Resilient Fusion Navigation Techniques: Collaboration in Swarm 9811983704, 9789811983702

Citation preview

Unmanned System Technologies

Rong Wang Zhi Xiong Jianye Liu

Resilient Fusion Navigation Techniques: Collaboration in Swarm

Unmanned System Technologies

Springer’s Unmanned System Technologies (UST) book series publishes the latest developments in unmanned vehicles and platforms in a timely manner, with the highest of quality, and written and edited by leaders in the field. The aim is to provide an effective platform to global researchers in the field to exchange their research findings and ideas. The series covers all the main branches of unmanned systems and technologies, both theoretical and applied, including but not limited to: . Unmanned aerial vehicles, unmanned ground vehicles and unmanned ships, and all unmanned systems related research in: . Robotics Design . Artificial Intelligence . Guidance, Navigation and Control . Signal Processing . Circuit and Systems . Mechatronics . Big Data . Intelligent Computing and Communication . Advanced Materials and Engineering The publication types of the series are monographs, professional books, graduate textbooks, and edited volumes.

Rong Wang · Zhi Xiong · Jianye Liu

Resilient Fusion Navigation Techniques: Collaboration in Swarm

Rong Wang Navigation Research Center College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

Zhi Xiong Navigation Research Center College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

Jianye Liu Navigation Research Center College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

ISSN 2523-3734 ISSN 2523-3742 (electronic) Unmanned System Technologies ISBN 978-981-19-8370-2 ISBN 978-981-19-8371-9 (eBook) https://doi.org/10.1007/978-981-19-8371-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To our families for their love and support

Preface

Aerial swarms are one of the future trends in the development of aeronautics technology, with the advantages of a wide operating range and the ability to perform multiple missions. Ensuring the mission completion rate and the safety and stability of swarm vehicles in complex and dynamic environments requires the support of navigation technology with high accuracy and reliability. Although unmanned swarms are receiving increasing attention, both through theoretical research and through increasing participation in industrial developments, the enhancement of navigation by means of effective and efficient collaboration remains largely unexplored. While my scholarly work has explored various aspects related to adaptive navigation systems (such as the navigation systems of robots and ground vehicles, aircraft, aerospace vehicles, and unmanned aerial vehicles), such as modelling, error characteristics, fusion algorithms, and fault detection and isolation, the present book presents a specialized investigation of navigation with resilient characteristics, which allows performance to be maintained through essential collaboration among members of a swarm in a global navigation satellite system (GNSS)-challenged environment. The lack of theories and algorithms related to resilient navigation fusion under multiplatform collaboration conditions is a bottleneck limiting the development of self-healing capabilities for unmanned swarm navigation. The traditional methods of fault-tolerant navigation theory have been proposed mainly to address the performance requirements of single-vehicle airborne navigation systems. Usually, the sources of redundant information for airborne augmentation systems are concentrated on the carrier platforms, while the sources of redundant information for ground-based augmentation systems generally consist of stations with fixed known locations. The sources of redundant information for unmanned swarm navigation consist of the collection of sensors on each platform scattered among the swarm members, and the redundancy of the information used is derived from a “crowd-sourced” body of navigation information obtained through independent and collaborative measurement. The mapping relationship between the navigation performance and observation error under cooperative conditions differs significantly from that for single-platform navigation: the observations are spatially distributed in an uncentralized manner, vii

viii

Preface

and the solutions of each platform’s navigation system are coupled and correlated rather than completely independent. The distributed nature of the redundant information sources for aerial swarms represents a significant difference compared to existing airborne and ground-based navigation augmentation approaches, and there is a lack of systematic research on quality control models for global swarm navigation information and optimal reconfiguration algorithms. This book addresses these issues by proposing that, unlike in traditional airborne and ground-based navigation augmentation, unmanned swarms can achieve resilient navigation fusion capabilities through intermember collaboration. In this book, we focus on resilient navigation fusion technology based on cooperation in the swarm and present research on six topics, namely collaborative fusion frameworks, modelling methods for collaboration, collaborative positioning fusion algorithms, collaborative geometry optimization algorithms, collaborative integrity augmentation algorithms, and collaborative fault-tolerant algorithms, to improve the comprehensive fault tolerance of swarm vehicle navigation. The frameworks of and modelling for collaborative resilient navigation fusion are discussed in this book. The algebraic and geometric fundamentals of collaborative resilient navigation fusion are introduced; various collaborative navigation structures of an aerial swarm and information fusion frameworks are introduced in Chap. 2, and 3 addresses the necessary modelling for resilient fusion in collaborative navigation. Positioning and geometry optimization algorithms for collaborative resilient navigation fusion are also proposed in this book. This part of the book focuses on improving the navigation accuracy of an aerial swarm in an environment with insufficient GNSS observations. Positioning algorithms for collaborative resilient navigation fusion are introduced, including the collaborative localization-based approach in Chap. 4 and the collaborative observation-based approach in Chap. 5. Furthermore, geometry optimization algorithms to improve navigation accuracy in resilient fusion for collaborative navigation are introduced in Chap. 6. Furthermore, integrity augmentation and fault-tolerant algorithms for collaborative resilient navigation fusion are proposed in this book. This part of the book focuses on improving the navigation robustness of an aerial swarm in an environment with insufficient GNSS observations. Optimal collaborative integrity augmentation methods to improve navigation integrity protection level in collaborative resilient navigation fusion are introduced in Chap. 7. On this basis, the collaborative fault detection methods to realize fault identification and exclusion in resilient fusion for collaborative navigation are introduced in Chap. 8. This book combines research results on resilient navigation fusion with hot topics in unmanned swarm development and will serve as a reference for those who are engaged in research on unmanned swarm collaboration theory, robust positioning, and navigation augmentation technology. The target audience includes postgraduate students, scholars, and general readers interested in navigation technologies and their application in autonomous collaborative swarms. This book is also relevant to

Preface

ix

people studying the operation of autonomous collaborative swarms with a focus on guidance, navigation, and control techniques. Nanjing, China October 2022

Rong Wang Zhi Xiong Jianye Liu

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (62073163, 61703208, 61873125), the Natural Science Foundation of Jiangsu Province (BK20170815), and the Qing Lan Project (2022). We would like to express our sincere thanks to the National Science Foundation of China. It has been a great pleasure to work with the colleagues of Springer. The support and help from Mr. Wayne Hu (the Project Editor) are greatly appreciated. We would like to acknowledge the contribution of Mr. Junnan Du, Miss Xin Chen, Miss Huiyuan Zhang, Mr. Li Liu, Mr. Hui He, and Mr. Weicheng Zhao, six students at the Nanjing University of Aeronautics and Astronautics, who helped conduct the tasks and preparing parts of the content. Discussion with members of the Navigation Research Center, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, is appreciated.

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Rise of Aerial Swarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Sensors for Relative Observation in an Aerial Swarm . . . . . . . . . . . . 1.2.1 Collaborative Navigation Based on Radio Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Collaborative Navigation Based on Visual Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Development of Collaborative Resilient Navigation . . . . . . . . . . . . . 1.3.1 Collaborative Navigation Research Focusing on the Suppression of INS Error Dispersion . . . . . . . . . . . . . . 1.3.2 Collaborative Navigation Research Focusing on Improving Positioning Accuracy in GNSS-Challenged Environments . . . . . . . . . . . . . . . . . . . . . 1.3.3 Collaborative Navigation Research Focusing on Improving the Collaborative Configuration of a Swarm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Fault-Tolerant Navigation Technology with Collaboration . . . . . . . . 1.4.1 Collaborative Fault-Tolerant Navigation Research Involving Integrity Protection Level Assessment and Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Collaborative Fault-Tolerant Navigation Research Involving Navigation Fault Detection and Identification Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Organization of Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3

10 11 12 14

2 Collaborative Resilient Navigation Frameworks . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Leader–Follower Collaborative Navigation Structure . . . . . . . . . . . . 2.3 Parallel Collaborative Navigation Structure . . . . . . . . . . . . . . . . . . . . .

19 19 22 22

3 4 5 6

6

7 8

9

xiii

xiv

Contents

2.4 Hierarchical Collaborative Navigation Structure . . . . . . . . . . . . . . . . 2.5 Collaborative Localization-Based Fusion Framework for Resilient Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Collaborative Observation-Based Fusion Framework for Resilient Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3 Modelling for Resilient Navigation via Collaboration . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Coordinate Frames Used in Resilient Navigation via Collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Ranging-Based Collaborative Observation Modelling . . . . . . . . . . . . 3.3.1 Relative-Ranging-Based Observation Geometry . . . . . . . . . . 3.3.2 Relative-Ranging-Based Observation Model . . . . . . . . . . . . . 3.3.3 Relative-Ranging-Based Error Model . . . . . . . . . . . . . . . . . . . 3.4 Range-Difference-Based Collaborative Observation Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Relative-Range-Difference-Based Observation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Relative-Range-Difference-Based Observation Model I ................................................... 3.4.3 Relative-Range-Difference-Based Observation Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Relative-Range-Difference-Based Error Model . . . . . . . . . . . 3.5 Bearing-Only Collaborative Observation Modelling . . . . . . . . . . . . . 3.5.1 Relative-Bearing-Only Observation Geometry . . . . . . . . . . . . 3.5.2 Relative-Bearing-Only Observation Model . . . . . . . . . . . . . . . 3.5.3 Relative Bearing Error Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Vector-of-Sight-Based Collaborative Observation Modelling . . . . . . 3.6.1 Relative-VOS-Based Observation Geometry . . . . . . . . . . . . . 3.6.2 Relative VOS Observation Model in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Relative VOS Error Model in Spherical Coordinates . . . . . . 3.6.4 Relative VOS Observation Model in Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Relative VOS Error Model in Cartesian Coordinates . . . . . . 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29

4 Collaborative Localization-Based Resilient Navigation Fusion . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Collaborative Localization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Least Squares Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Chan–Taylor Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Spherical Interpolatio Algorithm . . . . . . . . . . . . . . . . . . . . . . .

65 65 67 67 68 71

25 25 26 27

31 33 33 33 37 39 39 40 42 43 44 44 45 49 51 51 51 55 57 61 62 62

Contents

4.3 Online Estimation of the Collaborative Localization Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 LS Estimation Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 CT Estimation Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 SI Estimation Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Resilient Fusion Algorithm with the Collaborative Localization Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Resilient Fusion Model with the Collaborative Localization Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Resilient Fusion Process with the Collaborative Localization Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Simulation of CT and SI Collaborative Localization-Based Resilient Fusion . . . . . . . . . . . . . . . . . . . . 4.5.2 Simulation of LS Collaborative Localization-Based Resilient Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Collaborative Observation-Based Resilient Navigation Fusion . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Collaborative Observation-Based Navigation Algorithm with the Hierarchical Collaborative Navigation Structure . . . . . . . . . 5.2.1 Resilient Fusion Model with Collaborative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Resilient Fusion Process with Collaborative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Collaborative Observation-Based Navigation Algorithm with the Parallel Collaborative Navigation Structure . . . . . . . . . . . . . 5.3.1 Resilient Fusion Model with Collaborative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Resilient Fusion Process with Collaborative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Simulation of Hierarchical Collaborative Observation-Based Resilient Fusion . . . . . . . . . . . . . . . . . . . . 5.4.2 Simulation of Parallel Collaborative Observation-Based Resilient Fusion . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

72 72 72 73 73 73 75 78 78 84 89 90 93 93 95 95 98 100 100 102 104 104 110 115 115

xvi

Contents

6 Collaborative Geometry Optimization in Resilient Navigation . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Geometric Dilution of Precision in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Geometric Configuration in Collaborative Navigation . . . . . 6.2.2 Geometric Dilution of Precision in Collaborative Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Geometric Influence on Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Geometric Influence on Collaborative Resilient Fusion in a GNSS-Augmented Situation . . . . . . . . . . . . . . . . . 6.3.2 Geometric Influence on Collaborative Resilient Fusion in a GNSS-Denied Situation . . . . . . . . . . . . . . . . . . . . . 6.4 Geometry Optimization for Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Geometry Optimization Based on a Geometric Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Geometry Optimization Based on an Algebraic Search Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Simulation of Geometric-Analysis-Based Collaboration Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Simulation of Algebraic-Search-Based Collaboration Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119

7 Collaborative Integrity Augmentation in Resilient Navigation . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Integrity in Collaborative Resilient Navigation Fusion . . . . . . . . . . . 7.2.1 Geometry in Collaborative Navigation Integrity Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Integrity Protection Level in Collaborative Navigation . . . . . 7.3 Influence of Integrity in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Influence of Integrity in Collaborative Resilient Fusion in a GNSS-Augmented Situation . . . . . . . . . . . . . . . . . 7.3.2 Influence of Integrity in Collaborative Resilient Fusion in a GNSS-Denied Situation . . . . . . . . . . . . . . . . . . . . . 7.4 Integrity Optimization in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Integrity Optimization Based on the Geometric Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Integrity Optimization Based on the Algebraic Search Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 151

123 123 124 126 126 128 129 129 131 134 134 142 146 147

151 152 155 155 160 161 161 163

Contents

xvii

7.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Simulation of Cooperative Partner Optimization . . . . . . . . . . 7.5.3 Simulation of Navigation Integrity Augmentation . . . . . . . . . 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 165 166 169 170

8 Collaborative Fault Detection in Resilient Navigation . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Fault Detection Based on Integrity Augmentation in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . 8.2.1 Collaboration-Augmented Fault Detection . . . . . . . . . . . . . . . 8.2.2 Collaboration-Augmented Fault Identification and Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fault Detection Based on MHSS in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Multifault Detection Scheme with Collaborative Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Collaboration-Augmented Fault Detection . . . . . . . . . . . . . . . 8.3.3 Collaboration-Augmented Fault Identification and Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Simulation of Single-Fault Detection in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Simulation of Multifault Detection in Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Comprehensive Simulation of Collaborative Resilient Navigation Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173

9 Summary and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Development Trends and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 210 212

175 175 177 178 178 180 181 182 182 186 193 204 204

Abbreviations

3D AP CA-MHSS CGDOP CHPL CT DGPS DSRC ECEF EKF ENU FDE GDOP GNSS GPS HAL HC HDOP HPL IMU INS MEMS MHSS NC NLOS P2P PC RAIM RMSE ROI SAR

Three-dimensional Proportions of Anchors Collaborative Augmented Multiple Hypothesis Solution Separation Collaborative Geometric Dilution of Precision Collaborative Horizontal Protection Level Chan–Taylor Algorithm Differential Global Positioning System Dedicated Short-Range Communications Earth-centred-Earth-fixed Extended Kalman Filter East–North–Up Fault Detection and Exclusion Geometric Dilution of Precision Global Navigation Satellite System Global Positioning System Horizontal Alarm Limit Hierarchical Collaborative Navigation Structure Horizontal Dilution of Precision Horizontal Protection Level Inertial Measurement Unit Inertial Navigation System Micro Electro-Mechanical System Multiple Hypothesis Solution Separation Non-collaborative Non-line of sight Peer-to-Peer Paralleled Collaborative Navigation Structure Receiver Autonomous Integrity Monitoring Root Mean Square Error Regional of Interest Search and Rescue xix

xx

SI SLAM SOP TDOA UAV UGV UWB VAL VOS

Abbreviations

Spherical Interpolation Algorithm Simultaneous Localization and Mapping Single of Opportunity Time Difference of Arrival Unmanned Aerial Vehicles Unmanned Ground Vehicle Ultra-Wide Band Vertical Alarm Limit Vector of sight

Chapter 1

Introduction

Abstract Aerial swarms are a development trend for future unmanned systems, and they show significant advantages for improving navigation accuracy and reliability. Unlike in traditional airborne and ground-based navigation augmentation, unmanned swarms can achieve resilient navigation fusion capabilities through intermember collaboration. This chapter discusses the advantages of an aerial swarm and the benefits of introducing collaboration in the swarm. The sensors used for relative observation in an aerial swarm are introduced. The existing research work related to collaborative resilient navigation as well as fault-tolerant navigation technology with collaboration is introduced through a literature overview. The motivations of this book are discussed, and the organization of the content is described. Keywords Aerial swarm · Collaborative resilient navigation · Relative observation · Inertial navigation system error suppression · GNSS-challenged environments · Collaborative configuration · Fault-tolerant navigation

1.1 The Rise of Aerial Swarms A swarm is a social group of many individuals organized in a certain way with a common goal, who can communicate and collaborate and who show clear group intelligence. The concept of “swarms” originated from biologists’ studies of animal groups, in which individuals can accomplish complex tasks far beyond their own capabilities through organized and disciplined group behaviours, such as the migration of a flock of birds or the attack of a wolf pack, a phenomenon that has led to the study of “biological swarms” [1]. It was from the study of such “bioswarming” that aircraft swarming technology was developed. Due to environmental complexity and task diversity in applications such as emergency search and rescue (SAR) in canyons or urban areas with extensive infrastructures, it may be difficult for a single unmanned unit to perform effective and efficient tracking due to its limited sensing range and search scope. However, by sharing information and collaborating with each other, the ability of multiple

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_1

1

2

1 Introduction

such aircraft to execute complex missions could be effectively improved [2]. Therefore, aerial swarming is a development trend for future unmanned systems, with the following major advantages. 1. Distribution of functions for completing tasks. For a geodesy remote sensing mission, members of an aerial swarm carrying different types of loads can simultaneously complete multipoint acquisition and measurement tasks. For a SAR task, members of an aerial swarm carrying different equipment can simultaneously perform reconnaissance, search, rescue, and other operational tasks [3]. 2. Stronger robustness and fault tolerance. An aerial swarm is spatially, temporally, and functionally distributed. If the functioning of a single aircraft is impaired, the swarm system can re-establish synergy as necessary for redistribution; if this can be achieved within a short period, such a loss of function will not cause the failure of the entire system, and collaborative fault diagnosis among the multiple aircraft can support the reconfiguration of such a distributed system [4]. 3. Lower cost and greater economic advantages. Aircraft swarm systems allow expensive and complex individual systems to be decomposed into multiple lowcost, small, collaborative platforms. Not only can the use of smaller platforms reduce production costs, but such platforms are also relatively easy to maintain, secure, and manage, resulting in greater economic benefits [5]. Due to the advantages of aerial swarms, they can be applied in a wide range of complex and dynamic environments to ensure mission completion while guaranteeing safety and stability. The reliability and flexibility of a swarm navigation system are critical to the ability to construct an effective aerial swarm. Collaborative resilient navigation techniques make use of the interoperability of the multiple units in a swarm to improve not only the accuracy but also the reliability of the airborne navigation system. The study of collaborative technology originated from the theory of synergy established by the physicist Harken, which states that the coordinated work of multiple kinematic platforms can produce a “1 + 1 > 2” performance improvement [6]. By utilizing multivehicle information synergy combined with information fusion technologies, the specific navigation advantages that can be gained through collaboration among the members of a swarm are as follows. 1. Improving navigation accuracy. Based on the differences among the swarm members in the working environment during a mission, through information sharing and exchange, members with favourable navigation performance can provide synergistic observation information to members with poor navigation performance to help improve their navigation performance and even that of the whole aerial swarm [7]. 2. Augmenting navigation integrity monitoring. The redundant relative observations of multiple members can be analysed to utilize the distributed and related nature of the information to predict navigation sensor signal degradation and to monitor the health of the navigation system by computing statistical data,

1.2 Sensors for Relative Observation in an Aerial Swarm

3

thus guaranteeing the fault tolerance of the navigation system in complex environments [8]. 3. Facilitating fault identification and isolation. Collaborative observation among members in a swarm can provide redundant measurement information for the navigation system, and in the case of navigation sensor equipment failure, collaborative fault identification models can be established through the fusion of these distributed sources of information, thus assisting in navigation fault detection and isolation [9]. In summary, aerial swarming is a major future trend in aircraft development, with the advantages of a large operating range and the ability to perform multiple tasks. Navigation accuracy and reliability are critical for ensuring the flight stability and mission completion rate of an aerial swarm. Information collaboration among the members of the swarm can improve the navigation accuracy and fault tolerance of the navigation system. Therefore, research on the collaborative resilient navigation of swarms is of great theoretical significance and application value for the development of swarm vehicles.

1.2 Sensors for Relative Observation in an Aerial Swarm After the physicist Harken proposed the theory of synergy, collaborative navigation technology began to receive attention and was gradually applied to mobile robots, underwater submersibles, flying vehicles, and other devices [8]. In collaborative navigation positioning, relative observation information between vehicles can be obtained through radio navigation, visual navigation, etc. [9–11]. On this basis, corresponding collaborative navigation algorithms can be used to improve the overall navigation accuracy of the system.

1.2.1 Collaborative Navigation Based on Radio Measurements Radio measurement is one means of obtaining relative observations for the collaborative navigation of aerial swarms; in particular, relative range, angle, or speed measurements can be obtained in this way [12, 13]. Other types of relative measurements similar to radio measurements include radar range measurements and laser range measurements [14]. For determining the relative attitude between aircraft, researchers at West Virginia University proposed a method of estimating the relative attitudes of an aerial swarm using radio range measurements as well as aircraft onboard navigation information [15]. Their method does not require a priori information about the attitude of each vehicle and uses a non-linear least squares approach to estimate the attitude of a

4

1 Introduction

vehicle by considering the range and relative displacement information, and simulations have verified the effectiveness of the algorithm. Researchers at Brigham Young University proposed generating a compressed image of a radar echo signal for estimating the relative motion of a vehicle by using an extended Kalman filter (EKF) to fuse airborne inertial measurement unit (IMU) data and effectively reduce the drift error of the IMU data when the satellite signal disappears [16]. For determining the relative position between aircraft, researchers at Vanderbilt University obtained the relative position information of multiple aircraft nodes from satellite navigation position data and proposed a differential tracking method based on satellite navigation, which effectively improves the position accuracy compared to that achieved using the absolute position coordinates of the nodes [17]. Researchers at West Virginia University proposed an algorithm for fusing carrier-phase differential Global Positioning System (GPS) data, intervehicle ranging information, and inertial navigation system (INS) data, which is achieved by tightly combining the position of each aircraft based on its relative range. This algorithm improves the robustness of three-dimensional (3D) relative positioning by fusing each respective aircraft’s tight combination position information through relative navigation filters and using radio ranging information for baseline separation [18]. Researchers at the French Aerospace Laboratory proposed a method of achieving collaborative multicraft positioning and tracking by measuring the time-varying distance and relative and absolute velocity information among multiple aircraft and then using the measured relative distance and velocity information to estimate the relative positions of neighbouring craft and fuse them with the communication positions of the craft to achieve accurate position tracking [19]. A collaborative positioning scheme designed to improve the absolute position accuracy of a vehicle in a swarm by introducing range and directional information within the swarm vehicle system has also been proposed by researchers at Central South University for a swarm vehicle whose relative position accuracy is significantly higher than its absolute position accuracy [20].

1.2.2 Collaborative Navigation Based on Visual Measurements Visual measurement sensors are small in size and easy to use and can provide a wealth of information [21]. Visual sensors can observe the relative positions and attitudes of other members in a swarm and thus allow a swarm member to estimate its own position and attitude by matching relevant observations to correct its onboard navigation results [22]. Researchers at the Georgia Institute of Technology presented a method for distributed vision-aided cooperative localization and navigation for multiple intercommunicating autonomous vehicles based on three-view geometry constraints. A graph-based approach was applied for calculating the correlation terms between the

1.3 Development of Collaborative Resilient Navigation

5

navigation parameters associated with images participating in the same measurement. Experiments showed that such cooperative three-view-based vision-aided navigation may considerably improve the performance of an inferior INS [23]. Researchers at Embry–Riddle Aeronautical University proposed an architecture for solving the problem of state tracking estimation for the primary vehicle in a swarm in closeformation flight based on visual measurements and machine learning principles [24]. Researchers at West Virginia University proposed an algorithm for estimating the relative attitudes of multiple vehicles in a satellite-denied environment. The algorithm estimates the relative attitudes of vehicles by collecting information on the measured point-to-point distances between vehicles, computer vision measurements, and IMU and magnetometer data and constructs a non-linear Kalman filter to estimate the relative attitudes of the vehicles, thus achieving a high attitude estimation accuracy [25]. Researchers at the Korea Advanced Institute of Technology proposed a visual-sensorbased method of estimating the attitude of a vehicle. The algorithm first performs noise suppression on the vehicle beacon position and constructs a relative navigation filter consisting of the relative position and relative attitude, and its estimation accuracy and applicability are improved compared with conventional algorithms. This algorithm relies on a sigma-point Kalman filter to estimate the relative position, velocity, and attitude information of the vehicle by fusing inertial navigation information and line-of-sight measurement information [26]. The algorithm effectively achieves improved navigation accuracy and convergence speed. In summary, research works on relative position or attitude measurement algorithms based on radio and vision sensors have proven beneficial in introducing relative observations, providing a foundation for further research on resilient fusion navigation techniques for collaboration in aerial swarms.

1.3 Development of Collaborative Resilient Navigation Since the theory of synergy was proposed, scholars have also gradually started to investigate collaborative navigation technology, which was first applied to unmanned underwater vehicles and later extended to multiple types of mobile platforms, such as unmanned aircraft and mobile robots [27–29]. Many applications of vehicle swarms often require working in low-altitude environments where global navigation satellite system (GNSS) signals will suffer from anomalies due to multipath effects or signal absorption and obstruction by surrounding obstacles such as buildings, hills, bridges, and vegetation. These scenarios are often referred to as GNSS-challenged environments [30]. In a GNSSchallenged environment, the integrated INS/GNSS navigation system of a single vehicle will often suffer from insufficient satellite signal availability and a poor satellite configuration [31, 32]; consequently, reliable navigation performance cannot be guaranteed, which may cause severe degradation in navigation accuracy for the whole swarm. However, by obtaining relative observations such as distances and angles between members in the swarm and then fusing them with the information

6

1 Introduction

measured by the members individually through filter or graph theory, collaborative resilient navigation can be realized. In this way, INS error dispersion can be suppressed, and the collaborative configuration, as well as the positioning accuracy in GNSS-challenged environments, can be improved.

1.3.1 Collaborative Navigation Research Focusing on the Suppression of INS Error Dispersion Generally, integrated navigation systems utilize GNSS measurements to suppress INS errors. In complex environments where satellite signals are limited, relative observation information among multiple members of an aerial swarm can be used to build a collaborative navigation model to effectively suppress INS error dispersion. Researchers at the University of New South Wales proposed a collaborative inertial navigation method for use in a GNSS-challenged environment, in which an unmanned vehicle receives relative observations from neighbouring vehicles and fuses them with its own IMU data to reduce inertial guidance errors, improving the accuracy of position estimation [33]. Researchers at Brigham Young University addressed the problem of the collaborative navigation of multiple unmanned aerial vehicles (UAVs) with limited satellite signals by using the onboard radar systems of the UAVs for distance measurements and constructing an EKF to fuse these distance measurements with IMU data, thereby reducing inertial guidance drift errors [16]. Researchers at Northwestern Polytechnic University proposed a method for INS error correction in the collaborative relative navigation of multiple UAV platforms, which can effectively slow down INS error dispersion by establishing a mutual observation model among the UAVs [20].

1.3.2 Collaborative Navigation Research Focusing on Improving Positioning Accuracy in GNSS-Challenged Environments The use of relative observation information obtained by swarm vehicles during communication and interaction for positioning assistance is a new idea for achieving high-accuracy positioning in GNSS-challenged environments. Researchers at the University of Naples Federico II proposed a collaborative multivehicle flight navigation method based on position broadcasting and visual tracking, in which vehicles in a GNSS-challenged environment use auxiliary measurements of one or more collaborative UAVs with good GPS coverage and each vehicle applies the EKF to fuse all measurements to estimate its position. In this way, a vehicle is able to achieve metre-level positioning accuracy despite limited or absent satellite coverage [34].

1.3 Development of Collaborative Resilient Navigation

7

Researchers at West Virginia University proposed a method for relative vehicle navigation that fuses GPS/INS and ultra-wideband (UWB) ranging, in which the additional UWB ranging information leads to a significant improvement in vehicle positioning performance when a vehicle has poor satellite visibility and a poor geometry under fast dynamics [18]. Researchers at the Korea Advanced Institute of Science and Technology proposed a multipath-resistant collaborative GNSS navigation method for use in urban vehicle networks and established a multipath-resistant hybrid collaborative EKF model to improve the localization accuracy and robustness for all vehicles [35]. Researchers at Shanghai Jiaotong University proposed a collaborative navigation method using point-to-point ranging to assist GNSS-based systems and established a relative navigation model based on GNSS pseudorange double-difference observations and point-to-point ranging, which was shown to significantly reduce the positioning error in a GNSS-challenged environment [36].

1.3.3 Collaborative Navigation Research Focusing on Improving the Collaborative Configuration of a Swarm To effectively improve the positioning accuracy achieved through collaboration, collaborative geometry optimization has also gradually received increasing attention. A large number of redundant observations in a swarm not only increase the computational complexity and hardware cost but also affect the enhancement of positioning accuracy. Therefore, the study of geometric configuration optimization strategies for collaborative navigation systems can enable better resource utilization while maintaining favourable navigation accuracy. Researchers at the University of Naples Federico II used a “parent” UAV with good satellite coverage to support a “child” UAV in a GNSS-challenged environment, introducing a geometric accuracy factor to evaluate the positioning accuracy of the “child” UAV based on GNSS observations and collaborative vehicle measurements in order to achieve the optimal formation geometry and navigation performance [37]. Researchers at West Virginia University improved the satellite geometry configuration of UAVs by optimally designing the motion and collaboration strategy of ground-based UAVs to optimally assist in UAV–UAV collaborative navigation. Later, flight tests of the UAV–UAV collaborative navigation strategy verified that the positioning accuracy for a UAV could be successfully improved after the introduction of optimal UAV ranging updates under poor geometric configurations [38]. Researchers at Northwestern Polytechnic University proposed a collaborative navigation method to resist GPS signal loss, in which vehicles in GPS-constrained locations select their optimal sets of collaborative UAVs by minimizing the horizontal dilution of precision (HDOP) [39]. Researchers at the National University of Defense Technology proposed a distributed entropy-based measurement selection method for multirobot localization. To reduce the computational burden while maintaining real-time localization capabilities, the measurements

8

1 Introduction

that yield the greatest information gain in estimating the robots’ locations are selected from all measurements obtained by the robot group to update the pose estimates and the covariance matrix for the whole group. This method ensures the necessary localization accuracy while reducing the computational burden to improve the reliability and real-time performance of localization [40]. Researchers at Nanjing University of Aeronautics and Astronautics proposed a collaborative navigation position error analysis method based on a collaborative accuracy factor and designed a collaborative information screening strategy that can be used to effectively evaluate the collaborative positioning performance [41]. In summary, many scholars have carried out research on algorithms for collaborative positioning and configuration optimization in GNSS-challenged environments, providing a basis for the further study of collaboration-assisted fault-tolerant navigation algorithms.

1.4 Fault-Tolerant Navigation Technology with Collaboration In the complex swarm flight environment, fault diagnosis and fault tolerance of the navigation system are important for the collaborative navigation of the swarm vehicles. Fault detection and fault-tolerant navigation methods for a single vehicle have been widely studied, and the traditional methods can be divided based on their navigation structures, into fault detection algorithms with single filter structures and federal filter structures; based on the fault types targeted, into detection algorithms for abrupt and slowly changing faults [42]; and based on the number of navigation faults targeted, into detection algorithms for single and multiple faults [43]. Because the probability of multiple faults occurring simultaneously is higher in a GNSSchallenged environment, methods for detecting multiple faults take on particular importance; these methods can be mainly divided into multiple hypothesis solution separation (MHSS) algorithms, group separation algorithms, and parity vector methods. However, the above methods mainly use the internal information of a vehicle to build fault identification models, and the availability and accuracy of fault detection are seriously compromised in the case of insufficient available satellites [44]. In the context of vehicle swarming, the collaborative utilization of multivehicle information to improve the fault tolerance of navigation performance of the whole swarm has become a new focus of research direction in recent years and has led to the emergence of fault-tolerant navigation methods such as collaborative integrity monitoring algorithms. Collaborative fault-tolerant navigation algorithms have been developed to extend the concept of navigation integrity from independent platforms to collaborative multiplatform applications by using the redundancy of information among multiple platforms combined with their own sensor measurements to achieve collaborative navigation integrity monitoring capabilities [45]. One approach is the calculation of the

1.4 Fault-Tolerant Navigation Technology with Collaboration

9

protection level, a confidence indicator for a positioning solution obtained through collaborative navigation, which aims to assess the reliability of positioning results and the availability of fault detection [46, 47]. Another is the detection of faults occurring in the navigation system, i.e., the use of various types of fault detection and exclusion (FDE) algorithms to guarantee the fault tolerance of the swarm system [48, 49].

1.4.1 Collaborative Fault-Tolerant Navigation Research Involving Integrity Protection Level Assessment and Improvement In research on integrity protection level algorithms, since the horizontal protection level (HPL) method was proposed, much of the effort has focused on improving the availability of receiver autonomous integrity monitoring (RAIM) to support various navigation modes and on improving and optimizing the protection level [50, 51]. To address the problem that the protection level needs to be re-evaluated and improved after the introduction of collaborative observations in swarm vehicle integrity monitoring, researchers at the University of Illinois proposed a distributed collaborative simultaneous localization and mapping (SLAM) integrity monitoring algorithm for urban GNSS-challenged environments. This algorithm derives the level of protection by calculating the worst-case feature slope in a collaborative SLAM framework, achieving lower localization errors and tighter protection levels. Researchers at the University of Toronto proposed an optimized visual localization integrity monitoring algorithm and computed protection levels based on position state estimation for the task safety of multiple UAVs relying on visual perception, providing more reliable protection limits than traditional 3σ methods [52]. Researchers at the Korea Institute of Science and Technology proposed a vehicle navigation system that integrates airborne sensors and local GNSS technology to achieve higher integrity; its functions include the calculation of the corresponding protection level under a designed integrity requirement to effectively limit the position error during flight testing [53]. Researchers at Beijing University of Aeronautics and Astronautics proposed a collaborative integrity monitoring algorithm for the scenario of a limited number of navigation satellites, using the measurement data of different vehicles to aggregate fault detection statistics, and calculated the radial protection levels under different numbers of vehicles, effectively reducing the leakage rate of fault detection [54]. At present, the research on protection levels in collaborative integrity monitoring algorithms is in a stage of evaluation, and further optimization and improvement of such protection level methods under collaborative navigation conditions are still needed, considering the practical significance of the protection level and the integrity monitoring functions after the introduction of collaborative information.

10

1 Introduction

1.4.2 Collaborative Fault-Tolerant Navigation Research Involving Navigation Fault Detection and Identification Algorithms In the area of collaborative fault detection, researchers at the Queensland University of Technology proposed a relative positioning system with a real-time integrity monitoring framework based on dedicated short-range communications (DSRC) and GPS technology, which can provide safety alerts for GPS system error mismeasurement to ensure system integrity [55]. Researchers at Tufts University proposed a collaboration-enhanced receiver integrity monitoring algorithm that uses multiple mobile GNSS receivers to form common-mode residuals and detect satellite faults to achieve high fault detection sensitivity [45]. Researchers at the University of Seville investigated the problem of fault detection and identification for multiple UAVs when using cameras for collaborative applications, using redundant information provided by the visual sensors to detect faults in internal UAV sensors that could not be detected by traditional algorithms, and using position estimates obtained through visual tracking to substitute for damaged sensors [56]. Researchers at Xi’an University of Electronic Science and Technology proposed a two-layer filterbased multidimensional-scale collaborative fault-tolerant localization algorithm, constructing a filter based on multiple temporal motion states and geometric configurations that can effectively identify abnormal errors in swarm network distance measurements [57]. Researchers at Northwestern Polytechnic University proposed an integrity monitoring algorithm based on collaborative GNSS positioning, combining satellite pseudorange observations and node ranging measurements using a fullnetwork EKF and designing a fault detection algorithm using an MHSS strategy that can monitor all faults in the whole network simultaneously [58]. Current collaborative fault-tolerant navigation techniques are mainly aimed at the detection and troubleshooting of single-fault scenarios in navigation systems. However, because of the typically harsh and complex application environments of aerial swarms, multiple-fault scenarios and fault detection algorithms need further consideration in this context. In summary, research institutions in various countries have carried out research on collaborative fault-tolerant navigation algorithms for GNSS-challenged environments. The above research work has laid a foundation for the development of collaboration-assisted fault-tolerant navigation technology for aerial swarms; however, several key issues related to collaborative positioning and fault-tolerant navigation in the various aspects mentioned above remain unresolved and need further research.

1.5 Motivations

11

1.5 Motivations Aerial swarms will have great value in the future, and their reliable and efficient navigation performance in complex and harsh environments will be critical for achieving successful operations. Collaborative resilient navigation techniques can make full use of the redundant information captured by mutual observations between aircraft and can be used to improve the overall fault-tolerant navigation performance of an aerial swarm. From the above analysis of the current research status, it can be seen that many key problems remain in the existing navigation technology based on the fusion of collaborative information, including the construction of a collaborative navigation model and the optimization of the collaborative configuration of an aerial swarm in a GNSS-challenged environment; the prediction and optimization of the integrity protection level and monitoring capabilities after the introduction of collaborative observations; and the augmentation of the ability to detect multiple failures through collaboration to overcome the insufficient amount of available observations in a GNSS-challenged environment. In a GNSS-challenged environment, some of the vehicles will have insufficient satellite observations or a poor geometric configuration, resulting in a serious degradation in positioning accuracy. The existing methods usually utilize the relative observations of all vehicles directly, without analysing the impact of the intervehicle geometry on navigation performance and without considering the data transfer complexity and computational consumption of large-scale swarms. Therefore, there is a need to establish a model for the collaborative navigation of an aerial swarm based on the differences in navigation performance of the different members and to design a comprehensive cooperation strategy for the swarm members under dynamic geometric configuration conditions so as to optimize the cooperative observation configuration of the aerial swarm and improve navigation accuracy with the aid of the most helpful collaborative information. The integrity monitoring capabilities of the vehicles in a swarm are weak in a GNSS-challenged environment, and the level of integrity protection provided by a single aircraft alone cannot meet the requirements of navigation reliability, resulting in a loss of efficacy in monitoring failure cases. Traditional collaborative integrity monitoring algorithms are usually designed to evaluate the influence of the geometric configuration of the GNSS satellites on the protection level indicator. However, some of the available auxiliary collaborative information may provide little help in improving the integrity monitoring capabilities while causing an increase in the relative communication and measurement burdens. Therefore, it is necessary to consider the relationship between the protection level and the swarm configuration and then study corresponding protection level optimization strategies to enhance the integrity monitoring capabilities of the swarm. In a GNSS-challenged environment where a swarm member has an insufficient number of available satellite observations, multiple failures in the navigation system are difficult to detect, which can lead to a large positioning deviation. Due to the collaborative interaction of information among the members of the swarm, failures

12

1 Introduction

Fig. 1.1 Research content, key problems to be solved, and research objectives of this book

that occur in individual members will influence the navigation performance of the whole swarm. Therefore, it is necessary to analyse the impact of faults in the collaboration of aerial swarms, develop a suitable framework for collaborative resilient fusion-based navigation, and design collaborative resilient fusion-based navigation algorithms capable of detecting and identifying multiple observation faults to guarantee the fault detection and troubleshooting capabilities of an aerial swarm navigation system in complex environments. The research content, the key problems to be solved, and the research objectives of this book are schematically shown in Fig. 1.1. In summary, traditional fault-tolerant navigation technology cannot meet the requirements for the high-accuracy and reliable navigation of swarm vehicles in GNSS-challenged environments. The relevant performance can be further improved with collaborative resilient fusion-based navigation technology to enhance the fault tolerance of aerial swarm navigation systems. Therefore, it is of great significance and value to carry out research work on resilient navigation techniques for aerial swarm vehicles based on collaboration, which takes into account the differences in navigation performance among the members of a swarm. This book proposes a series of optimal reliable and efficient collaborative resilient fusion-based navigation algorithms for the implementation of aerial swarm navigation systems in complex environments and can serve as a reference for subsequent research on the design of aerial swarm navigation technology.

1.6 Organization of Content This book focuses on resilient navigation techniques against the background of collaboration in a swarm. The key techniques of collaborative resilient navigation are proposed, including the collaboration framework, collaborative observation modelling, geometry optimization, integrity augmentation, and fault detection.

1.6 Organization of Content

13

Experiments carried out to validate the effectiveness of the corresponding techniques are also presented. In the first half of this book, the applications of elastic navigation technology in collaborative aerial swarms are used as the research background, and the framework, model and system structures, and fusion algorithms for the implementation of elastic navigation for collaborative aerial swarms operating in various modes are studied, with the aim of achieving the effective fusion of airborne navigation information and relative observation information and improving the navigation performance of aircraft in collaborative aerial swarms. First, various approaches to collaborative elastic navigation are discussed in relation to the shortcomings of traditional leader– follower and parallel collaborative structures, and a dynamic hierarchical collaborative structure for aerial swarms is proposed. Based on this, two fusion frameworks, one for loosely coupled collaborative positioning-based elastic navigation and one for tightly coupled collaborative observation-based elastic navigation, are analysed. The modelling of elastic navigation under synergetic conditions is then investigated, and models employing line-of-sight distances, line-of-sight distance differences, lineof-sight angles, and line-of-sight vectors as synergetic observation information are analysed. On this basis, loosely coupled and tightly coupled elastic navigation fusion algorithms based on collaborative positioning and collaborative observation, respectively, are investigated in detail. In the chapter on loosely coupled collaborative localization, the collaborative localization algorithm and its covariance are investigated along with an airborne elastic fusion algorithm based on the results of collaborative localization; in the chapter on tightly coupled observation-based navigation, the error propagation characteristics, the adaptive hybrid observation model, and the elastic fusion algorithm under tight coupling are investigated. Throughout the first half of the book, the problem of navigation fusion solutions for collaborative airborne swarms is addressed. The second half of the book focuses on the robustness enhancement and efficiency improvement of resilient navigation techniques for collaborative aerial swarms, and research is conducted on distribution configuration optimization for resilient navigation, navigation integrity enhancement, and parallel navigation fault detection under collaborative conditions, with the aim of optimizing the collaboration in aerial swarms and achieving improved navigation accuracy and reliability through the novel and efficient on-demand collaboration strategies. First, a collaborative navigation structure scheme is designed for the different members’ own available observation differences in complex environments, and a collaborative navigation model and geometric configuration optimization method are proposed to adapt to the differences in observation characteristics, providing a basis for subsequent research on collaborative fault-tolerant navigation algorithms. Second, because the integrity protection level index of an individual swarm vehicle cannot meet the demands of collaborative navigation for ensuring the integrity performance and the availability of fault detection, a method of integrity protection level assessment under collaborative assistance conditions is proposed. In addition, in the context of fault detection and troubleshooting, a hierarchical collaborative fault-tolerant navigation architecture is

14

1 Introduction

proposed, a range-assisted inertial/satellite fault-tolerant navigation model is developed, and a collaboration-assisted MHSS algorithm is used to detect and troubleshoot multisatellite observations in the absence of sufficient swarm navigation observations. Throughout the second half of the book, the problem of improving navigation accuracy, efficiency, and reliability in collaborative aerial swarms is addressed.

References 1. Chen W, Liu J, Guo H, Kato NJIN (2020) Toward robust and intelligent drone swarm: challenges and future directions. PP (99):1–6 2. Montalbano NG, Humphreys TE (2020) Intercepting unmanned aerial vehicle swarms with Neural-network-aided game-theoretic target assignment. 2020 IEEE/ION Position, Location and Navigation Symposium. https://doi.org/10.1109/plans46316.2020.9110234 3. Arnold R, Mezzacappa E, Jablonski J, Abruzzo B (2020) Multi-Role UAV Swarm Behaviors for Wide Area Search using Emergent Intelligence. In: 4th World Conference on Smart Trends in Systems, Security and Sustainability (WorldS4), Electr Network, 2020 4. Jul 27–28 2020. pp 255–261 5. Wang H, Chen M, Fu P (2021) A distributed fault-tolerant mechanism for mission-oriented unmanned aerial vehicle swarms. International Journal of Communication Systems 34 (8). https://doi.org/10.1002/dac.4789 6. Bhanderi D, McCue L (2020) Low cost unmanned surface vehicle swarm formation control using a potential field. global Oceans 2020: Singapore - U.S. Gulf Coast. https://doi.org/10. 1109/ieeeconf38699.2020.9389058 7. Clark S, Usbeck K, Diller D, Schantz RE (2020) CCAST: A Framework and practical deployment of heterogeneous unmanned system swarms. GetMobile: Mob Comput Commun 24 (4):17–26. doi:https://doi.org/10.1145/3457356.3457362 8. Xiong Y, Wu N, Shen Y, Win MZ (2022) Cooperative localization in massive networks. IEEE Trans Inf Theory 68(2):1237–1258. https://doi.org/10.1109/tit.2021.3126346 9. Harris ZJ, Whitcomb LL (2021) Cooperative acoustic navigation of underwater vehicles without a DVL utilizing a dynamic process model: Theory and field evaluation. J Field Robot 38(5):700–726. https://doi.org/10.1002/rob.22008 10. Moafipoor S, Bock L, Fayman JA, Conroy E, Inst N (2020) Vision-Based collaborative navigation for UAV-UGV-Dismounted units in GPS challenged environments. In: 33rd International Technical Meeting of the Satellite-Division-of-The-Institute-of-Navigation (ION GNSS), Electr Network, 2020 11. Sep 21–25 2020. Institute of navigation satellite division proceedings of the international technical meeting. pp 573–584. https://doi.org/10.33012/2020.17684 12. Chakraborty S, Soc IC (2022) Effects of asymmetric locomotion methods on collaborative navigation and wayfinding in shared virtual environments. In: IEEE Conference on Virtual Reality and 3D User Interfaces (IEEE VR), Electr Network, 2022 13. Mar 12–16 2022. pp 952–953. https://doi.org/10.1109/vrw55335.2022.00327 14. Zhu J, Kia SS (2022) Decentralized cooperative localization with LoS and NLoS UWB InterAgent ranging. IEEE Sens J 22(6):5447–5456. https://doi.org/10.1109/jsen.2021.3083724 15. Wymeersch H, He J, Denis B, Clemente A, Juntti M (2020) Radio localization and mapping with reconfigurable intelligent surfaces: challenges, opportunities, and research directions. IEEE Veh Technol Mag 15(4):52–61. https://doi.org/10.1109/mvt.2020.3023682 16. Smaoui N, Heydariaan M, Gnawali O (2021) Anchor-oriented Time and Phase-based Concurrent Self-localization using UWB Radios.In: 2021 IEEE 46th Conference on Local Computer Networks. https://doi.org/10.1109/lcn52139.2021.9524990

References

15

17. Jianan Z, Kia SS (2020) An UWB-based communication protocol design for an infrastructurefree cooperative navigation. 2020 IEEE/ION Position, Location and navigation symposium. https://doi.org/10.1109/plans46316.2020.9110154 18. Strader J, Gu Y, Gross JN, Petrillo MD, Hardy J Cooperative relative localization for moving UAVs with single link range measurements. In: 2016 IEEE/ION Position, Location and Navigation Symposium (PLANS), 11–14 April 2016 2016. pp 336–343. https://doi.org/10.1109/ PLANS.2016.7479718 19. Quist EB, Beard RW (2016) Radar Odometry on Fixed-Wing Small Unmanned Aircraft. IEEE Trans Aerosp Electron Syst 52(1):396–410. https://doi.org/10.1109/taes.2015.140186 20. Hedgecock W, Maroti M, Sallai J, Volgyesi P, Ledeczi A (2013) High-accuracy differential tracking of low-cost GPS receivers. In: Paper presented at the Proceeding of the 11th annual international conference on Mobile systems, applications, and services, Taipei, Taiwan, 21. Gross JN, Gu Y, Rhudy MB (2015) Robust UAV Relative Navigation With DGPS, INS, and Peer-to-Peer Radio Ranging. IEEE Trans Autom Sci Eng 12(3):935–944. https://doi.org/10. 1109/tase.2014.2383357 22. Sarras I, Marzat J, Bertrand S, Piet-Lahanier H (2018) Collaborative multiple micro air vehicles’ localization and target tracking in GPS-denied environment from range-velocity measurements. Int J Micro Air Veh 10(2):225–239. https://doi.org/10.1177/1756829317745317 23. Zhong R, Chen Q (2020) Cooperative positioning method using distance measurement within a cluster and direction finding of a target. Hangkong Xuebao/Acta Aeronautica et Astronautica Sinica 41. doi:https://doi.org/10.7527/S1000-6893.2019.23768 24. Pengxiang Z, Yulin Y, Wei R, Guoquan H (2021) Cooperative Visual-Inertial Odometry. In: 2021 IEEE International Conference on Robotics and Automation. https://doi.org/10.1109/icr a48506.2021.9561674 25. Zhu P, Geneva P, Ren W, Huang G, Ieee (2021) Distributed visual-inertial cooperative localization. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Electr Network, 2021 26. Sep 27-Oct 01 2021. In: IEEE International Conference on Intelligent Robots and Systems. pp 8714–8721. doi:https://doi.org/10.1109/iros51168.2021.9636031 27. Indelman V, Gurfil P, Rivlin E, Rotstein H (2012) Distributed vision-aided cooperative localization and navigation based on three-view geometry. Robot Auton Syst 60(6):822–840. https:// doi.org/10.1016/j.robot.2012.02.003 28. Irigireddy S, Moncayo H (2020) Vision based relative navigation for close-formation flight missions. AIAA Scitech 2020 Forum:14 pp.-14 pp. 29. Hardy J, Strader J, Gross JN, Gu Y, Keck M, Douglas J, Taylor CN, Ieee (2016) Unmanned aerial vehicle relative navigation in GPS Denied Environments. In: IEEE/ION Position, Location and Navigation Symposium (PLANS), Savannah, GA, Apr 11–14 2016. IEEE-ION Position Location and Navigation Symposium. pp 344–352 30. Park J-S, Lee D, Jeon B, Bang H, Ieee (2013) Robust vision-based pose estimation for relative navigation of unmanned aerial vehicles. In: 13th International Conference on Control, Automation and Systems (ICCAS), Gwangju, SOUTH KOREA, 2013 31. Oct 20–23 2013. pp 386–390 32. Tanveer MH, Sgorbissa A, Thomas A (2020) An IPM approach to multi-robot cooperative localization: pepper humanoid and wheeled robots in a shared Space. In: 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Porto, PORTUGAL, 2020 33. Jul 29–31 2018. Lecture Notes in Electrical Engineering. pp 429–447. https://doi.org/10.1007/ 978-3-030-31993-9_21 34. Miller A, Rim K, Chopra P, Kelkar P, Likhachev M, Ieee (2020) Cooperative perception and localization for cooperative driving. In: IEEE International Conference on Robotics and Automation (ICRA), Electr Network, 2020 35. May 31-Jun 15 2020. In: IEEE International Conference on Robotics and Automation ICRA. pp 1256–1262

