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Princeton Satellite Systems, Inc.

SPACECRAFTATTITUDE and ORBIT

CONTROL

Michael Paluszek, Pradeep Bhatta, Paul Griesemer,

Joseph Mueller and Stephanie Thomas

S E C O N D E D I T I O N

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Spacecraft Attitude and Orbit Control2nd Edition

Michael Paluszek, Pradeep Bhatta, Paul Griesemer,Joseph Mueller and Stephanie Thomas

Princeton Satellite Systems, Inc.

6 Market Street, Suite 926, Plainsboro NJhttp://www.psatellite.com

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Spacecraft Attitude and Orbit Control, 2nd Edition.

Copyright c© 1996-2009 by Princeton Satellite Systems, Inc. All rights reserved. No portion of this book may be reproduced inany form without the written permission of the publishers, with exception of brief quotations in reviews.

MATLAB R© is a registered trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant theaccuracy of the text or exercises in this book. This books use or discussion of MATLAB R© software or related products does notconstitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB R©software.

Other product or brand names are trademarks or registered trademarks of their respective holders.

ISBN 978-0-9654701-0-0

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ACKNOWLEDGEMENTS

This book evolved over 15 years starting when Princeton Satellite Systems was incorporated. It encompasses thecollective experience of the authors in spacecraft attitude and orbit control. This started with work on the NASASpace Shuttle at the Charles Stark Draper Laboratory and continued with work on numerous spacecraft at RCA AstroElectronics (which became GE Astro Space and is now part of Lockheed Martin.) Most of the recent work discussedin this book relates to work done by Princeton Satellite Systems on numerous Air Force, NASA and commercialcontracts.

The authors would like to thank their many colleagues with whom they have worked over the past 30 years who insome way influenced the work in this book. This includes collaborators on projects and managers of projects onwhich Princeton Satellite Systems worked. This list is not necessarily complete and we apologize to anyone we haveoverlooked!

We would like to thank Professor Manuel Martinez-Sanchez, who was Mr. Paluszek’s and Ms. Thomas’ thesis advisorat MIT, Mr. Paul Zetocha, manager of numerous PSS contracts at AFRL, Dr. Barbara Sorensen of the U.S. AirForce, Bjorn Jakobsson of the Swedish Space Corporation with whom we have worked on the PRISMA project, Mr.Douglas Freesland who awarded us the Indostar attitude control system contract, Dr. Alfred Ng of the Canadian SpaceAgency, Mr. Douglas Bender of Boeing with whom we worked on TDRS, Mr. Rich Burns of NASA with whom wecollaborated on TechSat-21, Dr. Russell Carpenter of NASA who managed one of our formation flying contracts, Dr.Neil Goodzeit of Lockheed Martin, Bruce Campbell, and Andy Heaton of NASA with whom we have worked on solarsails, Dr. Robert McKillip of Continuum Dynamics who collaborated with us on fault detection, and Mr. ChristianPhillipe of ESTEC. We would also like to thank former members of the company who are no longer at PrincetonSatellite Systems: Dr. James Frueh, Mr. Derek Surka, Ms. Kerri L. Kusza, Ms. Wendy Sullivan, Ms. Margarita Brito,and Mr. Mike Miller.

One of the ideals we strive for at Princeton Satellite Systems is a work environment that encourages a balancedlifestyle. We recognize that our success at work is only possible through the support of our families, and we aregrateful to them.

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CONTENTS

Acknowledgements iii

Contents v

List of Examples xii

1 Introduction 11.1 How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Spacecraft Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Control System Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 What A Spacecraft Control Engineer Needs To Know . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Product Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Design Tutorial 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Conceptual Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Link Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Thermal and Optical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Spacecraft Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Attitude Control System Design and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.8 Power and Thermal Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.9 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Control System Design Process 273.1 Design Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Preliminary Designs 314.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Requirements Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Satellite Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Math 415.1 Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Time 496.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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7 Coordinate Systems 537.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 Selenographic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.3 Areocentric Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.4 Heliocentric Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8 Kinematics 578.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.3 Transformation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.4 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9 Attitude Dynamics 699.1 Inertia Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.2 Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729.3 Gyrostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759.4 Modeling Flexible Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769.5 A Simple Flexible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

10 Multibody Dynamics 8310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8310.2 Topological Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8310.3 Two Body Translational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8510.4 Translating Stage with Reaction Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8910.5 Pivoted Momentum Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

