Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles: Methods and Applications (Springer Aerospace Technology) [1st ed. 2023] 9819943108, 9789819943104

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Springer Aerospace Technology

Runqi Chai · Kaiyuan Chen · Lingguo Cui · Senchun Chai · Gokhan Inalhan · Antonios Tsourdos

Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles Methods and Applications

Springer Aerospace Technology Series Editors Sergio De Rosa, DII, University of Naples Federico II, Napoli, Italy Yao Zheng, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, Zhejiang, China Elena Popova, Air Navigation Bridge Russia, Chelyabinsk, Russia

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Runqi Chai · Kaiyuan Chen · Lingguo Cui · Senchun Chai · Gokhan Inalhan · Antonios Tsourdos

Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles Methods and Applications

Runqi Chai School of Automation Beijing Institute of Technology Beijing, China

Kaiyuan Chen Vanke School of Public Health Tsinghua University Beijing, China

Lingguo Cui School of Automation Beijing Institute of Technology Beijing, China

Senchun Chai School of Automation Beijing Institute of Technology Beijing, China

Gokhan Inalhan School of Aerospace, Transport and Manufacturing Cranfield University Cranfield, Bedfordshire, UK

Antonios Tsourdos School of Aerospace, Transport and Manufacturing Cranfield University Cranfield, Bedfordshire, UK

ISSN 1869-1730 ISSN 1869-1749 (electronic) Springer Aerospace Technology ISBN 978-981-99-4310-4 ISBN 978-981-99-4311-1 (eBook) https://doi.org/10.1007/978-981-99-4311-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

The development of trajectory optimization, guidance and control methods has become increasingly important due to its wide applications in both military and civil fields. A well-planned trajectory, together with an improved guidance and control system, is key to the success of a mission. As a result, the aim of this research is usually to work on the design of advanced algorithms which can circumvent the limitation brought by classical approaches and improve the implementation performance for various mission scenarios. This book focuses on the design, application and verification of advanced trajectory optimization, guidance and control (G&C) techniques for aerospace vehicles. Part I of the book focuses on the introduction of constrained aerospace vehicle trajectory optimization problems, with a particular focus on the design of high fidelity trajectory optimization methods, bio-inspired optimization-based strategies and fast convexificaton-based algorithms. In Part II, various optimization theory/artificial intelligence (AI)-based methods, such as model predictive control-based methods and deep neural network-based algorithms, are constructed and presented. The key aspects of applying these approaches, including their main advantages and inherent challenges, are detailed and discussed. Following that, some practical implementation considerations, together with a number of future research topics, are summarized. The comprehensive and systematic treatment of practical issues in aerospace vehicle trajectory optimization, guidance and control problems is one of the major features of the book, which is particularly suited for readers who are interested in learning practical solutions in aerospace vehicle trajectory optimization, guidance and control. The book will be beneficial to researchers, engineers and graduate students in the fields of G&C systems, engineering optimization, applied optimal control theory, etc.

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All the authors have carefully reviewed the content of this book before the printing state. However, it does not mean that this book is completely free from any possible errors. Hence, the authors would be very grateful to readers who will call attention to mistakes they might discover. Beijing, China Beijing, China Beijing, China Beijing, China Cranfield, UK Cranfield, UK April 2023

Runqi Chai Kaiyuan Chen Lingguo Cui Senchun Chai Gokhan Inalhan Antonios Tsourdos

Acknowledgements

The authors would like to thank Prof. Guo-Ping Liu, from Southern University of Science and Technology, for his constructive comments on both the theoretical part and practical parts of this book. The authors would like to express their sincere appreciation to Runsheng Wang and our graduate students, Qinghua Zhu, Peiyuan Lv, Jinsong Chen, Yi Hao, Yichen Jiang, Changhong Wu, Bingrun Jiang and Hankun Jiang for their help in drafting the book. Moreover, the authors would like to thank all the staff from Institute of Intelligent Information Processing and Control, School of Automation, Beijing Institute of Technology for everything they have done to make things easier for us throughout the preparation of this work. Finally, the authors would like to thank Beijing Institute of Technology, School of Automation, and Cranfield University, School of Aerospace, Transport and Manufacturing for providing us with support to make the work a reality.

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Contents

Part I

Advanced Trajectory Optimization Methods

1 Review of Advanced Trajectory Optimization Methods . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mathematical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 1.2.1 Continuous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Variable/Path Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mission Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Overall Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Numerical Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Gradient-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Evolutionary-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Convexification-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Dynamic Programming-Based Methods . . . . . . . . . . . . . . . . . 1.4 Multi-objective Spacecraft Trajectory Optimization . . . . . . . . . . . . . 1.4.1 Multi-objective Evolutionary Algorithms . . . . . . . . . . . . . . . . 1.4.2 Multi-objective Transcription Methods . . . . . . . . . . . . . . . . . . 1.5 Stochastic Spacecraft Trajectory Optimization . . . . . . . . . . . . . . . . . . 1.5.1 Chance-Constrained Spacecraft Trajectory Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Chance-Constrained Spacecraft Trajectory Optimization: Stochastic Dynamics . . . . . . . . . . . . . . . . . . . . . 1.6 Recent Practical Applications of the Optimized Trajectory . . . . . . . . 1.6.1 Design of Integrated Spacecraft Guidance and Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Design of Spacecraft/Satellite Formation Control Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Database-Based Online Guidance Strategy . . . . . . . . . . . . . . . 1.7 Conclusions and Future Development . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 7 8 8 9 10 10 12 13 16 17 18 19 20 23 26 26 29 31 32 32 33 33 34

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2 Heurestic Optimization-Based Trajectory Optimization . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Biased Particle Swarm Optimization Approach . . . . . . . . . . . . . . . . . 2.2.1 Unconstrained Multi-objective Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 MOPSO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 ε-Bias Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Local Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Evolution Restart Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Overall Algorithm Framework . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constrained Atmospheric Entry Problem . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Entry Phase Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Test Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Test Case Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Performance of Different Methods . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Convergence Analysis for Evolutionary Methods . . . . . . . . . 2.4.4 Computational Performance of Different Methods . . . . . . . . 2.4.5 Impact of the Bias Selection Strategy and Local Exploitation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Impact of the Restart Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 46 46 48 49 51 52 53 55 55 56 59 59 59 60 64 67 68 71 72 73

3 Highly Fidelity Trajectory Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Time-Optimal Reconnaissance Maneuver Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.1 Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.2 Flight Constraints and Objective . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.3 Overall Trajectory Optimization Formulation . . . . . . . . . . . . 82 3.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 Radau Pseudospectral Method . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3.2 A Pipelined Optimization Strategy . . . . . . . . . . . . . . . . . . . . . . 84 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.1 Simulation Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.2 Optimized Results of Using Different Models . . . . . . . . . . . . 90 3.4.3 Results with and Without Mesh Adaptive Process . . . . . . . . . 92 3.4.4 Comparative Results and Analysis . . . . . . . . . . . . . . . . . . . . . . 94 3.4.5 Case Studies with Noise-Perturbed Initial Conditions . . . . . . 99 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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4 Fast Trajectory Optimization with Chance Constraints . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Atmospheric Entry Optimal Control Problem . . . . . . . . . . . . . . . . . . . 4.2.1 Hypersonic Vehicle Dynamics and Constraints . . . . . . . . . . . 4.2.2 Atmospheric Entry Optimal Control Model . . . . . . . . . . . . . . 4.3 Nonconvex Chance-Constrained Optimization Approach . . . . . . . . . 4.3.1 Handling the Probabilistic Constraint . . . . . . . . . . . . . . . . . . . 4.3.2 Deterministic NCCO Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Convex Chance-Constrained Optimization Approach . . . . . . . . . . . . 4.4.1 Convex Relaxation of Dynamics and Hard Constraints . . . . 4.4.2 Convex Approximation of Control Chance Constraint . . . . . 4.4.3 Overall CCCO Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Parameters and Mission Cases Specification . . . . . . . . . . . . . 4.5.2 NCCO Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 CCCO Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fast Generation of Chance-Constrained Flight Trajectory for Unmanned Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Trajectory Planning Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Unmanned Vehicle System Equations . . . . . . . . . . . . . . . . . . . 5.2.2 Geometric Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Relationship Between Geometric Constraints and Vehicle Actual Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Control Chance Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Probabilistic Collision Avoidance Constraints . . . . . . . . . . . . 5.2.6 Objective and Optimization Model . . . . . . . . . . . . . . . . . . . . . 5.3 Convex-Programming-based Trajectory Planning Approach . . . . . . 5.3.1 Convexification of System Equations and Constraints . . . . . 5.3.2 Convex Trajectory Optimization Model . . . . . . . . . . . . . . . . . 5.4 Deterministic Chance-Constrained Trajectory Planning Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Convex Approximation of Control Chance Constraints . . . . 5.4.2 Convex Approximation of Probabilistic Collision Avoidance Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Overall Algorithm Framework . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Unmanned Vehicle Trajectory Generation . . . . . . . . . . . . . . . 5.5.2 Comparative Case Study: Without Chance Constrains . . . . . 5.5.3 Chance-Constrained Unmanned Vehicle Trajectory Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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107 107 109 109 111 112 112 113 114 115 116 117 117 118 119 123 127 129 131 131 134 134 135 135 137 137 137 138 138 140 142 142 144 147 149 149 151 152

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5.5.4 Comparative Case Studies: With Control Chance Constrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Comparative Case Studies: With Control and Obstacle Chance Constrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

153 155 159 162 163

Advanced Guidance and Control Methods for Aerospace Vehicles

6 Review of Advanced Guidance and Control Methods . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Organisation of the Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Types of Guidance and Control Systems . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Integrated Guidance and Control System . . . . . . . . . . . . . . . . 6.2.2 Partially Integrated Guidance and Control System . . . . . . . . 6.3 Review of Stability Theory-Based G&C Methods . . . . . . . . . . . . . . . 6.3.1 Design and Applications of Robust G&C Algorithms . . . . . . 6.3.2 Design and Applications of Stochastic G&C Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Potential Issues and Challenges of Stability Theory-Based G&C Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Design and Applications of Data-Driven G&C Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Review of Optimisation-Based G&C Methods . . . . . . . . . . . . . . . . . . 6.4.1 Design and Applications of Dynamic Programming-Based G&C Methods . . . . . . . . . . . . . . . . . . . . 6.4.2 Design and Applications of Model Predictive Control-Based G&C Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Challenges of Using Optimisation Theory-Based G&C Methods in Space/Aerospace Applications . . . . . . . . . 6.5 Review of AI-Based G&C Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Connection Between AI and Guidance and Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Design and Applications of AI-Based G&C Methods . . . . . . 6.5.3 Potential Issues and Challenges of AI-Based G&C Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions and Future Developments . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Continuing Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Optimization-Based Predictive G&C Method . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Missile-Target Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 2-D Missile Target Engagement . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 3-D Missile Target Engagement . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Receding Horizon Pseudospectral Control . . . . . . . . . . . . . . . . . . . . . 7.3.1 Discrete Approximation Model . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Moving Horizon Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Receding Horizon Pseudospectral Control . . . . . . . . . . . . . . . 7.3.4 NLP Optimality and Approximated KKT Conditions . . . . . . 7.3.5 Implementation Consideration . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Parameter Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Interception Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Effect of Parameter Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 209 209 211 213 213 216 217 218 221 222 222 223 227 230 232 233

8 Robust Model Predictive Control for Attitude Control Tracking . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Motivations and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Spacecraft Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Control Problem Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Assumptions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Design of the TRMPC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Outer-Loop TRMPC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Inner-Loop TRMPC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Overall Algorithm Framework . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Analysis of Feasibility and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Recursive Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Control Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Parameter Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Tracking Performance Evaluation and Comparative Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Case Studies on Algorithm Parameters . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Appendix: Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 235 237 238 239 239 239 240 241 242 246 248 250 250 251 251 251 252 252 256 257 258 259

About the Authors

Runqi Chai received his Ph.D. degree in aerospace engineering from Cranfield University, Cranfield, U.K., in August 2018. He was a Research Fellow at Cranfield University from 2018 to 2021. He was also a Visiting Researcher at The University of Manchester, Manchester, U.K., in 2021. He is currently a Professor at Beijing Institute of Technology, Beijing, China. His research interests include trajectory optimization, networked control systems, multiagent control systems and autonomous vehicle motion planning and control.

Kaiyuan Chen received her Ph.D. degree from Swansea University in 2022. She serves as a research fellow at Vanke School of Public Health, Tsinghua University. Currently, she is a member of Institute of Electrical and Electronics Engineers (IEEE) and Chinese Association of Automation (CAA). Her research interests include optimal control, smart health care, global health intelligent governance, the application of unmanned systems in public health and other relevant fields.

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About the Authors

Lingguo Cui received her Ph.D. degree in computer science and engineering from Beijing Institute of Technology, Beijing, China, in March 2007. She was a Visiting Researcher at Cranfield University, Cranfield, U.K., in 2016, and at University of Florida, U.S.A., from 2011 to 2012. She is currently an Associate Professor at Beijing Institute of Technology, Beijing, China. Her research interests include information security, wireless sensor networks, optimal control and optimization theory.

Senchun Chai received his B.S. and masters degrees in control science and engineering from Beijing Institute of Technology, Beijing, China, in 2001 and 2004, respectively, and his Ph.D. degree in networked control system from University of South Wales, Pontypridd, U.K., in 2007. He was a Research Fellow at Cranfield University, Bedford, U.K., from 2009 to 2010, and was a Visiting Scholar at University of Illinois at Urbana-Champaign Urbana, Champaign, IL, USA, from January 2010 to May 2010. He is currently a Professor at the School of Automation, Beijing Institute of Technology. He has authored and coauthored more than 100 journal and conference papers. His current research interests include flight control systems, networked control systems, embedded systems and multiagent control systems. Gokhan Inalhan received his B.Sc. degree in aeronautical engineering from Istanbul Technical University, in 1997, and M.Sc. and Ph.D. degrees in aeronautics and astronautics from Stanford University, in 1998 and 2004, respectively. He is BAE Systems Chair, Professor of Autonomous Systems and Artificial Intelligence, and Deputy Head of Autonomous and Cyber-physical Systems Centre at Cranfield University, Bedford, U.K. He has previously served as Director General at ITU Aerospace Research Centre, Istanbul, Turkey. He has authored or coauthored over 200 papers, book chapters, proceedings and technical reports in his areas of interest. His research interests include advanced controls, optimization and modeling aspects associated with autonomy and artificial intelligence for air, space,

About the Authors

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defence and transportation systems (urban air mobility, air traffic management/unmanned aerial systems (UAS) traffic management) with seminal works on multiple aircraft coordination and decentralized control of UASs. Antonios Tsourdos received his M.Eng. degree in electronic, control and systems engineering from the University of Sheffield, Sheffield, U.K., in 1995, an M.Sc. degree in systems engineering from Cardiff University, Cardiff, U.K., in 1996, and a Ph.D. degree in nonlinear robust autopilot design and analysis from Cranfield University, Bedford, U.K., in 1999. In 1999, he joined the Cranfield University as a Lecturer, was appointed Head of the Centre of Autonomous and Cyber-Physical Systems in 2007, a Professor of Autonomous Systems and Control in 2009, and the Director of Research Aerospace, Transport and Manufacturing in 2015. He currently leads the research team on autonomous systems within School of Aerospace, Transport and Manufacturing, Cranfield University. He has diverse expertise in both unmanned and autonomous vehicles as well as networked systems. His research interests include the fields of guidance, control and navigation for single and multiple unmanned autonomous vehicles as well as research on cyber-physical systems.

Part I

Advanced Trajectory Optimization Methods

Chapter 1

Review of Advanced Trajectory Optimization Methods

Abstract When encountering atmospheric or exo-atmospheric spacecraft flight, a well-designed trajectory is essential for making the flight stable and enhancing the guidance and control of the vehicle. Much research has focused on how to design suitable spacecraft trajectories available for various mission profiles. To optimize the flight trajectory, researchers have designed numerous useful tools successfully. Nevertheless, it is only in the last five years that the interest in how to plan flight trajectories and consider numerous mission goals and different model errors/uncertainties simultaneously has grown greatly. Note that for various practical guidance, navigation and control systems for spacecraft, during the trajectory planning process, the frequent consideration of multiple performance indices and various forms of uncertainty is necessary. Consequently, the multi-objective spacecraft trajectory optimization methods and stochastic spacecraft trajectory optimization algorithms are successfully proposed with the help of the requirements mentioned above. The core aim of this chapter is to provide a wide overview of current developments in numerical multi-objective trajectory optimization algorithms and stochastic trajectory planning approaches for spacecraft flight operations. First, we will briefly describe the process of how the problem is formulated mathematically. Then several optimization strategies for addressing spacecraft trajectory planning problems, such as gradient-based methods, convexification-based methods, and evolutionary/metaheuristic methods, are discussed. Besides, we will overview the formulation process of the multiobjective spacecraft trajectory optimization problem, as well as multiple types of multi-objective optimization algorithms. The significant features, for example, the merits and demerits of the newly-proposed multi-objective approaches, are summarized. Furthermore, we will pay some attention to the extension of the original deterministic problem to a stochastic form. To handle the stochastic trajectory planning formulation, several robust optimization algorithms are also outlined. Additionally, applications of the optimized trajectory proposed recently will be especially focused on. Finally, we will draw some conclusions and discuss further research about strategies for multi-objective and stochastic trajectory optimization.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_1

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Nomenclature NLP FONC SQP IP IPSQP LP SOCP SDP DP DDP SDDP GA DE VLDE PSO PPPIO AC ABC SA TS MOTO NSGA-II I-NSGA-II NSGA-III MOEA/D MOPSO MOAPSO MOAGPSO MOABC NPGA SPPSO ADEMGT WS PP FPP IFPP GP FGP FSGP ASM EMO MOEA

nonlinear programming first order necessary condition sequential quadratic programming interior point interior point sequential quadratic programming linear programming second order cone programming semidefinite programming dynamic programming differential dynamic programming stochastic differential dynamic programming genetic algorithm differential evolution violation learning differential evolution particle swarm optimization predator-prey pigeon-inspired optimization ant colony artificial bee colony simulate annealing tabu search multi-objective trajectory optimization nondominated sorting genetic algorithm II improved nondominated sorting genetic algorithm II nondominated sorting genetic algorithm III multi-objective evolutionary algorithm Based on decomposition multi-objective particle swarm optimization multi-objective adaptive particle swarm optimization multi-objective adaptive gradient particle swarm optimization multi-objective artificial bee colony niched pareto genetic algorithm strength pareto particle swarm optimization adaptive differential evolution and modified game theory weighted-sum physical programming fuzzy physical programming interactive fuzzy physical programming goal programming fuzzy goal programming fuzzy satisfactory goal programming adaptive surrogate model evolutionary multi-objective optimization multi-objective evolutionary algorithm

1.1 Introduction

MOT SOP CC RO CCO SDE SQF PDF gPC MCMC x u t b g J  L E I M ξ  wk

5

multi-objective transcription single-objective problem chance constraint robust optimization chance-constrained optimization stochastic differential equation stochastic quadrature formula probability density function generalized polynomial chaos Markov chain Monto Carlo state variable control variable time boundary function path function objective function Mayer cost process cost number of equality constraints number of inequality constraints number of objective functions uncertain parameter acceptable probability of occurrence weighted parameter

1.1 Introduction To bring humans into space, we have witnessed a large number of achievements and fantastic efforts in the past several decades. Significant influences have been brought into the military field as well as scientific and engineering applications by aerospace science and technologies. Among the influences, the improvement in the field of spacecraft technology has gained considerable attention [1, 2]. Up to now, researchers have designed, produced and launched several types of spacecraft and implemented them in various mission profiles successfully, for example, communications [3], interplanetary travel [4], regional reconnaissance [5], environmental monitoring [6], etc. Due to high operational costs, limited resources and the long development cycle, aerospace engineers usually require that the space vehicle could meet the basic mission requirements with an optimal or near-optimal optimization of some performance metrics. To reach this goal, a suitable handling of the space vehicle’s flight trajectory is naturally necessary, which promotes the development of trajectory optimization approaches. Many academic articles have demonstrated that for the flight stability and control improvement of the space vehicle, the trajectory design component is extremely

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important [7–10]. Conway provided a detailed discussion of the reason for the implementation of trajectory optimization in numerous space missions in 2011 [2], wherein other related trajectory optimization techniques were discussed as well. Besides, several significant practical scenarios were emphasized in this chapter and the references, including orbital transfer problems [11, 12], the spacecraft rendezvous and docking [13, 14], and the planetary entry [15–19]. These problems were summarised in a general form and handled as optimal control problems [20]. Note that the term “optimal control problems” and “trajectory optimization problems” are frequently used interchangeably in the literature, according to Betts [1]. For a more detailed discussion of the differences between these two statements, readers are referred to [21, 22]. According to the current development of optimal control theory on the whole, there are mainly two trends that are led by the development/application of numerical trajectory optimization stratigies designed for atmospheric or exo-atmospheric spacecraft flight circumstances. One is related to the increasing reliability and adaptivity of the system discretization, which makes the system discretization able to capture the features of the dynamical system maximally [20, 23–25]. And the other one is that the improvement of the solution optimality and the real-time capability can be achieved since optimization becomes more precise and computationally friendly. Based on the order of discretization and optimization, there are generally two types of numerical trajectory optimization techniques: the direct methods (“discretization then optimization”), and the so-called indirect methods (“optimization then discretization”) [23]. The first technique helps to seek the solution to the first-order necessary conditions for optimality with respect to spacecraft trajectory optimization problems. In the literature, successful examples of tackling issues without taking inequality constraints into account have been reported [26, 27]. The first-order necessary conditions in these works were transformed into two-point boundary value differential-algebraic equations. Nevertheless, when considering problems, including inequality constraints, this strategy may be ineffective. This is because determining the switch points at which the inequality restrictions become active is challenging, which restricts the practical implementation of this technique. As for the direct methods, discretizing the control or the state and control variables is the first step, and the original formulation is transformed into a static nonlinear programming problem (NLP). Following that, faced with the resulting static problem, we find that numerous optimization methods that are developed well could be utilized to obtain its optimal solution. Apparently, the direct method is much more convenient to be applied to the spacecraft trajectory design problem than the indirect method. Additionally, the way of formulating constraints becomes more straightforward. As a result, more attention is paid to the application of the “discretization then optimization” technique in engineering practice. This chapter mainly aims to show the latest improvements that have been accomplished during the process of developing techniques for spacecraft trajectory optimization problems. To be specific, we mainly focus on the newly-proposed optimization techniques applied in constrained trajectory optimization problems, multiobjective trajectory optimization problems, and stochastic trajectory optimization

1.2 Mathematical Formulation of the Problem

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problems. Thus because the discretization process is less important in comparison to the optimization process, we will just present a brief discussion in the following sections. A comprehensive and serious attempt was made to categorize discretization algorithms for spacecraft trajectory design in [12, 22], where readers can learn more details about the discretization process. Based on the findings presented in recently-published works, one of the intended goals of this chapter is to summarize the main advantages and disadvantages of different optimization techniques when using them to solve spacecraft trajectory optimization problems. It should be noted that, with the same optimization strategy being utilized, the result might vary dramatically from one specific mission to another. Moreover, when extending the problem to the multi-objective or stochastic version, the further it extends, the more obvious this phenomenon will be. Nevertheless, owing to the pioneering work of many researchers in the field of aerospace, numerous competitive results of different benchmark problems have become available now. As a consequence, obtaining a better acknowledgment of how these various optimization methods work when facing complicated mission circumstances. The remaining parts are organized as follows: Sect. 1.1 provides a brief description of the mathematical formulation of the spacecraft trajectory optimization problem. Following that, Sect. 1.2 summarizes many different optimization methods that are effective in tackling the problem. Specifically, the gradient-based methods, the evolutionary-based (i.e., heuristic/metaheuristic) methods, the convexification-based methods, and the dynamic programming-based methods are included. Section 1.3 will then give a review of the multi-objective spaceship trajectory optimization formulation as well as various kinds of multi-objective optimization techniques. And we also discuss the major characteristics, for instance, the advantages and disadvantages of the applications of the multi-objective trajectory planning techniques proposed recently. The extension of the deterministic problem formulation to a stochastic one is provided in Sect. 1.4. To solve this stochastic version of the problem, several robust optimization algorithms are summarized and analyzed. Section 1.5 additionally provides a high-level overview of current applications of optimal flight trajectories. Section 1.6 draws some conclusions and discusses how the multi-objective and stochastic trajectory optimization algorithms will be further developed.

1.2 Mathematical Formulation of the Problem Overall, for the majority of spacecraft flight missions, during the trajectory optimization process of their profiles, we should determine how to derive the mathematical modeling of the problem, select effective methods, and apply heuristics, and the goal of which is able to be found from a finite or even an infinite set of potential candidates. We can give the main goal for optimizing spacecraft trajectory as follows: for a considered vehicle, which will start at some point, a viable trajectory or path should be obtained, such that it can reach the preassigned target point with a predefined performance index optimized. According to Conway [2] and Betts [23],

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for designing the trajectory optimization formulation, making good use of optimal control theory is considered as the most common and convenient method. To be more specific, we will define a vector of time-dependent variable x(t) ∈ Rn x , which stands for the state of the spacecraft, along with a vector of time-dependent control variable u(t) ∈ Rn u , which can regulate the vehicle states. Several fundamental learning will be introduced in the following subsections, including the continuous dynamical systems, the variable/path constraints, and the performance index. After that, the total formulation of the trajectory optimization issue, which we consider in this chapter, is then shown in detail.

1.2.1 Continuous Dynamical Systems At present, the literature includes plenty of dynamical systems through which the movement of space spacecraft can be depicted for various missions. And fortunately, despite the diversity of mission profiles and vehicle types, we still can generalize a common form which can be applied to most of instances. As an example, it is common for the movement of the N x -dimensional state variable x ∈ R Nx to be represented by using a set of differential equations, which can be written as [28]: x(t) ˙ = f (x(t), u(t), t)

(1.1)

in which f stands for a function regarding time t ∈ R, the state x ∈ R Nx and control u ∈ R Nu variables. Generally speaking, f : R Nx × R Nu × R → Rn x is nonlinear. Mostly, the state trajectory x(t) starts at a given point x0 at a given time instant t0 . In addition, we need several state variables to arrive at a predetermined final point x f at the final time instant t f . We design an inequality to describe the above boundary condition, which can be denoted as: b L ≤ b(x0 , t0 , x f , t f ) ≤ bU in which b L , bU ∈ R Nb respectively stand for the lower and upper bounds of b(·, ·, ·, ·). b : R Nx × R × R Nx × R → R Nb .

1.2.2 Variable/Path Constraints In addition to taking the vehicle dynamical system into account, we should also consider all kinds of variable/path constraints so that the mission-dependent conditions can be satisfied, or the structural integrity of the vehicle can be preserved. Generally, we call these conditions as path constraints which we can describe in a common form (i.e., a common inequality with lower and upper bounds):

1.2 Mathematical Formulation of the Problem

9

g L ≤ g(x(t), u(t), t) ≤ gU

(1.2)

in which g L , gU ∈ R Ng respectively stand for the lower and upper bounds of the path function g(·, ·, ·). g : R Nx × R Nu × R → R Ng . As is often the case, diverse spacecraft flight mission profiles can include all kinds of path constraints. For example, the authors of [29] established a reusable launch vehicle trajectory optimization model, where we should fulfill three flight path constraints throughout the whole optimization process: load factor, aerodynamic heating and dynamic pressure. Subsequently, in [30], the authors constructed and solved a three-degree-of-freedom spacecraft reentry model while taking into account the no-fly zone constraints which put restrictions on the lateral and longitudinal positions of the vehicle. Moreover, the authors of [31] also took into account a spacecraft rendezvous and docking problem. In their study, considering the thrust direction restrictions, they applied a path constraint on the control variables. However, we can categorize all these constraints into the aforementioned two types which are denoted as Eq. (1.2). Based on Eq. (1.2), clearly, we need to satisfy the path constraints throughout the whole time domain (e.g., for ∀t ∈ [t0 , t f ]), which may become pure state constraints (i.e., functions of state variables) [32], pure control path constraints (i.e., functions of control variables) [33], or mixed path constraints (i.e., functions of state and control variables) [34].

1.2.3 Mission Objectives We define a viable flight trajectory as a solution to the space vehicle dynamical system (1.1) with all the path constraints fulfilled. Generally, for a particular mission profile, we can have a considerable quantity of viable flight trajectories. To select the special solution from the viable set, we usually specify a proper performance index as a reference, which makes the magnitude of goal achievement measurable. The goals should also be given by us. Besides, we usually describe the mission objective (cost function), which needs to be optimized, with the Bolza form shown as follows: t f J = (x0 , t0 , x f , t f ) +

L(x(t), u(t), t)dt

(1.3)

t0

According to Eq. (1.3), the cost function includes two parts. One term  : R Nx × R × R Nx × R → R stands for the Mayer cost, while L : R Nx × R Nu ×R → R denotes the process cost. Similar as the case of path constraints discussed above, different missions may require various definitions of performance index. However, we can describe the majority of them by the general form which Eq. (1.3) provides. As an example, in [35], the authors took into account the spacecraft reentry mission, making the final latitude maximized as the main objective so that a greater cross range value can be obtained. Apart from that, making the total amount of aero-

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dynamic heating minimized [18], making the final kinetic energy maximized [36], making the smoothness of the derived flight path, etc. minimized [37] are also the possible goals for this mission. Besides, the authors of [38] considered a low-thrust interplanetary mission while choosing to make the delivered mass maximized as the primary task. Moreover, in [12], a multiple-pass orbital transfer issue for the aero assisted spacecraft was studied. In this research, due to some subsegments which the mission possibly includes, the authors chose to make the fuel cost throughout the transfer (mass fraction) minimized as the primary goal. Additionally, It is important to highlight that the Mayer cost term  from Eq. (1.3) contains both the final time cost and the initial time cost, which, in fact, is significant for several particular missions. For instance, when dealing with orbital transfer issues or using the launch vehicle, we need to optimize the initial values of the state and time [39, 40].

1.2.4 Overall Formulation Since we have presented the vehicle dynamics, different types of constraints, and the mission performance index, we will outline a representative spacecraft trajectory optimization or optimal control model as search the control variable u(t) and the associated state x(t). In this way, the spacecraft is able to be controlled from its starting pose to a predetermined target pose, and make the mission-dependent objective function optimized with the dynamic and path constraints satisfied. The total formulation can be denoted as [28]: t f minimize J = (x0 , t0 , x f , t f ) +

L(x(t), u(t), t)dt t0

subject to ∀t ∈ [t0 , t f ] x(t) ˙ = f (x(t), u(t), t) (dynamic constraints) b L ≤ b(x0 , t0 , x f , t f ) ≤ bU (boundary conditions) g L ≤ g(x(t), u(t), t) ≤ gU (path constraints)

(1.4)

1.2.5 Numerical Solution Approach As we aforementioned in the introduction section of this chapter, to solve the spacecraft trajectory optimization issues, two commonly used techniques are given, which are called indirect approaches and direct approaches. To be exact, for the indirect approaches, we apply the calculus of variations, and the first order necessary conditions (FONCs) for optimality can be obtained. And then, we transform the original optimal control formulation to a Hamiltonian boundary-value issue, by which

1.2 Mathematical Formulation of the Problem

11

Fig. 1.1 Numerical solution approaches

an “optimization + discretization” solution-finding structure is generated. Different from indirect approaches, for direct approaches, we will apply a “discretization + optimization” method. The detail is shown as follows. To begin with, we use certain parametrization methods, such that the continuous-time system is transformed to a static system. Subsequently, we reformulate the original problem formulation as a static nonlinear programming issue. Then, we apply well-developed optimization algorithms, through which the optimal solution is obtained. In Fig. 1.1, we present a graphical diagram, which demonstrates numerical solution techniques that were developed to deal with the spacecraft trajectory optimization issues in the last twenty years. It is noteworthy that, for an indirect approach, the cost of constructing the FONCs is usually very expensive due to the complexity of the vehicle dynamics and a variety of constraints that need to be met. Moreover, when using indirect approaches, we need to deal with plenty of the resulting decision variables related to the indirect approach. Thus, from the perspective of realization, it could be better to consider the second class of approach (i.e., direct methods). In this chapter, we desire to look back on the nearest outcomes of optimization approaches, through which the optimal spacecraft flight trajectories can be produced reliably and efficiently. However, a

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detailed introduction and analysis for various discretization strategies is out of the extent of this book. We refer to [22] for such a broad review. Also, it is significant to mention that in the literature [41–43], some other geometric-based trajectory planning techniques were also introduced. These geometric-based approaches can obtain acceptable performance for generating viable trajectories, which are primarily applied as the motion planners for aircraft, UAVs, or autonomous ground vehicles. Nevertheless, as the primary objective of this chapter is to review trajectory optimization techniques relevant to the space vehicle literature, the geometric-based algorithm is omitted from the algorithm tree, which is illustrated in Fig. 1.1.

1.3 Optimization Algorithms As is shown in the discussion stated above, it is obvious that using optimization methods to solve the static nonlinear programming problem (NLP) is an essential procedure with an aim to address the spacecraft trajectory design problem. A standard NLP problem can be then described as [44]: Find decision variables x = [x1 , x2 , . . . , xn ] Minimize objective function J (x) subject to xmin ≤ x ≤ xmax Hi (x) = 0 G j (x) ≤ 0 (i = 1, 2, . . . , E) ( j = 1, 2, . . . , I )

(1.5)

in which E and I are the dimensionality of the equality and inequality constraints, respectively. In recent years, there have been a variety of effective optimization methods which can be helpful in solving the NLPs. Speaking of the primary goal of this section, we intend to give an overview of the state-of-the-art optimization algorithms proposed in the literature for producing the optimal spacecraft flight trajectories. In order to fully understand the performance and behaviors of various algorithms for handling different space vehicle flight missions, lots of practical results are required. Luckily, owing to the fact that there are numbers of them have been reported, this concern soon disappeared. Furthermore, with guiding readers to improve these techniques possible, it is hopeful that some of the constraints imposed by classic methods can be eventually avoided. Generally, four classes of optimization algorithms, namely the gradient-based, convexification-based, dynamic programming-based, and derivative-free (heuristicbased) optimization techniques, are often used in the literature to handle spacecraft trajectory optimization problems. To put it more specifically, the primary task of them is to obtain the optimal time history with respect to the spacecraft state and control variables. Furthermore, these algorithms can also be categorized as deter-

1.3 Optimization Algorithms

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Table 1.1 Popular deterministic optimization algorithms available for trajectory optimization problems Deterministic optimization algorithms Sequential quadratic programming (SQP) [11] Interior point method (IP) [45] Interior point sequential quadratic programming (IPSQP) [46] Linear programming (LP) [47] Second order cone programming (SOCP) [48] Semidefinite programming (SDP) [49] Dynamic programming (DP) [50] Differential dynamic programming (DDP) [51] Stochastic differential dynamic programming (SDDP) [52]

Table 1.2 Popular stochastic optimization algorithms available for trajectory optimization problems Stochastic optimization algorithms Genetic algorithm (GA) [15] Differential evolution (DE) [53] Violation learning differential evolution (VLDE) [54] Particle swarm optimization (PSO) [14] Predator-prey pigeon-inspired optimization (PPPIO) [55] Ant colony (AC) [56] Artificial bee colony (ABC) [57] Simulate annealing (SA) [58] Tabu search (TS) [59]

ministic and stochastic techniques, and the most popular optimization methods of each category are summarized and reported in Tables 1.1 and 1.2. It should be noted that this table does not contain all optimization methods in each category. Alternatively, in the following subsections, only some noteworthy instances are presented, and these strategies are discussed in detail. In related works, a significant number of numerical simulations were carried out. The findings confirmed an important fact that these newly-proposed optimization methods are effective enough to provide feasible solutions for addressing the constrained space vehicle trajectory design problems.

1.3.1 Gradient-Based Methods The classic gradient-based method is one of the most often used optimization strategies for generating the optimal spacecraft flight trajectory. Among all these gradient-

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based methods, the sequential quadratic programming (SQP) method, as well as the interior point (IP) method, are applied quite effectively to obtain the solution of large scale NLP problems [60]. Moreover, by applying a pseudospectral discretization method, a fuel-optimal aeroassisted spacecraft orbital transfer problem was first transformed to a static NLP in [61]. Then, the static NLP was addressed by using the standard SQP method to produce the fuel-optimal flight trajectory. Similarly, the SQP method was used as the main optimizer to obtain the time-optimal flight trajectory of a low-thrust orbital transfer problem in [11]. As for the objective for the SQP method, commonly, it is to reduce the original problem to a series of quadratic programming subproblems. Specifically, in this process, the most important procedure is to approximate the augmented Lagrangian quadratically and use Taylor expansion to linearize the constraints using Taylor expansion. To put it more precisely, each Newton iteration of the SQP loop relies on the solution of a quadratic programming subproblem involving Jacobian and Hessian matrix. The following are the solution-finding steps of SQP algorithm: Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Step 11

Construct the augmented Lagrangian function. Apply the quadratic model to approximate the augmented Lagrangian. Input the initial guess value xk . Use the Newton method to calculate the step direction gk . Calculate the step length αk based on the sufficient decrease conditions. Check the stopping optimality tolerance  of the current solution. If the stopping condition cannot be satisfied. Update xk+1 = xk + αk dk . Set k = k + 1 and go back to Step 4. If the stopping condition can be satisfied. Terminate the algorithm and output the optimal solution.

A graphical illustration of the general steps of applying gradient-based methods to address problem (1.5) is shown in Fig. 1.2. It should be highlighted that were it not for the line search algorithm, the enhanced robustness of the algorithm (as indicated in Step 4 to Step 9, and Fig. 1.2) would be usually hard to be achieved in the process of handling the optimization problem. Until now, numbers of line search algorithms have been proposed and implemented. In Chap. 3 of [44], more details of these algorithms are offered to readers who are willing to learn more about them. Basically, varieties of line search algorithms all share a similar philosophy, which is the good news for us to get a better understanding of them. Then specifically speaking, the aforementioned philosophy is that at each solution-finding iteration k, a search direction dk is firstly generated via the Newton, quasi-Newton, or gradient directions. Following that, the step length αk is determined along the pre-specified searching direction. In particular, the Armijo condition, the Wolfe condition, and the Goldstein condition [44, 46] are usually used to choose such an appropriate αk that the objective can be improved sufficiently. To optimize the optimal flight trajectory, SQP methods is usually a good choice owing to its effectiveness. However, there still exists an important problem. That is,

1.3 Optimization Algorithms

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Fig. 1.2 General steps of using gradient-based method

the accurate solution of the subproblem is required in most of the SQP implementations. And in the case of that, it is the computational burden of the solver that may probably be increased greatly [46]. Besides, as inequality constraints are handled by using the active set strategy in most SOP methods, the computational burden may be further increased unless the active set can be initialized appropriately. After the introduction and discussion of the SQP method, it is time to talk about another gradient-based method, namely the interior point (IP) method whose investigations can be found in a significant amount of work in the last decade. While utilizing this method, we all hope that the problem can be addressed in a simpler

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form. To achieve this, usually we need to transcribe the former inequality constraints to equality constraints with some particular slack variables introduced. In [45], the author applied the IP method to space vehicle trajectory design, wherein a space shuttle atmospheric reentry problem was studied and discretized by using a shooting method and finally based on the IP method, the resulting static NLP problem was then solved. Simulation results presented in this research proved that it is an effective way to use the IP method to solve NLP problem. Nevertheless, it is worth highlighting that in order to evaluate the quality of the optimization process, it is necessary for the IP method to define the penalty functions and initialize the penalty factor in the augmented merit function. In [46], for addressing the aeroassisted spacecraft trajectory design problem, the authors proposed a two-nested gradient-based method, which is namely the so-called interior point sequential quadratic programming (IPSQP) in which the advantages of the SQP and IP methods are combined. It is worth mentioning that in the algorithm framework of this technique, we can find an inner solution finding loop which was embedded in by design, and as a result, the QP subproblem was allowed to be solved inexactly. Thanks to this solution finding loop, not only the algorithm efficiency can be increased to some extent, but also the design may have more flexibility to manage the optimization process. Besides, the authors provided many simulation results as well as comparative studies which indicated that this improved gradient-based method is feasible and reliable enough.

1.3.2 Evolutionary-Based Methods Sometimes in an optimization problem, the classic gradient-based method may no longer be reliable or available due to the complexity of getting the gradient information of the objective functions or constraints (i.e., because of the high nonlinearity involved in these functions), in which case the evolutionary-based methods, also known as global optimization methods, are left to be the only way to generate the optimal solution, as no derivative information is required when applying an evolutionary method. And in the meantime, thanks to the fact that the difficulties of calculating the Jacobian and the Hessian matrix have been circumvented, it will be much easier to solve the problem. “Survival of the fittest” is basically the principle of global optimization methods or evolutionary algorithms, wherein it is adopted to a population of elements which represents candidate solutions [2, 17, 62]. It is worth noting that since the population is initialized randomly, no initial guess value is required by evolutionary algorithms, which is a big change in contrast to classic gradient-based algorithms. In terms of locating the global minimum, evolutionary algorithm is superior to classic gradient methods as its own nature enable itself to be more powerful in global optimization [14]. Up to now, various classes of evolutionary algorithms are available for engineers to generate the optimal solution of optimization problem. For instance, the agent-

1.3 Optimization Algorithms

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based class such as the pigeon-inspired optimization (PIO) and the particle swarm optimization (PSO), the generic class of evolutionary algorithms such as differential evolution (DE) and the genetic algorithm (GA), and the colony-based class of algorithms such as the artificial bee colony (ABC) algorithm and the ant colony optimization (ACO). Contributions made to develop or utilize these global optimization techniques in spacecraft trajectory design can be found in the literature. In [63], a modified GA played an important role in solving a constrained space capsule reentry trajectory design problem. Similarly, Kamesh et al. [64] combined a collocation approach and a hybrid GA so that it could solve an Earth-Mars orbital transfer problem. And more, the optimal trajectory for a space robotic manipulator is generated via a standard PSO algorithm in [62]. With an aim to develop a bi-level structural optimal control method, Conway et al. [2] incorporated global optimization methods and standard gradient-based algorithms. Moreover, in their latest study, a modified particle swarm optimization algorithm was effectively applied by Pontani and Conway [14] to locate the global optimal flight trajectory of a cycling spacecraft. We have previously worked on a space maneuver vehicle entry problem, wherein approximating the optimal flight trajectory [54] is the main objective. In this work, we constructed an enhanced differential evolution method which is merged with a violation-degree based constraint handling strategy. Furthermore, it was to improve the diversity of the current population that we embedded a simplex-based direct search mechanism in the former algorithm framework. Besides, considering the premature convergence of the algorithm, a learning technique was then utilized to circumvent it. Additionally, an optimization algorithm, which is inspired by ant colony, was established in [65] to produce a multi-phase space vehicle orbital flight trajectory. In [66], the authors applied an automated approach based on genetic algorithm and monotonic basin hopping to solve a launch vehicle interplanetary trajectory problem. Based on the aforementioned results, obviously, it is feasible to utilize heuristicbased algorithms to solve spacecraft trajectory design problems. However, it tends to be difficult to valid the optimality of the obtained solution. Moreover, it should be noted that in the heuristic optimization process, computational complexity tends to be quite high as well [67]. Therefore, it is currently difficult to regard heuristicbased approaches as a “standard” optimization algorithm which can be employed to address general spacecraft trajectory planning problems. And it should be noted that the computational performance of this kind of method is still expected to be improved.

1.3.3 Convexification-Based Methods In order to producing the optimal spacecraft flight trajectories, convexification-based algorithms, which have attracted more attention recently [68], is a good choice to make. Because there is a key feature of utilizing this type of strategy worthy enough

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to be noted. That is, it can guarantee both the solution and computational efficiency theoretically when carried out. However, there are too many kinds of spacecraft trajectory optimization problems. In practical terms most of them are usually nonconvex, which indicates that we need find some convexification methods to transform the original problem formulation to a convex one if we still want to take the convex optimization approach. And luckily there are numerous of them. Alternatively, the transforming process can be also considered as adopting a specific convex optimization model to approximate the original nonconvex formulation. The following are three kinds of convex optimization which can be usually found in the literature: 1. Linear programming model (LP). 2. Second-order cone programming model (SOCP) 3. Semi-definite programming model (SDP) Firstly, as for the LP model, just as every model has its time to be not reliable and sufficient to approximate the original problem formulation, the LP model might do so as well when the considered problem is complex enough (i.e., the nonlinearity of the system dynamics, objectives or constraints is high). Then in terms of the SDP model, compared with the other two models mentioned above, it has the most accurate approximation ability, but its transformed convex formulation is usually not so well-scaled, consequently leading to an increase with regard to the computational complexity. Nevertheless, there is a fact to the contrary that it is possible to balance both the computational complexity and the approximation accuracy by adopting the SOCP model. To put it more specifically, a second order cone is used in this strategy to approximate the problem constraints so that even a relatively small computing power can address the transformed problem. With an aim to utilize convexification-based optimization algorithms in space vehicle trajectory design, many relative works are widely carried out in the literature. For instance, under the consideration of nonconvex thrust magnitude constraints, the authors in [69, 70] take the convex optimization approach to effectively solve the planetary landing problem. Similarly, the optimal trajectory of the spacecraft entry planning problem was generated by using the SOCP model in [71], where both navigation uncertainties and the nonconvex collision-avoidance constraints were taken into account and rewritten as convex constraints during the optimization phase.

1.3.4 Dynamic Programming-Based Methods Naturally, there are numbers of local optimal solutions in nonlinear optimal control problems. The engineers are all hoping that there could be a good method to obtain the global optimal solution. And then, owing to the enhanced ability in dealing with local optimal solution and attaining stable performance, dynamic programmingbased methods have attracted significant attention. In this subsection, two commonly used dynamic programming-based methods, namely the standard dynamic program-

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ming (DP) method as well as the differential dynamic programming method (DDP), are discussed. In terms of DP, it is the Bellman’s principle of optimality that contributes to the birth and the implementation of DP in engineering optimization practice [50]. Basically speaking, the Bellman?s principle ensures that the optimal solution will not diverge, even if other points on the original optimal solution are selected as the starting point to retrigger the optimization process. However, it should be noted that DP produces the optimal solution for every possible decision variable by following this very principle. Hence, it is very likely to result in the curse of dimensionality [52]. Under the background of being puzzled by the fundamental problem in the application of the standard DP, the DDP approach arose at an opportune time [72]. In this strategy, the solution-finding process is performed locally in a small neighbourhood of a reference trajectory. Following that, in order to calculate the local optimal solution, this technique uses a backward and a forward sweep repeatedly until the solution converges. Up to now, in the literature, there are many successful implementations of the DDP algorithm in obtaining the optimal solution of some space missions. For instance, the authors in [73, 74] provided a comprehensive theoretical development of the DDP approach. Moreover, some numerical evaluation and practical implementation were included as well. In [72], in order to calculate the rendezvous trajectory to near Earth objects, the authors proposed a DDP-based optimization method. And in their work, the method was implemented successfully as well. However, regarding the recent DDP work, the model uncertainties and noises are not taken into account in most of the solution finding process. As a result, it tends to be hard to generate a nominal solution which has the ability to protect the feasibility of the current solution all along the trajectory from the negative effect of both model errors and uncertainties. As indicated in current works, we could easily draw a conclusion that for most existing optimization algorithms to say, after obtaining the results which can already be accepted as near-optimal solutions, there is still a long way to go and room for improvement in the implementation of these algorithms.

1.4 Multi-objective Spacecraft Trajectory Optimization If there is a melody through the whole research on space vehicle trajectory design problems, commonly “focusing on one single objective” must be the theme in early studies. For instance, minimizing the fuel consumption, maximizing the lading cross range, minimizing the time duration, etc. After learning about the relative literature, we are surprised to find that only in the last five years has the flight trajectories design under the consideration of multiple mission objectives attracted more attention than ever before. In a significant number of practical trajectory design problems, during the decision making phase, multiple performance measures should be frequently con-

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sidered, which successfully promotes the development of multi-objective trajectory optimization (MOTO) [75–77]. Just as the definition provided by Eq. (1.5), a standard multi-objective optimization problem can be described in the form of: Find decision variables x = [x1 , x2 , . . . , xn ] Minimize objective function J (x) = [J1 (x), J2 (x), . . . , JM (x)] subject to xmin ≤ x ≤ xmax Hi (x) = 0 G j (x) ≤ 0 (i = 1, 2, . . . , E) ( j = 1, 2, . . . , I )

(1.6)

in which M denotes the number of mission objectives considered in the problem. The main point of this section focuses on the latest development of multi-objective optimization algorithms for obtaining the optimal trajectories of different spacecraft flight missions. Moreover, the key features such as the advantages and disadvantages of adopting these recently-developed multi-objective optimization approaches are also discussed. After that, in terms of the performance and behaviours of different multi-objective optimization algorithms for solving different space vehicle flight mission problems, lots of reported results will be provided to readers. In this way, we hope that readers can eventually acquire a better understanding of these performance and behaviours. Furthermore, motivated by specific space vehicle trajectory design problems, it is much more possible for the readers to design their own algorithm or contribute to the improvement of these classic techniques. Typically, the MOTO methods can be classified into two categories, which are the multi-objective evolutionary algorithm-based strategy as well as the multi-objective transcription method-based techniques. In Tables 1.3 and 1.4, readers can find the most popular strategies among these two categories. It should be noted that this table is still not complete enough to list all the MOTO techniques under each category out. In other words, just several important and typical examples are discussed in detail in the following subsections.

1.4.1 Multi-objective Evolutionary Algorithms In order to address the MOTO problem, varieties of multi-objective algorithms have been proposed and even utilized in practice [95, 96]. In particular, adopting the principle of “pareto-optimal” is usually an effective way to generate the solution of the MOTO problem [97–99]. Only in the case that no other solutions are superior to the pareto-optimal solution in the current searching space with all objectives considered can this pareto-optimal solution be optimal [100, 101]. Given the fact that one single solution can hardly optimize all the objectives in the meantime, it

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Table 1.3 Popular multi-objective evolutionary algorithms available for trajectory design problems Multi-objective evolutionary algorithm-based techniques Nondominated sorting genetic algorithm II (NSGA-II) [78] Improved nondominated sorting genetic algorithm II (I-NSGA-II) [79] Nondominated sorting genetic algorithm III (NSGA-III) [80] Multi-objective evolutionary algorithm Based on decomposition (MOEA/D) [81] Multi-objective particle swarm optimization (MOPSO) [82] Multi-objective adaptive particle swarm optimization (MOAPSO) [83] Multi-objective adaptive gradient particle swarm optimization (MOAGPSO) [84] Multi-objective artificial bee colony (MOABC) [85] Niched pareto genetic algorithm (NPGA) [86] Strength pareto particle swarm optimization (SPPSO) [87] Adaptive differential evolution and modified game theory (ADEMGT) [88]

Table 1.4 Popular multi-objective transcription methods available for trajectory design problems Multi-objective transcription-based techniques Weighted-sum method (WS) [89] Physical programming method (PP) [90] Fuzzy physical programming (FPP) [37] Interactive physical programming (IPP) [91] Interactive fuzzy physical programming (IFPP) [92] Goal programming (GP) [93] Fuzzy goal programming (FGP) [79] Fuzzy satisfactory goal programming (FSGP) [36] Adaptive surrogate model (ASM) [94]

would be rather interesting to obtain every pareto-optimal solution so as to find the pareto-optimal set. To establish the pareto-optimal set, a detailed analysis of the evolutionary multiobjective optimization (EMO) methodology has been carried out. And finally, the results indicate that not only is the EMO a promising approach which can visualize the relationships between objectives, but it can also calculate the pareto-front [81]. The following are the typical steps of applying EMOs to address MOTO problems: 1. Initialize the population and other control parameters of the EMO algorithm. 2. For each candidate among the population/swarm, calculate the objective function values J and the constraint violation value. 3. Generate offspring population/swarm by using various evolutionary strategy. 4. Combine the offspring population/swarm with the previous population.

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5. Calculate the fitness value for each candidate using the information of objective function values and constraint violation value. 6. Assign all non-dominated ranks using the pareto dominant rule. 7. According to the selection operator, select the best set of individuals as the candidates of the new generation. 8. Repeat steps 2–7 until the maximum iteration number is reached. The past decades have witnessed the theoretical development and the widespread implementation of new EMO methods in aerospace engineering [102, 103]. Regarding the theoretical development, for example, in [67], an optimal path control technique was designed to handle general multi-objective optimization problems. Apart from that, Ji et al. [104] proposed a modified NSGA-II algorithm to solve a multiobjective allocation problem. Additionally, in [105], a decomposition-based sorting strategy for solving benchmark multi-objective problems was designed. In terms of the practical applications, the authors in [15] successfully utilized a constrained multi-objective evolutionary algorithm (MOEA) to address a bi-objective reentry trajectory design problem. In [106], a low-thrust gravity assist trajectory design problem was considered. In this work, two mission objectives that apparently contradict with each other, namely minimizing the fuel consumption and minimizing the flight time duration, were both taken into account. And owing to the NSGA-II algorithm, the pareto front was successfully generated. Moreover, because of the poor performance of the former NSGA-II algorithm, the authors in [107] designed an enhanced version, wherein they embedded a particular migration scheme in the original algorithm framework. Following that, this enhanced algorithm then played an important part in addressing an Earth-Jupiter orbital transfer problem under the consideration of different fly-by sequences. In [4], the standard PSO algorithm was extended to a multi-objective one, thereby establishing a so-called multi-objective PSO (MOPSO) method. Then MOPSO played an active role in handling an Earth-Jupiter-Saturn orbital transfer problem. The results provided in their work revealed the reliability as well as the feasibility of utilizing this very algorithm. In [80], considering constraints brought by no-fly zone or the path itself, an extended NSGA-III algorithm aiming at solving problems of the multi-objective spacecraft reentry trajectory design was then proposed. In order to get the evolutionary direction under control, the strategy introduced a series of reference points to do this job. Besides, with this algorithm implemented in practice, it tends to be very likely to reflect relationships between different mission objectives which obviously contradict with each other. Additionally, the produced pareto front usually tends to be more optimal as well as well-distributed after introducing reference points mentioned above. Regardless of effectively producing the pareto set as well as successfully reflecting the relationship between objectives which apparently contradicts with each other, there is such a great computational burden that the solution-finding process may be extremely hard to be performed. Additionally, MOEAs may encounter an inevitable obstacle when there are more than three objectives in problems we consider. To

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explain why the dimensionality is restricted, we should put our eyes into the MOEA framework more deeply. As a matter of fact, it is the current domination principle embedded in the algorithm framework of MOEA that accounts for the disability of MOEAs. Specifically speaking, neither can this principle emphasize feasible solutions, nor will it provide an adequate selection pressure [78, 81], thereby resulting in the obstacle mentioned above. Alternatively, it is nearly impossible to equably allocate the selection pressure to every pre-specified objective. As a result, as far as the pareto front is concerned, a decrease in the diversity of representation arises.

1.4.2 Multi-objective Transcription Methods It should be noted that attention in majority of the current studies is mainly paid to develop or implement MOEAs so as to solve general MOPs more effectively and efficiently [81, 103, 108–110]. Although this kind of method is specialized in not only producing the pareto front but also analyzing the relationships between contradicting objectives, the computational complexity may be inevitably increased. To explain this, it is necessary to point out that it is the optimization iteration and rank sorting process that commonly involved every objective considered in a specific problem, thus resulting in the negative effects mentioned above. Additionally, once it is required to fulfill different kinds of preference requirements, an interactive process will then become an extremely important procedure for the MOEA-based strategy, which may put an unbearable burden on the decision maker. Because of these stumbling blocks, this subsection will mainly focus on the multiobjective transcription (MOT) method. Aiming to address the multi-objective optimization problem, this kind of algorithm can be formulated in the form of Find decision variables x = [x1 , x2 , . . . , xn ] Minimize objective function J (x) = [J1 (x), J2 (x), . . . , JM (x)] subject to xmin ≤ x ≤ xmax Hi (x) = 0 G j (x) ≤ 0 J (x) ∈ P (i = 1, 2, . . . , E) ( j = 1, 2, . . . , I )

(1.7)

where P = {J (x)|P(Ji (x)) > P(J j (x))}, and in this formulation P(·) represents priority factors of different performance metrics. The inequality P(Ji (x)) > P(J j (x)) is a typical constraint that the priority of the jth objective is lower than the ith objective. Based on the formulation stated above, an effective algorithm is waiting to be designed so as to significantly reduce the computational complexity as well as efficiently handle the mission-dependent preference constraints. Generally speaking, to utilize the MOT methods is to transform the former multi-objective formulation to a single-objective problem (SOP). In contrast with MOEA strategies

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Fig. 1.3 Preference region illustration

discussed previously, the MOT strategies have the ability to fulfill the preference requirement like the priority constraints without relying on the rank sorting process which may consume a considerable amount of time. Typically, among all those MOT methods, it is in the weighted-sum strategy that different criterions are transformed into only one single objective by using weight coefficients, thereby leading to its widespread implementation. Although the weight coefficients are really convenient in practice, the investigation in [92] indicated that it sometimes may be incapable of representing the priorities or the true preferences. Besides, in order to carry out a multi-objective spacecraft rendezvous mission, the optimal control command was successfully computed by Gao et al. [13]. The key procedure of their study is that they transcribed the original multi-objective optimal control problem into a convex version whose constraints are mainly linear matrix inequalities. However, it can be extremely difficult to extend this formulation so as to handle the problem of the multi-objective optimal control while considering priority requirements in the meantime. To break free from the aforementioned drawbacks brought by weight coefficients as well as priority, a physical programming (PP) method was designed by Messac to convert the objectives in 1996 [90]. In this strategy, different with the former perspective of regarding the objective value as a whole, this value was deliberately divided into several different preference regions as depicted in Fig. 1.3 so that the original problem formulation can be replaced by a newly-constructed physical optimization model. In their follow-up research [91], they proposed and successfully embedded an interactive strategy in the original PP framework, thus establishing an interaction physical programming method (IPP). But due to the strong nonlinearity of the formulated optimization model, it is still difficult to formulate the preference model as well as select the preference function.