16

1 Introduction

36. Yang Y, Xiao Y, Li T (2021) A survey of autonomous underwater vehicle formation: performance, formation control, and communication capability. IEEE Commun Surv Tutor 23(2):815–841. https://doi.org/10.1109/comst.2021.3059998 37. Zekavat S, Buehrer RM, Durgin GD, Lovisolo L, Zhonghai W, Shu Ting G, Ghasemi A (2021) An overview on position location: past, present, future. Int J Wireless Inf Networks 28(1):45–76. https://doi.org/10.1007/s10776-021-00504-z 38. Nagai K, Fasoro T, Spenko M, Henderson R, Pervan B (2020) Evaluating GNSS navigation availability in 3-D mapped urban environments. 2020 IEEE/ION position, Location and navigation symposium. https://doi.org/10.1109/plans46316.2020.9109929 39. Humphreys TE, Murrian MJ, Narula L (2020) Deep-Urban Unaided Precise Global Navigation Satellite System Vehicle Positioning. IEEE Intell Transp Syst Mag 12(3):109–122. https://doi. org/10.1109/mits.2020.2994121 40. Alam N, Kealy A, Dempster AG (2013) Cooperative inertial navigation for GNSS-Challenged vehicular environments. IEEE Trans Intell Transp Syst 14(3):1370–1379. https://doi.org/10. 1109/tits.2013.2261063 41. Vetrella AR, Opromolla R, Fasano G, Accardo D, Grassi M (2017) Autonomous flight in GPSchallenging environments exploiting Multi-UAV cooperation and Vision-aided navigation. AIAA information systems - AIAA Infotech@Aerospace 42. Ko H, Kim B, Kong S-H (2015) GNSS Multipath-resistant cooperative navigation in urban vehicular networks. IEEE Trans Veh Technol 64(12):5450–5463. https://doi.org/10.1109/tvt. 2015.2481509 43. Shen J, Wang S, Zhai Y, Zhan X (2021) Cooperative relative navigation for multi-UAV systems by exploiting GNSS and peer-to-peer ranging measurements. IET Radar Sonar Navig 15(1):21– 36. https://doi.org/10.1049/rsn2.12023 44. Causa F, Vetrella AR, Fasano G, Accardo D Multi-UAV formation geometries for cooperative navigation in GNSS-challenging environments. In: 2018 IEEE/ION Position, Location and Navigation Symposium, PLANS 2018, April 23, 2018 - April 26, 2018, Monterey, CA, United states, 2018. 2018 IEEE/ION Position, Location and Navigation Symposium, PLANS 2018 - Proceedings. Institute of Electrical and Electronics Engineers Inc., pp 775–785. https://doi. org/10.1109/PLANS.2018.8373453 45. Sivaneri VO, Gross JN (2018) Flight-testing of a cooperative UGV-to-UAV strategy for improved positioning in challenging GNSS environments. Aerosp Sci Technol 82–83:575–582. https://doi.org/10.1016/j.ast.2018.09.035 46. Qu Y, Zhang Y (2011) Cooperative localization against GPS signal loss in multiple UAVs flight. J Syst Eng Electron 22(1):103–112. https://doi.org/10.3969/j.issn.1004-4132.2011.01.013 47. Wang L, Shao J-X, Wan J-W, Liu Y-H (2007) Distributed entropy based relative observation selection for multi-robot localization. Tien Tzu Hsueh Pao/Acta Electronica Sinica 35(2):333– 336 48. Wang R, Chen X, Xiong Z, Zhang H, Liu J (2022) A collaborative navigation method based on partnership optimisation for clustered aircraft. IET Radar Sonar Navig 16(6):1042–1052. https://doi.org/10.1049/rsn2.12241 49. Lee YC Investigation of extending Receiver Autonomous Integrity Monitoring (RAIM) to combined use of Galileo and modernized GPS. In: 17th International Technical Meeting of the Satellite Division of the Institute of Navigation, ION GNSS 2004, September 21, 2004 September 24, 2004, Long Beach, CA, United states, 2004. Proceedings of the 17th International Technical Meeting of the Satellite Division of the Institute of Navigation, ION GNSS 2004. Institute of Navigation, pp 1691–1698 50. Blanch J, Walter T, Enge P (2010) RAIM with optimal integrity and continuity allocations under multiple failures. IEEE Trans Aerosp Electron Syst 46(3):1235–1247. https://doi.org/ 10.1109/taes.2010.5545186 51. Jiang Y, Wang J (2016) A new approach to calculate the horizontal protection level. J Navig 69(1):57–74. https://doi.org/10.1017/s0373463315000545 52. Rife J, Ieee (2012) Collaboration-Enhanced receiver integrity monitoring with common residual estimation. In: IEEE/ION Position Location and Navigation Symposium (PLANS), Myrtle Beach, SC, 2012

References

17

53. Apr 23–26 2012. IEEE-ION Position Location and Navigation Symposium. pp 1042–1053 54. Ochieng WY, Sauer K, Walsh D, Brodin G, Griffin S, Denney M (2003) GPS integrity and potential impact on aviation safety. J Navig 56(1):51–65. https://doi.org/10.1017/s03734633 02002096 55. Maaref M, Khalife J, Kassas ZM (2021) Aerial vehicle protection level reduction by fusing gnss and terrestrial signals of opportunity. IEEE Trans Intell Transp Syst 22(9):5976–5993. https://doi.org/10.1109/tits.2021.3095184 56. Reddy PS, Sarma N, Das RK (2008) Enhancing fault-tolerance in a distributed mutual exclusion algorithm. In: 2006 9th International Conference on Information Technology. 57. Cheng Q, Sun R, Wang J (2020) A new historical data based GNSS fault detection and exclusion algorithm for UAV positioning. In: 2nd IEEE International Conference on Civil Aviation Safety and Information Technology (ICCASIT), Wuhan, PEOPLES R CHINA, 2020 58. Oct 14–16 2020. pp 1144–1149. https://doi.org/10.1109/iccasit50869.2020.9368839 59. Ni Z, Betaille D, Marais J, Berbineau M (2018) Extended Kalman Filter (EKF) Innovationbased Integrity Monitoring Scheme with C/N0 Weighting. 2018 IEEE 4th International Forum on Research and Technology for Society and Industry. https://doi.org/10.1109/rtsi.2018.854 8512 60. Kropp V, Berz G, Ieee (2014) Optimized MHSS ARAIM user algorithms: assumptions, protection level calculation and availability analysis. In: IEEE/ION Position, Location and Navigation Symposium (PLANS), Monterey, CA, 2014 61. May 05–08 2014. IEEE-ION Position location and navigation symposium. pp 308–323 62. Chengyao L, Waslander SL (2019) Visual measurement integrity monitoring for UAV Localization arXiv. arXiv:8 pp.-8 pp 63. Lee J, Kim M, Lee J, Inst N (2016) Integration of onboard sensors and Local Area DGNSS to Support High Integrity Unmanned Aerial Vehicles (UAV) Navigation. In: 29th International Technical Meeting of The-Satellite-Division-of-the-Institute-of-Navigation (ION GNSS+), Portland, OR, 2016 64. Sep 12–16 2016. Institute of navigation satellite division proceedings of the international technical meeting. pp 1477–1484 65. Liu Y, Zhu Y, Ieee (2013) A collaborative integrity monitor algorithm for low space aviation under limited number of navigation satellites. In: 2nd International Conference on Connected Vehicles and Expo (ICCVE), Las Vegas, NV, 2013 66. Dec 02–06 2013. In: International Conference on Connected Vehicles and Expo. pp 910–915. https://doi.org/10.1109/iccve.2013.133 67. Ansari K, Feng YM, Tang ML (2015) A Runtime integrity monitoring framework for RealTime relative positioning systems based on GPS and DSRC. IEEE Trans Intell Transp Syst 16(2):980–992. https://doi.org/10.1109/tits.2014.2349011 68. Suarez A, Heredia G, Ollero A (2018) Cooperative virtual sensor for fault detection and identification in Multi-UAV applications. J SensS 2018. https://doi.org/10.1155/2018/4515828 69. Fan Y, Qi X, Liu L (2021) Fault-Tolerant cooperative localization of 3D mobile networks via Two-Layer Filter multidimensional scaling. IEEE Sens J 21(6):8354–8366. https://doi.org/10. 1109/jsen.2020.3046736 70. Zhuang C, Zhao H, Hu S, Sun C, Feng W, Inst N (2019) Integrity monitoring algorithm for GNSS-based cooperative positioning applications. In: 32nd International Technical Meeting of the Satellite-Division-of-The-Institute-of-Navigation (ION GNSS), Miami, FL, 2019 71. Sep 16–20 2019. Institute of navigation satellite division proceedings of the international technical meeting. pp 2008–2022. https://doi.org/10.33012/2019.16881

Chapter 2

Collaborative Resilient Navigation Frameworks

Abstract There are several possible structures for the organization of the members of an aerial swarm, indicating different collaborative relationships. Meanwhile, there are different frameworks for fusing observations in collaborative navigation. These structures and frameworks determine the various approaches for realizing resilient navigation fusion through collaboration in an aerial swarm. This chapter discusses the leader–follower, parallel, and hierarchical collaborative navigation structures. Additionally, collaborative localization-based and collaborative observation-based fusion frameworks for resilient navigation are proposed. Keywords Leader–follower structure · Parallel structure · Hierarchical structure · Collaborative localization-based fusion framework · Collaborative observation-based fusion framework

2.1 Introduction In recent years, aerial swarm technology has been widely studied because of its considerable application potential. Aerial swarms are flexible, mobile, and can perform multiple tasks or subtasks simultaneously [1, 2]. With the development of technology for relative measurement and information sharing among UAVs, it has become possible to achieve coordination among UAVs in swarm tasks, such as coordinate searching and surveying [3]. Therefore, UAV swarm technology is one of the future development directions of UAV technology [4]. At present, UAV navigation mainly relies on inertial measurement units (IMUs) and global navigation satellite system (GNSS) receivers. However, an IMU typically has a continuous drift error, and GNSS signals are susceptible to external interference and often unreliable. Researchers have considered the relative navigation of a pair of aircraft. Lee et al. proposed an adaptive INS/GPS integrated navigation method for formation flight [5]. Hedgecock et al. proposed a relative attitude estimation method for two UAVs at the same altitude using ranging radar and airborne navigation system information [6]. Hardy et al. proposed the robust H-inf filtering algorithm for relative navigation systems based on GPS/INS integration under the condition

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_2

19

20

2 Collaborative Resilient Navigation Frameworks

of unknown noise characteristics [7]. Zhang et al. designed a visual relative navigation system based on a cubature filter to fuse the pitch angle, yaw angle, and other visual information [8]. Gross et al. proposed a UAV relative navigation method with Differential GPS (DGPS), INS, and peer-to-peer radio ranging [9]. Oh and Johnson proposed a visual navigation model to estimate the relative attitude between UAVs [10]. Zhu et al. proposed a fully parallel distributed relative navigation framework and improved the relative navigation accuracy by sharing collaborative information between aircraft [11]. Fosbury and Crassidis proposed a leader–follower collaborative navigation method that estimates the relative position, relative velocity, and relative attitude between a leader aircraft and a follower aircraft based on visual sensor measurements [12]. Wang et al. proposed a leader–follower relative navigation algorithm based on a vision sensor and ranging radar to determine the relative position and velocity of UAVs in formation flight [13]. Irigireddy and Moncavo proposed a method for solving the estimation and tracking problem of the leader aircraft through a visual algorithm for leader–follower formation flight structures [14]. Meng et al. proposed a method to estimate the attitude of the leader in formation flight through visual sensors [15]. The research above on relative navigation involving a pair of aircraft provides a foundation for the collaborative navigation of aerial swarm. However, the relative navigation technology mainly focused on the relationship between a pair of aircraft; thus, such methods are unsuitable for fusing both relative and absolute sensor measurements from multiple aircraft. To overcome the problems above, this chapter presents three collaborative navigation structures for UAV swarms, namely, a leader–follower collaborative navigation structure, a parallel collaborative navigation structure, and a hierarchical collaborative navigation structure. In the leader–follower collaborative navigation structure, aircraft with low position accuracy are designated as followers, while an aircraft with a high position accuracy is designated as a leader. The follower aircraft can receive relative measurement information from the leader aircraft collected by sensors to maintain their navigation performance. In the parallel collaborative navigation structure, all aircraft in the swarm are considered to be of equal importance and broadcast their own relative measurement information and that of other UAVs. The hierarchical collaborative navigation structure is an improved leader–follower structure. The aircraft in the swarm are divided into two layers, the leader layer and the follower layer, in accordance with their online-estimated positioning accuracy. With a dynamic collaborative navigation model based on this hierarchical collaborative navigation structure, the navigation system can collaboratively recover from both leader and follower failure. With the development of multi-information fusion technologies, a focus has been placed on collaborative navigation technology that could fuse information from both absolute and relative measurements for UAVs in formation. Sarras et al. proposed a collaborative positioning and tracking method for multiple micro aircraft in a satellite rejection environment, and the velocity of the target was estimated by measuring the relative and absolute velocity information between aircraft and combining local non-linear observers [16]. Strader et al. proposed a relative navigation method by

2.1 Introduction

21

combining carrier-phase differential GPS and radio ranging information with a lowcost inertial sensor, which enhanced the robustness of relative navigation based only on ultrawideband radio ranging technology [17]. Guo et al. proposed an indirect collaborative positioning method based on ultrawide-band information in the GPS rejection environment [18]. Dehghani and Menhai described a collaborative aircraft control method suitable for leader–follower structures and designed a cascade loop to adjust the relative distance and angle between the leader aircraft and the follower aircraft to increase the stability of collaborative navigation [19]. Duan and Luo proposed a method of sensing the relative position and direction of two UAVs based on a binocular vision sensor in leader–follower twin-aircraft flight mode, and the visual sensor was used to extract and identify the feature points of the aircraft to detect the relative position [20]. Collaborative navigation technology utilizes sensor information from multiple aircraft to achieve favourable navigation performance [21, 22]. However, for collaborative navigation methods that rely on GNSS receivers, a GNSS signal failure or outage may affect the reliability of navigation [23]. Moreover, traditional single leader–follower collaborative navigation schemes with constant cooperation have low reliability if the leader is in failure [24]. In this case, research on improved collaborative navigation methods should be performed to increase the potential navigation reliability of a collaborative UAV swarm. Based on the collaborative navigation structures proposed above, two collaborative information fusion frameworks are presented for resilient navigation. One is a collaborative localization-based fusion framework, and the other is a collaborative observation-based fusion framework. The collaborative localization-based fusion framework combines information from observation sensors, an INS, and an integration filter. The integration filter uses only the position information calculated from relative observations, which means that when there are not enough relative observations to calculate the aircraft’s position, collaborative navigation cannot be properly accomplished. The collaborative observation-based fusion framework also combines information from observation sensors, inertial instruments, and an integration filter. The integration filter accepts relative observations, such as relative distances, azimuths, and elevation angles, from other aircraft, allowing collaborative navigation to be performed even with a small number of relative observations. This chapter introduces the various collaborative navigation structures for a swarm as well as the collaborative information fusion frameworks. Sections 2.3, 2.4, and 2.5 describe the leader–follower, parallel and hierarchical collaborative navigation structures, respectively, and Sects. 2.5 and 2.6 describe the collaborative localization-based and collaborative observation-based fusion frameworks for resilient navigation.

22

2 Collaborative Resilient Navigation Frameworks

Fig. 2.1 Single leader–follower navigation structure

2.2 Leader–Follower Collaborative Navigation Structure Figure 2.1 shows the traditional single leader–follower structure for collaborative navigation with a constant cooperation relationship. In Fig. 2.2, the collaborative relationship between the leader and follower aircraft is constant. A leader aircraft, acting as an anchor member in the swarm, shares its position with its follower aircraft. The follower aircraft, acting as label members in the swarm, whose navigation systems need to be assisted, receive relative measurement information from the corresponding leader aircraft collected by sensors to maintain their navigation performance. However, when a leader aircraft suffers from navigation failure (denoted by A1 in Fig. 2.1), its label aircraft (L1-1 and L1-2 in Fig. 2.1) will be affected, and their navigation performance will decline significantly. Therefore, the reliability of this system is low. Additionally, in an aerial swarm flight environment, the traditional single leader–follower structure cannot fully utilize collaborative measurements from multiple aircraft.

2.3 Parallel Collaborative Navigation Structure The leader–follower collaborative navigation structure requires all follower aircraft to be within the communication range of the corresponding leader. Therefore, to achieve heterogeneous collaborative navigation in a large-scale, wide-area swarm,

2.4 Hierarchical Collaborative Navigation Structure

23

Fig. 2.2 Parallel collaborative navigation structure

it is necessary to study a parallel collaborative navigation architecture, as shown in Fig. 2.2. In the parallel collaborative navigation structure, the members of the aerial swarm broadcast their own relative measurement information and that of other members. This measurement information is then received by other members to be fused with their own internal sensor information. The parallel collaborative navigation structure can improve the overall computing efficiency of the collaborative navigation system for a heterogenous swarm. The navigation system of each member integrates the collaborative relative measurements transmitted by other members by constructing a relative navigation filter, leading to the parallel calculation of collaborative observation information.

2.4 Hierarchical Collaborative Navigation Structure To overcome the shortcomings of the traditional leader–follower navigation structure, a hierarchical collaborative navigation structure is proposed, as shown in Fig. 2.3. In the improved hierarchical collaborative navigation structure, all aircraft in the swarm system are divided into anchor aircraft and label aircraft in accordance with their current positioning accuracy. All anchor aircraft can transmit collaborative navigation information to label aircraft. When the navigation sensors of an anchor aircraft suffer failure (denoted by A1 in Fig. 2.3), the label aircraft continue to work normally because they can still receive collaborative navigation information from other anchor

24

2 Collaborative Resilient Navigation Frameworks

Fig. 2.3 Hierarchical collaborative navigation structure

aircraft (A2 and A3 in Fig. 2.3). Therefore, the hierarchical collaborative navigation structure has high reliability and robustness. In addition, a label aircraft can simultaneously receive collaborative navigation information from multiple anchor aircraft. Compared with the traditional single leader–follower structure, in which a given label aircraft can receive relative information from only one anchor aircraft, the hierarchical collaborative navigation structure can better maintain the navigation performance of the follower aircraft in a failure case. Two situations are considered here: the situation in which the relative measurement between an anchor aircraft and a label aircraft is blocked by another member between them and the situation in which the relative measurement between an anchor aircraft and a label aircraft is influenced by non-line-of-sight (NLOS) interference errors. For the scenario discussed here, if the relative angle and relative distance are measured, then the line-of-sight (LOS) vectors from followers to leaders can be obtained. Based on the geometry of the hierarchical collaborative navigation structure shown in Fig. 2.3, when a label aircraft is cooperating with several anchors, the position of the label aircraft can be determined as long as the VOS to at least one leader and that leader’s position are available. This means that the hierarchical collaborative navigation structure possesses high flexibility because at least one cooperative

2.6 Collaborative Observation-Based Fusion Framework for Resilient …

25

Estimated INS Errors

INS

...

Positions of Anchors

State Equation Estimation of INS Relative Observation Calculation Label Member Location Solving

A1 Measured Relative Observation Decompose

...

A2

Relative Range and/or Bearing

Collaborative Localization Error Covariance Estimation

Time Update Measurement Update

Measurement Equation Establishment by Collaborative Location

Collaborative Localization

Collaborative Fusion Filter

Fig. 2.4 Framework of collaborative localization-based fusion

anchor is effective for cooperative positioning, making the system robust to cases in which intervehicle connectivity is partially blocked.

2.5 Collaborative Localization-Based Fusion Framework for Resilient Navigation Based on the hierarchical structure for collaborative navigation established above, a hierarchical collaborative localization-based navigation framework can be designed as one approach for realizing collaborative fusion-based navigation, as shown in Fig. 2.4. The label aircraft can simultaneously receive different types of collaborative measurement information, such as relative distances and relative azimuth and elevation angles, from multiple anchor aircraft (denoted by A1, A2, … in Fig. 2.4). In accordance with the received information, the positions of the label aircraft in a relative coordinate system can be calculated. Then, these relative positions can be converted into coordinates in a geographic coordinate system. On this basis, the state equation and measurement equation can be established. Finally, a collaborative fusion filter can be designed to integrate a follower aircraft’s position with the measurements of its airborne navigation system, leading to navigation performance with enhanced reliability and accuracy.

2.6 Collaborative Observation-Based Fusion Framework for Resilient Navigation Based on the hierarchical structure for collaborative navigation established above, a hierarchical collaborative observation-based navigation framework can be designed

26

2 Collaborative Resilient Navigation Frameworks Estimated INS Errors

INS

...

Positions of Anchors

State Equation Estimation of INS Relative Observation Calculation

A1 Measured Observation Decompose

...

A2

Relative Range and/or Bearing

Relative Observation Error Modelling

Observation Error Covariance Estimation

Time Update Measurement Update

Measurement Equation Establishment by Relative Observation

CoIIaborative Observation Modelling

Collaborative Fusion Filter

Fig. 2.5 Framework of collaborative observation-based fusion

as another approach for realizing collaborative fusion-based navigation, as shown in Fig. 2.5. The label aircraft can simultaneously receive different types of collaborative observation information, such as relative distances and relative azimuth and elevation angles, from multiple anchor aircraft (denoted by A1, A2, … in Fig. 2.5). In accordance with the received information, the relative observation can be calculated, and the measured relative observation decomposition can be obtained; then, corresponding collaborative observation models can be established. On this basis, the state equation and measurement equation can be established, and observation estimation can be performed for the label aircraft in accordance with the collaboratively observed relationships. Then, a collaborative fusion filter can be designed to integrate the collaborative measurement information provided by the anchor aircraft with the measurements of the airborne navigation system equipped on each label aircraft in the swarm.

2.7 Conclusions This chapter discusses frameworks for collaborative resilient navigation. Compared with the single leader–follower collaborative navigation structure and the parallel collaborative navigation structure, the hierarchical collaborative navigation structure shows greater potential for flexibility and robustness in dealing with failures and degradation in the navigation measurements of swarm members. On this basis, an airborne navigation system with a resilient fusion strategy can be designed based on either a collaborative localization-based fusion framework or a collaborative observation-based fusion framework, which organize the available local measurements and relative observation information in different ways. These approaches provide the fundamental basis for studying corresponding modelling and fusion algorithms, which will be discussed in the following chapters.

References

27

References 1. Sharma RK, Ghose D (2009) Collision avoidance between UAV clusters using swarm intelligence techniques. Int J Syst Sci 40(5):521–538. https://doi.org/10.1080/002077209027 50003 2. Sharma R, Ghose D, IEEE (2007) Swarm intelligence based collision avoidance between realistically modelled UAV clusters. In: 26th American control conference, New York, NY, 2007 Jul 09–13 2007. Proceedings of the American control conference, p 223 3. Qiu H, Wei C, Dou R, Zhou Z (2015) Fully autonomous flying: from collective motion in bird flocks to unmanned aerial vehicle autonomous swarms. Sci China-Inf Sci 58(12). https://doi. org/10.1007/s11432-015-5456-x 4. Duan H, Yang Q, Deng Y, Li P, Qiu H, Zhang T, Zhang D, Huo M, Shen Y (2019) Unmanned aerial systems coordinate target allocation based on wolf behaviors. Sci China-Inf Sci 62(1). https://doi.org/10.1007/s11432-018-9587-0 5. Lee JY, Kim HS, Choi KH, Lim J, Chun S, Lee HK (2016) Adaptive GPS/INS integration for relative navigation. GPS Solutions 20(1):63–75. https://doi.org/10.1007/s10291-015-0446-4 6. Hedgecock W, Maroti M, Sallai J, Volgyesi P, Ledeczi A High-accuracy differential tracking of low-cost GPS receivers. In: 11th annual international conference on mobile systems, applications, and services, MobiSys 2013, June 25, 2013–June 28, 2013, Taipei, Taiwan, 2013. MobiSys 2013—Proceedings of the 11th annual international conference on mobile systems, applications, and services. Association for Computing Machinery, pp 221–234. https://doi.org/ 10.1145/2462456.2464456 7. Hardy J, Strader J, Gross JN, Gu Y, Keck M, Douglas J, Taylor CN, IEEE (2016) Unmanned aerial vehicle relative navigation in GPS denied environments. In: IEEE/ION position, location and navigation symposium (PLANS), Savannah, GA, 2016 Apr 11–14 2016. IEEE-ION position location and navigation symposium, pp 344–352 8. Xu Z, Naigang C, Xiaogang W, Hutao C (2017) The vision-based relative navigation using improved adaptive cubature huber-based filtering. In: 21st AIAA international space planes and hypersonics technologies conference, 10 pp 9. Gross JN, Gu Y, Rhudy MB (2015) Robust UAV relative navigation with DGPS, INS, and peerto-peer radio ranging. IEEE Trans Autom Sci Eng 12(3):935–944. https://doi.org/10.1109/tase. 2014.2383357 10. Oh S-M, Johnson EN (2007) Relative motion estimation for vision-based formation flight using unscented Kalman filter. In: AIAA guidance, navigation, and control conference 2007, August 20, 2007–August 23, 2007, Hilton Head, SC, United states, 2007. Collection of technical papers—AIAA guidance, navigation, and control conference 2007. AIAA International, pp 5365–5381. https://doi.org/10.2514/6.2007-6866 11. Zhu Y, Sun Y, Zhao W, Wu L (2020) A novel relative navigation algorithm for formation flight. Proc Inst Mech Eng Part G-J Aerosp Eng 234(2):308–318. https://doi.org/10.1177/095441001 9866060 12. Fosbury AM, Crassidis JL (2008) Relative navigation of air vehicles. J Guid Control Dyn 31(4):824–834. https://doi.org/10.2514/1.33698 13. Wang Y, Yang S, Hu H, Chen K, Ji Q, IEEE (2012) Relative navigation algorithm based on robust filter for UAV formation flight. In: International conference on control engineering and communication technology (ICCECT), Shenyang, Peoples R China, 2012 Dec 07–09 2012, pp 249–252. https://doi.org/10.1109/iccect.2012.178 14. Irigireddy S, Moncayo H (2020) Vision based relative navigation for close-formation flight missions. AIAA Scitech 2020 Forum 15. Ding M, Wei L, Wang B (2011) Vision-based estimation of relative pose in autonomous aerial refueling. Chin J Aeronaut 24(6):807–815. https://doi.org/10.1016/s1000-9361(11)60095-2 16. Sarras I, Marzat J, Bertrand S, Piet-Lahanier H (2018) Collaborative multiple micro air vehicles’ localization and target tracking in GPS-denied environment from range-velocity measurements. Int J Micro Air Veh 10(2):225–239. https://doi.org/10.1177/1756829317745317

28

2 Collaborative Resilient Navigation Frameworks

17. Strader J, Gu Y, Gross JN, De Petrillo M, Hardy J, IEEE (2016) Cooperative relative localization for moving UAVs with single link range measurements. In: IEEE/ION position, location and navigation symposium (PLANS), Savannah, GA, 2016 Apr 11–14 2016. IEEE-ION position location and navigation symposium, pp 336–343 18. Guo K, Qiu Z, Meng W, Xie L, Teo R (2017) Ultra-wideband based cooperative relative localization algorithm and experiments for multiple unmanned aerial vehicles in GPS denied environments. Int J Micro Air Veh 9(3):169–186. https://doi.org/10.1177/1756829317695564 19. Dehghani MA, Menhaj MB (2018) Stability of cooperative unmanned aerial vehicles based on relative measurements. Proc Inst Mech Eng Part G-J Aerosp Eng 232(15):2784–2792. https:// doi.org/10.1177/0954410017716477 20. Duan H, Luo Q, IEEE (2016) Integrated localization system for autonomous unmanned aerial vehicle formation flight. In: 12th IEEE international conference on control and automation (ICCA), Kathmandu, Nepal, 2016 Jun 01–03 2016. IEEE international conference on control and automation ICCA. pp 395–400 21. Tianshu C, Qingzhen Z, Yaolei Z (2011) A new method of cooperative localization for a long range flight formation. In: Proceedings of the 2011 first international conference on instrumentation, measurement, computer, communication and control. https://doi.org/10.1109/imccc.201 1.235 22. Stacey G, Mahony R (2016) A passivity-based approach to formation control using partial measurements of relative position. IEEE Trans Autom Control 61(2):538–543. https://doi.org/ 10.1109/tac.2015.2446811 23. Zhang Y, Mehrjerdi H, IEEE (2013) A survey on multiple unmanned vehicles formation control and coordination: normal and fault situations. In: International conference on unmanned aircraft systems (ICUAS), Atlanta, GA, 2013 May 28–31 2013. International conference on unmanned aircraft systems, pp 1087–1096 24. Zhang L, Xu D, Liu M (2011) Cooperative navigation algorithm for two leaders GUUV. In: 2011 international conference on intelligent computation technology and automation. https:// doi.org/10.1109/icicta.2011.528

Chapter 3

Modelling for Resilient Navigation via Collaboration

Abstract The modelling of observations and errors is the foundation of resilient navigation fusion. The relative observations that can be introduced into collaborative resilient fusion vary with the available relative measurement sensors. Therefore, it is necessary to perform modelling in accordance with the constraint properties of different relative observations of navigation parameters. This chapter discusses the modelling of collaborative observations as well as the corresponding error covariance, presenting approaches based on range measurements, range difference measurements, bearing-only measurements, and vector-of-sight measurements. In these analyses, modelling is carried out for both the hierarchical and parallel navigation structures of an aerial swarm. Keywords Collaborative navigation · Resilient fusion · Collaborative observation modelling · Ranging · Range difference · Bearing-only · Vector of sight

3.1 Introduction Based on the research on collaborative navigation structures presented in Chap. 2, this chapter studies observation modelling for collaborative navigation among members of a swarm. This lays a foundation for the collaborative navigation information fusion algorithm presented in Chap. 4. This chapter addresses the mutual observation relationship between a label member and an anchor member in a swarm. The geometric representation of this relationship is derived, and the actual measured and calculated relative observations are each modelled. By analysing the relationship between the actual measured and calculated observations, an observation model for aerial swarm collaborative navigation is finally established. When the members of a swarm are in flight, relative observations between aircraft can be obtained through various types of sensors, such as radio sensors [1]. The overall navigation accuracy of the swarm system can be improved by designing a corresponding collaborative navigation algorithm using the relative observations between a label aircraft with low airborne navigation accuracy and an anchor aircraft with high airborne navigation accuracy. The different members of an aerial swarm © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_3

29

30

3 Modelling for Resilient Navigation via Collaboration

have different purposes, types, and functions during flight [2, 3]. The different situations of the swarm members lead to differences in the positioning accuracies of their airborne navigation systems. Moreover, in an actual aerial swarm, the airborne photoelectric platforms carried by each member are different, and the navigation performance requirements may vary for members who play different roles in the swarm [4, 5]. In some cases, a label member may not be able to obtain relative range observations to an anchor, instead obtaining only relative bearing observations [6–9]. When only relative bearing observations between members can be obtained, the positions of the label members in the swarm can still be corrected by designing a collaborative navigation model and algorithm based on bearing observations. A collaborative navigation algorithm based on relative distance or bearing measurements only has high requirements on the number of available anchors. Collaborative navigation that relies on a single kind of relative observation is especially susceptible to environmental disturbances and sensor failures. In contrast, an aerial swarm equipped with a variety of sensors can simultaneously obtain multiple kinds of relative observations, such as relative distance and angle, allowing the swarm to achieve higher robustness in its overall navigation performance. In collaborative navigation and positioning, relative observations between members can be obtained through radio navigation technology, visual navigation technology, and other means. On this basis, a corresponding collaborative observation model can be established to improve navigation performance. One example is a method of estimating the relative attitude of an unmanned aerial vehicle (UAV) using only airborne radio frequency (RF) signals. Another is a method of realizing reliable relative positioning by using a signal of opportunity (SOP) in combination with inertial and visual data [10]. It is also possible to achieve cooperation by exchanging navigation data (i.e., available global navigation satellite system (GNSS) observation data) and using a monocular camera system for tracking based on relative vision, and the reliability and integrity of such a navigation algorithm have been proven [11]. The distances between platform nodes (users) can also be used to enhance navigation performance, and a method of establishing distance vectors from nodes with known or more accurate positions to unknown positions has been reported [12]. In Sect. 3.2, the coordinate frames used in resilient navigation via collaboration are provided. In Sects. 3.3 and 3.4 of this chapter, collaborative navigation models based on relative distances and distance differences, respectively, are studied. In Sect. 3.5, a collaborative navigation algorithm based on relative bearing information is studied. In Sect. 3.6, since multiple types of navigation information can be obtained during aerial swarm flight, collaborative navigation based on multisensor information fusion is studied. In Sects. 3.3, 3.4, 3.5 and 3.6, different collaborative observation geometries based on different relative observations are designed for each navigation type, and the observation models for collaborative navigation are established.

3.2 Coordinate Frames Used in Resilient Navigation via Collaboration

31

3.2 Coordinate Frames Used in Resilient Navigation via Collaboration In resilient navigation via collaboration, there are several geometric relationships and transformations that must be represented, as shown in Fig. 3.1. Figure 3.1 illustrates the two main coordinate frames used in resilient fusion-based navigation via collaboration: one is the global coordinate frame, and the other is the local coordinate frame. The global coordinate frame, which remains static during the whole navigation process, is used to represent the positions of all members in the aerial swarm. For example, the earth-centred-earth-fixed (ECEF) coordinate frame could be selected as the global coordinate frame [13]. The ith label member and the jth anchor member in the aerial swarm can be represented by vectors pi and p j , respectively, in the global coordinate frame. Then, the positions of all members in the swarm can be obtained from the latitudes, longitudes, and altitudes indicated by their equipped airborne navigation systems:

Fig. 3.1 Coordinate frames used in resilient navigation via collaboration

32

3 Modelling for Resilient Navigation via Collaboration

⎡

⎤ (R N + h i ) cos L i cos λi piGlobal (λi , L i , h i ) = ⎣ (R N + h i ) cos L i sin λi ⎦ [R N (1 − f )2 + h i ] sin L i

(3.1)

The local coordinate frame is used to represent the relative geometric relationships between the members of the aerial swarm. The origin of the local coordinate frame is at the location of the label member of current interest. For example, the east– north–up (ENU) coordinate frame, in which the coordinate axes correspondingly point east, north, and up, could be selected as the local coordinate frame. Obviously, the local coordinate frame moves with the manoeuvring of the label member, and each label member has its own local coordinate frame. The transformation matrix from the global frame to the local frame is [14] ⎡

⎤ cos λi 0 − sin λi Local C Global (λi , L i , h i ) = ⎣ − sin L i cos λi − sin L i sin λi cos L i ⎦ cos L i cos λi cos L i sin λi sin L i

(3.2)

In Eq. (3.2), the ECEF and ENU frames are selected as the global and local coordinate frames, respectively, as examples. For different coordinate frame selections, the transformation matrix should be changed accordingly. The relative vector of sight (VOS) can be represented as r i→ j = p j − pi Local = C Global (λi , L i , h i )[ pGlobal (λ j , L j , h j ) − piGlobal (λi , L i , h i )] j

(3.3)

It is important to note that these coordinates can be expressed in two ways: as spherical coordinates or as Cartesian coordinates. In spherical coordinates, the relative VOS is represented by the relative azimuth θi→ j , the relative elevation ϕi→ j , and the relative range ri→ j , as shown in Eq. (3.4); in Cartesian coordinates, the relative VOS is represented by the components along the x-, y-, and z-axes of the local coordinate frame, as shown in Eq. (3.5): ]T [ r i→ j = ri→ j θi→ j ϕi→ j

(3.4)

]T [ r i→ j = Δxi→ j Δyi→ j Δz i→ j

(3.5)

3.3 Ranging-Based Collaborative Observation Modelling

33

Fig. 3.2 Relative-ranging-based observation geometry

3.3 Ranging-Based Collaborative Observation Modelling 3.3.1 Relative-Ranging-Based Observation Geometry A multiple anchor–label collaborative navigation model can be established based on relative ranging. A label member of the swarm can obtain its relative distance from an anchor by means of airborne radar, a data link, or another kind of ranging sensor. The corresponding observation model is established in the label member’s local coordinate frame. A diagram of the relative-ranging-based observation geometry is shown in Fig. 3.2. Based on Fig. 3.2, in accordance with the relative geometric relationship between the label member and the high-precision anchor member, the relative range ri→ j can be obtained, and the corresponding observation model can then be established.

3.3.2 Relative-Ranging-Based Observation Model Measured Relative Range For aircraft with access to sufficient GNSS observations, their position errors and airborne clock errors can be estimated and compensated. Meanwhile, aircraft that do not have access to sufficient GNSS observations, as in the case of an inadequate number of observed GNSS satellites or a poor geometric configuration, may need assistance from other cooperative aircraft to augment their navigation performance. In this case, the aircraft that can successfully complete

34

3 Modelling for Resilient Navigation via Collaboration

GNSS-based positioning can be utilized to cooperate with other aircraft in the swarm. When the ith label member in an aerial swarm needs to be assisted, it utilizes a radio ranging system to measure the distance to its jth collaborative anchor member in the swarm, and the measured range can be represented as | | r˜i→ j = | pi − p j | + c · δti→ j + εr = [(xi − x j )2 + (yi − y j )2 + (z i − z j )2 ]1/2 + δti→ j + εr = ri→ j + (δti − δt j ) + εr

(3.6)

where pi = [ xi yi z i ]T and p j = [ x j y j z j ]T represent the actual values of the positions of the label member and the corresponding collaborative anchor member, respectively, in the local coordinate frame; δti and δt j are the equivalent airborne clock biases of the label and anchor aircraft, respectively, in the relative range measurement; and εr is the equivalent noise in the relative range measurement. Calculated Relative Range For the ith label member in the swarm and its jth collaborative anchor, their relative range can also be calculated in accordance with their positions as indicated by their equipped navigation systems: | | ⁀ | ⁀ |⁀ r i→ j = | pi − p j | ⁀

⁀

⁀

⁀

⁀

⁀

= [(x i − x j )2 + ( y i − y j )2 + (z i − z j )2 ]1/2 ∂ri→ j ∂ri→ j = ri→ j + δ pi + δ pj ∂ pi ∂ pj ⁀

(3.7)

⁀

where pi = [ x⁀ i ⁀y i ⁀z i ]T and p j = [ x⁀ j ⁀y j ⁀z j ]T represent the positions of the label member and the corresponding collaborative anchor member, respectively, in the local coordinate frame as provided by their airborne navigation systems; the direction cosine of the position error is [ ⁀ ∂ri→ j = ∂ r i→ j ∂ xi ∂ pi [ xi −x j = ri→ j

⁀

⁀

∂ r i→ j ∂ r i→ j ∂ yi ∂z i yi −y j z i −z j ri→ j ri→ j

]

]

] [ ⁀ ⁀ ⁀ ∂ri→ j = − ∂ r i→ j − ∂ r i→ j − ∂ r i→ j ∂ xi ∂ yi ∂ zi ∂ pj ] [ x −x y −y z −z = − rii→ j j − rii→ j j − rii→ j j

(3.8)

(3.9)

Obviously, the following relationship holds: ∂ri→ j ∂ri→ j =− ∂ pi ∂ pj

(3.10)

3.3 Ranging-Based Collaborative Observation Modelling

35

It can be concluded from Eq. (3.7) that the position errors of both the anchor and label members in this cooperative relationship will be mapped to the calculated relative range. The mapping coefficients depend on the direction cosine between each member and the line of sight (LOS). Relative-Ranging-Based Observation Model for the Hierarchical Collaboration Structure In the hierarchical collaborative navigation structure, for a label aircraft with a sufficient number of observed satellites and a favourable geometric configuration, its clock error, which is modelled using filter states, can be estimated and corrected. In addition, the round-trip timing technique and the Precision Time Protocol are adopted in the data link established for coordination, whose relative time synchronization accuracy can consequently be high [15, 16]. In the model of enhanced collaboration-augmented positioning, the residual of the anchor aircraft’s clock error, which represents the error remaining after the above correction, is considered as the equivalent ranging error. Then, the states to be estimated in the collaborative fusion process are ]T [ x i = δ pi δti

(3.11)

By taking the difference between Eqs. (3.6) and (3.7) above, the relative-rangingbased observation model for a pair of collaborating aircraft can be established as follows: ⁀

δri→ j = r˜i→ j − r i→ j ⁀

⁀

∂ r i→ j ∂ r i→ j δ pi + δti − δ p j − δt j + εri→ j ∂ pi ∂ pj [ ⁀ ] ][ δ p ] [ ⁀ ∂ r i→ j i ∂ r i→ j + − δ p j − δt j + εri→ j = − 1 ∂ pi δti ∂ pj

=−

≙ hi→ j x i + vi→ j

(3.12)

When the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.12) can be combined to obtain ⎡

.. .

⎤

⎡

⎤ .. . ⎥ ⁀ − r i→ j ⎥ ⎦ .. . ⎤

⎢ ⎥ ⎢ ⎥ = ⎢ r˜ yi→ = ⎢ δr i→ j ⎣ ⎦ ⎣ i→ j .. . ⎡ ⎡ ⎤ .. .. . ⎢ ⎢ . ⎥ ⎥ ⎢ ⎥ ⎥ =⎢ ⎣ hi→ j ⎦ x i + ⎣ vi→ j ⎦ .. .. . .

36

3 Modelling for Resilient Navigation via Collaboration

≙ H i→ x i + vi→

(3.13)

According to Eq. (3.13), when the measured and calculated relative ranges between the anchor and label members in the swarm are obtained, the observation vector of the ith label member, denoted by yi→ , can be constructed by accumulating the differences between the measured and calculated relative ranges to the corresponding collaborative anchor members, denoted by . . . , δri→ j , . . .; additionally, the observation noise vector of the ith label member, denoted by vi→ , can be constructed by accumulating the observation noise related to the corresponding collaborative anchor members, denoted by . . . , vi→ j , . . .. Then, given the covariance of the observation noise, the position error and clock bias of the ith label member, which are taken as the observation states, can be estimated. The observation matrix of the ith label member, denoted by H i→ , is constructed by accumulating the direction cosine vectors related to the corresponding collaborative anchor members, denoted by . . . , hi→ j , . . ., and takes the form shown below ⎡

.. .

.. .

.. .

⎢ xi −x j yi −y j zi −z j H i→ = ⎢ ⎣ − ri→ j − ri→ j − ri→ j .. .. .. . . .

.. ⎤ . ⎥ 1⎥ ⎦ .. .

(3.14)

Relative-Ranging-Based Observation Model for the Parallel Collaboration Structure In the parallel collaborative navigation structure, all the navigation errors of the collaborating members are collected as the states: ]T [ x swar m = · · · x i · · · x j · · · x k · · ·

(3.15)

]T ]T ]T [ [ [ where x i = δ pi δti , x j = δ p j δt j , and x k = δ pk δtk . By taking the difference between Eqs. (3.6) and (3.7), the following relativeranging-based observation model for a pair of collaborating aircraft can be established: ⁀

δri→ j = r˜i→ j − r i→ j ⁀

⁀

∂ r i→ j ∂ r i→ j δ pi + δti − δ p j − δt j + εri→ j =− ∂ pi ∂ pj ⎤ ⎡ δ pi [ ] ⎢ δti ⎥ ⁀ ⁀ ∂ r i→ j j ⎥ ⎢ = − ∂ ∂r i→ 1 − −1 ⎣ δ p ⎦ + εri→ j pi ∂ pj j δt j

(3.16)

the relationship in Eq. (3.10) and adopting the notation hi→ j = ] [ By⁀ considering ∂ r i→ j − ∂ p 1 , Eq. (3.16) can be represented as i

3.3 Ranging-Based Collaborative Observation Modelling

37

⁀

δri→ j = r˜i→ j − r i→ j

[ ] [ ] xi + εri→ j = hi→ j −hi→ j xj

(3.17)

When the relationships between multiple pairs of collaborating aircraft have been established, the corresponding observation models similar to Eq. (3.16) can be combined to obtain ⎡

yswar m

.. .

⎤

⎡

.. . ⁀ − r i→ j .. .

⎤

⎢ ⎥ ⎢ ⎥ ⎢ δr ⎥ ⎢ ⎥ ⎢ i→ j ⎥ ⎢ r˜i→ j ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⁀ = ⎢ δri→k ⎥ = ⎢ r˜i→k − r i→k ⎥ ⎢ . ⎥ ⎢ ⎥ .. ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ δr j→k ⎥ ⎢ r˜ j→k − r⁀ j→k ⎥ ⎣ ⎦ ⎣ ⎦ .. .. . . ⎡ ⎤⎡ ⎤ ⎡ ⎤ .. .. .. . ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ ⎢··· h ⎥⎢ x ⎥ ⎢ ε ⎥ · · · −h · · · i→ j i→ j ⎢ ⎥⎢ i ⎥ ⎢ ri→ j ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ .. ⎢ ⎥⎢ ... ⎥ ⎢ ... ⎥ . ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ = ⎢ · · · hi→k ··· −hi→k · · · ⎥⎢ x j ⎥ + ⎢ εri→k ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ .. ⎢ ⎥⎢ .. ⎥ ⎢ .. ⎥ ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ . ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ x k ⎥ ⎢ εr j→k ⎥ h j→k · · · −h j→k ⎣ ⎦⎣ ⎦ ⎣ ⎦ .. .. .. . . . ≙ H swar m x swar m + vswar m

(3.18)

3.3.3 Relative-Ranging-Based Error Model Relative-Ranging-Based Error Model for the Hierarchical Collaboration Structure To obtain the solution for the states, which represent the estimated position error and clock bias of the label member of interest, the statistical properties of the observation noise should be known in advance. According to Eq. (3.12), the observation noise for the ith label member in collaboration with the jth anchor member in an aerial swarm is ⁀

vi→ j = −

∂ r i→ j δ p j − δt j + εr ∂ pj

38

3 Modelling for Resilient Navigation via Collaboration

= −hi→ j δ p j − δt j + εr

(3.19)

Then, the corresponding covariance of the observation noise for the ith label member in collaboration with the jth anchor member is ] [ ] [ T cov vi→ j = E vi→ j vi→ j ] [ T 2 2 = E hi→ j δ p j δ p j hi→ j + δt j + εr ] T [ 2] [ [ 2] = hi→ j E δ p j δ p j hi→ j + E δt j + E εr T 2 2 = hi→ j σ 2p j hi→ j + σt j + σr

(3.20)

where σ 2p j is the position uncertainty of the jth anchor member, σt2j is the standard deviation of the clock bias of the jth anchor member, and σr2 is the measurement noise of the range sensor. For multiple pairs of collaborating aircraft, according to Eqs. (3.12), (3.13), and (3.20), the observation noise model can be derived as follows: [ ] cov[vi→ ] = diag{ · · · cov vi→ j · · · } ] [ T 2 2 = diag · · · hi→ j σ 2p j hi→ j + σt j + σr · · ·

(3.21)

It can be concluded from Eq. (3.21) that the error in relative ranging is influenced not only by the precision of the range sensor but also by the positioning and timing accuracies of the anchor members in the swarm, which will ultimately influence the collaborative navigation fusion performance. Relative-Ranging-Based Error Model for the Parallel Collaboration Structure For the parallel collaborative navigation structure, according to Eq. (3.18), the covariance is ] [ ] [ T 2 2 2 cov vswar m vswar m = diag · · · E(εi→ j ) · · · E(εi→k ) · · · E(ε j→k ) · · · [ ] = diag · · · σr2 · · · (3.22) where σr2 is the measurement noise of the range sensor.

3.4 Range-Difference-Based Collaborative Observation Modelling

39

3.4 Range-Difference-Based Collaborative Observation Modelling 3.4.1 Relative-Range-Difference-Based Observation Geometry A multiple anchor–label collaborative navigation model can be established based on relative range differences. A label member can obtain the relative distances to several anchor members by means of airborne radar, data links, or other ranging sensors. For ranging based on the time of arrival of radio signals, if the radio rangers equipped on the swarm members are unsynchronized, a differential measurement approach (e.g., the time difference of arrival) should be adopted to reduce the influence of clock errors [17–19]. A corresponding observation model can then be established in the label member’s local coordinate frame. A diagram of the relative-range-difference-based observation geometry is shown in Fig. 3.3. Based on Fig. 3.3, in accordance with the relative geometric relationships between the label member and two high-precision anchor members, the relative ranges ri→ j and ri→ j+1 can be obtained, and then, the range difference can be obtained as follows: Δri→ j, j+1 = ri→ j − ri→ j+1

Fig. 3.3 Relative-range-difference-based observation geometry

(3.23)

40

3 Modelling for Resilient Navigation via Collaboration

3.4.2 Relative-Range-Difference-Based Observation Model I Single Relative Range Difference For collaborative navigation, a set of range difference measurements is obtained to determine the distance differences between the label member and several anchor members: Δri→ j, j+1 = ri→ j − ri→ j+1 | | | | = | p j − pi | − | p j+1 − pi |

(3.24)

where p j = [ x j y j z j ]T and p j+1 = [ x j+1 y j+1 z j+1 ]T are the positions of the jth and ( j + 1)th anchor members, respectively, as indicated by their equipped airborne navigation systems, and pi = [ xi yi z i ]T is the position of the label member to be estimated: Δri→ j, j+1 = ri→ j − ri→ j+1 | | | | = | p j − pi | − | p j+1 − pi |

(3.25)

Based on the position coordinates of each member in the relative coordinate system established above and the relative distance difference information from the label member to be assisted to each anchor member, a collaborative navigation model based on the range difference principle is established as follows: Δri→ j, j+1 = ri→ j − ri→ j+1 = [(x j − xi )2 + (y j − yi )2 + (z j − z i )2 ]1/2 − [(x j+1 − xi )2 + (y j+1 − yi )2 + (z j+1 − z i )2 ]1/2

(3.26)

Squaring both sides of Eq. (3.26) yields 2 2 2 2 Δri→ j, j+1 = [(x j − x i ) + (y j − yi ) + (z j − z i ) ]

− [(x j+1 − xi )2 + (y j+1 − yi )2 + (z j+1 − z i )2 ] = (x 2j − 2xi x j + y 2j − 2yi y j + z 2j − 2z i z j ) − (x 2j+1 − 2xi x j+1 + y 2j+1 − 2yi y j+1 + z 2j+1 − 2z i z j+1 ) = (x 2j + y 2j + z 2j ) − (x 2j+1 + y 2j+1 + z 2j+1 ) − 2xi (x j − x j+1 ) − 2yi (y j − y j+1 ) − 2z i (z j − z j+1 ) The above Eq. (3.27) can be converted into matrix form as follows: 2 Δri→ j, j+1 + 2Δri→ j, j+1 ri→ j

(3.27)

3.4 Range-Difference-Based Collaborative Observation Modelling

41

⎡

= −2[ x j − x j+1 y j − y j+1

⎤ xi z j − z j+1 ]⎣ yi ⎦ + p 2j − p 2j+1 zi

(3.28)

where p j = (x 2j + y 2j + z 2j )1/2 and p j+1 = (x 2j+1 + y 2j+1 + z 2j+1 )1/2 are calculated from the coordinates of the jth and ( j + 1)th anchor member in the aerial swarm. Relative-Range-Difference-Based Observation Model The position of the ith label member, denoted by pi = [ xi yi z i ]T , and its range to the main anchor member, denoted by ri→ j , are taken as the states to be estimated: x i = [ pi ri→ j ]T = [ xi yi z i ri→ j ]T

(3.29)

Multiple range difference measurements can form a set of hyperbolic equations expressed in terms of the position of the label member. The position of the label member can then be estimated by solving this set of equations. Based on the obtained distance information, the effective relative positions of the members can be determined, and a corresponding collaborative positioning model can be established to obtain a collaborative positioning solution: ⎡

yi→

⎤ .. . ⎢ 2 ⎥ 1⎢ Δri→ j, j+1 + p j − p j+1 ⎥ ⎢ ⎥ = ⎢ 2 2 ⎣ Δri→ j, j+2 + p j − p j+2 ⎥ ⎦ .. . ⎡ ⎡ ⎤ ⎤ .. .. .. .. .. ⎤ ⎡ . . . . . x i ⎢ ⎢ ⎥ ⎥ ⎢ x j − x j+1 y j − y j+1 z j − z j+1 Δri→ j, j+1 ⎥⎢ yi ⎥ ⎢ vi→ j, j+1 ⎥ ⎥+⎢ ⎥⎢ ⎥ =⎢ ⎢ x j − x j+2 y j − y j+2 z j − z j+2 Δr ⎥⎣ z i ⎦ ⎢ v ⎥ i→ j, j+2 ⎦ ⎣ i→ j, j+2 ⎦ ⎣ .. .. .. .. .. ri→ j . . . . . ≙ H i→ x i + vi→

(3.30)

As shown in Eq. (3.30), when the relative differences between pairs of anchors and the label members in the swarm are obtained, a collaborative observation model can be established. The observation vector of the ith label member, denoted by yi→ , can be constructed by accumulating the constructed observations corresponding to the pairs of collaborative anchor members, denoted by . . . , yi→ j, j+1 , yi→ j, j+2 , . . ., 2 yi→ j, j+1 = Δri→ j, j+1 + p j − p j+1 ; moreover, the observation noise vector of the ith label member, denoted as vi→ , can be constructed by accumulating the observation noise related to the corresponding collaborative anchor member pairs, denoted by . . . , vi→ j, j+1 , vi→ j, j+2 , . . .. Then, given the covariance of the observation noise, the position error of the ith label member, which is taken as the observation state, can

42

3 Modelling for Resilient Navigation via Collaboration

be estimated. The observation matrix of the ith label member, denoted by H i→ , is constructed by accumulating the direction cosine vectors related to the corresponding collaborative anchor members, denoted by . . . , hi→ j, j+1 , hi→ j, j+2 , . . ., as follows: hi→ j, j+1 = [ x j − x j+1 y j − y j+1 z j − z j+1 Δri→ j, j+1 ].

