1 2 3 4 5 6 Orbital mechanics and 7 8 formation fl ying 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 41R 2244330077..iinnddbb ii 1188//0022//1122 11::1155 PPMM Related titles: Aerodynamic Measurements: From physical principles to turnkey instrumentation (ISBN 978-1-84569-992-5) Aerodynamic Measurements will present to its reader a comprehensive treatment of the theoretical bases which experimental techniques used in aerodynamics are based on. Limitations of each method in terms of accuracy, response time and complexity will also be addressed. This book will serve as a guide in choosing the most pertinent technique for each type of fl ow fi eld including: 1D, 2D, 3D, steady or unsteady, subsonic, supersonic or hypersonic. This is currently no book that presents as many techniques as those presented in this volume, with such a critical review of the various methods offered. Aerodynamic Measurements will help guide the reader to choose the most appropriate in each case and for each technique the evolution from old-fashioned mechanical process to the modern computerized version is reported in order to allow students and practitioners to understand what is behind the results shown on the coloured screen of their computers. Leveraging Information Technology for Optimal Aircraft Maintenance, Repair and Overhaul (ISBN 978-1-84569-982-6) Aimed at professionals in the aviation industry and students of Aircraft Maintenance, this book covers the use and management of technology in the Aviation Maintenance, Repair, and Overhaul (MRO) context. This book will provide a backdrop of current trends in the industry where Airlines are tending to retain their aircraft longer on the one hand, and rapidly introducing a new genre of aircraft such as the A380 and B787 into service. This book aims to provide industry professionals and students of Aviation MRO (at all levels) with the necessary principles, approaches and tools to respond effectively and effi ciently to the constant development of new technologies - both in general and within the Aviation MRO profession. It highlights the unique information requirements for Aviation MRO and delves into the detailed aspects of information needs from within the industry such as Engineering and Maintenance Planning, Aircraft/Engine/Component Confi gurations, Materials Management, Logistics, and Technical Documentation. It will highlight the difference between on-wing and off-wing maintenance with their unique information processing needs. This book is designed as a primer on IT services for Aircraft Engineering professionals and a handbook for IT professionals servicing this niche industry. Modelling and Simulation of Integrated Systems in Engineering: Issues of methodology, quality, testing and applications (ISBN 978-1-85709-078-2) This book places particular emphasis on issues of model quality and ideas of model testing and validation. Mathematical and computer-based models provide a foundation for explaining complex behaviour, for decision-making, for engineering design and for real-time simulators for research and training. Many engineering design techniques depend on suitable models. Assessment of the adequacy of a given model for an intended application is therefore critically important. Generic model structures and dependable libraries of sub-models that can be applied repeatedly are increasingly important. Applications are drawn from the fi elds of mechanical, aeronautical and control engineering, and mainly involve non-linear lumped- parameter models described by ordinary differential equations. Topics considered repeatedly in the context of these applications include: system identifi cation and parameter estimation; parameter sensitivity analysis; model optimisation; and inverse simulation. Details of these and other Woodhead Publishing books can be obtained by: visiting our web site at www.woodheadpublishing.com ● contacting Customer Services (e-mail: [email protected]; fax: +44 (0) 1223 ● 832819; tel.: +44 (0) 1223 499140; address: Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK) If you would like to receive information on forthcoming titles, please send your address details to Customer Services, at the address above. Please confi rm which subject areas you are interested in. 2244330077..iinnddbb iiii 1188//0022//1122 11::1155 PPMM Orbital mechanics and formation fl ying A digital control approach P A. C -L EDRO APÓ UGO and P M. B ETER AINUM 2244330077..iinnddbb iiiiii 1188//0022//1122 11::1155 PPMM Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 19102-3406, USA Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First published 2011, Woodhead Publishing Limited © P. A. Capó-Lugo and P. M. Bainum, 2011 The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfi lming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specifi c permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifi cation and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Control Number: 2011919191 Woodhead Publishing ISBN: 978-0-85709-054-6 (print) ISBN: 978-0-85709-387-5 (online) Typeset by Refi neCatch Ltd, Bungay, Suffolk Printed in the UK and USA 2244330077..iinnddbb iivv 1188//0022//1122 11::1155 PPMM 1 2 3 4 5 6 7 To my wife, my newborn son, and my family, thanks for their patience 8 and support. 