1−e2 ∂R ⎪ 2 ⎪ dt na na2 e ∂ω √ e ∂M ⎪ ⎪ ⎨ dω = 1−e2 ∂R − √cos i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂R dt na2 e ∂e na2 1−e2 sin i ∂i di ∂R ∂R i 1 = na2 √cos − na2 √1−e 2 sin i ∂ dt 1−e2 sin i ∂ω d ∂R 1 = na2 √1−e 2 sin i ∂i dt 2 dM 2 ∂R ∂R = n − 1−e − na dt na2 e ∂e ∂a

(2.85)

Equation (2.85) presents the famous Lagrange planetary perturbation equations. Once the specific expression of perturbing potential is known, it can be substituted into these equations to obtain the osculating elements at any instant. Thus, the position and velocity of the satellite can be solved as a two-body problem.

2.5.3 Gauss Perturbation Equations Lagrange equations are only applicable to situations in which the perturbing potential can be expressed analytically. To describe more general situations, we need to adopt Gauss perturbation equations. The perturbing acceleration acting on the satellite at any instant is divided into three components, S, T, and W, that are perpendicular to each other. T is along the satellite radius direction, W is normal to the orbital plane, and S satisfies the righthand rule with the other two components. Gauss perturbation equations correlate the rates of change of the orbital elements with these three components and can be expressed in the following form:  ⎧ da √2 Te sin f + S Pr = ⎪ 2 dt ⎪ n 1−e  √ ⎪ 2 ⎪ de ⎪ = √1−e T sin f + S(cos E + cos f ) ⎪ ⎪ dt na  ⎪ ⎨ dω = 1−e2 S sin f !1 + r " − T cos f − W r√cos i sin u dt nae P na2 1−e2 sin i d W ⎪ √ r sin u = ⎪ 2 2 dt ⎪ na 1−e sin i ⎪ ⎪ u di ⎪ = naW2 r√cos ⎪ dt ⎪ 1−e2 #   ⎩ dM 1 T 2re = n − nae − (1 − e2 ) cos f + S sin f (1 − e2 ) + dt a

(2.86)

r a

$

The eccentricity e appears on the RHS of Eq. (2.86). The equations may become less useful when e is very small. To circumvent this problem, it is common practice to replace e, ω, and M with ex = e cos ω, ey = e sin ω, and λ = M + ω, respectively. Thus, the corresponding differential equations in Eq. (2.87) become

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√ ⎧ de #  $ 1−e2 r r x ⎪ ⎨ dt = √ na 2 #Tsin u + S (1 + P ) cos u + P ex $ + dey S (1 + Pr ) sin u + Pr ey − T cos u − = 1−e dt na  √ ⎪ ⎩ dλ = n − 1 T 2r + √1−e2 e cos f − S(1 + r ) dt

na

a

1+

d cos iey dt d cos iex dt √ 1−e2 √ e sin f P 1+ 1−e2

1−e2



−

W r√cos i sin u na2 1−e2 sin i

(2.87) Thus, the impact of the small eccentricity is successfully eliminated. When the eccentricity and inclination are both small, as in the case of a geostationary orbit, we need new orbital elements to avoid singularity. The details of the definitions and expressions of these elements can be found in Ref. [2].

2.5.4 Numerical Integration Methods The previously discussed Lagrange and Gauss perturbation equations are both systems of differential equations, the solutions to which are normally obtained using numerical methods. Once the orbital elements at the initial instant are given, the position/velocity or the instantaneous orbit of the satellite can be calculated stepwise at the next instant based on the differential equations of these elements. Neither analytical expressions nor complex formula derivations are needed in the process of numerical integration. As long as the order and step size are properly set, numerical integration, in principle, can satisfy any accuracy requirement. Many methods can be used for numerical integration. They can be categorized into two groups: single-step methods and multi-step methods. A single-step method only relies on the value of one point to calculate the value of the next point, whereas a multi-step method requires knowledge about multiple points to solve for the next point. The family of Runge–Kutta methods are representative single-step methods. Assume a differential equation dy = f (t, y) dt

(2.88)

The Runge–Kutta method is given by yn+1 = yn +

m %

Wi ki

(2.89)

i=1

where Wi is the weight and m is the number of stages, i.e., the number of times needed in a single step for the method to calculate the function f on the RHS of Eq. (2.88). ki is defined as

2.5 Orbital Perturbation Equations and Their Solutions

⎛ ki = hf ⎝tn + Ci h, yn +

63 i−1 %

⎞ Aij kj ⎠

(2.90)

j=1

where Wi , Ci , and Aij can be obtained from Taylor series and h is the step size. The classical Runge–Kutta fourth-order (RK4) method is sufficient for regular accuracy requirements, and it is given by yn+1 = yn + 16 (k1 + 2k2 + 2k3 + k4 ) ⎧ k1 = hf (t ⎪ ⎪ " ! n , yn )1 ⎨ k2 = hf !tn + 2 h, yn + 21 k1 " 1 1 ⎪ ⎪ k3 = hf tn + 2 h, yn + 2 k2 ⎩ k4 = hf (tn + h, yn + k3 )

(2.91)

The strength of single-step methods is that once the initial value is known, the numerical solution at any integer step can be calculated using the recurrence relation; hence, the step size is easily controlled in the calculation. Additionally, compared to the classical form, the Gill form of RK4 yields reduced truncation errors. In theory, higher-order single-step integration methods can yield higher accuracies, provided that, of course, the solution is sufficiently smooth. Here, the popular error-embedded RKF7(8) single-step integrator is used. It uses the difference between the calculations of the Runge–Kutta methods of m and m + 1 order to evaluate the truncation error. Linear multi-step methods can avoid repeated calculations of the function in the single-step methods; the computational cost is reduced by making use of the information from preceding steps. Commonly used multi-step techniques include Adam– Cowell and Krogh–Shampine–Gordon (KSG) integrators. If the accumulation of round-off error is not the primary concern, practical experience suggests that the regular Adam–Cowell integrator is a better choice. In conventional (i.e., non-symplectic) algorithms, the local truncation errors are dissipative. However, as the arc to be calculated increases in length, the along-track error accumulates. For shorter arcs, because the local truncation error is small and the round-off error of the calculation can be neglected (double-accuracy calculation), the accuracy can still be ensured. The rule of thumb in practical applications is to use higher-order integrators for high-accuracy problems and lower-order integrators for low-accuracy problems.

2.5.5 Perturbation Methods Based on the approach of variation of constants, the perturbation methods can yield analytical solutions to the equations of perturbed motion in a simple way. Suppose an equation of perturbed motion is as follows:

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dσ = f0 (a) + f1 (σ, t, ε) dt

(2.92)

where σ is one of the six elements (e.g., mean anomaly M) of the instantaneous orbit,  T f0 (a) = δn, δ = 0 0 0 0 0 1 , and |f1i (σ, t, ε)| = O(ε), i = 1, 2, . . . , 6. Thus, the small-parameter power series expansion of the orbital element with respect to ε is σ = σ (0) (t) + σ (1) (t, ε) + σ (2) (t, ε2 ) + · · · + σ (l) (t, εl ) + · · ·

(2.93)

where σ (0) (t) = σ0 + δn0 (t − t0 ) is the solution for unperturbed motion and σ0 is the orbital element for the epoch t0 . Substituting Eq. (2.93) into Eq. (2.92) and expanding fi near σ (0) (t) d  (0) σ (t) + σ (1) (t, ε) + σ (2) (t, ε2 ) + · · · + σ (l) (t, εl ) + · · · dt ∂f0  (1) 1 ∂ 2 f0  (1) a + a(2) + · · · + a + a(2) + · · · = f0 (a) + ∂a 2 ∂a2 6  % ∂f1  (1) + f1 (σ, t, ε) + σj + σj(2) + · · · ∂σj j=1

+ ···

  1 % % ∂ 2 f1  (1) σj + σj(2) + · · · σk(1) + σk(2) + · · · + · · · 2 j=1 ∂σj ∂σk 6

+

2

6

k=1

(2.94) If the expanded power series of σ converges, rearranging the terms with the same power (εl ) on both sides of Eq. (2.94) yields σ (0) (t) = σ0 + δn0 (t − t0 )σ (1) (t)  *t  ∂n (1) δ a + f1 (σ, t, ε) = dtσ (2) (t) ∂a (0) σ =σ t0 ⎡ ⎤  %  *t 2 ! " ∂ n ∂f 1 ∂n 2 1 = ⎣δ + a(2) + a(1) σj(1) ⎦ ∂a 2 ∂a2 ∂σ j j t0

dt

(2.95)

σ =σ (0)

Equation (2.95) is the recursive formula of the small-parameter power series, where the higher-order perturbations are computed based on the lower-order perturbations. The above derivation is essentially a series expansion on the unperturbed orbit and, thereby, within the scope of classical perturbation methods. When it is used for solving short-arc problems, the accuracy of the solution is not significantly affected.

2.5 Orbital Perturbation Equations and Their Solutions

65

However, for long arcs, the effects of long-term items and long periods are very different, leading to potential distortion of the perturbed solution. For example, with conservative forces, a, e, and i should be free of long-term changes, but this may not be true when the classical perturbation method is used to calculate the small-parameter power series solution. Therefore, when searching for the perturbed solution, the two modified versions of the classical perturbation method are adopted, namely, the mean element method and the quasi-mean element method [1]. Meanwhile, the RHS function of the perturbation equation is decomposed into three parts: a perturbation function including a, e, and i; a periodic function involving  or ω; and a periodic function involving M. We then search for the perturbed solution that has similar properties. f1l = 0 is needed to ensure the validity of the mean element method. This requirement is usually satisfied or approximately satisfied in orbital mechanical systems. 1.

Small-parameter power series solution constructed using the mean element method σ = σ (t) + σl(1) (t, ε) + · · · + σs(1) (t, ε) + · · ·

(2.96)

where σ (t) = σ (0) (t) + σ1 (t − t0 ) σ (0) (t) = σ 0 + δn(t − t0 )   σ 0 = σ (t0 ) = σ0 − σl(1) (t0 ) + · · · + σs(1) (t0 ) + · · ·

2.

(2.97)

This suggests that the corresponding reference orbit is no longer an unperturbed solution; instead, it is an orbit taking long-term changes into account. Small-parameter power series solution constructed using the quasi-mean element method Different commensurable singularities (i.e., small denominators) may occur in the periodic terms (i.e., long-period terms) of perturbation changes, resulting in the failure of the perturbed solution. To avoid this problem, we may use the second modified version of the classical perturbation method. In this modification, while making full use of the advantages of the mean element method, a reasonable modification is made to the definition of the mean element set such that the reference orbit is changed from the mean orbit to a quasi-mean orbit σ (t) with only the short-period terms (σs(1) (t), σs(2) (t), …) separated. σ = σ (t) + σs(1) (t, ε) + · · · where σ (t) = σ (0) (t) + σc (t) + σl(1) (t) + · · ·

(2.98)

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σc (t) = σ1 (t − t0 ) + σ2 (t − t0 ) + · · · σl(1) (t) = σl(1) (t) − σl(1) (t0 ) σ (0) (t) = σ 0 + δn(t − t0 )  σ 0 = σ0 − σs(1) (t0 ) + · · ·

(2.99)

2.6 Sources of Orbital Perturbations 2.6.1 Earth’s Non-sphericity The perturbing potential of the Earth’s non-sphericity is described by V =

%

Vlm

l,m l

l GMe % % ae  = Plm (sin ϕ)(Clm cos mλG + Slm sin mλG ) r r m=0 l≥2

=

l GMe % ae l GMe % % ae l Jl Plm (sin ϕ) + Jlm Plm (sin ϕ) cos(mλ) r r r r m=1 l≥2

l≥2

(2.100) where, r ϕ , and λG are the geocentric distance, latitude, and longitude of the spacecraft in Earth-fixed coordinates, respectively; ae is the Earth’s equatorial radius; Pl and Plm are Legendre polynomials; and Clm and Slm are the gravitational constants of the Earth. The perturbing potential can be divided into zonal and tesseral harmonic terms, representing the impacts of the oblateness and ellipticity of Earth, respectively. The recursive formula for Legendre polynomials is given by +

⎧ 2 2 1/2 n + Slm ) ⎨ Jlm = (Clm Pl (μ) = 2n1n! dzd n (z 2 − 1)n , m mλ = arctan(S /Clm ) lm lm d Plm (μ) = (z 2 − 1)m/2 dzm Pm (z) ⎩ λ = λG − λlm

(2.101)

Next, we perform a nondimensional calculation and normalize the Plm (sin ϕ) terms in Legendre polynomials and the corresponding coefficients Clm and Slm , including distance, mass, and time. Let the geocentric gravitational constant μ = GMe = 1. We have ⎧ (ae is the equatorial radius of the Earth ellipsoid) ⎨ [L] = ae [M ] = Me (Me is the Earth mass, numerically substituted by GMe ) ⎩ [T ] = (ae /GMe )1/2

2.6 Sources of Orbital Perturbations

67

The normalized V is written as V =

l  %% 1 l≥2 m=0

r

l+1

P lm (sin ϕ)(C lm cos mλG + S lm sin mλG )

(2.102)

where P lm (μ) = Plm (μ)/Nlm , |μ| ≤ 1 C lm = Clm · Nlm , S lm = Slm · Nlm

(2.103)

⎧ 1/2 ! " (l+m)! 1 ⎪ ⎨ Nlm = 1+δ (2l+1)(l−m)! + 0m=0 ⎪ ⎩δ = 1 m = 0

(2.104)

Equation (2.103) can be again divided into zonal and tesseral harmonic terms; the specific coefficients can be found in Appendix III of Ref. [1]. For spacecraft orbiting Earth, the first zonal term (the J2 term) is dominant. Therefore, in engineering applications, only the perturbing effects of J2 are considered. Thus, V2 = −

J2 (3 sin2 ϕ − 1) 2r 3

(2.105)

where J 2 = 1.08263 × 10–3 . By using the perturbation method, we now calculate the first-order long-term rate of change of the orbital elements. ⎧ a1 = e1 = i1 = 0 ⎪ ⎪ ⎨ 1 = − PA22 n!cos i " A2 5 2 ⎪ ⎪ ω1 = − P2 n !2 − 2 sin i "√ ⎩ M1 = − PA22 n 1 − 23 sin2 i 1 − e2

(2.106)

where A2 = 23 J2 . For most orbits around the Earth, in the long term, it is sufficient to only consider J2, which represents the gravitational force due to the Earth’s non-sphericity. For some special orbits, however, it is necessary to include other perturbation terms. This is especially true for various RGTOs in which the small perturbing effects from some terms may accumulate and have special impacts. For geostationary orbits, we know from M 1 that J2 makes the semi-major axis of the orbit 2089 m longer than the theoretical semi-major axis of the nominal two-body orbit. If we include other perturbing effects of the Earth’s non-sphericity, the semi-major axis of the

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2 Spacecraft Orbits and Orbital Dynamics

geostationary orbit is 42166.289 km. If the gravitational forces of the Sun and Moon are taken into account as well, this value will be 42,165.7 km on average. The zonal terms affect the orbits of spacecraft including satellites primarily in the following two aspects. 1.

2.

Precession of the orbital plane. The Earth is an ellipsoid with an equatorial bulge and uneven mass distribution. The bulge can be simplified as a ringshaped mass band on the equator, which exerts a torque M on the orbital plane. According to the principle of gyro precession, this moment makes the orbital plane unable to remain stationary; instead, it rotates about the Earth’s axis. If we imagine a line starting from the Earth’s center that is normal to the orbital plane, the precession of the plane can be described as a pyramid, the axis of symmetry of which is the Earth’s axis. The direction of precession is a function of inclination. When the inclination is less than 90°, the orbital plane rotates in the direction opposite to the Earth’s rotation, and the ascending node moves westward. When the inclination is 90°, the gravitational force of the equatorial mass ring acting on the spacecraft is in its orbital plane; consequently, the plane does not rotate. When the inclination is greater than 90°, the direction of the torque is the opposite; that is, the orbital plane rotates in the same direction as the Earth’s rotation, and the ascending node moves eastward. In summary, the orbital plane rotates in the direction opposite to the spacecraft’s motion; in other words, the ascending node moves on the Earth’s equator in the direction opposite to the spacecraft’s motion. It should be noted that during the precession of the orbital plane, only the right ascension of the ascending node changes, while the inclination remains constant. Rotation of the major axis on the orbital plane. This makes the perigee move continuously in the orbital plane as well. In the special situation of i = 63.4°, the perigee basically stays above a certain latitude, which is useful for the Earth observation of some satellites. This angle is called the critical inclination. In most cases, the Earth’s non-sphericity is the primary perturbation for the orbit of a spacecraft. Of course, with the increase in orbit altitude (i.e., further from Earth), this perturbing effect decays. When the geocentric distance of a spacecraft is significantly larger than the size of the Earth itself, the Earth’s oblateness can be neglected. The first tesseral term J 22 is a small quantity of second order (J 22 = 1.81222 × 10–6 ), primarily affecting the short-period terms. For geostationary satellites, however, this perturbation is a long-period term; therefore, geostationary orbit is a kind of “Resonance Orbit”. It causes the satellite to drift toward the co-vertices of the equatorial plane (75.5° E, 104.5° W). The two semi-minor axes of the equatorial ellipse satisfy sin(2λG − 2λ22 ), where λ22 = −14.545◦ . The semimajor axis shows clear long-period features. However, for fixed-point satellites, this perturbation is equivalent to a long-term impact.

2.6 Sources of Orbital Perturbations

69

2.6.2 Atmospheric Drag Near Earth Atmospheric drag is a perturbing force generated by the Earth’s atmosphere that acts on a spacecraft. The magnitude of this drag is related to a series of factors such as the atmospheric density, spacecraft shape, and spacecraft velocity. Based on our knowledge in aerodynamics, we can determine the drag using D = −CD S

ρv2 2

(2.107)

where S is the characteristic surface area of the spacecraft, which normally adopts the maximum cross section of the spacecraft in the plane normal to the velocity vector, ρ the atmospheric density, v the velocity of the spacecraft relative to the atmosphere, and CD the drag coefficient. CD is a function of the spacecraft shape, flight attitude, and velocity. At a high Mach number, it can be considered constant. At an altitude of 150–500 km, the ambient flow is free molecular flow. In this case, the rebounding mechanism of the molecules off the surface of spacecraft determines the magnitude of CD , which usually takes a value in the range of 2–2.5. The negative sign in Eq. (2.108) suggests that the direction of the drag force is opposite to the flight direction of the spacecraft. D only acts in the orbital plane and, therefore, does not alter the orientation of the plane. The most significant long-term impact of atmospheric drag on the orbit is reflected in the variations of a and e, which make the elliptical orbit shrink continuously toward a circle. Below 110–120 km, owing to the high density, the spacecraft is unable to complete the next cycle. It falls in a spiral trajectory and eventually enters the denser layers of the atmosphere. A circular orbit with a period of 86.5–86.7 min and an altitude of 110–120 km is usually referred to as a critical orbit. The period from the time of reaching the target orbit to the time of reentry to the Earth’s atmosphere is the operation lifetime of a spacecraft. Atmospheric drag is the leading factor influencing the operation lifetime of low-Earth-orbit spacecraft. The impacts of aerodynamic forces drastically decay with the increase in orbit altitude. Atmospheric drag is considered the dominant perturbing force below 200 km. However, above 1,000 km, it can be neglected in most cases.

2.6.3 Gravitational Forces of the Sun and Moon Perturbations from the gravitational forces of the Sun and the Moon are due to the difference in gravitational accelerations they cause on the spacecraft and the Earth. A higher orbit altitude leads to greater perturbations. Below 1,600 km, these perturbations can be neglected, but they can be considerable for spacecraft in higher orbits. In a geostationary orbit, these perturbations must be considered to keep the

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satellite over a fixed point. Above 5,000 km, solar/lunar perturbations exceed the perturbations due to the Earth’s oblateness and become the leading perturbing factor. When the trajectory of a spacecraft deviates from the Earth’s center, the gravitational forces per unit mass of the Sun and Moon exerted on the spacecraft and the Earth are different. This difference leads to orbital perturbations. The perturbing force of the Sun and Moon exerted on a spacecraft near the Earth is given by  δ¨r = μs

rs − r rs − 2 rs |rs − r|2



 + μm

rm − r rm − 2 rm |rm − r|2

(2.108)

where r, rs , and rm are, respectively, the geocentric radii of the satellite, Sun, and Moon in geocentric coordinates and μs and μm are, respectively, the gravitational constants of the Sun and Moon. Hereafter, we let r1s = rs − r and r1m = rm − r. Because perturbations from the Sun and Moon have similar mechanistic principles, we proceed with the Sun as an example. Let the mean orbiting speed of the  21

s , where ros denotes the mean distance from the Earth around the Sun ns = GM ros Earth–Moon center of mass to the Sun (the mean orbiting speed of the Moon around  21

m . Thus, after rearranging, the the Earth is slightly different: nm = (Mm /(MGM m +Me ))rm perturbing acceleration of the satellite due to the Sun is  δ¨rs =

n2s

rs r1s



2 · r1s − rs

(2.109)

Expanding the above equation using Legendre polynomials to obtain the firstorder approximation (assuming r1s ≈ rs and eliminating second- and higher-order rs terms in the expansion), we arrive at r1s   rs r δ¨rs ≈ r · n2s 3 cos ξ · − rs r

(2.110)

where ξ is the angle between the position vectors of the Sun and satellite. Under the effects of the gravitational forces exerted by the Sun and Moon, the Earth’s surface is subject to elastic deformation, manifested in the form of Earth, ocean, and atmospheric tides. These tides also affect the orbit. Hence, they can be considered as auxiliary solar and lunar perturbations. Among them, Earth tides are the most influential. The effects of ocean tides are one order of magnitude lower than those of Earth tides, whereas the effects of atmospheric tides are two orders of magnitude lower than those of ocean tides. The effects of these tides are inversely proportional to r 5 , where r is the geocentric radius of the orbit. For low-Earth-orbit satellites, the effects of the tides and the gravitational forces exerted by the Moon are basically on the same order of magnitude. The tidal effects may be virtually ignored for high-flying satellites.

2.6 Sources of Orbital Perturbations

71

2.6.4 Solar Radiation Pressure When photons from the Sun are incident on an object, they exert a pressure force on the object that is inversely proportional to the square of the distance from the object to the Sun. Near the Earth, the solar pressure force per unit area exerted normal to the direction of solar radiation is k = 4.56 × 10−6 (1 + γ )N/m2

(2.111)

where γ represents the emissivity of the object surface, which is 0 for full absorption and 1 for full emission. The perturbing acceleration caused by solar radiation pressure is given by δ¨r =

kA S m

(2.112)

where A is the equivalent surface area of the spacecraft normal to the direction of solar radiation, m the spacecraft’s mass, and S the direction vector of solar radiation. The direction of solar pressure force is opposite to that of gravitational force. Solar radiation pressure also produces perturbations on the Earth’s orbit. However, because the Earth has a very small area-to-mass ratio (A/m), these effects can be ignored. Although perturbations from solar radiation pressure have long-term or longperiod impacts on all the six orbital elements, because most spacecraft have small values of A/m, the generated acceleration is equivalent to that caused by the Earth’s non-sphericity perturbation J2, which is a second-order small quantity. Therefore, in orbit design, perturbations due to solar radiation pressure can usually be neglected. In some cases, however, orbital changes resulting from these perturbations can be important factors in orbit determination or control. For example, for GSO satellites, eccentricity variations caused by solar pressure perturbations are a major concern for high-accuracy east–west position control. They are also the largest source of uncertainty in accurate orbit extrapolation for middle- and high-flying satellites. For solar panels, on the other hand, they are the primary driving force.

2.6.5 The Moon’s Non-sphericity The gravitational potential of the Moon’s non-sphericity takes the same form as Eq. (2.101), except that the gravitational constant is now the Moon’s gravitational constant (GMm ) and the equatorial radius of the reference ellipsoid is now am = 1, 738 km.

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2 Spacecraft Orbits and Orbital Dynamics l

l GMm % % am  V = Plm (sin ϕ)(Clm cos mλG + Slm sin mλG ) r r m=0

(2.113)

l≥2

Following Eq. (2.102), we can also obtain the normalized form of the perturbing potential, where the coefficients are C lm and S lm . Currently, the commonly used gravitational constants of the Moon are from two models, LP75G and LP165, developed by the NASA Jet Propulsion Laboratory (JPL). We can obtain the following useful information by analyzing these coefficients. 1.

2.

The dynamic oblateness J2 is rather small (10–4). The odd zonal term C2l−1,0 (l ≥ 2) that heavily affects eccentricity is close to J2 in terms of the order of magnitude (difference within 10–1), resulting in large-amplitude long-period variations in the eccentricity of the orbit around the Moon. The perturbation in eccentricity may lead to orbital lifetime issues. That is, when the perigee altitude is smaller than the equatorial radius of the Moon, the spacecraft will collide with its primary and terminate its life. Owing to the slow rotation of the Moon, the effects of the tesseral terms on the orbits of spacecraft moving around the Moon are clearly different from those on low-Earth-orbits. For example, the magnitude of J22 here is 10–5 . In the perturbation solution, J22 contains both short- and long-period terms.

2.6.6 Mars’ Non-sphericity The first satellite to measure the mass and oblateness of Mars was Mariner-4. The study of the global gravity field of Mars commenced with Mariner-9 (M9). However, the detailed gravity models GMMl and Mars50C were obtained from the data collected by M9 and the Viking probes (V01 and V02) only in the 1990s. The perturbation model of Mars’ non-sphericity is also described by the gravitational potential within the body-fixed frame. Its expression is identical to Eq. (2.101); therefore, it will not be repeated here. We present a few important parameters of Mars: the gravitational constant GMars = 42,828.44 km3 /s2 , mean equatorial radius = 3,394.2 km, and oblateness f = 0.0052083. Details about the coefficients of Legendre polynomials can be found in Ref. [3]. The orbital information of the Viking probes revealed low-resolution gravity fields of spherical harmonics having an order ranging from 6 to 12. Owing to the high-altitude orbits of M9 and the Viking probes near the poles, as well as the uneven distribution of orbital data in space, it was difficult to obtain high-resolution gravity models. Based on the orbital variation data from the Mars Global Surveyor (MSG) launched on November 11, 1996; Odyssey launched in 2001; and the Mars Reconnaissance Orbiter launched in 2005, various new models were obtained, such as GMM-1, GMM-2B, MARS50C, MGS75D, MGS95J, and MR0110B. Among them, GMM-2B, MGS75D, MGS95J, and MR0110B are considered more representatives [4].

2.6 Sources of Orbital Perturbations

73

2.6.7 Atmospheric Drag Near Mars The Martian atmosphere is primarily composed of CO2 (95.7%), N2 (2.7%) and Ar (1.6%), and it is remarkably different from the Earth’s atmosphere. Its density is only approximately 1% of the Earth’s atmospheric density. Therefore, in terms of aerodynamic forces and heating, the entry of a spacecraft into the atmosphere of the two planets will be very different. Based on the composition, temperature, isotope features, and physical properties of species, the Martian atmosphere is generally divided into three layers: upper (>200 km), middle (45–200 km), lower ( e2 , with two impulses in the same direction. a In Fig. 3.3, the three vectors e, e1 , and e2 are drawn in the same plane. According to the law of cosines e22 = e12 + e2 − 2e1 e cos(θ1 − θ0 ) Substituting e2 = 2



a μ (V

V1 =

(3.56)

− V1 ) into the above equation yields a 2 a

− V 2

−e cos(θ1 − θ0 ) +

a a

(3.57)

Clearly, V1 is a function of θ1 , where θ1 can be chosen at any position on the orbit. We have Fig. 3.3 Changes in the eccentricity vector caused by two tangential impulses in the same direction

96

3 Orbit Control

 e1 = ζ1 + jη1 = 2

a V1 ejθ1 μ

(3.58)

Considering e = e1 + e2 , we obtain V2 and the orbit control position. V2 =

θ2 =

1 2

   a a − V1 μ a

⎧ ⎨ arccos √

ζ2 ζ22 +η22 ζ2

⎩ − arccos √

(3.59)

(η2 ≥ 0)

ζ22 +η22

(η2 < 0)

(3.60)

where ζ2 = ξ − ξ1 η2 = η − η1 (ii)

a 2 a

< e2 , with two impulses in opposite directions.

According to Fig. 3.4, when the phase difference between the two impulses is 180°, the efficiency of correcting eccentricity is the highest, i.e., |e| = |e1 | + |e2 |. With θ1 = θ0 or θ0 + π , we arrive at V1 =

Fig. 3.4 Changes in the eccentricity vector caused by two tangential impulses in opposite directions

1 4

   a a + δ · e μ a

(3.61)

3.3 Orbit Control for Typical Spacecraft

97

where

δ=

1 When : θ1 = θ0 −1 When : θ1 = θ0 + π

Following the second control execution, the position and magnitude of the velocity change, θ2 and V2 , respectively, are given by θ2 = θ1 +π

− δ · e V2 = 41 μa a a (2)

 (3.62)

Inclination-keeping strategies

The variation in the inclination of an SSO satellite due to perturbations is rather small, i.e., normally 0.03◦ /a. Therefore, for satellites with a short lifetime, it is common practice to reduce the perturbing effects on inclination by placing an initial offset, without any special inclination correction. In cases where the satellite has a long lifetime or a high orbit control accuracy is required, proper inclination corrections are necessary. For nearly circular orbits, the control equations of the inclination and RAAN are cos u di = W dt na d sin u = W dt na sin i

(3.63)

It can be seen from the above equation that a force normal to the orbital plane can simultaneously change the inclination and RAAN. Therefore, to avoid changing RAAN, the control impulse should be exerted in the arcs near the ascending or descending nodes (u = 0◦ or u = 180◦ , respectively). We can also derive the relationship between the velocity change and inclination change from the above equation as  Vz =

Vs · i(u = 0◦ ) −Vs · i(u = 180◦ )

(3.64)

The velocity change is in the same direction as the positive normal vector of the orbit when Vz is positive, and it is in the opposite direction if Vz is negative.

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3 Orbit Control

3.3.2 Perturbation Analysis and Stationkeeping of HEO Spacecraft HEO satellites are mainly geostationary satellites, including those moving along regular geostationary orbits and inclined geosynchronous orbits. The stationkeeping of geostationary satellites uses the most typical stationkeeping technology, where the latitude and longitude deviations of a satellite with respect to a fixed point are constrained to a predefined error range. This is especially important for broadcast satellites and spot beam communication satellites because any position error of the satellite will result in decay in the gain of the ground receiving antenna. Geostationary satellites are normally required to have a high stationkeeping accuracy of 0.05° to 0.1°. The realization of such accuracy relies heavily on the onboard control system. 1.

Perturbed motion of HEO spacecraft

Geostationary spacecraft are primarily subject to three types of orbital perturbations: perturbations due to the Earth’s non-spherical gravitational field, gravitational forces from the Sun and Moon, and solar radiation pressure forces. The acceleration terms generated by these perturbations relative to the Earth’s gravitational acceleration are the zonal terms of the Earth’s non-sphericity, which are on the order of 10–4 ; tesseral terms of the Earth’s non-sphericity, which are on the order of 10–7 ; solar and lunar perturbations, which are on the order of 10–4 to 10–5 ; and solar radiation pressure term, which is on the order of 10–7 . In order to avoid the challenges in numerical computation and analysis that are associated with a small eccentricity and small inclination, we introduce a new set of non-singular orbital elements, namely, a, ix , iy , ex , ey , and λ, where ix and iy are the two components of the inclination vector i, ex and ey the two components of the eccentricity vector, and λ the mean longitude of the satellite. Their correlations with the elements of the Keplerian elliptical orbit are given by a=a

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

ix = sin i cos  iy = sin i cos  ex = e cos(ω + )⎪ ⎪ ⎪ ⎪ ⎪ ey = e sin(ω + ) ⎪ ⎪ ⎪ ⎭ λ = SA − S

(3.65)

where SA = ω +  + M is the mean right ascension of the satellite and S is the Greenwich hour angle. Considering that ix , iy , ex , and ey are all small quantities, we rewrite the perturbation equations in the following approximate form:

3.3 Orbit Control for Typical Spacecraft

2 ∂R da = dt Vs ∂λ dix −1 ∂R = dt Vs as ∂iy diy 1 ∂R = dt Vs as ∂ix −1 ∂R dex = dt Vs as ∂ey dey 1 ∂R = dt Vs as ∂ex dλ 2 ∂R = n − ne − dt Vs ∂a

99

(3.66A)

or alternatively, 2 T ne ˙ix = 1 cos( + u)W Vs ˙iy = 1 sin( + u)W Vs 1 2 e˙ x = sin( + u)S + cos( + u)T Vs Vs −1 2 e˙ y = cos( + u)S + sin( + u)T Vs Vs 1 λ˙ = n − ne − S Vs a˙ =

(3.66B)

where Vs is the velocity of thegeostationary orbit, u = ω + f , as the radius of the 2

geostationary orbit, and ne = k ams e the angular velocity of the Earth’s rotation. For the perturbations discussed above, it is usually unrealistic to obtain analytical solutions to their equations. In practice, therefore, we often employ approximation methods. Now, let us analyze the three types of perturbations by using the mean element method. (1)

Perturbations due to Earth’s non-sphericity

(a)

Zonal harmonic perturbations

For zonal harmonic perturbations, we only consider the J2 term. Its perturbation function R is

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3 Orbit Control

 2   Re 3 2 −k 2 me 1 J2 sin ϕ − R= r r 2 2

(3.67)

We substitute sin ϕ = sin i sin(f + ω) into the above equation and average M . By using  a 3 r

− 3  a 3 = 1 − e2 2 , cos 2f = 0 r

(3.68)

we obtain the long-term and short-period terms, R and Rs , respectively. R = R + Rs

(3.69A)

  − 23 3 k 2 me R2e 1 1 2 1 2  2 2 1 − e J2 − i i − − e (3.69B) x y 2 a3 3 2 x 2 y    − 23   1 1 3 k 2 me R2e a 2  2 2 2 − sin i Rs = R − R = J2 · − 1 − ex − ey 2 a3 r 3 2    1  a 3  2 ix − iy2 cos 2u1 − 2ix iy sin 2u1 (3.69C) + 2 r R=

where u1 = ω +  + f . Substituting R into the perturbation equations yields the long-term effects of the zonal term. Considering a small eccentricity and small inclination, we know that zonal harmonic perturbation only has long-term effects on the mean longitude λ:  2 Re dλ 3 = n − ne + J2 ne dt 2 a

(3.70)

In the above equation, n − ne represents the change in mean longitude caused by the difference between the mean angular velocity of the orbit and the angular velocity of the Earth’s rotation. Obviously, the other term describes the change in mean longitude caused by the zonal term:  2  Re 3 J2 ne = 9.8(◦ ) a 2 as

(3.71)

This also suggests that the role of the J2 term for a stationary satellite is to elevate the theoretical semi-major axis, thereby reducing n and, in turn, making the right-hand side (RHS) of Eq. (3.70) zero. Substituting Rs into the perturbation equation and using

3.3 Orbit Control for Typical Spacecraft

 a 3 r

101





≈ 1 + 3e cos f + · · · ≈ 1 + 3 cos λ + ne t ex + sin λ + ne t ey

(3.72)

we find the short-period perturbation on eccentricity exerted by the zonal term, with an amplitude of  2 3 Re J2 = 3.7 × 10−5 2 as (b)

(3.73)

Tesseral harmonic perturbations

In the potential function of the Earth’s gravitational field, longitude-related terms are called tesseral terms. For satellites in general orbits, they only generate short-period perturbations. However, when the satellite’s mean angular velocity and the Earth’s angular velocity are nearly commensurable, e.g., in the case of a geostationary orbit and GPS orbit, these terms result in long-period perturbations with large amplitudes. For a geostationary orbit, if we neglect the second- and higher-order small variables in the perturbation equations while only considering the J22 term, we have  2 Re k 2 me 3J22 cos 2(λ − λ22 ) R=− r r R=−

k 2 me 2 R 3J22 cos 2(λ − λ22 ) a3 e Rs = R − R

(3.74)

(3.75) (3.76)

Substituting R into the perturbation equations, we know that the tesseral term imposes long-term perturbations on the semi-major axis a and mean longitude λ: 2k 2 me R2e da = 6J22 sin 2(λ − λ22 ) dt na3

(3.77)

dλ 6k 2 me R2e = n − ne − 3J22 cos 2(λ − λ22 ) dt na5

(3.78)

Owing to the long-term perturbation on a and thereby on n, λ experiences an accelerated drift, which is a major source of perturbation that needs to be overcome in longitude stationkeeping. For a fixed-point satellite in a geostationary orbit, the third term on the RHS of Eq. (3.78) is a constant that determines the nominal semi-major axis of the satellite’s orbit. Differentiating both sides of Eq. (3.78) yields 3 dn d2 λ =− = dt 2 dt 2



3n da k 2 me 1 da =− a3 a dt 2a dt

(3.79)

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3 Orbit Control

Fig. 3.5 Relationship ..

λ&&

between λ(λ) and λ

λ

Taking the nominal values of a geostationary orbit for n and a, we find the relationship between the perturbation acceleration of mean longitude and the fixed-point longitude λ: d2 λ 3n2 R2 = − e2 e 6J22 sin 2(λ − λ22 ) 2 dt as

(3.80)

Converting the unit to (◦ )/d 2 , we obtain the following formula: d2 λ R2e = −2160π 6J22 sin 2(λ − λ22 ) dt 2 as2

(3.81)

..

Based on the above analysis, we can calculate the λ(λ) curve, which is shown ..

in Fig. 3.5, where the solid line represents λ(λ) caused by multiple perturbation terms including J22–J44. It can be easily seen that J22 is dominant. Moreover, at ..

..

λ = −105◦ , −11.5◦ , 75.1◦ , 161.9◦ , λ = 0, whereas at λ = 117◦ , λ takes the ..

maximal value, i.e., λ = 1.98 × 10−3 (◦ )/d 2 . Among the four equilibrium points, λ = −105◦ and 75.1° are stable, whereas λ = −11.5◦ and 161.9° are unstable. (2)

Perturbations due to gravitational forces from the Sun and Moon

According to the three-body perturbation equations, the solar/lunar perturbation function can be written as

3.3 Orbit Control for Typical Spacecraft

! R = k mj 2

" 1 r cos θ − (j = m, s) rj2 r − rj

103

(3.82)

where rj is the heliocentric/selenocentric distance vector from the Earth’s center, mj the mass of the Sun/Moon, and cos θ = rj0 · r 0 , with rj0 and r 0 being the unit vectors of rj and r. 1 Normally, we expand |r−r | as a Legendre polynomial such that j

    2   3  r 3 5 r 1 1 3 2 3 1+ cos (θ ) − + cos (θ ) − cos(θ ) R = k mj rj rj 2 2 rj 2 2 #  4   r 35 30 3 cos4 (θ ) − cos2 (θ ) + + ··· (3.83) rj 6 8 8 2



cos θ = (cos u cos  − sin u sin  cos i) · cos uj cos j − sin uj sin j cos ij

+ (cos u sin  + sin u cos  cos i) · cos uj sin j + sin uj cos j cos ij + sin u sin i sin uj sin ij (3.84) Let  =  − j and sin write

ij 2

= ε. Only considering the first-order terms of i, we







cos θ = ε2 · cos u + uj +  + 1 − ε2 · cos u − uj +  



1 + sin i sin ij cos u − uj − cos u + uj 2

(3.85)

Similarly, we can also obtain formulas for cos2 θ and cos3 θ . Let R = R0 + R2 + R3 + R4 . From Eq. (3.83), we have R0 =

k 2 mj rj

    k 2 mj r 2 3 1 cos2 θ − · rj rj 2 2   3  2 k mj r 5 3 3 cos θ − cos θ · R3 = rj rj 2 2   4  2 k mj r 35 30 3 cos4 θ − cos2 θ + · R4 = rj rj 8 8 8

R2 =

(3.86)

With Eq. (3.86), if we assume e ≈ 0 and i ≈ 0, the mean elements R2 , R3 , and R4 can be calculated. For lunar perturbations, considering that the eccentricity of the Moon’s orbit, em = 0.0549, is not negligible, we can expand em and keep terms up

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3 Orbit Control

to the second order as the long-term perturbations and the first-order terms as longperiod perturbations. If we let R = R0 + R2 + R3 and substitute it into Eq. (3.66A), both long-term and long-period perturbations from the Sun/Moon can be obtained. Here, we will only focus on the long-term perturbations. (a)

Long-term perturbations on inclination due to solar and lunar gravitational forces

Based on the apparent motion of the Sun, s = 0 and is = 23.44◦ . The orbital plane of the Moon undergoes precession with respect to the ecliptic poles, the period of which is 18.6 years. Therefore, m and im both exhibit periodic variations, which should be considered in the perturbation analysis. Let I (= 5.145◦ ) be the angle between the Moon’s orbit and ecliptic and βm the ecliptic longitude of the ascending node of the Moon on the ecliptic. βm = 259.183◦ − 0.05295◦ · t

(3.87)

where t denotes the Julian day starting from 12:00 on January 1, 1900. Substituting these values into the perturbation equations, we have 

dix = ˙i sin α = −0.134 sin βm + 0.003 sin 2βm ◦ a dt 

diy = ˙i cos α = 0.859 + 0.107 cos βm − 0.0025 cos 2βm ◦ a dt

(3.88)

where ˙i and α are the amplitude and direction angle of the rate of change of the inclination vector, respectively. Since the period of βm is 18.6 years, Fig. 3.6 shows the variations of ˙i and α in 18 years. The amplitude of the inclination perturbation rate is 0.75(◦ )/a ∼ 0.96(◦ )/a. The offset between the perturbation direction and the vernal equinox direction (i.e., the y-axis) is within 9°. In other words, the orbital normal approach approximately to the vernal equinox direction. (b)

Perturbations on mean longitude due to solar and lunar gravitational forces

Solar and lunar gravitational forces exert long-term and long-period perturbations on the satellite’s mean longitude:   

Aj 3 dλ = n − ne − 4 − 24ε2 + 24ε4 1 + ej2 dt 3 2

2 4 + 12ej 1 − 6ε + 6ε cos Mj



+ 24ε2 1 − ε2 cos 2Mj + 2ωj



+ 84ej ε2 1 − ε2 cos 3Mj + 2ωj



− 12ej ε2 1 − ε2 cos Mj + 2ωj

(3.89)

105 10

Amplitude of rate of change of inclination vector /deg

1 Amplitude Direction angle

0

0.8

0.6

0

2

4

6

10 8 Time / a

12

14

16

-10 18

Direction angle of rate of change of inclination vector / deg

3.3 Orbit Control for Typical Spacecraft

Fig. 3.6 Magnitude and direction angle of the rate of change of the inclination vector i

where sin 2j = ε. It can be seen from the above equation that solar and lunar gravitational forces impose long-term perturbations on the mean longitude, with the solar part being −0.00501 ∼ −0.00386(◦ )/d and the lunar part being −0.00205(◦ )/d . The lunar effects are shown in Fig. 3.7 as an example. The figure shows that the half-moon period term is dominant, with an amplitude of ±0.003(◦ )/d . (3)

Perturbations due to solar radiation pressure

The magnitude of perturbation due to solar radiation pressure is dependent on the satellite’s area-to-mass ratio. At present, all HEO satellites have large solar panels installed, possessing high area-to-mass ratios. Therefore, eccentricity perturbations caused by solar radiation pressure are an important concern in stationkeeping. Assume that the total effective area of solar panels and the satellite is A, altitude angle towards the Sun β, satellite mass ms , reflectance σ , and solar constant K = 1353 W/m2 . The perturbation acceleration caused by solar radiation pressure is given by a = (1 + σ )

A K cos β −rs0 ms

(3.90)

where rs0 represents the direction from the satellite to the Sun or, approximately, the unit vector from the Earth’s center to the Sun.

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3 Orbit Control -3

Rate of change /deg/day

-1

23.5+5 23.5-5

-2 -3 -4 -5 -6 -7

Periodic variation of rate of change/deg/day

x 10

0

50

100

150

200

250

300

350

400

-3

3

x 10

23.5+5 23.5-5

2 1 0 -1 -2 -3

0

50

100

300 250 200 150 Mean argument of perigee of the Moon’s orbit /deg

350

400

Fig. 3.7 Perturbations of the satellite’s mean longitude caused by the Moon’s gravitational forces

In most cases, the solar panels follow the Sun, but their normal does coincide with the radiation direction. Hence, when using Eq. (3.90) to calculate solar radiation pressure, the effective area should be the projected area of radiation on the panels. Let the unit vectors in the radial, transverse, and normal directions of the satellite’s orbit be S 0 , T 0 , W 0 , respectively. Project the perturbation acceleration in Eq. (3.90) in the three axes to obtain aS , aT , aW and then substitute them into the perturbation equations. It can be seen that solar radiation pressure imposes no long-term perturbation  on any orbital element. Only the eccentricity vector ex , ey experiences long-period perturbations due to radiation pressure, whereas the other elements only experience short-period perturbations. Long-period perturbations on eccentricity caused by solar radiation pressure can be calculated from    3 3(1 + σ )KA 1 dex =− cos is 1 − sin2 is sin us + sin2 is sin 3us dt 2nams 8 8    dey 3 2 3(1 + σ )KA 1 2 1 − sin is cos us + sin is cos 3us (3.91) =− dt 2nams 8 8

3.3 Orbit Control for Typical Spacecraft

107

(4)

Summary

(a)

Long-term and long-period perturbations on the inclination that are caused by solar/lunar gravity (mainly the frequency term 2ns ) are the primary factors affecting north–south stationkeeping. East–west stationkeeping of the satellite depends on the mean longitude and eccentricity. There are constant mean longitude drift terms in solar/lunar perturbations and the Earth’s non-sphericity perturbations, which can be counter-balanced by properly adjusting the nominal semi-major axis of the geosynchronous orbit, i.e.,

(b)

as∗ = ask + as = ask −

2 ask np 3 ne

(3.92)

where $ ask =

3

k 2 me = 42164.17 km n2e

  Re 2 5k 2 me 2 ne − np = − J2 R 3J22 cos 2(λ − λ22 ) 3 as ne as5 e       3 2 3k 2 mm 2 3 2 2k 2 ms 2 2 2 − − sin is 1 + es − − sin im 1 + em 3ne as5 3 2 2ne am5 3 2 By substituting the values into the above equations, we arrive at as∗ = 42165.063+ 0.69 sin2 im km. The accelerated drift of the mean longitude caused by tesseral harmonic perturbations is a major concern in the east–west stationkeeping of a satellite. The long-term effects of solar and lunar gravitational forces on the mean longitude should also be considered in high-accuracy stationkeeping. For satellites with high area-to-mass ratios that demand accurate stationkeeping, solar radiation pressure is also a major source of perturbations on eccentricity. 2.

Impulsive stationkeeping control for HEO spacecraft

According to the perturbation analysis, orbital elements have short-period components that are approximately daily, long-period components that have the solar or lunar frequency (or its integer multiples), and long-term components. Clearly, for the purpose of energy optimization and operation convenience, the control period must be larger than the short period. Furthermore, the control variables should not include the short-period terms; therefore, they need to be separated from the orbital elements. The primary goal of orbital station keeping is to eliminate long-term and long-period components. If the control accuracy allows, it is also possible to preserve

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3 Orbit Control

the long-period components, e.g., the solar/lunar long-period perturbations on the mean longitude. The onboard control engines are usually chemical thrusters generating a thrust on the order of newtons. Each control maneuver occurs rapidly relative to the 24-h period of the geostationary orbit; therefore, it can be considered impulsive. Another possible type of control engine is the electric thruster. Electric thrusters generate rather small thrusts; therefore, errors associated with the impulsive approximation tend to be large. Their control strategies will be described later in this section. For impulsive orbit transfer, integrating the perturbation equations in Eq. (3.66B) yields 3VT 2a VT or λ˙ = − Vs rs 1 2 ex = sin SA · VR + cos SA · VT Vs Vs 1 2 ey = − cos SA · VR + sin SA · V Vs Vs Vw ix = cos SA Vs Vw iy = sin SA Vs 2 λ = − VR Vs

a =

(3.93)

where VT , VR , and VW are the tangential, radial, and normal components of the change in velocity, respectively, and Vs is the velocity of the geostationary orbit. Next, we present stationkeeping control strategies for both east–west and north– south directions. (1)

East–west longitude stationkeeping strategies

(a)

Correction for drift rate

East–west longitude control mainly refers to the correction for the drift rate. That is, by properly selecting an initial value of the drift rate, the satellite is always controlled within the error box during a certain period of time. In most cases, east–west control only needs to consider the long-term mean longitude perturbations caused by the tesseral terms. However, in cases where the fixed-point accuracy requirement is stringent, e.g., 0.05°, the solar/lunar long-term effects on the mean longitude should also be taken into account when determining the control strategy. If the satellite has a small area-to-mass ratio, the eccentricity perturbations will also be small and can be neglected in the control strategy; they only need to be considered in error distribution. Satellites having a large area-to-mass ratio require the combined control of the semi-major axis and eccentricity. When determining the control strategy, the first step is to make the desirable stationkeeping accuracy λ clear. Then, through error distribution, the short-period

3.3 Orbit Control for Typical Spacecraft

109

terms of mean longitude perturbations, longitude oscillation amplitudes caused by the eccentricity, mean longitude measurement errors, and control output errors constitute the outer boundary of the control envelope. The remaining part is the allowable ring of drift, denoted by λd . ..Take, for example, a satellite at 125°E experiencing a perturbing acceleration of λ = −1.991 × 10−3 (◦ )/d 2 . Owing to the initial drift rate and mean longitude perturbing acceleration, the satellite exhibits mean longitude drift. When the satellite drifts close to the edge of the drift ring, it is possible to execute control and invert the drift. Consequently, the satellite will always be in the designated longitude range. To achieve the minimum fuel consumption and. maximum effective

control period, the satellite should be controlled within the λ, λ drift ring shown in Fig. 3.8. With all types of errors taken into account, the satellite, in reality, orbits within the drift region between CBA and C’B’A’. Because λ and λ are very close to each other, we use only λ in the subsequent analysis. Let λ = λf + α, where λf denotes a fixed-point longitude. Expanding f (λ) near λf yields



α¨ = λ¨ = f λf + f λf · α = g1 + g2 · α

(3.94)

where  



Re · 6J22 sin 2 λf − λ22 g1 = f λf = −2160π as  

Re · 12J22 cos 2 λf − λ22 g2 = f λf = −2160π as Fig. 3.8 Schematic of the ..

drift ring with λ > 0

(3.95A) (3.95B)

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3 Orbit Control

Fig. 3.9 Perturbation of the inclination vector

iy

i*

iN

ix

i0

By multiplying both sides of Eq. (3.94) with 2α˙ and integrating, we obtain the relationship between α and α: ˙ 2g1 α + g2 α 2 − α˙ 2 = 2g1 α0 + g2 α02 − α˙ 02

(3.96)

If g2 is negative, zero, or positive, the phase plot of α versus α˙ will be an ellipse, a parabola, or a hyperbola, respectively. We now calculate the relevant parameters with the drift ring shown in Fig. 3.8 as an example. If we assume that point C on the western side has the coordinates (α0 , α˙ 0 ), then point B corresponds to (−α0 , 0). Based on Eq. (3.96), we write −2g1 α0 + g2 α02 = 2g1 α0 + g2 α02 − α˙ 02

(3.97)

√ α˙ 0 = −sgn(g1 )2 g1 α0 (Point C takes positive values)

(3.98)

Hence,

It can be seen that α0 should take the same sign as g1 . To calculate the period T, because α only has an insignificant impact on α, ¨ we assume g2 = 0. Thus, integration yields √ 2 g1 α0 T =− g1

(3.99)

After a drift period, the satellite returns to the initial position. At this point, a driftrate correction should be executed, and the target velocity change can be directly obtained from Eq. (3.93):

3.3 Orbit Control for Typical Spacecraft

111

1 1 VT = − rs α˙ = − rs α˙ s − α˙ f 3 3

(3.100)

where α˙ f is the measured mean longitude drift rate. (b)

Simultaneous correction for drift rate and eccentricity

When the order of magnitude of eccentricity perturbation is outside the error circle, in addition to drift rate control, it is necessary to control eccentricity. The control is similar, except that in the east–west longitude drift ring, the daily oscillations due to eccentricity should also be considered now. |α + 2e sin M | ≤

1 λd 2

(3.101)

Alternatively, (α + 2e sin M )max − (α + 2e sin M )min ≤ λd

(3.102)

The on-orbit correction for the mean longitude drift rate usually relies on tangential velocity control. In some special cases (e.g., when perturbations due to solar radiation pressure are not significantly large), single impulse control can simultaneously correct for both variables. In more general cases, double impulse control is necessary. Double impulse control with the minimum fuel consumption features two control maneuvers that are separated by half an orbital cycle. With the two maneuvers, both the drift rate and eccentricity gain the desired changes. However, when the control quantity of the eccentricity is large, an intermediate drift rate is generated after the first control maneuver. The consequent half-day drift could be considerable and must, therefore, be considered [1]. Assuming an interval of time of N days between two control executions,

the eccentricity perturbation during this interval can be determined as e ex , ey . If the half-day drift is neglected, we have 1 e0 = − e 2

(3.103)

eN = e0 + e = −e0

(3.104)

Thus,

(2)

North–south stationkeeping strategies

Stationkeeping in the north–south direction primarily aims to keep the inclination smaller than a given value. Let imax be the maximum allowable inclination: i ≤ |imax |

(3.105)

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3 Orbit Control

When determining the north–south stationkeeping strategy, factors such as shortperiod perturbing terms of inclination and orbit measurement errors should be deducted in the error distribution. This suggests that the mean inclination can drift within an even smaller range (a circle with a radius of i∗ ). From the long-term and long-period perturbing effects of solar and lunar gravitational forces on the inclination, we know that both the direction and rate of inclination drift are related to βm , as described by Eq. (3.88). The corresponding inclination drift rate is rather low, which can be regarded as a constant in one north–south stationkeeping cycle. Thus, in one di control cycle, didtx and dty can be taken as constants (Fig. 3.9). Let dix dt diy dt

.

=  i · sin α 0 .

=  i · cos α

(3.106)

0

where $

2



diy dt   ix α = arctan iy

˙i =

dix dt

2

+

(3.107) (3.108)

Given the range of mean inclination i∗ , it is easy to determine the initial target inclination with the longest correction period as ix0 = −i∗ sin α iy0 = −i∗ cos α

(3.109)

After N days, the inclination reaches i∗ again under perturbations: ixN = i∗ sin α = −ix0 iyN = i∗ cos α = −iy0

(3.110)

Clearly, this control strategy also ensures the minimum fuel consumption. In other words, the study of inclination correction aims to, by properly choosing a target inclination, maintain the inclination after control within the allowable range for the longest possible period of time, or to achieve the minimum fuel consumption for inclination control during the mission cycle of the satellite [1]. 3.

Low-thrust stationkeeping for HEO spacecraft

The essential principle of electric thrusters is to convert electrical energy into the kinetic energy of the propellent so that the accelerated exhaust gas generates thrust.

3.3 Orbit Control for Typical Spacecraft

113

Various forms of electric propulsion have been proposed since the commencement of research in this field in the 1950s. Many practical electric propulsion systems have been employed in satellites and space probes. Even today, novel electric propulsion ideas are emerging. Electric propulsion can be roughly categorized into the following three types. • Electrothermal. This type is similar to chemical propulsion, but it utilizes electrical energy to heat the propellent and increase its enthalpy, resulting in a higher specific impulse. • Electrostatic. In this type, a propellent with relatively low ionization energy is ionized and then accelerated in a static electric field. This technology generates the highest specific impulse among all the electric propulsion types. • Electromagnetic. In this type, the propellent atoms form plasma through electron bombardment, which is then accelerated under the combined action of both electric and magnetic fields. The specific impulse generated by this type of propulsion is normally higher than that of electrothermal propulsion but lower than that of electrostatic propulsion. Electric propulsion has been extensively used in the control of orbital maneuver and long-term stationkeeping for geostationary communication satellites. For instance, the Boeing 702 platform and ESA Artemis have four electric thrusters (xenon ion or Hall thrusters) installed to achieve multiple control objectives including north–south stationkeeping, combined east–west and north–south stationkeeping, and transfer orbit control. Other examples include the SSL 1300 series, which adopts electric propulsion for north–south stationkeeping, and Russian communication satellites such as Ekspress, which use electric propulsion for both north– south and east–west stationkeeping. In this section, we will also use an example of east–west and north–south stationkeeping with four electric thrusters to introduce low-thrust stationkeeping control strategies. The common electric propulsion stationkeeping strategy for geostationary satellites is to take an orbit measurement cycle (N days) as the most basic control element, in which the first N1 days are dedicated to orbit measurement, whereas the later N2 days are dedicated to stationkeeping. More specifically, the electric thrusters are shut during the first N1 days of orbit measurement, whereas stationkeeping and angular momentum unloading are conducted every day during the remaining N2 days. This control scheme provides angular-momentum unloading coupled to stationkeeping control with prior information, facilitating the planning of angularmomentum unloading and reducing the required number of actuations of the thrust vectoring mechanism. (1)

Configuration of electric thrusters

In a cone arrangement, four electric thrusters are installed at the four corners of the satellite’s side, facing away from the Earth in a symmetric manner: two are at the northern side (NW and NE), and the other two are at the southern side (SW and SE), as shown in Fig. 3.10. Figure 3.11 shows the installation angles of a thruster. For each

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3 Orbit Control

Fig. 3.10 Schematic of configuration of electric thrusters

Fig. 3.11 Definitions of thruster installation angles

Thruster

thruster, α is defined as the absolute value of the angle between the projection of the installation vector on the xoz plane of body-fixed frame and the -z axis, whereas θ is defined as the absolute value of the angle between the projection of the installation vector on the yoz plane of body-fixed frame and the -y axis. Ideally, the installation of the electric propulsion system should have the following features.

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115

Table 3.1 Polarity of the electric thrusters Component of unit velocity change

NW

NE

SW

SE

VT

+et

−et

+et

−et

VN

−en

−en

+en

+en

VR

−er

−er

−er

−er

(a) (b)

The four thrusters are installed in pairs, symmetric about the xoz and yoz planes. In other words, each thruster has the same α and θ . By adjusting the thrust vectoring mechanism, it is guaranteed that the extended line of thrust vector will pass through the center of mass of the satellite, generating no additional torque.

Converting α and θ to unsigned unit vectors in the directions of R, N, and T, we write ⎤ ⎡ ⎤ ⎡√ tan α·tan θ 2 2 2 et tan α·tan θ+tan θ+1 ⎥ ⎢ (3.111) e = ⎣ en ⎦ = ⎣ √tan2 α·tan21θ+tan2 θ+1 ⎦ tan θ √ er 2 2 2 tan α·tan θ+tan θ+1

Polarity of the thrusters is given in Table 3.1. Assume an impulsive velocity change vector V = [VR , VT , VN ]. At of 1, by considering the changes in orbital elements , + a mean longitude λ, D, ex , ey , ix , iy , the governing equations of the stationary orbit can be written as ⎧ 3VT ⎪ D = − · ωE ⎪ ⎪ ⎪ Vs ⎪ ⎪ ⎪ ⎪ 2VR ⎪ ⎪ λ = − ⎪ ⎪ ⎪ Vs ⎪ ⎪ ⎪ ⎪ V VT R ⎪ ⎪ sin l + 2 cos l ⎨ ex = Vs Vs VR VT ⎪ ⎪ ⎪ ey = − cos l + 2 sin l ⎪ ⎪ Vs Vs ⎪ ⎪ ⎪ ⎪ VN ⎪ ⎪ ⎪ ix = cos l ⎪ ⎪ Vs ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iy = VN sin l Vs (2)

(3.112)

North–south stationkeeping strategies

The primary objective of north–south stationkeeping for a geostationary satellite is to reduce inclination using normal thrust such that the geographic longitude of the satellite remains in the dead zone.

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Perturbations on the inclination include long-term, half-year periodic, and halfmonth periodic terms. Electrically propelled satellites can adjust the control quantity of north–south stationkeeping based on the values of the thrust angles α and θ . Small values of α and θ only eliminate the long-term components of inclination perturbations, whereas the half-year components are partially eliminated or not eliminated at all. On the other hand, when α and θ are sufficiently large, both half-year and longterm components can be eliminated, resulting in a higher accuracy of north–south stationkeeping. Not controlling the half-year term may also have some benefits, e.g., a more uniform control quantity within each cycle and a less stringent requirement on the installation angles of the electric thrusters. In the current control cycle, the daily control quantities within N2 days of stationkeeping are 

ix ctr1 = ix1 +ix2

iyctr1 = iy1 when : iy ≥ 0 or iyctr1 = iy1 +iy2 when : iy < 0 - N day

where ix1 = −

0 day

( didtx )Day

N 2 day

(3.113)

- N day  diy 

and iy1 = −

0 day

dt

N 2 day

Day

are the daily mean perturbation i

y x and iy2 = − N 2 day are the terms to be eliminated within N days and ix2 = − N i 2 day additional control quantities caused by inclination exceeding the threshold. When iy2 > 0, there is no need to eliminate this term, because in this case, the satellite can take advantage of the kinematic tendency of the inclination. The angle of deviation of the thruster firing position with respect to a right ascension of 90° and 270° is given by

κ = a tan 2 −iyctrl , −ixctrl − 90◦

(3.114)

The variable κ is mostly used to determine the firing position for stationkeeping. (3)

Eccentricity control strategies

The primary objective of eccentricity control for a geostationary satellite is to decrease eccentricity using radial and tangential thrust such that the periodic daily oscillations of the geographic longitude of the satellite can be reduced. Geostationary satellites with an electric propulsion system normally have a larger area-to-mass ratio than regular satellites, leading to a larger perturbation circle. For such large perturbation circles, the corresponding control circles can be viewed to play the role of maintaining the eccentricity near the origin. Therefore, eccentricity control can be divided into two parts: (a) eliminating the motion of the eccentricity perturbation circle and (b) eliminating the long-term components of eccentricity perturbation, offset of the perturbation circle caused by shading, and other error terms such as the eccentricity-control deviation caused by the burn-time calculation for north–south stationkeeping. The first term of the daily control quantity is written as

3.3 Orbit Control for Typical Spacecraft

117

⎧ (cos(as + ns · N day) − cos as ) ⎪ ⎪ ⎨ exctrl1 = −R · cos is · N2day + n · N day) − sin αs ) (sin(α ⎪ s s ⎪ ⎩ eyctrl1 = −R · N 2day

(3.115)

The second term of the daily control quantity is ⎧ ex0 ⎪ ⎪ ⎨ exctrl2 = − N 2day e ⎪ ⎪ ⎩ eyctrl2 = − y0 N 2day

(3.116)

where ex0 and ey0 are the components of the mean initial eccentricity within N2 days of stationkeeping. (4)

Mean longitude control strategies

The primary objective of mean longitude control is to maintain the mean longitude near the desired value by using the tangential thrust, such that the mean longitude of the satellite is within the dead zone where the daily oscillations in longitude are removed. Here, we will introduce mean longitude control using a low-thrust drift ring, the process of which is shown in Fig. 3.12. A drift ring is different from chemically propelled stationkeeping. If the orbit determination period, N days, is regarded as the operation period of the mean longitude drift ring, then the first N1 days of the orbit determination segment are not controlled, whereas the remaining N2 days are controlled. N2 is an even number, and N 2 N 1. Since the radial component always exists when the thrusters are engaged, we can say that jumping occurs in the mean longitude drift rate of the controlled segment. We can further examine this with an example, where α = 12.9◦ , θ = 58.7◦ , single engine thrust F = 0.1 N, and the satellite mass is 4453.1 kg. If we remove the long-term components of inclination and only keep the half-year periodic components, the required V ≈ 0.27 m/s. This corresponds to a radial component of Fig. 3.12 Ideal drift ring curve

Controlled segment Orbit determination segment Target control point

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Table 3.2 Directions of the control quantities of the electric propulsion system Firing position

l1 = 90◦ + κ

l2 = 270◦ + κ

Thruster

NW

NE

SW

SE

iN

–

–

–

–

eR (ex )

eT ey

–

–

+

+

+

–

–

+

DT

–

+

–

+

Vr ≈ 0.22 m/s (negative sign). To eliminate these inclination perturbations, the total burn time of the four thrusters will be 3.3 h. Based on the governing equation for impulsive thrust of the stationary orbit, λ = −2 · Vr /VS , the daily mean longitude drift generated by the radial force is λ = 0.008◦ , or the daily drift rate D = 0.008◦ /day. The drift-rate jumping can be observed in Fig. 3.12. (5)

Thrust allocation strategies for stationkeeping

During north–south stationkeeping control of the satellite, every time the thrusters fire, forces are generated in the radial, tangential, and normal directions. They can control inclination while simultaneously adjusting the eccentricity and mean longitude drift. Table 3.2 lists all the control directions of inclination, eccentricity, and mean longitude drift rate from each thruster. Given the daily control quantities of inclination, eccentricity, and mean longitude drift rate, it is easy to use Eq. (3.112) to obtain the positive velocity change in the normal direction required of each thruster. VN (NW) = −A · D + (−B1 − B2 ) · ex + (−C1 + C2 ) · ey − D · iy VN (NE) = A · D + (−B1 + B2 ) · ex + (−C1 − C2 ) · ey − D · iy VN (SW) = −A · D + (B1 + B2 ) · ex + (C1 − C2 ) · ey − D · iy VN (SE) = A · D + (B1 − B2 ) · ex + (C1 + C2 ) · ey − D · iy (3.117) VS ·cos κ VS ·sin κ 1 · tan α·tan , B1 = 41 · VStan , B2 = 18 · tan , C1 = 41 · where A = 12 θ·ωE θ α·tan θ VS ·cos κ Vs 1 1 C2 = 8 · tan α·tan θ , and D = 4 · cos κ . From the normal control quantity, we obtain the total velocity change as

V =



tan2 α · tan2 θ + tan2 θ + 1 · VN

Now, we obtain the burn time of each thruster.   V (NW) Mass · Ve · 1 − e− Ve tN W = F

VS ·sin κ , tan θ

(3.118)

3.3 Orbit Control for Typical Spacecraft

tNE = tSW = tSE =

119

  V (NE) Mass · Ve · 1 − e− Ve  F V (SW) Mass · Ve · 1 − e− Ve  F V (SE) Mass · Ve · 1 − e− Ve F

(3.119)

where Ve denotes the engine exhaust velocity. It is calculated from Ve = g · Isp, where g is the gravitational acceleration in m/s2 and Isp is the engine-specific impulse in s. The solvable range of stationkeeping is very small. In practical applications, therefore, it is necessary to determine if the required control quantities of stationkeeping are beyond the capabilities of the thrusters. Based on the thrust allocation of the stationkeeping, we know that the range of east–west control is entirely dependent on the magnitude of the north–south control quantity, iy . That is, a larger north–south control quantity leads to a higher capability of east–west control. At initial stationkeeping, the east–west control quantity might be too large. In this case, the north– south control quantity in the current control cycle should be increased. Alternatively, irregular stationkeeping methods can be used for control.

3.3.3 Orbit Transfer of HEO Spacecraft Currently, there are three types of orbit insertion approaches for geostationary satellites. The first type, i.e., the conventional approach, makes use of the onboard chemical propulsion system to achieve multiple orbit transfer maneuvers at apogee. The satellite enters the stationary orbit with an altitude of 42,164 km and an inclination of 0° from a large elliptic orbit that has a perigee altitude of hundreds of kilometers. The second type utilizes the upper stage of the rocket to lift the satellite from the launch orbit to the vicinity of the operation orbit. The control principle is similar to that of the first type. That is, there is normally one orbit control maneuver at apogee before orbit insertion. The third type is based on an electric propulsion system, where the smallthrust technology helps achieve orbit acquisition. This type of orbit insertion takes advantage of the high specific impulse of electric thrusters, significantly reducing the insertion mass of the satellite. However, the associated challenges include a long duration of the transfer orbit and complex orbital control designs. 1.

Optimal strategies for high-thrust orbit transfer

High-thrust orbit transfer control refers to multiple orbit transfer maneuvers at apogee using a chemical (liquid) propulsion system. With predetermined orbit insertion parameters of the satellite (based on multiple factors including the rocket capabilities,

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3 Orbit Control

satellite mass, total engine impulse, and attitude determination capabilities), multiple apogee orbit transfer (if the insertion orbit is a super-synchronous orbit, perigee orbit transfer is normally conducted) is achieved by maneuvers without the radial component. Orbit transfer strategies include the determination of the number of orbit transfer maneuvers as well as the velocity change and time for each maneuver, with the ultimate goal of fuel optimization control. In engineering applications, the optimal strategies are usually determined through the following procedure. Based on orbit insertion parameters and the distribution of ground stations, the number and positions of orbit transfer should be properly selected. That is, the firing time and geographic longitude of each transfer point should satisfy various design constraints. For instance, (a) the burn time of each maneuver should not exceed the single operation duration of the thrusters; (b) the geographic longitude of the firing position should meet the ground control and measurement constraints (i.e., it should be within the ground control and measurement range sometime before and after firing); (c) there should be no overshooting of the orbital period; and (d) the number of maneuvers and intermediate orbits should be minimized. In each orbit transfer calculation, the optimization objective is always the total fuel consumption. That is, the objective function is the sum of three fuel consumption terms: the fuel consumption for the current transfer v1, fuel consumption for the subsequent transfer v2, and fuel consumption for station acquisition v3. The optimization parameters include the instant of firing, instant of shut down, and firing attitude. The technical conditions of each firing should also satisfy the requirements of the current plan in terms of the geographic longitude and target orbit drift rate after control. The equation of motion for the orbit transfer optimization problem is given in Eq. (3.120). The Earth’s non-sphericity term J2 is included, which can improve the accuracy of transfer orbit computation (valid for short-term orbit transfer processes). ⎛

⎞

⎛

x ⎜ ⎜y⎟ ⎜ ⎜ ⎟ ⎜ ⎛ ⎞ ⎜z⎟ ⎜ r ⎟ ⎜ d⎜ d⎜ . ⎟ −μ rx3 ⎜ ⎟ ⎜ ⎝ 2r ⎠ = ⎜ x˙ ⎟ = ⎜ dt dt ⎜ ⎟ ⎜ −μ ry3 ⎜ y˙ ⎟ ⎜ m ⎜ ⎟ ⎜ ⎜ ⎝ z˙ ⎠ ⎜ −μ z r3 ⎝ m

x˙ y˙  z˙ 2  xR2e + + J2 μ r 5 −1.5 + 7.5 rz  z 2  yR2e + + J2 μ r 5 −1.5 + 7.5 r  z 2  zR2e + + J2 μ r 5 −4.5 + 7.5 r − Ispfg0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ fx ⎟ m⎟ fy ⎟ ⎟ m⎟ ⎟ fz ⎟ m ⎠

(3.120)

where fx , fy , and fz the three components of the satellite engine thrust in the geocentric inertial frame; f the engine thrust; and Isp the engine specific impulse. =  The orbit transfer parameters to be optimized are X f1 ψ1 f2 ψ2 . . . . . . fn ψn , where n is the number of burns, fi the true anomaly of

3.3 Orbit Control for Typical Spacecraft

121

each firing point, and ψi the yaw angle during each burn, which is usually defined as the angle between the thrust and local due east. In engineering applications, we commonly use the method of exhaustion. This method involves the selection of a representative set of samples for each parameter within a reasonable range. Then, random combinations of different sampled parameters from a feasible solution space are examined, and the solution that minimizes the objective function represents the optimal orbit transfer parameters. The advantage of exhaustion is that the partial derivatives of the constraint and objective functions need not be solved. However, a large number of samples must be used to find the global optimal solution, making it computationally expensive. A different optimization approach is to use planning methods, such as the simplex algorithm and the penalty method. The penalty method [2] first converts orbit transfer constraints to equality constraints. By solving a quadratic-function minimization problem under equality constraints, the descent direction of the penalty function is obtained. In this manner, the constraints are satisfied, and the objective function is minimized simultaneously. However, it is noted that partial derivatives must be solved in this method. Hence, for multiple orbit transfer problems, numerical differentiation schemes must be used to obtain partial derivatives. Accompanying the computational cost, there also exists a possibility of convergence to local optimal solutions. 2.