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Fig. 1.4 Graphical illustration of the interactive decision-making process

Luckily there have been several reported techniques which may be useful to address the problem mentioned above. For example, inspired by the definition of fuzzy set, an enhanced PP algorithm was designed in [37]. The feasibility of this enhanced algorithm was then validated in the process of handling a problem of multiobjective space maneuver vehicle trajectory design. Moreover, the authors in [92] proposed an interactive process and embedded it in the original algorithm framework. More specifics about this process can be found in the illustration depicted in Fig. 1.4. Specifically, it is by adjusting preference functions and the aspiration level that the interactive process is then achieved. In this way, the decision makers can take control of it with more flexibility. It is reported as a theory that the proposed method has the capability to drive mission objectives into pre-specified desirable regions. To confirm this, relative researchers carried out numbers of simulations and finally the results validated the theory above. But just as every coin has its two sides, there is still a serious problem in the method as well. That is, the IFPP algorithm may not be able to act as the theory says once the decision maker only has a poor physical knowledge of the very problem. Also, this algorithm may run inefficiently with simultaneous consideration of preference requirements (i.e., may require several interactive trials). From the discussion above, it is obvious that preference constrains like priority requirements are particularly hard for current methods to deal with. To cope with this, the fuzzy goal programming (FGP) strategy, which is essentially a multi-objective transcription algorithm, was proposed in [79]. As the aforementioned theory above, the PP-based strategies along with its variances have the talent of driving mission objectives into pre-specified preference regions. Yet, unless the designer has a good knowledge of the specific problem, the PP-based strategies may be still not superior to the method discussed in [79]. And more, the difficulties in handling priority constraints, along with strong sensibility to not only the preference regions but also the aspiration levels, are all challenging problems in the process of utilizing the PP-based strategies. After listing all those shortcomings of the PP-based methods as a contrast, now it is time to talk about the FGP strategy. Firstly, if we take the goal attainment of different objectives as a long journey, this strategy is able to show how far we have

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travelled. Moreover, there is no need to worry about the poor physical knowledge any more as none of this is required in the FGP method. Besides, this strategy is really easy and convenient to use in practice as well. However, having all these advantages, this method still owns its drawbacks. The following are a brief conclusion about the merits and demerits of the FGP method. On the one hand, it is effective in handling preference requirements as well as performs well in generating the multi-objective optimal flight trajectory. On the other hand, the convergence ability of this method is poor as too many design variables are introduced.

1.5 Stochastic Spacecraft Trajectory Optimization The majority of the optimization-based approaches, which we have discussed before, can reliably and effectively produce optimal flight trajectories (especially, for optimal control sequences and spacecraft state). However, they only focus on deterministic models, where numerous model or actuator uncertainties are ignored. It is important to highlight that when planning trajectory for the actual mission scenarios, we generally should take into account numerous model or actuator uncertainties. Consequently, it is definitely necessary for the constraints and dynamics, which the stochastic variables can influence, to be adjusted appropriately. This can also promote to the improvement of stochastic spacecraft trajectory optimization [111–118]. In this section, for handling the issue of space vehicle trajectory optimization while taking into account chance constraints (CCs) and stochastic dynamics, several different computational frameworks, which the literature includes, will be studied and analyzed. To be exact, the chance-constrained spacecraft trajectory optimization with deterministic dynamic model will be illustrated in the following subsection. We will study the techniques, which are the feasible tools to handle stochastic variables involved in the constraints, in detail. Subsequently, Sect. 7.2 will review the chanceconstrained spacecraft trajectory optimization issue with stochastic dynamics, which can contribute to a better understanding of the basic definitions, solution methods, and some current difficulties of the stochastic spacecraft trajectory design issues.

1.5.1 Chance-Constrained Spacecraft Trajectory Optimization To begin with, in this subsection, we will discuss the issue of chance-constrained spaceship trajectory optimization with deterministic dynamics or equations of motion, in which the uncertain variables can only influence the flight path constraints. If we introduce the uncertain variable and probabilistic constraints in Eq. (1.4), we can denote the formulation of this kind of issue as:

1.5 Stochastic Spacecraft Trajectory Optimization

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t f minimi ze J = (x0 , t0 , x f , t f ) +

L(x(t), u(t), t)dt t0

subjectto ∀t ∈ [t0 , t f ] x(t) ˙ = f (x(t), u(t), t) (dynamic constraints) (boundary conditions) b L ≤ b(x0 , t0 , x f , t f ) ≤ bU Pr {b L ≤ b(x0 , t0 , x f , t f ) ≤ bU } ≥ 1 − φ (terminal CCs) Pr {g L ≤ g(x(t), u(t), t; ξ ) ≤ gU } ≥ 1 − g (path CCs)

(1.8)

where, by ignoring loss of generality, ξ ∈  ⊂ R N p is assumed to be an uncertain parameter with a known probability density function (PDF). Besides, we denote the noise-perturbed path function as g : R Nx × R Nu × R ×  → R Ng . In Eq. (1.8), Pr (·) indicates the probability operation, while  denotes the acceptable probability of occurrence. It is important to highlight that the path chance constraint in Eq. (1.8) is a joint chance constraint, which emphasizes the truth that any individual component of the vector function g need to probabilistically guarantee g iL ≤ g i ≤ gUi , i = 1, 2, . . . , N g , such that the joint event {g L ≤ g(x(t), u(t), t; ξ ) ≤ gU } can be satisfied. In order to deal with the joint chance constraint, usually it can be decomposed into individual scalar chance constraints, which is a conservative but common method with more advantages in calculation. To be exact, through Boole’s inequality, we can have a sufficient condition of the original joint chance constraint [112]: Pr {g i (ξ ) < g iL } ≤ 1,i ,

Pr {g i (ξ ) > gUi } ≤ 2,i

 Ng Then, with i=1 (1,i + 2,i ) < g applied, we can achieve the joint chance constraint. Usually, it is more interesting to use standard optimal control solution strategy to optimize the state and control trajectories than to design a absolutely novel solution strategy. Nevertheless, due to the fact that we rarely can carry out the valuation of Eq. (1.11), it is impossible to solve the formulation provided in Eq. (1.8) in its current form. Thus, we need to appropriately deal with the constraint, which stochastic parameters influence, such that we can transform solving the chance-constrained optimization issue into tackling the non-deterministic chance constraints. One of the typical methods usually applied is to transform the probabilistic constraint in Eq. (1.11) into a deterministic constraint, which contributes to the improvement of both chance-constrained optimal path design [112] and robust trajectory planning [111]. Typically, we can plan the robust trajectory through robust optimization (RO) method, which has the main benefit in easy understanding and convenient application. Over the past few years, numerous studies have occurred in this field [116, 119–122]. Specially, in [120], Li and Shi developed a robust distributed model predictive control method for a class of nonlinear multi-agent system, where a robustness constraint is brought in the optimization model to deal with the model uncertainty. The authors of [121] designed a differential evolution-based method to address the minimax optimization issues which naturally exist in practical robust designs. In

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[122], Wang and Pedrycz developed an adaptive data-driven RO algorithm, such that a class of optimization issue, where the parameter uncertainty are taken into account, can be handled. Besides, the authors of [116] proposed a novel robust optimization method which they called active robust optimization. As we all know, the RO formulation dedicates to obtain the solution of the worst-case optimization scenario, which implies that the computed solution should fulfill each constraint regarding achieve the stochastic parameters. That is to say, for an RO formulation, we cannot go against any constraints. As an alternative, based on chance-constrained optimization (CCO) techniques, we can design the chance-constrained optimal path. It is worth mentioning that by applying this kind of technique, constraint violations are permitted to exist only when it is less than a user-specified risk parameter. In [123] and the references therein, different CCO methods are reviewed in detail. In [124], a CCO-based model predictive control method was developed such that the movement of the ego vehicle can be optimized. Given the fact that there exist some uncertainties, along with constraints in the system state, the authors of [125] proposed a hybrid CCO algorithm, and an autonomous vehicle motion planning issue is addressed successfully by using it. Although it is more likely to have the strongest solution feasibility by using RO approaches, the CCO methods contribute to the less conservatism. Nevertheless, one of the difficulties of applying the CCO approaches is that we cannot directly compute the probabilistic functions as well as their derivatives. But luckily, there is an available method to deal with this problem. Specifically, this method is to replace or approximate these constraints by applying deterministic functions or samples [126–128]. Since the approximation-based methods is able to handle general probability distributions for the uncertainty and preserving feasibility of approximation solutions, we are inspired to apply this method. So far, several approximation algorithms have been designed according to the Bernstein approach [112, 126], the constraint tightening method [129], the scenario approximation [130], and so on. Commonly, a discretization method, which can parameterize uncertain variables and generate the trajectory ensemble, was applied to designing the chanceconstrained optimal path, which has been presented in these studies. Then, with standard optimal control solvers employed, one can address the resulting discretized version of the issue. There are various chance constraint approximation techniques, which were developed to replace the original probability constraints, such that the optimization method can obtain safe gradient information. The developed method was subsequently applied to search the optimal trajectories for various spacecraft flight trajectory planning scenarios, in which probabilistic constraints are also taken into account. Through simulation outcomes and comparative studies, it has been proved that the performances of these developed chance constraint solution methods are better than those of other existing robust optimization-based approaches. Moreover, for various chance-constrained spacecraft trajectory planning issues, the safe and less conservative solutions can also be obtained by applying these computational frameworks as well.

1.5 Stochastic Spacecraft Trajectory Optimization

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According to the aforementioned discussion, we can outline a typical solutionfinding procedure, which can solve the chance-constrained spacecraft trajectory design issue, with the following four stages: 1. 2. 3. 4.

Decompose the joint chance constraint. Approximate the chance constraints. Transcribe the original problem formulation into a deterministic version. Solve the problem by applying standard trajectory optimization solver.

As the CCO-based methods presented above are viable for taking place of the probabilistic constraints, several open challenges still exist. As an instance, a primary problem is that these methos usually contribute to a high conservatism as well as a challenge in control. Moreover, the differentiability, smoothness and convergence characteristics of the approximation method usually is difficult to be maintained.

1.5.2 Chance-Constrained Spacecraft Trajectory Optimization: Stochastic Dynamics The issue of optimizing a spacecraft trajectory under a chance constraint with noiseperturbed system dynamics is initially described in this subsection. Its formulation is simple to achieve by extending Eq. (1.8) and adding an unknown variable to the dynamics. It can be written specifically as follows: t f minimize J = J = E[(x0 , t0 , x f , t f ) +

L(x(t), u(t), t)dt] t0

subject to ∀t ∈ [t0 , t f ] x(t) ˙ = f (x(t), u(t), t) + ξ (noise-perturbed dynamics) (boundary conditions) b L ≤ b(x0 , t0 , x f , t f ) ≤ bU Pr {b L ≤ b(x0 , t0 , x f , t f ) ≤ bU } ≥ 1 − φ (terminal CCs) Pr {g L ≤ g(x(t), u(t), t) ≤ gU } ≥ 1 − g (path CCs)

(1.9)

The performance index J is written in expectation form because of the characteristics of stochastic dynamics and limitations. The terms  and L stand for the anticipated Mayer and process costs, respectively. Equation (1.9) makes a significant adjustment to the issue formulation given by Eq. (1.8) by introducing noise-perturbed system dynamics. It should be emphasized that the motion in this noise-perturbed system model is rather straightforward and may be thought of as conditionally deterministic. This shows that the system motion is deterministically dependent on the knowledge of the uncertain parameter ξ since, if the uncertain parameter ξ were known, the system motion would be known at all future time instants. Despite the fact that conditionally deterministic motion tends to be more constrained than stochastic motion, there are numerous real-world mission cases where the system equations are explicitly tied to some uncertain factors [131].

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By thinking of the uncertain effect as a stochastic process, as opposed to Eq. (1.9), a more general form of the stochastic spacecraft trajectory optimization model can be defined. This suggests that a set of stochastic differential equations are used to build the system dynamics (SDEs). The formulation of the whole problem is then represented in Eq. (1.10) [132]: t f minimize J = J = E[(x0 , t0 , x f , t f ) +

L(x(t), u(t), t)dt] t0

subject to ∀t ∈ [t0 , t f ] d x(t) = f (x(t), u(t), t)dt + G(x(t), u(t), t)dξ(t) (SDEs) b L ≤ b(x0 , t0 , x f , t f ) ≤ bU (boundary conditions) Pr {b L ≤ b(x0 , t0 , x f , t f ) ≤ bU } ≥ 1 − φ (terminal CCs) Pr {g L ≤ g(x(t), u(t), t) ≤ gU } ≥ 1 − g (path CCs)

(1.10)

in which f and G represent the drift and diffusion components of the random state process, and ξ(t) denotes the stochastic process. The uncertainty is now seen as a dynamic variable in the problem formulation (1.10). Therefore, a deterministic control sequence is no longer the appropriate response to this issue. Although there have been some studies of this topic in the literature, they have all focused on a streamlined linear quadratic variant [132]. The development of practical numerical algorithms for the general stochastic spaceship trajectory optimization problem is still in its early stages. An important step in solving problem (1.9), is to estimate or discover the unknown variables that appear inside the dynamics. This can be accomplished by using a specific stochastic quadrature formula (SQF) to approximate the data with  the required N wk F(ξk ) ≈ F(ξ )R(ξ )dξ error order (raft of convergence). If the equation k=1 holds, a SQF of degree N can be thought of as a collection of weighted parameters {wk }, k ∈ {1, 2, . . . , N } and N p -dimensional uncertain variables {ξk }, k ∈ {1, 2, . . . , N }. R(ξ ) is the PDF of ξ in this case. The addition of SQF allows for the construction of an approximation of the stochastic integral that E[F(x, u, ξ )] ≈

N 

wk F(x, u, ξk )

(1.11)

k=1

in which E(·) denotes the expectation operator. There have been numerous successful SQF methods and their variants described up to this point for quantifying the unknown variables. Due to their capability of breaking down the stochastic variables into a convergent series of polynomials, generalized polynomial chaos (gPC) theory-based techniques have stood out among the others. Numerous aerospace engineering applications have successfully used gPC algorithms [111, 133]. In order to derive the expression of stochastic systems, this type of technique generally uses deterministic orthogonal polynomials and coefficients. The effectiveness of gPC-based approaches for optimum control problems

1.6 Recent Practical Applications of the Optimized Trajectory

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with a limited set of stochastic variables was demonstrated in [133]. The gPC-based methods can be used to interpret the uncertain consequences, for instance, if the dynamic equations described in (1.9) only include a few stochastic variables. To address the probabilistic constraints present in Eq. (1.9), the majority of chanceconstraint handling techniques can still be used. However, this sort of solution becomes computationally expensive and is no longer appropriate for capturing the uncertainty for the problem for the uncertain trajectory optimization problem with a relatively large number of unknown parameters (1.9). The sampling-based methods are a different class of SQF approaches with a lot of development. The Markov chain Monte Carlo (MCMC) method and quasiMonto Carlo techniques are common examples. The simplicity of sampling-based approaches and the fact that the approximation error order is independent of the dimension of ξ are the driving forces behind their use. Considering MCMC as an N can be created by selecting random example, a stochastic variable ensemble {ξ }k=1 N ∼ R(ξ samples from the probability distribution (e.g., {ξ }k=1 √)). Each sample will −1 have an equal weight (e.g., wk = N ), resulting in an O(1/ N ) convergence rate in terms of the approximation error. As a result, the MCMC technique was used to model the unknown parameters in the majority of published papers. However, a propagation mechanism might be required to address the dynamics that have been impacted by noise. The system model’s nonlinearity makes it difficult to carry out this approach. Additionally, when the dynamics spread, all system states must abide by the probability restrictions as given in Eq. (1.9). The computational complexity and time will definitely grow as a result [112]. Additionally, the accumulation of constraint violations and conservatism may impair the solution’s optimality and the optimization algorithm’s capacity for convergence. Consequently, increased efforts are necessary to address the previously mentioned problems. Furthermore, there is still much opportunity for advancement in terms of applying existing techniques or developing fresh ones for problems involving the optimization of spacecraft trajectories under conditions of stochastic dynamics and noise perturbation.

1.6 Recent Practical Applications of the Optimized Trajectory The advancement of trajectory optimization techniques has led to the expansion of several new practical application domains in recent years. This section focuses on the most recent high-level applications of the ideal flying trajectories. Different application scenarios are examined for which the ideal trajectories have been successfully achieved or are the subject of in-depth scientific research. The design of integrated spacecraft guidance and control systems, the design of satellite or spacecraft formation control schemes, and the design of an online guiding strategy based on a database are a few examples of these scenarios.

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1.6.1 Design of Integrated Spacecraft Guidance and Control Systems The ability of the ideal flight trajectory to influence the design of online guidance and control systems for spacecraft is a crucial feature [134–136]. The most current articles easily contain works on advancing this subject. For the Mars entry capsule, a segmented predictor-corrector navigation technique was developed in [137]. In this work, a number of way-points were chosen as the segmentation goals together with a pre-planned optimal flying path. The conventional predictor-corrector methodology was then used for these parts. The simulation demonstrated that this methodology can shorten calculation time while maintaining the benefits of the conventional predictorcorrector method. To build a terminal sliding mode control-based guidance legislation for the Mars landing problem, Dai and Xia’s study [138] implemented an optimal Mars entrance trajectory similarly. The terminal sliding mode control technique used in their research was created to eliminate any differences between the real flight trajectory and the reference trajectory that was already pre-designed to be the best one. Furthermore, the uncertain term in the vehicle dynamics was measured using an extended state observer. Spacecraft navigation and control systems are typically created as two separate loops [139]. To track the angular command produced by the outer-loop guiding method, an inner loop autopilot is constructed. However, a two-loop design like this typically necessitates numerous design iterations and underutilizes the connections between various subsystems, leading to subpar performance [140]. The development of combined guidance law and flight control systems has gained popularity recently. To manage various types of spacecraft in close to real-time, Tian et al. [8] and Liu et al. [141] presented the integrated trajectory and attitude coordination control systems. Their system is composed primarily of three components: a spacecraft attitude controller, an online optimal feedback guiding component, and an offline trajectory optimization component. A reference flight trajectory is initially constructed in the offline trajectory optimization component using advanced trajectory optimization techniques discussed in the preceding sections. The reference is then given to the online optimal feedback guidance component, which uses a reference-tracking algorithm to create the control increment that will be utilized as the reference control commands for the inner attitude control system. This integrated design’s primary goal is to guide the vehicle so that it can fly along the predetermined reference path.

1.6.2 Design of Spacecraft/Satellite Formation Control Schemes The goal of spacecraft/satellite formation control is to guide a fleet of small spacecraft/satellites along a predetermined trajectory while maintaining a desired pattern. It is one of the most well-liked avenues that can aid in the advancement of space tech-

1.7 Conclusions and Future Development

33

nology and space travel in the future [142–144]. The literature contains a wide variety of efficient formation control strategies. The leader-follower approach has garnered the most attention among them due to its aptitude for handling challenging jobs. This kind of strategy can be explained by the fact that some of the fleet’s spacecraft function as the leaders, while the other spacecraft take on the role of the followers. In order to maintain the formation and carry out other mission requirements, it is often intended to compel the followers to follow the leader’s course. Therefore, for steady flight and better system control, a well-designed flight trajectory is essential, especially for the leading vehicle. In the past year, a significant amount of study has been done on this type of issue by combining attitude tracking control approaches and trajectory optimization techniques. To examine the issue of orbital maneuver for a formation system, for instance, the authors of [145] integrated trajectory optimization with configuration control. Additionally, a constrained trajectory optimization approach and a coordinative control strategy were suggested and effectively used to a micro space vehicle formation flying problem in [146], taking into account the communication distance and the ground projection area as limitations. In addition, a deep-space dual-spacecraft formation flying problem was investigated in [147], where the ideal flying trajectory for avoiding obstacles was created using a common trajectory optimization technique.

1.6.3 Database-Based Online Guidance Strategy Lately, there has been an increase in interest in creating an online guidance approach based on a sizable pre-planned set of optimal trajectories. In the literature [148, 149], contributions to this kind of method are mentioned. For the space vehicle reentry problem, for instance, a database-based online guidance strategy was created and used in [149]. A substantial database of optimal trajectories was originally constructed for this guidance technique. The onboard algorithm then generated the optimal command solution corresponding to a subset of initial-conditions variations. The key benefit of utilizing such a design is that real-time applicability can be easily obtained and that it is more dependable to be used online when compared to other guidance systems based on control theory. This kind of approach can also manage dispersions throughout the flying phase. As a result, we anticipate seeing more success stories in the application of database-based online coaching tactics over the next couple of decades.

1.7 Conclusions and Future Development The design of a spacecraft’s trajectory is usually acknowledged as an optimal control problem, and the choice of optimization technique has a significant impact on the final flight trajectory. This chapter provides an overview of recently created optimization techniques that can be used to solve constrained spacecraft trajectory design

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issues. This chapter reviews various spacecraft trajectory optimization techniques and results. It is crucial to note that there have lately been many numerical trajectory design approaches as a result of advancements in the fields of nonlinear programming and artificial intelligence-based optimization. Even when various restrictions need to be taken into account, using these innovative methodologies can help satisfy difficult mission requirements and increase the quality of the generated solution. The original problem formulation should be expanded to a multi-objective trajectory planning version or a stochastic trajectory planning version because in practice, more than one mission objective and various types of vehicle or environmental uncertainty may occur during the path planning phase. In this chapter, we evaluated the most recent theoretical advances in trajectory optimization for spacecraft, focusing on several competing mission performance indices, stochastic variables, restrictions, and handling techniques. In addition to providing instructions for the creation of trustworthy multi-objective and stochastic spacecraft trajectory optimization algorithms, the key elements of these recently created techniques, such as their benefits and drawbacks, were explained. To demonstrate conclusively that these methods and findings have already started to be widely used, some current applications of the improved trajectory were also covered. In the coming decades, we expect to hear about many more applications of multiobjective trajectory optimization and stochastic trajectory optimization techniques in areas including interplanetary travel, rendezvous and docking, formation flying, and planetary exploration. Novel multi-objective and stochastic trajectory optimization techniques are urgently required for many of these new spacecraft missions to handle various mission- or vehicle-dependent performance indices and model errors/uncertainties. Additionally, studies validating the efficiency, dependability, and effectiveness of trajectory optimization algorithms are quite likely to be published. This will make the algorithm credible in the eyes of the aircraft engineers.

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Chapter 2

Heurestic Optimization-Based Trajectory Optimization

Abstract Conventional optimization methods have certain problems in finding the optimal solution. The feasible solution space of a trajectory optimization model may be constrained to a relatively limited corridor due to numerous mission-related constraints, easily leading to local minimum or infeasible solution identification. This section focuses on an attempt to use a biased particle swarm optimization method to solve the constrained trajectory design problem. By adding a normalized objective that reflects the entire quantity of constraint violations, the suggested method reformulates the original issue into an unconstrained multi-criteria version. The algorithm also includes a local exploration operation, a novel ε-bias selection method, and an evolution restart strategy to speed up progress during the evolutionary process. The success of the suggested optimization technique is confirmed by numerical simulation experiments that were generated from a confined atmospheric entry trajectory optimization example. Executing a number of comparative case studies also demonstrates the main benefits of the suggested strategy.

2.1 Introduction Constrained optimal trajectory planning problem exists widely in various situations and profoundly affects the decision of optimal path. A suitable design of the optimal maneuver trajectory is significantly required for situations involving constrained trajectory planning in order to achieve improved guidance and control performance. It should be emphasized that there are two primary groups of approaches to solving this kind of problem: indirect and direct methods [1, 2]. The first-order optimality requirements for the differential-algebraic equations (DAE)-constrained systems are directly generated and solved in an indirect way using the “optimize then discretize” strategy. Applying this kind of methodology has received numerous significant contributions [3–6]. As an illustration, Yang and Hexi [3] used an indirect method to generate the energy-optimal trajectory for an irregular asteroid landing mission. An indirect heuristic approach was created by Pontani and Conway in [4] and successfully used to solve a low-thrust orbital transfer problem. Additionally, this indirect © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_2

43

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2 Heurestic Optimization-Based Trajectory Optimization

strategy was expanded in their later work [5] to address more complex trajectory design issues with higher numerical precision. Although the outcomes of an indirect method might be regarded as the theoretically ideal answers, indirect techniques are typically restricted to relatively lowdimensional problems, and it frequently fails when dealing with issues with intricate path constraints. In contrast, a direct approach employs a “discretize then optimize” strategy, where state and/or control variables are first discretized to transform the original issue into a static parameter optimization problem with a finite number of choice variables [7–9]. The ability to combine this kind of technique easily with sophisticated parameter optimization solvers to address the resulting NLP is one of its main benefits. Additionally, different kinds of system constraints are portrayed in a comparatively simple way. As a result, we give “discretize then optimize” approaches more consideration when implementing them. In the domain of flight vehicle trajectory optimization and robotic motion planning issues in challenging environments, direct methods have seen substantial recent application [10–12]. For instance, the optimal flight trajectory for an entry vehicle during the Mars entry phase was determined using the direct collocation method in [10]. Path constraints were put in place so that the vehicle could travel along a restricted corridor. A direct shooting method was used by the authors of [11] to plan the movements of a manipulator. In their research, the optimization model was made significantly more difficult by taking into account both the higher-order dynamics and the contact/friction force constraints. Moreover, a direct trajectory optimization framework for general manipulation platforms was developed in [12], where a number of mission-related constraints, including environmental constraints, collision avoidance constraints, and some geometric constraints, were involved in the formulation of the problem and taken into consideration during the optimization. For resolving constrained trajectory planning problems, heuristic methods such as particle swarm optimization (PSO), genetic algorithms (GA), and differential evolution (DE) have also attracted a lot of attention in addition to direct methods. The chief reason for using these particular optimization algorithms is that heuristic methods are more likely to find the globally optimal solution. In contrast, numerical gradient-dependent optimization algorithms, such as the interior-point method (IPM) [13], sequential quadratic programming (SQP) [14, 15], and other modified versions [16, 17], only guarantee convergence toward local optima. Numerous relevant works have emphasized this benefit, and as a result, various heuristic approaches have been suggested to solve constrained space vehicle trajectory optimization problems [18–21]. For instance, an improved PSO algorithm was published in [19] to address the issue of designing the reentry trajectory for reusable launch vehicles. This method relies on a modified mutation mechanism that was created to speed up evolution. The authors of [20], which addressed an Earth-to-Mars interplanetary problem together with a multiple-impulse rendezvous mission, proposed a parallel optimization framework including PSO, GA, and DE. The simulation results also showed that the proposed method’s performance tends to be problem dependent, despite the fact that the two problems were satisfactorily solved by the hybrid approach. Additionally, in [21], a segmented PSO method was recommended to investigate the optimal

2.1 Introduction

45

control sequence with a bang-bang structure for a time-optimal slew maneuver task by defining the number of switches as the optimization parameters. This PSO-based trajectory planner, however, may not be able to yield promising findings or may fail to discover viable solutions if a maneuver planning problem involves singular arcs or the optimal control sequence does not possess a bang-bang structure. Generally, each member of the swarm represents a potential answer to an optimization issue. Based on the given experimental results, the majority of researchers or engineers came to the conclusion that the PSO has the potential to eliminate local minima for various trajectory design problems with a suitable choice of algorithm parameters [18, 22]. This can be explained by the fact that both the experience of the group of particles (e.g., the so-called social component) and the experience of each individual (e.g., the so-called cognitive component) are taken into consideration in the swarm update formula. The PSO can also gain by having fewer function evaluations than other evolutionary optimization methods like the GA and DE, making it more effective. The research described in [23, 24] validated this important discovery. The application of PSO and its expanded versions to diverse engineering optimization issues can be valued and encouraged thanks to the major features highlighted above. The constraint management strategy typically plays a vital role. It can have a big impact on how well the optimization process works when using bio-inspired optimization methods to solve trajectory planning problems. It is important to note that a penalty function (PF) technique is frequently used in the majority of PSObased trajectory optimization solvers to deal with a variety of parameter constraints adhered to the optimization model [19, 25–27]. In other words, the fitness function is augmented with a new term (e.g., the so-called penalty term), indicating the violation of the constraint, and the particle with the lower fitness value is viewed as being a better individual than the others in the current swarm. Understanding and using this technique are both fairly simple. However, it could be challenging to strike a balance between the mission’s objective and the penalty terms. In order to solve the constrained trajectory planning problems, we employ a locally enhanced multi-objective PSO (MOPSO) method in order to avoid assigning penalty functions and additional penalty factors. The constrained optimization issue is then reformulated as an unconstrained multi-criteria version by first specifying the total quantity of constraint violations as an additional objective. Then, all possible solutions are ranked using the traditional non-dominant sorting technique. Noting that applications of this strategy to constrained engineering optimization problems have drawn significant attention (e.g., a detailed review can be found in [28]), the use of a PSO algorithm in constrained problems using a multi-objective approach has been reported in some significant works [29, 30]. However, the pareto dominance was the only strategy used by the majority of published efforts to remedy the issue. A direct application of MOPSO and pareto dominance rules to the altered multi-objective trajectory optimization problem may not have search bias in terms of the mission constraints, according to our prior studies [25, 26]. Consequently, a biased search should be implemented toward the feasible region; else, the algorithm performance may suffer greatly.

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The main contribution of this section lies in the following four aspects: 1. We make an effort to address the problem of insufficient bias for the basic MOPSO algorithm by incorporating a (CV)-bias selection technique based on constraint violation. Then, this approach is broadened to a more generic version, known as the ε-bias selection approach, making it more adaptable to simultaneously optimize the objective function and reduce the unavailability of the solution. 2. The biased MOPSO algorithm has an evolution restart method built into it, giving it an improved ability to avoid becoming stuck in nearby unfeasible regions. 3. A constrained atmospheric entrance trajectory design problem, similar to the one investigated in [25], is solved using the suggested method, with the exception that more constraints are modeled and incorporated into the optimization model. Case studies and detailed analysis are offered to underline the significance of the suggested bias selection procedure and the evolution restart strategy. 4. The proposed approach is compared with additional evolutionary algorithms and a commercial numerical optimum control solver (named CASADI). Comparative results demonstrate the main benefit of using the intended method while highlighting the key feature. To the best of the authors’ knowledge, the confined reentry trajectory optimization problem is addressed in this chapter by first combining the extended bias selection approach with the evolution restart strategy in the locally enhanced-MOPSO. The remainder of this section is organized as follows. The planned ε-bias selection approach, the locally enhanced-MOPSO, and the evolution restart strategy are introduced in Sect. 2.1. The optimal control formulation of the constrained atmospheric entry trajectory design problem is presented in depth in Sect. 2.2. In Sect. 2.3, examples of numerical simulation experiments and other comparison investigations are presented. Finally, Sect. 2.4 brings this article to a conclusion.

2.2 Biased Particle Swarm Optimization Approach 2.2.1 Unconstrained Multi-objective Optimal Control Problem A direct transcription method is adopted to solve the constrained optimal control problem proposed by general trajectory optimization formulation. In other words, the Nk −1 , control variable is parameterized over a finite collection of temporal nodes {ti }i=0 where Nk denotes the size of the temporal set. After that, the discretized control sequence is given by the notation u = (u 0 , . . . , u Nk −1 ). The static version of the problem (2.1) can be expressed as follows after control discretization and numerical integration for ordinary differential equations:

2.2 Biased Particle Swarm Optimization Approach

min J1 =

N k −1 

u(ti )

47

L(x(ti ), u(ti ))dt + (x(t Nk ), u(t Nk ))

i=0

s.t. ∀ti , i ∈ {0, 1, . . . , Nk } s  b j f (x j , u i j ) x(ti+1 ) = x(ti ) + h x j = x(ti ) + h

s 

(2.1)

j=1

a jm f (xm , u im )

m=1

g(x(ti ), u(ti )) ≤ 0 x(t Nk ) = x f

where the values for the intermediate state and control on [ti , ti+1 ] are represented by xm and u im . The discretization coefficients b j and a jm are defined by the used numerical integration method, and h is the step length. As an illustration, if we consider the fourth order Runge-Kutta technique, (b1 , b2 , b3 , b4 ) can be set to ( 16 , 13 , 13 , 16 ). The non-zero members of the a jm can be assigned the values (a21 , a32 , a43 )=( 21 , 21 , 1), respectively. A modest alteration to the problem formulation may be equally as successful as directly addressing the constrained optimization problem (2.1). The barrier function, for instance, is applied and the scalarized variant of the issue is solved using the conventional interior-point approach (as well as its improved versions). In order to convert issue (2.1) into an unconstrained multi-objective version that may be optimized by the MOPSO algorithm proposed in the following subsections, we introduce an additional normalized objective function in this subsection. The constraint violation value of a candidate solution (x, u) for the ith inequality constraint and jth terminal constraint can be stated as follows if the optimization problem comprises m inequality constraints and n terminal constraints:

μig

j

μx f

=

⎧ ⎨ 0,i ⎩

g (x,u) , g¯ i

1,

g i (x, u) ≤ 0; 0 ≤ g i (x, u) ≤ g¯ i ; g i (x, u) ≥ g¯ i .

⎧ ⎪ 1, ⎪ ⎪ j j ⎪ x (t N )−x ⎪ ⎪ ⎪ x¯ j −xkj (t f) , ⎪ Nk ⎨ f = 0, ⎪ j ⎪ x f −x j (t Nk ) ⎪ ⎪ , ⎪ j j ⎪ x f −x f ⎪ ⎪ ⎩ 1,

(2.2)

j

x j (t Nk ) ≥ x¯ f ; j

j

x f ≤ x j (t Nk ) ≤ x¯ f ; j

x j (t Nk ; u) = x f ; j xf j xf

≤ x (t Nk ) ≤ j

(2.3)

j xf;

≥ x j (t Nk ).

The maximum violation value of the ith inequality constraint in the current searching space is expressed in Eqs.(2.2) and (2.3) as g¯ i =max(g i (x, u)). Analogously, the terms j j x¯ f and x f can be defined. The jth planned terminal state value is represented by

48

2 Heurestic Optimization-Based Trajectory Optimization

j

x f . Constrained functions are assumed to have scalar values in Eqs. (2.2) and (2.3) j

for the sake of simplicity. Each member in the vector should be divided by g¯ i , x¯ f or j xf,

respectively, to carry out the division operation if the constraint functions of a candidate solution become a vector. j j Be aware that the major applications of g¯ i , x¯ f and x f are to normalize each violation of a constraint. It is clear from the specification of Eqs. (2.2) and (2.3) j that the values of μig , μx f ∈ [0, 1] are capable of reflecting the severity of constraint violation for the constraints given in general trajectory optimization formulation. Since the normalized constraint violation function may be added based on Eqs. (2.2) and (2.3), the resultant scalar constraint violation J2 ∈ [0, 1] has the form: 1 i 1 j J2 = μg + μ m i=1 n j=1 x f m

n

(2.4)

The violation of the scalar constraint is regarded as a separate objective function that must be minimized. The original problem formulation has been changed to an unconstrained biobjective variant by minimizing the objective functions provided by Eq. (2.4) and general objective function. This unrestricted bi-objection optimum control model is expressed as follows in a compact form: 

tf

min J1 = u(t)

min J2 = u(t)

L(x(t), u(t))dt + (x(t f ), t f )

t0 1 m

m  i=1

1 j μ n j=1 x f n

μig +

(2.5)

2.2.2 MOPSO Algorithm Certain parameter optimization algorithms should be used to create a control sequence that optimizes the two mission objectives taken into account in Eq. (2.5) The design and testing of a modified MOPSO algorithm are the main topics of this chapter. MOPSO is a typical multi-objective optimization algorithm with biological inspiration [31]. Each particle in the swarm represents a potential control sequence for the problem at hand, consisting of a position vector z and a velocity vector v: z(s) = [u1 (s), u2 (s), . . . , u N j (s)] v(s) = [v1 (s), v2 (s), . . . , v N j (s)]

(2.6)

The size of the swarm and the index of the most recent iteration is, respectively, denoted in Eq. (2.6), by N j and s = 1, 2, . . . , Ns . For simplicity, we refer to the j-th component of z(s) in the remaining sections of the study as z j , where j =

2.2 Biased Particle Swarm Optimization Approach

49

1, 2, . . . , N j . The particle investigates the search space by establishing a recurrence relation during the optimization iteration: z(s + 1) = z(s) + v(s + 1)

(2.7)

v(s + 1) = w · v(s) + c1 r1 · (p(s) − z(s)) + c2 r2 · (g(s) − z(s))

(2.8)

where v(s + 1) is given by:

Below is a list of variables that appeared in Eq. (2.8): w: p(s): g(s): c1 , c2 : r1 , r2 :

The inertia weight factor; The personal best position in the sth iteration; The global best position in the sth iteration; Factors reflecting strength of attraction; Two random constants on (0, 1].

This vector should be updated by using the jth particle’s personal best position p j (s): ⎧ ⎨ rand{p j (s − 1), z j (s)} if z j (s) ⊀ p j (s − 1) if z j (s) ≺ p j (s − 1) (2.9) p j (s) = p j (s − 1) ⎩ if z j (s)  p j (s − 1) z j (s) wherep j (s − 1) is the j-th particle’s personal best position at the (s − 1)-th iteration. In this case, the notation ≺ denotes the dominating relation as indicated by the Pareto optimal concept, and z1 ≺ z2 denotes that z1 is subordinate to z2 . The symbol ⊀ denotes the mutually dominating relation in Eq. (2.9). In this instance, the algorithm chooses one of these two vectors at random. Then, the nondominated solutions are gathered to create an external archive A(s) = [z1 (s), z2 (s), . . . , z Na (s)], where |A(s)| = Na denotes the current number of nondominated solutions in the archive, which will change as the evolutionary process progresses. Take note that Na ≤ N A , where N A is the archive’s maximum size as set by the designer. The following algorithm (for instance, Algorithm 1) is used to update A(s). Take note that the symbol | · | denotes a set’s size in Algorithm 1. After performing A(s) = A(s) ∪ A(s − 1) in Algorithm 1, if the size of A(s) exceeds N A , we eliminate the least feasible or ideal option based on the value of J1 or J2 , until the size hits N A .

2.2.3 ε-Bias Selection According to prior research [25, 26], if specific steps are not made to emphasize the searching direction toward the feasible region, the performance of applying a

50

2 Heurestic Optimization-Based Trajectory Optimization

Algorithm 1 Archive update process Input: A(s − 1) and p(s); Output: A(s); /*Main update process*/ for j := 1, 2, ..., |p(s)| do for m := 1, 2, ..., |A(s − 1)| do if zm ≺ p j then Remove zm from A(s − 1) Set an indicator I nd = 1 else Break end if end for if I nd = 1 then Add p j to A(s) end if end for Perform A(s) = A(s) ∪ A(s − 1) Output A(s) /*End archive update process*/

//*No individual in A(s − 1) dominates p j *//

heuristic method to solve trajectory optimization problems may be severely reduced. As a result, we create the ε-bias selection strategy to update the external archive further. An abbreviated version of this technique is initially offered to convey the idea of bias selection before going into further depth about the ε-bias selection strategy. This condensed form will then be expanded into a more generic version. Let’s analyze two candidate particles from the set A(s), z1 and z2 . It would seem that the total degree of constraint violation J2 has a higher priority than J1 and should be biased. A constraint violation (CV)-based bias selection technique can be created if we represent the value of J1 and J2 for z1 and z2 as J1 (z1 ), J1 (z2 ), J2 (z1 ), and J2 (z2 ). In other words, the particle z1 is regarded as superior to z2 if and only if the following CV-dominance conditions are met: 1. (J1 (z1 ) < J1 (z2 )) ∧ (J2 (z1 ) = J2 (z2 ) = 0); 2. 0 < J2 (z1 ) < J2 (z2 ); 3. (J2 (z1 ) = 0) ∧ (J2 (z2 ) > 0). The CV-based selection rules recommend that the biased objective should (e.g. J2 ) be closely followed while comparing particles. This allows for the continual preservation of the nondominated viable candidate until a more optimal choice is found. The CV-based bias selection strategy is expanded upon in the following to a broader ε-bias selection method. In this method, The particle z1 is specifically preferred over another contender z2 if the subsequent ε-dominance conditions are met: 1. (z2 ≺ z1 ) ∧ (J2 (z1 ) ≤ ε) ∧ (J2 (z2 ) ≤ ε); 2. ε < J2 (z1 ) < J2 (z2 ); 3. (J2 (z1 ) ≤ ε) ∧ (J2 (z2 ) > ε).

2.2 Biased Particle Swarm Optimization Approach

51

According to Eq. (2.5), it is obvious that J2 ∈ [0, 1]. The ε-dominance conditions become the CV-dominance conditions if ε = 0. However, if ε = 1, the ε-dominance criteria are comparable to standard Pareto-dominance rules ≺(no bias case) which are extensively implemented (see e.g., [29, 30]). A balancing parameter that can change the degree between these two extreme examples is the value of ε ∈ [0, 1]. Here, we provide a straightforward adaptive formula to set ε: ε = J2max − J2min

(2.10)

where J2max and J2min denote, separately, the maximum and minimum J2 values found in the archive. According to Eq. (2.10), ε tends to be small early in the evolution when all of the particles are relatively far from the feasible border, allowing for greater focus to be placed on constraint violation. On the other hand, ε tends to be larger such that J1 and J2 can be taken into account simultaneously when some of the particles are near the feasible border or they are already in the feasible region. The extended ε-bias selection strategy provides more flexibility to concurrently optimize the objective function and lessen the infeasibility of the solution than the CV-based bias selection strategy. At the conclusion of each iteration, the external archive A(s) will be updated using the ε-bias selection strategy.

2.2.4 Local Exploration MOPSO has a strong global exploration capability, according to early research on its development [28, 29]. A gradient-assisted operation can be used to emphasize the searching process’ local exploration as well. It is important to note that a number of earlier works [32, 33] have combined an evolutionary algorithm with a gradientbased method to enhance local search. Based on the provided results, it was confirmed that a local update method like this could raise the quality of the finished product. As a result, we present this method to update the archive’s elements and make greater advancements in each iteration. Let us denote the directional derivative of the two objectives along em as 

J1 (zm +  · em ) − J1 (zm ) →0   J2 (zm +  · em ) − J2 (zm ) ∇em J2 (zm ) = lim →0  ∇em J1 (zm ) = lim

(2.11)

in which m = 1, . . . , Na , and zm ∈ A(s). As demonstrated in [33],  denotes the length of the step. The two directional derivatives mentioned above can be further written as follows: ∇em J1 (zm ) = (∇ J1 (zm ))T · em (2.12) ∇em J2 (zm ) = (∇ J2 (zm ))T · em

52

2 Heurestic Optimization-Based Trajectory Optimization

In Eq. (2.12), ∇ J (zm ) stands for the gradient of J with respect to zm . A direction vector em which descends both J1 and J2 can be acquired by using: 

∇ J2 (zm ) m) + ω em = − ω1 ∇∇ JJ11 (z 2 (zm ) ∇ J2 (zm )

(2.13)

where the two weight coefficients hold ω1 + ω2 = 1, and ω1 < ω2 . As a consequence, the elements of the current archive are updated via zˆ m = zm +  · em

(2.14)

It’s vital to note that it is evident from examining at the definitions of μg and μx f that the resulting objective function J2 may not always be differentiable. Therefore, in practical situations, we swap out their equations with a piecewise smooth version. More specifically, Eq. (2.19), where λ and κ are positive constants, results in the repair of μg . Note that an analogous method can be used to acquire the repair of μx f (e.g., (μx f , λ, κ)). In this work, every E generation undergoes the gradient update procedure. ⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎨ (−μ2g + 2(1 + κ)μg − (κ − 1)2 )/4κ, (μg , λ, n) = μg , ⎪ ⎪ ⎪ (μg + λ)2 /4λ, ⎪ ⎩ 0,

μg > 1 + n; 1 − κ ≤ μg ≤ 1 + κ; λ < μg < 1 − κ; −λ ≤ μg ≤ λ; μg < −λ. (2.19)

2.2.5 Evolution Restart Strategy Complex constraints may be part of the problem formulation for some practical constrained optimization situations. The viable searching space may be severely constrained due to the high nonconvexity or nonlinearity of these constraints, and the PSO algorithm is likely to become stuck in one of the local infeasible regions. We offer an evolution restart technique to address this issue. Finding out if the current archive is already stuck in an unreachable area is the crucial step in this restart strategy. In actuality, this can be demonstrated by looking at J2 values for all of the particles. The current swarm is likely to converge to an infeasible location if the difference in J2 values between particles is small. More specifically, the two conditions listed below can be used as a warning sign for becoming stranded in inaccessible areas: 1. ∀zm ∈ A(s), J2 (zm ) = 0, 2. The variance of J2 (zm ) is less than a restart threshold μ.

2.2 Biased Particle Swarm Optimization Approach

53

The evolution restart approach will be utilized if these conditions are fulfilled. In other words, every particle in the swarm is randomly recreated in the search space. Even though evolution histories may provide useful information and offer feedback to help with optimization, it can be difficult to tell whether these historical data are promising. Additionally, a sizable portion of space should be set up in advance to keep these historical data. As a result, we make a decision just to throw away these data and start the evolution over by randomly re-initializing each particle in the swarm.

2.2.6 Overall Algorithm Framework The updated archive A(s) is used to choose the global best particle g(s) for the proposed method. The primary mission objective to be optimized is J1 in the converted problem formulation, whereas the solution’s violation of a constraint is J2 . As a result, g(s) can be chosen by adhering to the steps in Algorithm 2. Algorithm 2 g(s) selection process Input: A(s); Output: g(s); /*Main process*/ Initialize F(s) = {} and IF(s) = {} for m := 1, 2, ..., |A(s)| do if J2 (zm ) > 0 then IF(s) = IF(s) ∪ zm else F(s) = F(s) ∪ zm end if end for if F(s) = ∅ then g(s) = arg minzm ∈IF(s) J2 (zm ) else g(s) = arg minzm ∈F(s) J1 (zm ) end if Output g(s) /*End the process*/

In conclusion, Fig. 2.1 depicts the conceptual block diagram of the proposed MOPSO-based trajectory optimization algorithm. The general phases are summarized in the pseudocode in order to clearly demonstrate how the optimization process is carried out (see Algorithm 3). Notably, the evolutionary process is stopped in Step 9 of Algorithm 3 when one of the two following rules can be triggered:

54

2 Heurestic Optimization-Based Trajectory Optimization

Fig. 2.1 Conceptual block diagram of the proposed algorithm

1. s ≥ Ns ; 2. ∀zm ∈ A(s), J2 (zm ) = 0 and the difference of E(J2 ) between two consecutive iterations (e.g., the s-th and the (s − 1)-th iteration) is less than a tolerance value .

Algorithm 3 General steps for the optimization process Input: The algorithm parameters w, r1 , r2 , c1 , c2 , ω1 , ω2 , , N j , s = 1, and Ns ; Output: The final archive A(s); /*Main optimization iteration*/ Step 1: Randomly initialize the position and velocity vectors of the particles; Step 2: Obtain the state trajectory using numerical integration; Step 3: Calculate the two objective values for all particles among the current swarm; Step 4: Apply the nondominant sorting and Algorithm 1 to construct and update the archive A(s); Step 5: Update A(s) via the gradient-assisted local exploration; Step 6: Perform the ε-bias selection process to update A(s); Step 7: Search the global best particle g(s) from A(s) via Algorithm 2; Step 8: Update the velocity and position vectors of the particles; Step 9: Check whether the termination condition is triggered? if not, set s = s + 1 and return back to Step 2. Step 10: Terminate the optimization and output the final archive A(s); /*End optimization iteration*/

2.3 Constrained Atmospheric Entry Problem

55

The second rule indicates that E(·) outputs the expectation value and there is no room for improvement among workable alternatives.

2.3 Constrained Atmospheric Entry Problem This section describes the constrained atmospheric entry trajectory design problem’s optimal control formulation. In particular, Sect. 2.2.A formulates the system dynamics that are employed to characterize the motion of the spacecraft. A few entrance boundary and path constraints are then built in Sect. 2.2.B after that. Lastly, Sect. 2.2.C introduces the mission objectives chosen to evaluate the effectiveness of the entrance maneuver.

2.3.1 System Model The motion of the entering vehicle can be described by the first-order differential equations shown below: ⎡

⎤

⎡ ⎤ 0 V sin ψ cos γ ⎥ ⎢ ⎢ 0 ⎥ ⎥ ⎢ r cos φ ⎢ ⎥ ⎥ ⎢ V cos ψ cos γ ⎢ 0 ⎥ ⎥ ⎢ r ⎢ ⎥ ⎥ ⎢ ⎥ − mD − g sin γ x˙ = ⎢ ⎥x +⎢ ⎢ 0 ⎥u ⎥ ⎢ 2 V −gr ⎢ ⎥ L cos σ ⎥ ⎢ ⎢ 0 ⎥ ⎥ ⎢ L sin σmV +V ( r V ) cos γ ⎣ 0 ⎦ ⎣ mV cos γ + r sin ψ cos γ tan φ ⎦ −K σ Kσ V sin γ

(2.20)

where the system state variables are defined asx = [x p , xa , σ ]T ∈ R7 . Here, the 3-D position of the entry vehicle, consisting of the altitude h, longitudeθ , and latitudeφ, is determined by the expression x p = [h, θ, φ]T ∈ R3 . The radius distance r can be attained via r = h + Re where Re is the radius of the Earth. The entry vehicle’s velocity, flight path angle (FPA), and heading angle are each represented by a component of xa = [V, γ , ψ]T ∈ R3 . The actual and demanded bank angle profiles are denoted by [σ, σc ], and the control variable is given the value u = σc . Table 2.1 contains the physical definitions of various variables/parameters used in Eq. (2.20), as well as their values or calculation equations.

56

2 Heurestic Optimization-Based Trajectory Optimization

Table 2.1 Variable Definitions Variables

Calculation/values g = rμ2 r = h + Re ρ = ρ0 ex p(−h/H ) D = 21 ρC D V 2 L = 21 ρC L V 2 C D = C D0 + C D1 α + C D2 α 2 CL = CL0 + CL1 α S = 250 m2 Re = 6371.2 km ρ0 = 1.2256 kg/m3 H = 7.25 km m = 92073 kg μ = 398603.2 km3 /s2

g: gravity r : radius distance ρ: atmospheric density D: drag force L: lift force C D : drag coefficient: C L : lift coefficient: S: reference area Re : radius of the Earth ρ0 : sea-level air density H : density scale height m: mass μ: gravitational parameter

2.3.2 Entry Phase Constraints The following four categories of constraints must be satisfied throughout the planetary entry flight: 1. 2. 3. 4.

Safety corridor constraints; Variable terminal boundary constraints; State and control path constraints; Angular rate constraints.

2.3.2.1

Safety Corridor Constraints

The aerodynamic heat transfer rate Q, the dynamic pressure P, and the load factor N L must be contained within specific safety corridors for the duration of the flight in order to safeguard the entry vehicle’s structure. We could express this by: 0 ≤ Q(r, V, α) ≤ Q¯

(2.21)

0 ≤ P(r, V ) ≤ P¯

(2.22)

0 ≤ N L (r, V ) ≤ N¯

(2.23)

2.3 Constrained Atmospheric Entry Problem

57

¯ P, ¯ N¯ ) are set to (125, In Eqs. (2.21)–(2.23), the permissable peak values of ( Q, 280, 2.5). While the dynamic pressure P and load factor N L are primarily influenced by the radial distance r and the velocity V , the heat transfer rate Q is a function of the radial distance r , velocity V , and angle of attack (AOA)α. The following equation [34] can be used to get the value of α(in degree):  α=

40 − w1 (V − Vˆ )2 /3402 , if V < Vˆ ; 40, if V ≥ Vˆ .

(2.24)

in which Vˆ = 4570 m/s, and the value of w1 is equal to 0.20705. It is significant to mention that the heat transfer rate Q in Eq. (2.21) has two main parts: Q(r, V, α) = Q r (α) · Q d (r, V ) (2.25) in which Q r (α) denotes the aerodynamic heat flux and is calculated by Eq. (2.26), whereas Q d (r, V ) stands for the radiation heat transfer and is derived by Eq. (2.27). Q r (α) = q0 + q1 α + q2 α 2 + q3 α 3

(2.26)

Q d (r, V ) = k Q ρ 0.5 V 3.07

(2.27)

In Eq. (2.26), (q0 , q1 , q2 , q3 ) = (1.067, −1.101, 0.6988, −0.1903). Additionally, using Eqs. (2.28) and (2.29), you may determine the dynamic pressure and load factor, respectively. (2.28) P(r, V ) = 21 ρV 2 N L (r, V ) =

2.3.2.2

√

L 2 +D 2 mg

(2.29)

Boundary Constraints

To start the terminal area energy management phase, the terminal boundary constraints are set so that the flight states can achieve particular values at time t f [13, 18]. In particular, the height and flight path angle must be satisfied. |h(t f ) − h f | ≤ εh f |γ (t f ) − γ f | ≤ εγ f

(2.30)

where h f = 30 km and γ f = −5◦ are the targeted terminal altitude and flight path angle values, respectively. εh f = 500 m and εγ f = 0.1◦ denote the permissable errors. Furthermore, the terminal velocity is required to meet:

58

2 Heurestic Optimization-Based Trajectory Optimization

V fmin ≤ V (t f ) ≤ V fmax where V fmin and V fmax are set to 900 and 1100 m/s, resulting in V (t f ) ∈ [900, 1100]m/s.

2.3.2.3

State and Control Path Constraints

The system state and control variables can be confined within tolerated zones for the entire entry flight according to the state and control path constraints (e.g., ∀t ∈ [0, t f ]). These constraints are expressed as follows: h ≤ h(t) ≤ h¯ φ ≤ φ(t) ≤ φ¯ γ ≤ γ (t) ≤ γ¯ σ ≤ σ (t) ≤ σ¯

θ ≤ θ (t) ≤ θ¯ V ≤ V (t) ≤ V¯ ψ ≤ ψ(t) ≤ ψ¯ σ c ≤ σc (t) ≤ σ¯ c

(2.31)

in which x = [h, θ , φ, V , γ , ψ, σ ] and u = σ c denote the lower bounds of x and u, ¯ θ¯ , φ, ¯ V¯ , γ¯ , ψ, ¯ σ¯ ] and u¯ = σ¯ c represent the upper bounds with respect while x¯ = [h, to x and u, respectively.

2.3.2.4

Angular Rate Constraints

Early research revealed that angular variable trajectories present more oscillations than position and velocity profiles, which is often undesirable [13, 18]. Therefore, in contrast to some previous research studies [13, 18, 19, 27], the angular rate constraints are also taken into account in this work to allow for a smoother evolution of the related angular variables. These constraints can be modeled as follows: γ˙ ≤ γ˙ (t) ≤ γ¯˙ ˙ ψ˙ ≤ ψ(t) ≤ ψ¯˙ σ˙ ≤ σ˙ (t) ≤ σ˙¯

(2.32)

¯˙ σ¯˙ ] are assigned as [−0.5◦ , −0.5◦ , −0.5◦ ]/s ˙ σ˙ ] and [γ¯˙ , ψ, in which the values of [γ˙ , ψ, ◦ ◦ ◦ and [0.5 , 0.5 , 0.5 ]/s, respectively. For some specific purposes of the entry vehicle, such as regional reconnaissance [16] and payload delivery [35], imposing these constraints may be beneficial. The information collecting of inaccessible places or the delivery of high-precision payload tends to be considerably easier because the angular trajectories are less likely to have instantaneous variations.

