SPACECRAFT ATTITUDE DETERMINA TlON AND CONTROL

ASTROPHYSICS AND SPACE SCIENCE LIBRARY

VOLUME 73

SPACECRAFT

ATTITUDE DETERMINATION

AND

CONTROL

Edited by

JAMES R. WERTZ

Microcosm, Inc., Torrance ,CA

Written by

Members of the Technical Staff Attitude Systems Operation

Computer Sciences Corporation

Preparation of this material was supported by the Attitude Determination and Control Section, Goddard Space Flight Center, National Aeronautics and Space Administration under Contract No. NAS 5-11999 and by the System Sciences Division, Computer Sciences Corporation.

KLUWER ACADEMIC PUBLISHERS DORDRECHT f BOSTON f LONDON

library of Congress Cataloging in Publication Data

Computer Science Corporation. Attitude Systems Operation. Spacecraft attitude determination and control.

(Astrophysics and space science library ; v. 73) 'Contract no. NAS 5-11999.' Bibliography: p. Includes index. 1. Space vehicles-Attitude control systems. 2. Space vehicles-Guidance

systems. I. Wertz, James R. II. Title. III. Series. TL3260.C65 1978 629.47'42 78-23657 ISBN-13: 978-90-277-1204-2 e-ISBN-13: 978-94-009-9907-7 DOl: 10.1007/978-94-009-9907-7

Published by D. Reidel Publishing Company, P. O. Box 17, Dordrecht, Holland.

Sold and distributed in the U.S.A., Canada and Mexico by D. Reidel Publishir,g Company, Inc.

Lincoln Building, 160 Old Derby Street, Hingham, Mass. 02043, U.S.A.

Preparation of this material was supported by the Attitude Determination and Control Section, Goddard Space Flight Center, National Aeronautics and Space Administration under Contract No. NAS 5-11999 and by the System Sciences Division, Computer Sciences Corporation.

(Reprinted 1980, 1984, 1985, 1986, 1988, 1991, 1994, 1995, 2000, 2002)

All Rights Reserved Copyright © 1978 by D. Reidel Publishing Company, Dordrecht, Holland

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical

including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner

LIST OF CONTRIBUTING AUTHORS

All the authors are members of the technical staff in the Attitude Systems Operation, System Sciences Division, Computer Sciences Corporation. Sections written by each author are in brackets.

John Aiello-B.S. (Astronomy), Villanova University [5.2, Appendix G]

Jawaid Bashir-Ph.D. (Aerospace Engineering), M.S. (Electrical Engineering), Uni­versity of Colorado; B.S. (Electrical Engineering), Karachi University, Pak­istan [I8.I]

Robert M. Beard-M.S. (Mathematics), B.S. (Physics), Auburn University [16.3]

Bruce T. Blaylock-M.S. (Chemistry), University of Virginia; 8.S. (Chemistry), Eastern Montana College [6.3]

Lily C. Chen-'-Ph.D. (Physics), University of Wisconsin, Madison; M.S. (Physics), University of Cincinnati; B.S. (Physics), National Taiwan University [7.1, 11.3, 11.4, 11.5, Chapter 10]

Roger M. Davis-M.S. (Mechanical Engineering), Northeastern University; B.S. (Mechanical Engineering), University of Connecticut [16.4]

Demosthenes Dialetis-Ph.D. (Physics), University of Rochester; B.Sc. (Physics), University of Athens, Greece [16.4]

Lawrence Fallon, III-Ph.D., M.S. (Materials Science), University of Virginia; B.S. (Engineering Physics), Loyola College [6.4, 6.5, 7.6, 7.8, 13.4, 13.5, 21.3, Appendix D]

B. L. Gambhir-Ph.D. (Physics), University of Maryland; M.Sc. (Physics), B.Sc. (Physics, Mathematics, English), Punjab University, India [6 . .1, 19.1]

David M. Gottlieb-Ph.D. (Astronomy), University of Maryland; B.A. (Mathema­tics), Johns Hopkins University [5.3, 5.6, 7.7]

Mihaly G. Grell-M.S. (Physics), University of Sciences, Budapest [19.2]

Dale Headrick-Ph.D., M.S. (Physics), Yale University; B.S. (Physics) Louisiana State University [6.6, 18.2, 19.4]

Steven G. Hotovy-Ph.D. (Mathematics), University of Colorado; B.S. (Mathema­tics), University of Notre Dame [7.2, 13.2, 13.3]

