Springer Tracts in Mechanical Engineering Panfeng Huang Fan Zhang Theory and Applications of Multi-Tethers in Space Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA Springer Tracts in Mechanical Engineering (STME) publishes the latest develop- ments in Mechanical Engineering - quickly, informally and with high quality. 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More information about this series at http://www.springer.com/series/11693 Panfeng Huang Fan Zhang (cid:129) Theory and Applications of Multi-Tethers in Space 123 Panfeng Huang FanZhang National Key Laboratory of Aerospace National Key Laboratory of Aerospace Flight Dynamics, Schoolof Astronautics, Flight Dynamics, Schoolof Astronautics, Research Centerfor Intelligent Robotics Research Centerfor Intelligent Robotics Northwestern Polytechnical University Northwestern Polytechnical University Xi’an,Shaanxi, China Xi’an,Shaanxi, China ISSN 2195-9862 ISSN 2195-9870 (electronic) SpringerTracts inMechanical Engineering ISBN978-981-15-0386-3 ISBN978-981-15-0387-0 (eBook) https://doi.org/10.1007/978-981-15-0387-0 ©SpringerNatureSingaporePteLtd.2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface Theory and Applications of Multi-Tethers in Space provides a comprehensive overview of the recently developed space multi-tethers, including maneuverable space tethered net and space tethered formation, with detailed system description, dynamics modeling and analysis, and controller design. For each application of space multi-tethered system, detailed derivatives are given to describe and analyze the mathematical model of the system, and then, different control schemes are designed and proved for different problems of the application. In the textbook, Newton and Lagrangian mechanics are used for dynamics modeling, Hamilton mechanics and Poincare surface of section are introduced for dynamics analysis, and both of centralized and distributed controllers are employed to figure out the formation question of the multi-tethered system. Besides the equations and words, 3D design drawing, schematic diagram, control scheme block, Table et al. is used for easy reading and understanding. The graduate students in related research area can systematically learn space multi-tethered system and its applications, and we hope other researchers could be inspired by this technical book and make much more contribution to this topic. This book is not prepared as a collection of existing papers on dynamics and controlofmulti-space tethers; rather,wefirstestablishthestructurethat thetheory and applications of multi-space tethers should follow, and then fill in the details. The book is organized and presented in a widely accessible fashion. For example, the related control problems of maneuverable space tethered net are introduced under a space mission sequence, which has been described at the very beginning ofthepart.Inordertohelpthereaderwhoisinterestedaboutthefieldandwantsto studytheresultsbyhimselfortoexploitthespacetether’sapplicationsforhisown needs, the calculations and derived equations are detailed explained in this text. Thisbookprovidesthereaderswith extensive material fromthefirstconceptof spacetetheruptothecutting-edgeartintheareaoftheapplicationsofmulti-tethers in space presented in a pedagogical fashion. In addition to the emphasis on prac- ticality, many theoretical equations and theorems (which may have practical rele- vance and importance) are numerically verified using MATLAB/Simulink software. This book, which is based on a major developed at the Northwestern v vi Preface Polytechnical University of Space Science and Technology, is appropriate as a textbook for graduate and senior students, or as a self-study reference book for practicing engineers, who are with a basic knowledge of theoretical mechanics, classical control theory, and some knowledge of spacecraft orbital dynamics. This book is intended to be intelligently challenging for students. Organization This book consists of nine chapters and two appendixes which consisted of three parts including the Introduction (Chap. 1), Maneuverable Space Tethered Net (Chaps. 2–5), and Space Tethered Formation (Chaps. 6–9), in which two parts represent two classic applications of multi-tethers in space. Chapter1startswithabriefhistoricaloverviewofspacetether.Themotivations ofthe space tether’sstudy andpreviousresearchincluding theclassicsingle space tether and multi-tethers have been introduced. Chapter 2 opens the second part of the book, which is focused on the space tethered net. This part is organized based on a space mission of active debris removal.InChap.2,acomplexspacetetherednetsystemisdescribedindetailand modeled by mathematical equations. Then in Chaps. 