2.4 Test Results and Analysis

59

2.3.3 Objectives It is established that the atmospheric entry mission is an optimization problem. In objective functions, many performance indices that indicate the quality of the entry flight are established. For instance: • An efficiency-related measure can be proposed by minimizing the flight time duration (e.g., the terminal time instant t f ), which is: Obj1 = min t f

(2.33)

• A safety-related measure can be chosen by minimizing the total amount of aerodynamic heat. That is,  tf

Obj2 = min

Qdt

(2.34)

t0

• An entry capability-related measure can be designed by maximizing the cross range (e.g. the terminal φ(t f )). That is, Obj3 = max φ(t f )

(2.35)

• An energy-related measure can be designed by minimizing the terminal kinetic energy, which is written as (2.36) Obj4 = min V (t f ) All of the aforementioned objectives will be taken into account separately in the section following the simulation results.

2.4 Test Results and Analysis 2.4.1 Test Case Specification First, parameters relating to the algorithms are assigned in order to run the simulations. r1 and r2 are specifically created at random on the interval [0, 1]. [c1 , c2 ] is set to [1.49445, 1.49445], while w is calculated via w = (1 + r1 )/2. ω1 and ω2 are set to 0.3 and 0.7, respectively. [N j , Ns , Nk ] is assigned as [40, 2000, 100]. = 10−6 . The demanded bank angle is randomly initialized within the region σc ∈ [−90, 1]◦ . A PC with an Intel Quar-Core i7-4790 CPU (8GB RAM) is used to run the suggested algorithm. Four test cases are investigated for the considered atmospheric entry problem. In case i the goal is to minimize J1 = Obji , i ∈ {1, 2, 3, 4} while concurrently satisfying all different kinds of constraints. Four unconstrained bi-objective formulations are

60

2 Heurestic Optimization-Based Trajectory Optimization

produced by applying the reformulation procedure depicted in Fig. 2.1 to the total amount of constraint violation value, which is taken into account as an additional objective function J2 for each mission case. Comparative comparisons between the biased MOPSO technique and other sophisticated trajectory optimization algorithms were conducted to demonstrate the benefit of employing the proposed design. For the comparison research, a PSO-based trajectory optimization approach suggested in [18, 19] and a trajectory planning algorithm based on an artificial bee colony reported in [27] are both used. It is important to note that the penalty function strategy is used for these two evolutionary methods to address optimization model constraints. These methods also employ the evolution restart strategy described in Sect. 2.1.F. These two procedures are referred to as PFPSO and PFABC, respectively.

2.4.2 Performance of Different Methods In this subsection, the optimal solutions determined by using different heuristic trajectory optimization algorithms for a single experiment are first given and discussed. The four entry mission cases’ optimized state/control evolutions and corresponding path constraint history are shown in Figs. 2.2, 2.3, 2.4 and 2.5.

Fig. 2.2 State/control/constraint evolutions: Case 1

2.4 Test Results and Analysis

61

Fig. 2.3 State/control/constraint evolutions: Case 2

The pre-specified entry terminal boundary constraints and safety-related path constraints can be satisfied for all of the mission cases, as can be shown from Figs. 2.2, 2.3, 2.4 and 2.5, demonstrating the viability of the explored heuristic methods. Both the multi-objective transformation-based and penalty function-based constraint handling strategies can drive the search direction toward the viable region. In terms of flight trajectories, the same trend can be seen in the results produced by various heuristic algorithms for all the mission cases taken into consideration. The three evolutionary approaches can also yield nearly identical solutions for the third mission case. Additionally, it is evident by viewing the system state and control profiles that the generated trajectories are generally smooth. The differential equation applied to the actual bank angle variable σ is to account for this. This equation indirectly restricts the rate of the actual bank angle and can also be understood as a first-order filter. The proposed approach and another numerical optimum control solver called CASADI [36] are contrasted in order to resolve the constrained atmospheric entry problem (the interior point solver IPOPT [37] is applied in CASADI). This solver has gained popularity and has been used in the literature to solve a variety of motion planning and trajectory optimization issues [38, 39]. With = 10−6 serving as the optimization tolerance, CASADI is used to solve the four mission cases. Figures 2.2, 2.3, 2.4 and 2.5 show the optimized trajectories in visual form.

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2 Heurestic Optimization-Based Trajectory Optimization

Fig. 2.4 State/control/constraint evolutions: Case 3

As seen in Figs. 2.2, 2.3, 2.4 and 2.5, the trajectories generated by the proposed method and CASADI are comparable and generally follow the same pattern. But CASADI has its own distinctive qualities. For instance, CASADI is able to provide substantially smoother bank angle profiles for all the mission cases when compared to the developed approach. This is demonstrated by Figs. 2.2, 2.3, 2.4 and 2.5, which show that there are more oscillations visible on the bank angle trajectories produced by the proposed method and other heuristic methods. This is mainly caused by the evolutionary process’ unpredictability. Quantitative results for cases 1–4 are listed in Table 2.2 to show the performance attained using various optimization techniques in a clear way. It should be highlighted that in cases where solutions may be impractical, comparing only J1 is not particularly pertinent. If there are possible solutions, it is straightforward that the objective function value can be increased. As a result, J1 and J2 are pairwise compared in various ways in the comparison provided in Table 2.2 (e.g., a J1 value is associated with a J2 value, making it worthwhile to compare this pair with other pairs). Although the relative changes are generally not very large, it is clear from the solution pairs shown in Table 2.2 that utilizing the suggested approach results in better solutions with more appropriate objective values for all the mission cases taken into consideration. Be aware that the goal of mission case 3 is to maximize the final latitude value. For this mission situation, a higher objective value is desired.

2.4 Test Results and Analysis

63

Fig. 2.5 State/control/constraint evolutions: Case 4 Table 2.2 Optimal results obtained via different methods CaseNo. PFPSO PFABC Proposed J1 J2 J1 J2 J1 J2 Case.1 Case.2 Case.3 Case.4

977.85 40212 16.066 947.50

0 0 0 0

945.32 40613 16.060 964.88

0 0 0 0

908.23 39591 16.068 936.16

0 0 0 0

CASADI J1 J2 1061.15 40018 15.839 1086.81

0 0 0 0

The supplied solution pairs and trajectory profiles show that none of the constraints listed in Sect. 2.2.B are broken, guaranteeing the efficacy of both the CASADI and the suggested approaches. More crucially, based on these findings, one can rule out that the better J1 values aren’t the product of infeasible solutions. To evaluate the convergence potential and robustness of the suggested approach and the CASADI, numerous trials were also conducted. Specifically, the proposed approach with randomly initiated swarms was tested in 100 independent trials. Similarly, 100 CASADI trials were carried out with various initial estimate values. The histograms (shown in Figs. 2.6 and 2.7) were created using the solution pairs that

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Fig. 2.6 Histograms of J1 for the two methods

resulted from these two methods in order to demonstrate the relative differences between them in terms of J1 and J2 . The blue and red vertical lines in Fig. 2.6 represent the lower and upper outlier boundaries for the proposed method and CASADI, respectively. It is clear from the findings shown in Fig. 2.7 that the proposed method can move the candidate solution to the workable range for all trials. On the other hand, outliers may be seen in the J2 histograms obtained for CASADI, showing that CASADI repeatedly converges to the local infeasible solution. In addition, a number of outliers can be found by looking at the corresponding J1 histograms. It can be attributed as a result of impractical solutions. As a result, CASADI tends to be sensitive to starting guess values for the problem under consideration and has a higher likelihood of convergent to local infeasible solutions. From this perspective, the proposed approach’s robustness is superior to its counterpart since it takes advantage of the evolution restart strategy outlined in Sect. 2.1.F.

2.4.3 Convergence Analysis for Evolutionary Methods The convergence performance analysis of the various evolutionary trajectory optimization techniques investigated in this research is the main topic of this subsection. Particular focus is placed on the evolution histories of J1 and J2 for the considered

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Fig. 2.7 Histograms of J2 for the two methods

four mission cases. First, Fig. 2.8 shows the average value of J2 for each optimization iteration. As shown in Fig. 2.8, when compared to the PFABC and the PFPSO algorithms, the proposed multi-objective strategy tends to produce faster J2 convergence histories for the four mission cases. In particular, using the suggested strategy, the average value of J2 may be steered to zero for all mission cases with fewer than or equal to 10 optimization iterations. While this amount almost doubles when using alternative techniques. The J2 evolution trajectories emphasize how rapidly the feasible solution may be found using the suggested approach, which then propels the present swarm/population in that direction. Next, Fig. 2.9 displays the average value of J1 for each optimization iteration. Similar to the outcomes shown in Fig. 2.8, the proposed multi-objective strategy has the potential to generate J1 convergence histories for all the considered mission cases more quickly than its competitors. More specifically, the proposed algorithm’s average J1 cost value converges to a more optimal stable value in fewer optimization iterations. In subsequent optimization iterations, this average J1 value essentially stays the same and does not drop significantly. Convergence results for mission case 4 are partially extracted and presented to highlight this behavior clearly. For instance, by restricting the maximum number of iterations to 60, the last subfigure of Fig. 2.9 displays the evolution of the average J1 value. This subfigure makes it clear that the proposed approach may be used to solve the constrained atmospheric entry trajectory optimization problem with improved convergence performance.

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Fig. 2.8 J2 evolutions: Cases 1–4

Fig. 2.9 J1 evolutions: Cases 1–4

2 Heurestic Optimization-Based Trajectory Optimization

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2.4.4 Computational Performance of Different Methods Performance comparisons between the new algorithm and other approaches should be made in terms of computational cost, in addition to outcomes that are compared in terms of algorithm iterations. To accomplish this, initial consideration is given to how long CASADI and the proposed approach will need to compute. The index of optimization tolerance is a significant parameter that might affect the resulting computational durations, it should be noted. Mission cases 1 to 4 are repeated with various levels specified, and the average computational outcomes of 50 successful runs are summarized in Table 2.3. With the exception of mission cases 1 and 3, where is set to 10−6 , it is clear from the data in Table 2.3 that the computing times needed by the proposed method are often shorter than those of CASADI. Additionally, compared to the suggested approach, CASADI’s computational performance is more sensitive to . In other words, as gets tighter, the computation durations needed by CASADI are likely to increase significantly. In contrast, the reported results for the proposed global exploration-based strategy only indicate a minor increase in computing time. In order to create comparative experiments for the various evolutionary algorithms examined in this research, we introduced three indicators. These indications, which can be viewed from various angles, can show the computational cost needed by the evolutionary algorithms: • T1 : The average computation time required for different evolutionary algorithms to find the first feasible solution. • T2 : The average computation time required for different evolutionary algorithms to drive the entire population to the feasible region. • T3 : The average computation time required for different evolutionary algorithms to drive the average J1 value of all feasible solutions to reach a certain level Jˆ1 . We assign the Jˆ1 values as follows for mission cases 1–4: (1000, 41000, 16, 1000). For each of the four mission cases, 50 separate runs were conducted, and the average results are listed in Table 2.4. Actually, the proposed method involves more steps and is generally more costly with each iteration because it incorporates the local exploration process, ε-bias selection method, and evolution restart strategy. However, by doing these extra procedures,

Table 2.3 Computational performance of CASADI and the proposed method (in seconds) CaseNo. Proposed CASADI = 10−6 = 10−7 = 10−8 = 10−6 = 10−7 = 10−8 Case.1 Case.2 Case.3 Case.4

102.24 32.38 50.14 19.42

108.37 34.33 54.56 21.21

113.15 36.72 58.25 23.08

87.86 39.98 41.62 25.47

144.34 79.98 82.63 53.86

208.87 114.52 125.56 82.35

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Table 2.4 Computational performance of different evolutionary methods CaseNo. Proposed PFABC PFPSO T1 (s) T2 (s) T3 (s) T1 (s) T2 (s) T3 (s) T1 (s) T2 (s) Case.1 Case.2 Case.3 Case.4

0.64 0.78 0.73 0.77

5.23 7.44 6.86 6.89

63.25 26.68 25.74 17.91

1.62 2.78 2.36 2.29

8.64 15.53 8.89 13.21

64.36 37.50 31.26 18.72

2.05 1.82 1.44 1.58

43.27 15.15 8.58 13.26

T3 (s) 125.43 51.15 30.08 18.36

some advantages can be attained. For all mission cases, the first feasible solution and moving the entire population to the viable zone tend to be found faster with the suggested approach than with the alternatives, as shown in Table IV. Additionally, the proposed method can quickly push the candidate solution set to a desired level in comparison to existing algorithms. As a result, it is possible to comprehend the importance of taking these extra actions and their effectiveness.

2.4.5 Impact of the Bias Selection Strategy and Local Exploitation Process It has been demonstrated in earlier subsections that using the suggested method as opposed to other trajectory optimization planners allows us to obtain better final solutions. It is yet unclear, though, if employing the proposed bias selection strategy and the gradient-based local exploration approach will be able to aid in problemsolving. In order to further investigate the effects of using the bias selection strategy and the local exploration method, new experiments are being developed. There are two more tests conducted: • Experiment 1: We compare the results produced by applying the proposed algorithm with and without the bias selection strategy. • Experiment 2: We compare the results produced by applying the proposed algorithm with and without the local exploration method. For mission case 1 and case 2, the average histories of constraint violation (e.g., J2 evolutions) are shown in Figs. 2.10 and 2.11, respectively. The exhibited J2 trajectories clearly show that the proposed algorithm performs inferiorly to the one employing the bias selection method when using that strategy alone. More precisely, the swarm still contains infeasible solutions after performing numerous iterations. In addition, if a search bias is not included in the multi-objective trajectory optimization procedure, the J2 convergence history often tends to be substantially slower. We may therefore draw the conclusion that the application of the bias selection strategy has the potential to positively direct the multi-objective optimization process toward discovering more promising solutions.

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Fig. 2.10 Experiment 1: J2 results for mission case 1

Fig. 2.11 Experiment 1: J2 results for mission case 2

Regarding experiment 2, the matching J1 evolution results for entrance mission cases 1 and 2 are shown in Figs. 2.12 and 2.13, respectively. It is evident from the trajectories shown in Figs. 2.12 and 2.13 that MOPSO equipped with the local

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Fig. 2.12 Experiment 2: J1 results for mission case 1

Fig. 2.13 Experiment 2: J1 results for mission case 2

exploration approach can swiftly steer J1 to a more ideally stable value for the considered atmospheric entry mission cases. We can therefore conclude that updating the candidate set during the optimization iteration using the gradient-based local exploration method is advantageous. Be aware that identical findings for experiments

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Fig. 2.14 Failure case examples

1 and 2 can also be drawn for mission cases 3 and 4. For space considerations, we thus skip over the presentation of their outcomes.

2.4.6 Impact of the Restart Strategy The effectiveness of the restart strategy (RS) proposed in Sect. 2.1.F will be examined and tested in this subsection. By contrasting the outcomes of the suggested algorithm with and without this tactic, experiments were created. It should be noted that empirical tests were conducted, and the restart threshold value was set to 10−3 for the duration of the simulation. Table 2.5 lists the statistical results from 100 independent runs for each of the four mission cases, including the average value of the primary objective (denoted as mean(J1 )), the average constraint violation value of failure cases (denoted as mean(J2 )), the times at which the infeasible solution converged (denoted as Td), and the success rate (calculated by using rs = 1 − Td /100). Additionally, Fig. 2.14 presents instances of convergence failure for each of the four mission cases (taken from Table 2.5). The J2 evolution trajectories for various mission cases converge to a value that is above zero, as can be shown in Fig. 2.14. This further demonstrates that varied reentry mission cases may cause the evolution process to become trapped in regionally infeasible locations. When the restart strategy is included in the proposed algorithm, superior solution pairs (e.g., the J1 and J2 values) are obtained, according to an analysis of the findings

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Table 2.5 Results obtained with and without RS CaseNo. Proposed method with RS Mean (J1 ) Mean (J2 ) Case.1 Case.2 Case.3 Case.4 CaseNo. Case.1 Case.2 Case.3 Case.4

944.26 0 39801 0 15.952 0 963.35 0 Proposed method without RS Mean (J1 ) Mean (J2 ) 967.86 0.031 40168 0.028 15.902 0.027 983.77 0.025

Td

rs (%)

0 0 0 0

100 100 100 100

Td 9 9 6 13

rs (%) 91 91 94 87

shown in Table V. On the other hand, the proposed algorithm is more likely to converge to local infeasible regions if the restart strategy is not used. The reported success rate and instances of infeasible solutions converging are the key indicators of this. In conclusion, the restart method can be used in the suggested algorithm to provide more optimal solutions and improve convergence performance, according to the presented findings. In other words, the restart strategy’s contributions to the proposed methodology can be appreciated.

2.5 Conclusion In this chapter, a biased MOPSO method is proposed to address the problems associated with constrained trajectory optimization. The original problem is first reformulated using the suggested approach as an unconstrained multi-objective optimization model. The optimal option for the changed model is then sought using a locally enhanced evolutionary process, a ε-bias selection method, and an evolution restart strategy. Numerical tests on the solution of a constrained atmospheric entry maneuver planning problem were conducted to assess the efficacy of the proposed methodology. Studies that compared our method to other popular trajectory optimization algorithms were also carried out and presented. Based on the simulations that were run, we have concluded that: • If the evolution restart method is applied to the suggested algorithm, it is expected to produce more optimal solutions and better convergence performance, according to an analysis of the comparison results. • Applying the bias selection-based non-dominant sorting method has advantages. • By utilizing the local line search operation, one can locally explore the solution space and advance the evolutionary process.

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As a result, we think the developed approach will be valuable to the trajectory planner design community and can be a viable option that provides promising results for the considered reentry trajectory optimization problem.

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Chapter 3

Highly Fidelity Trajectory Optimization

Abstract In this chapter, the problem of time-optimal reconnaissance trajectory design for the aeroassisted vehicle is taken into account. Unlike the vast majority of previously reported works, the feasibility of using the dynamic model of high-order aeroassisted vehicle to plan the optimal flight trajectory is discussed, so as to narrow the gap between the simulation model and the real system. A highly-constrained optimal control model including six-degree-of-freedom vehicle dynamics is built. Besides, a pipelined optimization strategy is proposed in order to solve the formulated high-order trajectory planning model. Based on the variable order Radau pseudospectral method, this approach shows that the mesh grid used for discretizing the continuous system goes through several adaption iterations. This strategy can be used to smooth the flight trajectory and improve the convergence ability of the algorithm. The numerical simulation results show a few key features of the optimized flight trajectory. The effectiveness of the applied method and the high-order trajectory planning model are verified by a number of comparative studies.

3.1 Introduction Recently, considerable attention in aerospace industry has been attracted by aeroassisted vehicles because of their potential and reliability for long-endurance, lowenergy and propellent-free applications [1–3]. This type of vehicle has the flexibility to only use aerodynamic forces to execute maneuvers [4], which is one important feature that can complete various atmospheric flight missions [5, 6]. Recently, the problem of inaccessible area reconnaissance has been considered as an important research topic [7, 8] in these applications. Considering different mission-related requirements [9], the observation flight trajectory planning is becoming more and more crucial in the success of the mission [10, 11]. The reason for this phenomenon is that a well-planned maneuver trajectory is especially important to ensure safety and provide enhanced control. Over the past few decades, researchers and aerospace engineers have put a lot of effort into developing promising methods for trajectory planning. By consulting the literature, many effective trajectory planners can be found. Among them, reports of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_3

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space vehicle atmospheric flights are especially substantial. For example, the authors in Ref. [12] studied and solved the hypersonic vehicle atmospheric entry problem. A global collocation method was conducted and an emphasis on multiple no-fly zone constraints formulate the problem in this paper. Likewise, Ref. [13] proposed a constrained Particle Swarm Optimization-based (PSO) trajectory planner in order to search the end-to-end trajectory for hypersonic reentry vehicles. For the purpose of rapidly constructing the landing footprint, an interpolated model was also used to plot the bank angle profile. An improved PSO-based trajectory planning algorithm was constructed in Ref. [14] for the sake of obtaining enhanced convergence performance of the optimization process. Then, this approach was useful in optimizing the glide trajectory of the hypersonic reentry vehicle. A Gauss pseudo spectral method-oriented trajectory generator, whose purpose was to take multiple process constraints and terminal constraints into consideration, was proposed in Ref. [15] for a solar-powered aircraft. A hyper-heuristic trajectory generator was also proposed in Ref. [16] to optimize the flight trajectory of the satellite launch vehicle by combining genetic algorithms, PSO, and simulate annealing. It should be emphasized that the majority of these published studies use the socalled direct transcription technique, which discretizes the search space by applying a finite set of mesh grids to connect the beginning and terminal poses. The optimal vehicle state/control solutions are then investigated at these temporal nodes using computational optimization techniques. According to the presented simulation results, it is obvious that each of the aforementioned planners is capable of investigating plausible and almost ideal flight routes for aircraft or spacecraft. Nevertheless, the majority of them only used the Three-Degree-Of-Freedom (3-DOF) dynamic model to frame the trajectory optimization problems, which results in a drop in model fidelity. Model fidelity can be defined as the extent to which a model accurately captures the features of an actual system [17]. Researchers and engineers frequently use lower-fidelity dynamics to explain the motion of a flight vehicle in atmospheric trajectory optimization challenges [18, 19]. For instance, it’s typical to think of the vehicle as a point mass. In order to approximate solutions to higher-order dynamics, the obtained solutions are employed for preliminary analysis. Despite designing feasible trajectories with regard to the vehicle position and velocity variables that may be appropriate when using a lower-fidelity dynamic model, these trajectories may be impractical for a physical system (e.g., with respect to forces and moments). Moreover, compared to high-fidelity models, a potential safety issue with low-fidelity atmospheric trajectory planning algorithms exits. So, in this section, we are keen to see whether it is possible to predict the optimal trajectory by using a high-fidelity vehicle dynamic model. By using a common global collocation method, we explored the 3-DOF aeroassisted vehicle reconnaissance mission in Refs. [7, 20]. (e.g., the Radau pseudospectral method developed in Ref. [21]). On a set of fixed mesh grids, the applied approach constructs the optimal flight trajectory between the pre-assigned initial point and the target position. The model quality, solution accuracy, and outcome analysis are all

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improved by this study. More specifically, the primary contributions of this study and the specific research object might be summed up as follows: 1. We extend the original problem formulation to a reconnaissance trajectory planning model for the aeroassisted vehicle with six degrees of freedom (6-DOF) and flight time minimization. 2. We use a pipelined optimization strategy based on a variable order Radau pseudospectral method to address the problem given the complexity of vehicle dynamics and constraints. 3. To examine the differences between the two models and to demonstrate the viability of using the variable order pseudospectral approach for the considered problem, comprehensive comparative results are provided. The remainder of this article is divided into the following sections. We construct the problem for the time-optimal reconnaissance maneuver optimization in Sect. 3.1. The aeroassisted vehicle’s rotational and translational dynamics are introduced. This section also formulates the main objective function and a number of constraints. The Radau pseudospectral approach and the mesh grid adaptive strategy are reviewed in Sect. 3.2. In Sect. 3.3, a thorough simulation analysis with the identified ideal trajectory and comparison outcomes will be given. Finally, Sect. 3.4 provides concluding remarks.

3.2 Time-Optimal Reconnaissance Maneuver Optimization Problem The 3-DOF and 6-DOF models of the aeroassisted vehicle are first described in this section using two sets of differential equations. Mission restrictions that must be taken into account during the time-optimal reconnaissance maneuver are then introduced. Finally, a general concept for trajectory optimization is developed.

3.2.1 Model Dynamics The aeroassisted vehicle’s dynamic model is created using the three basic presumptions: Assumption 3.1 By treating the aeroassisted vehicle as a rigid body, distortional effects (e.g. the elastic DOF introduced by the flexible body) are eliminated. Assumption 3.2 Since we are searching for a strictly gliding descent trajectory during the reconnaissance maneuver phase, it is assumed that the engine is off and that no push is being applied.

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Assumption 3.3 We disregard the impact brought on by Earth’s rotation and view the planet as a symmetrical sphere. The 3-DOF motion of the aeroassisted vehicle can therefore be described by the differential equations shown below [7]: ⎧ dr ⎪ = V sin γ ⎪ dt ⎪ ⎪ dφ = V cos ψ cos γ ⎪ ⎪ dt r ⎪ ⎪ dθ ψ cos γ ⎨ = V sin dt r cos φ

dV = − SρC D2m(α)V − g sin γ ⎪ dt ⎪ ⎪ 2 2 dγ cos σ ⎪ ⎪ = SρC L (α)V + ( V V−rg ) cos γ ⎪ dt 2V m 2 r ⎪ ⎪ ⎩ dψ = SρC L (α)V sin σ + V tan φ cos γ sin ψ dt 2V m cos γ r 2

(3.1)

Equation (3.1) is also referred to the translational Equations of Motion (EMOs). Variables mentioned in this equation are r , φ, θ , V , γ , ψ, α, σ , S, C D , g, C L , ρ, representing the radius, latitude, longitude, speed, Flight Path Angle (FPA), azimuth angle, Angle Of Attack (AOA), bank angle, reference area, drag coefficient, gravity, lift coefficient, and atmosphere density, respectively. The consequences of factoring in forces and moments have not yet been completely examined if the flight trajectory is constructed using the 3-DOF model provided by Eq. (3.1). The accuracy of the outcome could deteriorate as a result. The rotating EMOs for the aeroassisted vehicle, which can be expressed as the following set of differential equations, should be adhered to in order to build a high-order 6-DOF model of the aeroassisted vehicle: ⎧ dα sin σ ˙ = q − p tan β cos α − ν tan β sin α + cos (ψ cos γ ⎪ dt β ⎪ ⎪ ⎪ ˙ ˙ ˙ ⎪ −φ sin ψ sin γ + θ cos φ cos ψ sin γ − θ sin φ cos γ ) ⎪ ⎪ σ ⎪ ⎪ (γ˙ − φ˙ cos ψ − θ˙ cos φ sin ψ) − cos ⎪ cos β ⎪ ⎪ dσ ⎪ = − p cos α cos β − q sin β − ν sin α cos β + α˙ sin β ⎪ dt ⎪ ⎪ ⎪ −ψ˙ sin γ − φ˙ sin ψ cos γ + θ˙ sin φ sin γ ⎪ ⎪ ⎪ ⎪ +θ˙ cos ψ cos φ cos γ ⎪ ⎪ ⎪ ⎪ dβ = p sin α − ν cos α + sin σ [γ˙ − φ˙ cos ψ + θ˙ cos φ sin ψ ⎪ ⎨ dt + cos σ (ψ˙ cos γ − φ˙ sin ψ sin γ (3.2) ⎪ ˙ cos φ cos ψ sin γ − θ˙ sin φ cos γ )] ⎪ − θ ⎪ ⎪ ⎪ I x x +Izz −I yy Mz dp Mx ⎪ ⎪ ⎪ dt = Ix x Izz −Ix2z Izz + Ix z ( Ix x Izz −Ix2z + qp Ix x Izz −Ix2z ) ⎪ ⎪ (I I −I I −I ) ⎪ ⎪ +qν yy Izzx x Izzzz−Izz2 x z ⎪ ⎪ xz ⎪ ⎪ dq = Ix z (ν 2 − p 2 ) + M y + pν Izz −Ix x ⎪ ⎪ dt I yy I yy I yy ⎪ ⎪ qν(I yy −I x x −Izz ) ⎪ Mx dν ⎪ = I [ + ] ⎪ x z 2 2 I x x Izz −I x z I x x Izz −I x z ⎪ dt ⎪ ⎪ I x2x −I yy I x x +I x2z Mz ⎩ +Ix x Ix x Izz −I 2 + pq Izz Ix x −I 2 xz

xz

3.2 Time-Optimal Reconnaissance Maneuver Optimization Problem

81

Table 3.1 Definition of aeroassisted vehicle-related parameters Notation Physical meaning β p q ν m S Mi (i = x, y, z) Ii j (i, j = x, y, z)

The sideslip angle of the vehicle The roll rate of the vehicle The pitch rate of the vehicle The yaw rate of the vehicle The mass of the vehicle The reference area of the vehicle The angular moments act on the vehicle The inertia moments act on the vehicle

Rotational variables appeared in these EMOs are β, p, q, and ν, respectively. The notations and physical meanings of various aeroassisted vehicle-related characteristics are compiled in Table 3.1 for clearer appreciation.

3.2.2 Flight Constraints and Objective The following three path constraints specify a safe corridor in which the aeroassisted vehicle should only move throughout the reconnaissance mission:

kn =



kq = kc ρ 0.5 V 3.15 ≤ kqmax

(3.3)

k p = 0.5ρV 2 ≤ k max p

(3.4)

L (α)V ( SρC D2m(α)V )2 + ( SρC 2m )2 ≤ knmax 2

2

(3.5)

max In Eqs. (3.3)–(3.5), kc = 9.4369 × 10−5 , while kqmax , k max denote the upper p , and kn bounds of kq , k p , and kn , respectively. Variable physical limits must also be taken into account in addition to the flight path constraints. The system-related variables, then, should only vary within their acceptable ranges. Thus, we have the subsequent box constraints: ⎧ r ∈ [r min , r max ], φ ∈ [φ min , φ max ] ⎪ ⎪ ⎪ min max ⎪ V ∈ [V min , V max ] θ ∈ [θ , θ ], ⎪ ⎪ ⎪ min max ⎪ γ ∈ [γ , γ ], ψ ∈ [ψ min , ψ max ] ⎪ ⎪ ⎨ β ∈ [β min , β max ] α ∈ [α min , α max ], (3.6) min max p ∈ [ p min , p max ] σ ∈ [σ , σ ], ⎪ ⎪ ⎪ min max min max ⎪ σ˙ ∈ [σ˙ , σ˙ ] α˙ ∈ [α˙ , α˙ ], ⎪ ⎪ ⎪ min max ⎪ , q ], ν ∈ [ν min , ν max ] q ∈ [q ⎪ ⎪ ⎩ min max Mi ∈ [Mi , Mi ], t ∈ [0, t max ]

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3 Highly Fidelity Trajectory Optimization

⎧ ⎨ r (t f ) = r f , φ(t f ) = φ f , θ (t f ) = θ f γ (t f ) = γ f , V (t f ) = V f , β(t f ) = β f ⎩ p(t f ) = p f , q(t f ) = q f , ν(t f ) = ν f

(3.7)

The main goal of the reconnaissance mission is to reduce total flight time while overflying a target location defined by the final boundary conditions (e.g., in Eq. (3.7)). As a result, the function of the mission objective is provided by: minimize J = t f

(3.8)

3.2.3 Overall Trajectory Optimization Formulation An overall trajectory optimization formulation that has the broad form can be created by compiling all the mission-related data given in earlier subsections. ⎧ Search x = x ∗ (t), u = u∗ (t) ⎪ ⎪ ⎪ ⎪ minimize J = (x(t f ), t f ) ⎪ ⎪ ⎪ ⎪ subject to ∀t ∈ [t0 , t f ] ⎪ ⎪ ⎨ dx = f (x(t), u(t)) dt g(x(t), u(t)) ≤ 0 ⎪ ⎪ ⎪ ⎪ , t , b(x ⎪ 0 0 x f,tf) = 0 ⎪ ⎪ ⎪ x ∈ [x min , x max ] ⎪ ⎪ ⎩ u ∈ [umin , umax ]

(3.9)

in which , f (·, ·), g(·), b(·) denote the objective function Eq. (3.8) in Mayer form, EMOs, path constraints, and boundary constraints given by Eqs. (3.1)–(3.6), respectively. x = [r, φ, θ, V, γ , ψ, α, σ, β, p, q, ν] ∈ R12 is the system state vector and u = [Mx , M y , Mz ] ∈ R3 is the control vector. (x ∗ (t), u∗ (t)) stands for the optimal solution to be searched.

3.3 Solution Method An optimal control problem is one that involves finding the optimal flying path for the aeroassisted vehicle reconnaissance mission that was built in Sect. 3.1. The pseudospectral direct transcription strategy is one technique that has been frequently used to resolve problem Eq. (3.9) [22, 23]. This method takes use of its adaptability for a variety of aerospace applications and uses the orthogonal collocation technique to increase the accuracy of polynomial approximation [22, 23]. Numer-

3.3 Solution Method

83

ous pseudospectral approaches have been developed in the literature using various collocation points [24, 25]. In this study, we investigate the potential of the Radau Pseudospectral Method (RPM) for the considered high-order trajectory optimization problem.

3.3.1 Radau Pseudospectral Method A crucial transcribing step in the RPM is to define the continuous-time state and control variables using a special polynomial. Prior to parameterizing the system state and control variables, a time domain transformation should be carried out. That is, the original time interval t ∈ [t0 , t f ] is mapped to τ ∈ [−1, 1) by using t−t 0 τ = t f −tf0 + tt−t . Now assume that there is a finite set of Legendre-Gauss-Radau f −t0 (LGR) temporal points (e.g., τ ∈ {τ0 , τ2 , . . . , τ Nk }, where Nk indicates the size of the temporal set.) The system state and control variables can be approximated as follows: ⎧ Nk  ⎪ ⎪ x(τk )L k (τ ) ⎨ x(τ ) ≈ k=0 (3.10) Nk  ⎪ ⎪ ⎩ u(τ ) ≈ u(τk )L k (τ ) k=1

in which L k denotes the Lagrange polynomials. With the use of Eq. (3.10), the following can be used to approximate the derivative of the state: k k   dx dk ≈ = x(τk ) x(τk )D jk (τk ) dτ dτ k=0 k=0

N

N

(3.11)

in which D jk stands for the differentiation matrix pertaining to the LGR nodes in the above example. The dynamic constraint is therefore equivalent to: Nk  k=0

x(τk )D jk (τk ) −

t f − t0 f (x k , uk ) = 0 2

(3.12)

In more detail, the two sets of EMOs of the aeroassisted vehicle are converted to algebraic equality constraints for the considered high-fidelity reconnaissance trajectory problem. The translational EMOs, for instance, are further written as:

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3 Highly Fidelity Trajectory Optimization

⎧N k  t f − t0 ⎪ ⎪ D jk rk − Vk sin γk = 0 ⎪ ⎪ 2 ⎪ k=0 ⎪ ⎪ Nk ⎪ t f − t0 V cos ψ cos γk ⎪ ⎪ =0 ⎪ D jk φk − 2 · k r k ⎪ k ⎪ k=0 ⎪ ⎪ Nk ⎪  t −t ⎪ V cos γ sin ψ ⎪ ⎨ D jk θk − f 2 0 · k r cosk φ k = 0 k

k=0

k

Nk

 ⎪ t f − t0 SρC D (αk )Vk2 ⎪ D jk Vk + − g sin γk = 0 ⎪ ⎪ 2 2m ⎪ k=0 ⎪ ⎪ Nk ⎪  ⎪ SρC L (αk )Vk2 cos σk t −t t f − t0 Vk2 − rk g ⎪ ⎪ D jk γk − f 2 0 · cos γk = 0 − · ⎪ 2Vk m 2 Vk rk ⎪ ⎪ ⎪ k=0 Nk ⎪ ⎪  SρC L (αk )Vk2 sin σk t −t t f − t0 Vk ⎪ ⎩ D jk ψk − f 0 · − tan φk cos γk sin ψk = 0 ·

2

k=0

2Vk m cos γk

2

(3.13)

rk

The corresponding approximation equations for rotational EMOs can be obtained analogously. The path constraints can also be approximated similarly to the system dynamics. The original problem is reformulated as a finite-dimensional NonLinear Programming (NLP) problem, which is solvable by using mature nonlinear optimization algorithms.

3.3.2 A Pipelined Optimization Strategy Directly applying RPM to the considered problem may be numerically challenging due to the complexity of vehicle dynamics and limitations. To address this issue, a pipelined optimization approach based on a variable order RPM is discussed in this section.

3.3.2.1

Pre-solve Process

The pre-solve process, the main optimization process, and the mesh grid update process make up the pipelined optimization strategy’s three steps. The initial guess value at temporal nodes is derived for the majority of numerical optimal control software by linear interpolation between the user-specified boundary guess value. Nevertheless, this could lead to huge constraint violation values, severely limiting the search space. This problem is attempted to be addressed by the addition of a presolve process, which optimizes the subsequent unconstrained optimization model using the PSO-based optimum control approach provided in Ref. [26]: min J = max{g(x(τi ; u), u), 0} + (x(τ Nk ; u)) u

(3.14)

where (x(τ Nk ; u)) = (x(τ Nk ; u) − x f )2 , while g(x, u) ≤ 0 is the compressed form of the path constraints defined in Sect. 3.1. In Eq. (3.14), x(τi ; u) signifies the

3.3 Solution Method

85

state that is obtained through the numerical integration of the system equations and control discretization. A feasible flight trajectory for the aeroassisted vehicle is swiftly generated by minimising the objective function specified in Eq. (3.14), which is identical to minimising the constraint violation of the original problem. (e.g.,u = (u0 , u1 , . . . , u Nk −1 ) and x = (x 0 , x 1 , . . . , x Nk )). The primary optimization process, which is described in detail in the next subsections, will then be warmly triggered using this solution.

3.3.2.2

Variable Order Adaptive Process

The mesh grid utilized to discretize the continuous time system becomes more sensitive with respect to the quality of the flight trajectory as a result of the vehicle dynamics becoming significantly more complex in contrast to the widely used 3DOF model. To illustrate the flight trajectory, large-scale mesh grids are typically preferred. Large-scale fixed mesh grids, however, could lead to algorithm convergence problems. Nonetheless, while a tiny size mesh grid helps enhance the convergence of the optimization process, it may miss out on important details of the optimal flight trajectory. As a result, the mesh grid adaptation plays a crucial part in the success of problem solving. In this study, the mesh adaptive process is fulfilled by further utilizing the Legendre polynomial series as described in Refs. [20, 27], which is different from the mesh adaptive technique employed in Refs. [24, 25]. To approximate the function x(τ ), a Legendre polynomial of Nk order is applied: x(τ ) ≈

Nk 

l i Ri (τ )

(3.15)

i=0

where Ri (τ ) serves as the Legendre polynomial’s foundation. This mesh adaptive strategy’s main principle is to use the decay rate l i to evaluate the function x(τ ). If x(τ ) is a smooth bounded function, then it should be noted that it can be precisely expressed by: ∞  x(τ ) ≈ l¯i Ri (τ ) (3.16) i=0

Once we use the Nk order approximation given by (3.15), the error term e can be conveyed as: Nk ∞      e = l i Ri (τ ) l¯i Ri (τ ) − i=0

i=0

Nk ∞       = (l i − l¯i )Ri (τ ) l¯i Ri (τ ) + i=Nk +1

i=0

(3.17)

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3 Highly Fidelity Trajectory Optimization

By using the triangle inequality, we can obtain: Nk ∞          ¯ e≤ l i Ri (τ ) +  (l i − l¯i )Ri (τ ) i=Nk +1

(3.18)

i=0

In Eqs. (3.17) and (3.18), the norm is defined by the inner product. Such as,   1   e =< e, e > 2 =



1 −1

e(τ )2 dτ

Nk ∞          Let e1 =  l¯i Ri (τ ) and e2 =  (l i − l¯i )Ri (τ ). We have the following i=Nk +1

i=0

relationship based on the Legendre polynomials’ orthogonality property: < Ri (τ ), R j (τ ) >= 

where δi j =

2 δi j 2i + 1

1i= j 0 i = j

(3.19)

(3.20)

Then the two error terms (e1 and e2 ) can be expressed by   ∞   e1 =  i=Nk

2l¯i 2i + 1 +1 2

(3.21)

  Nk  2(l i − l¯i )2 e2 =  2i + 1 i=0

(3.22)

From Refs. [25, 28], it was shown that l i ≈ l¯i , which implies that e2 can be disregarded because it is negligibly small in contrast to e1 [28]. Furthermore, by applying the convergence result shown in Refs. [24, 25] we can get an estimation of l¯i = c10−σ i , σ > 0. This estimation enables the error e = e1 + e2 to be further expressed as       ∞  ∞ 2 ¯2  2 li ≤  e≤ (3.23) l¯i ≤ e¯ 2i + 1 i=N +1 i=N +1 k

k

3.3 Solution Method

87

√ where e¯ = c10−σ (Nk +1) / 1 − 10−2σ . As a result, we discover an upper bound e¯ for the error term e. It should be noted that even though Lagrange interpolation is used to represent x(τ ) in the preceding part, the Legendre coefficient l i can be obtained by using (3.24) Rl i = x(τi ) where

⎡

⎤ R1 (τ1 ) · · · R Nk (τ1 ) ⎢ R1 (τ2 ) · · · R Nk (τ2 ) ⎥ ⎢ ⎥ R=⎢ ⎥ .. . . ⎣ ⎦ . . R0 (τ Nk +1 ) R1 (τ Nk +1 ) · · · R Nk (τ Nk +1 ) R0 (τ1 ) R0 (τ2 ) .. .

(3.25)

Suppose a tolerance value σ¯ is provided by the user. We state that x(τ ) is smooth if σ > σ¯ and vice versa. The current mesh interval is divided into subintervals if it is determined that a mesh interval is not smooth. On the other hand, if the present mesh is smooth, extending the polynomial degrees will further enhance the approximation. More specifically, once σ > σ¯ , we have the intention to increase the number of nodes in the current mesh interval and this number is calculated by Nk(H +1) = Nk(H ) + lg

ek(H )  σ

(3.26)

where H denotes the index of the mesh grid, while is the user-specified accuracy tolerance. If σ < σ¯ , the mesh grid currently in place will be divided. Two steps are taken to achieve this. Firstly, we determine the new mesh grid’s total node count using e(H )  σ¯ (3.27) N¯ k = Nk(H ) + lg k

Secondly, we calculate the number of subintervals Ns : Ns = N¯ k /Nk(H ) 3.3.2.3

(3.28)

Design Parameter Selection

It is correct that the variable order pseudospectral method implementation will include some design parameters that could affect the algorithm’s performance and capacity to converge. Therefore, in order to begin the solution-finding iteration, an appropriate design parameter selection process is required. In the actual application, this technique becomes even more crucial for trajectory optimization. Nevertheless, choosing the right design characteristics, (e.g., σ¯ and ), may rely on the specific situation. Now, we offer an interactive method for choosing design parameters. The issue can be resolved by providing an initial value σ¯ , and the designer will be shown

88

3 Highly Fidelity Trajectory Optimization

the solutions. The design parameters may be changed if the designer is not satisfied with the planning outcomes based on how the discovered solution behaves. More specifically, it would be advantageous to reassign a larger σ¯ and vice versa if the existing results oscillate greatly or contain any discontinuities. On the contrary, can have a larger initial value (e.g., = 1 × 10−3 ). This will increase the rate at which the optimization process converges. The designer can then tighten the tolerance level and start with the low-accuracy approach to produce outcomes that are more accurate.

3.4 Simulation Results 3.4.1 Simulation Setting To build the high-fidelity aeroassisted vehicle reconnaissance trajectory optimization model, several task-related and vehicle-dependent parameters should be assigned, as stated in Sect. 3.1. The variable initial conditions and desired final states, for instance, are shown in Tables 3.2 and 3.3, respectively.

Table 3.2 Variable initial conditions Variable Altitude, h 0 (km) Longitude, θ0 (◦ ) Latitude, φ0 (◦ ) Velocity, V0 (km/s) FPA, γ0 (◦ ) Azimuth, ψ0 (◦ ) AOA, α0 (◦ ) Bank angle, σ0 (◦ ) Slide slip angle, β0 ((◦ )/s) Roll rate, p0 ((◦ )/s) Pitch rate, q0 ((◦ )/s) Yaw rate, ν0 ((◦ )/s)

Value 79.24 0 0 7.80 −1 90 17 −75 0 0 0 0

3.4 Simulation Results Table 3.3 Variable terminal values Variable Altitude, h f (km) Longitude, θ f (◦ ) Latitude, φ f (◦ ) Velocity, V f (km/s) FPA, γ f (◦ ) Slide slip angle, β f (◦ ) Roll rate, p f ((◦ )/s) Pitch rate, q f ((◦ )/s) Yaw rate, ν f ((◦ )/s)

89

Value 50 17.88 4.30 4.27 0 0 0 0 0

The variable physical limits, rate/path constraints are set as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

50 km ≤ h ≤ 80 km, −180◦ ≤ φ ≤ 180◦ −180◦ ≤ θ ≤ 180◦ , 4 km/s ≤ V ≤ 8 km/s −180◦ ≤ ψ ≤ 180◦ −10◦ ≤ γ ≤ 10◦ , 0◦ ≤ α ≤ 40◦ , −0.5◦ ≤ β ≤ 0.5◦ −90◦ ≤ σ ≤ 0◦ , −5 (◦ /s) ≤ p ≤ 5 ((◦ )/s) ⎪ ⎪ −1 ((◦ )/s) ≤ σ˙ ≤ 1 ((◦ )/s) −1 ((◦ )/s) ≤ α˙ ≤ 1 ((◦ )/s), ⎪ ⎪ ⎪ ◦ ◦ ⎪ )/s) ≤ q ≤ 5 (( )/s), −5 ((◦ /s)) ≤ ν ≤ 5 ((◦ )/s) −5 (( ⎪ ⎪ ⎪ 5 5 ⎪ −1 × 10 lb · ft ≤ M ≤ 1 × 10 lb · ft, 0 ≤ kq ≤ 150 Btu/ft2 /s ⎪ i ⎪ ⎩ 2 0 ≤ k p ≤ 280 slug/ft , 0 ≤ kn ≤ 2.5 g Note that 1 lb = 0.45359 kg, 1 ft = 0.3048 m, 1 Btu/ft2 /s = 1.135654 W/cm2 /s, 1 slug/ft2 = 1.35582 kg/m2 , g = 9.81 m/s2 and 1 lb· ft = 1.35582 N· m The vehicle parameters are set as follows: ⎧ ⎪ ⎪ ⎨

m = 92079 kg, S = 250 m2 Ix x = 588791.0 kg · m2 , Ix z = 23211.6 kg · m2 I yy = 1303212.2 kg · m2 , Izz = 1534163.6 kg · m2 ⎪ ⎪ ⎩ Ix y = 0 kg · m2 , I yz = 0 kg · m2 The simulations were performed under Windows 7 and Intel(R) i7-4790 CPU, 2.90 GHz, with 8.00 GB RAM, while the MATLAB version is R2019a. The initial mesh grid for the simulation has a total of 40 points, and the accuracy tolerance parameter is set to = 10−6 .

90

3 Highly Fidelity Trajectory Optimization 3-DOF results

3-DOF-Con results

6-DOF results

20

Longitude ( )

Altitude (km)

80 70 60 50

15 10

40

5 0

0

100

200

300

400

0

100

Time (s) 6

300

400

300

400

8

Speed (km/s)

Latitude ( )

200

Time (s)

4 2 0

7 6 5 4

0

100

200

Time (s)

300

400

0

100

200

Time (s)

Fig. 3.1 Position and velocity profiles: 3-DOF model and 6-DOF model

3.4.2 Optimized Results of Using Different Models This subsection begins by analyzing the optimal results produced with both the high-fidelity 6-DOF model and the low-fidelity 3-DOF model. More specifically, Figs. 3.1 and 3.2 show these two models’ optimal translational state and control profiles, respectively. Figure 3.3 depicts the related heating rate, dynamic pressure, and normal load profiles. The rotational state profiles and corresponding control moment curves for the results of the 6-DOF model are also presented in Figs. 3.4 and 3.5, respectively. Both cases successfully meet the initial and terminal boundary conditions while obtaining the same accuracy threshold level, as shown in Figs. 3.1, 3.2, 3.3, 3.4 and 3.5. However, variations in the found solutions can be found. These distinctions are especially clear in the path constraint profiles for the high-fidelity and low-fidelity formulations. This suggests that the high-fidelity system might not be able to realize the low-fidelity trajectory findings, which highlights the significance of taking the forces and moments (e.g., rotational variables) into account throughout the trajectory optimization process. Planning plausible paths with regard to the vehicle translational variables may be possible using a lower-fidelity model, but this might potentially be dangerous. Therefore, it is advised to use the analyzed high-fidelity model to plan the aeroassisted vehicle reconnaissance trajectory in order to come up with a promising solution.

3.4 Simulation Results

91 3-DOF results

3-DOF-Con results

0 -1 -2

80 60 40

0

100

200

300

400

0

100

Time (s)

200

300

400

300

400

Time (s) -40

50 40

Bank ( )

AOA ( )

6-DOF results

100

Azimuth ( )

Path ( )

1

30 20 10

-60 -80 -100

0

100

200

Time (s)

300

400

0

100

200

Time (s)

Fig. 3.2 Path, azimuth, AOA and bank angle profiles: 3-DOF model and 6-DOF model

Additionally, a comparison between the pseudospectral approach’s trajectory optimization and a convex optimization method is conducted. It is important to note that this combinational method has grown in popularity recently [29, 30]. The results are shown in Figs. 3.1, 3.2 and 3.3 (denoted as “Con results”). The performance of the algorithm and the simplicity of the optimization technique have both been analyzed. By contrasting the necessary calculation time, it is conceivable to see how simple the optimization approach is. Particularly, the convex optimization requires only 2.07 s while the adaptive pseudospectral technique requires 5.42 s. The polynomial complexity of the convex optimization allows for a reduction in computational time of approximately 68.81. The answers developed using these two methodologies typically follow the same pattern in terms of algorithm performance. However, tiny variations in the state trajectories can be seen. This can be explained by several factors, including the linearization technique used to convexify the issue and the final mesh grid produced by the method described in Sect. 3.2.2. We have also evaluated how well these two approaches function in the 6-DOF instance. This approach, however, was unsuccessful due to convergence problems with the strategy based on convex optimization. One such explanation can be found. That is, for higher-order systems, the consecutive linearization procedure may lead to a significant buildup of mistakes. Therefore, even if convex optimization-based approaches have clear advantages, specific solutions for the convergence problem

92

3 Highly Fidelity Trajectory Optimization

Fig. 3.3 Path constraint profiles: 3-DOF model and 6-DOF model

should be developed, and further research in this area is possible, especially for high-order trajectory optimization issues.

3.4.3 Results with and Without Mesh Adaptive Process This subsection tests and analyzes the algorithm’s performance both with and without the mesh adaptive procedure. It should be highlighted that a direct application of the pseudospectral approach may encounter numerical difficulties when attempting to solve the 6-DOF reconnaissance trajectory optimization problem based on our experiments. A small-scale temporal set is subjected to the pre-solve method described in Sect. 3.2 in order to address this problem and enhance algorithm convergence. Two test cases were then run on the 6-DOF aeroassisted vehicle reconnaissance trajec-

3.4 Simulation Results

93 1

Roll rate ( /s)

Slide angle ( )

0.01 0.005 0 -0.005 -0.01

0.5 0 -0.5 -1

0

100

200

300

400

0

100

Time (s) 0.5

300

400

300

400

1

Yaw rate ( /s)

Pitch rate ( /s)

200

Time (s)

0

-0.5

0.5 0 -0.5 -1

0

100

200

300

400

0

Time (s)

100

200

Time (s)

M x (10 5 )

Fig. 3.4 Angular rate profiles: 6-DOF model solutions 1 0 -1 0

50

100

150

200

250

300

350

250

300

350

250

300

350

M y (10 5 )

Time (s) 0.5 0 -0.5 0

50

100

150

200

M z (10 5 )

Time (s) 1 0 -1 0

50

100

150

200

Time (s)

Fig. 3.5 Control moment profiles: 6-DOF model solutions

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3 Highly Fidelity Trajectory Optimization 20

Results without mesh update Results with mesh update

70

Longitude ( )

Altitude (km)

80

60

50

15

10

5

0

40 0

100

200

300

0

400

100

5

300

400

300

400

8

Speed (km/s)

4

Latitude (

200

Time (s)

Time (s)

3 2 1

7

6

5

4

0 0

100

200

300

400

0

Time (s)

100

200

Time (s)

Fig. 3.6 Position and velocity profiles: with and without mesh updates

tory optimization model with and without the mesh adaptive technique described in Sect. 3.2. The results are shown in Figs. 3.6, 3.7, 3.8, 3.9 and 3.10. The translational state patterns for the two examples are shown in detail in Figs. 3.6 and 3.7, while the related constraint evolutions are shown in Fig. 3.8. The rotating state profiles are shown in Fig. 3.9, and the optimal control moments for the two cases are shown in Fig. 3.10. One important finding from the results that have been presented is that the heating and dynamic pressure constraints tend to keep the vehicle from descending throughout the whole mission. To reduce the generated heat load and dynamic pressure and safeguard the structural integrity, the aeroassisted vehicle hops at about 210 s. It is evident from a comparison of the outcomes for the two situations that a mesh adaptive process and a sizable number of mesh grid points are required to produce smooth and promising reconnaissance trajectories. For the rotating state variables displayed in Fig. 3.9, this is more obvious. As a result, the benefits of using the mesh adaptive method can be demonstrated.

3.4.4 Comparative Results and Analysis To better understand the benefits of using the decay rate-based mesh refinement strategy, we compare the applied algorithm in this subsection to other high-precision

95

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3.4 Simulation Results

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Fig. 3.7 Path, azimuth, AOA and bank Angle profiles: with and without mesh updates

pseudospectral-based methods that are already known to exist in the literature. To form the comparative study, the ph method recommended in Ref. [31] and the hp method stated in Ref. [24] are specifically chosen. Early research revealed that these two techniques could deliver highly precise solutions for a variety of aerospacerelated trajectory optimization problems. The comparative results were obtained by assigning the mesh adaptive tolerance values as = ( 1 , 2 , 3 , 4 ) = (10−5 , 10−6 , 10−7 , 10−8 ). Table 3.4 shows the results for several approaches, where Na stands for the number of mesh adaptive iterations, Nk for the total number of nodes in the final mesh grid, and tc for the algorithm’s computation time in seconds. Note that in this table, the notation “−” indicates a failure (e.g., the algorithm fails to converge). Table 3.4 shows that the ph technique tends to use a somewhat high number of mesh iterations for the 1 and 2 situations. Additionally, the convergence of the ph approach requires much more calculation time than the others. Furthermore, the ph method even fails to converge for cases with tighter mesh adaptive tolerance (such as

3 and 4 ). This can be explained by the fact that increasing the polynomial degrees in the ph method is mostly responsible for reducing the polynomial inaccuracy. This technique has a number of drawbacks, including the potential for a gradual decline in error and the generation of a significant number of pointless collocations. Similar to this, the hp approach typically requires more adaptive iterations and computation time to attain the necessary mesh accuracy than the methodology shown in this study. This is obvious for 1 , 2 , and 3 cases. Due to a cautious assessment of the necessary polynomial degree, the mesh adaptation process in the hp method may add

3 Highly Fidelity Trajectory Optimization Dynamic pressure (slug/ft 2 ) Aerodynamic heating Q (Btu/ft 2 /s)

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Fig. 3.8 Path constraint profiles: with and without mesh updates

too many node points in a sub-interval. Inevitably, this will lengthen the time needed for algorithm convergence or perhaps cause failure. The data presented in Table 3.4’s associated trajectory profiles are shown visually. In particular, Fig. 3.11 shows the position and velocity profiles, while Fig. 3.12 shows the path, azimuth, AOA, and bank angle profiles. It is clear from looking at the trajectory profiles shown in Figs. 3.11 and 3.12, that the method proposed in this study can successfully converge to a solution for all cases that have been pre-specified. The resulting trajectory profiles, along with the mesh and computational results shown in Table 3.4, support the viability of the suggested methodology. It is also possible to evaluate the improved computing performance of this method compared to other high precision-methods.

3.4 Simulation Results 1

Results without mesh update Results with mesh update

0.005

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Table 3.4 Comparative results High precision methods Method in Ref. [31] Method in Ref. [24] Proposed method High precision methods Method in Ref. [31] Method in Ref. [24] Proposed method

ph-

1

hp-

1 Na

tc

10 6 5

3 Na

285 164 152

11.11 4.24 3.18

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14 9

239 210

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Nk

1

ph-

hp-

2

Nk

tc

412 261 208

37.2 5.92 4.47

tc

24 11 9

4 Na

Nk

tc

7.73 5.42

10

213

5.78

Proposed-

2

2

hp-

3

Proposed-

Proposed-

3

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20

80 75

15

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3.4 Simulation Results ph-

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Fig. 3.12 Path, azimuth, AOA and bank angle profiles: comparative study Table 3.5 Case specifications Test cases h 0 (km) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8

79.08 78.85 78.20 78.71 78.97 78.19 78.20 79.37

V0 (km/s)

γ0 (◦ )

ψ0 (◦ )

7.83 7.81 7.82 7.88 7.75 7.77 7.83 7.91

−1.01 −1.08 −0.92 −1.01 −0.90 −1.10 −0.96 −1.04

90.57 90.81 89.77 89.61 89.06 90.77 89.54 90.41

3.4.5 Case Studies with Noise-Perturbed Initial Conditions To further validate the efficacy and robustness of the employed method, case studies with noise-perturbed vehicle initial conditions were created and carried out in this section. Eight test cases with perturbed initial conditions were taken into consideration, and Table 3.5 lists their specific settings. The findings for various scenarios are achieved using the variable-order RPM approach and are shown in Figs. 3.13, 3.14, 3.15, 3.16 and 3.17.