3.4.3 Relative-Range-Difference-Based Observation Model II The relative range ri→ j+1 between the ith label member and the ( j + 1)th anchor member is calculated from the position coordinates of the members in the relative coordinate system as follows: | | ri→ j+1 = | p j+1 − pi | |( ) ( )| = | p j+1 − p j − pi − p j |

(3.31)

Squaring both sides of Eq. (3.31) yields | |2 2 | | ri→ j+1 = p j+1 − pi |( ) ( )| = | p j+1 − p j − pi − p j | | | = | r j→ j+1 − r j→i | | | |2 |2 = | r j→ j+1 | − 2r Tj→ j+1 r j→i + | r j→i |

(3.32)

Substituting Eq. (3.23) into Eq. (3.32) then yields | | |2 |2 (ri→ j − Δri→ j, j+1 )2 = | r j→ j+1 | − 2r Tj→ j+1 r j→i + | r j→i | 2 2 2 T 2 ri→ j + Δri→ j, j+1 − 2ri→ j Δri→ j, j+1 = r j→ j+1 − 2r j→ j+1 r j→i + ri→ j

(3.33)

Considering the actual observation environment, there are observation errors in each distance measurement, and Eq. (3.33) also contains errors. The spherical interpolation algorithm replaces the observation errors by introducing distance difference equation errors vi→ j, j+1 , as follows: 2 T r 2j→ j+1 − Δri→ j, j+1 + 2ri→ j Δri→ j, j+1 − 2r j→ j+1 r j→i = vi→ j, j+1

(3.34)

Multiple range difference measurements can form a set of hyperbolic equations expressed in terms of the position error of the label member. Equation (3.34) can be converted into matrix form as follows:

3.4 Range-Difference-Based Collaborative Observation Modelling

43

⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎤ .. .. .. .. . . . . ⎢ T ⎢ ⎢ 2 ⎢ ⎥ ⎥ ⎥ ⎥ 2 ⎢ ⎢ ⎢ r j→ j+1 − Δri→ ⎢ Δri→ j, j+1 ⎥ ⎥ ⎥ ⎥ j, j+1 ⎥ ⎢ ⎢ ⎥ − 2⎢ r j→ j+1 ⎥ r j→i = ⎢ vi→ j, j+1 ⎥ + 2r i→ j ⎢ 2 T ⎢ ⎢ ⎢r2 ⎥ ⎥ ⎥ ⎥ ⎣ r j→ j+2 ⎦ ⎣ vi→ j, j+2 ⎦ ⎣ j→ j+2 − Δri→ j, j+2 ⎦ ⎣ Δri→ j, j+2 ⎦ .. .. .. .. . . . . ⎡

≙ ηi→ j + 2ri→ j Δr i→ j,... − 2 R j→ r j→i = vi→ j,...

(3.35)

3.4.4 Relative-Range-Difference-Based Error Model By selecting the position error of the ith label member in the swarm, denoted by δx i = [δxi δyi δz i ]T , as the state to be estimated and then taking partial derivatives of Eq. (3.26), one obtains δyi→ j, j+1 = Δri→ j, j+1 − ri→ j + ri→ j+1 ∂Δri→ j, j+1 = + δvi→ j, j+1 ∂ pi ∂ri→ j ∂ri→ j+1 = δ pi − δ pi + δvi→ j, j+1 ∂ pi ∂ pi =

[

x j −xi ri→ j

−

x j+1 −xi y j −yi ri→ j+1 ri→ j

−

y j+1 −yi z j −z i ri→ j+1 ri→ j

−

z j+1 −z i ri→ j+1

⎡ ⎤ ] δxi ⎣ δyi ⎦ δz i

+ δvi→ j, j+1 ≙ δhi→ j, j+1 δx i + δvi→ j, j+1

(3.36)

Multiple range difference measurements can form a set of hyperbolic equations expressed in terms of the position error of the label member. The estimated position of the label member can be obtained by solving this set of equations; then, ⎡ δ yi→

.. .

⎤

⎢ ⎥ ⎢ δyi→ j, j+1 ⎥ ⎢ ⎥ =⎢ ⎥ ⎣ δyi→ j, j+2 ⎦ .. . ⎡ ⎡ ⎤ ⎤ .. .. . . ⎢ ⎢ ⎥ ⎥ ⎢ δhi→ j, j+1 ⎥ ⎢ δvi→ j, j+1 ⎥ ⎢ ⎢ ⎥ ⎥ =⎢ ⎥δx i + ⎢ δv ⎥ ⎣ δhi→ j, j+2 ⎦ ⎣ i→ j, j+2 ⎦ .. .. . .

44

3 Modelling for Resilient Navigation via Collaboration

≙ δ H i→ δx i + δvi→

(3.37)

3.5 Bearing-Only Collaborative Observation Modelling 3.5.1 Relative-Bearing-Only Observation Geometry Compared with relying on the time of radio signal arrival, cooperative navigation based on bearing-only observations reduces the requirement for clock synchronization and offers certain advantages. In accordance with the principle of LOS intersection, bearing-only collaborative positioning is less dependent on the amount of observation information available [20–23]. Thus, this type of collaborative positioning is studied in this section. The corresponding diagram of the relative-bearingbased observation geometry is established as shown in Fig. 3.4. The hemisphere in Fig. 3.4 is the unit hemisphere, and the coordinate system is the relative rectangular coordinate system with the label member of interest as the coordinate origin. Based on Fig. 3.4, in accordance with the relative bearing relationships between the label member and a high-precision anchor, the relative bearing is decomposed

Fig. 3.4 Relative-bearing-only observation geometry

3.5 Bearing-Only Collaborative Observation Modelling

45

into the relative azimuth θi→ j and the elevation ϕi→ j in the label member’s local coordinate frame.

3.5.2 Relative-Bearing-Only Observation Model Measured Relative Bearing The bearing observation sensor on the label aircraft can observe the relative bearing of each anchor within the LOS. The anchor aircraft can be represented in the local horizontal coordinate frame centred at the location of the label aircraft, where pi = [ xi yi z i ]T is the position of the ith label member, p j = [ x j y j z j ]T is the position of the jth anchor aircraft, and the LOS vector between them can be represented as r i→ j = p j − pi [ ]T = x j − xi y j − yi z j − z i [ ]T ≙ Δxi→ j Δyi→ j Δz i→ j

(3.38)

The LOS from the label member to the anchor member can be represented in terms of the azimuth and elevation angles: (

) ( θi→ j = arctan Δyi→ j /Δxi→ j ] [ 2 2 1/2 ϕi→ j = arctan Δz i→ j /(Δxi→ j + Δyi→ j )

(3.39)

where θi→ j is the relative azimuth angle and ϕi→ j is the elevation angle. The following equation holds: 2 2 1/2 (Δxi→ = ri→ j cos ϕi→ j j + Δyi→ j )

≙ di→ j

(3.40)

The bearing measurement sensor of the label member can observe the relative direction of the LOS between the label member and the anchor member in the aerial swarm: ( θ˜i→ j = θi→ j + εθi→ j (3.41) ϕ˜i→ j = ϕi→ j + εϕi→ j where εθ and εϕ are the measurement noise of the angular sensor in the relative azimuth and elevation directions, respectively. Calculated Relative Bearing The calculated relative bearing can be obtained from the coordinates of the label member and the anchor member as indicated by their equipped navigation systems. Accordingly, the calculated VOS is

46

3 Modelling for Resilient Navigation via Collaboration ⁀

⁀

⁀

r i→ j = p j − pi ]T [ = x⁀ j − x⁀ i ⁀y j − ⁀y i ⁀z j − ⁀z i ]T [ ≙ Δx⁀ i→ j Δ ⁀y i→ j Δ⁀z i→ j

(3.42)

The calculated relative bearing can then be derived in accordance with the direction of the VOS between the label member and the anchor member in the aerial swarm: ⎧⁀ ) ( ⁀ ⁀ ⎪ θ i→ j = arctan Δ y i→ j /Δx i→ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂θi→ j ∂θi→ j ⎪ ⎪ δ pi + δ pj = θi→ j + ⎪ ⎪ ⎨ ∂δ pi ∂δ p j [ ] (3.43) ⁀ ⁀2 ⁀2 ⁀ ⎪ 1/2 ⎪ ϕ = arctan Δ z /(Δ x + Δ y ) ⎪ i→ j i→ j i→ j i→ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ϕi→ j ∂ϕi→ j ⎪ ⎪ = ϕi→ j + δ pi + δ pj ⎩ ∂δ pi ∂δ p j ⁀

⁀

where θ i→ j and ϕ i→ j are the calculated relative azimuth and elevation, respectively, ⁀

⁀

⁀

⁀

⁀

⁀

and (x i , y i , z i ) and (x j , y j , z j ) are the coordinates of the label and anchor aircraft, respectively, as provided by their equipped navigation systems. Taking the partial derivatives of the angular functions yields ] ∂θi→ j 1 [ = 2 Δyi→ j −Δxi→ j 0 ∂δ pi di→ j [ ] ∂ϕi→ j 1 2 = Δz Δy Δz −d Δx i→ j i→ j i→ j i→ j i→ j 2 ∂δ pi di→ j ri→ j

(3.44) (3.45)

It can be derived from Eqs. (3.42) and (3.43) that ∂θi→ j ∂θi→ j =− ∂δ pi ∂δ p j

(3.46)

∂ϕi→ j ∂ϕi→ j =− ∂δ pi ∂δ p j

(3.47)

Relative-Bearing-Based Observation Model for the Hierarchical Collaboration Structure Considering that the errors contained in the rough position of the label member are significantly larger than the errors of the anchor member, which is equipped with a high-accuracy navigation system, the errors in the position of the label member are selected as the states to be estimated:

3.5 Bearing-Only Collaborative Observation Modelling

47

x i = δ pi

(3.48)

By taking the difference between Eqs. (3.41) and (3.43), the following observation model can be obtained: ] [ ] [ ] [⁀ θ˜i→ j δθi→ j θ i→ j = − ⁀ δϕi→ j ϕ˜i→ j ϕ i→ j [ ∂θi→ j ] ∂θ − ∂δ p δ pi − ∂δi→p j δ p j + εθi→ j i j = ∂ϕ j ∂ϕ j δ pi − ∂δi→ δ p j + εϕi→ j − ∂δi→ pj pj [ ∂θi→ j [ ∂θi→ j ] ] − ∂δ p δ p j + εθi→ j − ∂δ p j i = ∂ϕ j δ pi + ∂ϕ j − ∂δi→ δ p j + εϕi→ j − ∂δi→ pj p j

≙ hi→ j x i + vi→ j

(3.49)

When the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.49) can be combined to obtain ⎡ . ⎤ ⎡ . ⎤ ⎡ . ⎤ .. . . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⁀ ⎥ ⎢ δθi→ j ⎥ ⎢ θ˜i→ j ⎥ ⎢ ⎢ θ ⎥=⎢ ⎥ − ⎢ i→ j ⎥ yi→ = ⎢ ⎥ ⎢ δϕi→ j ⎥ ⎢ ϕ˜i→ j ⎥ ⎢ ⁀ ⎣ ⎦ ⎣ ⎦ ⎣ ϕ i→ j ⎥ ⎦ .. .. .. . . . ⎡ ⎡ ⎤ ⎤ .. .. ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎥ =⎢ + x h ⎣ i→ j ⎦ i ⎣ vi→ j ⎦ .. .. . . ≙ H i→ x i + vi→

(3.50)

Based on Eq. (3.13), when the measured and calculated relative bearings between the anchor and label members in the swarm are obtained, the collaborative observation model can be established. The observation vector of the ith label member, denoted by yi→ , can be constructed by accumulating the differences between the measured and calculated relative bearings to the corresponding collaborative anchor members, denoted by . . . , δri→ j , . . .; additionally, the observation noise vector of the ith label member, denoted by vi→ , can be constructed by accumulating the observation noise related to the corresponding collaborative anchor members, denoted by . . . , vi→ j , . . .. Then, given the covariance of the observation noise, the position error of the ith label member, which is taken as the observation state, can be estimated. The observation matrix of the ith label member, denoted by H i→ , is constructed by accumulating the direction cosine vectors related to the corresponding collaborative

48

3 Modelling for Resilient Navigation via Collaboration

anchor members, denoted by . . . , hi→ j , . . ., and takes the form shown below ⎡

H i→

.. .

⎤

⎥ ⎢ ] [ ⎥ ⎢ − d 21 Δyi→ j −Δxi→ j 0 ⎥ ⎢ i→ j [ ] ⎥ ⎢ =⎢ 1 2 Δxi→ j Δz i→ j Δyi→ j Δz i→ j −di→ j ⎥ 2 ⎥ ⎢ − di→ j ri→ j ⎦ ⎣ .. . ⎡ ⎤ .. .. .. . . . ⎥ ⎢ Δxi→ j ⎢ − Δyi→ j ⎥ 0 ⎢ ⎥ 2 2 di→ di→ j j ⎢ = ⎢ Δxi→ j Δzi→ j Δyi→ j Δzi→ j di→ j ⎥ ⎥ − d r2 2 2 ⎢ − di→ j ri→ ⎥ ri→ i→ j i→ j j j ⎦ ⎣ .. .. .. . . .

(3.51)

Relative-Bearing-Based Observation Model for the Parallel Collaboration Structure In the parallel collaborative navigation structure, all the navigation errors of the collaborating members are collected as the states: [ ]T x swar m = · · · δ pi · · · δ p j · · · δ pk · · ·

(3.52)

By taking the difference between Eqs. (3.41) and (3.43), the following observation model can be obtained: ] [ ] [ ] [⁀ θ˜i→ j δθi→ j θ i→ j = − ⁀ δϕi→ j ϕ˜i→ j ϕ i→ j [ ∂θi→ j ] ∂θ − ∂δ p δ pi − ∂δi→p j δ p j + εθi→ j i j = ∂ϕ j ∂ϕ j − ∂δi→ δ pi − ∂δi→ δ p j + εϕi→ j pj pj [ ∂θi→ j ] [ ] [ ] ∂θ − ∂δ p − ∂δi→p j δ pi εθi→ j i j = + (3.53) ∂ϕ j ∂ϕ j δ pj εϕi→ j − ∂δi→ − ∂δi→ p p j

j

Consider the relationships in Eqs. (3.46) and (3.47) and the notation ] [ ∂θ ∂ϕ j T hi→ j = − ∂δi→pij − ∂δi→ pj Then, Eq. (3.53) can be represented as [

δθi→ j δϕi→ j

]

[

θ˜ = i→ j ϕ˜i→ j

]

[⁀ ] θ i→ j − ⁀ ϕ i→ j

(3.54)

3.5 Bearing-Only Collaborative Observation Modelling

[

= hi→ j

49

] [ ] [ ] δ pi εθi→ j + −hi→ j δ pj εϕi→ j

(3.55)

When the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.55) can be combined to obtain ⎡

yswar m

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

.. .

δθi→ j δϕi→ j .. . δθi→k δϕi→k .. . δθ j→k δϕ j→k .. .

⎤

⎡

⎤

.. .

⎢ ⁀ ⎥ ⎢ ˜ ⎥ ⎢ θi→ j − θ i→ j ⎥ ⎢ ⎥ ⎢ ϕ˜i→ j − ϕ⁀ i→ j ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⁀ ⎥ ⎢ ˜ ⎥ ⎢ θi→k − θ i→k ⎥=⎢ ⎥ ⎢ ϕ˜i→k − ϕ⁀ i→k ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⁀ ⎥ ⎢ ⎥ ⎢ θ˜ j→k − θ j→k ⎥ ⎢ ⎦ ⎢ ϕ˜ j→k − ϕ⁀ j→k ⎣ .. .

⎡ ⎢ ⎢··· h i→ j ⎢ ⎢ ⎢ ⎢ ⎢ = ⎢ · · · hi→k ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

.. . · · · −hi→ j .. . ··· .. .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡

⎤ ⎢ ⎢ εθi→ j .. ⎢ ⎥⎢ . ⎥ ⎢ ε ⎥⎢ δ p ⎥ ⎢ ϕi→ j ⎥⎢ i ⎥ ⎢ . ⎥⎢ . ⎥ ⎢ .. ⎥⎢ .. ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ε · · · ⎥⎢ δ p j ⎥ + ⎢ θi→k ⎥⎢ . ⎥ ⎢ εϕi→k ⎥⎢ . ⎥ ⎢ . ⎥⎢ . ⎥ ⎢ . ⎥⎢ ⎥ ⎢ ⎥⎢ δ pk ⎥ ⎢ . ⎦⎣ ⎦ ⎢ εθ j→k ⎢ .. ⎢ εϕ j→k . ⎣ .. . ⎤⎡

··· −hi→k

h j→k · · · −h j→k .. .

≙ H swar m x swar m + vswar m

.. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.56)

3.5.3 Relative Bearing Error Model Relative-Bearing-Based Error Model for the Hierarchical Collaboration Structure To obtain the solution for the states, which represent the estimated position error

50

3 Modelling for Resilient Navigation via Collaboration

and clock bias of the label member of interest in the swarm, the statistical properties of the observation noise should be known in advance. According to Eq. (3.49), the observation noise for the ith label member in collaboration with the jth anchor member in the aerial swarm is [ ∂θi→ j ] − ∂δ p δ p j + εθi→ j j vi→ j = ∂ϕ j − ∂δi→ δ p j + εϕi→ j pj [ ∂θi→ j ] ] [ − ∂δ p εθi→ j j = + δ p ∂ϕ j j εϕi→ j − ∂δi→ p j

= hi→ j δ p j + ε θϕi→ j

(3.57)

Then, the corresponding covariance of the observation noise for the ith label member in collaboration with the jth anchor member is ] [ ] [ T cov vi→ j = E vi→ j vi→ j ] [ T T = E hi→ j δ p j δ p j hi→ j + ε θϕ ε θϕ T 2 = hi→ j σ 2p j hi→ j + σ θϕ

(3.58)

For multiple pairs of collaborating aircraft, in accordance with Eqs. (3.49), (3.50), and (3.58), the observation noise model can be derived as follows: ] [ cov[vi→ ] = diag{ · · · cov vi→ j · · · } [ ] T 2 = diag · · · hi→ j σ 2p j hi→ + σ · · · j θϕ

(3.59)

It can be concluded from Eq. (3.59) that the error in the relative bearing is influenced not only by the precision of the bearing sensor but also by the positioning accuracy of the anchor members in the swarm, which will ultimately influence the collaborative navigation fusion performance. Relative-Bearing-Based Error Model for the Parallel Collaboration Structure For the parallel collaborative navigation structure, according to Eq. (3.56), the covariance is ] [ ] { [ ] [ T 2 2 2 2 cov vswar m vswar m = diag · · · E εθi→ j εϕi→ j · · · E εθi→k εϕi→k [ ] } · · · E εθ2 j→k εϕ2 j→k · · · [ ] 2 = diag · · · σθ/ϕ (3.60) ··· 2 where σθ/ϕ is the measurement noise covariance of the bearing sensor.

3.6 Vector-of-Sight-Based Collaborative Observation Modelling

51

Fig. 3.5 Relative-VOS-based observation geometry

3.6 Vector-of-Sight-Based Collaborative Observation Modelling 3.6.1 Relative-VOS-Based Observation Geometry An improved multiple anchor–label collaborative navigation model is based on both relative distance and relative bearing information. A label aircraft can obtain the relative distance and relative bearing information from an anchor by means of airborne radar and laser sensors [24–27]. The observation model is established in the label member’s local coordinate frame. The relative vector-of-sight (VOS)-based observation geometry is shown in Fig. 3.5. Based on Fig. 3.5, in accordance with the relative distance and bearing relationships between the label member and the high-precision anchor, the relative distance d ji is decomposed into the X-, Y-, and Z-axis directions in the label member’s local coordinate frame.

3.6.2 Relative VOS Observation Model in Spherical Coordinates Measured Relative VOS The VOS can be represented as a combination of the measured range and bearing from the ith label member to the jth anchor member in a spherical coordinate frame, in accordance with Eqs. (3.6) and (3.41):

52

3 Modelling for Resilient Navigation via Collaboration

{

r˜ i→ j

⎡

} S

⎤ ⎡ ⎤ r˜i→ j ri→ j + (δti − δt j ) + εr ⎦ = ⎣ θ˜i→ j ⎦ = ⎣ θi→ j + εθ ϕ˜i→ j ϕi→ j + εϕ

(3.61)

Calculated Relative VOS Similarly, the calculated VOS can be obtained as a combination of the calculated range and bearing from the ith label member to the jth anchor member in a spherical coordinate frame, in accordance with Eqs. (3.7) and (3.43): {

⎡ ⁀ ⁀ ⁀ ⁀ ⁀ ⁀ 2 1/2 ⎤ ⎡⁀ ⎤ [( x i − x j )2 + ((y i − y j )2 + ( z i − ) z j) ] r i→ j ⎢ ⎥ { } ⁀ ⁀ ⁀ ⁀ ⁀ ⎢ ⎥ ⎢⁀ ⎥ arctan Δ y i→ j /Δ x i→ j r i→ j = ⎣ θ i→ j ⎦ = p i − p j =⎢ ] ⎥ [ S S ⎣ ⎦ ⁀ ⁀2 ⁀2 ⁀ arctan Δ z i→ j /(Δ x i→ j + Δ y i→ j )1/2 ϕ i→ j }

⎡ ∂r i→ j ⎤ ⎢ ∂ pi ri→ j ∂θ ⎢ ⎥ ⎢ i→ j = ⎣ θi→ j ⎦ + ⎢ ⎢ ∂δ pi ⎣ ∂ϕ ϕi→ j i→ j ⎡

∂δ pi

⎤

⎡ ∂ri→ j ⎤

⎢ ∂ pj ⎥ ⎢ ∂θ ⎥ ⎥δ pi + ⎢ i→ j ⎢ ∂δ p j ⎥ ⎣ ∂ϕ ⎦ i→ j

⎥ ⎥ ⎥δ p j ⎥ ⎦

(3.62)

∂δ p j

Relative VOS Observation Model for the Hierarchical Collaboration Structure In this model of enhanced collaboration-augmented positioning, the residual of the anchor member’s clock error, which represents the error remaining after the above correction, is considered as the equivalent ranging error. Then, the states to be estimated in the collaborative fusion process are ] [ x i = δ pi δti

(3.63)

By taking the difference in spherical coordinates between Eqs. (3.61) and (3.62) above, the following relative LOS vector observation model for a pair of collaborating aircraft can be established: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡⁀ r i→ j δri→ j r˜i→ j { } ⎥ ⎢⁀ δr i→ j S = ⎣ δθi→ j ⎦ = ⎣ θ˜i→ j ⎦ − ⎣ θ i→ j ⎦ ⁀ δϕi→ j ϕ˜i→ j ϕ i→ j ⎡ ⁀ ⎤ ⎡ ⁀ ⎤ ∂ r i→ j ∂r j − δ p − δt + ε ] [ − ∂ i→ 1 j r j j ⎢ ∂ p ∂θ ⎥ ⎢ ∂θ pi ⎥ δ pi i→ j ⎢ − i→ j δ p + ε ⎥ ⎥ + =⎢ − 0 θ j ⎣ ⎦ ⎣ ∂δ pi ⎦ δti ∂δ p j ∂ϕi→ j ∂ϕi→ j − ∂δ p 0 − δ p + ε ϕ j ∂δ p j j

≙ H i→ j x i + vi→ j

(3.64)

When the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.64) can be combined to obtain

3.6 Vector-of-Sight-Based Collaborative Observation Modelling

53

⎤ ⎧⎡ ⎤ ⎡ . ⎤⎫ .. .. ⎪ ⎪ . ⎪ ⎪ ⎢ { . } ⎥ ⎨⎢ . ⎥ ⎢ ⁀ . ⎥⎬ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ = ⎣ δr i→ j S ⎦ = ⎣ r˜ i→ j ⎦ − ⎣ r i→ j ⎦ ⎪ ⎪ ⎪ ⎪ .. .. .. ⎭ ⎩ . . . S ⎡ ⎡ ⎤ ⎤ .. .. ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎥ = ⎣ H i→ j ⎦ x i + ⎢ ⎣ vi→ j ⎦ .. .. . . ⎡

yi→

≙ H i→ x i + vi→

(3.65)

Based on Eq. (3.65), when the measured and calculated relative LOS vectors between the anchor and label members in the swarm, which combine the relative range and bearing information, are obtained, the corresponding collaborative observation model can be formed. The observation vector of the ith label member, denoted by yi→ , can be constructed by accumulating the differences in spherical coordinates between the measured and calculated relative VOSs to the corresponding collabora{ } tive anchor members, denoted by · · · , δr i→ j S , · · · ; additionally, the observation noise vector of the ith label member, denoted by vi→ , can be constructed by accumulating the observation noise related to the corresponding collaborative anchor members, denoted by · · · , vi→ j , · · · . Then, given the covariance of the observation noise, the position error of the ith label member, which is taken as the observation state, can be estimated. The observation matrix of the ith label member, denoted by H i→ , is constructed by accumulating the direction cosine vectors related to the corresponding collaborative anchor members, denoted by · · · , H i→ j , · · · , and takes the form shown below ⎡ ⎤ .. .. .. .. . . . . ⎢ ⎥ y −y z −z ⎢ − xi −x j − rii→ j j − rii→ j j 1 ⎥ ⎢ ⎥ ri→ j ⎢ ⎥ Δyi→ j Δxi→ j ⎢ ⎥ 0 0 − 2 2 (3.66) H i→ = ⎢ di→ di→ ⎥ j j ⎢ Δxi→ j Δzi→ j Δyi→ j Δzi→ j di→ j ⎥ − d r2 0⎥ ⎢ − di→ j r 2 2 ri→ i→ j i→ j i→ j j ⎣ ⎦ .. .. .. .. . . . . Relative VOS Observation Model for the Parallel Collaboration Structure In the parallel collaborative navigation structure, all the navigation errors of the collaborating members are collected as the states: ]T [ x swar m = · · · x i · · · x j · · · x k · · · [ [ [ ]T ]T ]T where x i = δ pi δti , x j = δ p j δt j , and x k = δ pk δtk .

(3.67)

54

3 Modelling for Resilient Navigation via Collaboration

By taking the difference between Eqs. (3.61) and (3.62), the following relative VOS observation model for a pair of collaborating aircraft can be established: ⎤ ⎤ ⎡ ⎤ ⎡⁀ r i→ j δr r ˜ i→ j i→ j { } ⎥ ⎢⁀ δr i→ j S = ⎣ δθi→ j ⎦ = ⎣ θ˜i→ j ⎦ − ⎣ θ i→ j ⎦ ⁀ δϕi→ j ϕ˜i→ j ϕ i→ j ⎤⎡ ⎤ ⎡ ⁀ ⁀ ∂r j ∂r j ⎡ ⎤ δ pi 1 − ∂ i→ −1 − ∂ i→ εri→ j p p j ⎥ ⎢ ⎥ ⎢ ∂θ i ∂θi→ j i→ j ⎥⎢ δti ⎥ ⎣ ⎦ =⎢ ⎣ − ∂δ pi 0 − ∂δ p j 0 ⎦⎣ δ p ⎦ + εθi→ j j ∂ϕi→ j ∂ϕi→ j εϕi→ j − ∂δ p 0 − ∂δ p 0 δt j j j ⎡

(3.68)

Consider the relationships in Eqs. (3.10), (3.46), and (3.47) and the notation ⎡ hi→ j =

⎤

⁀

∂ r i→ j ⎢ ∂θ∂ pi ⎢ − i→ j ⎣ ∂δ pi ∂ϕ j − ∂δi→ pj

−

1

⎥ ⎥ ⎦

(3.69)

Then, Eq. (3.68) can be represented as {

δr i→ j

⎤ ⎤ ⎡ ⎤ ⎡⁀ r i→ j δri→ j r˜i→ j ⎥ ⎢⁀ = ⎣ δθi→ j ⎦ = ⎣ θ˜i→ j ⎦ − ⎣ θ i→ j ⎦ ⁀ δϕi→ j ϕ˜i→ j ϕ i→ j ⎤ ⎡ ⎡ ⎤ δ pi εri→ j ]⎢ δti ⎥ [ ⎥ ⎣ ⎦ = hi→ j −hi→ j ⎢ ⎣ δ p ⎦ + εθi→ j j εϕi→ j δt j ⎡

} S

(3.70)

When the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.70) can be combined to obtain

3.6 Vector-of-Sight-Based Collaborative Observation Modelling

⎡

yswar m

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

.. .

δri→ j δθi→ j δϕi→ j .. . δri→k δθi→k δϕi→k .. .

δr j→k δθ j→k δϕ j→k .. .

55

⎡

⎤ .. . ⎢ ⎥ ⎥ ⁀ ⎥ ⎢ r ˜ − r ⎢ ⎥ i→ j i→ j ⎥ ⎢ ⎥ ⁀ ⎥ ⎢ ⎥ ⎢ θ˜i→ j − θ i→ j ⎥ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ϕ˜i→ j − ϕ⁀ i→ j ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ .. ⎥ ⎥ ⎢ ⎥ . ⎥ ⎢ ⎥ ⎥ ⎢ ⁀ ⎥ ⎢ r˜i→k − r i→k ⎥ ⎥ ⎥ ⎢ ⁀ ⎥ ⎥ = ⎢ θ˜ ⎥ ⎢ i→k − θ i→k ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ϕ˜i→k − ϕ⁀ i→k ⎥ ⎥ ⎥ ⎢ ⎥ .. ⎥ ⎢ ⎥ ⎥ ⎢ . ⎥ ⎥ ⎢ ⎥ ⁀ ⎥ ⎢ r˜ ⎥ ⎢ j→k − r j→k ⎥ ⎥ ⎥ ⎢ ⁀ ⎥ ⎢ θ˜ j→k − θ j→k ⎥ ⎥ ⎥ ⎢ ⎥ ⁀ ⎦ ⎢ ϕ˜ ⎥ − ϕ j→k ⎦ ⎣ j→k .. . ⎤

⎡ ⎡

⎡ ⎢ ⎢··· h i→ j ⎢ ⎢ ⎢ ⎢ ⎢ = ⎢ · · · hi→k ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

.. . · · · −hi→ j .. . ··· .. .

··· −hi→k

h j→k · · · −h j→k .. .

⎤⎢ ⎢ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ · · · ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎢ ⎢ ⎢ ⎣

.. . δ pi δti .. . δ pj δt j .. . δ pk δtk .. .

.. .

⎢ ⎢ε ⎢ ri→ j ⎥ ⎢ εθi→ j ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ εϕi→ j ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ε ⎥ ⎢ ri→k ⎥ ⎢ ⎥ + ⎢ εθi→k ⎥ ⎢ ⎥ ⎢ εϕi→k ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ εr j→k ⎥ ⎢ ⎦ ⎢ εθ j→k ⎢ ⎢ εϕ j→k ⎣ .. . ⎤

≙ H swar m x swar m + vswar m

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.71)

3.6.3 Relative VOS Error Model in Spherical Coordinates Relative VOS Error Model for the Hierarchical Collaboration Structure To obtain the solution for the states, which represent the estimated position error and clock bias of the label member of interest in the swarm, the statistical properties

56

3 Modelling for Resilient Navigation via Collaboration

of the observation noise should be known in advance. According to Eq. (3.49), the observation noise for the ith label member in collaboration with the jth anchor member in the aerial swarm is ⎤ ⎡ ∂r j δ p j − δt j + εr − ∂ i→ pj ⎥ ⎢ ∂θi→ j ⎥ vi→ j = ⎢ ⎣ − ∂δ p j δ p j + εθ ⎦ ∂ϕ j − ∂δi→ δ p j + εϕ pj ⎡ ∂r ⎤ j ⎡ ⎡ ⎤ ⎤ − i→ pj −1 εr ⎢ ∂θ∂ i→ ⎥ j ⎥ ⎣ ⎣ ⎦ ⎦ − δ p =⎢ + + δt ε 0 j θ ⎣ ∂δ p j ⎦ j ∂ϕi→ j ε 0 ϕ − ∂δ p j

= H i→ j δ p j + ht δt j + εr θϕ ⎡ ⎤ ] δ pj [ = H i→ j ht I ⎣ δt j ⎦ εr θϕ

(3.72)

With the notation Di→ j = [ H i→ j ht I ], the corresponding covariance of the observation noise for the ith label member in collaboration with the jth anchor member is ] [ ] [ T cov vi→ j = E vi→ j vi→ j ]} { [ T 2 2 = Di→ j E diag δ p j δ pTj δt 2j εr2 εθ/ϕ Di→ εθ/ϕ j ] [ T 2 2 Di→ = Di→ j diag σ 2p j σt2 σr2 σθ/ϕ σθ/ϕ j

(3.73)

For multiple pairs of collaborating aircraft, according to Eqs. (3.66), (3.72), and (3.73), the observation noise model can be derived as follows: [ ] cov[vi→ ] = diag{ · · · cov vi→ j · · · }

(3.74)

It can be concluded from Eq. (3.74) that the error in the relative bearing is influenced not only by the precision of the range and bearing sensors but also by the positioning accuracy of the anchor members in the swarm, which will ultimately influence the collaborative navigation fusion performance. Relative VOS Error Model for the Parallel Collaboration Structure For the parallel collaborative navigation structure, according to Eq. (3.71), the covariance is ] { [ ] [ T 2 2 2 = diag · · · E cov vswar m vswar ε ε ε m ri→ j θi→ j ϕi→ j · · · [ ] 2 2 2 · · · E εri→k ··· εθi→k εϕi→k [ ] } · · · E εr2j→k εθ2 j→k εϕ2 j→k · · ·

3.6 Vector-of-Sight-Based Collaborative Observation Modelling

[ ] 2 2 = diag · · · σr2 σθ/ϕ ··· σθ/ϕ

57

(3.75)

2 where σθ/ϕ is the measurement noise covariance of the bearing sensor.

3.6.4 Relative VOS Observation Model in Cartesian Coordinates Measured Relative VOS According to the relative-VOS-based observation geometry in Fig. 3.4, when the relative range and relative bearing can be obtained simultaneously, the VOS from the ith label member to the jth anchor member in the swarm can be decomposed in the label member’s local Cartesian coordinate reference frame as follows: ⎧ ⎪ Δx˜ = r˜i→ j cos(ϕ˜i→ j ) cos(θ˜i→ j ) ⎪ ⎨ i→ j (3.76) Δ y˜i→ j = r˜i→ j cos(ϕ˜i→ j ) sin(θ˜i→ j ) ⎪ ⎪ ⎩ Δ˜z = r˜ sin(ϕ˜ ) i→ j

i→ j

i→ j

]T [ where r˜ i→ j = Δx˜i→ j Δ y˜i→ j Δ˜z i→ j is the VOS in the label member’s local coordinate reference frame, r˜i→ j is the relative range measured by the ranging sensor, and θ˜i→ j and ϕ˜i→ j are the relative azimuth and elevation angles measured by the angular sensor. Considering that the measured relative range r˜i→ j and the measured relative bearing angles θ˜i→ j and ϕ˜i→ j contain measurement noise, substituting Eqs. (3.6) and (3.41) into Eq. (3.76) yields ⎧ = [ri→ j + (δti − δt j ) + εr ] cos(ϕi→ j + εθ ) cos(θi→ j + εϕ ) Δx˜ ⎪ ⎨ i→ j Δ y˜i→ j = [ri→ j + (δti − δt j ) + εr ] cos(ϕi→ j + εθ ) sin(θi→ j + εϕ ) ⎪ ⎩ Δ˜z i→ j = [ri→ j + (δti − δt j ) + εr ] sin(ϕi→ j + εθ )

(3.77)

Considering that the measurement errors of the relative distance and angles are generally small values, the errors associated with the measured values of the three axial components of the VOS are ⎤ Δx˜i→ j − Δxi→ j ⎥ ⎢ ei→ j = ⎣ Δ y˜i→ j − Δyi→ j ⎦ Δz i→ j − Δ˜z i→ j ⎡ ⎤ (εr + δti→ j ) cos ϕi→ j cos θi→ j − εθ ri→ j sin θi→ j cos ϕi→ j ⎢ ⎥ −εϕ ri→ j sin ϕi→ j cos θi→ j ⎢ ⎥ ⎢ ⎥ (εr + δti→ j ) cos ϕi→ j sin θi→ j + εθ ri→ j cos θi→ j cos ϕi→ j ⎥ =⎢ ⎢ ⎥ ⎢ ⎥ −εϕ ri→ j sin ϕi→ j sin θi→ j ⎣ ⎦ (εr + δti→ j ) sin ϕi→ j + εϕ ri→ j cos ϕi→ j ⎡

58

3 Modelling for Resilient Navigation via Collaboration ⎤⎡ ⎤ cos ϕi→ j cos θi→ j −ri→ j sin θi→ j cos ϕi→ j −ri→ j sin ϕi→ j cos θi→ j ε ⎥⎢ r ⎥ ⎢ ⎥ =⎢ ⎣ cos ϕi→ j sin θi→ j ri→ j cos θi→ j cos ϕi→ j −ri→ j sin ϕi→ j sin θi→ j ⎦⎣ εθ ⎦ εϕ sin ϕi→ j 0 ri→ j cos ϕi→ j ⎤ ⎡ cos ϕi→ j cos θi→ j ⎥ ⎢ ⎥ +⎢ ⎣ cos ϕi→ j sin θi→ j ⎦δti→ j sin ϕi→ j ⎡

(3.78)

≙ E i→ j εr θϕ + T i→ j δti→ j

Then, the measured VOS from the ith label member to the jth anchor can be represented as r˜ i→ j = r i→ j + ei→ j = r i→ j + E i→ j εr θϕ + T i→ j δti→ j

(3.79)

Calculated Relative VOS The VOS from the ith label member to the jth anchor can also be calculated from the positions indicated by their airborne navigation systems as follows: ⁀

⁀

⁀

r i→ j = p j − pi = ( p j + δ p j ) − ( pi + δ pi ) = ( p j − pi ) + (δ p j − δ pi ) = r i→ j + δ p j − δ pi

(3.80)

Relative VOS Observation Model for the Hierarchical Collaboration Structure The states to be estimated in the collaborative fusion process are similar to Eq. (3.63). By taking the difference between Eqs. (3.79) and (3.80), the following observation model can be obtained: ⁀

δr i→ j = r˜ i→ j − r i→ j = (r i→ j + E i→ j εr θϕ + T i→ j δti→ j ) − (r i→ j + δ p j − δ pi ) = δ pi − δ p j + E i→ j εr θϕ + T i→ j (δti − δt j ) ][ δ p ] [ i + (−δ p j − T i→ j δt j + E i→ j εr θϕ ) = I T i→ j δti ≙ H i→ j x i + vi→ j

(3.81)

Equation (3.81) represents the relationship among the measured and calculated VOSs, the errors of the relative observation sensors, and the errors of the positions provided by the airborne navigation systems of the anchor and label members. When the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.81) can be combined to yield

3.6 Vector-of-Sight-Based Collaborative Observation Modelling

⎤ ⎡ . ⎤ .. . ⎢ ⎥ ⎢ . ⎥ ⎢⁀ . ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ =⎢ ⎣ δr i→ j ⎦ = ⎣ r˜ i→ j ⎦ − ⎣ r i→ j ⎦ .. .. .. . . . ⎡ ⎡ ⎤ ⎤ .. .. ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎥ =⎢ + x ⎣ H i→ j ⎦ i ⎣ vi→ j ⎦ .. .. . . ⎡

yi→

59

.. .

⎤

⎡

≙ H i→ x i + vi→

(3.82)

Based on Eq. (3.82), when the measured and calculated relative VOSs between the anchor and label members in the swarm, which combine the relative range and bearing information, are obtained, the corresponding collaborative observation model can be formed. The observation vector of the ith label member, denoted by yi→ , can be constructed by accumulating the differences in spherical coordinates between the measured and calculated relative }VOSs to the corresponding collaborative anchor { members, denoted by . . . , δr i→ j S , . . .; additionally, the observation noise vector of the ith label member, denoted by vi→ , can be constructed by accumulating the observation noise related to the corresponding collaborative anchor members, denoted by . . . , vi→ j , . . .. Then, given the covariance of the observation noise, the position error of the ith label member, which is taken as the observation state, can be estimated. The observation matrix of the ith label member, denoted by H i→ , is constructed by accumulating the direction cosine vectors related to the corresponding collaborative anchor members, denoted by . . . , H i→ j , . . ., and takes the form shown below ⎡

H i→

.. ⎢. ⎢1 ⎢ ⎢ = ⎢0 ⎢ ⎢0 ⎣ .. .

.. . 0 1 0 .. .

⎤ .. .. . . ⎥ 0 cos ϕi→ j cos θi→ j ⎥ ⎥ ⎥ 0 cos ϕi→ j sin θi→ j ⎥ ⎥ ⎥ 1 sin ϕi→ j ⎦ .. .. . .

(3.83)

Relative VOS Observation Model for the Parallel Collaboration Structure In the parallel collaborative navigation structure, all the navigation errors of the collaborating members are collected as the states as shown in Eq. (3.67). By taking the difference between Eqs. (3.79) and (3.80), the relative VOS observation model for a pair of collaborating aircraft can be established as follows: ⁀

δr i→ j = r˜ i→ j − r i→ j

60

3 Modelling for Resilient Navigation via Collaboration

⎡ ⎤ ⎡ ⎤ δ pi εri→ j ]⎢ δt ⎥ [ i ⎥ ⎣ ⎦ = I T i→ j −I −T i→ j ⎢ ⎣ δ p j ⎦ + E i→ j εθi→ j εϕi→ j δt j ⎤ ⎡ ⎡ ⎤ δ pi εri→ j [ ]⎢ δt ⎥ i ⎥ ⎣ ⎦ ≙ hi→ j −hi→ j ⎢ ⎣ δ p j ⎦ + E i→ j εθi→ j εϕi→ j δt j

(3.84)

] [ where hi→ j = I T i→ j . Then, when the relationships between multiple pairs of collaborating aircraft are established, the corresponding observation models similar to Eq. (3.84) can be combined to obtain ⎡

yswar m

.. .

⎤

⎡

⎢ ⎥ ⎢ ⎢ δr ⎥ ⎢ ⎢ i→ j ⎥ ⎢ r˜ i→ j ⎢ . ⎥ ⎢ ⎢ .. ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ = ⎢ δr i→k ⎥ = ⎢ r˜ i→k ⎢ . ⎥ ⎢ ⎢ . ⎥ ⎢ ⎢ . ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ δr j→k ⎥ ⎢ r˜ ⎣ ⎦ ⎣ j→k .. .

.. . ⁀ − r i→ j .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⁀ − r i→k ⎥ ⎥ .. ⎥ ⎥ . ⎥ ⁀ − r j→k ⎥ ⎦ .. .

⎡

.. . ⎢ ⎢··· h · · · −h ··· i→ j i→ j ⎢ ⎢ . ⎢ .. ⎢ ⎢ = ⎢ · · · hi→k ··· −hi→k ⎢ .. ⎢ ⎢ . ⎢ ⎢ h j→k · · · −h j→k ⎣ .. .

≙ H swar m x swar m + E swar m vswar m

⎡

.. .

⎢ ⎢ε ⎢ ri→ j ⎤⎡ ⎤ ⎡ ⎤T ⎢ ⎢ εθi→ j .. .. ⎢ . . εϕi→ j ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ x ⎥ ⎢ E ⎥ ⎢ ⎢ ⎥⎢ i ⎥ ⎢ i→ j ⎥ ⎢ ... ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥⎢ .. ⎥ ⎢ . ⎥ ⎢ ε ⎥⎢ ⎥ ⎢ . ⎥ ⎢ ri→k ⎥⎢ ⎥ ⎢ ⎥ · · · ⎥⎢ x j ⎥ + ⎢ E i→k ⎥ ⎢ εθi→k ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ εϕi→k ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥⎢ x k ⎥ ⎢ E j→k ⎥ ⎢ .. ⎦⎣ ⎦ ⎣ ⎦ ⎢ .. .. ⎢ εr j→k . . ⎢ ⎢ εθ j→k ⎢ ⎢ εϕ j→k ⎣ .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.85)

3.6 Vector-of-Sight-Based Collaborative Observation Modelling

61

3.6.5 Relative VOS Error Model in Cartesian Coordinates Relative VOS Error Model for the Hierarchical Collaboration Structure To obtain the solution for the states, which represent the estimated position error and clock bias of the label member of interest in a swarm, the statistical properties of the observation noise should be known in advance. According to Eq. (3.81), the observation noise for the ith label member in collaboration with the jth anchor member in the aerial swarm is vi→ j = −δ p j − T i→ j δt j + E i→ j εr θϕ ⎡ ⎤ ] δ pj [ = −I −T i→ j E i→ j ⎣ δt j ⎦ εr θϕ

(3.86)

With the notation Di→ j = [ −I −T i→ j E i→ j ], the corresponding covariance of the observation noise for the ith label member in collaboration with the jth anchor member is ] [ ] [ T cov vi→ j = E vi→ j vi→ j ]} { [ T 2 2 = Di→ j E diag δ p j δ pTj δt 2j εr2 εθ/ϕ Di→ εθ/ϕ j ] [ T 2 2 Di→ = Di→ j diag σ 2p j σt2 σr2 σθ/ϕ σθ/ϕ j

(3.87)

For multiple pairs of collaborating aircraft, according to Eqs. (3.82), (3.86), and (3.87), the observation noise model can be derived as follows: } { ] [ cov[vi→ ] = diag · · · cov vi→ j · · ·

(3.88)

It can be concluded from Eq. (3.88) that the error in the relative bearing is influenced not only by the precision of the range and bearing sensors but also by the positioning accuracy of the anchor members in the swarm, which will ultimately influence the collaborative navigation fusion performance. It can be seen that Eqs. (3.73) and (3.87) have a similar form because the error sources do not change with the coordinates used for representation. However, the mapping matrix Di→ j is different between spherical coordinates and Cartesian coordinates, indicating the different relationships between the error sources and the observations. Relative VOS Error Model for the Parallel Collaboration Structure For the parallel collaborative navigation structure, according to Eq. (3.85), the covariance has the same form as Eq. (3.75). It should be noted that according to Eq. (3.85), the relative ranging and bearing measurement noise is mapped by the coefficient matrix E i→ j and indirectly influences the relative VOS observation, in contrast to the direct influence of the measurement noise in the VOS observation model of Eq. (3.71) in spherical coordinates.

62

3 Modelling for Resilient Navigation via Collaboration

3.7 Conclusions Relative range and bearing are two parameters that can be observed cooperatively by a pair of members in a swarm. Based on different available measurements that can be obtained by the swarm members, this chapter has established collaborative observation models based on ranging, range difference, bearing, and VOS measurements. Furthermore, for relative VOS observations, corresponding models have been derived in both spherical coordinates and Cartesian coordinates, providing a foundation for the further development of collaborative resilient fusion algorithms.