James S. Legg, Jr.-M.S. (Physics), University of North Carolina; A.B. (Physics, Mathematics), Washington and Lee University [8.1,8.3,8.4,9.1]

Gerald M. Lemer-Ph.D. (Physics), University of Maryland; B.A. (Physics), Johns Hopkins University [6.1, 6.2, 6.9, 7.1, 7.5, 9.2, 9.3, 12.2, 12.3, 18.1, 18.3, 19.5, Appendix F]

Menachem Levitas-Ph.D. (Physics), University of Virginia; B.S. (Physics), Uni­versity of Portland [7.3, 17.3]

K. Liu-B.S. (Physics), National Taiwan University [4.3]

vi LIST OF CONTRIBUTING AUTHORS

F. L. Markley-Ph.D. (Physics), University of California, Berkeley; B.E.P. (Engi­neering Physics), Cornell University [7.4, 7.9: 12.1, 15.2, 16.1, 16.2, 17.1, Appendix C]

Prafulla K. Misra-Ph.D., M.S. (Electrical Engineering), University of Maryland; B.Tech. (Electrical Engineering), Indian Institute of Technology, India [6.9]

Janet Niblack-M.A. (Mathematics), University of Texas; B.A. (Mathematics), Florida State University [S.2]

Michael Plett-Ph.D. (Physics), University of Virginia; B.S. (Physics), University of Cincinnati [5.1, 16.3, Appendix H]

Paul V. Rigterink-Ph.D. (Astronomy), University of Pennsylvania; B.A. (Mathematics), Carleton College [13.4]

John N. Rowe-Ph.D., M.S. (Electrical Engineering), Pennsylvania State Univer­sity; M.A. (Physics), Western Michigan University; B.A. (Physics), Oakland University, Michigan [4.4, 5.4, 5.5]

Ashok K. Saxena-M.S. (Aerospace Engineering), Virginia Polytechnic Institute and State University; B.E. (Mechanical Engineering), Jadavpur Univer­sity, India [1S.4, Appendix I]

Myron A. Shear-M.S. (Physics), University of Illinois; B.A. (Physics, Chemistry), Harvard University [20.1, 20.3, 21.1]

Malcolm D. Shuster-Ph.D. (Physics), University of Maryland; S.B. (Physics), Massachusetts Institute of Technology [19.2]

Peter M. Smith-Ph.D. (Chemistry), Georgetown University; M.Sc. (Spectroscopy), B.Sc. (Chemistry), Manchester University, England [11.1, 11.2, 21.4]

Des R. Sood-D. Eng. Sc. (Mechanical Engineering), Columbia University; M.S. (Mechanical Engineering), Roorkee University, India; B.S. (Mechanical En­gineering), Delhi University, India [6.7, 19.1]

C. B. Spence, Jr.-Ph.D., M.S. (Physics), College of William and Mary; B.S. (Physics), University of Richmond [17.1, 17.2]

Conrad R. Sturch-Ph.D. (Astronomy), University of California, Berkeley; M.S., B.A. (Physics), Miami University, Ohio [Appendix J]

Gyanendra K. Tandon-Ph.D. (Physics), Yale University; M.Sc. (Electronics), B.Sc. (Physics, Mathematics), Allahabad University, India [17.4, 21.2, Appendix E]

Vincent H. Tate-M.S., B.S. (Aerospace Engineering), Pennsylvania State Univer­sity [15.3]

James R. Wertz-Ph.D. (Physics), University of Texas, Austin; S.B. (Physics), Massachusetts InstituteofTechnology[4.1,4.2,9.4, 11.3, 11.4, 11.5, 13.1, 15.1; Chapters 1,2, 3, 10, 14, 22; Appendices A, B, K, L, M]

Robert S. Williams-Ph.D. (Physics), University of Maryland; B.S. (Physics), California Institute of Technology [6.S, 7.10, 19.3]

Kay yong-Ph.D., M.S. (Mechanical Engineering), Rensselaer Polytechnic Insti­tute; B.S. (Mechanical Engineering), National Cheng-Kung University, Tai­wan. [5.2]

FOREWORD

Roger D. Werking Head, Attitude Determination and Control Section National Aeronautics and Space Administration/

Goddard Space Flight Center

Extensiye work has been done for many years in the areas of attitude determination, attitude prediction, and attitude control. During this time, it has been difficult to obtain reference material that provided a comprehensive overview of attitude support activities. This lack of reference material has made it difficult for those not intimately involved in attitude functions to become acquainted with the ideas and activities which are essential to understanding the various aspects of spacecraft attitude support. As a result, I felt the need for a document which could be used by a variety of persons to obtain an understanding of the work which has been done in support of spacecraft attitude objectives. It is believed that this book, prepared by the Computer Sciences Corporation under the able direction of Dr. James Wertz, provides this type of reference.