11 Orbits 9511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9511.2 Representations of Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9511.3 Propagating Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10011.4 Gravitational Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.5 Linearized Orbit Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

12 Orbit Maneuvers 10712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.3 Impulsive Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10812.4 Low-Thrust Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.5 Close Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.6 Maneuvers with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

13 Formation Flying 11913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11913.2 Coordinate Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12113.3 Relative Orbit Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12613.4 Geometric Parameters for Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13613.5 Relative Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14513.6 Formation Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

14 Launch and Reentry 16114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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14.2 Two Dimensional Optimal Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16114.3 Two Dimensional Flat Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16414.4 3D Cartesian Spherical Rotating Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16714.5 3D Bank and Flight Path Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16814.6 Launch Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17014.7 Lambert Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

15 Trajectory Optimization 17515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17515.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17515.3 Minimization Subject to a Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17815.4 Problem Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18015.5 Zermelo’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18115.6 Solar Polar Imager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

16 Budgets 18716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18716.2 Pointing Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18716.3 Propellant Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18916.4 Mass Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18916.5 Power Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

17 Sensors 19317.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19317.2 Types of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19317.3 Planet Optical Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19417.4 Gyros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19917.5 Other Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

18 Rendezvous Sensors 20118.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20118.2 Rendezvous Sensor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20218.3 RADAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20618.4 LADAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21318.5 Optical Sensor Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

19 Optical Sensors 22119.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22119.2 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22119.3 Radiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23119.4 Imaging Chips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

20 Actuators 24120.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24120.2 Types of Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24120.3 Reaction Wheel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24220.4 Control Moment Gyro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24520.5 Thrusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24820.6 Magnetic Torquers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25020.7 Solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25120.8 Stepping Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

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21 Propulsion 25921.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25921.2 Physics of Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26021.3 Nozzle Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26221.4 Chemical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26321.5 Low Power Electric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26421.6 High Power Electric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26521.7 Multiple-Stage Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

22 Disturbances 27122.1 External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27122.2 Internal Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28322.3 Fourier Series Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

23 Simulation 28523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28523.2 Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28523.3 Nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28623.4 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28623.5 Simulations for Control System Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29023.6 Simulation of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

24 Control Design 29524.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29524.2 Simple Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29524.3 The General Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29724.4 Fundamental Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29824.5 Tracking Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30124.6 State Space Closed Loop Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30224.7 Approaches to Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30324.8 Single-Input Single-Output Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30824.9 Digital Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31224.10 Continuous to Discrete Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31824.11 Flexible Structure Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32224.12 Model Following Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33024.13 Double Integrator Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

25 Spacecraft Attitude Control 34125.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34125.2 Attitude Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34225.3 Propagation of Quaternions from Gyro Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34325.4 Gravity Gradient Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34425.5 Nutation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

26 Command Distribution 34926.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34926.2 The Optimal Torque Distribution Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34926.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

27 Attitude Estimation 35727.1 Introduction to Estimation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

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27.2 Gyro Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35927.3 Conversion from Continuous to Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36027.4 The Kalman Filter Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36027.5 Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36127.6 Batch Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36427.7 Vector Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36627.8 Disturbance Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36727.9 Spin Axis Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36827.10 Stellar Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37527.11 Static Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37527.12 Recursive Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

28 Orbit Estimation 38528.1 Recursive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38528.2 Batch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39128.3 Autonomous Navigation of Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39328.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

29 Geosynchronous Control 41329.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41329.2 The Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41429.3 Transfer Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41429.4 Mission Orbit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41429.5 The Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41629.6 Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41629.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41729.8 A Mission Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41729.9 Design Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41829.10 Spacecraft Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41829.11 Spinning Transfer Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42029.12 Acquisition Using The Dual Spin Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42429.13 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42529.14 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42529.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

30 Sun Nadir Pointing Control 43530.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43530.2 Coordinate Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43530.3 Sun Nadir Pointing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43630.4 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43830.5 Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44030.6 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

31 Solar Sails 44531.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44531.2 Solar Pressure Force Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44531.3 Propellantless Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44731.4 Flexible Structure Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45331.5 Sail Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

32 ISS Proximity Operations 461

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32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46132.2 ISS Rendezvous Aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46132.3 Proximity Operations Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46432.4 Proximity Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46832.5 Risk Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

33 ACS Testing 47133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47133.2 Industry Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47133.3 A Testing Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