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Fig. 3.13 Position and velocity profiles: noise-perturbed cases

Table 3.6 summarizes the detailed results in addition to the optimized trajectories for various cases. J ∗ denotes the optimized final time value, while Na and Nk again denote the total number of temporal nodes and the number of mesh adaptive iterations in the final adaptive iteration, respectively. Vl stands for the final solution’s value of constraint violation. One can see from the presented trajectory profiles (e.g., Figs. 3.13, 3.14, 3.15, 3.16 and 3.17) that the aeroassisted vehicle reconnaissance trajectory is affected by the changing of the initial conditions. However, if the variable-order RPM approach is applied, all cases can be properly optimized. This can be seen from several facts: all test cases’ optimization processes can effectively converge, and the necessary accuracy tolerance can be satisfied with constraint violation negated (see Table 3.6). Additionally, for the entirety of the flight missions, all trajectory profiles are maintained reasonably smoothly. This further demonstrates the robustness and efficacy of the utilized approach for the considered problem.

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Fig. 3.14 Path, azimuth, AOA and bank angle profiles: noise-perturbed cases

3.5 Conclusions Using a variable order pseudospectral approach, we looked into and resolved a highfidelity reconnaissance trajectory optimization problem for aeroassisted vehicles in this chapter. The aerodynamic, atmospheric, and different constraints are all included in the formulated trajectory optimization model, which further reduces the distance between it and the real system. To investigate the optimal situation, a pipelined strategy based on a variable order Radau pseudospectral method was developed. After examining the simulation’s outcomes, we discovered that: 1. As the low-fidelity 3-DOF model can potentially present a safety hazard, it is advantageous to incorporate both the translational and rotational dynamics in the trajectory planning phase.

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Roll rate ( /s)

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Fig. 3.15 Path constraint profiles: noise-perturbed cases

2. The 6-DOF reconnaissance trajectory optimization problem may present numerical challenges when the pseudospectral approach is applied directly. A potential recovery can be made by gently beginning the optimization process on an initial, small-scale mesh grid and gradually updating the mesh grid to meet the necessary accuracy level 3. If the variable order strategy can take the role of the fixed mesh grid-based method, the intended state and control trajectories tend to be smoother. Future research should focus on solving the convergence problem and enhancing the convergence rate of convex optimization-based trajectory design techniques. This is particularly interesting because using this kind of strategy can result in a reduction in calculation time (as indicated in Sect. 3.3.2). Additionally, a more sophisticated and organized design parameter selection strategy is preferred so that the algorithm’s performance and convergence capability can be further improved.

Dynamic pressure (slug/ft 2 )

Aerodynamic heating Q (Btu/ft 2 /s)

3.5 Conclusions

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Table 3.6 Results of different cases Test cases J ∗ (s) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8

331.55 332.70 330.35 330.39 330.94 333.36 329.86 332.65

Na

Nk

Vl

9 9 12 11 9 12 10 13

217 193 223 235 219 237 237 238

0 0 0 0 0 0 0 0

References 1. Conway BA (2012) A survey of methods available for the numerical optimization of continuous dynamic systems. J Optim Theory Appl 152(2):271–306. https://doi.org/10.1007/s10957-0119918-z 2. Sun L, Zhou Q, Jia B, Tan W, Li H (2020) Effective control allocation using hierarchical multiobjective optimization for multi-phase flight. Chin J Aeronaut. https://doi.org/10.1016/j.cja. 2020.02.020 3. Waslander S, Inalhan G, Tomlin CJ (2004) Decentralized optimization via Nash bargaining. In: Grundel D, Murphey R, Pardalos P (eds) Theory and algorithms for cooperative systems. World scientific series on computers and operation research, vol 4 4. Duan H, Li S (2015) Artificial bee colony based direct collocation for reentry trajectory optimization of hypersonic vehicle. IEEE Trans Aerosp Electron Syst 51(1):615–626. https://doi. org/10.1109/TAES.2014.120654 5. Tian B, Fan W, Su R, Zong Q (2015) Real-time trajectory and attitude coordination control for reusable launch vehicle in reentry phase. IEEE Trans Ind Electron 62(3):1639–1650. https:// doi.org/10.1109/TIE.2014.2341553 6. Yu J, Dong X, Li Q, Ren Z, Lv J (2020) Cooperative guidance strategy for multiple hypersonic gliding vehicles system. Chin J Aeronaut. https://doi.org/10.1016/j.cja.2019.12.003 7. Chai R, Savvaris A, Tsourdos A, Chai S, Xia Y (2018) Optimal fuel consumption finite-thrust orbital hopping of aeroassisted spacecraft. Aerosp Sci Technol 75:172–182. https://doi.org/10. 1016/j.ast.2017.12.026 8. Chai R, Savvaris A, Tsourdos A, Chai S, Xia Y (2018) Optimal tracking guidance for aeroassisted spacecraft reconnaissance mission based on receding horizon control. IEEE Trans Aerosp Electron Syst 54(4):1575–1588. https://doi.org/10.1109/TAES.2018.2798219 9. Gao H, Yang X, Shi P (2009) Multi-objective robust h-infinity control of spacecraft rendezvous. IEEE Trans Control Syst Technol 17(4):794–802. https://doi.org/10.1109/TCST. 2008.2012166 10. Pontani M, Conway BA (2017) Optimal trajectories for hyperbolic rendezvous with earth-mars cycling spacecraft. J Guid Control Dyn 41(2):360–376. https://doi.org/10.2514/1.G002984 11. Huo M, Fan Z, Qi N, Song Z, Shi X (2020) Fast cooperative trajectory optimization and test verification for close-range satellite formation using finite Fourier series method. Chin J Aeronaut. https://doi.org/10.1016/j.cja.2020.02.006 12. Zhao J, Zhou R (2015) Pigeon-inspired optimization applied to constrained gliding trajectories. Nonlinear Dyn 82(4):1781–1795. https://doi.org/10.1007/s11071-015-2277-9 13. Zhao J, Zhou R (2015) Particle swarm optimization applied to hypersonic reentry trajectories. Chin J Aeronaut 28(3):822–831. https://doi.org/10.1016/j.cja.2015.04.007

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Chapter 4

Fast Trajectory Optimization with Chance Constraints

Abstract This chapter investigates the optimal flight of aero-assisted reentry vehicles during the atmospheric entry flight phase while taking into account both deterministic and control chance constraints. We construct a chance-constrained optimal control model in order to depict the mission profile. However, standard numerical trajectory planning methods cannot be directly used to solve the problem due to the existence of probabilistic constraints (chance constraints). Therefore, to make the optimal control model solvable for standard trajectory optimization algorithms, we introduce an approximation-based strategy such that the probabilistic constraint is replaced by deterministic version. To achieve improved computational performance, we provide an alternative optimal control formulation that incorporates the convex-relaxed technique. This involves convexifying the vehicle nonlinear dynamics and constraints, as well as incorporating a convex probabilistic constraint handling approach. The effectiveness of the two chance-constrained optimization strategies and their corresponding probabilistic constraint handling methods is validated through numerical simulations.

4.1 Introduction Around the past 10 years, atmospheric entry trajectory optimization problems for hypersonic vehicles (HVs) have been an active research area and will continue to be a major concern of the aerospace industry [1–4]. In general, the goal of this kind of problem is to find a set of control profiles that will allow the HV to be guided from a specific starting position to a targeted final position with some particular performance metrics to be optimized. It is also required that multiple process constraints and/or waypoint constraints are satisfied during the planning stage, which will make the problem more challenging. Researchers and engineers have invested a lot of time and energy into developing a feasible and effective trajectory planner [5, 6]. The development of classical ”discretization + optimization” mode-based trajectory optimization methods, such as the well-known Gauss pseudospectral method (GPM) [7], the hp-adaptive pseudospectral approach [8], and several alternative improved © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_4

107

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versions [9–11], has received a lot of attention. Besides, literature [12–17] reported many bio-inspired optimization approaches to approximate the optimal control trajectory, and developed numerous evolutionary methods to further enhance the effectiveness and robustness of these algorithms. It has been shown that the studies mentioned are highly crucial for the development of entry trajectory optimization approaches. The effectiveness of the designed methods has been demonstrated by the results provided in those works. These researchers, however, primarily viewed the problem as a deterministic optimization model, whereas some constraint functions or system equations might be affected because of uncertain parameters in practice [18–20]. Because of this, the issue is no longer deterministic, resulting in the inability to search the optimal solutions using typical trajectory optimization methods directly. Consequently, in this chapter, we investigate trajectory optimization methods that can handle noise-perturbed constraints for system variables, which is distinct from most current studies that only take deterministic models into account. Notably, the control constraints or noise-perturbed state are commonly modeled as chance constraints, which have a significant impact on the behavior of the optimal flight trajectory. The robust optimization (RO) [21, 22] is a strategy for solving issues with this constraint. while using the RO, any constraint violation is not allowed, which is a key feature. As a result, it typically increases the difficulty of tackling the optimization problem since the feasible solution space is severely restricted. Furthermore, as the RO always takes the worst-case scenario of the problem caused by the uncertain parameter, the solution optimality might be reduced. Alternatively, chance-constrained optimization (CCO) has attracted increased attention in recent years in aerospace-related applications as an effective method. By applying this kind of algorithm, the probabilistic constraints are replaced by an approximated deterministic version. Different from the algorithms based on RO, the CCO accepts constraint violation to be less than a specified parameter, which results in a specific removal of restrictions on the feasible solution space. Reports on the application of CCO were reviewed in [23–25]. In particular, a dual-loop CCO algorithm was presented by the authors of [24] and benchmark CCO problems were solved successfully by applying the proposed algorithm. The authors in [25] designed a kernel density estimationbased CCO algorithm. It has been shown that this method can be used to handle a class of trajectory optimization missions in aerospace field. Additionally, Zhao and Kumar [26] proposed the split Bernstein approach, which is a new approximationbased method. In their research, after modifying the original version of the Bernstein approach by using a piecewise smooth approximation function, this method has the capability in dealing with typical chance-constrained optimal control problems. Typically, the original problem is transformed into a deterministic yet non-convex optimization model after applying probabilistic constraints approximation methods. However, one challenge of the use of CCO model is that its computation time is typically high and unpredictable. Fortunately, many published works have proposed a set of potentially-effective approaches to handle this issue. Among these, convex relaxation-based technique is a potential strategy. In the last couple of years, there has been increasing attention to the application of convex relaxation or convex opti-

4.2 Atmospheric Entry Optimal Control Problem

109

mization methods in the investigation of optimal flight path [27–30]. For example, Liu et al. [27] designed a convex relaxation-based strategy to search for the optimal intercept trajectory of an aerodynamically controlled missile while taking into account the dynamic pressure constraints and impact angle. Similar to literature [27], a sequential convex programming method was developed by Wang and Grant [28] to achieve planetary entry trajectory planning while optimizing the terminal velocity. Moreover, in Ref. [29], Zhao and Song designed a multi-phase convex optimization method in order to achieve optimal flight trajectories planning for reentry vehicles while satisfying the waypoint constraints and non-fly zone simultaneously. However, the resulting convex optimization problem via convex relaxation approach can guarantee the complexity of polynomial-time, which is the main motivation for the use of this approach, while other non-convex optimization formulations are generally NP-hard. As a result, in order to solve the chance-constrained HV atmospheric entry problem, we also focus on implementing convex relaxation-based approaches. We have two main objectives for this chapter, which are as follows. For one thing, we are focused on designing a strategy based on chance-constrained optimization (CCO), which can provide the optimal control command for the HV atmospheric entry flight with the simultaneous existence of multiple path constraints and probabilistic constraints. For another thing, we intend to present an approach based on CCO, which is computationally friendly, as a solution to address the rising need for online computation. To accomplish this, the original optimal control model is transcribed into a convexified version and a convex chance constraint approximation strategy is introduced by devoting significant amount of effort. In order to invest the operation and effectiveness of the two proposed algorithms, then, a number of simulation experiments were performed on the considered HV atmospheric entry problem. The structure of this chapter is outlined as follows. Section 4.2 establishes the mathematical formulation of the HV atmospheric entry problem, which includes different kinds of constraints. After introducing a chance constraint handling method based on approximation, a non-convex chance-constrained trajectory optimization method is constructed in Sect. 4.3. Then, Sect. 4.4 presents a different approach, which makes use of convex optimization. Numerical results are demonstrated in Sect. 4.5. Finally, Sect. 4.6 summarizes some concluding remarks for this chapter.

4.2 Atmospheric Entry Optimal Control Problem 4.2.1 Hypersonic Vehicle Dynamics and Constraints Table 4.1 provides a pre-summary for the definitions/notations of the parameters and variables, which will appear in the remaining portions of this section. The first step is to establish the equations of motion for the HV to characterize the atmospheric entry flight. The equations can be expressed as:

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4 Fast Trajectory Optimization with Chance Constraints

Table 4.1 Notations for variables x, u System state and control variables r, θ, φ Radial distance, longitude and latitude of the HV V, γ , ψ Velocity, flight path angle and azimuth angle of the HV σ, σc Actual and demanded bank angles S, m Reference area and mass of the HV CL , CD Aerodynamic lift and drag coefficients ρ, g Density of the atmosphere and gravity acceleration ρ0 , h s Sea-level air atmosphere density and density scale height Re , t Radius of the Earth and mission time L, D Lift and drag forces t0 , t f Starting and ending time points

x˙ = f (x) + B(x)u

(4.1)

in which x = [r, θ, φ, V, γ , ψ, σ ] ∈ R7 and u = σc . The expressions for the terms f (x) ∈ R7 and B(x) ∈ R7 are given by: ⎡

V sin γ

⎤

V sin ψ cos γ ⎥ ⎢ ⎥ ⎢ r cos φ ⎥ ⎢ V cos ψ cos γ ⎥ ⎢ r ⎥ ⎢ D − m − g sin γ f (x) = ⎢ ⎥ ⎥ ⎢ 2 V −gr L cos σ ⎥ ⎢ + ( ) cos γ ⎥ ⎢ L sin σmV V r V ⎣ mV cos γ + r sin ψ cos γ tan φ ⎦ Kσ

 T B(x) = 01×6 −K σ

(4.2)

(4.3)

In particular, the actual bank angle profile can be smoothed effectively by adding the demanded bank angle variable σc [28, 31]. The HV engine is turned off during the entry maneuver. The motion is therefore primarily influenced by the gravity acceleration g, and the aerodynamic forces like the lift L and drag D. The equations, which are required to compute these forces, and the isothermal atmospheric model are outlined below: ⎧ r −Re ⎪ ⎨ ρ = ρ0 e− hs (4.4) L = CC L D ⎪ ⎩ D = 1DρV 2 C S D 2 The entry trajectory should be constrained in a relatively-narrow corridor to ensure ˙ a safe flight. Three hard path constraints g(x) ∈ R3 , including the heating rate Q,

4.2 Atmospheric Entry Optimal Control Problem

111

dynamic pressure Pd , and load factor N L , are imposed to achieve this. The term g(x) is in the form of: ⎡

⎤ ⎡ ⎤ ⎤ ⎡ ˙ k Q ρ 0.5 V 3.15 Q¯ Q(x) 1 2 ⎦ ≤ ⎣ P¯d ⎦ g(x) = ⎣ Pd (x) ⎦ = ⎣ √2 ρV L 2 +D 2 n L (x) N¯ L mg

(4.5)

¯ P¯d , N¯ L ] denotes the permissible limits. Additionally, in order to solve where g¯ = [ Q, the problem under consideration, the following constraints must be met: state and control boundary constraints, as well as box constraints, which are represented by Eqs. (4.6) and (4.7):  x(t0 ) = x0 (4.6) x(t f ) = x f 

x ≤ x ≤ x¯ u ≤ u ≤ u¯

(4.7)

In Eq. (4.6), x0 ∈ R7 is defined as [r0 , θ0 , φ0 , V0 , γ0 , ψ0 , σ0 ], while x f ∈ R3 is specified as [r f , V f , γ f ]. It is important to note that in practical scenarios, some noises ξ may impact the control signal. For instance, we might not have a fixed maximum attainable control actuation level, which would lead to probabilistic control path constraint. we use the following formulation to describe this kind of constraint: ¯ ≥ Pr {C(u, ξ ) ≤ C}

(4.8)

In Eq. (4.7), Pr {·} is the probability operator, which indicates the probability of violating/meeting a noise-perturbed inequality. For instance, C(u, ξ ) ≤ C¯ is below/above a specific risk level . The bank angle chance constraint for the problem under consideration is expressed as: Pr {σc + ξσ ≤ σ¯ c } ≥ σ

(4.9)

in which the maximum value of σc is σ¯ c . ξσ and σ respectively represent the uncertain parameter acted on the demanded bank angle and the allowable risk level.

4.2.2 Atmospheric Entry Optimal Control Model It is important to provide the mission objective function J in addition to simulating the system dynamics and constraints. If we want to finish the full entrance maneuver as quickly as possible, in which the terminal time point t f should be minimized, the objective function can be described as:

112

4 Fast Trajectory Optimization with Chance Constraints

J1 = t f

(4.10)

If an entry trajectory with the maximum cross-range is required, the value of latitude φ f at the terminal time point t f (φ f = φ(t f )) should be set to maximum. That is: J2 = −φ f

(4.11)

Using Eqs. (4.1)–(4.11), therefore, the overall HV atmospheric entry trajectory optimization problem can be written in the form of: minimi ze J1 or J2 subjectto x˙ = f (x) + B(x)u x(t0 ) = x0 x(t f ) = x f x ≤ x ≤ x¯ g(x) ≤ g¯ ¯ ≥ Pr {C(u, ξ ) ≤ C}

(4.12)

4.3 Nonconvex Chance-Constrained Optimization Approach In this section, a nonconvex chance-constrained optimization (NCCO) method is presented, which is able to search for the optimal solution for the nondeterministic atmospheric entry trajectory optimization problem constructed in the last section.

4.3.1 Handling the Probabilistic Constraint A number of effective trajectory optimization algorithms have recently been suggested in the literature. It is crucial to note that the majority of these proposed algorithms cannot be used to directly search the solution of the optimization model (4.12). This is because due to the presence of probabilistic constraint, problem (4.12) is non-deterministic, whereas existing algorithms or solvers focus primarily on the deterministic optimization model. As a result, in order to address the considered problem, the noise-perturbed constraint (4.8) or (4.9) should be properly treated, which is motivated by [23]. We are able to construct the chance-constraint handling strategy process: Step 1. Take a preliminary transformation on Eq. (4.8) via P(u) = Pr {C(u, ξ ) ≤ ¯ = 1 − Pr {C(u, ξ ) > C}. ¯ C} Step 2. Consider C(u, ξ ) ≤ C¯ as an event so that P(u) can be rewritten in the form of (4.13):

4.3 Nonconvex Chance-Constrained Optimization Approach

113

P(u) = 1 − E(H (C(u, ξ )))

(4.13)

in which E represents the expectation, while the H function is in the form of (4.14).  1 if C(u, ξ ) ≤ C¯ H (C(u, ξ )) = (4.14) 0 if C(u, ξ ) > C¯ Step 3. The original probabilistic constraint is reformulated as follows: E(H (C(u, ξ ))) ≤ 1 −

(4.15)

Step 4. Define an approximation function (k, C(u, ξ )) as:

(k, C(u, ξ )) =

m2

m1 + k −kC(u,ξ ) e

+k

(4.16)

Step 5. Replace H (·) in Eq. (4.15) by (k, C(u, ξ )) and approximate the control chance constraint in the form of (4.17). E( (k, C(u, ξ ))) ≤ 1 −

(4.17)

Remark 4.1 In steps 4 and 5, the constructed approximation function is used to replace the unit step function H (·) in Eq. (4.15). It is demonstrated that (k, C(u, ξ )) approximates H (C(u, ξ )) aggressively and (k, C(u, ξ )) is strictly greater than H (C(u, ξ )). Additionally, the approximation function (4.16) is smooth and differentiable over the solution space, which tends to provide fewer numerical challenges for gradient-based optimization algorithms. This is another benefit of applying approximation function (4.16) instead of H (·).

4.3.2 Deterministic NCCO Model The Markov chain Monte-Carlo (MCMC) method [32], which is widely-used, is applied to calculate the expectation value of (k, C(u, ξ )) (e.g., E( (k, C(u, ξ )))). according to Eq. (4.17), and we have  E( (k, C(u, ξ ))) =



(k, C(u, ξ ))R(ξ )dξ

(4.18)

in which  is a measurable set of ξ , while R(ξ ) denotes the probability density function of ξ . By generating a collection of {ξi }iNm , the integral term of Eq. (4.18) can be approximated in the form of (4.19).

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4 Fast Trajectory Optimization with Chance Constraints

E( (k, C(u, ξ ))) =

Nm 1 

(k, C(u, ξi )) Nm i=1

(4.19)

in which Nm represents the size of the random sample. By using the transformation processes which is described in the previous subsection and the MCMC sampling method, the control chance constraint can be approximated and we have Nm 1 

(k, C(u, ξi )) ≤ 1 − (4.20) Nm i=1 As a result, the non-deterministic HV entry trajectory optimization model is replaced by a deterministic version, which can be formulated as: minimi ze J1 or J2 subjectto x˙ = f (x) + B(x)u x(t0 ) = x0 x(t f ) = x f x ≤ x ≤ x¯ g(x) ≤ g¯ Nm 1 

(k, C(u, ξi )) ≤ 1 − Nm i=1

(4.21)

Equation (4.21) is a typical optimal control model. Using the “discretization + optimization” mode, it can be tackled by well-developed trajectory optimization algorithms.

4.4 Convex Chance-Constrained Optimization Approach Notably, the original problem has been translated, according to Eq. (4.21), into a deterministic yet nonconvex optimization model. Nevertheless, the computational burden may be high and unpredictable while solving a nonconvex optimization problem, which is a potential problem. Fortunately, the convex relaxation-based algorithms are able to tackle this problem, as demonstrated by many related works. Hence, we concentrate on developing a convex chance-constrained optimization approach (CCCO) that can address the considered HV entry problem capably. The remainder of this section is structured as follows. The convex relaxation of system dynamics and hard constraints is derived in Sect. 4.2.1. In Sect. 4.2.2, a convex chance constraint handling approach is proposed, which is different from the approach reported in Sect. 3.1. Afterwards, Sect. 4.3.1 presents the overall CCCO model.

4.4 Convex Chance-Constrained Optimization Approach

115

4.4.1 Convex Relaxation of Dynamics and Hard Constraints In order to address the problem (4.21) by way of convex optimization, convexifying the nonconvex terms, which exist in the optimal control model, is a crucial step. Firstly, we consider the terminal cost, which is provided by Eqs. (4.10) and (4.11), in a more general form (x(t f )). A noteworthy point is that the terminal cost (x(t f )) might exist nonconvexity in some cases. Accordingly, convex relaxation is executed on (x(t f )) to address this concern. More precisely, with respect to a reference terminal point xr (t f ), a first-order Taylor expansion can be used for (x(t f )). We have (4.22) (x(t f )) ≈ (xr (t f )) + F (xr (t f ))(x(t f ) − xr (t f )) where F (x(t f )) =

∂ (x(t f )) ∂ x(t f )

The nonlinear function f (x), in addition to the terminal cost function, is a term that needs to be convexified in Eq. (4.10) as well. The first-order Taylor expansion is performed on f (x) with respect to a given reference trajectory xr (t) to implement this convex relaxation, which is similar to the terminal cost function. That is, f (x) ≈ f (xr ) + A(xr )(x − xr )

(4.23)

In Eq. (4.23), by partially differentiating f (x) with respect to x, the matrix A(x) ∈ R7×7 can be obtained. This can be written as: A(x) = ⎡

0 ⎢ a21 ⎢ ⎢ a31 ⎢ A(x) = ⎢ ⎢ a41 ⎢ a51 ⎢ ⎣ a61 0

0 0 0 0 0 0 0

0 a23 0 0 0 a63 0

∂ f (x) ∂x a14 a24 a34 a44 a54 a64 0

a15 a25 a35 a45 a55 a65 0

(4.24) 0 a26 a36 0 0 a66 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ a57 ⎥ ⎥ a67 ⎦ a77

(4.25)

In the Appendix of this paper, the nonzero terms of ai j , i, j = 1, ..., 7, which in Eq. (4.25), are defined. As a result, the transformed nonlinear dynamics has the form of (4.26) x˙ ≈ f (xr ) + A(xr )(x − xr ) + Bu It is clear via Eq. (4.5) that the path constraint functions are nonlinear as well. Therefore, researchers have devoted efforts to convexifying these equations. The

116

4 Fast Trajectory Optimization with Chance Constraints

convexified path constraint equations can be transcribed by analogically applying Taylor expansion theory. That is, g(x) ≈ g(xr ) + Fg (xr )(x − xr )

(4.27)

In Eq. (4.27), by partially differentiating g(x) with respect to x, the matrix Fg (x) ∈ R3×7 can be obtained and we have ⎡ ⎤ b11 0 0 b14 0 0 0 Fg (x) = ⎣ b21 0 0 b24 0 0 0 ⎦ (4.28) b31 0 0 b34 0 0 0 in which the nonzero terms bi j , i = 1, ..., 3 and j = 1, ..., 7 are defined in the Appendix of this paper.

4.4.2 Convex Approximation of Control Chance Constraint It is clear that (k, C(u, ξ )) is non-convex from the formulation of the chance constraint approximation function, which is provided by Eq. (4.16). It is needed to find a different convex approximation function c (C(u, ξ )) that can take the place of the function H (C(u, ξ )) in Eq. (4.15) to maintain the convexity of the problem. We propose a convex function, which is in the form of (4.29).

c (C(u, ξ )) = max(C(u, ξ ) + 1, 0)

(4.29)

in which the term max(C(u, ξ ) + 1, 0) returns the value that is greater between C(u, ξ ) + 1 and 0. It can be demonstrated that the convex function, which is provided by Eq. (4.29), is strictly greater than the unit step function H . Hence, an upper estimation of the original chance constraint is formed. Equations (4.30) and (4.31), respectively, can be used to approximate the expectation value of c (C(u, ξ )) and the original control chance constraint, which is similar to the process that has been described in detail in Sect. 3.2. Nm 1  E( c (C(u, ξ ))) =

c (C(u, ξi )) (4.30) Nm i=1 Nm 1 

c (C(u, ξi )) ≤ 1 − Nm i=1

(4.31)

4.5 Performance Evaluation

117

4.4.3 Overall CCCO Model According to the convex relaxation processes, which are executed respectively on the nonlinear dynamics, together with path constraints, objective functions, and control chance constraints, a CCCO model can be formulated as Eq. (4.32). minimi ze (xr (t f )) + F (xr (t f ))(x(t f ) − xr (t f )) subjectto x˙ = f (xr ) + A(xr )(x − xr ) + Bu x(t0 ) = x0 x(t f ) = x f x ≤ x ≤ x¯ g(xr ) + Fg (xr )(x − xr ) ≤ g¯ Nm 1 

c (C(u, ξi )) ≤ 1 − Nm i=1 x − xr  ≤ 

(4.32)

It should be noted that the convexity of the optimization model (4.32) is guaranteed by the convexity of inequality (4.29), locally linearized objectives, system dynamics and constraints. In addition, to guarantee the effectiveness of the linearization process executed for the objective, system dynamics and constraints, a trust-region constraint x − xr  ≤  is applied in Eq. (4.32). A successive convex trajectory optimization steps based on the CCCO model are as follows: Step 1. Assign system state variables x(t0 ) = x0 as the initial condition. Step 2. Using x0 and the initial control reference u rk , calculate the initial state reference trajectory xr(k) . Step 3. Generate a collection of random samples {ξi }iNm , then construct the convex approximation function (4.29). Step 4. Determine the optimization model (4.32)’s optimal solution (x k , u k ). Step 5. Determine the termination condition by using the constraint x k − xr(k)  ≤ ε. If the criteria cannot be met, assign xr(k) = x k , u r(k) = u k , set k = k + 1 and jump to Step 4. Otherwise, jump to Step 6. Step 6. Output the solution (x k , u k ).

4.5 Performance Evaluation The performance and effectiveness of the suggested NCCO and CCCO algorithms for addressing the chance-constrained HV entry optimization problem, respectively, are shown in this section.

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4 Fast Trajectory Optimization with Chance Constraints

4.5.1 Parameters and Mission Cases Specification The vehicle and environment-related parameters are comparable to the configurations examined in [3]. In particular, the spacecraft’s mass and reference area are set to 92079 kg and 249.91m2 , respectively. The atmospheric flight is simulated using the isothermal atmospheric model provided by (4.4). We primarily take into account the following two test cases in the simulation: • Mission case 1: Cross-range optimal trajectory planning; • Mission case 2: Time-optimal trajectory planning. The initial and terminal conditions, which both of the above two mission cases apply, and other mission-related parameters such as the maximum allowable variable bounds and path constraint values are all shown in Table 4.2. The following equation provides an approximation for aerodynamic parameters such as the lift and drag coefficients (C L , C D ) [3, 33]: C D = C D0 + C D1 α + C D2 α 2 CL = CL0 + CL1 α

(4.33)

where α represents the angle of attack (AOA) and its value can be obtained via [28].  α=

40 − w1 (V − 4570)2 /3402 , if V < 4570m/s; 40, if V ≥ 4570m/s.

(4.34)

in which w1 = 0.20705. From Eq. (4.20), the bank angle chance constraint is translated to Eq. (4.35) for the control chance constraint: Eξσ ((k, σc + ξσ − σ¯ c )) ≤ 1 − σ

(4.35)

where the maximum allowable constraint violation level 1 − σ is assigned as 10%, and uncertain parameters are designed to follow an exponential distribution

Table 4.2 Parameter assignment Parameters Values h0, m θ0 , deg φ0 , deg V0 , m/s γ0 , deg ψ0 , deg σ0 , deg ¯ Btu Q,

80 0 0 7800 –1 0 –75 200

Parameters P¯d , N/m 2 N¯ L , g hf,m V f , m/s γ f , deg σ , deg σc , deg t, s

Values/ranges 18000 2.5 25 750 –5 [–90, 1] [–90, 1] [0, 2000]

4.5 Performance Evaluation

119

R(z; λ) = λe−λz , in which z ≥ 0 and λ = 0.5. m 1 and m 2 are set to 1.0 and 0.5, respectively, for the non-convex chance-constraint approximation strategy. Moreover, Nk = 100 and Nm = 2 × 104 , respectively, are listed as the sizes of the temporal set and MCMC sample set.

4.5.2 NCCO Results and Discussions In this subsection, the performance of applying the NCCO formulation developed in Sect. 4.4.3 is firstly evaluated. The NCCO and CCCO algorithms tackle the same entry mission cases, which should be stated. In NCCO the original system dynamics, path constraints and the approximated chance constraint (4.20) are included in the optimization process, while in CCCO the convexified dynamics (4.26), path constraints (4.27) and chance constraint (4.31) are considered. Figures 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6 show the optimal results for both test cases, which are generated on the basis of the parameters and setting detailed in the previous subsection. More specifically, the obtained system state evolutions, path constraints profiles, and the control and chance constraint violation histories for mission case 1 are shown in Figs. 4.1, 4.2 and 4.3, respectively, whereas Figs. 4.4, 4.5 and 4.6 demonstrate the corresponding optimal results for mission instance 2. According to the state and path constraint trajectories (e.g., Figs. 4.1, 4.2, 4.4 and 4.5), it can be seen that the path limitations are able to remain in the allowable corridors and all the targeted final conditions can be successfully achieved. As a result, the validity of the obtained results can be verified. Besides, it can be observed that the violation rates of σc are always smaller than the level parameter 1 − σ according to the evolution profiles, which are presented in Figs. 4.3 and 4.6. Hence, the effectiveness of the chance constraint handling method can be confirmed. The constant k is an important parameter that could affect the performance of the algorithm, as stated in Sect. 4.4.3. As a result, a study is executed in order to further evaluate the sensitivity of k in relation to the optimal solutions. k = [k1 , k2 , k3 , k4 ] is assigned as [50, 200, 350, 500], then simulations for both cases 1 and 2 were executed. Figures 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6 illustrate the calculated trajectory profiles as well. Detailed results, which include the maximum path constraint/violation rate values achieved, and the algorithm execution time t p , are listed in Table 4.3. According to the displayed results shown in Figs. 4.1, 4.2 and 4.3 and Table 4.3, it can be observed that there is no sensitivity between the variance of k and the cross-range optimal results. It is interesting to note that the displayed results (e.g., see Figs. 4.4, 4.5 and 4.6 and Table 4.3, the time-optimal case 2) of mission case 2 reflect more noticeable differences. We can conclude that the control chance constraint violation history tends to become more aggressive, which means that more conservatism will be reduced during the optimization process, as the value of k goes higher. The dynamic pressure evolution profiles, At the same time, can reflect This aggressive behavior. As shown in Fig. 4.5, the Pd value reaches its peak value P¯d at about 560 seconds by setting k = k1 . However, this time is brought forward to about

120

4 Fast Trajectory Optimization with Chance Constraints 8

Speed (km/s)

Altitude (km)

80

60

40 CCCO Result NCCO Result

20 0

6 4 2 0

500

1000

0

500

1000

Time (s) 20

5

Bank Angle (deg)

Flight Path Angle (deg)

Time (s)

0

-5

0 -20 -40 -60 -80

-10 0

500

0

1000

500

1000

Time (s)

Time (s)

Fig. 4.1 System state evolutions: case 1 (NCCO results)

Heating (Btu/ft 2 · s)

150

k1 result k2 result

100

k result 3

k result 4

50

0

2

500

1000

1500

Time (s)

× 104

2

Load factor (g)

Dynamic pressure (Pa)

0

1.5 1 0.5 0

1.5 1 0.5 0

0

500

1000

1500

0

Time (s)

Fig. 4.2 Path constraints profiles: case 1 (NCCO results)

500

1000

Time (s)

1500

4.5 Performance Evaluation

σc (deg)

50

121

k1 result k2 result

0

k3 result k4 result

-50 -100 0

200

400

600

800

1000

1200

1400

1000

1200

1400

400

600

violation rate for σ

Time (s) 0.1

0.05

0 0

200

400

600

800

Time (s)

Fig. 4.3 Control/Chance constraint histories: case 1 (NCCO results) 8

80 70

k result

60

k3 result

2

Speed (km/s)

Altitude (km)

k1 result

k4 result

50 40

6

4

2

30 0

20 0

200

400

600

0

200

Time (s)

Time (s) 0

Bank Angle (deg)

Flight Path Angle (deg)

0

-2

-4

-6

-8

-20 -40 -60 -80 -100

0

200

400

600

0

Time (s)

Fig. 4.4 System state evolutions: case 2 (NCCO results)

200

400

Time (s)

600

122

4 Fast Trajectory Optimization with Chance Constraints

Heating (Btu/ft 2 · s)

200

k1 result k result

150

2

k result 3

100

k result 4

50 0 200

× 104 × 104 1.8

2 1.5

400 3

1.6 460500540580

1

600

Time (s)

Load factor (g)

Dynamic pressure (Pa)

0

0.5 0

2

1

0 0

200

400

600

0

200

Time (s)

400

600

Time (s)

Fig. 4.5 Path constraints profiles: case 2 (NCCO results) 50

k result 1

σc (deg)

k result 2

0

k3 result k result 4

-50

-100 0

100

200

300

400

500

600

400

500

600

Time (s) violation rate for σ

0.1

0.05

0 0

100

200

300

Time (s)

Fig. 4.6 Control/Chance constraint histories: case 2 (NCCO results)

510 seconds for k4 -based results. Therefore, there is a trend that a higher value k tends to result in a more optimal objective value. On the other hand, assigning k as a higher value is not always a good strategy for the NCCO algorithm. According to the processing time results, which are tabulated in Table 4.3, it is obvious that the algorithm execution time will increase when k is used at a higher value. Multiple factors including the numerical difficulty or tightness of the feasible solution space may contribute to this situation.

4.5 Performance Evaluation

123

Table 4.3 Detailed results for NCCO NCCO results Case 1 k1 k2 max Q, Btu max Pd , N/m 2 max N L , g max V io, % J1∗ , deg tp, s NCCO results max Q, Btu max Pd , N/m 2 max N L , g max V io, % J2∗ , s tp, s

129.79 16813 1.71 0.81 16.34 24.37 Case 2 k1 185.89 18000 2.50 6.14 624.70 55.58

k3

k4

129.33 16958 1.72 5.34 16.35 28.52

129.85 16943 1.72 7.47 16.35 27.17

130.13 16942 1.72 9.32 16.36 35.56

k2 186.61 18000 2.50 6.75 624.64 59.90

k3 189.07 18000 2.50 8.07 619.37 61.12

k4 189.41 18000 2.50 9.56 616.92 77.55

In conclusion, from these displayed results, a major benefit of applying the NCCO approach, which is detailed in Sect. 4.4.3, is that the probabilistic constraint applied to the control variable can be aggressively guaranteed. As a consequence, it is feasible to reduce the conservatism during the optimization process and increase the optimality of the obtained solution.

4.5.3 CCCO Results and Discussions We assess the performance of the CCCO method, which is proposed in Sect. 4.4 to solve the considered problem. Besides, we perform further studies, thereby the obtained results of the CCCO method are compared against the results of the NCCO method. According to the comparative time histories of the system state variables and path constraints for the two mission cases, which are shown in Figs. 4.7, 4.8 and 4.10, 4.11, it can be observed that the optimal solutions obtained via the CCCO are similar to the NCCO. Besides, it can be seen from Figs. 4.8 and 4.11 that the path constraint profiles obtained, which use the designed NCCO and CCCO methods, can always be restricted in their tolerant corridors. This can ensure the safety of flight. Figures 4.9 and 4.12 illustrate the actual and demanded bank angle [σ, σc ], respectively. The control evolution files obtained applying NCCO and CCCO reflect significant differences and part of the reason is the chance constraint approximation strategy used in the two optimization models. It can be determined that the chance constraint

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4 Fast Trajectory Optimization with Chance Constraints 8

Speed (km/s)

Altitude (km)

80

60

40 CCCO Result NCCO Result

20 0

6 4 2 0

500

0

1000

500

20

5

Bank Angle (deg)

Flight Path Angle (deg)

1000

Time (s)

Time (s)

0

-5

-10

0 -20 -40 -60 -80

0

500

1000

0

500

Time (s)

1000

Time (s)

Fig. 4.7 System state evolutions: case 1 (CCCO results) Heating (Btu/ft 2 · s)

150

CCCO Result NCCO Result

100

50

0

2

500

1000

Time (s)

× 104

2

Load factor (g)

Dynamic pressure (Pa)

0

1.5 1 0.5 0

1.5 1 0.5 0

0

500

1000

0

Time (s)

Fig. 4.8 Path constraints profiles: case 1 (CCCO results)

500

Time (s)

1000

4.5 Performance Evaluation

σc (deg)

50

125 CCCO Result NCCO Result

0 -50 -100 0

200

400

600

800

1000

1200

1000

1200

violation rate for σ

Time (s) 0.1

0.05

0 0

200

400

600

800

Time (s)

Fig. 4.9 Control/Chance constraint histories: Case 1 (CCCO results) 8

CCCO Result NCCO Result

Speed (km/s)

Altitude (km)

80

60

40

6 4 2 0

20 0

200

400

0

600

200

600

0

Bank Angle (deg)

0

Flight Path Angle (deg)

400

Time (s)

Time (s)

-2 -4 -6

-20 -40 -60 -80 -100

-8 0

200

400

600

0

Time (s)

Fig. 4.10 System state evolutions: case 2 (CCCO results)

200

400

Time (s)

600

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4 Fast Trajectory Optimization with Chance Constraints

Heating (Btu/ft 2 · s)

200

CCCO Result NCCO Result

150 100 50 0 200

× 104

2

400

600

Time (s) 3

Load factor (g)

Dynamic pressure (Pa)

0

1.5 1 0.5 0

2

1

0 0

200

400

600

0

200

Time (s)

400

600

Time (s)

Fig. 4.11 Path constraints profiles: case 2 (CCCO results) 50

σc (deg)

CCCO Result NCCO Result

0 -50 -100 0

100

200

300

400

500

600

400

500

600

violation rate for σ

Time (s) 0.1

0.05

0 0

100

200

300

Time (s)

Fig. 4.12 Control/Chance constraint histories: case 2 (CCCO results)

handling method, which is used in the NCCO formulation, can issue in more optimal objective value and more aggressive approximation performance. Nevertheless, the amount of time needed for calculation to reach convergence will remarkably increase. In particular, Table 4.4 displays the specific execution performance of using CCCO for the two mission cases. Two major conclusions can be drawn from a comparison of the results presented in Tables 4.3 and 4.4. Firstly, the violation rate of Eq. (4.35) can be effectively guaranteed to be less than the level parameter ε = 0.1 (10%)

4.6 Conclusion

127

Table 4.4 Detailed results for CCCO Results Case 1 max Q, Btu max Pd , N/m 2 max N L , g max V io, % J2∗ , s tp, s

130.45 17125 1.77 5.03 16.29 6.47

Case 2 183.74 18000 2.5 5.09 631.98 7.32

by applying the convex approximation-based chance constraint handling strategy. Secondly, even though there is a trend that the CCCO algorithm issues in higher conservatism compared with NCCO, the computational performance of using the CCCO is significantly better in comparison to the NCCO. This is caused by the guaranteed polynomial-time complexity of applying the CCCO, while the processing complexity of NCCO is unpredictable. Additionally, there is a point that should be noted from the comparative results, which can be observed according to these control evolution profiles. That is, there is a trend that the CCCO algorithm issue in a σc control sequence, which has instant variations or less oscillations, to further smooth the actual bank angle profile. When applying the proposed methods to handle mission case 2, this trend becomes more evident. This is because for this mission case, the optimal control structure should ideally maintain a bang-singular-bang mode. It is worth mentioning that although using the convexified formulation provided by Eq. (4.32) can result in better computational performance, transcribing a nonconvex trajectory optimization model into a convex form is not always beneficial. The convex relaxation process can lead to serious mismatches between the original system and the convexified one for specific trajectory optimization problems, which will affect the reliability and effectiveness of the calculated results. Furthermore, the commonly used NLP solvers and the nonconvex chance-constrained formulation (4.21) can both be effective solutions to these flight path design problems with uncertain parameters.

4.6 Conclusion In this chapter, the problem of hypersonic vehicle atmospheric entry under a complex environment has been explored. The trajectory optimization model becomes difficult to solve as a result of the coexistence of probabilistic and hard path constraints. Two chance-constrained trajectory optimization methods were presented and implemented in order to efficiently explore the optimal control profile. By using either of the above methods, the original chance-constrained optimal control problem can be replaced by a deterministic version. In this way, convex optimization methods and

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4 Fast Trajectory Optimization with Chance Constraints

general NLP solvers can both solve the above problems. Several numerical simulations were executed to show the Key features of the presented designs. It can be seen from the obtained results that the presented methods are capable of producing near-optimal solutions while meeting various sorts of constraints, and the optimal solution has a bang-bang structure. Moreover, the computational performance of the algorithm will be enhanced by applying the convex-relaxed design in general. In this way, the gap between offline simulation and online implementation will be bridged. Hence, we recommend that the presented designs are used for flight trajectory planning for the hypersonic vehicle during the atmospheric entry flight phase. The proposed NCCO and CCCO model will contain more complex mission-related chance constraints in future research. For example, it is desirable to consider the uncertain no-fly zone constraints, which are widely present in practice. In order to maintain the convexity of the CCCO model, as a result, it is necessary to properly address this sort of constraint to reformulate it into a convex form.

Appendix Nonzero components of A(x) are defined in the following equations: a14 a21 a24 a26 a34 a36 a44

= sin γ γ sin ψ = − V cos r 2 cos φ cos γ sin ψ = r cos φ γ cos ψ = V cos r cos φ = cos γ rcos ψ = − V cos γr sin ψ = −2kC D V

a15 a23 a25 a31 a35 a41 a45

= V cos γ sin ψ tan φ = V cos rγ cos φ V cos γ sin ψ = − r cos φ = − V cosrγ2 cos ψ = − V cos γr cos ψ 2 γ = kCHD V + 2g sin r = −g cos γ

(A.1)

cos γ γ ) + 2g Vcos r2 r gr +V 2 cos γ V 2r

σ + a51 = −V ( kC Lhcos s

a54 = kC L cos σ + 2 a55 = sin γ grV−V r σ a57 = − L cos V kC L sin σ a61 = −V ( H cos γ + cos γ sinr 2ψ tan θ ) a63 = V cos γr sin ψ (1 + tan2 θ ) L sin σ a64 = kCcos + cos γ sinr ψ tan θ γ V tan γ kC L sin σ ψ tan θ a65 = − V sin γ sin cos γ r θ cos ψ a66 = V cos γ tan r L cos σ a67 = V cos γ a77 = K k = ρ S/2m

(A.2)

References

129

Besides, the nonzero components of Fg (x) are defined in Eq. (A.3) b11 b14 b21 b24

√ = 21 k Q V 3.15 ρ/ h s 2.15 √ = 3.15k Q V ρ = 21 ρV 2 / h s = ρV 

b31 =

1 ρV 2 S 2

C L2

+

(A.3)

C D2 /mgh s

b34 = S C L2 + C D2 /mg

References 1. Smirnov NN (2019) Space flight safety: experiments and supercomputing. Acta Astronaut 163:1–5. https://doi.org/10.1016/j.actaastro.2019.08.021 2. Pan L, Peng S, Xie Y, Liu Y, Wang J (2020) 3d guidance for hypersonic reentry gliders based on analytical prediction. Acta Astronaut 167:42–51. https://doi.org/10.1016/j.actaastro.2019. 07.039 3. Chai R, Savvaris A, Tsourdos A (2017) Violation learning differential evolution-based hpadaptive pseudospectral method for trajectory optimization of space maneuver vehicle. IEEE Trans Aerosp Electron Syst 53(4):2031–2044. https://doi.org/10.1109/TAES.2017.2680698 4. Chawla C, Sarmah P, Padhi R (2010) Suboptimal reentry guidance of a reusable launch vehicle using pitch plane maneuver. Aerosp Sci Technol 14(6):377–386. https://doi.org/10.1016/j.ast. 2010.04.001 5. Chai R, Savvaris A, Tsourdos A, Chai S, Xia Y (2017) Improved gradient-based algorithm for solving aeroassisted vehicle trajectory optimization problems. J Guid Control Dyn 40(8):2093– 2101. https://doi.org/10.2514/1.G002183 6. Zhou H, Wang X, Cui N (2019) Glide trajectory optimization for hypersonic vehicles via dynamic pressure control. Acta Astronaut 164:376–386. https://doi.org/10.1016/j.actaastro. 2019.08.012 7. Hager WW, Hou H, Rao AV (2016) Convergence rate for a gauss collocation method applied to unconstrained optimal control. J Optim Theory Appl 1–24. https://link.springer.com/article/ 10.1007/s10957-016-0929-7 8. Darby CL, Hager WW, Rao AV (2011) Direct trajectory optimization using a variable low-order adaptive pseudospectral method. J Spacecr Rocket 48(3):433–445. https://doi.org/10.2514/1. 52136 9. Fahroo F, Ross IM (2008) Pseudospectral methods for infinite-horizon nonlinear optimal control problems. J Guid Control Dyn 31(4):927–936. https://doi.org/10.2514/1.33117 10. Garg D, Patterson M, Hager WW, Rao AV, Benson DA, Huntington GT (2010) A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46(11):1843–1851. https://doi.org/10.1016/j.automatica.2010.06.048 11. Mao Y, Zhang D, Wang L (2017) Reentry trajectory optimization for hypersonic vehicle based on improved gauss pseudospectral method. Soft Comput 21(16):4583–4592. https://doi.org/ 10.1007/s00500-016-2201-3 12. Conway BA (2012) A survey of methods available for the numerical optimization of continuous dynamic systems. J Optim Theory Appl 152(2):271–306. https://doi.org/10.1007/s10957-0119918-z 13. Duan H, Li S (2015) Artificial bee colony based direct collocation for reentry trajectory optimization of hypersonic vehicle. IEEE Trans Aerosp Electron Syst 51(1):615–626. https://doi. org/10.1109/TAES.2014.120654

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14. Englander JA, Conway BA (2017) Automated solution of the low-thrust interplanetary trajectory problem. J Guid Control Dyn 40(1):15–27. https://doi.org/10.2514/1.G002124 15. Kim JJ, Lee JJ (2015) Trajectory optimization with particle swarm optimization for manipulator motion planning. IEEE Trans Industr Inf 11(3):620–631. https://doi.org/10.1109/TII.2015. 2416435 16. Pontani M, Conway BA (2014) Minimum-fuel finite-thrust relative orbit maneuvers via indirect heuristic method. J Guid Control Dyn 38(5):913–924. https://doi.org/10.2514/1.G000157 17. Yokoyama N, Suzuki S (2005) Modified genetic algorithm for constrained trajectory optimization. J Guid Control Dyn 28(1):139–144. https://doi.org/10.2514/1.3042 18. Zou W, Shi P, Xiang Z, Shi Y (2020) Consensus tracking control of switched stochastic nonlinear multiagent systems via event-triggered strategy. IEEE Trans Neural Netw Learn Syst 31(3):1036–1045. https://doi.org/10.1109/TNNLS.2019.2917137 19. Wu Z, Karimi HR, Shi P (2019) Practical trajectory tracking of random lagrange systems. Automatica 105:314–322. https://doi.org/10.1016/j.automatica.2019.04.006 20. Xu J, Shi P, Lim C, Cai C, Zou Y (2019) Reliable tracking control for under-actuated quadrotors with wind disturbances. IEEE Trans Syst, Man, Cybern: Syst 49(10):2059–2070. https://doi. org/10.1109/TSMC.2017.2782662 21. Chan T, Mar P (2017) Stability and continuity in robust optimization. SIAM J Optim 27(2):817– 841. https://doi.org/10.1137/16M1067512 22. GonzAlez-Arribas D, Soler M, Sanjurjo-Rivo M (2017) Robust aircraft trajectory planning under wind uncertainty using optimal control. J Guidance, Control, Dyn 1–16. https://doi.org/ 10.2514/1.G002928 23. Chai, R, Savvaris, A, Tsourdos, A, Chai, S, Xia, Y, Wang, S (2019) Solving trajectory optimization problems in the presence of probabilistic constraints. IEEE Trans Cybern 1–14 (2019). https://doi.org/10.1109/TCYB.2019.2895305 24. Geletu A, Hoffmann A, Kloppel M, Li P (2017) An inner-outer approximation approach to chance constrained optimization. SIAM J Optim 27(3):1834–1857. https://doi.org/10.1137/ 15M1049750 25. Caillau JB, Cerf M, Sassi A, Trelat E, Zidani H (2018) Solving chance constrained optimal control problems in aerospace via kernel density estimation. Optim Control Appl Methods 39(5):1833–1858. https://doi.org/10.1002/oca.2445 26. Zhao Z, Kumar M (2017) Split-bernstein approach to chance-constrained optimal control. J Guid Control Dyn 40(11):2782–2795. https://doi.org/10.2514/1.G002551 27. Liu X, Shen Z, Lu P (2016) Exact convex relaxation for optimal flight of aerodynamically controlled missiles. IEEE Trans Aerosp Electron Syst 52(4):1881–1892. https://doi.org/10. 1109/TAES.2016.150741 28. Wang Z, Grant MJ (2017) Constrained trajectory optimization for planetary entry via sequential convex programming. J Guid Control Dyn 40(10):2603–2615. https://doi.org/10.2514/1. G002150 29. Zhao DJ, Song ZY (2017) Reentry trajectory optimization with waypoint and no-fly zone constraints using multiphase convex programming. Acta Astronaut 137:60–69. https://doi.org/ 10.1016/j.actaastro.2017.04.013 30. Ure NK, Inalhan G (2012) Autonomous control of unmanned combat air vehicles: design of a multimodal control and flight planning framework for agile maneuvering. IEEE Control Syst Mag 32(5):74–95. https://doi.org/10.1109/MCS.2012.2205532 31. Chai R, Savvaris A, Tsourdos A (2016) Fuzzy physical programming for space manoeuvre vehicles trajectory optimization based on hp-adaptive pseudospectral method. Acta Astronaut 123:62–70. https://doi.org/10.1016/j.actaastro.2016.02.020 32. Constantine P, Kent C, Bui-Thanh T (2016) Accelerating markov chain monte carlo with active subspaces. SIAM J Sci Comput 38(5):A2779–A2805. https://doi.org/10.1137/15M1042127 33. Jorris TR, Cobb RG (2009) Three-dimensional trajectory optimization satisfying waypoint and no-fly zone constraints. J Guid Control Dyn 32(2):551–572. https://doi.org/10.2514/1.37030 34. Kemal Ure N, Inalhan G (2012) Autonomous Control of UCAVs Design of a multi modal control and flight planning framework for agile maneuvering. IEEE Control Syst Mag 32(5):74–95

Chapter 5

Fast Generation of Chance-Constrained Flight Trajectory for Unmanned Vehicles

Abstract In this chapter, a fast chance-constrained trajectory generation strategy is presented that uses convex optimization and convex approximation of chance constraints to settle the problem of unmanned vehicle path planning. A path-lengthoptimal trajectory optimization model is developed for unmanned vehicles, taking into account pitch angle constraints, curvature radius constraints, probabilistic control actuation constraints, and probabilistic collision avoidance constraints. Afterward, the convexification technique is applied to convert the nonlinear problem into a convex form. To handle probabilistic constraints in the optimization model, convex approximation techniques are used to replace probabilistic constraints with deterministic ones while maintaining the convexity of the optimization model. The proposed approach has been proven effective and reliable through numerical results from case studies. Comparative studies have also shown that the proposed design generates more optimal flight paths and has improved computational performance compared to other chance-constrained optimization methods.