Optimal strategies for low-thrust orbit transfer

Low-thrust orbit transfer is also based on electric propulsion, where the system configuration is similar to what we have discussed in Sects. 3.3.2–3.3.3. However, the use of the propulsion system is quite different. Specifically, low-thrust orbit transfer is a multi-lap firing process, in which the optimal orbit transfer control has no analytical solution; it is necessary to solve a two-point boundary value problem to obtain the global optimal control parameters. Studies on this topic have been ongoing for many years, and the technology has been practically applied in the control of transfer from a geostationary transfer orbit to a geostationary orbit. For example, the US newgeneration Advanced Extremely High Frequency (AEHF) military communications satellites employ a hybrid propulsion system consisting of apogee engines, attitude control engines, and a dual-mode Hall electric propulsion sub-system. Boeing also released their new-generation electrically propelled geosynchronous satellite bus 720SP, a more advanced platform of the 702 family. With the adoption of the all-electric propulsion system, the payload ratio of the satellites has remarkably increased. The technology used by this satellite bus not only ensures continuous low-thrust control during orbit transfer, but also realizes on-orbit low-thrust control for east–west and north–south stationkeeping throughout the satellite lifetime. (1)

Optimization problem of electrically propelled orbit transfer

If the modified equinoctial orbital elements are adopted to describe the orbital motion of the spacecraft, the integration speed and accuracy can be improved while avoiding the potential occurrence of singularity. The relationship between the modified equinoctial orbital elements and the classical Keplerian orbital elements is given

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3 Orbit Control

by ⎧

p = a 1 − e2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ex = e cos( + ω) ⎪ ⎪ ⎨ ey = e sin( + ω) 

⎪ hx = tan i 2 cos  ⎪ ⎪ ⎪ 

⎪ ⎪ hy = tan i 2 sin  ⎪ ⎪ ⎪ ⎩ L=+ω+θ

(3.121)

Further, we define the following auxiliary variables: ⎧ ⎪ ⎨ W = 1 + ex cos L + ey sin L Z = hx sin L − hy cos L ⎪ ⎩ C = 1 + h2x + h2y

(3.122)

The orbital dynamic equations expressed with modified equinoctial orbital elements are written as x˙ = A + Ba  T x = p, ex , ey , hx , hy , L

(3.123)

where A=

  p μ W

00000

W 3μ p2

T (3.124)

⎡

⎤ 0 2p 0 ⎥   ⎢ ⎢ W sin L (W + 1) cos L + ex −Zey ⎥ ⎥ p μ⎢ Zex  ⎥ ⎢ −W cos L (W + 1) sin L + ey B= ⎢ ⎥ 0 0 C cos L 2 ⎥ W ⎢ ⎢ ⎥ ⎣ 0 0 C sin L 2 ⎦ 0 0 Z  T a = ar au ah = aeng + adis

(3.125)

(3.126)

Here, a is the acceleration vector of external forces other than the two-body T  gravitational force, ar au ah the components of a in the geocentric-orbit reference frame, aeng the engine-thrust acceleration vector, and adis the acceleration vector of other perturbations. Assuming a spacecraft mass m, an engine thrust magnitude T , and a constant specific impulse Isp , the orbital dynamic equations can be rewritten as [3]

3.3 Orbit Control for Typical Spacecraft

T Bu + BaJ 2 m T u m ˙ =− Isp g0 aeng u =  , u ≤ 1 T m

123

x˙ = A +

(3.127)

-t The performance target of the minimum-time orbit transfer problem is Min t0f dt, where the transfer time is -unknown. In the minimum-fuel orbit transfer problem, the t performance target is Min t0f udt, where the transfer time is usually predetermined. For convenience, we will call these two scenarios the TF problem and MF problem in the subsequent text, respectively. (2)

Optimal control of electric propulsion

The optimal control problem is solved using Pontryagin’s maximum principle. Introduce a costate variable λ = [λp , λex , λey , λhx , λhy , λL ]T corresponding to the modified equinoctial orbital elements and another costate variable λm that corresponds to mass. The Hamiltonian of the TF problem is ˙ H = 1 + λT x˙ + λm m

(3.128)

The Hamiltonian of the MF problem is ˙ H = u + λT x˙ + λm m

(3.129)

The costate variables in both the TF and MF problems satisfy the following differential equations: . ∂  T ∂H ∂ =− λ x˙ + λm m λ=− ˙ = − [λ · (A + Ba) + λm m] ˙ ∂x ∂x ∂x   ∂(A + Ba)T ∂(λm m) ˙ λ+ =− ∂x ∂x . ∂H (3.130) λm = − ∂m

Through a series of derivations, we can obtain the optimal control laws for the TF and MF problems [3]. The optimal control problem is eventually converted to a two-point boundary value problem. (3)

Solving the two-point boundary value problem

When solving the two-point boundary value problem with the conventional indirect method, the single shooting method based on the theory of Newton’s iteration is

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usually adopted. To use this method, the first step is to guess the values of λ and λm at an instant t0 . Then, integrating the differential equations of state and costate variables yields the values of these variables at an instant tf . If the error S between these final values of integration and the given boundary values at tf is greater than the error tolerance, construct gradient matrices for S with respect to λ(t0 ) and λm (t0 ). Iteratively adjust λ(t0 ) and λm (t0 ) using Newton’s method and perform an integral test for the final values, until S = 0. The challenging part of the single shooting method is the guess of initial values of the costate variables and the derivation of the gradient matrices. Even if the gradient matrices are constructed using numerical methods, it is still possible, in some problems, to encounter ill-conditioned matrices during iteration, leaving the problem unsolvable. In other words, this technical difficulty is inevitable as long as the method is based on gradient information, irrespective of the iterative correction method used. To overcome this challenge, a feasible approach is to convert the two-point boundary value problem of optimal control to a parameter estimation problem, which can then be solved using methods such as the unscented Kalman filter (UKF) [3]. The convergence of the UKF parameter estimation method relies on probability estimation theory, rather than gradient information. Hence, it is considered to show excellent convergence performance.

3.3.4 Return Orbit Control for Lunar Exploration Orbit design is a key step in the design of lunar exploration. It also serves as the objective of orbit control. The design problem is equivalent to solving a two-point boundary value problem with all kinds of constraints in an N-body problem. Owing to the complexity of N-body problems, they are often simplified with various mathematical models, leading to different orbit design approaches. Most orbit design methods for deep-space exploration are based on two-body models and adopt patched conic approximation [4]. This is followed by the use of more complex models and numerical search to obtain the nominal orbit. This procedure is the conventional orbit design approach based on impulsive transfer. The successful completion of CE-1 and CE-2 missions [5] has validated the Chinese orbit design for lunar exploration, and the technology has reached maturity. In this section, we focus on the design and control of a cislunar return trajectory, which is also a hot research topic.

3.3.5 Orbit Dynamic Models for Lunar Exploration 1.

Coordinate systems relative to lunar exploration

The following coordinate systems will be used in this book.

3.3 Orbit Control for Typical Spacecraft

(1)

125

Geocentric lunar-orbit inertial frame Oe Xlo Ylo Zlo

The geocentric lunar-orbit inertial frame has its origin Oe at the Earth’s center, its Oe Xlo axis in the lunar orbit plane and pointing towards the Moon’s center, and its Oe Zlo axis perpendicular to the lunar orbit plane, in line with angular velocity vector of the Moon’s revolution around the Earth. The Oe Ylo axis is perpendicular to both Oe Xlo and Oe Zlo , forming a right-handed Cartesian system. This reference frame is used to describe the parameters of the Moon’s motion relative to the Earth. (2)

Geocentric ecliptic inertial frame Oe Xeo Yeo Zeo

The geocentric ecliptic inertial frame has its origin Oe at the Earth’s center, its Oe Xeo axis in the ecliptic plane and pointing towards the vernal equinox, and its Oe Zeo axis perpendicular to the ecliptic plane, in line with angular velocity vector of the Earth’s revolution around the Sun. The Oe Yeo axis is perpendicular to both Oe Xeo and Oe Zeo , forming a right-handed Cartesian system. This reference frame is used to describe the parameters of the Sun’s motion relative to the Earth. (3)

Selenocentric equatorial inertial frame om xi yi zi

The selenocentric equatorial inertial frame has its origin om at the Moon’s center, its om xi axis in the lunar equatorial plane and pointing towards the vernal equinox, and its om zi axis perpendicular to the lunar equatorial plane, in line with angular velocity vector of the Moon’s rotation. The om yi axis is perpendicular to both om xi and om zi , forming a right-handed Cartesian system. This reference frame is used to describe the selenocentric orbital parameters of the spacecraft. (4)

Selenocentric equatorial fixed frame om xm ym zm

The selenocentric equatorial fixed frame has its origin om at the Moon’s center; its om xm axis in the lunar equatorial plane, passing through Sinus Medii on the Moon, and pointing towards the mean position of the Earth; and its om zm axis perpendicular to the lunar equatorial plane, in line with angular velocity vector of the Moon’s rotation. The om ym axis is perpendicular to both om xm and om zm , forming a right-handed Cartesian system. This reference frame is used to describe the lunar geographic latitude/longitude and the Moon’s non-sphericity perturbations. (5)

Selenocentric lunar-orbit inertial frame om xlo ylo zlo

The selenocentric lunar-orbit inertial frame has its origin om at the Moon’s center, its om xlo axis in the lunar orbit plane and pointing towards the Earth’s center, and its om zlo axis perpendicular to the lunar orbit plane, in line with angular velocity vector of the Moon’s revolution around the Earth. The om ylo axis is perpendicular to both om xlo and om zlo , forming a right-handed Cartesian system. This reference frame is used to describe the parameters of the Earth’s motion relative to the Moon.

126

(6)

3 Orbit Control

Selenocentric translating inertial frame om xei yei zei

The selenocentric translating inertial frame has its origin om at the Moon’s center and its om xei , om yei , and om zei axes in line with the Oe Xi , Oe Yi , and Oe Zi axes of the geocentric equatorial inertial frame, respectively. The om yi axis is perpendicular to both om xi and om zi , forming a right-handed Cartesian system. That is, this frame is translated from the geocentric equatorial inertial frame. It is used for conversion between selenocentric and geocentric orbital parameters. (7)

Perifocal frame OXp Yp Zp

The perifocal frame has its origin O at the primary body’s center, its OXp axis in the orbital plane of the spacecraft and pointing towards the pericenter, and its OZp axis perpendicular to the orbital plane, in line with angular velocity vector of the spacecraft’s revolution around the primary. The OYp axis is perpendicular to both OXp and OZp , forming a right-handed Cartesian system. 2.

Orbital dynamic equation

For a lunar probe orbit, the Moon and Earth are the primary bodies in the selenocentric and geocentric segments of the orbit, respectively. Therefore, the following perturbing forces need to be considered: the N-body perturbing force f N , perturbing force due to the Moon’s non-sphericity f nsm , perturbing force due to the Earth’s nonsphericity f nse , perturbing force due to solar radiation pressure f solar , and perturbing force due to atmospheric drag f drag . In the geocentric equatorial inertial reference frame, the complete orbital dynamic equation can be given in the following form: ! " rmp rep d2 rep rem f = −μe 3 − μm 3 + 3 + N 2 dt rep rmp rem m +

f drag f f f nsm + nse + solar + m m m m

(3.131)

where rep and rmp are, respectively, the spacecraft’s position vectors with respect to the Earth and Moon; μe and μm , respectively, the standard gravitational parameters of the Earth and Moon; m the spacecraft mass; and rem the Moon’s position vector with respect to the Earth. The first and second terms on the RHS of this equation represent the gravitational acceleration of the Earth and Moon as the primary, respectively. Now, let us examine each perturbing force in the complete orbital dynamic equation. (1)

Perturbing force from N-body gravity

The perturbing celestial objects here include the Sun and planets in the solar system other than the Earth. The perturbing force of their gravities f N exerted on a spacecraft with mass m is given by

3.3 Orbit Control for Typical Spacecraft

fN = −m

N 4 i=1

127

! μi

rei − rep rei +5 5 5rei − rep 53 rei3

" (3.132)

where N is the number of perturbing celestial bodies, μi the standard gravitational parameter of the ith perturbing body, and rei the position vector of the ith perturbing body in the geocentric equatorial inertial frame, which can be obtained from the ephemerides. (2)

Perturbing force from the Moon’s non-sphericity

In the selenocentric equatorial fixed frame, the mathematical model of the Moon’s non-spherical gravitational field can be expressed by a spherical harmonic series Vnsm :  l  N   m μm 4 4 Rm m = P ln (sin ϕm ) C ln cos nλm + S ln sin nλm r r n=0 l

Vnsm

(3.133)

l=2

where Rm is the lunar equatorial radius; r, λm , ϕm the spherical components of the spacecraft in the selenocentric equatorial fixed frame representing the selenocentric distance, selenocentric longitude, and selenocentric latitude, respectively; and N the order of the Moon’s non-spherical gravity model. Currently, LP165 and LP75G are m two lunar gravity models with satisfactory accuracy for practical applications. C ln m and S ln are two constants in the models. The perturbing force due to the Moon’s non-sphericity f nsm is calculated from f nsm = m(Smei )T ∇Vnsm

(3.134)

where Smei is the coordinate transformation matrix from the selenocentric translating inertial frame to the selenocentric equatorial fixed frame and ∇Vnsm is the gradient of the spherical harmonic series of the Moon’s non-sphericity with respect to the Cartesian coordinates in the selenocentric equatorial fixed frame. (3)

Perturbing force from Earth’s non-sphericity

In the geocentric equatorial fixed frame, the mathematical model of the Earth’s non-spherical gravitational field can be expressed by a spherical harmonic series Vnse :  l  N   e μe 4 4 Re e e = P ln (sin ϕe ) C ln cos nλe + S ln sin nλe R R n=0 l

Vnse

l=2

(3.135)

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3 Orbit Control

where Re is the equatorial radius of the Earth; R, λe , ϕe the spherical components of the spacecraft in the geocentric equatorial fixed frame representing the geocentric distance, geocentric longitude, and geocentric latitude, respectively; and N the order of the Earth’s non-spherical gravity model. Currently, JGM-3 and WGS84 are two e Earth gravity models with satisfactory accuracy for practical applications. C ln and e S ln are two constants in the models. The perturbing force due to the Earth’s non-sphericity f nse is calculated from f nse = m(T ei )T ∇Vnse

(3.136)

where T ei is the coordinate transformation matrix from the geocentric equatorial inertial frame to the geocentric equatorial fixed frame and ∇Vnse is the gradient of the spherical harmonic series of the Earth’s non-sphericity with respect to the Cartesian coordinates in the geocentric equatorial fixed frame. (4)

Perturbing force from solar radiation pressure

The perturbing force due to solar radiation pressure f solar is calculated from fsolar = −kPSus

(3.137)

where k is the surface coefficient, which is in the range of 0–2; P the atmospheric radiation pressure, approximated as a constant 4.56 × 10−6 N/m2 near the Earth; S the effective area of the spacecraft under radiation; and us the unit vector pointing from the spacecraft towards the Sun, which is obtained from us =

res − rep res − rep

(3.138)

Here, res and rep denote the position vectors of the Sun and spacecraft with respect to the Earth, respectively.

3.3.6 Design of Cislunar Return Trajectory The accurate design of the cislunar return trajectory in a direct manner is a very complicated task. On the other hand, as the foundation of spacecraft orbital dynamics, the two-body problem analysis is a mature technique but suffers from insufficient accuracy. However, if we patch the orbits from multiple two-body problems, we will be able to obtain an approximate solution for the actual operation orbit, which is known as the method of patched conic approximation. As pointed out in Ref. [6], in the preliminary analysis of a flight mission, patching two conic orbits at the boundary of the sphere of influence of the Moon is a satisfactory approximation.

3.3 Orbit Control for Typical Spacecraft

129

Hence, it is common practice to adopt patched conic approximation based on the double two-body model in the preliminary design of the cislunar return trajectory. For the orbit design of deep-space exploration missions, the initial values obtained from preliminary orbit design are critical in the sense that they directly influence the complexity and convergence of numerical calculations in the accurate orbit design. Relatively accurate orbital initial values significantly reduce the time and number of iterations required in the search for precise orbital design parameters. On the other hand, the improper selection of initial values is likely to result in the divergence of parameters in the eventual precise orbit design. Moreover, in the preliminary design phase, only the major factors affecting the probe motion are considered, and the calculations are conducted using simple formulas based on reasonable assumptions. Hence, the large volume of computation and qualitative analysis of various transfer orbit characteristics enable the identification of the primary factors influencing orbital properties, which serve as an important guide for the practical orbit design. 1.

Double two-body model for the cislunar return trajectory

The double two-body model is an extension of the conventional two-body model in which the concepts of the sphere of influence and orbit patching are introduced. The complete double two-body model is often simplified by only considering the gravity of the primary. Thus, during the return flight of the spacecraft, a cislunar return trajectory based on the double two-body model is only subject to the Moon’s gravity within the sphere of influence of the Moon, resulting in a hyperbolic trajectory in the selenocentric segment; outside the sphere of influence of the Moon, the spacecraft is only affected by the Earth’s gravity, resulting in an elliptic trajectory in the geocentric segment. The sphere of influence is defined as the range in which the first primary’s ratio of perturbed acceleration to unperturbed acceleration is equal to that of the second primary. The radius of the sphere of influence of the Moon, ρ, is usually calculated from  ρ=

mm me

 25 REL

(3.139)

where mm is the Moon’s mass, me the Earth’s mass, and REL the Earth–Moon distance. To simplify the double two-body model for a cislunar return trajectory, the following assumptions are adopted. (1)

When the spacecraft is within the sphere of influence of the Moon, it is only subject to the Moon’s gravity; the Earth’s gravity and other perturbations are not considered. The corresponding orbital dynamic equation is rmp d2 rmp = μm 3 2 dt rmp

(3.140)

130

3 Orbit Control

where rmp is the position vector of the spacecraft with respect to the Moon and μm is the standard gravitational parameter of the Moon. (2)

When the spacecraft is outside the sphere of influence of the Moon, it is only subject to the Earth’s gravity; the Moon’s gravity and other perturbations are not considered. The corresponding orbital dynamic equation is rep d2 rep = μm 3 2 dt rep

(3.141)

where rep is the position vector of the spacecraft with respect to the Earth and μe is the standard gravitational parameter of the Earth. (3) (4) (5)

The Moon is in a uniform circular motion around the Earth’s center with a radius equal to the mean Earth–Moon distance. The conic sections of the geocentric and selenocentric segments are patched at the boundary of the sphere of influence of the Moon. The Moon’s state is calculated using approximate ephemerides.

The double two-body model can be used in the preliminary design and analysis of the cislunar return trajectory, the results from which provide initial values for the subsequent precise orbit design. 2.

Selection of six intuitive parameters

Figure 3.13 shows a schematic of the selenocentric segment of the cislunar return trajectory based on the double two-body model. This figure corresponds to the situation in which the escape point is in the ascending phase of the selenocentric segment. For the opposite situation, where the escape point is in the descending phase of the selenocentric segment, the calculation of the orbital parameters follows the same procedure, which will not be discussed here. In Fig. 3.13, om xp yp zp is the selenocentric perifocal frame; om xlo ylo zlo the selenocentric lunar-orbit inertial frame; Oe and

Fig. 3.13 Schematic of the selenocentric segment of the cislunar return trajectory based on the double two-body model

3.3 Orbit Control for Typical Spacecraft

131

om , respectively, the centers of the Earth and Moon; A and C, respectively, the insertion and escape points; AC the selenocentric segment of the cislunar return trajectory; iL the angle between the selenocentric orbit and lunar orbit plane (i.e., inclination); δ the angle between the line connecting the escape point C and the Moon’s center om and the line connecting the Earth’s center Oe and the Moon’s center om ; and ϕ the corresponding latitude of the escape point on the sphere of influence of the Moon. We now adopt the following six intuitive parameters: the selenocentric distance of the lunar parking orbit rAL , inclination iL , angle between the Earth–Moon line and the Moon-escape-point line δ, latitude of the escape point ϕ, selenocentric true anomaly of the escape point fCL , and track angle of the accelerating point βAL . With these six independent parameters, we can directly solve for the orbital parameters of the selenocentric segment of the trajectory, laying a good foundation for the computation of the entire trajectory. 3.

Selection of orbital parameters in the selenocentric segment

We use the following system of equations to calculate the semi-latus rectum pL and eccentricity eL of the selenocentric segment of trajectory: ⎧ ⎪ ⎨ rAL =

pL 1 + eL cos 0◦ pL ⎪ ⎩ρ = 1 + eL cos fCL

(3.142)

where ρ = 66200 km is the radius of the sphere of influence of the Moon. Thus, the semi-major axis aL of the selenocentric segment is obtained from aL =

pL 1 − eL2

(3.143)

The flight time of the selenocentric segment is $ TCL =

(−aL )3 (eL sinh ECL − ECL ) μL

(3.144)

where hyperbolic eccentric anomaly corresponding to escape point C, ECL , is obtained from !$ " eL − 1 fCL tan ECL = 2 arctan h (3.145) 1 + eL 2 In the om xp yp zp frame, the position vector rAL and velocity vector vAL of the insertion point A are

132

3 Orbit Control

T  rAL = rAL 0 0  vAL = 0

μL (eL +1) √ μL pL

0

(3.146) T (3.147)

Hence, the required change in velocity at insertion point A is  VA = |vAL | − 4.

μL rAL

(3.148)

State of the escape point

In the om xp yp zp frame, the position vector rCL and velocity vector vCL of the escape point C are T  rCL = ρ cos fCL ρ sin fCL 0 vCL =



−μL sin fCL μL (eL +cos fCL ) √ √ μL pL μL pL

0

(3.149)

T (3.150)

Figure 3.14 shows a schematic of the escape point of the cislunar return trajectory based on the double two-body model, where om xlo ylo zlo is the selenocentric lunarorbit inertial frame; om the Moon’s center; A and C, respectively, the insertion and escape points; AC the selenocentric segment of the cislunar return trajectory; D the intersection point of Earth–Moon line and the sphere of influence; E the intersection point of the great circle of C and lunar orbit plane; F the intersection point of the Fig. 3.14 Schematic of the escape point of the cislunar return trajectory based on the double two-body model

3.3 Orbit Control for Typical Spacecraft

133

selenocentric orbit plane and lunar orbit plane on the sphere of influence; iL the inclination; δ the angle between the Earth–Moon line and Moon-escape-point line; ϕ the latitude of the escape point; u=CF ; λ1=DE ; and λ2=EF . Based on the simplification formulas for right spherical triangles, we have  cos δ tan ϕ sin u = sin ϕ sin iL , cos λ1 = , sin λ2 = cos ϕ tan iL

(3.151)

From Fig. 3.14, we know that the RAAN ΩL and argument of perilune ωL of the selenocentric orbit should satisfy ΩL = 180◦ + λ1 + λ2 , ωL = 180◦ − u − fCL

(3.152)

The transformation matrix from the selenocentric lunar-orbit inertial frame om xlo ylo zlo to the selenocentric perifocal frame om xp yp zp , T plo is T plo = RZ (ωL )RX (iL )RZ (L ) Hence, in the selenocentric lunar-orbit inertial frame om xlo ylo zlo , the position and velocity vectors of the insertion point (RAL and V AL , respectively) and escape point (RCL and V CL , respectively) are

T

T RAL = T plo rAL , V AL = T plo vAL

(3.153)

T

T RCL = T plo rCL , V CL = T plo vCL

(3.154)

Assuming that the Moon is in circular motion around the Earth with a radius of 384,400 km, in the selenocentric lunar-orbit inertial frame om xlo ylo zlo , the position vector REL and velocity vector V EL of the Earth are  T REL = 384400 0 0

(3.155)

 T V EL = 0 1.018 0

(3.156)

Thus, the position vector RCE and velocity vector V CE of the escape point C with respect to the Earth’s center are

5.

RCE = RCL − REL

(3.157)

V CE = V CL − V EL

(3.158)

Selection of orbital parameters in the geocentric segment

134

3 Orbit Control

With RCE and V CE , we can obtain the orbital elements and other relevant parameters of the geocentric segment in the geocentric lunar-orbit inertial frame Oe Xlo Ylo Zlo . The specific angular momentum H CE and semi-latus rectum pE are  H CE = HCEx HCEy HCEz = RCE × V CE pE =

|H CE |2 μE

(3.159)

(3.160)

The orbital elements of the geocentric segment, semi-major axis aE , eccentricity eE , inclination iE , RAAN ΩE , and argument of perigee ωE are given by aE =

1 |V CE |2 μE

−



eE =

(3.161)

pE aE

(3.162)

HCEz |H CE |

(3.163)

1−

iE = arccos

2 |RCE |

⎧

Nx ⎪ ⎪ Ny ≥ 0 ⎨ arccos |N| ΩE =

N ⎪ ⎪ ⎩ 360◦ − arccos x Ny < 0 |N| ⎧ N · eE ⎪ ⎪ (eEz ≥ 0) ⎨ arccos |N||eE | ωE = ⎪ ⎪ 360◦ − arccos N · eE (e < 0) ⎩ Ez |N||eE |

(3.164)

(3.165)

where  T  T N = Nx Ny Nz = K × H CE , K = 0 0 1

(3.166)

  T  RCE 1 V CE × H CE − μE eE = eEx eEy eEz = |RCE | μE

(3.167)

Other orbital parameters of the geocentric segment may include the geocentric distance of perigee rJE , the geocentric sweep angle of the escape point fCE , and the geocentric eccentric anomaly of the escape point ECE : rJE =

pE 1 + eE

(3.168)

3.3 Orbit Control for Typical Spacecraft

135



  1 pE fCE = arccos −1 eE rE !$ " 1 − eE fCE ECE = 2 arctan tan 1 + eE 2

(3.169)

(3.170)

With the acquired orbital parameters of the geocentric segment, we can further obtain the position and velocity vectors of perigee (RJE and V JE , respectively) or the reentry point (RZE and V ZE , respectively) within the geocentric lunar-orbit inertial frame Oe Xlo Ylo Zlo . The flight time of the geocentric segment of an indirect return trajectory, TCJ , is calculated from $ aE3 TCJ = (3.171) (ECE − eE sin ECE ) μE The change in velocity VJ required at perigee is  VJ = |V JE | −

μE rJE

(3.172)

The total flight time TAJ and total required velocity change VAJ of a direct return trajectory are TAJ = TAC + TCJ

(3.173)

VAJ = VA + VJ

(3.174)

For a direct return trajectory, the geocentric sweep angle fZE and geocentric eccentric anomaly EZE corresponding to a reentry point that has a geocentric distance of rZE are given by   1 pE −1 eE rZE !$ " 1 − eE fZE = 2 arctan tan 1 + eE 2 

fZE = arccos

EZE

(3.175)

(3.176)

Thus, the flight time of the geocentric segment of a direct return trajectory, TCZ , is calculated from $ aE3 TCZ = (3.177) (ECE − EZE − eE sin ECE + eE sin EZE ) μE

136

3 Orbit Control

The reentry angle of the direct return trajectory, θ , can be expressed as ⎛ θ = − arcsin⎝ 

⎞ eE sin fZE

⎠

(3.178)

1 + 2eE cos fZE + eE2

The total flight time TAZ and total required velocity change VAZ of a direct return trajectory are

6.

TAZ = TAC + TCZ

(3.179)

VAZ = VA

(3.180)

Selection of geographic parameters

Based on the instant at which the probe arrives at the escape point TC , we can use the approximation formula to obtain the position and velocity of the Moon at this instant, which further result in the state transition matrix (T loi )T from the geocentric lunarorbit inertial frame Oe Xlo Ylo Zlo to the geocentric equatorial inertial frame Oe Xi Yi Zi . Thus, within Oe Xi Yi Zi , the position vector rJE and velocity vector vJE of perigee are rJE = (T loi )T RJE , vJE = (T loi )T V JE

(3.181)

With rJE and vJE , we can calculate the orbital elements of the geocentric segment within Oe Xlo Ylo Zlo , as well as the latitude and longitude of perigee. The right ascension θJE and latitude ϕJE of perigee can be acquired from the following equations with a known rJE : θJE = arctan

rJEy rJEz , ϕJE = arcsin |rJE | rJEx

(3.182)

where T  rJE = rJEx rJEy rJEz

(3.183)

With the instant of the escape point TC , the instant of reentry point is TC + TCZ . Convert the instant of perigee to Julian day (JD). Then, decompose JD into J0 and UT , with the former denoting the Julian day of 0:00 coordinated universal time (UTC) and the latter denoting the hour past 0:00 UTC. JD = J0 +

UT 24

(3.184)

3.3 Orbit Control for Typical Spacecraft

137

The right ascension of Greenwich when the spacecraft arrives at perigee is calculated from Eq. (3.185) [7]: UT 24 (3.185)

θG = 100.4606184 + 36000.77004T02 + 0.000387933T02 + 360.98564724

where T0 =

J0 − 2451545 36525

(3.186)

We also obtain the perigee longitude from λJE = θJE − θG

(3.187)

The same method can be used to calculate the position vector rZE , velocity vector vZE , orbital elements, and longitude λJE and latitude ϕJE of the reentry point for a direct return trajectory. So far, we have completed the calculation of the orbital parameters of a cislunar return trajectory based on the double two-body model. We have learned that by choosing the six new independent parameters that have clear physical meanings, we are able to directly obtain the orbital parameters of the selenocentric segment and conveniently patch the orbits at the escape point. The calculation is rather simple, and the return trajectory parameters can be computed with analytical formulas. In the next section, we will focus on the effects of the six intuitive parameters on the characteristics of the cislunar transfer orbit, with an attempt to determine how the six parameters and the instant of the escape point are qualitatively related to the final constraints. To obtain the desired orbit, it is necessary to analyze the relationship between the six intuitive parameters and the constraints. Owing to the complexity of this problem, it is impossible to find the quantitative relationship. Instead, we must search for qualitative correlations to determine the parameter candidates. Hence, the orbital properties must be analyzed. Since the return trajectory must be a geocentric prograde orbit (iE ≤ 90◦ ), we will not analyze the situation of iE > 90◦ . In the following analysis, we select 0:00 on September 1, 2015 as the instant of escape. Considering that the mean length of a sideral month is 27.32166 mean solar days, this assumption has some generality. (1)

Effects of selenocentric distance of lunar parking orbit

Let us assume that the inclination of the lunar parking orbit of the probe iL = 40◦ , angle between the Earth–Moon line and the Moon-escape-point line δ = 70◦ , latitude of the escape point ϕ = 3◦ , and track angle of the accelerating point βAL = 90◦ . We are now required to adjust the true anomaly of the escape point fCL according

138

3 Orbit Control

Table 3.3 Effects of the selenocentric distance of the lunar parking orbit on the return trajectory ◦

rAL /(km) fCL /(◦ ) iE / λJE /(◦ ) ϕJE /(◦ ) VA /(km/s) TCJ /(h) 1838

141.9753

18.74

98.85

3.13

0.79

109.79

1938

141.0748

18.37

97.35

3.12

0.77

109.92

2038

140.2041

18.00

94.89

3.09

0.76

110.05

2238

138.5439

17.28

93.03

3.05

0.73

110.29

2538

136.2303

16.24

88.90

2.99

0.70

110.65

2738

134.7892

14.57

86.23

2.95

0.68

110.88

to the selenocentric distance of the lunar parking orbit, such that the Earth-parkingorbit altitude of the probe rJE = 6600 km. Table 3.3 summarizes the effects of the selenocentric distance of the lunar parking orbit on the parameters of the cislunar return trajectory. The following conclusions can be drawn from Table 3.3. (a)

(b)

(c) (2)

A larger selenocentric distance of the lunar parking orbit leads to a smaller required true anomaly of the escape point, slightly smaller required velocity change, and slightly increased flight time. A larger selenocentric distance of the lunar parking orbit leads to a slightly reduced inclination of the selenocentric orbit and latitude/longitude of the perigee. Overall, the variation of the selenocentric distance of the lunar parking orbit has no significant effects on the orbital characteristics of the spacecraft. Effects of inclination of lunar parking orbit

Assume rAL = 1938km, δ = 70◦ , ϕ = 3◦ , and βAL = 90◦ . We are now required to adjust fCL according to the inclination of the lunar parking orbit such that rJE = 6600 km. Table 3.4 summarizes the effects of the inclination of the lunar parking orbit on the return trajectory parameters. Table 3.4 Effects of the inclination of the lunar parking orbit on the return trajectory iL /(◦ )

fCL /(◦ )

iE /(◦ )

λJE /(◦ )

ϕJE /(◦ )

10

141.0006

27.80

VA /(km/s)

TCJ /(h)

82.49

3.48

0.77

110.89

20

141.0704

30

141.0925

24.46

84.41

3.34

0.77

110.73

21.24

90.44

3.22

0.77

40

110.39

141.0748

18.37

97.35

3.12

0.77

109.92

50

141.0256

14.91

104.86

3.03

0.77

109.32

60

140.9545

13.93

114.60

2.95

0.77

108.63

70

140.8705

12.50

126.22

2.91

0.77

107.87

80

140.7820

11.64

137.34

2.88

0.77

107.08

90

140.6953

11.39

148.58

2.90

0.77

106.28

3.3 Orbit Control for Typical Spacecraft

139

The following conclusions can be drawn from Table 3.4. (a)

(b)

(c)

(3)

Once the position of the escape point and the associated δ and ϕ are properly selected, return to Earth is guaranteed irrespective of iL as long as fCL is properly chosen. A larger iL leads to a smaller required fCL , an almost constant required change in velocity to reach the designated altitude above the Earth, and a slightly reduced flight time. A larger iL also leads to a slightly reduced inclination of the geocentric orbit, a slightly reduced latitude of the perigee, and a slightly increased longitude of the perigee. Effects of angle between spacecraft–Moon line and Earth–Moon line

Assume rAL = 1938 km, iL = 40◦ , ϕ = 3◦ , and βAL = 90◦ . We are now required to adjust fCL according to the angle between the spacecraft–Moon line and Earth–Moon line such that rJE = 6600 km. Table 3.5 summarizes the parameters of the cislunar return trajectory corresponding to different values of this angle. The following conclusions can be drawn from Table 3.5. (a)

(b)

With an increase in δ, the required fCL drastically increases and then slightly decreases, the flight time monotonically increases, and the required velocity change rapidly decreases before a slight increment. In other words, there exists a value of δ corresponding to the minimum change in velocity. For the current example, the minimum required change in velocity occurs around δ = 70◦ . These observations suggest that the escape-point parameter δ determines not only the change in velocity required by the lunar probe, but also the flight time. There exist blind spots in the angle between spacecraft–Moon line and Earth– Moon line, implying that a design of the cislunar return trajectory is not possible at these angles.

Table 3.5 Effects of the altitude of the lunar parking orbit on the return trajectory δ/(◦ )

fCL /(◦ )

iE /(◦ )

λJE /(◦ )

ϕJE /(◦ )

VA /(km/s)

TCJ /(h)

10

/

/

/

/

/

/

20

113.4033

33.78

176.60

10.40

1.29

30

124.3669

26.30

7.60

6.16

0.97

61.47

40

132.9357

22.60

−140.16

4.65

0.86

74.11

50

137.4864

20.46

74.39

3.94

0.80

84.73

60

140.0068

19.15

−82.13

3.50

0.78

97.34

70

141.0748

18.37

97.35

3.12

0.77

109.92

80

141.0047

17.98

−123.99

2.70

0.77

124.68

90

139.9448

17.95

19.92

2.18

0.78

138.65

46.40

140

3 Orbit Control

Table 3.6 Effects of the latitude of the escape point on the return trajectory ϕ/(◦ )

fCL /(◦ )

iE /(◦ )

λJE /(◦ )

ϕJE /(◦ )

VA /(km/s)

TCJ /(h)

−10

/

/

/

/

/

/

−5

139.0208

21.05

103.97

2.54

0.79

108.54

0

140.8285

4.06

97.93

2.99

0.77

109.76

3

141.0748

18.37

97.35

3.12

0.77

109.92

5

140.9518

27.30

97.93

3.17

0.77

109.83

10

139.3466

53.75

104.19

3.33

0.78

108.69

15

/

/

/

/

/

/

(c)

(d) (4)

With an increase in δ, the inclination of the geocentric orbit and latitude of perigee slightly decrease, but the longitude of perigee does not exhibit any significant change. δ directly affects the required velocity change and flight time, and it is a key regulating parameter in the optimization of return trajectories. Effects of latitude of escape point

Assume rAL = 1938km, iL = 40◦ , δ = 70◦ , and βAL = 90◦ . We are now required to adjust fCL according to the escape point latitude such that rJE = 6600 km. Table 3.6 summarizes the effects of latitude of the escape point on the cislunar transfer orbit. The following conclusions can be drawn from Table 3.6. (a)

(b) (c)

(5)

With an increase in ϕ, the required change in velocity decreases before a slight increment. That is, there exists a value of ϕ corresponding to the minimum change in velocity. For the current example, this value is ϕ = 3◦ . The flight time needed shows the opposite trend, i.e., a slight increase followed by a slight decrease. There exist blind spots in ϕ, implying that a design of the cislunar return trajectory is not possible at these values of the escape-point latitude. With an increase in ϕ, the inclination of the geocentric orbit shows a remarkable increase and the latitude of perigee slightly increases, but the longitude of perigee does not exhibit any significant change. Effects of the selenocentric true anomaly of escape point

Assume rAL = 1938km, iL = 40◦ , δ = 70◦ , and βAL = 90◦ . We are now required to obtain the return trajectory parameters based on the values of the selenocentric true anomaly of the escape point. Table 3.7 summarizes the effects of various selenocentric true anomaly values of the escape point on the parameters of the cislunar transfer orbit. The following conclusions can be drawn from Table 3.7. (a)

A larger fCL leads to a slight decrease in the required velocity change and a slightly reduced flight time.

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141

Table 3.7 Effects of the selenocentric true anomaly of the escape point on the return trajectory fCL /(◦ )

rJE /(km)

iE /(◦ )

λJE /(◦ )

ϕJE /(◦ )

VA /(km/s)

TCJ /(h)

136

667.53

24.58

151.08

2.62

0.82

104.17

137

1372.09

21.81

142.78

2.72

0.81

104.12

138

2314.07

20.36

133.33

2.83

0.79

106.16

139

3483.34

19.45

122.76

2.92

0.79

107.30

140

4871.55

18.84

111.09

3.01

0.78

108.51

141

6471.94

18.39

98.35

3.11

0.77

109.82

(b)

(c)

(6)

A larger fCL also results in a significantly increased altitude of the geocentric parking orbit, a somewhat increased longitude of perigee, and a slightly reduced latitude of perigee. Even a small variation in the selenocentric sweep angle would lead to a significant change in the altitude of the geocentric parking orbit of the probe. This imposes higher requirements on deep-space measurement and orbit control. Effects of track angle of accelerating point From the conservation of angular momentum, we know RAL VAL sin βAL = RCL VCL sin βCL

(3.188)

With other orbital parameters, the RAL of the accelerating point and the RCL , VCL , and βCL of the escape point can be determined. To achieve the minimum required VAL at the accelerating point, we should have βAL = 90◦ . That is, the acceleration direction of the probe in the lunar parking orbit should be the same as its velocity direction. In fact, this acceleration scheme is the most popular choice in engineering applications to ensure the minimum energy needed for acceleration. (7)

Effects of escape point time

We first analyze the effects of the escape point time on indirect return trajectories. Assume rAL = 1938 km, iL = 40◦ , δ = 70◦ , ϕ = 3◦ , fCL = 141.0748, and βAL = 90◦ . In this case, rJE = 6600 km. Table 3.8 summarizes the effects of different dates in September 2015 on the return trajectory parameters, and the effects on the perigee latitude are separately illustrated in Fig. 3.14 with a higher resolution. Table 3.8 summarizes the effects of different times between the 1st and 2nd days of September 2015 on the return trajectory parameters, and the effects on the perigee longitude are separately illustrated in Fig. 3.15 with a higher resolution (Fig. 3.16) (Table 3.9). The following conclusions can be drawn from Tables 3.8 and 3.9. (a) (b)

The selection of the escape point date does not affect the flight time and required velocity change of the return trajectory. The variation of the escape point date hardly affects the inclination of the return orbit, but it may considerably impact the geocentric latitude. For every

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Table 3.8 Effects of the different dates in September 2015 on the return trajectory parameters d /(UTC)

iE /(◦ )

1d

18.37

5d

18.72

149.37

−12.48

0.77

109.92

10 d

18.54

−148.96

−13.62

0.77

109.92

15 d

18.17

−114.86

−4.79

0.77

109.92

20 d

18.65

178.22

−14.75

0.77

109.92

25 d

18.59

104.38

−6.49

0.77

109.92

30 d

18.53

91.27

−4.09

0.77

109.92

Fig. 3.15 Effects of the different dates on the perigee latitude

Fig. 3.16 Effects of the different time on the perigee longitude

λE /(◦ ) 97.35

ϕE /(◦ ) 3.12

VA /(km/s)

TCJ /(h)

0.77

109.92

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143

Table 3.9 Effects of the different hours between the 1st and 2nd days of September 2015 on the return trajectory parameters h/(UTC)

iE /(◦ )

λE /(◦ )

2h

18.38

68.46

4h

18.39

39.57

6h

18.40

8h 10 h

ϕE /(◦ )

VA /(km/s)

TCJ /(h)

2.73

0.77

109.92

2.34

0.77

109.92

10.67

1.95

0.77

109.92

18.41

−18.23

1.56

0.77

109.92

18.42

−47.13

1.17

0.77

109.92

12 h

18.43

−76.03

0.78

0.77

109.92

14 h

18.44

−104.93

0.40

0.77

109.92

16 h

18.45

−133.83

0.01

0.77

109.92

18 h

18.46

−162.74

−0.37

0.77

109.92

20 h

18.47

168.36

−0.75

0.77

109.92

22 h

18.48

139.45

−1.13

0.77

109.92

24 h

18.49

110.55

−1.51

0.77

109.92

(c)

delay of 2 h in the escape point time, the corresponding change in longitude is approximately 30°. The return trajectory parameters exhibit periodic variations with the sidereal month cycles of the escape point time.

We have examined the design of cislunar return trajectories based on the double two-body model. Double two-body orbit design essentially searches for the proper escape point and the state at this point such that the probe can satisfy certain constraints of the lunar exploration mission under the separate action of the gravitational forces from the two primary bodies. Based on the analysis of the effects of six intuitive parameters and the escape point time on the cislunar transfer orbit, we can draw the following conclusions. (1) (2) (3) (4) (5)

iE and ϕE can be effectively controlled by adjusting iL . By adjusting δ, the flight time and required change in velocity can be selected, and ϕE and λE can be regulated as well. iE and ϕE can be adjusted by varying ϕ. By varying fCL , hE can be adjusted, and θ can be subsequently selected. ϕE can be adjusted by varying the escape point date, and λE can be adjusted by varying the time corresponding to the escape point date.

In the above points, λE , ϕE , and hE are the longitude, latitude, and altitude of the terminal point, respectively, and θ is the reentry angle. With the conclusions drawn from the above analyses on the return trajectory characteristics, we can summarize the design procedure of an indirect return trajectory as follows. (1)

Specify the altitude and inclination of the lunar parking orbit based on the mission requirements.

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(2)

Select a proper escape point instant. Compared to direct return trajectories, an indirect return trajectory has multiple return windows, making this selection more flexible. Based on the requirement on the inclination of the geocentric orbit, select a proper escape point latitude. Select a proper angle between the spacecraft–Moon line and Earth–Moon line. Based on the requirement on the geocentric distance of perigee, select a proper selenocentric true anomaly of the escape point. Iteratively test the satisfaction of the requirements on the geocentric distance of perigee and the inclination of the geocentric orbit. If this fails, return to step 3) and repeat.

(3) (4) (5) (6)

The design procedure of a direct return trajectory is summarized as follows. (1) (2)

(3) (4) (5) (6)

Specify the altitude and inclination of the lunar parking orbit based on the mission requirements. Based on the requirement on the geocentric latitude, first select a date corresponding to the escape point time and then select a time corresponding to the escape point date. Based on the requirement on the inclination of the geocentric orbit, select proper escape point latitude. Based on the requirement on the geocentric latitude, select a proper angle between the spacecraft–Moon line and Earth–Moon line. Based on the requirement on the geocentric distance of perigee, select a proper selenocentric true anomaly of the escape point. Iteratively test the satisfaction of the requirements on the geocentric distance of perigee, inclination of the geocentric orbit, and geocentric latitude/longitude. If this fails, return to step (2) and repeat.

Let us now examine the validity of the orbit design with a specific example. For an indirect return trajectory, the relevant parameters are listed in Table 3.10. The corresponding selenocentric and geocentric parameters are listed in Tables 3.11 and 3.12, respectively. The selected global parameters for a direct return trajectory are listed in Table 3.13, and the corresponding selenocentric and geocentric parameters are listed in Tables 3.14 and 3.15, respectively. In these tables, TA , TC , TJ , and TZ , respectively, denote the instants of the insertion point, escape point, perigee, Table 3.10 Parameter selection for an indirect return trajectory TB /(UTC)

rAL /(km)

iL /(◦ )

δ/(◦ )

ϕ/(◦ )

fCL /(◦ )

βAL /(◦ )

2015/09/21 15:43:18

1938

40

73.03

10.02

139.4930

90

Table 3.11 Selenocentric parameters of an indirect return trajectory TA /(UTC)

aL /(km)

eL

iL /(◦ )

L /(◦ )

ωL /(◦ )

VA /(km/s)

2015/09/20 20:57:36

−8848.64

1.23

40.00

264.92

24.80

0.78

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145

Table 3.12 Geocentric parameters of an indirect return trajectory TJ /(UTC)

aE /(km)

eL

iE /(◦ )

E /(◦ )

ωE /(◦ )

VJ /(km/s)

2015/09/25 14:02:09

193,209.26

0.97

43.00

54.96

206.95

3.12

Table 3.13 Parameter selection for a direct return trajectory TB /(UTC)

rAL /(km)

iL /(◦ )

δ/(◦ )

ϕ/(◦ )

fCL /(◦ )

βAL /(◦ )

2015/09/21 19:27:18

1938

40

74.31

8.28

140.2060

90

Table 3.14 Selenocentric parameters of a direct return trajectory TA /(UTC)

aL /(km)

eL

iL /(◦ )

L /(◦ )

ωL /(◦ )

VA /(km/s)

2015/09/21 00:21:42

−8930.14

1.22

40.00

264.14

26.85

0.78

Table 3.15 Geocentric parameters of a direct return trajectory TZ /(UTC)

aE /(km)

eL

iE /(◦ )

E /(◦ )

ωE /(◦ )

θ/(◦ )

λE /(◦ )

ϕE /(◦ )

2015/09/25 21:10:56

193,134.07

0.97

42.98

167.13

181.41

−6.01

42.30

4.89

and reentry point; aL , eL , iL , ΩL , and ωL , respectively, denote the semi-major axis, eccentricity, inclination, RAAN, and argument of perilune of the selenocentric orbit; aE , eE , iE , ΩE , ωE , respectively, denote the semi-major axis, eccentricity, inclination, RAAN, and argument of perigee of the geocentric orbit; and λE and ϕE , respectively, denote the longitude and latitude of the reentry point. Figure 3.17 shows the simulation results of the return trajectory, without considering whether it is a direct or indirect trajectory. For saving energy, we choose δ and fCL to be approximately equal to 70° and 140°, respectively. The predetermination of these ranges is very helpful for parameter search, in addition to the benefit of ensuring an energy-saving trajectory design. Because the selections of optional parameters are similar, the obtained orbital parameters for the two types of return trajectories are very close to each other. However, from the perspective of energy, the required change in velocity for the direct return trajectory is only about one-fifth of that for the indirect counterpart. The advantage of the indirect trajectory is its feasibility for LEO rendezvous and docking (RVD) missions. In this section, based on the two-body model and double two-body model, we introduced the methods and procedures associated with the preliminary design of both direct and indirect cislunar return trajectories, along with simulation examples. We presented a method using six intuitive parameters to solve for the preliminary trajectory based on the double two-body model. This rather intuitive and simple method enables the direct acquisition of orbital parameters while avoiding the necessity of solving complex equations. Furthermore, the selected parameters all have clear physical meanings, making them conducive to the analysis of orbital characteristics.

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Fig. 3.17 Simulation of a cislunar return trajectory based on the six intuitive parameters

For example, by constructing angles between characteristic lines, one will be able to obtain useful qualitative conclusions, e.g., regarding the flight time and required change in velocity. Therefore, it will be easy for the designer to further search for trajectories that satisfy the constraints. We also presented preliminary design examples for two types of return trajectories with minimum fuel consumption and compared their results. Thus, the validity of this method has been confirmed.

3.3.7 Precise Design of Cislunar Return Trajectory The dynamic equations for return trajectories based on the two-body or double twobody model only include the gravitational forces of the primary bodies. Although such an approximation is acceptable in many occasions, it is noted that based on the accuracy of the model, the obtained trajectory should not be viewed as a precise orbit. On the other hand, if all the perturbations, such as the N-body gravity, Moon’s nonsphericity, Earth’s non-sphericity, solar radiation pressure, and atmospheric drag, are included in the dynamic equation for trajectory calculation, the computational complexity will inevitably escalate, although this is clearly a more accurate approach. Therefore, a reasonable approach would be to select an appropriate dynamic equation while ensuring a good trade-off between computational accuracy and efficiency. Unlike the use of analytical formulas in the preliminary design, numerical integration is needed in the precise design. The choice of the numerical integration method also directly affects the computational complexity and efficiency. Once the mathematical model is selected, the problem is narrowed to solving for parameters at the orbit insertion point, for which forward search is usually adopted.

3.3 Orbit Control for Typical Spacecraft

147

Ref. [8] presents a precise design of a return trajectory for a lunar probe, but their target orbit is an Earth parking orbit, instead of an orbit for reentry to Earth. Ref. [9] combines the design of a cislunar return trajectory with a reentry simulation, thereby proposing a complete trajectory design with the final reentry. However, their search efficiency could be further improved. Moreover, we should pay attention to the design of indirect return trajectories for cases where malfunctions or special mission requirements are considered. In this section, based on the complete dynamic equation of the return trajectory, we will compare the results calculated by excluding only one type of perturbation at a time with those obtained by referring to the reentry point parameters, for two different termination conditions for integration, i.e., a fixed flight time and fixed geocentric distance. Subsequently, by comparing the performance of methods including the 4th-order Runge–Kutta (RK) method, explicit 4th-order Adams method, 4th-order predictor–corrector, 12th-order Adams-Cowell predictor–corrector, and RKF7(8) inorbit integration, we will choose the most appropriate model for the cislunar return trajectory design that can be used in practical applications. Next, for the two types of return trajectories, we will propose a precise design method for cislunar return trajectory with fast differential correction based on the B-plane parameters of a hyperbolic orbit, in which hyperbolic B-plane parameters at the escape point of the selenocentric segment are used as the target values of the first iteration and the constraint values of the geocentric segment are used as the target values of the second iteration. With this method, when the constraints are satisfied, we will be able to complete the precise design of both types of trajectories after acquiring the state of the orbit insertion point. These will be further demonstrated with simulation cases. 1.

Differential correction

Differential correction is essentially an iterative shooting method based on gradient information. It relies on the sensitivity of the constraints described by the state transition matrix on the perturbations of the regulating variable to iteratively adjust the regulating variable until the constraints satisfy the expected values. Let P be the regulating variable and Q the target variable. They can be expressed as a function: Q = f (P). Expanding this function using a Taylor series and ignoring the higher-order terms, we arrive at Q ∂f = ∂P P

(3.189)

We denote the state transition matrix by Φ, which is calculated from Φ=

∂f ∂P

(3.190)

The key to differential correction is the acquisition of the state transition matrix, which can be obtained using a closed-form expression, numerical integration, or

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finite difference. For complicated problems, the finite difference is often employed. In this method, an initial perturbation is added to every regulating variable before the calculation of the corresponding target variable. Subsequently, The difference between the current target variable and the global target value is calculated. The ratio of this difference to the initial perturbation is defined as the state transition matrix. The procedure of differential correction is described as follows. (1)

From the current regulating variable P, calculate the current target variable Q:

Q = f (P) (2)

Calculate the difference between the current target variable Q and the global target value Q∗ :

Q = Q − Q∗ (3)

f (P + ε) − f (P − ε) 2ε

(3.193)

From state transition matrix Φ, calculate the correction P for the regulating variable: P = Φ −1 Q

(5)

(3.192)

With the given perturbation ε of the regulating variable, calculate the corresponding state transition matrix Φ:

Φ= (4)

(3.191)

(3.194)

Based on the correction P for the regulating variable, update the regulating variable P:

P = P + P

(3.195)

Repeat from the first step. The iteration terminates when the difference between Q and Q∗ in the second step is smaller than a given value σ . 2.

B-plane parameters of hyperbolic orbit

Kizner discovered the linear relationship between the B-plane parameters of the target body and the orbital state deviation [10]. Search algorithms that employ Bplane parameters as the target parameters can describe the target distribution and

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149

Fig. 3.18 Schematic of the B-plane parameters of a hyperbolic orbit

have global convergence; therefore, they are extensively used in orbit design and optimization in deep-space exploration missions. The B-plane is defined as the plane passing through the center of the target body and perpendicular to the asymptote of the orbit insertion hyperbola. The associated B-plane reference frame has its origin at the center of the target body. The S axis is the asymptote of the orbit insertion hyperbola. If we define a reference direction N, which is the normal of the equatorial plane of the target body, the T axis is the cross product of S and N. The R axis satisfies the right-hand rule with the other two axes, S and T. A schematic of the frame is shown in Fig. 3.18. The B-vector, denoted by B, starts at the B-plane origin and points to the intersection point of the hyperbolicorbit asymptote and the B-plane. The magnitude of B equals the semi-minor axis of the hyperbola. Another two B-plane parameters, BT and BR, are defined as the components of B on the T and R axes, respectively: BT = B · T BR = B · R

(3.196)

Based on the above definitions, for a two-body orbit model, the B-plane parameters at any instant can be calculated as follows once the corresponding position vector r and velocity vector v of the spacecraft are known. With r and v, we can first calculate the unit vector of the orbital plane n and the eccentricity vector of the hyperbolic orbit e: r×v |r × v|   1 μ v2 − r − (r · v)v e= μ r n=

(3.197) (3.198)

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Fig. 3.19 Simulation plot of the direct return trajectory

The angle between the hyperbolic-orbit asymptote and the eccentricity vector, β, is calculated from 

β = cos−1 1 e

(3.199)

Then, the unit vector of the hyperbolic-orbit asymptote can be obtained from e×n e S = cos β + sin β |e × n| e

(3.200)

The unit vectors of the other two axes of the B-plane are T =S×N R=S×T

(3.201)

 T N= 001

(3.202)

where

Thus, we arrive at the semi-major axis a and semi-minor axis b of the hyperbolic orbit: 6 2  v μ μ − (3.203) a=− 2 2 r  b = −a e2 − 1 (3.204)

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151

The B-vector is expressed as B = b(S × n)

(3.205)

Let TTCA be the time of periapsis passage from the current instant. We can write TTCA =

H − e sin hH   −μ a3

(3.206)

where H represents the mean anomaly of the hyperbolic orbit at the current instant. The B-plane parameter vector without a prescribed time of arrival is expressed as  T Q = BT BR

(3.207)

The B-plane parameter vector with a prescribed time of arrival is expressed as T  Q = BT BR TTCA

(3.208)

Errors in the B-plane parameters are defined as follows: BT = BT − BT ∗ BR = BR − BR∗ ∗ TTCA = TTCA − TTCA

(3.209)

∗ where BT ∗ , BR∗ , and TTCA are the target values of the B-plane parameters. Thus, the B-plane parameter error vector without a prescribed time of arrival is

 T Q = BT BR

(3.210)

Similarly, the B-plane parameter error vector with a prescribed time of arrival is  T Q = BT BR TTCA 3.

(3.211)

Differential correction based on B-plane parameters of hyperbolic orbit

With BT and BR (hyperbolic B-plane parameters at the escape point of the selenocentric segment), as well as the flight time from the orbit insertion point to the escape T  point TAC , we can construct the target variable Q = BT BR TAC for differential correction. Compared to elliptic B-plane parameters, hyperbolic B-plane parameters are more extensively and maturely used. More importantly, with the hyperbolic B-plane parameters, we only have to perform integration for the selenocentric

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segment, instead of the entire return trajectory, resulting in a significantly reduced computational time. T T   Let P = P1 P2 P3 . . . be the regulating variable and Q = BT BR TAC the target variable. The state transition matrix Φ is given by ⎡ Φ=

∂Q ⎢ =⎣ ∂P T

∂BT ∂P T ∂BR ∂P T ∂TAC ∂P T

⎤

⎡

⎥ ⎢ ⎦=⎣

∂BT ∂BT ∂BT ∂P1 ∂P2 ∂P3 ∂BR ∂BR ∂BR ∂P1 ∂P2 ∂P3 ∂TAC ∂TAC ∂TAC ∂P1 ∂P2 ∂P3

⎤ ··· ⎥ ···⎦ ···

(3.212)

Φ can be calculated using the finite-difference method. Taking the first element in the matrix as an example, if we assume a small perturbation ε, we can write ∂BT BT (P1 + ε, P2 , P3 , . . .) − BT (P1 − ε, P2 , P3 , . . .) = ∂P1 2ε

(3.213)

The other elements in Φ can be calculated similarly. 4.

Precise design method for return trajectories

There are four ways to reach a designated point of the geocentric segment through the cislunar return trajectory: 1) the insertion point at the ascent phase of the selenocentric segment and terminal point at the ascent phase of the geocentric segment (i.e., an ascent-ascent trajectory), 2) insertion point at the ascent phase of the selenocentric segment and terminal point at the descent phase of the geocentric segment (i.e., an ascent-descent trajectory), 3) insertion point at the descent phase of the selenocentric segment and terminal point at the descent phase of the geocentric segment (i.e., descent-ascent trajectory), and 4) insertion point at the ascent phase of the selenocentric segment and terminal point at the descent phase of the geocentric segment (i.e., descent-descent trajectory). For a direct trajectory, the terminal point corresponds to the reentry point, whereas for an indirect trajectory, the terminal point corresponds to the perigee. Irrespective of the chosen trajectory type, the trajectory design can follow the procedure described below. Based on the conclusions from Ref. [11], choose an appropriate orbit insertion instant such that the latitude of the reentry point roughly satisfies the requirement; thereby, the initial values of the insertion point can also be obtained. Choose the RAAN of the selenocentric orbit L , argument of perilune ωL , and required velocity at the insertion point vA as the regulating variable P. Choose the hyperbolic B-plane parameters of the selenocentric segment BT and BR as well as the flight time from the insertion point to the escape point TAC as the target variable Q1 in the first iteration. Choose the terminal altitude rE , reentry angle θ , and return orbit inclination iE from the constraints as the target variable Q2 in the second iteration. Within the framework of differential correction, use the RKF7(8) algorithm to complete the precise design of the return trajectory. The results obtained from the first iteration will satisfy the requirements on the hyperbolic B-plane parameters, whereas the results of the second

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153

iteration will satisfy the requirements on the reentry altitude, reentry angle, and return orbit inclination. Since the terminal point of a direct return trajectory corresponds to the reentry point, there is an additional requirement on the longitude and latitude of the reentry point. In this case, fine tune the return orbit inclination to achieve the desired latitude of the reentry point, and then, adjust the orbit insertion instant such that the desired longitude of the reentry point is reached. Check the reentry latitude again and iteratively repeat this process until both the latitude and longitude reach the desired values. 5.

Simulation example

Owing to the restrictions of Chinese landing sites, the geocentric segment has to employ the ascent phase; yet, there is no special requirement on the selenocentric segment. Hence, in this simulation, we will adopt the ascent-ascent trajectory. Assume that the return will be executed in September 2015. By using a proper mathematical model for the trajectory and setting a flight time of T = 72h, the orbit insertion instant of the direct return trajectory is 00:38:57 on September 20, 2015. The position vector r and velocity vector v at the insertion point within the geocentric equatorial inertial frame can be calculated from Eq. (3.214); their units are m and m/s, respectively.

r = −128232752.16i − 353774402.36j − 117003737.91k v = 1061.54i + 2092.77j − 839.61k

(3.214)

The state of the reentry point can be obtained from Eq. (3.215). The parameters of the reentry point are listed in Table 3.16.

rZE = 3842099.26i + 5214274.23j + 547565.54k vZE = −7606.46i + 3416.28j + 7182.97k

(3.215)

The B-plane target parameter vector is   Q∗ = BT BR TAC = −4347900.82 −1223970.25 12.9862

(3.216)

where BT and BR have a unit of m and TAC has a unit of h. The size of the corresponding sphere of influence of the Moon is 58,599,068.64 m. Using differential correction based on the hyperbolic B-plane parameters, we can reach the target value Q∗ by adjusting the initial values only once. Moreover, since integration is Table 3.16 Reentry point parameters of the direct return trajectory TZ /(UTC)

rZE /(km)

θ/(◦ )

λE /(◦ )

ϕE /(◦ )

iE /(◦ )

2015/09/23 00:38:57

6500.02

−6.00

42.33

4.83

42.10

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Table 3.17 Perigee parameters of the indirect return trajectory

TZ /(UTC)

rZE /(km)

iE /(◦ )

θ/(◦ )

2015/09/23 00:38:57

6600.00

43.00

0.00

only needed for the selenocentric segment, the initial values for the second iteration can be rapidly obtained. The computational efficiency of the algorithm is satisfactory. For an indirect return trajectory, the longitude and latitude are not included in the constraints. Hence, the requirement on the orbit insertion time is less stringent, resulting in multiple available return windows. To facilitate comparison, we will use the same orbit insertion time (i.e., 00:38:57 on September 20, 2015) for the indirect return trajectory as well. Now, the position vector r and velocity vector v at the insertion point within the geocentric equatorial inertial frame are

r = −128232279.89i − 353779573.84j − 117001071.96k v = 1065.67i + 2091.14j − 846.08k

(3.217)

The state of perigee can be obtained from Eq. (3.218). The perigee parameters are listed in Table 3.17.

rJE = 2936716.34i + 5730564.77j + 1447874.51k (3.218) vJE = −8008.78i + 2322.49j + 7052.33k At perigee, through the Hohmann transfer orbit, an Earth parking orbit with an inclination iE = 43◦ , a semi-major axis a = 6600km, and an eccentricity e = 1 can be realized. The required change in velocity for orbit transfer is v = 2309.79i − 669.82j − 2033.95k

(3.219)

It can be easily seen that, compared to the direct trajectory, an extra velocity change of v = 3149.72 m/s is needed in the geocentric segment of the indirect trajectory. Based on these results, we plot the 3D simulated direct return trajectory in Fig. 3.19.

3.3.8 Impulsive Thrust Orbit Control for Cislunar Transfer On November 1, 2014, the Chang’e 5-T1 return capsule accurately landed at the target landing site, marking the success of the high-speed cislunar return mission, a phase III project of the Chinese Lunar Exploration Program. The guidance, navigation, and control (GNC) system of the service module ensured the successful delivery of the return capsule to the return corridor with a precise reentry angle and velocity, which was considered a key element of the entire mission. In this section, we introduce

3.3 Orbit Control for Typical Spacecraft

155

impulsive thrust orbit control for cislunar transfer based on the orbit control process of the high-speed cislunar return vehicle. 1.

Analysis of deviation of reentry angle

The reentry angle is defined as the angle between the velocity vector and the local horizontal line during reentry. This angle directly affects the maneuvering capabilities and aerodynamic heating of the return vehicle. To ensure good reentry conditions, it is necessary to analyze the factors influencing the deviation of the reentry angle. During the flight of the service module, these factors may include the orbit measurement accuracy, attitude adjustment before orbit control, orbit control errors, attitude adjustment after orbit control, and module-capsule separation velocity. Now, let us examine some influential components of the orbit control process. (1)

Accuracy of orbit control

Irrespective of whether 10 N thrusters or 490 N engines are used, owing to the use of a high-accuracy accelerometer for measuring the velocity change and the adoption of the velocity-change shut-down strategy, the error of the orbit control process is usually less than 10 cm/s. (2)

Effects of orbit control attitude establishment on the orbit

Subsequent to the transition from a cruise attitude to an orbit control attitude, the spacecraft is likely to deviate from the desired Sun-facing orientation, leading to the need for consuming the onboard battery. Therefore, the attitude adjustment process should be kept short. Normally, before the establishment of the orbit control attitude, the attitude adjustment is performed using 10 N thrusters simultaneously in three axes. Jetting from attitude adjustment affects the orbit. The establishment of the orbit control attitude is usually completed before orbit control, whereas orbit measurement to determine the orbit control strategy is completed about 4 h prior to orbit control. That is, the velocity change associated with orbit control attitude establishment is not part of the desirable velocity change of orbit control; this undesirable velocity change is, therefore, considered an error of the current orbit control. The change in velocity generated during attitude adjustment is mostly uncertain. For example, in CE-1 and CE-2 missions, the in-orbit attitude adjustment in most cases generated velocity changes of approximately 5 cm/s, but this number could reach 50 cm/s in some extreme cases. (3)

Effects of switching from orbit control attitude to cruise attitude on the orbit

The completion of orbit control is accompanied with the consumption of a large amount of onboard energy. Therefore, the spacecraft needs to rapidly switch to a Sun-tracking attitude to recharge. Such a switch, similar to the opposite switch we just discussed, is also executed by the 10 N thrusters simultaneously along the three axes of the spacecraft. Likewise, undesirable jets with the same order of magnitude will be generated during this process, influencing the orbit. The establishment of the

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Sun-tracking attitude is completed after orbit control. Thus, the associated velocity change is not part of the desirable velocity change of orbit control; this undesirable velocity change should also be considered an error of the current orbit control. Since this velocity change occurs at a time different from that of the switch from the cruise attitude to orbit control attitude, it tends to affect the orbit in a different way. (4)

Effects of jets in other processes on the orbit

Other processes during flight involving jets may include spinning initiation/termination and momentum-wheel unloading. The timing of these events is usually controllable. Therefore, they can be conducted from orbit measurement, and their effects on the orbit can be compensated for during the subsequent orbit control. 2.

Supporting measures for high-accuracy orbit control

In addition to the jet reduction measures including the use of momentum wheels for Sun tracking in the control system design, various techniques were used in jet management during the whole course of flight. In particular, the following measures were taken to ensure the control accuracy of the return trajectory. (1)

Attitude trajectory design before orbit control

Spacecraft flight requires multiple attitude maneuvers, which are usually performed using jets. The deviations generated in early-stage orbit control may be compensated for in midcourse correction, but for the 5th or 6th midcourse correction, the orbit control accuracy must be ensured because there are no further opportunities for compensation. In this situation, momentum wheels can be used for attitude adjustment. (2) (a)

(b)

(3)

Whole-course jet management In the cruise attitude, momentum wheels were used to avoid orbit perturbations due to jets. However, during cruise, owing to the environmental perturbing torque, angular-momentum accumulation is not completely avoidable, leading to momentum-wheel saturation. To reduce momentum-wheel unloading, the return vehicle adopted the three-axis zero-momentum scheme so that the system had a higher angular-momentum storage capacity with the minimum frequency of jet unloading. In the cruise attitude, the spacecraft slowly spun about the Sun-tracking (+x) axis to counter-balance most of the perturbing torque and avoid jet unloading. Furthermore, before each orbit transfer, the ground station commanded the spacecraft to terminate spinning stepwise using wheel control to again avoid jets. High-accuracy, high-reliability, closed-loop orbit control utilizing accelerometer data

Orbit control burn was initiated by the autonomous program with a predetermined firing time, whereas orbit control shut-down was controlled by the dual fail-safe

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157

measure of both velocity change and time. High-accuracy orbit control inevitably imposes a high accuracy requirement on the accelerometer, especially constant drift accuracy. Moreover, to ensure the accuracy of determining the orbit transfer strategy, in-orbit calibration is needed for the orbit control engines. (a)

On-orbit calibration of accelerometer

On-orbit accelerometer calibration can be divided into two aspects. The first is the calibration of the zero offset. Before each orbit control, examine data in a certain period during which no thrust acceleration exists. The average value is then used as the zero offset of the accelerometer. This will be compensated for in-orbit control when the accelerometer data are used to calculate the velocity change. The second aspect is the calibration of the scale coefficient of the accelerometer using orbit measurement data. After each orbit transfer, the ground measurement station provides the velocity change in the orbit transfer process, as well as the accumulated velocity change measured using the accelerometer. Subsequently, the impulse equivalent scale coefficient of the accelerometer can be calculated. This is used in the calculation of the next orbit transfer strategy to compensate for the orbit transfer velocity change, improving the orbit control accuracy. (b)

On-orbit calibration of engine thrust

After orbit insertion, the engine thrust varies with changes in the temperature and pressure of the propellant tank. If a constant thrust is used in the calculation of the orbit control strategy, large errors are expected, which affect the orbit transfer accuracy. Therefore, thrust calibration after each orbit transfer is an essential step in highaccuracy orbit transfer. Various thrust calibration methods have been proposed globally. Reentering vehicles mostly utilize orbit measurement data to calibrate thrust, with additional correction based on propellant tank parameters including pressure and temperature. (c)

Quasi-real-time calculation and calibration of control strategies

Before each orbit transfer with 490 N engines, ground support should determine the orbit transfer strategy based on the orbital information from the spacecraft. When doing so, the spacecraft mass reduction and extra velocity change due to PropellantSettling should be considered. The velocity change due to inertial attitude adjustment and Sun capturing is another term to be considered. After 490 N orbit transfer, the engine thrust and accelerometer scale factor should be calibrated based on orbit measurement results, and they serve as the input for the next orbit transfer.

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References 1. Yang J (2002) Spacecraft orbital dynamics and control II. China Astronautic Publishing House, Beijing 2. Hou L, Li H, Yang X (2008) Optimization platform for multiple orbit transfer control parameters under constraints. In: The 21st academic meeting of space exploration commission, chinese society of space research. Suifenhe, Heilongjiang, pp 101–106 3. Li J (2013) Continuous thrust orbital maneuver optimization. A Dissertation of Beihang University, Beijing 4. Yang J (1995) Spacecraft orbital dynamics and control I. China Astronautic Publishing House, Beijing 5. Yang W, Zhou W (2007) Orbit design for Lunar exploration satellite CE-1. Spacecraft Eng 16(6):16–24 6. Bate RR, Mueller DD, White JE (1971) Fundamentals of astrodynamics. Dover Publications, New York 7. Curtis HD (2005) Orbital mechanics for engineering students.Oxford: Elsevier Butterworth Heinemann 8. Gao Y (2008) Study on the cislunar transfer trajectory of the lunar probe. A Dissertation of National University of Defense Technology, Changsha 9. Zhang L, Dengyun Yu, Zhang He (2010) Design of Moon return trajectory with direct atmospheric reentry. Spacecraft Eng 19(5):50–55 10. Kizner W (1959) A method of describing miss distances for lunar and interplanetary trajectories. NASA Report: NASA-CR-63483 11. Zhang L, Dengyun Yu, Zhang He (2011) Preliminary design and characteristic analysis of Moon-to-Earth transfer trajectories. Chinese Space Sci Technol 31(3):62–69

Chapter 4

Spacecraft Attitude Kinematics and Dynamics

4.1 Introduction Spacecraft attitude describes the rotational motion of a spacecraft around its center of mass, often by the orientation or direction of its fixed body frame with respect to a reference frame. Attitude kinematics describes the laws of variables for rotation and usually characterizes the relationship among parameters such as attitude angles, angular velocity, and angular acceleration, without reference to the causes of motions. Attitude dynamics mainly describes the laws of rotation of a spacecraft around its center of mass with internal and external moments, as well as the relative motion of the interior of the spacecraft with internal moments.