9 Pedro A. Capó-Lugo 10 1 To my wife, my son, and his wife for their support and understanding. 2 Peter M. Bainum 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 41R 2244330077..iinnddbb vv 1188//0022//1122 11::1155 PPMM 2244330077..iinnddbb vvii 1188//0022//1122 11::1155 PPMM Contents List of fi gures xi List of tables xvii List of symbols xix Acknowledgements xxix Preface xxxi About the authors xxxiii 1 Introduction 1 1.1 Introduction to the book 1 1.2 Book division 2 1.3 References 4 2 Two body orbital motion 7 2.1 Introduction to orbital motion 7 2.2 Constraints and generalized coordinates 8 2.3 Lagrange’s equation 10 2.4 System of particles 11 2.5 Two body orbital motion problem 13 2.6 Orbital equations of motion 15 2.7 Energy and velocity of orbiting bodies 21 2.8 Escape velocity 24 2.9 Earth coordinate inertial (ECI) system 25 2.10 Period of an orbit 27 2.11 Development of Kepler’s equation 29 2.12 Suggested problems 31 2.13 References 34 vii 2244330077..iinnddbb vviiii 1188//0022//1122 11::1155 PPMM Orbital mechanics and formation fl ying 3 Orbital perturbations in the two body motion 37 3.1 Introduction to disturbance effects 37 3.2 Lagrange planetary equations 38 3.3 Perturbation due to the earth oblateness 41 3.4 The near-Earth atmosphere effects 52 3.5 Solar radiation pressure force 58 3.6 Other disturbance effects 67 3.7 Suggested problems 69 3.8 References 71 4 Frame rotations and quaternions 75 4.1 Introduction to rotations and quaternions 75 4.2 Two-dimensional frame rotations 76 4.3 Three-dimensional frame rotations 78 4.4 Example of frame rotations 80 4.5 Quaternion defi nition and rotations 85 4.6 Quaternion to Euler angle relations 88 4.7 Suggested problems 90 4.8 References 90 5 Rigid body motion 93 5.1 Introduction to attitude dynamics 93 5.2 Rate of change of a vector 95 5.3 Moment of inertia 97 5.4 Principal moments of inertia 101 5.5 Energy formulation 102 5.6 Rate of change of a quaternion 108 5.7 Ares V equations of motion 109 5.8 Suggested problems 117 5.9 References 119 viii 2244330077..iinnddbb vviiiiii 1188//0022//1122 11::1155 PPMM Contents 6 Environmental and actuator torques 121 6.1 Introduction to torque formulation 121 6.2 Environmental torques 122 6.3 Actuator (or control) torques 134 6.4 Suggested problems 151 6.5 References 152 7 Continuous and digital control systems 155 7.1 Introduction to methods of designing continuous and discrete control systems 155 7.2 Ares V equations of motion for fi rst stage fl ight 157 7.3 Continuous control formulation 158 7.4 Discrete control formulation 213 7.5 Adaptive and intelligent controls 233 7.6 Suggested problems 238 7.7 References 243 8 Example 247 8.1 Introduction to examples in spacecraft attitude dynamics and control 247 8.2 Nanosatellite problem defi nition 248 8.3 B-dot controller for fast corrections 253 8.4 Linear quadratic regulator for attitude correction 255 8.5 Linear quadratic regulator control weight design 262 8.6 Suggested problems 270 8.7 References 273 9 Formation fl ying 277 9.1 Introduction to formation fl ying 277 9.2 Tschauner–Hempel formulation 278 9.3 Clohessy–Wiltshire formulation 286 ix 2244330077..iinnddbb iixx 1188//0022//1122 11::1155 PPMM Orbital mechanics and formation fl ying 9.4 Earth oblateness and solar effects in formation fl ying 286 9.5 Lawden solution 293 9.6 Discrete optimal control problem for formation fl ying 294 9.7 Formation fl ying controller implementation 301 9.8 Suggested problems 316 9.9 References 318 10 Deployment procedure for a constellation 321 10.1 Introductory comments 321 10.2 Desired conditions of the satellites in the proposed tetrahedron constellation 322 10.3 Transfer from a circular orbit to the elliptical orbit (stage 1) 323 10.4 Station-keeping procedure (stage 2) 328 10.5 Deployment procedure for the tetrahedron constellation 329 10.6 Remarks 335 10.7 Suggested problems 335 10.8 References 336 11 Reconfi guration procedure for a constellation 339 11.1 Introduction to the reconfi guration process of a constellation 339 11.2 Data mining process of the Lagrange Planetary equations 341 11.3 Fuzzy logic controller 344 11.4 Phase I to II in-plane motion fuzzy logic control system 348 11.5 Phase II to III in-plane motion fuzzy logic controller 351 11.6 Out-of-plane motion correction 353 11.7 Some solutions for the reconfi guration procedures 356 11.8 Implementation of the fuzzy logic controller 365 11.9 Adaptive control scheme for reconfi guration procedure 365 11.10 Remarks 372 11.11 Suggested problems 373 11.12 References 375 x 2244330077..iinnddbb xx 1188//0022//1122 11::1155 PPMM