5.1 Introduction The design of flight trajectories for unmanned vehicles has been a focus of research for the past 20 years due to its growing applications, such as regional detection, urgent medical supply delivery, rescue, and formation flying [1, 2]. An effective path generator is a self-sufficient method to generate a feasible flight trajectory between the initial and target positions of a vehicle while taking into account various missionspecific vehicle and environment requirements modeled as constraints during the planning phase. In the past, geometric path planning was deemed the primary method for plotting the movement of Unmanned Aerial Vehicles (UAVs). A significant amount of research has been done on geometric-based path planners for UAV path generation problem [3–6], including the pioneering work in [3, 4]. A 2-D dynamic trajectory generation approach [3] was presented to guide UAVs along a set of pre-defined waypoints. In [4], a 3-D Dubins curve-based trajectory generation method was designed that considered both pitch angle and curvature radius constraints. However, this algo© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_5

131

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5 Fast Generation of Chance-Constrained Flight Trajectory …

rithm was only effective for long-distance flights. Yang and Sukkarieh [5] proposed a Bezier curve-based path generation method with obstacle avoidance. Although most geometric-based motion planners can generate feasible paths, they often fail to take into account the optimality of the path. Additionally, mission constraints are often difficult to incorporate, and the way they are handled may need to be re-designed for various problems. Recent years have seen significant attention given to optimization-based path planners for unmanned vehicles [7, 8], which formulate and solve a trajectory optimization model and incorporate mission requirements as constraints. A number of papers have been published on the development and application of optimization-based path generation techniques [9–11]. In [9], a genetic algorithm-based path generator was proposed and applied to the optimization of 3-D flight paths for military UAVs with no-fly zone constraints. Kim and Lee proposed a heuristic path generator using particle swarm optimization in [10] to determine the optimal motion of a manipulator. In [11], the authors developed a multi-layer optimization-based trajectory generator for calculating the optimal movement of an autonomous space vehicle. Tang and Hauser designed a data-driven indirect algorithm and sensitivity analysis tool in [12] to quickly generate optimal control solutions for different benchmark problems. Although these optimization-based approaches have been shown to be effective, one major challenge in implementing these path planners is that their computational or convergence performance may not be fast enough for real-time applications. This becomes even more challenging when the vehicle dynamics are highly nonlinear or complex constraints such as path and control chance constraints must be satisfied. Wang et al. [13, 14] adopted a dynamic Dubins-Helix method to create constrained trajectories for unmanned vehicles. This technique extended the task scenario from two dimensions to three dimensions and was able to achieve minimum curvature radius and pitch angle constraints in real time. However, there are three main drawbacks to this technique. First, the trajectory produced by the algorithm has points with discontinuous curvature because the algorithm divides the flight into separate segments and designs the path for each segment independently. These points with discontinuous curvature may cause significant position errors for the path tracker. Second, the optimality of the flight path produced by this method is not as good as other optimization techniques for a variety of mission cases. Finally, this motion planner does not have the ability to handle control chance constraints and probabilistic collision avoidance constraints. It is worth mentioning that in many real-world situations involving UAV flights, the constraint information may not be precisely known and could be affected by uncertainties [15]. For example, uncharacterized temperature/weather influences and errors in localization and mapping can cause uncertainty in the maximum attainable actuation level constraints as well as obstacle avoidance constraints. Therefore, it is important to incorporate chance constraints in the design of flight trajectories, necessitating new chance-constrained unmanned vehicle trajectory planning methods to provide effective tools for finding solutions. In this chapter, we aim to design a trajectory planning algorithm that minimizes the flight path length of a fixed-wing UAV while satisfying deterministic path constraints,

5.1 Introduction

133

probabilistic control constraints, and obstacle avoidance constraints, all in real time. To achieve these goals, we use convex optimization and convex approximation of chance constraints. There are many previous works in the literature that have applied convex optimization methods to aerospace guidance and control problems, such as in the planetary-entry problem in [16]. This approach was improved and adapted for use in other areas in [17, 18]. Additionally, an asteroid landing problem was considered in [19], where convex optimization was used to explore time-optimal and minimum error landing trajectories. In addition to the previously mentioned works, there is a significant amount of literature about UAV path planning problems which uses convex optimization [20, 21]. For example, in [20], a convex optimization-oriented path planning method was developed and tested on a quad-rotor to assess and verify the performance of the developed method. In [21], a sequential convex optimization method modified by a line search update process was used to solve a UAV and unmanned ground vehicle (UGV) rendezvous problem. Simulation outcomes reported show that this approach achieved enhanced convergence performance and real-time capability. The use of convex optimization is motivated by the fact that convex programs can be solved in polynomial time if the problem can be formulated as a convex program [22]. Nevertheless, earlier methods show no possibilities as the direct solutions to the problem at hand because of the presence of chance constraints, which make the probabilistic functions non-deterministic. In [23], a chance-constrained optimization strategy was developed and applied to a similar UAV path planning problem. However, two issues are still unresolved. Firstly, the approach used a non-convex function to approximate the probabilistic constraints, resulting in non-convexity in the transformed trajectory optimization model and increased computational burden. Secondly, the formulation did not account for uncertain obstacles, limiting its practical applicability. So in this work, we aim to design a convex chance constraint approximation strategy that can transcribe the probabilistic control and obstacle avoidance constraints into a deterministic and convex form, preserving the convexity of the optimization model. This approximation approach is then integrated into a convexified path planning framework, which makes it a potentially appropriate solution for solving the chance-constrained UAV trajectory planning task alternatively. In summary, the work in this chapter makes three main contributions. First, the chance-constrained UAV trajectory planning problem considered in [23] is further extended so as to take the uncertain effects caused by both the actuator and obstacles into account. This extended problem formulation is more comprehensive and representative than the previous version. Second, we design a convex approximation strategy that allows us to reformulate the extended nonlinear UAV chance-constrained trajectory planning model into a deterministic and convex one. Third, lots of comparative case studies are conducted to demonstrate the effectiveness of the proposed fast chance-constrained trajectory generation method and its improved computational performance. The structure of this chapter is shown below. In Sect. 5.2, the mathematical formulation of the trajectory planning problem is illustrated. In Sects. 5.3 and 5.4, the convexified version of the unmanned vehicle trajectory optimization model and

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the convex chance constraint approximation method are presented, respectively. In Sect. 5.5, numerical results are presented, including optimal flight trajectories and comparative case studies. Finally, Sect. 5.6 is the conclusion.

5.2 Trajectory Planning Formulation This section outlines the research study’s unmanned vehicle trajectory planning model. Specifically, Sect. 5.2.1 shows the nonlinear system equations of the unmanned vehicle. Then, in Sects. 5.2.2 and 5.2.3, the geometric constraints imposed on the model, along with the relationship between geometric constraints and actual vehicle constraints, are introduced. Sections 5.2.4 and 5.2.5 respectively illustrate the control chance constraints and the probabilistic collision avoidance constraints. Based on the system model and mission constraints, a path length-optimal unmanned vehicle trajectory optimization formulation is established in Sect. 5.2.6.

5.2.1 Unmanned Vehicle System Equations We construct the equations of motion of a fixed-wing unmanned vehicle as [4, 13]: ⎧ dp x ⎪ = cos γ (s) cos φ(s) ⎪ ds ⎪ dp ⎪ y ⎪ ⎨ ds = cos γ (s) sin φ(s) dpz = sin γ (s) ds ⎪ dφ ⎪ ⎪ = μ1 ⎪ ds ⎪ ⎩ dγ = μ2 ds

(5.1)

In the above equation, p = ( px , p y , pz ) ∈ R3 represents the position of the unmanned aerial vehicle (UAV) in three-dimensional space and s is the curvilinear abscissa along the path. The heading angle is denoted by φ and the pitch angle by γ . μ1 and μ2 represent the derivative values with respect to the heading and pitch angles and will act as control variables in the trajectory optimization model defined in Sect. 5.2.6. This chapter differs from most previous studies [4, 5, 24] in that it requires a variety of mission constraints to be considered during flight. These constraints are divided into three categories: geometric constraints, control chance constraints, and probabilistic collision avoidance constraints.

5.2 Trajectory Planning Formulation

135

5.2.2 Geometric Constraints One of the geometric constraints is the minimum curvature radius of the unmanned vehicle. This constraint is imposed to smooth out the system state and control profiles. In addition, certain requirements must be placed on the pitch angle for safety in 3-D space. This is done by restricting the pitch angle to a certain range. Therefore, the geometric constraints that must be considered during flight include: 1. The curvature radius R(s) should satisfy |R(s)| > R min . 2. The pitch angle should satisfy γ min ≤ γ ≤ γ max . Note that in 1., R(s) is calculated by:  R(s) = 1/ μ21 (s) cos2 γ (s) + μ22 (s)

(5.2)

5.2.3 Relationship Between Geometric Constraints and Vehicle Actual Constraints Unlike the model described by Eq. (5.1), there are other forms of UAV dynamics that have been reported in the literature [8, 21]. It is important to note that there are connections between the model given by (5.1) and these more complex nonlinear fixed-wing UAV dynamics that are commonly used in other works (e.g., [8, 21]). This will become clearer through the analysis of the relationships between the geometric constraints defined in 4.2.2 and the actual constraints on the vehicle (e.g., structural, propulsive, and envelope constraints). To start, we will revisit the dynamics model applied in [8, 21]: − g sin γ V˙ = (T −D) m L sin σ φ˙ = mV (5.3) cos γ −mg cos γ γ˙ = L cos σ mV ˙ and pitch angle (γ˙ ) where the derivatives of vehicle velocity (V˙ ), heading angle (φ), ˙ ˙ with respect to time t are represented by V , φ, and γ˙ . The thrust (T ), vehicle mass (m), gravity (g), and roll angle (σ ) are denoted by other parameters T , m, g, and σ , respectively. The aerodynamic lift and drag forces, L and D, are defined as follows: L = ρV 2SC L 2 D = ρV 2SC D 2

(5.4)

Here, ρ and S symbolize the atmospheric density and reference area of the vehicle, respectively. The lift coefficient is represented by C L , while the drag coefficient, C D , is modeled as a function of C L .

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5 Fast Generation of Chance-Constrained Flight Trajectory …

Typically, in order to meet both the vehicle’s propulsive and size constraints, limitations can be imposed on the variables σ and T . To fulfill additional physical restrictions, such as structural constraints, the value of the vertical load factor, n z , may be limited. This can be expressed as: nz =

L+T mg

(5.5)

Based on Eqs. (5.4) and (5.5), two components of the curvature radius (such as the horizontal and vertical components) can be expressed in the following form: Rh = Rv =

2 V = n z gVsin σ cos γ φ˙ 2 V = n z g cos Vσ −g cos γ γ˙

(5.6)

From Eqs.(5.3)-(5.6), it is possible to find functional relationships for V , γ , Rh , and Rv such that: V 2 = FV (n z , T ) γ = Fγ (n z , T ) (5.7) Rh = FRh (n z , T, σ ) Rv = FRv (n z , T, σ ) Equation (5.7) can be recast into a more concise representation: ⎛

⎞ V2 ⎜ γ ⎟ ⎜ ⎟ = F(n z , T, σ ) ⎝ Rh ⎠ Rv

(5.8)

The above equation shows that we can map a domain governed by UAV physical variables (e.g., n z , T , and σ ) to another domain governed by geometric variables. Remark 5.1 It is straightforward to establish the following relationship by comparing the definition of R(s) in Eq. (5.2) with Eq. (5.6):  R(s) = 1/ μ21 cos2 γ + μ22  ˙ = 1/ ( Vφ )2 cos2 γ + ( Vγ˙ )2  = 1/ R12 + R12 h

Here, V =

ds , dt

μ1 =

dφ , ds

and μ2 =

dγ ds

.

v

(5.9)

5.2 Trajectory Planning Formulation

137

5.2.4 Control Chance Constraints In this chapter’s consideration of the trajectory planning task, the maximum and minimum achievable control actuation levels of the unmanned vehicle are not considered to be fixed. It is assumed that the control limitations will be affected by some uncertain variables (such as ξμ1 and ξμ2 ):

Pr {|μ1 + ξμ1 | ≤ μmax 1 } ≥ μ 1 Pr {|μ2 + ξμ2 | ≤ μmax 2 } ≥ μ 2

(5.10a) (5.10b)

where ξμ1 and ξμ2 are two uncertain parameters with known probability density functions (PDFs). μ1 and μ2 are the acceptable probabilities of occurrence. μmax 1 and μmax represent the maximum allowable values for μ1 and μ2 , respectively. 2

5.2.5 Probabilistic Collision Avoidance Constraints Collision avoidance path constraints are also included in the planning model. In this chapter, obstacles can be modeled as polygons or polyhedrons. For example, a safe region can be defined for a polyhedron as:  = {p ∈ R : 3

Mj No  

T amn p + bmn + ξmn > 0}

(5.11)

n=1 m=1

where p = ( px , p y , pz ). No and M j denote the number and the number of obstacles  of faces of the jth obstacle, respectively. The symbols and represent the logical ”and” and the logical ”or” relations. amn ∈ R3 and bmn ∈ R, while ξi j ∈ R is the uncertain variable. The region occupied by the set  consists of areas outside all of the obstacles in the environment. The pose of unmanned vehicle must remain within the safe region to avoid any potential collisions. Since there is uncertainty in the safe set , we model the collision avoidance constraint as a probabilistic inequality in the form of: (5.12) Pr { p ∈ } ≥ o

5.2.6 Objective and Optimization Model The objective of the unmanned vehicle trajectory planning problem is to create a seamless flight path linking the starting pose P0 ( px , p y , pz ) and the final pose P f ( px , p y , pz ). The aim is to optimize the vehicle’s flight path s, taking into consideration geometric constraints and various chance constraints. Therefore, a path-

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5 Fast Generation of Chance-Constrained Flight Trajectory …

length-optimal nonlinear chance-constrained unmanned vehicle trajectory optimization formulation can be formulated as follows:  s ⎧ ⎪ minimize J = ds ⎪ ⎪ ⎪ z(s) 0 ⎪ ⎪ dp x ⎪ s.t. ⎪ = cos γ (s) cos φ(s) ⎪ ds ⎪ dp y ⎪ ⎪ = cos γ (s) sin φ(s) ⎪ ds ⎪ ⎪ dpz ⎪ = sin γ (s) ⎪ ⎪ ds ⎪ dφ ⎪ ⎪ = μ 1 (s) ⎪ ds ⎨ dγ = μ2 (s) ds (5.13) ⎪ ) = x0 , x(s f ) = x f x(s ⎪ 0 ⎪ ⎪ ⎪ |R(s)| > R min ⎪ ⎪ ⎪ ⎪ γ min ≤ γ (s) ≤ γ max ⎪ ⎪ ⎪ max ⎪ |μ1 | ≤ μmax ⎪ 1 , |μ2 | ≤ μ2 ⎪ ⎪ max ⎪ Pr {|μ + ξ | ≤ μ } ≥ μ 1 ⎪ 1 μ1 1 ⎪ ⎪ max ⎪ Pr {|μ + ξ | ≤ μ } ≥ μ 2 ⎪ 2 μ2 2 ⎩ Pr { p ∈ } ≥ o In Eq. (5.13), s ∈ (s0 , s f ) and z(s) = (x(s), u(s)), in which the system state variables are defined as x(s)=[ px (s), p y (s), pz (s), φ(s), γ (s)]T . The boundary conditions are given by x(s0 ) = x0 and x(s f ) = x f , respectively. Here, the boundary variables are denoted as x0 and x f . The control variable u(s) consists of μ1 (s) and μ2 (s). That is, u(s) = [μ1 (s), μ2 (s)]T .

5.3 Convex-Programming-based Trajectory Planning Approach There are currently many numerical algorithms (e.g., the pseudospectral method [25] and the multiple shooting method [26]) that can be used to solve standard nonlinear optimal control problems. However, a major challenge in using these techniques is that they tend to be inefficient due to the need to solve nonlinear programming problems directly. To address this issue, the fast trajectory generation approach proposed in this chapter suggests transcribing the original formulation into a convex program, which allows for faster convergence and avoids the time-consuming nonlinear optimization process.

5.3.1 Convexification of System Equations and Constraints To reformulate the optimization model, the first step is to linearize the system dynamics. Eq. (5.1) can be simplified as ddsx = f (x, u). Here, f (x, u) has the following form:

5.3 Convex-Programming-based Trajectory Planning Approach

139

⎡

⎤ cos γ (s) cos φ(s) ⎢ cos γ (s) sin φ(s) ⎥ ⎢ ⎥ ⎥ sin γ (s) f (x, u) = ⎢ ⎢ ⎥ ⎣ ⎦ μ1 (s) μ2 (s)

(5.14)

A linear unmanned vehicle system model can then be obtained by taking the derivative of f (x, u) with respect to x and u, which can be expressed as: dx = A(x r , u r )x + B(x r , u r )u + c(x r , u r ) ds

(5.15) r

where (x r , u r ) is the reference state and control pair and it satisfies ddsx = f (x r , u r ). c(x r , u r )= f (x r , u r ) − A(x r , u r )x r − B(x r , u r )u r . In Eq. (5.15), A(x r , u r ) and B(x r , u r ) are, respectively, the partial derivative of f (·, ·) with respect to x and u at the operating point (x r , u r ). That is, ∂ f (x, u)  x=x r ∂x u=u r  ∂ f (x, u)  r r B(x , u ) = x=x r ∂u u=u r A(x r , u r ) =

(5.16)

For the unmanned vehicle system given by Eq. (5.1), A(x r , u r ) and B(x r , u r ) can be written as: r r A(x , u ) ⎡ ⎤ 0 0 0 sin φ(s) cos γ (s) − cos φ(s) sin γ (s) ⎢ 0 0 0 cos φ(s) cos γ (s) − sin φ(s) sin γ (s) ⎥ ⎢ ⎥ 0 cos γ (s) = ⎢0 0 0 ⎥ ⎣0 0 0 ⎦ 0 0 000 0 0 x=x r

⎡

0 ⎢0 ⎢ B(x r , u r ) = ⎢ ⎢0 ⎣1 0

(5.17)

u=u r

⎤

0 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1 x=x rr

(5.18)

u=u

Up to this point, in the original optimization model (5.13), the nonlinear dynamics have been written into a linear form. Then we should reformulate the nonconvex path constraint (e.g., the curvature radius constraint). The transformation method is similar to that of the dynamics. Specifically, R(x, u) can be transformed into a convex form as: R(x, u) = R(x r , u r ) + Rx (x r , u r )(x − x r ) (5.19) +Ru (x r , u r )(u − u r )

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where Rx and Ru are partial derivatives of R with respect to x and u respectively. This transformation is easy to implement and can be applied to various constraints.

5.3.2 Convex Trajectory Optimization Model To address the problem, discretization methods must be utilized to transform the continuous-time problem into a static convex program. The chapter employs the pseudospectral method (PS) as the discretization technique. Detailed formulations and convergence properties of the PS method can be found in references such as [25, 27], but are omitted here for brevity. It is worth mentioning that combining the pseudospectral method with convex optimization for solving aerospace optimal control problems has been explored in previous studies, such as [28, 29]. According to the reported results, this combination of methods has been proven to produce more accurate results compared to other standard methods while also keeping the computation time feasible. Hence, this chapter employs this hybrid approach to formulate the convex trajectory optimization model for the current problem. By applying the PS method, the linearized dynamics can be further written as: N  i=0

Dik xi =

s f − s0 (A(xkr , u rk )xk 2

(5.20)

+ B(xkr , u rk )u k + c(xkr , u rk )) where the initial and final values of s are represented by s0 and s f respectively. The index k takes on values from 1 to N , and Dik refers to the differentiation matrix. By transforming the problem into a convex form and discretizing the system dynamics and path constraints, a convex model for unmanned vehicle trajectory planning is established: ⎧ minimize J = s N ⎪ ⎪ zk ⎪ ⎪ ⎪ ⎪ s.t. ∀k ∈ {1, ..., N } ⎪ ⎪ ⎪ N ⎪  ⎪ s f − s0 ⎪ ⎪ (A(xkr , u rk )xk Dik xi = ⎪ ⎪ 2 ⎪ ⎪ i=0 ⎪ ⎪ ⎪ +B(xkr , u rk )u k + c(xkr , u rk )) ⎨ r r |R(xk , u k ) + Rx (xkr , u rk )(xk − xkr ) (5.21) ⎪ r r r min ⎪ +R (x , u )(u − u )| > R ⎪ u k k k k ⎪ ⎪ ⎪ x(s0 ) = x0 , x(s N ) = x f ⎪ ⎪ ⎪ ⎪ x min ≤ xk ≤ x max ⎪ ⎪ ⎪ ⎪ |u k | ≤ u max ⎪ ⎪ ⎪ ⎪ |xk − xkr | ≤ δx ⎪ ⎪ ⎩ |u k − u rk | ≤ δu

5.3 Convex-Programming-based Trajectory Planning Approach

141

Equation (5.21) is a typical convex program, as the mission objective, equality and inequality constraints are all convex functions. In Eq. (5.21), x(sk ) and u(sk ) are abbreviated as xk and u k , respectively. Linear approximation is used to convert the nonlinear dynamics and constraints into a convex form. However, we know that linearization is only accurate in the vicinity of the reference trajectory. Therefore, in Eq. (5.21), the two trust-region constraints, |xk − x r ∗ k| ≤ δ ∗ x and |u k − u r ∗ k| ≤ δ ∗ u, are introduced to control the fluctuation of the state and control profiles and improve the linearization process. Our experiments have shown that these simple trust-region constraints with fixed radii (δx and δx ) can effectively handle the problem. However, it is possible to improve the optimization process by incorporating trust-region constraints with variable radii. For instance, motivated by a related work [30], the optimization program can be revised by adding quadratic trust-region constraints: ⎧ min J = s N + wx r x 2 + wu ru 2 ⎪  ⎪ zk ⎪ ⎪ ⎪ ⎪ s.t. ∀k ∈ {1, ..., N } ⎪ ⎪ ⎪ N ⎪  ⎪ s f − s0 ⎪ ⎪ ⎪ (A(xkr , u rk )xk Dik xi = ⎪ ⎪ 2 ⎪ ⎪ i=0 ⎪ ⎪ ⎨ +B(xkr , u rk )u k + c(xkr , u rk )) r r |R(xk , u k ) + Rx (xkr , u rk )(xk − xkr ) ⎪ ⎪ ⎪ +Ru (xkr , u rk )(u k − u rk )| > R min ⎪ ⎪ ⎪ ⎪ x(s0 ) = x0 , x(s N ) = x f ⎪ ⎪ ⎪ ⎪ x min ≤ xk ≤ x max ⎪ ⎪ ⎪ ⎪ |u k | ≤ u max ⎪ ⎪ ⎪ ⎪ [xk − xkr ]T [xk − xkr ] ≤ r xk ⎪ ⎩ [u k − u rk ]T [u k − u rk ] ≤ ru k

(5.22)

where wx and wu stand for positive weighting parameters. r x = [r x1 , r x2 , ...r x N ]T and ru = [ru 1 , ru 2 , ...ru N ]T , respectively.  · 2 denotes the 2-norm. In this formulation,  the optimization variable is redefined as z k =(xk , u k , r xk , ru k ). Besides, the trust region radius is adjusted during the iteration, thereby further enhancing the convergence and robustness of the optimization process. The solution to the optimization problem (5.21) is usually obtained by iteratively solving a sequence of convex optimization problems. Nevertheless, the use of linear approximations may result in a critical issue referred to as “artificial infeasibility” during the solution-finding process [21, 30, 31]. This issue arises when a solution to the original non-convex problem does not meet the requirements of the linearized dynamics equations and path constraints outlined in (5.21), or vice versa. Therefore, certain measures should be taken to reduce or eliminate these inconsistencies caused by the linearization errors, or the optimization process may suffer from slow convergence or even fail to converge. The efforts of Acikmese [31] and Wang [21, 30] are significant in addressing the problem. Acikmese [31] eliminated artificial infeasibility by introducing a virtual

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5 Fast Generation of Chance-Constrained Flight Trajectory …

control term in the linear dynamics, allowing any feasible state to be reached as no constraint was imposed on the virtual control variable. Wang [30] proposed a linesearch strategy to reduce artificial infeasibility and enhance the convergence of the sequential optimization process. This line-search method was later applied to solve UAV and UGV rendezvous problems in [21], resulting in a reduction of error from linear approximation and improved convergence in successive optimization. It should be noted that the Eqs. (5.21) and (5.22) are formulated without taking into account the chance constraints specified in Eqs. (5.10) and (5.12). In the following section, a method for approximating chance constraints in a convex manner will be introduced and applied to formulate a fast chance-constrained trajectory planning model.

5.4 Deterministic Chance-Constrained Trajectory Planning Formulation In this section, we aim to find a suitable method for incorporating the probabilistic control and collision avoidance constraints (Eq. (5.10)) into the convex formulation (Eq. (5.21) or Eq. (5.22)). The method should be easy to implement and maintain the convexity of the optimization model. One potential approach for the control chance constraints is to use the min-max strategy, which is commonly used in robust optimization tasks [32]. However, this strategy does not allow for any constraint violations, which can restrict the search space of the optimization process and decrease the performance and optimality of the solution. Therefore, we propose an alternative convex chance constraint approximation strategy in this study, which will be discussed in more detail in the next subsection.

5.4.1 Convex Approximation of Control Chance Constraints Let us consider the control chance constraints (5.10) in a general form Pr {g(u, ξ ) > 0} ≤ . If we treat the inequality g(·, ·) > 0 as an event, then the probability of occurrence can be calculated via:  Pr {g(u, ξ ) > 0} = E(H (g(u, ξ ))) = H (g(u, ξ ))L(ξ )dξ (5.23) 

where E(·) is the expectation function. L(ξ ) is the PDF of ξ . H (·) is the Heaviside function, which has the form of:  1 if g(u, ξ ) > 0 H (g(u, ξ )) = (5.24) 0 if g(u, ξ ) ≤ 0

5.4 Deterministic Chance-Constrained Trajectory Planning Formulation

143

The core idea behind the convex chance constraint approximation strategy is to use a convex function, (g(u, ξ )), to replace H (g(u, ξ )) in Eq. (5.23). The convex function proposed in this study has a simple form: (g(u, ξ )) = (g(u, ξ ) + 1)+

(5.25)

The symbol “+” in the superscript represents the maximum value (e.g. max g(u, ξ ) + 1, 0). The following theorem shows that using Eq. (5.23) gives an upper approximation of the original chance constraint. Theorem 5.1 Given that the convex approximation function in the form of Eq. (5.25), if an integral quantity can be defined in the form of  Vc (u) =



(g(u, ξ ))L(ξ )dξ

(5.26)

then, Vc is an upper bound of the control chance constraint Pr {g(u, ξ ) > 0}. Proof According to the definition of (g(u, ξ )), it is obvious that ≥ 0 for any g(u, ξ ) and > 1 for g(u, ξ ) > 0. Hence, ∀ξ ∈ , we have  Vc (u) ≥

(g(u, ξ ))L(ξ )dξ

{ξ ∈,g(u,ξ )>0}



≥

L(ξ )dξ

{ξ ∈,g(u,ξ )>0}

=Pr {ξ ∈ , g(u, ξ ) > 0} which completes the proof.



Theorem 5.1 can be used to create a relaxation of the control chance constraint. For instance, Vc (u) ≤  (5.27) Satisfaction of Eq. (5.27) ensures that the original chance constraint is also satisfied. To calculate the integral in Eq. (5.26), the Markov chain Monte-Carlo (MCMC) sampling strategy as proposed in [23] is employed due to its simplicity. First, a set Nξ are generated, and then Vc (u) can be calculated as of random parameters {ξ q }q=1 follows: Nξ 1  Vc (u) ≈ (g(u, ξ q )) (5.28) Nξ q=1

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Thus, the chance-constrained unmanned vehicle trajectory planning model is formulated as follows: ⎧ minimize ⎪ zk ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

J = sN ∀k ∈ {1, ..., N } N  s f − s0 Dik xi = (A(xkr , u rk )xk 2 i=0 +B(xkr , u rk )u k + c(xkr , u rk )) r r |R(xk , u k ) + Rx (xkr , u rk )(xk − xkr ) +Ru (xkr , u rk )(u k − u rk )| > R min Nξ  1 (g(u k , ξ q )) ≤  Nξ

(5.29)

q=1

x(s0 ) = x0 , x(s N ) = x f x min ≤ xk ≤ x max |u k | ≤ u max |xk − xkr | ≤ δx |u k − u rk | ≤ δu

The following theorem demonstrates that Eq. (5.27) is a convex constraint and the resulting chance-constrained formulation (5.29) is a convex optimization problem. Theorem 5.2 The chance constraint approximation inequality (5.27) is a convex constraint and the chance-constrained optimization model given by Eq. (5.29) is a convex program. Proof It is clear from the definition of the control constraints that g(u, ξ ) is convex in relation to u for any fixed value of ξ within the set . Furthermore, the function (g(u, ξ )) is both convex and non-decreasing. Therefore, (g(u, ξ )) is convex in relation to u for ∀ξ ∈ .  Since L(ξ ) is a non-negative probability measure on , the integral  (g(u, ξ ))L(ξ )dξ is also convex. The convexity of inequality (5.27) combined with the convexity of the optimization model (5.21) confirms the convexity of the optimization model (5.29), thus completing the proof. 

5.4.2 Convex Approximation of Probabilistic Collision Avoidance Constraints As to solve the optimization model while taking into account the probabilistic obstacle avoidance constraints, we reformulate inequality (5.12) into a more manageable form. This is accomplished by going through two steps. First,  we use the big-M technique to convert the logical disjunction (OR) symbol ( ) appeared in the safe

5.4 Deterministic Chance-Constrained Trajectory Planning Formulation

145

 region (5.11) to a logical conjunction (AND) symbol ( ). Then, we use Boole’s inequality-based decomposition method to separate the joint constraint into individual chance constraints, which preserves the convexity of the optimization model and simplifies the process of finding a solution. Consider the disjunction shown in Eq. (5.11), its satisfaction is equivalent to Mj 

T amn p + bmn + ξmn > 0

m=1

⇔

Mj 

T amn p + bmn + ξmn + Mz mn > 0

(5.30)

m=1

z mn ∈ {0, 1}, M > 0 where z mn is a binary integer variable. To ensure that one of the elements of the mthM j z mn < M j must be imposed. M indexed z is zero, an additional constraint of m=1 is a positive constant. Using this big-M technique, constraint (5.12) can be rewritten as: Mj No   T (5.31) amn p + bmn + ξmn + Mz mn > 0} ≥ o Pr { n=1 m=1

In order to split the joint constraint in (5.31) for a series of events E m , Boole’s inequality can be used. This results in a conservative yet simple form of the chance constraint for obstacle avoidance: For ∀n ∈ {1, ..., No }, ∀m ∈ {1, ..., M j } T Pr {amn p + bmn + ξmn + Mz mn > 0} ≥ mn M N j o  mn ≥ o

(5.32)

n=1 m=1

z mn ∈ {0, 1}, M > 0 Up to now, the original probabilistic obstacle avoidance constraint has been broken down into individual chance constraints. The convex function approximation method developed earlier can be directly applied to approximate the probabilistic term in (5.32). Similar to (5.29), we can construct the following mixed-integer convex program:

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5 Fast Generation of Chance-Constrained Flight Trajectory …

⎧ ⎪ ⎪ minimize z k ,z mn ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

J = sN ∀k ∈ {1, ..., N }, ∀n ∈ {1, ..., No }, ∀m ∈ {1, ..., M j } N  s f − s0 Dik xi = (A(xkr , u rk )xk 2 i=0 +B(xkr , u rk )u k + c(xkr , u rk )) r r |R(xk , u k ) + Rx (xkr , u rk )(xk − xkr ) +Ru (xkr , u rk )(u k − u rk )| > R min Nξ  1 (g(u k , ξ q )) ≤  Nξ q=1 Nξ  T q 1 (amn pk + bmn + ξmn + Mz mn ) ≤ mn Nξ

(5.33)

q=1 Mj No  

mn ≥ o

n=1 m=1

z mn ∈ {0, 1}, M > 0 x(s0 ) = x0 , x(s N ) = x f x min ≤ xk ≤ x max |u k | ≤ u max |xk − xkr | ≤ δx |u k − u rk | ≤ δu

where pk represents the current 3-D position of the unmanned vehicle at node k. The convexity of the problem (5.33) is a direct result of the definition of polyhedral obstacles, the use of Boole’s inequality-based decomposition, and Theorem 2. Unlike the formulation given by Eq. (5.29), the mixed-integer convex program can be solved using a branch-and-bound strategy [33]. Remark 5.2 Similar to Eq. (5.22), an alternative mixed-integer convex optimization formulation with varying trust-region radii r x and ru can be formulated by introducing the weighting parameters wx and wu :

5.4 Deterministic Chance-Constrained Trajectory Planning Formulation

⎧ minimize ⎪ ⎪ ⎪ zk ,zmn ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

J = s N + wx r x 2 + wu ru 2 ∀k ∈ {1, ..., N }, ∀n ∈ {1, ..., No }, ∀m ∈ {1, ..., M j } N  s f − s0 (A(xkr , u rk )xk Dik xi = 2 i=0 +B(xkr , u rk )u k + c(xkr , u rk )) r r |R(xk , u k ) + Rx (xkr , u rk )(xk − xkr ) +Ru (xkr , u rk )(u k − u rk )| > R min Nξ  1 (g(u k , ξ q )) ≤  Nξ q=1 Nξ

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

147

 T q 1 (amn pk + bmn + ξmn + Mz mn ) ≤ mn Nξ

(5.34)

q=1 Mj No  

mn ≥ o

n=1 m=1

z mn ∈ {0, 1}, M > 0 x(s0 ) = x0 , x(s N ) = x f x min ≤ xk ≤ x max |u k | ≤ u max [xk − xkr ]T [xk − xkr ] ≤ r xk [u k − u rk ]T [u k − u rk ] ≤ ru k 

The variable z k is defined as (xk , u k , r xk , ru k ) above. The optimization process in Eq. (5.34) takes into account varying-radius trust region constraints, which may improve its convergence and robustness.

5.4.3 Overall Algorithm Framework The overall algorithm framework is summarised in Algorithm 5.1. It should be noted that in Algorithm 5.1, Step 6 involves finding the solution to the mixed-integer convex optimization model represented by Eq. (5.33). As a result, the sequential mixed-integer convex programming method, as described in [33], is chosen as the primary optimization method for determining the solution. Inspired by prior studies [21, 30], a line search technique can be implemented in Step 8 of Algorithm 5.1 to address the problem of artificial infeasibility for the problem at hand. For the sake of thoroughness, the specifics of this technique are provided in Algorithm 5.2. In Algorithm 5.2, h represents the equality constraints, such as the linearization of the dynamics, and g represents the inequality constraints, such as the path constraints. h k (z r +1 ) and gk+ (z r +1 ) denote the error caused by the linearization process and the

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5 Fast Generation of Chance-Constrained Flight Trajectory …

Algorithm 5.1 Chance-constrained trajectory planning process Input: Algorithm/mission-dependent parameters: N , Nξ , x r , u r , x0 , x f , and R min ; /*Main Iteration*/ Step 1: Perform the convexification process with respect to system dynamics and constraints; Step 2: Generate the differential matrix Dik and discretize the convexified problem via PS; q N Step 3: Generate the random parameter {ξ q , ξi j }q=1 ; Step 4: Formulate the chance-constrained convex trajectory planning model via Eq. (5.33); Step 5: Address the chance-constrained convex optimization model; Step 6: Obtain the solution pair and update the reference pair (x r +1 , u r +1 ); Step 7: Check the stopping condition for convergence: max |x r +1 − x r | ≤ x , max |u r +1 − u r | ≤ u ; Step 8: If the stopping criteria are met, then output the solution. If not, perform a line search process and go back to Step 1. Output: The optimal trajectory pair (x ∗ , u ∗ );

Algorithm 5.2 A line-search process Input: z r = (x r , u r ), z r +1 and line-search parameters: ν1 , ν2 ∈ (0, 0.5), and 0 < c1 < c2 < 1; /*Main line-search process*/ Step 1: Calculate pr +1 = z r +1 − z r ; Step 2: Define the merit function M in the form of: M(z r +1 ; ν1 , ν2 ) = ν1

N 

h k (z r +1 )1 + ν2

k=1

N 

gk+ (z r +1 )1

k=1

where gk+ (z r +1 ) = max{gk (z r +1 ), 0}; Step 3: Calculate the directional derivative of M via: (M(z r +1 ; ν1 , ν2 ); pr +1 ) =   M(z r +1 + εpr +1 ; ν1 , ν2 ) − M(z r +1 ; ν1 , ν2 ) lim ε→0 ε Step 4: Search αr +1 such that the following condition holds true: M(z r +1 ; ν1 , ν2 ) + c1 αr +1 (M(z r +1 ; ν1 , ν2 ); pr +1 ) ≤ M(z r +1 + αr +1 pr +1 ; ν1 , ν2 ) ≤ M(z r +1 ; ν1 , ν2 ) + c2 αr +1 (M(z r +1 ; ν1 , ν2 ); pr +1 ) Step 5: Execute z r +1 = z r + αr +1 pr +1 ; Output: The updated pair (x r +1 , u r +1 );

5.5 Numerical Results

149

magnitude of violation of the path constraint at node k, respectively. The values of h k (z r +1 ) and gk (z r +1 ) can be computed using the following equation: h k (z

r +1

)=

N  i=0

Dik xir +1 −

s f − s0 ( f (xkr +1 , u rk+1 ) 2

gk (z r +1 ) = R min − R(xkr +1 , u rk+1 )

(5.35)

(5.36)

The l1 merit function M(z r +1 ; ν1 , ν2 ) uses positive penalty factors ν1 and ν2 to penalize the overall violation of the nonconvex constraints as stated in Eqs. (5.35) and (5.36). Theorem 5.3 Now there are mainly two types of chance constraint approximation methods: convex approximations [34, 35] and nonconvex approximations [23]. Most nonconvex methods try to aggressively approximate the H function, which reduces conservatism and improves the optimality of solutions. However, using such techniques can compromise the convexity of the original problem, resulting in increased computational complexity. Therefore, in this work, we focused on convex approximations. It has been shown in [15] that among other convex approximation functions, the function tends to result in the least conservative bounds when solving standard uncertain control problems. Therefore, we use this function to handle the chance constraints present in the trajectory planning task.

5.5 Numerical Results 5.5.1 Unmanned Vehicle Trajectory Generation The proposed trajectory planning approach based on convex programming is first validated by conducting a case study for a long distance scenario where chance constraints (e.g., Eqs. (5.10a) and (5.10b)) are not taken into consideration. The initial and terminal boundary settings for this particular case are set as x0 = [500m, 100m, 300m, 15◦ , 240◦ ], and x f = [−100m, 400m, 0m, 15◦ , 45◦ ] respectively. In terms of geometric constraints, the numerical results were obtained under the condition of R min = 40m and γ ∈ [−15◦ , 20◦ ], μ1 , μ2 ∈ [−1◦ /s, 1◦ /s]. In Fig. 5.1, the state and control trajectories of the unmanned system are shown by using the nonlinear pseudospectral algorithm (NPS) [27] and the convexificationbased trajectory planning approach developed in this chapter. The NPS is executed using the second generation general purpose optimal control software (GPOPS-II) [36]. For the convexified optimization model, a state-of-the-art solver developed in [37] using the primal-dual interior point algorithm with a precision of  = 10−8 is employed to find the optimal solution. The trust region constraints for x and u are set as δx = [1000, 1000, 1000, π, 2π ]T and δu = [1, 1]T respectively. The stopping con-

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5 Fast Generation of Chance-Constrained Flight Trajectory …

Fig. 5.1 Optimized trajectories for the unmanned vehicle

ditions are set as x = [0.1, 0.1, 0.1, 0.5π , 1π ]T and u = [0.01, 0.01]T respectively. 180 180 To start the optimization process for the sequential convex optimization process, an initial guess trajectory must be provided. For the case study, a linear interpolation between the state boundary conditions is used and the resulting state sequence is applied as the initial state guess profile x 0 . The initial control profile u 0 is set as zero. By setting x r = x 0 and u r = u 0 , the optimization process described in Algorithm 5.1 can be started. In Fig. 5.1, the trajectory of the unmanned vehicle is shown from different perspectives. Fig. 5.1a displays the projection of the unmanned vehicle trajectory on the y-z plane, while Fig. 5.1b shows the flight path in 3-D plane. The heading angle and pitch angle profiles of the vehicle are illustrated in Fig. 5.1c and d respectively. The control signals used for the vehicle, such as μ1 and μ2 , can be seen in Fig. 5.1e and f, respectively. The results show that both approaches can successfully produce flight paths without violating any constraints. The state profiles produced by these two algorithms have some variations. These differences may be due to factors such as the linearization process for system dynamics and path constraints. Additionally, the state and control profiles generated by the proposed trajectory planning algorithm tend to be smoother when compared to the solutions generated by the NPS algorithm. This is particularly noticeable in the control profiles, where high-frequency oscillations can

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Table 5.1 Settings for different test cases Case Initial pose ( p x , p y , pz ) γ φ 1 2 3 4 5 6

(120, −30, 250) (380, 230, 200) (−80, 10, 250) (400, −250, 600) (−200, −200, 450) (−200, 200, 250)

–10 0 0 0 0 15

100 30 20 350 340 240

Final pose ( p x , p y , pz )

γ

φ

(220, 150, 100) (280, 150, 30) (50, 70, 0) (600, −150, 300) (−300, −80, 100) (500, 800, 0)

–10 0 0 0 0 15

300 200 240 150 100 45

be seen in the NPS solutions, whereas the proposed approach generates more stable and achievable control evolution profiles, making it more suitable for real-world applications. The results also include details on the optimality and the computational performance of the algorithm. The optimal path length achieved by the NPS approach was 1203.32m(s N P S = 1203.32m), while the proposed approach achieved a path length of 1205.33m(s pr op = 1205.33m). However, the time required to generate a solution using the NPS is 6.37 s, which is significantly longer than the time required by the proposed method (0.77 s). Additionally, the proposed algorithm is able to converge successfully, guaranteeing the global optimality of its solution. In contrast, the solution obtained using the NPS can only be considered a local or near-optimal solution.

5.5.2 Comparative Case Study: Without Chance Constrains In this subsection, we conducted a series of comparative case studies without taking into account chance constraints. The parameters for the different test cases, including the initial and final positions of the unmanned vehicle, are listed in Table 5.1. These test cases were originally designed in [13]. To ensure fairness, we re-run these test cases using the proposed algorithm. It is worth noting that the selected test cases cover both short-distance and long-distance flying scenarios. In particular, the first three test cases tend to produce shorter flight paths, while the last three tests tend to result in relatively longer flight paths. The proposed approach is evaluated by comparing it to other established techniques found in literature, such as the NPS algorithm from [27], a waypoint-based algorithm examined in [24], and the multiple shooting-based (MS) method presented in [26]. The results, including the length of the trajectories s and the execution times t p for various flying scenarios, are summarized in Table 5.2. From Table 5.2, it can be seen that the optimal trajectory length values obtained by the proposed method and the NPS method are similar. The proposed fast trajectory

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5 Fast Generation of Chance-Constrained Flight Trajectory …

Table 5.2 Comparative results for different approaches Case no. Proposed NPS [27] Babaei et al. [24] s (m) t p (s) s (m) t p (s) s (m) t p (s) 1 2 3 4 5 6

580.75 668.19 977.18 1168.61 1360.80 1167.90

0.38 0.58 0.42 0.53 0.61 1.02

582.73 675.18 978.52 1168.61 1363.80 1169.95

5.43 5.85 5.64 5.72 6.31 6.07

750.82 875.10 1200.63 1375.14 1667.44 1401.84

6.29 7.45 7.27 6.69 6.83 6.76

MS-based [26] s (m) t p (min) 583.48 670.37 979.91 1169.74 1370.23 1168.83

1.11 1.08 1.23 1.00 1.47 2.05

planning method generally produces more optimal flight paths than the approach of Babaei et al. [24] and the MS-based method [26]. However, the execution time required by the strategy developed in this chapter is significantly shorter than its counterparts. It should be noted that in previous studies, such as [13, 14], the authors suggested a geometric approach for creating flight paths for unmanned vehicles. The approach was shown to produce feasible trajectories and had faster average execution time compared to the method presented in this study. However, it had a drawback of potentially having multiple disturbance points in the system state profiles. Additionally, experiments showed that the method presented in this chapter resulted in more optimal flight paths for some mission cases compared to the previously developed method.

5.5.3 Chance-Constrained Unmanned Vehicle Trajectory Generation In the analysis, the effect of control chance constraints on the optimal flight trajectory of an unmanned vehicle is studied. The equations in Eq. (5.10) assume that ξμ1 and ξμ2 follow an exponential distribution, represented by the probability density function f (x; λ) = λe−λx , x ≥ 0; f (x; λ) = −λe−λx , x < 0, with a rate parameter of λ = 70. Additionally, the risk parameter is set at 5% for both 1 − μ1 and 1 − μ2 . The fast trajectory generation approach is utilized to tackle the chance-constrained problem by incorporating the technique described in Sect. 4.4. The probabilistic constraints are transformed by selecting a relatively large sample size (e.g., N = 2 × 105 ). The effectiveness of this framework is demonstrated by conducting a short distance mission case (e.g., case 1 in Table 5.2). The optimized state and control profiles are shown in Fig. 5.2, where the blue line represents the solution considering chance constraints and the red line represents the solution without considering Eq. (5.10).

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153

Fig. 5.2 Optimal chance-constrained trajectories for the unmanned vehicle

The results show that the maximum violation rates for Eq. (5.10) are all less than 5%, indicating the reliability of the convex chance constraint approximation method. Additionally, Fig. 5.2 illustrates that the inclusion of control chance constraints leads to a slight increase in the path length. This is because, as seen in Fig. 5.2e and f, the control variables are not able to reach their maximum or minimum permissible values at certain times. Due to this lack of control, the unmanned vehicle may need to take a longer flight to reach the desired final position.

5.5.4 Comparative Case Studies: With Control Chance Constrains In this section, a comparison of case studies is conducted to evaluate the performance of the proposed trajectory generation approach that includes chance constraints. The standard trajectory design formula (Eq. (5.21)) is combined with three different methods for approximating chance constraints: the nonconvex chance constraint approximation (NCCA) method from [23] (with an accuracy index of  = 10−8 ), the

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Table 5.3 Chance-constrained results for different approaches Case no. Proposed NCCA [23] BM [34] s (m) t p (s) s (m) t p (s) s (m) t p (s) 1 2 3 4 5 6

582.62 669.64 978.06 1169.83 1362.75 1174.74

4.23 4.75 3.46 2.62 4.13 6.15

581.72 670.11 978.03 1169.65 1366.77 1181.58

42.23 44.65 57.92 47.78 53.72 56.63

591.25 673.24 981.58 1172.11 1368.85 1202.29

17.14 21.33 15.35 12.27 22.34 31.61

KF [35] s (m) t p (s) 610.36 679.24 987.79 1177.37 1375.24 1226.56

37.44 41.27 36.55 33.38 35.89 51.69

exponential-function based method (also known as the Bernstein method (BM)) from [34], and the kinship function-based (KF) approach from [35]. These three methods all involve approximating probabilistic constraints. The test cases listed in Table 5.1 were repeated using the four different chance-constrained trajectory planning methods. The results for path length and execution time are shown in Table 5.3. First, in comparison to the nonconvex chance constraint approximation (NCCA) method proposed in [23], the method developed in this chapter converges to a comparable objective with a faster convergence speed. The NCCA method in [23] aggressively aims to approximate the H function, which reduces conservatism and improves solution optimality. As a result, for some test cases, the NCCA method can produce slightly better solutions than the proposed algorithm. However, this approach sacrifices convexity of the original problem, resulting in a significant increase in computational burden for the optimization process. A potential solution is to relax the optimization tolerance value, which may decrease the accuracy of the obtained solution. Then, the results of the comparison between different convex approximationbased methods are analyzed. Table 5.3 shows that the proposed fast chanceconstrained trajectory generation strategy in this chapter produces a more optimal flight path in a shorter amount of time for various flight scenarios compared to the BM and KF methods. This suggests that the proposed convex chance constraint approximation method is less conservative than the ones reported in [34, 35], and is more suitable for addressing the unmanned vehicle trajectory design task. A convex approximation approach that is too conservative can result in a small feasible set, making it harder to find the optimal solution and degrading the computational performance of the algorithm.

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155

5.5.5 Comparative Case Studies: With Control and Obstacle Chance Constrains In this section, we examine the effect of including probabilistic collision avoidance constraints on the optimal trajectory of an unmanned vehicle. We re-run two flight scenarios, one short-range and one long-range, as outlined in Table 5.1 (cases 2 and 4) while taking into account the presence of uncertain obstacles in the environment. Specifically, we incorporate constraints on the x-y plane that represent uncertain nofly zones, with the form p y = amn px + bmn + ξmn . In the short-range flight case 2, we consider two obstacles with the following specifications: ⎧ a11 ⎪ ⎪ ⎨ a21 O1 ⎪ a31 ⎪ ⎩ a41 ⎧ ⎪ ⎪ a12 ⎨ a22 2 O a32 ⎪ ⎪ ⎩ a42

= 0.5 = 0.05 = −4.7 = −3.2

= 0.1 = 10 = −0.3 = 10

b11 b21 b31 b41

b12 b22 b32 b42

= 140 = 330 = 1650 = 1433

= 280 = −3620 = 505 = −4187

For flight case 4, two obstacles are considered: ⎧ ⎨ a13 = −13 b13 = 7590 b23 = −1150 O 3 a23 = 1 ⎩ a33 = −0.4 b33 = −134 ⎧ a14 ⎪ ⎪ ⎨ a24 O4 ⎪ a34 ⎪ ⎩ a44

= −1 =1 =1 = −1

b14 b24 b34 b44

= 620 = −970 = −1150 = 460

The uncertain parameter ξmn is assumed to follow a Gaussian distribution of N (0, 1.5). The maximum allowable violation probability is set to 0.1. Unlike the approach used in earlier sections, the optimized solutions listed in Table 5.2 (i.e., solutions obtained without taking into account probabilistic control and obstacle avoidance constraints) are chosen as the starting point for the iterative optimization process. It’s worth noting that a thorough examination of the impact of various initial guess generation methods will be presented in the following subsection. By solving the equation given in Eq. (5.33), the chance-constrained solutions are then determined. In Figs. 5.3 and 5.4, the changes in the objective function value during the sequential optimization process for the two flight cases are shown. Additionally, Figs. 5.5 and 5.6 further demonstrate the convergence history in relation to the merit func-

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5 Fast Generation of Chance-Constrained Flight Trajectory … 668 667.9

Objective value (m)

667.8 667.7 667.6 667.5 667.4 667.3 667.2 1

3

5

7

9

11

13

15

Iteration

Fig. 5.3 Convergence history of the objective function: case 2 with obstacles 1171.4

1171.2

Objective value (m)

1171

1170.8

1170.6

1170.4

1170.2

1170 1

3

5

7

9

11

13

15

Iteration

Fig. 5.4 Convergence history of the objective function: case 4 with obstacles

tion value. It is important to note that the objective value can indicate the solution’s optimality, while the merit function value measures the constraint violation. From Figs. 5.3 and 5.4, it can be seen that for the given cases, the objective function reaches a stable value and remains at that level until the specified tolerance is met (for example, both cases end after 15 iterations). However, it is interesting to note

5.5 Numerical Results

157

25

Evolution trajectory of the merit function

Merit function value

20

15

10

5

0

-5 1

3

5

7

9

11

13

15

Iteration

Fig. 5.5 Convergence history of the merit function: case 2 with obstacles 20

Evolution trajectory of the merit function

18 16

Merit function value

14 12 10 8 6 4 2 0 -2 1

3

5

7

9

11

13

15

Iteration

Fig. 5.6 Convergence history of the merit function: case 4 with obstacles

that the converged solutions do not have the minimum objective values. Specifically, the objective value does not always decrease and there are some intermediate iterations where the objective value is higher or lower than the converged solution.

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5 Fast Generation of Chance-Constrained Flight Trajectory … 400 Initial guess trajectory Final converged trajectory

350

Y (m)

300

250

200

150

100 250

300

350

400

450

500

X (m)

Fig. 5.7 Convergence history of the flight trajectory: case 2 with obstacles -100 Initial guess trajectory Final converged trajectory

-150 -200 -250

Y (m)

-300 -350 -400 -450 -500 -550 -600 300

400

500

600

700

800

900

1000

X (m)

Fig. 5.8 Convergence history of the flight trajectory: case 4 with obstacles

This is because, during the line search process outlined in Algorithm 5.2, an updated solution pair may be obtained that decreases the merit function value, but does not necessarily decrease the objective value. This is evident in the merit function trajectories shown in Figs. 5.5 and 5.6, which have a consistently decreasing pattern for the two cases. However, there is no guarantee that an updated solution pair will

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159

always result in a decrease or increase in the objective value. For example, in some intermediate iterations, the solution’s optimality may be compromised in order to make progress in terms of the merit function, resulting in oscillations in the objective value’s evolution history. Figures 5.7 and 5.8 show the evolution of the flight trajectory (projected onto the xy plane) during the optimization process. The initial trajectory is indicated by a black dashed line, while the flight trajectory at different optimization iterations is indicated by solid lines that change color from dark blue to red. As can be seen from these figures, the flight trajectories for these two cases tend to become more aggressive as the number of iterations increases. This means the vehicle tends to approach uncertain obstacles in order to achieve a better objective value. Additionally, based on our observations, the trajectories become very close after 5 iterations for case 2 and after 11 iterations for case 4. This can also be observed by looking at the evolution of the objective function and merit function values.

5.5.6 Sensitivity Analysis In this subsection, we first conduct a sensitivity study to examine the effect of various initial guess generation methods on the convergence performance of the proposed approach. The methods chosen for examination are: • Method A: The initial trajectory of the vehicle’s state, x 0 , is obtained by using the dynamics equation, Eq. (5.1), to move the vehicle from its initial boundary value, x0 , using a specified initial control trajectory, u 0 . In this test, u 0 is set to zero. • Method B: The initial state trajectory, x 0 , is obtained by interpolating linearly between the boundary conditions for the state, such as x0 and x f , while the initial control trajectory, u 0 , is set to zero. • Method C: The initial state and control guess trajectories are chosen to be the solutions of the convex optimization problem that do not take into account the probabilistic control and collision avoidance constraints. Figures 5.9 and 5.10 show the evolution of the merit function for mission cases 2 and 4, respectively. It is clear from the results that for both mission cases, using the third initial guess generation method leads to faster convergence in the sequential optimization process. In fact, as shown in Fig. 5.10, using the first initial guess generation method results in a convergence problem for mission case 4, where the current solution does not meet the prescribed convergence condition even after reaching the maximum allowable iteration number of 50. These comparative results demonstrate that the convergence performance of the proposed approach is sensitive to the initial guess trajectories. Therefore, it is recommended to use Method C to generate initial guess trajectories when applying the optimization model given by Eq. (5.33). In order to examine the effect of the maximum allowable violation probability o for the probabilistic obstacle avoidance constraints on the optimized results, different

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5 Fast Generation of Chance-Constrained Flight Trajectory … 400

Evolution trajectory via Method A Evolution trajectory via Method B Evolution trajectory via Method C

350

Merit function value

300 250 200 150 100 50 0 -50 0

5

10

15

20

25

30

35

Iteration

Fig. 5.9 Merit function histories for case 2 using different guess trajectories

Evolution trajectory via Method A Evolution trajectory via Method B Evolution trajectory via Method C

500

Merit function value

400

300

200

100

0 0

5

10

15

20

25

30

35

40

45

Iteration

Fig. 5.10 Merit function histories for case 4 using different guess trajectories

50

5.5 Numerical Results

161

400

350

Y (m)

300

250

200

150

100 200

Allowable probability 1 Allowable probability 2 Allowable probability 3

250

300

350

400

450

500

550

600

X (m)

Fig. 5.11 Optimal chance-constrained trajectories: case 2 with obstacles Table 5.4 Results with uncertain obstacles Allowable Case 2 results probability s (m) t p (s) Level 1 Level 2 Level 3

674.23 675.35 675.82

5.81 6.98 8.33

Case 4 results s (m)

t p (s)

1170.22 1191.53 1192.15

5.25 6.38 7.47

values of o were assigned for testing. For instance, o was assigned to three levels (Level 1 = 0.1, Level 2 = 0.05, Level 3 = 0.01) to test the performance of the proposed convex probabilistic collision avoidance constraint approximation strategy. Figure 5.11 shows the planned flight trajectory for mission case 2 on the x-y plane with different allowable constraint violation probabilities, and Fig. 5.12 presents the simulated trajectory results for mission case 4. As can be seen in Figs. 5.11 and 5.12, as the maximum allowable constraint violation probability decreases, the optimal chance-constrained flight trajectory tends to become more cautious. This is particularly evident in the result of mission case 4, where the unmanned vehicle switches its maneuver from going through the two obstacles to executing a much safer but longer flight outside of the obstacles. Detailed results, such as the path length and execution time, are presented in Table 5.4. As can be seen from Table 5.4, the execution time required by the strategy developed in this chapter may slightly increase as the probabilistic collision avoidance constraint becomes more strict. However, all the trials still converge in a short amount of

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5 Fast Generation of Chance-Constrained Flight Trajectory … -100 -150

Allowable probability 1 Allowable probability 2 Allowable probability 3

-200 -250

Y (m)

-300 -350 -400 -450 -500 -550 -600 300

400

500

600

700

800

900

1000

X (m)

Fig. 5.12 Optimal chance-constrained trajectories: case 4 with obstacles

time. These results further confirm the effectiveness of the proposed mixed-integer convex chance-constrained optimization model, given by Eq. (5.33), for planning the flight trajectory of the unmanned vehicle while taking into account probabilistic control and collision avoidance constraints.

5.6 Conclusion In this chapter, the nonlinear trajectory optimization problem becomes more complex, a fast algorithm for generating unmanned vehicle trajectories is presented, incorporating convex optimization and approximations of probabilistic constraints. The optimization model is deterministic and convex, even with probabilistic control and obstacle avoidance constraints. The proposed algorithm is found to have two main advantages over other methods: smooth calculated system state and control profiles, and improved computational performance. These advantages have been validated through case studies, and the approach is believed to be useful for those working on chance-constrained optimization and unmanned vehicle trajectory planning tasks.