4.2 Attitude and Attitude Kinematics 4.2.1 Attitude Description To describe the attitude of a spacecraft, a spatial reference coordinate system and a spacecraft-fixed body coordinate system need to be defined, where the angle relationship between the two describes the attitude state. Different types of representations can be used for the attitude. Common representations include direction cosine matrices, Euler angles, Euler angle–axis representations, and quaternions [1–4]. In practice, it is recommended to choose the representation reasonably according to the application and the characteristics of attitude parameters. 1.

Direction cosine matrix A direction cosine matrix describes the relationship between two coordinate systems in the most direct manner. Let x 0b , y0b , and z 0b be the unit coordinate vectors in the fixed coordinate system, and x 0r , y0r and z 0r be the unit coordinate vectors in the reference coordinate system. The relationship between the vectors

© Beijing Institute of Technology Press 2022 Y. Xie et al., Spacecraft Dynamics and Control, Space Science and Technologies, https://doi.org/10.1007/978-981-33-6448-6_4

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4 Spacecraft Attitude Kinematics and Dynamics

is expressed as follows: ⎤ ⎡ 0⎤ ⎡ ⎤⎡ 0 ⎤ xr C11 C12 C13 xr x 0b ⎣ y0 ⎦ = C br ⎣ y0 ⎦ = ⎣ C21 C22 C23 ⎦⎣ y0 ⎦ r r b z 0b z 0r C31 C32 C33 z 0r ⎡

where C br is the direction cosine matrix mapping vectors from the reference frame to the fixed frame, also called the attitude matrix. The matrix contains elements Ci j (i, j = 1, 2, 3) called direction cosines, which are the scalar products of the unit vectors in the two coordinate systems. Given C Tbr Cbr = I, nine equations for the direction cosine matrix should be derived. However, the matrix consists of mutually orthogonal unit vectors; therefore, there are only three independent attitude parameters. In this chapter, I denotes the identity matrix of the corresponding dimension, unless otherwise stated. 2.

Euler angles Any two coordinate systems can coincide with three consecutive rotations about different axes of one of the systems, and the angle of each rotation is called the Euler angle. An attitude can be described intuitively and conveniently using the Euler angles. The fixed frame is rotated relative to the reference frame by the three commonly used Euler angles: φ, about the xb axis; θ , about the yb axis; and ψ, about the z b axis (see Fig. 4.1). Primitive rotation matrices can then be obtained as ⎡

⎡ ⎡ ⎤ ⎤ ⎤ 1 0 0 cos θ 0 − sin θ cos ψ sin ψ 0 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ C x (φ) = ⎣ 0 cos φ sin φ ⎦, C y (θ ) = ⎣ 0 1 0 ⎦, C z (ψ) = ⎣ − sin ψ cos ψ 0 ⎦ 0 − sin φ cos φ sin θ 0 cos θ 0 0 1

When using the Euler angles to describe an attitude, the corresponding direction cosine matrix is related to the sequence of the three Euler rotations. The sequences or conventions can be categorized into two types: (a) the axes of the three rotations are different, and (b) the axes of the first and third rotations are

Fig. 4.1 Primitive rotation matrices

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161

the same. In a Euler angle sequence, the axes xb , yb , and z b are often denoted by 1, 2, and 3, respectively. Their corresponding primitive rotation matrices are denoted by C 1 (φ), C 2 (θ ), and C 3 (ψ), respectively. For brevity, cos φ, cos θ , and cos ψ are denoted as cφ , cθ , and cψ , respectively, and sin φ, sin θ , and sin ψ are denoted as sφ , sθ , and sψ , respectively. Thus, the direction cosine matrices for the first type of Euler angle conventions are ⎤ cψ cθ cψ sθ sφ + sψ cφ −cψ sθ cφ + sψ sφ C 123 = C 3 (ψ)C 2 (θ )C 1 (φ) = ⎣ −sψ cθ −sψ sθ sφ + cψ cφ sψ sθ cφ + cψ sφ ⎦ sθ −cθ sφ cθ cφ ⎤ ⎡ cθ cψ cθ sψ cφ + sθ sφ cθ sψ sφ − sθ cφ ⎦ C 132 = C 2 (θ )C 3 (ψ)C 1 (φ) = ⎣ −sψ cψ cφ cψ sφ sθ cψ sθ sψ cφ − cθ sφ sθ sψ sφ + cθ cφ ⎤ ⎡ sψ −cψ sθ cψ cθ C 231 = C 1 (φ)C 3 (ψ)C 2 (θ ) = ⎣ −cφ sψ cθ + sφ cθ cφ cψ cφ sψ sθ + sφ cθ ⎦ sφ sψ cθ + cφ sθ −sφ cψ −sφ sψ sθ + cφ cθ ⎤ ⎡ sψ sφ sθ + cψ cθ sψ cφ −cψ sθ + sψ sφ cθ C 213 =C 3 (ψ)C 1 (φ)C 2 (θ ) = ⎣ −sψ cθ + cψ sφ sθ cψ cφ sψ sθ + cψ sφ cθ ⎦ cφ sθ −sφ cφ cθ ⎤ ⎡ cθ cψ − sθ sφ sψ cθ sψ + sθ sφ cψ −sθ cφ C 312 = C 2 (θ )C 1 (φ)C 3 (ψ) = ⎣ −cφ sψ cφ cψ sφ ⎦ ⎡

C 321

sθ cψ + cθ sφ sψ sθ sψ − cθ sφ cψ cθ cφ

⎤ cθ sψ −sθ cθ cψ = C 1 (φ)C 2 (θ )C 3 (ψ) = ⎣ −cφ sψ + sφ sθ cψ cφ cψ + sφ sθ sψ sφ cθ ⎦ sφ sψ + cφ sθ cψ −sφ cψ + cφ sθ sψ cφ cθ ⎡

For the second convention, the angles of the three rotations are denoted by θ1 , θ2 , and θ3 , and ci and si (i = 1, 2, 3) are denoted by cos θi and sin θi , respectively. Thus, the direction cosine matrices for the six different conventions of the second type are ⎤ s2 s1 −s2 c1 c2 = C 1 (θ3 )C 2 (θ2 )C 1 (θ1 ) = ⎣ s3 s2 c3 c1 − s3 c2 s1 c3 s1 + s3 c2 c1 ⎦ c3 s2 −s3 c1 − c3 c2 s1 −s3 s1 + c3 c2 c1 ⎤ ⎡ s2 s1 s2 s1 c2 = C 1 (θ3 )C 3 (θ2 )C 1 (θ1 ) = ⎣ −c3 s2 c3 c2 c1 − s3 s1 c3 c2 s1 + s3 c1 ⎦ s3 s2 −s3 c2 c1 − c3 s1 −s3 c2 s1 + c3 c1 ⎤ ⎡ c3 c1 − s3 c2 s1 s3 s2 −c3 s1 − s3 c2 c1 ⎦ = C 2 (θ3 )C 1 (θ2 )C 2 (θ1 ) = ⎣ s2 s1 c2 s2 c1 s3 c1 + c3 c2 s1 −c3 s2 −s3 s1 + c3 c2 c1 ⎡

C 121

C 131

C 212

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4 Spacecraft Attitude Kinematics and Dynamics

⎤ c3 c2 c1 − s3 s1 c3 s2 −c3 c2 s1 − s3 c1 ⎦ = C 2 (θ3 )C 1 (θ2 )C 2 (θ1 ) = ⎣ −s2 c1 c2 s2 s1 s3 c2 c1 + c3 s1 s3 s2 −s3 c2 s1 + c3 c1 ⎤ ⎡ c3 c1 − s3 c2 s1 c3 s1 + s3 c2 c1 s3 s2 = C 3 (θ3 )C 1 (θ2 )C 3 (θ1 ) = ⎣ −s3 c1 − c3 c2 s1 −s3 s1 + c3 c2 c1 c3 s2 ⎦ s2 s1 −s2 c1 c2 ⎤ ⎡ c3 c2 c1 − s3 s1 c3 c2 s1 + s3 c1 −c3 s2 = C 3 (θ3 )C 2 (θ2 )C 3 (θ1 ) = ⎣ −s3 c2 c1 − c3 s1 −s3 c2 s1 + c3 c1 s3 s2 ⎦ s2 c1 −s2 s1 c2 ⎡

C 232

C 313

C 323

3.

Euler axis–angle parameterization According to Euler’s theorem, any displacement of a rigid body around a fixed point can be obtained by rotating the frame by a certain angle about an axis passing through that point. ex , e y , ez , which are the components of the unit vector along the e axis in the reference coordinate system, and the angle of rotation around the axis  can be used as the attitude parameters. In particular, the components of the vector along the e axis in the body frame are the same as those of the reference frame. The e axis is called the Euler axis, and the angle  is called the Euler angle. Such a representation of the attitude is called the Euler axis–angle parameterization. Given the constraint ex2 + e2y + ez2 = 1, the representation contains three independent parameters. The Euler axis–angle parameterization describes direction cosine matrices for the attitude as ⎡

⎤ c + e2x (1 − c ) ; ex e y (1 − c ) + ez s ; ex ez (1 − c ) − e y s ⎢ ⎥ Cbr = ⎣ ex e y (1 − c ) − ex s ; c1 + e2y (1 − c ) ; e y ez (1 − c ) + ex s ⎦ 2 ex ez (1 − c ) + e y s ; e y ez (1 − c ) − ex s ; c + ez (1 − c ) = c I + (1 − c )eeT − s e×

where c and s represent cos  and sin , respectively, and e× is the skewsymmetric matrix of e: ⎡

⎤ 0 −ez e y e× = ⎣ ez 0 −ex ⎦ −e y ex 0 For a known direction cosine matrix C, its corresponding Euler axis–angle parameters are

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163

⎡

⎤ C23 − C32 e = 2 sin1  ⎣ C31 − C13 ⎦ C12 − C21 cos  = 21 (trC − 1) where Ci j (i, j = 1, 2, 3) are the elements of the direction cosine matrix C and trC is the trace of C. 4.

Quaternions According to the Euler axis–angle representation, a quaternion of an attitude is defined as  T

qv q= = ex sin 2 e y sin 2 ez sin 2 cos 2 q4 T

where q v = q1 q2 q3 is the vector part of the quaternion and q4 is a scalar. These four parameters follow the constraint equation below: q12 + q22 + q32 + q42 = 1 Using the trigonometric formulas cos  = 2 cos2 2 − 1 and sin  = 2 sin 2 cos 2 , the relationship between the direction cosine matrix C and the quaternion q is expressed as ⎡

⎤ q42 + q12 − q22 − q32 2(q1 q2 + q4 q3 ) 2(q1 q3 − q4 q2 ) C(q) = ⎣ 2(q1 q2 − q4 q3 ) q42 − q12 + q22 − q32 2(q2 q3 + q4 q1 ) ⎦ 2(q1 q3 + q4 q2 ) 2(q2 q3 − q4 q1 ) q42 − q12 − q22 + q32 Quaternions can also be described in another form:  q=

q0 qv



= cos 2 ex sin

 2

e y sin

 2

ez sin

  T 2

where q0 is the scalar part of the quaternion. In this book, quaternions are expressed in the first form, unless otherwise stated.

4.2.2 Attitude Kinematics Attitude kinematics describes the relationship between the variation of a spacecraft’s attitude parameters and its angular velocity. The attitude kinematics equation for a direction cosine matrix C is

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dC = −ω× C dt where ω is the velocity of rotation in the body frame with respect to the reference frame and ω× is the skew-symmetric matrix of ω. To describe the attitude in terms of the Euler angles, attitude kinematics equations can be obtained based on the sequence of the Euler rotations. Taking the abovementioned “1–2–3” sequence as an example, we have ⎛⎡

⎤ ⎛⎡ ⎤ ⎡ ⎤⎞⎞ φ˙ 0 0 ⎝ ⎣ ⎦ ⎝ ⎣ ⎦ ⎣ ˙ ω = C 3 (ψ) θ + C 1 (φ) 0 ⎦⎠⎠ 0 + C 2 (θ ) ˙ ψ 0 0 A more compact form of the above equation is T

ω = S φ˙ θ˙ ψ˙ where ⎡

⎤ cos ψ cos θ sin ψ 0 S = ⎣ − sin ψ cos θ cos ψ 0 ⎦ sin θ 0 1 Similarly, the body angular velocity can be obtained according to the time derivatives of the Euler angles as follows:

φ˙ θ˙ ψ˙

T

= S−1 ω

where S−1 is the inverse of S and is expressed as ⎡

S−1

⎤ cos ψ/ cos θ − sin ψ/ cos θ 0 =⎣ sin ψ cos ψ 0⎦ − cos ψ tan θ sin ψ tan θ 1

The kinematics equations for other Euler angle sequences can be obtained in a similar manner. S and S−1 for sequences of rotations about different axes are presented below. 1.

3–2–1 sequence: ⎡

⎡ ⎤ ⎤ 1 0 − sin θ 1 sin φ tan θ cos φ tan θ S = ⎣ 0 cos φ sin φ cos θ ⎦, S−1 = ⎣ 0 cos φ − sin φ ⎦ 0 − sin φ cos φ cos θ 0 sin φ/ cos θ cos φ/ cos θ

4.2 Attitude and Attitude Kinematics

2.

165

3–1–2 sequence: ⎡

⎡ ⎤ ⎤ cos θ 0 − cos ϕ sin θ cos θ 0 sin θ ⎦, S−1 = ⎣ tan φ sin θ 1 − tan φ cos θ ⎦ S=⎣ 0 1 sin φ sin θ 0 cos φ cos θ − sin θ/ cos φ 0 cos θ/ cos φ 3.

2–1–3 sequence: ⎡

⎤ ⎡ ⎤ cos ψ sin ψ cos φ 0 cos ψ − sin ψ 0 S = ⎣ − sin ψ cos ψ cos φ 0 ⎦, S = ⎣ sin ψ/ cos φ cos ψ/ cos φ 0 ⎦. 0 sin θ 1 tan φ sin ψ tan φ cos ψ 1 4.

1–3–2 sequence: ⎡ ⎤ ⎤ cos θ sec ψ 0 sin θ sec ψ cos θ cos ψ 0 − sin θ S = ⎣ − sin ψ 1 0 ⎦, S−1 = ⎣ cos θ tan ψ 1 sin θ tan ψ ⎦. − sin θ cos ψ 0 cos θ − sin θ 0 cos θ ⎡

5.

2–3–1 sequence: ⎡ ⎤ ⎤ 1 − cos φ tan ψ sin φ tan ψ 1 − sin ψ 0 S = ⎣ 0 cos φ cos ψ sin φ ⎦, S−1 = ⎣ 0 cos φ sec ψ − sin φ sec ψ ⎦. 0 sin φ cos φ 0 − sin φ cos ψ cos φ ⎡

For the second type of conventions, where the axes of the first and third rotations are the same, the rotational velocity can be expressed as T

ω = S θ˙1 θ˙2 θ˙3 S and its inverse S−1 for each rotation sequence are given below. 1.

1–2–1 sequence: ⎡

⎤ ⎡ ⎤ cos θ2 0 1 0 sin θ3 csc θ2 cos θ3 csc θ2 S = ⎣ sin θ3 sin θ2 cos θ3 0 ⎦, S−1 = ⎣ 0 cos θ3 − sin θ3 ⎦. 1 − sin θ3 cot θ2 − cos θ3 cot θ2 cos θ3 sin θ2 − sin θ3 0 2.

1–3–1 sequence: ⎡ ⎤ ⎤ 0 − cos θ3 csc θ2 sin θ3 csc θ2 0 1 cos θ2 S = ⎣ − cos θ3 sin θ2 sin θ3 0 ⎦, S−1 = ⎣ 0 sin θ3 − cos θ3 ⎦. 1 − cos θ3 cot θ2 − sin θ3 cot θ2 sin θ3 sin θ2 cos θ3 0 ⎡

166

3.

4 Spacecraft Attitude Kinematics and Dynamics

2–1–2 sequence: ⎤ ⎡ ⎤ sin θ3 sin θ2 cos θ3 0 sin θ3 sin θ2 0 − cos θ3 sin θ2 ⎦. S=⎣ 0 1 ⎦, S−1 = ⎣ cos θ3 0 sin θ3 cos θ2 − sin3 cot θ2 1 cos θ3 cot θ2 − cos θ3 sin θ2 sin θ3 0 ⎡

4.

2–3–2 sequence: ⎡

⎡ ⎤ ⎤ cos θ3 sin θ2 − sin θ3 0 cos θ3 csc θ2 0 sin θ3 csc θ2 ⎦. S = ⎣ cos θ2 0 1 ⎦, S−1 = ⎣ − sin θ3 0 cos θ3 − cos θ3 cot θ2 1 − sin θ3 cot θ2 sin θ3 sin θ2 cos θ3 0 5.

3–1–3 sequence: ⎡ ⎤ ⎤ sin θ3 csc θ2 cos θ3 csc θ2 0 sin θ3 sin θ2 cos θ3 0 S = ⎣ cos θ3 sin θ2 − sin θ3 0 ⎦, S−1 = ⎣ − sin θ3 0 ⎦. cos θ3 0 1 cos θ2 − sin θ3 cot θ2 − cos θ3 cot θ2 1 ⎡

6.

3–2–3 sequence: ⎡ ⎤ ⎤ − cos θ3 csc θ2 sin θ3 csc θ2 0 − cos θ3 sin θ2 sin θ3 0 S = ⎣ sin θ3 sin θ2 cos θ3 0 ⎦, S−1 = ⎣ cos θ3 0 ⎦. sin θ3 0 1 cos θ2 cos θ3 cot θ2 − sin θ3 cot θ2 1 ⎡

The attitude kinematics equation for the quaternion q can be expressed as ⎤⎡ ⎤ ⎡ ⎤ q1 0 ωz −ω y ωx q˙1 ⎢ q˙2 ⎥ 1 ⎢ −ωz 0 ωx ω y ⎥⎢ q2 ⎥ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎣ q˙3 ⎦ 2 ⎣ ω y −ωx 0 ωz ⎦⎣ q3 ⎦ q˙4 −ωx −ω y −ωz 0 q4 ⎤⎡ ⎤ ⎡ ωx q4 −q3 q2 q1 ⎥⎢ ω y ⎥ 1⎢ q q −q q 3 4 1 2 ⎥⎢ ⎥ = ⎢ 2 ⎣ −q2 q1 q4 q3 ⎦⎣ ωz ⎦ −q1 −q2 −q3 q4 0 ⎡

A more compact form of the above equation is       × 1 −ω× ω 1 ω qv qv q˙ = q4 I + q= T T −q v 0 0 2 −ω 0 2

4.3 Attitude Dynamics

167

4.3 Attitude Dynamics 4.3.1 Attitude Dynamics of Rigid-Body Spacecraft A schematic diagram of a rigid body B that moves in inertial space with an angular velocity vector ω is shown in Fig. 4.2. P is any point in the rigid body, R P is its position vector relative to an inertial point of reference O, and it is fixed with a basis

T vector f b = i j k . Furthermore, dm is the mass of a differential element on B, and it radius vectors relative to the points P and O are r and R = R P + r, respectively. The absolute velocity of the differential element dm is   ˙ P + (˙r )I V = R˙ = R I   ˙ P is the velocity of P relative to the inertial frame and (˙r )I is the velocity where R I of dm relative to P in the inertial space. As the differential element dm in the rigid body has no relative motion with respect to the point P, the velocity of the differential element dm can be expressed as   ˙ P +ω × r = VP +ω × r V = R b

(4.1)

  ˙ P is the velocity of the point P relative to the fixed frame. where R b The momentum of the differential element dm is V dm. Its moment of momentum relative to the reference point P is r × V dm. The moment of momentum of the entire rigid body relative to the reference point P is  hP =

r × V dm B

ω

Fig. 4.2 Schematic diagram of rigid-body motion B

zb

O

xI

RP

yI

r

k

R

zI

dm

i

P j

xb

yb

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4 Spacecraft Attitude Kinematics and Dynamics

 where B represents the integral over the entire rigid body B. Inserting Eq. (4.1) into the above equation, we obtain 



hP =

r × (ω × r)dm − V P × B

rdm

(4.2)

B

When the reference point P coincides with the center of mass of the rigid body C, we have B rdm = 0. Hence,  h =

r × (ω × r)dm

C

B





r × (ω × r)dm = B



 r · rI − r r T dmω

(4.3)

(4.4)

B

The integral on the right-hand side of Eq. (4.4) can be written as 



 r T rI − r r T dm =

B



 T r × r × dm

(4.5)

B

T

By inserting the coordinate vector r = r x r y r z for the vector r in the fixed frame in Eq. (4.5), we have   T  r T rI − r r T dm = r × r × dm B ⎡ B ⎤ r y2 + r z2 −r x r y −r x r z ⎢ ⎥ = ⎣ −r y r x r x2 + r z2 −r y r z ⎦dm . B −r z r x −r z r y r x2 + r y2 ⎡ ⎤ Jx −Jx y −Jx z = ⎣ −Jyx Jy −Jyz ⎦

J=



−Jx z −Jyz

(4.6)

Jz

For simplicity, we hereafter assume that h is the angular momentum about the center of mass of the spacecraft and omit the superscript “C,” unless otherwise stated. According to the theorem on the moment of momentum, when the center of mass of a system C is taken as the reference point, the time derivative of the angular momentum h is equal to the external moments T acting on the center of mass of the rigid body. According to the relation between the time derivatives of vectors in different coordinate systems, we have 

dh dt



 =

I

dh dt

 +ω×h = T b

(4.7)

4.3 Attitude Dynamics

169

where the subscripts I and b denote the time derivatives in the inertial coordinate system and the fixed coordinate system for the rigid body, respectively. Given that   dh ˙ + J ω˙ = Jω dt b The moment of inertia matrix is constant in the fixed frame, i.e., J˙ = 0. Equation (4.7) can then be written as 

dh dt

 = J ω˙ + ω × h = T

(4.8)

I

T

where T = Tx Ty Tz is the coordinate vector in the fixed frame. T T



Inserting the coordinate vectors ω = ωx ω y ωz and T = Tx Ty Tz in the fixed frame into Eq. (4.8), the projection of the vectors can be expressed as ⎡

⎤⎡ ⎤ ⎡ ⎤× ⎡ ⎤ ⎡ ⎤ Jx −Jx y −Jx z ω˙ x ωx Jx −Jx y −Jx z Tx ⎣ −Jyx Jy −Jyz ⎦⎣ ω˙ y ⎦ + ⎣ ω y ⎦ ⎣ −Jyx Jy −Jyz ⎦ = ⎣ Ty ⎦ −Jx z −Jyz Jz ω˙ z ωz −Jx z −Jyz Jz Tz

(4.9)

When the basis vector f b coincides with the inertial body frame, Eq. (4.9) can be simplified as   Jx ω˙ x − Jy − Jz ω y ωz = Tx Jy ω˙ y − (Jz − Jx )ωx ωz = Ty   Jz ω˙ z − Jx − Jy ωx ω y = Tz

4.3.2 Attitude Dynamics of Flexible Spacecraft For the schematic of the structure of a spacecraft with large flexible solar-panel arrays shown in Fig. 4.3, we introduce the derivation process of the basic equations of dynamics of the flexible spacecraft. The origin of the satellite-axis frame O xb yb z b is the center of mass of the system, and the origin of the flexible appendage frame P xa ya z a is at the hinge point of the appendage P. The inertia matrix of the spacecraft’s central body B in the satellite-axis frame is J b , and the inertia matrices of appendage A in the satellite-axis frame and the appendage frame are J as and J a , respectively. The transition matrix from the satellite-axis frame to the appendage frame is A. The distance vector from the hinge point of the appendage to the center of mass of the system is r a , and the distance vector from the center of mass of the appendage to the

170

4 Spacecraft Attitude Kinematics and Dynamics

Fig. 4.3 Schematic diagram of a flexible spacecraft

mj u A j r0

ρj

yb

ra

P

B zb

O

xb

hinge point is r 0 . The angular velocity of the center body relative to space is denoted by ω, and the angular velocity of the appendage relative to the center body is ωa . The distance vector of the mass m in the appendage coordinate system to the hinge point is r j , and its elastic displacement vector is u j . The distance vector of any mass m j to the center of mass of the system is ρ j , and it can be expressed as ρ j = r a + r j + u j . Let T e be the external moment acting on the body, T a be the hinge moment for the interaction between the body and the appendages, and F j be the elastic stress and damping force acting on the appendage mass m j . Based on the theorem on momentum and angular momentum as well as rotational dynamics of the satellite system, the basic equations of dynamics of the flexible spacecraft can be obtained as [3] ⎧  T T × ⎪ ¨ j = Te J s ω˙ + J a AT ω˙ a + m j (r × a A + A r j )u ⎪ ⎪ ⎪ A ⎨  T AT J a ω˙ a + J a ω˙ + AT m j r×j u¨ j = T a ⎪ A ⎪  T  T ⎪   ⎪ ⎩ m j u¨ j + m j r × ω˙ a + m j (A r × T + r × A)ω˙ = F j ( j ∈ A) a j j

(4.10)

where Js = Jb + J a =





 ×  × r a + AT r j m j r a + AT r j

A

 T × m jr× j rj

A

  T × T J a = ma r × A + AT J a A a A ro Here, J s is the inertia matrix of a satellite when its flexible appendages are fixed. Dynamics equations comprised of the Euler parameters and flexible modal coordinates can be formulated by normalizing the dynamics equations of the flexible spacecraft using the vector of mode shape, matrix of mode shape, and modal coordinates

4.3 Attitude Dynamics

171

of the flexible body: J s ω˙ + ω× J s ω + FsR η + Ra ω˙ a = T e J a ω˙ a + F a η¨ + RTa ω˙ = T a η¨ + 2ζ η˙ + 2 η + FsT ω˙ = 0 In the equations above, J a = A J a ;  is the diagonal array of modal frequencies of solar-panel wings; η is the modal coordinate array of solar-panel wings; ζ is the modal damping coefficient of solar-panel wings, which is generally 0.005; F s is the matrix of flexible coupling coefficients for the vibration of solar-panel wings on the rotation of the body; F a is the matrix of flexible coupling coefficients for the vibration of solar-panel wings on the rotation of the wings; and Ra is the matrix of rigid coupling coefficients for the rotation of solar-panel wings on the rotation of the satellite. Ignoring the rotational dynamics of solar arrays, the dynamics equation of the flexible spacecraft can be simplified as J s ω˙ + ω× J s ω + F s η¨ = T e η¨ + 2ζ η˙ + 2 η + F Ts ω˙ = 0 When the solar array rotates, the system frequency in the above-mentioned flexible mode changes with the rotation angle of the solar array. By ignoring the gyroscopic torque and external moments and expressing the moment of inertia matrix of the rigid body as J R = J s − F s F Ts , the above equations become J R ω˙ = 2ζ F s η˙ + F s 2 η    2  ˙ + I + F Ts J −1 η¨ + 2ζ I + F Ts J −1 R F s η R Fs  η = 0

T Given X η = ηT η˙ T , we have X˙ η = Aη X η where  Aη =

0 I  2    T −1  F −2ζ I + F − I + F Ts J −1 s s J R Fs  R



With specific values of the matrix Aη , its eigenvalues can be obtained to determine the system frequency of each flexible mode with the corresponding rotation angle of the solar-panel wing. The modal fundamental frequency of the flexible spacecraft with a pair of solar wings is 0.47 Hz. Figure 4.4 presents the results of the analysis of

172

4 Spacecraft Attitude Kinematics and Dynamics 0.52 0.515 0.51

flex. mode 1 /Hz

0.505 0.5 0.495 0.49 0.485 0.48 0.475 0.47

0

50

100

150

200 arfa /deg

250

300

350

400

Fig. 4.4 Variation of flexible modal system frequency with the rotation angle of the solar array

the relationship between the system frequency of the flexible mode and the rotation angle of the solar array.

4.3.3 Attitude Dynamics of Liquid-Filled Spacecraft Liquid is another important factor contributing to the complexity of spacecraft dynamics [5]. To study the motion of a liquid-filled spacecraft, the following basic simplifying assumptions are often made [1, 6–8]: (1) (2) (3) (4)

(5)

Except for the liquid, all parts of the spacecraft are rigid bodies. The liquid is homogeneous and incompressible. The liquid is almost an ideal fluid, and the effect of its viscosity is a first-order small quantity. The surface tension of the liquid is ignored. For a three-axis stabilized spacecraft with spin stabilization and acceleration, the range of acceleration is between 0.02 g and 5 g, where g is the acceleration due to gravity. In this instance, the Bond number is greater than 100. Therefore, the effect of surface tension is negligible. Under low-gravity conditions in which the Bond number is less than 100, the motion of the liquid has little influence on the spacecraft. The liquid exhibits micromovement relative to the container.

4.3 Attitude Dynamics

1.

173

Fundamental equations and boundary conditions

Let O xb yb z b be the fixed frame of the spacecraft body, the origin and axes of which are fixed to the rigid body of the spacecraft. x b , yb , z b are the unit vectors along the axes; m R , r RC , and J R denote the mass of the rigid body, position vector, and inertia tensor of the center of mass to the point O, respectively; vR and aR are the absolute velocity and absolute acceleration of the point O, respectively; ωR is the absolute angular velocity of the rigid body; F L and T L are the force of the liquid on the rigid body and the moment about the point O, respectively; and F E and T E are the external forces and moments acting on the rigid body, respectively. The dynamics equations for the translation and rotation of the rigid body can be expressed as m R [aR + ω˙ R × r RC + ωR × (ωR × r RC )] = F L + F E

(4.11)

m R r RC × aR + J R · ω˙ R + ωR × J R · ωR = T L + T E

(4.12)

The motion of the liquid is described using the following parameters. V denotes the volume occupied by the liquid in one or more containers (i.e., fuel tanks) in the spacecraft; Sw is the solid wall interface of the liquid; SF refers to the free surface of the liquid; F(r, t) = 0 is the equation of the free surface; ξ is the surface wave height perpendicular to SF ; n is the outward-pointing normal (vector) of the solid wall or free surface; r represents the position vector of a liquid particle in the body frame; u is the relative velocity in the system; p is the pressure; ρ is the density; v refers to the coefficient of viscosity of motion; and ∂t∂ denotes the relative partial derivative in the system Ob1 b2 b3 . Given the assumptions previously stated, the momentum equation of incompressible fluids (i.e., the Navier–Stokes equation) can be expressed as 1 ∂u + (u · ∇)u + 2ωR × u + ω˙ R × r + ωR × (ωR × r) = −aR − ∇ p + νu ∂t ρ (4.13) and the continuity equation is ∇·u=0

(4.14)

where ∇ = b1

∂ ∂ ∂ + b2 + b3 ∂ x1 ∂ x2 ∂ x3

is the Hamiltonian operator. In the following sections,  = ∇ · refers to the Laplace operator.

174

4 Spacecraft Attitude Kinematics and Dynamics

The boundary condition for the solid wall is n · u = 0 (on SW )

(4.15)

The kinematic free-surface boundary condition is n · u = ξ (on SF )

(4.16)

The dynamic free-surface boundary condition is p = constant (on SF )

(4.17)

The force and torque of the liquid F L and T L , respectively, on the spacecraft can be expressed as  FL =  TL =

 pnd S + ρv

SW

(4.18)

r × (n · ∇)[n × (n × u)]dS

(4.19)

Sw



r × pndS + ρv SW

(n · ∇)[n × (n × u)]dS

SW

The first terms in Eqs. (4.18) and (4.19) are the pressure and pressure moment, respectively, while the second terms are the viscous shear force and shear moment, respectively. If the liquid exhibits a vortex motion in the container and the curl of the mass force is zero, from the vorticity on both sides of Eq. (4.13), the Helmholtz equation can be derived by taking the curl on both sides of Eq. (4.13). Hence, ∂ωL + ωR × ωL = (ωL · ∇)v + vωL ∂t

(4.20)

where ωL = 21 ∇ × u is the curl and v = vR + ωR × r + u is the absolute velocity of the liquid at r. 2.

Equivalent mechanical models

Similar to the dynamics model of a flexible spacecraft, partial differential equations obtained by the method of mechanics of continuous media can be used to describe the dynamics model of a liquid-filled spacecraft. However, such an approach cannot be easily applied to engineering design and practice. Therefore, it is necessary to simplify the model. A simplified model should have the following features.

4.3 Attitude Dynamics

(1) (2) (3) (4)

175

It should generally reflect the action of the liquid on the spacecraft (including forces and moments). It should consist entirely of ordinary differential equations with discrete coordinates, and the order of a differential equation should be as low as possible. The parameters of the motion of a rigid body should be retained. The simplified motion of the liquid should be comparable with a simple mechanical system, such as a spring–mass system or a simple pendulum, and both should have the same differential equations.

Feature (4) is physically intuitive. A simplified model with the above features is called an equivalent mechanical model in engineering. A three-dimensional pendulum is used to represent the effect of liquid sloshing. The following are assumed. (1)

(4)

The effect of liquid sloshing in a liquid-filled spacecraft is equivalent to that of a simple pendulum. The parameters of the liquid-filled spacecraft, namely, the mass of the pendulum, fixed point, and pendulum length, remain unchanged. The pendulum is a particle fixed to the body of the spacecraft by a lightweight link. The point of suspension of a pendulum is on the body axis xb .

(1)

Three-dimensional equivalent mechanical model

(2) (3)

A dynamics model of a three-dimensional pendulum [9, 10] can be established using the Newton–Euler method, as shown in Fig. 4.5. The radius vector of the pendulum in the inertial frame is Rp = Rb + r t + r p where Rb is the position vector of the dry center of mass O of the spacecraft relative to the inertial frame OI xI yI z I , r t is the position vector of the fixed point of the pendulum Oc relative to the dry center of mass O of the spacecraft, and r p is the position vector of the center of mass of the pendulum relative to the fixed point of the pendulum Oc . xb

Fig. 4.5 Three-dimensional pendulum model for a liquid-filled spacecraft

O

yb xI yI

Oc Rp

OI

zb

rt

Rb

zI

rp p

176

4 Spacecraft Attitude Kinematics and Dynamics

Fig. 4.6 Pendulum-axis frame

Given the derivative of the coordinates of r p in the inertial frame, we have r˙ p = r p + ωR × r p where ωR denotes the angular velocity of the body frame with respect to the inertial frame and r p is the derivative of r p in the body frame. The coordinates of the radius vector r p in the pendulum-axis frame are [−L p 0 0]T . Its derivative is zero. Therefore, r p = ωp × r p where ωp is the angular velocity of the pendulum-axis frame (see Fig. 4.6) with respect to the body frame. As r t is fixed to the spacecraft body, we have r˙ t = ωR × r t Therefore, the first derivative of Rp is ˙ b + ωR × r t + (ωR + ωp ) × r p ˙p = R R and the second derivative of Rp is R¨ p = R¨ b + ω˙ R × r t + ωR × (ωR × r t ) + (ω˙ R + ω˙ p ) × r p + (ωR × ωp ) × r p + (ωR + ωp ) × [(ωR + vp ) × r p ] According to Newton’s second law of motion, the total external force on the system is

4.3 Attitude Dynamics

177

F a = m p R¨ p + m R R¨ b = (m p + m R ) R¨ b + m p ω˙ R × r t + ωR × (ωR × r t ) + (ω˙ R + ω˙ p ) × r p ! +(ωR × ωp ) × r p + (ωR + ωp ) × [(ωR + ωp ) × r p where F a = F E + (m p + m R )g In the above equation, F E denotes the external force acting on the rigid body, (m p + m R )g is the force of gravity on the spacecraft, and m p is the pendulum mass. Given the total external moments on the system and Euler’s theorem, the total external moments with respect to the center of mass of the rigid body O is T a = J R · ω˙ R + ωR × J R · ωR + (r t + r p ) × m p R¨ p where J R is the inertia tensor of the rigid body. Therefore, T a = T E + m p (r t + r p ) × g where T E is the external moment acting on the rigid body and m p (r t + r p ) × g is the moment induced by the gravity of the pendulum relative to O. With respect to the point of suspension Oc , the external moment on the pendulum is T p = r p × m p R¨ p + T d − r p × m p g where T d is the damping torque and can be obtained using the following equation: T d = C p ωp In the above equation, C p = 2ξ f p m p L 2p ( yb yb + z p z p ), yb yb and z p z p are dyads, yb and zp are unit vectors along the axes of the"two rotations of the pendulum-axis frame with respect to the body frame, f p = Fx /[(m p + m R )L p ], and Fx is the component of the thrust force FE along the xb axis. The above vector equation is projected to the body frame and expressed using coordinates. The vectors that need to be projected are g g, Rb , ωR , r t , r p , ωp , r b , F E , T E , p1 , p2 , among which the coordinates of Rb , ωR , r t , F E , T E , p1 , and p2 in the body frame can be obtained directly. The transition matrix C bi from the inertial system to the body frame can be obtained based on the current attitude of the spacecraft. The gravitational acceleration in the body frame is expressed as

178

4 Spacecraft Attitude Kinematics and Dynamics

⎡

⎤ ⎡ ⎤ gx −g g = ⎣ g y ⎦ = C bi ⎣ 0 ⎦ gz 0 where g is the acceleration due to gravity. The pendulum-axis frame rotates twice with respect to the body frame: once around the yb axis by α1 and once around the z p axis by α2 . Therefore, the transition matrix from the body frame to the pendulum-axis frame is ⎡

Cpb

⎤ ⎤⎡ cos α2 sin α2 0 cos α1 0 − sin α1 ⎦ = ⎣ − sin α2 cos α2 0 ⎦⎣ 0 1 0 0 0 1 sin α1 0 cos α1

Thus, in the body frame, r p is expressed as ⎡

⎤ − cos α1 cos α2 ⎦ r p = CTpb [−L p , 0, 0]T = L p ⎣ − sin α2 sin α1 cos α2 The angular velocity ωp of the pendulum can be expressed as ωp = α˙ 1 yb + α˙ 2 z p The coordinates in the body frame are ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 cos α1 0 sin α1 0 sin α1 α˙ 2 ωp = α˙ 1 ⎣ 1 ⎦ + α˙ 2 ⎣ 0 1 0 ⎦⎣ 0 ⎦ = ⎣ α˙ 1 ⎦ cos α1 α˙ 2 − sin α1 0 cos α1 0 1 Taking the derivative of r p , it can be verified that r˙ p = ω× p rp As the point of suspension Oc is on the body axis x b , r t in the body frame is expressed as ⎡

⎤ rtx rt = ⎣ 0 ⎦ 0 In summary, the dynamics equations for the three-dimensional pendulum in a matrix form are given below:

4.3 Attitude Dynamics

179

m R aR + m p ap − (m p + m R )g = F E

(4.21)

× I R ωR + ω× R I R ωR + m p (r t + r p ) (a p − g) = T E

(4.22)

× mp r × p ap − m p r p g + T d = 0

(4.23)

where aR is the projection of the acceleration R¨ b of the spacecraft’s rigid body in the body frame and ap is the projection of the acceleration R¨ p of the pendulum in the body frame. (2)

Two-dimensional equivalent mechanical model

Three-dimensional dynamic models of the liquid-filled spacecraft can be very complicated and unconducive to data analysis and controller design. This section introduces two-dimensional dynamics equations. The motion of the x Oz plane and the x O y plane is symmetrical; therefore, only one plane needs to be considered. Liquid sloshing has little influence on the motion of the y Oz plane; therefore, it can be ignored in data analysis and modal design. Vector equations, namely, Eqs. (4.21)–(4.23), are projected to the body frame, and it is assumed that the spacecraft only moves in the x Oz plane, as shown in Fig. 4.7. The coordinates of a vector are ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ Rbx rtx −L p cos α1 0 0 ⎦, ωR = ⎣ θ˙y ⎦, ωp = ⎣ α˙ 1 ⎦ Rb = ⎣ 0 ⎦, r t =⎣ 0 ⎦, r p = ⎣ 0 Rbz

0

L p sin α1

0

Based on the above expressions, the coordinate equations are

Fig. 4.7 Two-dimensional pendulum model of a liquid-filled spacecraft on the xOz plane

0

180

4 Spacecraft Attitude Kinematics and Dynamics

# $ %& m R ( R¨ bx − gx ) + m p ( R¨ bx − gx ) − θ˙y2 rtx + L p (α˙ 1 + θ˙y )2 cos α1 + (α¨ 1 + θ¨y ) sin α1 = Fx

(4.24)

#

$ %& m R ( R¨ bz − gz ) + m p R¨ bz − gz − rtx θ¨y + L p −(α˙ 1 + θ˙y )2 sin α1 + (α¨ 1 + θ¨y ) cos α1 = Fz $

%

2 )θ¨ − m r (JRy + m p rtx ˙ 1 + θ˙y )2 sin α1 + (α¨ 1 + θ¨y ) cos α1 ) = Ty y p tx R¨ bz − gz + L p ((α

(4.25) (4.26)

L p (α¨ 1 + θ¨y ) + ( R¨ bx − gx ) sin α1 + ( R¨ bz − gz ) cos α1 − rtx (θ˙y2 sin α1 + θ¨y cos α1 ) + 2ξ L p f p α˙ 1 = 0

(4.27) In the above equations, Fx and Fz are the components of the thrust force FE along Ty is the component of the control moment T E along the xb and z b axes, respectively; " the yb axis; f p = Fx /[(m R + m p )L p ] denotes the sloshing frequency; ξ denotes the sloshing damping; and gx = −g cos θ y and gz = −g sin θ y are the components of gravitational acceleration along the xb and yb axes, respectively. According to the symmetry of planes, dynamics equations for the x O y plane can be obtained : # $ %& m R ( R¨ bx − gx ) + m p ( R¨ bx − gx ) − θ˙z2 rtx + L p (α˙ 2 + θ˙y )2 cos α2 + (α¨ 2 + θ¨z ) sin α2 = Fx

(4.28)

# $ %& m R ( R¨ bz − gz ) + m p R¨ by − g y − rtx θ¨z + L p −(α˙ 2 + θ˙z )2 sin α2 + (α¨ 2 + θ¨z ) cos α2 = Fy # $ %& 2 )θ¨ − m r ˙ 2 + θ˙y )2 sin α2 + (α¨ 2 + θ¨z ) cos α2 = Tz (JRz + m p rtx z p tx R¨ by − gz + L p (α

L p (α¨ 2 + θ¨z ) + ( R¨ bx − gx ) sin α2 + ( R¨ by − g y ) cos α2 rtx (θ˙z2 sin α2 + θ¨z cos α2 ) + 2ξ L p f p α˙ 2 = 0

(4.29) (4.30)

(4.31)

where gx = −g cos θz , g y = g sin θz are the components of gravitational acceleration along the xb and z b axes, respectively.

4.3.4 Attitude Dynamics of Multi-Body Spacecraft A spacecraft is usually equipped with multiple moving appendages, such as antennas and solar arrays, which have different patterns of movement. If an appendage has a large mass, the spacecraft tends to have variable parameters, uncertainties, and coupling effects. In addition, large spacecraft may be constructed through an onorbit assembly of multiple cabins. Combined bodies may include mechanical arms or multiple cabins, which greatly increase the complexity of the structure and motion of the spacecraft.

4.3 Attitude Dynamics

181

Fig. 4.8 Schematic diagram of a spacecraft with moving appendages

A

r1 ra

B

r0

O1

O

zI

xI

1.

OI

yI

Attitude dynamics of spacecraft with moving appendages [11, 12]

To establish a complete dynamics model for a spacecraft with antennas, ignoring the influence of flexibility, this type of spacecraft can be abstracted as a system composed of two rigid bodies connected by an attachment point, as shown in Fig. 4.8. The antenna has two rotational degrees of freedom: around its x-axis, and around its y-axis. The rotation of the antenna is generally described by the reference frame and the antenna-axis frame, where the antenna-axis frame moves with respect to the reference frame. Without loss of generality, the reference frame is set to coincide with the body frame, and the antenna-axis frame is set to coincide with the reference frame when the antenna is at the nominal position, as shown in Fig. 4.9. It is assumed that the antenna first rotates around its X-axis by φx and then rotates around its Y-axis by φ y . The transition matrix from the antenna-axis frame to the body frame is ⎤ 0 sin φ y cos φ y = ⎣ sin φx sin φ y cos φx − sin φx cos φ y ⎦ − cos φx sin φ y sin φx cos φx cos φ y ⎡

C 01

Given φ = [ φx φ y ]T , the relationship between the angular velocity ω1 and rotation angle φ of the antenna can be expressed as Fig. 4.9 Schematic diagram of the spacecraft body and antenna z1 x1

y1

O1

O

y

x

z

182

4 Spacecraft Attitude Kinematics and Dynamics

ω1 = S1 φ˙

(4.32)

where  S1 =

cos φ y 0 sin φ y 0 1 0

T

The antenna’s reference angle of rotation is denoted by φ d , and φ e = φ − φ d is the rotation error. Finding the second time derivative of φ e and using Eq. (4.32), we have φ¨ e = ST1 ω˙ 1 − (ST1 S2 + φ¨ d )

(4.33)

where ⎡

⎤ − sin φ y ⎢ ⎥ 0 ⎦ S2 = φ˙ x φ˙ y ⎣ cos φ y It is assumed that the spacecraft moves on a circular orbit with an altitude of H. The orbital radius is Rs = Re + H , where Re denotes the radius of the Earth. The orbital inclination is i, and λ is the right ascension of the ascending node. t0 is defined as the initial time when the spacecraft passes the ascending node, λt0 is the longitude of the nadir point at t0 , and the geographic latitude and longitude of the ground target are denoted by (λE , λN ). r (i) , r (e) , r (o) , r (b) represent the coordinates of the vector r in the inertial coordinate system, Earth coordinate system, orbit coordinate system, and body coordinate system, respectively. To calculate the antenna’s reference rotation angle, it is vital to first obtain the expression for the antenna’s axis of pointing in the body frame, as shown in Fig. 4.10. The coordinates of the ground target r (e) d in the inertial frame are (e) r (i) d = C ie r d

(4.34)

Fig. 4.10 Geometric relationship between the antenna and ground target rs

ra

rd

4.3 Attitude Dynamics

183

where ⎡

r (e) d

⎤ cos λN cos λE ⎢ ⎥ = Re ⎣ cos λN sin λE ⎦. sin λN

C ie is the direction cosine matrix from the Earth coordinate system to the inertial coordinate system and is expressed as ⎡

⎤ cos(ωie t + λ) − sin(ωie t + λ) 0 C ie = ⎣ sin(ωie t + λ) cos(ωie t + λ) 0 ⎦ 0 0 1 where ωie is the angular velocity of the Earth’s rotation, and λ = λ − λE . The position r (i) s of the spacecraft in the inertial coordinate system can be calculated in real time based on its orbital position: T (o) r (i) s = C oi r s

(4.35)

T where r (o) s = [0 0 − Rs ] and C oi is the direction cosine matrix from the inertial coordinate system to the orbital coordinate system. (i) (i) Let r (i) a = r d − r s . Then, the coordinates of the antenna’s axis of pointing in the body coordinate system are (i) r (b) a = C bi r a

(4.36)

where C bi is the direction cosine matrix from the inertial coordinate system to the body coordinate system. By unitizing r (b) a , we have 1 ua = ' ' r (b) ' (b) ' a 'r a '

(4.37)

It is known that the coordinates of the antenna’s axis of pointing in the antenna-axis

T frame are r (1) a = 0 0 1 . Then, their expression in the body frame is ua = C 01 r (1) a

(4.38)

By determining the solution to simultaneous Eqs. (4.37) and (4.38), the antenna’s reference angle of rotation can be obtained :

184

4 Spacecraft Attitude Kinematics and Dynamics



φdx φdy



 =

  arctan − uu a2 a3 arcsin(u a1 )

(4.39)

In the above equation, u a,i (i = 1, 2, 3) is the ith component of ua , and both φdx and φdy fall within the range of (−90◦ , +90◦ ). From Eqs. (4.34) and (4.35), we have (e)× (e) (i) T (o)× (o) (i) (i) r s , r˙ (i) r˙ (i) a = r˙ d − r˙ s d = C ie ωie r d , r˙ s = C oi ωo

From Eq. (4.36), we have (i) (b)× C bi r (i) r˙ (b) a = C bi r˙ a − ω a

where ω = ωbo + C bo ωo , ω˙ = ω˙ bo − ω× C bo ωo , and C bo is the direction cosine matrix from the inertial frame to the body frame. From Eq. (4.37), we have 1 u˙ a = ' ' (I3 − ua uTa )˙r (b) a ' (b) ' 'r a ' From Eq. (4.39), the angular velocity of the reference rotation angle φ˙ d is obtained as ⎡ u u˙ − u˙ u ⎤ a2 a3 a2 a3 ⎢ u 2a2 + u 2a3 ⎥ ⎢ ⎥ φ˙ d = ⎢ ⎥ 1 ⎣ ( u˙ a1 ⎦ 1 − u 2a1 The dynamics equation for a rigid-body spacecraft with a motion antenna can be expressed as M(φ) X˙ + N(X, φ)X = T c + T d

(4.40)

In the equation above, M(φ) is the generalized inertia matrix of the system, which is nonlinear in the antenna’s rotation angle φ; X = [ωT , ωT1 ]T represents the angular velocity of the body and antenna; N(X, φ) is the complex nonlinear term of the system; T c = [uT , uT1 ]T denotes the control moment of the system; and T d denotes the moment of external interference.

4.3 Attitude Dynamics

185 

M(φ) =

J − mρ × ρ ×T

J 01 − m 1 ρ × C 01 ρ ×T 1

T ×T J 10 − m 1 ρ × 1 C 01 ρ



m 21 × ×T ρ ρ

J 1)− m 1 1 * ⎤ m 1 (ρ − r a )× (2ω× C 01 × ∗ T × × ×T ⎥ ⎢ ω ( J − mρ ρ ) + C 01 ω1 J 1 C 01 × × ⎥ ⎢ +C 01 ω× 1 )ρ 1 + C 01 ω1 J 1 ⎥ ) * N(X, φ) = ⎢ ⎥ ⎢   × T × × 2 m m ρ C ω (ρ − r ) ⎣ a × × × ×⎦ 1 1 01 T × 1 ω1 J 1 + m ρ 1 2(C 01 ω) + ω1 ρ 1 × ∗ T T T × +(C 01 ω) J 1 C 01 + ω1 J 1 C 01 ⎡

  ρ = m1 m 0 ρ 0 + m 1 C 01 ρ 1 a + m 1 ra is the coordinate vector of the center of mass of the system in the body frame during the rotation of the appendage. When the appendage is at the nominal position, ρ = 0. ρ 0 is the coordinate vector of the center of mass of the system in the body frame during the rotation of the appendage. ρ 1 is the coordinate vector of the center of mass of the appendage in the appendage axis frame. m = m 0 + m 1 is the total mass of the system. m 0 and m 1 denote$ the mass of the body and appendage, respec-%  × T  × T T  × T T + r× C 01 + C 01 ρ × tively. J = J 0 +C 01 J 1 C T01 +m 1 r × a ra a C 01 ρ 1 1 C 01 r a denotes the moment of inertia of the entire satellite. J 0 and J 1 are the inertia matrices of the body and appendage, respectively, with respect to the origin of their coordinate systems. The mixed inertia matrix for the body and appendage is denoted by  × T ∗   . J 1 = 2 J 1 + J1x + J1y + J1z I; J1x , J1y , J 01 = J T10 = C 01 J 1 + m 1 r × a C 01 ρ 1 and J1z are the principal diagonal elements of the matrix J 1 . As expressed in Eq. (4.40), the model of the system is quite complex and has high nonlinearity, strong coupling, and variable parameters. As the mass and inertia of the antenna increase, the dynamic coupling between the body and the antenna also increases. It should be noted that C 01 is a matrix containing trigonometric functions of the antenna’s rotation angle φ, which is time-varying when the antenna is rotating. Given that C 01 is included in J, J 01 , ρ, the generalized inertia matrix of the system M(φ) is also time-varying and is represented as a nonlinear function of the antenna’s angle of rotation φ. 2.

Attitude dynamics of the combined body [13]

Figure 4.11 shows the structure of a combined body composed of a base spacecraft b, a payload spacecraft e, and an N-link space manipulator with flexible joints. The manipulator has an open-chain structure. Each link has a rotary joint that has one degree of freedom. The base spacecraft can perform translational and rotational motion in three-dimensional space. The payload is a passive rigid body fixed to the N-link manipulator. In total, the combined body has N + 6 degrees of freedom. Here, OI xI yI z I denotes the Earth-centered inertial frame. The subscript “I” is omitted without causing ambiguity. The following coordinate systems are established. (1)

Coordinate system Oc xc yc z c for the center of mass of the system: The origin Oc is located at the center of mass of the system c. The Oc xc , Oc yc , and Oc z c axes are parallel to and have the same orientation as the respective axes of the Earth-centered inertial frame.

186

4 Spacecraft Attitude Kinematics and Dynamics

Fig. 4.11 Model of a combined system of rigid bodies

(2)

(3)

(4)

Body coordinate system for the base spacecraft Ob xb yb z b : The origin Ob is located at the center of mass of the base spacecraft c0 . The Ob xb , Ob yb , and Ob z b axes are in the direction of the principal axes of the moment of inertia matrix of the base spacecraft. Body coordinate system for the payload spacecraft Oe xe ye z e : The origin Oe is located at the center of mass of the payload ce . The Oe xe , Oe ye , and Oe z e axes are in the direction of the principal axes of the moment of inertia matrix of the payload. Body coordinate system for the rigid link Ok xk yk z k (k = 1, . . . , N ): The origin Ok is located at the center of mass ck of the kth link. The Ok xk , Ok yk , and Ok z k axes are in the direction of the principal axes of the moment of inertia matrix of the link k.

Hereafter, a superscript on the right of the element symbol indicates “relative to the coordinate system,” and a superscript on the left of the element symbol indicates “in the coordinate system.” The default superscript on the left of the element symbol indicates “in the inertial frame.” To facilitate research, this section introduces the following assumptions. (1) (2)

(3)

The joints and links of the base spacecraft, payload spacecraft, and manipulator are all rigid bodies. The combined body is in an ideal zero-gravity condition, and environmental torques such as gravity-gradient moments and solar radiation pressure are ignored. Orbiting is not considered.

4.3 Attitude Dynamics

187

To present expressions clearly, the rigid bodies and joints of the combined body are all numbered. The base spacecraft b is marked as the 0th rigid body; the k(k = 1, . . . , N − 1)-link manipulator is the k(k = 1, . . . , N − 1)th rigid body; and the payload spacecraft e with the N-link manipulator is the Nth rigid body. The joint between rigid body k − 1 and rigid body k is called joint k. First, the position of any particle m on the rigid body k(k = 0, . . . , N ) of the combined body is derived. As shown in Fig. 4.11, the vector Rk,m from the origin of the inertial frame O to the particle m on rigid body k is Rk,m = Rk + r k,m (k = 0, . . . , N )

(4.41)

where Rk is the vector from the origin of the inertial frame O to the center of mass of rigid body k and r k,m is the vector from the center of mass of rigid body k to the particle m. Rk can be further expressed as Rk = r c + ρ k (k = 0, . . . , N )

(4.42)

where r c is the vector from the origin of the inertial frame O to the center of mass of the combined body c and ρ k is the vector from the center of mass of the combined body c to the center of mass ck of rigid body k. ρ k is determined by the configuration of the system, which implies that the position of the center of mass ck of any rigid body k of the combined body with respect to the center of mass of the system c is independently determined by the configuration of the system. According to the above definition of the vector, it is easy to find that adjacent rigid bodies have the following relationship: ρ k − ρ k−1 = r k−1 − l k (k = 1, . . . , N ) where r k is the vector from the center of mass ck of rigid body k to joint k and l k is the vector from the center of mass ck of rigid body k to joint k − 1. Hence, ρ k = ρ k−1 + r k−1 − l k = ρ0 +

k 

(r i−1 − l i ) (k = 1, . . . , N )

(4.43)

i=1

According to the definition of center of mass, we have N  k=0

mk ρk = 0

(4.44)

188

4 Spacecraft Attitude Kinematics and Dynamics

where m k represents the mass of the k th rigid body. Inserting Eq. (4.43) into Eq. (4.44), we have Mρ 0 +

N  k=0

mk

k 

(r i−1 − l i ) = 0

(4.45)

i=1

N m k is the total mass of the combined body. where M = k=0 ρ 0 can be derived from Eq. (4.45) as follows: ρ0 = −

N k  mk  k=0

=−

N 

M

(r i−1 − l i ) = −

i=1

N 

(r i−1 − l i )

i=1

N  mk k=i

M

(r i−1 − l i )(1 − μi )

i=1

In the above equation, μi denotes the distribution of mass, which is defined as ⎧ 0 i =0 ⎪ ⎪ i−1 ⎨  mj i = 1, . . . , N μi  M ⎪ ⎪ ⎩ j=0 1 i = N +1 ρ k can then be expressed as a mass-weighted resultant: ρk =

k  i=1

(r i−1 − l i )μi −

N 

(r i−1 − l i )(1 − μi ) (k = 0, . . . , N )

(4.46)

i=k+1

Let ci be the center of mass of the rigid body i. Then, the augmented vector is defined as follows. The original rigid body i can be changed to the augmented body i* using a particle with an added mass of Mμi on joint i and a particle with an added mass of M(1 − μi+1 ) on joint i + 1, as shown in Fig. 4.12. The center of mass of the augmented body i ∗ is ci∗ . In general, ci and ci∗ do not coincide. ci∗ , r i∗ and l i∗ are the vectors of ci∗ pointing to ci , joint i, and joint i + 1, respectively. As shown in Fig. 4.11, the model follows the following geometric relationships: ci∗ = −ci r i∗ = r i − ci l i∗ = l i − ci The augmented vector vik (i, k = 0, . . . , N ) is then defined as

4.3 Attitude Dynamics

189

Fig. 4.12 Schematic diagram of the augmented vector

⎧ ∗ ⎨ ri i < k vik  ci∗ i = k ⎩ ∗ li i < k

(4.47)

Based on Eqs. (4.47), (4.46) can be written in the more compact general form: ρk =

N 

vik (k = 0, . . . , N )

(4.48)

i=0

In Eq. (4.48), ρ k is decomposed into N + 1 independent fixed vectors of the rigid body. The method for decomposing the vector is described as follows. ρ k is the sum of all vectors k before rigid body l i∗ , vectors c∗k of rigid body k, and vectors ci∗ after rigid body k. In addition, the above equation shows that the position of the center of mass of the kth rigid body is related to the positions of all rigid bodies. This is different from the ground-fixed manipulator, the center of mass of which is only related to the positions of the first k − 1 rigid bodies, reflecting the location coupling effects of the space manipulator. Inserting Eq. (4.48) into Eqs. (4.41) and (4.42), we obtain Rk,m = r c + ρ k + r k,m = r c +

N 

vik + r k,m

i=0

= rc +

N 

vik,m

(4.49)

i=0

where vik,m = vik + δik r k,m (i, k = 0, . . . , N )

(4.50)

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4 Spacecraft Attitude Kinematics and Dynamics

δik is the Kronecker delta. For the payload e fixed to the rigid body N, it is noted that r N ,e = r N . Then, the end effector’s position vector r e can be written using Eqs. (4.49) and (4.50) as follows: re = rc +

N 

vi N ,e

(4.51)

i=0

Thus, an analytical closed-form expression for the position of the combined body is obtained. vik is a fixed vector of rigid body i. Following the time derivation of Eq. (4.48), we have ρ˙ k =

N 

ωi × vik (k = 0, . . . , N )

i=0

˙ k,m of any point m on rigid body k and the velocity of Similarly, the velocity R the center of mass of the payload r˙ e can be obtained from Eqs. (4.49) and (4.51), respectively, as follows: ˙ k,m = r˙ c + R

N 

ωi × vik,m

(4.52)

i=0

r˙ e = r˙ c +

N 

ωi × vi N ,e

(4.53)

i=0

The next step is to determine the angular velocity of the rigid bodies of the combined body. denotes the angular velocity of rigid body i relative to rigid body i − 1, and ωi−1 i the absolute angular velocity of rigid body k can be expressed as ωk = ω0 +

k 

ωi−1 (k = 1, . . . , N ) i

(4.54)

i=1

where ω0 is the absolute angular velocity of the base spacecraft. The angular velocity of the payload is ωe = ω0 +

N  i=1

ωi−1 i

(4.55)

4.3 Attitude Dynamics

191

Equations (4.52)–(4.55) yield the linear and angular velocity vectors of any point ˙ k,m , r˙ e , ωk , and ωe . They are used to construct a function for in the combined body: R the relative angular velocity of the rigid body ωi−1 i , the absolute angular velocity of the base spacecraft ω0 , and the velocity of the center of mass of the combined body r˙ c . The above function can then be used to derive the Jacobian matrix (kinematics differential equation). The three coordinates of the vector vik in the coordinate system i (i.e., i vik ) are all constant and follow the transformation relationship below: vik = T i i vik = T 0 0 vik 0

vik = 0 T i i vik

(4.56)

In the above equations, T i ∈ R3×3 is the transformation matrix describing the rotation of the body frame of the ith rigid body with respect to the inertial frame, which can be expressed as T i (e, n, q1 , . . . , qi ) = T 0 (e, n)0 T i (q1 , . . . , qi ) 0

T i (q1 , . . . , qi ) = 0 A1 (q1 ) . . . i−1 Ai (qi )

where i−1 Ai (qi ) is the transformation matrix that transforms the column vectors in the i coordinate system into the i −1 coordinate system. It is a function of the relative joint angles qi in the two coordinate systems. The transformation matrix T 0 can be obtained by calculating the Euler parameters e and n: T 0 (e, n) = (n 2 − eT e)I + 2eeT + 2ne×

(4.57)

e(a, θ ) = a sin

θ 2

(4.58)

n(a, θ ) = cos

θ 2

(4.59)

where a is the unit vector of the momentary rotation axis, θ is the momentary rotation angle, the superscript “×” denotes an antisymmetric matrix, and I represents a 3 × 3 identity matrix. As the system moves, T 0 needs to be updated:  1 × e + nI 0 ω0 2 1 n˙ = − eT 0 ω0 2 e˙ =

192

4 Spacecraft Attitude Kinematics and Dynamics

With the transformation of T 0 , the position of the center of mass of the payload can be obtained using Eq. (4.51): re = rc + T 0

N 

0

T i i vi N ,e

(4.60)

i=0

The angular velocity of any rigid body can be obtained from Eq. (4.54): ωk = ω0 + ω0k = ω0 + T 0

k 

0

(4.61)

T i i ui q˙ i

(4.62)

i=1

= ω0 + T 0 0 F k q˙ (k = 1, . . . , N )

(4.63)

In the equations above, i ui is the unit vector in the i coordinate system that is parallel to the axis of rotation passing through joint i, and 0 F k is a 3 × N matrix defined as 0

F k  [0 T 1 1 u1 , 0 T 2 2 u2 , . . . , 0 T k k uk , 0] (k = 1, . . . , N )

where 0 is a 3 × (N − k) zero matrix and q = [q1 , q2 , . . . , qk , . . . , q N ]T is the vector of the joint angle. Based on Eqs. (4.56)–(4.63), the velocity and angular velocity of the particle m on rigid body k of the combined body can be obtained using Eqs. (4.52) to (4.55): ˙ k,m = r c + T 0 (0 J 11k,m 0 ω0 + 0 J 12k,m q) ˙ R

(4.64)

˙ ωk = T 0 (0 ω0 + 0 J 22k,m q)

(4.65)

where 0

J 11k,m  −

N 

0

T i i vik,m

×

i=0 0

J 12k,m  −

N 

0

T i i vik,m

× 0

Fi

i=0 0

J 22k,m  0 F k

It can be seen from Eqs. (4.64) and (4.65) that the velocity of the particle m on rigid body k consists of two parts: the velocity produced by the motion of the joint,

4.3 Attitude Dynamics

193

and the velocity produced by the motion of the base spacecraft. 0 J 11k,m is a 3 × 3 antisymmetric matrix having elements corresponding to the expression for the vector from the center of mass of the combined body to the point m in the base-spacecraft coordinate system. Both 0 J 12k,m and 0 J 22k,m are 3 × N matrices that describe the influence of the joint motion on the velocity m. Matrices 0 J 11k,m , 0 J 12k,m , and 0 J 22k,m only depend on the configuration of the system q. Simultaneous Eqs. (4.64) and (4.65) can be expressed as  x˙ k,m =

˙ k,m R ωk



⎤ r˙ c = J ∗k,m ⎣ 0 ω0 ⎦ q˙ ⎡0

(4.66)

where J ∗k,m (e, n, q) = diag(T 0 , T 0 )0 J ∗k,m (q)   0 I J 11k,m 0 J 12k,m 0 ∗ J k,m (q) = 0 0I J 22k,m

(4.67)

In the above equation, 0 J ∗k,m is a function of the augmented vector, which is a function of the mass, geometric parameters, and configuration q of all rigid bodies of the combined body. The Jacobian matrix J ∗k,m relates the controllable configuration q to the motion of the particle m on rigid body k. It should be noted that the Jacobian matrix is a 6 × (N + 6) non-square matrix, which indicates the inherent redundancy of the system. Putting k = N and m = e, the equation for the motion of the payload spacecraft can be obtained: ˙ r˙ e = r˙ c + T 0 (0 J 11 0 ω0 + 0 J 12 q)

(4.68)

˙ ωe = T 0 ( 0 ω0 + 0 J 22 q)

(4.69)

where 0

J 11  −

N 

0

T i i vi N ,e

×

i=0 0

J 12  −

N 

0

i=0 0

J 22  0 F N

T i i vi N ,e

× 0

Fi

194

4 Spacecraft Attitude Kinematics and Dynamics

After solving simultaneous Eqs. (4.68) and (4.69), the kinematics model can be obtained: ⎡0 ⎤   r˙ c r˙ e = J ∗ ⎣ 0 ω0 ⎦ x˙ = ωe q˙ where J ∗ (e, n, q) = diag(T 0 , T 0 )0 J ∗ (q)   0 I J 11 0 J 12 0 ∗ J (q) = 0 0I J 22 J 11 is a 3 × 3 antisymmetric matrix showing the expression for the vector from the center of mass of the combined body to the center of mass of the payload in the base-spacecraft coordinate system, 0 J 12 is a 3 × N matrix that includes the vector of the joint to the center of mass of the payload, and 0 J 22 denotes the Jacobian matrix for the payload when the base spacecraft is assumed to be fixed. All elements in 0 J ∗ (q) are dependent only on the configuration q. It should be noted that the rank of the matrix J ∗ is 6 because its first six columns contain six independent coordinates. In this regard, if a joint of the manipulator does not rotate, its end effector can also reach any position in space through the movement of the base spacecraft. The linear momentum of the combined body with respect to the origin O of the inertial space is 0

p = M r˙ c =

N 

˙k mk R

(4.70)

k=0

The equation for the translational motion of the center of mass of the system can be obtained through the time derivation of Eq. (4.70): p˙ = M r¨ c = f ext

(4.71)

In Eq. (4.71), f ext is the total external force acting on the system: f ext =

N  

f k,m

k=0 m

where f k,m denotes the external force acting on the particle m on the kth rigid body. The angular momentum of the combined body relative to the origin of the inertial frame O is

4.3 Attitude Dynamics

195

h=

N 

˙ k) (H k · ωk + m k Rk × R

(4.72)

k=0

where H k represents the inertia tensor of rigid body k with respect to its center of mass. Substituting Eqs. (4.42) and (4.44) into Eq. (4.72), we have h = r c × p + hc

(4.73)

where hc is the angular momentum of the center of mass of the system relative to the combined body: hc =

N 

(H k · ωk + m k ρ k × ρ˙ k )

(4.74)

k=0

Substituting Eqs. (4.48) and (4.42) into Eq. (4.74), we obtain hc =

N 

H k · ωk +

k=0

=

N  N N  

m k vik × ω j × v jk

j=0 i=0 k=0

N  N N  

Di jk · ω j

(4.75)

j=0 i=0 k=0

where

 Di jk = H i δi j δ jk + m k (v jk · vik )I − v jk sik (i, j, k = 0, . . . , N ) In the above equation, the dyad Di jk is a function for the inertia distribution of the system and the augmented vector vik . δi j and δ jk are the Kronecker deltas. The dyad Di j is defined as follows: ⎧ −M((l ∗j · r i∗ )I − l ∗j r i∗ ) i< j ⎪ ⎪ ⎨ N  Di j  H i + m k ((vik · vik )1 − vik vik ) i = j ⎪ k=0 ⎪ ⎩ −M((r ∗j · l i∗ )I − r ∗j l i∗ ) i> j

(4.76)

where I denotes the unit dyad. Equation (4.76) can be further simplified as hc =

N N   j=0 i=0

Di j · ω j

(4.77)

196

4 Spacecraft Attitude Kinematics and Dynamics

where Di j is a function only of the augmented vector. When i = j, Dii is the inertia tensor of the ith augmented body, i.e., the moment of inertia of the ith rigid body with respect to the center of mass of its augmented body. By taking the time derivative of Eq. (4.73), we obtain h˙ = ntot = r c × f ext +

N  N 

N  N 

Di j · ω˙ j +

j=0 i=0

ω j × D ji · ω j +

j=0 i=0

N  N 

' '2 dˆ i j 'ω˙ j '2

j=0 i=0

(4.78) In the equation above, next denotes the external moments acting on the combined body, ntot is the total moment that includes the moments generated by external forces, and dˆ i j is a function only of the augmented vector. These variables are expressed as follows: next =

N  

nk,m

k=0 m

ntot = next +

N  

Rk,m × f k,m

k=0 m

⎧ ∗ ∗ ⎪ ⎨ −M(l j × r i ) i < j di j  0 i= j ⎪ ⎩ −M(r ∗ × l ∗ ) i > j i j

+

The linear momentum equation can be easily obtained using Eqs. (4.70) and (4.71). Equation (4.77) can be written as   hc = T 0 0 D 0 ω0 + 0 Dq q˙ where 0

Dj 

N 

0

Di j ( j = 0, . . . , N )

0

Dj

0

Dj 0Fj

i=0 0

D

N  i=0

0

Dq 

N  i=0

4.3 Attitude Dynamics

197

0

Dqq 

N 

0

F Tj 0 D j 0 F j

i=0

In the above equations, 0 D is the expression of the moment of inertia of the combined body relative to the center of mass of the system in the base-spacecraft 0 Di j is in the matrix form body frame, which is a 3 × 3 positive-definite matrix.   defined by Eq. (4.76) and is a function of qi , . . . q j (i < j). 0 Dii is a function of (q1 , . . . qi ). In addition, we have 0

DiTj = 0 D ji (i, j = 0, 1, ..., N )

With Eqs. (4.63) and (4.42), the derivative of the angular momentum relative to the center of mass of the system is 0 ˙ h˙ c = T 0 (0 D0 ω˙ 0 +0 Dq q¨ + C + 1 (q, ω0 , q))

(4.79)

where C + 1 is a 3 × 1 matrix with a nonlinear term with respect to the velocity. According to the assumptions, the potential energy of the system is ignored. In this sense, the total energy is equal to the total kinetic energy: 1 1 T = M r˙ c · r˙ c + (ωk · H k · ωk + m k ρ˙ k · ρ˙ k ) 2 2 k=0 N

1 1  M r˙ c · r˙ c + ωi · D i j · ω j 2 2 j=0 i=0 N

=

N

The first term in the above equation denotes the translational kinetic energy of the system, and the second term represents the rotational kinetic energy, i.e., the kinetic energy of the system with respect to its center of mass. The kinetic energy can be expressed as ⎡0 ⎤ r˙ 1 0 T 0 T T  + ⎣0 c ⎦ T = r˙ c ω0 q˙ H (q) ω0 2 q˙ where ⎡

⎤ MI 0 0 H+ (q) = ⎣ 0 0 D(q) 0 Dq (q) ⎦ 0 0 Dq (q)T 0 Dqq (q)

(4.80)

198

4 Spacecraft Attitude Kinematics and Dynamics

In Eq. (4.80), I is the identity matrix. The inertial matrix H + (q) is a (N + 6) × (N + 6) positive-definite matrix that depends on the mass and inertia of the system. It is a function of the configuration q and is not related to the attitude of the base spacecraft. Generalized forces can be obtained according to the principle of virtual work. The virtual work acting on the system is δW = τ T δq +



0

f k,m 0 nk,m

T

0

δx k,m

k,m

= Q rT 0 δr c + Q Tθ aδθ + Q qT δq

(4.81)

Specifically, τ is an N × 1 moment vector, a is the instantaneous axis of rotation, δθ is the angle of rotation, Q r is the generalized force of translation of the combined body, Q θ is the generalized force of rotation of the combined body about the center of mass, Q q represents the generalized forces under configuration changes, 0 f k,m represents the forces acting on the particle m of the kth rigid body in the body coordinate system of the base spacecraft, 0 nk,m represents the moments acting on the particle m of the kth rigid body in the body coordinate system of the base spacecraft, and 0 δx k,m denotes the virtual displacement of the particle m of the kth rigid body and is expressed as ⎡0

0

δx k,m

⎤ δr c = 0 J ∗k,m ⎣ aδθ ⎦ δq

(4.82)

J ∗k,m is the Jacobian matrix defined by Eq. (4.67). Based on Eqs. (4.81) and (4.82), the generalized force in the inertial coordinate system can be obtained: 0

⎡

⎤ ⎡ ⎤   Qr N  0  f 0 ∗ Q  ⎣ Qθ ⎦ = ⎣ 0 ⎦ + J k,m (e, n, q)T k,m nk,m k=0 m Qq τ The generalized force Q obtained here is a coupled function of the attitude (e, n) and the configuration q of the base spacecraft. It can be decoupled in some special cases, such as zero external force in a free-floating environment. Lagrange’s equations cannot be adopted to solve dynamics problems here because the three-dimensional rigid body rotates. A quasi-Lagrangian formulation may be used, but the derivation of the kinetic energy T and the angular velocity 0 ω0 can be quite complicated. Therefore, the system dynamics equations are obtained from the moment equation, i.e., Eq. (4.79):

4.3 Attitude Dynamics

199

⎡0

⎤ r¨ c ˙ = Q H ∗ (q)⎣ 0 ω˙ 0 ⎦ + C ∗ (q,0 ω0 , q) q¨

(4.83)

In the above dynamics equations, the inertia matrix H ∗ (q) is given by Eq. (4.80), and the nonlinear term is expressed as follows: ⎡

0

⎤

˙ = ⎣ C ∗1 (q,0 ω0 , q) C + (q,0 ω0 , q) ˙ ⎦ ∗ 0 ˙ C 2 (q, ω0 , q) Equation (4.83) is composed of N + 6 equations, which describe the relationship among external forces, external moments, internal joint contact forces, and the motion of a free-flying combined rigid-body system. The first term on the left-hand side of the equation represents the product of the inertia matrix and the acceleration vector. The second term contains the centripetal force and Coriolis force. In particular, the first three rows of the term represent translational motion and are expressed by Eq. (4.71) in the body coordinate system of the base spacecraft. When all external forces are expressed in the same coordinate system or are all zero, the translation and rotation can be decoupled. The next three equations are the Euler equations for the multi-body system. The last N equations describe the motion of the space manipulator. If a base spacecraft is fixed, the equation for the space manipulator is the same as that of the ground-fixed manipulator.