References 1. Wallace JW, Mahmood A, Jensen MA, Mehmood R (2020) Cooperative relative UAV attitude estimation using DoA and RF polarization. IEEE Trans Aerosp Electron Syst 56(4):2689–2700. https://doi.org/10.1109/taes.2019.2952701 2. Abdelkader M, Guler S, Jaleel H, Shamma JS (2021) Aerial swarms: recent applications and challenges. Curr Robot Rep 2(3):309–320. https://doi.org/10.1007/s43154-021-00063-4 3. Tahir A, Boling J, Haghbayan M-H, Toivonen HT, Plosila J (2019) Swarms of unmanned aerial vehicles—a survey. J Ind Inf Integr 16. https://doi.org/10.1016/j.jii.2019.100106 4. Arnold R, Abruzzo B, Korpela C, IEEE (2019) Towards a heterogeneous swarm for object classification. In: IEEE national aerospace and electronics conference (NAECON), Dayton, OH, 2019 Jul 15–19 2019. IEEE national aerospace and electronics conference, pp 139–147 5. Ferreira Filho EB, Pimenta LCA, IEEE (2020) Segregation of heterogeneous swarms of robots in curves. In: IEEE international conference on robotics and automation (ICRA), Electr Network, 2020 May 31–Jun 15 2020. IEEE international conference on robotics and automation ICRA, pp 7173–7179 6. Lee S, Park C, Lee S-Y, Jeon JH, Lee D, IEEE (2021) UWB based relative navigation and leader-follower formation for UAVs using maneuvering of a follower. In: 21st international conference on control, automation and systems (ICCAS), South Korea, 2021 Oct 12–15 2021. International conference on control automation and systems, pp 239–243. https://doi.org/10. 23919/iccas52745.2021.9649880 7. Ming L, Shudi W, Gengyue S, Nan W, Yanshun Z (2020) A relative navigation method based on wireless ranging for UAV in GPS denied environment. In: Proceedings of 2nd international conference on electrical, communication, and computer engineering. https://doi.org/10.1109/ icecce49384.2020.9179243 8. Mendes P, Batista P (2021) A study on cooperative navigation of AUVs based on bearing measurements. In: OCEANS 2021: San Diego—Porto. https://doi.org/10.23919/oceans44145. 2021.9705838 9. Santos D, Batista P (2020) Cooperative navigation based on bearing and range measurements to different vehicles. In: 21st IFAC world congress on automatic control—meeting societal challenges, Electr Network, 2020 Jul 11–17 2020, pp 14552–14557. https://doi.org/10.1016/j. ifacol.2020.12.1460 10. Souli N, Makrigiorgis R, Kolios P, Ellinas G, IEEE (2021) Cooperative relative positioning using signals of opportunity and inertial and visual modalities. In: IEEE 93rd vehicular technology conference (VTC-Spring), Electr Network, 2021 Apr 25–28 2021. IEEE vehicular technology conference VTC. https://doi.org/10.1109/VTC2021-Spring51267.2021.9449064 11. Causa F, Opromolla R, Fasano G (2021) Cooperative navigation and visual tracking with passive ranging for UAV flight in GNSS-challenging environments. In: 2021 international conference on unmanned aircraft systems. https://doi.org/10.1109/icuas51884.2021.9476681

References

63

12. Lee JK, Grejner-Brzezinska DA, Toth C (2012) Network-based collaborative navigation in GPS-denied environment. J Navig 65(3):445–457. https://doi.org/10.1017/s03734633120 00069 13. Rahman F, Aghapour E, Farrell JA (2021) Vehicle ECEF position accuracy and reliability in the presence of DGNSS communication latency. IEEE Intell Transp Syst Mag 13(4):262–272. https://doi.org/10.1109/mits.2019.2953500 14. Noureldin A, Karamat TB, Georgy J (2013) Basic navigational mathematics, reference frames and the earth’s geometry. In: Fundamentals of inertial navigation, satellite-based positioning and their integration. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 21–63. https://doi. org/10.1007/978-3-642-30466-8_2 15. Niranjayan S, Molisch AF (2012) Ultra-wide bandwidth timing networks. In: 2012 IEEE international conference on ultra-wideband. https://doi.org/10.1109/icuwb.2012.6340488 16. Batstone K, Oskarsson M, Astrom K, IEEE (2016) Robust time-of-arrival self calibration and indoor localization using wi-fi round-trip time measurements. In: IEEE international conference on communications (ICC), Kuala Lumpur, Malaysia, 2016 May 23–27 2016. IEEE international conference on communications workshops, pp 26–31 17. Chen H, Ballal T, Saeed N, Alouini M-S, Al-Naffouri TY (2020) A joint TDOA-PDOA localization approach using particle swarm optimization. IEEE Wireless Commun Lett 9(8):1240–1244. https://doi.org/10.1109/lwc.2020.2986756 18. Delcourt M, Le Boudec J-Y (2021) TDOA source-localization technique robust to timesynchronization attacks. IEEE Trans Inf Forensics Secur 16:4249–4264. https://doi.org/10. 1109/tifs.2020.3001741 19. Hubacek P, Vesely J, Olivova J (2022) The complete analytical solution of the TDOA localization method. Def Sci J 72(2):227–235. https://doi.org/10.14429/dsj.72.16933 20. Kim J (2022) Locating an underwater target using angle-only measurements of heterogeneous sonobuoys sensors with low accuracy. Sensors 22(10). https://doi.org/10.3390/s22103914 21. Molloy TL, Perez T, Williams BP (2020) Optimal bearing-only-information strategy for unmanned aircraft collision avoidance. J Guid Control Dyn 43(10):1822–1836. https://doi. org/10.2514/1.G004896 22. Molloy TL, Perez T, Williams BP (2020) An optimal bearing-only-information strategy for unmanned aircraft collision avoidance arXiv. arXiv:35, 35 pp 23. Shalev H, Klein I (2021) BOTNet: deep learning-based bearings-only tracking using multiple passive sensors. Sensors 21(13). https://doi.org/10.3390/s21134457 24. Anh Tuyen L, Le Chung T, Xiaojing H, Ritz C, Dutkiewicz E, Bouzerdoum A, Franklin D (2020) Hybrid TOA/AOA localization with 1D angle estimation in UAV-assisted WSN. In: 2020 14th international conference on signal processing and communication systems. https:// doi.org/10.1109/icspcs50536.2020.9310043 25. Fokin G, Vladyko A (2021) Vehicles tracking in 5G-V2X UDN using range, bearing and inertial measurements. In: 2021 13th international congress on ultra modern telecommunications and control systems and workshops. https://doi.org/10.1109/icumt54235.2021.9631627 26. Chengeng S, Gong Z, Jun L (2021) Research on the influence of pseuado-range biases on precise orbit determination and clock error calculation for Beidou navigation satellites. In: China satellite navigation conference. https://doi.org/10.1007/978-981-16-3138-2_46 27. Rogel N, Raphaeli D, Bialer O (2021) Time of arrival and angle of arrival estimation algorithm in dense multipath. IEEE Trans Signal Process 69(5907):5919. https://doi.org/10.1109/tsp. 2021.3121635

Chapter 4

Collaborative Localization-Based Resilient Navigation Fusion

Abstract The collaborative localization-based framework is one of the typical approaches used to realize resilient navigation fusion. In this approach, the relative observations from collaborating anchor members are first utilized to solve for the locations of label members who suffer from navigation degradation, and the results are then fused with the local measurements of these label members to realize navigation augmentation. This chapter discusses collaborative localization algorithms. The online estimation of the collaborative localization covariance is investigated. Resilient fusion models and processes for obtaining collaborative localization solutions are introduced with simulated examples. Keywords Collaborative · Least squares · Chan–Taylor · Spherical interpolation · Covariance estimation · Resilient fusion

4.1 Introduction Aerial swarm technology originates from the behaviour of biological swarms in nature. Swarm technology results not in the simple formation flight of multiple swarm members but rather in a highly ordered intelligent group of multiple members that collaborate and autonomously adjust their behaviours to accomplish complicated and variable tasks [1, 2]. The members of the swarm are assigned different tasks in accordance with their various types and functions; consequently, there are differences in the positioning precision of the members of an aerial swarm. For example, the area in which a search and rescue operation must be performed is often complex, and numerous uncontrollable factors exist. This may lead to a sudden decline in the positioning accuracy of some members in particular locations at specific times [3, 4]. Failing to perform effective positioning in an area without satellite navigation signals will greatly affect the search and rescue efficiency. Traditional airborne inertial navigation equipment is not subject to external environmental interference, but it does suffer from drift errors, meaning that its accuracy will quickly degrade, necessitating the use of information from other navigation systems to assist; satellite navigation technology is more accurate but is also © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_4

65

66

4 Collaborative Localization-Based Resilient Navigation Fusion

vulnerable to electromagnetic interference and less stable, meaning that it cannot be used as a reliable source of navigation information during tasks [5, 6]. Therefore, aerial swarms impose high requirements on the accuracy and reliability of navigation systems. Collaborative navigation technology can make up for the reliability and accuracy limitations of single-member navigation and thus is an effective way to improve the overall positioning accuracy of the members of a swarm. In the case that relative observations between members can be obtained through radio and other sensors, the swarm system’s overall navigation accuracy can be improved by using the relative observations between auxiliary members with lower on-board navigation accuracy and reference members with higher on-board navigation accuracy in a corresponding collaborative navigation algorithm. In recent years, many scholars worldwide have conducted research on passive airborne positioning. Regarding the relevant positioning systems, comparatively mature positioning techniques include directional cross-positioning [7, 8], time difference positioning [9, 10], and frequency difference positioning [11, 12]. Because the positioning speed and positioning accuracy are both important indicators, passive positioning is most commonly used. Among the available positioning methods, the time difference of arrival (TDOA) method is frequently applied. With the ongoing development of modern communication technology, TDOA positioning techniques based on multiple sensors have become one of the most commonly used means of radiation source positioning. This type of positioning system is suitable for broadband signals and can achieve high positioning accuracy. Reference [13] analysed and simulated the time difference accuracy and geometric factor accuracy of TDOA positioning. The results showed that the TDOA estimation performance is positively correlated with the signal bandwidth, signal accumulation time, and signal-to-noise ratio, while the TDOA localization accuracy is positively correlated with the TDOA estimation accuracy and the distance from the receiving station. A two-step TDOA localization technique was proposed in reference [14], which can effectively reduce the influence of clock errors. It uses a known source to define a weight for each pair of sensors to reflect their degree of time synchronization. Then, a weighted least squares estimator is used with the newly created weights and the TDOA observations received from an unknown source. Localization solution can be broadly classified into three major categories: Parameter search methods, iterative methods, and closed-form methods. Parameter search methods include grid-by-point search methods [15] and importance sampling methods [16]. Iterative methods include convex optimization methods [17], Taylor series iterative methods [18], constrained overall least squares estimation methods [19], and constrained weighted least squares estimation methods [20, 21]. Although the positioning accuracy of both parameter-search-type methods and iterative-type methods can approach the Cramér–Rao lower bound, the former has higher computational complexity and take more time, while the latter is sensitive to initial values and thus are prone to local convergence, which will eventually lead to large errors. It is difficult for all of the above algorithms to fully consider the impacts of clock and sensor errors on the final positioning results. In general, closed-form methods have higher computational efficiency. They are more suitable for satisfying the rapidity

4.2 Collaborative Localization Algorithms

67

requirements of collaborative navigation positioning in special environments. The distance measurement error of each member in a swarm system usually does not obey a Gaussian distribution because of the nonvisual ranging error of the mutual distance observations caused by the positions of the swarm members and other sources of interference in the intermember range observations. In this chapter, we introduce collaborative navigation algorithms for swarm members based on collaborative localization-based resilient fusion. Based on the collaborative localization model and structures established in the previous chapter and the related observation data, the position of a swarm member can be solved by using a collaborative localization algorithm. First, a least squares algorithm based on the TDOA has low complexity but weak resistance to nonvisual ranging errors, and the accuracy of this algorithm is low in the case of large errors in the TDOA observations. A Chan–Taylor-based solution algorithm is proposed for the collaborative navigation model, in which the Chan algorithm solves the collaborative navigation model to obtain the position coordinates of the member to be assisted by means of a twofold least squares algorithm and the solution result is then used as the initial value for the Taylor iteration algorithm, a corresponding iteration threshold is set, and the position of the member to be assisted is finally obtained with high accuracy in the relative coordinate system. A weighted spherical interpolation method [22] is used in this chapter, and the observation error is restricted and linearized to solve for the positioning information using the least squares approach. Compared to traditional methods, this method improves the positioning accuracy in an environment with nonGaussian observation errors to a certain extent. The spherical interpolation algorithm needs only a simple matrix operation to calculate the position of the member to be assisted; thus, its computational intensity is relatively low, as it does not require a large number of iterations or searches, and the efficiency of the algorithm is high. The covariance of the above collaborative localization algorithm is estimated online, and finally, the algorithmic solution results of the collaborative localization scheme are fused with those of the airborne navigation equipment using a filter equation to correct the target position results.

4.2 Collaborative Localization Algorithms 4.2.1 Least Squares Algorithm In accordance with the modelling for resilient navigation via collaboration established in Chap. 3, the relationship between the relative observation information and the position of the label member to be assisted can be obtained as follows: yi→ = H i→ x i + v i→

(4.1)

68

4 Collaborative Localization-Based Resilient Navigation Fusion

where the forms of the corresponding observation vector and matrix depend on the type of collaborative observations used, as shown in Eq. (3.13), Eq. (3.50), Eq. (3.65), or Eq. (3.82). The least squares (LS) algorithm, which was proposed by Karl Gauss, is a classic method of solving for the desired results in accordance with multiple observations and corresponding models. As discussed in Chap. 3, the observations from a given label member to different anchor members vary in their noise levels because they are influenced by the position uncertainties of the anchor members, which vary under different global navigation satellite system (GNSS) signal availability conditions. Therefore, some weighting among the collaborative observations is needed to balance their contributions. Accordingly, the objective function for optimization is J ( xˆ i ) = ( yi→ − H i→ xˆ i )T W i→ ( yi→ − H i→ xˆ i )

(4.2)

where xˆ i represents the estimated states of the ith label member in the aerial swarm and W i→ is the weighting matrix for the collaborative observations. When the T −1 ] . The Markov estimation weighting strategy is adopted, W i→ = cov[v i→ v i→ T collaborative observation noise covariance cov[v i→ v i→ ], which reflects the noise level of the collaborative observations, can be obtained in accordance with Eq. (3.21), Eq. (3.59), or Eq. (3.88) depending on the type of collaborative observations used and may be influenced not only by the precision of the range and bearing sensors but also by the positioning accuracy of the anchor members in the swarm. Then, the states can be estimated as follows: T T xˆ i = (H i→ W i→ H i→ )−1 H i→ W i→ yi→

(4.3)

4.2.2 Chan–Taylor Algorithm Based on the range difference observation model, this section discusses a joint Chan– Taylor (CT) iterative co-navigation model solution algorithm by analysing ranging noise that obeys a Gaussian distribution. Chan’s algorithm solves the hyperbolic TDOA equations by means of a dual-weighted LS approach, whereas the weighted Taylor iterative algorithm estimates the position coordinates of the label member of interest and corrects these position coordinates by applying the squared error principle. Thus, the dependence on the choice of the initial values is reduced. The CT collaborative localization algorithm studied here is based on the idea of combining the characteristics of the Chan and Taylor algorithms. Specifically, the dual-weighted LS approach used in the Chan algorithm is adopted to obtain the initial value for the Taylor algorithm, which is then iterated to obtain a higher localization accuracy. This algorithm achieves high accuracy in the case of ranging noise with a Gaussian distribution [23–25].

4.2 Collaborative Localization Algorithms

69

In accordance with the collaborative observation model of Eq. (3.30), appropriate estimated states that contain the estimated position of the label member to be assisted can be obtained. The LS objective function is adopted for optimization [26, 27]: (0)

xˆ i

(0)

(0)

(0) = arg min[( yi→ − H i→ xˆ i )T W i→ ( yi→ − H i→ xˆ i )] (0) (0) T T W i→ H i→ )−1 H i→ W i→ yi→ = (H i→

(4.4)

(0) Here, W i→ is the weighting matrix of the range difference observations, which is calculated as (0) W i→ = Q −1

(4.5)

where Q = diag[ · · · σi→ j, j+1 σi→ j, j+2 · · · ] is composed of the covariances of the range measurements. Then, the estimation precision is iteratively improved based on the solution xˆ i(0) of Eq. (4.4), yielding (1)

xˆ i

(1)

(1)

(1) = arg min[( yi→ − H i→ xˆ i )T W i→ ( yi→ − H i→ xˆ i )] (1) (1) T T W i→ H i→ )−1 H i→ W i→ yi→ = (H i→

(4.6)

(1) where W i→ is the weighting matrix of the range difference observations, which is calculated as (1) W i→ = (B Q B)−1

(4.7)

(0) (0) Here, B = diag[ · · · rˆi→ j+1 rˆi→ j+2 · · · ] is composed of the ranges from the label member to certain other members in the aerial swarm. Its components take the following form:

| | | | (0) (0) ˆ p rˆi→ = − p | j+1 | j+1 i

(4.8)

(0) where pˆ i(0) = xˆ i→ {1 : 3, 1} is the estimated position of the ith label member in the aerial swarm according to Eq. (4.4) and p j+1 is the position of the ( j + 1)th anchor member as indicated by its equipped airborne navigation system. The covariance corresponding to Eq. (4.6) is (1) T cov( xˆ i(1) ) = (H i→ W i→ H i→ )−1

(4.9)

Then, a second iteration is performed to improve the estimation precision based (1) of Eq. (4.6), yielding on the solution xˆ i→ (2)

xˆ i

(2)

(2)

(2) = arg min[(g i→ − G xˆ i )T W i→ (g i→ − G xˆ i )] (2) (2) G)−1 G T W i→ g i→ = (G T W i→

(4.10)

70

4 Collaborative Localization-Based Resilient Navigation Fusion

] [ (1)2 (1)2 (1)2 (1) where g i→ = Δxi→ j Δyi→ j Δz i→ j ri→ j is calculated from the estimated states (1) (1) (1) ˆ i(1) − p j , with pˆ i(1) = xˆ i(1) {1 : 3, 1}, and as follows: [ Δxi→ j Δyi→ j Δz i→ j ] = p (1) (1) ri→ j = xˆ i {4, 1}. The corresponding matrix and vectors are

⎡

1 ⎢0 G=⎢ ⎣0 1

0 1 0 1

⎤ 0 0⎥ ⎥ 1⎦ 1

(4.11)

(2) (1) W i→ = 4B ' cov( xˆ i→ )B '

(4.12)

(1)2 (1)2 (1)2 (1) B ' =diag[ Δxi→ j Δyi→ j Δz i→ j ri→ j ]

(4.13)

Ultimately, the position coordinates of the label member to be assisted in the relative coordinate system are obtained as shown below xˆ i(3) = ±( xˆ i(2) )1/2 + r i→ j

(4.14)

where r i→ j denotes the position coordinates of the main anchor member expressed in the local coordinates of the ith member of the swarm. Finally, based on the calculated distance from the main anchor member, the position coordinates with the smallest distance are selected to obtain the solution of the Chan algorithm as the estimated position of the label member in the local coordinate system. This solution provided by Chan’s algorithm is used as the initial value of Taylor’s iterative algorithm, and a threshold is set for Taylor’s algorithm such that the algorithm will iterate until the set threshold is reached [28–30], at which time the estimated position of the label member to be assisted is obtained with high accuracy. (4)

δ xˆ i

(4)

(4)

(3) = arg min[(δ yi→ − δ H i→ δ xˆ i )T W i→ (δ yi→ − δ H i→ δ xˆ i )] (4) (4) T T W i→ δ H i→ )−1 δ H i→ W i→ δ yi→ = (δ H i→

(4.15)

(4) where W i→ is the weighting matrix of the range difference observations, which is calculated as (4) W i→ = Q −1

(4.16)

Through continuous iteration as expressed in Eq. (4.15), the position coordinates of the label member to be assisted in the relative coordinate system are finally obtained by correcting the solution of Eq. (4.14) as follows: xˆ i(4) = xˆ i(3) + δ xˆ i(4) .

(4.17)

4.2 Collaborative Localization Algorithms

71

4.2.3 Spherical Interpolatio Algorithm The above joint CT algorithm is mainly designed for the situation in which the positioning accuracy is high when the range error is dominated by Gaussian noise. However, nonvisual ranging errors will also arise in swarming flight due to the geometric relationships between the swarm members. In environments with nonvisual ranging errors and other types of interference, the ranging errors of each member of the swarm system will not obey a Gaussian distribution [31, 32]. To address this situation, a collaborative localization algorithm based on an SI algorithm is proposed in this section to improve the collaborative localization solution accuracy in the case of non-Gaussian observation errors to some extent. The spherical interpolation (SI) algorithm is efficient because it requires only relatively simple matrix operations with low computational intensity and does not require a large number of iterations or searches. This algorithm replaces the traditional observation error by introducing a distance equation error in the observation equation and uses this as an optimization basis to convert the original nonlinearly constrained problem into a linearly constrained problem. Finally, the location is solved by using a dual LS algorithm, which effectively improves the accuracy of localization in a non-Gaussian noise environment. The collaborative navigation model solved by the SI algorithm is based on the relative range differences. The equations describing the relationships between label and anchor members of the swarm on the basis of the relative range differences are obtained in accordance with Eq. (3.35). The position coordinates in the relative coordinate system of the label member are solved for using the LS method as follows: rˆ j→i = arg min[v i→ j,... ] =

1 T (R R j→ )−1 R Tj→ (ηi→ j + 2ri→ j Δr i→ j,... ) (4.18) 2 j→

For verification, the above solution result satisfies the nonlinear constraint that rˆ Tj→i rˆ j→i = ri→ j

(4.19)

Substituting Eq. (4.18) into Eq. (3.35) yields (ηi→ j + 2ri→ j Δr i→ j,... )[I − R j→ (R Tj→ R j→ )−1 R Tj→ ] = v i→ j,...

(4.20)

Then, Eq. (4.20) is solved using the LS approach as follows: ri→ j = arg min[v i→ j,... ] =

T T 1 Δr i→ j,... E Eηi→ j T T 2 Δr i→ j,... E EΔr i→ j,...

(4.21)

where E = I − R j→ (R Tj→ R j→ )−1 R Tj→ . By substituting the solution given in Eq. (4.21) into Eq. (4.19), the position of the label member is obtained as follows:

72

4 Collaborative Localization-Based Resilient Navigation Fusion

rˆ j→i = arg min[v i→ j,... ] =

T T Δr i→ . 1 T j,... E Eηi→ j ( R j→ R j→ )−1 R Tj→ (ηi→ j + 2Δr i→ j,... ) T T 2 Δr i→ j,... E EΔr i→ j,...

(4.22)

4.3 Online Estimation of the Collaborative Localization Covariance 4.3.1 LS Estimation Covariance For the LS algorithm, the estimation covariance is cov( xˆ i − x i ) = E[( xˆ i − x i )( xˆ i − x i )T ] T = (H i→ W i→ H i→ )−1

(4.23)

where the observation matrix H i→ varies with the type of collaborative observations used and is described by Eq. (3.14), Eq., Eq. (3.66), or Eq.. The observation weighting matrix W i→ = cov[v i→ ]−1 and the collaborative observation noise covariance cov[v i→ ], which reflect the evaluated noise level of the collaborative observations, are obtained in accordance with Eq. (3.21), Eq., Eq. (3.59), or Eq. depending on the type of collaborative observations used and may be influenced not only by the precision of the range and bearing sensors but also by the positioning accuracy of the anchor members in the swarm.

4.3.2 CT Estimation Covariance To evaluate the accuracy of estimation, the Cramér–Rao lower bound (CRLB) is introduced. The CRLB establishes a lower bound on the variance of all unbiased parameter estimates and provides a criterion for comparing the performance of unbiased estimates [33, 34]. This bound can be used when calculating the TDOA localization error in the case of signals with smooth Gaussian noise: cov( xˆ i − x i ) = E[( xˆ i − x i )( xˆ i − x i )T ] T = c2 (δ H i→ W i→ δ H i→ )−1

(4.24)

where δ H i→ is the error observation matrix obtained from Eq. (3.36) and Eq. (3.37), c is the speed of the radio signal, W i→ = Q −1 is the weighting matrix of the range difference observations, and the covariance of the range difference observation noise is

4.4 Resilient Fusion Algorithm with the Collaborative Localization Solution

⎤⎫ . . .. .. . . ⎪ ⎪ . . . . ⎪ ⎥⎪ ⎢ ⎬ ⎥ ⎢ 1 ··· 1 1 ···⎥ I+ ⎢ Q = σr2 ⎥ ⎪ 2⎢ ⎪ ⎪2 ⎣ · · · 1 1 · · · ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ . . . . . . .. .. . . ⎧ ⎪ ⎪ ⎪ ⎪ ⎨1

73

⎡

(4.25)

where σr2 is the variance of the ranging noise.

4.3.3 SI Estimation Covariance For the SI algorithm, the estimation covariance is 2 T −1 cov( xˆ i − x i ) = ri→ j Q(U U )

(4.26)

where Q is the weighting matrix of the range difference observations from Eq. (4.25) and the matrix U is calculated as U = Δr i→ j,...

r Tj→i ri→ j

+R j→

(4.27)

where the corresponding parameters are obtained from Eq. (3.35).

4.4 Resilient Fusion Algorithm with the Collaborative Localization Solution 4.4.1 Resilient Fusion Model with the Collaborative Localization Solution State Equation of the Airborne Navigation System The error model of the INS is selected as the state equation of the airborne navigation system. For the ith label member in a swarm, the state equation is X i (k)=Φi (k,k − 1)X i (k − 1)+Γ i (k,k − 1)W i (k − 1)

(4.28)

where Φ i is the state transition matrix of the INS error, Γ i is the noise input matrix of the INS, W i is the noise vector of the INS with covariance Q i , and X i is the error vector of the INS, which can be defined as [35] ]T [ X i (k) = ϕ i δv i δ piL L A ε b εr ∇

(4.29)

74

4 Collaborative Localization-Based Resilient Navigation Fusion

where ϕ i = [ϕ E ϕ N ϕU ] are the platform error angles; δv i = [δv E δv N δvU ] are the velocity errors; δ piL L A = [δL[ δλ δh] are ] the latitude, [ longitude, and] altitude errors, respectively; ε b = εbx εby εbz and εr = εr x εr y εr z are the constant drift [errors and first-order Markov drift errors, respectively, of the ] gyroscope; and ∇ = ∇x ∇ y ∇z is the acceleration bias. Measurement Equation based on the Collaborative Localization Solution According to the collaborative localization algorithm proposed in Sect. 3, the collaborative localization solution for the ith label member in the aerial swarm can be represented as ⎤ LC LL A pCi = ⎣ λC ⎦ = piL L A + v iL L A hC ⎡ ⎤ ⎡ vN ⎤ L ⎢ REM ⎥ = ⎣ λ ⎦ + ⎣ R N vcos L ⎦ h vU ⎡

(4.30)

where piL L A = [L λ h]T is the true position of the ith label member in the aerial swarm represented in terms of the latitude, longitude, and altitude and R M and R N are the radii of curvature of the meridian circle and the prime vertical, respectively: R M = Re (1 − 2 f + 3 f sin2 L)

(4.31)

R N = Re (1 + f sin2 L)

(4.32)

where Re is the Earth’s equatorial radius and f is the Earth’s flatness ratio [36]. Meanwhile, the position of the ith label member in the swarm can be indicated by the equipped INS and represented as ⎡

p LI iL A

⎤ LI = ⎣ λ I ⎦= piL L A + δ piL L A hI ⎡ ⎤ ⎡ ⎤ δL Li = ⎣ λi ⎦ + ⎣ δλ ⎦ hi δh

(4.33)

Then, by taking the difference between Eq. (4.30) and Eq. (4.33), the following measurement model is obtained for resilient fusion:

4.4 Resilient Fusion Algorithm with the Collaborative Localization Solution

75

⎡

⎤ ⎡ ⎤ (L C − L I )R M δL I R M + v N Y i (k) = ⎣ (λC − λ I )R N cos L I ⎦ = ⎣ δλ I R N cos L I + v E ⎦ hC − h I δh I + vU ⎡ ⎤ .. ⎡ ⎤ ⎡ ⎤ . vN RM ⎢ ⎥ LL A ⎥ + ⎣ = ⎣ 03×6 δ p R N cos L I 03×9 ⎦⎢ vE ⎦ ⎣ i ⎦ . 1 vU ..

(4.34)

≙ H i (k)X i (k) + V i (k)

4.4.2 Resilient Fusion Process with the Collaborative Localization Solution On the basis of the filtering model established above, a collaborative navigation algorithm based on resilient fusion can be designed. The steps of this algorithm are as follows. STEP 1 Local Filter Time Update On the basis of the state equation of the airborne navigation system in Eq. (4.29), the filter time updates for the ith label member are obtained as follows: Xˆ i (k/k − 1)=Φi (k,k − 1) Xˆ i (k − 1/k − 1) P i (k/k − 1)=Φi (k,k − 1) P i (k − 1/k − 1)ΦiT (k,k − 1) +Γ i (k,k − 1) Q i (k − 1/k − 1)Γ iT (k,k − 1)

(4.35)

(4.36)

where Xˆ i (k/k − 1) and P i (k/k − 1) are the predicted error state and corresponding covariance, respectively, during the current epoch and Xˆ i (k − 1/k − 1) and P i (k − 1/k − 1) are the estimated error state and corresponding covariance, respectively, in the previous epoch. STEP 2 Hierarchical Categorization of Swarm Members The aircraft in the swarm are categorized in accordance with their status. The anchor members with normally functioning navigation systems form the anchor layer in the hierarchy of the swarm. The number of aircraft in the anchor layer, denoted by n, is determined. The label members and any anchor members that are experiencing faults are categorized as belonging to the label layer in the established hierarchy [37]. The members in the anchor layer share their positions with the other members of the swarm. STEP 3 Relative Observation The relative distance ri→ j and/or relative azimuth angle ϕi→ j and relative elevation angle θi→ j between the ith label member and the jth cooperative anchor member ( j = 1, 2, · · · , n) are measured. The range difference measurements are observed in accordance with Eq. (3.24). Moreover, as shown in Eq. (3.61) and Eq. (3.76), the measured vector of sight (VOS) is obtained

76

4 Collaborative Localization-Based Resilient Navigation Fusion

from the relative distance ri→ j and/or the relative azimuth angle ϕi→ j and relative elevation angle θi→ j . STEP 4 Dynamic Relative Observation Modelling The collaborative observation model is established considering the positions provided by the airborne navigation system of the ith label member and shared by the jth cooperative anchor member ( j = 1, 2, · · · , n). The observation model and corresponding error model for collaborative localization are obtained in accordance with the type of relative observations available. Thus, the observation models for the cooperative navigation filter are established. STEP 5 Collaborative Localization In accordance with the corresponding collaborative localization algorithm, the position of the label member in the aerial swarm is obtained based on the relative observations received from the collaborative anchor members in the aerial swarm, as discussed in Sect. 2. For the case in which relative ranging observations and/or relative bearing observations are available, the LS algorithm is adopted to collaboratively obtain the localization solution for the label member. For the case of relative ranging difference observations, the CT algorithm or the SI algorithm may be adopted to collaboratively obtain the localization solution for the label member. STEP 6 Collaborative Localization Covariance Estimation The covariance of the errors of collaborative localization based on the relative observations is estimated. The estimation method for the collaborative localization covariance varies with the localization algorithm adopted, as discussed in Sect. 3. Then, the covariance matrix of the current collaborative localization errors, Ri (k) = cov( xˆ i − x i ), is obtained for the cooperative navigation fusion filter of the ith label member. STEP 7 Local Filter Measurement Updates The measurement updating process of the cooperative navigation filter is performed by the ith label member as follows [38]: Xˆ i (k/k)= Xˆ i (k/k − 1) + K i (k)[Zi (k) − H i (k) Xˆ i (k/k − 1)]

(4.37)

K i (k) = P i (k/k − 1)H i (k)T [H i (k) P i (k/k − 1)H i (k)T + Ri (k)]−1

(4.38)

P i (k/k)=[I − K i (k)H i (k)] P i (k/k − 1)

(4.39)

where Xˆ i (k/k) and P i (k/k) denote the estimated error states and corresponding covariance, respectively, in the current epoch. Then, the estimated INS error states of the ith label member are fed back to the INS procedure for error compensation. Finally, the compensated positions and corresponding covariance values are shared with other members in the aerial swarm to support cooperative navigation in the next epoch.

4.4 Resilient Fusion Algorithm with the Collaborative Localization Solution

77

Fig. 4.1 Collaborative localization-based resilient fusion process

According to the description of the hierarchical collaborative navigation fusion process given above, two state update processes are performed at different frequencies: Time updating and measurement updating. The overall cooperative navigation fusion process for a label aircraft is shown in Fig. 4.1. The time update in the navigation system is designed to occur at a high frequency of fifty to one hundred hertz in order to track the highly dynamic variations in the INS error states [39] without utilizing any intervehicle transmitted data. The measurement update in the fusion process, in which the intervehicle transmitted data are utilized, needs only to correct the error of the time update at a low frequency of one to ten hertz. Here, only the position of the cooperative anchor in the epoch of the measurement update (occurring at one to ten hertz) needs to be transmitted via intervehicle communication. The other measurement updates are executed by the corresponding sensors equipped on the label aircraft itself. Thus, the communication burden incurred under the hierarchical collaborative navigation structure is significantly less than that of the parallel structure. Moreover, in the collaborative localization-based resilient fusion framework, only a sufficient number of relative observations are necessary to achieve successful localization; thus, this framework is suitable for aerial swarms with a large number of members and sufficient relative observations.

78

4 Collaborative Localization-Based Resilient Navigation Fusion

Fig. 4.2 Aerial swarm flight trajectories

4.5 Simulation Examples 4.5.1 Simulation of CT and SI Collaborative Localization-Based Resilient Fusion 4.5.1.1

Simulation Setup

Simulation of CT and SI collaborative localization-based resilient fusion was carried out with the range-difference-based observation model. For the simulation, a sevenmember swarm was considered, with a simulated flight time of 3600 s. The flight trajectories of the aerial swarm are shown in Fig. 4.2, and the relative ranges between label member 1 and the other members are shown in Figs. 4.3 and 4.13. The gyroscope drift was 0.1 deg/h, the accelerometer bias was g*10–4 , and the positioning accuracy of each member acting as an anchor was 3 m. Since the ranging errors are subsequently considered for both Gaussian and non-Gaussian measurement error environments, the ranging error simulation parameters will be given separately for the different ranging error environments.

4.5.1.2

Analysis of Navigation Performance

In these simulations, the resilient fusion algorithm with the collaborative localization solution was executed in accordance with the process designed in Sect. 4. Here, label member 1 is selected as an example to compare the position errors of its low-accuracy airborne navigation system after estimation and compensation with

4.5 Simulation Examples

79

Fig. 4.3 Relative ranges between label member 1 and other members

different methods. The root mean square error (RMSE) is also used as a criterion for evaluating the position error correction effect for this label member. Gaussian Measurement Error Environment In this simulation, the ranging error was specified to follow a Gaussian distribution, and the variance was set to 1 m. Figure 4.4 shows the curves describing the variations in the position errors of the label member under the Gaussian ranging noise environment when the CT and SI collaborative localization-based resilient fusion algorithm are used as well as in the noncollaborative (NC) case with the low-accuracy airborne navigation system only. The statistical results for the corresponding position RMSEs in the Gaussian environment are shown in Table 4.1. It can be concluded from Fig. 4.4 and Table 4.1 that introducing relative range difference observations to perform collaborative resilient fusion helps reduce the position error of a label member of a swarm with low-accuracy airborne navigation equipment. Furthermore, in the Gaussian noise measurement environment, the performance achieved when using the CT algorithm for resilient fusion is slightly Fig. 4.4 Comparison of position errors (Gaussian environment)

80 Table 4.1 Comparison of position RMSEs (Gaussian environment)

4 Collaborative Localization-Based Resilient Navigation Fusion RMSE of position (m)

NC

CT

SI

Lon

16.89

3.32

5.18

Lat

17.96

3.30

5.67

Alt

22.32

4.16

7.50

better than that achieved with the SI algorithm, with the former reaching an accuracy closer to the CRLB. Non-Gaussian Measurement Error Environment In a Gaussian ranging noise environment, the CT collaborative localization-based resilient fusion algorithm has a favourable effect in improving the position accuracy of a label member; however, in the case of a nonvisual ranging situation due to the position relationships between the swarm members, the estimation precision may degrade. In this section, we study and analyse the solution accuracy of the CT and SI collaborative localization algorithms based on relative range differences in a nonvisual ranging environment with the following non-Gaussian ranging error function [40]: v(k) = e(k) + e(k − 1)/2

(4.40)

where e(k) denotes Gaussian noise. Figure 4.5 shows the position errors of the label member in the non-Gaussian ranging error environment when the CT and SI collaborative localization-based resilient fusion algorithms are used as well as in the NC case with the low-accuracy airborne navigation system only. The statistical results for the corresponding position RMSEs in the non-Gaussian environment are shown in Table 4.2. According to Fig. 4.5 and Table 4.2, when the ranging errors obey a non-Gaussian distribution, the CT collaborative localization-based resilient fusion algorithm shows Fig. 4.5 Comparison of position errors (non-gaussian environment)

4.5 Simulation Examples Table 4.2 Comparison of Position RMSEs (Non-Gaussian environment)

81 RMSE of position (m)

NC

CT

SI

Longitude

15.64

10.23

5.13

Latitude

16.89

9.79

6.07

Height

22.82

12.24

7.26

degradation in its localization accuracy, whereas the SI collaborative localizationbased resilient fusion algorithm improves the solution accuracy in the case of nonGaussian measurement noise. Therefore, it can be concluded that in an environment with Gaussian range measurement noise, the CT collaborative localizationbased resilient fusion algorithm shows higher solution accuracy and better position accuracy. However, environments necessitating nonvisual range measurements, in which the relative distance measurement noise obeys a non-Gaussian distribution, occur frequently in actual aerial swarm flight. In this case, the SI collaborative localization-based resilient fusion algorithm is more effective in improving the navigation performance of an aerial swarm.

4.5.1.3

Analysis of Performance-Influencing Factors

In this section, the main factors affecting the solution accuracy of collaborative localization algorithms based on relative range differences are analysed through simulation, including the relative ranging measurement accuracy, the anchor positioning accuracy, and the number of anchor members. Analysis of the Influence of the Relative Ranging Error Level First, the simulationbased analysis of the influence of the relative ranging error level is presented. Relative ranging errors at different levels were considered in these simulations. In addition, considering the typical swarm flight environment, the relative ranging noise was specified to be non-Gaussian. The variation in the position error with the ranging accuracy is shown in Fig. 4.6, in which both the CT and SI collaborative localization algorithms are represented. The position errors of the CT and SI collaborative localization algorithms under different ranging accuracies are compared in Fig. 4.7. From the simulation results in Figs. 4.6, 4.7, it can be seen that the relative ranging error level has a significant influence on both the CT and SI collaborative localizationbased resilient fusion algorithms, and the position error increases with the relative ranging error level. In a measurement environment with non-Gaussian ranging noise, the SI collaborative localization-based resilient fusion algorithm is less affected by the ranging error than the CT algorithm is. Analysis of the Influence of the Anchor Positioning Accuracy Next, the simulation-based analysis of the anchor member positioning accuracy on collaborative localization-based resilient fusion using the range difference observation model is presented. By specifying different levels of anchor positioning accuracy, the position RMSEs of a swarm in a non-Gaussian measurement error environment were

82

4 Collaborative Localization-Based Resilient Navigation Fusion

Fig. 4.6 Variation in position error with ranging accuracy

Fig. 4.7 Comparison of position errors under different ranging accuracies

studied in the cases of fusion with the CT and SI algorithms. The variation in the position error with the anchor positioning accuracy is shown in Fig. 4.8, and the position errors under different anchor positioning accuracies are compared in Fig. 4.9. In these figures, both the CT and SI collaborative localization-based resilient fusion algorithms are represented. From the simulation results in Figs. 4.8, 4.9 it can be seen that the anchor member positioning accuracy has a significant influence on both the CT and SI collaborative localization-based resilient fusion algorithms with the relative range difference observation model, and the position error decreases as the anchor positioning accuracy improves. In the simulated non-Gaussian error environment, the SI collaborative localization-based resilient fusion algorithm is less affected by variations in the anchor positioning accuracy than the CT algorithm is. Analysis of the Influence of the Number of Anchors Finally, the simulationbased analysis of the influence of the number of anchor members on the swarm navigation performance is presented. By varying the number of anchor members and

4.5 Simulation Examples

83

Fig. 4.8 Variation in position error with anchor positioning accuracy (by CT and SI)

Fig. 4.9 Comparison of position errors under different anchor positioning accuracies (by CT and SI)

carrying out resilient fusion with different collaborative localization algorithms in a non-Gaussian ranging noise environment, the resulting position errors of the aerial swarm were compared. The variation in the position error with the number of anchors is shown in Fig. 4.10, and the position errors under different numbers of anchors are compared in Fig. 4.11. Both the CT and SI collaborative localization-based resilient fusion algorithms are represented in these figures. From the simulation results in Figs. 4.10, 4.11 above, it can be seen that as the number of anchor members increases, the position error of the aerial swarm rapidly decreases, which effectively improves the navigation performance for a label member with low airborne navigation accuracy. However, once the number of anchor members reaches a certain level, both the CT and SI algorithms are less effective in improving the navigation performance of the aerial swarm. These findings can serve as a reference for choosing an efficient number of anchors in practical applications of aerial swarms.

84

4 Collaborative Localization-Based Resilient Navigation Fusion

Fig. 4.10 Variation in position error with number of anchors (by CT and SI)

Fig. 4.11 Comparison of position errors under different numbers of anchors (by CT and SI)

4.5.2 Simulation of LS Collaborative Localization-Based Resilient Fusion 4.5.2.1

Simulation Setup

Simulation of LS collaborative localization-based resilient fusion was carried out with the bearing-only observation model as an example. For the simulations, a sixmember swarm was considered, with a simulated flight time of 3600 s. The flight trajectories of the members of the aerial swarm were the same as those of members A1–A5 and L1 in Fig. 4.2, and the relative bearings between label member 1 and the other members are shown in Figs. 4.12 and 4.13. The gyroscope drift was 0.1 deg/h, the accelerometer bias was g*10–4 , and the positioning accuracy of each member acting as an anchor was 3 m.

4.5 Simulation Examples

85

Fig. 4.12 Relative Elevations Between Label Member 1 and Other Members

Fig. 4.13 Relative Azimuth Angles Between Label Member 1 and Other Members

4.5.2.2

Analysis of Navigation Performance

In these simulations, the resilient fusion algorithm with the collaborative localization solution was executed in accordance with the process designed in Sect. 4. Here, label member 1 is selected as an example to compare the position errors of its low-accuracy airborne navigation system after estimation and compensation with different methods; meanwhile, the RMSE is used as the criterion for evaluating the position error correction effect for the label member. Figure 4.14 shows the curves describing the variations in the position errors of the label member when the LS collaborative localization-based resilient fusion algorithm is used and in the NC case with the low-accuracy airborne navigation system only. The statistical results for the corresponding position RMSEs in the Gaussian error environment are shown in Tables 4.2, 4.3

86

4 Collaborative Localization-Based Resilient Navigation Fusion

Fig. 4.14 Comparison of position errors

Table 4.3 Comparison of position RMSEs (Gaussian environment)

RMSE of position (m)

NC

LS

Lon

40.25

12.23

Lat

43.13

13.90

Alt

48.48

15.42

It can be concluded from Fig. 4.14 and Table 4.2 that introducing relative bearing observations to perform collaborative resilient fusion helps reduce the position error of a label member in the swarm with low-accuracy airborne navigation equipment.

4.5.2.3

Analysis of Performance-Influencing Factors

In this section, the main factors affecting the solution accuracy of the collaborative localization algorithm based on relative bearing observations are analysed through simulation, including the relative bearing measurement accuracy, the anchor positioning accuracy, and the number of anchor members. Analysis of the Influence of the Relative Bearing Error Level First, the simulationbased analysis of the influence of the relative bearing error level is presented. Relative bearing errors at different levels were considered in the simulations. The variation in the position error with the bearing measurement accuracy is shown in Fig. 4.15, in which the LS collaborative localization algorithm is represented. The position errors of the LS collaborative localization algorithm under different bearing measurement accuracies are compared in Fig. 4.16. As seen from the above simulation results, the relative angle measurement error has a large impact on the solution effect of the algorithm. As the relative angle measurement error increases, the position solution error of the cooperative navigation algorithm based on relative angle information gradually increases, and the rate of increase also gradually increases. Therefore, in actual engineering applications, the

4.5 Simulation Examples

87

Fig. 4.15 Variation in position error with bearing accuracy

Fig. 4.16 Comparison of position errors under different bearing accuracies

measurement accuracy of the relative angle sensor should be carefully preserved to maintain a reasonable level. Analysis of the Influence of the Anchor Positioning Accuracy Next, the simulation-based analysis of the anchor member positioning accuracy on collaborative localization-based resilient fusion with the bearing-only observation model is presented. By specifying different levels of anchor positioning accuracy, the position errors of the swarm with the bearing-only observation model and the corresponding LS algorithms were studied. The variation in the position RMSE with the anchor positioning accuracy is shown in Fig. 4.17, and the position errors under different anchor positioning accuracies are compared in Fig. 4.18. Both of these figures present the results of LS collaborative localization-based resilient fusion. From the simulation results in Figs. 4.17, 4.18, it can be seen that the anchor member positioning accuracy has a significant influence on the LS collaborative

88

4 Collaborative Localization-Based Resilient Navigation Fusion

Fig. 4.17 Variation in position error with anchor positioning accuracy (by LS)

Fig. 4.18 Comparison of position errors under different anchor positioning accuracies (by LS)

localization-based resilient fusion algorithm with the relative-bearing-only observation model, and the position error decreases as the anchor positioning accuracy improves. Analysis of the Influence of the Number of Anchor Numbers Finally, the simulation-based analysis of the influence of the number of anchor members on the swarm navigation performance is presented. By varying the number of anchor members and carrying out resilient fusion with the collaborative localization algorithm based on the relative-bearing-only model, the position errors of the aerial swarm were compared. The variation in the position error with the number of anchors is shown in Fig. 4.19, and the position errors under different numbers of anchors are compared in Fig. 4.20, in which the results of LS collaborative localization-based resilient fusion are presented. As seen from the simulation results in Figs. 4.19, 4.20 above, as the number of anchor members gradually increases, the collaborative localization-based resilient

4.6 Conclusions

89

Fig. 4.19 Variation in position error with number of anchors (by LS)

Fig. 4.20 Comparison of position errors under different numbers of anchors (by LS)

fusion algorithm based on relative bearing observations can make use of more reference information, and the navigation performance gradually improves; however, as the number of reference aircraft increases, the reduction in the magnitude of the position RMSE decreases, and the performance enhancement eventually becomes negligible. Therefore, when using this algorithm, the number of anchor members should be properly optimized.

4.6 Conclusions This chapter has mainly studied collaborative localization-based resilient navigation fusion, for which localization by means of the LS, CT, and SI algorithms has been adopted. The hierarchical cooperative navigation structure was adopted, and

90

4 Collaborative Localization-Based Resilient Navigation Fusion

a collaborative resilient fusion model and a corresponding process were designed. Then, collaborative localization algorithms based on the CT algorithm, the SI algorithm, the LS method, and on-board error correction were studied. Simulations of aerial swarms with relative range difference observations and relative-bearing-only observations were presented as examples. The simulation results show that collaborative localization-based resilient navigation fusion can effectively improve the navigation accuracy of lower-precision members of a swarm system. For the relative range difference observation model, the CT algorithm offers higher solution accuracy when the ranging error follows a Gaussian distribution, and the SI algorithm offers higher solution accuracy when the ranging error follows a non-Gaussian distribution. Moreover, the influencing factors of the relative ranging or bearing measurement accuracy, the positioning accuracy of the anchor members, and the number of anchor members were analysed through simulation, and the findings can serve as a reference for the selection of appropriate parameters in practical applications of aerial swarms.