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19. Yang H, Bai X, Baoyin H (2017) Rapid generation of time-optimal trajectories for asteroid landing via convex optimization. J Guid Control Dyn 40(3):628–641. https://doi.org/10.2514/ 1.G002170 20. Szmuk M, Pascucci CA, Acikmese B (2018) Real-time quad-rotor path planning for mobile obstacle avoidance using convex optimization. In: 2018 IEEE/RSJ international conference on intelligent robots and systems (IROS), pp 1–9. https://doi.org/10.1109/IROS.2018.8594351 21. Wang Z, McDonald ST (2020) Convex relaxation for optimal rendezvous of unmanned aerial and ground vehicles. Aerosp Sci Technol 99:105756. https://doi.org/10.1016/j.ast.2020. 105756 22. Boyd S, Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press 23. Chai R, Savvaris A, Tsourdos A, Chai S, Xia Y, Wang S (2019) Solving trajectory optimization problems in the presence of probabilistic constraints. IEEE Trans Cybern 1–14 (2019). https:// doi.org/10.1109/TCYB.2019.2895305 24. Babaei AR, Mortazavi M (2010) Three-dimensional curvature-constrained trajectory planning based on in-flight waypoints. J Aircr 47(4):1391–1398. https://doi.org/10.2514/1.47711 25. Garg D, Patterson M, Hager WW, Rao AV, Benson DA, Huntington GT (2010) A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46(11):1843–1851. https://doi.org/10.1016/j.automatica.2010.06.048 26. Hota S, Ghose D (2014) Optimal trajectory planning for path convergence in three-dimensional space. Proc Inst Mech Eng, Part G: J Aerospace Eng 228(5):766–780. https://doi.org/10.1177/ 0954410013479714 27. Guo T, Li J, Baoyin H, Jiang F (2013) Pseudospectral methods for trajectory optimization with interior point constraints: verification and applications. IEEE Trans Aerosp Electron Syst 49(3):2005–2017. https://doi.org/10.1109/TAES.2013.6558034 28. Sagliano M (2017) Pseudospectral convex optimization for powered descent and landing. J Guid Control Dyn 41(2):320–334. https://doi.org/10.2514/1.G002818 29. Sagliano M (2019) Generalized hp pseudospectral-convex programming for powered descent and landing. J Guid Control Dyn 42(7):1562–1570. https://doi.org/10.2514/1.G003731 30. Wang Z (2019) Optimal trajectories and normal load analysis of hypersonic glide vehicles via convex optimization. Aerosp Sci Technol 87:357–368. https://doi.org/10.1016/j.ast.2019.03. 002 31. Mao Y, Szmuk M, Acikmese B (2016) Successive convexification of non-convex optimal control problems and its convergence properties. In: 2016 IEEE 55th conference on decision and control (CDC), pp 3636–3641. https://doi.org/10.1109/CDC.2016.7798816 32. Marantos P, Bechlioulis CP, Kyriakopoulos KJ (2017) Robust trajectory tracking control for small-scale unmanned helicopters with model uncertainties. IEEE Trans Control Syst Technol 25(6):2010–2021. https://doi.org/10.1109/TCST.2016.2642160 33. Kronqvist J, Bernal DE, Lundell A, Grossmann IE (2019) A review and comparison of solvers for convex minlp. Optim Eng 20(2):397–455. https://doi.org/10.1007/s11081-018-9411-8 34. Nemirovski A, Shapiro A (2006) Convex approximations of chance constrained programs. SIAM J Optim 17(4):969–996. https://doi.org/10.1137/050622328 35. Feng C, Dabbene F, Lagoa CM (2011) A kinship function approach to robust and probabilistic optimization under polynomial uncertainty. IEEE Trans Autom Control 56(7):1509–1523. https://doi.org/10.1109/TAC.2010.2099734 36. Patterson MA, Rao AV (2014) Gpops-ii: a matlab software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming. ACM Trans Math Softw 41(1):1–37. https://doi.org/10.1145/2558904 37. Domahidi A, Chu E, Boyd S (2013) Ecos: an socp solver for embedded systems. In: 2013 European control conference (ECC), pp 3071–3076. https://doi.org/10.23919/ECC.2013.6669541

Part II

Advanced Guidance and Control Methods for Aerospace Vehicles

Chapter 6

Review of Advanced Guidance and Control Methods

Abstract Over the past few decades, how to design a sophisticated guidance and control the (G&C) system for space and aerospace vehicles has been widely researched, which has increasingly drawn attention from all over the world and will continue to do so. As is known to all, there are various model uncertainties and environmental disturbances in G&C system. Therefore, robust and stochastic controlbased methods have unsurprisingly played a key role in the system design. Furthermore, a large number of researchers have proposed and successfully established several algorithms which can effectively guide and steer the motion of space/aerospace vehicles. In addition to these stability theory-focused techniques, a major trend in recent years has been the development of optimisation theory- and artificial intelligence (AI)-based controllers for space and aeronautical vehicles in an effort to address the demand for greater system performance. According to related studies, in terms of practical application, these recently established strategies are more advantageous, and they may be suitable for the onboard decision-making system as well. In this chapter, the latest algorithms were analyzed systemically. The chapter begins with a succinct summary of issues with space/aerospace vehicle guidance and control. The discussion of a wide range of scholarly papers pertaining to G&C approaches based on stability theory follows the summary, which examines and explains the potential inherent problems. Then, a summary of different recently proposed optimisation theory-based methods is provided. These methods are expected to generate the optimal guidance and control commands, such as dynamic programming-based, model predictive control-based methods, and other modified versions. This chapter also covered the discussion of their main benefits and inherent drawbacks, which are important in terms of their applications. We also noticed that the combination of AI techniques and the optimal control of vehicle systems has been a new research direction. Therefore, in the following part, we gave a special focus on the recent effort to discuss the feasibility of its application. The key points of the analysis demonstrate how these AI models may be useful for solving space/aerospace vehicle control issues. Finally, a list of potential future study subjects is provided, along with a few concerns for practical implementation.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_6

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6.1 Introduction 6.1.1 Background More than ever, it is clear to witness how space and aerospace activities affect our modern world. Numerous outstanding accomplishments in space/aerospace flight missions have been made throughout the previous few decades, including deep space exploration [1], satellite surveillance [2], interplanetary travel [3], Mars landings [4], space debris removal [5], atmospheric re-entry or hopping [6], missile-target engagement [7], spacecraft rendezvous and docking [8], spacecraft or unmanned aerial vehicle (UAV) swarms [9], and multi-spacecraft formation flying [10]. A graphic representation of a few such examples is shown in Fig. 6.1. The development of sophisticated guidance and control (G&C) methods has significantly contributed to and is particularly significant for these successful projects [11]. A promising guidance and control system may successfully generate instructions and carry out operations for a space or aeronautical vehicle, allowing the craft to reliably complete the mission objectives [12]. Two well-known tendencies in the advancement of sophisticated guidance and control methods can be found by looking into the literature. The creation of resilient or stochastic control-based methods was the first significant advancement. Numerous successful robust and stochastic control algorithms have been developed in recent

Fig. 6.1 Typical examples: a Interplanetary travel [3]; b Mars landings [4]; c Space debris removal [5]; d Atmospheric re-entry or hopping [6]; e Missile-target engagement [7]; f Spacecraft rendezvous and docking [8]

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years to direct and control the movements of spacecraft and aircraft in a range of missions [13–15]. A fundamental design requirement should always be to increase system robustness and fault tolerance. This is mostly due to the fact that in actual reality, numerous types of model uncertainty, environmental disturbances, sensor measurement noise, and actuator malfunctions are usually present. The entire system is likely to become unstable and the flight mission will fail if particular remedies with regard to these adverse consequences are not carried out. Since stability theory is the primary source of the majority of robust and stochastic control algorithms, it makes sense to include these methods under stability theory-based methods. The second popular approach is to use optimisation theory-based methods to guide and steer the motion of space/aerospace vehicles in addition to the former method [16, 17]. This kind of approach, which has gained popularity recently, can be seen as the direct result of fusing the ideas of control and optimization. It is important to note that this kind of combination may have benefits for the development of algorithms for guidance and control. The following traits are the main indicators of this method: • Improved system performance: A given performance index can be partially optimized throughout the control process as the control issue is reformulated for an optimisation task. This results in improved system performance. The requirement for higher system performance can be satisfied by this means. • Enhanced algorithm flexibility and functionality: Optimisation theory-based control methods can be easily combined with tools such as artificial neural networks (ANNs) [18], adaptive methods [19], and disturbance observers [20] Optimisation theory-based control methods can be easily combined with tools like artificial neural networks, adaptive methods, and disturbance observers, improving the controller’s capacity to recognize system uncertainties and reject disturbances. This leads to increased algorithm flexibility and functionality. Additionally, variable and process restrictions can also be managed by this kind of algorithm. Several efficient optimisation theory-based control methods, including (heuristic) dynamic programming-based methods, model predictive control-based methods, and other improved versions, have recently been published in the literature. Furthermore, a significant amount of research has been put into investigating the circumstances that might offer theoretical assurances regarding the stability of the system as well as the viability of the optimization process [21, 22]. In this way, the robustness of these methods for space/aerospace vehicle guidance and control that is based on optimisation theory is further enhanced. A recent interest in using ANNs or deep neural networks (DNNs) to achieve the online guiding and control of space/aerospace vehicles can be found in the literature, thanks to the impressive advancements in machine learning (ML), artificial intelligence (AI), and deep learning technologies. These models are able to maintain the evident benefits of optimisation theory-based control methods while also keeping a good computational overhead in real time, which is a key benefit of utilizing them [23, 24]. This kind of algorithm specifically attempts to train some neural network models that can produce the best guidance and control commands, constituting the onboard decision-making system for various mission profiles. Numbers of network

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models have been developed and studied over the past ten years, and successful investigations of these methods can be found in a variety of aerospace- and spacerelated applications, such as low-thrust orbital transfer [25], spacecraft atmospheric re-entry [26], and planetary pinpoint landing [27]. These applications all have a similar design concept, despite variations in the applied network models and training methods. Consequently, for the sake of clarity, these types of methods are referred to as AI-based G&C methods throughout this study.

6.1.2 Motivation Maybe there isn’t a single method that can solve all space/aerospace vehicle G&C issues with performance that is generally acceptable. However, creating accurate guiding and control algorithms is a rapidly evolving domain, and each method has certain benefits and drawbacks in terms of functionality, efficiency, adaptability, and complexity. This shows that choosing a suitable G&C algorithm typically depends on the problem. This article’s main goal is to give a thorough analysis of the most recent research findings as well as potential future research avenues in this area. In addition, it’s crucial to comprehend the distinguishing characteristics of various approaches and to be aware of any potential difficulties and difficulties in employing these algorithms. More precisely, our intentions are as follows: • sort the existing approaches that have been published recently into categories based on their design philosophies; • list the benefits and drawbacks of various G&C approaches; • analyze the present difficulties in applying various methods for aerospace and spaceflight applications; • present some principles for the creation of guidance and control methods that are based on AI, optimisation theory, and stability theory.

6.1.3 Organisation of the Article The structure of this review article is as follows: A basic summary of space/aerospace vehicle guidance and control systems is given in Sect. 6.2. The extensive collection of scholarly works investigating guiding and control techniques based on stability theory is described in Sect. 6.3. In Sect. 6.4, a variety of newly established optimisation theory-based guiding and control methods are discussed. These methods include dynamic programming-based methods, model predictive control-based methods, and other improved variants. Important elements like the primary benefits and inherent difficulties of implementing these approaches are also thoroughly covered. After that, Sect. 6.5 focuses especially on current initiatives to investigate potential applications

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of AI-oriented methodologies in relation to the G&C of various space/aerospace vehicles. At last, Section 6 presents some final observations and potential research areas. We are aware that it can be challenging to review, examine, and analyse all the significant works in this domain. Therefore, we focus mostly on works that have been released since the start of this century. Additionally, current works published within the last ten years are given priority.

6.2 Types of Guidance and Control Systems The separation principle may have played a significant role in the early development of the G&C systems [28, 29]. More specifically, the conceptual design in Fig. 6.2a illustrates the fundamental concept of a so-called separate guidance and control (SGC) system for space/aerospace vehicles.

Fig. 6.2 Typical space/aerospace vehicle guidance and control systems

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The G&C systems are divided into two loops in an SGC framework, enabling independent design of both loops. Actually, engineers and researchers still prefer to separate the G&C issues, and a variety of literature contains research reports that claim to have contributed to the development of SGC [29, 30]. However, according to some investigations [31–33], A SGC scheme may experience a number of problems, including high-frequency oscillations, expensive parameter tweaking, and large time lags. Since these problems are typically undesirable for practical applications, integrated guidance and control (IGC) systems have been developed as a result [34–36] and guidance and control (PIGC) systems are partially integrated [31–33].

6.2.1 Integrated Guidance and Control System Researchers have put a lot of effort into developing IGC systems over the past ten years, with applications ranging from guided missiles to autonomous space vehicles [34–36]. In contrast to the SGC, the IGC seeks to combine the guidance and control loops (for instance, as seen in Fig. 6.2b). One significant benefit of the IGC design is that an integrated system of this kind can fully leverage the vast data of six-degree-offreedom (6-DOF) vehicle dynamics, partially solving the time lag issue. We briefly examine some advancements in this area and highlight certain elements of existing IGC designs as more published studies on developing IGC systems become available in the body of literature. In the work presented by Luo et al. [34], in order to conduct an air-to-air autonomous attack mission, an IGC system was built. The authors initially introduced the guided missile and the nonlinear dynamics of unmanned combat aerial vehicles. The main controller was then suggested using a robust control algorithm and a parameter adaptive strategy. According to the authors’ simulation results, there are certain benefits to using the proposed IGC as opposed to the conventional SGC. For instance, the suggested IGC can achieve a high attack precision while saving roughly 25% of the control effort. Similarly, in [35], for the issue of 3-D hypersonic vehicle interception, a quick and reliable IGC system was suggested. In order to speed up the system response, a compound control law was developed, including signal compensation. Numerous comparison research and numerical simulations were conducted and presented, based on the which it was confirmed that the suggested IGC system can concurrently improve the interceptor’s robustness and response capacity.

6.2.2 Partially Integrated Guidance and Control System In recent years, a new concept about advanced G&C systems was introduced, which is known as PIGC [31–33].Combining the strengths of SGC and IGC, this approach seeks to overcome the drawbacks of one-loop IGC systems, such as their inability to

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fully utilise time scale separation and their relatively high cost of parameter adjustment [31]. The PIGC system formulates the issue using an inner-outer loop structure, much like the SGC system does. The 6-DOF dynamics will be partly manipulated by each loop. Figure 6.2c, which illustrates the PIGC in more detail, makes it clear that the body rate commands will be made and sent to the inner loop from the outer loop. The 6-DOF dynamics’ nonlinear body rate equations will then be used by the inner loop to track the body rate command profiles. The works presented by Padhi et al. [31, 32] are particularly significant in the context of PIGC. Specifically, in [31], a PIGC scheme for a missile-target engagement mission was proposed by the authors, while in the work reported in [32], a PIGC scheme was successfully developed for a UAV formation flying mission. The advantages of the IGC and the traditional SGC were successfully inherited by the produced PIGC, according to the presented simulation findings. It was demonstrated that PIGC can lead to smaller miss distance values than IGC in the missile-target engagement operation. In terms of the UAV formation flying mission, the PIGC has the ability to quickly construct and maintain a large number of UAVs in a pre-specified formation. It was also demonstrated that problems brought on by the separation of the translational and rotational vehicle dynamics’ time scales were satisfactorily resolved. Notably, interesting branches including cost-effective IGC/PIGC systems, adaptive IGC/PIGC systems, and networked IGC/PIGC systems are beginning to appear in the literature [37]. It is outside the purview of this article to classify and describe these G&C systems in detail. For such an exhaustive review, readers who are interested are referred to [37]. As an alternative, we concentrate on reviewing the most recent advancements in G&C algorithms that are useful and accessible for various space/aerospace vehicle G&C systems in this study.

6.3 Review of Stability Theory-Based G&C Methods The majority of recently established G&C methods for aerospace/spacecraft are built on the foundation of stability theory. This can be due to the growing theoretical progress made in various disciplines of control theory since the turn of the century, including robust control, adaptive control, stochastic system theory, and data-driven control. This kind of method still has theoretical and practical difficulties, which has sparked additional studies on the subject.

6.3.1 Design and Applications of Robust G&C Algorithms When deterministic G&C policies is applied, unsurprisingly, the occurrence of numerous model uncertainties or environmental disruptions may result in the failing of achieving the anticipated control performance. Therefore, one of the main goals in the design of reliable G&C systems is to deal with or reject these detrimental

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impacts [13, 14]. The motion of various space/aerospace vehicles has been successfully guided and controlled by a number of robust G&C algorithms that have recently been disclosed in the literature. A few common techniques in the design of G&C systems are given in the ensuing subsections. These techniques consist of backstepping-based schemes, dynamic inversion-based schemes, fractional-order control-based schemes, finite-time sliding mode control (SMC) theory-based schemes, and other reliable ones.

6.3.1.1

Finite-Time SMC Theory-Based Scheme

Among various methods, the variable structure control method based on SMC has gained much popularity in recent years [38]. It has been widely employed for space/aerospace vehicle G&C systems due to its insensitivity to disturbances and uncertainties [38, 39]. It is worth noticing that the asymptotic stability and convergence for the linear hyperplane-based SMC designs described in [38, 39] may only apply to the sliding manifold. To put it another way, the system errors might not reach an equilibrium position in a finite time. Therefore, this problem has stimulated researchers to further develop the G&C methods based on finite-time SMC theory [40–42]. The terminal sliding mode control (TSMC) strategy is a variant of traditional SMC. It is noteworthy that for the advantage of achieving finite-time stability, TSMCbased algorithms have been the subject of intensive research recently, particularly in the area of space/aerospace vehicle G&C systems. For the Mars atmospheric entry challenge, for instance, the authors of [40] developed a TSMC algorithm for the steering commands. They developed a hybrid structure by combining TSMC with a second-order differentiator that calculates the overall external disturbance. The control accuracy can be higher by using the proposed idea, which is proven by the simulation results. The control accuracy can be higher by using the proposed idea, which is proven by the simulation results. Additionally, The tracking error between the desired references and the actual spacecraft state trajectories is solved by the newly developed approach, which can be steered to a small neighbourhood of zero within finite time. The singularity problem may affect the TSMC-based G&C algorithms, which was emphasized in [41, 42]. To address this issue, the authors of these two papers focused their efforts on designing non-singular TSMC (NTSMC) approaches. In particular, in [41], an NTSMC system was created to generate control commands for a small satellite in real time. More specifically, bounded external disturbances were taken into account when formulating the problem, and by implementing the authors’ design, the inherited singularity problem of the conventional TSMC approach was successfully avoided. Furthermore, a flexible spaceship attitude tracking problem was solved in [43] by using a non-singular fast TSMC (NFTSMC) control approach. Additionally, an adaptive strategy was put forth in order to estimate the boundary of the unidentified external disturbances. By combining the benefits of NTSMC and traditional SMC,

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the researchers achieved the goal, which is to speed up the algorithm convergence. Compared with the given simulation results, the convergence speed can be faster while the controlled system’s finite-time stability can still be ensured. The chattering problem, on the other hand, is a significant issue that must be properly taken into account in the design of SMC G&C laws. This phenomenon makes it difficult to realise the G&C system due to the fact that it can stimulate the high-frequency dynamic characteristics of the system. Numerous G&C algorithms based on finite-time SMC theory have been tested to improve and generalise standard SMC in order to successfully solve this issue [44–48]. For example, in [44], for the purpose of solving a rigid spaceship attitude tracking control problem, a higher-order SMC (HOSMC) method was suggested. The chattering phenomenon was successfully eliminated by raising the order of the sliding mode (for instance, by setting it higher than the relative degree of the spacecraft system). To bolster the main argument of this chapter, simulation studies were offered. The authors of [45] suggested a two-layer control system that included a TSMC manifold and a HOSMC to successfully synchronise the attitude of spacecraft in finite time while also somewhat lightening the communication load. By running a number of numerical simulations, it was possible to verify the validity of this hybrid design as well as the chatter-free performance. In addition to the HOSMC, the chattering issue can also be alleviated by some integral SMC (ISMC)-based guidance and control systems. For example, in [46], when it comes to actuator uncertainties, the key to tracking spaceship attitude is an integral TSMC (ITSMC) system, which is suggested by the authors. According to major theoretical findings and simulation tests, it is suggested that adding an integral term to the sliding surface design could reduce steady-state error and solve the chattering issue while simultaneously maintaining finite-time stability. In [47], By using a modified ISMC approach, a spacecraft attitude control system was obtained, which took actuator saturation and external disturbances into account. This method could be used to produce robust tracking performance and reduce trajectory tracking error to zero in a limited amount of time. In addition, the authors of [48] also promoted a composite spacecraft attitude stabilisation system that combines an ISMC controller and a disturbance observer. To demonstrate the efficacy and improved disturbance rejection performance of their idea, simulation examples were run.

6.3.1.2

Fractional-Order Control-Based Schemes

Designing fractional-order control (FOC)-based strategies for space/aerospace vehicle navigation and control issues has become increasingly popular in recent years [49, 50]. When it comes to adjusting the closed-loop response and fulfilling desired performance standards, methods of fractional order control play an important role by leveraging their flexibility. Such an approach typically uses fractional-order derivatives and integrals of the state for feedback, giving it more latitude to match the desired behavior of the controlled plant. It is worth noticing that several fractional-order

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control-based methods have been designed and successfully used [35, 49–51]. To stabilize the rigid spacecraft rotational dynamics, for instance, two fractional-order proportional integral derivative (FOPID) methods were suggested in [49]. These two suggested FOPID controllers differ from integer feedback control schemes in that they all apply the fractional derivative and integral feedback terms with adjustable fractional orders. The experimental simulations revealed that the fractional derivative and integral orders can be tuned to simultaneously decrease the settling time and control effort. A fractional-order control scheme was put up by the authors in [50] to stabilise a spacecraft attitude system. The main goal of this research is to create a fractionalorder integral sliding mode control (FOISMC) algorithm for the problem by combining the benefits of fractional feedback control and ISMC. By examining the given comparative results, it was possible to recognise the primary benefit of using FOISMC over pure ISMC. In fact, issues like integral saturation and a slow convergence rate could affect the ISMC scheme. The fundamental cause is the existence of integral action, which can impose a big impact on how well the control scheme works. Similarly, in [35], a FOISMC method was proposed by the authors and disturbances were removed by established reliable compensating signals. They also built a parallel control framework to carry out the sliding phases, which accelerated the system’s response time. It was demonstrated that this method might be used to design guidance and control systems for spacecraft. Additionally, a FOISMC scheme was proposed in [51], which took into account the difficulty of small satellite attitude control. The effectiveness and benefit of utilising their approach to produce improved steady-state performance were shown by theoretical findings and numerical simulations.

6.3.1.3

Dynamic Inversion-Based Scheme

For a number of space/aerospace vehicles, it has been widely acknowledged that nonlinear dynamic inversion (NDI)-based algorithms are effective guiding and control strategy, which often eliminates the nonlinearity of the system by converting the original nonlinear dynamics into a fully or partially linear counterpart, allowing for the use of conventional linear control techniques [52, 53]. Although there have been some attempts to use NDI methods for flight control [52, 53], a major issue is that accurate dynamic inversion may automatically suffer from a lack of resilience. In order to overcome this problem, researchers have worked hard, and a better version known as incremental NDI (INDI) control has been successfully proposed [54–57]. An INDI-based method was used by the writers of [55] to act as an aircraft’s trajectory tracking controller. This method took actuator errors and model uncertainties into consideration and dealt with it by gradually introducing control inputs. The proposed design’s ability to carry out the tracking mission in the presence of uncertainties and actuator defects was confirmed by simulated experiments. Additionally, comparison studies with the NDI method further supported the benefit of utilising INDI.

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In [56], INDI’s resilience and stability were rigorously examined utilising the Lyapunov method and nonlinear system perturbation theory. By doing numerous Monte-Carlo simulation tests on an aircraft command tracking task while taking into account model uncertainties and outside disruptions, it was shown that the fundamental theoretical conclusion was validated. In addition, In addition, a robust hovering controller for the quadcopters was built using a cascaded INDI scheme in [57] to serve as a robust hovering controller for the quadcopters. Based on the reported experimental findings, it was established that a PID controller can be significantly improved in terms of control accuracy and disturbance rejection performance compared to a traditional PID controller.

6.3.1.4

Backstepping-Based Scheme

Another well-liked method for developing reliable guidance and control algorithms is backstepping-based (BS) approaches [58, 59]. This kind of method typically consists of two steps. A virtual subsystem control signal is designed in the first step. The controller will then use this virtual input to construct the control command for the actual subsystem. Numerous published papers provide typical instances of designing or using backstepping-based guidance and control algorithms for diverse space/aerospace systems [59–61]. In order to solve the trajectory tracking issue for air-breathing hypersonic vehicles, Hu and Meng proposed [59] an adaptive BS control scheme. In their research, the controller design took input saturations and aerodynamic uncertainty into account. An effort was made to combine the benefits of dynamic surface control and traditional BS in order to prevent repetitive differentiations of the virtual control variables. Through the use of computer simulations, their proposal’s efficacy was verified. A brand-new BS-based guidance algorithm for the missile-target engagement system was put forth in [60]. The capability of the proposed BS guiding law to fulfill the seeker’s field-of-view constraints while still attaining the intended impact time is a key component. The suggested BS-based guidance method may successfully complete the interception mission at the specified impact time, according to the results of numerical simulations.

6.3.1.5

Other Robust Control Schemes

Additionally, there are other resilient control schemes that have also been proposed and used in the designing of spacecraft G&C systems [49, 62–64]. In [65], the fault tolerant control-based guidance scheme was proposed. and in [62], the H∞ -based guidance algorithm was developed. Both of them are relevant examples. Besides, in 2016, a nonlinear tracking controller and an attitude control strategy were also developed for spacecraft carrying heavy objects by Bandyopadhyay et al. [63]. One key aspect of the suggested design is the ability to guarantee both global exponential convergence to the reference attitude trajectory as well as bounded tracking errors

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in the presence of uncertainty. The feasibility of using this reliable control scheme in upcoming asteroid capture missions was verified by carrying out case studies. The construction of an ordinary differential equation (ODE) Cpartial differential equation (PDE) feedback controller for exact attitude trajectory tracking and slewing in the face of bounded disturbances was done in a manner similar to that in [64]. In this study, the unique equations of motion for the rotation of spacecraft with strainactuated solar arrays in one degree of freedom were established. To further illustrate the efficiency of the suggested design, thorough stability analyses and proofs were offered. However, the majority of these approaches are focused on conventional robust control theory, and numerous simulation experiments and analyses have demonstrated their effectiveness in controlling disturbances.

6.3.2 Design and Applications of Stochastic G&C Algorithms Deterministic systems are the main support with regard to the majority of the contributions previously stated. However, errors in the system state equations or input sequences may exist in practical space/aerospace vehicle systems, leading to stochastic systems [66]. The concept of a stochastic process must be introduced in these situations, which encourages the creation of stochastic guiding and control algorithms. Combining stochastic theory with optimal control to create stochastic optimal guidance and control algorithms is a common design approach for stochastic guidance and control systems. In the literature, a number of efforts have been put out to create or implement this kind of technique in a number of spaceflight missions. For instance, in the work of Kota et al. [67], to direct the manoeuvres of a spaceship while taking into account the probabilistic uncertainties in the system equations, a stochastic optimal control law was presented. Similar to Kota’s approach, a stochastic optimal guidance and control technique was used by Duta et al. [68] to carry out spaceship manoeuvres. In their research, the spread of uncertainty in the equations of the spacecraft system was completely taken into account, and a Monte Carlo simulations framework was developed to identify the types of manoeuvres and the stabilisation of uncertain systems. Uncertain space or aerospace vehicle systems can also be handled by stochastic sliding mode-based guidance and control algorithms. This approach, which integrates stochastic theory and SMC, has been effectively used in space missions. To direct control the direction of the motion of a space manipulator, for instance, the authors in [69] devised a stochastic SMC-based controller. They modeled the plant as a nonlinear stochastic Markovian jump system that took unknown time-varying delays into account. A fuzzy observer which can estimate the uncertain state was built to cope with the uncertain terms effectively, by which stochastic stability can be ensured. Moreover, by utilising stochastic NFTSMC theory, a terminal guidance law was developed in [70]. In this study, stochastic noise was taken into account, and the outcomes demonstrated the value of using a hybrid approach.

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It is important that considerable research has concentrated on how to set up finitetime or stochastic adaptive guidance and control algorithms [71, 72]. Stochastic theory and adaptive or finite-time control were combined into this kind of approach, and it has been demonstrated that they are capable of addressing the uncertainties present in aerospace vehicle systems. A stochastic adaptive IGC framework that can steer hypersonic missiles during exo-atmospheric and atmospheric flights was developed in Chen’s work, for instance [72]. This work, in contrast to previous findings, studied stochastic uncertainties in the system equations and developed a novel adaptive law to guarantee the stochastic stability of the system. Notably, the majority of the mentioned stochastic guiding and control algorithms were based on the concept of stability. In other words, an equilibrium point or a particular trajectory was used to determine the algorithm’s convergence. In recent years, an increasing number of researchers have focused on the control techniques using contraction-based incremental stability analysis [73–75]. All system trajectories of a dynamical system must converge to one another in order for incremental stability to exist, which is crucial for observer designs for stochastic systems [75] as well as synchronisation issues involving swarms of satellites and other spacecraft [73]. A typical example can be found in the work by Chung et al. [73], where a phase synchronization controller was suggested for a class of networked systems in the face of stochastic uncertainties. The controller was generated via the contraction analysis. Then, based on this controller,a large hive of spacecraft operating in low Earth orbit was reconfigured. An optimal feedback tracking controller was developed in their subsequent study [75] for a class of Itoˆ nonlinear stochastic systems. In this work, minimizing the upper bound of the steady-state tracking error was the objective function, and it was used to solve a convex optimisation problem to optimize the feedback gain as well as other controller parameters. An uncertain spacecraft attitude tracking control example was used to test the effectiveness of this system. The proposed design’s advantage over competing options was supported by simulation results.

6.3.3 Potential Issues and Challenges of Stability Theory-Based G&C Algorithms In the preceding subsections, we have discussed the different methods, based on which we can summarize in detail the primary benefit of each type of approach. Regarding robust guidance and control algorithms, Table 6.1 outlines the primary merit of some typical techniques. Although there are some benefits to using these stability theory-based methods for space/aerospace vehicle guidance and control, numerous theoretical and practical concerns still need to be further explored. The following list of issues provides a more detailed summary.

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Table 6.1 Main advantages of stability theory-based G&C algorithms Method Main advantage TSMC: NTSMC: NFTSMC: HOSMC: ISMC: FOC: NDI INDI BS H∞ :

It can stabilise a system with guaranteed finite-time convergence It can alleviate singularity issues A faster convergence rate can be achieved compared with NTSMC The chattering issue can be successfully resolved effectively alleviating the chattering problem It can reduce the steady-state error, alleviate the chattering by completely utilizing higher-derivative information It is adaptable when it comes to achieving desired performance requirements and fine-tuning the closed-loop response It allows the application of traditional linear control techniques INDI offers further system robustness improvement over traditional NDI The flexibility to deal with uncertainties can be increased by using the recursive design approach It can ensure control stability while optimising some performance indices

• TSMC: Typically, convergence moves slowly toward the equilibrium point. When TSMC methods are used, singularity issues could also readily appear. • NTSMC: The system state variables tend to converge slowly when the system is far from the equilibrium point. Additionally, issues like chattering, slow convergence and “stagnation of convergence” may be found, if the unit vector control law is designed in conjunction with the global arrival condition. • NFTSMC: Similar to NTSMC, issues like chattering and “convergence stagnation” need to be further taken into account, despite the fact that the convergence speed can be increased. • HOSMC: It may be difficult to collect the higher-order derivative information of the system in engineering practice. Additionally, an accurate system model must be utilized to establish the HOSMC controller. • ISMC: This approach could result in a significant overshoot and a protracted regulation period in the circumstance of a substantial initial mistake, which would worsen transient performance. When the control input is limited, this condition would be even worse. • FOC: In most cases, the fractional order must be chosen and adjusted carefully. This value’s selection can have a negative impact on the system’s robustness and stability if the selection of this value is poor. Furthermore, the restricted FOC law’s design has received scant attention. Be aware that numerous practical limits may exist in real-world applications and necessitate appropriate solutions. • NDI: For this kind of method to obtain an exact dynamic cancellation, the controlled system often needs to be precisely understood. Due to numerical mistakes, environmental uncertainties, external perturbations, and model simplifications, the ideal condition is challenging to achieve in real-world applications.

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• BS: According to [76], with regard to numerical inaccuracies, the control effectiveness of BS laws is usually sensitive. Keep in mind that a practical implementation often requires an onboard processor to run the guidance and control algorithm numerically. This phenomenon is quite unpleasant as a result. A significant problem of several finite-time SMC-based methods, in addition to the difficulties listed above, is that, in most practical situations, the finite time cannot typically be specified by a user. Therefore, despite the fact that methods based on finite-time SMC theory have begun to emerge, there is still a long way to go for its maturity. One benefit of stochastic guiding and control approaches is that they can handle many kinds of uncertainty in the system equations. However, it is also clear that there may be certain difficulties: • It is challenging to analyze system uncertainty if the state variables of a practical guidance and control system can not be observed or the feedback control includes a time delay. • Because a lot of assumptions may need to be introduced into the design of stochastic guidance and control methods, the outcome is likely to be conservative. Despite being acceptable in theoretical analysis, it would be hard to achieve practical stability in practical usage. • It is impossible to analyze different systems with the same strategy, even if adaptive strategies can be useful for estimating uncertain variables and mitigating their adverse effects. If the estimating procedure cannot be done efficiently, the control accuracy may suffer greatly, which is likely to occur in a practical application. Moreover, it should be highlighted that by using the stability theory-based guidance and control algorithms, specific challenges to these difficulties can be found for various space/aerospace engineering applications, such as spacecraft swarms and multi-spacecraft formation flying [9, 10, 77]. For instance, the desired swarm behaviour is driven by both the nonlinear time-varying attitude dynamics and the orbital dynamics. Therefore, one difficulty is to achieve the desired condition and keep an optimal and robust control performance at the same time [77]. In real applications, it would be more complex because numerous actuator, environmental, and communication uncertainties also need to be taken into account [73]. Additionally, a centralized stability theory-based guidance and control system may also find it challenging to control a large swarm of vehicles due to the high communication and computing costs [73]. A few significant works have been described in the literature to address these open topics [78–81]. For a wide scale of space robotic systems, the writers of [79] for example, utilized an inhomogeneous Markov chain-based probabilistic swarm guidance algorithm to achieve the desired swarm behaviour. The robustness and scalability aspects of the suggested approach were supported by rigorous proofs. The proposed design’s potential to achieve robust control performance was further demonstrated by multiple simulation results, supporting the theoretical conclusions. A decentralized algorithm for multi-spacecraft formation flying was proposed by the authors in [80]. The fact that the computational complexity of the suggested algo-

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rithm stays the same as the number of spacecraft in the swarm grows is a significant benefit. Several hardware-in-the-loop tests were conducted to confirm the efficacy of the suggested algorithm, increasing the likelihood of future real-world applications.

6.3.4 Design and Applications of Data-Driven G&C Algorithms Importantly, all of the aforementioned algorithms are model-based control methods [82]. In other words, the guiding or control law is made by using the system equations’ knowledge and information [83]. However, as space and aerospace vehicle system advance, it is inevitable that the dynamic model becomes more complex than before. Consequently, new challenges for the development of guidance and control methods would be created. Thus, researchers have proposed data-driven guidance and control (DDGC) algorithms to solve the issue without relying on the understanding of the vehicle’s mathematical model. More precisely, the following circumstance used datadriven guiding and control algorithms: 1. Modeling uncertainty and external disturbances that affect the vehicle system is challenging, or they can have a big impact on how model-based guidance and control algorithms are developed. 2. A unified mathematical model is difficult to use to describe the motion of an aerospace or space vehicle. 3. The vehicle’s mathematical model is too intricate to create a robust guidance and control system. Recent years have witnessed significant development and application of datadriven control methods in the design of guidance and control systems. Data-driven control methods’ core principle is that we only modify input and output signals. In other words, this method establishes an input-to-output model, which is updated by using the gathered input and output datasets. By doing this, it is not necessary to get the information of the system model. The primary challenge becomes building a controller that allows for the tracking of the desired output signals. A closed-loop adaptive process can be seen as the core of a data-driven guiding and control algorithm. Figure 6.3 depicts a simplified conceptual diagram of this kind of approach. A number of researchers have made their contributions to the development of this type of method, and the related literature is also available. For example, in [84], in order to solve the issue of attitude control of spacecraft while taking noncooperative targets into account at the same time, the authors proposed a model-free data-driven control scheme to generate the most effective control commands for the attitude system. A number of simulation tests were used to validate the suggested data-driven control’s efficacy and stability. Similarly, in [85], a combined spacecraft attitude control problem was solved with the assumption that both the spacecraft model and the exterior disturbances were

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Fig. 6.3 A simplified conceptual diagram of data-driven control

unknown. To handle this uncertain challenge, the researchers proposed a predictionbased data-driven model-free control scheme. Based on a number of comparison investigations, the researchers came to the conclusion that using their proposal, as opposed to the conventional model-free adaptive control method, would result in improved control performance. In their follow-up work [86], they examined whether integrating the data-driven control method with an external state observer to produce reliable tracking control for combined spacecraft is feasible or not. A notable advantage of data-driven control approaches is that it can make full use of the input and output data of the system to complete the controller design work, with no need for model information. However, the following are the key problems: • It is hard to find a unified theoretical framework to match this type of approach due to the fact that most DDGC methods are established independently. • With regard to the robustness of DDGC methods, the theoretical analysis has not been fully developed. Furthermore, data noise and data dropout may have a significant impact on practical stability.

6.4 Review of Optimisation-Based G&C Methods Stability theory-oriented methods have been discussed above. In recent years, the demand for superior system performance has been increasingly growing. Consequently, Designing optimisation theory-based controllers for space/aerospace vehicles is gaining more and more popularity. According to studies, from both the theoretical and application aspects, these newly established optimisation theory-based strategies enjoy a number of advantages A promising optimisation theory-based control approach still needs to overcome some obstacles before it can be made available for utilization in onboard decision-making, stimulating more investigation into this subject.

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6.4.1 Design and Applications of Dynamic Programming-Based G&C Methods Using methods which are based on dynamic programming and approximate dynamic programming is one common example of optimisation theory-based control. Based on the Bellman’s principle of optimality, dynamic programming (DP), a traditional feedback type of control, may identify the best solution to a variety of control problems [87]. Traditional DP, on the other hand, has two drawbacks: (1) the controller is difficult to modify once its structure is fixed; and (2) the operation necessitates a significant amount of processing time, which is too expensive to afford in realtime applications. As a result, scientists have looked into new ways to improve the traditional DP so that it can be used as computational guidance and control methods. By looking into related literature, we can find that a number of scientists have been working on whether differential dynamic programming (DDP) can be used to compute the local optimal control actions [88]. DDP can be viewed as a second-order approximation of standard DP. Specifically, consider a discrete dynamic system in the form of xk+1 = f (xk , u k ), k = 1, ..., N with the cost function J given by J ({u k }) =  N +1 (x N +1 ) +

N 

L k (xk , u k )

(6.1)

k=1

Here, (xk , u k ) denotes the state and control pair at time tk .  N +1 (·) and L k (·, ·) are the terminal and process costs, respectively. In DP, a cost-to-go function is introduced: Vk (xk , {u k , u k+1 , ..., u N }) =  N +1 (x N +1 ) +

N 

L i (xi , u i )

(6.2)

i=k

The aim of DP is to find an optimal control sequence {u ∗k , u ∗k+1 , ..., u ∗N } such that Vk is minimized. If we denote the optimal cost-to-go function as Vk∗ (xk ) and substitute Eq. (6.1) into it, one can obtain ∗ Vk∗ (xk ) = min[Vk+1 (xk+1 ) + L k (xk,u k )] uk

(6.3)

Then, DDP performs a second-order approximation of Eq. (6.3) around the reference ({x¯k }, {u¯ k }) and nullifies the expanded Vk∗ (xk ) function with respect to δu k . Here δu k = u k − u¯ k . This process results in an optimal feedback law: δu ∗k = αk + βk δxk

(6.4)

∗ (xk+1 ) and its first and second order derivain which αk and βk are functions of Vk+1 tives. A backward sweep can be used to compute their values. In order to plan a new control moment, DDP would iteratively perform reverse actions on the nominal

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Fig. 6.4 A commonly used structure of ADP

state trajectory. The new state trajectory is then updated or evaluated by performing a forward action. The fact that DDP may attain a quadratic convergence speed is one of its key characteristics. However, because a Hessian matrix must be computed (e.g., due to second-order approximation), the real-time implementation becomes more challenging. Applying the finite difference technique is one possible recovery procedure. Convergence problems, however, could result from it. Approximate dynamic programming is a newly developed technique. It can serve as an ideal controller for a number of different control systems [89]. In certain publications, this technique is also known as adaptive dynamic programming (ADP) or heuristic dynamic programming (HDP) [90, 91]. The main idea of this newlydeveloped system is to approximate the optimal control actions by carrying either an offline iteration process or an online learning update process, which is different from the classic DP and DDP methods. Figure 6.4 displays an ADP structure that is frequently utilized. The controller module, as seen in Fig. 6.4, generates control actions to direct the system states through interaction with the environment. Through an evaluation function, an evaluation module is integrated by analysing the performance of the system to adjust the control. In the literature, numerous DP-oriented control methods have been proposed. In recent years, some classical approaches have been reported in the literature. What we aim to do here is to condense the main advantages of these approaches. Methods summarised in Table 6.2 have been applied or may be applied to space/aerospace vehicle G&C issues. The following is a brief list of the main characteristics and benefits of the methods in Table 6.2: • ADP: The policy-iteration or value-iteration strategies form the foundation of the ADP method and its variations. When it comes to linear or nonlinear issues as well as continuous or discrete system variables, these strategies are often appropriate. • IADP: By combining the advantages of incremental nonlinear control techniques and linear approximate DP, this approach can better handle unknown, nonlinear systems as well as time-varying references.

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Table 6.2 Popular DP-oriented methods reported in recent years DP-oriented G&C methods Adaptive dynamic programming (ADP) [92] Incremental approximate dynamic programming (IADP) [93] Iterative neural dynamic programming (INDP) [94] Stochastic differential dynamic programming (SDDP) [95] Action-dependent heuristic dynamic programming (ADHDP) [96] Sparse Gauss-Hermite quadrature differential dynamic programming (SGHQDDP) [97]

• INDP: This method, which is built on the IADP framework, is a data-driven optimal control algorithm, which can handle inaccurate and unmodelled systems. • SDDP: The foundation for the SDDP approach is the framework of DDP. By including the unscented transform method and linear feedback control policies, it alters the original algorithm’s structure. It has been demonstrated that this method can solve low-thrust orbital transfer issues with more reliable solutions and fewer penalties. • ADHDP: When it comes to linear or nonlinear problems, using the ADHDP approach is often appropriate, which is similar to ADP. Additionally, the learning and control performance of this method can be greatly enhanced by taking advantage of its interaction process. • SGHQDDP: This method is a compound computational guidance algorithm, which takes advantage of the merits of DDP and the Gauss-Hermite quadrature rule. By applying this approach, enhanced guidance performance and reduced computational time are likely to be achieved.

6.4.2 Design and Applications of Model Predictive Control-Based G&C Methods Additionally, model predictive control (MPC) and receding horizon control (RHC)based G&C methods have also drawn a lot of interest. In [98], there is a recent assessment of MPC’s uses in various aircraft systems. The driving force behind the adoption of MPC-oriented methods is their potent ability to manage mission-related limitations. An MPC-based approach typically constructs and solves an online finitehorizon optimum control problem to generate an optimal control sequence. The plant is then subjected to a portion of the control actions from the optimal control sequence. The design problem in terms of optimal tracking guidance was settled in [99]. In order to guide the aero-assisted spacecraft to follow predetermined trajectories during the atmospheric entry phase, the goal of this work was to devise an MPC algorithm to output guiding commands in real time. Figure 6.5 illustrates the functionality of the

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Algorithm 6.1 The online operation of the MPC-based tracking guidance algorithm 1: At each time point k := 0, 1, ..., do: 2: (1). Compute x(k) for the plant. 3: (2). Construct the following optimal control formulation: minimise J =

N

+ j|k)Qx(k + j|k) + + j|k)Ru(k + j|k) subjectto ∀ j ∈ [1, 2, ..., N ] x(k + j + 1|k) = f (x(k + j|k), u(k + j|k)) x(k|k) = xk xmin ≤ x(k + j + 1|k) ≤ xmax u min ≤ u(k + j + 1|k) ≤ u max h(x(k + j + 1|k), u(k + j + 1|k)) ≤ 0 x j=1

T (k

N −1 T j=0 u (k

(6.5)

where xmin , xmax , u min , and u max represent the lower and upper bounds of the state and control variables, respectively. xk is the current state, and 4: (3). Solve the problem (6.5) to obtain: u ∗ (k) = [u ∗ (k|k), u ∗ (k + 1|k), ..., u ∗ (k + N − 1|k)]. 5: 6: 7:

(6.6)

(4). Apply u k = u ∗ (k|k) to the plant until the next sampling instant. (5). Assign k = k + 1. (6). Return to (1).

Fig. 6.5 The operation of the MPC tracking guidance algorithm

suggested MPC tracking guiding algorithm, with the optimization process depicted in the lower dashed portion. Algorithm 6.1 summarizes some of the crucial steps of this MPC-based tracking guiding algorithm. It is important to note that the optimisation process is a crucial factor that affects whether MPC-oriented guiding and control algorithms can be effective and high-efficient. Therefore, it also encourages the development of fast optimisation algorithms [100]. There are two possible directions for new MPC technology developments. The first mainly focuses on the algorithms’ resilience or the capacity to reduce disturbances,

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Table 6.3 Popular MPC-oriented methods reported in recent years MPC-oriented G&C methods Linear covariance-based MPC (LCMPC) [105] Model predictive static programming (MPSP) [101] Quasi-spectral model predictive static programming (QSMPSP) [103] Model predictive convex programming (MPCP) [106] Sampling-based stochastic model predictive control (SSMPC) [107] Tube-based robust model predictive control (TRMPC) [108] Linear pseudospectral model predictive control (LPMPC) [109]

which eventually results in the development of stochastic MPC (SMPC) and robust MPC (RMPC). The second path aims to make the algorithm better suited for practical guiding and control tasks by reducing the computational load of the online optimization process. This results in the development and application of model predictive static programming (MPSP)[101] and its various enhanced versions [102–104]. Many MPC-focused guiding and control methods have recently been introduced in the literature. Here, Table 6.3 summarizes various recently published MPC-oriented methods for space or aerospace vehicle guidance and control issues, highlighting some common approaches that have been published in recent years. The following is a brief summary of the main characteristics and benefits of the methods shown in Table 6.3: • LCMPC: The linear covariance updating mechanism of this method can be successfully used to produce a robust guidance performance. In terms of the Earth atmospheric re-entry issue, this approach has been used to produce navigation commands successfully. • MPSP: The only difference between this method and conventional MPC is the addition of a prediction-correction procedure to update the control history in a closed form. Furthermore, the problem’s scope can be shrunk to a manageable level. • QSMPSP: This method, which greatly improves computational efficiency, is built on the MPSP framework with smaller optimization parameters. • MPCP: Convex programming techniques are used in this method to handle online optimization. Numerous vehicle guiding issues have used this method, which has shown that the computational performance can be improved. • SSMPC: This method is based upon the stochastic MPC (SMPC) foundation [110], which incorporates a computationally efficient offline sampling strategy. The algorithm has been used to generate control actions for a spacecraft rendezvous and docking mission, and the outcomes on an experimental test bed have validated its effectiveness. • TRMPC: The TRMPC is built on the same structure as the traditional MPC. The difference is that tube techniques are used by TRMPC in order to deal with the impacts of outside disturbances, which ensures the resilience of the control

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process. The computational cost can be effectively reduced by employing a tube over the “min-max” structure, which is a significant advantage. This method has been used to successfully maneuver spacecraft during activities such as rendezvous and proximity. • LPMPC: A linear pseudospectral discretization scheme and the conventional MPC framework are integrated into this method in order to lower the number of optimisation variables, which will enhance the efficiency of online computation. It has proven that this method is potentially suitable for addressing a number of vehicle guiding issues.

6.4.3 Challenges of Using Optimisation Theory-Based G&C Methods in Space/Aerospace Applications Although there are some benefits of using various guidance and control methods based on control optimisation theory, their development is still in its early stages, and several theoretical and practical issues require more research. For DP-oriented guiding and control methods, specifically: • When there are several mission-related constraints in a single practical and control problem, most of DP-oriented methods’ control performance may be adversely impacted. • In terms of some more inclusive problems, the requirement of dividing them into sub-problems and storing intermediate results is a common feature of DP-oriented methods. In applications used for onboard guidance and control, this will cost a lot of memory. • The computing complexity of DP-oriented methods is typically considerable for their nature, making it challenging to use them in shipboard applications. When a stochastic process is incorporated into the algorithm, the problem gets worse. It can be noted that there are several difficulties with MPC-based guiding and control methods. One major issue is that when the scale of the system or restrictions, mission objectives, and model nonlinearity increase, the online computing complexity would increase rapidly. Researchers and engineers have worked very hard to find ways to efficiently lessen the computing load of the online optimization process [111–113]. For example, in [111], By using a sequential convex programming (SCP) approach, the goal of the authors is to reduce the difficulty of the online optimization procedure, by which they successfully offered an application of SCP in terms of optimal guiding and reconfiguration problem for swarms of spacecraft. Similar to this, the SCP-based optimum guiding and control algorithm was successfully used fot a multi-UAV optimal reconfiguration problem in [112]. The effectiveness of the suggested design was further supported by experimental findings. In the work of Rebecca et al. [113], the SCP approach was further adjusted to decrease processing time while enhancing algorithmic resilience. This study is built

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on and made use of the aforementioned two significant results. When it comes to the application on rigid spacecraft, this approach is also suitable. These studies demonstrated how the convex-relaxation strategy can improve computational speed while successfully resolving the complexity issue brought on by nonconvex constraints and nonlinear dynamics. The accuracy and reliability of the discovered solution, however, can be argued. Due to higher-order linearization errors and artificial infeasibility, poor convergence can be a challenge for these convexrelaxation methods [114, 115]. In addition to the problem of computing complexity, the guidance and control performance of MPC-based approaches can be significantly impacted by uncertain effects including model uncertainties, external disruptions and so on. Therefore, when designing MPC-based guiding and control algorithms, the researchers must take the resilience notion into consideration. The literature contains many reports of theoretical findings, analysis, and tools which are designed for better resilience. Robust design based on constraint tightening [21], min-max structures [116], stochastic tubes [117], and adaptive laws [118] are a few examples of possible solutions. However, they are still far from mature. Therefore, the already-established methods are more popular than these newly-proposed ones. Additionally, detrimental effects on the viability of the optimisation process may be generated by the use of additive strategies and tools, leading to poor guidance and control stability. It should also be mentioned that whether MPC-based approaches can be applied to practical guidance and control issues is particularly significant. However, a typical weakness for some of the existing MPC-oriented solutions is lacking realistic setups with regard to the problem formulation. For instance, researchers frequently assume that the information between neighbouring subsystems may be (partially) accessed when the structure of distributed or hierarchical MPC is designed [119–121]. However, due to the presence of communication faults or failures, this assumption may be not in accord with the real situation in various space/aerospace engineering applications, such as spacecraft formation flying and UAV swarms. Hence it is urgently necessary to find out alternative strategies so that this assumption can be relaxed or removed.

6.5 Review of AI-Based G&C Strategies In addition to the abovementioned success attained by applying stability theoryand optimisation theory-based guidance and control methods, combining Al-based techniques with optimal guidance and control of space/aerospace vehicle systems have become a new research direction for recent research. In terms of this type of strategy, deep learning is utilized to build an ideal guidance and control network. Early studies supported the viability of using this newly designed strategy, but more research is needed to address a number of theoretical and practical difficulties.

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6.5.1 Connection Between AI and Guidance and Control Problems We can create machines and procedures that are capable of performing in a more intelligent and effective way thanks to advancements in AI technology and processing resources. It is undeniable that there is a high interest in applying AI, and some connections between AI and optimal guidance and control problems have already been made, even though the kind of exploration is still in its early stages. Reviewing the literature, we can find four probable connections in particular.

6.5.1.1

AI and Vision-Based Pose Estimation for Spacecraft/UAV

Researchers have looked into the potential for using AI and vision-based algorithms to estimate the pose of various spacecraft or aerial robot flight missions in recent years. There are some classic examples in the body of literature already in existence [80, 122–126]. In particular, the issue of spacecraft proximity operations, such as formation flying and on-orbit servicing, was taken into consideration in [122]. To estimate the pose and shape of uncooperative orbiting objects simultaneously, researchers used various on-board monocular-based approaches. The filter-based simultaneous localization and mapping (SLAM) algorithms and architectures were further modified to produce the desirable robust estimation. To assess the effectiveness of the suggested vision-based designs, numerical simulations were made. A new monocular-based pose approximation approach for uncooperative spacecraft was also proposed by the authors of [123]. One distinctive aspect of their research is that this research can generate robust key points by assessing the effectiveness of existing localization algorithms. Then, using this set of key points, a convolutional neural network (CNN) was trained to provide specialized descriptors resistant to variations in illumination. To evaluate the merit of this specific design, comparative simulations were run. A novel monocular-based pose estimation approach for uncooperative spacecraft was also proposed by the authors of [125], the distinctive aspect of which is the ability to build a set of reliable key points by assessing the effectiveness of existing localization algorithms. Then, with the use of this set of key points, a convolutional neural network (CNN) was trained to provide specialised descriptors resistant to variations in illumination. To evaluate the merit of this specific design, comparative simulations were run.

6.5.1.2

Optimal Trajectory Generation

Applying AI algorithms to the creation of the optimal manoeuvre trajectories for spacecraft may be the most well-known connection between AI and optimum control [127, 128]. Applying AI-based techniques like evolutionary algorithms and tree

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search methods, in contrast to conventional trajectory optimization approaches [6, 100, 129, 130], boosts the chances of finding a mission’s overall optimal solution. More importantly, designers who are not equipped with strong background knowledge related to the problem can also make use of it, which means that the implementation of the type of strategy is relatively easy. In [131], the authors used a monotonic basin-hopping strategy with the evolutionary algorithm, by which a fully automated solution method was developed in order to solve the low-thrust interplanetary trajectory planning problem. Apart from this representative example of using evolutionary algorithms, researchers have also explored the feasibility of using different ML methods in the generation of optimal trajectories [25]. For example, the authors of [25] showed how they utilised a trained AI model to estimate the optimal multi-impulse transfer trajectories and the optimal transfer costs for a multi-target mission. Some mission cases demand fast estimation of optimal trajectories, and sometimes computing the mission performance index can be expensive. In this kind of scenario the use of a trained AI model is important.

6.5.1.3

ML/AI-Based Nonlinear Control and Estimation Design

A large number of publications with “neural network control” and “ML/AI-based control” as keywords can be found in the literature [132–135]. For instance, in [135], how to use AI-based methods to control the landing of drone was taken into account. A DNN was built to roughly represent the high-order effect in order to increase the applied model’s accuracy. The estimation model was then used to build a nonlinear feedback linearization controller. Results from experiments confirmed the stability and efficacy of an AI-based controller. The proposed method of using DNN with spectral normalisation is directly relevant to spacecraft control, even if this chapter is for a drone controller. However, neural networks are merely used as auxiliary tools in the aforementioned works to make up for unmodeled system terms. They are then integrated with other control strategies to direct the system’s motion. For various spaceflight missions or UAV swarms, the feasibility of using AI models as online motion controllers and state estimators have been looked into by researchers recently. The work completed by Tsukamoto and Chung [136] serves as one illustrative example in this area, the authors of which sampled a sequence of data points of the optimal contraction metric offline by solving a convex optimisation problem. After that, an online motion controller was created using a deep long short-term memory recurrent neural network that had been trained to roughly represent the sampled data. The suggested strategy was successfully used to tackle real-time issues, such as spacecraft optimum motion planning. For multi-vehicle motion control systems, an imitation learning-based approach was suggested in [137], firstly proposing to use a global planner to create a variety of demonstration trajectories. A local policy that can be successfully deployed in real time was then proposed by extracting local observations and learning them via deep imitation learning.