References 1. Tu S (1998) Satellite attitude dynamics and control. China Astronautic Publishing House, Beijing 2. Lv Z, Lei Y (2013) Satellite attitude measurement and determination. National Defense Industry Press, Beijing 3. Zhang R (1998) Attitude dynamics and control of satellite orbit. Beihang University Press, Beijing 4. Zhou J (2001) Principles of spacecraft control. Northwestern Polytechnical University Press, Xi’an 5. Wang Z (2017) Attitude control and sloshing suppression for a class of liquid-filled spacecraft. Degree Thesis of China Academy of Space Technology, Beijing 6. Ibrahim RA (2005) Liquid sloshing dynamics: theory and applications. Cambridge University Press, New York, USA 7. Abramson HN (1966) The dynamic behavior of liquids in moving containers. Washington D.C., USA: NASA 8. Chobotov VA (1991) Spacecraft attitude dynamics and control. Malabar Krieger Publishing Company, Florida 9. Wang Z, Zhang H, Jinchang Hu (2017) Passive control for a three-dimensional liquid-filled spacecraft. Aerosp Control Appl 43(3):15–20 10. Zhang H, Wang Z (2016) Attitude control and sloshing suppression for liquid-filled spacecraft in the presence of sinusoidal disturbance. J Sound Vib 38(3):64–75

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4 Spacecraft Attitude Kinematics and Dynamics

11. Li F, Lei Y (2014) Terminal sliding mode compound control method for multi-body Spacecraft attitude maneuver. Aerosp Control Appl 40(1):14–18 12. Li F (2014) Research on attitude control methods of spacecraft with motion antennas. Degree Thesis of China Academy of Space Technology, Beijing 13. Deng Y (2017) Research on modeling and coordinated control of combined body. Degree Thesis of China Academy of Space Technology, Beijing

Chapter 5

Spacecraft Attitude Determination

5.1 Introduction The main task in the attitude determination of a spacecraft is to accurately estimate the spacecraft’s attitude and angular velocity by gathering data with attitude sensors. The estimated attitude as the feedback information is provided to an attitude control system to promote the attitude control of the spacecraft, and can also be used to provide the reference for payloads. An attitude determination system is an essential part of a spacecraft’s attitude control system, and it plays a decisive role in determining the accuracy of the attitude control system and on-orbit payload stability.

5.2 Modeling of Attitude Sensor Errors Attitude determination using a combination of high-precision star sensors and gyroscopes (or gyros) is a basic approach to determine the attitude of high-performance spacecraft [1]. Accurate models of attitude sensors play an important role in improving the system design for achieving high-precision attitude determination and enhancing the reliability of mathematical simulations. Mathematical models that can accurately describe the characteristics of gyros and star sensors are prerequisites for the design of high-precision satellite attitude determination systems, and they must be investigated using computer simulations of spacecraft attitude determination and attitude control systems.

© Beijing Institute of Technology Press 2022 Y. Xie et al., Spacecraft Dynamics and Control, Space Science and Technologies, https://doi.org/10.1007/978-981-33-6448-6_5

201

202

5 Spacecraft Attitude Determination

5.2.1 Modeling of Random Errors of Gyroscopes Ideally, the output of a gyro should be proportional to the component of the rotational speed of the satellite body along the gyro input axis in the inertial frame. However, gyros may exhibit output errors in practice. Current models for gyro errors mainly include models of static errors, dynamic errors, and random errors. In particular, random errors form an important part of measurement errors of gyro outputs, and they are one of the main factors that affect the high-precision attitude determination of satellites. Given the limited methods for decomposing and quantifying different types of gyro errors, the random errors are largely simulated as the Gaussian white noise in past applications of gyros, which hinders the accurate description of the characteristics of the gyro outputs. In this section, fiber-optic gyroscopes (FOGs) are considered as an example to present the characteristics of gyro errors, methods for acquiring model parameters, and model verification [2]. 1.

Characteristics of gyro errors FOGs mainly have three types of random errors: angle random walk (ARW), zero-bias instability, and rate random walk (RRW). (1)

Angle random walk The ARW of an FOG mainly arises from the relative intensity noise of light sources, the electrical noise and shot noise, and other high-frequency noise, which appears as white noise in the angular rate of the gyro outputs. The influence of ARW can be limited by calculating the average of output data. The power spectral density (PSD) of ARW is PARW ( f ) = N 2

(2)

(3)

(5.1)

where f denotes the frequency of a sinusoidal signal and N is an ARW coefficient, in units of (°)/(hHz1/2 ); the units can be converted as (1°)/(hHz1/2 ) = (1/60)°/h1/2 . Zero-bias instability Zero-bias instability can be used to describe low-frequency drift in a gyro’s angular-rate measurement data. This type of error usually occurs because of the Faraday effect, temperature fluctuations, and other low-frequency environmental noise. In an FOG that has been operating for a long time (e.g., 1 h to dozens of hours), zero-bias instability is one of the main sources of angular errors. Rate random walk RRW is a part of the errors in the integrated white noise superimposed on the observed gyro angular rate. The variance of this type of error increases with the increase of the sampling period. Noise sources for RRW are still to be discovered. The PSD of RRW is

5.2 Modeling of Attitude Sensor Errors

203

 PRRW ( f ) =

K 2π

2

1 f2

(5.2)

where K is an RRW coefficient, in units of (°)/(h2 Hz1/2 ); the units can be converted as (1°)/(h2 Hz1/2 ) = (1/60)°/h3/2 . The value of K is not related to the sampling period, and its physical meaning will be explained later. All the above-mentioned types of errors can influence the precision of gyro measurements. The predominant error type is subject to the length of operation of the system. For example, in a satellite attitude determination system using gyros and star sensors, the periodic calibration of gyro drifts by star sensors requires a highly precise angular rate in the gyro output over a short period of time. In this instance, ARW is the main error type impacting system performance. On the other hand, in an inertial navigation system, the gyros usually operate independently for a long time. Therefore, in this case, zero-bias instability and RRW, which reflect long-term drift, are the main error types impacting the performance of the inertial navigation system. In addition, the gyro output contains other types of errors, such as quantization noise and rate ramp. 2.

Types of gyroscope models ARW and RRW are the main components of the random errors of a gyroscope. T  As shown in Fig. 5.1, ω = ωx ω y ωz denotes the angular rate of the gyro’s  × input axis. Δ and Ψ denote the scale-factor error and installation error matrix, respectively, which are expressed as ⎡ Δ=⎣

x

⎤

⎤ 0 −ϕz ϕ y δ y ⎦, [Ψ × ] = ⎣ ϕz 0 −ϕx ⎦ δz −ϕ y ϕx 0

Fig. 5.1 Model of a fiber-optic gyroscope

⎡

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5 Spacecraft Attitude Determination

Here, ωn represents the natural frequency of a second-order system, and ωout denotes the gyro output. The model shown in Fig. 5.1 illustrates the influence of the scale-factor error, installation error, random error, and frequency bandwidth on the gyro outputs, and it presents a comprehensive picture of the characteristics of gyro errors. Without considering the gyro’s dynamic characteristics, its output is denoted by ωm . The relationship between ωm and ω is mathematically expressed as

  ω m = I ++ Ψ × ω + b + d + wg

(5.3)

b˙ = 0

(5.4)

d˙ = wd

(5.5)

In the above equations, I is the identity matrix. In this chapter, “I” always denotes the identity matrix of the corresponding dimension, unless otherwise T  stated. b = bx b y bz is the constant bias of the gyro. d and wg denote the RRW and ARW, respectively, and the sum of the two is the random error of the gyro. wg and wd can be regarded as the Gaussian white noise with zero mean. To facilitate analysis, a component of the random error is expressed as a function of time t, i.e., sm (t) = d(t) + wg (t)

(5.6)

˙ = wd (t) d(t)

(5.7)

where sm (t) denotes the random error of the gyro and d(t), wg (t), and wd (t) represent the components of d, wg , and wd , respectively. Given that both wg (t) and wd (t) are the Gaussian white noise with zero mean, their statistical characteristics can be expressed as

E wg (t)wg (τ ) = σg2 δ(t − τ ), E{wd (t)wd (τ )} = σd2 δ(t − τ )

(5.8)

where τ denotes time; σg and σd are positive constants representing the standard deviation of noise wg (t) and wd (t), respectively; and δ(·) is the Dirac delta function. The following section will analyze the relationship of σg and σd with the ARW and RRW coefficients. Based on the variance expressed in Eq. (5.8), the PSDs of the white noise wg (t) and wd (t) are Pwg ( f ) = σg2 , Pwd ( f ) = σd2 Let Pwg ( f ) = PARW ( f ). Based on Eq. (5.1), we have

(5.9)

5.2 Modeling of Attitude Sensor Errors

205

σg = N

(5.10)

As can be seen from Eq. (5.7), the error signal d(t) is obtained by integrating the white noise wd (t) in Eq. (5.8), and the transfer function for the integration is H (s) =

1 s

(5.11)

The signal d(t) is obtained from the noise wd (t) through the linear system H (s), and its PSD is calculated using Eq. (5.12):  2 Pd ( f ) =  H (s)|s= j2π f  Pwd =

σd2 (2π f )2

(5.12)

Let Pd ( f ) = PRRW ( f ). Based on Eq. (5.2), we have σd = K

(5.13)

It is known from Eqs. (5.10) and (5.13) that the gyro’s ARW coefficient N and RRW coefficient K represent the standard deviation of the input white noise wg (t) and wd (t) in the model of random errors, respectively. The PSDs of random errors wg (t) and wd (t) in Eqs. (5.9) and (5.12) match the PSDs of ARW and RRW in Eqs. (5.1) and (5.2), respectively. Based on the above analysis, this section presents a method for simulating the gyro’s random errors. Signals processed by a computer should be discrete signals. The sampling period of the gyro is denoted by T s . By discretizing Eqs. (5.6) and (5.7), we obtain the following discretized expressions for the random errors: Smk = Dk + Wgk

(5.14)

Dk = Dk−1 + Wdk

(5.15)

where

Wgk

1 = Ts

t k +Ts

wg (t)dt

(5.16)

tk

From Eq. (5.8), we can readily obtain 

2 E Wgk



⎧ t +T ⎫ t k +Ts ⎨ 1 k s ⎬ σ2 1 g =E wg (t)dt · wg (τ )dτ = ⎩ Ts ⎭ Ts Ts tk

tk

(5.17)

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5 Spacecraft Attitude Determination

or   N2 2 = E Wgk Ts

(5.18)

Wdk is expressed as Wdk = k ·

t k +Ts

1 Ts

wd (t)dt

(5.19)

tk

where t k +Ts

k =

(t)dt

(5.20)

tk

Based on Eq. (5.7), we have (t) = e0·t = 1. Hence, k = Ts . The variance of Wdk is then obtained:

2

E Wdk

⎧ t +T ⎫ t k +Ts ⎨ k s ⎬ =E wg (t)dt · wg (τ )dτ = σd2 Ts ⎩ ⎭ tk

(5.21)

tk

or 2 E Wdk = K 2 Ts

3.

(5.22)

As mentioned previously, N and K are in units of (°)/h1/2 and (°)/h3/2 , respectively. However, the standard deviations of noise Wgk and Wdk obtained using Eqs. (5.18) and (5.22), respectively, are in units of (°)/h. It is noticeable that the variance of Wgk , which is used to describe ARW, decreases with the increase of the sampling period, while the variance of Wdk for generating RRW increases with the increase of the sampling period. Allan variance method The Allan variance method is used to analyze the frequency-domain characteristics in the time domain. The coefficients of gyro errors can be identified by processing a sample space composed of gyro output data. The Allan variance is calculated through the following steps. (1) (2)

Data collection. L samples of the FOG’s angular rate are logged with a fixed sample period Ts to obtain a sample space with length L. Creation of arrays. Every m (m = 1, 2,…, M, M < L/2) pieces of data in the sample space are divided into a group, and J independent arrays are obtained. J can be calculated as follows:

5.2 Modeling of Attitude Sensor Errors

207

Fig. 5.2 Curve showing the random errors of gyro measurements

J = [L/m],

(3)

(5.23)

where the operator [x] rounds x toward zero. Data averaging. By determining the average of each group of raw data (i.e., the group mean ωk (m)) as shown in Fig. 5.2, a set of random variables with group means as elements is obtained: ωk (m) =

m 1  ω(k−1)m+1 , k = 1, 2, . . . , J m i=1

(5.24)

ω , ω , . . . , ωm , ωm+1 , ωm+2, . . . , ω2m , . . . , ω L−m+1 , ω L−m+2 , . . . , ω L  1 2        k=1 k=J k=2 ↓ ↓ ↓ ω1 (m) ω J (m) ω2 (m) (4)

Calculation of the Allan variance. The duration of each array τm = mTs is defined as the correlation time, and the Allan variance can be calculated as follows:  1 (ωk+1 (m) − ωk (m))2 2(K − 1) k=1 K−1

σ 2 (τm ) =

(5.25)

For each specific correlation time τm , the Allan variance is obtained by multiplying the square of the difference in group means of adjacent groups by 1/2. For different values of m, let τm = mTs and

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5 Spacecraft Attitude Determination

τm = Ts , 2Ts , . . . , M Ts . Then, the σ (τm )−τm curve can be drawn in a log–log plot, which is called the Allan variance curve. The influence of ARW on the Allan variance σ (τm ) decreases with the increase in the correlation time τm , which reflects the number of elements used for finding the group mean ωk (m), while the influence of RRW increases as τm increases. Considering only the ARW and RRW of the gyro, the Allan variance can be expressed in a simplified form as follows: 2 2 + σRRW σ 2 (τm ) = σARW

(5.26)

where 2 = σARW

N2 2 K 2 τm , σRRW = τm 3

(5.27)

It is noticeable that the Allan standard deviations σARW and σRRW for the −1 2 1 2 ARW and RRW are proportional to τm / and τm/ , respectively. The coefficients of each gyro error can be easily estimated using the Allan variance. The total variance is expressed as σ 2 (τm ) = A−1 τm−1 + A1 τm

(5.28)

The Allan variance coefficients A-1 and A1 can be obtained through leastsquares fitting: θ = (Φ T Φ)−1 Φ T y

(5.29)

where ⎡

τ1−1   ⎢ τ −1 A−1 ⎢ 2 , = ⎢ . θ= A1 ⎣ ..

τ1 τ2 .. .

−1 τM τM

⎤

⎡

⎢ ⎥ ⎢ ⎥ ⎥, y = ⎢ ⎣ ⎦

σ 2 (τ1 ) σ 2 (τ2 ) .. .

⎤ ⎥ ⎥ ⎥ ⎦

σ 2 (τ M )

Assuming that the units of σ (τm ) and τm are (°)/h and s, respectively, the ARW and RRW coefficients can be calculated using the Allan variance coefficients as follows: √ !

 A−1 ◦ 1/2  ( )h , K = 60 3A1 (◦ )h3/2 (5.30) N= 60 It should be noted that if the sample space only has a small number of elements, the reliability of the obtained Allan variance is low, resulting in low credibility of estimation of the error coefficients. Therefore, the

5.2 Modeling of Attitude Sensor Errors

209

Fig. 5.3 Allan variance curve of the measured data

gyro measurement data should be sufficient to ensure the accuracy of the estimation. The sample length is usually several hours. 4.

Simulation analysis Figure 5.2 shows the data measured using an FOG. Following the instructions given in the previous section, the Allan variance has been calculated, and a log– log plot of the Allan standard deviation versus correlation time has been drawn, as shown in Fig. 5.3. The ARW and RRW coefficients of the FOG obtained through least-squares fitting are ◦

◦

N = 0.0085 /h1/2 , K = 0.3979 /h3/2 Based on Eqs. (5.4) and (5.5), the Gaussian white noise terms with zero mean and variances of σg2 and σd2 are taken as wg and wd , respectively. A curve of the gyro’s random error was simulated, as shown in Fig. 5.4. Figure 5.5 shows the Allan variance curve obtained using the Allan variance method with the simulated data of gyro errors. According to the Allan variance curve in Fig. 5.5, the corresponding ARW and RRW coefficients obtained through least-squares fitting are N = 0.0084 (°)/h1/2 and K = 0.3799 (°)/h3/2 . The calculated N and K are close to the ARW and RRW coefficients obtained from the measured data.

210

Fig. 5.4 Simulated curve of the gyro’s random error

Fig. 5.5 Allan variance curve with simulated data

5 Spacecraft Attitude Determination

5.2 Modeling of Attitude Sensor Errors

211

5.2.2 Modeling of Star-Sensor Measurement Errors Changes in the solar incidence angle can make the thermal environment alternate between hot and cold, resulting in the deformation of the star-sensor body and its mounting structures, which is the main cause of low-frequency errors of the star sensor. Because the solar incidence angle changes with the orbital period, the lowfrequency error of the star sensor can be regarded as a periodic signal, and its period approximately equals the orbital period of the satellite. Periodic low-frequency errors of the star sensor occur because of the alternating cold and hot thermal environment. Using the data observed using the star sensor for filtering and correction can lead to periodic changes in the estimated gyro drifts. Moreover, the low-frequency errors of the star sensor can cause periodic errors in attitude determination, affecting the precision of attitude determination. In the star-sensor coordinate system, the star sensor’s axes are ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 s x = ⎣ 0 ⎦, s y = ⎣ 1 ⎦, s z = ⎣ 0 ⎦ 0

0

1

B

x0 , B y0 , and B z0 are their corresponding nominal mounting orientations in the satellite frame, and the star sensor’s nominal mounting matrix M0 in the satellite frame is M0 =

B

x0 B y0 B z0

T

(5.31)

The nominal mounting matrix M0 can be obtained using high-precision groundbased measurements. The structural deformation of the star sensor in orbit can cause a deviation of its actual mounting matrix M from the nominal mounting matrix M0 in the satellite frame. Assuming that the direction cosine matrix C BI represents the actual attitude of the satellite frame with respect to the inertial frame, the direction cosine matrix for the star sensor frame with respect to the inertial frame is CSI = MCBI

(5.32)

The actual orientations of the star sensor’s measurement axes in the inertial space are I x, I y, and I z. Hence, CSI =

I

x Iy Iz

T

(5.33)

The deviation matrix M can be obtained from the three angles of rotation around the nominal mounting frame. The angles of rotation of the vector around the three axes are denoted by θ1 , θ2 , and θ3 . Given that they are small quantities, the deviation matrix M can be written as

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5 Spacecraft Attitude Determination

⎡

⎤ 1 θ3 −θ2 M ≈ ⎣ −θ3 1 θ1 ⎦ θ2 −θ1 1

(5.34)

Based on the deviation matrix M, the star sensor’s actual mounting matrix can be expressed in the following two forms: M = Mo · M

(5.35a)

M = M · M0

(5.35b)

and

For clarity, the above two forms are referred to according to their structure as the form with the deviation matrix on the right of the dot product and that with the deviation matrix on the left. The matrix M has different physical meanings in the two expressions. In the first expression, the matrix can be equivalent to the change of the satellite frame relative to its initial state, while in the second expression, the matrix can be regarded as the deformation of the star-sensor coordinate system with respect to the satellite coordinate system. Given the measurement error of the star sensor ⎡ ⎤ 1 ψ −θ (5.36) C ≈ ⎣ −ψ 1 φ ⎦ θ −φ 1 where the errors φ, θ , and ψ include the star sensor’s low-frequency errors φLF , θLF , and ψLF as well as random errors φn , θn , and ψn , i.e., φ = φLF + φn , θ = θLF + θn , ψ = ψLF + ψn Low-frequency errors can be expressed in the Fourier series as follows [3]: φLF =

n 



 AφLF,i sin ωLF,i t + BφLF,i cos ωLF,i t

i=1

θLF = ψLF =

n  i=1 n  i=1



 AθLF,i sin ωLF,i t + BθLF,i cos ωLF,i t



 AψLF,i sin ωLF,i t + BψLF,i cos ωLF,i t

5.2 Modeling of Attitude Sensor Errors

213

The random variables φn , θn , and ψn are the equivalent angles of measurement noise with respect to the three axes. In other words, they are equivalent to the small angle of each rotation about the actual measurement axis, and they are unrelated zero-mean white noise with the variances σ 2 [φn ] = σ 2 [θn ] = σx2y , σz2 [ψn ] = σz2 where σx y and σz are constants greater than zero, which are occasionally referred to as the optical-axis pointing accuracy and horizontal-axis pointing accuracy, respectively. Given the characteristics of attitude measurement of the star sensor with a narrow optical field of view, σz2 > σx2y usually holds. Taking into account the deformation and measurement errors of the spacecraft in orbit, the output of the star sensor is [4] ˆ SI = C

I

xˆ I yˆ I zˆ

T

T  = C MC B I = C I x I y I z

(5.37)

where I xˆ , I yˆ , and I zˆ are the unit vectors of the measurement axes of the sensor in the inertial frame.

5.3 Three-Axis Attitude Determination Based on State Estimation The star sensor is a high-precision attitude sensor. The inertial attitude of a satellite can be directly determined using star-direction data measured by a star sensor fixed to the satellite body. Owing to the complexity of acquiring and processing the star sensor’s attitude data, attitude parameters are often not produced in time, and the accuracy of attitude determination may also be affected by the random errors in starsensor measurements. The gyro has characteristics complementary to those of the star sensor. High-precision gyroscope measurements have a fast dynamic response and small random errors. In addition, gyro drifts can be calibrated using star-sensor measurements. Therefore, star sensors and gyros are usually used together for highprecision satellite attitude determination. 1.

Gyro and star-sensor measurement models Without considering exponential drifts of the gyro, the angular-velocity measurement model of the gyro can be expressed as g(t) = ω(t) + b + ng

(5.38)

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5 Spacecraft Attitude Determination

T  where the gyro’s measurement noise ng = n gx n gy n gz is the stable Gaussian white noise and   E ng = 0, E ng ngT = σg2 I The gyro constant drift b can be considered as a random-walk process driven by the Gaussian white noise, i.e., b˙ = nb

(5.39)

In the above equation, the noise nb is not correlated with ng ; further,

E{nb } = 0, E nb nbT = σb2 I According to the gyro’s output g(t), the measured value of the angular velocity T  of the satellite ω(t) ˆ = ωˆ x ωˆ y ωˆ z is ω(t) ˆ = g(t) − bˆ

(5.40)

where bˆ is the estimated gyro constant drift. The coordinates of the unit vector along the optical axis of the star sensor in the inertial frame are T  ZSIm = Z SIm,x Z SIm,y Z SIm,z The measurement model of the star-sensor output can be expressed as ZSIm = ZSI + ZSI

2.

(5.41)

where ZSI denotes the actual value of the unit vector along the optical axis of the error, which satisfies star sensor in the inertial frame. ZSI is the measurement  T = σs2 I − ZSI ZTSI . E{ZSI } = 0 and E ZSI ZSI Equations of state The attitude kinematics equation of a satellite is q˙ =

1 (ω)q 2

(5.42)

 T where q = q1 q2 q3 q4 is the quaternion of the satellite’s attitude and ω = T  ωx ω y ωz is the angular velocity of the satellite. Furthermore,

5.3 Three-Axis Attitude Determination Based on State Estimation

215

⎤ 0 ωz −ω y ωx ⎢ −ωz 0 ωx ω y ⎥ ⎥ (ω) = ⎢ ⎣ ω y −ωx 0 ωz ⎦ −ωx −ω y −ωz 0 ⎡

It is assumed that the errors in attitude estimation are small quantities and that θx , θ y , and θz are the components of errors in the satellite’s attitude estimates. In other words, θx , θ y , and θz can be regarded as three angles obtained from the rotation of the actual attitude with respect to the estimated attitude about the three axes of the satellite. In addition, bx , b y , and bz denote the errors in the estimation of the gyro constant drifts. Based on the above, the variable for the state-estimation error of the system can be expressed as T  δX = θx θ y θz bx b y bz

(5.43)

Based on the above gyro model and the attitude kinematics equation, the stateestimation error equation can be written as [5] d (δX) = F(t)δX + w(t) dt

(5.44)

where ⎡

0 ωˆ z −ωˆ y ⎢ −ωˆ 0 ωˆ z x ⎢ ⎢ ⎢ ωˆ y −ωˆ x 0 F(t) = ⎢ ⎢ 0 0 0 ⎢ ⎣ 0 0 0 0 0 0

−1 0 0 0 0 0

0 −1 0 0 0 0

⎤ 0 0 ⎥ ⎥ ⎥ −1 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ 0

The statistical characteristics of the system noise w(t) are expressed as  

E{w(t)} = 0, E w(t)wT (t) = diag σg2 , σg2 , σg2 , σb2 , σb2 , σb2 By discretizing the continuous system with the sampling time t, we obtain δXk+1 = δXk + Wk where  ≈ I + F(t)t

(5.45)

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5 Spacecraft Attitude Determination

⎡

⎤ 1 ωˆ z t −ωˆ y t −t 0 0 ⎢ −ωˆ t 1 ωˆ x t 0 −t 0 ⎥ z ⎢ ⎥ ⎢ ⎥ 1 0 0 −t ⎥ ⎢ ωˆ y t −ωˆ x t =⎢ ⎥ ⎢ 0 0 0 1 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 1 0 ⎦ 0 0 0 0 0 1

3.

The statistical characteristics  k are expressed as E{Wk } =  of the system noise W

T 2 2 2 2 2 2 0 and E Wk Wk = diag σg , σg , σg , σb , σb , σb t. Observation equations ˆ BI of The relationship between the measured value CBI and estimated value C the direction cosine matrix for the attitude of the satellite with respect to the inertial frame is   ˆ BI CBI ≈ I − (θ)× C

(5.46)

where ⎤ θx θ = ⎣ θ y ⎦ θz ⎡

and (θ)× is the anti-skew-symmetric matrix of θ, i.e., ⎤ 0 −θz θ y (θ)× = ⎣ θz 0 −θx ⎦ −θ y θx 0 ⎡

The output of the star sensor can thus be expressed as ZSIm = CTBI ZSB + ZSI   ˆ TBI I + (θ)× ZSB + ZSI ≈C   T ˆ SI − C ˆ BI (ZSB )× 0 δX + ZSI =Z

(5.47)

where ZSB is the direction of the vector along the optical axis of the star sensor ˆ TBI ZSB . Based on the above-mentioned equain the satellite frame and Zˆ SI = C tions for the output of the star sensor, three types of observation equations are presented below. (1)

Observation Eq. 1 [6] By projecting ZSIm to a plane perpendicular to Zˆ SI , it can be obtained that   ˆ TBI (ZSB )× 0 δX + ζ1 V1T ZSIm = V1T C

(5.48a)

5.3 Three-Axis Attitude Determination Based on State Estimation

  ˆ TBI (ZSB )× 0 δX + ζ2 V2T ZSIm = V2T C

217

(5.48b)

where V1 , V2 , and Zˆ SI constitute a right-handed orthogonal coordinate system; ζ1 = V1T ZSI and ζ2 = V2T ZSI ; and " E

ζ1 ζ2

#

" = 0, E

 #  ζ1  ζ1 ζ2 = σs2 I ζ2

When measuring data with the optical axes of two star sensors (labeled as A and B), the observation equations for their corresponding filters can be obtained using Eqs. (5.48a) and (5.48b). Star sensor A: Z1A = H1A δX + ζA

(5.49a)

where $ × &    T

T T V1A V1A ZSIm,A CBI ZSB,A 0 ζ1A = Z1A = = , H , ζ × 1A A T

T T V2A ZSIm,A ζ2A V2A CBI ZSB,A 0

E{ζA } = 0, E ζA ζAT = σs2 I 

%

%

Star sensor B: Z1B = H1B δX + ζB

(5.49b)

where $ × &    T

T T V1B ZSIm,B CBI ZSB,B 0 V1B ζ1B , H = = Z1B = , ζ × 1B B T

T T V2B ZSIm,B ζ2B V2B CBI ZSB,B 0

E{ζB } = 0, E ζB ζBT = σs2 I 

%

%

We rewrite the above observation equations in the following form. Star sensor A: ⎧  T  V1A ⎪ ⎪ Z ZSIm,A = ⎪ 1A ⎪ T ⎪ V ⎪ 2A ⎪ $ ⎪

× & ⎪ ⎨ T ˆT V1A CBI ZSB,A 0

× {Z1A , H1A , R1A } : H1A = (5.50a) T ˆT ⎪ V2A CBI ZSB,A 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R1A = σs2 I ⎪ ⎪ ⎩

218

5 Spacecraft Attitude Determination

Star sensor B: ⎧  T  V1B ⎪ ⎪ Z ZSIm,B = ⎪ 1B ⎪ T ⎪ V ⎪ 2B ⎪ $ ⎪

× & ⎪ ⎨ T ˆT V1B CBI ZSB,B 0

× {Z1B , H1B , R1B } : H1B = T ˆT ⎪ V2B CBI ZSB,B 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R1B = σs2 I ⎪ ⎪ ⎩ (2)

(5.50b)

Observation Eq. 2 [5] According to the first observation equation given by Eqs. (5.50a) and (5.50b), HA and HB are time-varying observation matrices. Based on the characteristics of the Kalman filter for a discrete time-invariant system, it is necessary to ensure a constant-coefficient observation matrix in addition to a constant-coefficient state equation (see Eq. (5.45)) to design a constant-gain Kalman filter. For two observation equations {Z1 , H1 , R1 } and {Z2 , H2 , R2 }, if the variance matrices obtained by updating the states based on the observations of the two observation equations are still equal with the same estimated state and covariance matrix given at any time, the two observation equations are thus equivalent. Sufficient conditions for the equivalence of the observation equations are as follows: H1T R1T H1 = H2T R2T H2

(5.51)

H1T R1−1 Z1 = H2T R2−1 Z2

(5.52)

Through the equivalent transformation of the observation equations, the following can be obtained: Star sensor A: ⎧ T ˆ Z2A = Z× ⎪ SB,A CBI V1A ZSIm,A ⎪ ⎪   ⎨ {Z2A , H2A , R2A } : H2A = −I 0 (5.53a) ⎪

 ⎪ 1 ⎪ R−1 = ⎩ I − ZSB,A ZTSB,A 2A σs2 Star sensor B: ⎧ T ˆ Z2B = Z× ⎪ SB,B CBI V1B ZSIm,B ⎪ ⎪   ⎨ {Z2B , H2B , R2B } : H2B = −I 0 ⎪  ⎪ 1

−1 ⎪ ⎩ R2B = 2 I − ZSB,B ZTSB,B σs

(5.53b)

5.3 Three-Axis Attitude Determination Based on State Estimation

219

where H2A and H2B are constant-coefficient observation matrices. For star sensor A, we have $

 & 1 T 0 2 I − ZSB,A ZSB,A T T T T σ s H1A R1A H1A = H2A R2A H2A = 0 0   × T ˆ BI V1,A ZSIm,A 0 ZSB,A C T T T T H1A R1A Z1A = H2A R2A Z2A = 0 0

(3)

Similar results can be derived for star sensor B. Observation Eq. 3 [7] The vectors along the horizontal axes of the two star sensors are denoted by XSB,A , XSB,B , YSB,A , and YSB,B . The measurements of the two star sensors have the following relationship. Star sensor A: T T ˆ BI ZSIm,A = −XSB,A θ + υ1A C YSB,A

(5.54a)

T T ˆ BI ZSIm,A = YSB,A C XSB,A θ + υ2A

Star sensor B: T T ˆ BI ZSIm,B = −XSB,B YSB,B C θ + υ1,B

(5.54b)

T T ˆ BI ZSIm,B = YSB,B C XSB,B θ + υ2,B

where & &  $ T  T YSB,A YSB,B CBI ZSI,B CBI ZSI,A υ1B = = , υB = T T υ2B XSB,A CBI ZSI,A XSB,B CBI ZSI,B



T T 2 E{υA } = E{υB } = 0, E υ A υ A = E υ B υ B = σs I 

υ1A υA = υ2A



$

%

%

%

%

Based on Eqs. (5.54a) and (5.54b), measurement equations can be expressed in the following form. Star sensor A: $ & ⎧ T ˆ BI ⎪ YSB,A C ⎪ ⎪ Z3A = ⎪ T ⎪ ˆ BI ZSIm,A C XSB,A ⎪ ⎪ ⎪ ⎪   ⎨ T 0 −XSB,A {Z3A , H3A , R3A } : H3A = (5.55a) T ⎪ YSB,A 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ R3A = σs2 I

220

5 Spacecraft Attitude Determination

Star sensor B: $ & ⎧ T ˆ BI ⎪ YSB,B C ⎪ ⎪ Z3B = ⎪ T ⎪ ˆ BI ZSIm,B C XSB,B ⎪ ⎪ ⎪ ⎪   ⎨ T 0 −XSB,B {Z3B , H3B , R3B } : H3B = T ⎪ YSB,B 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ R3B = σs2 I

(5.55b)

It can be determined from the previous description that H3A and H3B are constant-coefficient observation matrices. 4.

Kalman filter equations The Kalman filter equations can be designed according to the equation of state of the system (see Eqs. (5.54a) and (5.54b)) and the above three observation equations. For the second (see Eqs. (5.53a) and (5.53b)) and third (see Eqs. (5.55a) and (5.55b)) observation equations, the steady-state Kalman filter gain matrix K can be used to reduce the amount of calculation. For the second observation equation, the following Kalman filter equations are given. Filter estimate:   ˆ k/k = Kk Z2A (5.56) δX Z2B Updated estimate: ˆ k−1 + δX ˆ k/k ˆk = X X

(5.57)

Calculation of the Kalman gain:  T −1 T −1  Kk = Pk/k H2A R2A H2B R2B

(5.58)

Calculation of the prediction-error covariance matrix:

−1 −1 = Pk−1/k−1 T + Q Pk/k−1

(5.59)

Calculation of the filter-error covariance matrix:     1 I − ZSB,B ZTSB,B 0 1 I − ZSB,A ZTSB,A 0 −1 −1 + 2 (5.60) = Pk/k−1 + 2 Pk/k 0 0 0 0 σs σs Equation (5.56) for state estimation can also be expressed as

5.3 Three-Axis Attitude Determination Based on State Estimation

221

 ×  ˆ BI ZSIm,A + Z× ˆ BI ZSIm,B 1 C ZSB,A C SB,B ˆ δXk/k = 2 Pk/k σs 0 where the subscript “k” denotes the number of samples of the discrete system.

5.4 Calibration of Relative Error of Attitude Sensor The mounting measurement error and errors caused by structural deformation are very likely to occur in the mounting of attitude sensors, given various influencing factors such as restrictions in ground-based precision measurement conditions and measurement precision, emission vibrations, structural stress release in orbit, and the space environment. The errors are often different even for different sensors on the same spacecraft, resulting in different measurement references for sensors. Inconsistent references can directly affect the accuracy of satellite attitude determination by sensors, and the change in references caused by the varying combination of attitude sensors often produces a certain degree of fluctuation in the estimated attitude, orientation, and stability of the satellite. In severe cases, it even affects the loading of high-resolution observation satellites. In high-precision Earth remote sensing and astronomical observation, the influence of the change in the sensor’s relative deviation from the reference on the attitude measurement is typically compensated for by installing the sensors and payload using the same reference based on an integrated stable structure and high-precision temperature control. This is done to unify the references for attitude measurement and payload pointing. On-orbit relative reference calibration for attitude sensors including star sensors and gyros is an effective approach to eliminating on-orbit relative deviations from references [8]. At present, most filter-based calibration methods generally expand the dimensions of the sensor’s parameters to estimate relative deviations from references with the state and the satellite’s attitude. This method has been widely used in the post-processing of ground-derived data; however, it requires considerable calculation if directly applied to real-time on-orbit calibration.

5.4.1 Calibration of Relative References for Star Sensors To achieve real-time on-orbit calibration of the star sensor’s relative deviation from the reference, this section introduces two types of expressions to describe the mounting error parameters in the establishment of a star-sensor measurement model. Based on relevant observation equations, an estimator of relative deviation parameters has been designed using the Kalman filter method. Furthermore, this section explores methods to evaluate the performance of on-orbit calibration based on the actual evaluation requirements of on-orbit applications. Equations for star-sensor mounting and measurement errors are given in Sect. 5.2.2. As the low-frequency

222

5 Spacecraft Attitude Determination

error is a slow variation on the order of magnitude of the orbital period, it is regarded as a constant value and processed with the system error in this section. For brevity, the superscript “B” on the left of the symbol for a vector in the satellite coordinate system will be omitted in the following [4]. 1.

Calibration methods for measurement references for star sensors

For brevity, we assume that the measurement performance of any combination of two star sensors (labeled by i = 1 and 2) is consistent without loss of generality. In other words, the equivalent angles of noise φi , θi , and ψi are unrelated zero-mean white noise terms with the following variances: σ 2 {φi } = σ 2 {θi } = σx2y , σz2 {ψi } = σz2 Assuming that the measurement of star sensor 1 is taken as the attitude reference, the filters for estimating the deviation of the two mounting error models in Eq. (5.35) are designed to calibrate the deviation of star sensor 2 from the reference, and the differences between the two filters are compared. (1)

Measurement equations with the deviation matrix on the right side

Based on Eq. (5.37) and the actual mounting matrix of the star sensor (see Eq. (5.35a)), we have T ˆ TSI,1 = C2 M0,2 M2 M1T M0,1 ˆ SI,2 C CT1 C

(5.61)

Assuming that the mounting deviation angles are small quantities, we have M2 M1T ≈ I − (θ2 − θ1 )×

(5.62)

T  where I is the identity matrix of the corresponding dimension and θi = θi1 θi2 θi3 is the vector of the mounting deviation angles of the star sensor i. With Eq. (5.62), Eq.(5.61) can be expressed as T T ˆ SI,2 C ˆ TSI,1 C1 = M0,2 M0,1 − M0,2 (θ2 − θ1 )× M0,1 CT2 C

(5.63)

T ˆ SI,2 C ˆ TSI,1 C1 M0,1 CT2 C (θ2 − θ1 )× = I − M0,2

(5.64)

Hence,

Without considering the small second-order quantities, the second term on the right-hand side of Eq. (5.64) is expanded as follows:

5.4 Calibration of Relative Error of Attitude Sensor T ˆ SI,2 C ˆ TSI,1 C1 M0,1 = x0,2 I xˆ 2T I xˆ 1 x0,1 + y0,2 I yˆ 2T I yˆ 1 y0,1 + z0,2 I zˆ 2T I zˆ 1 z0,1 M0,2 CT2 C  



+ y0,2 I yˆ 2T I yˆ 1 z0,1 − z0,2 I zˆ 2T I zˆ 1 y0,1 φ1 + z0,2 I yˆ 2T I yˆ 1 y0,1 − y0,2 I zˆ 2T I zˆ 1 z0,1 φ2  



I TI I TI I TI I TI + z0,2 zˆ 2 zˆ 1 x0,1 − x0,2 xˆ 2 xˆ 1 z0,1 θ1 + x0,2 zˆ 2 zˆ 1 z0,1 − z0,2 xˆ 2 xˆ 1 x0,1 θ2  



+ x0,2 I xˆ 2T I xˆ 1 y0,1 − y0,2 I yˆ 2T I yˆ 1 x0,1 ψ1 + y0,2 I xˆ 2T I xˆ 1 x0,1 − x0,2 I yˆ 2T I yˆ 1 y0,1 ψ2

223

(5.65)

T ˆ SI,2 C ˆ TSI,1 C1 M0,1 . According to Eq. (5.65), the matrix S Let S = M0,2 CT2 C includes the measurement errors φi , θi , and ψi of the star sensor as well as the linear correlated term S and uncorrelated term S0 , where S0 is

S0 = x0,2 · I xˆ 2T I xˆ 1 x0,1 + y0,2 I yˆ 2T I yˆ 1 y0,1 + z0,2 I zˆ 2T I zˆ 1 z0,1 Based on Eqs. (5.64) and (5.65), the deviation of the two star sensors is ⎡

⎤ ⎡ ⎤ S0 (2, 3) S (2, 3) θ21 =θ2 − θ1 = ⎣ −S0 (1, 3) ⎦ + ⎣ −S (1, 3) ⎦ S0 (1, 2) S (1, 2)

(5.66)

where S0 (·) and S (·) are elements corresponding to the matrices S0 and S , respectively. According to Eq. (5.66), the observation equation for the attitude calibration system between star sensors can be established as ⎤ S(2, 3) Z = ⎣ −S(1, 3) ⎦ = Hθ21 + Z S(1, 2) ⎡

(5.67)

In Eq. (5.67), the observation-equation coefficient matrix H is an identity matrix, T  and the observation error is Z = −S (2, 3) S (1, 3) −S (1, 2) . Considering the characteristics of the output noise of the sensor and the expression of Z, it can be determined that the mean value of the error Z follows E{Z} = 0

measurement and that the variance matrix R = E ZZT is related to the output of the star sensor. (2)

Measurement equations with the deviation matrix on the left side

Based on the measurement outputs I xˆ i , I yˆ i , and I zˆ i (i = 1, 2) and nominal mounting, the following observation vector can be established: ⎡

⎤ T x0,1 x0,2 − I xˆ 1T I xˆ 2 T Z = ⎣ y0,1 y0,2 − I yˆ 1T I yˆ 2 ⎦ T z0,1 z0,2 − I zˆ 1T I zˆ 2 Given θ i1 = θi1 + φi , θ i2 = θi2 + θi , and θ i3 = θi3 + ψi , we have

(5.68)

224

5 Spacecraft Attitude Determination

⎡

⎤ 1 θ i3 −θ i2 Mi = Ci Mi ≈ ⎣ −θ i3 1 θ i1 ⎦ θ i2 −θ i1 1 Based on Eq. (5.37) and the actual mounting matrix of the star sensor (see Eq. (5.35b)), we obtain T T ˆ SI,1 C ˆ TSI,2 = M1 M0,1 M0,2 C M2

(5.69)

Without considering the small second-order quantities, the following expressions can be derived from Eq. (5.69) based on the observations from Eq. (5.68): T T T T T x0,2 − I xˆ 1T I xˆ 2 = z0,1 x0,2 θ 12 − y0,1 x0,2 θ 13 + z0,2 x0,1 θ 22 − y0,2 x0,1 θ23 x0,1 T T T T T y0,1 y0,2 − I yˆ 1T I yˆ 2 = −z0,1 y0,2 θ 11 − x0,1 y0,2 θ 13 + z0,2 y0,1 θ 21 − x0,2 y0,1 θ 23 T T T T T z0,1 z0,2 − I zˆ 1T I zˆ 2 = z0,1 x0,2 θ 22 − z0,1 y0,2 θ 21 − z0,2 x0,1 θ 13 − z0,2 y0,1 θ 11

 T Taking the measurement of star sensor 1 as the attitude reference, θ1 = 0 0 0 and θ21 = θ2 hold for the measured deviation from the reference. Based on the relational equations derived from Eq. (5.69), the measurement equation for the calibration system can be obtained as Z = Hθ21 + Z

(5.70)

where ⎡

⎤ T T 0 z0,2 x0,1 −y0,2 x0,1 T T H = ⎣ z0,2 y0,1 0 −x0,2 y0,1 ⎦ T T −y0,2 z0,1 x0,2 z0,1 0 ⎡ T ⎤ T T T z0,1 x0,2 θ1 − y0,1 x0,2 ψ1 + z0,2 x0,1 θ2 − y0,2 x0,1 ψ2 T T T T Z = ⎣ −z0,1 y0,2 φ1 − x0,1 y0,2 ψ1 + z0,2 y0,1 φ2 − x0,2 y0,1 ψ2 ⎦ T T T T z0,1 x0,2 θ2 − z0,1 y0,2 φ2 − z0,2 x0,1 ψ1 − z0,2 y0,1 φ1 The mean and variance matrix of Z are

E{Z} = 0, R = E ZZT According to the expression of Z and the measurement

model of the star sensor, it has zero mean, and the variance matrix R = E ZZT is a constant-coefficient matrix related to the nominal mounting and measurement precision of the star sensor. In addition, the variance matrix can be obtained directly from the expression of its error with simple operations. A detailed explanation is not presented here, because of the limited space.

5.4 Calibration of Relative Error of Attitude Sensor

(3)

225

Calibration of deviation based on the Kalman filter method

Assuming that the on-orbit deformation error of the star sensor remains unchanged or changes slowly with time, the equation of state can be approximated as θ21 (k + 1) = θ21 (k) + nθ (k)

(5.71)

where k is the number of samples in the discrete system, the state noise nθ (k) is zero-mean white noise with the variance R{nθ (k)} = σn2 I , and σn2 is usually a small quantity. For a stochastic system composed of the equation of state given by Eq. (5.71) and the observation equation given by Eq. (5.67) or Eq. (5.70), the Kalman filter method is used to design a filter to estimate the relative deviation from the reference. Filters for the two systems corresponding to different observation equations can be written as follows: Filter equation: ( ) θˆ 21 (k) = θˆ 21 (k − 1) + K(k) Z(k) − Hθˆ 21 (k − 1) Gain matrix: K(k) = P(k)HT R−1 (k) Prediction-error variance matrix: P(k/k − 1) = P(k − 1) + σn2 I Filter-error variance matrix: P(k) = (I − K(k)H)P(k/k − 1) The initial value can be set toP(0) = P0 > 0, θˆ 21 (0) = 0 where Z(k), H, and R(k) denote the observation vector of the measurement equation, coefficient matrix, and variance matrix of noise Z in Eq. (5.57) or (5.70), respectively. In addition to different expressions of parameters, the two-filter calibration systems also have differences in the following aspects. (1)

The noise variance matrix in the observation equation with the deviation matrix on the left side is a constant-coefficient matrix related only to the nominal mounting. For a fully controllable and observable linear time-invariant stochastic system, the filter-error variance matrix tends to be a positive constant

226

(2)

5 Spacecraft Attitude Determination

matrix, and the gain matrix tends to be a constant-coefficient matrix if the filtering duration is sufficiently long. In this sense, a steady-state Kalman filter can be designed for the stochastic system composed of Eqs. (5.70) and (5.71) to simplify the on-orbit calculation, and the corresponding steady-state gain matrix K can be obtained by solving the relevant algebraic Riccati equations. For the noise variance matrix in the observation equation with the deviation matrix on the right side, it can be determined from Eq. (5.56) that when star sensor 1 is replaced with star sensor 2 as the reference, the corresponding relative deviation θ12 satisfies the following relationship: θ12 =θ1 − θ2 = −θ21

The above relationship does not apply to the observation equation with the deviation matrix on the left side. According to the coefficient matrix in the observation equation (i.e., Eq. (5.70)) and the identity matrix in the state equation (i.e., Eq. (5.71)), it can be easily determined that the calibration system is not fully random nor observable when the determinant det(H) of the matrix H is zero under some specific relative mounting of sensors. To avoid this, it is advised to re-select the combinations of measurement axes with better observability. Alternatively, the information from the axes of the two star sensors in other combinations can be added to the observation vector in Eq. (5.58), and the corresponding observation equation can be directly derived from Eq. (5.59). In practice, the optical axes of star sensors are generally not mounted in parallel. The above issue can also be avoided by adjusting the direction of the horizontal axis by rotating along the optical axis. 2.

Evaluation method for the performance of relative deviation calibration

As the actual mounting deformation of an on-orbit star sensor is unknown, groundbased evaluation of the calibration accuracy needs to combine the results of calibration and measurement. In general, the deviation of the angle between the two axes calculated based on the output of the two star sensors from the corresponding angles calculated based on the calibrated mounting (i.e., the residual) is used as the basis for the evaluation. As the optical-axis pointing accuracy of the star sensor is much higher than the accuracy of the horizontal axis (i.e., σz2 > σx2y ), it is preferable to take the residual between the optical axes as the basis for the evaluation. Taking the angle between the optical axes of two star sensors as an example, it is assumed that of star sensor 2 from star sensor 1 is T  the estimated deviation θˆ21 = θˆ2 = θˆ21 θˆ22 θˆ23 , and the mounting direction of the optical axis of star sensor 2 is calibrated to zˆ 2 based on the deviation. After calibration, the angle between the optical axes of the two star sensors is B

T  zˆ 2 = α Z 12 + α + n bz αˆ Z 12 = arccos z0,1

(5.72)

5.4 Calibration of Relative Error of Attitude Sensor

227



where α Z 12 = arccos I z1T I z2 is the actual angle between the two optical axes, α is the determined term of the residual that includes system errors, and n bz denotes the random term of the residual with zero mean. Based on the unit vectors I zˆ 1 and I zˆ 2 along the optical axes of the two star sensors, the angle between the optical axes I αˆ Z 12 is obtained as I



αˆ Z 12 = arccos I zˆ 1T I zˆ 2 = α Z 12 + n I z

(5.73)

where n I z denotes the random term. Next, we analyze the description of the mounting error of the star sensor given in Eq. (5.35b). Given θ˜2i = θˆ2i − θ2i (i = 1–3), we have T B T T x0,2 + θ˜21 z0,1 y0,2 zˆ 2 − I z1T I z2 ≈ −θ˜22 z0,1 z0,1

Assuming that θ˜2i is uncorrelated, based on Eq. (5.72), we obtain     T T x0,2 + E θ˜21 z0,1 y0,2 α = −E θ˜22 z0,1  



 2   2 T T σ 2 {n bz } = z0,1 x0,2 σ 2 θ˜22 + z0,1 y0,2 σ 2 θ˜21

(5.74)

According to Eqs. (5.72) and (5.74), the deviation of the angle between the two optical axes is αˆ Z 12 = I αˆ Z 12 − B αˆ Z 12 = n I z − α − n bz

(5.75)

The deviation αˆ Z 12 is the deviation of the angle between the calibrated mounting axis and the measured optical axis, and its expected value is

E αˆ Z 12 = E{n I z } − α Because the measurement error of the star sensor is a small quantity, by using Eq. (5.73), we obtain I TI zˆ 1 zˆ 2



 



= cos α Z 12 + n I z ≈ cos α Z 12 − n I z sin α Z 12

(5.76)

Without considering the small second-order quantities of the measurement error of the sensor, we obtain the following from Eq. (5.37) and the characteristics of measurement noise: * + θ1 I x1T I z2 − φ1 I y1T I z2

 2 I TI 2 =2σx2y sin2 α Z 12 σ zˆ 1 zˆ 2 = σ I TI I TI +θ2 x2 z1 − φ2 y2 z1

228

5 Spacecraft Attitude Determination

Using the variance I zˆ 1T I zˆ 2 and Eq. (5.76), the mean and variance of the random noise n I z can be obtained: E{n I z } = 0, σ 2 {n I z } = 2σx2y According to Eq. (5.75) and the statistical characteristics of n I z , the mean and variance of the calibrated residual αˆ Z 12 of the star sensor are

E αˆ Z 12 = −α

σ 2 αˆ Z 12 = 2σx2y + σ 2 {n bz } ≥ 2σx2y

(5.77)

As can be seen from the above, the mean and variance of αˆ Z 12 obtained from the statistical analysis can be used to measure the agreement between the calibrated value and actual on-bit value of the relative deviation parameters of star sensors. The closer the mean α to zero and the closer the variance to 2σx2y , the higher the calibration accuracy of the relative deviation parameters from the references. 3.

Simulation analysis and application

(1)

Mathematical simulation analysis

The nominal mounting for two star sensors on the satellite body can be expressed as ⎡

M0,1

⎤ 0 −0.83867057 −0.54463904 ⎦ = ⎣ −1 0 0 0 0.54463904 −0.83867057

⎡

M0,2

⎤ 0 −0.66913061 −0.74314483 = ⎣ −0.73135370 0.50682355 −0.45634598 ⎦ 0.68199836 0.54350172 −0.48937115

The optical-axis pointing accuracy 3σx y is 1 (3σ), and the horizontal-axis pointing accuracy 3σz is 20 (3σ). Taking star sensor 1 as the reference, the relative deviation of different forms for the axes of star sensor 2 is simulated, calibrated, and verified using the mounting deviation model in Eq. (5.35b). According to Eqs. (5.75) and (5.77), when the calibrated deviation matches the actual deviation, the mean of the calibrated mounting and the measured residual of ˆ Z 12 is zero with a maximum precision of 1.414 the angle between optical axes α (3σ) (Fig. 5.6). With star sensor 1 as the reference, the deviation of the axes of star sensor 2 from a constant reference is set to –30 , 30 , and 0 , respectively, by using the expression for mounting deviation in Eq. (5.35b). The output of the star sensor is obtained using Eq. (5.37), which is then calibrated. The results are shown in Fig. 5.6. After the filter is stable, the estimated values of the relative deviation parameters for the three axes are all less than 0.1 (3σ).

5.4 Calibration of Relative Error of Attitude Sensor

229

x/arcsec.

-29.5

-30

-30.5 200

300

400

500

600

700

800

300

400

500

600

700

800

300

400

500 t/s

600

700

800

y/arcsec.

30.5

30

29.5 200

z/arcsec.

0.5

0

-0.5 200

Fig. 5.6 Parameters estimated through filtering with deviation from a constant reference

The estimated values are used to calibrate the mounting of star sensor 2. The residual of the angle between the optical axes calculated using measurement data is shown in Fig. 5.7. According to a statistical analysis of the data, the mean and the mean square error (multiplied by 3) of the calibrated residual are –0.01 and 1.424 , respectively. This result is very close to the limiting accuracy of 1.414 (3σ), which validates the calibration of deviation from the references. (2)

Application of calibration based on on-orbit data

An on-orbit satellite has up to seven orbital periods, each of which is approximately 5,960 s. Two star sensors of the satellite output telemetry data, of which one is used as a reference for measurements. On the ground, the filter calibration method for the mounting deviation model (see Eq. (5.35b)) is used to calibrate the relative deviation of the other star sensor. The estimated relative deviation parameters for the three axes of star sensor 2 are shown in Fig. 5.8. The results show that compared with the nominal mounting of the star sensor, the maximum deformation of on-orbit star sensors from the reference configuration is close to 100 with periodic fluctuations of approximately 16 in the amplitude. Spectral analysis was performed on the first set of data for the estimated relative attitude deviation from the reference in Fig. 5.8. The results are shown in Fig. 5.9. It can be seen that in addition to the constant-value terms, the estimates contain errors in the orbital period.

230

5 Spacecraft Attitude Determination

Fig. 5.7 Residual of the angle between the optical axes before and after calibration 80 60 40

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 4

x 10 0 -100 -200

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 4

x 10 10 0 -10

0

0.5

1

1.5

2.5 2 Time(s)

3

3.5

4

Fig. 5.8 Calibration of deviation from the reference based on on-orbit data (unit: arcsec)

4.5 4

x 10

5.4 Calibration of Relative Error of Attitude Sensor

231

14 X: 0.0001678 Y: 12.35

12

AngErr(")

10

8

6 X: 0.0003357 Y: 3.619

4

2

0

0

0.001

0.002

0.003

0.004

0.005 f(Hz)

0.006

0.007

0.008

0.009

0.01

Fig. 5.9 Spectral analysis of the estimated deviation based on on-orbit data

5.4.2 Calibration of Gyro Errors For three gyros, the input axes of which have nominal mounting directions orthogonal to each other, the gyro coordinate system O-x G yG zG (hereinafter referred to as the G system) can be established. The coordinate axes of the G system are parallel to the nominal directions of the input axes, as shown in Fig. 5.10. The three input axes can be denoted by Gx , Gy , and Gz . The occurrence of a mounting deviation implies that the input axes do not coincide with the coordinate axes of the gyros. For the gyro Gx , it is assumed that the direction zG

O

yG

xG Fig. 5.10 Schematic of the mounting error of the gyro’s head relative to the gyro components

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5 Spacecraft Attitude Determination

of its input axis in the G system is G pGx , and two misalignment angles are used to describe the direction G pGx in the G system: ⎡

G

pGx

⎤ cos αx ≈ ⎣ sin αx cos βx ⎦ sin αx sin βx

(5.78)

where αx is the misalignment between the actual gyro pointing angle of Gx and the nominal gyro pointing angle. Let Gx y = sin αx cos βx and Gx z = sin αx sin βx . Given that αx is generally a small quantity, Eq. (5.78) can be rewritten as ⎡ G

1

⎤

pGx ≈ ⎣ Gx y ⎦ Gx z

where Gx y and Gx z are small quantities. Similarly, for the other two gyros G y and G z , ⎤ ⎡ ⎤ Gyx Gzx G ≈ ⎣ 1 ⎦, pGz ≈ ⎣ Gzy ⎦ 1 Gyz ⎡

G

pGy

where Gyx and Gyz are the actual components of the gyro Gy on the x G axis and zG axis of the G system, respectively; Gzx and Gzy are the actual components of the gyro Gz on the x G axis and yG axis of the G system, respectively. By denoting the transition matrix between the G system and the satellite-body coordinate system (i.e., the B system) as M GB , the sensitive axis of Gx in the B system is expressed as ⎡

1

⎤

pGx = M TGB G pGx ≈ M TGB ⎣ Gx y ⎦ Gx z Similarly, the gyros Gy and Gz in the B system can be expressed as ⎤ ⎤ ⎡ Gyx Gzx ≈ M TGB ⎣ 1 ⎦, pGz ≈ M TGB ⎣ Gzy ⎦ 1 Gyz ⎡

pGy

5.4 Calibration of Relative Error of Attitude Sensor

233

Considering only the mounting errors of the gyro, when the satellite’s angular velocity is ω in the satellite body frame, its projections on the input axes of each gyro are ωGx = pTGx ω, ωGy = pTGy ω, ωGz = pTGz ω The satellite’s angular velocity ω˜ calculated based on the gyro’s output is ⎡ ⎤ ⎤ [ 1 Gx y Gx z ]M GB ω ωGx ⎢ ⎥ ω˜ = M TGB ⎣ ωGy ⎦ = M TGB ⎣ [ Gyx 1 Gyz ]M GB ω ⎦ ωGz [ Gzx Gzy 1 ]M GB ω ⎡

Ignoring the small second-order quantities, the above equation can be simplified to ⎡

1

ω˜ = M TGB ⎣ Gyx Gzx

⎤ Gx y Gx z 1 Gyz ⎦ M GB ω Gzy 1

(5.79)

Based on the above analysis of angular velocity under the gyro’s mounting errors and considering the gyro’s scale error, constant drifts, and other measurement errors, the gyro output is ωˆ gx = (1 + Kx ) pTGx ω + bx + dx + wgx 

ωˆ gy = 1 + Ky pTGy ω + b y + d y + wgy ωˆ gz = (1 + Kz ) pTGz ω + bz + dz + wgz where Kx , Ky , and Kz are the coefficients for the gyro scale-factor error; bx , b y , and bz are the constant drifts in the gyro scale; dx , d y , and dz are the exponential drifts in the gyro scale; and wgx , wgy , and wgz are the random drifts in the gyro scale. The satellite’s angular velocity ωˆ calculated based on the gyro measurements is ⎡

⎤ ωˆ gx ωˆ = M TGB ⎣ ωˆ gy ⎦ ωˆ gz ⎡ ⎤ (1 + Kx )[ 1 Gx y Gx z ]M GB ω + bx + dx + wgx  ⎢

⎥ = M TGB ⎣ 1 + Ky [ Gyx 1 Gyz ]M GB ω + b y + d y + wgy ⎦ (1 + Kz )[ Gzx Gzy 1 ]M GB ω + bz + dz + wgz Ignoring the small second-order quantities, the above equation can be simplified to

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5 Spacecraft Attitude Determination

⎡

⎤ Gx y Gx z ωˆ = M TGB ⎣ Gyx 1 Gyz ⎦ M GB ω Gzx Gzy 1 ⎡ ⎤ 0 K x 0 + M TGB ⎣ 0  K y 0 ⎦ M GB ω 0 0 K z ⎡ ⎤ bx + dx + wgx + M TGB ⎣ b y + d y + wgy ⎦ bz + dz + wgz 1

Thus, the measurement error in the satellite’s angular velocity is δω = ωˆ − ω ⎡

⎤ ⎡ ⎤ ⎡ ⎤  K x Gx y Gx z bx dx + wgx = M TGB ⎣ Gyx  K y Gyz ⎦ M GB ω + M TGB ⎣ b y ⎦ + M TGB ⎣ d y + wgy ⎦ Gzx Gzy  K z bz dz + wgz

It is clear that the measurement error in the angular velocity is affected by various factors, including the gyro’s scale-factor error, mounting deviation, constant drifts, ARW, and RRW. Let -T , M GB = m x m y m z where m x , m y , and m z are the theoretical mounting directions of Gx , Gy , and Gz , respectively, in the satellite body frame. The measurement error in the satellite’s angular velocity can then be expressed as ⎡

⎤ K x

 δω = m x (M GB ω)T ⎣ Gx y ⎦ + m x bx + m x dx + wgx Gx z ⎡ ⎤ Gyx

 + m y (M GB ω)T ⎣  K y ⎦ + m y b y + m y d y + wgy Gyz ⎡ ⎤ Gzx

 + m z (M GB ω)T ⎣ Gzy ⎦ + m z bz + m z dz + wgz K z

(5.80)

As can be seen from Eq. (5.80), errors in gyro measurements include deterministic and random errors. The deterministic errors include

5.4 Calibration of Relative Error of Attitude Sensor

235

⎡

bGx

⎤ ⎡ ⎤ ⎡ ⎤ K x Gyx Gzx ⎥ ⎥ ⎥ T⎢ T⎢ = bx + (M GB ω) ⎣ Gx y ⎦, bGy = b y + (M GB ω) ⎣  K y ⎦, bGz = bz + (M GB ω) ⎣ Gzy ⎦ Gx z Gyz K z T⎢

With a constant ω, the equivalent gyro constant drifts given above, namely, bGx , bGy , and bGz , should also be constants. The random error of the gyro measurement can be expressed as





 r Gx = m x dx + wgx , r Gy = m y d y + wgy , r Gz = m z dz + wgz Random drifts are not correlated with the satellite’s angular velocity ω. In fact, they are components of the random error of each gyro in the B system. Given the above analysis, it can be concluded that if the satellite’s angular velocity remains unchanged, the equivalent constant drift of the gyro is a fixed value. This means that if the satellite’s angular velocity remains constant, the gyro’s constant drifts, mounting error, and scale-factor error cannot be distinguished based on the expression for the measurement error in angular velocity. The equivalent gyro drift changes with the satellite’s angular velocity. According to this law, the gyro scale factor, mounting deviation, and actual constant drift can be calibrated by obtaining sufficient equivalent gyro drifts at different angular velocities of the satellite. To estimate a total of 12 parameters for gyro errors, the expression of the equivalent gyro drift when the satellite rotates at a constant angular velocity ω can be rewritten as ⎡ ˆ ⎤ K x ˆ Gx y ⎥ ⎢   ⎥ bˆ Gx1 = (M GB ω)T 1 ⎢ ⎣ ˆ Gx z ⎦ bˆ x ⎡ˆ ⎤ Gyx ˆ Ky ⎥ ⎢   ⎥ bˆ Gy1 = (M GB ω)T 1 ⎢ ⎣ ˆ Gyz ⎦ bˆ y ⎡ˆ ⎤ Gzx ˆ Gzy ⎥ ⎢   ⎥ bˆ Gz1 = (M GB ω)T 1 ⎢ ⎣ ˆ Kz ⎦ bˆ z The above expression includes three equations, each of which contains four unknown parameters. Therefore, the constant angular velocity sequence { ω1 ω2 . . . ωn } in different amplitudes is given, and the estimated equivalent gyro drifts for each angular velocity are obtained. Then, a set of linear equations with these unknowns can be established, as shown below.

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5 Spacecraft Attitude Determination

⎡ ˆ ⎤ ⎤ bGx1 1 ⎡ ˆ Kx ⎤ 1⎥ bˆ Gx2 ⎥ ˆ Gx y ⎥ ⎢ ⎥ ⎥⎢  ⎥=⎢ ⎢ .. ⎥⎢ .. ⎥ ⎦ ⎣ ˆ ⎣ ⎦ Gx z . . ⎦ ˆ T bGx (M GB ωn ) 1 bˆ Gx,n ⎡ ⎡ ⎤ ⎤ bˆ Gy1 (M GB ω1 )T 1 ⎡  ˆ Gyx ⎤ ⎢ bˆ Gy2 ⎥ ⎢ (M ω )T 1 ⎥⎢ ˆ GB 2 ⎢ ⎥ ⎥ K y ⎥ ⎥=⎢ ⎢ .. .. ⎥⎢ . ⎥ ⎣ ˆ Gyz ⎦ ⎢ ⎣ .. ⎦ ⎣ . .⎦  bˆ Gy (M GB ωn )T 1 bˆ Gyn ⎡ ⎡ ⎤ ⎤ bˆ Gz1 (M GB ω1 )T 1 ⎡  ˆ Gzx ⎤ ⎢ (M ω )T 1 ⎥⎢ ˆ ⎢ bˆ Gz2 ⎥ GB 2 ⎢ ⎥ ⎥ Gzy ⎥ ⎥=⎢ ⎢ ⎢ . ⎥ .. .. ⎥⎢ ⎦ ⎣ ˆ . ⎣ ⎣ ⎦ Kz . . . ⎦ ˆ T bGz (M GB ωn ) 1 bˆ Gzn

⎡

(M GB ω1 )T ⎢ (M ω )T GB 2 ⎢ ⎢ .. ⎣ .

We define the following unknown parameters xi (i = 1, 2, 3): ⎤ ⎤ ⎤ ⎡ ˆ ⎡ˆ ⎡ˆ K x Gyx Gzx ⎥ ⎥ ⎢ ⎢ ⎢ ˆ ˆ Gx y ⎥ ˆ ⎥ ⎢ K y ⎥, x3 = ⎢ Gzy ⎥ x1 = ⎢ ⎣ ⎣ ˆ Gx z ⎦, x2 = ⎣  ˆ Gyz ⎦ ˆ Kz ⎦ bˆ Gx bˆ Gy bˆ Gz Further, we define the coefficient matrix A and vectors B i (i = 1, 2, 3) as ⎡

⎡ ˆ ⎡ˆ ⎡ˆ ⎤ ⎤ ⎤ ⎤ bGx1 bGy1 bGz1 1 ⎢ bˆ Gx2 ⎥ ⎢ bˆ Gy2 ⎥ ⎢ bˆ Gz2 ⎥ 1⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ .. ⎥, B 1 = ⎢ .. ⎥, B 2 = ⎢ .. ⎥, B 3 = ⎢ .. ⎥ ⎣ ⎣ ⎣ ⎦ ⎦ ⎦ . . . . ⎦ T (M GB ωn ) 1 bˆ Gx,n bˆ Gyn bˆ Gzn

(M GB ω1 )T ⎢ (M ω )T GB 2 ⎢ A=⎢ .. ⎣ .

Then, we solve the following equation: Axi = B i (i = 1, 2, 3)

(5.81)

Thus, the estimated values of the parameters for gyro errors xˆ i = A+ B i can be obtained. For the above equations, to obtain the gyro error parameters, it is necessary to ensure that the matrix A has full column rank, and at least four different angular velocities of the satellite should be provided. Equivalent gyro constant drifts can be obtained through filtering using a combination of gyros and star sensors. However, the parameters of the coefficient matrix A in Eq. (5.81) contains the actual angular velocity of the satellite ω. Considering the gyro errors, it would be difficult to obtain the actual angular velocity accurately. Therefore, it is suggested to first determine

5.4 Calibration of Relative Error of Attitude Sensor

237

the inertial orientation of the satellite (i.e., ω = 0), obtain accurate estimates of gyro constant drifts, and then estimate errors such as the scale-factor error.