References 1. Arnold C, Brown J (2020) Performance evaluation for tracking a malicious UAV using an autonomous UAV swarm. In: 11th IEEE Annual ubiquitous computing, electronics and mobile communication conference (UEMCON), Electr Network Oct 28–31. pp 707–712 2. Duan H, Yang Q, Deng Y, Li P, Qiu H, Zhang T, Zhang D, Huo M, Shen Y (2019) Unmanned aerial systems coordinate target allocation based on wolf behaviors. Sci China-Inf Sci 62(1). https://doi.org/10.1007/s11432-018-9587-0 3. de Alcantara Andrade FA, Hovenburg AR, de Lima LN, Rodin CD, Johansen TA, Storvold R, Moraes Correia CA, Haddad DB (2019) Autonomous unmanned aerial vehicles in search and rescue missions using real-time cooperative model predictive control. Sensors 19(19). https:// doi.org/10.3390/s19194067 4. Kumar SA, Vanualailai J, Ieee (2017) A lagrangian UAV swarm formation suitable for monitoring exclusive economic zone and for search and rescue. In: 1st Annual IEEE Conference on control technology and applications, Hi Aug 27–30. pp 1874–1879 5. Alqurashi M, Jinling W (2015) Performance analysis of fault detection and identification for multiple faults in GNSS and GNSS/INS integration. J Appl Geod 9(1):35–48. https://doi.org/ 10.1515/jag-2014-0019 6. Rife J (2009) Influence of GNSS integrity monitoring on undetected fault probabilities for single and multiple fault events. Navig J Inst Navig 56(4):275–287 7. Zekavat SA, Kolbus A, Xiaofeng Y, Zhonghai W, Pourrostam J, Pourkhaatoun M (2008) A novel implementation of DOA estimation for node localization on software defined radios: achieving high performance with low complexity. In: 2007 IEEE international conference on signal processing and communications. https://doi.org/10.1109/icspc.2007.4728486 8. Zekavat SA, Kolbus A, Yang X, Wang Z, Pourrostam J, Pourkhaatoun M, Ieee (2007) A novel implementation of DOA estimation for node localization on software defined radios: achieving high performance with low complexity. In: IEEE International conference on signal processing and communications, Dubai, U Arab Emirates Nov 24–27. pp 983–986 9. Fu X, Tao W, Shixin W (2019) A novel method of improved three-dimensional fixed target multi-station passive time difference positioning. In: Journal of physics: Conference series 1325:012020 pp 012026-012020. https://doi.org/10.1088/1742-6596/1325/1/012020 10. Qiu Fei Y, Ding Sun F (2020) Research on Multi-platform time difference location technology. In: 2020 IEEE 5th International conference on signal and image processing. https://doi.org/10. 1109/icsip49896.2020.9339328

References

91

11. Guo X, Lu H, Zhang Y, Zeng B, Guo Y (2017) Study on joint passive localization using time difference of arrival and frequency difference of arrival to improve the accuracy of four-satellite localization. Adv Mech Eng 9(12). https://doi.org/10.1177/1687814017737451 12. Jingning D, Yang TC (2018) The effect of bottom attenuation on high frequency source localization using the frequency difference MFP. 2018 OCEANS—MTS/IEEE Kobe Techno-Oceans. https://doi.org/10.1109/oceanskobe.2018.8559336 13. Yang X, Ye Y-X (2021) Geometric dilution of precision for far-distance TDOA location of shortwave. In: 2nd Information communication technologies conference, ICTC 2021, May 7, 2021 - May 9, 2021, Nanjing, China, 2021. Institute of electrical and electronics engineers Inc., pp 60–64. https://doi.org/10.1109/ICTC51749.2021.9441633 14. Delcourt M, Le Boudec J-Y (2021) TDOA Source-localization technique robust to timesynchronization attacks. IEEE Trans Inf Forensics Secur 16:4249–4264. https://doi.org/10. 1109/tifs.2020.3001741 15. Ma F, Xu Y, Xu P (2019) 2D-TDOA passive location based on geodetic longitude and latitude. J Commun 40(5):2019118–2019111–2019118–2019118 16. Wang G, Chen H (2011) An importance sampling method for TDOA-Based source localization. IEEE Trans Wireless Commun 10(5):1560–1568. https://doi.org/10.1109/twc.2011.030311. 101011 17. Xu E, Ding Z, Dasgupta S (2011) Reduced complexity semidefinite relaxation algorithms for source localization based on time difference of arrival. IEEE Trans Mob Comput 10(9):1276– 1282. https://doi.org/10.1109/tmc.2010.263 18. Kocur D, Svecova M, Kazimir P (2019) Determining the position of the moving persons in 3D space by UWB sensors using taylor series based localization method. Acta Polytechnica Hungarica 16(10):45–63 19. Yang K, An J, Bu X, Sun G (2010) Constrained total least-squares location algorithm using time-difference-of-arrival measurements. IEEE Trans Veh Technol 59(3):1558–1562. https:// doi.org/10.1109/tvt.2009.2037509 20. Jiang R, Chen H, Chen Z (2018) An evaluation of preventive maintenance quality of buses using weighted least squares method. Ind Eng J 21(3):82–86. https://doi.org/10.3969/j.issn. 1007-7375.2018.03.010 21. Lee J (2009) A reformulation of weighted least squares estimators. Am Stat 63(1):49–55. https://doi.org/10.1198/tast.2009.0011 22. Noroozi A, Sebt MA (2016) Spherical interpolation method of emitter localisation using weighted least squares. IET Signal Proc 10(8):841–854. https://doi.org/10.1049/iet-spr.2015. 0237 23. Peng Q, Yue Y, Chang Z, Zeng Y, Liu T, Cai S, Ieee (2018) Comparison of wireless location algorithms in mobile communication networks. In: 5th International conference on systems and informatics (ICSAI), Nanjing, Peoples R China Nov 10–12 2018. pp 1259–1264 24. Chang X, Ye S, Jiang Y, Guan T, Wang J (2017) Three-Dimensional positioning of wireless communication base station. In: 2nd IEEE Advanced information technology, electronic and automation control conference (IAEAC), Chongqing, Peoples R China Mar 25–26 2017. pp 2727–2732 25. Fang S-H, Lin T-N (2008) Indoor location system based on discriminant-adaptive neural network in IEEE 802.11 Environments. IEEE Trans Neural Netw 19(11):1973–1978. https:// doi.org/10.1109/tnn.2008.2005494 26. Aiguo L, Fuzeng L (2018) An improved localization algorithm based on CHAN with high positioning accuracy in NLOS-WGN environment. In: 2018 10th International conference on intelligent Human-machine systems and cybernetics. https://doi.org/10.1109/ihmsc.2018. 00083 27. Zhang J-w, Yu C-l, Tang B, Ji Y-y (2008) Chan location algorithm application in 3-Dimension space location. In: International colloquium on computing, communication, control and management, Guangzhou, Peoples R China Aug 04–05 2008. pp 622–625. https://doi.org/ 10.1109/cccm.2008.76

92

4 Collaborative Localization-Based Resilient Navigation Fusion

28. Hua C, Zhao K, Dong D, Zheng Z, Yu C, Zhang Y, Zhao T (2020) Multipath map method for TDOA based indoor reverse positioning system with improved chan-taylor algorithm. Sensors 20(11). https://doi.org/10.3390/s20113223 29. Zhang H, Yu M, Cui X-r, Gulliver TA, Ieee (2011) An improved taylor series based location algorithm for IEEE 802.15.4a Channels. In: IEEE pacific rim conference on communications, computers and signal processing (PACRIM), Univ Victoria, Victoria, Canada Aug 23–26 2011. pp 499–503 30. Canclini A, Antonacci F, Sarti A, Tubaro S (2013) Acoustic source localization with distributed asynchronous microphone networks. IEEE Trans Audio Speech Lang Process 21(2):439–443. https://doi.org/10.1109/tasl.2012.2215601 31. Huang J, Wan Q (2012) Analysis of TDOA and TDOA/SS based geolocation techniques in a non-line-of-sight environment. J Commun Netw 14(5):533–539. https://doi.org/10.1109/jcn. 2012.00011 32. Weiguo G, Baochun L, Baoguo L, Pijie J (2013) A TDOA localization algorithm based on elman neural network for cellular networks. Inf Technol J 12(23):7143–7147. https://doi.org/ 10.3923/itj.2013.7143.7147 33. Dexiu H, Shiwen C, Hang B, Chuang Z (2019) Liping L (2019) CRLB for joint estimation of TDOA, phase, FDOA, and Doppler rate. J Eng 21:7628–7631. https://doi.org/10.1049/joe. 2019.0581 34. Vaghefi RM, Buehrer RM, Ieee (2016) On the CRLB of TDOA/FDOA estimation from MIMO signals. In: IEEE/ION position, location and navigation symposium (PLANS), Savannah, GA. Apr 11–14 2016 pp 772–778 35. Noureldin A, Karamat TB, Georgy J (2013) Modeling INS errors by linear state equations. In: Fundamentals of inertial navigation, Satellite-based positioning and their integration. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 201–223. https://doi.org/10.1007/978-3-642-304 66-8_6 36. Noureldin A, Karamat TB, Georgy J (2013) Basic navigational mathematics, reference frames and the earth’s geometry. In: Fundamentals of inertial navigation, Satellite-based positioning and their integration. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 21–63. https://doi. org/10.1007/978-3-642-30466-8_2 37. Xiong J, Cheong JW, Xiong Z, Dempster AG, Tian S, Wang R (2021) Integrity for multi-sensor cooperative positioning. IEEE Trans Intell Transp Syst 22(2):792–807. https://doi.org/10.1109/ tits.2019.2956936 38. Kim MJ, Bae SB, Joo MGA (2015) Simplified hybrid navigation system design of a mobile robot using kalman filter. Berlin, Heidelberg. Comput Sci Its Appl. Springer Berlin Heidelberg, pp 15–20 39. Kuznetsov AG, Portnov BI, Izmailov EA (2014) Two classes of aircraft strapdown inertial navigation systems on laser gyros: Development and test results. Gyroscopy Navig 5(4):187– 194. https://doi.org/10.1134/s2075108714040087 40. Varlamos PK, Heretakis II, Papakanellos PJ, Trakadas PT, Capsalis CN (2005) Measurements and simulation for a joint non-Gaussian fast-fading model in indoor-propagation environments. Microw Opt Technol Lett 45(6):515–519. https://doi.org/10.1002/mop.20866

Chapter 5

Collaborative Observation-Based Resilient Navigation Fusion

Abstract The collaborative observation-based framework is another typical approach for realizing resilient navigation fusion. In this approach, the relative observations from collaborating anchor members are directly fused with the local measurements without performing localization first. On the basis of collaborative observation modelling, this chapter discusses collaborative observation-based navigation algorithms with hierarchical and parallel collaborative navigation structures. Simulated examples for both situations are also provided. Keyword Collaborative Navigation · Relative Observation · Hierarchical Structure · Parallel Structure · Resilient Fusion

5.1 Introduction In aerial swarm tasks, navigation performance is critical. Satellite navigation and positioning technology have been applied in various fields, including aircraft navigation. Satellite positioning is based on pseudorange observations between a global navigation satellite system (GNSS) satellite and the antenna of the user’s receiver and relies on the known instantaneous positions of the satellite to determine the position of the user. In the open air, accurate positioning can be achieved by receiving sufficient satellite signals. However, in urban high-rise environments and canyons or in situations with satellite signal rejection and other challenges, satellite signals may be obscured, resulting in a lack of sufficient visible satellites for localization [1, 2]. In such cases, an aircraft must completely depend on its inertial navigation system (INS) to provide the required navigation accuracy. However, very high-accuracy INSs are not suitable for the members of aerial swarms due to their size and cost [3, 4]. Most of the microaircraft in aerial swarms employ inexpensive microelectromechanical system (MEMS) inertial measurement units (IMUs), and maintaining navigation accuracy in such circumstances, without the availability of georeferenced features, is a great technical challenge [5–9]. To solve this problem without collaborative localization in advance, this chapter adopts the approach of utilizing collaborative observations directly to assist an airborne navigation system in carrying out error estimation and compensation. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_5

93

94

5 Collaborative Observation-Based Resilient Navigation Fusion

The concept of synergetics was proposed by Professor Hermann Haken at the University of Stuttgart in Germany. The synergetic theory posits that the evolution of various natural and social systems from disorder to order is the result of the interaction and coordination of various elements of a system [10]. Compared with single-point positioning, cooperative positioning enhances the information transmission effect and the accuracy of point positioning and improves the positioning performance of the whole system [11, 12]. Researchers at the Swedish Defence Research Institute proposed the idea of introducing cooperation into the field of satellite positioning[13]. Researchers at West Virginia University presented a cooperative navigation algorithm to estimate the global poses of a group of unmanned aerial vehicles (UAVs) using intervehicle ranging and measurements of anomalies in the Earth’s magnetic field [14]. Researchers at Brigham Young University presented a way to improve navigation accuracy for multiple small UAVs by collaboratively sharing information in a Global Positioning System (GPS)-denied environment [15]. Vetrella et al. proposed a cooperative navigation method that integrates inertial measurements, magnetometer data, available satellite pseudorange information, cooperative UAV position data, and monocular camera information to effectively improve the navigation performance of a UAV swarm under GPS constraints [16]. Causa et al. studied a method of calculating the cooperative formation accuracy based on visual measurements, and experimental results showed that with the appropriate cooperative formation, the positioning accuracy of a UAV swarm could reach the metre level through visual measurement assistance [18]. Chen et al. proposed a belief-propagation-based UAV positioning algorithm by jointly using measurements from GNSS satellites, peer UAVs, ground control stations, and signals of opportunity [18]. Considering that the previously discussed collaborative localization-based navigation framework usually has a requirement on the number of anchor members, which cannot always be satisfied, this chapter studies collaborative observationbased resilient navigation fusion algorithms for aerial swarms with both hierarchical and parallel collaboration structures. For collaborative observation-based navigation with the hierarchical collaborative navigation structure, the relative observations between an anchor member and a label member are directly fused with the airborne INS measurements, thereby reducing the requirement on the number of anchors and allowing each collaborative observation to be utilized in a timely manner to augment the navigation performance of the label member. For collaborative observation-based navigation with the parallel collaborative navigation structure, a hybrid recursive network is designed for an environment in which the number of available GNSS satellites is insufficient. Through relative range measurements, communication, and information sharing among the members of the aerial swarm, constraint conditions among the observations are introduced to form a collaborative union, and the members of the swarm collaborate with other members in turn until the cooperation requirements are met. With this method, a large-scale aerial swarm can realize localization through collaboration even when each member of the aerial swarm observes only an insufficient number of GNSS satellites and fails to achieve independent localization, thus improving the navigation robustness of the aerial swarm.

5.2 Collaborative Observation-Based Navigation Algorithm …

95

5.2 Collaborative Observation-Based Navigation Algorithm with the Hierarchical Collaborative Navigation Structure 5.2.1 Resilient Fusion Model with Collaborative Observations State Equation of the Airborne Navigation System In the hierarchical collaborative navigation structure introduced in Sect. 2.4, the members of the aerial swarm are classified into label members to be assisted and anchor members to serve as references. The state equation is established from the perspective of the errors of the airborne navigation system equipped on a label member. The error model of the INS is selected as the state equation of the airborne navigation system. For the ith label member in a swarm, the state equation is [19]. X i (k) = Φi (k, k − 1)X i (k − 1) + Γ i (k, k − 1)W i (k − 1)

(5.1)

where Φ i is the state transition matrix of the INS errors, Γ i is the noise input matrix of the INS, Wi is the noise vector of the INS with covariance Q i , and Xi is the error vector of the INS, which can be defined as ]T [ X i (k) = ϕ i δv i δ piL L A δti ε b εr ∇

(5.2)

where ϕ i = [ϕ E ϕ N ϕU ] are the platform error angles; δv i = [δv E δv N δvU ] are the velocity errors; δ piL L A = [δL δλ δh] are[ the latitude,] longitude,[ and altitude] errors, respectively; δti is the clock error; ε b = εbx εby εbz and εr = εr x εr y εr z are the constant drift [errors and ]first-order Markov drift errors, respectively, of the gyroscope; and ∇ = ∇x ∇ y ∇z is the acceleration bias. Measurement Equation based on Relative Observation Here, the vector of sight (VOS) is selected as the relative observation as an example for discussion. The position in the global coordinate frame (e.g., the earth-centred–earth-fixed (ECEF) frame) can be transformed in accordance with Eq. (3.1) [20]: ) ( ⎧ ⁀ ⁀ ⁀ ⁀ ⎪ ⎪ x i = R N + h i cos L i cos λi ⎪ ⎪ ⎪ ⎪ ⎪ ) ( ⎨ ⁀ ⁀ ⁀ ⁀ y i = R N + h i cos L i sin λi ⎪ ⎪ ⎪ ] [ ⎪ ⎪ ⁀ ⁀ ⎪ ⁀ ⎪ ⎩ z i = R N (1 − f )2 + h i sin L i

(5.3)

96

5 Collaborative Observation-Based Resilient Navigation Fusion ⁀

⁀

⁀

where (λi , L i , h i ) represent the longitude, latitude, and altitude, respectively, provided by the airborne navigation systems of the label member and R N and f are the radius of the prime vertical and the ellipticity of the Earth, respectively. The measured longitude, latitude, and altitude of the ith member obtained by its airborne navigation system can be represented as follows: ⎧⁀ ⎪ λi =λi +δλi ⎪ ⎪ ⎨ ⁀

L i =L i + δL i ⎪ ⎪ ⎪ ⎩⁀ h i = h i + δh i

(5.4)

where (λi , L i , h i ) are the true values of the longitude, latitude, and altitude, respectively, of the ith label member, respectively, and (δλi , δL i , δh i ) are the errors of the airborne navigation system. Moreover, by taking the measurement errors associated with the longitude, latitude, and altitude provided by the navigation systems of the leader and follower into consideration and substituting Eq. (5.4) into Eq. (5.3), the errors associated with the positions provided by the airborne navigation system of the ith label member can be expressed as ⎧ ⁀ ⎪ δxi = x i − xi ⎪ ⎪ ⎪ ⎪ ⎪ = − δλi (R N + h i ) sin λi cos L i − δL i (R N + h i ) sin L i cos λi + δh i cos L i cos λi ⎪ ⎪ ⎪ ⎪ ⁀ ⎨ δyi = y i − yi ⎪ = δλi (R N + h i )cosλi cos L i − δL i (R N + h i ) sin L i sin λi + δh i cos L i sinλi ⎪ ⎪ ⎪ ⎪ ⁀ ⎪ ⎪ δz i = z i − z i ⎪ ⎪ ⎪ ⎩ = δL i [R N (1 − f )2 + h i ] cos L i + δh i sin L i (5.5) Then, Eq. (5.5) can be transformed into matrix form as follows: ⎤ δxi ⎣ δyi ⎦ δz i ⎡ ⎤⎡ ⎤ −(R N + h i ) sin λi cos L i −(R N + h i ) sin L i cos λi cos L i cos λi δλi = ⎣ (R N + h i )cosλi cos L i −(R N + h i ) sin L i sin λi cos L i sinλi ⎦⎣ δL i ⎦ 0 [R N (1 − f )2 + h i ] cos L i sin L i δh i ⎡

≙ δ pi = M i δ piL L A For the jth anchor member, a similar relationship holds:

(5.6)

5.2 Collaborative Observation-Based Navigation Algorithm …

δ p j =M j δ p Lj L A

97

(5.7)

In the relative VOS observation model of Eq. (3.81), there are three kinds of errors resulting from the difference between the measured and calculated VOSs: the relative observation sensor errors and the errors associated with the airborne navigation systems of the anchor and label members. Among these errors, the errors of the airborne navigation system of the label member, which is equipped with lowprecision sensors, are generally the largest. The errors of the airborne navigation system of the anchor member, which is equipped with high-precision sensors, are relatively small in magnitude. The errors of the relative distance and angles measured by the radar ranger or laser sensor are also relatively small [22, 22–24]. Therefore, the observation model can be established in accordance with Eq. (3.81), Eq. (5.2), and Eq. (5.6) as Y i→ j (k) = r˜i→ j − i → j = H i→ j x i + v i→ j = H i→ j (k)M i (k)X i (k) + v i→ j (k)

(5.8)

For the situation in which multiple anchor members cooperate with the ith label member, the following observation equation can be established: ⎡

⎤ .. . ⎢ ⎥ ⎢ r˜i→ j − ri→ j ⎥ ⎥ Y i→ (k) = ⎢ ⎢ r˜ ⎥ ⎣ i→ j+1 − ri→ j+1 ⎦ .. . ⎡ ⎤ .. . ⎢ ⎥ ⎢ H i→ j (k) ⎥ ⎥ =⎢ ⎢ H i→ j+1 (k) ⎥ M i (k)X i (k) + V i→ (k) ⎣ ⎦ .. .

(5.9)

Then, in accordance with the relative VOS error model established in Eq. (3.87), the covariance of the observation noise vector v i→ j (k) can be obtained as follows: ] [ T cov[V i→ j ] = E v i→ j v i→ j { [ ]} T = Di→ j E diag δ p j δ pTj δt 2j εr θϕ εr2θϕ Di→ j ] T [ 2 2 2 = Di→ j diag σ pj σt σr θϕ Di→ j ] T [ 2 = Di→ j diag Mi (k)σ pj MiT (k) σ 2t σ r2θϕ Di→ j Then, the covariance of the observation noise is

(5.10)

98

5 Collaborative Observation-Based Resilient Navigation Fusion

Ri→ (k) = cov[v i→ ] { [ ] } = diag . . . cov v i→ j ...

(5.11)

5.2.2 Resilient Fusion Process with Collaborative Observations On the basis of the filtering model established above, a hierarchical collaborative navigation algorithm can be designed. Taking the resilient fusion navigation based on relative VOS observation as an example, the steps are as follows. STEP 1 Filter Time Update On the basis of the state equation of the airborne navigation system in Eq. (5.1), the filter time updates for the ith label member are obtained; the equations are similar to Eq. (4.35) and Eq. (4.36). STEP 2 Hierarchical Categorization of the Swarm The aircraft in the swarm are categorized according to their status. The anchor members with normally operating navigation systems form the anchor layer in the hierarchy of the swarm [25]. The number of aircraft in the anchor layer, denoted by n, is determined accordingly. The label members and the anchor members that are experiencing faults are categorized as belonging to the label layer in the established hierarchy. The aircraft in the anchor layer share their positions with the other members of the swarm. STEP 3 Relative VOS Observation The relative distances r˜i→ j , relative azimuth angles θ˜i→ j , and relative elevation angles ϕ˜i→ j between the ith label member and the jth cooperative anchor member ( j = 1, 2, · · · , n) are measured. Then, the n relative VOSs thus obtained are decomposed in accordance with Eq. (3.76) to obtain the measured VOS values r˜ i→ j ( j = 1, 2, · · · , n). STEP 4 Relative VOS Calculation The relative VOSs are also calculated considering the position information provided by the airborne navigation system of the ith ⁀

⁀

⁀

label member, denoted by (λi , L i , h i ), and the positions shared by the jth coopera⁀

⁀

⁀

tive anchor member ( j = 1, 2, · · · , n), denoted by (λ j , L j , h j ), in accordance with ⁀ Eq. (3.3) and Eq. (3.80) to obtain the calculated VOS values r i→ j ( j = 1, 2, · · · , n). STEP 5 Observation Error Covariance Estimation The covariance of the errors in the relative VOS observations is calculated from the noise covariances of the previously obtained information collected by the ranging sensor (σr ) and the angular sensor (σθ/ϕ ) as well as the covariance of the positioning error (σ pj ) estimated and shared by the jth anchor member ( j = 1, 2, · · · , n), as expressed in Eqs. (5.10) and (5.11). Then, the covariance matrix of the current observation errors Ri→ (k) is obtained for the cooperative navigation fusion filter of the ith label member. STEP 6 Dynamic Observation Modelling The error projection coefficient matrix of the position error M i of the ith label member is calculated according to Eq. (5.6), and the observation matrix H i (k) for the cooperative navigation filter is then obtained

5.2 Collaborative Observation-Based Navigation Algorithm …

99

according to Eq. (3.83). Moreover, the error projection coefficient matrix of the position error M j of the jth anchor member ( j = 1, 2, · · · , n) is calculated according to Eq. (5.7), and the error projection coefficient matrices of the ranging and angular sensors (E i→ j and T i→ j ) are calculated according to Eq. (3.78). Then, the observation error propagation matrix D j (k) for the cooperative navigation filter is obtained according to Eq. (3.87). Thus, the observation models for the cooperative navigation filter are established. STEP 7 Filter Measurement Updates The measurement updating of the cooperative navigation filter is performed by the ith label member. The equations are similar to Eq. (4.37) and Eq. (4.39). Then, the estimated INS error states of the ith label member are fed back to the INS procedure for error compensation. The compensated positions and corresponding covariance values are shared with the other members of the aerial swarm to support cooperative navigation in the next epoch. According to the description of the hierarchical collaborative navigation fusion process given above, two state update processes are performed at different frequencies: time updating and measurement updating. The overall cooperative navigation fusion process for a label aircraft is shown in Fig. 5.1. The time update in the navigation system is designed to occur at a high frequency of fifty to one hundred hertz in order to track the highly dynamic variations in the INS error states without utilizing any intervehicle transmitted data [26, 27]. The measurement update in the collaborative fusion process, in which the intervehicle transmitted data are utilized, needs only to correct the error of the time update at a low frequency of one to ten hertz. Here, only the position of the cooperative anchor in the epoch of the measurement update (occurring at one to ten hertz) needs to be transmitted via intervehicle communication. The other relative observations are executed by the corresponding sensors equipped on the label aircraft itself. Thus, the communication burden incurred under the hierarchical collaborative navigation structure is significantly reduced. Moreover, in the collaborative observation-based resilient fusion framework, there is no requirement on the number of relative observations needed to perform measurement updates; thus, this framework is applicable for aerial swarms with a small number of members and insufficient relative observations.

Fig. 5.1 Collaborative observation-based resilient fusion process (hierarchical structure)

100

5 Collaborative Observation-Based Resilient Navigation Fusion

5.3 Collaborative Observation-Based Navigation Algorithm with the Parallel Collaborative Navigation Structure 5.3.1 Resilient Fusion Model with Collaborative Observations State Equation of the Aerial Swarm for Collaborative Navigation In the parallel collaborative navigation structure introduced in Sect. 2.3, the members of the aerial swarm do not need to be classified. Thus, the state equation is designed from the perspective of the errors of the airborne navigation systems in the whole swarm. All the errors of the collaborating members are collected as the states: ]T [ x swar m = · · · x i · · · x j · · · x k · · ·

(5.12)

]T ]T ]T [ [ [ where x i = δ pi δti , x j = δ p j δt j , and x k = δ pk δtk . GNSS Pseudorange Observation Equations All pseudoranges ρi p between the ith member of the aerial swarm in the selected collaborative union and the sth GNSS satellite whose signal is received are used to establish the pseudorange measurement equations for the system. The measured pseudorange can be formulated as [28, 30] ρ˜is = ρis + δti + vρs

(5.13)

where ρis is the true value of the distance between the sth satellite and the ith member of the aerial swarm, δti is the distance corresponding to the equivalent clock error of the sth GNSS satellite, and vρi is the pseudorange measurement noise. The calculated pseudorange is | | ⁀ |⁀ | ρ is = | pi − ps |

(5.14)

⁀

where pi is the position of the ith member of the aerial swarm indicated by its equipped airborne navigation system according to Eq. (3.1), represented in the global coordinate frame, and ps is the position of the sth GNSS satellite represented in the global coordinate frame, which is obtained from the GNSS constellation ephemeris [30, 32]. Then, Eq. (5.14) is expanded into a Taylor series relative to the position of the ith member, taking the pseudorange measurement difference of Eqs. (5.13) and (5.14): ⁀

δρis = ρ˜is − ρ is ⁀

=−

∂ ρ is δ p + δti + ερ ∂ pi i

is

5.3 Collaborative Observation-Based Navigation Algorithm …

=

[

⁀

− ∂∂ρpis i

1

][ δ p ]

≙ his x i + ερ

δti

i

+ ερ

101

is

is

(5.15)

where the Jacobi matrix is ⁀ ] [ ∂ ρ is is ∂ρis ∂ρis = ∂ρ ∂ xi ∂ yi ∂ z i ∂ pi ] [ s yi −ys z i −z s = xiρ−x ρis ρis is

(5.16)

Relative-Range-Based Observation Equations The measured and calculated relative ranges between the ith and jth members of the aerial swarm can be represented as shown in Eq. (3.6) and Eq. (3.7), respectively. According to Eq. (3.10) and Eq. (3.16), ⁀

δri→ j = r˜i→ j − r i→ j ⁀

⁀

∂ r i→ j ∂ r i→ j δ pi + δti − δ p j − δt j + εr i→ j =− ∂ pi ∂ pj ⎡ ⎤ [ ⁀ ] δ pi ⁀ ⎢ δti ⎥ ∂ r i→ j ∂ r i→ j ⎥ = − 1− −1 ⎢ ⎣ δ p ⎦ + εr i→ j ∂ pi ∂ pj j δt j [ ] ] xi [ + εr i→ j ≙ hi→ j −hi→ j xj

(5.17)

Measurement Equations with Hybrid Observations When signals are received from multiple GNSS satellites and multiple pairwise collaborative relationships are established, the corresponding observation models similar to Eqs. (5.15) and (5.17) can be combined to obtain ⎤ ⎡ ⎤ ⎡ .. .. . ⎥ ⎢ . ⎥ ⎢ ⁀ ⎥ ⎢ δρ ⎥ ⎢ ρ ˜ − ρ ⎢ is is ⎥ is ⎥ ⎢ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ρ˜ − ρ⁀ ⎥ δρ ⎢ js js ⎥ js ⎥ ⎢ ⎢ . ⎥ ⎢ ⎥ . ⎥ .. yswar m = ⎢ ⎥ ⎢ .. ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⁀ ⎢ ⎢ δri→ j ⎥ ⎢ r˜i→ j − r i→ j ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎥ ⁀ ⎢ δr j→k ⎥ ⎢ ⎢ ⎣ ⎦ ⎣ r˜ j→k − r j→k ⎥ ⎦ .. .. . .

102

5 Collaborative Observation-Based Resilient Navigation Fusion

⎡ ⎤ ⎤ .. .. . . ⎢ ⎥ ⎥⎡ . ⎤ ⎢ ⎢ ⎢··· h ⎥ ⎥ . ··· is ⎢ ⎥⎢ . ⎥ ⎢ ερ is ⎥ ⎢ ⎥⎢ ⎥ ⎢ . ⎥ . . ⎢ ⎥⎢ x i ⎥ ⎢ .. ⎥ .. .. ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎥⎢ .. ⎥ ⎢ ··· ··· h js ⎢ ⎥⎢ . ⎥ ⎢ ερ js ⎥ ⎢ ⎢ ⎥ ⎥ .. ⎥ .. .. ⎥ ⎥⎢ =⎢ xj ⎥ + ⎢ . . ⎢ ⎢ ⎥⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎥⎢ . ⎢ ⎥ ⎢ · · · hi→ j · · · −hi→ j ⎥⎢ .. ⎥ ⎢ εr i→ j ⎥ ··· ⎢ ⎥ ⎥⎢ ⎥ ⎢ .. .. .. ⎢ ⎥⎢ x ⎥ ⎢ .. ⎥ ⎢ ⎥⎣ k ⎦ ⎢ . ⎥ . . . ⎢ ⎢ ⎥ ⎥ . ⎢ εr j→k ⎥ ⎢ . ··· h j→k · · · −h j→k · · · ⎥ ⎣ ⎣ ⎦ ⎦ . .. .. .. . . . ⎡

≙ H swar m x swar m + v swar m

(5.18)

5.3.2 Resilient Fusion Process with Collaborative Observations On the basis of the filtering model established above, a parallel collaborative navigation algorithm can be designed. Taking the resilient fusion navigation based on relative range observation as an example, the steps are as follows. STEP 1 GNSS Pseudorange Observation The pseudoranges ρ˜is measured by the GNSS receiver equipped on the ith label member (i = 1, 2, · · · , n) for the sth GNSS satellite (s = 1, 2, · · · , m) are obtained, as shown in Eq. (5.13). STEP 2 Configuration of the Collaborative Relationships in the Swarm The errors of the navigation systems, including the positioning errors and clock errors of the members of the aerial swarm, are collected as the states of the swarm, as expressed in Eq. (5.12). STEP 3 Relative Range Observation The relative distances r˜i→ j between the ith label member and the jth cooperative anchor member ( j = 1, 2, · · · , n) are measured. STEP 4 Pseudorange and Relative Range Calculation The pseudorange can also be calculated from the position indicated by the airborne navigation system of the i th member (i = 1, 2, . . . , n) and the position of the sth GNSS satellite (s = 1, 2, . . . , m) indicated by the constellation ephemeris. The relative range can also be calculated from the positions indicated by the airborne navigation systems of the i th and jth members (i, j = 1, 2, . . . , n) in accordance with Eq. (3.7) to obtain the calculated ⁀ value of the relative range, r i→ j . STEP 5 Dynamic Hybrid Observation Modelling The GNSS pseudorange observation equations are established in accordance with the GNSS signal availability of each member as expressed in Eq. (5.15). The relative range observation equation for a pair of members is established in accordance with their collaborative relationship as

5.3 Collaborative Observation-Based Navigation Algorithm …

103

shown in Eq. (5.17). Then, the measurement equations with hybrid observations for the whole aerial swarm are established by combining the pseudorange and relative range observation equations. STEP 6 Joint Positioning with Hybrid Observations The positions of the members in the aerial swarm that are included in the collaborative relationships are solved jointly by means of the dynamic hybrid observation model and the least squares algorithm [32]. As discussed in Chap. 3, the pseudorange and relative range observations vary in their noise levels. Therefore, weighting among the collaborative observations is needed to balance their contributions. Accordingly, the objective function for optimization is J ( xˆ swar m ) = ( yswar m − H swar m xˆ swar m )T W swar m ( yswar m − H swar m xˆ swar m ) (5.19) When the Markov estimation weighting strategy is adopted, the weighting matrix T −1 for the estimation is W swar m = cov[v swar m v swar m ] . The noise covariance of the hybrid collaborative observations can be obtained by collecting the noise variance of the pseudorange as measured by the GNSS receiver, σρ , and the noise variance of the relative range as measured by the ranging sensor, σr : ] [ T cov[v i→ v i→ ] = diag · · · σr2 · · · σr2 · · · σρ2 · · · σρ2 · · ·

(5.20)

Considering Eqs. (5.19) and (5.20), the solution for the combined states of the whole swarm is T −1 T xˆ swar m = (H swar m W swar m H swar m ) H swar m W swar m yswar m

(5.21)

Then, the estimated INS error and clock error states of the collaborating members in the swarm are fed back to their corresponding INS procedures for error compensation. The compensated positions and corresponding covariance values are shared with the other members in the aerial swarm to support cooperative navigation in the next epoch. According to the description of the parallel collaborative navigation fusion process given above, two state update processes are performed at different frequencies: time updating and measurement updating. The overall cooperative navigation fusion process for a label aircraft is shown in Fig. 5.2. In Fig. 5.2, the availability of GNSS pseudoranges varies with different epochs and members. When the GNSS pseudoranges obtained by a single member in the aerial swarm are not sufficient for independent localization, collaborative relationships can be configured between members in the swarm, and the corresponding relative ranges can be introduced to construct a collaborative resilient fusion model. By solving this resilient fusion model based on hybrid collaborative observations, the error states of the collaborating members can be estimated and compensated, enhancing the

104

5 Collaborative Observation-Based Resilient Navigation Fusion

Fig. 5.2 Collaborative observation-based resilient fusion process (Parallel Structure)

navigation robustness of the aerial swarm in a GNSS-limited environment. It should be noted that the solvability of the collaborative resilient fusion model depends on whether it is of full rank, which also varies with different epochs and with the geometry of the GNSS satellites and the members of the aerial swarm. Therefore, error estimation and compensation will be performed for different members each time, and the frequency is also variable.

5.4 Simulation Examples 5.4.1 Simulation of Hierarchical Collaborative Observation-Based Resilient Fusion 5.4.1.1

Simulation Setup

To verify the effectiveness of the hierarchical collaborative observation-based resilient fusion algorithm and the correctness of the collaborative navigation model, simulations were performed. In these simulations, the number of aircraft participating in swarm flight was 30. The simulation time was 3600 s. The simulated flight trajectories of the aircraft in the swarm are shown in Fig. 5.3. The relative ranges, azimuth angles, and elevation angles between label member 1 and other members are shown in Figs. 5.4, 5.5, 5.6. The gyroscope drift was 0.1 deg/h; the accelerometer bias was g*10–4 ; the accuracies of the ranging and angular measurements were 5 m and 0.1 deg, respectively; and the positioning accuracy of the members acting as anchors was 2 m.

5.4 Simulation Examples

Fig. 5.3 Aerial swarm flight trajectories Fig. 5.4 Relative ranges between label member 1 and other members

Fig. 5.5 Relative elevation angles between label member 1 and other members

105

106

5 Collaborative Observation-Based Resilient Navigation Fusion

Fig. 5.6 Relative azimuth angles between label member 1 and other members

In the following simulation analyses in Sect. 5.4.1, unless otherwise stated in a certain subsection, the simulations were carried out using the scenarios and conditions specified above. Moreover, 100 Monte Carlo simulations were performed to analyse the influence of various factors on navigation performance.

5.4.1.2

Analysis of the Influence of the Relative VOS Accuracy

In this subsection, the influence of the relative ranging and angular observation accuracies on the navigation performance is analysed. The position errors under different ranging and bearing accuracies are compared in Figs. 5.7 and 5.9, respectively. The variations in the position error with the ranging and bearing accuracies are shown in Figs. 5.8 and 5.10, respectively. In Figs. 5.7 and 5.9, the positioning error curves for label member 1 at ranging accuracy values of 15 m and 5 m and at angular accuracy values of 1.0° and 0.1°, Fig. 5.7 Comparison of position errors under different ranging accuracies (RA)

5.4 Simulation Examples Fig. 5.8 Variation in position error with ranging accuracy

Fig. 5.9 Comparison of position errors under different bearing accuracies (BA)

Fig. 5.10 Variation in position error with bearing accuracy

107

108

5 Collaborative Observation-Based Resilient Navigation Fusion

respectively, are shown as examples, and the traditional single leader–follower (LF) collaboration structure and the proposed hierarchical collaborative fusion (HC) structure are compared. In Figs. 5.8 and 5.10, the curves of the average root mean square error (RMSE) of the position of label members are shown for different ranging and bearing observation accuracies, respectively, and the traditional and proposed algorithms are compared. As seen in Figs. 5.8 and 5.10, for a label member under either the proposed HC structure or the traditional LF structure, improving the ranging and angular observation accuracies reduces the positioning error. For the same ranging and angular observation accuracies, the positioning error of the label member based on the proposed HC structure is improved compared with that obtained using the traditional LF collaboration structure. As seen in Figs. 5.8 and 5.10, with increasing relative ranging and angular accuracies, the average position RMSE of the aircraft in the swarm decreases. For the same ranging and angular observation accuracies of the sensors on the aircraft, the proposed HC structure yields a 20–30% improvement in the average positioning accuracy of the swarm compared to that based on the traditional LF collaboration structure. According to the rates of increase of the curves, the performance of the proposed HC structure is less influenced by the ranging accuracy and angular observation accuracy than is that of the traditional LF collaboration structure. According to the relationships between the average positioning error and the relative ranging and angular accuracies shown in Figs. 5.8 and 5.10, equivalent navigation accuracy can be reached with relatively low sensor accuracy by deploying the HC structure instead of the traditional LF collaboration structure. This means that the proposed HC structure reduces the requirements imposed on the ranging and angular accuracies to meet certain navigation performance specifications.

5.4.1.3

Analysis of Anchor–Label Navigation Error Propagation

The number of anchors in the swarm and the ratio of the number of anchors to the overall swarm size also influence the positioning accuracy for the label members. The position errors of a label member for different proportions of anchors in the swarm are shown in Fig. 5.11. Figure 5.11 presents the positioning error curves for a label member under the LF collaboration structure and under the HC structure in the cases that 10% and 30% of the swarm members are anchors. Figure 5.12 shows how the average positioning accuracy for all members changes with the number of anchors in the swarm, which is represented by a proportion ranging from 10 to 55%. As seen in Fig. 5.11, for a label member under the proposed HC structure, increasing the proportion of anchors is helpful for reducing the positioning error, whereas the corresponding reduction achieved under the traditional LF collaboration structure is not obvious. This indicates that compared with the traditional LF collaboration structure, the proposed HC structure utilizes the cooperative information of the swarm more effectively to improve the navigation performance. With the

5.4 Simulation Examples

109

Fig. 5.11 Comparison of position errors under different proportions of anchors (AP)

Fig. 5.12 Variation in the position error with the proportion of anchors

same proportion of anchors in the swarm, the positioning error of a label member under the proposed HC structure is improved compared with that under the traditional LF collaboration structure. As seen in Fig. 5.12, as the proportion of anchors with high-precision airborne navigation sensors in the swarm increases, the average position RMSE of the swarm decreases. For the same proportion of anchors in the swarm, the proposed HC structure always results in smaller average positioning errors than the traditional LF collaboration structure. It can be concluded from Fig. 5.12 that with an increasing number of anchors in the swarm, the average positioning error of the members of the swarm decreases. Furthermore, the improvement in the positioning accuracy as the proportion of anchors increases from 10 to 30% is larger than that seen with a further increase from 30 to 50%. This finding indicates that more anchors in the swarm are helpful for improving the average navigation performance. However, when there is already a large proportion of anchors in the swarm, i.e., 40% or more, increasing the number of anchors is

110

5 Collaborative Observation-Based Resilient Navigation Fusion

not necessary because the marginal contribution to the navigation performance will be small.

5.4.2 Simulation of Parallel Collaborative Observation-Based Resilient Fusion 5.4.2.1

Simulation Setup

To verify the effectiveness of the parallel collaborative observation-based resilient fusion algorithm and the correctness of the collaborative navigation model, simulations were performed. In these simulations, the number of members participating in the swarm flight was eight. The flight time of the aerial swarm in the simulations was approximately 200 s. The airborne ranging module, communication module, data storage module, computer module, and GNSS receiver module were simulated for each member to realize mutual radio sensing, communication, and ranging. The number of GNSS satellites visible to each member of the aerial swarm is shown in Fig. 5.13. As seen in Fig. 5.13, all members of the aerial swarm could observe a sufficient number of GNSS satellites during the period of 0–120 s. However, most members were unable to receive signals from a sufficient number of GNSS satellites during the period of 120–200 s; consequently, the navigation performance needed to be augmented. Fig. 5.13 Number of GNSS satellites visible to swarm members

5.4 Simulation Examples

5.4.2.2

111

Analysis of the Construction of Collaborative Unions

The number of collaborating members among the members of the aerial swarm are shown in Fig. 5.14. The collaborative relationships in the aerial swarm in two selected epochs are shown in Fig. 5.15 and Fig. 5.16 as examples. As shown in Fig. 5.14, different members collaborated in different epochs to form the essential collaborative unions for positioning. In Fig. 5.15, Ss (s = 1, 2, · · · ) represents the number of GNSS satellites observed by the corresponding member. Each member could observe only three or fewer GNSS satellites, which is not sufficient for independent localization, thus necessitating collaboration. In the epoch represented in this figure, members M1, M3, and M7, which could each observe Fig. 5.14 Number of collaborating members in the swarm

Fig. 5.15 Collaborative relationship in the aerial swarm at 185 s

112

5 Collaborative Observation-Based Resilient Navigation Fusion

Fig. 5.16 Collaborative relationship in the aerial swarm at 190 s

three GNSS satellites, were chosen to form a collaborative union to prioritize localization. In the epoch depicted in Fig. 5.16, members M1, M2, M3, and M8 were chosen to form a collaborative union. M1 and M2 could each observe two GNSS satellites, and M3 and M8 could each observe three. As illustrated by these examples, the members of a swarm will combine with different partners in different epochs to form collaborative unions for localization. After the members of the current collaborative union realize localization, they can act as the anchor members for the other members of the swarm that have not yet achieved localization to determine their locations in accordance with the HC structure.

5.4.2.3

Analysis of Navigation Performance

The positioning performance achieved with the proposed parallel collaborative observation-based resilient fusion method was compared with the performance without collaboration through simulation. The positioning error curves for four selected members are shown in Fig. 5.17, 5.18, 5.19, 5.20 as examples. The position errors of members as obtained with the traditional navigation method without collaboration (NC) and the proposed method with paralleled collaboration (PC) are compared, respectively. As seen in Figs. 5.17, 5.18, 5.19, 5.20, independent localization failed during several episodes because an insufficient number of GNSS satellites were observed. Benefitting from the introduction of constraint conditions through mutual observation and information sharing among the members of the collaborative union, the proposed parallel collaborative observation-based resilient fusion method can successfully realize continuous positioning in an environment where the number of visible GNSS satellites is insufficient.

5.4 Simulation Examples

113

Fig. 5.17 Comparison of position errors (Member M3)

Fig. 5.18 Comparison of position errors (Member M4)

In Fig. 5.17, member M3 cannot be localized during the periods of 89~102 s, 116~130 s, and 166~210 s by using the traditional method, but it can be localized at these times by using the collaborative positioning algorithm. In fact, we can see from Fig. 5.19 that when the proposed method is used, member M3 can be localized throughout the whole flight time, and the positioning error curves lie within a small range, which indicates good performance. Similarly, as seen from Fig. 5.20, member

114

5 Collaborative Observation-Based Resilient Navigation Fusion

Fig. 5.19 Comparison of position errors (Member M6)

Fig. 5.20 Comparison of position errors (Member M8)

M8 cannot be localized during the periods of 95~105 s and 124~210 s by using the traditional method, but it can be localized at these times by using the collaborative positioning algorithm. Indeed, Fig. 5.20 shows that member M8 can be localized throughout the whole flight time by using the proposed method.

References

115

5.5 Conclusions This chapter has proposed collaborative observation-based resilient fusion navigation methods for aerial swarms with hierarchical and parallel structures. The simulation results show that compared with the traditional LF structure, the proposed hierarchical structure for cooperative fusion improves the positioning accuracy for low-precision label aircraft and causes the position error for label aircraft to be less affected by relative distance and angle errors. This approach also has the potential to improve the fault tolerance of collaborative navigation systems. When some anchors are damaged, the entire collaborative navigation system can continue to function, and the anchor positions can be recovered after failure, thus verifying the effectiveness of this method. In some complicated and challenging environments, such as areas with high buildings, canyons, or satellite signal rejection, the members of an aerial swarm may be unable to receive sufficient GNSS satellite signals to meet their positioning requirements. This chapter has also proposed a cooperative positioning method based on a hybrid recursive network to solve this problem. Through mutual radio sensing, communication, and ranging among the members of a collaborative union, all members of the union can be localized successfully. Then, the members that have achieved successful localization broadcast their own information to other members that cannot achieve independent localization to help obtain their positions. Finally, some or all members can be successfully localized even in a complex and challenging environment, which greatly improves the positioning ability and accurate task execution ability of the aerial swarm.

References 1. Wang L, Groves PD, Ziebart MK (2012) Multi-constellation GNSS performance evaluation for urban canyons using large virtual reality city models. J Navig 65(3):459–476. https://doi. org/10.1017/s0373463312000082 2. Zhang G, Hsu L-T (2021) Performance assessment of GNSS diffraction models in urban areas. Navig-J Inst Navig 68(2):369–389. https://doi.org/10.1002/navi.417 3. Deurloo R, Volckaert M, Huang B, Smolders K, Barreau V, Inst N (2018) Assessment of a robust MEMS-based RTK/INS system for UAV applications. In: 31st international technical meeting of the satellite-division-of-the-institute-of-navigation (ION GNSS), Miami, FL, 2018. Sep 24– 28 2018. Institute of navigation satellite division proceedings of the international technical meeting, pp 2750–2760. https://doi.org/10.33012/2018.16062 4. Kortunov VI, Dybska IY, Proskura GA, Kravchuk AS (2009) Integrated mini INS based on MEMS sensors for UAV control. IEEE Aerosp Electron Syst Mag 24(1):41–43. https://doi.org/ 10.1109/maes.2009.4772754 5. Jouybari A, Bagherbandi M, Nilfouroushan F (2022) Impact of GNSS signal outage on Eops using forward Kalman filter and smoothing algorithm. Int Arch Photogramm Remote Sens Spatial Inf Sci XLIII-B2-2022:59–64. https://doi.org/10.5194/isprs-archives-XLIII-B2-202259-2022

116

5 Collaborative Observation-Based Resilient Navigation Fusion

6. Du S, Sun W, Gao Y (2015) Integration of GNSS and MEMS-based rotary INS for bridging GNSS outages. In: 6th China satellite navigation conference (CSNC), Xian, PEOPLES R CHINA, 2015. May 13–15 2015. Lecture notes in electrical engineering, pp 659–676. https:// doi.org/10.1007/978-3-662-46632-2_58 7. Manickam S, O’Keefe K, Inst N (2016) Using tactical and MEMS grade INS to protect against GNSS spoofing in automotive applications. In: 29th international technical meeting of thesatellite-division-of-the-institute-of-navigation (ION GNSS+), Portland, OR, 2016. Sep 12–16 2016. Institute of Navigation Satellite Division Proceedings of the International Technical Meeting, pp 2991–3001 8. Rabiain AH, Kealy A, Morelande M (2013) Tightly coupled MEMS based INS/GNSS performance evaluation during extended GNSS outages. J Appl Geodesy 7(4):291–298. https://doi. org/10.1515/jag-2013-0056 9. Salabert P, Inst N (2017) The future of GNSS in Civil Aviation: opportunities and challenges ION GNSS+2017. In: 30th international technical meeting of the satellite-divisionof-the-institute-of-navigation (ION GNSS+), Portland, OR, 2017, Sep 25–29 2017. Institute of navigation satellite division proceedings of the international technical meeting, pp 2742–2757 10. Kurazume R, Hirose S (2000) Experimental study of a cooperative positioning system. Auton Robot 8(1):43–52. https://doi.org/10.1023/A:1008988801987 11. Causa F, Fasano G, Grassi M, Ieee (2019) Improving autonomy in GNSS-challenging environments by multi-UAV cooperation. In: IEEE/AIAA 38th digital avionics systems conference (DASC), San Diego, CA, Sep 08–12 2019. IEEE-AIAA Digital Avionics Systems Conference 12. Vetrella AR, Fasano G, Accardo D, Ieee (2016) Cooperative navigation in GPS-challenging environments exploiting position broadcast and vision-based tracking. In: International conference on unmanned aircraft systems (ICUAS), Arlington, VA, 2016, Jun 07–10 2016, pp 447–456 13. Garello R, Lo Presti L, Corazza G, Samson J (2012) Peer-to-peer cooperative positioning: part I: GNSS aided acquisition. Inside GNSS 14. Yang C, Strader J, Gu Y, Canciani A, Brink K (2020) Cooperative navigation using pairwise communication with ranging and magnetic anomaly measurements. J Aerosp Inform Syst 17(11):624–633. https://doi.org/10.2514/1.I010785 15. Ellingson G, Brink K, McLain T (2020) Cooperative relative navigation of multiple aircraft in global positioning system-denied/degraded environments. J Aerosp Inform Syst 17(8):470– 480. https://doi.org/10.2514/1.I010802 16. Vetrella AR, Fasano G, Accardo D (2016) Cooperative navigation in GPS-challenging environments exploiting position broadcast and vision-based tracking. In: 2016 international conference on unmanned aircraft systems, ICUAS 2016, June 7, 2016–June 10, 2016, Arlington, VA, United states, 2016. Institute of Electrical and Electronics Engineers Inc., pp 447–456. https:// doi.org/10.1109/ICUAS.2016.7502647 17. Causa F, Vetrella AR, Fasano G, Accardo D, Ieee (2018) Multi-UAV formation geometries for cooperative navigation in GNSS-challenging environments. In: IEEE/ION position, location and navigation symposium (PLANS), Monterey, CA, 2018. Apr 23–26 2018, pp 775–785 18. Chen X, Gao W, Wang J (2013) Robust all-source positioning of UAVs based on belief propagation. Eurasip J Adv Signal Process. https://doi.org/10.1186/1687-6180-2013-150 19. Zanetti R, D’Souza C (2021) Inertial navigation. In: Baillieul J, Samad T (eds) Encyclopedia of systems and control. Springer International Publishing, Cham, pp 993–999. https://doi.org/ 10.1007/978-3-030-44184-5_100036 20. Zhang P, Xu C, Hu C, Chen Y (2012) Coordinate transformations in satellite navigation systems. In: Berlin, Heidelberg, 2012. Advances in electronic engineering, communication and management, vol 2. Springer Berlin Heidelberg, pp 249–257 21. Christensen R, Geller D, Hansen M (2020) Linear covariance navigation analysis of range and image measurement processing for autonomous lunar lander missions. In: 2020 IEEE/ION position, location and navigation symposium. https://doi.org/10.1109/plans46316.2020.910 9838

References

117

22. Mo S, Jin X, Lin C, Zhang W, Xu Z, Jin Z (2021) Multi-satellite relative navigation scheme for microsatellites using inter-satellite radio frequency measurements. Sensors 21(11). https:// doi.org/10.3390/s21113725 23. Karimian A, Tron R (2022) Bearing-only navigation with field of view constraints. IEEE Control Syst Lett 6:49–54. https://doi.org/10.1109/lcsys.2020.3048802 24. Mok S-H, Pi J, Bang H (2020) One-step rendezvous guidance for improving observability in bearings-only navigation. Adv Space Res 66(11):2689–2702. https://doi.org/10.1016/j.asr. 2020.07.035 25. Xiong J, Cheong JW, Xiong Z, Dempster AG, Tian S, Wang R (2021) Integrity for multi-sensor cooperative positioning. IEEE Trans Intell Transp Syst 22(2):792–807. https://doi.org/10.1109/ tits.2019.2956936 26. Kaur A, Balsundar P, Kumar V, Mantri A, Raj ANJ, Ieee (2016) MEMS based inertial navigation system: an exploratory analysis. In: 5th international conference on wireless networks and embedded systems (WECON), Chitkara Univ, Rajpura, INDIA, 2016 Oct 14–16 2016, pp 77–82 27. Noureldin A, Karamat TB, Eberts MD, El-Shafie A (2009) Performance enhancement of MEMS-based INS/GPS integration for low-cost navigation applications. IEEE Trans Veh Technol 58(3):1077–1096. https://doi.org/10.1109/tvt.2008.926076 28. Chengeng S, Gong Z, Jun L (2021) Research on the influence of pseudo-range biases on precise orbit determination and clock error calculation for beidou navigation satellites. China Satellite Navig Conf. https://doi.org/10.1007/978-981-16-3138-2_46 29. Saito K, Sugiyama T, Ieee (2021) GPS pseudo range correction by the number of reflections and incident angle estimations. In: 12th international conference on ICT convergence (ICTC)— beyond the pandemic era with ICT convergence innovation, South Korea, 2021. Oct 20–22 2021. International conference on information and communication technology convergence, pp 1014–1018. https://doi.org/10.1109/ictc52510.2021.9620805 30. Chen H, Niu F, Su X, Geng T, Liu Z, Li Q (2021) Initial results of modeling and improvement of BDS-2/GPS broadcast ephemeris satellite orbit based on BP and PSO-BP neural networks. Remote Sens 13(23). https://doi.org/10.3390/rs13234801 31. Malys S, Solomon R, Drotar J, Kawakami T, Johnson T (2021) Compatibility of terrestrial reference frames used in GNSS broadcast messages during an 8 week period of 2019. Adv Space Res 67(2):834–844. https://doi.org/10.1016/j.asr.2020.11.029 32. Liu L, Wang R, Xiong Z, Liu J, Chen X (2021) Collaborative positioning method for UAV cluster based on hybrid recursive network. In: International conference on autonomous unmanned systems, ICAUS 2021, Sept 24, 2021–Sept 26, 2021, Changsha, China, 2022. Lecture notes in electrical engineering. Springer Science and Business Media Deutschland GmbH, pp 3233–3242. https://doi.org/10.1007/978-981-16-9492-9_317

Chapter 6

Collaborative Geometry Optimization in Resilient Navigation

Abstract The geometry of the collaborating partners is one of the factors that influence the effectiveness of collaborative resilient navigation. In this chapter, an improved geometric dilution of precision is introduced to quantitatively evaluate geometric configurations in collaborative resilient navigation fusion. The influence of geometry on the accuracy of collaborative resilient navigation fusion is discussed in both the GNSS-augmented and GNSS-denied situations. Geometry optimization algorithms based on a geometric analysis method and an algebraic search method are proposed, with simulated examples. Keyword Collaborative navigation · Geometry optimization · Geometric dilution of precision · Geometric analysis · Algebraic Search · GNSS-augmented · GNSS-denied resilient fusion

6.1 Introduction The aircraft swarm, which is composed of multiple cooperating members, has received increasing attention recently. The swarmed aircraft has prominent advantages in terms of the ability to perform complex and dangerous tasks compared with a single unmanned vehicle [1]. Navigation performance is critical for realizing coordinative autonomy of swarming aircraft [2]. In large-scale emergency search and rescue (SAR) tasks, establishing reliable positioning and navigation services rapidly in the search area is critical. As a popular class of navigation systems, global navigation satellite systems (GNSSs) represent a common navigation approach for unmanned vehicles. However, GNSS signals are susceptible to various issues affecting their performance, such as blocking and jamming [3]. As a result, the GNSS receiver equipped on an aircraft may have access to only a few visible satellites in a poor geometric configuration under critical conditions [4]. Emergency SAR tasks must often be performed in urban environments or canyons, where the blockage of GNSS signals commonly leads to degradation in positioning accuracy, affecting the operation of unmanned vehicles relying on GNSS technology [5, 6].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_6