This book can serve as a reference for individuals involved in mission planning, attitude determination, and attitude dynamics; an introductory textbook for stu­dents and professionals starting in this field; an information source for experimen­ters or others involved in spacecraft-related work who need information on spacecraft orientation and how it is determined, but who have neither the time nor the resources to pursue the varied literature on this subject; and a tool for encouraging those who could expand this discipline to do so, because much remains to be done to satisfy future needs.

The primary purpose of this book is to provide short descriptions of various aspects of attitude determination, prediction, and control with emphasis on the ground support which presently must be provided. The initial chapters provide the necessary background and describe environment models and spacecraft attitude hardware. The authors then present the fundamentals that are essential to a basic understanding of the activities in this area as well as flight-proven concepts which can be used as a basis for operational state-of-the-art activities or as a stepping stone to improved processes. In a limited fashion, Chapter 22 presents future activities which affect or are a part of spacecraft attitude support. It is not the intention of this book to advance the state of the art but rather to call attention to the work that has been done in the successful support of spacecraft attitude requirements and to stimulate future thinking.

PREFACE

The purpose of this book is to summarize the ideas, data, and analytic techniques needed for spacecraft attitude determination and control in a form that is readable to someone with little or no previous background in this specific area. It has been prepared for those who have a physics or engineering background and therefore are familiar with the elementary aspects of Newtonian mechanics, vector algebra, and calculus. Summaries of pertinent facts in other are~s are presented without proof.

This material has been prepared by 35 members of the technical staff of the Attitude Systems Operation of the System Sciences Division of Computer Sciences Corporation (CSC) for the Attitude Determination and Control Section of NASA's Goddard Space Flight Center. It necessarily reflects our experience in this area and therefore is concerned primarily with unmanned, Earth-orbiting spacecraft. None­theless, the basic principles are sufficiently broad to be applicable to nearly any spacecraft.

Chapters l, 2, 3, 10, and 15 provide introductory material at a more qualitative level than that of the other chapters. The suggested order of reading depends on the background and interest of the reader:

I. Those who are primarily concerned with mission planning and analysis and who would like a general overview should read Chapters I, 2, 3, 10, 15, and 22.

2. Those who are primarly interested in attitude determination should read Chapters I and 2, Sections 3.1 through 3.3, Chapter 10, Appendices A and B, and Chapters II through 14.

3. Those who are primarily interested in attitude dynamics and control should read Chapters I and 2, Sections 3.1 through 3.3 and 12.1, Chapters 15 through 19, and Appendices C through H.

4. Those who are primarily interested in the space environment, attitude hardware, and data acquisition should read Chapters I through 9 and Appendices G through J.

5. Those who are primarily interested in the development of mission related software should read Chapters I and 2, Sections 3.1 through 3.3, Chapters 20 and 21, Chapters 8 and 9, Sections Il.l and 11.2, and Chapters 12, 4, 5, and 7.

The International System of Units is used throughout the book and a detailed list of conversion factors is given in Appendix K. Because nearly all numerical work is now done with computers or hand calculators, all constants are given to essentially their full available accuracy. Acronyms have generally been avoided, except for spacecraft names. The full spacecraft names are listed in Appendix I, which also provides a cross-referenced list of the attitude hardware used on various spacecraft, including all those used as examples throughout the text.

_ Because much of the material presented here has not appeared in the open literature, maJlY of the references are to corporate or government documents of limited Circulation.- To -improve the exchange of information, Computer Sciences

PREFACE ix

Corporation reports referenced herein are available for interlibrary loan through your librarian by writing to:

Head Librarian Technical Information Center Computer Sciences Corporation 8728 Colesville Road Silver Spring, MD 20910

Standard computer subroutines for attitude analysis cited throughout the book are available from:

COSMIC Barrow Hall University of Georgia Athens, GA 3060 I

by asking for Program Number GSC12421, Attitude Determination and Control Utilities. Each of these subroutines is briefly described in Section 20.3.