5.5 Ground-Based Post-Event High-Precision Attitude Calibration Attitude determination systems based on star sensors and gyros have been widely used for satellites with high-precision pointing requirements. Actual star-sensor errors contain multiple error sources, including constant errors, short-period errors, and random errors. In particular, the short-period error, also called the low-frequency error, is mainly caused by factors such as the imaging field of view of the star sensor, exposure process, and thermal environment of space. Its period is mainly in the range of tens of seconds to several orbital periods. System errors can generally be calibrated in orbit. Random noise is generally suppressed by joint filtering with star sensors and gyros. Therefore, the short-period error is one of the key factors restricting the accuracy improvement of attitude determination. This section will introduce a ground-based post-posture attitude calibration method using a Kalman filter that is based on the estimation of equivalent constant drifts to identify low-frequency errors of the star sensor and improve the accuracy of satellite attitude determination [9]. 1.

Expression for gyro constant drift The attitude kinematics equation of the satellite can be expressed as q˙ =

1 Ω(ω)q 2

(5.82)

T  where q is the quaternion of the satellite’s attitude, ω = ωx ω y ωz denotes the angular velocity, and ⎤ 0 ωz −ω y ωx ⎢ −ωz 0 ωx ω y ⎥ ⎥ Ω(ω) = ⎢ ⎣ ω y −ωx 0 ωz ⎦ −ωx −ω y −ωz 0 ⎡

(5.83)

qg represents the estimated attitude quaternion based on the output angular velocity measured by the gyro, and δq denotes the deviation between the gyropredicted and actual attitudes. Thus, the relationship between the gyro-predicted attitude and the actual attitude of the satellite can be expressed as q = qg ⊗ δq or

238

5 Spacecraft Attitude Determination

δq = qg−1 ⊗ q where ⊗ denotes the multiplication of quaternions. The model of the gyro can be expressed as ωg = ω + b + n

(5.84)

where ωg is the output angular velocity measured by the gyro, b is the gyro constant drift, and n is the gyro measurement noise. The equation for attitude estimation through gyro measurements is q˙ g =

1 1 Ω(ωg )qg = Ω(ω + b + n)qg 2 2

(5.85)

Then, the following relationship holds: ) d ( −1 qg ⊗ q = q˙ g−1 ⊗ q + qg−1 ⊗ q˙ dt  1 = δq ⊗ ω − (ω + b + n) ⊗ δq 2  1 1 1 = δq ⊗ ω − ω ⊗ δq − b ⊗ δq − n ⊗ δq 2 2 2

δ˙q =

(5.86)



 ω , b, and n are represented in a similar form. 0 Let qd = δ˙q ⊗ δq−1 . According to the scalar (qd4 = 0), the expression for b can be obtained using Eq. (5.86): where ω =



2 ω + ωT δqv δqv − ω − n b = −2qdv + δq4

2.

(5.87)

where δqv and δq4 are the vector and scalar parts of δq , respectively. Hereafter, the quaternion parameter with the subscript “v represents the vector part of the quaternion. Equation (5.87) expresses the relationship of the gyro constant drift with the quaternion estimation error and its derivatives, the satellite’s angular velocity, and the gyro’s measurement noise. Because the measurement noise includes the measurement noise of the attitude sensor, the gyro constant drift is usually not calculated using Eq. (5.87) in practice. Rather, it is often estimated using filtering methods. Equivalent gyro constant drifts with low-frequency errors Because the actual attitude of a real satellite is unknown, the gyro constant drift given by Eq. (5.87) can only be calculated using the measurement output of the attitude sensor. After introducing star sensors, the measurement errors of the star sensors may also affect the calculation of gyro constant drifts.

5.5 Ground-Based Post-Event High-Precision Attitude Calibration

239

Because the low-frequency error of a star sensor is generally a small quantity, it can be expressed as qLEF =

1

1 1 ϕ θ ψ 2 LEF 2 LEF 2 LEF

1

T

(5.88)

where ϕLEF , θLEF , and ψLEF are the low-frequency error angles in the roll, pitch, and yaw directions, respectively. Given their periodic characteristics, they can be expressed in the Fourier series as follows: ϕLEF (t) =

N

[axk cos(kωo t) + bxk sin(kωo t)] k=1 N   a yk cos(kωo t) + b yk sin(kωo t) θLEF (t) = k=1 N   azk cos(kωo t) + bzk sin(kωo t) ψLEF (t) = k=1

(5.89)

where ωo denotes the satellite’s angular velocity, t is the time, N is the total order of the Fourier series modeled for the low-frequency error of the star sensor, and aik and bik (i = x, y, z) are the coefficients of the Fourier series to be determined for the low-frequency error. Taking into account the low-frequency measurement error, the measurement model for the star sensor is qs = q ⊗ qLEF

(5.90)

Then, the quaternion estimation error of the gyro measured by the star sensor can be expressed as δqs = qg−1 ⊗ qs = qg−1 ⊗ q ⊗ qLEF = δq ⊗ qLEF

(5.91)

The corresponding gyro constant drift can be expressed as bs = −2δq˙ s ⊗ δq˙ s−1 + δqs ⊗ ω ⊗ δ˙q˙ s−1 − ω − n

(5.92)

 d

δq ⊗ qLEF = δ˙q ⊗ qLEF + δq ⊗ q˙ LEF dt

(5.93)

where δq˙ s =

By inserting Eq. (5.86) into Eq. (5.92), we obtain bs = −2δ˙q ⊗ δq−1 −1 −1 + δq ⊗ (−2q˙ LEF ⊗ qLEF + qLEF ⊗ ω ⊗ qLEF ) ⊗ δq−1 − ω − n

(5.94)

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5 Spacecraft Attitude Determination

As can be seen from Eq. (5.94), the calculation of the equivalent gyro constant drift becomes quite complicated after introducing the low-frequency error of the star sensor. In addition, the short-period error has a coupling relationship with its derivative, the gyro estimation error, and the angular velocity of the satellite. Considering the small-angle assumption of short-period errors, for time signals on the order of magnitude of the orbital period, q˙ LEF should also have the features of small angles. Then, the following approximate expression holds [10]:  q˙ LEFv ≈ q˙ LEF ≈ q˙ LEF ⊗ 0 ⎤ ⎡ ωx − 2qLEF3 ω y + 2qLEF2 ωz ⎢ ω y + 2qLEF3 ωx − 2qLEF1 ωz ⎥ −1 ⎥ qLEF ⊗ ω ⊗ qLEF ≈⎢ ⎣ ωz − 2qLEF3 ωx + 2qLEF1 ω y ⎦ 0 −1 qLEF



Under the small-angle assumption, the influence of the low-frequency error of the star sensor on the estimation of the gyro constant drift can be determined by comparing Eqs. (5.87) and (5.94). The relationship among the gyro constant drift, the low-frequency error of the star sensor, and the satellite’s angular velocity is as follows: ⎡

⎤ −2qLEF3 ω y + 2qLEF2 ωz b˜ s = − 2q˙ LEFv + ⎣ +2qLEF3 ωx − 2qLEF1 ωz ⎦ −2qLEF3 ωx + 2qLEF1 ω y

3.

(5.95)

where b˜ s =bs − b. According to Eq. (5.95), when the low-frequency error of the star sensor is zero, bs = b holds. Therefore, b˜ s is the equivalent gyro constant drift caused by the low-frequency error of the star sensor, indicating that it has the same periodic properties as the low-frequency error of the star sensor. Calculation of equivalent gyro constant drifts and angular velocity of the satellite In order to determine the Fourier series coefficients for the low-frequency errors qLEF of the star sensor, we must first obtain b˜ s and the satellite’s angular velocity ω. Assuming that the gyro constant drift b does not contain information with the same frequency as qLEF , the estimated equivalent constant drift bs is obtained using joint filtering with the gyros and star sensors. Without causing confusion, the estimated value is still denoted by the variable bs , and other variables are handled in the same manner. Based on the analysis of the relationship among the angles between the optical axes of the star sensors, frequency information on the change in the optical axis of the star sensor can be obtained. In addition, the part information b˜ s of the estimated gyro constant bs , which have the same frequencies of the information on the change in the optical axis of the star sensor, can be extracted.

5.5 Ground-Based Post-Event High-Precision Attitude Calibration

241

Conventional filtering methods can be used to estimate the equivalent gyro constant drift. According to the satellite’s dynamics equation and gyro’s measurement equations, the following equations of state are established for the filtering process of the state-estimation system: 

δq˙ v δb˙





−ω× ˆ − 21 I 3×3 = 03×3 03×3



    1 δqv − 2 I 3×3 03×3 wg + δb 03×3 I 3×3 wb

(5.96)

where δqv is the scalar part of the quaternion of the attitude estimation error; δb is the residual of the estimate of the gyro constant drift; and wg and wb are both system noise with the variances Qwg and Qwb , respectively.   δ q˙ v By defining δX = , Eq. (5.96) can be written in a more compact form: δ b˙ ˙ = AδX + Bw(t) δX

(5.97)

where  A=

  1    − 2 I 3×3 03×3 wg −ω× ˆ − 21 I 3×3 ,B = , w(t) = 03×3 03×3 03×3 I 3×3 wb

By discretizing the continuous state equation with the filter period T , we obtain (k + 1, k) = eAT ≈ I + AT   T − 2 I 3×3 03×3 (k + 1, k) = BT = 03×3 T I 3×3 Taking two star sensors as an example, the measured value z can be established using the sensor’s measurement data: ⎤ ⎡ Y STB,A Cˆ B I Z SIm,A −X ST B,A ⎥ ⎢ T ⎢ YT Cˆ Z ⎥ ⎢X S B,A z = ⎢ ST B,A ˆ B I SIm,A ⎥ = 2⎢ ⎣ −X T ⎣ Y S B,B C B I Z SIm,B ⎦ S B,B Y STB,B X ST B,B Cˆ B I Z SIm,B ⎡

It should be noted that ⎡

−X ST B,A ⎢ YT S B,A H = 2⎢ ⎣ −X T S B,B Y STB,B

⎤ 0 0⎥ ⎥ 0⎦ 0

⎤ 0 0⎥ ⎥δ X + V 0⎦ 0

(5.98)

242

5 Spacecraft Attitude Determination T T where XSB,i , YSB,i , and i = A, B are the unit vectors of the horizontal axes of the star sensors in the satellite coordinate system; ZSIm,i is the measured value of the optical axis of the star sensor in the inertial system; and the direction ˆ BI represents the current estimated attitude of the star relative cosine matrix C to the inertial system. Therefore, according to the Kalman filter method, the update of the state estimate at the time k can be obtained as

ˆ k|k−1 + Kk zk ˆ k = δX δX

(5.99)

where Kk is the filter gain matrix. The orbital period of the satellite is denoted by T . According to Eq. (5.95), the estimated values of the equivalent gyro constant drift of N1 orbital  periods,  where N1 is an even number, are obtained as bs (t ∈ t0 , t0 + N1 T ), and time transfer is performed as follows: t = t − t0 −

N1 T 2

  where t ∈ − N21 T, N21 T . The components along the three axes of bs (t ∈   t0 , t0 + N1 T ) are extended to a function with a period of 2 T , which can be expressed in the Fourier series as follows: bs,i (t) = where ω N = equations:

2π . T

∞ 



 1 a0 + an cos nω N t + bn sin nω N t 2 n=1

Each coefficient can be calculated using the following

2 ak = N1 T 2 bk = N1 T

N1 T /2

 f (t) cos nω N t dt

−N1 T /2 N1 T /2

 f (t) sin nω N t dt

−N1 T /2

Hence, bs,i (t) =

      ∞  1 N1 N1 a0 + T + bk sin kω N t − t0 − T ak cos kω N t − t0 − 2 2 2 k=1

(5.100)

5.5 Ground-Based Post-Event High-Precision Attitude Calibration

243

Based on Eq. (5.100), the part of the Fourier series corresponding to the period of the low-frequency error of the star sensor can be obtained, which is the information on the gyro constant drift b˜ s . In the previous section, the estimated gyro equivalent constant drift bs and the corresponding equivalent angular velocity of the satellite ωs have been obtained using the conventional filter for attitude determination. The obtained angular velocity includes two parts: the actual angular velocity of the satellite body ω and the deviation of the gyro measurement caused by the low-frequency error ω. The estimated value of the constant drift of the gyro itself can be obtained using the equivalent constant drift caused by the low-frequency errors b˜ s and bs , which is obtained through filtering as follows: bˆ = bs − b˜ s

(5.101)

According to the kinematics equation, the estimated value of the actual angular velocity of the satellite is ωˆ = ωˆ s − bˆ 4.

(5.102)

Attitude calibration method based on equivalent gyro constant drifts With the satellite’s actual angular velocity ω and b˜ s , the coefficients for the qLEF model can be determined by inserting qLEF and the expression of its derivative into Eq. (5.95). The expression for the gyro equivalent constant drift (i.e., Eq. (5.95)) can be written in the component form as follows: b˜sx = −2q˙ LEF1 − 2qLEF3 ω y + 2qLEF2 ωz N = [axk kωo sin(kωo t) − bxk kωo cos(kωo t)] k=1 N   a yk ωz cos(kωo t) + b yk ωz sin(kωo t) + k=1 N   azk ω y cos(kωo t) + bzk ω y sin(kωo t) − k=1

b˜sy = −2q˙ LEF2 + 2qLEF3 ωx − 2qLEF1 ωz N   =− axk ωz cos(kωo t) + bxk ωz sin(kωo t) k=1 N   a yk kωo sin(kωo t) − b yk kωo cos(kωo t) + k=1 N   azk ωx cos(kωo t) + bzk ωx sin(kωo t) + k=1

b˜sz = −2q˙ LEF3 − 2qLEF3 ωx + 2qLEF1 ω y N   axk ω y cos(kωo t) + bxk ω y sin(kωo t) = k=1

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5 Spacecraft Attitude Determination

N   azk ωx cos(kωo t) + bzk ωx sin(kωo t) k=1 N   azk kωo sin(kωo t) − bzk kωo cos(kωo t) +

−

k=1



By defining a variable XLEF = ax1 bx1 a y1 b y1 az1 bz1 ax2 bx2 a y2 b y2 az3 . . . the above equation can be written in a more compact form as DXLEF = b˜ s

T

,

(5.103)

Let sk = sin(kωo t) and ck = cos(kωo t). The coefficient matrix D is ⎤ ⎡ ωz sk −ω y ck −ω y sk ωo s1 −ωo c1 ωz c1 ωz s1 −ω y c1 −ω y s1 . . . kωo sk −kωo ck ωz ck ⎥ ⎢ D = ⎣ −ωz c1 −ωz s1 ωo s1 −ωo c1 ωx c1 ωx s1 . . . −ωz ck −ωz sk kωo sk −kωo ck ωx ck ω x sk ⎦ ω y c1 ω y s1 −ωx c1 −ωx s1 ωo s1 −ωo c1 . . . ω y ck ω y sk −ωx ck −ωx sk kωo sk −kωo ck

After obtaining the measured values for m different times, it can be found using Eq. (5.103) that ⎤ ⎤# ⎡ b˜ s (1) D(1) ⎥ ⎢ ⎥ ⎢ = ⎣ ... ⎦ ⎣ ... ⎦ b˜ s (m) D(m) ⎡

XLEF

(5.104)

where []# denotes pseudo-inversion. Once the Fourier series of the low-frequency error is obtained, the estimated attitude of the satellite can be calibrated using Eq. (5.90) as follows: −1 qˆ = qˆ s ⊗ qLEF

(5.105)

When the satellite is in normal orbit, the drift angle is generally corrected, and the satellite’s angular velocity is small. Although the orbital angular velocity in the pitch direction is significant (usually 0.06◦ /s for low-orbit satellites), the maximum angular velocity along the other two axes is small at approximately 0.005◦ /s. In addition, a phase difference may be introduced to the estimated equivalent gyro constant drift and low-frequency error signal obtained through filtering. Therefore, when a certain error exists in the satellite’s angular velocity and the amplitude of the low-frequency error is small, the relative error in the calculated results may be large when using Eq. (5.104) directly. Based on the above, the results of Eq. (5.104) can be further calibrated by using the filter method to reduce the error in attitude determination. The Fourier series coefficient variables for the low-frequency error are expanded as the states of the filter system. According to the satellite’s dynamics equation and gyro’s measurement equations, the continuous state equation of the filter system is expressed as

5.5 Ground-Based Post-Event High-Precision Attitude Calibration

$

&

δq˙ ·

v

˙ LEF δX



−ωˆ × 0 = 0 0

&

$ δq ·

v

δXLEF

 +

− 21 I 3×3 0 0 I

245



wg wLEF



A more compact form of the above equation is ˙ = AδX + Bw(t) δX $ where δX =

&

δq ·

 ,A =

v

(5.106)

   1 −ω× ˆ 0 − 2 I 3×3 0 , and w(t) = ,B = 0 I 0 0

δXLEF  wg . wLEF Let q = qˆ −1 ⊗ qˆ s . We define the observed value z as 

z = qv −



Xˆ L E F

(5.107)

where ⎤ c1 s1 0 0 0 0 · · · ck sk 0 0 0 0 = ⎣ 0 0 c1 s1 0 0 · · · 0 0 ck sk 0 0 ⎦ 0 0 0 0 c1 s1 · · · 0 0 0 0 ck sk ⎡

Thus, the observation equation is   z = I δX + V

5.

(5.108)

where V denotes the equivalent measurement noise. Filters can be designed based on Eqs. (5.106) and (5.108) to estimate the attitude of the satellite and coefficients of the low-frequency error. To reduce the influence of the angular-velocity measurement error on the accuracy of attitude determination, Q2wg is generally set to be larger than the variance of the actual measurement error of the gyro. Simulation verification In the simulation, the random drift of the gyro is set to 0.06°/h; the gyro constant drifts are set to 5°/h, –3°/h, and 2°/h; and the variance of the measurement noise of the star sensor is set to 5 (1 = 1/3600°). In the low-frequency error model of the star sensor described in Eq. (5.36), let√N = 1. The coefficients of the model are listed in Table 5.1. In the table, n = a 2 + b2 represents the amplitude of the low-frequency error corresponding to the frequency, and a and b are the Fourier series coefficients. Table 5.1 also lists the coefficients for the low-frequency error estimation. As presented in the table, the estimation accuracy for the amplitude of lowfrequency error of the star sensor is approximately 10%, with the estimation accuracy in the pitch direction being the highest. As shown in Figs. 5.11 and

246

5 Spacecraft Attitude Determination

Table 5.1 Calculated coefficients of low-frequency error of the star sensor (unit:  ) ϕL E F Actual value

θL E F

ψL E F

Estimated value

Actual value

Estimated value

Actual value

Estimated value

a

10

8.54

−6

−5.67

3

2.61

b

2

3.18

−5

−5.35

10

11.49

n

10.20

9.11

7.81

10.44

11.78

7.80

Fig. 5.11 Low-frequency error of the star sensor and calibration results

5.12, the residual after the calibration of the low-frequency error and the induced error in the attitude determination are approximately 1". However, the accuracy of attitude determination has dropped significantly compared with the accuracy of 10 obtained before the calibration. To further improve the calibration accuracy of the low-frequency error of the star sensor, the filter method is adopted for calibration. The nominal orbital angular velocity ωo = 0.00111 rad/s. A model of the gyro is established with the ARW set to 10–4 °/h1/2 and the RRW set to 0.001°/h3/2 . The high-frequency noise of two star sensors is set to 1"/30 (optical axis/horizontal axis, 3 σ ). The angular frequency with short-period error for the three axes of star sensor 1 is ωo , ωo , and 0 with amplitudes of 2 , 2 , and 0 , respectively. The angular frequency with short-period error for the three axes of star sensor 2 is 2 ωo , 2 ωo , and 0 with amplitudes of 3 , 3 , and 0 , respectively. Two star sensors are used for attitude determination. The estimated satellite attitude and equivalent gyro constant drift obtained through filtering for an attitude determination system that does not consider the low-frequency error

5.5 Ground-Based Post-Event High-Precision Attitude Calibration

247

Fig. 5.12 Comparison of the accuracy of satellite attitude determination

are shown in Figs. 5.13 and 5.14, respectively, where the errors in attitude determination along the three axes are 1.75 , 3.57 , and 2 . The estimated gyro constant drift shows marked periodic variations in the orbit. The actual gyro constant drift can be obtained by removing the orbit period information from the estimated gyro constant drift of two orbit periods obtained by filtering. As shown in Fig. 5.15, the magnitude of error in the estimation of the gyro constant drift is controlled to the order of 0.001°/h, and no significant orbit period information exists in the estimation error of the gyro constant drift. Based on the constant drift, the angular velocity of the satellite can be estimated more accurately, as shown in Fig. 5.16. Based on data analysis, the estimated angular velocity of the satellite and the equivalent gyro constant drift with periodic variations in the orbit can be obtained. Subsequently, using Eq. (5.104), the Fourier series coefficients ai and bi for the predominant low-frequency error along the three axes of the satellite can be achieved, respectively, as (unit: ): 1.02 0.92; 0.99 −2.76; −0.78 0.83. The estimated amplitudes of the low-frequency error for the three axes are 1.38 , 2.93 , and 1.14 . The calculated Fourier series coefficients for the low-frequency error are used as the initial values of the filter, and the estimated attitude is used as the measurement input. Consequently, the errors in attitude determination along the three axes after filtering have significantly decreased. As shown in Fig. 5.17, the attitude errors along the three axes are all smaller than 1 .

248

5 Spacecraft Attitude Determination

Fig. 5.13 Attitude determination error without considering low-frequency error in filtering

5.6 Determination of Spin Angular Velocity with Abnormal Attitude To address the abnormal attitude of a three-axis-stabilized satellite in orbit with high spin, this section introduces a method to determine the spin angular velocity of the satellite based on the measurement of the Sun sensor [11]. The measurement accuracy of the angular spin rate is analyzed based on the sensor measurement error and measurement principles. In addition, a strategy is introduced to ensure the accuracy of determining different angular spin rates without changing the telemetry data of the Sun sensor. This approach can provide a basis for establishing despinning and precession control strategies for the on-orbit rescue of uncontrolled satellites and the evaluation of their effectiveness.

5.6.1 Principle of Determining Spin Angular Velocity of Satellites In general, the sun sensor uses two slit sensors installed perpendicular to each other for measuring the orientation of the sun vector in the sensor coordinate system. The

5.6 Determination of Spin Angular Velocity with Abnormal Attitude -3

249

Gyro bias error / (deg/hr)

x 10 5 0 -5

2000

6000

4000

8000

10000

12000

14000

12000

14000

-3

x 10 5 0 -5

2000

4000

6000

8000

10000

6000

8000 t/s

10000

-3

x 10 5 0 -5

2000

4000

12000

14000

Fig. 5.14 Deviation of estimated constant drift from the actual value without considering lowfrequency error in filtering

measurement coordinate system of the digital sun sensor is denoted by oxs ys z s , where the aiming axis of the sensor is z s . On the plane xs oys perpendicular to the z s axis, the two slits are parallel to the xs axis and ys axis, respectively. The measurement of the digital sun sensor is the tangent of the angle between the projection of the sun vector on the reference plane and the aiming axis, which are denoted by m x and m y , respectively. Based on the output of the sun sensor, the orientation of the solar unit vector in the sensor coordinate system is ⎡

⎤ mx S ⎣ my ⎦ S= . m 2x + m 2y + 1 1 1

(5.109)

The measurement of the vector in the satellite frame is T S = MSB · SS

(5.110)

In a short time, the change in the sun vector in inertial space is ignored; therefore, the variations in S measured at different times can all be considered to be caused by the rotation of the satellite body. If the sun vectors measured by the digital sun sensor at consecutive times t1 , t2 , and t3 are S1 , S2 , and S3 , respectively, in the satellite

250

5 Spacecraft Attitude Determination -3

estimated error of Gryo bias / (deg/hr)

x 10 1 0 -1

1000 2000 3000 4000 5000

6000 7000 8000 9000 10000 11000

-3

x 10 1 0 -1 1000

2000

3000

4000

5000

6000

7000

8000

9000 10000 11000

-3

x 10 1 0 -1

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 t/s

Fig. 5.15 Deviation between the estimated and actual values of constant drift after data processing -3

5

rate / (deg/s)

x 10

real fited

0 -5

0

2000

4000

6000

8000

10000

12000

2000

4000

6000

8000

10000

12000

2000

4000

6000 t/s

8000

10000

12000

-0.062 -0.064 -0.066

0 -3

5

x 10

0 -5

0

Fig. 5.16 The angular velocity of the satellite with the gyro constant drift

5.6 Determination of Spin Angular Velocity with Abnormal Attitude

251

attitude error / (arcsec) 1 0.5 0 -0.5 1000

2000

3000

4000

5000

6000

7000

8000

9000 10000 11000

0.5 0 -0.5 -1 2000

3000

4000

5000

6000

7000

8000

9000

10000 11000

1 0 -1 2000

3000

4000

5000

6000 t/s

7000

8000

9000 10000 11000

Fig. 5.17 Attitude determination error considering the low-frequency error in filtering

frame, the two variation vectors can be obtained as follows: S21 = S2 − S1 , S32 = S3 − S2 The unit vector of the satellite’s spin axis in the satellite frame is denoted by e, and the angular rate of the satellite’s spin about the spin axis is ω. The spin angular velocity of the satellite can then be expressed as ω = ωe. The attitude matrix at t is A(t), and the attitude matrix at t + t is A(t + t). Thus, we have A(t + t) = A(t)A(t) where A(t) = cos ϕ I + (1 − cos ϕ)eeT − sin ϕe× In the above equation, ϕ = ω · t, and ⎡

⎤ 0 −ez e y e× = ⎣ ez 0 −ex ⎦ −e y ex 0

(5.111)

252

5 Spacecraft Attitude Determination

Assuming that the satellite’s spin axis e and spin angular rate ω remain unchanged during the time period [t1 ,t3 ], the vectors S1 , S2 , and S3 should have the following relationship: S2 = A(t2 − t1 )S1 , S3 = A(t3 − t2 )S2

(5.112)

In Eq. (5.112), A(t2 − t1 ) and A(t3 − t2 ) are in the same form as A(t) in Eq. (5.111), where θ1 = ω · (t2 − t1 ) and θ2 = ω · (t3 − t2 ). From Eq. (5.112), we obtain

 eT S21 = eT A(t2 − t1 )S1 − S1

 

= eT (1 − cos ϕ1 ) eeT − I − sin ϕ1 e× S1 = 0 Similarly, eT S32 = 0. Therefore, S21 and S32 are perpendicular to the spin axis e, and their geometric relations are shown in Fig. 5.18. Specifically, θ is the angle between the vectors S21 and S32 , and o is the intersection of the spin axis e and the plane on which the vectors S21 and S32 are located. When the spin axis e is not parallel to B S1 , e=

S32 × S21 |S32 × S21 |

(5.113)

The angles between the spin axis e and the sun vectors B S1 , B S2 , and B S3 in the satellite frame are all equal at different times. Therefore, by using the three angles of the plane on which the vectors S21 and S32 are located, we obtain Fig. 5.18 Principle of the geometrical determination of spin angular velocity

5.6 Determination of Spin Angular Velocity with Abnormal Attitude

θ=

253

1 1 (θ1 + θ2 ) = ω(t3 − t2 ) 2 2

The spin angular rate ω is ω=

  2 ST21 S32 2θ = arccos |S21 | · |S32 | t3 − t1 t3 − t1

(5.114)

If the continuous measurement interval t of the digital sun sensor remains constant (i.e., θ = ϕ1 = ϕ2 ), the spin angular rate is ω=

  1 ST21 S32 θ = arccos |S21 | · |S32 | t t

(5.115)



The time series B Sk , k = 1, 2, . . . of the sun vector in the satellite body frame is obtained through sensor measurements. Thus, by using Eqs. (5.113) and (5.114), the spin angular velocity of the satellite in the satellite body frame can be obtained.

5.6.2 Strategies for Reducing the Determination Error of Spin Angular Rate and Improving the Determination Accuracy To account for the measurement errors of the sensor, let m x and m y be the measurement errors of m x and m y , respectively. Both are uncorrelated Gaussian white noise with zero mean, and their variance is   σ 2 [m x ] = σ 2 m y = rs2 (rs > 0) Considering that rs is a small quantity in practice, the measurement equation of the solar sensor given by Eq. (5.109) can be written as S = S S + S

(5.116)

T  where S ≈ m x m y 0 . The measurement error of the sensor has the following statistical characteristics given the mean and variance:  T E{S} ≈ 0 0 0 ⎡ ⎤ 100

E SST = rs2 ⎣ 0 1 0 ⎦ 000

(5.117)

254

5 Spacecraft Attitude Determination

Assuming that the continuous measurement interval t of the digital sun sensor is constant, according to the principle described in Fig. 5.18, θ = θ1 = θ2 , and S21 = S32 = 2|sin θse | sin

θ 2

(5.118)

where θse is the angle between the sun vector and the spin axis e. The measurement error of the spin angular rate is denoted by ω. When the sensor’s measurement and ω · t are small quantities, according to Eqs. (5.115) and (5.118), 4t sin2 θse sin2

θ 2

sin θ ω = −ST21 S32 − ST32 S21

(5.119)

where S21 and S32 are the deviation of the vectors S21 and S32 caused by the sensor measurement, respectively. Furthermore, S21 = S2 − S1 , S32 = S3 − S2 where Si is the deviation of the sun vector measured by the sensor at ti (i = 1, 2, 3). Si at different times can be considered uncorrelated. Without loss of generality, in the following analysis, it is assumed that the solarsensor coordinate system agrees with the satellite coordinate system. In other words, the mounting matrix MSB is an identity matrix. If the two coordinate systems do not coincide, the satellite coordinate system needs to be redefined according to the sun-sensor coordinate system. After completing operations in the redefined satellite coordinate system, it can be converted back to the original satellite coordinate system. The characteristics of the sun vector are calculated based on the sensor measurement as described in Eq. (5.117). For v = −ST21 S32 − ST32 S21 , E{v} = 0, and         T T T T T E v 2 = ST 21 E S32 S32 S21 − 2S21 E S32 S21 S32 +S32 E S21 S21 S32

Based on the expressions ⎡

⎤ 100 E S32 ST32 = E S21 ST21 = 2r 2 ⎣ 0 1 0 ⎦ 000 ⎡ ⎤ 100



E S32 ST21 = −E S2 ST2 = −r 2 ⎣ 0 1 0 ⎦ 000



the three terms on the right-hand side of Eq. (5.120) can be written as

(5.120)

5.6 Determination of Spin Angular Velocity with Abnormal Attitude

255



 T T S21 = 2r 2 S21 2 − S221z S21 E S32 S32

 T T S32 S32 = 2r 2 S32 2 − S232z E S21 S21

T T S21 S32 = −r 2 S21z S32z E S32 S21 where S21z and S32z are the third components of the vectors S21 and S32 , respectively. Equation (5.120) can then be written as

E v2 = 2r 2 4 sin2 θse sin2

θ 2

− S221z − S232z +S21z S32z



When θ is a small quantity, S21z ≈ S32z . Hence,

E v2 = 2r 2 4 sin2 θse sin2

θ 2

− S221z



As can be seen from Fig. 5.18, S21 moves periodically around e with the rotation of the satellite in the plane perpendicular to the spin axis e. Therefore, 0 ≤ S221z ≤ 4 sin2 θse sin2

θ 2

sin2 θze

(5.121)

where θze is the angle between the spin axis e and the z s axis of the sun sensor, and sin θze = ez . In the above equation, when S21 moves in the direction perpendicular to z s , S221z = 0. Furthermore, when S21 moves in the direction parallel to the plane on which the spin axis e and the z s axis are located, S221z = 4 sin2 θse sin2 θ2 sin2 θze . By using Eq. (5.121), E v2 can be expressed as 2 r2 r 2 cos2 θze ≤ ≤ E ω θ 2 2 2 2 2t 2 sin θse sin 2 sin θ 2t 2 sin θse sin2

θ 2

sin2 θ

(5.122)

When θ = ω · t is a small quantity, the above equation can be approximated as 2 2r 2 2r 2 cos2 θze ≤ ≤ E ω ω4 t 6 sin2 θse ω4 t 6 sin2 θse

(5.123)

During the despinning of the satellite, if the satellite’s spin angular rate becomes progressively smaller, it is known from Eq. (5.123) that the error will become progressively larger when the spin angular rate is determined by data from two consecutive intervals. The determination accuracy for a certain spin angular rate ω is inversely proportional to the cubic power of the measurement interval for the vectors. Therefore, while the telemetry data of the satellite remain unchanged, the determination accuracy can be effectively improved by increasing the interval of spin attitude data. Figure 5.19 shows the variations of the sun vector obtained by calculating the differentials of the sun vectors with multiple sampling intervals based on the spin angular velocity. In the figure, Sn , Sn+1 , …, Sm , Sm+1 , …, Sn+k , Sn+k+1 , … are

256

5 Spacecraft Attitude Determination

Fig. 5.19 Schematic of variations in the sun vector considering the influence of measurement accuracy

equally spaced sun-vector measurements in the satellite frame, where k ≥ 1 and m > n. According to the above-mentioned principle of the measurement of spin angular velocity along the spin axis, the unit vector and angular rate along the spin axis under different measurement intervals are calculated as follows: Sm+k,m × Sn+k,n  e = −  Sm+k,m × Sn+k,n  0 / T Sn+k,n Sm+k,m 1    ω= arccos   Sm+k,m  ·  Sn+k,n  (n − m)t

(5.124)

(5.125)

where Sm+k,m = Sm+k − Sm and Sn+k,n = Sn+k − Sn . When m − n = 1 and k = 1, Eqs. (5.124) and (5.125) are identical to Eqs. (5.113) and (5.114), respectively.

5.6.3 Simulation Verification and Application 1.

Verification through mathematical simulation The measurement accuracy of the sun sensor is set to 0.01° (3σ). The direction of the optical axis of the sensor is along the Y-axis of the satellite. The raw measurement data are collected at the sampling interval Ts = 0.5 s. The coordinate vector of the satellite’s spin axis in the satellite coordinate system is [0.183 0.913 0.365] T, and the angle between the spin axis and the sun vector is 24.1°. When the amplitude of the angular velocity of the satellite is 36°/s, the spin angular rate is determined using Eqs. (5.124) and (5.125) with m = 2, n = 1, and k = 1. By inserting t = 0.5 s, θse = 24.1◦ , r = 0.01◦ , and θ = t · ω = 18◦

5.6 Determination of Spin Angular Velocity with Abnormal Attitude

2.

257

into the right-hand side of the second inequality in Eq. (5.122), the theoretical determination error is calculated as 0.717°/s (3σ). The direction of the spin axis and spin angular rate determined and verified by mathematical simulations are shown in Figs. 5.20 and 5.21, respectively. It is known from Fig. 5.18 that the determination accuracy for spin angular velocity is approximately 0.7°/s (3σ), which is in agreement with the theoretical value. When the amplitude of the satellite’s angular velocity decreases to 3°/s, m = 2, n = 1, and k = 4 are selected in Eqs. (5.124) and (5.125). The time interval of the measured data of the sun sensor is determined to be four times the original sampling interval corresponding to the spin angular velocity. Similarly, the theoretical determination error can be calculated based on t = 4Ts = 2 s, θse = 24.1◦ , and θ = ωt = 6◦ as 1.583°/s (3 σ). The spin angle rate determined and verified through mathematical simulation is shown in Fig. 5.22. The determination error is found to be 1.58°/s (3σ), which is consistent with the theoretical result. Applications When the satellite’s angular velocity is approximately 36°/s, the spin angular velocity is measured directly using the telemetry data of the digital sun sensor at an interval of 0.5 s. Further, m = 2, n = 1, and k = 1 are used in Eqs. (5.124) and (5.125). The results are shown in Figs. 5.23 and 5.24. According to Fig. 5.24, the determination accuracy for the spin angular velocity is approximately 0.5° (3σ) without considering the outliers. The average value of the results clearly shows that the theoretical angular velocity of the satellite has reduced by 1°/s.

Fig. 5.20 Components of the spin axis in the satellite frame

258

Fig. 5.21 Spin angular rate of the satellite

Fig. 5.22 Spin angular rate of the satellite

5 Spacecraft Attitude Determination

5.6 Determination of Spin Angular Velocity with Abnormal Attitude

259

Fig. 5.23 Angles between the spin axis and the three axes of the satellite frame before and after despinning

Fig. 5.24 Spin angular rates of the satellite before and after despinning

260

5 Spacecraft Attitude Determination

Fig. 5.25 Spin angular rates of the satellite before and after despinning

Consider that the satellite’s angular velocity is reduced to approximately 3°/s, with m = 2, n = 1, and k = 4. Although the satellite’s spin angular velocity has decreased by an order of magnitude with respect to the initial angular velocity of 36°/s, the data interval of the sun sensor is four times the initial value. From Eq. (5.123), it is found that the determination error of spin angular rate is theoretically 1.56 times the determination error when the angular velocity is 36°/s. According to Fig. 5.25, the actual measurement error is approximately 0.8°/s (3σ), and the average value of the results clearly shows that the theoretical angular velocity of the satellite has reduced by 0.3°/s.

References 1. Lv Z, Lei Y (2013) Satellite attitude measurement and determination. National Defense Industry Press, Beijing 2. Xiong K, Lei Y, Zeng H (2010) Modeling and simulation of fiber optic gyros based on Allan variance method. Aerosp Control Appl 36(3):8–14 3. Xiong K, Tang L, Lei Y (2011) Multiple model Kalman filter for attitude determination of precision pointing spacecraft. Acta Astronaut 68:843–852 4. Lei Y (2017) Relative alignment error calibration method for star sensors of spacecraft in orbit. Aerosp Control Appl 43(6):1–7

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5. Li J, Chen Y (1996) Constant-gain information filter for attitude determination of precision pointing spacecraft. In: 47th international astronautical congress (IAF-96-A.5.08). Beijing, China 6. Lefferts EJ, Markley FL, Shuster MD (1982) Kalman filtering for spacecraft attitude estimation. J Guidance, Control Dyn 5(5): 417–429 7. Liu Y, Chen Y (2003) Star-sensor measurement model and its application to the spacecraft attitude determination system. J Astronaut 24(2):163–167 8. Pittelkau ME (2017) Survey of calibration algorithms for spacecraft attitude sensors and gyros. AAS 07–295 9. Lei Y, Tian K, Zong H et al (2014) A post-event attitude determination method based on calibration of low-frequency errors of star sensor. In: Proceedings of the 2014 Symposium on inertial technology by the professional group on inertial of the sciences and technology committee of the china aerospace science and technology corporation. Beijing 10. Tian K, Zong H, Lei Y (2014) A calibration method for low-frequency errors of star sensor based on gyro constant drift estimation. In: Proceedings of the 16th national conference on space and moving body control technology, pp 162–166 11. Lei Y, Li M (2018) Spinning angle velocity determination for in-orbit satellites with attitude anomaly. Aerosp Control Appl 44(1): 15–20

Chapter 6

Spacecraft Attitude Control

6.1 Introduction Spacecraft attitude control has a variety of requirements, which are mainly for attitude stabilization control and attitude maneuver control. Attitude stabilization control keeps the attitude of a spacecraft along a reference frame by overcoming the influence of the internal and external disturbance torque. Attitude maneuver control refers to the process of changing a spacecraft’s attitude from unknown to known or from one attitude to another; it may also refer to the tracking of the desired attitude. In addition to the attitude control of the spacecraft body, certain subsystems of the spacecraft must be controlled to complete a space mission, necessitating unified attitude control or the hierarchical control of the spacecraft. As spacecraft technology continues to advance, control objects tend to have an increasingly complex structure. At present, all large spacecraft are equipped with large solar arrays, large liquid fuel tanks, and rotating or movable parts such as antennas. Given the complexity of a spacecraft’s structure, various issues need to be considered in spacecraft control, such as vibration suppression for flexible spacecraft, sloshing suppression for liquid-filled spacecraft, and the control of multi-body systems. More attention to these issues have to be paid to the design of spacecraft control systems.

6.2 Attitude Control Based on Angular-Momentum Management Devices Devices for angular-momentum exchange include fixedly mounted flywheels, control moment gyroscopes (CMGs), and gimbaled momentum wheels. A fixedly mounted flywheel adjusts its speed of rotation or angular momentum to realize the exchange of angular momentum between the flywheel and the satellite. When the flywheel accelerates, it produces a reaction torque on the satellite body, i.e., the control © Beijing Institute of Technology Press 2022 Y. Xie et al., Spacecraft Dynamics and Control, Space Science and Technologies, https://doi.org/10.1007/978-981-33-6448-6_6

263

264

6 Spacecraft Attitude Control

torque. A CMG operates at a constant high speed, implying that it has a large constant angular momentum. It changes the direction of the angular-momentum vector through the motion of CMG gimbals and produces a gyroscopic reaction torque on the satellite body. Angular-momentum management systems are mainly classified into two types: the zero-momentum control system and the biased-momentum control system. Considering that biased-momentum control has been well established, only zero-momentum control is introduced in this book.

6.2.1 Spacecraft Attitude Stabilization Control For zero-momentum control, the yaw, pitch, and roll channels can be designed independently after introducing decoupling laws into the roll-yaw channel. Compared with the biased-momentum control, the zero-momentum control is more complex and enables high precision. Furthermore, the design of control laws for zero-momentum control is more flexible and convenient, which makes the zero-momentum control particularly suitable for the attitude maneuver control. The selection of the control law is a process of using information about attitude error to provide appropriate control instructions so that angular-momentum management devices can generate the control moments to suppress or eliminate the attitude error. The control laws are designed based on the task requirements for the function and performance of the attitude control system, and such design can be achieved using proportional-derivative (PD) control, proportional-integral-derivative (PID) control, or other methods. For a control system that describes attitude in Euler angle and its time derivative, PD controller can be expressed in the form as h˙ x = kpx φ + kdx φ˙ h˙ y = kpy θ + kdy θ˙ h˙ z = kpz ψ + kdz ψ˙ where φ, θ , and ψ are the roll, pitch, and yaw angles, respectively; hx , hy , and hz are the components of the resultant angular momentum of the momentum management devices in the satellite body frame; The system would operate stably and meet certain requirements when the appropriate control parameters kp and kd are selected for each channel. In the case of constant external disturbances on the system, an integral can be introduced into the control law to eliminate the steady-state attitude error. For a control system that describes attitude in quaternions and the angular velocity, quaternion feedback control can be directly used, and attitude kinematics can be written in the quaternion form as q˙ v =

1 q4 ω − ω × qv 2

6.2 Attitude Control Based on Angular-Momentum Management Devices

265

1 q˙ 4 = − ωT qv 2 where qv and q4 are the vector and scalar parts of the quaternion q, respectively. The attitude dynamics equation is J ω˙ + ω × Jω = T c where J is the moment of inertia of the spacecraft, the control torque T c = −h˙ w − ω × hw , and hw is the resultant momentum of the angular-momentum management device. Assuming that the gyroscopic torque can be precisely compensated for, the PD control law with quaternion feedback is T c = −K p qev − K d ω + ω × Jω

(6.1)

In Eq. (6.1), the control coefficient matrices are K p , K d > 0; qev is the vector part of the quaternion for attitude estimation error qe = q−1 r ⊗ q; and qr is the target attitude quaternion. The scalar part of qe should satisfy qe4 ≥ 0. With such control, the closed-loop system becomes asymptotically stable. If the gyroscopic torque cannot be accurately compensated for, we can use the following PD control law with quaternion feedback: T c = −K p qev − K d ω

(6.2)

Based on the control law in Eq. (6.2), to theoretically ensure the asymptotic stability of the closed-loop system, as shown by WIE parameters   [1], the control can be selected as K p = kI3 > 0 and K d = diag kdx kdy kdz > 0, or K p = (αI3 + βJ)−1 and K −1 p K d > 0, where α and β are non-negative numbers. In the above quaternion feedback control law, for the Earth-orienting spacecraft, the quaternion q and angular velocity ω need to be converted into the attitude and angular velocity of the satellite with respect to the orbital frame. With the attitude  T stabilization control with small attitude errors, we have qe ≈ 21 φ θ ψ ; then, quaternion feedback control can be equivalent to Euler-angle-based feedback control. In order to achieve high-performance attitude stabilization control, in addition to the use of the above feedback control laws based on attitude measurement, it may be necessary to estimate and compensate for attitude disturbances on the satellite. This section introduces an attitude stabilization control method for the attenuation of payload periodic disturbances [2]. 1.

Satellite dynamics

A spacecraft with large payloads moving periodically around the fixed axis of a satellite is affected by dynamic and static load imbalances. When a payload moves, it

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6 Spacecraft Attitude Control

produces disturbance torques on the satellite, affecting the performance of the attitude control system. Considering the central body of the satellite as a rigid body, the dynamics of the satellite, accounting for the payload disturbances, can be expressed as J ω˙ + ω× (Jω + hw ) = T c + T d

(6.3)

where ω denotes the absolute angular velocity of the satellite, J refers to the moment of inertia of the satellite, hw represents the resultant angular momentum of the angular-momentum exchange device, T c is the attitude control torque vector, T d is the disturbance torque generated from the movement of the payload on the satellite, and (·)× denotes the antisymmetric matrix formed by a vector. Since the motion of the payload is periodic, the disturbance torque produced by its movement can be expressed as T d = AL · sin(ωL t + βL )

(6.4)

where AL is the magnitude vector of disturbance torque in the satellite frame, ωL is the angular frequency of the load, and βL denotes the disturbance phase. We can further express the disturbance torque as T d = aL sin(ωL t) + bL cos(ωL t)

(6.5)

where aL and bL are constant coefficients. 2.

Attitude stabilization control with disturbance attenuation

The satellite’s attitude with respect to the orbital frame is described by Euler angles φ, θ , and ψ, and then the absolute angular velocity of the spacecraft ω can be expressed as ⎡ ⎤ 0 (6.6) ω = ωbo + C bo (φ, θ, ψ)⎣ −ωo ⎦ 0 where ωbo is the angular velocity in the satellite body frame with respect to the orbital frame; C bo (φ, θ, ψ) is the direction cosine matrix in the satellite body frame with respect to the orbital frame; and ωo denotes orbital angular rate, which can be regarded as a constant for near-circular orbits. When the attitude angles φ, θ , and ψ are small quantities, ωbo can be approximated  T as ωbo = φ˙ θ˙ ψ˙ ; thus, we have ⎤ ⎡ ⎤ φ¨ 0 ⎣ ⎦ C ω˙ = ⎣ θ¨ ⎦ + ω× ω o bo bo ¨ ψ 0 ⎡

(6.7)

6.2 Attitude Control Based on Angular-Momentum Management Devices

267

The controller can be designed as ⎡ ⎤ ⎡ ⎤ ⎤ φ˙ 0 φ ⎣ ⎦ T c = −ωn2 J ⎣ θ ⎦ − ζn ωn J ⎣ θ˙ ⎦ + ω× (Jω + hw ) + Jω× C bo bo ωo ˙ ψ 0 ψ ⎡

(6.8)

where ωn , ζn > 0 are control parameters, which determine the dynamic characteristics of the closed-loop system. Different parameters can be designed for different channels according to specific requirements for the control performance of the system. For brevity, we use the same parameters for all three channels. Based on Eq. (6.8), the dynamic equations of the closed-loop system can be obtained using Eqs. (6.3) and (6.7) ⎤ ⎡ ⎤ ⎡ ⎤ φ˙ φ¨ φ ⎣ θ¨ ⎦ + ζn ωn ⎣ θ˙ ⎦ + ωn2 ⎣ θ ⎦ = J −1 T d ψ˙ ψ¨ ψ ⎡

It can be determined from the above that a certain amount of attitude deviation will occur in the system under bounded payload disturbances. It is difficult to accurately compensate for disturbance torques, partly because the dynamic and static imbalances of large payloads are difficult to measure accurately on the ground, and partly because of the rotation-axis misalignment and on-orbit distortion of the payload structure. To suppress or attenuate the influence of the disturbances, we can estimate the disturbances in real time. Let T c = T c −ω× (Jω + hw ). Equation (6.3) can be written in a more compact form as J ω˙ = T c + T d If T˙ d = 0, we can establish the equation of the state x = follows:     ⎧ 0 ⎨ x˙ = 0 0 x + −1  I 0  J Tc ⎩ y=ω= 0I

(6.9) T

 T J −1 T d ωT as

(6.10)

Based on Eq. (6.10), the estimator for T d is designed as x˙ o = −Lxo − LJ −1 T c − L2 ω 

T d = J(xo + Lω)

(6.11)

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6 Spacecraft Attitude Control

where L is the selected positive definite coefficient matrix, which is usually set as L = LI (L > 0); I is the identity matrix of the corresponding dimensions; xo denotes the observed state; and T d is an estimate of disturbance torque. With disturbance estimation and compensation, we design the attitude control law as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ φ˙ φ 0 ⎣ ⎦ − Td C T c = −ωn2 J ⎣ θ ⎦ − ζn ωn J ⎣ θ˙ ⎦ + ω× (Jω + hw ) + Jω× ω o bo bo ψ˙ 0 ψ (6.12) 



The above estimator regards the disturbance torque T d as a constant or a slow time-varying signal. However, if the speed of the periodic motion of the payload is large, the above estimator may have a very large deviation in estimation. To address this issue, we can further consider the periodic characteristics of disturbance torques. For T d in Eq. (6.4), its second time derivative is T¨ d = −ωL2 AL sin(ωL t + βL ) = −ωL2 T d We can establish the following equation of

 T  −1 T T T by using Eqs. (6.9) and (6.13): J T˙ d ω J −1 T d ⎡ ⎧ ⎪ ⎪ ⎨ x˙ = ⎣

⎤ ⎤ ⎡ 0 I0 0 −ωL2 I 0 0 ⎦x + ⎣ 0 ⎦ ⎪ I 0 0 J −1 T c ⎪ ⎩ y=ω= 00I x

(6.13) the

state

x

=

(6.14)

Based on Eq. (6.14), we can design an estimator in a similar form as the estimator for T d in Eq. (6.11). Owing to space limitations, the design process will not be explained here. 3.

Simulation verification

The above-mentioned disturbance torque compensation method is tested and verified using a single-axis air-bearing platform, and the testing system is shown in Fig. 6.1. The system’s moment of inertia is 450 kg m2 . The simulated moment of inertia of the payload is 140 kg m2 . The nominal speed of rotation is 45 °/s. The platform is equipped with a momentum wheel of 125 Nm s, and the maximum output torque is 0.17 Nm. The system’s control period is 0.25 s. Without disturbance torque compensation, we obtained the platform (rotational) angle and angular velocity curve under PID control, as shown in Figs. 6.2 and 6.3. The error in attitude control is approximately 0.015°, and the error in angular velocity is 0.004 °/s.

6.2 Attitude Control Based on Angular-Momentum Management Devices

269

Fig. 6.1 Single-axis air-bearing platform for the verification of periodic-disturbance attenuation methods

Fig. 6.2 Attitude angle without disturbance torque compensation

-3

x 10

Attitude angle (°)

5 0 -5 -10 -15

Fig. 6.3 Angular velocity without disturbance torque compensation

0

200

400

600 Time (s)

800

1000

200

400

600

800

1000

-3

Angluar velocity (°/s)

4

x 10

2 0 -2 -4

0

Time (s)

For the disturbance estimators in Eqs. (6.5)–(6.10), the control effect after introducing the estimated disturbance torque into system compensation is shown in Figs. 6.4 and 6.5. The error in attitude control is approximately 0.008°, and the error in attitude angular velocity is 0.0022 °/s.

270

6 Spacecraft Attitude Control

Fig. 6.4 Attitude angle with disturbance estimation and compensation

-3

Attitude angle (°)

8 6 4 2 0 -2

0

150 Time (s)

50

100

50

100 150 Time (s)

200

250

300

200

250

300

-3

3

Angluar velocity (°/s)

Fig. 6.5 Angular velocity with disturbance estimation and compensation

x 10

x 10

2 1 0 -1 -2

0

6.2.2 Attitude Maneuver Control An attitude maneuver includes attitude reorientation or the tracking of certain attitude trajectories. The input of attitude reorientation control is a fixed target attitude relative to the reference frame, and the control performance during maneuvering is generally not required. A satellite needs to reach the target attitude at a specific time and become stabilized to meet the attitude requirements of payloads. In addition, attitude maneuvering is limited by various constraints in practice, such as the measurement of the system attitude, execution of control, and spacecraft dynamics. In practice, attitude tracking is usually adopted to achieve rapid attitude maneuvers. In general, attitude planning is based on the initial attitude of the satellite and the target attitude to allow smooth transition from the initial attitude to the target attitude. 1.

Error-quaternion equation for attitude-tracking control

This section briefly introduces the derivation of the error equation for a satellite. The attitude motion of a satellite is described by the quaternion for the rotation of T  the spacecraft body frame with respect to the inertial frame, q = qv q4 , and

6.2 Attitude Control Based on Angular-Momentum Management Devices

271

T  the angular velocity of the satellite, ω = ωx ωy ωz . Then, the kinematics of the rigid-body spacecraft is 1 1 q × ω + q4 ω 2 v 2 1 q˙ 4 = − ωT qv 2

q˙ v =

The satellite’s angular velocity can be expressed in terms of the quaternion as   ˙v ω = 2 q4 q˙ v − qv q˙ 4 − 2q× vq The direction cosine matrix from the inertial frame to the body frame is   C = q42 − qv qTv I3 + 2qv qTv − 2q4 q× v If actuators’ dynamics and external moments are not considered, we can obtain the rigid-body satellite’s attitude dynamics equation using Euler’s theorem as J ω˙ + ω × Jω = u The quaternion for the rotation of the target frame with respect to the inertial T  frame is expressed as qr = qrv qr4 , and the angular velocity is expressed as ωr =  T ωr1 ωr2 ωr3 . In a similar manner, the target angular velocity can be expressed using the target quaternion as   ˙ rv ωr = 2 qr4 q˙ rv − qrv q˙ r4 − 2q× rv q The transformation matrix from the inertial frame to the target frame is   2 − qrv qTrv I 3 + 2qrv qTrv − 2qr4 q× C r = qr4 rv Next, the transformation matrix from the target frame to the body frame can be obtained as   2 − qev qTev I 3 + 2qev qTev − 2qe4 q× C e = qe4 ev where qe = q−1 r ⊗ q is the quaternion for the rotation of the body frame with respect to the target frame and ωe = ω − C e ωr is the angular velocity in the body frame with respect to the target frame. As shown by COSTIC [3], the open-loop tracking error dynamics can be expressed as follows:

272

6 Spacecraft Attitude Control

q˙ e =

1 R(qe )ωe 2

(6.15)

˙ r) + u J ω˙ e = −(ωe + C e ωr )× J(ωe + C e ωr ) + J(ω× e C e ωr − C e ω

(6.16)

where  R(qe ) = 2.

q× ev + qe4 I 3 −qTev



Trajectory planning for attitude maneuvers

There are a few challenges for the rapid maneuvering of a spacecraft. First, a large initial attitude deviation during maneuvering causes the actuator torque to be easily saturated. Second, when a satellite quickly maneuvers from the current attitude to the target attitude, an attitude overshoot may make it difficult to stabilize the attitude of the satellite quickly enough to meet the payload requirements. Third, for a flexible spacecraft, flexible vibrations may be induced during rapid maneuvering, resulting in a long stabilization period. To solve the above issues, a common method is to connect the initial attitude and target attitude with a sufficiently smooth characteristic curve. According to Euler’s theorem, any displacement of a rigid body about a fixed point can be obtained by rotating it by a certain angle around an axis passing through that point. Let us posit that the initial attitude quaternion of the satellite is q0 and the target attitude quaternion is qt . Then, we have a fixed axis of rotation e and angle of rotation χm so that the satellite rotates about the fixed axis of rotation from the initial attitude to the target attitude. From q0 and qt , we obtain the quaternion for attitude maneuverable qm as qm = q−1 0 ⊗ qt where q−1 0 is the inverse of q0 and “⊗” denotes the product operator. qm4 and qmv are the scalar and vector part of qm , respectively. When the initial attitude does not coincide with the target attitude, we calculate the angle of rotation χm and axis of rotation e using qm as   χm = 2 arccos qm4 , e =

1    qmv sin χm 2

The three-axis attitude trajectory planning for attitude maneuvers is converted into one-dimensional variable planning for the angle of rotation χm . The time history of the angle of rotation χ (t) obtained by planning is a sufficiently smooth curve, and the following boundary constraints shall be met. Initial conditions: χ (t0 ) = 0, χ˙ (t0 ) = 0. Terminal conditions: χ (tm ) = χm ,χ˙ (tm ) = 0.

6.2 Attitude Control Based on Angular-Momentum Management Devices

273

where t0 and tm denote the start and end time of the maneuver, respectively. As the satellite’s attitude maneuverability is limited by the measurement of angular velocity, actuator torque outputs, angular momentum, and other factors, the time history of Euler angles χ (t) needs to satisfy the following conditions: ¨ ≤ amax max |χ˙ (t)| ≤ ωmax , max |χ(t)|

t∈[t0 tm ]

t∈[t0 tm ]

where ωmax and amax > 0. For practical applications and system implementation, when selecting a characteristic curve for trajectory planning, it is important to consider engineering constraints and hopefully obtain the analytical expressions of χ (t), χ˙ (t), and χ¨ (t). In addition to polynomial trajectory planning for robot control, this section introduces the following three typical trajectory-planning strategies. (1)

Time-optimal trajectory planning

The time-optimal trajectory is the time-optimal bang-coast-bang (BCB) trajectory under limited control. The maneuver trajectory is divided into three stages according to the change in angular acceleration: acceleration, null acceleration, and deceleration. The angular acceleration at the acceleration stage is a constant greater than zero, amax . The angular acceleration at the stage of null acceleration is zero. The angular acceleration at the deceleration stage is the negative of the maximum angular acceleration, −amax . Related parameters for the whole trajectory are shown in Fig. 6.6. The trajectory includes a total of four time-domain characteristic parameters, i.e., t0 ∼ t3 , which satisfy Fig. 6.6 Time-optimal trajectory planning

274

6 Spacecraft Attitude Control

t1 − t0 = t3 − t2 Except for the starting time of attitude maneuver t0 , other time-domain characteristic parameters are calculated independently according to the Euler angles χm and constraint conditions (ωmax and amax ). The acceleration of Euler angles during maneuvering χ¨ (t) is ⎧ ⎨ amax t ∈ [t  0 , t1) χ(t) ¨ = 0 t ∈ t1, t2 ⎩ −amax t ∈ [t2 , t3 ) From the first-order and second-order integration of the angular acceleration χ¨ (t), we can obtain the corresponding χ(t) ˙ and χ (t), respectively. (2)

Sinusoidal-acceleration trajectory planning [4]

According to the change in angular velocity, the sinusoidal acceleration profile comprises three phases: acceleration, constant velocity, and deceleration. Specifically, the angular acceleration in the acceleration and deceleration phases is in the form of a sine function. The maximum amplitude of the angular acceleration, amax , is zero. In the constant-velocity phase, the angular velocity is zero. Figure 6.7 shows the attitude-planning trajectory with sinusoidal acceleration profiles. The trajectory includes a total of four time-domain characteristic parameters, i.e., t0 ∼ t3 , which satisfy t 1 − t0 = t3 − t2 =

Fig. 6.7 Sinusoidal-acceleration trajectory planning

Tsin 2

6.2 Attitude Control Based on Angular-Momentum Management Devices

275

Fig. 6.8 Mixed sinusoidal-acceleration trajectory planning

where Tsin is a set parameter. Similar to time-optimal trajectory planning, except for the start time of attitude maneuver t0 , the characteristic parameters for other time points are calculated independently. The angular acceleration for each stage χ¨ (t) is ⎧ ⎪ ⎪ ⎨





t ∈ [t0 , t1 )   χ(t) ¨ = 0 t ∈ t1, t2   ⎪   ⎪ ⎩ −amax sin 2π t − t2 + 1 Tsin t ∈ [t2 , t3 ) Tsin 2 (3)

amax sin

2π Tsin (t

− t0 )

Mixed sinusoidal-acceleration trajectory planning [5]

As shown in Fig. 6.8, according to the change in velocity, mixed sinusoidalacceleration trajectory planning includes three phases: acceleration, constant velocity, and deceleration. In addition, the acceleration and deceleration phases each include three sub-phases: an increase in acceleration amplitude, constant acceleration amplitude, and a decrease in acceleration amplitude. Overall, the whole trajectory can be divided into seven phases. The trajectory includes a total of eight time-domain characteristic parameters, i.e., t0 ∼ t7 , which satisfy t 1 − t0 = t3 − t2 = t5 − t4 = t7 − t6 = where Tsin is a set parameter. The angular acceleration for each stage χ(t) ¨ is

Tsin 4

276

6 Spacecraft Attitude Control

χ¨ (t) =



⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

amax sin amax sin

2π (t Tsin

− t0 )

 amax 2π (t − t2 + Tsin



t ∈ [t0 , t1 ) t ∈ [t1 , t2 )



Tsin ) 4

t ∈ [t2 , t3 )

0   ⎪ ⎪ 2π ⎪ ⎪ a sin (t − t − t − t ) max 0 2 1 ⎪ Tsin ⎪ ⎪ ⎪ ⎪ −amax ⎪   ⎪ ⎪ ⎩ a sin 2π (t − t − t − 2t ) max

0

Tsin

2

1

t ∈ [t3 , t4 ) . t ∈ [t4 , t5 ) t ∈ [t5 , t6 ) t ∈ [t6 , t7 )

It can be seen from the form of acceleration that this trajectory combines the features of the time-optimal trajectory and sinusoidal-angular-acceleration trajectory with a short period and smooth trajectory. 3. (1)

Attitude maneuver control of rigid-body spacecraft Attitude maneuver based on PD control

Equations (6.15) and (6.16) are the kinematics and dynamics error equations of a rigid-body spacecraft, respectively. With known dynamic parameters and without considering external disturbances, based on quaternion error, we can design the attitude-tracking PD control law as ˙ r) u = −kp qev − kd ωe +(ωe +Cωr )× J(ωe +Cωr ) − J(ω× e Cωr − C ω Specifically, the first two terms on the right-hand side correspond to conventional PD control, while the next two terms represent nonlinear compensation based on the system dynamics parameters J, target attitude angular velocity ωr , and angular acceleration parameters ω˙ r of the planned trajectory; kp and kd are proportional and derivative coefficient matrices, which are generally taken as kp = 2k1 J and kd = 2k2 J, respectively, where k1 and k2 are positive real numbers. From the Lyapunov function V = 21 ωTe kp Jωe +2(1−qe4 ), it can be determined that the attitude-tracking control law can ensure the asymptotic stability of the closed-loop system based on Lyapunov’s stability theorem. Thus, we have  T  T lim qe = 0 0 0 1 , lim ωe = 0 0 0

t→∞

t→∞

In the verification of the simulation, the moment of inertia of the spacecraft’s target is (unit: kg m2 ) ⎡

⎤ 280 0 0 J = ⎣ 0 300 0 ⎦ 0 0 240 The initial attitude q0 and desired attitude qt are set to

6.2 Attitude Control Based on Angular-Momentum Management Devices

277

 T  T q0 = 0 0 0 1 , qt = 0.2120 0.5540 0.1485 0.7912 The control period in the simulation is Ts = 0.1 s, where the control-law parameters are selected as k1 = 0.01 and k2 = 0.08. According to the initial attitude q0 and the target attitude qt , we use the mixed sinusoidal-acceleration algorithm for trajectory planning, where the maximum angular acceleration amax = 0.009 rad/s2 , the maximum angular velocity ωmax = 3.2 °/s, and the period of the sine function Tsin = 12 s. The results of the control simulation are shown in Figs. 6.9 and 6.10. As can be seen from the results, the designed controller with known system parameters can ensure that the system completes the attitude maneuver within 50 s and stabilizes the angular velocity below 0.0002 °/s. Considering that the actual inertia parameter may be unknown or may vary substantially, we select the same control parameters as above and set the satellite’s Fig. 6.9 Satellite attitude (°)

80

Roll Pitch Yaw

Euler angle (°)

60

40

20

0

-20

0

10

20

30

40

50

Time (s)

Fig. 6.10 Attitude angular velocity

3

Roll Pitch Yaw

Angular velocity (°/s)

2.5 2 1.5 1 0.5

X: 49.8 Y: -0.0001094

0 -0.5

0

10

20

Time (s)

30

40

50

278

6 Spacecraft Attitude Control

inertia matrix as (unit: kg m2 ) ⎡

⎤ 140 0 0 J = ⎣ 0 150 0 ⎦ 0 0 120 The simulation results in Figs. 6.11 and 6.12 indicate that a large deviation in system parameters can have a great impact on the performance of attitude control. The maximum deviation in attitude during maneuvering is close to 8°, and the maximum deviation in attitude angular velocity is approximately 1.5 °/s. (2)

Adaptive control with linear parameters

Traditional PD-based attitude-tracking control methods are difficult to apply to systems with uncertain parameters. A large deviation in actual parameters from nominal parameters can have a significant impact on the control performance of a Fig. 6.11 Satellite attitude

70

Roll Pitch Yaw

Euler angle (°)

60 50 40 30 20 10 0

0

10

20

30

40

50

Time (s)

Fig. 6.12 Attitude angular velocity

4

Roll Pitch Yaw

Angular velocity (°/s)

3 2

X: 49.9 Y: 0.1914

1 0 -1 -2

0

10

20

Time (s)

30

40

50

6.2 Attitude Control Based on Angular-Momentum Management Devices

279

system. Consequently, it is considerably challenging to construct fixed-parameter controllers that adapt to strongly nonlinear systems. Therefore, it is necessary to adopt a more robust adaptive control method to improve the performance of attitude control. (1)

Controller design

The attitude dynamics equation for rigid-body spacecraft is J ω˙ + ω × Jω = u The moment of inertia is ⎤ J11 J12 J13 J = ⎣ J12 J22 J23 ⎦ J13 J23 J33 ⎡

The following linear operator is introduced: ⎡

⎤ a1 a2 a3 0 0 0 L(a) = ⎣ 0 a1 0 a2 a3 0 ⎦ 0 0 a1 0 a2 a3 T  Let ϑ = J11 J12 J13 J22 J23 J33 . Then, a model for parameter estimation in the linear parametrization [6] can be obtained as Ja = L(a)ϑ

(6.17)

As the moment of inertia J and the vector determined by its parameters ϑ cannot be accurately known, the truth values can be replaced by the estimated values J and ϑ . Without considering external disturbance torques, the control law can be written as follows: 







˙ r) − u = −kp qev − kd ωe +(ωe +Cωr )× J (ωe +Cωr ) − J (ω× e Cωr − C ω

1 J (q× ev + qe4 I3 )ωe 2 

where kp and kd are positive real numbers. Inserting Eq. (6.17) into the above equation yields 



˙ r )ϑ − u = kp qev − kd ωe + (ωe + Cωr )× L(ωe + Cωr )ϑ − L(ω× e Cωr − C ω 

= kp qev − kd ωe + (ωe + Cωr )× L(ωe + Cωr )ϑ + 

1 L(q× ev ωe + qe4 I3 ωe )ϑ 2



1 ˙ r − q× L(−2ω× e Cωr + 2C ω ev ωe − qe4 I3 ωe )ϑ 2

Putting  ϑ = ϑ − ϑ, for the positive definite matrix Q, we have



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6 Spacecraft Attitude Control

V =

1 1 T  (ωe + qev )T J(ωe + qev ) + 2(kp + kd )(1 − qe4 ) +  ϑ Qϑ > 0 2 2

By taking the time derivative of V, we obtain T ϑ Qϑ˙˜ V˙ = (ωe + qev )T J(ω˙ e + q˙ ev ) − 2(kp + kd )˙qe4 +  T ϑ Qϑ˙˜ = (ωe + qev )T (J ω˙ e + J q˙ ev ) + (kp + kd )ωTe qev + 

Based on the spacecraft dynamics equations and Eq. (6.15), we derive ˙ r )ϑ + u J ω˙ e = −(ωe + Cωr )× L(ωe + Cωr )ϑ + L(ω× e Cωr − C ω and q˙ ev =

1 × 1 (q + qe4 I3 )ωe = (q× ωe + qe4 ωe ) 2 ev 2 ev

The above equations can be summarized as 1  ˙ d − q× V˙ = (ωe + qev )T {(ωe +Cωd )× L(ωe + Cωd )ϑ˜ − L(2ω× e Cωd − 2C ω ev ωe − qe4 ωe )ϑ} 2 T T ˙˜ + (−kp qT ev qev − kd ωe ωe ) + ϑ Qϑ

Let (ωe + qev )T {(ωe +Cωr )× L(ωe + Cωr ) −

1  ˙ r − q× L(2ω× e Cωr − 2C ω ev ωe − qe4 ωe )}ϑ 2

ϑ = − ϑ Qϑ˙˜ = −ϑ˙˜ T Q T

i.e., Qϑ˙˜ = −{(ωe + Cωr )× L(ωe + Cωr ) −

1 T ˙ r − q× L(2ω× e Cωr − 2C ω ev ωe − qe4 ωe )} (ωe + qev ) 2

Then the parameter adaptation law can be obtained as ϑ˙˜ = −Q−1 {(ωe + Cωr )× L(ωe + Cωr ) −

1 T ˙ r − q× L(2ω× e Cωr − 2C ω ev ωe − qe4 ωe )} (ωe + qev ) 2

˙ˆ the update law for the moment of inertia parameters is Given that ϑ˙˜ = ϑ, 1 T ˙ r − q× ϑ˙ˆ = −Q−1 {(ωe + Cωr )× L(ωe + Cωr ) − L(2ω× e Cωr − 2C ω ev ωe − qe4 ωe )} (ωe + qev ) 2

6.2 Attitude Control Based on Angular-Momentum Management Devices

(2)

281

Simulation verification

The moment of inertia of spacecraft is expressed as (unit: kg m2 ) ⎡

⎤ 280 0 0 J = ⎣ 0 300 0 ⎦ 0 0 240 The initial attitude q0 and target attitude qt are the same as described in the previous section. In the simulation, the control period is Ts = 0.1 s, where the controllaw parameters are selected as k1 = 0.01, and k2 = 0.08. We have adopted the mixed sinusoidal-acceleration algorithm for trajectory planning and the parameters for trajectory planning are the same as in the previous section.  The initial estimated values of the moment of inertia parameters are set to ϑ 0 = 250 10 20 280 5 220 , the control parameters kp = 200 and kd = 160, and the positive definite matrix Q = 10−6 I6 . The simulation results are presented in Figs. 6.13 and 6.14. The controller shows good tracking performance for the attitude control system. 