119

120

6 Collaborative Geometry Optimization in Resilient Navigation

At present, sensors used in relative navigation system mainly include inertial measurement unit and satellite navigation system. However, the inertial measurement unit has constant value drift, and the satellite navigation system signal is susceptible to external interference, and its reliability is low. They cannot meet the accuracy requirements of cluster UAV relative navigation system. Collaborative navigation technology utilizes the sensor information between multiple aircrafts to achieve information collaboration. Combined with data fusion, it improves the positioning accuracy of multiple aircrafts, compensates for the errors of navigation sensors, and identifies, isolates and recovers possible faults. Collaborative navigation technology can improve the navigation accuracy of low-precision aircrafts by using the position information of high-precision aircrafts, and increase the reliability of cluster aircraft system. Collaborative navigation technology greatly expands the application range of UAV, improves the positioning accuracy of UAV and improves the stability, safety and reliability of UAV formation. Reference [7] proposed a collaborative positioning and tracking method for multiple micro-aircrafts in a satellite rejection environment. This method estimates the velocity of target by measuring the relative and absolute velocity information between aircraft and combining local non-linear observers. This method is not effective in locating and tracking targets with large velocity and distance. Reference [8] proposed an adaptive INS/GPS integrated navigation method in formation flight. This method estimates the covariance of INS and GPS through different filters, and finally estimates the relative velocity and attitude through adaptive Kalman filtering. It requires the error of navigation system to obey gaussian distribution and the absolute output information of INS of each aircraft. Reference [9] proposed a relative attitude estimating method for two UAVs at the same altitude using the information of ranging radar and airborne navigation system. This method mainly estimates the attitude of aircraft, and does not modify the position and velocity errors. Reference [10] proposed the robust filtering algorithm to solve the non-linear filtering problem of relative navigation system based on GPS/INS combined measurement under the condition of the uncertainties existing in relative navigation system model and the unknown of statistical characteristics of noise. Reference [11] proposed a relative navigation method by information fusion of carrier phase differential GPS and radio ranging and its own low-cost inertial sensor. This method enhanced the robustness of relative navigation by ultra-wideband radio ranging technology. Reference [12] designed a visual relative navigation system based on volume Kalman filter to fuse the pitch angle, yaw angle, and other visual information and realized the relative navigation between the leader and follower. This method is more robust to non-Gaussian error but has high requirements for imaging clarity of airborne visual sensor device. Reference [13] proposed a GPS relative tracking method to achieve the relative position information of multiple receivers. Reference [14] proposed a new visual navigation model to estimate the relative attitude between UAVs. Reference [15] proposed an indirect collaborative positioning method based on ultra-wideband in the GPS rejection environment. However, this method is only suitable for small quadrotor aircraft formation with low flying speed and close flying formation distance. Reference [16] proposed a fully parallel distributed collaborative navigation framework. It aims to make each aircraft a fusion

6.1 Introduction

121

centre. In addition, it enhances the relative navigation accuracy by sharing collaborative navigation information between aircrafts. Reference [17]described an aircraft control method suitable for leader–follower structure, and designed a cascade loop to adjust the relative distance and angle between the leader aircraft and the follower aircraft to increase the stability of the collaborative navigation. However, this method did not correct the position and velocity errors of the follower aircraft. Reference [18] proposed a leader–follower collaborative navigation method based on extended Kalman filter. It estimates the relative position, relative velocity and relative attitude between the leader aircraft and the follower aircraft by measuring information from the visual sensors. This method requires high accuracy navigation information of the leader aircraft and can only measure the relative position information of an aircraft at one time. It is not suitable for a large-scale cluster aircraft flight system. Reference [19] proposed a leader–follower relative navigation algorithm based on filtering to determine the relative position and velocity of UAV in formation flight. The algorithm mainly aims at model establishment of vision sensor and ranging radar system and filter parameter selection in non-Gaussian environment. However, this method has high requirements for aircraft airborne sensor equipment and the system model only considers the two-aircraft formation situation. Reference [20] proposed a method of sensing the relative position and direction of two UAVs based on binocular vision sensor in the leader–follower twin-aircraft flight mode. The visual sensor is used to extract and identify the feature points of the aircraft to detect the relative position. Its estimation accuracy is up to the centimetre level. However, this method is only applicable to small-sized and low-speed two-aircrafts. The sensing distance of two aircrafts is also limited, so it is not suitable for large-scale UAV cluster flight system. Reference [21] proposed a method for solving the estimation and tracking problem of the leader aircraft through a visual algorithm in the leader–follower formation flight structure. This method is based on machine learning algorithm and requires a large amount of raw data, and has large requirements for the velocity and flying distance of the aircraft. Reference [22] proposed a method to estimate the attitude of the leader in the formation flight through visual sensors. This method uses machine vision algorithm to achieve the purpose of attitude estimation but does not improve the position accuracy of leader. The introduction of intermember cooperation in swarm has shown potential in improving the reliability of navigation systems in GNSS-challenged environments [23, 24]. The members of swarm in GNSS-challenged environments can exploit supplementary relative observations or vision-based tracking results broadcast by one or more collaborative members of swarm flying under favourable GNSS coverage [25, 27]. Considering the problem of GNSS performance degradation in urban canyons, reference [28] proposed a collaborative relative navigation method based on peerto-peer (P2P) ranging observations to assist the GNSS and established a collaborative relative navigation model based on GNSS pseudorange double-difference DD observations and P2P ranging. The positioning error of the GNSS in some challenging environments was reduced significantly. The navigation accuracy was effectively improved. Reference [29] proposed a multiple unmanned aerial vehicle

122

6 Collaborative Geometry Optimization in Resilient Navigation

(UAV) formation geometries method for collaborative navigation in a GNSS challenging environment. A “parent UAV” under favourable satellite coverage is used to support a “child UAV” whose reception of satellites’ signal is partly blocked. The concept of generalized dilution of precision is introduced as a way to predict the positioning accuracy of child UAV resulting from available GNSS observations and collaborative observations, and can thus be used to individuate optimal formation geometries and navigation performance bounds. The proper formation geometries allow collaborative observations to provide metre-level positioning accuracy. Reference [30] proposed a collaborative robust navigation method for Unmanned Ground Vehicle (UGV) and UAVs. The UGV can assists UAVs through point-to-point radio ranging to improve the positioning geometry of UAVs. Two collaborative strategies and two different estimation strategies for the UGV assistance of UAVs were developed to reduce the 3D positioning error of the UAV from approximately 1 m to approximately 10 cm. Reference [13] proposed a relative navigation method for UAVs based on Differential Global Positioning System (DGPS)/Inertial Navigation System (INS) and Ultra Wide Band (UWB) point-to-point radio ranging. When the visibility/geometry of a satellite is poor for a small, dynamic UAV, the additional UWB ranging information can significantly improve the dynamic baseline estimation performance of DGPS/INS. Reference [31] proposed a Doppler/GNSS and Inertial Observation Unit (IMU) tightly coupled collaborative positioning method in a Vehicular Ad Hoc Network. In the full GPS coverage environment, the method’s improvement in the relative positioning precision outperforms differential GPS by 28%. In a GPS outage, the performance improvement can be up to 66%. Although introducing intermember cooperation in a swarm may be helpful, optimization of the partnerships established for such cooperation is essential to ensure that the navigation performance can be enhanced effectively and efficiently. On the one hand, utilizing the information from all members of the swarm may increase the burden of intermember observation and communication. On the other hand, introducing intermember observations with low accuracy or collected in poor geometrical configurations may contribute less to improving the navigation performance. The target service area, which is called the region of interest (ROI), is dynamically changing during a SAR task with the manoeuvring of the unmanned ground vehicles; therefore, a novel navigation augmentation approach appropriate for a dynamic ROI is needed. Thus, research on collaborative navigation based on partnership optimization should be carried out, with the goal of ensuring that intermember collaborative partnerships that offer the greatest possible contributions are adopted. To address the problems of low positioning performance for swarm members in complex environments and the inefficiency of fully connected collaboration, this chapter proposes a collaboration-augmented navigation method for swarm members. In this method, a collaboration reference ring is established to analyse the existing geometric distribution of the satellites, and an optimal collaborative subarea division and partner selection strategy is designed to realize comprehensive configuration optimization and navigation performance improvement. Moreover, a bearingonly cooperative navigation method based on genetic optimization is proposed for

6.2 Geometric Dilution of Precision in Collaborative Resilient Navigation …

123

dynamic air–ground navigation augmentation. By analysing the scenario of cooperative navigation augmentation for a SAR task, the relative-bearing-only observation geometry and its influence on the positioning accuracy are analysed. Then, a genetic optimization strategy for cooperative dynamic air–ground navigation augmentation is analysed. The effectiveness of the cooperative geometry optimization method is validated through simulation, and the cooperation-augmented navigation performance in dynamic air–ground SAR scenarios is analysed.

6.2 Geometric Dilution of Precision in Collaborative Resilient Navigation Fusion 6.2.1 Geometric Configuration in Collaborative Navigation The observation geometry of a member in a swarm can be represented in terms of the relative azimuth and elevation angles of the observed GNSS satellites and the other collaborative members of the swarm. For a given label member, all of its observed GNSS satellites and anchor members, regardless of their distance, are projected onto a unit sphere with the label member at the centre, as shown in Fig. 6.1. In the polar coordinate frame of the label member, the position of the sth GNSS satellite is represented by a pair of angles (ϕG(s) , θG(s) ) corresponding to the satellite’s azimuth and elevation, which represents the position of the satellite projected onto Z n (U )

G(s) V (i )

θG( s )

θV(i )

ϕV(i )

ϕG( s )

Yn ( N )

X n (E)

Fig. 6.1 Geometries of an anchor member and a satellite from the perspective of a label member

124

6 Collaborative Geometry Optimization in Resilient Navigation

the unit sphere centred on the position of the label member. The position of the ith collaborative member of the swarm is similarly expressed as (ϕV(i) , θV(i) ) in the polar coordinate frame of the label member.

6.2.2 Geometric Dilution of Precision in Collaborative Navigation In GNSS positioning applications, the geometric dilution of precision (GDOP) is used to represent the mapping relationship between measurement noise and positioning accuracy, reflecting the influence of the observation geometry. On the basis of the observation models presented in Chap. 3, collaborative positioning can be carried out. However, in the collaborative positioning scenarios considered here, the relationship between the measurement noise and the positioning error is more complex than it is in the case of GNSS positioning; thus, a new indicator should be established to extend the traditional concept of the GDOP. Collaborative Geometric Dilution of Precision for the Hierarchical Collaboration Structure In the hierarchical collaborative navigation structure, the positioning error covariance σ 2p j of the jth anchor member in the swarm, which represents the uncertainty of the position indicated by its airborne navigation system, changes with time. Meanwhile, the direction cosines in the observation model, which depend on the relative position between the ith label member and the jth anchor member, also vary with time. Influenced by the above factors, according to Eqs. (3.20), (3.58), (3.73), or Eq. (3.87), the covariance of the relative observations (cov[vi→ j ] for relative ranging observations or cov[vi→ j ] for relative bearing or vector-of-sight (VOS) observations) varies during the flight of the aerial swarm. Moreover, the covariance of the relative observations in collaborative fusion navigation is different from the covariance of the pseudorange noise, σρ . In accordance with Eq. (4.23), the collaborative geometric dilution of precision (CGDOP) can be established as follows: ⎡

cov( xˆ i − x i ) C G D O Pi = trace σr2e f

⎤1/2

T W i→ H i→ )−1 ]1/2 [trace(H i→ = σr e f

(6.1)

where the observation weighting matrix W i→ = cov[vi→ ]−1 reflects the evaluation of the noise level of the collaborative observations, which may be influenced not only by the precision of the range and bearing sensors but also by the positioning accuracy of the anchor members in the swarm; σr e f is the standard deviation for normalization and can be set to a constant value. In addition, when GNSS signals are available, the GNSS pseudoranges that are received by the label member are used together with the relative observations. In this

6.2 Geometric Dilution of Precision in Collaborative Resilient Navigation …

125

case, the calculation of the CGDOP should account for the geometric relationships with respect to both the GNSS satellites and the anchor members. In this case, the CGDOP equation has a similar form to Eq. (6.1), while the observation weighting matrix W i→ is different and takes the following form: ⎡

W i→

⎤−1

..

⎢ . ⎢ σρ2 I n 0 ⎢ ⎢ . ⎢ .. =⎢ ⎢ ⎢ cov[vi→ j ] ⎢ ⎢ 0 cov[vi→ j+1 ] ⎣

..

.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6.2)

where σρ2 is the variance of the GNSS pseudorange noise. Thus, the CGDOP can be used to quantify the effect of the geometric configuration in cooperation-augmented navigation on the basis of the GNSS pseudorange measurements. When all the label members’ observation geometries need to be evaluated and optimized globally, the following equation can be established: {n C G D O Pswar m =

wi C G D O Pi {n i=1 wi

i=1

(6.3)

where wi is the weight of the ith member of the aerial swarm in the geometry evaluation. CGDOP for the Parallel Collaborative Structure In the parallel structure, the positioning of all members in the collaborative union is carried out together, so the CGDOP of the collaborative union can be calculated as a whole. According to Eq. (5.21), the estimation covariance of collaborative positioning in the parallel structure is T −1 cov( xˆ swar m − x swar m ) = (H swar m W swar m H swar m )

(6.4)

Thus, the CGDOP can be calculated as ⎡

C G D O Pswar m

cov( xˆ swar m − x swar m ) = trace σr2e f

T −1 1/2 [trace(H swar m W swar m H swar m ) ] = σr e f

⎤1/2 (6.5)

In Eqs. (6.1), (6.3) and (6.5), the CGDOP describes the linear amplification relationship between the positioning error and the normalized observation error, where the different contributions of the relative observations from the anchor members in

126

6 Collaborative Geometry Optimization in Resilient Navigation

the swarm are also considered. This metric can be used as a tool to evaluate the influence on the positioning accuracy exerted by the geometric configuration formed by the satellites and the collaborating members of the swarm.

6.3 Geometric Influence on Collaborative Resilient Navigation Fusion 6.3.1 Geometric Influence on Collaborative Resilient Fusion in a GNSS-Augmented Situation When observations relative to anchor members are introduced to augment the navigation performance of a label member on the basis of GNSS positioning technology, the resulting hybrid observations are composed of pseudoranges and relative observations. In this situation, the geometric influence on cooperative navigation augmentation can be analysed. In this subsection, GNSS-augmented collaborative navigation based on relative ranging is discussed as an example. Since the position of each satellite relative to the label member of interest is determined, the optimization of the geometric configuration becomes a problem of optimizing the positions of the collaborating swarm members. The satellite distribution with respect to the label member can be modelled as a ring on the standard sphere, as shown in Figs. 6.2, 6.3 and 6.4. As shown in Figs. 4.2, 4.3 and 4.4, each observed GNSS satellite is projected onto the surface of the unit sphere as a point. When there are more than three visible

Fig. 6.2 Reference ring for collaborative augmentation optimization (on the Unit Sphere)

6.3 Geometric Influence on Collaborative Resilient Navigation Fusion

127

Fig. 6.3 Reference ring for collaborative augmentation optimization (on the XOZ Plane)

Fig. 6.4 Reference ring for collaborative augmentation optimization (on the XOY Plane)

satellites, not all satellites may be in the same plane. Based on the positions of the points projected onto the surface of the unit sphere, a fitted circle can be obtained, which is referred to as the GNSS satellite reference circle in this book. The points G (s) are the projected points representing the GNSS satellites’ positions on the surface of the unit sphere. Point C is the centre of the GNSS satellite reference circle, and point F is the symmetry point with respect to point C. Therefore, the line C F is the central axis of the GNSS satellite reference circle on the unit sphere. On the basis of the GNSS satellite reference circle, the distance between each satellite and the centre of the circle is obtained, and the maximum and minimum of these distances are used as the radii to construct two concentric circles. The area

128

6 Collaborative Geometry Optimization in Resilient Navigation

between these two concentric circles forms a ring, which is referred to as the GNSS satellite reference ring in this book, with spherical radii of I I I I ren = minI(ϕG(s) , θG(s) ) − (αC , βC )I spher e I I I (s) (s) I rem = maxI(ϕG , θG ) − (αC , βC )I

(6.6)

spher e

where ||spher e denotes the distance on the unit sphere and the coordinates (αC , βC ) represent the azimuth and elevation of the centre of the GNSS satellite reference circle. All satellites are distributed within this GNSS satellite reference ring, and thus, introducing additional observations near this ring will have little effect on improving the geometric configuration. Instead, the positions of the collaborating aircraft should be far from the GNSS satellite reference ring, as will be discussed in Sect. 6.4.

6.3.2 Geometric Influence on Collaborative Resilient Fusion in a GNSS-Denied Situation When observations relative to anchor members are adopted for positioning in a GNSS-denied environment, the anchor members’ geometry relative to the label member of interest can be analysed to determine its influence on the collaborative positioning performance. In this subsection, collaborative positioning based on relative-bearing-only observations is discussed as an example. It is known from Sect. 3.5 that the navigation accuracy based on bearing-only cooperative augmentation is related to several factors: the angular measurement precision, the geometric configuration, and the localization deviation of the anchor aircraft. Thus, for a given angular measurement precision and localization deviation of the anchor aircraft, the navigation accuracy can be improved by optimizing the configuration of the anchor aircraft. The relationship between the geometric configuration and the positioning accuracy is shown in Fig. 6.5. It can be concluded from Fig. 6.5 that when the lines of sight (LOSs) from two anchor members to the label member are nearly orthogonal, the area of the positioning uncertainty is smaller, indicating that a higher positioning accuracy can be achieved through cooperative augmentation. In contrast, when the LOSs of the two anchor members are nearly parallel (such as A2 and A3), the positioning accuracy achieved through cooperative augmentation is low. Given a certain angle between two LOSs, the closer the distance is between the label ground vehicle and an anchor member, the better the positioning accuracy of the label member (in this example, because A2 is closer to L1 than A3 is, collaborative positioning with A1 and A2 offers higher accuracy than collaborative positioning with A1 and A3 does). In addition, a larger localization deviation of an anchor member (such as A4) will lead to inflation of the cooperative positioning error. Thus, the influence of the geometry on bearing-only

6.4 Geometry Optimization for Collaborative Resilient Navigation Fusion Fig. 6.5 Effect of configuration on positioning accuracy (Bearing-only observations)

A4 Positioning (Actual Pos) Error

129

A2 A B

A4 (Nav Pos)

C A1

D

L1 Bearing measurement error

A3

cooperative augmentation is related to multiple factors and needs to be quantitatively evaluated.

6.4 Geometry Optimization for Collaborative Resilient Navigation Fusion 6.4.1 Geometry Optimization Based on a Geometric Analysis Method An aerial swarm that includes multiple cooperating members exhibits the potential for navigation performance augmentation. Considering the significant influence of geometry on navigation performance in collaborative resilient navigation fusion, optimization of the geometry is needed. By means of optimal collaborative partner selection, the configuration strategy for the anchor members that can enable the greatest enhancement of the navigation performance can be adopted dynamically. In this way, the effectiveness and efficiency of collaborative resilient fusion can be improved. The geometric analysis method, in which the best anchor member configuration strategy is determined by comparing the geometric relationships between observations, is one approach for realizing collaboration optimization. An example scenario of collaborative resilient fusion in a GNSS-augmented situation involving an aerial swarm is shown in Fig. 6.6. In this example scenario, the hierarchical collaborative navigation structure is adopted. Considering the differences in the positioning accuracy of each aircraft, the members of the swarm are divided into two types: anchor aircraft, with favourable navigation performance, and label aircraft, with unfavourable navigation performance. When the geometric configuration of the satellites for a label aircraft is

130

6 Collaborative Geometry Optimization in Resilient Navigation

…

…

… Anchor Member

…

GNSS observations

Anchor Member

Cooperated observations Label Member

Fig. 6.6 Scenario of collaborative resilient fusion in a GNSS-augmented situation

poor, an anchor aircraft can transmit collaborative observations to the label aircraft to assist in geometric configuration optimization. The workflow of the collaborative navigation augmentation algorithm for a swarm of aircraft is shown in Fig. 6.7. As shown in Fig. 6.7, a label member in the swarm that needs assistance first utilizes its airborne radio system to determine the number of available anchor aircraft and obtain the relative distance to each anchor aircraft and then calculates the relative azimuth and elevation angles with respect to these aircraft in accordance with the position and distance measurements obtained by its airborne sensors. Simultaneously, collaborative partner selection is carried out to select the optimal anchor aircraft to assist the label aircraft. The collaborative aircraft selection algorithm is based on analysing the GDOP in collaboration-augmented positioning, as established in Sect. 3.3. Collaborative aircraft selection is mainly implemented by selecting the optimal collaborative anchor aircraft that can help the label aircraft achieve the smallest CGDOP. The conclusions of the geometric configuration analysis for collaborative navigation presented in Sect. 3.2, which indicate that the selected optimal cooperative anchor aircraft should be far away from the GNSS satellite reference ring, are also applied here. Considering the differences in the characteristics of the satellite pseudorange measurements and the collaborative measurements, the label aircraft establishes a collaboratively assisted observation model by means of the weighted least squares method. Finally, the label aircraft realizes geometric configuration optimization and obtains its optimized positioning results.

6.4 Geometry Optimization for Collaborative Resilient Navigation Fusion

131

Mode for Collaborative Partner Selection

Start

Mode for Optimial Sub-area Division

Airborne INS Measurement

Sphere Coordinate Frame Establishment for Label Member

Available Anchor Member Search in Sub-area

Airborne GNSS Receiver Measurement

Anchor-label Relative Angle Transformation

Optimal Collaborative Partner Selection

GNSS visibility and Anchor Member Analysis

Sphere Coordinates Mapping of GNSS satellites and Anchor Members

Relative Observations Modelling for Selected Partner

Reference Circle Fitting According to all Visible Satellites

Relative Observations and Covariance Update

Reference Ring Extension According to Extreme Elevations of Visible Satellites

CGDOP Update According to Current Collaborative Partnership

Optimal Sub-area Division According to Reference Ring

CGDOP Meet Requirement

Collaboration Available N Mode for Alarm Insufficient Observation

Y

N

Y Collaborative Resilient Fusion with Optimal Selected Partner

Navigation End? N Y End

Fig. 6.7 Workflow of geometry optimization based on the geometric analysis method

6.4.2 Geometry Optimization Based on an Algebraic Search Method By optimizing the configuration strategy of the anchor members to achieve the greatest possible enhancement of navigation performance while minimizing the manoeuvre distance of the anchor members, the optimal locations of the anchor

132

6 Collaborative Geometry Optimization in Resilient Navigation

members can be determined dynamically. Then, the anchor members can dynamically manoeuvre to their optimal locations to improve the effectiveness and efficiency of collaborative resilient fusion. The algebraic search method, in which the best anchor member manoeuvring strategy is determined by comparing performance indicators between different manoeuvring strategies, is an approach for realizing collaboration optimization. An example scenario of collaborative resilient fusion in a GNSS-denied situation involving an aerial swarm is shown in Fig. 6.8. In this example scenario, the hierarchical collaborative navigation structure is adopted. As shown in Fig. 6.8, the label members manoeuvre along their onlineplanned trajectories to execute a SAR task. The areas along the navigation segments that require navigation augmentation are taken as the ROIs. The need for navigation augmentation may arise due to the terrain blocking GNSS signals, a detailed search emphasis in a certain scope, and other factors. Then, the anchor aircraft broadcast their positions and uncertainties to the users, i.e., label members, in the current ROI. Through cooperative observation of the relative bearings with respect to the anchors, the navigation performance of the users in the ROI can be augmented. As the SAR task progresses, the ground vehicles leave the current ROI (ROI_1 in Fig. 6.8) and will eventually enter the next ROI (ROI_2 in Fig. 6.8). At this time, the configuration of the anchor aircraft needs to be dynamically optimized to provide the desirable navigation performance enhancement, and the optimized localizations to which the anchor aircraft should manoeuvre need to be dynamically adjusted. As discussed in Sect. 3.5, collaborative navigation augmentation is carried out by the anchor aircraft and the label members. The inertial sensor unit (ISU) of each label aircraft establishes its local rectangular coordinate system, in which the positions of Fig. 6.8 Scenario of collaborative resilient fusion in a GNSS-denied situation

Y Moving Trajectories L3 L2

ISP Key ISP

L1

X O ROI1

A1

A2

ROI2

6.4 Geometry Optimization for Collaborative Resilient Navigation Fusion

133

the anchor aircraft are represented. The bearing-only measurement equation is established in accordance with the relative LOS direction. Then, the relative observations are fused with the data from the INSs equipped on the label members to augment their navigation performance. For cooperative navigation, the manoeuvring of the anchor aircraft should be bounded to save energy while airborne. Thus, excessive manoeuvring distances of the anchor aircraft to achieve only a small improvement in the CGDOP should be avoided. A genetic algorithm is adopted to optimize the geometry of the anchor members, as discussed below. Considering the influential factors identified above, a penalty term is set for the CGDOP as follows: ( 0 s(k, k − 1) ≤ slim P Rj = (6.7) s(k, k − 1) s(k, k − 1) > slim I I where s(k, k − 1) = I r j (k) − r j (k − 1)I is the manoeuvring distance of the jth anchor member according to the configuration strategy and slim is the limit on the manoeuvring distance. For each geometric distribution of the anchor aircraft, a corresponding fitness function can be designed. Considering the enhancement in navigation performance and the constraint on the anchor aircraft manoeuvring distance, the optimal geometric distribution of the anchor aircraft can be found. Since the direction of evolution is towards an increase in fitness, to determine the optimal configuration with the smallest feasible CGDOP, the fitness function in the genetic algorithm is designed as follows: ({ Fitness =

n i=1

w Li C G D O Pi {n + i=1 w Li

{m

j=1 w A j P R j {m j=1 w A j

)−1 (6.8)

where w Li is the weight of the jth label member, where a greater weight indicates greater importance and attention in configuration optimization, and w A j is the weight of the jth anchor member, where a greater weight indicates greater importance and attention in manoeuvring distance reduction. The workflow of the geometry optimization process based on the algebraic search method is shown in Fig. 6.9. In this way, the problem of anchor aircraft configuration optimization for air– ground cooperative navigation is transformed into a CGDOP minimization problem. The anchor aircraft configuration is selected based on a genetic algorithm in accordance with the fitness function given in Eq. (6.8), following the principle that the higher the fitness is, the greater the probability of being selected. This strategy ensures that good chromosomes (which are related to favourable cooperative navigation performance) are more likely to be passed on to the offspring, making each generation perform increasingly better.

134

6 Collaborative Geometry Optimization in Resilient Navigation Setup Trajectories of Label Members Start

Setup ROI Setup Maneuver Limitation of Anchor Members

ROI Update Optimal Algorithm Setup

ISPs Sampling

Calculate CGDOP for ROI According to Current Actual Anchor Distribution

Arrange ISPs Weight and Anchors Weight Anchor Members' Optimal Distribution Search

Calculate CGDOP in a Anchor Distribution Scheme for ith ISP

Calculate Optimal CGDOP for ROI According to Optimal Anchor Distribution Scheme

All ISPs Calculated? Y

N

Calculate Punishment Item in a Anchor Distribution Scheme for jth Anchor Maneuver Distance

N

Anchor Members Maneuver

N N

All Anchors Calculated? Y Accumulate CGDOP for all the ISPs and Punishment Item for all the Anchors in a Anchor Distribution Scheme

Swarm Mission Accomplished?

Y Anchor Members Maneuver as Distribution with Optimal CGDOP

Y Anchor Members Keep Current Distribution End

Calculate Fitness of a Anchor Distribution Scheme

Search Optimal Anchor Distribution Scheme

Fig. 6.9 Workflow of geometry optimization based on the algebraic search method

6.5 Simulation Examples 6.5.1 Simulation of Geometric-Analysis-Based Collaboration Optimization 6.5.1.1

Simulation Setup

To verify the effectiveness of the collaborative configuration optimization algorithm, simulations of swarm with 10 members are carried out. The accuracy of the airborne relative ranger was 5 m and the flight time was 3600 s. Figure 6.10 shows the simulated flight trajectories of the members in the swarm. Figures 6.11, 6.12 and 6.13 shows the relative distances, elevation angles, and azimuth angles between member 1 and the other 9 member during the flight of the swarm as an example. Figure 6.14 shows the visibility of GNSS satellites during the flight of label members.

6.5 Simulation Examples

135

Fig. 6.10 Flight trajectories of the swarmed members

Fig. 6.11 Relative distances between member 1 and the other members

6.5.1.2

Analysis of the Optimal Collaboration Partner Selection Strategy

Simulation of the optimal collaboration partner selection strategy was carried out first, in which two examples are provided. The pseudocolour maps of the CGDOP reduction values after the introduction of information from a collaborative member at different positions are shown in Figs. 6.15 and 6.16.

136

6 Collaborative Geometry Optimization in Resilient Navigation

Fig. 6.12 Relative elevation between member 1 and the other members

Fig. 6.13 Relative azimuth between member 1 and the other members

From the pseudocolour maps of Figs. 6.15 and 6.16, we can see that for a collaborative member, the area where the reduction in the CGDOP is small (the blue area) approximately forms a ring, and the area where the reduction in the CGDOP is large (the yellow area) is either outside or inside this circle. These results show that when the geometric configuration of the satellites is poor, the introduction of collaborative

6.5 Simulation Examples

137

information from the proper position can result in a good configuration optimization effect.

Fig. 6.14 Visibility of GNSS satellites during the flight of label members

Fig. 6.15 Reductions in the CGDOP when the collaboration partner is at different locations (Case 1)

138

6 Collaborative Geometry Optimization in Resilient Navigation

Fig. 6.16 Reductions in the CGDOP when the collaboration partner is at different locations (Case 2)

6.5.1.3

Analysis of the Influence of the Equivalent Ranging Accuracy

The influence of the equivalent ranging accuracy was also analysed. The pseudocolour maps of the reductions in the CGDOP when the equivalent ranging error is 3 and 15 m are shown in Figs. 6.17 and 6.18. The statistical results under the different equivalent ranging errors are shown in Fig. 6.19. As seen from Figs. 6.17 and 6.18, as the relative ranging error increases, the centre position of the satellite reference ring remains basically unchanged, but the area of the pseudocolour map in which the CGDOP reduction is small (the blue area) increases. These simulation results show that as the equivalent ranging error increases, the optimal collaborative area becomes smaller. From Fig. 6.19, we can see that as the equivalent ranging error increases, the margin of the CGDOP reduction decreases. This shows that as the ranging error increases, the degree of optimization that can be achieved decreases. Moreover, the area proportions for which the CGDOP is in the three selected ranges also rapidly decrease. This shows that when the observation accuracy is higher, the range from which optimal member can be selected is wider, and the effect of configuration optimization is better.

6.5.1.4

Analysis of Collaboration-Augmented Navigation Performance

On the basis of the validation of the partnership optimization strategy, a dynamic simulation of collaboration-augmented navigation with CGDOP optimization for an

6.5 Simulation Examples

139

Fig. 6.17 Reductions in the CGDOP with collaborative augmentation (Equivalent Ranging Error 3 m)

Fig. 6.18 Reductions in the CGDOP with collaborative augmentation (Equivalent Ranging Error 15 m)

140

6 Collaborative Geometry Optimization in Resilient Navigation

Fig. 6.19 Area proportions to certain CGDOP levels under different equivalent ranging errors

swarmed member was conducted. Figure 6.20 shows the optimal selected anchor members in the swarm for label member 1 during flight as an example. Figure 6.21 shows the geometric configurations for label member 1 at 1200 s, as an example. As shown in Fig. 6.21, a label member in the swarm can select an optimal collaborative anchor member at any time in accordance with the continuous changes in the geometric configuration of the satellites. In each case, the selected member is far away from the satellite reference ring and allows a good geometric configuration to be achieved.

Fig. 6.20 Optimal selected anchor members for label member 1 during flight

6.5 Simulation Examples

141

Fig. 6.21 Geometric configuration for label member 1 on epoch 1200 s

In order to evaluate the collaborative navigation performance of the member swarm, we define some statistical performance indicator. The number of members with CGDOP less than 5 is calculated and used to evaluate the geometric configuration of aircraft swarm. And the mean value of Root Mean Square Error (RMSE) position error of swarmed members is calculated during the flight. Comparison of navigation with different collaborative augmentation method, including the value of CGDOP and RMSE is shown in Figs. 6.22, 6.23 and 6.24. As shown in Fig. 6.23, introducing collaborative augmentation is helpful for improving the navigation performance of the whole swarm. Compared with the traditional random member augmentation, the optimal member collaborative augmented navigation method could provide better positioning accuracy for aerial swarm. And Fig. 6.22 Comparison of CGDOP of label member 1 with and without collaborative partner selection strategy

142

6 Collaborative Geometry Optimization in Resilient Navigation

Fig. 6.23 Comparison of average CGDOP of the aerial swarm with different collaborative partner selection strategy

Fig. 6.24 Comparison of position error with different collaborative partner selection strategy

the proportion of members that meet the performance indicators can be greatly increased, thereby improving the overall navigation performance.

6.5.2 Simulation of Algebraic-Search-Based Collaboration Optimization 6.5.2.1

Simulation Setup

The parameters in these simulations were established as follows to verify the proposed method. The drift of the gyroscope was 0.1 deg/h, the bias of the accelerometer was g * 10–4 , and the standard deviation of the angular measurement noise was 0.1 deg.

6.5 Simulation Examples

143

In this evaluation, 100 Monte Carlo simulations were performed to analyse the positioning error of the label members. The sampling points were evenly distributed along the flight paths, and the weight ratio between ordinary sampling points and key sampling points in the ROIs was 1:100. For comparison, simulations without configuration optimization were also carried out.

6.5.2.2

Analysis of Cooperative Geometry Optimization

Figure 6.25 shows the manoeuvring of the anchor aircraft, the trajectories of the label members to be assisted, and the CGDOP distributions along the trajectories after configuration optimization. As seen in Fig. 6.25, the dynamic configuration optimization method keeps the CGDOP at a low level throughout the task area, and the CGDOP is smaller in the ROIs. This is because the method makes the positioning precision higher at sampling points within the ROIs, which have a higher weight, at the expense of making the CGDOP outside the ROIs larger. Figures 6.26 and 6.27 show curves representing the variations in the relative distances and relative included angles, respectively, between the anchor aircraft and the label members to be assisted for two anchor members after configuration optimization. The aircraft to be assisted are in ROI1 during the period of 100–200 s and in ROI2 during the period of 400–500 s.

Fig. 6.25 CGDOP distributions along the label members’ trajectories and ROIs

144

6 Collaborative Geometry Optimization in Resilient Navigation

Fig. 6.26 Relative distances between anchor and label members

Fig. 6.27 Relative angles between label and anchor members

As seen in Figs. 6.26 and 6.27, with configuration optimization, the relative distances in the ROI tend to decrease, and the relative included angles tend to be perpendicular. These findings demonstrate that the configuration is optimal when the bearing between the observer and the target is perpendicular and when the observer is closer to the target.

6.5.2.3

Analysis of Cooperation-Augmented Navigation for Dynamic ROIs

On the basis of the cooperative geometry optimization analysis, cooperationaugmented navigation for dynamic ROIs was carried out. Figure 6.28 compares the positioning errors with and without configuration optimization. Figure 6.29 shows the position RMSEs achieved with different bearing measurement errors.

6.5 Simulation Examples

145

Fig. 6.28 Comparison of position errors of label members before and after optimization

Fig. 6.29 Position RMSEs with different bearing measurement errors

As seen in Fig. 6.28, the configuration optimization method reduces the positioning errors of the label members throughout the whole navigation period, although it sacrifices the precision in some periods in exchange for improving the precision in the ROIs. The magnitude of this sacrifice can be controlled by setting the weights of the sampling points. According to the discussion in Sect. 6.3.2, the larger the bearing measurement error is, the more obvious the amplification of the positioning error in terms of the CGDOP, and the more the configuration optimization method can reduce the navigation error. As seen in Fig. 6.29, when high-precision bearing measurements cannot be achieved in long-distance operation, the configuration optimization method is of great significance for improving the navigation precision. In addition, the improvement in navigation precision in the ROIs is significantly greater than that along the whole flight path, indicating the effectiveness of the proposed method.

146

6 Collaborative Geometry Optimization in Resilient Navigation

6.6 Conclusions To address the problem of a poor geometric configuration of GNSS satellites or the blockage of GNSS signals for the navigation of a swarm member in a complex environment, this chapter proposes collaborative geometry optimization for resilient navigation based on two methods: a geometric analysis method and an algebraic search method. First, a partner selection method for aerial swarms is proposed based on a geometric analysis method, in which a GNSS satellite reference ring is established to analyse the existing geometric distribution of satellites and design an optimal collaborative subarea division and partner selection strategy. Simulation results show that the proposed method can be used to effectively evaluate the optimal collaborative area and select an optimal anchor member in the swarm to assist any label member in the swarm, reducing the GDOP and positioning error compared to those achieved by relying on the GNSS only and significantly improving the navigation performance. Considering the requirements of navigation precision in real-time ROIs, a dynamic aircraft swarm configuration optimization method based on the CGDOP is also proposed. The optimized CGDOP distribution is analysed, and the levels of positioning precision inside and outside the ROI with and without configuration optimization are compared. The influence of the bearing measurement accuracy on the configuration optimization method is also analysed by comparing the position RMSEs under different measurement accuracies. The simulation results and analysis show that the proposed configuration optimization method for collaborative navigation in aerial swarms can dynamically adjust the positions of the reference aircraft in accordance with changes in the ROI and markedly improve the positioning precision of the aircraft to be assisted in the ROI. Notably, the precision improvement effect is better under a larger measurement error. Finally, a cooperative navigation method based on an algebraic search method relying on genetic optimization is proposed for dynamic air–ground navigation augmentation. The CGDOP distributions inside and outside the ROI with and without configuration optimization are compared. The simulation results and analysis show that the proposed method can optimize the manoeuvring of the anchor aircraft in accordance with variable ROIs, improving the positioning precision of the label members to be assisted in a SAR task. On the basis of the methods proposed in this chapter, additional constraint conditions could be introduced for optimization under a certain scenario in order to realize favourable navigation performance for different tasks.

References

147

References 1. Yang J, You X, Wu G, Hassan MM, Almogren A, Guna J (2019) Application of reinforcement learning in UAV cluster task scheduling. Future Gener Comput Syst Int J Esci 95:140–148. https://doi.org/10.1016/j.future.2018.11.014 2. Duan H, Yang Q, Deng Y, Li P, Qiu H, Zhang T, Zhang D, Huo M, Shen Y (2019) Unmanned aerial systems coordinate target allocation based on wolf behaviors. Sci China-Inf Sci 62(1). https://doi.org/10.1007/s11432-018-9587-0 3. Zimmermann F, Eling C, Klingbeil L, Kuhlmann H (2017) Precise positioning of Uavs— dealing with challenging RTK-GPS measurement conditions during automated UAV flights. In: International conference on unmanned aerial vehicles in geomatics, Bonn, Germany. ISPRS annals of the photogrammetry remote sensing and spatial information sciences, pp 95–102. https://doi.org/10.5194/isprs-annals-IV-2-W3-95-2017 4. Vetrella AR, Fasano G, Ieee (2016) Cooperative UAV navigation under nominal GPS coverage and in GPS-challenging environments. In: 2nd IEEE international forum on Research and Technologies for Society and Industry leveraging a better tomorrow (RTSI), Bologna, Italy, pp 313–317 5. Nieuwenhuisen M, Droeschel D, Beul M, Behnke S, Ieee (2015) Autonomous MAV navigation in complex GNSS-denied 3D environments. In: IEEE international symposium on safety, security, and rescue robotics, west lafayette, IA 6. Schleich D, Beul M, Quenzel J, Behnke S (2021) Autonomous flight in unknown GNSS-denied Environments for disaster examination. In: 2021 international conference on unmanned aircraft systems. https://doi.org/10.1109/icuas51884.2021.9476790 7. Sarras I, Marzat J, Bertrand S, Piet H (2018) Collaborative multiple micro air vehicles’ localization and target tracking in GPS-denied environment from range-velocity measurements. Int J Micro Air Veh 10(2):225–239. https://doi.org/10.1177/1756829317745317 8. Lee JY, Kim HS, Choi KH, Lim J, Chun S, Lee HK (2016) Adaptive GPS/INS integration for relative navigation. GPS Solutions 20(1):63–75. https://doi.org/10.1007/s10291-015-0446-4 9. Hedgecock W, Maroti M, Sallai J, Volgyesi P, Ledeczi A (2013) High-accuracy differential tracking of low-cost GPS receivers. In: 11th annual international conference on mobile systems, applications, and services, Mobisys 2013, June 25, 2013–June 28, 2013, Taipei, Taiwan; Proceedings of the 11th annual international conference on mobile systems, applications, and services. association for computing machinery, pp 221–234. https://doi.org/10. 1145/2462456.2464456 10. Hardy J, Strader J, Gross JN, Gu Y, Keck M, Douglas J, Taylor CN (2016) Unmanned aerial vehicle relative navigation in GPS denied environments. In: IEEE/ION position, location and navigation symposium, PLANS 2016, April 11, 2016–April 14, 2016, Savannah, Georgia; Proceedings of the IEEE/ION position, location and navigation symposium, PLANS 2016. Institute of Electrical and Electronics Engineers Inc., pp 344–352. https://doi.org/10.1109/ PLANS.2016.7479719 11. Strader J, Gu Y, Gross JN, De Petrillo M, Hardy J (2016) Cooperative relative localization for moving UAVs with single link range measurements. In: IEEE/ION position, location and navigation symposium, PLANS 2016, April 11, 2016–April 14, 2016, Savannah, Georgia, 2016; Proceedings of the IEEE/ION position, location and navigation symposium, PLANS 2016. Institute of Electrical and Electronics Engineers Inc., pp 336–343. https://doi.org/10. 1109/PLANS.2016.7479718 12. Xu Z, Naigang C, Xiaogang W, Hutao C (2017) The vision-based relative navigation using improved adaptive cubature huber-based filtering. In: 21st AIAA international space planes and hypersonics technologies conference, 10 pp–10 pp 13. Gross JN, Gu Y, Rhudy MB (2015) Robust UAV relative navigation with DGPS, INS, and peerto-peer radio ranging. IEEE Trans Autom Sci Eng 12(3):935–944. https://doi.org/10.1109/tase. 2014.2383357

148

6 Collaborative Geometry Optimization in Resilient Navigation

14. Oh S-M, Johnson EN (2007) Relative motion estimation for vision-based formation flight using unscented kalman filter. In: AIAA Guidance, navigation, and control conference 2007, August 20, 2007–August 23, 2007, Hilton Head, SC, United states, 2007; Collection of technical papers—AIAA guidance, navigation, and control conference 2007. AIAA international, pp 5365–5381. https://doi.org/10.2514/6.2007-6866 15. Guo K, Qiu Z, Meng W, Xie L, Teo R (2017) Ultra-wideband based cooperative relative localization algorithm and experiments for multiple unmanned aerial vehicles in GPS denied environments. Int J Micro Air Veh 9(3):169–186. https://doi.org/10.1177/1756829317695564 16. Zhu Y, Sun Y, Zhao W, Wu L (2020) A novel relative navigation algorithm for formation flight. Proceed Inst Mech Eng Part G-J Aerosp Eng 234(2):308–318. https://doi.org/10.1177/095441 0019866060 17. Dehghani MA, Menhaj MB (2018) Stability of cooperative unmanned aerial vehicles based on relative measurements. Proceed Inst Mech Eng Part G J Aerosp Eng 232(15):2784–2792. https://doi.org/10.1177/0954410017716477 18. Fosbury AM, Crassidis JL (2008) Relative navigation of air vehicles. J Guid Control Dyn 31(4):824–834. https://doi.org/10.2514/1.33698 19. Wang Y, Yang S, Hu H, Chen K, Ji Q, Ieee (2012) Relative navigation algorithm based on robust filter for UAV formation flight. In: International conference on control engineering and communication technology (ICCECT), Shenyang, Peoples R China, 2012 Dec 07–09 2012, pp 249–252. https://doi.org/10.1109/iccect.2012.178 20. Duan H, Luo Q, Ieee (2016) Integrated localization system for autonomous unmanned aerial vehicle formation flight. In: 12th IEEE International Conference on Control and Automation (ICCA), Kathmandu, NEPAL, 2016. Jun 01–03 2016. IEEE International Conference on Control and Automation ICCA, pp 395–400 21. Irigireddy S, Moncayo H (2020) Vision based relative navigation for close-formation flight missions. AIAA Scitech 2020 Forum 22. Ding M, Wei L, Wang B (2011) Vision-based estimation of relative pose in autonomous aerial refueling. Chin J Aeronaut 24(6):807–815. https://doi.org/10.1016/s1000-9361(11)60095-2 23. Qu Y, Wu J, Xiao B, Yuan D (2017) A fault-tolerant cooperative positioning approach for multiple UAVs. IEEE Access 5:15630–15640. https://doi.org/10.1109/ACCESS.2017.273 1425 24. Qu Y, Zhang Y (2011) Cooperative localization against GPS signal loss in multiple UAVs flight. J Syst Eng Electron 22(1):103–112. https://doi.org/10.3969/j.issn.1004-4132.2011.01.013 25. Vetrella AR, Fasano G, Accardo D, Ieee (2016) Cooperative navigation in GPS-challenging environments exploiting position broadcast and vision-based tracking. In: International Conference on Unmanned Aircraft Systems (ICUAS), Arlington, VA 26. Jun 07–10 2016. International conference on unmanned aircraft systems, pp 447–456 27. Vetrella AR, Opromolla R, Fasano G, Accardo D, Grassi M (2017) Autonomous flight in GPSchallenging environments exploiting multi-UAV cooperation and vision-aided navigation. In: AIAA information systems—AIAA Infotech@Aerospace 28. Shen J, Wang S, Zhai Y, Zhan X (2021) Cooperative relative navigation for multi-UAV systems by exploiting GNSS and peer-to-peer ranging measurements. IET Radar Sonar Navig 15(1):21– 36. https://doi.org/10.1049/rsn2.12023 29. Causa F, Vetrella AR, Fasano G, Accardo D, Ieee (2018) Multi-UAV formation geometries for cooperative navigation in GNSS-challenging environments. In: IEEE/ION position, location and navigation symposium (PLANS), Monterey, CA, pp 775–785 30. Sivaneri VO, Gross JN (2017) UGV-to-UAV cooperative ranging for robust navigation in GNSS-challenged environments. Aerosp Sci Technol 71:245–255. https://doi.org/10.1016/j. ast.2017.09.024 31. Shen F, Cheong JW, Dempster AG (2017) A DSRC Doppler/IMU/GNSS tightly-coupled cooperative positioning method for relative positioning in VANETs. J Navig 70(1):120–136. https:// doi.org/10.1017/s0373463316000436

Chapter 7

Collaborative Integrity Augmentation in Resilient Navigation

Abstract Navigation integrity is critical for the reliability of aerial swarms. The geometric configuration of the selected partners for collaboration has a significant impact on the effectiveness and efficiency of the collaborative integrity enhancement. In this chapter, an improved protection level is introduced to quantitatively evaluate navigation integrity in collaborative resilient navigation fusion. The influence of geometry on integrity in collaborative resilient navigation fusion is discussed in both the GNSS-augmented and GNSS-denied situations. Integrity optimization algorithms based on the geometric analysis method and the algebraic search method are proposed, with simulated examples. Keywords Collaborative navigation · Integrity augmentation · Geometric dilution of precision · Protection level · Geometric analysis · Algebraic search · GNSS-augmented · GNSS-denied · Resilient fusion

7.1 Introduction Aerial swarm shows enormous potential for various applications, including real-time multi-aircraft monitoring, search and rescue, precision agriculture, infrastructure inspection, and so on [1–3]. Accurate positioning is essential for aerial swarm, but the navigation performance of GNSS is seriously influenced by the working environment [4]. The integrity of the navigation system is critical for flight safety and the completion of tasks [5]. With sufficient redundant observation form enough pseudorange measurements with a favourable geometry, the receiver autonomous integrity monitoring technique could detect a failure that occurs in GNSS pseudorange and guarantee the error bound [6]. The protection level is used to evaluate the reliability of positioning results. When the observed GNSS satellites are in poor geometrical configuration, the protection level would show a large value, indicating that the failure in pseudorange is insignificant to be detected from current redundant observation, and the receiver autonomous integrity monitoring strategy could not exclude minor failure to guarantee the high positioning accuracy. More seriously, when the

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_7

149

150

7 Collaborative Integrity Augmentation in Resilient Navigation

number of observed GNSS satellites is insufficient, the receiver autonomous integrity monitoring strategy would lose efficacy and the failure could hardly be detected [7, 8]. Introducing collaborative observation from other members in the swarm shows the potential in augmenting navigation integrity [9, 10]. The collaboration among swarmed aircraft can provide additional observation to improve the integrity of its onboard navigation systems, especially in the situation where the observed GNSS satellites are insufficient. Reference [11] introduced the method of sharing information between receivers for integrity monitoring to predict the local degradation of GNSS signals. Reference [12] uses information from multiple cooperative aircraft to monitor the health of the navigation system. Reference [13] proposes a cooperative monitoring algorithm for distributed synergistic SLAM (Simultaneous localization and mapping) in the city GNSS-challenging environment, and the introduction of collaborative observation information in combination with swarm aircraft needs to be re-evaluated and improved. Reference [14] utilizes the observation data of cooperative aircraft to form fault detection statistics, and calculates the radial protection level under different aircraft numbers to effectively reduce the leak detection rate of fault detection. Although collaboration among swarm aircraft shows potential in navigation integrity augmentation, adopting all the observations between aircraft in a swarm would raise burdens of relative communication as well as real-time measuring, which is inefficient for navigation systems [15, 16]. Furthermore, the geometry of the collaborated partners is a key factor that influences the effectiveness of the navigation integrity augmentation, and introducing a collaborative observation that composes poor geometrical configuration may be helpless for improving the significance of failures as well as the integrity protection level [17, 18]. So the collaborative partnership needs to be optimized to provide fewer but most helpful collaborative observations for collaborative integrity augmented navigation. In this chapter, an integrity-augmented navigation method for aerial swarm based on collaborative partner optimization is proposed. The geometry constructed by the GNSS satellites and the cooperative partners in the aerial swarm are analysed dynamically, and the augmented integrity protection levels with different collaborative relationships are predicted. Then the collaborative partner, which makes a key contribution to integrity augmentation, is adopted in collaborative navigation integrity monitoring, improving the navigation integrity efficiently.