The preparation of this book was a cooperative effort on the part of many individuals. It is a pleasure to acknowledge the help of Robert Coady, Roger Werking, and Richard Nankervis of NASA's Goddard Space Flight Center, who initiated and supported this project. At Computer Sciences Corporation, direction, support, and help were provided by Richard Taylor, David Stewart, Michael Plett, and Gerald Lerner. In addition to the authors, who provided extensive review of each other's sections, particularly helpful reviews were provided by Peter Batay­Csorba, Stanley Brown, Charles Gray, Lawrence Gunshol, William Hogan, Whit­tak Huang, James Keat, Anne Long, J. A. Massart, Donald Novak, Franklin VanLandingham, Donna Walter, and Chao Yang.* Considerable assistance in obtaining reference material was supplied by Gloria Urban and the staff of the CSC Technical Information Center. Jo Border and the CSC Publications Depart­ment supplied a consistently high quality of support in editing, composition, and graphics. Jerry Greeson and the Graphics Department staff prepared nearly all of the 450 illustrations in the book. Figures 17-4, 18-19, and 19-15 are reproduced by permission of the American Institute of Aeronautics and Astronautics. Anne Smith edited the final version of the manuscript and Julie Langston, the publications editor for the manuscript, did an outstanding and professional job of translating multiple early drafts into grammatical English, handling the numerous details of producing a finished manuscript, and preparing the final layout.

Silver Spring, Maryland July 1978

James R. Wertz

*The editor would appreciate that any residual errors be brought to his attention at the following address: Computer Science Corporation, 8728 Colesville Road, Silver Spring, Maryland 20910.

x

STANDARD NOTATION

Standard notation developed in the first three chapters and used throughout the remainder of this book is given below. Unfortunately, notation, coordinate systems, and even definitions are frequently used differently for different spacecraft. The definition of attitude and the orientation of the roll, pitch, and yaw axes vary more than most quantities.

Alphabets and Type Styles

All arc lengths are lowercase Greek, ()

All rotation angles are uppercase Greek, A

These are not exclusive because common usage may require Greek characters for some non angular measure.

All n-vectors are bold face, E or x. All quaternions are boldface italics, q. Boldface is used exclusively for vectors, quaternions, or the identity matrix, 1.

Points on the sky are labeled with uppercase italic Roman, E. Antipodal points (the antipode is the point 180 deg away from a given point) have a - I superscript, E - I.

Vectors and Matrices

The treatment of vectors is illustrated by the Sun vector:

S S-I

S S lSI or S

point on the celestial sphere in the direction of the Sun point on the celestial sphere opposite the direction of the Sun vector from spacecraft to Sun unit vector from the spacecraft toward the Sun magnitude of the Sun vector. S is used if there is no possibility of ambiguity. Otherwise lSI is used either the ith component of the Sun vector or an arbitrary component of the Sun vector

Both uppercase and lowercase letters win be used· for vectors. Matrices will be represented by uppercase Roman letters with the following notation:

M or [Mij] det M or IMijl M- 1

MT

Mij

1 I

matrix or second-rank tensor determinant of M inverse of M transpose of M either an arbitrary component of M or the component in the ith row,jth column identity matrix moment of inertia tensor

STANDARD NOTATION

Coordinate Systems

Spacecraft-Centered Celestial Coordinates:

a right ascension /) declination

Body-Fixed Coordinates for Spinning Spacecraft:

cp A

or ()

azimuth elevation coelevation (measured from the spin axis)

xi

No standard definitions apply for body-fixed coordinates for three-axis stabilized spacecraft.

Roll, Pitch, Yaw:

Yaw axis, Y, toward the nadir Pitch axis, P, toward n~gati'ye ~rbit normal Roll axis, R, such that R = P X Y Unfortunately, the roll, pitch, and yaw axes do not have generally accepted meanings in spaceflight. Usage depends on the context.

~r roll angle, measured about R ~p pitch angle, measured about P ~y yaw angle, measured about Y

Positive roll, pitch, and yaw are right-handed rotations about their respective axes.

Standard Symbols

The letter "t" in all forms (Roman, Greek, uppercase, and lowercase) is used only for time or time intervals, except that superscript T is used to indicate the transpose of a matrix.