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Additionally, It is worth paying attention to the work of Carlos and Dario [24] who built deep neural networks to learn the optimal guidance and control laws. More importantly, the outcomes of a number of pinpoint-landing case studies in which the trained networks were used as feedback controllers in real time have proven that Carlos and Dario’s suggestion was effective. Initial stability analysis of this DNNbased control approach was presented in their subsequent research [138], which adds to the plausibility of using this direct mapping strategy as the primary tool to generate approximations of the ideal guidance commands for spacecraft. In this chapter, due to the possible benefits and applications, designing and applying MI/AI-based technology to accomplish online guiding and control is our main focus.

6.5.2 Design and Applications of AI-Based G&C Methods The advantages of sophisticated trajectory optimisation methods and deep learning techniques are combined in AI-based guidance and control methods, creating an integrated framework that can quickly generate the finest guidance and control orders. In an AI-based approach, there are essentially two steps [24]. Creating a sizable dataset of ideal flight trajectories for a certain mission profile is the first step. It should be emphasised that there usually isn’t an open dataset for a particular application involving space. As a result, this phase is crucial, and simulated findings are employed alternatively. The second step involves creating and training (deep) neural networks using the pre-generated dataset so that they can later be used to directly represent the optimal interaction between the state and the control actions. The work described in [139] is to better illustrate how the application of AI-based guidance and control approaches might improve a spacecraft-related mission. In order to steer the spacecraft to certain final conditions during the atmospheric entry phase, a deep neural network-driven system that generates control actions in real time is being developed in this work. Figure 6.6 depicts the generation of the ideal trajectory dataset, from which three primary processes can be extracted: Creating the perturbation values for the initial conditions is the first step. The second step is about building and solving the trajec-

Fig. 6.6 Illustration of trajectory dataset generation

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Table 6.4 Popular methods developed via different AI models in recent years Different AI models Artificial neural network (ANN) [140] Deep neural network (DNN) [24, 139, 141] Support vector machine (SVM) [142, 143] Reinforcement learning (RL) [144] Deep reinforcement learning (DRL) [145]

tory optimisation model. In the third step, compiling the optimal state and control outcomes into a dataset is required. Deep neural networks are built and trained after the dataset is generated. The control actions can then be produced in real time using the trained neural networks, creating an online feedback structure. Guidance and control methods developed via various AI models have been reported in the literature. Here, we aim to summarise the main features and advantages of some typical approaches reported in recent years. These methods are summarised below (see Table 6.4). It is important to note that all of the methods in Table 6.4 have been applied to space/aerospace vehicle guidance and control issues or have the potential to be. The following list of salient characteristics and benefits in terms of these methods is briefly provided: https://www.overleaf.com/project/62fdb7e39812340d84c9d9c0 • ANN: It is relatively simple for ANN-driven guiding and control algorithm to be designed and put into practice due to its straightforward structure. • DNN: The network structure of DNN is more intricate than that of ANN. For example, DNN has more hidden layers. Although the structure is deeper, it can be easier to obtain a better control performance by applying DNN, which has been demonstrated in [24, 139]. Furthermore, a significant amount of computation time can be saved by using DNN-driven control algorithm, when compared to optimisation theory-based guiding and control methods like MPC and DP. • SVM: Both linear and nonlinear systems can be handled by the SVM. The model is trained using kernel functions, which is more likely to get improved prediction accuracy. Applications of SVM models to orbital transfer issues have been effectively carried out, the outcomes of which have supported its efficacy. • RL: By creating a reward function and giving the system feedback, the ability to interact with the environment can further improve learning outcomes, which is one of the key benefits of adopting RL. As a result, an RL-driven model has a tendency to be more robust and adaptable in the face of unknowns or unforeseen circumstances. Successful RL applications have been made to autonomous lunar and martian landing issues. The outcomes validated the potential of using RL to create the guidance system. • DRL: DRL is comparable to the standard RL and can be seen as an approach that combines DNN and RL. The reason why it is used by researchers is that it enjoys

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more advantages for studying functional relationships in high-dimensional states and control spaces.

6.5.3 Potential Issues and Challenges of AI-Based G&C Methods Although adopting various AI-based guiding and control methods can obtain certain benefits, their development is still in its early stages, and several issues need to be further examined. The following list of problems and difficulties is more specific: • ANN: The network structure of ANN is relatively simple, which is hard to provide an adequate estimation. When it comes to nonlinear dynamic systems in real practice, the issue becomes more serious. • DNN: The network structure parameters of a DNN-based approach, such as the size of the dataset and the number of neurons or hidden layers, have a significant impact on how well it performs. It is challenging to set these parameters to ensure that the network performs at its optimal state, and the problems may vary according to different situations. Furthermore, it can be challenging to tell whether the DNN is trained properly when the designer lacks sufficient understanding of the issue. In addition, the trustworthy simulation results are not enough to certify the effectiveness of DNN-based methods. This is because when it to actual missions, it may lead to catastrophic failures if the initial conditions are significantly off from the norm. • SVM: The SVM has performed poorly when used in orbital transfer test situations. The SVM should therefore be well adapted for more mission scenarios, which need further treatments. • RL and DRL: The computational cost is much higher than that of the alternative methods as a result of the implementation of the interaction process. The threat posed by practical applications is increased as a result. This means that an incompletely optimised or even impractical solution might be accepted by the onboard processor. This could cause the mission to fail and is extremely unfavourable for practical applications. Although AI-based guidance and control algorithms still need more exploration and research, we think that over time, this kind of method will become more mature and popular for solving a range of vehicle guidance and control issues.

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6.6 Conclusions and Future Developments 6.6.1 Concluding Remarks Exo-atmospheric and atmospheric flight research have progressed significantly over the past ten years, and as a result, a variety of cutting-edge guidance and control algorithms for space/aerospace vehicles have been created. In this study, we looked into recently developed methods that could provide promising guidance and control performance for different flight missions. More specifically, the techniques reviewed in this chapter were divided into three main categories: stability theory-based methods, optimisation theory-based methods, and AI-based methods. We also analyzed the main characteristics of various types of algorithms, as well as the associated problems and difficulties. Recent applications of these approaches were given special attention in order to increase the depth and breadth of the literature review. Although the majority of the reviewed methods were developed to support various spaceflight and atmospheric missions, they can also be used to solve other engineering challenges, such as guidance and control issues for autonomous ground vehicles [146, 147], and unmanned surface vehicles [148, 149]. The following is a summary of some concluding reflections based on the work described in Sects. 3 to 5: • Algorithm implementation simplicity: From the aspect of implementation, stability theory-based solutions typically outperform both optimisation theory-based and AI-based methods. This is mostly because of their simple method of obtaining control laws and their simplicity of utilization. However, as it is mostly based on the stability theory, there might not generally be any room for optimisation. • Algorithm flexibility: The flexibility of optimization-based algorithms may be superior than that of the others. For instance, it is relatively simple to combine this kind of approach with other tools, such as disturbance observers, adaptive methods, and neural network identifiers to detect uncertain system parameters or reject the influence of disturbances. More crucially, by treating some problemspecific requirements and constraints as additional constraints and adding them to the optimisation model, some problem-dependent requirements and restrictions can be taken into account in the control process. • Algorithm efficiency: When it comes to efficiency, the performance of stability theory-based and AI-based methods is better than that of optimisation theorybased methods. This is due to the fact that an online re-planning process is typically needed in an approach based on optimisation theory, which considerably increases the computational load in real time. For particular mission scenarios, like spacecraft swarms and multi-spacecraft formation flying in the midst of weather and communication uncertainties, this problem becomes more critical. • Algorithm stability and robustness: Stability theory-based approaches and optimisation theory-based methods typically outperform AI-based algorithms in terms

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of operational stability and robustness. This is mainly because the theoretical conclusions and a variety of recovery techniques of the former are more advantageous.

6.6.2 Continuing Research Given the fact that the development of guidance and control algorithms for space/ aerospace vehicles is still in its early stage, future research can be conducted from the following perspectives: • Stability theory-based techniques may involve the insertion of new algorithmrelated parameters, which could have significant effects on the effectiveness of the control. These parameters are often assigned based mostly on the knowledge of an expert in the program. To improve control performance, the design of new optimal parameter-tuning techniques could gain popularity. • In order to meet numerous realistic limitations while maintaining the system’s stability as well as its capacity to attenuate uncertainty and disturbances, it is desirable to investigate more advanced constraint handling techniques. • Both model-based and data-driven strategies have certain benefits and drawbacks, as discussed in Sect. 6.3. It would also be beneficial to create complementary techniques that combine the advantages of the two approaches. • Researchers should focus on specific advancements to enhance the online computational performance of the optimisation theory-based approach in order to broaden its applicability. There are several possible ways that can be considered fast sampling strategies, efficient optimization algorithms, or simpler optimisation formulations. • It is typically necessary to consider performance index while designing the optimal guidance and control systems. Therefore, it is imperative to develop solutions that may successfully compromise or balance a variety of user-specified performance indexes. • The pre-generated dataset used to train an AI-based control system may quickly become out of date in practical situations. In this scenario, it is greatly desirable to propose specific mapping relationship adjustment procedures. • The fact that AI-based control approaches lack theoretical assurances of viability and closed-loop stability is a major issue. As a result, we think that future research will increasingly focus on the stability of AI-driven control systems. • More advanced validation tools and techniques are needed for AI-based guidance and control systems, which could offer space scientists and engineers more tools to deeper understand the credibility of an algorithm.

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Chapter 7

Optimization-Based Predictive G&C Method

Abstract This chapter focuses on the design of predictive control-based optimization method for addressing missile interception problems. Due to the nonlinearity or inherent limitations of the missile-target dynamics, it is often hard to design control algorithms with high accuracy and efficiency. To tackle this issue, a pseudo-spectral nonlinear receding horizon control (RHPC) scheme is developed and used to generate optimal control commands. The problem of state estimation in the presence of measurement noise is also solved by the Moving Horizon Estimation (MHE) algorithm. Since the RHPC and MHE algorithms solve the optimal open-loop control problem online at each sampling time, their associated computational cost may be high. Therefore, recently proposed sensitivity-based nonlinear programming (NLP) algorithms are used and integrated into the optimization framework to reduce the computational cost of the optimization process. Simulations and numerical analysis demonstrate the effectiveness of the proposed scheme.

7.1 Introduction The most important and difficult components of modern missile missions include the design of nonlinear missile interception guidance and control algorithms. This type of problem has been widely studied during the past decades [1–5]. But designing an optimal or near-optimal control strategy [6–8] is still very difficult. In these problems, The main theoretical and practical challenges are the inherent nonlinearities, including he missile-target dynamics, uncertainties in the aerodynamic model, target maneuver capability, measurement noises and variable/mission constraints. Various robust control algorithms have been investigated [9–11] to improve the performance of interception. For instance, Zhu et al. [9] applied a modified slidingmode control to generate the guidance law. In this situation, the target acceleration was handled by the extended state observer. Also, in [10], taking the model uncertainties and target movement into consideration, a stochastic optimal guidance law was designed from the Markov chain approximation technique. But the reported works do not address the inside constraint. For instance, the state and control limits © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_7

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or the velocity increment are not included. In reality, these requirements should be considered under the controller designs. In this research, the problem is a Receding Horizon Pseudospectral Control (RHPC) design to solve the integrated missile interception guidance and control problems. What’s more, missile guidance and control systems are designed separately as two loops [2, 4], which means that an inner loop autopilot is constructed in order to track the acceleration command generated by the outer-loop guidance algorithm. But a design usually leads to large design iterations. It also does not fully exploit the relationships between different subsystems, which results in suboptimal performance [12]. Recently, a growing interest in the design of integrated guidance law and flight control systems has appeared. For example, in [13] the authors proposed an integrated sliding-mode controller, which is designed to solve the guidance and control of interceptors. Furthermore, Panchal et al. [12] proposed a continuous-time predictive control-based integrated guidance and control algorithm. It can fulfill the 2-D missile-target interception mission, which shows that the end-game performance of the interceptor can be effectively enhanced by taking into account the coupling between the guidance and control dynamics. Because in a dual-control structure, additional degrees of freedom and more missile state information can be used, the performance is satisfying. As a result, the integrated design for this missile guidance and control system which is referred to as Integrated Guidance and Control (IGC) [14], is considered in this investigation. In this chapter, the missile-target IGC algorithm is mainly based on the implementation of Model Predictive Control (MPC). Based on its ability to deal with control and state constraints, the motivation for the use of Receding Horizon Control (RHC) or MPC naturally arises in practical applications [15, 16]. Contributions are made to apply MPC, which can be found in the literature [17–20]. Also, Li et al. [17] proposed a neural-network based robust MPC algorithm, which generates the optimal missile guidance law. Zhao et al. [18] designed an MPC-based algorithm, which can generate the multi-missile guidance law. Weiss et al. [19] implemented an MPC algorithm, which solves the spacecraft rendezvous and docking problems. Wen et al. [20] developed a specific MPC scheme with output feedback, which can solve the deorbiting electrodynamic tether system well. Recently, control algorithms based on pseudospectral methods are becoming more popular. It offers a promising alternative to MPC [16, 21, 22]. Pseudospectral methods can also be used to solve optimal control problems. Especially for problems under constraints using a specific discretization of the solution [23–25]. Pseudospectral methods have a main advantage, which is that a high approximation accuracy can be achieved with much fewer temporal nodes. It means that the size of the resulting static NLP problem can be decreased significantly. So the application of pseudospectral methods in MPC schemes can have positive influences on improving real-time computational performance. The optimization process [17, 25–27] is one of the key components of the RHC schemes. Because the RHPC algorithm solves an open-loop optimal control problem at each sampling instant, the optimization procedure influences the effectiveness and efficiency largely. To meet the high real-time requirements of the RHPC scheme con-

7.2 Missile-Target Nonlinear Model

209

structed in Section 3, we applied and embedded in the RHPC framework a recently proposed NLP sensitivity-based optimization technique [28]. This algorithm applies the implicit function theory. The optimal solution is found around a continuously updated reference solution. Also, we can find a detailed description of this near optimal gradient-based method in [28, 29]. The complicated solution-finding can be avoided by approximating the optimal solution inexactly, which indicates that the online computational performance of the proposed RHPC method has great improvement.. The main contributions of the work reported in this chapter are mainly about two sides. Firstly, prior to performing the MPC-based IGC algorithm, the influence of noise in the measurement of the model state is decreased by implementing an MHE technique. Although it is not a novel idea to use the MHE algorithm to estimate the unmeasured state, it has not been researched in the field of missile guidance and control, which means the present study is an attempt to address the concern. Secondly, the online MPC optimization model is solved using a specific pseudospectral method, which is different from the work carried out in [17]. As a result, it can improve the solution-finding accuracy. Moreover, the computational performance of the optimization process is enhanced, which is achieved by analyzing the NLP sensitivity of the solutions at instants from two consecutive update times. The rest of this chapter is organized as follows. Section 7.2 tells the overall interception strategy and the nonlinear dynamics of the three-dimensional missile-target system. Section 7.3 is about the main results, where a moving horizon state estimation is combined with an RHPC scheme to intercept the presence of measurement noises. Section 7.4 is about numerical simulations to demonstrate the effectiveness of the proposed IGC strategies. The concluding remarks are given in Sect. 7.5.

7.2 Missile-Target Nonlinear Model 7.2.1 2-D Missile Target Engagement Let us consider a standard 2-D geometry of planar interception scenario illustrated in Fig. 7.1. The corresponding nonlinear kinematics are given by [2, 4, 9]: r˙ = VT cos (θ − ϕT ) − VM cos (θ − ϕ M ) θ˙ = (−VT sin (θ − ϕT ) + VM sin (θ − ϕ M ))/r ϕ˙ M = VAMM ϕ˙ T = VATT

(7.1)

where r is the range along the Line-of-Sight (LOS). VT and A T are target velocity and acceleration, respectively. Correspondingly, VM and A M represent the missile velocity and acceleration. θ stands for the LOS angle. ϕ M and ϕT are the flight path angle of the missile and target.

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Fig. 7.1 Missile-Target engagement geometry

Then, by considering the normal acceleration as the control input, the following state-space model of missile-target engagement formulation can be constructed [4, 9]: r˙ = Vr V2 V˙r = rθ + A T r − A M sin (θ − ϕ M ) (7.2) θ˙ = Vrθ V˙θ = − VrrVθ + A T θ − A M cos (θ − ϕ M ) where Vr = VT cos (θ − ϕT ) − VM cos (θ − ϕ M ), Vθ = −VT sin (θ − ϕT ) + VM sin (θ − ϕ M ). Vθ can be treated as a transversal component of relative velocity rotating with the LOS. A T r = A T sin (θ − ϕT ) and A T θ = A T cos (θ − ϕT ). A T r and A T θ can be described as the projection components of the target acceleration. During the engagement, the target maneuver is considered to be given by the first order lag dynamics given by: A˙ T = (AcT − A T )/τT

(7.3)

where AcT is the commanded target acceleration, while τT is the time constant associated with the target dynamics. Subsequently, the pitch-plane dynamics for the missile should be constructed so as to describe the missile attitude related to the inertial frame. That is,

7.2 Missile-Target Nonlinear Model

α˙ = q − (L βα f 1 (α) + L δ f 2 (α + δ))/VM q˙ = Mαβ f 3 (α) + Mδ f 4 (α + δ) + Mq q δ˙ = (δc − δ)/τs

211

(7.4)

where α shows the angle of attack; q means for the pitch rate; δ and δc are, respectively, the actual and demanded deflection angles. Similarly to Eq. (7.3), δ is made by the first order dynamics with the time constant τs . L βα , L δ , Mαβ , Mδ and Mq are the aerodynamic forces and pitch moments on the missile, respectively. f i , i = 1, 2, 3, 4 are saturation functions for the nonlinear aerodynamic characteristics of the missile. Based on the engagement equations and pitch-plane dynamics, the integrated model is then established. Rewrite the dynamic equation and defining the state variable more compactly (e.g. x = [r, Vr , θ, Vθ , A T , α, q, δ]T =[x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ]T ). Then Eqs. (7.2)–(7.4) in the state space can be given by: x˙ = f (x(t), u(t), t)

(7.5)

where f ∈ 8 is the right hand side of the dynamic equations (7.2)–(7.4). u = δc is the control input. In this chapter, the aim at the integrated guidance and control law is to design the presence of model uncertainties and noise measurements of the state model. Designing an optimization-based predictive controller such that the state variables (given by Eq. (7.2)) is another objective, which can be stabilized to the origin.

7.2.2 3-D Missile Target Engagement Extending the mission scenario a 3-D case can better illustrate the 3-D engagement system. The equations of motion for the missile and target can be constructed separately as follows: ⎧ ⎨ X˙ M = VM sin ϕ Ma cos ϕ Me Missile : Y˙ M = VM cos ϕ Ma sin ϕ Me ⎩ ˙ ⎧ Z M = VM sin ϕ Me (7.6) ⎨ X˙ T = VT sin ϕT a cos ϕT e T arget : Y˙T = VT cos ϕT a sin ϕT e ⎩ ˙ Z T = VT sin ϕT e where ϕ Ma , ϕT a , ϕ Me and ϕT e are azimuth and elevation angles of the missile and target. Based on Eq. (7.6), the 3-D dynamic model of the missile-target engagement system is constructed as follows:

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Fig. 7.2 3-D Missile-Target engagement geometry

⎧ r˙ = VT cos(θ y − ϕT e ) cos(θz − ϕT a ) ⎪ ⎪ ⎪ ⎪ −VM cos(θ y − ϕ Me ) cos(θz − ϕ Ma ) ⎪ ⎪ ⎪ ⎪ ˙ θ = (VT sin(θz − ϕT a ) − VM sin(θz ⎪ z ⎪ ⎪ ⎪ −ϕ ⎪ Ma ))/r ⎪ ⎪ ⎪ ˙ θ = (V ⎨ y T cos(θz − ϕT a ) sin(θ y − ϕT e ) −VM cos(θz − ϕ Ma ) sin(θ y ⎪ ⎪ −ϕ Me ))/r cos θz ⎪ ⎪ ⎪ ⎪ ϕ ˙ ⎪ T a = A T y /VT ⎪ ⎪ ⎪ ϕ ˙ ⎪ T e = A T z /VT ⎪ ⎪ ⎪ ϕ ˙ = A M y /VM ⎪ ⎪ ⎩ Ma ϕ˙ Me = A M z /VM

(7.7)

where θ y and θz are the LOS angles. Equation (7.7) can be demonstrated using the 2-D engagement system given by Eq. (7.1). The target acceleration is modeled as: A˙ T y = (AcT y − A T y )/τT and A˙ T z = (AcT z − A T z )/τT , where A T y and A T z stand for the yaw and pitch lateral accelerations. More detailed description of the 3-D case missile-target interception geometry is shown in Fig. 7.2. Remark 7.1 Remarking is worth that to simplify the engagement formulation, also in some relative references and the missile-target dynamic model used in this chapter, the effect of gravity was omitted. The gravitational effects can be simply subtracting the gravity from the acceleration command to design the guidance and control command. This strategy considers the gravity implicitly. It might also result in some

7.3 Receding Horizon Pseudospectral Control

213

deviations from the real system. In order to explicitly incorporate gravity compensation into the missile-target engagement system, future works should be carried out. The effect of gravity can be optimally compensated.

7.3 Receding Horizon Pseudospectral Control 7.3.1 Discrete Approximation Model The multi-interval Legendre-Gauss-Radau (LGR) pseudospectral method is applied to parameterize the continuous time equations of state dynamics given by Eq. (7.5) [21, 23–25], because the numerical solutions of the RHPC problems are important. The motivation for the use of the pseudospectral algorithm is decided by its high accuracy in function approximation. In [23], a detailed introduction with respect to the different classes of pseudospectral methods can be found. The time horizon is divided into N˜ mesh intervals [ti , ti+1 ] for i = 1, ..., N˜ . Space the mesh grid points and the  is assumed to be the length of the mesh interval. The state and control variables are discretized by using the Lagrange interpolation. The result over the ith time interval is as: The multi-interval Legendre-Gauss-Radau (LGR) pseudospectral method for the numerical solutions of the RHPC problems is applied to parameterize the continuous time equations of state dynamics given by Eq. (7.5) [21, 23–25]. The motivation for the use of pseudospectral algorithm is determined by its high accuracy in function approximation. In [23], a detailed introduction with respect to the different classes of pseudospectral methods can be found. Divide the time horizon into N˜ mesh intervals [ti , ti+1 ] for i = 1, ..., N˜ . The mesh grid points are equally spaced. The  is assumed to be the length of the mesh interval. The state and control variables are discretized over the ith time interval by applying the Lagrange interpolation. The result is applied to parameterize the continuous time equations of state dynamics given by Eq. (7.5) [21, 23–25]. The motivation of pseudospectral algorithm relies on its high accuracy in function approximation. In [23], a detailed introduction with respect to the different classes of pseudospectral methods can be found. The time horizon is divided into N˜ mesh intervals [ti , ti+1 ] for i = 1, ..., N˜ . The mesh grid points are equally spaced. The  is assumed to be the length of the mesh interval. The state and control variables are discretized over the ith time interval by the Lagrange interpolation, as: x (i) (t) ≈ X (i) (t) = u (i) (t) ≈ U (i) (t) =

 Nk +1 j=1

 Nk j=1

(i) (i) (i) x (i) j L j (t) = x (i) (i) (i) u (i) j L j (t) = u

(7.8) (7.9)

where j = 1, 2, ..., Nk , Nk is the number of LGR collocation points. t j ∈ [ti , ti+1 ] can be obtained by solving PK −1 (t) + PK (t) = 0, where PK is the K th order Legendre

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7 Optimization-Based Predictive G&C Method

(i) (i) (i) polynomial. a(·) is a positive weight function. (i) = [L (i) 1 , L 2 , ..., L Nk ] where L j is the Lagrange interpolation basis function. One advantage of using pseudospectral approximation is that the derivative of the state equations (e.g. x(t) ˙ = f (x(t), u(t), t)) can be obtained by differentiating the approximation function:

x˙ (i) (t) =

d x (i) (t) dt

(i)

≈ d xdt (t)  k +1 = Nj=1

d a(t) (i) ( L (t))x (i) j dt a(t j ) j

(7.10)

a(t) Note that the term dtd ( a(t L j (t)) can be obtained at collocation points. It can also be j) compacted into a differentiation matrix. That means:

D jk =

d a(t) (i) ( L (t)) |t=t j dt a(t j ) j

(7.11)

where D jk denotes the elements of the Nk × (Nk + 1) differentiation matrix and can be calculated by the following equation:

D jk

⎧ −Nk (Nk +2) ⎪ ⎨ L (t4) , k = j = 0; N j 1 = L Nk (tk ) t j −tk , k = j, 1 ≤ k, j ≤ Nk ; k ⎪ ⎩ −1 , 1 ≤ k = j ≤ Nk . 2(1−t j )

(7.12)

To show the approximation accuracy of the Legendre-Gauss-Radau Pseudospectral Method (LGRPM), Fig. 7.3 demonstrates a comparison between the approximations of open-loop optimal control solutions, which uses LGRPM and Zero-OrderHold (ZOH) functions. It is commonly used in the MPC framework [17, 19]. This example is also a convergence analysis of an open-loop solution to the exact solution. The problem formulation associated with it is defined as follows:  1 [u(t)2 + x(t)u(t) + 1.25x(t)2 ]dt minimi ze J = 21 (7.13) 0 subject to x(t) ˙ = 0.5x(t) + u(t), x(0) = 1 The exact state and control trajectories of this problem are: (1−t) x ∗ (t) = cosh cosh (1) sinh cosh (1−t) u ∗ (t) = − (1−t)+0.5 cosh (1)

(7.14)

The approximation errors are measured using the maximum base ten logarithms of the state and control variables. That is E x = max log10 |x j − x ∗ (t j )| j

E u = max log10 |u j − u ∗ (t j )| j

(7.15)

7.3 Receding Horizon Pseudospectral Control

215

Fig. 7.3 Approximation comparison of an open-loop optimal control problem

LGRPM can produce almost identical results with the exact solution from Fig. 7.3. But ZOH functions cannot get such high accuracy. What is more, the algorithm will steer the approximation error to zero, because the number of basis functions increases. Remark 7.2 Pseudospectral optimal control is the choice of collocation points. The approximation to the dynamics may be poor if the current mesh grid is chosen improperly when it comes to the optimal control of underactuated nonlinear dynamical systems (e.g., the missile dynamical system), which brings the development of mesh refinement strategies. As a result, the current mesh grid will be updated several times to achieve higher accuracy. Many effective mesh refinement strategies embedded in the pseudospectral methods were recently developed. To solve the MHE and MPC formulation, this paper is interested in applying the pseudospectral method. A detailed analysis of the approximation error order of the pseudospectral method is out of the scope of this chapter, which is referred to [30] for such an analysis. Remark 7.3 Mesh refinement-based pseudospectral methods may result in several calls to the NLP solver and a significant computational cost is one important issue. It is usually hard to select a proper accuracy threshold for the mesh refinement process because of the lack of physical knowledge of the system dynamics and

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7 Optimization-Based Predictive G&C Method

the uncertainties/noises in the model. As a result, making a tradeoff between the approximation accuracy and real-time applicability needs the multi-interval LGR pseudospectral method with a fixed number of collocation points to be applied to produce a relatively dense mesh grid. In the simulation section, this mesh grid setting is given. And perturbations of this number will only result in negligible differences of the results.

7.3.2 Moving Horizon Estimation The nonlinear MPC constructs a series of optimal control problems to optimize a specified objective function while accounting for the system dynamics and constraints [26], which is also a technique based on numerical optimization. The assumption of full-state feedback is used in the design of optimization-based controllers. However, in some practical applications (e.g., the missile interception guidance and control), there might be some measurement noises. In the system, the state variables are not directly available [10, 12]. To solve this problem, an MHE technique is developed through constructing an online sub-optimization problem. Because of the measurement noise, the relationships between the measurable outputs and the integrated missile-target state variables are carefully defined, which is noted as y = h(x). h(·), and can be mapped from the missile-target state space. As a result, it is the measurable output space. In many practical scenarios, just a part of the states is available for measurement, where it is necessary to reconstruct the state information by using a limited number of measurements. The solution-finding is carried out using the latest N¯ measurements y ij , when it comes to the MHE optimization process. It can also be obtained at the sampling time instants t ij , where i = 1, ..., N¯ .The MHE subproblem can then be formulated as follows through the LGRPM method to approximate the dynamics: minimi ze JM H E =

 N¯ i=1

2 h(z (i) j ) − y j 2

subject to ∀t (i) j ∈ [ti , ti+1 ] N k +1  ti+1 − ti (i) (i) (i) f (z (i) D (i) jk z j = j , u j , tj ) 2 k=1 z j − z max ≤ 0 u j − u max ≤ 0

(7.16)

where z j stands for the state estimation at time instant t j , whereas h(z j ) is the actual measured value. The initial state estimation term should be introduced in the objective function. But when it comes to the missile-target intercept problem, it is assumed that there is a known initial state vector for the engagement system.

7.3 Receding Horizon Pseudospectral Control

217

The aim for the use of MHE over standard tools, including the Extended Kalman Filter (EKF), relies on its ability to dealing with highly nonlinear system dynamics (e.g., the nonlinear missile-target engagement system considered in this study). The EKF is computationally efficient, even though it requires the following conditions: firstly, the variances of the noises are small; secondly, the system nonlinearities are mild, so the linearization of dynamics can still be valid. At the same time, utilizing the MHE algorithm requires more computational efforts because it needs to solve the nonconvex nonlinear optimization problems which are connected to the MHE formulation. But only the latest N¯ measurements are taken into account instead of all the Nk measurements is one advantage of using the MHE formulation (7.16) is that. As a result, the computational complexity can be reduced significantly. At the same time, the online performance can be improved. ========================== Objective function Eq. (7.16) is a way to measure the missile-target state estima¯) ) is referred to as tion errors. The state estimate at the time instant tk+1 (e.g., z (NNk +1 the initial condition of the subsequent model predictive pseudospectral control step.

7.3.3 Receding Horizon Pseudospectral Control MPC can be regarded as an iterative optimization process which produces control moments. It achieves this by performing a moving horizon trajectory optimization [26, 27]. Periodically recalculating with the current state as an initial condition achieves the control, which provides a feedback action. The feedback action can improve robustness to uncertainties and disturbances. ¯) , the moving prediction horizon Through the updated initial condition x1(1) = z (NNk +1 . of the kth RHPC optimization problem becomes [tk+1 , tk+1 + T ], where T = N sampling It means that the moving horizon of the RHPC formulation consists of N intervals. We can get the following stage cost function because the control objective of the RHPC is to push the missile-target system given by Eq. (7.5) to the origin: J R H PC =

 N 

 tk+1 + N

(x (i) )T Qx (i) + (u (i) )T Ru (i) dt

(7.17)

i=1 tk+1

. Q ∈ 4×4 is a semi-definite matrix. R ∈ 1×1 is a symmetric where i = 1, ..., N positive definite matrix. Introducing ψ(x (i) , u (i) , t (i) ) = (x (i) )T Qx (i) + (u (i) )T Ru (i) and using a Gauss quadrature to approximate the integral term helps to rewrite the RHPC cost as: Nk N   (i) (i) ¯ (7.18) ω j ψ(x (i) J R H PC = j , u , tj ) i=1 j=1

where ω j is the LGR weight and defined as:

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7 Optimization-Based Predictive G&C Method

 ωj =

+1

−1

L j (t)dt

(7.19)

According to [23], Eq. (7.19) can be rewritten as:

ωj =

2 , (Nk +1)2 1−t j 1 , (Nk +1)2 [L Nk (t j )]2

j = 0; j = 0.

(7.20)

Through the above analysis, the RHPC formulation is considered as an online optimal control problem, whose minimum value of cost function is defined by Eq. (7.18) subject to the state, control, and nonlinear algebraic constraints. The RHPC optimization model can be given by: minimi ze J¯R H PC =

Nk N   (i) (i) ω j ψ(x (i) j , u j , tj ) i=1 j=1

subjectto

N k +1  k=1

(i) D (i) jk x j =

ti+1 − ti (i) (i) f (x (i) j , u j , tj ) 2

(7.21)

x (i) j − x max ≤ 0 u (i) j − u max ≤ 0

7.3.4 NLP Optimality and Approximated KKT Conditions The computational cost associated with it can be high and usually cannot be afforded online [17, 28] is one significant challenge of optimization-based control strategies. A NLP sensitivity-based optimization method is applied and embedded in the RHPC framework to deal with this problem. Solving an easier approximate problem improves the computational performance. Based on the constructed online optimization formulation shown in Eq. (7.21), the corresponding augmented Lagrange function is then given by: Nk N   (i) (i) (i) ( N¯ ) ω j ψ(x (i) L = j , u j , t j ) + λ1 (x 1 − z Nk +1 ) i=1 j=1 Nk N k +1    N (i) (i) (i) f (x (i) D (i) + λTj ( jk x j − j , u j , t j )) 2 j=1 k=1 Nk Nk   − x ) + μTj (u (i) + ν Tj (x (i) max j j − u max ) j=1 j=1

(7.22)

7.3 Receding Horizon Pseudospectral Control

219

where λ j , ν j , μ j , j = 1, ..., Nk are vectors of the Lagrange multipliers. To simplicity the presentation, the superscript representing the index of time interval is ignored. The optimal solution of the optimization problem (7.21) should satisfy the first-order optimality or Karush-Kuhn-Tucker (KKT) conditions given by: ¯

) =0 ∇λ1 L = x1 − z (NNk +1 N k +1   N ∇λ j L = D jk x j − fj = 0 2 k=1

∇x j L = ω j ∇x j ψ j + ∇u j L = ω j ∇u j ψ j −

N k +1 

 T λ j λk N Aj Dk j − + νj = 0 ωj 2 ωj

k=1  T λ j N B j ωj 2

(7.23)

+ μj = 0

where f j := f (x j , u j , t j ), ψ j := ψ(x j , u j , t j ), A Tj := ∇x j f j and B Tj := ∇u j f j , ¯

) respectively. By defining p := z (NNk +1 , the first-order nonlinear equations can be rewritten in a more condensed form:

ζ (s( p, Nk + 1), p) = 0

(7.24)

where s( p, Nk + 1) is the solution vector and is given by s( p, Nk + 1)T = [x1T , u 1T , λ1T , ν1T , μ1T x2T , u 2T , λ2T , ν2T , μ2T ..., x NT k , u TNk , λTNk , ν NT k , μTNk ]. The optimal solution is then defined as: s ∗ ( p, Nk + 1). NLP solvers based on Newton-iteration search for a given solution s ∗ ( p0 , Nk + 1) by successive linearization of Eq. (7.24) (e.g. firstorder Taylor expansion) around the current searching point s j ( p0 , Nk + 1), where j is the iteration index. This can be described as: K ( p0 , Nk + 1)s = −ζ (s( p0 , Nk + 1), p0 ) | K ( p0 , Nk + 1) = ∂ζ ∂s (s( p0 ,Nk +1), p0 )

(7.25)

where K is the KKT matrix. Equation (7.25), which is combined with suitable adjustments to monitor the step length s (e.g. line search or trust region techniques). It leads to the optimal solution s ∗ ( p0 , Nk + 1). To improve the online performance of the optimization algorithm we need to analyze the effect of perturbations on p around the nominal solution. These sensitivity results are used to approximate solutions to solve the neighbouring problems. The sensitivity-based optimization can be explained as exploiting the similarity between the solutions of the optimization problem at two consecutive update time instants. To do this, the following theory regarding NLP sensitivity is introduced [28, 29]. Theorem 7.1 ([28, 31]) Consider the RHPC optimization problem given by Eq. (7.21) with f (·) and ψ(·) , which are twice continuously differential in a neighborhood of the nominal solution s ∗ ( p0 , Nk + 1), when the nominal solution s ∗ ( p0 , Nk + 1) can satisfy the Linear Independence Constraint Qualifications (LICQ) [28, 31] and Second Order Sufficient Conditions (SOSC) [28, 31].

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7 Optimization-Based Predictive G&C Method

1. s ∗ ( p0 , Nk + 1) is an isolated local optimal solution of the problem and the associated Lagrange multipliers are unique. 2. For p in a neighborhood of p0 , there is a unique, continuous and differentiable vector function s ∗ ( p, Nk + 1), which is a local optimal solution satisfying the LICQ and SSOC conditions. 3. There exists positive constants c1 and c2 such that |s ∗ ( p, Nk + 1) − s ∗ ( p0 , Nk + 1)| ≤ c1 | p − p0 |; and the optimal values satisfy |JNk +1 ( p) − JNk +1 ( p0 )| ≤ c2 | p − p0 |. The results in Theorem 7.1 allow the application of the implicit function theory to Eq. (7.24) at the nominal solution point s ∗ ( p0 , Nk + 1): K ∗ ( p0 , Nk + 1)

∂s ∗ ∂ζ (s( p, Nk + 1), p) =− |s ∗ ( p0 ,Nk +1) ∂p ∂p

(7.26)

where K ∗ ( p0 , Nk + 1) is the KKT matrix calculated at s ∗ ( p0 , Nk + 1). The right k +1), p) |s ∗ ( p0 ,Nk +1) = [−In x , 0, ..., 0], where n x hand side term of Eq. (7.26) is ∂ζ (s( p,N ∂p is the degrees of freedom of the state equations. If the nominal solution s ∗ ( p0 , Nk + 1) satisfies SSOC and LICQ, the KKT matrix can be used to calculate the sensitivity matrix shown in Eq. (7.26). Based on this, the estimation of the neighboring problem can be approximated by: s˜ ( p, Nk + 1) = s ∗ ( p0 , Nk + 1) +

∂s ∗ ( p − p0 ) ∂p

(7.27)

where s˜ stands for the approximation of s ∗ ( p, Nk + 1). Based on the continuity and differentiability assumptions, there exists a positive constant c3 such that |˜s ( p, Nk + 1) − s ∗ ( p, Nk + 1)| ≤ c3 | p − p0 |2 . k +1), p) ) requires The calculation of the sensitivity matrix in Eq. (7.26) (e.g. ∂ζ (s( p,N ∂p n x backsolves. This process is usually expensive, especially when the size of the system becomes larger. To solve this, the step length s( p, Nk + 1) = s˜ ( p, Nk + 1) − s ∗ ( p0 , Nk + 1) is calculated by linearization of KKT conditions at the nominal solution point s ∗ ( p0 , Nk + 1). K ∗ ( p0 , Nk + 1)s( p, Nk + 1) = −ζ (s ∗ ( p0 , Nk + 1), p)

(7.28)

where ζ (s ∗ ( p0 , Nk + 1), p) corresponds to the KKT matrix in the nominal solution. s can be described as a Newton step starting from the nominal solution to the solution of the neighboring problem, which means s˜ ( p, Nk + 1) can satisfy Eq. (7.27). That only a single backsolve is required to compute the sensitivity matrix is its biggest advantage. This update costs negligible time than addressing the NLP problem to obtain new solutions. f (·) and ψ(·) are convex quadratic functions, s˜ ( p, Nk + 1) = s ∗ ( p, Nk + 1), which means the approximate solution is exactly equivalent.

7.3 Receding Horizon Pseudospectral Control

221

The change of the active sets about the inequality constraints may affect the results of the sensitivity analysis. When s( p, Nk + 1) = s˜ ( p, Nk + 1) − s ∗ ( p0 , Nk + 1) is large enough to result in a change with respect to the current active set, approximation of the KKT conditions will become non-smooth, which means that Eq. (7.27) does not hold true. And the updated solution s˜ ( p, Nk + 1) might violate the box constraints. Theorem 7.1 does not hold at the points where the change of active set occurs, which means the continuity and differentiability of s ∗ ( p, Nk + 1) with respect to p cannot be preserved. Using the generalized SOSC condition as well as the relaxed set of constraint qualifications [29] can tackle this.

7.3.5 Implementation Consideration In order to better present the proposed algorithm, the overall procedures of the MHE algorithm and the MPC method are summarised, which are shown in Algorithms 7.1 and 7.2. Algorithm 7.1 Framework of the MHE real-time loops /*Offline*/ ( N¯ ) Step 1: Initialize z 1 , Nk and N¯ ; k Step 2: Generate the LGR points {t j } Nj=1 , the differential matrix via Eq. (7.12), and the LGR weight coefficients via Eq. (7.20); /*Online (main Loop)*/ Step 3: At each time step j = 1, ..., Nk (a). Discretize the continuous system via Eq. (7.10); (b). Construct the NLP problem via Eq. (7.16); ( N¯ )

Step 4: Wait for a new measurement z j+1 ; Step 5: Calculate the derivative of the objective and constraints; Step 6: Solve the optimization problem (7.23) via the sensitivity-based method; Step 7: Update the primal and dual solutions via Eqs. (7.26)–(7.23); Step 8: Repeat Steps 3–7 until the next time step;

According to Algorithm 7.2, for the MPC loop, the control variable is also recalculated for each time step, providing feedback to reduce the effects which are caused by uncertainties or model errors. Except for the structure of the MPC algorithm, it is also significant to know how the MPC works. So a computational complexity analysis about the number of operations is also required to solve an iteration of MPC versus the dimensionality, where the number of collocation points and the time horizon of the problem are provided. If an optimal control problem contains the n x state variables, n u control variables, and Nk LGR points are applied to discretize the

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7 Optimization-Based Predictive G&C Method

Algorithm 7.2 Framework of the MPC real-time loops /*Offline*/ Step 1: Initialize x1 , Nk and N˜ ; k Step 2: Generate the LGR points {t j } Nj=1 , the differential matrix via Eq. (7.12), and the LGR weight coefficients via Eq. (7.20); /*Online (main Loop)*/ Step 3: At each time step j = 1, ..., Nk (a). Discretize the continuous system via Eq. (7.10); (b). Construct the NLP problem via Eq. (7.21); Step 4: Wait for a new state x j+1 ; Step 5: Calculate the derivative of the objective and constraints; Step 6: Solve the optimization problem (7.23) via the sensitivity-based method; Step 7: Update the primal and dual solutions via Eqs. (7.26)–(7.23); Step 8: Repeat Steps 3-7 until the next time step;

system. If the mesh grid which is consisted of N˜ sampling intervals and the length of the mesh interval is , it means that O( N˜ (n x (Nk + 1) + n u Nk )3 ) operations are required to solve the formulation [17, 32].

7.4 Simulation Studies 7.4.1 Parameter Specification To verify the effectiveness of the proposed RHPC about IGC approach, numerical simulations are carried out. All the simulation results were carried out on Matlab under Windows 7 and Intel (R) i7-3520M CPU, 2.90GHZ, with 4.00 GB RAM. The = 3, Nk = 4. Q and R are parameters of the RHPC algorithm are chosen as N¯ = 2, N obtained according to Bryson’s rule [33]. The lower and upper bounds of the state and control variables are relatively chosen as: r ∈ [0, 20000], Vr , Vθ ∈ [−5000, 5000], θ ∈ [−40, 40], α ∈ [−20, 20], A M ∈ [−350, 350], and δ ∈ [−20, 20]. What is more, the pitch and raw rate will vary in the region [−250, 250]. The initial positions of the missile are assigned as: X M (0) = 0 m, Y M (0) = 0, and Z M = 0. The missile’s initial flight path angle and velocity are chosen as: ϕ M = 35◦ and VM = 1800 m/s. What is more, the initial flight path angle and velocity of the target are set as ϕT = 50◦ and VT = 2000 m/s. The initial range along the LOS is r = 12000 m. The initial LOS angle is θ = 40◦ . The measurement of the LOS is referred to as a first-order-lag system. The target acceleration is given by A T = (150 + d AT ) sin π t (m 2 /s).

7.4 Simulation Studies

223 trajectory

8000

-3000

z/m

y/m

6000 4000 2000

-2000 -1000

0

0 0

5000

0

10000

5000

10000

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x/m trajectory

0

10000

y(Up)/m

-1000

z/m

trajectory

-4000

-2000

5000 0 -4000

-3000

10000

-2000

-4000 0

5000

y/m

10000

z(East)/m

0

5000 0

x(North)/m

Fig. 7.4 3-D Missile-Target engagement trajectory (no uncertainty and noise)

to evaluate the performance of the MHE approach against measurement noises, the measurements of the missile-target range r , target acceleration A T and the LOS angle (θ y and θz ) are assumed to be disturbed by d = [dr , d AT , dθ y , dθz ], where d is the zero mean Gaussian noise with standard deviation of 10m, 2m/s and 1mrad. The missile model-dependent parameters are: L¯ βα = L¯ α − L¯ δ , L¯ α = 1070.1m/s 2 , L¯ δ = 191.8m/s 2 , M¯ αβ = M¯ α − M¯ δ , M¯ α = −353.4s −2 , M¯ δ = − 283.3s −2 , M¯ q = −14.8s −1 and the trajectory update time step is set to 0.2 s. Because the RHPC optimization problem is demonstrated as a large-dimension NLP problem, it is important to obtain a robust and rapid convergence to the optimal solution. So all the optimization variables are scaled using the strategy suggested in [33]. The optimal solution is obtained via the sensitivity-based SQP algorithm formulated in Sect. 7.3.4.

7.4.2 Interception Results The performance of the proposed IGC design is evaluated for a sample run. In this case, we consider no model uncertainty and measurement noise. The missile-target engagement trajectories, together with the state measurement results, are displayed in Figs. 7.4 and 7.5. The corresponding estimation error evolutions are presented in Fig. 7.6. Through the result displayed in Fig. 7.4, the missile can engage the target successfully with a 0.032 m miss distance. The plant states can be estimated satisfactorily and the estimation error can be steered to a small neighbourhood of the origin from

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7 Optimization-Based Predictive G&C Method trajectory

8000

-3000

z/m

y/m

6000 4000 2000

-2000 -1000

0

0 0

5000

0

10000

5000

10000

x/m trajectory

x/m trajectory

0

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y(Up)/m

-1000

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trajectory

-4000

-2000

5000 0 -4000

-3000

10000

-2000

-4000 0

5000

y/m

10000

z(East)/m

0

5000 0

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Fig. 7.5 3-D Missile-Target engagement trajectory (with uncertainty and noise)

the state estimation results shown in Figs. 7.5 and 7.6, t. So these results demonstrate that the MHE algorithm can have a good performance for estimating state variables in the missile-target engagement system. For the computational performance, the average processing time is around 0.1721 s in this case, smaller than the trajectory update time. The result was performed by considering the measurement noise and the parameter ¯ M¯ were assumed to be uncertainty (e.g., the missile aerodynamic parameters L, varied randomly by ±10% from the model values). Figure 7.7 depicts the time history of the 3-D intercept geometry, which is obtained by applying the RHPC-based IGC method. The findings of the associated state estimate trajectory are then presented in Fig. 7.8. It should be emphasized that A T y and A T z represent the typical target acceleration profiles along the elevation plane and azimuth plane, respectively, in the last two images (such as Fig. 7.8h and i). These data show that the plant states may still be estimated satisfactorily in the presence of measurement noises by applying the MHE process and reducing its least-squares objective function provided by Eq. (7.16). The outcomes of the suggested control scheme are contrasted with those of more conventional missile guiding and control methods. For instance, the design of a Primal-Dual Neural Network-based (PDNN) predictive control scheme published in [17] and an integrated Sliding Mode Guidance and Control (SMGC) design investigated in [3]. Performance index weightings are used in the PDNN approach to account for the state and control input limitations. The control saturation function is used, however, by the SMGC control scheme to handle restrictions. The comparative time histories for the missile’s acceleration, control input, and attitude angles are plotted in Fig. 7.9, from which it can be seen that, in the absence

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of model uncertainties and measurement noises, the proposed RHPC-based IGC law can produce state and control trajectories without violating the pre-specified state and control constraints. (The control moments affecting the missile are shown in Fig. 7.9e–f). The SMGC results show that, despite the control requirements being ensured by the employment of saturation functions, the missile acceleration and angular rate trajectories exhibit constraints violations. The PDNN results reveal similar phenomena. The state constraints may not be satisfied by performance index weightings, leading to constraint breaches. Imposing state constraints might further restrict the allowable control regions implicitly. As shown in Fig. 7.9, the algorithm has to sacrifice its maximum allowable control moments to satisfy the missile acceleration and attitude angle constraints. Based on the findings presented in Figs. 7.7, 7.8 and 7.9, it can be concluded that the RHPC approach achieves an engagement time of 3.303 s, indicating that the interception mission examined in this study may be completed in a short amount of time. Additionally, the average computation time for the RHPC optimization process is 0.1729 s, and the miss distance for this sample run is 0.324 m. These two variables are slightly more important than when model uncertainty and measurement noise are not taken into account. This can be explained by the possibility that the perfor-

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Fig. 7.7 Estimation error profiles (no uncertainty and noise)

mance of the suggested control strategy will suffer as a result of taking these noises and uncertainties into account. The optimization process is repeated online at each sampling instant. However, in accordance with the RHPC scheme’s design, which is described in Sect. 7.3 of this work, and the final state values of the preceding process will be used as the initial conditions of the ongoing control loop. This regressive approach can offer feedbacks that considerably lessen the effects of uncertainty and model mistakes, strengthening the resilience of the control algorithm. The suggested optimization-based predictive IGC scheme has to undergo a significant number of Monte-Carlo simulations in order to further validate its effectiveness. It is generally recognized that the Monto-Carlo simulation, which enables the evaluation of the effects of various system noises and uncertainties, is a strong tool for analyzing the effectiveness and resilience of a design. In order to prepare for the missile-target engagement operation, 500 Monte-Carlo simulations were run. When stochastic disturbances and measurement noises are taken into account in the missiletarget system, simulation results demonstrate that the proposed optimization-based predictive control algorithm can reduce the state estimation error to a small value and achieve the hit in the majority of cases with an average miss distance of 0.0362 m and an average interception time of 3.305s. The average runtime per iteration of MPC graph is plotted in Fig. 7.10. The mean value of this runtime array is further calculated, which is about 0.1723s. It is again

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Fig. 7.8 State estimation performance (with uncertainty and noise)

smaller than the trajectory update time. Hence by applying the proposed control scheme and based on the results presented in Fig. 7.10, the real-time applicability can be preserved.

7.4.3 Comparative Study Studies comparing the missile intercept accuracy attained using the proposed IGC solver with different alternative MPC-based controllers were conducted. Examples are MPC controllers based on Direct Sequential Quadratic Programming (DSQP) and Differential Dynamic Programming (DDP). Additionally, all algorithm-dependant parameters are calibrated ideally to ensure a fair comparison, as recommended in related studies (see, for example, [20, 31, 34, 35]). It is important to note for comparison’s sake that the missile’s effectiveness depends heavily on how accurately it hits its target. As a result, this serves as the primary criterion for comparing the effectiveness of various controller architectures. The miss distance distribution produced using the various guidance and control systems is shown in Fig. 7.11. For each of the engagement scenarios, the cumulative

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miss distance statistics are displayed in the final subplot of Fig. 7.10. The cumulative miss distance chart, commonly referred to as Single-Shot-Kill-Probability (SSKP), is a useful tool for visualizing the Monto-Carlo performance of guided missile systems. Figure 7.11 shows that, in terms of the SSKP value, the DSQP-based technique outperforms both the suggested method and the DDP-based controller (a higher SSKP value can be obtained with a small value of miss distance). It should be emphasized that our investigations suggest that greater precision and fidelity may come at a cost in computational time for real-time performance. In order to improve the real-time applicability of various control schemes and to conduct a fair comparison, a relatively small accuracy index (for example, 1 × 10−4 ) is used in the optimization process and the comparative study. Each MPC optimization issue may be solved using the suggested approach, DSQP, or DDP in an average calculation time of 0.1723, 0.5743, and 0.6611 s, respectively. The optimization time is shorter than the trajectory update time, allowing the suggested approach to be applied in real time. As a result, it can be said that when compared to other algorithms examined in this research, the proposed RHPC-based algorithm can preserve the real-time applicability without scarifying the interception accuracy significantly (this is reflected

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by Fig. 7.11, where a relatively high SSKP value can be obtained by applying the proposed strategy).

7.4.4 Effect of Parameter Uncertainty In this subsection, the influence of parameter uncertainty for the computational time and the interception accuracy is worked. By taking the missile aerodynamic parameters are varied randomly through ±10%, ±15% and ±20% for the model values, the sensitivity results of the MPC-based controllers to modeling errors are displayed in Figs. 7.12, 7.13 and 7.14. Table 7.1 summarizes the maximum, average, and minimum Miss Distance (MD) values as well as the typical calculation time required for each MPC optimization procedure. The results shown in Figs. 7.12, 7.13 and 7.14 and Table 7.1 make it clear that an increase in uncertainty effect would lead to both an increase in computational time and a drop in interception accuracy. It is important to highlight that a different comparative study known as warm-start DSQP (wsDSQP) was conducted and is shown in Table 7.1. Using the previous time step’s solution as a starting point, the NLP problem is immediately solved in this approach. When adopting the wsDSQP, interception performance can be enhanced compared to that of the standard DSQP solution, both in terms of calculation and to some extent. However, the real-time applicability of wsDSQP is still not achieved.

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Table 7.1 Comparative results for different methods Indicators ±10% uncertainty DSQP-based wsDSQP-based max(MD) (m) min(MD) (m) mean(MD) (m) CPU (s) Indicators max(MD) (m) min(MD) (m) mean(MD) (m) CPU (s) Indicators max(MD) (m) min(MD) (m) mean(MD) (m) CPU (s)

0.2254 0.0281 6.3928E-05 0.5743 ±15% uncertainty DSQP-based 0.7929 0.0878 3.9464E-05 0.7229 ±20% uncertainty DSQP-based 2.1056 0.2893 1.1122E-04 0.8425

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Proposed method 2.9237 0.3476 1.5891E-04 0.1892

DDP-based 6.5470 2.2833 4.2029E-04 0.7070

To deal with all the uncertain cases, the proposed RHPC control scheme, as well as the sensitivity-based optimization method, is able to solve the real-time applicability and get a competitive interception accuracy. But it is investigated that the real-time applicability will lose if the uncertainty interval is increased to ±25%. And the average running time for solving the optimization problem will be increased to around 0.2738 s.

7.5 Conclusion In order to resolve the missile integrated guidance and control problem in the face of model parameter uncertainties and measurement disturbances, an optimizationbased predictive control technique was developed and put into practice in this study. Predictive control and state estimation for moving horizons were discretized using a multiple-interval pseudospectral approach. After that, the NLP formulation that was produced was solved using a sensitivity-based nonlinear programming strategy. The NLP sensitivity information was used to approximate the ideal solution to lower the computational complexity and match real-time requirements. It was demonstrated through numerical simulations how effective and reliable the suggested approach is. The findings demonstrate that the integrated guidance and control system examined

References

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in this research is capable of meeting the aforementioned conditions for the missile interception task. However, there are still some problems with applying the suggested control strategy to the missile-target interception issue. For instance, if substantial noises and model mistakes are taken into account, the performance of the sensitivity-based optimization strategy may be significantly impacted. The implementation of larger prediction horizons will be constrained as a result of the slower processing speed. More sophisticated aerodynamic models should be used since the current missile dynamic model is rather straightforward. As a more dense mesh grid is needed to provide an accurate approximation of the dynamics, it will obviously increase the computational workload of the optimizer. These problems will make up the bulk of our subsequent study.