(3)

Characteristic model-based golden-section adaptive control [7, 8]

In addition to uncertain parameters, there exist structural uncertainties such as modeling errors and unmodeled dynamics in an actual control system. Unlike conventional models, the characteristic model [9] does not only rely on the accurate dynamic analysis of objects, and it can be modeled according to the dynamics and control performance of objects. The characteristic model can compress all the relevant information into the model parameters, and it is equivalent to the actual output of the object in a simple form. The golden-section adaptive control method can be used to achieve Fig. 6.13 Satellite attitude (°)

80

Roll 70

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60

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50 40 30 20 10 0 -10

0

5

10

15

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25

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45

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282

6 Spacecraft Attitude Control 0.015

Error in angular velocity (°/s)

Fig. 6.14 Error in attitude angular velocity

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0.01

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the effective control of objects with unknown parameters. It has a simple design, sound robustness, and good adaptability. (1)

Characteristic model based on error quaternion

Let E(qe ) = as

1 2

(q× ev + qe4 I)

−1

. We rewrite the kinematic error equation in Eq. (6.15) q˙ ev = E(qe )−1 ωe 1 q˙ e4 = − qTev ωe 2

  By finding the time derivative of the angular-velocity error ωe = ω − C qe ωr , we have −1

q¨ ev = E˙ E˙qe + E−1 ωe ω˙ e = ω˙ − C ω˙ r + ω× e Cωr where E(qe ) and C(qe ) are abbreviated as E and C, respectively. Without considering external moments, by inserting the time derivative into the dynamics equation, we have ˙ r) + u J ω˙ e = − (ωe + Cωr )× J(ωe + Cωr ) + J(ω× e Cωr − C ω By multiplying both sides of the equation by ET from the left, we obtain M q¨ ev + N q˙ ev + G = ET u

(6.18)

6.2 Attitude Control Based on Angular-Momentum Management Devices

283

where M = ET JE −1 N = ET (JEE˙ + (E˙qev + Cωr )× J)E

G = ET (Cωr )× JCωr + ET JC ω˙ r Let y = qev and the control period be Ts . We discretize the derivatives in the following form: y(k + 1) − 2y(k) + y(k − 1) Ts2 y(k) − y(k − 1) y˙ ≈ Ts y¨ ≈

By substituting the above into Eq. (6.18), we obtain the second-order time-varying characteristic model of the satellite as y(k + 1) = F1 (k)y(k) + F2 (k)y(k − 1) + G0 u(k) + d 0 (k) where F1 (k) = 2I − Ts M −1 N, F2 (k) = −I + Ts M −1 N, G0 (k) = Ts2 M −1 ET , d 0 (k) = Ts2 M −1 G

It can be seen from the above expression that ET is a matrix related to the error quaternion. When the error in attitude tracking is small, it can be approximated to a positive definite diagonal matrix. If the tracking error exceeds a certain threshold, it cannot be guaranteed that ET is still positive definite. To ensure the positive definiteness of the control input coefficient matrix, we define generalized control inputs as follows: u (k) = ET u(k)

(6.19)

Without considering modeling errors, we obtain the second-order time-varying characteristic model under the newly defined control inputs y(k + 1) = F1 (k)y(k) + F2 (k)y(k − 1) + G0 u (k) where G0 (k) = Ts2 M −1 . In general, the inertia matrix is diagonally dominant, and the off-diagonal matrix elements differ greatly from the diagonal elements. When the control period is

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6 Spacecraft Attitude Control

sufficiently small, F1 (k), F2 (k), and G0 (k) can be approximated as diagonal matrices ⎡

⎡ ⎤ ⎤ f1,11 (k) 0 f2,11 (k) 0 0 0 F1 (k) = ⎣ 0 f1,22 (k) 0 ⎦, F2 (k) = ⎣ 0 f2,22 (k) 0 ⎦ 0 0 f1,33 (k) 0 0 f2,33 (k) ⎡ ⎤ 0 g11 (k) 0 G0 (k) = ⎣ 0 g22 (k) 0 ⎦ 0 0 g33 (k) (2)

Identification of characteristic model parameters

The second-order time-varying characteristic model is expressed as the following parameter estimation equation: y(k + 1) = θ T (k)ϕ(k) T T   where ϕ(k) = yT (k) yT (k − 1) uT (k) and θ (k) = F1 (k) F2 (k) G0 (k) . The algorithm for the identification of the characteristic model matrix is T  γ ϕ(k) y(k + 1) − θ T (k)ϕ(k) θ n (k + 1) = θ (k) + α + ϕ T (k)ϕ(k)





where 0 < γ < 1 and α > 0 are constants. Given the form of the characteristic model coefficients, if the control period Ts → 0, each parameter has the following characteristics: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

F1,ii (k) → 2 F1,ji (k) → 0 F2,ii (k) → −1 ⎪ ⎪ ⎪ F2,ji (k) → 0 ⎪ ⎪ ⎩ G  (k) → 0, G  (k) > 0 i i Further, the parameters are within the following reference ranges: ⎧ ⎨ F1,ii (k) ∈ [1.4331, 2) F (k) ∈ (−1, −0.5134) ⎩ 2,ii  Gi (k) ∈ (0, ∞) 

Finally, we obtain the estimation matrices for the characteristic parameters F1 (k),  F2 (k), and G0 (k).





6.2 Attitude Control Based on Angular-Momentum Management Devices

(3)

285

Golden-section adaptive control

Let  x(k) =

   y(k − 1) x1 (k) = x2 (k) y(k)

By inserting the above equation into the expression of the second-order timevarying characteristic model, we can rewrite the characteristic model into the following state-space form: x(k + 1) = A(k)x(k) + B(k)u (k) y(k) = C T x(k) where  A(k) =

     0 I 0 0 , B(k) = , C(k) = F2 (k) F1 (k) G0 (k) I 







We use the estimation matrices F1 (k), F2 (k), and G0 (k) that we obtained to replace the original matrices for characteristic parameters, F1 (k), F2 (k), and G0 (k). Then, the golden-section adaptive control law is 



ul (k) = −(C T B(k))−1 C T A(k)Lx(k)

(6.20)

where  

A(k) =

     0 0 I l2 I , B(k) = , l = 0.382, l = 0.618, L =  1 2 l1 I F2 (k) F1 (k) G0 (k) + λ(k)I 

















In Eq. (6.20), (C T B(k))−1 is (G0 (k) + λ(k)I)−1 . To avoid the singularity of G0 (k) in parameter identification, the anti-singularity factor λ(k) > 0 is introduced together  with G0 (k) to constitute the gain of the control system so as to adjust the transition time in the system. Different λ(k) values can be selected for different channels, and we have ⎡ ⎤ 0 0 gˆ 11 (k) + λ1 (k)  ⎦ G0 (k) + diag(λi (k)) = ⎣ 0 0 gˆ 22 (k) + λ2 (k) 



0

0

gˆ 33 (k) + λ3 (k)

In general, λ(k) or λi (k) is usually a small constant. After parameter identification runs for a period of time, the estimated parameters converge to truth values, following

286

6 Spacecraft Attitude Control

which they gradually decrease or even become null. Following the above logic, λ(k) is selected as a variable parameter related to attitude error: λ(k) = e−k(1−qe4 ) λ(k − 1) where k is a constant coefficient not less than zero. Based on Eqs. (6.19) and (6.20), we have the golden-section adaptive control law 



ul (k) = −(ET )−1 (C T B(k))−1 C T A(k)Lx(k) In order to achieve the stable movement transition of the system, we design the following logical differential controller according to the attitude deviation: ud (k) = −kd (y(k) − y(k − 1))   N     |y(k − N − i) − y(k − N + i − 1)| kd = d i=1

where d  denotes the logical differential control coefficient and the integer N ≥ 1. Based on the above, we obtain the total control quantity of the golden-section adaptive control as u(k) = ul (k) + ud (k) (4)

Simulation examples

The moment of inertia of the spacecraft’s target is (unit: kg m2 ) ⎡

⎤ 280 0 0 J = ⎣ 0 300 0 ⎦ 0 0 240 The initial attitude q0 and target attitude qt are set to  T  T q0 = 0 0 0 1 , qt = 0.2120 0.5540 0.1485 0.7912 In the simulation, the control period is Ts = 0.1 s, where the control-law parameters are selected as k1 = 0.01 and k2 = 0.08. Similar to the simulation in the previous section, we use the mixed sinusoidal-acceleration algorithm for trajectory planning, and the parameters are the same. For input–output decoupling, the initial value of the identification parameter and the values of anti-singularity factors are set to F1,ii (k) = 2, F2,ii (k) = −1, Gi (k) = 0

6.2 Attitude Control Based on Angular-Momentum Management Devices

287

⎧ 4 ⎨ λ1 (k) = e−1×10 (1−qe4 ) λ(k − 1), λ1 (0) = 3 × 10−5 3 λ (k) = e−9×10 (1−qe4 ) λ(k − 1), λ2 (0) = 2.4 × 10−5 ⎩ 2 3 λ3 (k) = e−7×10 (1−qe4 ) λ(k − 1), λ3 (0) = 3 × 10−5 The results of the simulation are presented in Figs. 6.15 and 6.16. It can be seen from the figures that the system with adaptive control still maintains good performance in maneuver tracking with the system structure and parameters of the controlled object unknown. 4.

Attitude maneuver control of flexible spacecraft [10]

Compared with conventional satellites, agile (Earth-observation) satellites have many advantages. They can achieve flexible target detection by quickly changing the attitude according to mission requirements. Especially in remote sensing, fast attitude maneuvering can greatly improve the flexibility of applications and observation efficiency of a satellite so as to achieve the fast and efficient collection of remote-sensing data of targets that are not at the nadir point. Given the application advantages of Fig. 6.15 Satellite attitude

Euler angle (°)

80

Roll Pitch Yaw

60 40 20 0 -20

0

10

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30

50

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3

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2.5 2 1.5 1

X: 39.9 Y: 9.306e-07

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10

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288

6 Spacecraft Attitude Control

small agile satellites in the field of remote sensing, rapid attitude maneuverability has begun to be realized in high-performance large flexible spacecraft. However, the large flexible solar arrays installed on large flexible spacecraft make the spacecraft’s dynamics characteristics complicated, posing challenges to high-performance attitude maneuver control. (1)

Control method based on frequency analysis

To address the issues in-arbita attitude maneuver control of flexible satellites with time delays in the system, we have designed an attitude maneuver tracking controller which combines PD control and compensation control, and the parameters of the controller are delibrately selected based on the classical frequency analysis method so as to guarantee the system stability. (1)

Design of attitude controller

The desired attitude quaternion and angular velocity for a sufficiently smooth trajectory of attitude maneuver tracking are qr and ωr , respectively. Based on the satellite’s current attitude q and angular velocity ω, we can obtain the error quaternion qe and the angular-velocity error ωe : qe = q−1 r ⊗q

(6.21)

  ω e = ω − C qe ω r

(6.22)

   2  T T × where   C qe = qe4 − qev qev I + 2qev qev − 2qe4 qev and I is a 3 × 3 identity matrix. C qe is abbreviated as C. Given the above, we obtain the kinematics and dynamics equation based on the error quaternion as 1 1 q˙ ev = − ωe × qev + qe4 ωe 2 2 1 q˙ e4 = − ωTe qev 2

(6.23)

and J ω˙ e +Fη¨ = T f η¨ + 2ξ Ω η˙ + Ω 2 η + FT ω˙ = 0 where ˙r T f = T c − (ωe + Cωr )× J(ωe + Cωr ) + Jω× e Cωr − JC ω

(6.24)

6.2 Attitude Control Based on Angular-Momentum Management Devices

289

In general, the flexible modes of a satellite η cannot be measured. If the dynamics parameters are known, the control method is similar to PD control with a stable attitude. Based on information about the attitude and angular velocity, we design the following attitude-tracking control law: T f = − 2K p qev sgn(qe4 ) − K d ωe

(6.25)

In the above equation, the control parameters K p and K d are proportional and derivative coefficient matrices, respectively. They are usually diagonal matrices, and sgn(·) is a sign function ⎧ ⎨ −1, x < 0 sgn(x) = 0, x = 0 ⎩ +1, x > 0 We defined the quasi-static deformation of structures in the flexible mode during the attitude motion of the satellite as ηr , which meets the following differential equations:   η¨ r + 2ξ Ω η˙ r + Ω 2 ηr = −FT C qe ω˙ r

(6.26)

Let ηe = η − ηr . Then, Eq. (6.24) can then be expressed as J ω˙ e +Fη¨ e = τ f + Fη¨ r η¨ e + 2ξ Ω η˙ e + Ω 2 ηe + FT ω˙ e = 0

(6.27)

Let θ = 2qev sgn(qe4 ). If the error in attitude tracking is small, qev ≈ 0 and qe4 ≈ 1. Then, Eq. (6.23) can be approximated as θ˙ ≈ ωe

(6.28)

and η¨ r + 2ξ Ω η˙ r + Ω 2 ηr ≈ −FT ω˙ r Let us posit that J is diagonally dominant so that the coupling effects between axes can be ignored. Based on the analysis and design of the attitude stabilization control of flexible satellites, we can analyze and design a control method for each channel of the tracking system, which is composed by Eqs. (6.27) and (6.28), using classical frequency-domain analysis. Fη¨ r is regarded as the external input disturbance on the system, and the maximum time delay of the system is Td . By taking the Laplace transform of Eqs. (6.27) and (6.28), we obtain

290

6 Spacecraft Attitude Control

⎛ ⎝Ji +



2 2 −Fi,j s

j

s2 + 2ξ j s + 2j

⎞ ⎠s2 θi (s) = e−Td s Tf,i (s)

(6.29)

In the above equation, the subscript i (i = 1, 2, 3) represents the ith channel; the subscript j (j = 1, . . . , m) represents the jth flexible mode; e−Td s is the transfer function of the pure time delay of the system; Ji is the principal inertia of the channel i; and the other variables are the corresponding components of the above-mentioned vectors or matrices. By taking the Laplace transform of Eq. (6.25), we obtain the transfer function of each channel in the control system as   Tf,i (s) = − Kp,i + Kd,i s θi (s)

(6.30)

where Kp,i and Kd,i are the components of K p and K d corresponding to channel i. According to the form of control shown in Eq. (6.30), we can select suitable control parameters K p and K d based on the selection of control parameters and the stability analysis of Eq. (6.29) by using classical frequency-domain analysis. The time delay e−Td s does not affect the characteristics of the system’s amplitude, given its frequency characters. However, it will introduce a phase lag, which has the following relationship with the frequency f : φ = 2π f · Td It can be seen from the above expression that the phase loss caused by time delays at different flexible modal frequencies has a linear relationship with the flexible modal frequency. Owing to the low damping of flexible modes, it is difficult for the flexible vibration induced by rapid maneuvering to be reduced by its own damping in a short time. Therefore, low-order flexible modes are designed to be phase-stabilized to provide active damping to the vibration through attitude control. Higher-order modes are designed as gain-stabilized, considering the system’s time delays and uncertainties in dynamics modeling. The system’s vibration can be reduced by the damping of its own structure. With the control law in Eq. (6.25) alone, if suitable control parameters cannot be selected for both broadband control and the stabilization of flexible modes, we can further introduce different forms of structural filters as shown by Wie and Byun [11] to reduce the gains in high-order unstable modes and appropriately increase gains in low-order modes. Based on the determined parameters K p and K d , we can calculate T f using Eq. (6.25). Taking into account the angular momentum of the angular-momentum management device, the system’s attitude control torque command is expressed as ˙ r + ω× h T c = −2K p qev − K d ωe + (ωe + Cωr )× J(ωe + Cωr ) − Jω× e Cωr + JC ω (6.31)

6.2 Attitude Control Based on Angular-Momentum Management Devices Fig. 6.17 Nichols chart of the open-loop transfer function for a low-frequency mode (phase-stabilized)

291

60

Amplitude (dB)

40

20

0

-20

-40

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-250

-200

-150

-100

-50

0

50

100

Phase (°)

where h is the resultant angular momentum of the angular-momentum management device. (2)

Verification results

The nominal frequencies of the first two orders of flexible modes of two solarpanel wings of a satellite are set to 0.25 Hz and 0.6 Hz, respectively. In the design of the controller, the constant for the system’s time delay is 0.25 s. We designed an attitude controller using Eq. (6.31) based on the given form of attitude control and the method for the selection of control-law parameters. Specifically, the phasestabilization method is adopted for the first two orders of flexible modes, and the gain-stabilization method is used for higher-order modes. Figure 6.17 shows the Nichols chart for the open-loop properties of the roll channel of the system with the designed controller when the rotation angle of the solar array is 45° and the gain of the first-order flexible mode is approximately 6 dB. The open-loop frequency characteristics of the pitch and yaw channels are similar to those of the roll channel; hence, they are not included here. Using the controller presented above, the satellite’s attitude angular velocity and angular-velocity error for maneuvering is calculated, as shown in Fig. 6.18. As can be seen from the results, the vibration of flexible modes during maneuvering has been effectively attenuated, and the angular velocity quickly converged to nearly zero after the desired attitude had been reached. With the same PD control parameters, the gains of the first two orders of flexible modes are both below 0 dB after introducing a trap filter, implying that all the flexible modes are gain-stabilized. The corresponding Nichols chart for the openloop properties of the roll channel of the system is shown in Fig. 6.19. The satellite’s attitude angular velocity and angular-velocity error for the roll maneuver is shown in Fig. 6.20. It can be seen from the figure that the flexible modes showed marked

6 Spacecraft Attitude Control

Angular velocity / ((° )/s)

292 1

Roll Pitch Yaw

0.5 0 -0.5 -1

50

350

300

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200

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100

t /s

Error / ((° )/s)

0.015 Roll Pitch Yaw

0.01 0.005 0 -0.005 -0.01 -0.015

50

350

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t /s Fig. 6.18 Attitude angular velocity and angular-velocity error

Fig. 6.19 Nichols chart of the open-loop transfer function for a low-frequency mode (gain-stabilized)

60

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vibration during maneuvering, which affected the stability of the attitude after the desired attitude is reached. (2)

Adaptive control based on characteristic modeling [12]

(1)

Establishment of the control model

Angular velocity / ((° )/s)

6.2 Attitude Control Based on Angular-Momentum Management Devices

293

1 Roll Pitch Yaw

0.5 0 -0.5 -1

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100

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t /s Fig. 6.20 Satellite’s attitude angular velocity and angular-velocity error

In the dynamics equation for a flexible spacecraft, if we regard the term Fs η¨ as disturbance, the system equation can be expressed as J ω˙ + ω× Jω = u˜

(6.32)

¨ where u˜ = u − Fs η. To facilitate the controller design, we can express the dynamics equation of a satellite without disturbance in the following form: J ω˙ + ω× Jω = u

(6.33)

As large-angle attitude maneuver may occur between the roll axis and pitch axis when the satellite is in orbit, we adopt the “1-3-2” sequence of rotation to establish the kinematics equation. The Euler angles corresponding to the three axes of the  T satellite are defined as X = ϕ θ ψ , where ϕ, θ , and ψ are the roll, pitch, and yaw angles of the spacecraft, respectively. Then, we arrive at the following kinematics equation: ω = SX˙ + C 132 ωo

(6.34)

294

6 Spacecraft Attitude Control

  where ωo = 0 −ωo 0 is the angular velocity in the orbital frame with respect to the inertial frame, which is assumed to be a constant, and C 132 is the direction cosine matrix for the attitude. ⎡ ⎤ cos θ sec ψ 0 sin θ sec ψ S−1 = ⎣ cos θ tan ψ 1 sin θ tan ψ ⎦ − sin θ 0 cos θ S can be approximated as an identity matrix when the angles are small. By taking the time derivative of Eq. (6.34), we obtain ω˙ = SX¨ + S˙ X˙ + C˙ 132 ωo We insert the above equation into Eq. (6.32) and multiplying both sides of the equation by ST from the left and use the following relations: M = ST JS  × N = ST J S˙ − ST JSX˙ S + ST (C 132 ωo )× JS − ST (JC 132 ωo )× S G = ST J C˙ 132 ωo − ST (JC 132 ωo )× C 132 ωo Consequently, we obtain M X¨ + N X˙ + G = ST u

(6.35)

where M is a positive definite matrix. According to the theory of characteristic modeling, we can establish a discrete characteristic model corresponding to that in Eq. (6.35) with the system’s sampling time Ts : X(k + 1) = f 1 (k)X(k) + f 2 (k)X(k − 1) + g0 u(k) + d 0

(6.36)

where f 1 (k), f 2 (k), and g0 (k) ∈ R3×3 are coefficient matrices. We discretize Eq. (6.36) in the following manner to approximate the characteristics of each parameter: X(k) − X(k − 1) X(k + 1) − 2X(k) + X(k − 1) ˙ ,X = X¨ = Ts2 Ts Hence,

6.2 Attitude Control Based on Angular-Momentum Management Devices

295

f 1 (k) = 2I − Ts M −1 N, f 2 (k) = −I + Ts M −1 N d 0 (k) = Ts2 M −1 G, g0 (k) = Ts2 M −1 ST where I is the identity matrix of the corresponding dimension. The orbital angular velocity ωo and sampling time Ts2 are both small quantities, and the satellite’s inertia matrix is generally diagonally dominant. Therefore, each coefficient matrix can be further expressed by a diagonal array to reduce the complexity of the controller. (2)

Identification of model parameters

In this section, we introduce an adaptive controller corresponding to the characteristic model. First, we use parameter identification to identify the parameters of the characteristic model. Equation (6.36) can be expressed as X(k + 1) = Θ T (k)Φ(k)

(6.37)

T  T  where Θ(k) = f 1 (k), f 2 (k), g0 (k) and Φ(k) = X T (k), X T (k − 1), uT (k) . Hereafter, the subscripts i = 1, 2, 3 denote the components of the three-axis attitude of the satellite: φ, θ, and ψ, respectively. Equation (6.37) in component form is X i (k + 1)) = Φ T (k)θ i (k), i = ϕ, θ, ψ.

(6.38)

We use a multivariate gradient algorithm to estimate the parameters in Eq. (6.38). Compared with the multivariate least-squares regression method, this algorithm can reduce the amount of calculation by at least half. The specific algorithm is 



θ (k + 1) = θ (k) +



  aΦ(k) T T X (k + 1) − Φ (k)θ (k) c + Φ T (k)Φ(k) 

where θ (0) can be assigned with any value, c > 0, and 0 < a < 2. In a satellite control system, noise due to measurement and actuators is inevitable. To avoid the influence of system noise on parameter identification, we adopt an identification method with a dead zone and adjust a according to the estimated system deviation, i.e., ! ! ⎧ ! T !  T ⎪ ! ! ⎪ 0, ⎨ !X (k + 1) − φ (k)θ (k)! ≤ 1 ! ! a= ! T !  ⎪ ⎪ T ! ⎩ C1 , ! !X (k + 1) − φ (k)θ (k) ! > 1

296

6 Spacecraft Attitude Control

where the constant C1 satisfies the condition 0 < C1 < 2 and the constant 1 is a dead-zone parameter. The specific values of the parameters must be determined according to the noise level of the system. In addition, according to the estimated range of parameters, we can project identified parameters according to the range estimated in the analysis of the previous model, i.e.,   θ (k + 1) = Pro θ (k + 1)





where Pro(•) is the specific projection algorithm. (3)

Design of attitude controller

Assuming that gˆ 0 (k) is non-singular, the control law of the multivariate goldensection adaptive controller based on the characteristic model is   −1  L1 fˆ 1 (k)e(k) + L2 fˆ 2 (k)e(k − 1) + uA (k) uG (k) = gˆ 0 (k) + β(k)I

(6.39)

where e(k) = X d (k) − X(k) is the output error, X d (k) is the desired attitude, L1 = 0.382, and L2 = 0.618. For adaptive control, X d (k) is zero; the control item to be designed, which is usually uA (k) = input  uA (k) is the compensation  ˆ ˆ ˆ L2 I − f 1 (k) − f 2 (k) e(k); and f 1 (k), fˆ 2 (k), and gˆ 0 (k) are the estimated values of the corresponding coefficient matrices in Eq. (6.38). For tracking, we design the following multivariate tracking and maintenance control law based on an all-coefficient adaptive control method:   −1  X d (k + 1) − fˆ 1 (k)X d (k) − fˆ 2 (k)X d (k − 1) uT (k) = gˆ 0 (k) + β(k)I In order to effectively eliminate system deviation and improve the dynamic performance of the system, the following logic integral controller is adopted for each axis: "  uI,i (k) = uI,i (k − 1) + K I,i ei (k) uI,i (k) = C 3,i uI,i (k) where the three axes of the spacecraft are denoted by the subscript i. The constant C 3,i > 0 can vary with the output error and rate of change of the error

K I,i

⎧ ⎨ kmax , ei (k)(ei (k) − ei (k − 1)) > 0, √ = |ei (k)| > 2 , |ei (k) − ei (k − 1)| > 22 ⎩ kmin , else

6.2 Attitude Control Based on Angular-Momentum Management Devices

297

where kmax >> kmin > 0 and 2 is a constant related to the level of measurement noise. The dead zone is set to avoid frequent false switching of the integral constant caused by noise in an actual control system, which can affect the steady-state performance of the system. When the attitude deviation is large, the damping of the system should be minimized to adapt to rapid adjustments or rapid maneuvers. On the other hand, the system’s damping should be increased by as much as possible to achieve high stability during stabilization and control. Considering the above, we can use the following logic differential controller, which is based on the deviation of attitude angle ud (k): Kd0 e˙ i (k) ud,i (k) = # # λ1 # $ # k # # ei (s)# + C2 # #s=k−m # ˙ Kd0 > 0 and where the deviation of attitude angular velocity e˙ (k) = X˙ d (k) − X(k), C2 > 0 are constants, 0 ≤ m ≤ k, and λ1 ≥ 0. According to the above expression of the logic differential controller, the differential coefficient increases as the system’s deviation angle decreases. In summary, the overall system control is based on the tracking and maintenance control, golden-section adaptive control law, logic integral control law, and logic differential control law. It is expressed as u = uT + uG + uI + ud (4)

Simulation verification

The components of the three-axis moment of inertia of the satellite are set to 11,000, 8700, and 6100 kg m2 . Two large flexible solar-panel arrays are installed on the satellite along the direction of the pitch axis. The natural frequency of the solar-panel array is 0.32 Hz. Sixth-order flexible modes are used in the model. To adapt to the requirements for rapid attitude maneuvering of the satellite, a CMG of 12 Nm/100 Nm s is used as the actuator. The sampling time of the simulation is 0.2 s, the accuracy of attitude angle −4 ◦ , the measured angular-velocity random walk coefmeasurement is 2.78 √ × 10 ◦ ficient is 0.001 / h,√and the mean-square deviation in the torque noise of each CMG is 0.00067 Nm s. The controller parameters are set as follows: Kd0 = 100, C2 = 0.005, λ1 = 0.5, kmax = 2, and kmin = 0.005. At the 300th second of the simulation, a maneuver started T  under steady-state control. The target attitude angle is 30◦ 0 0 . During the control process, trajectory planning is first performed for the target angle. The maximum angular velocity of the satellite’s roll axis is limited to 0.5 ◦ /s. The simulation results are presented in Figs. 6.21 and 6.22.

298

Fig. 6.21 Three-axis attitude angles

Fig. 6.22 Angular velocities of three-axis attitude

6 Spacecraft Attitude Control

6.3 Steering Strategies for Angular-Momentum Management Devices

299

6.3 Steering Strategies for Angular-Momentum Management Devices In actual spacecraft control systems, the most common angular-momentum management devices are flywheels and CMGs. Compared to attitude jets, these devices have high accuracy in output torques and do not consume any working medium; therefore, they can be used for high-precision and high-stability spacecraft. Among angularmomentum management devices, flywheels have small output torques and are easy to operate because the direction of the output torque is constant; therefore, they have been widely used in spacecraft that do not require high attitude maneuverability. A CMG can quickly change the direction of angular momentum through the rotation of the low-speed gimbal; therefore, it can produce a large output torque, which is suitable for spacecraft that require high attitude maneuverability. Despite that, the CMG outputs torque through the rotation of the gimbal, and the direction of the torque is related to the current gimbal angle, making system control complicated in practice.

6.3.1 Flywheel Control Strategies A satellite’s attitude can be controlled by adjusting the speed of rotation of the flywheel to allow the exchange of angular momentum between the flywheel and satellite body. Avoiding the angular-momentum saturation of the flywheel before reaching the momentum envelope has always been a special concern in the design of the control system for spacecraft with disturbances [13–15]. For a spacecraft with a given redundant flywheel configuration, this section proposes a torque distribution method [15] that applies pseudo-inverse and null motion based on the deliberately selected target momentum on the basis of the visualization analysis of the momentum envelope. The proposed method enables the system to make full use of the actual maximum momentum envelope and is suitable for real time and effective momentum management of spacecraft in orbit. 1. (1)

Angular-momentum envelope of flywheel systems and visualization analysis Angular-momentum envelope of flywheel systems

For a system composed of non-parallel N (N ≥ 3) flywheels, the resultant angular momentum can be regarded as a polyhedron mapped from an N-dimensional hypercube to three-dimensional space. The polyhedron is composed of vertices, edges, and faces. Specifically, the vertices are generated by the angular momentum of all flywheels in the saturated condition, the sides are the angular-momentum synthesis of all the flywheels except one flywheel in the saturated condition, the faces are the angular-momentum synthesis of all flywheels except two flywheels in the saturated condition, and the number of faces is N (N − 1).

300

6 Spacecraft Attitude Control

We define the flywheel i (i = 1.2, . . . , N ) as mounted in the direction hw,i , which is a unit vector. Its corresponding angular momentum Hw,i should meet the following constraints: Hw min,i ≤ Hw,i ≤ Hw max,i

(6.40)

where Hw min,i and Hw max,i are the minimum and maximum values of the flywheel’s angular momentum. In general, Hw max,i = −Hw min,i > 0. Let the angular momentum vector be T  H w = Hw1 Hw2 . . . Hw,N We can express the resultant angular momentum of the flywheel system as   H  = hw1 hw2 . . . hw,N H w = C w H w

(6.41)

  where C w = hw1 hw2 . . . hw,N is the mounting matrix of the flywheel system. The resultant angular momentum of the flywheel system can be regarded as the mapping from an N-dimensional hypercube to three-dimensional space. For any unit vector in the satellite frame, we have ⎡

⎤ cos α cos β S(α, β) = ⎣ sin α cos β ⎦ sin β

(6.42)

    where the parameters α ∈ 0 360◦ and β ∈ −90◦ 90◦ . By performing the matrix transformation ⎡ ⎤ cos β cos α cos β sin α sin β C = ⎣ − sin α cos α 0 ⎦ − sin β cos α − sin β sin α cos β we can convert S into  T S = CS = 1 0 0



Let H denote the projection of the resultant angular momentum H  of the flywheel system in the direction of S. Thus, we have C w H w = H S

(6.43)

By left-multiplying both sides of the above equation by the matrix C, we obtain

6.3 Steering Strategies for Angular-Momentum Management Devices



C hw1 hw2

⎡ ⎤ H  . . . hw,N H w = ⎣ 0 ⎦ 0

301

(6.44)

Let hˆ w,i = Chw,i . Equation (6.43) can be expressed as ⎡

hˆ w1 (1) ⎣ hˆ w1 (2) hˆ w1 (3)

⎤ ⎡ ⎤ hˆ w2 (1) . . . hˆ w,N (1) H hˆ w2 (2) . . . hˆ w,N (2) ⎦H w = ⎣ 0 ⎦ 0 hˆ w2 (3) . . . hˆ w,N (3)

(6.45)

where hˆ w,i (j) is the jth element of hˆ w,i . Equation (6.45) can be decomposed into the following two parts:  hˆ w1 (1) hˆ w2 (1) . . . hˆ w,N (1) H w = H

(6.46a)

   hˆ w1 (2) hˆ w2 (2) . . . hˆ w,N (2) 0 H w == 0 hˆ w1 (3) hˆ w2 (3) . . . hˆ w,N (3)

(6.46b)



and 

The problem of finding the maximum resultant angular momentum of the flywheel system in the direction of S is a linear programming problem under the constraints of Eqs. (6.40) and (6.46). max H (H w ) = max Hw

Hw

%

 & hˆ w1 (1) hˆ w2 (1) . . . hˆ w,N (1) H w

(6.47)

The constrained linear programming problem in Eq. (6.47) can be solved by converting it into a standard linear programming problem or by using the function LINPROG in MATLAB [16]. To construct the visual angular-momentum envelope of the flywheel system, we can let S be in the traverse direction by taking values in the domain of parameters α and β to find the maximum amplitude H∗ (α, β) of the resultant angular momentum of the flywheel system along this direction. (2)

Gear trains in typical configurations and visualization analysis of angularmomentum envelope

A zero-momentum system requires no less than three flywheels to realize the threeaxis attitude control of the satellite. The output torque performance of the torque flywheel at a low rotor speed is greatly affected by friction characteristics; therefore, at least four flywheels are often used to form a zero-momentum system with the angular-momentum offset of each flywheel. In addition, more flywheels may be mounted to meet the requirements for the satellite’s attitude maneuverability or system backups. This section presents the results of an angular-momentum

302

6 Spacecraft Attitude Control

envelope analysis of flywheel systems in four configurations: a three orthogonalflywheel configuration, 3 + 1S (three orthogonal and one-skewed) configuration, four-skewed-flywheel configuration, and five-flywheel-pyramid configuration. (1)

Three orthogonal flywheel system

The three orthogonal flywheel system consists of three orthogonal flywheels and is mounted parallel to the satellite’s main inertia axis. The three flywheels independently control the satellite’s attitude along three axes. Each flywheel is mounted under the satellite in the following direction:  T  T  T hw1 = 1 0 0 , hw2 = 0 1 0 , hw3 = 0 0 1 The flywheel system’s angular-momentum envelope is a cube with edges of unit length, as shown in Fig. 6.23. (2)

Flywheel system in a “3 + 1S” configuration

To the three orthogonal configuration, the flywheel system in the 3 + 1S configuration adds a fourth flywheel mounted along the rhumb line relative to the main inertia axis of the satellite, as shown in Fig. 6.24. During the attitude control of the satellite, the skewed flywheel is used to maintain angular momentum, while the other three flywheels form a zero-momentum system. Each flywheel of the “3 + 1S” flywheel system is mounted in the following direction:

Fig. 6.23 Angular-momentum envelope of the three orthogonal flywheel system

6.3 Steering Strategies for Angular-Momentum Management Devices

303

Fig. 6.24 “3 + 1S” flywheel system and its angular-momentum envelope

 T  T  T hw1 = 1 0 0 , hw2 = 0 1 0 , hw3 = 0 0 1

'√ '√ '√ T hw4 = 1 31 31 3 The flywheel system’s angular-momentum envelope is a dodecahedron, which has a larger angular-momentum space than the three orthogonal flywheel system. Its maximum amplitude is 1.5. (3)

Four-skewed flywheel system

The flywheel system in a four-skewed-flywheel configuration has four skewed flywheels, which are distributed at equal intervals on the cone surface of an inertia axis of the satellite. This type of configuration is also called the pyramid configuration. Figure 6.25 shows the configuration of the flywheel system when the principal axes of inertia of the flywheel system is the Z-axis of the satellite. In the figure, the

Fig. 6.25 Configuration of the four-skewed flywheel system

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6 Spacecraft Attitude Control

Fig. 6.26 Angular-momentum envelope of the four-skewed flywheel system

half-apex angle of the cone is γ , and the angle between the projection of the flywheel hw1 on the xoy plane and the x-axis of the satellite is η. Figure 6.26 shows the flywheel system’s angular-momentum envelope when γ = 57.47◦ and η = 0, which is a dodecahedron. Compared with the “3 + 1S” flywheel system, which also has four flywheels, the four-skewed flywheel system has a larger angular-momentum space, and its maximum value is 2.151. In summary, the fourskewed flywheel system has a large angular-momentum capacity and a strong ability to absorb external disturbances and accumulate angular momentum, which can be particularly useful for satellites that require a certain level of attitude maneuvering. (4)

Flywheel system in the five-flywheel-pyramid configuration

The flywheel system in the five-flywheel-pyramid configuration is similar to the four-skewed flywheel system. It consists of five flywheels that are distributed at equal intervals on the cone surface of an inertia axis of the satellite. Figure 6.27 shows the configuration of the flywheel system when the principal axis of inertia for the flywheel system is the Z-axis of the satellite. In the figure, the half-apex angle of the cone is γ , and the angle between the projection of the flywheel hw1 on the xoy plane and the x-axis of the satellite is η. Figure 6.28 shows the flywheel system’s angular-momentum envelope when γ = 57.47◦ and η = 0, which is an icosahedron. Compared with the four-flywheel system, the five-flywheel-pyramid system has a larger angular-momentum space, and its maximum value is 2.6887, which is close to that of a spatial sphere. 2.

Torque distribution method based on the optimization of the flywheel system’s power consumption indexes

The control torque command T c obtained by the attitude controller needs to be distributed to each flywheel based on the geometric relationship between flywheels.

6.3 Steering Strategies for Angular-Momentum Management Devices

305

Fig. 6.27 Five-flywheel-pyramid configuration

Fig. 6.28 Angular-momentum envelope of the five-flywheel-pyramid system

When the dimension of the controlled quantity is greater than the control dimension, the distribution method is not unique. Assuming that the angular momentum to be provided by the flywheel system corresponding to the control torque command T c is H  , we need to select a specific objective function to find the optimal solution to the extreme value problem. We select the following objective function for flywheel power consumption: J = H Tw W H w where W = W T > 0 is a weighted coefficient matrix. The total power consumption of the flywheel is the smallest when W = I N . Considering the constraints between the resultant angular momentum and the angular momentum of the flywheel system, we introduce the Lagrangian operator λ to

306

6 Spacecraft Attitude Control

convert the above-mentioned constrained extreme value problem into an optimization problem of the following objective function: J = H Tw W H w + λT (H  − C w H w ) By taking the partial derivation of the objective function J with respect to H w and λ, we obtain the optimal distribution law as  −1 H w = W −1 C Tw C w W −1 C Tw H 

(6.48)

Let the torque distribution matrix for the flywheel system be Dw =  −1 W −1 C Tw C w W −1 C Tw . For the command control torque T c , the torque distribution law for the flywheels is ˙ w = −Dw T c H

(6.49)

When W = IN (N × N unit matrix), the torque distribution matrix is  −1 Dw = C Tw C w C Tw As the above matrix is the pseudo-inverse of the flywheel system’s mounting matrix C w , the corresponding torque distribution method is called the pseudo-inverse distribution method. The torque flywheel is generally set to run at an offset speed, and its corresponding nominal angular-momentum vector is H w0 . When there are differences in the friction as well as the dynamics and control of the rotational speed among different flywheels, although the resultant angular momentum of the flywheel system has not changed with the torque distribution law in Eq. (6.48), there may be cases where the flywheel speed deviates from the nominal value. To address this issue, we can use the null motion of the flywheel to keep the rotational speed of the flywheel close to the nominal value. The momentum distribution law of the flywheel system with null motion is ˙ w = −Dw T c + kN (IN − Dw C w )(H w − H 0 ) H

(6.50)

where the adjustment coefficient kN ≥ 0 and IN is an N × N identity matrix. With a specific angular-momentum distribution method, when the flywheel’s angular momentum is saturated, the space reached by the flywheel system’s angular momentum is a subset of the actual angular-momentum space. To distinguish the two, we call the angular-momentum envelope with a specific distribution method “the system’s angular-momentum envelope with the specific distribution.” Assuming the upper and lower limits of the angular momentum of each flywheel are the same and satisfy

6.3 Steering Strategies for Angular-Momentum Management Devices

307

Hw max,i = −Hw min,i = Hw lim > 0 With the distribution matrix Dw , let the maximum angular momentum of the flywheel system along the direction of a given unit vector S(α, β) be Hs max : Hs max =

Hw lim Dw · S∞

(6.51)

where ·∞ is the L-∞ norm of the vector, which is the maximum value of the absolute value of the elements in the vector. Similar to the previous section, we can obtain the angular-momentum envelope of the flywheel system with the distribution matrix Dw by using Eq. (6.49). Figure 6.29 shows the angular-momentum envelope of the aforementioned flywheel systems in the three orthogonal configuration and four-skewed configuration under a pseudo-inverse distribution. Compared with the corresponding angular-momentum envelopes given in the previous section, it can be seen that the angular-momentum space reached under the pseudo-inverse distribution is a subset of the actual angularmomentum space. This means that a part of the angular-momentum space may not be reached under a specific momentum distribution. 3.

Torque distribution and angular-momentum management

In Eq. (6.40), the constraint condition for each flywheel’s angular momentum is a compact set, and Hw,i is a convex set. The plane polygon enveloped by the angular momentum is a convex polygon. For any two flywheels k and l, when the angular momentum of the other N–2 flywheels is saturated, the variation range of the resultant angular momentum of the corresponding flywheel system is a parallelogram region

(a) “3 + 1S” configuration

(b) Four-skewed configuration

Fig. 6.29 Angular-momentum envelope of the flywheel system under a pseudo-inverse distribution

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6 Spacecraft Attitude Control

when Hw,k and Hw,l change within their range, and the normal to this parallelogram is ! ! n = hw,k × hw,l !hw,k × hw,l ! Further, its two sides are parallel to hw,k and hw,l . In this chapter, · is the 2-norm, which denotes the length of the vector. When the projected angular momentum of the N–2 saturated flywheels reaches the maximum (or minimum) on the normal n, the corresponding parallelogram is on the angular-momentum envelope of the flywheel system, i.e.,  : A1 A2 A3 A4 . As shown in Fig. 6.30, the normal to the plane is n, which is taken as the outwardpointing normal. According  to the right-hand rule, the four vertices are A1 to A4 , and the coordinates of Ai are xi yi zi . For a flywheel system consisting of N flywheels, we can obtain the number of parallelograms on the envelope with any combination of two flywheels as N (N − 1). Any ith vertex Ai of the polygon  and its two adjacent vertices can form two −−−→ −−−→ −−→ −−→ vectors Ai Ai+1 and Ai Ai−1 (A4 A1 and A4 A3 if i = 4). From any two adjacent vectors, we can obtain the outward-pointing normal n of the plane  as ⎤ nx −−−→ −−−→ n = ⎣ ny ⎦ = Ai Ai+1 × Ai Ai−1 nz ⎡

(6.52)

For any vertex Ai , we can obtain the equation for the plane in which  is located nx (x − xi ) + ny (y − yi ) + nx (z − zi ) = 0

n

Fig. 6.30 Schematic of the parallelogram on the angular-momentum envelope

(6.53)

A1

S B A2 A3

O

A4

6.3 Steering Strategies for Angular-Momentum Management Devices



309

The equation of the straight line passing the origin along the vector S = T Sx Sy Sz is x = Sx t, y = Sy t, z = Sz t

(6.54)

where t is a parameter. We can obtain the equation for the parameter t by substituting each frame component of the equation of the straight line into Eq. (6.53):   nx (Sx t − xi ) + ny Sy t − yi + nz (Sz t − zi ) = 0

(6.55)

When nT S = 0, the solution is 

nT OA t=

1

(6.56)

nT S

If t > 0, there exists a point of intersection between the straight line from the origin along the unit vector S and the plane where the quadrilateral  is located, and the − → vector OB =  t · s. When a point of intersection exists between the line along the unit vector S and the plane where the quadrilateral  is located, the angular momentum of the corresponding two flywheels at the intersection point B is unsaturated. Let us ∗ ∗ define them as k and l, and let Hw,k and Hw,l denote their limiting angular momenta, ∗ respectively. The limiting angular momentum of the other saturated flywheels is Hw,i %  ∗ & Hw,i ∈ Hmin,i Hmax,i . Then, we can obtain the corresponding limiting angular momentum of the unsaturated flywheels as 

∗ Hw,k ∗ Hw,l





= hw,k hw,l

+

⎛ ⎝tB S −



⎞ ∗ Hw,i hw,i ⎠

(6.57)

i=k,l

where [·]+ is an operator that returns the Moore–Penrose (M–P) pseudo-inverse of the matrix. The angular-momentum envelope of the N-flywheel system consists of p = N (N − 1) parallelogram regions υ (υ = 1, . . . , p) that are surrounded by four vertices of the envelope polyhedron. When the saturation state of the angular momentum of the flywheel in each parallelogram region υ is known, we can then calculate the limiting angular momentum of the flywheel at each point of the envelope surface. Assume that the current flywheel’s angular-momentum vector is H w and that the determined flywheel’s nominal angular momentum vector is H w0 . Without loss of generality, let us posit that C w H w0 = 0, implying that the system is a zero-momentum system. The angular momentum of the flywheel system is

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6 Spacecraft Attitude Control

H = CwHw When external disturbance exists, a specific torque distribution of the flywheel system will be used to achieve attitude control. Without angular-momentum unloading, the external disturbance causes the amplitude of the resultant angular momentum of the flywheel system to increase gradually until it approaches the corresponding angular-momentum envelope. When H   = 0, we determine the unit vector in real time according to the resultant angular momentum of the flywheel system  S = H  H   After obtaining the limiting angular momentum H ∗w of the flywheel at the intersection of S with the angular-momentum envelope, we can obtain the limit of the resultant angular momentum of the flywheel system H ∗ . The deviation in the current angular momentum H w of the flywheel from the limiting angular momentum in the direction of S is H w = H ∗w − H w For the desired control torque T c , we can use the following flywheel torque distribution law        ˙ c = −C Tw C w C Tw −1 T c + kNull I − C Tw C w C Tw −1 C w H w H w  (6.58) H ˙ c is the flywheel’s command torque vector, I is the identity matrix of the where H corresponding dimension, and the coefficient kNull ≥ 0. In order to avoid division by zero when the angular momentum H w is close to the envelope surface, when H w  is less than a certain predetermined amount and  > 0, we only use the following pseudo-inverse distribution law:   ˙ c = −C Tw C w C Tw −1 T c H In the flywheel’s torque distribution law given by Eq. (6.58), the first part is the pseudo-inverse distribution law, which generates the desired control torque T c , and the second part is the null-motion distribution law, which uses the flywheel’s limiting angular momentum H ∗w (i.e., the limit H ∗ = H∗ S of the resultant angular momentum of the corresponding flywheel system along the S direction). As the angular momentum accumulates, the flywheel’s angular momentum will be adjusted in real time without the output of the resultant disturbance torques until the angular momentum of the flywheel system reaches the angular-momentum envelope. 4.

Simulation verification

6.3 Steering Strategies for Angular-Momentum Management Devices

311

We use an Earth-observation satellite orbiting at an altitude of 36,000 km as an example. The moment of inertia of the satellite body is  % & J = diag 3200 1820 2400 . kg · m2 The system is equipped with four flywheels with a torque of 0.1 Nm and an angular momentum of 10 Nm s in the pyramid configuration, as shown in Fig. 6.37. Zero-momentum control is adopted for the whole satellite, and the nominal offset angular momentum of the flywheel is  T H w0 = −5 5 −5 5 (Nms) With a known attitude error quaternion q(t) from the satellite frame to the reference frame and a known angular-velocity error ω(t), we use the following attitude control law: T c = K p q + K d ω + ω× (Jω + H  ) where J is the moment of inertia of the satellite, ω is the angular velocity of the satellite, ω× is the antisymmetric matrix of ω, and the control coefficient matrices K p and K d > 0 are % & % & K p = diag 3.2 2.8 2.4 , K d = diag 380 330 280 Environmental disturbances on the satellite are T  T d = 1 × 10−4 0 1 2 sin(ωo t) , (Nm) where ωo = 9.0993 × 10−5 rad/s. In order to compare and verify the effectiveness of the proposed strategies for angular-momentum management, we first have performed simulations using the common pseudo-inverse torque distribution method. It is posited that kNull = 0. The simulation results are shown in Figs. 6.31 and 6.32. Before the angular momentum of the flywheel system reaches the angular-momentum envelope at 120,000 s, the angular momenta of flywheels 1, 2, and 4 are saturated at approximately 60,000 s, 70,000 s, and 110,000 s, respectively. During this period, the attitude angles also fluctuate to a certain extent. Assuming that kNull = 0.01, the simulation using the angular-momentum management strategy proposed in this chapter is performed. According to the three-axis Euler angles of the satellite relative to the orbital frame in Fig. 6.33 and the angular momentum of the four flywheels in Fig. 6.34, it can be determined that the control performance of the flywheel system remains unaffected by the partial saturation of the angular momenta before the angular momentum of the flywheel system reaches

312

Fig. 6.31 Three-axis Euler angles

Fig. 6.32 Angular momentum of four flywheels

6 Spacecraft Attitude Control

6.3 Steering Strategies for Angular-Momentum Management Devices

Fig. 6.33 Three-axis Euler angles

Fig. 6.34 Angular momentum of four flywheels

313

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6 Spacecraft Attitude Control

the envelope, which is achieved by adjusting the flywheel’s angular momentum in real time during the process of absorbing the angular momentum. The system loses control of the attitude only when the angular momenta of the two flywheels saturate simultaneously or when the angular momentum of the flywheel system reaches the envelope.

6.3.2 CMG Control Strategies 1.

Working principle of a CMG system

A single-gimbal CMG (SGCMG) consists of a high-speed rotor, a gimbal frame, and a gimbal motor. As shown in Fig. 6.35, the high-speed rotor rotates at a constant speed to provide a constant angular momentum h, and the gimbal motor rotates along the direction of the axis of the gimbal frame g at an angular rate δ˙ to change the direction of angular momentum h in order to produce the desired output torque. The corresponding output torque can be expressed as [17] ˙ × h) = −δhc ˙ T cmg = −δ(g where h denotes the amplitude of the angular momentum of a CMG. Given that the gimbal motion of an SGCMG only has one degree of freedom, the control system needs at least three SGCMGs to meet the requirements of three-axis attitude control. The resultant angular momentum of the n-SGCMG system, which consists of n (n ≥ 3) SGCMGs, is the vector sum of the angular momentum of n SGCMGs Fig. 6.35 Schematic showing the working principle of SGCMG

6.3 Steering Strategies for Angular-Momentum Management Devices

H(δ1 , . . . , δn ) =

n 

315

hi (δi )

i=1

where δi is the gimbal angle of the ith SGCMG, hi (δi ) is the angular momentum of the ith SGCMG, and i = 1, . . . , n. For a combination of gimbal angles in the n-SGCMG system δ = [δ1 , δ2 , δ3 , . . . δn ]T ∈ n , the angular momentum that can be achieved H ∈ R3 . Assuming the amplitude of the angular momentum of each CMG is assumed the same in this chapter, the angular momentum hi (δi ) of a single CMG i in the satellite frame can be expressed as ⎡

⎡ ⎤ ⎤ ⎛⎡ ⎤ ⎞ a1i sin δi + b1i cos δi b1i a1i hi (δi ) = h · ⎣ a2i sin δi + b2i cos δi ⎦ = h · ⎝⎣ a2i ⎦ sin δi + ⎣ b2i ⎦ cos δi ⎠ a3i sin δi + b3i cos δi a3i b3i where a1i and b1i are parameters that are only related to the mounting of CMG i. Therefore, the resultant angular momentum H(δ1 , . . . , δn ) of n SGCMGs can be expressed in a compact form as H(δ1 , . . . , δn ) = h(A sin δ + B cos δ)I n×1  T where I n×1 = 1 . . . 1 an n-dimensional unit vector and the coefficient matrices A and B are ⎤ ⎡ ⎤ ⎡ b11 b12 . . . b1n a11 a12 . . . a1n A = ⎣ a21 a22 . . . a2n ⎦, B = ⎣ b21 b22 . . . b2n ⎦ a31 a32 . . . a3n b31 b32 . . . b3n The sine matrix sin δ and cosine matrix cos δ of the gimbal angle are ⎡ ⎢ sin δ = ⎣

sin δ1 0

..

0 . sin δn

⎤

⎡

⎥ ⎢ ⎦, cos δ = ⎣

cos δ1

..

0

The time derivative of H(δ1 , . . . , δn ) can be obtained as ˙ 1 , . . . , δn ) = ∂H(δ1 , . . . , δn ) δ˙ H(δ ∂δ = h(A cos δ − B sin δ)δ˙ = h · C · δ˙

0 . cos δn

⎤ ⎥ ⎦

316

6 Spacecraft Attitude Control

where C = A cos δ − B sin δ. Furthermore, the ith column of C, ci , is ⎤ ⎡ ⎤ a1i b1i ∂hi (δi ) ⎣ ⎦ ci = = a2i cos δi − ⎣ b2i ⎦ sin δi ∂δi a3i b3i ⎡

The first partial derivative of ci with respect to δi is ⎡ ⎤ ⎡ ⎤ a1i b1i ∂ci ∂ 2 hi (δi ) ⎣ a2i ⎦ sin δi − ⎣ b2i ⎦ cos δi = −hi (δi ) = = − ∂δi ∂δi2 a3i b3i The higher partial derivatives of ci with respect to δi can be deduced as ∂ 3 hi (δi ) ∂ 2 ci = = −ci (δi ) 2 ∂δi ∂δi3 ∂ 4 hi (δi ) ∂ 3 ci = = hi (δi ) ∂δi3 ∂δi4 When the rotational speed of the high-speed rotor changes, SGCMG can also generate control torque through acceleration or deceleration, similar to a flywheel. This type of SGCMG is called a variable-speed CMG (VSCMG). Hereafter in this chapter, CMG refers to SGCMG with a flywheel rotor at a constant speed. 2.

Analysis of angular-momentum envelope and singularity of CMGs

The differential of the angular momentum of the control system for the CMGs is dH =

n  i=1

dhi =

n  dhi i=1

dδi

dδi =

n 

ci dδi

i=1

where ci = ddhδii = gi × hi .   Given C = c1 c2 . . . cn , if the two gimbal axes are not coplanar and min(rank(C)) = 2, all ci coplanar. As shown in Fig. 6.36, there exists a vector u such that ci · u = 0. Fig. 6.36 Schematic of the singular vector

6.3 Steering Strategies for Angular-Momentum Management Devices

317

Fig. 6.37 Schematic of the relationship between the CMG momentum and a given singular direction

Hence dH · u =

n 

dhi · u =

i=1

n 

ci · udδi = 0

i=1

Therefore, all CMG output torques are in a plane perpendicular to the vector u, and the control torque along the vector u cannot be produced. The vector u that satisfies the above conditions is called a singular vector, and δ S = {δi } is called a singular point (or singular state). The singular vector is orthogonal to the gimbal axis ci and vector gi , and |ci | = 1. Thus, there is relationship among ci , ci and hi as g ×u # ci = gi × hi = ± # i #g i × u # and  gi × u × gi # hi = ci × g i = ± # #g i × u # 

* # #  2 # # Let ei = hi ·u = ± gi × u = ± 1 − gi · u . When ei = 0, u does not coincide with any gimbal axis. It can be seen from Fig. 6.37 that the corresponding singular vector u and the system’s angular momentum H S in the singular state δ S can be expressed as [18]      1  εi u − gi · u gi  * g × u × gi = H =  2 ei i i i 1 − gi · u S

where εi = sign(ei ) = sign(hi · u) = ±1.

318

6 Spacecraft Attitude Control

Fig. 6.38 Schematic of outer singularity (saturated singularity)

According to the distribution of the singular angular momentum H S of the CMGs, the singular states can be divided into two types [18]: outer singularity and inner singularity. An outer singularity is essentially a saturation singularity. When the resultant angular momentum of the CMGs reaches the momentum envelope of the SGCMG system, the projection of each gyro momentum in this direction reaches its maximum value, as shown in Fig. 6.38. The end point of angular momentum can only move in the direction of envelope tangents or the internal direction of the tangents, making the system lose controllability. Other singularities are called inner singularities. Therefore, CMGs cannot provide torque in the direction of the normal line N of the envelope and lose controllability in this direction. An inner singularity is a singularity of the gimbal’s configuration. The resultant angular momentum of CMGs is within the momentum envelope, but the projection of the angular momentum of each CMG in the direction of the command torque is the largest; therefore, the CMGs lose their ability to generate gyro moments in that direction. In the singular state, the torque vector of each CMG is perpendicular to the plane of the gimbal axis gi and the commanded torque ui . In other words, it is on the nodal line of the vertical plane of the gimbal rotation plane and the commanded-torque vector. Inner singularity can be further divided into explicit singularity (or elliptic singularity) and hidden singularity (or hyperbolic singularity) [17, 18]. Near a hidden singularity, the gimbal’s configuration can be adjusted without causing changes in the resultant angular momentum. This implies that this resultant angular momentum can reconfigure the gimbal of the CMGs to escape from the singular state without generating additional gyroscopic torques. Let us use the pyramid configuration in Fig. 6.39 as an example to analyze the distribution of hidden singular points. The inclination angle is β, and the directions corresponding to the four gimbal axes are  T  T g1 = sin β 0 cos β , g2 = 0 sin β cos β  T  T g3 = − sin β 0 cos β , g4 = 0 − sin β cos β Let β = 54.74°. We can then obtain the distribution of explicit and hidden singular points corresponding to the singular vector us of the configuration, as shown in Figs. 6.40 and 6.41. A plane is used to intercept the explicit singular points to obtain their distribution, and the results are shown in Figs. 6.53 and 6.54. It can be seen

6.3 Steering Strategies for Angular-Momentum Management Devices

Fig. 6.39 Pyramid configuration

Fig. 6.40 Explicit singularity surface in the pyramid configuration

319

320

6 Spacecraft Attitude Control

Fig. 6.41 Inner singularity surface in the pyramid configuration

from the figures that the singular plane inside the pyramid configuration is very complicated. In particular, the distribution of explicit singular points is not entirely close to the outer envelope (Figs. 6.42 and 6.43). For any direction of space u, there exist 2n groups of singular gimbal configurations. Changing the symbol of εi is equivalent to changing the singular gimbal angle δiS to δiS + π . In this case, the singular momentum surface formed by the Fig. 6.42 Distribution of explicit singular points on the XOY plane

3 2

Hy(Nms)

1 0 -1 -2 -3 -4

-2

0 Hx(Nms)

2

4

6.3 Steering Strategies for Angular-Momentum Management Devices Fig. 6.43 Distribution of explicit singular points on the XOZ plane

321

2

Hz(Nms)

1

0

-1

-2 -3

-2

-1

0 Hx(Nms)

1

2

3

sign of the singular configuration gimbal angle is exactly the same as the angularmomentum singular surface −H S (u) = H S (−u), which corresponds to the original singular configuration. Therefore, for any direction of space u, the actual number of singular surfaces of angular momentum is 2n−1 . When calculating the singular angular momentum H S of the system, there exist different combinations of configurations according to the values of the CMG angular momentum signs εi . For n + 1 configuration combinations of n CMG systems H 0 : {+ + · · · +}, H 1 : {− + · · · +}, H 2 : {+ · · · − +}, . . . , H n : {+ · · · − +} The above configurations can be used to describe the angular momentum envelope of the CMGs. Specifically, there exist windows in the angular-momentum envelope H0 of the gimbal configuration {+ + … + }, and the angular momentum for the other gimbal configurations H1, …, Hn smoothly fill the H0 surface along the edge of the hole; when m > n, the angular-momentum surface Hm is inside the angularmomentum envelope [19]. Figure 6.44 shows the distribution of the singular angular

Fig. 6.44 Singularity surfaces of the pyramid configuration

322

6 Spacecraft Attitude Control

momentum of the combination H0{ + + + + } and the combination H2{ + − + + } under the CMG pyramid configuration. In order to obtain the main performance indexes considered in the configuration analysis, it is necessary to solve for the minimum angular momentum on the momentum envelope, the minimum angular momentum of the singular points that cannot be escaped by null motion, etc. The optimal configuration should be determined by comprehensive consideration. In the configuration design, it is usually expected that the angular momentum envelope of the configuration can meet the task requirements and there are few explicit singular points, or it is expected that the explicit singular points are as close as possible to the angular momentum envelope. 3.

Steering laws for singularity avoidance

The steering laws for CMGs can be divided into local methods and global methods [20]. In the local method, the gimbal angular-rate command of the CMG system is calculated based on the current state of the system and the torque. The local steering law does not consider the impact of current decisions on the future state, and therefore it is difficult to ensure that all the singularities can be avoided in the CMG system. The SGCMG system, in particular, is very likely to be locked in the singular state because it has fewer degrees of freedom. The global method considers the long-term behavior of the system in the design of the steering law, integrates the CMG control into the first-level control of the system, and considers how to avoid singularities. Clearly, the global method is more comprehensive than the local method for the design of the steering law. However, it is often very complicated and requires a large amount of calculation; therefore, it is difficult to implement in practice. Three typical steering laws are the Moore–Penrose (MP) pseudo-inverse steering law, pseudo-inverse steering law with null motion, and singularity-robust (SR) inverse steering law. (1)

Typical steering laws

Let T c denote the desired CMG output torque. We have δ˙ = C + T c where C + is the pseudo-inverse of the Jacobian matrix C. (1)

Moore–Penrose (MP) inverse

We select the following objective function: Jδ˙ =

1 ˙T ˙ δ δ 2

Considering the constraints of gimbal movements, we can convert the problem under the constraint T r = C(δ)δ˙ into an extreme value problem of the following objective function by using the Lagrange multiplier λ:

6.3 Steering Strategies for Angular-Momentum Management Devices

Jδ˙ =

323

1 ˙T ˙ δ δ + λT (C(δ)δ˙ − T c ) 2

By taking the partial derivative δ˙ of λ and making the derivative zero, the MP inverse steering law can be obtained as δ˙ = C T (CC T )−1 T c The determinant of the matrix CC T , i.e. det(CC T ), is defined as singularity measure in this chapter. For the CMG pyramid configuration, it is posited that the CMGs constantly output  T the desired torque T c = 1 0 0 Nm, and the initial gimbal angles is set as δ = [30◦ 30◦ 30◦ 30◦ ]T . The results of numerical simulations are shown in Figs. 6.45, 6.46 and 6.47. The figures show the performance of the MP inverse method in the non-singular state of the gimbal angle until the momentum saturation of the system. Without changing any other parameters, we modified the initial gimbal angles to be in an elliptic singular state δ = [−90◦ 0 90◦ 0]T . The resulting singularity measure curve is shown in Fig. 6.48. Given that the MP inverse method cannot circumvent singularities, when the initial state is an elliptic singular state, the system cannot effectively output torques. The initial value of the singular measure is extremely small and tends to approach zero. 160 δ δ δ δ

140

1 2 3 4

CMG gimbal angles (°)

120 100 80 60 40 20 0 -20 0

5

10

15

20

25

Time (s) Fig. 6.45 CMGs’ gimbal angles

30

35

40

45

324

6 Spacecraft Attitude Control 30 dδ1/dt dδ2/dt

CMG gimbal angluar rates (°/s)

25

dδ3/dt dδ4/dt

20

15

10

5

0

-5

-10

0

5

10

15

20

25

30

35

40

45

25

30

35

40

45

Time (s) Fig. 6.46 SGCMGs’ gimbal angular rates 2 1.8

Singularity measure

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

5

10

15

20

Time (s) Fig. 6.47 Singularity measure

6.3 Steering Strategies for Angular-Momentum Management Devices

325

-7

6

x 10

Singularity measure

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 -3

Time (s)

x 10

Fig. 6.48 Singularity measure for an elliptic singular initial gimbal angles

(2)

Steering law with null motion

In the SGCMG steering law, a gimbal reconfiguration command that does not induce an additional output torque can be introduced. This adjustment of the gimbal configuration is called idling, and the reconfiguration is called null motion. The gimbal angular-rate command can be regarded as the combination of two parts: δ˙ = δ˙ T + δ˙ N where δ˙ T is the rotational speed command with torque output and δ˙ N is the idle command. They have the following relationship: C δ˙ T = T c , C δ˙ N = 0 Given the properties of a generalized inverse, a solution can be obtained as   δ˙ N = αNM In − C T (CC T )−1 C uM

326

6 Spacecraft Attitude Control

This solution is used as the idle command, where αNM refers to an undetermined parameter and In is an n × n identity matrix. The key point of idle control is to select the scalar αNM and the vector uM to ensure that the increment in gimbal configuration measurement D is positive so that singularities are effectively avoided. The null motion can be designed based on the gradient of the singularity measure of the gimbal configuration T T T ˙ = ∂D δ˙ = ∂D δ˙ T + ∂D δ˙ N D ∂δ ∂δ ∂δ

The first term on the right-hand side of the above equation determines the demand for the control command. Idle control can only affect the second term on the righthand side. The idle command can be designed as   ∂D δ˙ N = αNM In − C T (CC T )−1 C ∂δ Therefore, a gradient-based null-motion steering law [17] of the SGCMG system can be achieved as   ∂D δ˙ = C T (CC T )−1 T c + αNM In − C T (CC T )−1 C ∂δ where

∂D ∂δ

=



∂D ∂D ∂δ1 ∂δ2

...

∂D ∂δn

T .

Defining the elements of the matrix CC T as em,k and the partial derivative with  respect to δi as em,k,i = ∂e∂δm,ki (m, k = 1, 2, 3; i = 1, . . . , n), the components of the singular gradient are       ∂D  2  2  2 e22 e33 − e23 + e22i e11 e33 − e13 + e33i e11 e22 − e12 = e11i ∂δi    + 2e12i (e13 e23 − e12 e33 ) + 2e13i (e12 e23 − e22 e13 ) + 2e23i (e12 e13 − e11 e23 ) Given that C = A cos δ − B sin δ, the following equation can be obtained:   + + , , ∂ CC T ∂ cos δ T ∂ sin δ T ∂ cos δ ∂ sin δ = A −B A − B CT + C ∂δi ∂δi ∂δi ∂δi ∂δi    2   T T 2 = 2 sin δi cos δi Bi Bi − Ai Ai + sin δi − cos δi Ai BTi + Bi ATi where Ai and Bi are the i th columns of matrices A and B, respectively. (3)

Singularity-robust (SR) method

6.3 Steering Strategies for Angular-Momentum Management Devices

327

If the singularity measure D = det(CC T ) = 0, the gimbal angles of SGCMGs become singular, and no pseudo-inverse solution exists. Considering the characteristics of singular points, the occurrence of singularity does not indicate the absence of torque output; rather, it indicates the absence of torque output in the singular direction, which is the direction orthogonal to the column vector C(δi ) of the torque matrix. Therefore, when the command torque is not parallel to the singular direction, the inverse problem of the torque equation can be modified to an unconstrained optimization problem. The optimization index is Jδ˙ =

1 1 T αSR δ˙ δ˙ + (C(δ)δ˙ − T c )T (C(δ)δ˙ − T c ) 2 2

where αSR is a weight coefficient. By finding the partial derivative of each variable in the above equation and making them equal to zero, the SR steering law can be obtained δ˙ = C T (CC T + αSR I)−1 T c where I is a 3 × 3 identity matrix. As the weight coefficient αSR is introduced, the SR steering law contains a small amount of control torque error, but it remains controllable at singular points. Based on the characteristics of the weight coefficient and singular points, at non-singular points, let αSR = 0; when D = det(CC T ) approaches zero, we increase the weight coefficient αSR . In other words, αSR is assigned with different values according to the degree of singularity. It is generally taken as a continuous function of the singular measure. The SR method has two major disadvantages. The first is that the actual output torque deviates from the desired torque. The second is that when the direction of the desired torque is parallel to the singular direction, the value of the gimbal angular velocity is zero, causing a phenomenon called “gimbal lock.” Based on the SR steering law, we have C δ˙ = CC T (CC T + αSR I)−1 T c = T c − αSR (CC T + αSR I)−1 T c Hence   −1  T CC + αSR I C δ˙ = CC T + αSR I T c − αSR T c = CC T T c When the direction of the desired torque is parallel to the singular direction, C(δ)T T c = 0. By substituting it into the above equation, the following equation can be obtained   T CC + αSR I C δ˙ = 0

328

6 Spacecraft Attitude Control

As CC T + αSR I is positive definite, C δ˙ = 0, which implies that the solution of the gimbal angular-velocity command is zero, causing a gimbal lock. To address this issue, by introducing a small steering law error based on matrix perturbation an improved SR steering law [21] can be obtained as  −1 δ˙ = C T CC T + αSR (I + P) T c where ⎡

⎤ 0 ε3 ε2 P = ⎣ ε3 0 ε1 ⎦ ε2 ε1 0 and εi (i = 1, 2, 3) is an appropriately small quantity that changes periodically. (3)

Steering methods based on command torque adjustment.

Given the characteristics of singular points of a CMG, explicit singularities cannot be avoided merely through a gimbal reconfiguration that does not generate disturbance torques. The SR inverse steering law introduces an anti-singularity factor into the matrix to make the output torque deviate from the desired torque to avoid singularities. In contrast, the steering law based on the adjustment of the command torque directly adjusts the direction and size of the desired control torque to avoid singularities. (1)

Vector-adjusting robust steering law (VARS) [10]

Geometrically, when a CMG system is singular, the output torque cannot be produced along the singular direction. Mathematically, a gimbal lock occurs when the SR control law is used. In view of this, when the gimbal configuration is near or in a singular state, we can directly adjust the command torque through vector deflection control to escape from the singular state and avoid a gimbal lock. We define the vector-adjustment matrix as ⎤ 1 −υ3 υ2 = ⎣ υ3 1 −υ1 ⎦ −υ2 υ1 1 ⎡

AAd

where υi (i = 1, 2, 3) is the vector- adjustment coefficient of the command torque, which is set to 0   if D > Dsv υi = sat −kυ,i (D − Dsv ) else

6.3 Steering Strategies for Angular-Momentum Management Devices

329

Here, kυ,i is the gain coefficient, Dsv > 0 is the adjustment threshold, D = det(CC T ), and sat(·) is the saturation function ⎧ ⎨ −υLimt x < −υLimt sat(x) = x |x| ≤ υLimt ⎩ υLimt x > υLimt The saturation limit υLimt is generally taken as 0 ≤ υLimt < 1. For the control torque command T c , the corresponding adjustment algorithm is T Ad = AAd · T c

(6.59)

Based on the vector-adjustment algorithm described in Eq. (6.59), we introduce an anti-singularity factor similar to the factor in the conventional SR control law to prevent the inverse operation of CC T in the CMG system when approaching and passing through the singular state during singularity avoidance. In addition, considering that the null-motion method can effectively avoid hidden singularities without disturbance torques, it can be used to avoid hidden singularities when the singular measure value is small and before vector adjustment so that the attitude maneuvering performance of the satellite is not affected by singular states. Given the above, we can design the vector-adjusting robust steering law as follows: 1 δ˙ = − C T (C C T + αs2 I3 )−1 AAd · T c h + αs1 (I3 − C T (CC T + αs2 I3 )−1 C)

∂D(δ) ∂δ

(6.60)

where αs1 and αs2 are coefficients greater than zero, ∂D(δ) is the gradient of D with ∂δ respect to the vector δ, and the other variables are the same as before. The steering law in Eq. (6.60) consists of three parts: a singularity-avoidance algorithm based on null motion, a singularity-avoidance algorithm based on vector adjustment, and an anti-singularity factor. According to their different functions and effects on the system, they are generally designed to be introduced sequentially in the decreasing order of the singular measure D. In the process of approaching or moving away from the singular state of the system, the introduction or removal of discontinuities by different algorithms may lead to abrupt changes in the calculated gimbal angular-velocity command of the CMG. In order to avoid this issue, the coefficients αs1 and αs2 should be designed as continuous functions of D. With determined thresholds Dsv1 and Dsv2 (Dsv2 ≥ Dsv ≥ Dsv1 ), based on the order of algorithms and the continuity of coefficients, the coefficients αs1 and αs2 can be designed as

330

6 Spacecraft Attitude Control

Table 6.1 Simulation parameters

Parameter

Value

h

4 Nm s

kυ,i (i = 1, 2, 3)

2

υLimt

0.7

αs10

0

αs20

0.5

kαs1

0.18

kαs2

0.1

Dsv1

0.5

Dsv2

2

Dsv

0.7

β

54.74°

  ⎧ 1 1 ⎪ k +αs10 · − D < Dsv1 ⎪ Dsv2 ⎨ αs1 Dsv1  αs1 = kαs1 · 1 − 1 + αs10 Dsv1 ≤ D < Dsv2 D Dsv2 ⎪ ⎪ ⎩ 0 D ≥ Dsv2   "  1 1 sat kαs2 · D+10 D < Dsv1 −6 − D +10−6 sv1 αs2 = 0 else where kαs1 and kαs2 non-negative constants, αs10 is a non-negative constant offset, and the saturation limit of the saturation function sat(•) is a non-negative constant αs20 . Taking the pyramid configuration as an example, we compares and simulate the SR steering law with null motion and the vector-adjusting robust steering law. We set the  T  T desired torque to T c = 4 0 0 Nm and the initial gimbal angle to δ = 0 0 0 0 . The other parameters are listed in Table 6.1. The simulation results of the SR singularity avoidance algorithm with null motion are shown in Figs. 6.49, 6.50, 6.51, and 6.52. As can be seen from the results, an T  internal explicit singularity occurred at δ = −90◦ 0◦ 90◦ 0◦ . When the system is in the singular state, the gimbal angular rates generated by the command torque in the singular direction are zeros, and a gimbal lock occurs, implying that the system cannot automatically escape from the singular state or reach the surface of the angularmomentum envelope. Simulation results with the vector-adjusting robust steering law are presented in Figs. 6.53, 6.54, 6.55, and 6.56. It can be seen from Fig. 6.53 that the gimbal singular measure decreases during the simulation and triggers the vector-adjusting robust steering algorithm, which subsequently increases the singular measure. According to Fig. 6.56, when approaching the singular state, the vector adjustment of the desired control torque command can effectively avoid singularities without a gimbal lock,

6.3 Steering Strategies for Angular-Momentum Management Devices

331

Fig. 6.49 CMG gimbal angle

Fig. 6.50 CMG gimbal angular velocity

which occurs with conventional singularity-avoidance methods. Further, the vector adjustment can help the angular momentum of the system reach the envelope. (2)

Command-torque spiral-search steering law [8, 22]

In spacecraft attitude control, the deviation of the command torque needs to be as small as possible from the expected while ensuring effective singularity avoidance. To this end, we design a steering law based on the command torque in a spiral-search pattern. During vector adjustment, it searches for the direction of deflection torque that avoids singularities around the initial desired control torque command. The idea is to make the adjusted torque command T Ad take the direction of T c as its central axis

332

6 Spacecraft Attitude Control

Fig. 6.51 Singular measure

Fig. 6.52 CMG angular momentum

and to search continuously in the space with the predetermined deflection angle until it is out of the singular state. Specifically, this method is implemented as follows. The command torque T c is rotated by a certain angle κ about any axis of rotation r orthogonal to it, and we obtain a corresponding torque vector after the rotation t. The torque vector t is rotated by a certain angle about the desired command torque T c , and we obtain the adjusted desired control torque T Ad . For a fixed angle κ, the direction of the vector t varies when the axis of rotation r varies. As the axis of rotation changes, all the obtained torque vectors t are distributed on a cone surface, as shown in Fig. 6.57a. The angle of rotation κ = f (D) is designed to change monotonously with the singular measure. Under this condition, if the singular measure D of the system decreases when approaching a singularity, the trajectory of the adjusted desired torque T Ad is in a spiral shape, as shown

6.3 Steering Strategies for Angular-Momentum Management Devices Fig. 6.53 Gimbal angle with the VARS law

Fig. 6.54 Singularity measure with the VARS law

Fig. 6.55 Gimbal angular velocity with the VARS law

333

334

6 Spacecraft Attitude Control 14

12

CMG Momentum

10

8

6

4

2

0 0

1

2

3

4

5 6 Time(sec)

7

8

9

10

Fig. 6.56 Angular momentum with the VARS law

Fig. 6.57 Schematic of command-torque spiral search

in Fig. 6.57b. Therefore, the singularity-avoidance strategy based on this torque adjustment is called command-torque spiral-search steering law. The deflection process of the above rotation can be decomposed into two independent rotations, i.e. T Ad = R2 R1 T c In the above equation, R1 is the command-torque deflection matrix and R2 is the rotation matrix of t = R1 T c around the command torque T c , both of which are obtained from Rodrigues’ rotation formula. Further, R = R2 R1 is the vectoradjustment coefficient matrix.