3–5, it is organized by the mission sequence, namely, folding and storing, releasing, and approaching. The folding criteria of the flexible net are introduced in Chap. 3, and corresponding unfolding characteristics are studied. In Chap. 4, the control scheme during approaching for both configuration-keeping and configuration-maneuvering are studied. Different from the centralized control scheme studied in Chap. 4, dis- tributed control is addressed to solve the same approaching questions. The readers can learn and study further control strategies based these two chapters. Chapter 6 is the first of last four chapters, which compose the third part of the book.Inthispart,anotherimportantapplicationofmulti-spacetether,namelyspace tethered formation is researched. Two different formations, closed-chain and hub-spoke formations, are modeled and analyzed in Chap. 6. The control schemes of these two formations are presented in the following chapters. In Chap. 7, the formation-keepingcontrolschemeisstudied. Besides thetraditional thruster-based controlscheme,anovelthrusterandtether-tension-basedcontrollerisalsoproposed andanalyzed.Chapter8presentsthedeploymentandretrievalcontrolofhub-spoke system based on natural motion without any controller. In Chap. 9, advantage control schemes of the hub-spoke system for formation-keeping are proposed including error-based and time-based coordinated control. Xi’an, China Panfeng Huang 2019 Fan Zhang Acknowledgements This book is based on the research supported by the National Natural Science FoundationofChina.Forthefacilitiesanddaily-operationsupporting,thankstothe National Key Laboratory of Aerospace Flight Dynamics, Research Center for Intelligent Robotics, and School of Astronautics of Northwestern Polytechnical University. WeowespecialthankstoMr.KonstantinEduardovichTsiolkovskyforhismuch fruitful work on the creation of the theory of jet aircraft, multistage rockets, gas turbine engine, and of course, the SPACE TETHER. Sincere thanks to our friends for their sagacious companionship, colleagues for scientific and friendly support and in particular Zhongjie Meng, Yizhai Zhang, Zhengxiong Liu, Gangqi Dong, and Zhiqiang Ma. Thanks as well to Yakun Zhao, Ya Liu, Yongxin Hu, and other students in the Research Center for Intelligent Robotics, who helped in proofreading and providing valuable feedback. We are grateful to all their help. Finally, many thanks to our families for their encouragements and tender supports. vii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Form Single Tether to Multi-tethers. . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Research on Space Tethered Net. . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Tethered Space Net in Early Applications. . . . . . . . . . . . . 3 1.2.2 Tethered Space Net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Maneuverable Tethered Space Net . . . . . . . . . . . . . . . . . . 5 1.3 The Research on Space Tethered Formation. . . . . . . . . . . . . . . . . 6 1.3.1 Structure and Configuration . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Dynamics and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Formation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Scope and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Part I Maneuverable Space Tethered Net 2 Dynamics Modeling and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Mission and System Description . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Mission Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Dynamics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Coordinate Description. . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Dynamics Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Dynamics Analysis of the Flexible Net . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Dynamics Simplification . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Dynamics Analysis of the Symmetrical Flexible Net . . . . . 25 2.3.3 Dynamics Analysis of the Asymmetrical Flexible Net . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ix x Contents 3 Folding Pattern and Releasing Characteristics . . . . . . . . . . . . . . . . . 33 3.1 Folding Pattern of the Stored Flexible Net . . . . . . . . . . . . . . . . . . 33 3.1.1 Selection Criteria of the Folding Pattern . . . . . . . . . . . . . . 33 3.1.2 Square Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Star Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.4 Cross-Shaped Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Natural Flight of the System Without Control . . . . . . . . . . . . . . . 42 3.3 Shooting Conditions of the Maneuverable Units. . . . . . . . . . . . . . 44 3.3.1 Shooting Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Shooting Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4 Centralized Deployment Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Configuration-Keeping Control After Releasing . . . . . . . . . . . . . . 47 4.1.1 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.3 Stability Analysis of the Closed-Loop System. . . . . . . . . . 55 4.1.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Configuration-Keeping Control Under the Unknown Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.1 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.4 Stability Analysis of the Closed-Loop System. . . . . . . . . . 75 4.2.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 Coordinated Control for the Maneuverable Space Tethered Net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.1 Improved Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5 Distributed Deployment Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 Distributed Configuration-Maneuvering Control After Releasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.1.1 Orbital Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.1.2 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1.4 Stability Analysis of the Closed-Loop System. . . . . . . . . . 121 5.1.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.2 Distributed Configuration-Maneuvering Control with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2.1 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133