Orbital elements:

a semimajor axis e eccentricity

inclination w argument of perigee n right ascension of the ascending node Mo mean anomaly at epoch To epoch time P orbital period

xii STANDARD NOTATION

Vectors:

L angular momentum N torque B magnetic field E nadir vector S Sun vector A attitude vector H horizon crossing vector; HI and Ho for in-crossing and out-crossing to) angular velocity vector x m-dimensional state vector y n-dimensional observation vector

Angles:

fJ Sun angle = Sun/attitude angular separation 1/ Nadir angle = Earth center/attitude angular separation <I> Rotation angle from the Sun to the center of the Earth about the attiutde 1J; Sun/Earth center angular separation

Astronomical Symbols:

EEl Earth 0) Sun <y' Vernal Equinox

Additional astronomical symbols are defined in Fig. 3-10.

Miscellaneous

Il indicates an arbitrary interval, as Ilt = t2 - t 1

6 indicates an infinitesimal interval in which first-order approximations may be used, as L=Lo+N6t

A dot over any symbol indicates dffferentiation with respect to time, i.e., x= dx/dt.

The Kronecker Delta, 6/, is defined as

6/=1 fori=j

6/ = 0 for i =1= j

The Dirac delta function, 6D (x - xo), is defined by

6D (x-xo)=O for x =1= Xo

List of Contributing Authors Foreword Preface Standard Notation

TABLE OF CONTENTS

PART I-BACKGROUND

Xlll

V

VII

Vlll

X

1. INTRODUCTION 1 1.1 Representative Mission Profile 3 1.2 Representative Examples of Attitude Determination and Control 10 1.3 Methods of Attitude Determination and Control 16 1.4 Time Measurements 18

2. ATTITUDE GEOMETRY 22 2.1 The Spacecraft-Centered Celestial Sphere 22 2.2 Coordinate Systems 24 2.3 Elementary Spherical Geometry 31

3. SUMMARY OF ORBIT PROPERTIES AND TERMINOLOGY 36 3.1 Keplerian Orbits 36 3.2 Planetary and Lunar Orbits 48 3.3 Spacecraft Orbits 52 3.4 Orbit Perturbations 62 3.5 Viewing and Lighting Conditions 71

4. MODELING THE EARTH 82 4.1 Appearance of the Earth at Visual Wavelengths 83 4.2 Appearance of the Earth at Infrared Wavelengths 90 4.3 Earth Oblateness Modeling 98 4.4 Modeling the Structure of the Upper Atmosphere 106

5. MODELING THE SPACE ENVIRONMENT 113 5.1 The Earth's Magnetic Field 113 5.2 The Earth's Gravitational Field 123 5.3 Solar Radiation and the Solar Wind 129 5.4 Modeling the Position of the Spacecraft 132 5.5 Modeling the Positions of the Sun, Moon, and Planets 138 5.6 Modeling Stellar Positions and Characteristics 143

PART II-ATTITUDE HARDWARE AND DATA ACQUISITION

6. ATTITUDE HARDWARE 6.1 Sun Sensors 6.2 Horizon Sensors 6.3 Magnetometers 6.4 Star Sensors

155 155 166 180 184

XIV TABLE OF CONTENTS

6.5 Gyroscopes 196 6.6 Momentum and Reaction Wheels 201 6.7 Magnetic Coils 204 6.8 Gas Jets 206 6.9 Onboard Computers 210

7. MATHEMATICAL MODELS OF ATTITUDE HARDWARE 217 7.1 Sun Sensor Models 218 7.2 Horizon Sensor Models 230 7.3 Sun Sensor/Horizon Sensor Rotation Angle Models 237 7.4 Modeling Sensor Electronics 242 7.5 Magnetometer Models 249 7.6 Star Sensor Models 254 7.7 Star Identification Techniques 259 7.8 Gyroscope Models 266 7.9 Reaction Wheel Models 270 7.10 Modeling Gas Jet Control Systems 272

8. DATA TRANSMISSION AND PREPROCESSING 278 8.1 Data Transmission 278 8.2 Spacecraft Telemetry 293 8.3 Time Tagging 298 8.4 Telemetry Processors 304

9. DATA V ALIDA TION AND ADJUSTMENT 310 9.1 Validation of Discrete Telemetry Data 312 9.2 Data Validation and Smoothing 315 9.3 Scalar Checking 328 9.4 Data Selection Requiring Attitude Information 334