6.3 Steering Strategies for Angular-Momentum Management Devices

335

When T c  = 0, R1 and R2 are chosen in the following manner. Command-torque rotation matrix R1 : R1 = cos κI + (1 − cos κ)rrT + sin κr×

(6.61)

where the axis of rotation r can be selected as any unit vector orthogonal to the command torque T c , and the angle of rotation κ can be designed as κ=

+ , 1 π −ηD2 ∈ 0, π e 2 2

where the coefficient η is an appropriate positive number. Rotation matrix of t = R1 T c around the command torque, R2 : R2 = cos ϕI + (1 − cos ϕ)rrT + sin ϕr×

(6.62)

where the axis of  rotation r is a unit vector along the direction of the command torque T c , i.e., r = T c T c , and the angle of rotation ϕ can be taken as ϕ=

μ πt 360

Here, t is the time elapsed in the search, and the coefficient μ is an appropriate positive integer. ϕ determines the angular interval of two rotations in adjacent time. After obtaining the deflection torque command T Ad during each control period, we substitute it into Eq. (6.60) and then obtain the CMG gimbal angular-velocity command. Taking the pyramid configuration (β = 54.74°) as an example, the effect of the singularity-avoidance method based on command-torque spiral search is demon T strated by the numerical simulation. Let the desired command torque T c = 4 0 0 Nm, and the nominal angular momentum of each CMG h = 4 Nm s. The initial gimbal  T angle is selected as δ = 0 0 0 0 . The settings of parameters are listed in Table 6.2. Simulation results are presented in Figs. 6.58, 6.59, 6.60 and 6.61. As can be seen from Figs. 6.58 and 6.59, the system doesn’t encounter a gimbal lock. The command-torque spiral-search method can effectively prevent the decrease in the singular measure. It can be seen from Fig. 6.61 that the system can quickly escape from a singular state at approximately 1.2 h and reach the envelope surface with an angular momentum of 3.2 h. Based on a three-axis air-bearing platform, the command-torque spiral-search steering law is verified through physical simulation. The actuators of the system are CMGs in a pyramid configuration with the skew angle β = 65°, and its configuration is shown in Fig. 6.62.

336 Table 6.2 Simulation parameters

Fig. 6.58 Gimbal angle with the command-torque spiral-search method

Fig. 6.59 Singular measure with the command-torque spiral-search method

6 Spacecraft Attitude Control Parameter

Value

αs20

0.1

αs10

0

kαs1

0.18

kαs2

0.1

Dsv

0.7

Dsv1

0.5

Dsv2

2

μ

4

η

1

6.3 Steering Strategies for Angular-Momentum Management Devices

337

Fig. 6.60 Gimbal angular rate with the commanded-torque spiral-search method

Fig. 6.61 Angular momentum with the command-torque spiral-search method

With this pyramid configuration, explicit singularities can be observed at the maximum angular-momentum envelope (454 Nm s) and at the angular momentum (near 220 Nm s) in the Xb OYb plane, as shown in Fig. 6.63. The initial gimbal angle vector of the CMGs is set to [45° 225° 45° 225°]T so that the initial resultant angular momentum of the actuators is zero. The test results of a maneuver from 0° to 20° and a reversing maneuver to 0° are presented in Figs. 6.64, 6.65, 6.66 and 6.67. According to the results of physical simulations, during the round-trip maneuver, when the CMG gimbal configuration approaches the singular state where the angular momentum is approximately 220 Nm s, the singular measure decreases rapidly. However, from the attitude results, the system can successfully pass through the singularity with the proposed steering law.

338

6 Spacecraft Attitude Control

Xb

G3 G4

H3

Zb

H4

Ob

H2

G1

G2 H1

Yb

Fig. 6.62 Schematic of the pyramid configuration

400

X: 2.896e-14 Y: 454.8

Hy(Nms)

200

0

-200

-400 -600

-400

-200

0 200 Hx(Nms)

400

600

Fig. 6.63 Distribution of singular points in the Xb OYb plane with the pyramid configuration

6.3.3 Steering Strategies for Hybrid Actuators In this section, we introduce a joint control strategy for the rapid single-axis maneuvering of large-scale satellites based on CMGs and a flywheel-based hybrid actuation system. Specifically, the CMGs are mainly used for large-angle maneuvering attitude control of the satellite, and the flywheels are mainly used for the attitude stabilization of the satellite. The flywheel-based operation adopts the torque distribution method

6.3 Steering Strategies for Angular-Momentum Management Devices

339

Fig. 6.64 Attitude angle

Fig. 6.65 Attitude angular velocity

based on the optimization of the flywheel system’s power consumption indexes. As the method has been introduced in the previous section, it will not be explained again here. 1.

Analysis of hybrid configuration

Let us posit that the satellite’s maneuver axis is the X-axis and that four flywheels are mounted in a pyramid configuration around the X-axis of the satellite, where the

340

6 Spacecraft Attitude Control

Fig. 6.66 Singularity measure

Fig. 6.67 Angular momentum of the CMG system

angle between the direction of the flywheel’s axis of rotation and the X-axis is β. The gimbal axes gi (i = 1, 2) of two single-gimbal CMGs are mounted parallel to the ± Z-axis of the satellite so that their initial angular momentum is parallel to the ± Y-axis of the satellite. The hybrid configuration of the CMGs and the flywheels is shown in Fig. 6.68, where the CMG’s angular momentum is H cmg,i (i = 1, 2) and the flywheel’s angular momentum is H w,i (i = 1 ∼ 4).

6.3 Steering Strategies for Angular-Momentum Management Devices

341

Fig. 6.68 Schematic of the hybrid configuration of CMGs and flywheels

Given the characteristics of the hybrid configuration of CMGs and flywheels, the flywheels output three-axis control torque for attitude stabilization control of the satellite and the CMG gimbal is locked at the initial position. During attitude maneuvering, the gimbal movements of two CMGs jointly output the maneuver torque of the maneuver axis and the flywheels maintain the attitude of the other two non-maneuver axes. For attitude maneuvering of the roll axis, during the increase in the satellite’s maneuver angular velocity, the angular-momentum direction of the two CMGs approach the ± X-axis from the initial direction through the gimbal angular movements. At the stage of maintaining the satellite’s maneuver angular velocity, the two CMGs’ gimbal angles are maintained. In the maneuver deceleration phase, the two CMGs’ gimbal angles change so that the CMGs’ angular momentum can return to the initial position. In this process, the resultant angular momentum generated by the CMGs is always along the satellite’s maneuver axis. The movement of the two CMGs is shown in Fig. 6.69. During the whole attitude maneuver, the angular momentum of the two CMGs moves like scissors; therefore, the configuration of the CMGs is called the scissors configuration. Let us posit that the angles of rotation of the two CMG gimbals at the initial position are zero and that the angles of rotation along the gimbals are δ1 and δ2 . Then the resultant angular momentum is ⎡

H cmg

⎤ sin δ1 + sin δ2 = hcmg ⎣ cos δ1 − cos δ2 ⎦ 0

(6.63)

where hcmg is the amplitude of angular momentum of the CMG. Based on the above equation, we obtain the Jacobian matrix as

342

6 Spacecraft Attitude Control

Fig. 6.69 Schematic of angular-momentum exchange between two CMGs in a scissors configuration

⎡

C cmg

⎤ cos δ1 cos δ2 = ⎣ − sin δ1 sin δ2 ⎦ 0 0

(6.64)

The resultant torque T cmg generated by the CMGs is ˙ cmg = −hcmg C cmg δ˙ T cmg = −H

(6.65)

 T where δ˙ = δ1 δ2 is the gimbal angular-velocity vector. Based on the ideal law of motion of CMG angular momentum during attitude maneuvering in Fig. 6.69, it can be determined that the components of angular momentum H cmg and torque T cmg acting on the satellite in the direction of the non-maneuver axis are zero if the gimbal angular motion satisfies δ1 = δ2 during maneuvering. According to the expression of C cmg , a singular state of the gimbal configuration occurs δ1 = δ2 = ±90◦ ; therefore, the gimbal angular motion  when ◦ ◦ must satisfy δi ∈ −90 90 (i = 1, 2). For a particular moment of inertia of the satellite’s maneuver axis Jx , based on the exchange of angular momentum, we can determine that the maximum amplitude of the satellite’s maneuver angular velocity, ωmax , must satisfy the following condition: ωmax
0 and Kd > 0 are parameters of the PD controller. It can be noticed that the denominator of the transfer function in the above equation contains 0.01s + 1, which is actually a low-pass filter. The purpose of introducing it is to make the controller achievable, and the influence of this item on the tested frequency band is negligible. The filters are designed as second-order filters [11] in series Gfil (s) =

nf / s2 /ωzi2 + 2ξzi s/ωzi + 1 2 s2 /ωpi + 2ξpi s/ωpi + 1 i=1

where nf is the number of second-order filters and the second-order filter parameters ωzi , ξzi , ωpi , and ξpi (i = 1, 2, . . . , nf ) are the zero frequency, zero damping, pole frequency, and pole damping, respectively. The spacecraft parameters are listed in Table 6.3. By inserting them into Eq. (6.81), we obtain Gys (s) = e−0.1s

3×

60s2 + 3.645s + 519 + 182.3s3 + 2.759 × 105 s2

104 s4

Table 6.3 Spacecraft parameters Lp

rtx

mR

mp

JRx m2

m

m

kg

kg

kg

0.2

−1.2

300

20

350

JRy kg

m2

500

JRz kg

ξ

g m2

500

kg

m/s2

1.622

– 0.001

350

6 Spacecraft Attitude Control

The design aims to achieve a shear frequency greater than 3 rad/s. Considering that the spacecraft parameters given in Table 6.3 make the transfer-function pole greater than zero, it can be determined that the spacecraft is upright and that sloshing will cause a phase advance and an increase in amplitude. The parameters of the PD controller can be designed as follows: Kp = ωk2 , Kd = 2ξk ωk where ωk , ξk > 0 are design parameters. According to the requirements of shear frequency, we design ωk = 3, ξk = 1.4 The Nichols diagram of the open-loop transfer function GPD Gys under the PD control alone is shown in Fig. 6.74. The corresponding shear frequency is 8.44 rad/s. The system’s gain margin is 4.2 dB. The phase margin is 29.21°. It is necessary to further design the filters to increase the gain margin and phase margin to meet the requirements. Nichols Chart 100

Open-Loop Gain (dB)

80

60

40

20

0

-20 -225

-180

-135

-90

-45

Open-Loop Phase (deg)

Fig. 6.74 Nichols diagram of open-loop transfer function Gys with PD control

0

45

6.4 Liquid-Filled Spacecraft Control

351

As can be seen from Fig. 6.74, to improve the phase margin, the phase angle near the shear frequency needs to be increased. To increase the gain margin, the shear frequency needs to be reduced. It can be observed that the lower frequency band of the system has a higher gain; therefore, reducing the phase in the lower frequency band will not cause system instability. Therefore, we design a lag filter in the lower frequency band to reduce the gain near the system shear frequency, and we set the minimum phase value of the lag filter in the lower frequency band to avoid system instability. In addition, we design an advanced filter near the shear frequency to increase the phase of the system so as to increase the phase-angle margin. The designed filter is Gfil (s) =

2 / s2 /ωzi2 + 2ξzi s/ωzi + 1 2 s2 /ωpi + 2ξpi s/ωpi + 1 i=1

The relevant parameters are listed in Table 6.4. The Nichols diagram of the openloop transfer function for the system GPD Gfil Gys is shown in Fig. 6.75. The shear frequency of the system is 5.84 rad/s. The gain margin is 7.67 dB. The phase margin is 62.26°. Based on the above, we design the final controller of the y-axis as Gcy (s) =

JRy (2ξk ωk s + ωk2 ) Gfil 0.01s + 1

For the z-axis, by substituting the parameters in Table 6.3 into Eq. (6.82), we obtain Gzs (s) = e−0.1s

60s2 + 0.3645s + 519 3 × 104 s4 + 182.3s3 + 2.759 × 105 s2

Thus, the z-axis controller can be designed as Gcz (s) =

JRz (2ξk ωk s + ωk2 ) Gfil (s) 0.01s + 1

In other words, the same PD controller and filter parameters for the y- axis are used here, and only the gain of the controller is modified. The Nichols diagram of the open-loop transfer function of the z-axis of the system is shown in Fig. 6.76. It can be seen from the diagram that the system’s shear frequency is 5.84 rad/s. The gain margin is 7.67 dB, while the phase margin is 62.26°. Table 6.4 Filter parameters ωz1

ωz2

ωp1

ωp2

ξz1

ξz2

ξp1

ξp2

0.3

4

0.2

5

0.5

0.5

0.5

0.5

352

6 Spacecraft Attitude Control Nichols Chart 100

80

Open-Loop Gain (dB)

60

40

20

0

-20

-40 -225

-180

-135

-90

-45

0

45

90

Open-Loop Phase (deg)

Fig. 6.75 Nichols diagram of open-loop transfer function Gys with PD control and filters

As liquid sloshing has little effect on the rotation about the x-axis, the controller for the x-axis is designed as a simple PD controller Gcx (s) =

IRx (2ξk ωk s + ωk2 ) 0.01s + 1

6.5 Multi-Body Spacecraft Attitude Control 6.5.1 Hybrid Attitude Control with Moving Antenna In response to the requirements for rapid platform response and rapid largeangle maneuvering, on-orbit services and spacecraft technology for space-based early warnings have started to demand hybrid control techniques for simultaneous maneuver of the platform and antenna. The essence of attitude maneuver is strongly coupled nonlinear control, which may be affected by factors such as external disturbances and uncertainties in the inertia. To achieve high accuracy in attitude tracking,

6.5 Multi-Body Spacecraft Attitude Control

353 Nichols Chart

100

80

Open-Loop Gain (dB)

60

40

20

0

-20

-40 -225

-180

-135

-90

-45

0

45

90

Open-Loop Phase (deg)

Fig. 6.76 Nichols diagram of open-loop transfer function Gzs with PD control and filters

the attitude control system needs to be robust to resist internal and external uncertainties. This section is based on the attitude dynamics model of a spacecraft with a moving antenna established in Sect. 4.3.4. We will introduce an attitude control design method that takes into account satellite attitude control and antenna pointing control. 1.

Terminal sliding-mode control (TSMC) [25, 26]

Sliding-mode control (SMC) is a robust nonlinear control method that is completely invariant to internal and external disturbances in the sliding mode. Based on the terminal SMC (TSMC) method, we design a hybrid control system that satisfies the requirements for the simultaneous maneuver of the spacecraft body and antenna so that the spacecraft body can suppress the disturbance torque caused by the antenna’s movement within a limited time. Based on the body’s attitude dynamics model, we combine the limited-time disturbance observer with the TSMC method and design an attitude controller suitable for large-angle maneuvering of the spacecraft body. To reduce the influence of body maneuvering on antenna pointing, we introduce approximate compensation for the angular acceleration of the body into the antenna controller, and the pointing angle of the antenna is corrected kinematically.

354

(1)

6 Spacecraft Attitude Control

Limited-time disturbance observer

The attitude dynamics equation of the spacecraft body, given by Eq. (4.40), can be expressed as (J + J)ω˙ + ω× (J + J)ω = u + d

(6.84)

Irrespective of the orbital angular velocity ωo , with the target quaternion qd and the desired angular velocity ωd , we find the time derivative of the attitude angularvelocity error ωe = ω + C q ωd , and based on C˙ q = −ω× e C q , we obtain ˙d ω˙ e = ω˙ + ω× e C q ωd − C q ω

(6.85)

 T   T where C q = 1 − 2qTev qev I + 2qev qTev − 2qe4 q× is the attitude ev and qev = qev qe4 error quaternion. −1 J. By inserting Eq. (6.84) into Eq. (6.85), we obtain Let (J + J)−1 = J +  −1 ω˙ e = f (t) + J u + d˜

f (t) = −J

−1

˙d ω× J ω + ω× e C q ωd − C q ω

−1 −1 d˜ = − Jω× (J + J)ω − J ω× Jω −  Ju + (J +  J)d

(6.86) (6.87) (6.88)

where f (t) is the definite term and d˜ is the unknown bounded comprehensive disturbance term. Let x ∈ Rn and 0 < α < 1. We define the function sig(x)α as sig(x)α = [|x1 |α sgn(x1 ), . . . , |xn |α sgn(xn )]T On this basis, let 0 < δ < 1, and we define the function fal(x, α, δ) as " fal(x, α, δ) =

sig(x)α , x ≥ δ  x δ 1−α , x < δ

˜ For the system (in Eq. 6.86), Let z1 be the estimate of ωe and z2 be the estimate of d. the limited-time disturbance observer is e1 = z 1 − ω e z˙ 1 = v1 = −λ1

. sgn(e1 )dτ − β1 sig(e1 )1/ 2 −1

e2 = z2 + f (t) + J u − v1

6.5 Multi-Body Spacecraft Attitude Control

355

. z˙ 2 = −λ2

sgn(e2 )d τ − β2 sig(e2 )1/ 2

(6.89)

where λi , βi > 0 (i = 1, 2). Equation (6.89) is composed of two robust, accurate differentiators. Without measurement noise, after the first differentiator operates for a limited time ts1 , let ˜ z1 = ωe and v1 = ω˙ e . At ts1 , the expression of e2 is equivalent to e2 = z2 − d, implying that the structures of the two differentiators are identical. By selecting appropriate parameters, we can make the second differentiator reach z2 = d˜ after a limited time ts2 . Then, the estimated time of the total disturbance is ts = ts1 + ts2 . (2)

TSMC-based attitude controller

Based on Eq. (6.86), we select the following fast terminal sliding surface: s = ωe + k1 qev + k2 sig(qev )γ

(6.90)

where s ∈ R3 is the sliding-mode variable, k1 , k2 > 0, and 0 < γ < 1. Let τ1 , τ2 > 0, 0 < α < 1. The reaching law is chosen as s˙ = −τ1 s − τ2 sig(s)α

(6.91)

Considering external-environment disturbances and inertia uncertainties, we design the fast TSMC-based controller using the sliding-mode surface (see Eq. 6.90) and the reaching law (see Eq. 6.91):  # #  γ −1 q˙ ev − J(f + z2 + τ1 s + τ2 sig(s)α ) u = −J k1 I + γ k2 diag #qev #

(6.92)

This controller can make the error quaternion qev converge to 0 within a limited time. (3)

Controller for antenna pointing

During the attitude maneuvering of a spacecraft, the required angle of rotation of an antenna will be varying for the same pointing target. From the perspective of kinematics, the antenna’s angle of rotation is relative to the body frame, while the trajectory of the pointing target is usually relative to the orbital frame. Therefore, we need to introduce a correction of the body attitude into the antenna’s angle of rotation, as shown in Fig. 6.77. Let us posit that the angle between the target orientation and the xo Oyo plane in the orbital coordinate system Oxo yo zo is φp1 and that the angle between the projection in the xo Oyo plane and the Ox-axis is φp2 . Then, the component of the target orientation in the orbital frame is

356

6 Spacecraft Attitude Control

Fig. 6.77 Schematic of antenna pointing angle

⎡

cos φp1 cos φp2

⎤

⎥ ⎢ vo = ⎣ cos φp1 sin φp2 ⎦ sin φp1 It is known that the coordinates of the antenna’s pointing axis are va = [ 0 0 1 ]T , (b) (b) (b) T and the components of vo in the body frame v(b) o = [ vo1 vo2 vo3 ] can be expressed as v(b) o = C bo vo = C 01 va

(6.93)

Therefore, the reference rotational angle of the antenna is φ d . 

φdx φdy



 =

 (b)   v arctan − o2(b) vo3

(b) arcsin(vo1 )

(6.94)

It can be seen from the antenna’s dynamics equation that the angular acceleration of the body’s attitude and the angular acceleration of the antenna are coupled to each other. To reduce the pointing deviation of the attitude angular acceleration to the antenna, we can apply approximate compensation to the antenna’s dynamics equation. To facilitate the design of the antenna controller, we combine the antenna’s dynamics equation and its kinematics equation. Based on Eq. (4.33), we have φ¨ = ST1 (ω˙ 1 − S2 )

(6.95)

where the expressions of S1 , S2 are ⎡

⎡ ⎤ ⎤ cos φy 0 − sin φy ⎦ S1 = ⎣ 0 1 ⎦, S2 = φ˙ x φ˙ y ⎣ 0 cos φy sin φy 0 Let uφ = ST1 u1 ∈ R2 . S1 and J 01 are the values of S1 and J 01 at φ = 0. By substituting Eq. (6.95) into Eq. (4.40), we obtain

6.5 Multi-Body Spacecraft Attitude Control

357

φ¨ = M 1 uφ − M 10 ω˙ + d 1 T

T

(6.96)

T

−1 where M 1 = S1 J −1 1 S1 , M 10 = S1 J 1 J 01 , and the other terms are included in d 1 . As the disturbance observer (see Eq. 6.89) has an accurate estimate of ω˙ e , ω˙ can ˙ d . The controller for the antenna before be replaced by ω = v1 − ω× e C q ωd + C q ω integrating the PD controller is designed as

˙ uφ = M −1 1 (−kp1 φ e − kd1 φ e + M 10 ω)

(6.97)

where kp1 and kd1 are the proportional coefficient and differential coefficient, respectively. (4)

Mathematical simulations

To verify the effectiveness of the proposed control method, the spacecraft body’s attitude is simulated to track and maneuver a certain moving target, and the antenna is simulated to point accurately to a certain fixed target. Let the initial attitude angle of the spacecraft body be [−10◦ 30◦ 20◦ ]T . With the 3−2−1 sequence of Euler angle attitude description, the initial attitude quaternion is q(0) = [ −0.1277 0.2393 0.1893 0.9437 ]T The initial target quaternion is qd (0) = [0 0 0 1]T , and the desired angular velocity is + , + ,  + ,  2π t 3π t T πt sin sin ωd = 0.05 sin (rad s) 50 50 50 Irrespective of the orbital movement, let us posit that the initial position of the antenna is the nominal position, i.e., φ(0) = 0. With the target’s pointing angle of φ p = [20◦ −20◦ ]T , the process of movement of the antenna from the initial angle of rotation to the target angle of rotation is planned by the command preprocessor [27]. In the simulations, the step size is set to Ts = 0.02 s, and the parameters of the disturbance observer are selected as β1 = 5, β2 = 2, λ1 = 2.5, λ2 = 0.5 The parameters of the body attitude controller and antenna PD controller are set to. k1 = k2 = 0.8, τ1 = τ2 = 0.2, γ = 0.8, α = 0.5, kp1 = 100, kd1 = 50 The simulation results are shown in Figs. 6.78, 6.79, 6.80, 6.81, and 6.82. It can be seen from the simulation results that the body attitude can quickly track the given

358

6 Spacecraft Attitude Control

0.25 q

e1

0.2

q

e2

0.15

q

e3

q

ev

0.1 0.05 0 -0.05 -0.1 -0.15

0

10

20

30

40

50

t/s Fig. 6.78 Attitude-quaternion tracking error

6

ωe1

4

ωe2

2

ωe3

ωe (°/s)

0 -2 -4 -6 -8 -10

0

10

20

30 t/s

Fig. 6.79 Angular-velocity tracking error

40

50

6.5 Multi-Body Spacecraft Attitude Control

359

0.5 s

0.4

s 0.3

s

1 2 3

s

0.2 0.1 0 -0.1 -0.2 -0.3

0

10

20

t/s

30

40

50

Fig. 6.80 Sliding surface curve

φ x (°)

100

φ dx

50

φx

0 -50

0

10

20

0

10

20

30

40

50

30

40

50

80

φ y (°)

60 40 20 0

t/s Fig. 6.81 Reference angle of rotation and actual angle of rotation of the antenna

target attitude, and the tracking error is less than 1×10−3 °. Owing to the approximate compensation of the angular acceleration of the body in the PD controller, the errors of the antenna in the maneuver phase and stabilization phase are significantly reduced, and the pointing error is less than 0.01◦ .

360

6 Spacecraft Attitude Control

60

φ ex

50

φ ey

40

φe

30 20 10 0 -10

10

0

30

20

40

50

t/s Fig. 6.82 Tracking error in the angle of rotation of the antenna

2.

Hybrid sliding-mode variable structure control based on the characteristic model [26, 28]

Without considering the environmental disturbance torque d and the coupling term J, the attitude dynamics equations of the spacecraft body in Eq. (4.40) is J ω˙ + ω× Jω = u

(6.98)

Similar to Sect. 6.3.3, we can obtain the second-order time-varying characteristic model based on the above equation y(k + 1) = F1 (k)y(k) + F2 (k)y(k − 1) + G0 (k)u(k) + d 0 (k)

(6.99)

The characteristic model without modeling errors d 0 (k) is y(k + 1) = F1 (k)y(k) + F2 (k)y(k − 1) + G0 (k)u(k)

(6.100)

For sampling with small time intervals, it is evident that F1 (k), F2 (k) can be approximated as a diagonal matrix. The parameter matrix of the characteristic model can be expressed as ⎡

⎡ ⎤ ⎤ f11 (k) 0 0 f21 (k) 0 0 F1 (k) = ⎣ 0 f12 (k) 0 ⎦, F2 (k) = ⎣ 0 f22 (k) 0 ⎦ 0 0 f13 (k) 0 0 f23 (k)

6.5 Multi-Body Spacecraft Attitude Control

361

⎡

⎤ g11 (k) g21 (k) g31 (k) G0 (k) = ⎣ g12 (k) g22 (k) g32 (k) ⎦ g13 (k) g23 (k) g33 (k) Let  x(k) =

   x1 (k) y(k − 1) = ∈ R6 x2 (k) y(k)

Then Eq. (6.100) can be expressed in the following state-space form: "

x(k + 1) = A(k)x(k) + B(k)u(k)

(6.101)

y(k) = C T x(k) where      0 0 0 I , B(k) = A(k) = ,C = F2 (k) F1 (k) G0 (k) I 





To facilitate derivation in the later part, we use G0 (k) to represent G0 (k) + λI. Then, the golden-section adaptive control law can be expressed as 



ul (k) = (C T B(k))−1 C T A(k)Lx(k)

(6.102)

In Eq. (6.102),      0 0 I l I ,L = 1 , B(k) = A(k) = l2 I F2 (k) F1 (k) G0 (k)













The estimated values of the coefficient matrix of the characteristic model can be obtained using the multivariate recursive least-squares identification algorithm or the gradient identification algorithm. (1)

Sliding-mode control based on characteristic model

Given the convergence errors in the parameter identification process, we can regard the characteristic model (see Eq. 6.101) as a linear time-varying system with model uncertainties or time-varying disturbances "





x(k + 1) = A(k)x(k) + B(k)u(k) + d(k) s(k) = C T x(k)

(6.103)

362

6 Spacecraft Attitude Control 

where the sliding-mode variable s(k) ∈ R3 , C T = [ c1 I c2 I ], C T B(k) is not singular, d(k) = [ 0T d T2 (k) ]T , d 2 (k) ∈ R3 , and d(k) evidently does not satisfy the matching condition. The characteristic model parameters have slow time-varying characteristics at a certain interval, and the control variables and output variables in an actual project are bounded. It can be determined that d 2 (k) also has slow time-varying characteristics and is bounded. Based on the above, let us posit that before the characteristic model parameters converge to the truth value, d 2 (k) in Eq. (6.103) satisfies the following condition: |d2i (k) − d2i (k − 1)| ≤ εi , (i = 1, 2, 3)

(6.104)

where ε = [ ε1 ε2 ε3 ]T . For the time being, the uncertainty term d(k) is not considered. Based on Eq. (6.103), we design an attenuation-based sliding-mode control law. For s(k +1) = ρs(k), ρ(|ρ| < 1) is selected as the attenuation coefficient. Then, we have   s(k + 1) = C T x(k + 1) = C T A(k)x(k) + B(k)u(k) = ρs(k) 



The attenuation-based sliding-mode control law is   uρ (k) = −(C T B(k))−1 C T A(k) − ρI x(k) 



Based on Eqs. (6.105) and (6.102), if C T = [ 0 l2 I ] and ρI =



(6.105) 

l2 −l1 F1 (k) l2

=

lρ F1 (k), then ul (k) = uρ (k). This means that the golden-section adaptive control law can be equivalent to the time-varying attenuation-based sliding-mode control law to some extent. For a discrete-time system (see Eq. 6.103), we design the attenuation-based sliding-mode control law as 



u(k) = (C T B(k))−1 [C T B(k − 1)u(k − 1) + (ρ − 1)s(k) 



− C T (A(k)x(k) − A(k − 1)x(k − 1))]

(6.106)

where ρ(|ρ| < 1)is the attenuation coefficient, and C needs to asymptotically stable C T x(k) = 0 . With this control law, the state of the system will reach global consistency and be eventually bounded. For Eq. (6.103), x1 (k + 1) = x2 (k). Under the conditions that s(k) = c1 x1 (k) + x1 (k + 1) and |c1 | < 1, we have y(k) = x1 (k) ≤

|c2 | 1 ε 1 − |c1 | 1 − |ρ|

(6.107)

6.5 Multi-Body Spacecraft Attitude Control

363

Compared with the golden-section adaptive control method, the attenuation-based sliding-mode control method (see Eq. 6.106) adds an adjustable attenuation coefficient. While the dynamic characteristics of the system have improved without introducing a transition process, the controlled quantity is still large, which can easily lead to the saturation of the actuator. We can introduce the attenuation-based power reaching law into the sliding-mode control strategy to improve the dynamic characteristics of the system. A discrete power reaching law can be designed as s(k + 1) = (1 − τ1 T )s(k) − τ2 T sig(s(k))α

(6.108)

where 0 < τ1 T , τ2 T < 1. It can be seen from Eq. (6.108) that the discrete power reaching law is essentially the discretization of the continuous power reaching law s˙ = −τ1 s − τ2 sig(s)α . For a discrete-time system (see Eq. 6.103), we design the sliding-mode control law based on the power reaching law as 



u(k) = (C T B(k))−1 [C T B(k − 1)u(k − 1) − τ1 T s(k) − τ2 T sig(s(k))α 



− C T (A(k)x(k) − A(k − 1)x(k − 1))]   where C needs to be asymptotically stable C T x(k) = 0 . With this law, the state of the system will reach global consistency and eventually be bounded. Owing to the existence of the sign function, chattering is likely to occur after s(k) converges to δ. To reduce the effect of chattering on the controllability of the system, the sign function can be replaced by a saturation function. (2)

Mathematical simulations

In the simulations, we set the initial attitude angle to [−10◦ 30◦ 20◦ ]T . The initial angular velocity is zero, and the target attitude angle is a set of sine functions       πt  T 10◦ × sin πt − sin πt sin 30 40 30 The relevant parameters of the attitude controller for the spacecraft body are as follows: T = 0.02s, λ = 5 × 10−5 , α = 0.5, τ1 = τ2 = 0.05, C T = [ 0.5 1 ]I, and δ = 10−4 . Without considering the body’s attitude correction, let the target angle of rotation of the antenna be [20◦ − 20◦ ]T . Then, the PD control parameters are kp1 = 100 and kd1 = 15. The simulation results are shown in Figs. 6.83, 6.84, 6.85, 6.86, 6.87, and 6.88. According to the simulation results, with certain controller parameters, after approximately 20 s, the attitude angle smoothly changed to the target attitude angle, and the tracking accuracy is higher than 0.005◦ . In practice, it is not necessary for the identified values of the model parameters to converge strictly to the truth value.

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φ (°)

20

φd 0 -20

φ 0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

θ (°)

50 0 -50

ψ (°)

50 0 -50

t/s Fig. 6.83 Attitude-angle tracking curve 15 10

φe

5

θe ψe

0 -3

y

e

-5

5

x 10

-10 -15 0 -20 -25 -5 20

-30 -35

0

10

20

40 30

Fig. 6.84 Attitude-angle tracking error

60 40

80 50 t/s

60

100 70

80

90

100

6.5 Multi-Body Spacecraft Attitude Control

365

ω1(°/s)

5 0 -5

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

ω2(°/s)

5 0 -5

ω3(°/s)

2 0 -2

t/s Fig. 6.85 Attitude angular velocity

φ x (°)

30

φ xd

20

φx

10 0

0 10

20

40

20

40

60

80

100

60

80

100

φ y (°)

0 -10 -20 -30

0

t/s Fig. 6.86 Antenna’s angle of rotation

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6 Spacecraft Attitude Control

10

u

x

u

5

y

u u (N.m )

z

0

-5

-10

-15

0

20

40

60

80

100

t/s Fig. 6.87 Satellite’s control torque

0.05

u1x

1

u (N.m )

0

u1y

-0.05

-0.1 -0.15

-0.2

0

60

40

20

80

100

t/s Fig. 6.88 Antenna’s control torque

They only need to converge within the bounded interval of the model parameters, and it is important to ensure that G0 (k) is non-singular. 

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6.5.2 Hybrid Attitude Control of Combined Body Based on the attitude dynamics of the combined body described in Sect. 4.3.4 and under the presumption that the actuator used in spacecraft can generate a force acting on the center of mass and a torque about the center of mass, we design a corresponding hybrid attitude controller [29]. For simplicity, it is posited that the external force is generated by an air jet and the force acting on the end effector. The nozzle is fixed with respect to the spacecraft, and the force acting on the end of the nozzle is fixed relative to the inertial frame. Then, the dynamics equation of a free-flight system can be written as H + (q)¨z0 + C + (q, 0 ω0 , q˙ ) = Q

(6.109)

where z˙ 0 is the velocity vector, i.e. ⎡

⎤ r˙c z˙ 0 = ⎣ 0 ω0 ⎦ q˙ 0

(6.110)

The above equation describes the dynamics of a free-flight system subjected to external forces and torques. The generalized force Q can be decomposed into the disturbance force Qd and the control force Qc : Q = Qc + Qd

(6.111)

The control force includes the jet force of the spacecraft and the joint moments of mechanical arms. The control force can be written as ⎡0

⎤ ⎡ 0 ⎤T ⎡ 0 ⎤ fc fc I J 11,c 0 J 12,c Qc = J Tq ⎣ 0 nc ⎦ = ⎣ 0 I 0 ⎦ ⎣ 0 nc ⎦ τ τ 0 0 I

(6.112)

where 0 f c and 0 nc are the expressions of the jet force and moment acting on the spacecraft in the spacecraft coordinate system, respectively, and J q is a reversible square matrix. Similarly, if we posit that the disturbance force only acts on the end effector, then the disturbance force can be expressed as +



Qd = J (e, n, q)

T

fe ne

 (6.113)

where f e and ne are the force and moment acting on the end effector, respectively.

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Considering the similarity between Eq. (6.112) and the equation for the groundfixed manipulator, the latter’s control law may be applied to the former. However, Eq. (6.112) has two features. The first is that it requires an appropriate spacecraft attitude. The second is that the system is redundant because the spacecraft is not fixed. Its redundancy allows it to complete additional tasks, which includes the control of the position and attitude of the base spacecraft and the coordinated control of the manipulator and spacecraft. Moreover, by planning the motion of the spacecraft, the end effector can reach the desired point while the manipulator reaches the desired configuration. As mentioned in Chap. 4, J + is a 6 × 12 matrix, which is still an irreversible nonsquare matrix, despite the fact that the manipulator has 6 degrees of freedom. We can use the additional matrix 0 J + S to control the position and attitude of the spacecraft. ⎡ 0 ⎤ ⎤ I J 11 0 J 12 r˙e 0 ⎢0 I ⎢ ωe ⎥ J 22 ⎥ ⎢ ⎥ ⎥ z˙ 1 = ⎢ ⎣ R˙ c ⎦ = diag(T 0 , T 0 , T 0 , T 0 )⎣ I 0 J 11,c 0 J 12,c ⎦z˙ 0 = J z z˙ 0 ωc 0 I 0 ⎡

(6.114)

where z˙ 1 is the output velocity vector. J z is a 12 × 12 reversible matrix. The following gives a transposed Jacobian controller with inertial feedback. First, by substituting Eq. (6.114) into Eq. (6.112), we obtain a kinetics equation with z˙ 1 as the output:  T  z1 +  C = J −1 Qc H¨ z  is expressed as In the above equation,  C contains nonlinear terms, and H T + −1   = J −1 H Jz H z  is reversible. We define the error e as If J z is not singular, the inertial matrix H e = z1,des − z1 where z1 is obtained through inertial feedback and z1,des is the desired inertial point. It is posited that both the inertial position and attitude of the spacecraft and the end effector can be measured. Therefore, if the control force      K p e + K d e˙ + z¨ 1,des +  C Qc = J Tz H where K p and K d are symmetric matrices that are positive definite, then for the error dynamics, we have

6.5 Multi-Body Spacecraft Attitude Control

369

e¨ + K d e˙ + K p e = 0 which means that the error gradually approaches zero. A simplified transposed Jacobian controller can be obtained by using a sufficiently high gain:   Qc = J Tz K p e + K d e˙ Finally, the control jet force and joint torque through the inversion of Eq. (6.112) can be obtained as ⎡0 ⎤ fc  −1   ⎣ 0 nc ⎦ = J Tq J Tz K p e + K d e˙ τ The above equation offers an approach to the coordinated control of the spacecraft and manipulator based on the inertial position and attitude of the spacecraft and the end effector. If these quantities are not measurable, we can still use an estimate of the error e for coordinated control, though model uncertainties may be introduced.

References 1. Wie B (1998) Space vehicle dynamics and control. American Institute of Aeronautics and Astronautics, Inc. 2. Li M, Lei Y, Xiaogang Mu (2019) Satellite attitude control method with load disturbance suppression and physical test. Chinese Space Sci Technol 39(1):73–77 3. Costic BT, Dawson DMW, de Queiroz MS et al (2000) A quaternion-based adaptive attitude tracking controller without velocity measurements. In: Proceedings of the 39th IEEE conference on decision and control, pp 2424–2429 4. Tan S, Lei Y (2015) Logic differential control method for agile maneuvers of spacecraft. Chinese Opt. 8:18–22 5. Tan S, He Y, Wei C et al (2011) Mixed sine maneuvering path guiding method for high-paddle fundamental frequency satellite. ZL201110409470.X. 6. Slotine J-JE, Li W (2004) Applied nonlinear control. Pearson Education Asia Limited and Chia Machine Press 7. Sun Y, Yuan L, Lei Y (2017) Characteristic model-based large angle maneuver attitude control for spacecraft. In: The 36th Chinese control conference. Dalian, China 8. Sun Y (2017) Research on satellite attitude control methods based on control moment gyroscope. Degree Thesis of China Academy of Space Technology, Beijing 9. Hongxin Wu, Jun Hu, Xie Y (2009) Intelligent adaptive control based on characteristic model. China Science and Technology Press, Beijing 10. Yuan Li, Lei Y, Yao N et al (2018) Attitude maneuver control and verification for flexible satellites based on the SGCMG system. J Astronaut 39(1):43–51 11. Wie B, Byun K-W (1989) New generalized structural filtering concept for active vibration control synthesis. J Guidance 12(2):147–154 12. Lei Y, Tan S, Liu Y (2010) Rapid maneuver and stabilization control of spacecraft attitude. Chinese Space Sci Technol 30(5):48–53

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13. Landis Markley F, Reynolds RG, Liu FX et al (2010) Maximum torque and momentum envelopes for reaction wheel arrays. J Guidance Control Dyn 33(5): 1606–1614. 14. Yoon H, Seo HH, Choi H-T (2014) Optimal uses of reaction wheels in the pyramid configuration using a new minimum infinity-norm solution. Aerosp Sci Technol 39: 109–119. 15. Lei Y, Li Y (2017) Momentum envelope analysis and momentum management for spacecraft with a flywheel array. Chinese Space Sci Technol 37(6):1–9 16. Zhang D (2007) Matlab numerical analysis and applications. National Defense Industry Press, Beijing 17. Zhang R (1998) Attitude dynamics and control of satellite orbit. Beihang University Press, Beijing 18. Wie B (2004) Singularity analysis and visualization for single-gimbal control moment gyro system. J Guid Control Dyn 27(2):126 19. Margulies G, Auburn NN (1978) Geometric theory of single-gimbal control moment gyro system. J Astronaut Sci 26(2):159–191 20. Zhong Wu, Hongxin Wu (2000) Survey of steering laws for single gimbal control moment gyroscope system. J Astronaut 21(4):140–145 21. Wie B (2005) Singularity escape/avoidance steering logic for control moment gyro system. J Guid Control Dyn 28(5):948–956 22. Sun Y, Yuan Li, Lei Y (2016) SGCMG singularity avoidance method based on command torque vector helix search. Aerosp Control Appl 42(6):26–30 23. Zong H, Lei Y (2007) Research on methods for rapid attitude maneuver control of satellite based on control moment gyroscopes. Aerosp Control Appl 33(5):11–16 24. Wang Z (2017) Attitude control and sloshing suppression for a class of liquid-filled spacecraft. Degree Thesis of China Academy of Space Technology, Beijing 25. Li F, Lei Y (2014) Terminal sliding mode compound control method for multi-body spacecraft attitude maneuver. Aerosp Control Appl 40(1):14–18 26. Li F (2014) Research on Attitude control methods of spacecraft with movable antennas. Degree Thesis of China Academy of Space Technology, Beijing 27. Gawronski W, Almassy WT (2002) Command preprocessor for radio telescopes and microwave antennas. IEEE Antennas Propag Mag 44(2):30–37 28. Li F, Lei Y (2014) Variable structure control for spacecraft attitude tracking based on characteristic model. In: Proceedings of the 33rd Chinese control conference. Nanjing, China 29. Deng Y (2017) Research on modeling and coordinated control of combined body. Degree Thesis of China Academy of Space Technology, Beijing

Chapter 7

Autonomous Guidance, Navigation, and Control of Spacecraft

7.1 Introduction The autonomous guidance, navigation, and control (GNC) system is an essential component of a spacecraft. It serves the purposes of determining the attitude and orbital parameters of the spacecraft, guiding the spacecraft to move on the intended orbit or towards the desired target, and controlling the attitude and orbit of the spacecraft according to mission requirements without ground-station support. Autonomous navigation involves the real-time measurement of attitude- and orbitrelated parameters using onboard instruments of the spacecraft, as well as the calculation of navigation parameters based on the requirements of the flight mission. Depending on the choice of the reference frame, autonomous spacecraft navigation can be categorized into absolute navigation and relative navigation. The former is associated with the determination of kinematic parameters with respect to a specific inertial frame, whereas the latter is implemented with respect to a body-fixed frame. On the basis of the current position and velocity of a spacecraft, autonomous guidance using orbital mechanics enables the spacecraft to vary orbital maneuver times as well as the magnitudes and directions of velocity to reach the desired location and velocity at a set time or move on the desired orbit. This is performed in an on-orbit manner. Typically, autonomous guidance includes (a) Clohessy– Wiltshire (CW) guidance, line-of-sight (LOS) guidance, and optimal multi-pulse guidance for rendezvous and docking; (b) standard ballistic guidance and predictive guidance for reentry; and (c) optimal variable-thrust guidance, nominal orbit guidance, explicit guidance, and gravity-turn landing guidance for soft landing on extraterrestrial bodies. Autonomous control is the process in which a spacecraft control its attitude or orbit according to the guidance requirements, current motion status, and attitudecontrol requirements by using attitude actuators and orbital maneuvering engines to achieve attitude and orbit control goals.

© Beijing Institute of Technology Press 2022 Y. Xie et al., Spacecraft Dynamics and Control, Space Science and Technologies, https://doi.org/10.1007/978-981-33-6448-6_7

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7.2 Absolute Autonomous Navigation Based on the working principles, absolute autonomous spacecraft navigation typically includes inertial, celestial, satellite, and terrestrial navigation. In this section, we limit our focus to the strapdown inertial navigation system (SINS) and Global Positioning System (GPS) integrated autonomous navigation based on navigation and attitude sensors, which has been applied in practice and has shown good application prospects.

7.2.1 SINS-GPS Integrated Navigation In SINS-GPS integrated navigation systems, onboard gyroscopes and accelerometers feed data to computers to obtain navigation information. The computers can calculate the attitude information of the spacecraft in real time by using the angular velocity data measured by the gyroscopes. Such information, together with the data from the accelerometers, can be processed by the computers to determine the position and velocity of the spacecraft at any time. GPS navigation is realized by receiving position data sent by GPS satellites using onboard GPS receivers and computing the spacecraft position with respect to the desired reference frame based on the GPS ephemeris data. Typically, the absolute position of the spacecraft can be estimated by solving the navigation equation with distances between the spacecraft and GPS satellites obtained using pseudo-random code or carrier-phase measurements. SINS is highly dynamical and autonomous, but its navigation errors accumulate with time. On the other hand, GPS navigation has high positioning accuracy with small errors, but it is susceptible to disturbances. Thus, SINS-GPS integrated navigation is applied to achieve good performance. We now will introduce this method in detail by considering the example of atmosphere reentry navigation. 1.

Strapdown inertial navigation and its real-time solution We select the geocentric 1st equatorial system (inertial coordinate system) OE X I YI Z I as the navigation frame of reference and the measurement frame of the inertial measurement unit O X b Yb Z b as the return-vehicle body frame. (1)

Initialization At the initial moment of reentry, we must determine (a) the center of mass of the return vehicle with frame OE X I YI Z I , (b) the   respect to the navigation velocity components X, Y, Z , Vx , Vy , Vz , and (c) attitude quaternions (q0 , q1 , q2 , q3 ) of the return-vehicle body frame O X b Yb Z b with respect to the navigation frame OE X I YI Z I . This is done with the support of the attitude measurement system and GPS receiver of the return vehicle, or the ground monitoring and control network.

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373

With the input from a ground monitoring and control network, X, Y, Z , Vx , Vy , Vz can be obtained directly. The position and velocity components obtained by the GPS receiver,X G , YG , Z G , VG X , VGY , VG Z , are associated with the World Geodetic System 1984 (WGS-84) coordinate system, which is an Earthfixed system. Therefore, a transformation from this system to the geocentric 1st equatorial system is needed. By using the attitude angle obtained by the onboard attitude measurement system, we can calculate the direction cosine matrix Cob of the returnvehicle body frame O X b Yb Z b with respect to the orbit frame O xo yo z o . Furthermore, by using the position and velocity of the center of mass of the return vehicle, we can calculate the direction cosine matrix Coi of the frame O xo yo z o with respect to the geocentric 1st equatorial system (inertial system) OE X I YI Z I . Thus, the direction cosine matrix of O X b Yb Z b with respect to OE X I YI Z I should be Cib = Cio Cob . The initial values of the quaternions, q0, q1, q2, q3, are determined using the following equation: ⎡

⎤ 2(q1 q3 + q0 q2 ) q02 + q12 − q22 − q32 2(q1 q2 − q0 q3 ) C bi = ⎣ 2(q1 q2 + q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 − q0 q1 ) ⎦ 2(q2 q3 + q0 q1 ) q02 − q12 − q22 + q32 2(q1 q3 − q0 q2 ) (2)

Navigation equation and its real-time solution The basic navigation equation in an inertial system is R¨ = g + a

(7.1)

where R is the radius vector from the Earth’s core to the spacecraft’s center of mass, g the gravitational acceleration of the spacecraft’s center of mass, and a the apparent acceleration of the spacecraft’s center of mass. In the geocentric 1st equational system, Eq. (7.1) can be written as 

V˙ X V˙Y V˙ Z

T

T  T  = g X gY g Z + a X a Y a Z

Data measured by the accelerometer are the projection of the apparent gravitational acceleration of the spacecraft with respect to the inertial system in the return-vehicle body frame. The transformation matrix from the return-vehicle body frame to the 1st equational system, C ib , is calcu T lated based on ωx1 ω y1 ωz1 measured by the gyroscope and the initial values of quaternions. Hence, we have 

V˙ X V˙Y V˙ Z

T

T  T  = g X gY g Z + C ib ax1 a y1 az1

(7.2)

where Vx , Vy , and Vz can be obtained through integration. The position components X, Y, and Z can be obtained by taking the integral of

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7 Autonomous Guidance, Navigation, and Control of Spacecraft



X˙ Y˙ Z˙

T

T  = VX VY VZ

C ib is obtained from

C bi = C ib

T

⎤ 2(q1 q3 − q0 q2 ) q02 + q12 − q22 − q32 2(q1 q2 + q0 q3 ) ⎥ ⎢ = ⎣ 2(q1 q2 − q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 + q0 q1 ) ⎦ 2 2 2 2 2(q2 q3 − q0 q1 ) q0 − q1 − q2 + q3 2(q1 q3 + q0 q2 ) ⎡

(7.3)

where d 1 [ q0 q1 q2 q3 ]T = Ω[ q0 q1 q2 q3 ]T dt 2 ⎤ ⎡ 0 ωz1 −ω y1 ωx1 ⎢ −ωz1 0 ωx1 ω y1 ⎥ ⎥ Ω=⎢ ⎣ ω y1 −ωx1 0 ωz1 ⎦ −ωx1 −ω y1 −ωz1 0  T Here, the initial value of q0 q1 q2 q3 is determined during initialization. Givenattitude quaternions (q0 , q1 , q2 , q3 ), position and velocity components X, Y, Z , Vx , Vy , Vz within the inertial system O E X I Y I Z I , and the sidereal time at the initial moment of navigation (λc0 ), we can compute the altitude and longitude/latitude of the sub-satellite point of the return vehicle, as well as other parameters of interest. 2.

SINS-GPS integrated navigation solution. (1)

(2)

3.

Position and velocity of SINS corresponding to GPS time It is known that data provided by GPS receivers were obtained at past moments, rather than the present. Therefore, in SINS-GPS integrated navigation, it is necessary to compare the GPS data against the SINS data at the same instant. This is achieved by calculating SINS data corresponding to a GPS time using interpolation. Comparison of GPS data with SINS data at the same instant to determine whether the GPS data are usable If the difference between the two is within a reasonable range, then we assume that the GPS data are reliable.

SINS-GPS integrated navigation filtering. Assuming the GPS data are usable as measurement values, errors in the attitude quaternions, position, velocity, and inertial measurement unit of SINS-GPS integrated navigation can be estimated and corrected using a Kalman filter.

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7.2.2 Attitude-Sensor-Based Autonomous Navigation In this section, we present a method of autonomous orbit determination using onboard attitude sensors in the context of geosynchronous equatorial orbit (GEO) satellites. This method employs a measurement system consisting of one Earth sensor and twostar sensors, with the mathematical aid of Hill’s equations with a perturbation term and Kalman filter. By using this method, the spacecraft orientation and motion trends in space can be obtained in an online and reliable manner without the requirement of additional satellite hardware, providing necessary orbit information for satellite control. 1.

State equation Assuming that a satellite moving in a nominal GEO is free from perturbations, it will orbit Earth once per day with the sub-satellite point position maintained constant with respect to the ground. For a perturbed satellite moving in a GEO, missions normally require the position deviations of the satellite in both the eastwest and north-south directions to be within ± 0.1°. Since these deviations are negligibly small, the satellite’s equations of motion can be linearized near the nominal GEO. First, we need to establish the nominal GEO coordinate system, Oxyz, as shown in Fig. 7.1. The filled rectangle in this figure represents the perturbed satellite, whereas the filled ellipse represents an imaginary satellite. The origin of the coordinate system, O, is at the center of mass of the imaginary satellite, with the z axis pointing towards Earth’s core and the y axis being normal to the orbit. The three axes constitute a right-handed system. E is the center of the Earth, and EA represents the projection of the radial vector of the real satellite on the equatorial plane. λ and β are the longitude and latitude deviations of the real satellite relative to the imaginary satellite within the Earth-centered inertial (ECI) frame, respectively. Now, we arrive at the classic equations of relative motion, i.e., Hill’s equations

Fig. 7.1 Motion of a real GEO satellite relative to an imaginary one

Earth E

Real GEO satellite λ

β

z

o maginary GEO satellite

y

x

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

x¨ − 2ω0 z˙ = ax y¨ + ω02 y˙ = a y z¨ + 2ω0 x˙ −

(7.4)

3ω02 z

= az

x With the longitude deviation λ = rgeo , latitude deviation β = z deviation r = rgeo , we perform the following transformations:

⎧ ⎪ ⎨

λ¨ − 2ωor˙ = β¨ + ωo2 β =

ax rgeo ay rgeo

⎪ ⎩ r¨ + 2ω λ˙ − 3ω2 r = o o

y , rgeo

and radial

(7.5) az rgeo

where rgeo is the GEO radius, ω0 is the orbital angular velocity of the GEO satellite, and ax , a y , az denote the perturbation acceleration of the satellite in each of the three directions. The state vector is defined as x = [ λ β r λ˙ β˙ r˙ ]T

(7.6)

The disturbance vector is  b= 000

a x a y az rgeo rgeo rgeo



The state matrix is obtained from Eq. (7.5) as follows: ⎡

0 ⎢0 ⎢ ⎢ ⎢0 A=⎢ ⎢0 ⎢ ⎣0 0

0 0 0 0 −ωo2 0

0 1 0 0 0 0 0 0 0 0 3ωo −2ωo

0 1 0 0 0 0

⎤ 0 0 ⎥ ⎥ ⎥ 1 ⎥ ⎥ 2ωo ⎥ ⎥ 0 ⎦ 0

The state equation of the system is x˙ = Ax + b + Γ

2.

(7.7)

where Γ is the model error, consisting of the error of the perturbation acceleration and the linearization error Hill’s equation. Measurement equation Three-axis stabilized satellites often adopt Earth sensors and star sensors for attitude measurements. Here, we use one Earth sensor and two-star sensors to obtain a part of the orbital information, in addition to a precise onboard clock and computer. The attitude sensor measures the unit vector from the Earth’s

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core to the satellite, E, the coordinate of which in the ECI frame is (E)I , as well as the unit vectors from the two stars to the satellite, S1 and S2. The star sensors determine the spacecraft attitude by referring to the unit vectors of the stars with respect to the ECI frame, S1I and S2I, with their coordinates in the ECI frame being S1I and S2I, respectively. Hence, the star-fixed frame includes two angles, η1 and η2 , which are obtained from cos η1 = E · S1 cos η2 = E · S2

(7.8)

Similarly, the ECI frame also includes two angles, η1 and η2 , which are obtained from cos η1 = Z · S1I cos η2 = Z · S2I

(7.9)

where Z is the unit vector within the ECI frame pointing from the Earth’s core to O, which is the origin of the nominal GEO frame. Its coordinate in the ECI frame is (Z)I . Thus, the measurement vector y is obtained from y = [ cos η1 − cos η1 cos η2 − cos η2 ]T The measurement equation is written as y = h(x, t) + Ψ

(7.10)

where Ψ is the measurement noise. Assuming that the current right ascension of the imaginary satellite is θ , in the ECI frame, we have ⎧  T ⎪ (E) I = cos(λ + θ ) cos β sin(λ + θ ) cos β sin β ⎪ ⎪ ⎪ ⎪  ⎪ ⎨ (Z) = cos β sin θ 0 T I (7.11) T  ⎪ ⎪ (S1 ) I = S1x S1y S1z ⎪ ⎪ ⎪ T ⎪ ⎩ (S ) =  S2x S2y S2z 2 I By combining Eqs. (7.10) and (7.11), we obtain  T h(x, t) = (E)I · (S1 )I − (Z)I · (S1 )I , (E)I · (S2 )I − (Z)I · (S2 )I   S1x cos(λ + θ ) cos β + S1y sin(λ + θ ) cos β + S1z sin β − S1x cos θ − S1y sin θ = S2x cos(λ + θ ) cos β + S2y sin(λ + θ ) cos β + S2z sin β − S2x cos θ − S2y sin θ

(7.12)

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

Navigational algorithm In the previous two subsections, we established the state equation and measurement equation of the satellite orbit, respectively, which are both nonlinear functions. Hence, we can use the extended Kalman filter (EKF) to estimate the orbital elements. Moreover, because the perturbation acceleration divided by the orbital radius in the state equation is negligible, an approximation can be reasonably applied when using the Kalman filter. Combining Eqs. (7.11) and (7.12), we express the system model as 

x˙ = Ax + b +  y = h(x,t )+ ψ

(7.13)

The discretization of Eq. (7.13) yields x(k + 1) = (I + AT )x(k) + T b(k) + Γ k where T is the sampling period and Γ k represents the discrete Gaussian white noise of the state error. Γ k satisfies   E{Γ k } = 0, W Γ k Γ lT = Q k δkl where Q k is the variance matrix of the state error and δkl is the Kronecker delta. The discrete form of the measurement equation, Eq. (7.13), is written as y(k) = h(x(k), k) + ψ k

(7.14)

where ψ k denotes the discrete Gaussian white noise of the measurement error. ψ k satisfies     E ψ k = 0, W ψ k ψ lT = Rk δkl where Rk is the variance matrix of the measurement error.

7.3 Relative Autonomous Navigation In spacecraft missions such as rendezvous and docking (RVD), relative navigation estimates the kinematic parameters of the spacecraft for rendezvous guidance and control. The design of relative navigation demands the selection of appropriate sensors for the relative measurement system based on the specific requirements of rendezvous guidance and control. The goal of relative navigation is to acquire all the required kinematic parameters using filtering algorithms, the measured data, and the dynamical and kinematic laws of the spacecraft.

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379

There are two approaches to achieve relative navigation. When the target orbit is given, filters can be designed to directly estimate the state of relative motion based on the observations from a differential satellite navigation device, microwave radar, and laser radar, which are the relative measurement sensors, together with a relative motion model treated as the state equation. On the other hand, when the target orbit is unknown, we need to use the orbit measurement information of the chaser and the observations from the relative measurement sensors to estimate the orbital parameters of the chaser and target prior to the computation of the state of relative motion between the two spacecraft.

7.3.1 Relative State Estimation In an Earth-orbit RVD mission, the acquirement of high-precision orbital information of the target usually relies on a ground measurement and control system. Hence, based on the relative motion models and observations from the relative measurement sensors, it is possible to design filters to directly estimate the state of relative motion. 1.

State equation of relative motion For a near-circular-orbit RVD mission, consider the relative position and relative velocity of the chaser in the RVD frame with respect to the target as the state  T vector, i.e., X = x y z x˙ y˙ z˙ . The CW equation is used as the state equation: X(k+1) = (k+1,k)X(k) + G(k+1,k)u(k) + w(k)

(7.15)

where ⎤ 0 −2(1 − cos φ)/ωoT 1 0 6(sin φ − φ) (4 sin φ − 3φ)/ωoT ⎥ ⎢ cos φ 0 0 sin φ/ωoT 0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢0 0 4 − 3 cos φ 2(1 − cos φ)/ωoT 0 sin φ/ωoT ⎥ ⎢ (k + 1, k) = ⎢ ⎥ 0 −6ωoT (1 − cos φ) 4 cos φ − 3 0 −2 sin φ ⎥ ⎢0 ⎥ ⎢ ⎦ ⎣ 0 −ωoT sin φ 0 0 cos φ 0 0 0 3ωoT sin φ 2 sin φ 0 cos φ ⎡ 2 − 2T /ω ⎤ 0 2 sin φ/ωoT (−4 cos φ/ωoT + 4/ωoT − 3ωoT T 2 /2)/ωoT oT ⎥ ⎢ 2 0 −(cos φ − 1)/ωoT 0 ⎥ ⎢ ⎥ ⎢ 2 2 ⎥ ⎢ 2T /ωoT − 2 sin φ/ωoT 0 (1 − cos φ)/ωoT ⎥ ⎢ G(k + 1, k) = ⎢ 4 sin φ/ωoT − 3T 0 2(cos φ − 1)/ωoT ⎥ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ 0 sin φ/ωoT 0 (2 − 2 cos φ)/ωoT 0 sin φ/ωoT ⎡

Here, φ = ωoT T , and T is the discretization period. The variance matrix of the process noise is

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

 Q(k) = E w(k)w(k)T ⎡ 3 2 3 σx2 T3 0 0 σx2 T2 0 2σx2 ωoT T3 3 2 ⎢ 0 σ y2 T3 0 0 σ y2 T2 0 ⎢ 2 ⎢ T3 2 T3 2 0 0 σz 3 −2σz ωoT 3 0 σz2 T2 ⎢   =⎢ 2 3 2 T3 ⎢ σx2 T2 0 −2σz2 ωoT T3 σx2 T + 4σz2 ωoT 0 σx2 − σz2 ωoT T 2 3 ⎢ 2 2T 2 ⎣ 0 σy 2 0 0 σy T 0  2  3 2 2 T3 2σx2 ωoT T3 0 σx − σz2 ωoT T 2 0 σz2 T + 4σx2 ωoT σz2 T2 3

2.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

where σx , σ y , and σz are the standard deviations of process noise in each axis and are determined by the magnitudes of model linearization error, orbit perturbation error, and measurement error of the accelerometer. Relative navigation using differential satellite navigation device A satellite navigation device is used by a moving carrier to receive signals from a navigation satellite and perform signal processing. In a low-Earth-orbit (LEO) RVD mission with a cooperative target, the pseudo-range differential or carrierphase differential calculated from the measurements of the satellite navigation devices of two spacecraft contains relative position information and can thereby be used as the relative measurement information. Moreover, the device can use information on the pseudo-range, carrier phase, or carrier-phase smoothed pseudo-range, along with the velocity information (pseudo-range rate), to calculate and export the relative position and velocity of the chaser with respect to the target. (1)

Measurement equation based on pseudo-range single difference We set the measured values of pseudo-range of the chaser and target with respect to the jth navigation satellite at the same instant as ρ˜cj (t) = ρcj (t) + Cδtcj (t) + δρion,c (t) + δρtrop,c (t) + vρ,c j

j

j

ρ˜t (t) = ρt (t) + Cδtt (t) + δρion,c (t) + δρtrop,c (t) + vρ,t

(7.16)

where c is the speed of light; ρ j (t) the actual distance between the spacecraft and the jth navigation satellite; δt j (t) the difference between the receiver and satellite clock errors; δρion and δρtrop the measurement errors caused by delay in the thermosphere and troposphere, respectively; and vρ the random measurement error. If the pseudo-range single difference is adopted, we arrive at

 j j j ∇ ρ˜ct (t) = ρcj (t) − ρt (t) + c δtcj (t) − δtt (t) + δρion,c (t) − δρion,c (t) + δρtrop,c (t) − δρtrop,c (t) + vρ,c − vρ,t j

j

Thus, δtc (t) − δtt (t) = δtc (t) − δtt (t). That is, the satellite clock error is counterbalanced. In addition, because the two spacecraft are rather close,

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381

their paths in space to the navigation satellite and their speed of light errors caused by the thermosphere are identical. Therefore, differentiation should effectively minimize the impacts of these sources of error. The orbital position of the target can be determined using its pseudorange measurement signals. After transformation, its coordinate in the T  RVD frame is xT yT z T . Subsequently, the RVD coordinate of the jth navigation satellite position can be calculated using the navigation ephemeris data. The measurement equation is then written as  Zj = −



j

j

j

(xT + x − x g )2 + (yT + y − yg )2 + (z T + z − z g )2 j

j

j

(xT − x g )2 + (yT − yg )2 + (z T − z g )2

+ c(δtc − δtt ) + vρ ( j = 1, 2, . . . , n gps )

(7.17)

where vρ =δρion,c (t)−δρion,c (t)+δρtrop,c (t)−δρtrop,c (t) is the measurement noise. c(δtc − δtt ) in this equation is the clock error term of the two receivers, which should be treated as the state to be estimated. Assume c(δtc − δtt ) = d, where 

(2)

d˙ = f + wd f˙ = wf

(7.18)

Here, wd and wf are zero-mean white Gaussian white noise terms. Measurement equation based on carrier-phase single difference The single difference of carrier-phase measurements between the target and chaser with respect to the jth navigation satellite is j

λ∇ϕ j (t) = λϕcj (t) − λϕt (t) 

j j j = ρcj (t) − ρt (t) + c δtcj − δtt + λ∇ Nct + vϕ

(7.19)

where ϕ j (t) is the phase difference with respect to the jth navigation satellite and λ the carrier-phase wavelength. The carrier-phase single difference is distinct from the pseudo-range single difference in that the latter has j an extra term λ∇ Nct , which is the difference in cycle ambiguity of the two receivers with respect to the jth navigation satellite. The cycle ambiguity needs to be determined using special algorithms. Once determined, it becomes a known integer.

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

The corresponding measurement equation is  Zj =

j

j

j

(xT + x − x g )2 + (yT + y − yg )2 + (z T + z − z g )2  j j j − (x T − x g )2 + (yT − yg )2 + (z T − z g )2 

j j + c δtcj − δtt + λ∇ Nct + vϕ ( j = 1, 2, . . . , n gps )

(3)

Measurement equation based on relative position and velocity In WGS-84, based on the position measurements of two spacecraft with respect to a common-view satellite; the pseudo-range, carrier phase, or carrier-phase smoothed pseudo-range; and the velocity measurements (pseudo-range rate), the relative position and velocity of the chaser relative T  to the target, xWGS84 yWGS84 z WGS84 x˙WGS84 y˙WGS84 z˙ WGS84 , can be computed using a least-squares iterative algorithm in a single-difference or double-difference manner. The computed relative position and velocity are transformed to the RVD frame as follows: ⎤ ⎤ ⎡ xˆ xWGS84 ⎢ ⎥ ⎥ ⎢ T T T ⎣ yˆ ⎦ = Cr_To CTo_I CCD C z (θG )⎣ yWGS84 ⎦+Cr_To CTb_To lgps,T − Cr_To CTo_I CCb_I lgps,C zˆ z WGS84 ⎡

⎡ ⎤ ⎤ ⎡ xˆ˙ x˙WGS84 − yWGS84 ωe ⎣ yˆ˙ ⎦ = Cr_ CTo_I CTCD Cz (θG )⎣ y˙WGS84 + xWGS84 ωe ⎦ z˙ WGS84 zˆ˙ where θG is the real-time Greenwich hour angle (GHA) of a star, CCD the transformation matrix from the J2000 mean equator inertial frame to the J2000 instantaneous equator inertial frame, CTo_I the transformation matrix from the inertial frame to the target’s orbital frame, Cr_To the transformation matrix from the target’s orbital frame to the RVD frame, Cz the transformation matrix for the z-axis, and lgps,C the installation position vector of the GPS receiver with respect to the target body frame.  T Setting Z = xˆ yˆ zˆ xˆ˙ yˆ˙ zˆ˙ as the variable to be observed, we can obtain the following measurement equation: ⎡ ⎤ x ⎢ y⎥ ⎢ ⎥ ⎢ ⎥ ⎢z⎥ ⎥ Z = I⎢ ⎢ x˙ ⎥ + vgps ⎢ ⎥ ⎢ ⎥ ⎣ y˙ ⎦ z˙

(7.20)

7.3 Relative Autonomous Navigation

383

where vgps is the measurement noise. 3.

Relative navigation based on rendezvous radar Rendezvous radars can be categorized into microwave radars and laser radars. Rendezvous radars can provide measurements of the relative range ρrad , LOS elevation angle αrad , and LOS azimuth angle βrad . Microwave rendezvous radars can measure the rate of change of relative range in addition to the three parameters measured by laser rendezvous radars. We can adopt two methods for constructing the measurement equation for navigational filters: direct filtering and indirect filtering. The former uses the data measured by the rendezvous radar as the observations for filtering, whereas the latter adopts the relative position data in the RVD frame, which are transformations of the data measured by the rendezvous radar, as the observations for filtering. (1)

Direct measurement model If the relative range ρrad , LOS elevation angle αrad , and LOS azimuth angle T  βrad are directly used as the measured data, i.e., Z = ρrad αrad βrad , the measurement equation can be written as ⎡



⎤ 2 2 2 xrad + yrad + z rad ⎢ ⎥   ⎢ ⎥ Z = ⎢ arc sin −z rad / x 2 + y 2 + z 2 ⎥ + vrad rad rad rad ⎣ ⎦

(7.21)

arc tan(yrad /xrad ) where ⎡ ⎤ ⎡ ⎡⎡ ⎤ ⎤ ⎤ xrad x ⎣ yrad ⎦ = −⎣Crad_ Cb CCb_I CTrI ⎣⎣ y ⎦ − CTTb_r lref ⎦+Crad_ Cb lrad ⎦ z rad z Here, CrI represents the direction cosine matrix from the inertial frame to the RVD frame, CCb_I the direction cosine matrix from the inertial frame to the chaser body frame, Crad_ Cb the direction cosine matrix from the chaser body frame to the rendezvous radar installation frame, CTb_r the direction cosine matrix from the target body frame to the RVD frame, lref the position vector of the cooperative target mounted on the target spacecraft with respect to the target body frame, and lrad the installation position vector of the rendezvous radar with respect to the target body frame. vrad is the measurement noise of the rendezvous radar, the variance of which satisfies ⎤ ⎡ σρ2 0 0 ⎥ ⎢ Rραβ = ⎣ 0 σα2 0 ⎦ 0 0 σβ2

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

(2)

where σρ , σα , and σβ are the standard deviations of the distance, LOS angle, and measurement noise, respectively. Indirect measurement model By using the data measured by the rendezvous radar, the position of the radar’s cooperative target with respect to the rendezvous radar measurement frame can be obtained using the following equation: ⎧ ⎪ ⎨ xradm = ρrad cos αrad cos βrad yradm = ρrad cos αrad sin βrad ⎪ ⎩ z radm = −ρrad sin αrad The relative center of mass position of the two spacecraft in the RVD frame can be obtained through a transformation based on the rendezvous-radar installation, attitude, and orbital parameters ⎤ ⎡ ⎡ ⎤ ⎤ xradm xm T T ⎣ ym ⎦ = −CrI CCb_I ⎣Crad_ Cb ⎣ yradm ⎦ + lrad ⎦ + CTTb_r lref zm z radm ⎡

(7.22)

The indirect measurement equation is ⎡ ⎤ x ⎢ ⎡ ⎤⎢ y ⎥ ⎥ 100000 ⎢ ⎥ z⎥ ⎢ ⎥⎢ ⎥ Z = ⎣ 010000⎦⎢ ⎢ x˙ ⎥ + v˜ rad ⎢ ⎥ 001000 ⎢ ⎥ ⎣ y˙ ⎦

(7.23)

z˙ T  where Z = xm ym z m and v˜ rad denotes measurement noise. Hence, the variance matrix of the indirect measurement noise (˜vrad ) is obtained as ˜ = Cr_rad Cmραβ Rραβ CTmραβ CTr_rad R

(7.24)

where ⎡

Cmραβ

⎤ cαrad · cβrad −ρrad · sαrad · cβrad −ρrad · cαrad · sβrad = ⎣ cαrad · sβrad −ρrad · sαrad · sβrad ρrad · cαrad · cβrad ⎦ −sαrad −ρrad · cαrad 0

Cr_ rad = CrI CTCb_I CTrad_ Cb , cαrad = cos αrad , sαrad = sin αrad , cβrad = cos βrad , sβrad = sin βrad .