References 1. Yu W, Chen W (2014) Guidance law with circular no-fly zone constraint. Nonlinear Dyn 78:1953–1971 2. Shtessel YB, Shkolnikov IA, Levant A (2007) Smooth second-order sliding modes: Missile guidance application. Automatica 43(8):1470–1476 3. Shtessel YB, Tournes CH (2009) Integrated higher-order sliding mode guidance and autopilot for dual control missiles. J Guid Control Dyn 32(1):79–94 4. Shtessel YB, Shkolnikov IA, Levant A (2009) Guidance and control of missile interceptor using second-order sliding modes. IEEE Trans Aerosp Electron Syst 45(1):110–124 5. Zhao J, Zhou R (2015) Unified approach to cooperative guidance laws against stationary and maneuvering targets. Nonlinear Dyn 81:1635–1647 6. He S, Lin D, Wang J (2015) Robust terminal angle constraint guidance law with autopilot lag for intercepting maneuvering targets. Nonlinear Dyn 81:881–892 7. Zhao Y, Sheng Y, Liu X (2016) Impact angle constrained guidance for all-aspect interception with function-based finite-time sliding mode control. Nonlinear Dyn 85:1791–1804 8. Zhao Q, Dong X, Liang Z, Ren Z (2017) Distributed group cooperative guidance for multiple missiles with fixed and switching directed communication topologies. Nonlinear Dyn 90:2507– 2523 9. Zhu Z, Xu D, Liu J, Xia Y (2012) Missile guidance law based on extended state observer. IEEE Trans Ind Electron 60(12):5882–5891 10. Hexner G, Shima T (2007) Stochastic optimal control guidance law with bounded acceleration. IEEE Trans Aerosp Electron Syst 43(1):71–78 11. Inalhan G, Stipanovic D, Tomlin C (2002) Decentralized optimization, with application to multiple aircraft coordination. In: Proceedings of the 41st IEEE conference on decision and control, 2002, vol 1 (2002), pp 1147–1155. https://doi.org/10.1109/CDC.2002.1184667 12. Panchal B, Mate N, Talole S (2017) Continuous-time predictive control-based integrated guidance and control. J Guid Control Dyn 40(7):1579–1595 13. Shima T, Idan M, Golan OM (2006) Sliding-mode control for integrated missile autopilot guidance. J Guid Control Dyn 29(2):250–260 14. Wang X, Wang J, Gao G (2015) Partial integrated missile guidance and control with state observer. Nonlinear Dyn 79:2497–2514 15. Chai R, Savvaris A, Tsourdos A, Chai S, Xia Y (2018) Optimal tracking guidance for aeroassisted spacecraft reconnaissance mission based on receding horizon control. IEEE Trans Aerosp Electron Syst 54(4):1575–1588

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16. Genest R, Ringwood JV (2016) Receding horizon pseudospectral control for energy maximization with application to wave energy devices. IEEE Trans Control Syst Technol 25(1):29–38 17. Li Z, Xia Y, Su CY, Deng J, Fu J, He W (2014) Missile guidance law based on robust model predictive control using neural-network optimization. IEEE Trans Neural Netw Learn Syst 26(8):1803–1809 18. Zhao J, Zhou S, Zhou R (2016) Distributed time-constrained guidance using nonlinear model predictive control. Nonlinear Dyn 84:1399–1416 19. Weiss A, Baldwin M, Erwin RS, Kolmanovsky I (2015) Model predictive control for spacecraft rendezvous and docking: Strategies for handling constraints and case studies. IEEE Trans Control Syst Technol 23(4):1638–1647. https://doi.org/10.1109/TCST.2014.2379639 20. Wen H, Zhu ZH, Jin D, Hu H (2016) Model predictive control with output feedback for a deorbiting electrodynamic tether system. J Guid Control Dyn 39(10):2455–2460 21. Fahroo F, Ross IM (2008) Pseudospectral methods for infinite-horizon nonlinear optimal control problems. J Guid Control Dyn 31(4):927–936. https://doi.org/10.2514/1.33117 22. Williams P (2004) Application of pseudospectral methods for receding horizon control. J Guid Control Dyn 27(2):310–314 23. Williams P (2004) Jacobi pseudospectral method for solving optimal control problems. J Guid Control Dyn 27(2):293–297 24. Yang L, Zhou H, Chen W (2014) Application of linear gauss pseudospectral method in model predictive control. Acta Astronaut 96:175–187 25. Chai R, Savvaris A, Tsourdos A (2016) Fuzzy physical programming for space manoeuvre vehicles trajectory optimization based on hp-adaptive pseudospectral method. Acta Astronaut 123:62–70 26. Ahn CK, Shi P, Wu L (2014) Receding horizon stabilization and disturbance attenuation for neural networks with time-varying delay. IEEE Trans Cybern 45(12):2680–2692 27. Lian C, Xu X, Chen H, He H (2015) Near-optimal tracking control of mobile robots via receding-horizon dual heuristic programming. IEEE Trans Cybern 46(11):2484–2496 28. Zavala VM, Biegler LT (2009) The advanced-step nmpc controller: optimality, stability and robustness. Automatica 45(1):86–93 29. Biegler L, Yang X, Fischer G (2015) Advances in sensitivity-based nonlinear model predictive control and dynamic real-time optimization. J Process Control 30:104–116 30. Hager W, Hou H, Rao A (2015) Convergence rate for a Radau collocation method applied to unconstrained optimal control. arXiv:1508.03783 31. Nocedal J, Wright SJ (2006) Numerical optimization. springer series in operations research. Siam J Optim 32. Wang Y, Boyd S (2009) Fast model predictive control using online optimization. IEEE Trans Control Syst Technol 18(2):267–278 33. Tian B, Fan W, Su R, Zong Q (2015) Real-time trajectory and attitude coordination control for reusable launch vehicle in reentry phase. IEEE Trans Ind Electron 62(3):1639–1650. https:// doi.org/10.1109/TIE.2014.2341553 34. Tassa Y, Erez T, Smart W (2007) Receding horizon differential dynamic programming. In: Advances in neural information processing systems, vol 20 (2007) 35. Kouvaritakis B, Cannon M (2016) Model predictive control, vol 38. Springer International Publishing, Switzerland

Chapter 8

Robust Model Predictive Control for Attitude Control Tracking

Abstract In this chapter, we study the optimal time-varying attitude control problem for rigid spacecraft with unknown system constraints and additive perturbations. A dual-loop cascaded tracking control framework is established by designing a new nonlinear tube-based robust model predictive control (TRMPC) algorithm. The proposed TRMPC algorithm explicitly considers the effect of disturbances and applies tightened system constraints to predict the motion of the nominal system. The computed optimal control is combined with a nonlinear feedback method so that the actual system trajectory can always be controlled in a tubular region around the nominal solution. To promote the recursive feasibility of the optimization process and to ensure the input-state stability tracking control process, a terminal controller and corresponding terminal invariant set are also constructed. The effectiveness of the proposed two-loop TRMPC control scheme for reference trajectory tracking problem is verified through an experimental study. Several comparative studies are performed and the results obtained show that the proposed scheme is more promising for constraint handling and attitude tracking than other recently developed schemes considered in this study.

8.1 Introduction 8.1.1 Literature Review Throughout the past few decades, as a basic but significant problem, how to develop an advanced attitude control system for rigid spacecraft has received much attention from both academia and the aerospace industry [1, 2], which has important significance for stable (exo) atmospheric explorations and operations. It is crucial to remember that attitude control systems are primarily designed to allow spacecraft to stabilize their attitude or track the target successfully. So far, numerous contributions have been noted in order to achieve this difficult task in the field of attitude stabilization [3, 4]. For instance, by utilizing both disturbance observers and terminal sliding mode control (SMC), the authors of [3] developed a composite attitude © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Chai et al., Advanced Trajectory Optimization, Guidance and Control Strategies for Aerospace Vehicles, Springer Aerospace Technology, https://doi.org/10.1007/978-981-99-4311-1_8

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stabilization method. Similarly, in [4], an adaptive fixed-time sliding mode controller was developed to deal with the issue of attitude stabilization for flexible spacecraft. Besides attitude stabilization, how to control the attitude motion to track a preassigned reference trajectory is another crucial component of space/aerospace vehicle control. In order to accomplish this aim, over the past few years, a variety of robust tracking control methods, which take into account system uncertainties and environmental disturbances, have been designed [5, 6]. Among these methods above, SMC-oriented approaches and finite-time control algorithms (and their expansions) have been regarded as two of the most promising contenders [7, 8]. To be specifical, the authors of [7] researched a reusable launch vehicle attitude tracking problem by developing and proposing a multi-variable second-order SMC controller such that the accurate tracking performance can be guaranteed. In addition, a distributed tracking algorithm for a group of rigid spacecraft, which are working under an uncertain environment, was designed in [8]. In order to robustly follow the dynamic leader, the authors of [8] combined a distributed finite-time control method with an adaptive neural-network-based observer in their design. Nevertheless, the majority of the aforementioned researches and methods either simply assume that the controlled system is unconstrained or did not sufficiently take into account the mechanical restrictions, such as the state and input constraints. Since the aforementioned two kinds of situations may bring poor tracking performance in engineering practice, it is significant for tracking algorithms to consider system constraints. Moreover, the state and input constraints tend to become harder to meet when system equations contain unknown external disturbances, which will further impair control performance. Model predictive control (MPC) is a powerful substitute that is able to handle constrained systems. And numerous literature works have acknowledged and highlighted the benefit of using MPC [9–11]. For instance, a non-linear MPC method for reference signals tracking was developed by Limon et al. [10], which was tested successfully on an industrial four-tank process control platform. In addition, the reference tracking issue for autonomous underwater vehicles was researched in [11]. In order to improve the tracking performance, the authors of [11] developed a Lyapunovbased MPC method, which was combined with a unique contraction constraint, in their study. Besides, some significant MPC-based results in terms of spacecraft or unmanned aerial vehicle (UAV) attitude control have been presented [12–14]. For example, to control a linearized spacecraft model subject to system constraints, an explicit MPC method was developed in [12]. The authors of [12] consider that the computational cost needed by their method is low, which makes it appropriate for systems with finite CPU resources. This is one critical feature of their idea. Additionally, the authors of [13] took spacecraft rendezvous and docking problems into account. They successfully developed an MPC method and used it to control the linearized attitude dynamics. Nevertheless, it should be noted that the practical systems tend to be unfeasible when employing a reduced dynamic model. To deal with this problem, the modeling approach for UAV trajectory tracking issues, which consider the presence of external disturbances, was studied in [15–17], wherein different UAV system models, including the autoregressive exogenous model and the non-linear Hammerstein model, were built and researched. Furthermore, the authors of [18]

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illustrated that installing a three-axis gimbal system on a UAV is able to enhance the trajectory tracking performance of the MPC effectively. In their subsequent work, a meta-heuristic swarm optimization algorithm was designed and applied by authors with the goal of optimizing the parameters that exist in the controller. According to the authors of the works described above, the majority of these developed methods are able to obtain reliable attitude control performance. However, their results are mostly supported by simulation or experimental research, which means that it is still challenging to carry out rigorous mathematical analysis that involves closed-loop stability and recursive feasibility. It should be noted that, in order to control the attitude of rigid spacecraft, a non-linear MPC method was developed in [19]. Specifically, by Constructing a homogeneous Lyapunov function and a locally stable terminal set, the authors could ensure the recursive feasibility and closed-loop stability in their study. Nevertheless, these theoretical results may easily become useless if additive disturbances perturb the controlled system. Therefore, how to derive explicit conditions that can ensure these crucial properties continues to be a challenging problem, which is, in particular, difficult for the non-linear robust model predictive attitude control method.

8.1.2 Motivations and Contributions So far, many academics and engineers have developed various robust MPC (RMPC) methods in order to actively handle the disturbances or uncertainty that the operating environment contains. Among those methods, applying the min-max structure is a typical scheme. To be more specific, with all potential disturbance realizations taken into account, the online optimization procedure is able to generate a conservative robust solution [20, 21]. But it should be noted that the computational cost of this procedure is normally high. Besides that, with the prediction horizon growing larger, its computational complexity tends to increase significantly. In addition to min-max structure, applying the so-called tube techniques is another common method [22, 23], which has received a lot of research in terms of linear systems stabilization. Specifically, the optimized control policy and the feedback control policy are the two parts that make up the robust control law, where the feedback control policy is in charge of keeping the actual state trajectory inside a tube centered around the nominal system trajectory, whereas the optimal control policy is in charge of guiding the nominal states to the origin. This strategy has a number of potential benefits. For example, applying this method can improve the robustness and ensure the stability. Nevertheless, because the reference trajectory and attitude dynamics are normally non-linear, it could be challenging to directly apply this method to the considered tracking control issue. As a result, we will make attempts in this chapter to extend tube-based MPC (TMPC) so that the considered problem can be solved by using it. More specifically, our goal is to develop and apply a non-linear tube-based robust MPC (TRMPC) method for rigid spacecraft, which will allow us to track the expected time-varying angular trajectories successfully even if additive disturbances

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and system constraints are taken into account. In conclusion, the following four parts present the primary contribution of this chapter which is dissimilar from that of other published studies: 1. This chapter proposes a new spacecraft attitude tracking control framework by designing a non-linear TRMPC method. In spite of disturbances and system constraints, the developed solution, where a dual-loop cascaded control structure is used, is still able to track time-varying angular signals successfully. 2. This chapter designs a compound feedback law, which is combined with tightened state and control constraints. By using this compound feedback law, the spacecraft is able to accomplish the robust constraint satisfaction with all admissible disturbances, 3. To promote the recursive feasibility of the TRMPC optimization process, this chapter develops the terminal controllers and the associated terminal invariant sets. 4. This chapter derives the explicit conditions which can ensure the input-to-state stability (ISS) of the dual-loop tracking system. A number of experimental studies have been carried out and the explicit conditions can be confirmed. In conclusion, this study presents a novel attempt at the rigid spacecraft attitude tracking control issues. That is, one can develop and apply a dual-loop TRMPC method with ensured feasibility and stability properties for dealing with rigid spacecraft attitude tracking control issues.

8.1.3 Organization The rest of this chapter is divided into the following sections. The tracking control objective and the non-linear attitude dynamics of spacecraft are discussed in Section II. The design and the demonstration of TRMPC for the outer-loop and inner-loop tracking systems are discussed in Sect. refsec8.3. Then, Sect. refsec8.4 demonstrates several significant theoretical properties in detail. Via a range of experimental investigations and analysis, the efficiency of the developed method is then confirmed in Sect. refsec8.5. Lastly, Sect. refsec8.6 make conclusions and comments. Notation: Let R and Rn represent the real space and n-dimensional Euclidean space, respectively. The diagonal matrix is represented by diag{x1 , ..., xn }, in which x = [x1 , ..., xn ]T , |x| = [|x1 |, ..., |xn |]T x1 , ..., xn stand √ for the entries. For a vector √ T T and x = x x. Similarly, x P = x Px stands for the P-weighted norm, where P  is positive definite. For an n-by-n matrix Z, its 2-norm is denoted as Z = λmax (ZT Z), where λmin (·) and λmax (·) represent the minimum and maximum eigenvalues, respectively. Take into account two sets A ⊂ Rn and B ⊂ Rn ; we have the operations A ⊕ B = {x + y|x ∈ A, y ∈ B} and A  B = {z ∈ Rn |{z} ⊕ B ⊂ A}.

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8.2 Problem Formulation 8.2.1 Spacecraft Attitude Dynamics Use the form of [9, 24, 25] to represent The attitude dynamics of a rigid spacecraft, which is shown as follows: ˙ =Rω +  f  −1

(8.1a) −1

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(8.1b)

In Eq. (8.1),  = [α, β, σ ]T is composed of three angular variables: angle of attack α, sideslip angle β, and bank angle σ . ω = [ p, q, ν]T is composed of three angular rate variables: roll rate p, pitch rate q and yaw rate ν. M = [Mx , M y , Mz ]T , in which Mi , i = (x, y, z) stands for the roll, pitch and yaw moments, respectively.  f ∈ R3 and d ∈ R3 represent the additive disturbances that are caused from environmental noise and model simplification. The matrix terms R, I, and  ∈ R3×3 are shown as follows: ⎡ ⎤ − tan β cos α 1 − tan β sin α sin α 0 − cos α ⎦ R=⎣ (8.2) − cos β cos α − sin β − cos β sin α ⎡

⎤ ⎡ ⎤ Ix x 0 −Ix z 0 −ν q I = ⎣ 0 I yy 0 ⎦ ,  = ⎣ ν 0 − p ⎦ −q p 0 −Ix z 0 Izz

(8.3)

In Eq. (8.3), Ii j (i, j = x, y, z) stands for the inertia moments.

8.2.2 Control Problem Objectives When constructing the attitude controller, the following two key control goals are anticipated to be met in this chapter: • Control the actual attitude profiles  to follow the expected spacecraft attitude angle profiles r = [αr , βr , σr ]T with the influence of unknown disturbances  f and d . • Make tracking performance optimized with the system state and control constraints satisfied. In terms of the first goals, if the tracking error vectors are denoted as Eθ =  − r ∈ R3 and Eω = ω − ωr ∈ R3 , subsequently, we can obtain the tracking error dynamics for the two subsystems in the form of

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8 Robust Model Predictive Control for Attitude Control Tracking

˙r E˙ θ = Rω +  f −  −1 E˙ ω = −I Iω + I−1 M + d − ω˙ r

(8.4a) (8.4b)

It should be noted that we can produce a nominal version of Eq. (8.4) by disregarding the influence of disturbances: ˙˜ = R() ˜ ω˜ −  ˙r E θ ˙˜ = −I−1 (ω)I ˜ − ω˙ r ˜ ω˜ + I−1 M E ω

(8.5a) (8.5b)

˜ ω˜ where the nominal angle, angular rate and control variables are represented by , ˜ respectively. The nominal tracking error vectors in the inner and outer loops and M, are denoted as E˜ ω and E˜ θ , respectively. In terms of the second goal, we can describe the admissible sets of the angular ¯ and Xω = {ω : ω ≤ ω}, ¯ in and angular rate variables as X = { :  ≤ } ¯ and ω¯ are two known positive constants. In terms of the control moment M, which ¯ in which M¯ is a known we can define its admissible set as U M = {M : M ≤ M}, positive constant. Since a spacecraft can only be manoeuvred within its mechanical restrictions, we need to take into account these kinds of system constraints frequently for practical applications. Remark 8.1 In the developed attitude tracking control method, an inner-outer loop structure is used. More specifically, the outer loop is equivalent to the attitude angle subsystem (8.1a), where a virtual control input ωc is introduced. By using the developed outer-loop TRMPC method (detailed in Sec III.A), we can derive the ωc . Then, we take this value as the reference instruction (e.g., ωr = ωc ) and send it to the inner loop, where the angular rate subsystem (8.1b) should be taken into account, and the inner-loop TRMPC method is used to derive the real control moment M.

8.2.3 Assumptions and Preliminaries We need to make several assumptions in order to develop the TRMPC method [6, 25]. Assumption 8.1 The system variables , ω and M are measurable, and the rotation of Earth is disregarded in the mission.  Assumption 8.2 Take the attitude angle reference, the expected angular rate and ¯ 0, their derivatives to be bounded and smooth, which can also be denoted as r  ≤ ˙ ¯  r  ≤ 1 , ωr  ≤ ω¯ 0 , and ω˙ r  ≤ ω¯ 1 . Assumption 8.3 During the flight mission, the case of |β| = X , where β stands for the sideslip angle.

π 2

is excluded from 

8.3 Design of the TRMPC Algorithm

241

Assumption 8.4 Take the external disturbances to be bounded, which can also be  denoted as  f  ≤ η1 and d  ≤ η2 . When deriving the main theoretical results of the developed TRMPC, we should use these assumptions. Additionally, via the subsequent lemma, we will show that the 2-norm of R and R−1 has lower and upper bounds with proper specification of . Lemma 8.1 If Assumption 8.3 is satisfied, the 2-norm of R() and R−1 () is bounded, which means that we obtain r ≤ R() ≤ r¯ and r ≤ R−1 ()  ≤ r¯ .  Proof Through R() = λmax (R T R), we can compute the 2-norm of R(). It should be noted that R T R and RR T have same non-zero eigenvalues. By solv1 can be derived. Therefore, for ing |λI − RR T | = 0, λ1 = 1 + tan2 β, λ2 = λ3 =  π π any β ∈ (− 2 , 2 ), we can obtain r = 1 and r¯ = 1 + tan2 β. Furthermore, from Eq. (8.2), we obtain R−1 , whose matrix terms can be denoted as ⎡

R−1

⎤ − cos α sin β cos β sin α − cos α cos β cos2 β 0 − sin β ⎦ =⎣ − sin α sin β cos β − cos β − sin α cos β

(8.6)

 T λmax (R−1 R−1 ) stands for its 2-norm. By addressing |λI − (R−1 )T R−1 | = 0, λ1 = 2 cos β, λ2 = λ3 = 1 can be derived. Therefore, r = r¯ = 1 is obtained. The proof is finished. 

8.3 Design of the TRMPC Algorithm Based on TMPC, a novel TRMPC design will be shown in this section. Some principles can be found from previous studies about designing TMPC for linear system regulation issues [26, 27]. Nevertheless, because the reference trajectory and attitude dynamics are naturally non-linear, these tube-based methods cannot be directly used to the considered tracking control issue. Therefore, in order to reach both convergent tracking performance and robust constraint satisfaction for non-linear spacecraft attitude tracking systems, our energies can be directed to broaden the concept of tightening system constraints, building the terminal controller and invariant set, and developing the feedback law. Apply the proposed TRMPC method as the main controller such that the actual tracking errors Eθ and Eω can be stabilized toward the origin.

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8 Robust Model Predictive Control for Attitude Control Tracking

8.3.1 Outer-Loop TRMPC Design In the outer attitude angle control loop, The objective function at time instant tk ∈ {tk }k∈N can be denoted as ˜ θ (tk )) = Jθ (E˜ θ (tk ), U



tk +T

˜ θ (τ |tk ))dτ L θ (E˜ θ (τ |tk ), U

tk

+gθ (E˜ θ (tk + T |tk ))

(8.7)

˜ θ (τ |tk )) = E˜ θ (τ |tk )2 + U ˜ θ (τ |tk )2 with U ˜ θ (τ |tk ) = In Eq. (8.7), L θ (E˜ θ (τ |tk ), U P Q 2 ˜ ˙ ˜ ˜ ˜ |tk ) − r and gθ (Eθ (tk + T |tk )) = Eθ (tk + T |tk ) R . Choose the R((τ |tk ))ω(τ matrices P, Q, R ∈ R3×3 to be positive definite and diagonal. Note that the pro3 3 , Q = diag{qi }i=1 and cess of determining their components (e.g., P = diag{ pi }i=1 3 R = diag{ri }i=1 ) should be based on Lemma 8.2, . The predictive horizon is represented by T . The total control scheme, as mentioned earlier, consists of two stages. To begin with, by dealing with the nominal system-based optimization model, we can obtain a central trajectory and an optimized control action. The nominal system-based optimization model is shown as follows: min

˜ ∗ (τ |t ) ω˜ ∗ c (τ |tk ), k

s.t.

where

˜ θ (tk )) Jθ (E˜ θ (tk ), U ∀τ ∈ [tk , tk + T ] ˜ k |tk ) ⊕ Oθ (tk ) ∈ (t ˙˜ |t ) = R() ˜ ω˜ c (τ k ˙E ˜ ω˜ c −  ˙r ˜ θ (τ |tk ) = R() tube ˜ (τ |tk ) ∈ Xθ ω˜ c (τ |tk ) ∈ Utube ω ˙˜ (t + T |t ) ∈ tube E θ k k θ

⎧ tube ⎪ ˜ ˜ ¯ tube ⎪ ⎪Xθ = { :  ≤ } ⎪ √ ⎪ ⎪ 3 ⎪ ¯ 1 , k¯ = max{1/ki }i=1 ¯ tube = ¯ − 3kη ⎪ ⎪ ⎪ ⎪ tube ⎪ tube ⎪ ⎪ ⎨Uω = {ω˜ c : ω˜ c  ≤ ω¯ c } √ tube ω¯ c = R˜ ω¯ − 3η1 , R˜ = r /¯r ⎪ ⎪ ⎪ ⎪ ⎪ tube = {E˜ θ : E˜ θ  R ≤ θ } ⎪ θ ⎪ ⎪ ⎪

tube ⎪  ¯1 √ ⎪ ω¯ c − r¯ ⎪ ⎪ ¯ ¯ ¯ , −

= λ (R) · min − 3 kη ⎪ min 0 1 ˜ ⎩ θ ¯r K

(8.8)

(8.9a) (8.9b) (8.9c) (8.9d) (8.9e) (8.9f)

In Eqs. (8.8) and (8.9), the tightened state and input constraints are represented by Xtube and Utube ω , respectively. In addition, we introduce an initial state set Oθ and θ ˜ the values of which can be obtained offline by two gain matrices (e.g., K and K), employing the outcomes obtained in the next two lemmas. In particular, Lemma 8.2

8.3 Design of the TRMPC Algorithm

243

gives some guidance on how to build a terminal controller ω˜ cf and the associated terminal invariant set tube θ . Before Lemma 8.2 is introduced, Definition 8.1 should be presented firstly: Definition 8.1 If the following cases keep true for any τ ∈ [tk + T, tk+1 + T ], then for the error system (8.5a), ω˜ cf and tube stand for the terminal controller and the θ terminal invariant set: ⎧ tube ˜ ⎪ ⎪ ⎨(τ |tk ) ∈ Xθ ω˜ cf (τ |tk ) ∈ Utube ω ⎪ ⎪ ⎩ ˜ θ (τ |tk )) ≤ 0 g˙ θ (E˜ θ (τ |tk )) + L θ (E˜ θ (τ |tk ), U

(8.10a) (8.10b) (8.10c)

Lemma 8.2 tube θ , which is described in Eqs. (8.9e)–(8.9f), stands for a terminal invariant set for the system (8.5a) arising from the terminal controller: ˜ K ˜  ˙r ˜ E˜ θ + R−1 () ω˜ cf = R−1 () ˜ = diag{k˜i }3 . Moreover, for the parameters, pi qi < in which K   √ √i=1 ri − ri2 −4 pi qi ri2 + ri2 −4 pi qi should be satisfied. k˜i ∈ , 2qi 2qi

(8.11) ri2 4

and 

˜ should be taken note of when it Proof To begin with, the nominal state variable  ˜ ˜ enters tube . From (8.5a), we obtain  = E +  θ r and θ ˜ = 

˜ √Eθ  R λmin (R)

+ r 

(8.12)

Because E˜ θ  R ≤ θ and Eq. (8.9f) hold, we obtain

θ + r  λmin (R) √ ¯ 1= ¯ − 3kη ¯ tube ≤

˜ ≤√ 

(8.13)

In addition, based on the equation of ω˜ cf that is provided by (8.11) and Lemma 8.1, we have ˜ ˜ ˜ ˙ r ˜ √Eθ  R + R−1 () K  ω˜ cf  ≤R−1 () λmin (R) (8.14) ˜ √ θ ¯ 1 ≤ ω¯ tube ≤¯r K + r¯ λmin (R) Then, if ω˜ cf is applied over τ ∈ [tk + T, tk+1 + T ), the derivative of Jθ is obtained to be

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8 Robust Model Predictive Control for Attitude Control Tracking

˜ E˜ θ (τ |tk )) g˙ θ (E˜ θ (τ |tk )) + L θ (E˜ θ (τ |tk ), −K ˜ E˜ θ (τ |tk )2 + E˜ θ (τ |tk )2 ˜ E˜ θ (τ |tk )2 − K = K Q R R ˜ 2 E˜ θ (τ |tk )2 − K ˜ E˜ θ (τ |tk )2 ≤ K Q R +E˜ θ (τ |tk )2

(8.15)

P

Because k˜i ∈



√

ri2 −

ri2 −4 pi qi 2qi

,

√

ri2 +

ri2 −4 pi qi 2qi

 , it is clear that

− ri k˜i + pi + qi k˜i2 < 0

(8.16)

If Eq. (8.16) is substituted into Eq. (8.15), we obtain the stability condition g˙ θ + L θ < 0. All conditions that are made in Definition 8.1 have thus far been confirmed represent the terminal controller and the by us, which proves that ω˜ cf and tube θ corresponding invariant set for system (8.5a).  Since we have determined the optimized control law and the resulting nominal ˜ ∗ (τ |tk ), respectively, for τ ∈ [tk , tk + T ]), trajectory (denoted as ω˜c ∗ (τ |tk ) and  next, We will develop a robust control law, which we can use to the real angle system. Because of the influence of disturbances, it should be noted that a difference between the optimized one and the actual angular profile may be caused by using ω˜c ∗ in the real system over τ ∈ [tk , tk+1 ). The deviation can be represented by Oθ (τ ) = ˜ ∗ (τ |tk ). Make Oθ (τ ) differentiated, we obtain (τ ) −  ˜ ∗ )ω˜ ∗c (τ |tk ) +  f O˙ θ (τ ) = R()ωc (τ ) − R(

(8.17)

Next, the robust control law can be developed in the form of   ˜ ∗ )ω˜ ∗c (τ |tk ) − KOθ (τ ) ωc (τ ) = R−1 () R(

(8.18)

3 , ki > 0. The following lemma demonstrates that using in which K = diag{ki }i=1 robust control law (8.18) can guarantee the actual angular trajectory inside the tube region centered around the nominal solution with the actual state and input constraints satisfied.

Lemma 8.3 If ωc (τ ) is used to control the actual attitude angle system (8.1a) over τ ∈ [tk , tk+1 ), along with  f is upper bounded by η1 , we will obtain: 1. It is available for the system state and input constraints to be satisfied, which can be denoted as (τ ) ∈ X and ωc (τ ) ∈ Xω , respectively; ˜ ∗ (τ |tk ) ⊕ Oθ , 2. The actual attitude angle trajectory can remain within (τ ) ∈   in which Oθ = {Oθ (τ ) : |Oθ | ≤ [−η1 /k1 , −η1 /k2 , −η1 /k3 ]T }. Proof For (1), if ωc (τ ) is substituted into O˙ θ (τ ), we can obtain O˙ θ (τ ) =  f (τ ) + KOθ (t), Oθ (0) = 0

(8.19)

8.3 Design of the TRMPC Algorithm

245

We can have the solution of (8.19) in the form of 

τ

Oθ (τ ) = eKτ Oθ (0) +

eK(τ −t)  f (t)dt

(8.20)

0

Because of  f  ≤ η1 , it holds that Oθ (τ ) ≤ η1 [1/k1 , 1/k2 , 1/k3 ]T  ≤

√ ¯ 1 3kη

(8.21)

According to the definition of Oθ (τ ), we can obtain that ∗

˜ (τ |tk ) + Oθ (τ ) (τ ) = ∗

˜ (τ |tk ) + Oθ (τ ) ≤ √ ¯ 1= ¯ tube + 3kη ¯ ≤

(8.22)

which denotes that  ∈ X . Furthermore, by defining  ωˆ c = R()ωc ωˆ ∗c

we subsequently get

∗

˜ = R(

)ω˜ ∗c

ωˆ c = ωˆ ∗c + KOθ

(8.23a) (8.23b) (8.24)

Then, two sets should be defined in advance:  ˆ ω = {ωˆ c : ωˆ c  ≤ r ω} U ¯ ˆ tube U = {ωˆ ∗c : ωˆ ∗c  ≤ r¯ ω¯ ctube } ω

(8.25a) (8.25b)

From (8.21), it is obvious that Oθ ={Oθ (τ ) : Oθ (τ ) ≤ η1 [1/k1 , 1/k3 , 1/k3 ]T } √ ¯ 1} ⊂{Oθ (τ ) : Oθ (τ ) ≤ 3kη

(8.26)

ˆ ω and U ˆ tube After looking into Oθ , U ω , it can be obtained that ˆ ˆ tube U ω ⊕ KOθ ∈ Uω

(8.27)

ˆ tube ˆ tube ˆ ∗c ∈ U Therefore, if ω˜ ∗c ∈ U ω , according to (8.23)–(8.27), it is evident that ω ω ˆ and ωˆ c ∈ Uω . After that, according to (8.23a) and (8.9d), it can be concluded that ωc ∈ Xω . ˜ ∗ (τ |tk ) + Oθ (τ ), it is obvious that (τ ) ∈ For (2), according to (τ ) =  ˜ ∗ (τ |tk ) ⊕ Oθ , which follows that the system state will remain inside a tube region  centered around the nominal solution. 

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8 Robust Model Predictive Control for Attitude Control Tracking

8.3.2 Inner-Loop TRMPC Design In this section, the inner-loop tracking controller will be designed, where the reference command ωr is introduced and corresponds to the outcomes from the outer loop. The inner-loop objective function, which is similar to that of the outer loop, can be defined in the form of  tk +T ˜ ω (tk )) = ˜ ω (τ |tk ))dτ L ω (E˜ ω (τ |tk ), U Jω (E˜ ω (tk ), U (8.28) tk +gω (E˜ ω (tk + T |tk )) ˜ ω (τ |tk )) = E˜ ω (τ |tk )2 + U ˜ ω (τ |tk )2 with U ˜ ω (τ |tk ) = where L ω (E˜ ω (τ |tk ), U P Q −1 −1 ˜ ˜ ω(τ ˜ |tk ) − ωr + I M(τ |tk ) and gω (E˜ ω (tk + T |tk )) = E˜ ω (tk + T |tk )2R . −I (ω)I Similar to Eq. (8.7), the process of determining the components of the matrices P, Q, R is based on Lemma 8.4. Next step, we should determine the optimization model in the inner loop. In the inner loop, its optimization model is denoted as: min

˜ ∗ (τ |t ) ω˜ ∗ (τ |tk ),M k

s.t.

where

˜ ω (tk )) Jω (E˜ ω (tk ), U ∀τ ∈ [tk , tk + T ] ˜ k |tk ) ⊕ Oω ω(tk ) ∈ ω(t ˙˜ |tk ) = −I−1 (ω)I ˜ ˜ ω˜ + I−1 M ω(τ ˙E ˜ ω (τ |tk ) = −I−1 (ω)I ˜ − ω˙ r ˜ ω˜ + I−1 M tube ˜ |tk ) ∈ Xω ω(τ ˜ |tk ) ∈ Utube M(τ M ˙E ˜ ω (tk + T |tk ) ∈ tube ω

⎧ tube ˜ ≤ ω¯ tube } Xω = {ω˜ : ω ⎪ ⎪ ⎪ √ ⎪ ⎪ 3 ⎪ ¯ 2 , k¯ = max{1/ki }i=1 ω¯ tube = ω¯ − 3kη ⎪ ⎪ ⎪ ⎪ ⎪ tube ⎪ ˜ ˜ ¯ tube ⎪ ⎨U M = {M : M ≤ M } √ M¯ tube = M¯ − c¯ + d¯ − 3Iη2 ⎪ ⎪ ⎪ ⎪ ⎪tube = {E˜ ω : E˜ ω  R ≤ ω } ⎪ ω ⎪ ⎪ ⎪ ¯ tube ¯

⎪  ⎪ M − d − ω¯ 1 I tube ⎪ ⎪ = λ (R) · min − ω ¯

, ω ¯ ⎩ ω min 0 ˜ KI

(8.29)

(8.30a) (8.30b) (8.30c) (8.30d) (8.30e) (8.30f)

It is obvious that the optimization model in the inner loop is similar to Eq. (8.8). Then, ˜ f the following lemma should be given and used to develop the terminal controller M and the terminal invariant set tube . ω Lemma 8.4 tube ω , which is defined by Eqs. (8.30e)–(8.30f), stands for a terminal invariant set for the system (8.5b) that arises from the terminal controller:

8.3 Design of the TRMPC Algorithm

247

˜ f = (ω)I ˜ E˜ ω M ˜ ω˜ + Iω˙ r + IK ˜ = diag{k˜i }3 . Moreover, for the parameters, pi qi < in which K   √ √i=1 ri − ri2 −4 pi qi ri2 + ri2 −4 pi qi should be satisfied. ki ∈ , 2qi 2qi

(8.31) ri2 4

and 

˜ f in (8.31), we have the following inequality Proof According to the formula of M ˜ f  ≤(ω)I ˜ E˜ ω  M ˜ ω ˜ + Iω˙ r  + IK

(8.32)

It should be noted that based on (8.3), we can obtain the term (ω)Iω = F(ω) in the form of ⎤ ⎡ ⎤ ⎡ − pq I x z − qν I yy + qν Izz F1 (ω) F(ω) = ⎣ F2 (ω) ⎦ = ⎣ pν I x x − p 2 I x z − ν 2 I x z − pν Izz ⎦ F3 (ω) − pq I x x + pq I yy + qν I x z

(8.33)

Moreover, if we define that c¯ = arg maxω∈{ω:ω≤ω} ¯ F(ω) d¯ = arg maxω∈X F(ω) ˜ tube ω Then, we can further denote Eq. (8.32) as ˜ E˜ ω  ˜ f  ≤d¯ + ω¯ 1 I + IK M ˜ ˜ √Eω  R ≤ M¯ tube ≤d¯ + ω¯ 1 I + IK λmin (R)

(8.34)

˜ f ∈ Utube . We will skip the remaining process for evidencing which indicates M M Lemma 8.4, considering its proof has a similar format to that of Lemma 8.2.  The next step is similar to the situation for the outer loop. After the proof of Lemma 8.4, we will study Oω (τ ), which stands for the difference between the nom˜ |tk ) is used inal angular rate trajectory and actual angular rate trajectory when M(τ in the system (8.1b) over [tk , tk+1 ). To be more specific, we get O˙ ω (τ ) = − I−1 (ω)Iω(τ ) + I−1 M(τ ) ˜ ∗ (τ |tk ) + d ˜ ω˜ ∗ (τ |tk ) − I−1 M + I−1 (ω)I

(8.35)

Subsequently, from (8.35), we can obtain a compound angular rate control law in the form of: ˜ ∗ (τ |tk ) − IKOω (τ ) M(τ ) = (ω)Iω(τ ) − (ω˜ ∗ )Iω˜ ∗ (τ |tk ) + M (8.36) Subsequently, for the actual angular rate system, we apply the robust control rule (8.36) to it. Besides, some insights about the actual angular rate trajectory and the robust constraint fulfillment are given in Lemma 8.5.

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8 Robust Model Predictive Control for Attitude Control Tracking

Lemma 8.5 If M(τ ) is used to control the actual angular rate system (8.1b) over τ ∈ [tk , tk+1 ), along with d is upper bounded by η2 , the following results can be obtained: 1. It is available for the constraints in angular rate and control moment to be satisfied, which can be denoted as ω(τ ) ∈ Xω and M(τ ) ∈ U M , respectively; 2. It is available for the actual angular rate trajectory to remain inside ω(τ ) ∈ ω˜ ∗ (τ |tk ) ⊕ Oω , where Oω = {Oθ (τ ) : |Oθ | ≤ [−η2 /k1 , −η2 /k2 , −η2 /k3 ]T }.  Proof The evidence is omitted here because the process of handling it is similar to that of Lemma 8.3. We will skip the process for evidencing Lemma 8.5, considering its proof has a similar format to that of Lemma 8.3. 

8.3.3 Overall Algorithm Framework For the spacecraft attitude control structure, Fig. 8.1 depicts an illustrative diagram of the spacecraft attitude system under control. Then, we demonstrate in more detail about how the TRMPC method executes in Fig. 8.2, where we select the inner angular rate control loop in Fig. 8.1 and detail it. Moreover, in Algorithm 1, we outline the associated implementation steps. Remark 8.2 It is important to note that the developed controller for dual-loop TRMPC tracking are robust against external disturbances. The system variable constraints is the primary factor that influences the robustness of the controller. By tightening the system variable constraints in the nominal optimization model, the influence of disturbances could be taken into account specifically, and the robustness can be accomplished. Next, as Algorithm 1 says, the robust feedback control law is then developed by applying the optimal solution. To be more precise, the robust formula

Fig. 8.1 Schematic diagram of the entire system

8.3 Design of the TRMPC Algorithm

249

Fig. 8.2 Implementation of the TRMPC scheme

Algorithm 8.1 TRMPC for angular rate control loop 1: procedure 2: Step 1: At tk , obtain the actual state ω(tk ); ˜ k |tk ) ⊕ Oω 3: Step 2: Establish the initial condition ω(tk ) ∈ ω(t 4: and the optimization model (8.29); ˜ ∗ (τ |tk ); 5: Step 3: Address (8.29) to obtain ω˜ ∗ (τ |tk ) and M 6: Step 4: Calculate the robust control law M(τ ) via (8.36); 7: Step 5: Use M(τ ) to steer system (8.1b) over τ ∈ [tk , tk+1 ); 8: Step 6: Set tk ← tk+1 and go back to Step 1; 9: end procedure

˜ ∗ , ω˜ ∗c , ) of the developed inner and outer TRMPC can be denoted as ωc (τ ) = κω ( ˜ ∗ , ω) respectively, where the solutions to optimization modand M(τ ) = κ M (ω˜ ∗ , M ˜ ∗ , ω˜ ∗c ) and (ω˜ ∗ , M ˜ ∗ ), respectively. As mentioned in els (8.8) and (8.29) represent ( Lemmas 8.3 and 8.5, if ωc (τ ) and M(τ ) are used to regulate the systems (8.1a) and (8.1b), we can accomplish robust constraint satisfaction. Besides, the actual state trajectory is controlled by the developed compound control law to remain inside a tube region. Note that the optimal state trajectory, the feedback gain matrix K, and the upper bounds of the external disturbances (η1 or η2 ) all work together to define the tube region. Remark 8.3 In prior comparable studies [22, 23], the authors acquired the robust feedback law by handling another MPC, such that the influence caused by additive disturbances is reduced. In contrast to these studies [22, 23], the feedback linearization of (8.17) and (8.35) are respectively used to offline derive the feedback laws, which are obtained from (8.18) and (8.36). It implies that we only need to solve a nominal system-based open-loop optimization issue in real time, at time instant

250

8 Robust Model Predictive Control for Attitude Control Tracking

tk . One significant benefit of this approach is that we don’t need other extra optimization procedures, which in fact, considerably reduces the computational burden that must be carried out online. Nevertheless, it should be noted that when applying this approach, establishing tightened state/control constraints may bring us trouble. Furthermore, when addressing a variety of engineering situations in practice, it is possible to make it easier to apply the algorithm to practical issues.

8.4 Analysis of Feasibility and Stability 8.4.1 Recursive Feasibility We will study the primary theoretical characteristics of the developed TRMPC in this section. More exactly, in order to demonstrate the recursive viability of using the developed TRMPC control method, we will develop and evidence Theorem 8.1. Theorem 8.1 If a viable control solution for the optimization problems (8.8) and (8.29) at time point tk can be identified, considering the outer and inner tracking error systems are denoted as (8.4a) and (8.4b) respectively, the optimization problems are recursively viable for all tk+1 > tk . Proof To begin with, we consider the outer tracking system (8.4a) and optimization issues (8.8). And we presume that through (8.8), we can obtain an optimal control solution ω˜ ∗c (τ |tk ) successfully. Subsequently, we calculate the robust control law (8.18) and bring its result into the actual attitude angle system over ˜ ∗ (tk+1 |tk ) ⊕ Oθ , which implies that [tk , tk+1 ]. Lemma 8.3 brings us (tk+1 ) ∈  ∗ ˜ k+1 ) =  ˜ (tk+1 |tk ) is a viable nominal attitude angle value for optimization issue (t (8.8) at time instant tk+1 . With those primary conditions, a potential control sequence for issue (8.8) at tk+1 could be written as ω˜ c (τ |tk+1 ) =

ω˜ ∗c (τ |tk ) if τ ∈ [tk+1 , tk + T ) ω˜ cf (τ |tk ) if τ ∈ [tk + T, tk+1 + T )

(8.37)

It should be noted that the tail of ω˜ ∗c and the terminal control law provided by (8.10) are the two components, which make up this potential control sequence. We should recollect Lemma 8.2, which demonstrates that at tk+1 , ω˜ c (τ |tk+1 ) is a viable solution. It is evident that E˜ θ (τ |tk+1 ) can be directed into tube by ω˜ c (τ |tk+1 ) for τ ∈ θ is [tk+1 , tk + T ) with the control constraints and tightened state guaranteed. As tube θ a terminal invariant set, it follows that E˜ θ (τ |tk+1 ) ∈ tube under the terminal control θ ˜ |tk+1 ) ∈ Xtube . As ω˜ c (τ |tk+1 ) over [tk + T, tk+1 + T ), which also indicates that (τ θ a result, for (8.8), ω˜ c (τ |tk+1 ) is a viable solution at tk+1 . Finally, we accomplish the proof through induction.

8.5 Performance Evaluation

251

We will skip the evidencing process for the inner tracking system (8.4b) and optimization issue (8.29), considering its proof has similar steps to those of the outer loop. 

8.4.2 Control Stability The subsequent theorem (e.g., Theorem 8.2) demonstrates that the applying developed control method can ensure the ISS of the tracking error system, which further guarantees that the system with developed control method can successfully track the time-varying attitude angle profiles. Theorem 8.2 If the optimization process of the dual-loop TRMPC-based attitude control algorithm is workable at tk , then the outer and inner tracking error systems have ISS. Proof The Appendix includes the proof.



8.5 Performance Evaluation 8.5.1 Parameter Assignment In this section, we will carry out and show a variety of experimental investigations and analyses that were conducted to verify the usefulness and performance of the developed dual-loop TRMPC method. To begin with, we will provide the following details in terms of the attitude-dynamics-related parameters: Ix x = 434270slug · ft2 , Ix z = 17880slug · ft2 , I yy = 961200slug · ft2 , Izx = 17880slug · ft2 and Izz = 1131541slug · ft2 . And we define other items, which I contains, to zero. Besides, we should guarantee that the following constraints can be fulfilled by thesystem variables: X = { :  ≤ 45deg}, Xω = {ω : ω ≤ 5deg/s} and U M = M : M ≤ 1 × 105 lb · ft . Then, an attitude tracking scenario is constructed, in which we denote the reference angular signals as αr = 10 + 2.5 sin(0.5t), βr = 0, and σr = −30 + 5 cos(0.5t) respectively. We design the initial values for several variables, which are denoted as (0) = [7.5, 10, −30]T , ω(0) = [0, 0, 0]T and M(0) = [0, 0, 0]T . Right now, we have ensured the initial viability. Next, in terms of the TRMPC method, we determine the weight and local gain matrices for the inner-loop and outer-loop TRMPC as P = diag{8, 8, 8}, Q = diag{2, 2, 2}, ˜ = diag{2.5, 2.5, 2.5}, by applying Lemmas 8.2 and 8.4. We R = diag{1, 1, 1} and K set η1 = 0.5 and η2 = 0.1 to bound  f and d , respectively. For the two loops, we determine K = diag{5, 5, 5} as the feedback gain matrix. With the prediction horizon T = 20 s, we select the sampling period as 0.2 seconds.

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8 Robust Model Predictive Control for Attitude Control Tracking

8.5.2 Experimental Setup We will construct a hardware-in-the-loop platform in which the experimental tests can be carried out. Specifically, as Fig. 8.3 illustrates, the system test platform consists of an inertial measurement unit (IMU), a controller unit (ADVANTECH 610L with I58500/4G/1TB and PCI-1723-BE D/A converting module), a 3-axis rotating platform, and a spaceship model simulator (NI PXIe-8820 2.2 GHz Celeron 1020E Dual-Core, NI PXI-6723 D/A converting module and PXI-6224 A/D converting module). The operating principle of the platform is depicted in Fig. 8.4. We use the NI PXI real-time controller with NI LabVIEW real-time module version 8.0 to carry out the spacecraft model, in which attitude angle dynamics and angular velocity are taken into account. Then, by utilizing a D/A converter module, the 3-axis rotating platform can be driven by our controller. The developed control law can be produced by dealing with the spacecraft parameters (angle of attack; sideslip and bank angles; and roll, pitch and yaw angular rates), which are collected from the IMU and then delivered to the controller unit through RS232.

8.5.3 Tracking Performance Evaluation and Comparative Studies Figure 8.5 illustrates the tracking performance, which the developed dual-loop TRMPC method provides with the existence of constraints and disturbances. This figure makes it evident that a promising tracking performance for the discussed issue can be provided by using the developed method, which means that we

Fig. 8.3 System test platform

8.5 Performance Evaluation

Fig. 8.4 Working principal of the platform

Fig. 8.5 Attitude tracking and attitude tracking error evolution

253

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8 Robust Model Predictive Control for Attitude Control Tracking

Fig. 8.6 Angular rate evolution trajectories

can successfully track the expected angle profiles in a short amount of time and rapidly control the tracking errors of  to a tiny neighborhood of the origin. The related angular rate evolution trajectories and the input profiles are shown in Figs. 8.6 and 8.7, respectively. In these two figures, it is evident that during the whole closed-loop control process, all the variable constraints (represented by the red dashed line) are fulfilled, which demonstrates that in the TRMPC optimization formulation, limiting the input variables and the nominal system state to a tightened region works effectively. However, it should also be noted that the generated angular rate profiles have several irregularity in the shape, which may, in fact, be induced by a variety of factors, including the periodic and nonlinear properties of the reference angular trajectories and the consideration of additive disturbances working on the angular rate dynamics. But by comparing other results given by different methods, it is obvious that the trajectory curves are relatively-smooth. Moreover, it is also important to note that when looking at the input evolution profiles shown in Fig. 8.7, a few minor oscillations can be seen. Actually, the non-linear feedback component of the developed control law is the primary reason of bringing about the oscillations. In conclusion, the method is attempting to limit the state deviations between the noise-perturbed system and the nominal system. Subsequently, we carry out comparative works between the developed method and other newly established methods to demonstrate the benefits of implementing TRMPC. Specifically, we choose the non-linear MPC (NMPC)-based attitude tracking approach presented in [19] and the sliding mode control (SMC)-based tracking approach presented in [3] as the schemes to be compared. It should be noted that

8.5 Performance Evaluation

255

Fig. 8.7 Input evolution trajectories

because the chosen NMPC disregards the disturbances in dynamics and only depends on the built-in robustness of deterministic MPC, we can consider the chosen NMPC to be a non-robust method. Figs. 8.5, 8.6 and 8.7 demonstrate the related attitude angle tracking trajectories evolution profiles, angular rate evolution profiles, and control input  t evolution1 profiles, respectively. Then, we use the performance measure Ind1 = ( 0 f Eθ 2 dt) 2 by which the reference tracking (steady-state) performances of various methods can be objectively evaluated. For easy demonstration, Table 8.1 provides specific data, such as the scaled Ind1 and the peak constraint values. It is obvious that a superior reference tracking performance than that of the competitors (e.g., smaller Ind1 value) can be obtained by the developed TRMPC approach, as evidenced by the outcomes displayed in Figs. 8.5, 8.6 and 8.7 and Table 8.1. Besides, the NMPC- and SMC-based tracking methods also bring out harder control evolution profiles and harder angular rate evolution profiles. More crucially, from Fig. 8.6, it is obvious that with the influence of additive disturbances, we are pre-

Table 8.1 Comparative results of different methods Method Ind1 Peak constraint max(| p|) TRMPC NMPC [19] SMC [3]

131.0882 137.4215 131.869

5.0000 8.3778 9.3319

max(|q|)

max(|ν|)

3.6108 3.7954 5.1652

4.2630 6.2475 7.1271

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8 Robust Model Predictive Control for Attitude Control Tracking

vented from maintaining the actual angular rate profiles in the specified constraint zones by using the NMPC and SMC-based approaches. As a result, we can draw the conclusion that the developed TRMPC control method provides benefits when addressing physical constraints and disturbance rejection.

8.5.4 Case Studies on Algorithm Parameters O is a crucial variable that we need to precisely study. The value of O indicates the difference between the optimized nominal trajectory and the controlled trajectory, such as the trajectory with and without the robust feedback control component incorporated in the developed TRMPC. We should make sure that O can remain within an invariant tube region based on the TRMPC design. The Fig. 8.8 presents the evolution trajectories of O = [αe , βe , σe ]T , such that we can further demonstrate this idea. Moreover, according to Lemma 8.3, we can represent the invariant tube region as ⎡ ⎤ ⎡ 1 ⎤  |αe |

10  1 ⎦ O = [αe , βe , σe ]T ∈ R3  ⎣ |βe | ⎦ ≤ ⎣ 10 1 |σe | 10 The blue dashed lines in Fig. 8.8 represent the tube boundaries. From Fig. 8.8, the outcome makes it clear that the boundary lines do not cut the trajectories of αe ,

Fig. 8.8 Angular variable deviation trajectories

8.6 Conclusions

257

Table 8.2 Results for different cases Case no. Performance Ind1 Case 1 Case 2 Case 3

131.0882 129.7358 119.1921

Ind2 2.8285 2.9950 4.8494

βe , and σe at any point during the evolution, which demonstrates that the developed TRMPC can keep the state error value in the invariant tube region. Then, we will carry out comparative case works so that effects of using various gain matrices K on the tracking performance can be examined. To be specifical, we will discuss three instances: Case 1 K1 = diag{10, 10, 10}, resulting in O1 = [0.1, 0.1, 0.1]T ; Case 2 K2 = diag{15, 15, 15}, resulting in O2 = [0.067, 0.067, 0.067]T ; Case 3 K3 = diag{20, 20, 20}, resulting in O3 = [0.05, 0.05, 0.05]T . The related state deviation outcome values are presented in Fig. 8.8, in which the blue, red, and black dashed lines, respectively indicate the tube boundaries for Cases 1–3. We quantify and provide the specific tracking outcomes in Table  t 8.2. To make 1 the control cost measurable, we introduce a new indicator Ind2 = ( 0 f M2 dt) 2 in Table 8.2 such that we can assess the control cost. According to the state deviation curves in Fig. 8.8 and the outcomes in Table 8.2, it is obvious that we can have a better angle tracking performance by adopting a bigger feedback gain value. Besides, a bigger feedback gain value can also lessen the difference between the optimized nominal solutions and the actual state trajectories. Nevertheless, Lemma 8.3 states that the size of the state deviation tube tends to become small with a bigger feedback gain value. As a result, we should pay more control costs in order to remain the actual state variable within a relatively-tight tube region.

8.6 Conclusions In this study, we first built a dual-loop control framework, in which the expected spacecraft attitude reference trajectory can be well-followed, despite disruptions and system constraints. Second, we developed a novel TRMPC method and applied it as the controller to output the control actions for the spacecraft angular and angular rate systems. In this way, the tracking performance can be improved and optimized with robust constraint fulfillment maintained. Third, we obtained the specific requirements relating to the recursive feasibility and ISS through the development of an invariant set and a terminal control law. Forth, according to the given experimental outcomes,

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the ability of the developed TRMPC method to rapidly guide the attitude motion to track the expected reference trajectory with state and input limitations satisfied was confirmed. Besides, compared to the other approaches examined in this study, the developed TRMPC is able to produce a superior attitude tracking performance, which was confirmed in the comparative studies. As a result, we conclude that TRMPC is a promising control algorithm, which can be trusted to use the developed attitude control method to solve the considered problem about reference tracking. It is important to note that the primary drawback of the developed TRMPC method is that several degrees of conservativeness are introduced as a result of the use of a robust feedback law, which will reduce the optimality of the solution. As a result, future research may be executed to improve the TRMPC method in order to lessen the conservativeness with system robustness maintained.

8.7 Appendix: Proof of Theorem 2 Proof The whole evidencing process includes two stages. To begin with, it should be demonstrated that the tracking error of the nominal attitude system is able to be directed to its origin. Next, the demonstration should be given to show that the robust feedback law could be used to guarantee the real state trajectory to converge to an invariant tube area, in which the center corresponds to the optimized trajectory of the ˜ θ∗ (tk )) nominal system. To be specific, we select the optimum objective Jθ (E˜ θ∗ (tk ), U as the Lyapunov function for the outer attitude angle tracking error system. Subsequently, the following inequality must be fulfilled, where Vθ denotes the difference in this Lyapunov function at tk and tk+1 : ˜ θ (tk+1 )) − Jθ (E˜ θ∗ (tk ), U ˜ θ∗ (tk )) Vθ ≤ Jθ (E˜ θ (tk+1 ), U  tk +T ˜ θ∗ (τ |tk )2Q )dτ (E˜ θ∗ (τ |tk )2P + U =−  +

tk tk+1 +T

tk+1

˜ θ (τ |tk+1 )2Q )dτ (E˜ θ (τ |tk+1 )2P + U

− E˜ θ∗ (tk + T |tk )2R + E˜ θ (tk+1 + T |tk+1 )2R  tk+1 +T ˜ θ (τ |tk+1 )2Q )dτ = (E˜ θ (τ |tk+1 )2P + U +

tk +T  tk

tk+1

(8.38)

˜ θ∗ (τ |tk )2Q )dτ (E˜ θ∗ (τ |tk )2P + U

− E˜ θ∗ (tk + T |tk )2R + E˜ θ (tk+1 + T |tk+1 )2R ˜ θ (tk )) stands for the objective value of the designed solution In (8.38), Jθ (E˜ θ (tk ), U (8.37). It should be noted that from Lemma 8.2, it is obtained that g˙ θ (E˜ θ (τ |tk )) + ˜ θ (τ |tk )) ≤ 0. Then, the equation should be integrated from tk + T to L θ (E˜ θ (τ |tk ), U

References

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tk+1 + T , and its result needs to be applied to (8.38). Finally, the inequality Vθ ≤ 0 is gotten, which implies that the angle tracking error vector E˜ θ is asymptotically converged to the origin. According to the convergence property of E˜ θ , there exists a K function , which can ensure that for each t > 0, we can obtain E˜ ∗ (t) ≤ K(E∗ (0), t). Based on Theorem 8.1 and Lemma 8.3 (2), Oθ (t) ∈ Oθ can be derived, which indicates the existence of a K∞ function, which can ensure that for every t > 0, Oθ (t) ≤ K∞ (η1 ). Therefore, it follows that for every t > 0, the inequality E(t) ≤ K(E∗ (0), t) + K∞ (η1 ) can be satisfied, which indicates that the outer tracking error system has ISS and also implies that the actual attitude angle tracking error is ultimately bounded. Here, we will skip the evidencing process for the inner tracking system, considering that its proof has a similar format to that of Lemma 8.2. 

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