7.2 Integrity in Collaborative Resilient Navigation Fusion

151

7.2 Integrity in Collaborative Resilient Navigation Fusion 7.2.1 Geometry in Collaborative Navigation Integrity Augmentation An aircraft swarm includes multiple aircraft in cooperation and provides the potential for navigation integrity augmentation. An example of cooperative navigation involving an aircraft swarm is shown in Fig. 7.1. As Fig. 7.1 shows, when the protection level provided by the GNSS cannot meet the requirements of navigation performance, cooperative observations relative to other anchor aircraft could be introduced for navigation integrity augmentation. In order to meet the cooperative navigation integrity requirements, there is a position protection zone for each member, i.e., the practical significance of the integrity protection level indicator provided to guarantee that the cooperative navigation system meets the probability of failure detection. The anchor member is able to provide a good level of position integrity protection for itself due to the high number of available satellites. The label member, on the other hand, has fewer available satellites and is of a poorer configuration, so the horizontal and vertical protection levels provided by itself alone are greater than the corresponding Horizontal Alarm Limit (HAL) or Vertical Alarm Limit (VAL) and are outside the position protection area. If the fault is not detected and identified, the navigation integrity is poor. In this case, the level of cooperative integrity protection is obtained by the residual test method through GNSS Satellites

Acceptable Error

Actual Position

Fault Identification

Anchor Member 1

2VAL Label Member 1

Cooperative Observations

Positioning Solution without Cooperative Integrity Augmentation

HAL Protection Area for Positioning

Anchor Member 3

Anchor Member 2

Positioning Solution with Cooperative Integrity Augmentation

Fig. 7.1 The scenario of navigation integrity augmentation based on cooperation

152

7 Collaborative Integrity Augmentation in Resilient Navigation

the introduction of some of the cooperative observation information between the reference aircraft in a suitable position, which can be within the acceptable error range to improve the cooperative navigation system integrity and thus enhance the whole cluster navigation integrity monitoring capability.

7.2.2 Integrity Protection Level in Collaborative Navigation Collaborative Integrity Protection Level for the Hierarchical Collaboration Structure In the cooperative navigation integrity monitoring algorithm scheme for the hierarchical collaborative navigation structure, the label member of interest can be normalized for heteroskedasticity using the weighted least squares approach, and the integrity protection level is calculated based on residual test theory. The collaborative positioning residual is defined as εi→ = yi→ − H i→ xˆ i

(7.1)

where xˆ i denotes the estimated states obtained in accordance with Eq. (4.3); then, substituting Eq. (4.3) into Eq. (7.1) yields T T εi→ = yi→ − H i→ (H i→ W i→ H i→ )−1 H i→ W i→ yi→ T T = (I − H i→ (H i→ W i→ H i→ )−1 H i→ W i→ ) yi→ T T = (I − H i→ (H i→ W i→ H i→ )−1 H i→ W i→ )(H i→ x i + vi→ ) T T = (I − H i→ (H i→ W i→ H i→ )−1 H i→ W i→ )vi→

/\ Si→ vi→

(7.2)

Then, the sum of the squares of the residual errors of collaborative positioning is C SS E i→ = ( yi→ − H i→ xˆ i )T W i→ ( yi→ − H i→ xˆ i ) T = εi→ W i→ εi→ T T = vi→ Si→ W i→ Si→ vi→

(7.3)

According to the chi-square distribution, when the sum of the squares of the pseudorange residuals is used as the detection statistic, we can obtain the following hypotheses: • H0: No fault occurs in the observations of the ith member of the aerial swarm. Then, E(vi→ ) = 0, and C SS E i→ ∼ χ 2 (m i + n i − oi ). • H1: Faults occur in observations of the ith member of the aerial swarm. Then, E(vi→ ) /= 0, and C SS E i→ ∼ χ 2 (m i + n i − oi , λi ).

7.2 Integrity in Collaborative Resilient Navigation Fusion

153

Here, λi = E(C SS E i→ ) is the decentralization parameter, m i is the number of observations obtained from the GNSS satellites visible to the ith member, n i is the number of anchor members collaborating with the ith member of the aerial swarm, and oi is the dimensionality of the states to be estimated in the least squares algorithm and depends on the states chosen, as expressed in Eqs. (3.11), (3.48), or (3.63). When there is no fault, the system is in the normal state, and an alarm occurs; it is a false alarm. Given the false alarm rate PF A , we can obtain the following probability equation [19, 20]: { P(C SS E i→ < T ) =

T

f χ 2 (m i +ni −oi ) (x)d x = 1 − PF A

0

(7.4)

The detection threshold T for the statistically detectable quantity C SS E i→ can be obtained from the above formula. When a fault occurs, C SS E i→ should be higher than T ; if it is lower than T , then the fault will be missed. Given the missed alarm rate PM D , the following probability equation can be obtained [21]: {

T

P(C SS E i→ < T ) = 0

f χ 2 (m i +ni −oi ,λ) (x)d x = PM D

(7.5)

In the collaborative positioning of the ith member of the aerial swarm, the horizontal positioning error caused by a fault b j is C R P E i, j =

/

2 2 Ai→(1, j ) + Ai→(2, j ) b j

(7.6)

T T where Ai→ = (H i→ W i→ H i→ )−1 H i→ W i→ and the subscripts (1, j ) and (2, j ) represent the elements of the matrix in the first and second rows, respectively, and in the jth column. When the sum of the squares of the residual errors C SS E i→ is used as the test statistic, the characteristic slope of the association between the horizontal positioning error and the test statistic can be obtained for each observation: / 2 2 Ai→(1, j) + Ai→(2, j) S L O P E i, j = (7.7) Si→( j, j)

If the observation with the maximum characteristic slope does not produce a missed detection in the event of a fault, then the navigation system can guarantee a given probability of missed detection. On this basis, the horizontal positioning protection level of collaborative navigation (CHPL) is calculated as follows: ( ) / C H P L i = max S L O P E i, j · λi j

(7.8)

154

7 Collaborative Integrity Augmentation in Resilient Navigation

In the case that all label members’ integrity protection levels need to be evaluated and optimized globally, the following equation can be established: {n C H P L swar m =

wi C H P L i {n i=1 wi

i=1

(7.9)

where wi is the weight of the ith member of the aerial swarm in geometry evaluation. Collaborative Integrity Protection Level for the Parallel Collaboration Structure In the parallel structure, the positioning of the members of the collaborative union is carried out together, so the horizontal protection level (HPL) of the collaborative union can be calculated as a whole. According to the observation model for the parallel collaborative navigation structure given in Eqs. (3.18), (3.56), (3.71), or (3.85), the collaborative positioning residual in the parallel structure is ε swar m = yswar m − H swar m xˆ swar m

(7.10)

where xˆ i represents the estimated states obtained according to Eq. (5.21); then, substituting Eq. (5.21) into Eq. (7.10) yields T −1 T ε swar m = yswar m − H swar m (H swar m W swar m H swar m ) H swar m W swar m yswar m T −1 T = (I − H swar m (H swar m W swar m H swar m ) H swar m W swar m )vswarm

/\ Sswar m vswar m

(7.11)

Then, the sum of the squares of the residual errors is C SS E swar m = ( yswar m − H swar m xˆ swar m )T W swar m ( yswar m − H swar m xˆ swar m ) T = εi→ W swar m ε swar m T T = vswar m Sswar m W swar m Sswar m vswar m

(7.12)

According to the chi-square distribution, when the sum of the squares of the pseudorange residuals is used as the detection statistic, we can obtain the following hypotheses: • H0: No fault occurs in the observations of all members of the aerial swarm. Then, E(vswar m ) = 0, and C SS E swar m ∼ χ 2 (m swar m + n swar m − oswar m ). • H1: Faults occur in the observations of all members of the aerial swarm. Then, E(vswar m ) /= 0, and C SS E swar m ∼ χ 2 (m swar m + n swar m − oswar m , λswar m ). Here, λswar m = E(C SS E swar m ) is the decentralization parameter, m swar m is the number of observations obtained from visible GNSS satellites, n swar m is the number of relative observations between collaborating members in the aerial swarm, and oswar m is the dimensionality of the states of the collaborative union in the aerial swarm, which is estimated using the least squares algorithm depending on the states

7.3 Influence of Integrity in Collaborative Resilient Navigation Fusion

155

chosen and the composition of the collaborative union, as expressed in Eqs. (3.15), (3.52), or (3.67). When there is no fault, the system is in the normal state, and if an alarm occurs, it is a false alarm. Given the false alarm rate PF A , we can obtain a probability equation that is similar to Eq. (7.4). When a fault occurs, C SS E swar m should be higher than T ; if it is lower than T , the fault will be missed. Given the missed alarm rate PM D , a probability equation can be obtained that is similar to Eq. (7.5). In the collaborative positioning of the aerial swarm, the horizontal positioning error caused by a fault b j is C R P E swar m, j =

/

A2swar m(1, j ) + A2swar m(2, j) b j

(7.13)

T −1 T where Aswar m = (H swar m W swar m H swar m ) H swar m W swar m and the subscripts (1, j ) and (2, j) represent the elements of the matrix in the first and second rows, respectively, and in the jth column. When the sum of the squares of the residual errors C SS E swar m is used as the test statistic, the characteristic slope of the association between the horizontal positioning error and the test statistic can be obtained for each observation: / A2swar m(1, j) + A2swar m(2, j) S L O P E swar m, j = (7.14) Sswar m( j, j)

If the observation with the maximum characteristic slope does not produce a missed detection in the event of a fault, then the navigation system can guarantee a given probability of missed detection. On this basis, the CHPL is calculated as ( ) / C H P L swar m = max S L O P E swar m, j · λswar m j

(7.15)

7.3 Influence of Integrity in Collaborative Resilient Navigation Fusion 7.3.1 Influence of Integrity in Collaborative Resilient Fusion in a GNSS-Augmented Situation In this chapter, an integrity-augmented navigation method for aerial swarms based on collaborative partner optimization is proposed. The geometry constructed by the GNSS satellites and the cooperation partners in the aerial swarm is analysed dynamically, and the augmented integrity protection levels with different collaborative relationships are predicted. Then, the collaborative partners adopted in collaborative

156

7 Collaborative Integrity Augmentation in Resilient Navigation

navigation integrity monitoring make a key contribution to integrity augmentation, effectively improving the ability to guarantee navigation integrity. Principle of Collaborative Navigation Integrity Augmentation The evaluation and optimization of the integrity protection level metric in a swarm of aerial vehicles ensure the integrity protection capability of collaborative navigation and help to avoid the possibility of no significant improvement in integrity due to the introduction of poor collaborative observation information from the reference vehicles. The level of protection is determined by the characteristic slope of each observation, i.e., it is related to the accuracy of each observation and the geometric distribution the observations form together. This chapter investigates the optimization of the protection level in relation to the geometric distribution with respect to the vehicle to be assisted in order to improve the integrity monitoring capability of collaborative navigation. Figure 7.2 shows the fault detection distributions of the vehicle to be assisted under collaboration-assisted conditions compared to the fault detection distributions under independent positioning [22]. In the case of independent positioning, there are five satellites visible to the vehicle to be assisted, and the second satellite has the largest characteristic slope, S L O P E 2I nd . The observation with the largest characteristic slope of the anchor member has the smallest test statistic for the horizontal position error specified by the cooperative navigation system. In independent positioning, the HPL exceeds the HAL, and a missed detection will occur in the event of the failure of the second satellite; consequently, the integrity guarantee is poor in the case of independent positioning [23]. With the introduction of a suitable reference vehicle, the characteristic slope of the corresponding observation (S L O P E 2Co ) is reduced compared with the case of independent positioning, and the CHPL is less than the alarm limit, keeping the position deviation within the specified range. Horizontal Positioning Error

SLOPE2Ind

HPL HAL

Miss Alarm

SLOPE5Ind SLOPE2Co

CHPL

SLOPE5Co

0

T Tr

Tw

Test statistics

Fig. 7.2 Comparison of fault detection distributions under independent and collaborative positioning

7.3 Influence of Integrity in Collaborative Resilient Navigation Fusion

157

Generally, in addition to the characteristic slope, the level of integrity protection can also be described in terms of the amount of change in the accuracy factor, and the two methods have been shown to be equivalent. The magnitude of the change in the horizontal dilution of precision (HDOP) is defined as [24, 25] δ H D O Pj =

/

H D O P j2 − H D O P 2 = S L O P E j

(7.16)

where H D O P is the horizontal dilution of precision for all observations and H D O P j is the horizontal accuracy factor for the remaining observations after the removal of the jth observation of the label member. The problem of optimizing the level of integrity protection can thus be linked to the method of optimizing the configuration with respect to the vehicle to be assisted, as discussed in Chap. 6, where a synergistic regional distribution model for optimizing the level of protection was investigated in the standard spherical coordinate system of the vehicle to be assisted. As shown in Fig. 7.1, introducing an appropriate cooperative observation helps improve the navigation integrity protection level. The observation geometry is reconstructed due to the additional observation relative to the collaboration partner, which also influences the integrity protection level. To improve the integrity protection level, several conditions should be satisfied after the collaborative observation is introduced: the slopes of the characteristic curves of the satellites should be reduced, and the slopes of the characteristic curves of the anchor aircraft should also be smaller than the slopes of the characteristic curves of the observed satellites. Therefore, optimization of the cooperative partnership is essential for guaranteeing the effectiveness of navigation integrity monitoring, as will be discussed in the following sections. Geometric Influence on the Collaboration-Augmented Integrity As discussed above, the anchor members’ positions exert an important influence on the improvement of the HPL. The variation in the precision factor, δ H D O P j , is related to the geometric relationships among the label member, the anchor members, and the GNSS satellites. Consider first the case of a label member employing an anchor member for collaborative navigation, where HPL optimization is actually a problem of varying the maximum value of the change in the horizontal accuracy factor corresponding to a series of observations, as follows: δH PL = H PL − C H PL ( ) √ = H P L − max S L O P E j · λ j ( ) √ = H P L − max δ H D O P j · λ j

(7.17)

where H P L is the HPL under independent positioning of the label member and δ H D O P j is the amount of change in the horizontal accuracy factor for the jth observation of the label member. If an observation contains deviations, it will provide inaccurate position information if the rest of the measurements constitute a poor

158

7 Collaborative Integrity Augmentation in Resilient Navigation

Fig. 7.3 Optimal collaborative areas for navigation integrity augmentation (case 1)

observation geometry, making faults in the observations more difficult to detect and prone to missed detection [26, 27]. In this chapter, the protection level is optimized by “homogenizing” the geometric distribution to reduce the variation in the horizontal accuracy factor based on an evaluation of the collaborative area of the label member using the GNSS satellite reference circle introduced in Sect. 6.3.1. Because the protection level is directly influenced by the satellite with the highest variation in the horizontal accuracy factor, the optimal collaborative area for the protection level is determined by discarding this satellite before fitting the GNSS satellite reference circle. A schematic diagram of the poor collaborative region and the optimal collaborative regions for protection level optimization to be performed by the anchor member is shown in Fig. 7.3. The practical significance of δ H D O P j is that if the jth measurement is subject to a deviation and the geometric configuration of the other measurements is poor, the position information may be inaccurate, and detection of the failures will be difficult. Therefore, fault detection is most difficult for the satellite with the largest δ H D O P and easier for other satellites with smaller δ H D O P values. Based on this relation, it can be concluded that the effect of adding auxiliary aircraft near a satellite with a small δ H D O P j is limited, and the optimization effect on the HPL is generally weak. In other areas, including near the satellite with the largest S L O P E j (i.e., corresponding to the satellite for which a fault is most difficult to detect), adding auxiliary aircraft is more effective, and the optimization effect on the CHPL is better.

7.3 Influence of Integrity in Collaborative Resilient Navigation Fusion

159

Therefore, the least squares circle fitting method is also used to determine the optimal collaborative area for a given integrity protection level. Unlike the collaborative configuration optimization method, in which all satellites are used to fit the reference circle, it is necessary to select the most suitable satellites to fit the reference circle in CHPL optimization. Let the number of visible navigation satellites be n; then, the satellite with the largest δ H D O P j is removed, and the remaining (n − 1) satellites are used to fit the satellite reference circle. Then, the distances from these (n − 1) satellites to the centre of the circle are calculated, and the maximum and minimum distances are taken as the radii to construct the satellite reference ring, as shown in Fig. 7.5. Let the area inside the reference ring be D1 . D1 is the area with the poorest optimization effect for the integrity protection level; thus, auxiliary aircraft should not be selected in this area. The optimal collaborative areas lie inside the inner radius of the reference ring and outside the outer radius of the reference ring. It should be noted that sometimes the δ H D O P j values of two satellites are almost the same, and both are very large. In such a case, it will be difficult to detect a fault in either satellite, and two circles will need to be fitted to obtain two reference rings, as shown in Fig. 7.4. Specifically, the number of visible satellites is set to n. Then, the satellite with the largest δ H D O P j is removed, and the remaining (n − 1) satellites are fitted to obtain a reference ring D1 . Next, the satellite with the second largest δ H D O P j value is removed instead, and the remaining (n − 1) satellites are fitted to obtain a reference ring D2 . D1 and D2 are the areas with poor optimization effects, and auxiliary aircraft should not be selected in these areas. That is, the total area of poor coordination is D = D1 ∪ D2 . The optimal collaborative areas are outside this area.

Fig. 7.4 Optimal collaborative areas for navigation integrity augmentation (case 2)

160

7 Collaborative Integrity Augmentation in Resilient Navigation

7.3.2 Influence of Integrity in Collaborative Resilient Fusion in a GNSS-Denied Situation When observations relative to anchor members are adopted for positioning in a GNSS-denied environment, the anchor members’ geometry relative to the label member can be analysed to determine the influence on the navigation integrity. In this section, collaborative positioning based on relative-bearing-only observations is discussed as an example. It is known from Sect. 7.3.1 that the integrity of collaborative navigation is related to the relative geometry, which is influenced by several factors: the angular measurement precision, the geometric configuration, and the localization deviation of the anchor aircraft. Thus, given a certain angular measurement precision and localization deviation of the anchor aircraft, the navigation integrity can be improved by optimizing the configuration of the anchor aircraft. The relationship between the configuration and the positioning accuracy is shown in Fig. 7.5. It has been concluded in Sect. 6.3.2 that when the lines of sight (LOSs) from two anchor members to the label member are nearly orthogonal, the area of the positioning uncertainty is smaller, indicating that a higher positioning accuracy can be achieved through cooperative augmentation. In Fig. 7.5, when there is a fault bias in the relative observations of anchor members A1 and A2, the collaborative positioning results will indicate an incorrect position of L1. To enable the detection of the failure, more anchors should be added to provide redundant relative observations. In this case, when another anchor member is added, its range and bearing relative to the label member influence the navigation integrity protection level. Figure 7.5 also shows that when the added anchor member (such as A3) is nearly parallel with one of the existing anchor members (such as A2), it can provide little help in detecting a fault bias in A1/A2 in collaborative positioning. If A3 is added on the basis of A1/A2, Fig. 7.5 Effect of configuration on navigation integrity (bearing-only observations)

A2 A B L1(Wrong Pos)

C A1

L1(Actual Pos)

A4

Fault Bias Bearing measuremen t error

A3

D

7.4 Integrity Optimization in Collaborative Resilient Navigation Fusion

161

the incorrect position of L1 cannot be distinguished. This can also be explained by Eq. (7.17): when A3 is added on the basis of A1/A2, since A2 and A3 are nearly parallel, the influence on the HDOP in cooperative augmentation is low, which means that there is little impact on the integrity protection level. In contrast, when anchor member A4 is added, which is not parallel with any other existing anchor members relative to the label member, the incorrect position of the label member becomes evident, reflecting the influence of the fault bias in A1. According to the relationship between the HPL and the HDOP in Eq. (7.17), given a certain angle between two LOSs, the closer the distance is between an anchor member and the label member, the more significant the influence on the HDOP, leading to a greater impact on the integrity protection level. In addition, a larger localization deviation of an anchor member will reduce the weight of the corresponding observation in the positioning process, resulting in less impact on the integrity protection level. Thus, the influence of geometry on cooperative integrity augmentation is related to multiple factors, and their effects need to be evaluated quantitatively.

7.4 Integrity Optimization in Collaborative Resilient Navigation Fusion 7.4.1 Integrity Optimization Based on the Geometric Analysis Method An aerial swarm that includes multiple cooperating members has access to redundant observations for navigation integrity augmentation. An example scenario of collaborative resilient fusion in a GNSS-augmented situation involving an aerial swarm, similar to the scenario considered in Sect. 6.4.1, is discussed here with regard to the topic of navigation integrity optimization in collaborative resilient navigation fusion. In this example scenario, the hierarchical collaborative navigation structure is adopted. When the navigation integrity protection level of GNSS positioning for a label aircraft cannot meet the given requirement, an anchor aircraft can transmit collaborative observations to the label aircraft to assist in integrity augmentation. The workflow of collaborative navigation integrity augmentation for an aerial swarm is shown in Fig. 7.6. As shown in Fig. 7.6, the GNSS pseudoranges and the relative ranges obtained from the radio frequency (RF) module of the label member, which are used to optimally select an anchor member, are fused by means of navigation integrity augmentation algorithms. First, the label member uses its airborne radio system to obtain relative observations from other members and calculates the relative azimuth and elevation angles in accordance with the position and distance obtained by the airborne inertial navigation system (INS). Simultaneously, the mapping of the GNSS satellites and anchor members in the local spherical coordinate system is obtained. The visible satellites that make the largest contributions to the horizontal dilution of precision

162

7 Collaborative Integrity Augmentation in Resilient Navigation

Start

Mode for Optimial Sub-area Division

Airborne INS Measurement

Sphere Coordinate Frame Establishment for Label Member

Airborne GNSS Receiver Measurement

Anchor-label Relative Angle Transformation

Available Anchor Member Search in Sub-area

Optimal Collaborative Partner Selection

Sphere Coordinates Mapping of GNSS Satellites and Anchor Members

GNSS visibility and Anchor Member Analysis Collaboration Available

Mode for Collaborative Partner Selection

Y

N Mode for Alarm Integrity Protection Failed

Relative Observations Modelling for Selected Partner

Identify the Largest Contributors to HDOP from Visible Satellites

Relative Observations and Covariance Update

Reference Circle Fitting According to Visible Satellites Excluding the Largest Contributors to HDOP Reference Ring Extension According to Extreme Elevations of Visible Satellites Excluding the Largest Contributors to HDOP

CHPL Update According to Current Collaborative Partnership CHPL Meet Requirement N Y Collaborative Integrity Augmentation with Optimal Selected Partner

Optimal Sub-area Division According to Reference Ring

Navigation End?

N Y End

Fig. 7.6 Workflow of integrity augmentation based on the geometric analysis method

(HDOP) are identified. Then, the reference rings are estimated based on the visible satellites, excluding the largest contributors to the HDOP. Next, the label member uses a collaborative partner selection algorithm to identify the optimal anchor member for navigation integrity augmentation. Considering the difference between satellite pseudorange measurements and cooperative measurements, the least squares principle is used to overcome heteroskedasticity. An observation model of assisted satellite navigation for cooperative aircraft is established, and the cooperative integrity

7.4 Integrity Optimization in Collaborative Resilient Navigation Fusion

163

protection level is obtained based on residual detection theory. Finally, relative observations from the optimal selected collaborative partner are used to assist the satellite system in detecting and identifying faults. After the removal of a faulty observation, navigation results with higher integrity are obtained.

7.4.2 Integrity Optimization Based on the Algebraic Search Method By optimizing the integrity protection strategy of the aerial swarm to enhance the navigation integrity as much as possible while minimizing the manoeuvring distance of the anchor members, the optimal locations of the anchor members can be determined dynamically. Then, the anchor members can dynamically manoeuvre to their optimal locations, improving the integrity and efficiency of collaborative resilient fusion. The algebraic search method, in which the best anchor member manoeuvring strategy is determined by comparing performance indicators between different manoeuvring strategies, is one approach for realizing collaborative optimization. An example scenario of collaborative resilient fusion in a GNSS-denied situation involving an aerial swarm, similar to the scenario considered in Sect. 6.4.2, is discussed here with regard to the topic of navigation integrity optimization in collaborative resilient navigation fusion. In this example scenario, the hierarchical collaborative navigation structure is adopted. The configuration of the anchor aircraft needs to be optimized dynamically to achieve desirable enhancement of the navigation integrity, and the optimized locations to which the anchor members should manoeuvre need to be adjusted dynamically. For cooperative navigation, the manoeuvring of the anchor aircraft should be bounded to save energy while airborne. Thus, excessive manoeuvring distances of the anchor aircraft to achieve only a small improvement in the CHPL should be avoided. A genetic algorithm is adopted to optimize the geometry of the anchor members. Considering the influential factors identified above, a penalty term is set as defined in Eq. (6.7). For each geometric distribution of the anchor aircraft, a corresponding fitness function can be designed. Considering the enhancement in navigation performance and the constraint on the anchor aircraft manoeuvring distance, the optimal geometric distribution of the anchor aircraft can be found. Since the direction of evolution is towards an increase in fitness, to determine the optimal configuration with the smallest feasible CHPL, the fitness function in the genetic algorithm is designed as follows: ({ Fitness =

n i=1

w Li C H P L i {n + i=1 w Li

{m

j=1 w A j P R j {m j=1 w A j

)−1 (7.18)

164

7 Collaborative Integrity Augmentation in Resilient Navigation

where w Li is the weight of the ith label member, where a greater weight means greater importance and attention in integrity augmentation, and w A j is the weight of the jth anchor member, where a greater weight means greater importance and attention in manoeuvring distance reduction. The workflow of the geometry optimization process based on the algebraic search method is shown in Fig. 7.7. In this way, the problem of integrity augmentation for cooperative navigation is transformed into a fitness minimization problem. The anchor aircraft configuration is selected based on a genetic algorithm in accordance with the fitness function given in Eq. (7.18), following the principle that the higher the fitness is, the greater is the probability of being selected. This strategy ensures that good chromosomes (which are related to a favourable cooperative navigation integrity protection level) are more likely to be passed on to the offspring, making each generation perform increasingly better. It should also be noted that based on the discussions in Chap. 6 and this chapter, the problems of geometry optimization and integrity augmentation are different. A good geometric configuration for improving the positioning accuracy and reducing Setup Trajectories of Label Members Start

Setup ROI Setup Maneuver Limitation of Anchor Members

ROI Update Optimal Algorithm Setup

ISPs Sampling

Calculate CHPL for ROI According to Current Actual Anchor Distribution

Arrange ISPs Weight and Anchors Weight Anchor Members' Optimal Distribution Search

Calculate CHPL in a Anchor Distribution Scheme for ith ISP

N

Calculate Optimal CHPL for ROI According to Optimal Anchor Distribution Scheme

All ISPs Calculated? Y

Calculate Punishment Item in a Anchor Distribution Scheme for jth Anchor Maneuver Distance

N

Anchor Members Maneuver Y N

N

All Anchors Calculated? Y Accumulate CHPL for all the ISPs and Punishment Item for all the Anchors in a Anchor Distribution Scheme

Swarm Mission Accomplished?

Anchor Members Maneuver as Distribution with Optimal CHPL

Y Anchor Members Keep Current Distribution End

Calculate Fitness of a Anchor Distribution Scheme

Search Optimal Anchor Distribution Scheme

Fig. 7.7 Workflow of integrity augmentation based on the algebraic search method

7.5 Simulation Examples

165

the collaborative geometric dilution of precision (CGDOP) does not always improve the integrity and reduce the CHPL. Therefore, in the cooperative navigation of aerial swarms, the objectives of improving the accuracy and integrity need to be considered simultaneously, and a comprehensive fitness function combining Eqs. (6.8) and (7.18) should be constructed to obtain the globally optimal distribution of members.

7.5 Simulation Examples 7.5.1 Simulation Setup To verify the effectiveness of the cooperative integrity augmentation method proposed in this chapter, simulations were carried out. The positioning accuracy of the anchor members’ airborne navigation systems was 10 m. The equivalent ranging error of the airborne ranging sensor was 5 m. The number of members of the aerial swarm was 10, and the simulated flight time was 3600 s. The simulated flight traces of the aerial swarm are similar to those in Sect. 7.3.1, as shown in Figs. 6.10–6.13.

7.5.2 Simulation of Cooperative Partner Optimization By comparing the reduction in value between the CHPL after the introduction of collaborative augmentation and the original CHPL, the optimal collaborative area can be analysed, as shown in Figs. 7.8, 7.9, 7.10 and 7.11. In Figs. 7.8 and 7.10, the pseudocolour images show the areas of poor HPL improvement obtained via the traversal method as one or two blue rings, whereas areas of greater improvement are shown in yellow. When the cooperative anchor member is within a blue ring, the HPL decreases only slightly, or an increase could even occur. After the removal of the satellite with the largest δHDOP value, the position of the satellite ring fit based on the remaining satellites is roughly the same as that of the blue ring in the pseudocolour image. The degree of optimization of the HPL is different for different satellite distributions; the improvement in the HPL is poor in some areas and good in others. As shown in Figs. 7.9 and 7.11, compared with the value achieved without the assistance of an anchor member, the δHDOP of the original satellite with the largest δHDOP is greatly reduced with the assistance of the optimal anchor, and the δHDOP values of the other satellites are also smaller in this case; therefore, the HPL can be reduced to a large extent. In contrast, with assistance from the worst aircraft, the δHDOP of the original satellite with the largest δHDOP value does not notably decrease and may even increase.

166

7 Collaborative Integrity Augmentation in Resilient Navigation

Fig. 7.8 δHPL with the introduction of anchor members in different orientations (group 1)

Fig. 7.9 δHDOP with the introduction of assistance from different anchor members (group 1)

7.5.3 Simulation of Navigation Integrity Augmentation On the basis of the optimal collaborative area analysis, a simulation-based analysis of the protection level improvement was carried out. In the process of cooperative navigation, the optimal anchor member for improving the integrity protection level can be selected dynamically, as shown in

7.5 Simulation Examples

167

Fig. 7.10 δHPL with the introduction of anchor members in different orientations (group 2)

Fig. 7.11 δHDOP with the introduction of assistance from different anchor members (group 2)

Fig. 7.12. The δHDOP contributions before and after the introduction of assistance from anchor members are compared in Figs. 7.13 and 7.14. A comparison of the HPL and CHPL for navigation integrity is shown in Fig. 7.15. As seen in Figs. 7.12, 7.13 and 7.14, with assistance from the optimal anchor members, the δHDOP value of the original satellite with the largest δHDOP is greatly reduced, resulting in a meaningful contribution to improving the navigation protection level. As Fig. 7.15 shows, the largest δHDOP among the satellites is

168

7 Collaborative Integrity Augmentation in Resilient Navigation

Fig. 7.12 Optimal anchor member selection for CHPL improvement for label member 1

Fig. 7.13 δHDOP contributions without anchor members’ assistance

reduced with the anchor members’ assistance; consequently, the HPL of the system is greatly reduced, improving the ability to detect minor faults as well as the integrity of navigation.

7.6 Conclusions

169

Fig. 7.14 δHDOP contributions with anchor members’ assistance

Fig. 7.15 Comparison of HPL and CHPL for navigation integrity

7.6 Conclusions Collaboration among the members of an aerial swarm can make available additional observations to improve the integrity of their onboard navigation systems. For this purpose, partnership optimization is critical to ensure the effectiveness and efficiency of navigation integrity augmentation. This chapter proposes an integrity-augmented navigation method for aerial swarms based on collaborative partner optimization. The

170

7 Collaborative Integrity Augmentation in Resilient Navigation

results show that with the integrity augmentation level prediction strategy, the anchor member with the highest benefit for navigation integrity is selected to optimize the cooperative partnership. Furthermore, the effectiveness of collaborative partnership optimization for integrity augmentation is verified, illustrating the superiority of the proposed method compared with the traditional independent integrity framework in improving the integrity protection level and fault detection capability.

References 1. Shakhatreh H, Sawalmeh AH, Al-Fuqaha A, Dou Z, Almaita E, Khalil I, Othman NS, Khreishah A, Guizani M (2019) Unmanned aerial vehicles (UAVs): a survey on civil applications and key research challenges. IEEE Access 7:48572–48634. https://doi.org/10.1109/access.2019.290 9530 2. Yang J, You X, Wu G, Hassan MM, Almogren A, Guna J (2019) Application of reinforcement learning in UAV cluster task scheduling. Future Gener Comput Syst Int J Esci 95:140–148. https://doi.org/10.1016/j.future.2018.11.014 3. Chung S-J, Paranjape AA, Dames P, Shen S, Kumar V (2018) A survey on aerial swarm robotics. IEEE Trans Rob 34(4):837–855. https://doi.org/10.1109/tro.2018.2857475 4. Alam N, Kealy A, Dempster AG (2013) Cooperative inertial navigation for GNSS-challenged vehicular environments. IEEE Trans Intell Transp Syst 14(3):1370–1379. https://doi.org/10. 1109/tits.2013.2261063 5. Egea-Roca D, Seco-Granados G, Lopez-Salcedo JA (2017) Comprehensive overview of quickest detection theory and its application to GNSS threat detection. Gyroscopy Navig 8(1):1–14. https://doi.org/10.1134/S2075108717010035 6. Zabalegui P, De Miguel G, Perez A, Mendizabal J, Goya J, Adin I (2020) A review of the evolution of the integrity methods applied in GNSS. IEEE Access 8:45813–45824. https://doi. org/10.1109/ACCESS.2020.2977455 7. Gargiulo G, Leonardi M, Zanzi M, Varacalli G (2010) GNSS integrity and protection level computation for vehicular applications 8. Zhu N, Marais J, Betaille D, Berbineau M (2018) GNSS position integrity in urban environments: a review of literature. IEEE Trans Intell Transp Syst 19(9):2762–2778. https://doi.org/ 10.1109/tits.2017.2766768 9. Ducatelle F, Di Caro GA, Foerster A, Bonani M, Dorigo M, Magnenat S, Mondada F, O’Grady R, Pinciroli C, Retornaz P, Trianni V, Gambardella LM (2014) Cooperative navigation in robotic swarms. Swarm Intell 8(1):1–33. https://doi.org/10.1007/s11721-013-0089-4 10. Shen J, Wang S, Zhai Y, Zhan X (2021) Cooperative relative navigation for multi-UAV systems by exploiting GNSS and peer-to-peer ranging measurements. IET Radar Sonar Navig 15(1):21– 36. https://doi.org/10.1049/rsn2.12023 11. Margaria D, Falletti E, IEEE (2014) A novel local integrity concept for GNSS receivers in urban vehicular contexts. In: IEEE/ION Position, location and navigation symposium (PLANS), Monterey, CA, May 05–08 2014, pp 413–425 12. Yang L, Rife J, Inst N (2016) Estimating covariance models for collaborative integrity monitoring. In: 29th International technical meeting of the-satellite-division-of-the-institute-ofnavigation (ION GNSS+), Portland, OR, Sep 12–16 2016, pp 1103–1113 13. Bhamidipati S, Gao GX, Inst N (2019) Distributed cooperative SLAM-based integrity monitoring via a network of receivers. In: 32nd International technical meeting of the satellite-division-of-the-institute-of-navigation (ION GNSS), Miami, FL, Sep 16–20 2019, pp 2023–2034. https://doi.org/10.33012/2019.16882 14. Liu Y, Zhu Y, IEEE (2013) A collaborative integrity monitor algorithm for low space aviation under limited number of navigation satellites. In: 2nd International conference on connected

References

15.

16.

17.

18.

19.

20.

21. 22.

23.

24.

25. 26. 27.

171

vehicles and expo (ICCVE), Las Vegas, NV, Dec 02–06 2013, pp 910–915. https://doi.org/10. 1109/iccve.2013.133 Gross JN, Gu Y, Rhudy MB (2015) Robust UAV relative navigation with DGPS, INS, and peerto-peer radio ranging. IEEE Trans Autom Sci Eng 12(3):935–944. https://doi.org/10.1109/tase. 2014.2383357 Ansari K, Feng Y, Tang M (2015) A runtime integrity monitoring framework for real-time relative positioning systems based on GPS and DSRC. IEEE Trans Intell Transp Syst 16(2):980– 992. https://doi.org/10.1109/tits.2014.2349011 Causa F, Vetrella AR, Fasano G, Accardo D, IEEE (2018) Multi-UAV formation geometries for cooperative navigation in GNSS-challenging environments. In: IEEE/ION Position, location and navigation symposium (PLANS), Monterey, CA, Apr 23–26 2018, pp 775–785 Salos D, Martineau A, Macabiau C, Bonhoure B, Kubrak D (2014) Receiver autonomous integrity monitoring of GNSS signals for electronic toll collection. IEEE Trans Intell Transp Syst 15(1):94–103. https://doi.org/10.1109/tits.2013.2273829 Khanafseh S, Joerger M, Pervan B, Ieee (2012) Integrity risk of cycle resolution in the presence of bounded faults. In: IEEE/ION Position location and navigation symposium (PLANS), Myrtle Beach, SC, Apr 23–26 2012, pp 664–672 Min D, Kim Y, Lee J (2021) Position error analysis of carrier-based DGNSS systems under ephemeris fault conditions. J Position Navig Timing 10(4):263–269. https://doi.org/10.11003/ jpnt.2021.10.4.263 Rife J (2009) Influence of GNSS integrity monitoring on undetected fault probabilities for single and multiple fault events. Navig J Inst Navig 56(4):275–287 Blanch J, Walter T (2021) Fast protection levels for fault detection with an application to advanced RAIM. IEEE Trans Aerosp Electron Syst 57(1):55–65. https://doi.org/10.1109/taes. 2020.3011997 Gerbeth D, Martini I, Rippl M, Felux M, Inst N (2016) Satellite selection methodology for horizontal navigation and integrity algorithms. In: 29th International technical meeting of thesatellite-division-of-the-institute-of-navigation (ION GNSS+), Portland, OR, Sep 12–16 2016, pp 2789–2798 Zhang P (2019) Research on satellite selection algorithm in ship positioning based on both geometry and geometric dilution of precision contribution. Int J Adv Robot Syst 16(1). https:// doi.org/10.1177/1729881419830246 Jwo DJ (2001) Efficient DOP calculation for GPS with and without altimeter aiding. J Navig 54(2):269–279. https://doi.org/10.1017/s0373463301001321 Jiang Y, Wang J (2016) A new approach to calculate the horizontal protection level. J Navig 69(1):57–74. https://doi.org/10.1017/s0373463315000545 Kim E, Song J, Shin Y, Kim S, Son P-W, Park S, Park S (2022) Fault-free protection level equation for CLAS PPP-RTK and experimental evaluations. Sensors 22(9). https://doi.org/10. 3390/s22093570

Chapter 8

Collaborative Fault Detection in Resilient Navigation

Abstract Fault diagnosis is a key component of navigation integrity assurance for aerial swarms. The redundancy of a navigation system is enhanced with the introduction of observations from optimally configured collaborative partners, thus improving the capabilities of fault detection, identification, and isolation. In this chapter, fault detection, identification, and exclusion based on collaborative integrity augmentation are discussed. Multifault detection based on multiple hypothesis solution separations in collaborative resilient navigation fusion for an aerial swam is proposed, with simulated examples. Keywords Collaborative navigation · Fault detection · Fault identification and exclusion · Multiple hypothesis solution separation · Resilient fusion

8.1 Introduction During the flight of an aerial swarm, the fault diagnosis and fault tolerance of the navigation systems of the swarm members are important. A global navigation satellite system (GNSS) can provide users with accurate positioning information, but GNSS performance is severely limited in many of the typical working environments of aerial swarms, such as urban areas and canyons [1]. Meanwhile, the probability of the simultaneous failure of multiple sources of satellite observation information is also increased [2]. Such failure will not only reduce the positioning accuracy of an individual member but also affect the reliability of the collaborative information shared between members, thereby affecting the overall accuracy of collaborative navigation. In the process of aerial swarm navigation, fault detection, identification, and isolation are based on the guarantee of a cooperative navigation integrity requirement for each member. Traditional multifault detection algorithms mainly utilize the internal information of sensors to build fault identification models; such algorithms include the multiple hypothesis solution separation (MHSS) algorithm [3], the group separation (GS) algorithm [4], and the maximum likelihood ratio method [5]. As presented in [6], by calculating the spatial distances between different subsets of member pairs, multiple faults can be excluded without an iterative search for faulty signals. In [7], © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_8

173

174

8 Collaborative Fault Detection in Resilient Navigation

measures of reliability and separability (correlation coefficients between fault detection statistics) were used to measure the capability of fault detection and exclusion (FDE) algorithms. The problem of GNSS fault detection was described as a parametric quadratic programming (PQP) problem [8]. However, in the above methods, the detection sensitivity will be severely decreased in a GNSS-limited environment. Since the probability of multiple observations failing simultaneously is increased in a GNSS-challenged environment and traditional fault diagnosis methods mainly use a vehicle’s own sensor information to build the fault identification model, which will lead to reduced detection availability and detection errors when the amount of available observations is insufficient, the overall navigation accuracy will be seriously compromised by the occurrence of multiple faults. Information collaboration between vehicles provides additional redundant measurement information for a vehicle’s navigation system, which can assist in the detection and identification of multiple faults. This approach is expected to solve the problem of the difficulty of detecting multiple faults in complex environments and to extend the concept of fault-tolerant navigation from a single vehicle to multiple vehicles [9, 10]. In an aerial swarm, there is the potential for the availability of multiple observations between members, which can be shared and combined to implement the collaborative characterization and prediction of GNSS signal degradation [11]. A collaborative test statistic can be constructed to monitor the integrity of the system [12]. Recently, collaborative integrity monitoring (CIM) algorithms fusing ultra-wideband (UWB) ranging [13, 14] or dedicated short-range communications (DSRC) [15, 16] with GNSS technology have been applied in vehicle networks. The collaboration-enhanced receiver integrity monitoring (CERIM) algorithm uses multiple mobile GNSS receivers to detect common-mode satellite faults [17, 18]. For applications with multiple members, a hybrid collaborative positioning system (COPS) and Global Positioning System (GPS) fault-tolerant navigation algorithm was developed using a chi-square residual test scheme to accommodate GPS faults [19]. In [20], observations of the same satellite from different members were combined to form fault detection statistics and discriminate observation outliers. Nevertheless, the above techniques consider only the single-fault situation and are mostly used in vehicle networks. In an aerial swarm, when the protection level provided by the GNSS module of a label member cannot meet the requirements of navigation integrity, the CIM algorithm for the label member mainly consists of three parts: collaborative integrity protection level assessment, collaborative integrity protection level optimization, and collaborative fault detection and elimination. First, the label member calculates the slope of each satellite and estimates the best collaborative area by analysing its own GNSS satellite distribution, and it then selects the anchor member with the best position as its collaborative partner through a comprehensive selection strategy. Then, the relative observations between the label member and the selected anchor member are fused with the satellite pseudorange observations to construct a collaborative resilient navigation model, which takes the heteroskedasticity of the different observations into consideration and calculates the collaborative integrity protection level by using residual test theory, with the ultimate goal of improving the navigation integrity

8.2 Fault Detection Based on Integrity Augmentation in Collaborative …

175

protection level. The methods above have been discussed in Chap. 7. Finally, on the premise of ensuring the integrity of the collaborative navigation system and the availability of fault detection, this chapter discusses how fault detection and identification are carried out with the aid of the relative observations from the selected anchor member and how the fault-tolerant navigation results are output after eliminating the faulty observations. To enhance the fault tolerance of the navigation systems of swarm members in challenging environments, a collaborative fault-tolerant navigation method is proposed. In this method, a collaborative navigation model is built, and an improved MHSS algorithm is used to exclude faulty GNSS measurements. In short, the proposed method is designed to collaboratively perform multifault detection in navigation. It extends the concept of fault-tolerant navigation from a single member of the swarm to the collaborative multimember case. The collaboration-augmented framework is constructed by fusing information from other members through a fusion filter to support multifault detection. The collaboration-augmented MHSS algorithm can construct the necessary test statistic even under insufficient satellite availability, thus improving the applicability of multifault detection. The proposed method can be applied in collaborative navigation to guarantee the fault tolerance of the swarm members.

8.2 Fault Detection Based on Integrity Augmentation in Collaborative Resilient Navigation Fusion 8.2.1 Collaboration-Augmented Fault Detection Aerial swarms commonly operate in harsh environments where their measurements can fail, resulting in navigation system failure through the influence of propagation over intervehicle communication links during collaboration. Satellite pseudorange failures affect the reliability of the GNSS equipment of members receiving faulty satellite signals, leading to large positioning deviations. When observation information containing a deviation is received by a pending anchor member, this, in addition to its own impaired positioning accuracy, also causes detection or identification errors in the fault diagnosis results and a serious reduction in inspection sensitivity and accuracy. Collaboration-Augmented Fault Detection Under the Hierarchical Collaboration Structure In cooperation-augmented fault detection under the hierarchical collaborative navigation structure, the label members can be normalized for heteroskedasticity using the weighted least squares approach, and faults are detected based on residual test theory. After a label member has been optimized for its integrity protection level to obtain a guarantee of fault detection availability, the observations of the selected anchor member are further utilized to assist in the detection of navigation system

176

8 Collaborative Fault Detection in Resilient Navigation

faults. Under the assumption that there are no faults in the relative range, the posttest weight error of the residual sum of squares C SS E i→ is calculated in accordance with Eq. (7.3) for the ith member of the aerial swarm and is used as the test statistic for the navigation system: / C SS E i→ m i + n i − oi

σˆ i→ =

(8.1)

where m i is the number of observations obtained from the GNSS satellites visible to the ith label member of the aerial swarm, n i is the number of anchor members collaborating with the ith member, and oi is the dimensionality of the states to be estimated in the least squares algorithm, which depends on the states chosen, as expressed in Eqs. (3.11), (3.48), or (3.63). Given the false alarm rate PF A , the detection threshold T for the statistically detectable quantity C SS E i→ can be obtained in accordance with Eq. (7.4), and the detection limit for the test statistic σˆ i→ can be calculated as / σT =

T m i + n i − oi

(8.2)

In the navigation process, the real-time calculated test statistics σˆ i→ and σT are compared. If σˆ i→ > σT , then there is considered to be a fault in the current observations of the navigation system of the ith label member in the swarm. Collaboration-Augmented Single-Fault Detection for the Parallel Collaboration Structure In the parallel collaborative navigation structure, the positioning of all members of the collaborative union is carried out simultaneously, so the residuals of the collaborative union can be calculated as a whole, as shown in Eq. (7.10). After the collaborative union in an aerial swarm has been optimized for its integrity protection level to obtain a guarantee of fault detection availability, the relative observations are further utilized to assist in the detection of navigation system faults in the collaborative union. Under the assumption that there are no faults in the relative range, the posttest weight error of the residual sum of squares C SS E swar m is calculated in accordance with Eq. (7.12) for the members of the collaborative union and is used as the navigation system test statistic: / σˆ swar m =

C SS E swar m m swar m + n swar m − oswar m

(8.3)

where m swar m is the number of observations obtained from the visible GNSS satellites, n swar m is the number of relative observations between collaborating members in the aerial swarm, and oswar m is the dimensionality of the states of the collaborative union as estimated in the least squares algorithm, which depends on the states chosen and the composition of the collaborative union, as expressed in Eqs. (3.15), (3.52), or Eq. (3.67). Given the false alarm rate PF A , the detection threshold T for

8.2 Fault Detection Based on Integrity Augmentation in Collaborative …

177

the statistically detectable quantity C SS E swar m can be obtained in accordance with Eq. (7.4), and the detection limit for the test statistic σˆ i→ can be calculated as / σT =

T m swar m + n swar m − oswar m

(8.4)

In the navigation process, the real-time calculated test statistics σˆ swar m and σT are compared. If σˆ swar m > σT , then there is considered to be a fault in the current observations of the navigation systems of the members of the collaborative union in the aerial swarm.

8.2.2 Collaboration-Augmented Fault Identification and Exclusion Collaboration-Augmented Fault Identification and Exclusion for the Hierarchical Collaboration Structure It is necessary to reconstruct the detection statistics for satellite system fault identification. The residual vector is used to construct a detection statistic to determine whether gross error exists in a residual; this statistic can be expressed as follows [21, 22]: di, j = /

I I Iεi→( j ) I

(8.5)

W i→( j, j) Si→( j, j )

There are m i + n i statistic values in the corresponding observation model. Given the total false alarm rate PF A , the false alarm rate for a single statistic is PF A /(m i + n i ). Then [23, 24], (

P di, j > Td

)

2 = 2π

{∞ Td

x2

e− 2 d x =

PF A m i + ni

(8.6)

The detection limit Td can be obtained from the above formula. The corresponding statistic di, j for the jth observation is compared against the detection limit Td . If di, j > Td , then the jth observation is considered to be faulty and should be isolated from the observations of the ith member of the aerial swarm. Collaboration-Augmented Fault Identification and Exclusion for the Parallel Collaboration Structure It is necessary to reconstruct the detection statistics for satellite system fault identification. The residual vector is used to construct a detection statistic to determine whether gross error exists in a residual; this statistic can be expressed as follows:

178

8 Collaborative Fault Detection in Resilient Navigation

dswar m, j = /

I I Iε swar m( j ) I W swar m( j, j) Sswar m( j, j )

(8.7)

There are m swar m +n swar m statistic values in the corresponding observation model. Given the total false alarm rate PF A , the false alarm rate for a single statistic is PF A /(m swar m + n swar m ). Then, (

P dswar m, j > Td

)

2 = 2π

{∞ Td

x2

e− 2 d x =

PF A m swar m + n swar m

(8.8)

The detection limit Td can be obtained from the above formula. The corresponding statistic dswar m, j for the jth observation is compared against the detection limit Td . If dswar m, j > Td , then the observation is considered to be faulty and should be isolated from the observations of the collaborative union in the aerial swarm.

8.3 Fault Detection Based on MHSS in Collaborative Resilient Navigation Fusion 8.3.1 Multifault Detection Scheme with Collaborative Augmentation To realize collaborative resilient navigation fusion for the navigation systems of the members of an aerial swarm, this chapter designs a two-tier fault detection architecture for the swarm members and uses the fault-free cooperative observations to assist in navigation system fault detection. In an aerial swarm, the anchor members and the label members form a two-layer fault detection structure. In the first layer, all the anchor members use the traditional MHSS algorithm to detect and eliminate possible satellite faults of their own, obtain fault-free navigation results, and share their own positioning results with the label member, while the label members do not output erroneous navigation results due to the insufficiency of their own available measurements for detecting faults in their observation information. In the second layer of detection, each label member acquires the observation information from the fault-free anchor members in the previous layer and fuses it with its own insufficient satellite measurement information. Then, a multistage fusion filter is constructed through a cooperative MHSS algorithm to detect and eliminate multiple satellite observation faults. Finally, the label members output their own positioning results, and together with the results for the anchor members, fault-tolerant navigation results are obtained for the whole aerial swarm. This chapter introduces the use of collaborative ranging information and the MHSS algorithm to detect multiple faults. Figure 8.1 shows a schematic diagram of the process of multiple-fault detection.