PART III-A TIITUDE DETERMINATION

10. GEOMETRICAL BASIS OF ATTITUDE DETERMINATION 343 10.1 Single-Axis Attitude 344 10.2 Arc-Length Measurements 346 10.3 Rotation Angle Measurements 349 10.4 Correlation Angles 353 10.5 Compound Measurements-Sun to Earth Horizon Crossing

Rotation Angle 357 10.6 Three-Axis Attitude 359

II. SINGLE-AXIS ATTITUDE DETERMINATION METHODS 362 Il.l Methods for Spinning Spacecraft 363 11.2 Solution Averaging 370 11.3 Single-Axis Attitude Determination Accuracy 373 11.4 Geometrical Limitations on Single-Axis Attitude Accuracy 389 11.5 Attitude Uncertainty Due to Systematic Errors 402

TABLE OF CONTENTS xv

12. THREE-AXIS ATTITUDE DETERMINATION METHODS 410 12.1 Parameterization of the Attitude 410 12.2 Three-Axis Attitude Determination 420 12.3 Covariance Analysis 429

13. STATE ESTIMATION ATTITUDE DETERMINATION METHODS 436 13.1 Deterministic Versus State Estimation Attitude Methods 436 13.2 State Vectors 438 13.3 Observation Models 443 13.4 Introduction to Estimation Theory 447 13.5 Recursive Least-Squares Estimators and Kalman Filters 459

14. EV ALUA TION AND USE OF STATE ESTIMATORS 471 14.1 Prelaunch Evaluation of State Estimators 471 14.2 Operational Bias Determination 473 14.3 Limitations on State Vector Observability 476

PART IV-ATTITUDE DYNAMICS AND CONTROL

15. INTRODUCTION TO ATTITUDE DYNAMICS AND CONTROL 487 15.1 Torque-Free Motion 487 15.2 Response to Torques 498 15.3 Introduction to Attitude Control 502

16. ATTITUDE DYNAMICS 510 16.1 Equations of Motion 510 16.2 Motion of a Rigid Spacecraft 523 16.3 Spacecraft Nutation 534 16.4 Flexible Spacecraft Dynamics 548

17. ATTITUDE PREDICTION 558 17.1 A ttitude Propagation 558 17.2 Environmental Torques 566 17.3 Modeling Internal Torques 576 17.4 Modeling Torques Due to Orbit Maneuvers 580

18. ATTITUDE STABILIZATION 588 18.1 Automatic Feedback Control 588 18.2 Momentum and Reaction Wheels 600 18.3 Autonomous Attitude Stabilization Systems 604 18.4 Nutation and Libration Damping 625

19. ATTITUDE MANEUVER CONTROL 636 19.1 Spin Axis Magnetic Coil Maneuvers 636 19.2 Spin Plane Magnetic Coil Maneuvers 642 19.3 Gas Jet Maneuvers 649 19.4 Inertial Guidance Maneuvers 655 19.5 Attitude Acquisition 661

xvi TABLE OF CONTENTS

PART V-MISSION SUPPORT

20. SOFTWARE SYSTEM DEVELOPMENT 681 20.1 Safeguards Appropriate for Mission Support Software 681 20.2 Use of Graphic Support Systems 686 20.3 Utility Subroutines 690

21. SOFTWARE SYSTEM STRUCTURE 696 21.1 General Structure for A ttitude Software Systems 696 21.2 Communications Technology Satellite Attitude Support System 700 21.3 Star Sensor Attitude Determination System 703 21.4 Attitude Data Simulators 709

22. DISCUSSION 714

PART VI-APPENDICES

APPENDIX A-SPHERICAL GEOMETRY 727 APPENDIX B-CONSTRUCTION OF GLOBAL GEOMETRY PLOTS 737 APPENDIX C-MATRIX AND VECTOR ALGEBRA 744 APPENDIX D-QUATERNIONS 758 APPENDIX E-COORDINATE TRANSFORMATIONS 760 APPENDIX F-THE LAPLACE TRANSFORM 767 APPENDIX G-SPHERICAL HARMONICS 775 APPENDIX H-MAGNETIC FIELD MODELS 779 APPENDIX I-SPACECRAFT ATTITUDE DETERMINATION

AND CONTROL SYSTEMS 787 APPENDIX J -TIME MEASUREMENT SYSTEMS 798 APPENDIX K-METRIC CONVERSION FACTORS 807 APPENDIX L-SOLAR SYSTEM CONSTANTS 814 APPENDIX M-FUNDAMENTAL PHYSICAL CONSTANTS 826 Index 830