7.3 Relative Autonomous Navigation

4.

385

Filter design For relative navigation problems, considering that the CW equations are linear and time-invariant, if the measurement equation based on the relative position and velocity is adopted, then the measurement equation is linear as well. Hence, the Kalman filter can be used. For other cases, nonlinear filtering methods are required for filter design owing to the nonlinearity of the measurement equation. Typically, we use deterministic sampling and filtering algorithms such as EKF and the unscented Kalman filter (UKF) for nonlinear filtering.

7.3.2 Autonomous Orbit Determination and Relative State Estimation For RVD missions in deep-space exploration, the ground measurement and control system often has a limited capacity and may fail to supply the precise orbit of the target spacecraft. Therefore, based on the measured information of orbits and relative measurements, the chaser has to estimate the orbits for the target and itself before the state of relative motion between the two can be indirectly computed. During a space fly-around phase in deep-space missions, the probe can capture clear images of the central celestial body using autonomous optical navigation sensors (e.g., visible-light cameras and UV sensors). Subsequently, the contour of the central object can be obtained using image-processing algorithms such as edge detection and extraction, based on which information including the apparent semi-diameter and body center can be calculated. With the information acquired from the autonomous optical navigation sensors and relative sensors, navigation filters can be designed to simultaneously estimate the orbits of the two spacecraft. 1.

State equation T  We use the positions and velocities and the target and chaser, Xt = RtT VtT T T   and Xc = RcT VcT , respectively, as the state vector, i.e., X = XcT XtT . The orbital dynamic equation of the state vector is ˙ = X



 ft (Xt ) + w(t) fc (Xc )

(7.25)

where w(t) is the system noise representing themodeling error, the statistical  features of which include E{w(t)} = 0 and E w(t)wT (t) = Q(t). For the convenience of filter design, we use  only the orbital dynamic equation of the J2 term as the state equation. f j X j (j = c,t) can be written in the following form:

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

⎧ dx j ⎪ = v jx ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ dy j ⎪ ⎪ ⎪ = v jy ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dz j ⎪ ⎪ = v jz ⎪ ⎪ dt ⎪ ⎪  ⎨     2  xj dv j x zj 3J2 Rm 2 = −μ 3 1 + 1−5 + wv j x ⎪ ⎪ dt 2 R R Rj j j ⎪ ⎪ ⎪  ⎪     2  ⎪ ⎪ yj dv j y zj 3J2 Rm 2 ⎪ ⎪ ⎪ 1−5 + wv j y ⎪ dt = −μ 3 1 + 2 ⎪ Rj Rj Rj ⎪ ⎪ ⎪  ⎪     2  ⎪ ⎪ ⎪ zj dv j z zj 3J2 Rm 2 ⎪ ⎪ = −μ 3 1 + 3−5 + wv j z ⎩ dt 2 Rj Rj Rj  where R j = x 2j + y 2j + z 2j and Rm is the equatorial radius of the central object. Because the orbital dynamic equation is a nonlinear differential equation, linearization and discretization are required to use EKF as the navigation filter. First, we obtain the linearized equation as  F(X) =

F c (Xc ) 0 0 F t (Xt )

 (7.26)

where ⎡

0 0 0 ⎢ 0 0 0 ⎢   ⎢ 0 0 0 ⎢ Fj Xj = ⎢ ⎢ F j (4, 1) F j (4, 2) F j (4, 3) ⎢ ⎣ F j (5, 1) F j (5, 2) F j (5, 3) F j (6, 1) F j (6, 2) F j (6, 3)

1 0 0 0 0 0

0 1 0 0 0 0

⎤ 0 0⎥ ⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 0⎦ 0

The detailed form of F j can be found in Ref. [8]. Using the filtering period t, we arrive at the discretized disturbance equation: δ Xˆ k|k−1 = k|k−1 δ Xˆ k−1 + W k−1  k|k−1 =

   I 6 +F c Xc,k−1 · t 0  0 I 6 +F t Xt,k−1 · t

where k|k−1 updates the covariance matrix of state prediction and W k−1 is a random white noise sequence, the variance of which is  Q k−1 =  k|k−1 Q(tk−1 )Tk|k−1 ; here, Q(tk−1 ) = diag 01×3 σ2vc 01×3 σ2vt , σ2v j =

7.3 Relative Autonomous Navigation

387

 2.

σv2 σv2 σv2 , j = c,t, and σv2 is determined by the magnitude of perturbation acceleration and the accelerometer precision. Measurement equation Consider a Mars exploration mission as an example. Observations from the autonomous optical navigation sensor installed on the chaser include the direction and apparent semi-diameter. Together with the results of inertial attitude determination, we obtain the direction and distance to the center of Mars in the inertial frame. The corresponding measurement equation is written with the function h1 (Xk ). The direct observations from the laser radar are the relative range and LOS angle. With the installation matrix and attitude information of the laser radar, the relative position in the inertial frame is obtained, with the corresponding measurement equation written with the function h2 (Xk ). Thus, the overall measurement equation is  ⎤ xc /xc2 + yc2 + z c2 ⎢ y / x 2 + y2 + z2 ⎥ ⎢ c  c c c⎥ ⎡ ⎤   ⎢ / xc2 + yc2 + z c2 ⎥ vdir,k ⎢ z c ⎥ h1 (Xk ) ⎢ ⎥ =⎢ Zk = xc2 + yc2 + z c2 ⎥ + ⎣ vρ,k ⎦ ⎢ ⎥ h2 (Xk ) ⎢ ⎥ xt − xc vrel,k ⎢ ⎥ ⎣ ⎦ yt − yc zt − zc k ⎡

(7.27)

where vdir,k , vρ,k , and vrel,k are the measurement noise of the direction to the center of Mars, distance to the center of Mars, and relative position, respectively. The measurement noise has a mean of zero, whereas its variance is set according to the sensor performance. When an EKF is used in the filter design, the measurement equation needs to be linearized as  ∂h1 (X)  ∂h(X) (7.28) = ∂h∂X 2 (X) ∂X ∂X where ⎡

 3 3 yc2 + z c2 /Rc3  −xc yc /R  c 3 −xc z c /Rc3 3 2 2 ⎢ ∂h1 (X) ⎢ −yc xc /Rc xc + z c /Rc  −yc z c /R 03×9  c =⎣ yc2 + xc2 /Rc3 −z c xc /Rc3 −z c yc /Rc3 ∂X xc /Rc yc /Rc z c /Rc 01×9  Rc = xc2 + yc2 + z c2  ∂h2 (X)  = −I3×3 03×3 I3×3 03×3 ∂X

⎤ ⎥ ⎥, ⎦

388

3.

7 Autonomous Guidance, Navigation, and Control of Spacecraft

With the estimated Xˆ k , the relative state in the RVD frame can be calculated as ˆ t,k and rˆ˙ = CTrI ( Vˆ c,k − Vˆ t,k ) − ωrI × rˆ k , where CrI and ωrI are ˆ c,k − R rˆ k =CTrI R the attitude matrix and angular velocity of the RVD frame with respect to the inertial frame, respectively. CrI and ωrI can also be calculated using Xˆ k . Filter design Because the state equation and measurement equation are both nonlinear, we must use nonlinear methods in the filter design, such as deterministic sampling filters including EKF and UKF.

7.4 Guidance and Control for Rendezvous and Docking By comprehensively considering RVD mission requirements, orbital motion models, and constraints such as fuel consumption and docking time, rendezvous guidance aims to realize a reasonable rendezvous trajectory and simultaneously provide the magnitudes and directions of velocity increments needed for transfer based on certain algorithms. Rendezvous control based on the navigation information and requirements of guidance and attitude control uses certain algorithms to calculate the control forces and torques that need to be exerted on a spacecraft (usually the chaser), which are realized using the actuators (e.g., engines) installed on the spacecraft. It is not difficult to imagine that the requirements for flight time, control precision, fuel consumption, and safety vary in the different phases of an RVD mission. Hence, the guidance and control strategies also vary. In the far-range guidance phase, fuelconsumption optimization is the primary concern because the two spacecraft are far from each other. In the close-range autonomous control phase, the control precision, flight time, and mission safety increase in priority as the two spacecraft approach each other. In the final translation and berthing phase, in addition to the position control of the two spacecraft, the strategy should also consider attitude control, i.e., the six-degrees-of-freedom (6DOF) control of the relative position and relative attitude. In this section, we first introduce the guidance and control requirements of each flight phase in a typical RVD mission, following which we present several rendezvous guidance and control methods that are important for engineering applications.

7.4.1 Flight Phases and Mission Requirements The goal of RVD guidance is to guide the chaser to approach the target and then dock. RVD is a process of both approaching, in terms of distance, and refinement, in terms of precision. Based on the mission requirements, orbital design, configuration of the ground monitoring station, and RVD measurement devices of the chaser, the RVD process is usually divided into four phases: long-range guidance, homing, approach, and final translation and berthing. The last three phases are collectively called the autonomous control phase, as shown in Fig. 7.2. Moreover, the safety

7.4 Guidance and Control for Rendezvous and Docking

389

Orbital altitude

Target orbit

V

(1) Long-range guidance (2) Homing R (3) Approaching (4) Final translation & berthing (4)

(3)

(2)

(1)

Fig. 7.2 Phases of rendezvous and docking

of RVD missions is extremely critical. Therefore, connecting zones or hold points are designed at the connections between neighboring phases, and a safety zone is reserved near the target. 1.

2.

3.

4.

Long-range guidance phase The task in the long-range guidance phase is to guide the chaser from the initial insertion orbit to the orbit that will be the start point of the autonomous control phase. Through multiple orbital control steps, it is desirable to increase the chaser orbit altitude, adjust the phase difference between the two spacecraft, decrease the distance between the two spacecraft, and eliminate the orbital plane errors of the two spacecraft. Homing phase The task in the homing phase is to guide the chaser to the terminal point of this phase through autonomous orbital maneuvers. This terminal point is typically co-planar with the target at the same altitude but a few kilometers away from the target. The chaser can rely on the output data from its onboard relative navigation sensors to complete the calculation of relative navigation, guidance law, and control quantities as well as to achieve autonomous orbital control. Approach phase The task in the approach phase is to capture the docking corridor. From the hold point at the end of the homing phase, the chaser gradually approaches the target through single or multiple orbital control steps until the chaser is guided to the entrance of the target docking corridor. Translation and berthing phase The task in the final translation and berthing phase is to create the initial condition for docking. In the docking corridor, through the 6DOF control of relative position and relative attitude, the chaser should approach the target along its docking axis until safe contact is achieved.

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

7.4.2 Rendezvous and Docking Guidance The first RVD missions were launched in the 1960s. Typical RVD guidance methods include Lambert guidance, which is based on absolute orbital parameters, as well as CW guidance, LOS guidance, and TH guidance, which are based on relative motion parameters. All the US RVD missions in the early days, such as Project Apollo and space shuttles, used Lambert guidance for rendezvous orbital control. Lambert guidance laws are designed on the basis of absolute orbital parameters of the spacecraft, which can be used to solve complicated orbital control problems while considering both the far-range guidance phase and autonomous control phase. Since the beginning of the twenty-first century, CW guidance has been widely used by multiple space agencies worldwide for the autonomous control phase of RVD. Although the computation of the guidance laws is rather simple, CW guidance relies on the estimation of the relative position and velocity by the navigation systems, making it suitable only for RVD missions with nearly circular orbits. LOS guidance has been successfully applied in the USSR/Russian and Chinese RVD missions. LOS guidance has a low requirement on the sensor field of view, high guidance precision, and high fuel consumption. Hence, it is a good candidate for close-range relative position control. TH guidance is normally implemented based on elliptical-orbit, close-range, linearized, dynamic models, and it can be used for the guidance of closerange rendezvous with elliptical orbits. This type of guidance is more computationally expensive than CW guidance, and its computation needs the true anomaly of the target’s orbit. In the autonomous control phase, during a relatively short transfer period, the precision of TH guidance is close to that of Lambert guidance. 1.

Lambert guidance The rendezvous of spacecraft can be formalized as a Lambert problem. That is, as shown in Fig. 7.3, given the initial position R1 , terminal position R2 , and time of transfer flight t, the flight orbit must be determined. The Lambert problem, also known as the Gauss problem, is essentially a two-point boundary value problem of solving differential equations. The Lambert time of flight theorem states that the time of flight for a conic curve passing through two points in space, t, is dependent only on the semi-major axis a, sum of distances R1 + R2 , and chord length c. We now introduce a typical method for solving the Lambert problem. By selecting an independent iteration variable and writing the control equations according to the time of flight theorem, solutions with a certain degree of precision can be obtained through iterative algorithms. Commonly

Fig. 7.3 Schematic of Lambert rendezvous

7.4 Guidance and Control for Rendezvous and Docking

391

used iterative algorithms include the Gauss–Seidel iteration, universal-variable algorithm, Battin’s iterative algorithm, and Gooding’s algorithm. In Battin’s algorithm, x = cos α2 is adopted as the independent iteration variable, where α is the Lagrange multiplier. The corresponding system of equations is  ⎧   ⎪ y = 1 − λ2 1 − x 2 ⎪ ⎪ ⎪ ⎪ ⎪ η = y − λx ⎨ z = 1−λ−xη 2  ⎪ ⎪ ⎪ Q = 43 F 3, 1; 25 ; z ⎪  ⎪ ⎪ ⎩ μ3 ttran = η3 Q + 4ηλ a m

By numerically solving for x using the Newton–Raphson method, the iterative function is obtained as f (x) = η3 Q + 4ηλ − f  (x) =

μ ttran am3

 ηλ  3 4 + 3(2q − 1) 5 3η Q + 4λ η Q+ 4q(1 − q)y y

where ttran = t2 − t1 is the time of orbital transfer, q = 21 (1 − x), λ = √ 1 R1 R2 cos θ , θ is the angle between R1 and R2 , s = 0.5(R1 + R2 + c), s 2   and F 3, 1; 25 ; z is the hypergeometric function obtained based on the appendix of Ref. [24]. After x is obtained, we can obtain the semi-latus rectum of the orbit as p=

R1 R2 sin2 θ 2 2

   2 0.5s 1 − λ 1 − x 2 − λx

from which we achieve the required initial velocity V1+

2.

√ =

μp

R1 R2 sin

θ 2

  R2 R2 − R1 + (1 − cos θ )R1 p

The velocity increment needed for the guidance is VLM = V1+ − V1 , where V1 is the orbital velocity at R1 before orbital control. CW guidance Usually, LEO target spacecraft move in nearly circular orbits. When the relative distance is far lower than the orbital altitude, with the target’s orbital frame, as the reference frame, the orbital motion of the chaser with regard to the target can be described by the CW equations. Then, CW double-pulse guidance laws can be designed based on the equations. Given initial conditions, analytical

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

solutions to the kinematic equations on the orbital plane can be obtained using double-pulse control. 

   x(t) x(t) ˙ ρ(t) = , ρ(t) ˙ = z(t) z˙ (t) The solution to the CW equations is 

        A(t) B(t) ρ(t0 ) B(t) 0 ρ(t f ) = + v(t0 ) ˙ f) ˙ 0) ρ(t D(t) C(t) D(t) ρ(t v(tf )

(7.29)

where 

1 6 sin(ωoT t) − 6ωoT t A(t) = 0 4 − 3 cos(ωoT t)  B(t) =



4 sin(ωoT t)/ωoT − 3t −2/ωoT + 2 cos(ωoT t)/ωoT 2/ωoT − 2 cos(ωoT t)/ωoT sin(ωoT t)/ωoT   0 −6ωoT + 6ωoT cos(ωoT t) C(t) = 0 3ωoT sin(ωoT t)   4 cos(ωoT t) − 3 −2 sin(ωoT t) D(t) = cos(ωoT t) 2 sin(ωoT t)



 ˙ 0 ) , the method of double-pulse Given the initial position and velocity ρ(t0 ), ρ(t guidance control should be used to determine the two velocity pulses at the initial and terminal moments for determining the relative position and velocity,  ˙ f ) , within the desired time t = tf − t0 . If tan(ωoT t/2) = 3/8ωoT t, ρ(tf ), ρ(t the matrix B(τ ) is inversible, and the double-pulse control has the following solution: ˙ 0) v(t0 ) = B −1 (t)[ρ(tf ) − A(t)ρ(t0 )] − ρ(t ˙ f )C(t)ρ(t0 ) D(t)ρ(t ˙ 0 ) D(t)v(t0 ) v(tf )ρ(t The out-of-plane double-pulse control algorithm is V1y = ωoT (y(tf ) − y0 cos(ωoT t))/ sin(ωoT t) − y˙0 V2y = −( y˙0 + V1y ) cos(ωoT t) + y0 ωoT sin(ωoT t) 3.

LOS guidance The equation of relative motion with respect to the LOS frame in LOS guidance was introduced in Sect. 2.7. By omitting the higher-order terms, the relative dynamic equation with respect to the LOS frame can be approximated as

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Fig. 7.4 Control on the dead band of rotational angular velocity

F

ωoff − ωon

− ωoff

ωon

ω −F

!

ρ¨ − ρωζ2 = aξ 2ρω ˙ ζ + ρ ω˙ ζ = aη

(7.30)

For rendezvous, LOS guidance should fulfill two tasks, that is, it should make the rotational velocity of the relative range in space zero and the relative range rate in the LOS direction negative. We call the control of the angular velocity of LOS rotation lateral control and that of the velocity along the LOS direction longitudinal control. (1)

(2)

Lateral control Lateral control, realized through on–off control, is designed to eliminate the angular velocity of LOS rotation. As shown in Fig. 7.4, ωon and ωoff denote on and off thresholds, respectively, which are determined by the measurement precision, number of control on–off, propellant consumption, and minimum work time. Longitudinal control On the basis of lateral control, the goal of longitudinal control is to facilitate a gradual approach between the two spacecraft along a set trajectory or corridor. Using the 1st equation of Eq. (7.30), we can approximately achieve ρ¨ = aξ , a second-order integral, after the elimination of LOS rotation. ˙ + k1 ρ + k2 ρ˙ Sρ = ρ˙ 2 sgn(ρ) where k1 > 0 and k2 > 0. The control law is aξ = −Aξ sgn(Sρ ) where the constant amplitude Aξ > 0.

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7.4.3 Rendezvous and Docking Control The main challenges associated with RVD control include the attitude-stabilization control of the chaser during orbital transfer, high-precision 6DOF control of the relative position and relative attitude in the translation and berthing phase, and longperiod relative position control at hold points. The first two can be formalized as a robust high-precision position and attitude- control problem. 1.

Robust high-precision position and attitude control

There exist various methods for spacecraft position and attitude control, and proportional-integral-derivative (PID) control and phase-plane control are the most commonly used methods in engineering applications. In addition, robust H∞ /H2 control, linear quadratic regulator/linear quadratic Gaussian (LRQ/LQG) control, and adaptive control are used in practice. Feature-model-based intelligent adaptive control was originally generalized and developed from all-coefficient adaptive control and is now considered a practically powerful adaptive control method. Based on the physical principles, dynamic features, and environmental characteristics of spacecraft or controlled industrial objects, this method starts with the construction of a feature model for the object by considering control objectives and performance requirements. With the feature model, intelligent control and adaptive control theories can be used to design the controller based on the complexity and control-performance requirements of the object. Previous theoretical research has shown that closed-loop systems consisting of controllers designed using this method and the object are stable under certain conditions. Here, we focus on the use of feature-model-based intelligent adaptive control in RVD applications. (1)

Feature modeling of the controlled object

Feature modeling constructs a model based on the dynamic features and controlperformance requirements of the controlled object, instead of a precise dynamic analysis of the object. It is a powerful modeling approach proposed for high-order objects with unknown parameters, critical to the modeling of flexible bodies. Some of the characteristics of feature models are summarized below: (a)

(b) (c) (d)

With the same input control, the outputs of the object’s feature model and the actual object are equivalent, i.e., within the allowable range of output error in the dynamical process. In a stable situation, the outputs should be equal. The form and order of the feature model are mostly determined by the controlperformance requirements, in addition to the object features. The constructed feature model should be simpler than the original dynamic equation of the object for convenient realization in engineering applications. Unlike reduced-order models for high-order systems, feature models compress the high-order information into feature variables without information loss. Feature models are typically described by slowly varying differential equations.

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A spacecraft usually has two flexible solar panels, which can be described by a flexible dynamic model of a complex spacecraft featuring a central rigid body and flexible accessories. The details are given in Sect. 4.3. For this type of spacecraft, the feature model of the controlled object can be developed from its dynamic equation through three steps: representation by a rigid body coupled with flexible panels, conversion from a high-order system to a low-order system, and precise discretization. With further theoretical analysis, the feature model can utilize a second-order slowly varying differential equation to describe each channel of roll, pitch, and yaw y(k + 1) = f 1 (k)y(k) + f 2 (k)y(k − 1) + g0 (k)u(k) + g1 (k)u(k − 1) Typically, g1 (k) is much smaller than go (k). (2)

Feature-model-based intelligent adaptive control

(a)

Golden-section adaptive control

Assume that the differential equation of a time-invariant second-order plant with unknown parameters after discretization is y(k + 1) = f 1 (k)y(k) + f 2 (k)y(k − 1) + go (k)u(k) + e(k) where e(k) is the zero-mean white noise. A golden-section adaptive control law is designed as follows: u g (k) = −

1 [L 1 fˆ1 (k)ye (k) + L 2 fˆ2 (k)ye (k − 1)] g0 (k) + λc

Here, ye (k) = y(k) − ym (k), where ym (k) is the output set value, L 1 = 0.382, L 2 = 0.618 (i.e., the golden ratios), and λc is a small constant added to ensure that the denominator is defined. When the golden-section adaptive control law is applied, although the variance of the steady-state output error of the closed-loop system is not guaranteed to be minimal when the estimated value equals the true value, the system still features good dynamic quality. In addition, when the estimated and true values are not equal, the closed-loop system remains stable. Therefore, this controller is of high practical value as the second-best robust control method. (b)

Logic derivative control

The main function of logic derivative control is to provide negative velocity feedback in the system response process, as well as to adjust the differentiation strategy based on the system operation status and control requirements to increase system damping, suppress overshoot, and minimize steady-state micro-amplitude oscillations. This control law can be flexibly expressed in various forms and can be designed in an

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ad-hoc manner according to the specific control tasks in different scenarios. Two commonly used forms of this law are as follows: (i)

Logic derivative control law I: u d1 (k) = −kd1 | y˙ (k)|ed1

(7.31)

where  kd1 =

c1 sgn( y˙ )

∀τ ∈ [t − Ts , t], |y(τ )| ≤ r

c1 (0.5|y| + cd1 )−1/2 sgn( y˙ )

∀τ ∈ [t − Ts , t], |y(τ )| >r

 , ed1 = ρ( max [ y˙ (τ )] t−T ≤τ 0 ki− , ye (k) y˙e (k) ≤ 0

Combining the above control laws, the total control variable is u(k) = u g (k) + u d (k) + u i (k) (3)

Golden-section and phase-plane integrated adaptive control

Phase-plane control is a classic jet control method that has been extensively used in the attitude control of satellites and spaceships. However, this control method has a

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fairly large number of design parameters that need to be empirically determined. By incorporating the concept of golden-section adaptive control in phase-plane control, a second-order nonlinear controller, the parameter design problem of phase-plane control can be solved. Reference [10] proposed golden-section and phase-plane integrated control as well as logic-differential and phase-plane integrated control to solve the problem of parameter design in phase-plane control. This method has been applied in orbit for multiple RVD missions of the Shenzhou manned spacecraft and Tianzhou cargo spacecraft. The method successfully solved the problems of attitude-stabilization control during orbital transfer and high-precision 6DOF control of the relative position and relative attitude in the translation and berthing phase, exhibiting excellent control performance. 2.

Long-period relative position control at the hold point

With a long holding period, the relative position control strategy at the hold point primarily aims to minimize the propellant consumption of the chaser while satisfying the demand of relative position holding precision. Therefore, the major concern when designing such control parameters is the trade-off between control precision and propellant consumption. Let us discuss this issue in two situations: out of and within the orbital plane. (1)

Out of the orbital plane

The dynamic equation of the out-of-plane relative motion is y¨ + ω02 y = a y

(7.34)

where the trajectory is given by 

y(t) = ωy˙00 sin(ω0 t) + y0 cos(ω0 t) y˙ (t) = y˙0 cos(ω0 t) − y0 ω0 sin(ω0 t)

That is, the curve out of the orbital plane is sinusoidal and not coupled with the in plane curve. Hence, the maximum out-of-plane deviation should be y02 + ( yω˙0 )2 . For instance, if the initial position and velocity errors are 330 m and 0.3 m/s, respectively, the maximum out-of-plane deviation is calculated to be 450 m. For fairly distant (e.g., 5 km) hold points, in one orbital period, a control-free approach out of the orbital plane is acceptable. Consequently, safety issues can be avoided, and the position control criteria can be met while securing propellant savings. If a longer holding time or a higher holding precision is desired, the outof-plane control should be performed at the zero position for decreasing propellant consumption.

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(2)

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Within the orbital plane

The dynamic equation of the in-plane relative motion is 

x¨ + 2ω0 z˙ = ax z¨ − 2ω0 x˙ − 3ω02 z = az

(7.35)

There is a constant disturbance term 3ω02 z in the z direction, which may considerably increase with time. This suggests that, when designing the control parameters, if we attempt to set a greater margin for the position holding error to achieve fuel savings, the result might be counterproductive. On the other hand, a higher control precision in the z direction will result in a lower propellant consumption, provided frequent jetting can be avoided.

7.5 Guidance and Control for Reentry Spacecraft return and reentry can be categorized into ballistic, semi-ballistic (or semi-lifting), and lifting reentry based on different aerodynamic characteristics of the spacecraft in the reentry phase. In the Earth’s atmosphere, two major aerodynamic forces act on the spacecraft: a drag opposite to the velocity direction and a lift perpendicular to the velocity direction. By dividing the lift by drag, we obtain an important parameter called the lift-to-drag ratio, L/D. Ballistic, semi-ballistic, and lifting reentry correspond to the reentry of spacecraft with a lift-to-drag ratio of 0–0.1, 0.1–0.5, and > 0.5, respectively. Owing to the severe aerodynamic heating and braking overload during reentry, the aerodynamic configuration, structure, and return trajectory of spacecraft reentering the Earth’s atmosphere must be specially designed according to specific reentry conditions.

7.5.1 Ballistic Reentry The lift-to-drag ratio of a spacecraft reentering the Earth’s atmosphere through ballistic reentry is zero or close to zero (L/D = 0–0.1). That is, the spacecraft traveling through the atmosphere is only subject to drag, with no or limited uncontrollable lift. Hence, once the spacecraft leaves the original orbit, it returns to the ground along the predetermined trajectory free of control. Because this motion is similar to the warhead motion of ballistic missiles, we name it ballistic reentry. This method was a popular choice for returnable spacecraft in the early days of space exploration. One of the advantages of ballistic reentry is that a spacecraft returning in this manner is free of aerodynamic lift. Consequently, the spacecraft design can adopt

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simple shapes such as axisymmetric rotators with a blunt end like spheres and cones. Another benefit associated with ballistic reentry is the relatively short period that the spacecraft has to spend in the atmosphere. The short period results in a low amount of total aerodynamic heating, which allows for a simpler thermal protection system. In addition, ballistic reentry spacecraft are the most technically achievable, which is why all the US and USSR returning spacecraft in the early days adopted this technology with no exceptions. Its major disadvantage is that the spacecraft’s motion in the atmosphere is uncontrollable, implying that it is impossible to correct the fall-point location error during reentry. This leads to an inevitably large recovery zone. Furthermore, owing to the rapid deceleration of the spacecraft, ballistic reentry generates the largest peaks of heat flux and braking overload among the three reentry scenarios, although the total heat generated is small. To reduce the peak heat flux, the inflow-facing end of the spacecraft has to be designed as part of a sphere with a considerably large diameter. The main method to reduce the peak overload is to very carefully control the reentry flight-path angle; this is especially critical in manned spacecraft. Ballistic reentry spacecraft can be further divided into two categories, depending on whether the drag coefficient is constant or variant. The latter is still in the conceptual stage and has not been realized in engineering practice. Based on the idea of a varying drag coefficient, during the starting phase of reentry in the region where the atmosphere is thin, the flight-path angle can be set to be larger to ensure that the spacecraft does not exit the atmosphere. As the spacecraft enters deeper into the atmosphere, the drag coefficient can be adjusted to alleviate heating and overload.

7.5.2 Semi-ballistic Reentry A certain amount of lift can help spacecraft return by increasing the width of its reentry corridor. While other conditions are kept the same, a higher lift-to-drag ratio also reduces the peaks of braking overload and heat flux. In addition, a spacecraft in semi-ballistic reentry can change the direction of the lift through roll control and thereby adjust, to some extent, its trajectory in the atmosphere. Because the spacecraft has such a maneuvering capacity, a relatively small fall-point dispersion can be expected, enabling control of the landing site within a reasonable area. However, spacecraft with lift naturally spend a longer period in the atmosphere with a longer flight path compared to ballistic reentry spacecraft; therefore, they are subjected to a greater amount of total heat. Semi-ballistic reentry has the advantages of ballistic reentry spacecraft, namely, a simple structure and easy thermal protection, while overcoming some of their disadvantages to some extent by properly making use of the lift. In terms of the structure, the lift offsets the center of gravity from the central axis by a small distance, as shown in Fig. 7.5. Thus, the spacecraft (with the exception of those with spherical shapes) will have a certain angle of attack (AOA) generating lift, which is called the trim AOA. Of course, this lift is relatively small at less than half of the drag. If the

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Fig. 7.5 Effect of the center of gravity offset on the lift

spacecraft is rotated about its geometric axis by an angle ϕ, the lift generated by the trim AOA can be decomposed into a vertical component (upward force) and a horizontal component (lateral force). During reentry, these two components of lift can be adjusted by controlling the roll angle ϕ using a logic program. In this manner, the trajectory of the spacecraft in the atmosphere, and thereby its fall points, can be controlled to a certain degree. Hence, this type of reentry using ballistic return vehicles with limited controllable lift is referred to as the semi-ballistic reentry or semi-lifting reentry. This technology was first used by the US Gemini spacecraft to achieve a certain lift-to-drag ratio by employing a trim AOA. The spacecraft was able to adjust its fall point within a tongue-shaped zone with a length of 1,200 km and width of 110 km. The USSR/Russian Soyuz spacecraft and the Chinese Shenzhou manned spacecraft also adopted this type of reentry. With this technology, the fallpoint zone can be within several kilometers in dimensions, which is smaller than that of traditional ballistic reentry. Another type of semi-ballistic reentry is skip reentry. It consists of alternating ballistic flight and skipping flight. At a certain time after the start of skip reentry, the spacecraft bounces out of the atmosphere with a change of lift. This is followed by another period of ballistic flight before the spacecraft enters the atmosphere again. This process can be repeated multiple times. Every time the spacecraft reenters, it is decelerated by the atmosphere. Because the return orbit experiences drastic variations in altitude, it is called a skipping orbit. Even the orbit of a return vehicle that does not bounce out of the atmosphere after reentry but still experiences large variations in altitude can be called a skipping orbit. A spacecraft returning with the second cosmic velocity usually adopts a skipping orbit to reduce overload and adjust the fall point in a large range. These are the two major advantages of skip reentry, which greatly improve the reentry conditions. This type of reentry was adopted by the US Apollo lunar spacecraft to return to Earth after its Moon exploration mission as well as the Chinese Chang’e 5-T1 experimental robotic spacecraft. In terms of shape, semi-ballistic reentry spacecraft essentially retain the simple structure of conventional ballistic spacecraft. Because spacecraft adopting these two reentry technologies have no or limited lift, vertical landing is the only option. When approaching the ground, a parachute system must be used to ensure a safe landing.

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7.5.3 Lifting Reentry When horizontal landing is desired, a sufficiently large lift must be exerted on the spacecraft so that the reentry orbit, especially the landing section, is sufficiently gentle. The main purpose of horizontal landing is to achieve non-destructive and precision landing, making the spacecraft reusable. Spacecraft capable of horizontal landing typically have a lift-to-drag ratio larger than 1, implying that the lift is greater than drag during reentry. Such a high lift will not be achievable by offsetting the center of mass from the axis of symmetry as discussed above. Therefore, a lifting reentry spacecraft cannot be designed as a rotator. Instead, the spacecraft must have an asymmetric lifting body. The lifting body could be winged or wingless, as long as the lift-to-drag ratio is greater than unity. However, a wingless lifting body generates lift solely from its aerodynamic shape, which will be compromised by the complicated shape and increased size/mass. Hence, wingless lifting bodies not commonly used. All the existing lifting spacecraft, such as the US space shuttles, and those under development have winged lifting bodies and are similar to aircraft in shape. Owing to the maneuvering flexibility and the ability for horizontal landing, lifting reentry spacecraft are successful in avoiding some of the problems associated with ballistic and semi-ballistic reentry, with the additional advantage of low overload. However, they have new problems. For instance, because lifting reentry is gentler, its range and time are much longer than those of ballistic and semi-ballistic reentry. Although the peaks of heat flux and deceleration overload are lower, the total amount of heating and the heating period is higher. Furthermore, because lifting spacecraft tend to have much more complicated geometries, in addition to the requirement of reuse, many aspects such as control, aerodynamics, thermal protection, and structures could be quite complex.

7.5.4 Reentry Guidance 1.

Coordinate systems of reentry The return frame Oxyz, fixed on Earth, is the reference system for describing the return- vehicle orbit and attitude movement during the entire returning process. The origin O is the point where the line passing through the center of mass of the return vehicle and the Earth’s core intersects with the surface of the standard Earth ellipsoid at the initial instant of return, and it rotates with the Earth. At the initial instant of return, the Oy axis passes through the Earth’s core and the center of mass of the returning vehicle, and the upward direction is defined as its positive direction. At the initial instant of return, the Oz axis is normal to the original orbit of the returning vehicle, pointing inward. The Ox axis is determined by the right-hand rule. Figure 7.6 shows a schematic of this return frame.

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Fig. 7.6 Definition of the return frame

x

y o OE

z

The reentry frame E xe ye z e , also an Earth-fixed frame, is used for the convenience of guidance law design. The origin E is the point where the line passing through the nominal reentry point and the Earth’s core intersects with the surface of the standard Earth ellipsoid, and it rotates with the Earth. The E ye axis passes through the Earth’s core, and its positive direction points upward. The included angle between the E xe axis and the north is Ae , which is measured as positive in the clockwise direction when viewing from the North Pole. Ae is also referred to as the velocity azimuth angle of the nominal reentry point. The E z e axis is determined by the right-hand rule. Figure 7.7 shows the definitions of Ae , λe , e , λ f , and f , where λe , e , λ f , and f denote the longitude of the nominal reentry point, latitude of the nominal reentry point, longitude of the nominal deployment point, and latitude of the nominal deployment point, respectively. The return-vehicle frame O1 x1 y1 z 1 is fixed to the returning spacecraft. The origin O1 is fixed at the center of mass of the vehicle. The positive O1 x1 axis Fig. 7.7 Definitions of Ae , λe , e , λ f , and f

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is parallel to the longitudinal geometric axis of the vehicle and points towards the larger end of the vehicle. The O1 y1 axis is on meridian plane III, i.e., the half plane formed by the longitudinal geometric axis and the center of mass. The O1 z 1 axis is determined by the right-hand rule. This frame is used for the following reasons. Firstly, it is the installation frame of SINS. That is, gyroscopes and accelerometers are all installed with respect to this frame. Secondly, all the aerodynamic parameters obtained from ground wind-tunnel tests are given in this frame. Thirdly, in kinematic equations, the attitude of this frame with respect to the return frame O x yz is defined as the return vehicle attitude. The 2nd return-vehicle frame O1 x2 y2 z 2 is obtained by rotating the 1st returnvehicle frame O1 x1 y1 z 1 about the O1 z 1 axis by an angle an . This new frame mainly has three uses. Firstly, O1 x2 y2 z 2 is simple and convenient to use to describe the inertia tensor of the return vehicle because O1 x2 , O1 y2 , and O1 z 2 are approximated as the principal inertia axes of the return vehicle when designing its mass distribution. Secondly, it is the installation frame of the attitude- control engine, which is typically installed in such a manner that a torque is present with respect to only one axis to minimize the three-axis coupling of attitude dynamics. Thirdly, an is designed to equal the nominal trim AOA under some flight conditions so that the O1 x2 axis can be approximated as the direction of the velocity V in reentry control. Figure 7.8 shows the definitions of the body frames of the return vehicle.

Fig. 7.8 Definitions of the body frames of the return vehicle

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2.

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Dynamic equation of the reentry phase ⎤ ⎡ ˙ ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎧⎡ ˙ ⎤ Vx Wx gx x Vx ⎪ ⎪ ⎪ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎪ T ⎪ ⎣ V˙ y ⎦ = [A10 ] ⎣ W˙ y ⎦ + [A1 ]⎣ y + r0 ⎦ + [B1 ]⎣ Vy ⎦ − ⎣ g y ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ z Vz gz ⎪ V˙z W˙ z ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ ⎪ x˙ Vx ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎪ ⎪ ⎣ y˙ ⎦ = ⎣ Vy ⎥ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ Vz z˙ ⎪ ⎪ ⎪ ⎡⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎤ ⎨⎡ ⎤ ω˙ x f x1 Mr x1 Md x1 Mcx1 (7.36) ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥ −1 ⎢⎢ ⎪ ⎪ ⎣ ω˙ y ⎦ = [I1 ] ⎣⎣ f y1 ⎦ + ⎣ Mr y1 ⎦ + ⎣ Mdy1 ⎦ + ⎣ Mcy1 ⎦⎦ ⎪ ⎪ ⎪ ⎪ ω˙ z f z1 Mr z1 Mdz1 Mcz1 ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ q0 q˙0 ⎪ ⎪ ⎪ ⎥ ⎥ ⎢ ⎢ ⎪ ⎪ ⎢ q1 ⎥ ⎢ q˙1 ⎥ 1 ⎪ ⎪ = [] ⎥ ⎥ ⎢ ⎢ ⎪ ⎪ ⎪ ⎣ q2 ⎦ ⎣ q˙2 ⎦ 2 ⎪ ⎪ ⎪ ⎪ ⎪ q˙3 q3 ⎪ ⎪ ⎩ m˙ = − f (Fx , Fy , Fz )

3.

T  where W˙ x W˙ y W˙ z are apparent accelerations of the three axes of the returnvehicle frame and are determined by the engine thrust and aerodynamic forces exerted on the return vehicle; [A10] is the direction cosine matrix from the reentry frame to the return-vehicle frame; [A1] and [B1] are the matrices associated with centrifugal acceleration and Coriolis acceleration, respectively; gx , g y , and gz are the projections of gravitational acceleration in the return frame T  Oxyz; I1 is the moment of inertia in the return-vehicle frame; f x1 f y1 f z1 are the gyroscope torques generated by the rotation of the return vehicle; T  Mr x1 Mr y1 Mr z1 are the aerodynamic torques acting on the return vehicle;  T Mdx1 Mdy1 Mdz1 are the aerodynamic damping torques acting on the return  T   vehicle; Mcx1 Mcy1 Mcz1 are the attitude-control torques; and f Fx , Fy , Fz is the rate of mass change due to the action of actuators working in different directions. Reentry guidance law Reentry guidance is mainly performed based on the nominal trajectory method or predictive guidance method. (1)

Nominal trajectory method The nominal trajectory method involves the pre-loading of reentry orbit information into the computer of the return vehicle. During reentry, however, the actual trajectory may deviate from the nominal one owing to initial-condition errors, atmospheric changes, and variations in the aerodynamic coefficients. At this point, the guidance system produces an error

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signal by comparing the actual and nominal trajectories and provides information about the required attitude angle based on the error signal. With this information, the attitude-control system makes the actual trajectory follow the nominal trajectory. The nominal trajectory method primarily consists of two parts, nominal trajectory planning and the online tracking of the reference trajectory. Through offline planning, a nominal trajectory satisfying various requirements in terms of the overload, thermal protection, and precision is obtained. The state variables of the nominal trajectory are stored in the onboard computer of the return vehicle. By comparing these state variables with the current state variables calculated by the navigation system, the vehicle acquires the error signal, based on which a guidance law is produced. This method requires the stored state variables to be functions of given independent variables, which could be time, velocity, or range. Here, we take a further look at this method with an example in which time is used as the independent variable. The associated guidance law can be divided into longitudinal and lateral parts. The longitudinal guidance law takes the overload of nominal trajectory with time n, rate of attitude change h˙ x , range L, and range rate L˙ as the state variables. The return vehicle pre-loads these state variables with the nominal trajectory and compares them with the corresponding actual values computed by the navigation system during flight. The generated ˙ can then be used to calculate the state errors, δn, δ h˙ x , δL, and δ L, ∗ commanding bank angle γ VT , which is the angle of the total lift deviation from the E xe ye plane of the reentry frame. A simple linear feedback is written as * = k1 δn + k2 δ h˙ x + k3 δL + k4 δ L˙ (L/D) cos γ VT

(7.37)

where L/D is the lift-to-drag ratio and k1,k2,k3,k4 are the gains of the guidance law. The main objective of longitudinal guidance law design is to determine the values of gain. Lift control can only change the direction of the lift. Because longitudinal range control uses cos γ ∗V T times the actual total lift, lateral range control is only left with sin γ ∗V T times the lift, where the magnitude of γ ∗V T is predetermined by the longitudinal guidance law. Hence, the lateral range control merely changes the sign of bank angle. In order to keep the lateral range error asymptotically within an allowable range, an appropriate “funnel” can be placed along the longitudinal reentry plane E xe ye , with the spout end of the “funnel” set at the standard fall point. When the return vehicle moves laterally to the boundary of the “funnel,” the bank angle changes its sign. Consequently, the motion of the vehicle is always confined within the “funnel.”

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(2)

Predictive guidance method The predictor–corrector guidance method focuses on the elimination of the error between the predicted fall point of the actual orbit and the preset fall point. Unlike the nominal trajectory approach, this method tracks the error between the fall point corresponding to the actual reentry orbit and the theoretical fall point at every instant. The control demand is then generated based on the error information while considering the constraints on the maximum overload and heat flux. Thus, the orbital control of the spacecraft is realized. Predictive guidance can achieve a higher precision of the fall point than the nominal trajectory method. In addition, it is not sensitive to the initial conditions of reentry. Its major restriction is the difficulty of real-time in-orbit computation. However, with the improvement of onboard computational power, predictor–corrector guidance is being increasingly used in practical applications. There are two major issues concerning predictor–corrector guidance: the prediction of the fall-point location and selection of the guidance method. For fall-point prediction, two options are available: fast numerical integration and closed-loop analysis. In the former, the kinematic equations are numerically integrated by the onboard computer. It can process any possible flight condition with high precision. Parameters such as range, acceleration, and heat flux can also be predicted. However, numerical integration has high requirements for computational power and storage. On the other hand, closed-loop analysis aims to obtain an explicit solution by reasonably simplifying the kinematic equations of the return vehicle. Owing to the complexity of reentry motion, many assumptions are usually adopted to solve the problem. Hence, the obtained result is only an approximate solution to the kinematic equations; it is also impossible to obtain the analytical solution under all conditions. Although the closed-loop analysis method has more constraints and a lower prediction precision, in addition to poor versatility and an inability to handle any flight condition, it is less computationally expensive, with lower requirements for computational power and storage.

7.6 Guidance and Control for Soft Landing on Extraterrestrial Bodies 7.6.1 Soft Landing on the Moon The requirements for soft landing on the Moon can be divided into three levels from low to high: the requirement of essential landing safety without consideration of the fall-point location, requirement of a certain degree of fall-point control as well as obstacle avoidance, and requirement of high-precision fall-point control with

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accurate landing. In the history of Moon exploration, these three levels have been realized step by step. In this section, we present the guidance and control approaches to meet each requirement. 1.

Guidance and control for a safe soft landing The basic requirement for safe landing is that the lunar lander should be able to land on the Moon at a low vertical velocity and zero horizontal velocity in the upright attitude; the precision of the fall point is not considered. Unmanned lunar probes launched before the 1960s were, in general, at this level. This was because the safe landing was technically easy to achieve. Another important reason was that knowledge about the Moon was fairly limited at that time; thus, there were no specific expectations about the landing area. For lunar landers with this level of landing requirements, the gravity-turn method is the most commonly used guidance approach [24], in which the direction of the braking acceleration is always kept opposite to that of the velocity vector by imposing attitude control on the vehicle during landing. The gravity-turn process is illustrated in Fig. 7.9. Here, we assume that the Moon’s surface below the flight trajectory is a uniform gravitational field with a gravitational acceleration of g, and the thrust direction is always strictly opposite to the velocity direction during the turning process. With the probe velocity v, thrust-to-weight ratio of the engine u, and included angle between the velocity and gravity ψ, we write the dynamic equations of the center of mass of the lander in the turning process as follows: 

v˙ = −gu + g cos ψ vψ˙ = −g sin ψ

Fig. 7.9 Schematic of the process of gravity-turn landing

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7 Autonomous Guidance, Navigation, and Control of Spacecraft

It can be proved that if u > 0, the angle ψ approaches zero over time. Consequently, the lander’s attitude naturally turns upright. Hence, the self-turning of the probe into the desired vertical attitude during deceleration is the working principle and the most distinct feature of the gravity-turn method. However, the guidance law must also satisfy the control requirements of attitude h and velocity v. This is achieved by tuning the control variable u to form a closed-loop gravity-turn guidance law. Normally, the calculation of the control variable u requires the tracking of a nominal trajectory. There are multiple options available for the selection of such a nominal trajectory, e.g., the altitude– velocity curve [25], slant range–velocity curve [26], and time–altitude curve [27]. Here, we use the time–altitude curve as an example to briefly introduce the closed-loop gravity-turn method. Using the state variables × 1 = v−vf, x2 = , and x3 = h, where vf denotes the terminal target velocity, we construct a model for the controlled object ⎤ ⎤ ⎡ g cos x2 − gu x˙1 ⎥ g sin x2 ⎣ x˙2 ⎦ = ⎢ ⎦ ⎣ − x1 +v f x˙3 −x1 cos x2 ⎡

(7.38)

y = x3 Taking the second derivative of the output equation, we have y¨ = x¨3 = −g(1 −

vf sin2 x2 ) + gu cos x2 x1 + vf

(7.39)

If the control input satisfies u=

    v f sin2 x2 1 g 1− + h¨ d − c2 ( y˙ − h˙ d ) − c1 (y − h d ) (7.40) g cos x2 x1 + v f

where c1 and c2 are positive constants, then the output equation becomes y¨ = h¨ d − c2 ( y˙ − h˙ d ) − c1 (y − h d )

(7.41)

We can set proper values for c1 and c2 to make the above equation stable. It can be seen from this control process that, because this is a continuous tracking approach, the output of u is generally continuous as well. This means the engine should be able to continuously vary thrust. The closed-loop gravity-turn guidance law does not consider propellant consumption, but optimization can be approximately ensured through the design of the trajectory to be tracked. With minimum propellant consumption during the gravity turn as the optimization objective, analysis using optimal control theory shows that the optimal gravity-turn guidance law is a bang-bang controller. The

7.6 Guidance and Control for Soft Landing on Extraterrestrial Bodies

2.

409

engine only needs to be switched on and off, and the thrust need not be varied; further, switching needs to be performed once at most [28]. According to the gravity-turn equations, the landing-site location is not one of the objectives of guidance and control, suggesting that this type of guidance law can only be applied to basic exploration missions without requirements on the fall-point location. Guidance and control for soft landing with fall-point control capabilities Since the Apollo age, engineers have developed the ability to control the landing site, as well as the ability of terminal obstacle avoidance. In order to achieve this, the flight trajectory has been made more complicated, and the flight is decomposed into various phases according to the guidance laws. Figure 7.10 shows the flight-trajectory composition of the Apollo Lunar Module descent [29]. The entire powered descent process is divided into three phases: braking, approach, and terminal descent. The braking phase starts from an altitude of 15 km and ends at an altitude of 2.3 km. This is an autonomous phase dominated by the GNC system, with the main objective of deceleration. The approach phase follows, during which the lunar module’s pitch attitude is first adjusted to have a 9° offset from the plumb line. It then keeps this attitude and descends with a constant acceleration along an approximately straight trajectory. In this phase, the pilot can observe the Moon’s surface through the porthole and reset the landing site. When the altitude reaches 46 km, the lander enters the terminal descent phase, during which the pilot implements attitude inclination control to achieve translation and the descent velocity is controlled by the guidance law until the lander contacts the Moon’s surface. This phase can also switch to the GNC autonomous vertical descent control. The polynomial guidance method was used in both the main braking and approach phases of Apollo descent. In this method, the engine thrust acceleration is assumed to be a quadratic function of time. Based on the current and target positions and velocities, a parameter of the acceleration function satisfying the terminal conditions is computed. The thrust acceleration commands can then be determined based on this parameter [30]. Figure 7.11 shows the definition of the guidance frame. OG is located at the target point. XG points to the sky. ZG is on the horizontal plane and points to Powered descent initialization Approach phase (low gate) Terminal descent phase Ignition

Touchdown Fig. 7.10 Flight-phase composition of the Apollo descent

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7 Autonomous Guidance, Navigation, and Control of Spacecraft XG

ZG

YG OG

Fig. 7.11 Definition of the guidance frame

the flight direction of the probe. YG and the other two perpendicular axes form a Cartesian coordinate system. In the computation of the guidance law, we first assume a reference flight trajectory within the guidance frame, the thrust acceleration of which is a quadratic function of time. Conducting integration twice, we arrive at r ref (t) = r tG + vtG (t − tt ) + atG (t − tt )2 /2 + j tG (t − tt )3 /6 + stG (t − tt )4 /24

(7.42)

where t is the current instant, tt the terminal target instant, r tG the terminal target position, vtG the terminal target velocity, atG the terminal target acceleration, j tG the terminal target jerk, and stG the terminal target snap. We define tgo = tt − t, which denotes the remaining guidance time. Then, Eq. (7.42) becomes 2 3 4 /2 − j tG tgo /6 + stG tgo /24 r ref (t) = r tG − vtG tgo + atG tgo

(7.43)

We next take the derivative of Eq. (7.43) with respect to t. The velocity and acceleration corresponding to the current reference trajectory are 2 3 /2 − stG tgo /6 vref = vtG − atG tgo + j tG tgo

(7.44)

2 aref = atG − j tG tgo + stG tgo /2

(7.45)

Based on position and velocity feedback, we obtain the commanded acceleration of the current instant

7.6 Guidance and Control for Soft Landing on Extraterrestrial Bodies

acmd,G = aref − (vG − vref )

KV Kr − (r G − r ref ) 2 tgo tgo

411

(7.46)

where r G and vG are the position and velocity of the lander within the guidance frame. By substituting Eqs. (7.43)–(7.45) into Eq. (7.46) and carefully setting the parameters, the commanded acceleration can be readily calculated as 2 −6vtG +vG /tgo +atG acmd,G = 12r tG −r G /tgo

(7.47)

where tgo is the only parameter to be determined. Theoretically, tgo can take any value. However, it should satisfy a certain constraint, such as a jerk constraint or acceleration constraint, in the context of soft landing. Hence, tgo can be calculated according to its constraint. The polynomial guidance equation indicates that by constraining the 3D terminal position, landing-site control can be realized. Moreover, obstacle avoidance control is conveniently achievable by resetting the landing site. However, this type of guidance law, by default, does not optimize propellant consumption. In order to solve this problem, polynomial guidance in the braking phase tends to be replaced with an explicit guidance law capable of fuel-consumption optimization in real engineering applications, i.e., a time-varying function of the magnitude and direction of thrust is used to minimize the propellant consumption "tf m(t)dt ˙ → min

J= to

This problem usually needs to be solved using numerical optimization methods, but the associated computational costs are challenging for onboard computers. Hence, engineers have developed several explicit calculation methods for the thrust commands based on various assumptions. Now, let us examine a simple second-best explicit guidance method [31]. This method involves two assumptions: (a) the engine thrust is fixed, and (b) the gravitational field is constant, i.e., the ground plane assumption. The definition of the guidance frame for this method is shown in Fig. 7.12, where ζ is the local plumb line, η is within the local horizontal plane and normal to the orbital plane, and ξ is the other orthogonal axis within the horizontal plane and points to the flight direction. The projections of the spacecraft velocity on these three axes are u, v, and w, respectively. F is the engine thrust. θ and  denote the vertical and horizontal attitude angles, respectively. The terminal constraints of this guidance law include the three-component velocity and altitude

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Fig. 7.12 Definition of the guidance frame for a second-best explicit guidance method

⎧ u(tgo ) = u f ⎪ ⎪ ⎪ ⎨ v(t ) = v go f ⎪ w(t ) = w go f ⎪ ⎪ ⎩ ζ (tgo ) = ζf

(7.48)

Because the horizontal position is not constrained, the attitude angles θ and  satisfying time or fuel optimization can be written as 

tan ψ = tan ψo tan θ = κ1 + κ2 t

(7.49)

where 0, κ1, and κ2 are constants. During descent, the engine thrust is mostly used for deceleration; therefore, the attitude angle θ is typically small. Approximately, we have 

cos θ ≈ 1 sin θ = κ1 + κ2 t

(7.50)

If the thrust acceleration of the engine is assumed to be constant, by neglecting the η and ξ components of the gravitational acceleration, we can write the kinematic equations of the probe in the following form:

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413

⎧ ζ˙ = w ⎪ ⎪ ⎪ ⎨ u˙ = a cos ψ N 0 ⎪ v ˙ = a sin ψ N 0 ⎪ ⎪ ⎩ w˙ = aN (κ1 + κ2 t) + gζ

(7.51)

where aN is the thrust acceleration. The integration of Eq. (7.51) yields the transformation from initial conditions to terminal constraints, based on which the 4 parameters, 0, κ1, κ2, and tgo, can be determined. Finally, the attitude angles are obtained ψ = arctan  θ = arcsin

vf − v0 uf − u0

aV − (u 2 + v2 )/ζ − gζ aN

(7.52)  (7.53)

where tgo =



# (u f − u 0 )2 + (vf − v0 )2 aH

# 2 aV = [6(ζf − ζ0 ) − 2(wf + 2w0 )tgo ] tgo

3.

(7.54) (7.55)

Here, aH is the horizontal component of aN. The above guidance law has 4 constraints, including the terminal position and velocity. Theoretical analysis has shown that a maximum of 5 terminal variables can be constrained in fixed-thrust guidance. Therefore, more complicated guidance laws may have the 3-component velocity, altitude, and out-of-plane lateral position error as the constrained variables, whereas the range is not constrained. Chang’e 3, the first Chinese soft-landing lunar rover, used exactly this type of 5terminal-state guidance law in its main braking phase. Specifically, an adaptive fixed-thrust explicit guidance method was used [32]. Guidance and control for accurate soft landing After the year 2000, in the context of the Constellation Program, which aims at send humans to the Moon, NASA explicitly stated that their goal is to improve the accuracy of lunar landing to 100 m, i.e., realizing the concept of “precision lunar landing” [33]. In order to achieve this goal, with the premise of optimal propellant consumption, the guidance law must have full control over the 6 components of terminal position and velocity. Many researchers have conducted extensive studies since the proposal of accurate landing. Based on a theoretical analysis, the optimal guidance for accurate landing was obtained: the engine should be in an on–off state, i.e., “min–max” or “max–min-max” state [34]. On the basis of this conclusion, reference flight trajectories and engine-thrust curves have been calculated for some specific problems using numerical optimization. However, the numerically obtained

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results are only of theoretical value. In other words, the real-time guidance problem has been converted to a trajectory design problem, which cannot be executed on the onboard computer of the lander. Based on the existing fixed-thrust guidance law, varying the thrust is considered an effective means for range control [35]. The above-mentioned fixed-thrust explicit guidance method can constrain five state variables except range. The actual flight range is determined by engine thrust. With the same initial mass, altitude, and velocity, a higher retrorocket thrust will lead to a shorter actual flight range, and vice versa. Hence, we can use the thrust of one of the engines as our input and compare the predicted fall-point range from the guidance law with the target range. Based on the results, we can adjust the retrorocket thrust. This method generates a series of continuous thrust-varying commands, based on which range control is also achieved. However, it is worth noting that continuous thrust variation apparently does not comply with the optimal bang-bang control obtained from theoretical research; therefore, fuel consumption will not be optimal. Therefore, the practical application of the optimal control theory is an important topic for current research.

7.6.2 Soft Landing on Mars 1.

Phases of soft landing on mars Mars is a planet with an atmosphere. Although its atmospheric density is remarkably lower than that of Earth, the existence of an atmosphere suggests that the landing process is overall dominated by aerodynamic deceleration. Hence, the retrorockets are only ignited during the final phase, which results in low fuel consumption. In general, Mars atmospheric entry can be divided into three phases: (1) aerodynamic deceleration, (2) parachute deployment, and (3) obstacle avoidance and soft landing relying on the main engines in the terminal descent phase after parachute release. Therefore, soft landing on Mars can be characterized as a combination of Earth atmospheric reentry and the final phase of soft landing on the Moon. The aerodynamic deceleration of Mars’ atmospheric entry mostly adopts the ballistic method. That is, there is no active control other than navigation, and the landing-site accuracy dispersion can be up to hundreds of kilometers. Curiosity used the semi-ballistic method. That is, with the involvement of the GNC system, the landing dispersion ellipse was tightened to 18.1 km × 6.9 km, and the actual landing deviation was only 2.2 km. During parachuting, the GNC system only provides continuous navigation measurements, without the implementation of active control. After parachute release, there are three options for Mars rovers to land. The first method is the use of airbags, as in the case of Spirit and Opportunity. With airbags as a cushion, the rover can land on Mars after bouncing. This is a fairly simple method that does not require complex GNC equipment. The second method involves the use of conventional retrorockets, similar to

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Table 7.1 Landing parameters of three representative Mars probes Viking

Phenix

Curiosity

Mission phase

Altitude (km)

Mission phase

Altitude (km)

Mission phase

Altitude (km)

Lifting entry

125–6

Ballistic entry

125–12.9

Attitude hold

126−52

Lift control

52−14 k

Parachuting

14−2 k

Parachuting

6–1.4

Parachuting

12.9–0.94

Free fall (2 s)

1.4–1.3

Free fall (3 s)

0.927–0.76 k Powered approach

Gravity turn

1.3–1.82 × Gravity turn 10−2

Constant-speed 1.82 × descent 10−2 −0

0.76–5.2 × 10−2

Constant-speed 5.2 × descent 10−2 −0

2−1.05 × 10−1

Constant-speed 1.05 × descent 10−1 −0.7 × 10−1 Decelerating descent

0.7 × 10−1 −1.86 × 10−2

Suspension and 1.86 × 10−2 separation

2.

the terminal descent phase of lunar landing. There have been studies proposing active obstacle avoidance during this phase. However, thus far, this method has not been successfully realized with any Mars probe. The third method makes use of a sky crane, which has been applied to the latest Curiosity probe. The lander was suspended at some altitude, following which the sky crane released the rover onto the Martian surface. Table 7.1 lists the landing parameters of three representative Mars probes. Guidance and control algorithms for soft landing on mars (1)

Aerodynamic deceleration The guidance and control methods for atmospheric entry can be divided into two categories: reference trajectory tracking and state-predictionbased trajectory correction. While the reference trajectory tracking method is easy to realize, it is a linear method. If a large difference exists between the actual and reference trajectories, the linear assumption no longer holds, leading to increased guidance and control errors. Considering our current knowledge about the terrain of Mars and the data-processing capabilities of the onboard computers, the first method is more appropriate for Mars entry missions in the short term. For the nominal trajectory design, the reference profile must be optimized to offer a sufficient margin to handle various environmental dispersion issues. Furthermore, the maneuvering capability of the vehicle must be considered so that the guidance law will not saturate the bank angle and lose the ability to converge the range error in the presence of uncertain

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environmental factors. The limit of the nominal bank-angle profile is estimated based on the dispersion of atmospheric density ρ and aerodynamic coefficient Cd [36] σmin nom = arccos(1 − ρ − Cd )

(7.56)

Bank angle

(⁰)

The typical dispersions of atmospheric density and aerodynamic coefficient are 25% and 10%, respectively. Thus, the allowable minimum initial bank angle is calculated to be 49º. Similarly, the maximum bank angle is 131º. If the nominal profile of the bank angle cannot satisfy this range constraint, then the lift to drag ratio should be increased or the ballistic coefficient should be decreased. For Mars landing, if the vehicle has a large ballistic coefficient or the atmosphere is thin near the landing site, a constant reference bank-angle profile will usually fail to simultaneously satisfy the constraints of landing accuracy and parachute deployment. In this case, a variable bank-angle reference profile can be adopted to enhance performance. Reducing the bank angle of low-velocity flight can prolong the time spent in the denser region of the atmosphere. Previous research has shown that for the most feasible variable reference bank-angle profiles, the component of lift on the vertical plane is rather small during the initial phase of entry. The bank angle is set as σ2 (e.g., 80º), which then linearly decreases, as shown in Fig. 7.13, until the above-mentioned minimum value σ1 is reached (e.g., 50º). A variable bank-angle profile can achieve the maximum decrease rate of altitude, resulting in a satisfactory balance between the altitude and position of parachute deployment. Once the reference back angle profile is determined, the performance of closed-loop dispersion simulation can be used to evaluate the need for iterative optimization of the profile or the need for adjustment of the guidance law gain and cross range control logic in order to improve entry performance.

σ2

σ1 V1

V2 Velocity magnitude (m/s)

Fig. 7.13 Open-loop profile of the bank angle

7.6 Guidance and Control for Soft Landing on Extraterrestrial Bodies

(2)

417

Powered descent The Viking probe used the gravity-turn method for the guidance of powered descent [30]. This guidance strategy employs two altitudevelocity modes to constrain the solution space, where the low-velocity curve is written as H = 3.06 × 10−7 + 1.53 × 10−1 V + 8.795 × 10−5 V 2 − 1.029 × 103 V 3

(7.57)

The high-velocity curve is written as H = 3.23 × 10−7 + 1.20 × 10−4 V + 1.856 × 10−4 V 2 − 2.479 × 104 V 3

(7.58)

Both the low- and high-velocity modes start from the same altitude, but the former needs less thrust and thereby has a longer ignition transient than the latter. The constant-velocity mode follows gravity-turn guidance. At an altitude of 16.8 m, the lander enters the terminal descent phase. To achieve precision Mars landing (tens to hundreds of meters from the exact target point), guidance in the powered descent phase is critical. It needs to accomplish two major tasks: (1) the calculation of a fueloptimized nominal trajectory that satisfies all necessary constraints and (2) the guidance of the lander using retrorockets, after it is released from the parachute, to safely land at the target point.

7.6.3 Soft Landing on Asteroids 1.

Characteristics of soft landing on asteroids The soft-landing exploration of asteroids is different from that of large celestial bodies, such as the Moon and Mars, in the following aspects: (1)

(2)

Weak gravity. Asteroids generally have smaller sizes than large celestial bodies such as the Moon and Mars, with a diameter ranging from tens of meters to tens of kilometers. Their surface gravity is very weak (on the order of 10–5–10–3 m/s2). Even a slight collision between the probe and asteroid surface could lead to the escape of the probe. Hence, a higher control precision is needed for landing. High uncertainty. Owing to the small volume of asteroids, it is very difficult to observe their topographic and geomorphological details from Earth. In addition, most asteroids to be visited have never been explored, and prior knowledge is usually not available for reference. Therefore, surface rendering reconnaissance must be conducted by the probes in a close range to the asteroids to determine their geomorphological features in detail and estimate their kinematic characteristics including the spin axes. The guidance and control strategy for asteroid landing is more demanding,

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(3)

2.

requiring on-orbit intelligent control. That is, the landing strategy needs to be adjusted in an adaptive manner based on the real-time, on-orbit measured information. Fast rotation. Asteroids are usually spin-stabilized with a rotation period of several hours or even several quarters. Consequently, a probe will experience rapid changes in lighting on the asteroid surface during landing. It is impossible to choose a suitable sunlight condition as in the case of landing on the Moon or other large bodies. Therefore, the landing sensors are expected to be able to adapt to the quickly changing lighting conditions. Furthermore, guidance and control schemes must be robust and adaptive. In summary, the targets of asteroid exploration have uncertain and complicated environments, which require the probe control systems to be intelligent, autonomous, robust, and adaptive so that the probes can safely land and stay on the asteroid surface.

Guidance and control strategy for far-range rendezvous The primary objective of far-range guidance and control is to accurately identify, lock, and approach the target asteroid. When the probe is a few thousand kilometers away from the asteroid, the optical navigation sensors can identify the asteroid with the help of the ground radio tracking system. The orbit determination data can then be corrected. When the distance to the asteroid is less than 100 km, the distance sensors acquire the distance information, and the optical navigation sensors measure the LOS direction information of the asteroid. At this moment, the far-range rendezvous autonomous relative GNC phase begins. The guidance and control strategy used in this phase is primarily LOS guidance. The LOS frame ρ 0 q 0 k describes the relative motion between the probe and asteroid. When the probe is sufficiently close to the asteroid and is in the same orbital plane, LOS guidance can be confined to the orbital plane of relative motion. Figure 7.14 shows the definition of this LOS frame. The 2D equations of the orbital plane (ρ 0 q 0 ) within the LOS frame can be written as ⎧ ⎨ ρ¨ − ρ q˙ 2 + 2ωρ q˙ − 3ρω2 sin2 q = f ρ (7.59) ⎩ ρ q¨ + 2ρ˙ q˙ − 2ωρ˙ − 3 ρω2 sin 2q = f q 2 The parallel rendezvous scheme can be used in the LOS frame. Let the angular velocity of relative range be zero, i.e., q˙ = 0, and the relative velocity in the LOS direction be negative, i.e., ρ˙ < 0. Control along the LOS angular velocity is lateral control, whereas control along the LOS direction is longitudinal control. Schemes decoupling the control in the channels of range and LOS angle can be adopted as well. The phase-plane method is a feasible choice; alternatively, a

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419

Fig. 7.14 LOS frame of relative motion

3.

4.

PID controller can be used separately in the range channel and the LOS angle channel. Guidance and control strategy for intermediate-range fly-around inspection When the probe is kilometers or tens of kilometers away from the asteroid, it must conduct a fly-around inspection for the topographic mapping of the asteroid surface and estimation of the asteroid’s kinematic characteristics, including its spin axis. This lays the foundation for the subsequent landing and touchdown. Based on Hill’s equations of relative dynamics, the guidance and control strategy for the fly-around phase can be divided into two categories. The first category features natural fly-around based on the characteristics of Hill’s equations for relative motion. This scheme is suitable in situations where the target asteroid orbit is nearly circular. For elliptical orbits, timely active control is necessary. The other category is forced fly-around under active control. Natural fly-around is difficult to achieve in reality because the orbital period of closed fly-around is equal to the revolution period of the target asteroid, which is generally very long (e.g., several years). For surveying, a feasible method that takes full advantage of the asteroid rotation is to select an appropriate position relative to the asteroid to implement rest control and observation. In forced fly-around, it is possible to design the shape and period of the trajectory according to specific needs. This is a more flexible method than natural flyaround, but it requires higher fuel consumption. Guidance and control strategy for intermediate-range transfer and approach Once the accurate mapping of the asteroid surface and estimation of kinematic information (e.g., spin axis) are completed in the fly-around phase, the probe can choose an initial candidate landing site on the asteroid surface. Subsequently, guidance and control for large-range transfer should be applied to guide the probe to reach the area above the target touchdown site.

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Fig. 7.15 Definition of the heliocentric orbit frame

CW guidance can be adopted in this phase. This double-pulse open-loop controller is simple and requires low fuel consumption, but it suffers from low control accuracy. The reference coordinate system for CW guidance is the heliocentric orbit frame of the target (asteroid). Figure 7.15 shows the definition of this frame. The heliocentric orbits of typical asteroids are elliptical. Hence, the equations for relative motion can be written as ⎧ μ ⎪ ˙ − ω2 x = − 3 x + f x ⎪ ⎪ x¨ + 2ω y˙ + ωy r ⎪ m ⎪ ⎪ ⎨ 2μ 2 y¨ − 2ω x˙ − ωx ˙ − ω y = 3 y + fy (7.60) ⎪ rm ⎪ ⎪ ⎪ μ ⎪ ⎪ ⎩ z¨ = − 3 z + f z rm

5.

where ω is the angular velocity in the elliptical orbit, which is no longer a constant. The angular acceleration ω˙ is nonzero. Consequently, it is difficult to acquire the analytical solution. However, considering that the period of circumsolar revolution is quite long (e.g., several years), the angular velocity can be regarded as a constant if the time interval for orbit transfer is much smaller than the revolution period. The induced error can be corrected using the third pulse. Guidance and control strategy for close-range descent and touchdown when the probe gets very close to the asteroid, the optical imaging sensors will only be able to capture a local region of the asteroid. At this point, a relative reference frame with respect to the local region needs to be used. The rotation of the asteroid should be fully considered. Moreover, the probe should have zero transverse velocity relative to the asteroid, approaching slowly along

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421

the longitudinal direction to ensure a safe, reliable, and precise landing on the asteroid surface while avoiding the risk of rebound. The primary guidance and control strategy in this phase is reference trajectory guidance. It provides an ideal reference descent trajectory and requires the position and velocity of the probe relative to the asteroid to satisfy certain constraints. The reference trajectory can be a simple straight line or an optimized curve. The probe uses real-time closed-loop control to track the reference trajectory. The commanding acceleration ac is written as ac = kp (r − r c ) + kd (˙r − r˙ c )

(7.61)

where r, r˙ and r c , r˙ c are the actual and reference positions and velocities of relative motion, respectively. The key to this phase is the accurate control of the relative position and velocity to ensure safe landing. The reference trajectory can be designed as an acceleration– coasting–braking trapezoid trajectory.

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