8.3 Fault Detection Based on MHSS in Collaborative Resilient Navigation …

179

Navigation result

GNSS satellite n Anchor member 1

Sat 1,2

Sat n-1,n

dX1 dP1

Xu

Sub-filter u

dXu dPu

Sub-sub- X1,1 filter 1,1

dX1,1, dP1,1

Sub-sub- X1,s filter 1,s

dX1,s, dP1,s

Sat n-1,n, n-2

Sub-subfilter u,1

Sub-subfilter u,s

Xu,1

T1,s

dXu,1, dPu,1 du,s

Xu,s

dXu,s, dPu,s

Fault Identification

Sat n-1,n,1

Tu

d1,s

Anchor member m Sat 1,2,n

du

Normal

du > Tu Fault

Fault Exclusion

Sat 1,2,3

X1

Sub-filter 1

Fault Detection

GNSS satellite 1

X0

Main filter

SINS

du,s < Tu,s

Tu,s

Fig. 8.1 Scheme of multifault detection based on MHSS with cooperation

In this figure, the notation “/⊂ Sat” indicates “does not contain satellites”, that is, some satellite observations are excluded. There are three levels of filters in this scheme. The first level is the main filter level, which uses all observations of the satellites and cooperating members. The second level is the subfilter level, where each subfilter excludes the ath set of observations, which includes two satellites. The third level is the sub-subfilter level, at which the bth satellite is additionally excluded relative to the subfilter level. Then, the difference in the positioning solutions between the main filter and subfilter levels is used to perform the first layer of fault detection, and the difference in the positioning solutions between the subfilter and sub-subfilter levels is used to perform the second layer of faulty satellite identification. Filter States in MHSS For the layered filters for multifault detection as depicted in Fig. 8.1, the states are selected in accordance with the states to be estimated, as expressed in Eq. (5.2) for the hierarchical collaborative navigation structure or in Eq. (5.12) for the parallel collaborative navigation structure. Filter Observations in MHSS The filter observation model for the main filter is shown in Eq. (5.9) for the hierarchical structure and in Eq. (5.18) for the parallel structure. For convenience, the observations in the measurement equation for the main filter are represented as

180

8 Collaborative Fault Detection in Resilient Navigation

y0 = [ · · · y (u1) · · · y (u2) · · · ]T

(8.9)

For subfilter Fu , the observations used are equivalent to y0 excluding the pair of observations y (u1) and y (u2) ; thus, yu = [ · · · y (u1−1) y (u1+1) · · · y (u2−1) y (u2+1) · · · ]T

(8.10)

For sub-subfilter Fu,s , the observations are equivalent to yu excluding the observation y (s) ; thus, yu,s = [ · · · y (u1−1) y (u1+1) · · · y (u2−1) y (u2+1) · · · y (s+1) y (s+1) · · · ]T

(8.11)

Estimated States in MHSS In the filter structure of the improved MHSS algorithm, the system model of each subfilter and sub-subfilter has a similar form. Nevertheless, the observation models, which combine the GNSS pseudorange observations and the relative observations between members, will differ between these subfilters and sub-subfilters. The states estimated by the main filter, which utilizes all the observations of Eq. (8.9), are denoted by xˆ 0 . The states estimated by the subfilter that excludes the uth set of observations, as shown in Eq. (8.10), are denoted by xˆ u , and the states estimated by the sub-subfilter that excludes the sth observation in addition to the uth set of observations, as shown in Eq. (8.11), are denoted by xˆ u,s . Fault detection and identification can be realized by appropriately designing the strategy applied to compare the states estimated by the main filter, the subfilters, and the sub-subfilters, as will be discussed in the following sections.

8.3.2 Collaboration-Augmented Fault Detection In this work, the positioning solutions of the subfilters are obtained from n − 2 observations. The separation vector between the solutions of the main filter and the uth subfilter is d x u = xˆ 0 − xˆ u

(8.12)

where xˆ 0 is the estimate from the main filter and xˆ u is the estimate from the subfilter that excludes the uth set of satellite measurements. The number of subfilters is gu = Cn2 . The covariance matrix of the solution separation vector is d P u = E(d x u · d x Tu ) oss cr oss T = P 0 − P cr 0 |u − ( P 0 |u ) + P u

(8.13)

8.3 Fault Detection Based on MHSS in Collaborative Resilient Navigation …

181

oss where P 0 , P u , and P cr 0 |u represent the covariance matrices of the corresponding state vectors [19]. hpos The horizontal position solution separation vector is d x u = d x u [1 : 2]; accordingly, the test statistic is constructed as

du = |d x uhpos |

(8.14)

where | · | represents the modulus of a vector. The covariance matrix of the horizontal position solution separation vector is d P uhpos = d P u [1 : 2, 1 : 2]. Because d P uhpos is a nondiagonal matrix and the be the maximum eigenvalue of dominant element is the largest eigenvalue, let λmax u d P uhpos . For a given false alarm probability PF A , the detection threshold Ti corresponding to the test statistic du can be calculated as [25, 26] Tu =

/ λmax · er f −1 (1 − PF A /2gu ) u

(8.15)

where er f −1 ( ) is the inverse error function. Then, fault detection can be performed based on the a test statistics and the corresponding thresholds, with the following criteria: Normal: if du < Tu for any u, u = 1, 2, . . . , a; Fault: if du > Tu for at least one u, u = 1, 2, . . . , a.

8.3.3 Collaboration-Augmented Fault Identification and Exclusion Fault identification is performed based on the second-layer statistics. Similar to the above analysis, the separation vector between the solutions of a subfilter and one of its sub-subfilters is d x u,s = xˆ u − xˆ u,s

(8.16)

where xˆ u is the estimate from the uth subfilter and xˆ u,s is the estimate from the sub-subfilter that excludes the sth satellite observation relative to the corresponding subfilter. gu,s is the number of sub-subfilters under the uth subfilter, and gu,s = n −2. The covariance matrix of d x u,s is T d P u,s = E(d x u,s · d x u,s ) oss cr oss T = P u − P cr u |s − ( P u |s ) + P u,s

(8.17)

oss where P u , P u,s , and P cr u |s represent the covariance matrices of the corresponding state vectors.

182

8 Collaborative Fault Detection in Resilient Navigation

The horizontal position solution separation vector of each sub-subfilter is hpos d x u,s = d x u,s [1 : 2]; accordingly, the test statistic of each sub-subfilter is constructed as hpos du,s = |d x u,s |

(8.18)

where | · | represents the modulus of a vector. Similar to the calculation method for the threshold of a subfilter, the detection threshold Tu,s corresponding to the test statistic du,s can be calculated as Tu,s =

/ −1 λmax (1 − PF A /2gu,s ) u,s · er f

(8.19)

hpos hpos where λmax u,s is the maximum eigenvalue of the matrix d P u,s , d P u,s = d P u,s [1 : 2, 1 : 2]. Fault identification is performed to determine whether the remaining observations are free of faults. The ith set of satellites excluded in the subfilter is presumed to be faulty if the test statistics satisfy

du,s < T u,s , ∀ s, s = 1, 2, . . . , gu,s

(8.20)

Then, the fusion filter is reconstructed from the observation information of this subfilter, excluding the faulty observations, and the inertial navigation system (INS) is modified to obtain the final navigation result based on the fault-free observations. The workflow of fault detection based on MHSS is shown in Fig. 8.2.

8.4 Simulation Examples 8.4.1 Simulation of Single-Fault Detection in Collaborative Resilient Navigation Fusion 8.4.1.1

Simulation Setup

To compare the fault detection ability of the traditional satellite fault detection algorithm with that of the optimal auxiliary aircraft fault detection algorithm for different horizontal protection levels (HPLs), the residual test method was used for verification and analysis. The No. 23 satellite was assumed to have a 35 m bias failure in the pseudorange; specifically, a pseudorange fault was added for a visible satellite in different periods of the navigation process, as shown in Table 8.1.

8.4 Simulation Examples

183

GNSS and Relative Observing

GNSS Observation Modelling Relative Observation Modelling

Maximum Number of Detectable Faults Calculation

Sub-sub-filters Construction by Exclude One More Observation

Main Filter Construction by All Observations

Solution Separation Vector Calculation for Sub-sub-filters Detection Statistics Calculation for Subsub-filters

Sub-filter Construction by Exclude Observations

Observation Sufficiency Analysis Collaborative FDI Capability Available?

Mode for Fault Identification

Mode for Multifault Detection

Start

Detection Shoulder Calculation for Filters at All Levels

Solution Separation Vector Calculation for Sub-filters

Y

Analysis the Sub-subfilters in Normal and Corresponding Excluded Observation

Detection Statistics Calculation for Subfilters

N Mode for Alarm Fault Detection Failed

Identification of Faulty Observations

Detection Shoulder Calculation for Filters at All Levels

Faulty Observations Isolation Y

Any Statistics Exceed Shoulder? N Collaboration Resilient Fusion for Navigation

N

Navigation End? Y End

Fig. 8.2 Workflow of Fault Detection based on MHSS Table 8.1 Comparison of average HPL/CHPL during fault injection

Time (s)

HPL (m)

CHPL (m)

1

1000–1100

486

120

2

2500–2600

93

72

184

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.3 Test statistics for the traditional fault detection algorithm

8.4.1.2

Analysis of Fault Detection Performance

In accordance with the simulation conditions, the detection statistics and threshold values of the traditional fault detection algorithm and the fault detection algorithm with optimal anchor member collaboration in the navigation process were obtained, as shown in Figs. 8.3 and 8.4, respectively. As shown in Figs. 8.3 and 8.4, in the period of 2500–2600 s, both algorithms can detect and isolate the satellite fault, and the system positioning error remains unchanged. However, in the fault period of 1000–1100 s, the traditional fault detection algorithm cannot detect the satellite fault, and the position error of the satellite system notably increases, whereas the optimal auxiliary aircraft detection algorithm can still detect and eliminate the faulty satellite. Because the horizontal positioning protection level of collaborative navigation (CHPL) is large in the period of 1000–1100 s, missed detection may occur. Therefore, it is impossible to detect and identify faulty satellites in all cases. In this case, the CHPL value of the system is greatly reduced after the introduction of optimal aircraft assistance, and no alarm limits are exceeded.

8.4.1.3

Analysis of Fault-Tolerant Navigation Performance

The GNSS position error curves in two fault periods and the periods before and after the faults are compared, and the position errors obtained with no fault detection, the traditional fault detection algorithm, and the optimal fault detection algorithm with

8.4 Simulation Examples

185

Fig. 8.4 Test statistics for the traditional fault detection algorithm with optimal anchor member collaboration

optimal anchor member collaboration are also compared, as shown in Figs. 8.5 and 8.6. As shown in Fig. 8.6, during the period of 2500–2600 s, both the traditional fault detection algorithm based on the individual redundancy of a member and the fault detection algorithm based on collaboration-augmented redundancy can successfully identify and isolate the failure, guaranteeing the navigation accuracy. However, as seen in Fig. 8.5, during the period of 1000–1100 s, the traditional fault detection algorithm based on the individual redundancy of a member fails to identify and isolate the failure, and the position error is significantly biased. Meanwhile, benefitting from collaboration-augmented redundancy, the proposed fault detection algorithm can successfully identify and isolate the failures, allowing the navigation performance to be maintained. Therefore, the proposed collaborative fault detection approach can meet the system requirements in terms of the missed detection rate, accurately detect a faulty satellite, and effectively improve the fault tolerance of the navigation system.

186

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.5 Position errors based on different fault detection methods during 900–1200 s

8.4.2 Simulation of Multifault Detection in Collaborative Resilient Navigation Fusion 8.4.2.1

Simulation Setup

To verify the effectiveness of the collaborative fault-tolerant navigation algorithm, simulations of a swarm with six members were carried out. Figure 8.7 shows the simulated flight trajectories of the swarm members. All members could obtain relative measurement information through their airborne radio ranging systems, and the equivalent ranging error was 5 m. In the simulations of the INSs of all members, the gyroscope accuracy was 0.1°/h, and the accelerometer bias was 300 μg. The different GNSS satellites visible to each member of the swarm were simulated by setting different masked elevation angles. The equivalent pseudorange error of the GNSS √ was 3 m. Table 8.2 shows the GNSS satellites visible to the swarm members; “ ” means that the satellite is always visible during flight, and “–” means that the satellite is visible only in some time periods. Figure 8.8 shows the pseudorange errors affected by satellite failures for different members, where a numeral preceded by the symbol “#” in the legend indicates the serial number of the failed satellite.

8.4 Simulation Examples

Fig. 8.6 Position errors based on different fault detection methods during 2400–2700 s

Fig. 8.7 Trajectories of the members of the aerial swarm

187

188

8 Collaborative Fault Detection in Resilient Navigation

Table 8.2 Visibility of GNSS satellites to swarm members during flight Swarm member

Number of visible satellites

Member 1

4–5

Member 2

5

Member 3

8

Member 4

4

Member 5

7–8

Member 6

4–5

Serial number of satellite 1 √

2 √

√

√

√

√

√ √

3 √ √

4 √

5

√

– √

√

√

6

√

√

√

– √

√

8

√ √

√ √

7

√ √

√

√

√

√

√

√ –

Fig. 8.8 Pseudorange errors affected by satellite failures for different members

Because the swarm members are divided into anchor members and label members and the label members need assistance from their collaborative anchor members, we will present the simulation results from the perspectives of single label members and all members in the aerial swarm.

8.4.2.2

Analysis of Fault Detection Performance

This section verifies the detection performance of the collaboration-augmented MHSS (CA-MHSS) algorithm and compares it with that of the traditional MHSS algorithm. Swarm member 1 is taken as an example of a label member that needs assistance. Member 3 and member 5 have sufficient visible satellites to achieve good positioning performance and thus can be used as anchor members. For label member 1, from Fig. 8.8, we can see that the pseudorange observations fail in the periods of 600–800 s and 1200–1400 s. Because there are at most five

8.4 Simulation Examples

189

visible satellites, 10 subfilters are constructed when detecting two faulty satellites, and each subfilter has at most three sub-subfilters. Figures 8.9 and 8.10 show the test statistics and corresponding thresholds of subfilters 1–10 for the traditional MHSS and proposed CA-MHSS methods. During the period of 600–800 s, the detection results of both methods are consistent in the case of five satellites. Faults exist in the satellite pseudorange because more than one subfilter’s test statistics exceed the corresponding thresholds. However, during 1400–1600 s, the traditional MHSS algorithm cannot detect the faults because there are only four visible satellites, and the very few available observations lead to an incorrect subset solution. In contrast, the CA-MHSS algorithm can still detect the faults in this case through the introduction of two collaborative observations. These findings show that this method offers higher detection availability for GNSS pseudorange faults in the case of a few satellites. Due to the large number of sub-subfilters, we show data for only some of them. Figures 8.11, 8.12, 8.13 and 8.14 show the test statistics and corresponding thresholds of the sub-subfilters under subfilter 1 and subfilter 2 for MHSS and CA-MHSS. In the period of 1200–1400 s, since the traditional MHSS algorithm cannot detect the faults, all the test statistics of the sub-subfilters are below the thresholds, so this algorithm also cannot identify the faulty satellites. However, for CA-MHSS, in the

Fig. 8.9 Test statistics and corresponding thresholds for subfilters 1, 2, 3, 4, and 5, each excluding two satellites, for MHSS and CA-MHSS

190

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.10 Test statistics and corresponding thresholds for subfilters 6, 7, 8, 9, and 10, each excluding two satellites, for MHSS and CA-MHSS

periods of 600–800 s and 1200–1400 s, only the test statistics of sub-subfilters 1,1, 1,2, and 1,3 under subfilter 1 are below the thresholds, which shows that satellite observations of subfilter 1 are free of faults. Therefore, the first and second satellites, which are excluded in subfilter 1, are the faulty satellites. This method accurately identifies the faulty satellites and then excludes them, improving the ability to detect multiple errors in parallel even when a few satellites are observed. Thus, by using CA-MHSS, the swarm members can maintain small position errors.

8.4.2.3

Analysis of Fault-Tolerant Navigation Performance

This section verifies the overall fault-tolerant navigation performance for the swarm. During flight, all members use FDE algorithms. The anchor members with many visible satellites use the MHSS algorithm to detect and exclude faults, whereas most label members, which have few visible satellites, incorporate collaborative observations from the anchor members and use the CA-MHSS algorithm to detect satellite faults. According to the visibility and fault conditions of the GNSS satellites for each member as shown in Table 8.1 and Fig. 8.4, members 1, 2, 4, and 6 are the label members, and members 3 and 5 are the anchor members.

8.4 Simulation Examples

191

Fig. 8.11 Test statistics and corresponding thresholds for sub-subfilters 1,1, 1,2, and 1,3, each excluding three satellites, for MHSS and CA-MHSS

The root mean square error (RMSE) metric is adopted to evaluate the positioning accuracy of each member. Figures 8.15, 8.16 and 8.17 show the position RMSEs of the swarm members without FDE and when using the traditional MHSS and CA-MHSS algorithms during flight. Table 8.3 shows the position RMSEs of the members during their respective faulty time periods for MHSS, CA-MHSS, and the case without FDE (non-FDE), where “/” means that a cell has no data. As seen in Fig. 8.16, the traditional MHSS algorithm can be used to detect and exclude the failures that influence members 2, 3, and 6 in the non-FDE case depicted in Fig. 8.15. However, with the traditional MHSS algorithm, the position errors of members 1, 2, and 4 still diverge to 25 m during their faulty episodes because of an insufficient fault detection ability. More seriously, the position errors continue to diverge after these faulty episodes due to the failure tracking effect. As seen in Fig. 8.17, when the CA-MHSS algorithm is used, the position errors of all members remain within 5 m during flight and are not affected by faulty GNSS pseudorange observations. As seen in Table 8.3, for the label members with few visible satellites, the CAMHSS algorithm can detect two satellite faults that cannot be detected by the traditional MHSS algorithm. For example, in the period of 430–510 s, the position error of member 4 is only 1.50 m when collaborative observations are considered. However, its position error is 22.57 m without assistance. In some cases, both the CA-MHSS

192

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.12 Test statistics and corresponding thresholds for sub-subfilters 2,1, 2,2, and 2,3, each excluding three satellites, for MHSS and CA-MHSS

and MHSS algorithms can detect and exclude multiple faults; however, the positioning accuracy achieved using CA-MHSS is slightly higher than that achieved using the MHSS algorithm. The reason is the increase in the number of available observations. After two faulty satellites are excluded, the number of available satellite observations has decreased. However, when the CA-MHSS algorithm is used, the collaborative observations can continue to assist the label members in achieving better positioning performance. To better evaluate the fault-tolerant navigation performance of the swarm, we calculate the number of swarm members with a position error less than 3 m (that is, δp < 3 m) and use this quantity as a statistical performance indicator. Figure 8.18 shows the number of members with a position error of less than 3 m during the faulty time periods for the MHSS, CA-MHSS, and non-FDE cases. As seen in Fig. 8.18, when the collaborative augmentation method is not used, not all members’ position errors are within the specified range. Multiple faults can be detected in only a few cases. However, when using the CA-MHSS algorithm, all six members can keep their position errors below 3 m by excluding faulty satellites. The simulation results show that when only a few satellites are visible, the introduction of collaborative information can greatly improve the fault-tolerant navigation capability of an aerial swarm.

8.4 Simulation Examples

193

Fig. 8.13 Test statistics and corresponding thresholds for sub-subfilters 3,1, 3,2, and 3,3, each excluding three satellites, for MHSS and CA-MHSS

8.4.3 Comprehensive Simulation of Collaborative Resilient Navigation Fusion 8.4.3.1

Simulation System Construction

To simulate the navigation process for an aerial swarm, a parallel simulation architecture was designed. Compared with independent aircraft navigation simulations, the simulation of collaborative resilient navigation fusion in an aerial swarm must account for the temporal logic of each navigation phase of each member. The airborne INS and satellite navigation system of each member is simulated simultaneously; then, the exchange of relative measurement information among multiple aircraft is conducted; and finally, the cooperative fault-tolerant navigation results are output. Based on this, the parallel navigation architecture for simulating the cooperative fault-tolerant navigation of an aerial swarm was designed as shown in Fig. 8.19. The implementation of the aerial swarm navigation process in the parallel simulation architecture is divided into the following phases. • Individual member navigation and fault injection. Trajectory simulation is started for each member of the swarm, simulating both the inertial guidance system

194

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.14 Test statistics and corresponding thresholds for sub-subfilters 4,1, 4,2, and 4,3, each excluding three satellites, for MHSS and CA-MHSS

Fig. 8.15 Position RMSEs of swarm members without FDE

8.4 Simulation Examples

Fig. 8.16 Position RMSEs of swarm members when using traditional MHSS

Fig. 8.17 Position RMSEs of swarm members when using CA-MHSS

195

196

8 Collaborative Fault Detection in Resilient Navigation

Table 8.3 Comparison of position RMSEs in faulty time periods Member Member 1 Member 2

Faulty time period

Position RMSE (m) Non-FDE

MHSS

CA-MHSS 1.08

600–800 s

30.08

2.91

1200–1400 s

28.92

28.94

1.21

750–850 s

20.13

20.08

1.15

Member 3

1250–1360 s

17.25

0.94

/

Member 4

430–510 s

22.89

22.57

1.52

Member 5

No fault

/

/

Member 6

1000–1110 s

2.35

1.06

0.80 26.72

Fig. 8.18 Number of members with a position error less than 3 m for the MHSS, CA-MHSS, and Non-FDE cases

module and the GNSS module; then, the GNSS module of each member is simulated with and without fault states, with specific fault values set by the fault injection module. • Categorization of the members’ role in the swarm. Based on the number of navigation observations available in each vehicle’s own GNSS module, vehicles with insufficient available observations are identified as label members, while members with sufficient observations are identified as anchor members. Each label member outputs fault-tolerant navigation results through a multistage filter module, which are then used for further cooperative fault-tolerant navigation.

8.4 Simulation Examples

197 Start

Independent Navigation Measurement Simulation of Member i Simulative Fault Injection for Member i

…

Independent Navigation Measurement Simulation of Member j

…

Simulative Fault Injection for Member j

Independent Navigation Measurement Simulation of Member k Simulative Fault Injection for Member k

Swarm Hierarchical Categorization

Anchor Member j Fault Detection and Fault-tolerant Navigation Results Sharing

…

Label Member i

… Anchor Member k

Relative Observation and Interactive Communication Simulation

Collaborative Information

Fault Detection and Fault-tolerant Navigation Results Sharing

Collaborative Information

Collaborative Configuration Optimization Collaborative Integrity Augmentation Collaborative Fault Detection Navigation Performance Evaluation of Aerial Swarm

Navigation End?

End

Fig. 8.19 Workflow for the simulation of collaborative resilient navigation fusion in a swarm

• Simulation of relative measurements and data transfer. The anchor members perform relative measurement data simulation and data transmission via their relative observation modules, and the label members obtain the relative distance to each anchor member and the position of the anchor member. • Cooperative fault-tolerant navigation of the label members. Each label member performs data fusion and comprehensive processing of the relative observation

198

8 Collaborative Fault Detection in Resilient Navigation

and GNSS observation information through its cooperative observation information processing module and then uses its geometric configuration optimization module, cooperative integrity augmentation module, and cooperative faulttolerant navigation module to perform cooperative resilient navigation fusion and output cooperative fault-tolerant navigation results in accordance with the system requirements. • Evaluation of the navigation performance of the aerial swarm. After all members have obtained their final navigation outputs for the current moment, statistical performance evaluation indexes related to the geometric accuracy factor, integrity protection level, and position error of cooperative navigation are calculated to monitor and analyse the navigation performance of the swarm in real time. The parameter settings include the aerial swarm parameter settings, airborne inertial measurement unit (IMU) error parameter settings, and the GNSS observation and fault parameter settings for each member. First, the simulation flight time, the number of aircraft in the swarm, and the relative observation noise are set. The error parameters of the airborne inertial sensors are also set. The GNSS variance settings include satellite equivalent pseudorange error settings, masked angle settings for visible satellites, and masked azimuth and range settings. The GNSS satellite fault settings indicate whether a vehicle experiences a satellite pseudorange fault and, in the case of a fault, the continuous duration of the fault, the number of faulty satellites, and the sizes of the pseudorange deviations are also set.

8.4.3.2

Simulation Setup

This section verifies the integrated fault-tolerant navigation performance of swarm vehicles in a complex application scenario with a poor GNSS geometry and multiple satellite failures when using a cooperative navigation configuration optimization algorithm, a CIM algorithm, and a cooperative fault-tolerant navigation algorithm. The number of swarm vehicles in these simulations was set to 10, and the flight time was 3600 s. The trajectory, onboard IMU error parameters, satellite system equivalent pseudorange error, and radio ranging system error for each vehicle were the same as those in Sect. 6.5.1.1. To simulate complex swarm navigation scenarios, different satellite cut-off altitude angles were set for each member’s GNSS module such that some members had insufficient satellites available and poor geometric configurations, and satellites 1 and 2 were set to experience faults for part of the time period, resulting in different degrees of pseudorange faults. The variations in the geometric dilution of precision (GDOP) and the satellite pseudorange fault injection characteristics for the GNSS module of each vehicle in the swarm with the above simulation parameters are shown in Figs. 8.20 and 8.21, respectively.

8.4 Simulation Examples

199

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

Time Fig. 8.20 GDOP variations for the GNSS module of each vehicle in the swarm

Fig. 8.21 Satellite pseudorange fault injection in the GNSS module of each vehicle in the swarm

8.4.3.3

Comprehensive Verification of the Cooperative Fault-Tolerant Navigation Performance of Swarm Vehicles

The number of label members and anchor members in an aerial swarm will change dynamically during navigation, depending on the number of visible satellites and their geometric configuration for each member. The label members use some of the

200

8 Collaborative Fault Detection in Resilient Navigation

anchor members to assist in the optimization of their own navigation performance indicators as well as for multifault detection and troubleshooting. A curve showing the evolution of the number of label members during the navigation process is shown in Fig. 8.22 and the series number of the member act as label member in swarm is shown in Fig. 8.23.

Fig. 8.22 Number of label members in the swarm

Fig. 8.23 Serial number of members acting as label members in the swarm

8.4 Simulation Examples

201

Fig. 8.24 Serial number of selected collaboration partners (member 5)

In the process of collaborative navigation, to ensure that the label members have sufficient collaborative observation information to assist in the completion of multifault detection and identification and to ensure the navigation integrity for fault detection, the cooperative navigation configuration optimization algorithm and the protection level optimization algorithm introduced in Chaps. 6 and 7 are first applied to select two to three reference vehicles. Then, the multifault detection algorithm based on collaborative observation information that has been introduced in this chapter is applied for failure detection, identification, and isolation. Thus, the overall framework of collaborative resilient navigation fusion in an aerial swarm is constructed. Member 5, which acts as a label member for most of the simulated duration, is taken as an example here, and its selected anchor members during the navigation process are shown in Fig. 8.24. As shown in Fig. 8.24, member 5 is able to select optimal anchor members based on the changes in the geometric distribution of its visible satellites during navigation, thereby ensuring the integrity of its own navigation system and enabling collaboration-assisted multifault detection. As described in Sect. 5.5.3.2, for each vehicle suffering GNSS satellite pseudorange fault injection, during the faulty time period of 1800–2100s, members 6 and 8 act as the anchor members using the traditional MHSS algorithm for fault detection and troubleshooting, while members 1, 5, and 10 act as the label members, which perform multifault detection using information from the selected anchor members and obtain cooperative fault-tolerant navigation results. The position RMSE curves for all members affected by satellite failure after the application of the conventional MHSS algorithm and CA-MHSS algorithm are shown in Figs. 8.25 and 8.26, respectively.

202

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.25 Position RMSEs when using the conventional MHSS algorithm (members 1, 5, 7, 8, and 10)

Fig. 8.26 Position RMSEs when using the CA-MHSS algorithm (members 1, 5, 7, 8, and 10)

As seen from Fig. 8.25, the position errors of members 1 and 5, which are the label members, are still greater than 20 m with the traditional MHSS algorithm, and the influence of the faults is not completely eliminated; only the two anchor members and the one other label member are not affected by the failures. In contrast, when the CA-MHSS algorithm is used, the position errors of all five members remain within

8.4 Simulation Examples

203

Fig. 8.27 Comparison of the average CGDOP for the members of the swarm

3 m, indicating that the positioning performance is not affected by the failures and shows good fault tolerance. After 3600 s of simulation, the results of cooperative resilient navigation fusion of the aerial swarm were obtained, and two statistical performance indicators, the average collaborative geometric dilution of precision (CGDOP) and average CHPL of the members of the aerial swarm, were calculated and compared with the results in the noncooperative case, as shown in Figs. 8.27 and 8.28. As seen from Figs. 8.27 and 8.28, with the cooperative navigation configuration optimization algorithm and the cooperative integrity monitoring algorithm, the geometric accuracy factor (CGDOP) and the CHPL remain at a low level for all vehicles requiring assistance; specifically, the CGDOP remains below 5 and the CHPL remains below 100 m during navigation, representing more than a three-fold performance improvement compared to the case without cooperation. Both statistical performance metrics are able to meet the requirements of cooperative navigation. In summary, the cooperative navigation configuration optimization algorithm and the cooperative integrity monitoring algorithm enable better collaboration and ensure the integrity of cooperative navigation, while the cooperation-assisted multifault detection algorithm improves the availability of multifault detection by introducing cooperative observation information, thereby ensuring the fault-tolerant navigation capability of the aerial swarm.

204

8 Collaborative Fault Detection in Resilient Navigation

Fig. 8.28 Comparison of the average CHPL for the members of the swarm

8.5 Conclusions This chapter has proposed a collaborative fault-tolerant navigation method for aerial swarms. A collaboration-augmented navigation model is established by fusing GNSS pseudorange observations and member ranging observations through a fusion filter. Within this framework, the CA-MHSS algorithm is applied, and faulty satellite observations are detected using statistics from subfilter solutions based on the subsets of the satellites. Simulation results show that in the case of a few GNSS satellites, the CA-MHSS algorithm has a better fault detection ability than the traditional MHSS algorithm. The availability and accuracy of multifault detection for label members are greatly improved. In addition, the positioning error of the entire swarm is kept small. The proposed method improves the reliability and fault tolerance of the navigation systems of the members of an aerial swarm.

References 1. Alam N, Kealy A, Dempster AG (2013) Cooperative inertial navigation for GNSS-challenged vehicular environments. IEEE Trans Intell Transp Syst 14(3):1370–1379. https://doi.org/10. 1109/tits.2013.2261063 2. Angus JE (2006) RAIM with multiple faults. Navig J Inst Navig 53(4):249–257. https://doi. org/10.1002/j.2161-4296.2006.tb00387.x 3. Blanch J, Walter T, Enge P (2010) RAIM with optimal integrity and continuity allocations under multiple failures. IEEE Trans Aerosp Electron Syst 46(3):1235–1247. https://doi.org/ 10.1109/taes.2010.5545186

References

205

4. Lee YC (2004) Investigation of extending receiver autonomous integrity monitoring (RAIM) to combined use of Galileo and modernized GPS. In: 17th International technical meeting of the satellite division of the institute of navigation, ION GNSS 2004, September 21–24, 2004, Long Beach, CA, United States. Institute of Navigation, pp 1691–1698 5. Sun S (2010) RAIM algorithm for detection of multiple fault satellites based on maximum likelihood ratio. J Chin Inert Technol 19(3):312–315 6. Kewei Z, Papadimitratos P (2021) Fast multiple fault detection and exclusion (FM-FDE) algorithm for standalone GNSS receivers. IEEE Open J Commun Soc 2:217–234. https://doi.org/ 10.1109/ojcoms.2021.3050333 7. Yang T-Y, Sun D (2020) Global navigation satellite systems fault detection and exclusion: a parameterized quadratic programming approach. IEEE Trans Aerosp Electron Syst 56(4):2862–2871. https://doi.org/10.1109/taes.2019.2956624 8. Alqurashi M, Jinling W (2015) Performance analysis of fault detection and identification for multiple faults in GNSS and GNSS/INS integration. J Appl Geodesy 9(1):35–48. https://doi. org/10.1515/jag-2014-0019 9. Liu J, Cai B, Lu D, Wang J, IEEE (2018) A local weighting method for GNSS receiver autonomous integrity monitoring using pseudorange residuals. In: 21st IEEE International conference on intelligent transportation systems (ITSC), Maui, HI, Nov 04–07 2018, pp 3067– 3074 10. Zhuang C, Zhao H, Hu S, Sun C, Feng W, Inst N (2019) Integrity monitoring algorithm for GNSS-based cooperative positioning applications. In: 32nd International technical meeting of the satellite-division-of-the-institute-of-navigation (ION GNSS), Miami, FL, Sep 16–20 2019, pp 2008–2022. https://doi.org/10.33012/2019.16881 11. Margaria D, Falletti E (2014) A novel local integrity concept for GNSS receivers in urban vehicular contexts. In: 2014 IEEE/ION Position, location and navigation symposium, PLANS 2014, May 5–8 2014, Monterey, CA, United States. Institute of Electrical and Electronics Engineers Inc., pp 413–425. https://doi.org/10.1109/PLANS.2014.6851399 12. Bhamidipati S, Gao GX, Inst N (2019) Distributed cooperative SLAM-based integrity monitoring via a network of receivers. In: 32nd International technical meeting of the satellite-division-of-the-institute-of-navigation (ION GNSS), Miami, FL, Sep 16–20 2019, pp 2023–2034. https://doi.org/10.33012/2019.16882 13. Masiero A, Toth C, Gabela J, Retscher G, Kealy A, Perakis H, Gikas V, Grejner-Brzezinska D (2021) Experimental assessment of UWB and vision-based car cooperative positioning system. Remote Sens 13(23). https://doi.org/10.3390/rs13234858 14. Xin J, Xie G, Yan B, Shan M, Li P, Gao K (2021) Multimobile robot cooperative localization using ultrawideband sensor and GPU acceleration. IEEE Trans Autom Sci Eng. https://doi.org/ 10.1109/tase.2021.3117949 15. Ansari K, Feng Y, Tang M (2015) A runtime integrity monitoring framework for real-time relative positioning systems based on GPS and DSRC. IEEE Trans Intell Transp Syst 16(2):980– 992. https://doi.org/10.1109/tits.2014.2349011 16. Ansari K, Wang C, Lei W, Yanming F (2013) Vehicle-to-vehicle real-time relative positioning using 5.9 GHz DSRC media. In: 2013 IEEE 78th Vehicular technology conference. https://doi. org/10.1109/VTCFall.2013.6692454 17. Yang L, Rife J, Inst N (2016) Estimating covariance models for collaborative integrity monitoring. In: 29th International technical meeting of the-satellite-division-of-the-institute-ofnavigation (ION GNSS+), Portland, OR, Sep 12–16 2016, pp 1103–1113 18. Rife J, IEEE (2012) Collaboration-enhanced receiver integrity monitoring with common residual estimation. In: IEEE/ION Position location and navigation symposium (PLANS), Myrtle Beach, SC, Apr 23–26 2012, pp 1042–1053 19. Qu Y, Wu J, Xiao B, Yuan D (2017) A fault-tolerant cooperative positioning approach for multiple UAVs. IEEE Access 5:15630–15640. https://doi.org/10.1109/ACCESS.2017.273 1425

206

8 Collaborative Fault Detection in Resilient Navigation

20. Liu Y, Zhu Y (2013) A collaborative integrity monitor algorithm for low space aviation under limited number of navigation satellites. In: 2013 2nd IEEE International conference on connected vehicles and Expo, ICCVE 2013, December 2–6 2013, Las Vegas, NV, United States. IEEE Computer Society, pp 910–915. https://doi.org/10.1109/ICCVE.2013.6799925 21. Xin L, Kun F, Xiao L, Jichao D, Zhipeng W (2021) Fault identification method of GNSS/INS integrated navigation system based on the fusion of chi-square test and multiple solution separation algorithm. China Satell Navig Conf. https://doi.org/10.1007/978-981-16-3146-7_52 22. Ling Y, Knight NL, Yong L, Rizos C (2013) Optimal fault detection and exclusion applied in GLASS positioning. J Navig 66(5):683–700. https://doi.org/10.1017/s0373463313000155 23. Wang S, Zhan X, Zhai Y, Liu B (2020) Fault detection and exclusion for tightly coupled GNSS/INS system considering fault in state prediction. Sensors 20(3). https://doi.org/10.3390/ s20030590 24. Ammiche M, Kouadri A, Ieee (2017) Constant false alarms rate for fault detection. In: 5th International conference on electrical engineering—Boumerdes (ICEE-B), Boumerdes, Algeria, Oct 29–31 2017 25. Xin C, Rong W, Zhi X, Jianye L, Weixing Q, Huiyuan Z (2021) Collaborative fault-tolerant navigation method for unmanned swarm based on MHSS. In: 2021 IEEE International conference on unmanned systems. https://doi.org/10.1109/icus52573.2021.9641417 26. Kropp V, Berz G, IEEE (2014) Optimized MHSS ARAIM user algorithms: assumptions, protection level calculation and availability analysis. In: IEEE/ION Position, location and navigation symposium (PLANS), Monterey, CA, May 05–08 2014, pp 308–323

Chapter 9

Summary and Scope

Abstract Resilient fusion techniques for collaborative navigation in aerial swarms is an exciting area of research, and much work remains to be explored in the future. In this chapter, the research presented in this book is summarized, and the conclusions are comprehensively discussed. An outlook on the trends of development in research on collaborative resilient navigation fusion is discussed, and recommendations for future work are provided. Keywords Aerial swarm · Collaborative navigation · Summary and conclusions · Development trend and future work · Resilient fusion

9.1 Summary and Conclusions Aerial swarm technology has the characteristics of high reliability, high mission completion rate, and high adaptability to complex missions and is one of the future directions of aircraft technology development [1, 2]. Based on the typical flight environments and characteristics of aerial swarms, this book has investigated the resilient fusion technique for collaborative navigation in a swarm. Collaborative navigation fusion utilizes relative range, relative bearing, and vector-of-sight information to address the differences in the configurations of the navigation sensors of the aircraft in a swarm. Based on the typical complex working environments and characteristics of aerial swarms, this book has presented research on six topics, including collaborative fusion frameworks, modelling methods for collaboration, collaborative positioning fusion algorithms, collaborative geometry optimization algorithms, collaborative integrity augmentation algorithms, and collaborative fault-tolerant algorithms. The main research work presented in this book is summarized as follows: 1. Frameworks and Modelling for Collaborative Resilient Navigation Fusion In this part, the algebraic and geometric fundamentals of collaborative resilient navigation fusion were introduced, including various collaborative navigation structures of aerial swarms and the frameworks and models for resilient fusion in collaborative navigation. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Wang et al., Resilient Fusion Navigation Techniques: Collaboration in Swarm, Unmanned System Technologies, https://doi.org/10.1007/978-981-19-8371-9_9

207

208

9 Summary and Scope

– Collaborative Navigation Structures of Aerial Swarms Various collaborative navigation structures, which reflect the relationships among the members of an aerial swarm, were discussed. Compared with the single leader–follower structure and the parallel structure for collaborative navigation, the hierarchical collaborative navigation structure shows greater potential for flexibility and robustness in dealing with failures and degradation in the navigation measurements of the swarm members. – Frameworks for Resilient Fusion in Collaborative Navigation An airborne navigation system with a resilient fusion strategy can be designed via either a collaborative localization-based fusion framework or a collaborative observation-based fusion framework, in which the local measured information and the relative observed information are organized in different ways. An understanding of these approaches provides the fundamental basis for studying the corresponding modelling and fusion algorithms. – Models for Resilient Fusion in Collaborative Navigation Relative range and relative bearing are the two kinds of information that can be observed cooperatively by a pair of members in a swarm. Based on the available measurements that can be obtained by the members, collaborative observation models based on range, range difference, bearing, and vector-of-sight observations were established. Furthermore, for relative vector-of-sight observations, formulations in both spherical coordinates and Cartesian coordinates were derived, providing a foundation for the development of collaborative resilient fusion algorithms. 2. Positioning and Geometry Optimization Algorithms for Collaborative Resilient Navigation Fusion This part of the book focused on improving the navigation accuracy of aerial swarms in environments with insufficient availability of global navigation satellite system (GNSS) observations. Positioning algorithms for collaborative resilient navigation fusion were introduced, including collaborative localizationbased and collaborative observation-based methods. Furthermore, geometry optimization algorithms for resilient fusion in collaborative navigation were introduced to improve the navigation accuracy. – Collaborative Localization-Based Resilient Navigation Fusion Collaborative localization-based resilient navigation fusion was discussed for cases in which localization is performed by means of the least squares, Chan– Taylor, and spherical interpolation algorithms. Simulation results showed that collaborative localization-based resilient navigation fusion can effectively improve the navigation accuracy of the lower-precision members of a swarm system. Moreover, the influential factors of the relative ranging or bearing measurement accuracy, the positioning accuracy of the anchor members, and the number of anchor members were analysed through simulation, providing a reference for the selection of effective parameters in practical applications of aerial swarms.

9.1 Summary and Conclusions

209

– Collaborative Observation-Based Resilient Navigation Fusion The approach of collaborative observation-based resilient navigation fusion, in which relative observations are directly fused with local measurements without performing localization first, was discussed in relation to both the hierarchical and parallel collaborative navigation structures. A cooperative positioning method based on a hybrid recursive network was proposed to realize resilient fusion in a GNSS-denied environment. Through mutual radio sensing, communication, and ranging among the members of a collaborative union, all members can be localized successfully based on their shared information. This approach also has the potential to improve the fault tolerance of collaborative navigation systems. Even when some anchor members are damaged, the entire collaborative navigation system can continue to function, and the anchor positions can be recovered after failure, thus verifying the effectiveness of this method. – Collaborative Geometry Optimization in Resilient Navigation Two algorithms for optimizing the cooperative navigation configuration of an aerial swarm were introduced, one based on geometric analysis and the other based on an algebraic search method. Simulation results showed that such an algorithm can achieve the desired configuration optimization effect compared to the all-selected case by selecting only one or two suitably positioned anchor members and that proper screening of the possible collaboration strategies among the aircraft in a swarm can ensure a better collaborative relationship, thereby effectively improving the overall navigation accuracy of the swarm in a GNSS-challenged environment. 3. Integrity Augmentation and Fault-Tolerant Algorithms for Collaborative Resilient Navigation Fusion This part of the book focused on improving the navigation robustness of an aerial swarm in an environment with insufficient GNSS observations. The collaborative integrity augmentation in resilient navigation is introduced, including integrity optimization based on the geometric analysis method and the algebraic search method. Furthermore, collaborative fault detection in resilient navigation was introduced to support fault identification and isolation, improving the navigation reliability of the aerial swarm. – Collaborative Integrity Augmentation in Resilient Navigation Integrity augmentation methods for aerial swarm navigation with cooperative assistance were investigated, the relationship between the protection level and the swarm configuration was clarified, and a protection level optimization method and a cooperation-assisted fault detection algorithm were proposed. Simulation results showed that the proposed algorithm can avoid the selection of an anchor member that offers less improvement in the protection level for a given label member, leading to a better integrity protection level for the swarm aircraft. In this way, the possibility of fault detection and the integrity protection capability of the navigation systems of the swarm members can be

210

9 Summary and Scope

ensured in the scenario of cooperative navigation with a given probability of missing detection. – Collaborative Fault Detection in Resilient Navigation A cooperative fault-tolerant navigation algorithm based on multiple hypothesis solution separation for swarm aircraft was designed by taking advantage of the redundant observation information among members of the swarm. This approach enables the use of the multiple hypothesis solution separation algorithm to construct test statistics in order to detect and identify multiple satellite failures even when a label member has only a small number of visible satellites. At the same time, the accuracy of the cooperative auxiliary information provided by the anchor members is guaranteed under a two-level fault detection structure. The position error of the whole aerial swarm is kept low even in the case of multiple satellite failures, thus effectively improving the fault-tolerant navigation capabilities of aerial swarms in GNSS-challenged environments.

9.2 Development Trends and Future Work In this book, research on the technique of resilient fusion for collaborative navigation in aerial swarms has been investigated, including the topics of frameworks and modelling, positioning and geometry optimization algorithms, and integrity augmentation and fault-tolerant algorithms. The objective of such research work is to improve the accuracy and reliability of navigation by means of effective and efficient collaboration among the members of an aerial swarm in a GNSS-challenged environment. On this basis, further in-depth research on the resilient navigation fusion technique still needs to be carried out, mainly focusing on the following aspects. 1. Modelling Considering Additional Factors – This book has studied the impact of several factors on an algorithm’s solution accuracy. The high communication density in an aerial swarm with a large number of members will increase the time delay of information exchange, affecting the solution accuracy. Subsequent work can further analyse the impact of the mutual communication between aircraft on an algorithm’s solution performance and the design of corresponding cooperative navigation algorithms to reduce the high dependence on the communication systems of the aircraft [3, 4]. – The interflight clock difference has been treated as part of the observation error in the modelling of cooperative navigation in an aerial swarm. However, a detailed analysis of the interflight clock difference modelled as a state quantity has not been carried out. Subsequent work can further improve the understanding of the impact of interflight clock differences on cooperative navigation and investigate the possibility of expanding the state modelling problem [5–7].

9.2 Development Trends and Future Work

211

– A simulation platform for the cooperative navigation of aerial swarms has been established based on simulated relative observations. In subsequent research, we can further study and verify the algorithm performance for a large number of vehicles in different motion states and with different motion trajectories and reduce the algorithm error by optimizing the structure of the cooperative navigation system [8, 9]. 2. Deep Optimization for Improving the Navigation Accuracy – Because multiple types of navigation information can be obtained by vehicles flying in a swarm, a collaborative navigation algorithm based on only a single type of relative measurement information cannot make full use of the redundant constraint conditions of the aerial swarm. Therefore, algorithms need to be developed to further improve the navigation accuracy by fusing multiple types of sensor information when the number of high-precision reference vehicles is low in order to overcome the low reliability of cooperative navigation systems based on single-sensor information [10, 11]. – The effects of several factors, such as the high communication density in a swarm consisting of a large number, on the navigation performance should be investigated. The presence of multiple independent and coupled measurements in an aerial swarm, so there are particular complex failure modes in the cooperative navigation system than in an individual aircraft, which requires further analysis [12, 13]. 3. Enhancing Robustness to More Complex Failures – The simulation validations presented in this book mainly focused on the overall navigation performance of an aerial swarm under differential GNSS observation conditions and abrupt satellite pseudorange faults. In subsequent research, the cooperative fault-tolerant navigation performance of an aerial swarm under additional constraints and more complex fault modes can be further validated [14, 15]. – Relative measurement information can be obtained when aircraft are flying in a swarm. However, the problem arises that not all sensor information obtained from the same direction necessarily belongs to the same aircraft. Therefore, further research will be needed to reduce the impact of overlapping measurements on the cooperative navigation performance and to design a cooperative navigation algorithm based on the relative information of single aircraft to improve the navigation accuracy for the lower-precision aircraft in a swarm system. In summary, the resilient fusion technique for collaborative navigation in an aerial swarm shows advantages in enhancing the accuracy and reliability of navigation in GNSS-challenged environments, and the research presented here provides a theoretical reference for subsequent research on the autonomous operation of aerial swarms. Modelling considering additional factors, deep optimization for improving the navigation accuracy, and enhancing robustness to more complex failures are expected

212

9 Summary and Scope

to be the main future development trends of work on resilient fusion for navigation. It is foreseeable that with the development of resilient navigation fusion technology, future aerial swarms will become more intelligent and adaptive [16] and will have broader application prospects in civil engineering [17], transportation [18], agriculture [19], and other fields.

References 1. Buerkle A, Segor F, Kollmann M (2011) Towards autonomous micro UAV swarms. J Intell Rob Syst 61(1–4):339–353. https://doi.org/10.1007/s10846-010-9492-x 2. Tahir A, Boling J, Haghbayan M-H, Toivonen HT, Plosila J (2019) Swarms of unmanned aerial vehicles—a survey. J Ind Inf Integr 16. https://doi.org/10.1016/j.jii.2019.100106 3. Bouachir O, Abrassart A, Garcia F, Larrieu N, IEEE (2014) A mobility model for UAV Ad hoc Network. In: International conference on unmanned aircraft systems (ICUAS), Orlando, FL, 2014 May 27–30 2014. International conference on unmanned aircraft systems. pp 383–388 4. Yuan Z, Jin J, Sun L, Chin K-W, Muntean G-M (2018) Ultra-reliable IoT communications with UAVs: a swarm use case. IEEE Commun Mag 56(12):90–96. https://doi.org/10.1109/mcom. 2018.1800161 5. Yuan W, Wu N, Etzlinger B, Wang H, Kuang J (2016) Cooperative joint localization and clock synchronization based on Gaussian message passing in asynchronous wireless networks. IEEE Trans Veh Technol 65(9):7258–7273. https://doi.org/10.1109/tvt.2016.2518185 6. Vaghefi RM, Buehrer RM (2015) Cooperative joint synchronization and localization in wireless sensor networks. IEEE Trans Signal Process 63(14):3615–3627. https://doi.org/10.1109/tsp. 2015.2430842 7. Azar AT, Serrano FE, Kamal NA, Koubaa A, Ammar A (2021) Robust decentralized asynchronous control of unmanned aerial vehicles swarm with fast convergence switching topology. In: Advanced machine learning technologies and applications. Proceedings of AMLTA 2021. Advances in intelligent systems and computing. https://doi.org/10.1007/978-3-030-697174_62 8. Honig W, Preiss JA, Kumar TKS, Sukhatme GS, Ayanian N (2018) Trajectory planning for quadrotor swarms. IEEE Trans Rob 34(4):856–869. https://doi.org/10.1109/tro.2018.2853613 9. Nigam N, Bieniawski S, Kroo I, Vian J (2012) Control of multiple UAVs for persistent surveillance: algorithm and flight test results. IEEE Trans Control Syst Technol 20(5):1236–1251. https://doi.org/10.1109/tcst.2011.2167331 10. Dong L, Sun D, Han G, Li X, Hu Q, Shu L (2020) Velocity-free localization of autonomous driverless vehicles in underground intelligent mines. IEEE Trans Veh Technol 69(9):9292– 9303. https://doi.org/10.1109/tvt.2020.2970842 11. Fascista A, Ciccarese G, Coluccia A, Ricci G (2018) Angle of arrival-based cooperative positioning for smart vehicles. IEEE Trans Intell Transp Syst 19(9):2880–2892. https://doi.org/10. 1109/tits.2017.2769488 12. Arafat MY, Moh S (2019) Localization and clustering based on swarm intelligence in UAV networks for emergency communications. IEEE Internet Things J 6(5):8958–8976. https://doi. org/10.1109/jiot.2019.2925567 13. Zheng J, Yang T, Liu H, Su T, Wan L (2021) Accurate detection and localization of unmanned aerial vehicle swarms-enabled mobile edge computing system. IEEE Trans Industr Inf 17(7):5059–5067. https://doi.org/10.1109/tii.2020.3015730 14. Hu J, Niu H, Carrasco J, Lennox B, Arvin F (2022) Fault-tolerant cooperative navigation of networked UAV swarms for forest fire monitoring. Aerosp Sci Technol 123. https://doi.org/10. 1016/j.ast.2022.107494

References

213

15. Wang H, Chen M, Fu P (2021) A distributed fault-tolerant mechanism for mission-oriented unmanned aerial vehicle swarms. Int J Commun Syst 34(8). https://doi.org/10.1002/dac.4789 16. Wang H, Zhao H, Zhang J, Ma D, Li J, Wei J (2020) Survey on unmanned aerial vehicle networks: a cyber physical system perspective. IEEE Commun Surv Tutorials 22(2):1027– 1070. https://doi.org/10.1109/comst.2019.2962207 17. Shakhatreh H, Sawalmeh AH, Al-Fuqaha A, Dou Z, Almaita E, Khalil I, Othman NS, Khreishah A, Guizani M (2019) Unmanned aerial vehicles (UAVs): a survey on civil applications and key research challenges. IEEE Access 7:48572–48634. https://doi.org/10.1109/access.2019.290 9530 18. Golestan K, Sattar F, Karray F, Kamel M, Seifzadeh S (2015) Localization in vehicular ad hoc networks using data fusion and V2V communication. Comput Commun 71:61–72. https://doi. org/10.1016/j.comcom.2015.07.020 19. Ju C, Son HI (2018) Multiple UAV systems for agricultural applications: control, implementation, and evaluation. Electronics 7(9). https://doi.org/10.3390/electronics7090162