34 Fault Detection 48734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48734.2 Satellite Fault Detection and Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48734.3 Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48834.4 Fault Detection Background and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49034.5 Discrete Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49134.6 Voting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49134.7 Expert Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49234.8 Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49234.9 Analytical Monitoring Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49234.10 Reasoning Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49634.11 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49734.12 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49734.13 Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49834.14 Detection Filters for Attitude Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49934.15 Detection Filters for Orbit Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50134.16 Example Using Detection and Parity Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

35 Flight Operations 51135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51135.2 Elements of Flight Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51135.3 Mission Operations Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51135.4 Mission Operations Entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51135.5 Mission Operations Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51335.6 Mission Operations Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51435.7 Mission Control Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51535.8 Mission Operations Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

36 Mission Planning 51736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51736.2 Orbit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51736.3 Launch Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52136.4 Observation Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52336.5 Ground Coverage for Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52736.6 Attitude Profile Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

37 Thermal 53537.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53537.2 Radiation from a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53637.3 Dynamic Thermal Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

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37.4 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53737.5 Radiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53837.6 Heat Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53837.7 Multilayer Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53937.8 Isothermal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54137.9 Thermal Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54137.10 Solar Panel Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

38 Power 54738.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54738.2 Power Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54738.3 Power Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55138.4 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

39 Communication Links 55339.1 Communication System Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55339.2 Information Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55439.3 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55439.4 Link Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55839.5 Line of Sight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56339.6 Intersatellite Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

40 Glossary of Acronyms 575

A Probability 579A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579A.2 Axiomatic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579A.3 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580A.4 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580A.5 Evaluating Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583A.6 Combining Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583A.7 Multivariate Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584A.8 Random Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584A.9 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

B Laplace Transforms 587

C Standard Atmosphere 589

Bibliography 591

Index 602

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Introduction1.1 Simple code example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Attitude jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Design Tutorial2.1 Link as a function of pointing error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Preliminary Designs4.1 Control resolution example showing multiple ways of handling finite pulsewidth . . . . . . . . . . . 36

Attitude Dynamics9.1 Rigid body dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739.2 Momentum bias spacecraft nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Orbits11.1 Orbit simulation of impulsive inclination change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.2 Linearized orbit frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Orbit Maneuvers12.1 Rendezvous trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.2 Low thrust trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11212.3 Optimal planar maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11312.4 Frequency response of x-force to x-position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.5 A* 2D maneuver with obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.6 A* 3D maneuver with stayout zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11612.7 A* 3D maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Formation Flying13.1 Example Trajectory Found with HillsEqns.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 13013.2 Example Simulation Using RelativeOrbitRHS.m . . . . . . . . . . . . . . . . . . . . . . . . 13013.3 Example of Periodic Motion and Lawden’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 13213.4 Example Periodic Relative Trajectory in an Eccentric Orbit . . . . . . . . . . . . . . . . . . . . . . 14113.5 Example Periodic Relative Trajectory in an Eccentric Orbit, with Symmetric Cross-Track Motion . . 14113.6 Trajectories for the Eccentric Tetrahedron Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 14413.7 Trajectories of the reconfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Launch and Reentry14.1 Minimum time orbit injection from the moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16514.2 Three stage to orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16714.3 Single stage to orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16914.4 Newtonian lift and drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Preliminary Designs15.1 Zermelo solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18315.2 Zermelo cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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15.3 SPI trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Sensors17.1 Earth sensor chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19517.2 Scanning earth sensor with standard chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Rendezvous Sensors18.1 Radar range equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21318.2 CNR vs. pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21518.3 Ladar transmit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21619.1 Diffraction limited resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22819.2 Airy pattern for a point source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22919.3 Light gathering ability of a telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22919.4 Visible stars for an ideal aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22919.5 Circle of confusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23019.6 Pinhole camera star image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23119.7 Earth, moon and sun spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Actuators20.1 Reaction wheel frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24420.2 Bristle friction and Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24620.3 Double gimbal CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24720.4 Blowdown curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25020.5 Magnetic field at geo orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25120.6 Magnetic torque at geo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25220.7 Stepping motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Propulsion21.1 Payload to fuel ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26221.2 Electric propulsion system masses for LEO/GEO roundtrip. . . . . . . . . . . . . . . . . . . . . . . 26721.3 Optimal exhaust velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26821.4 Exhaust velocity in electron volts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26821.5 Payload Ratio and Burnout Velocity in Multiple Stage Rockets . . . . . . . . . . . . . . . . . . . . 270

Disturbances22.1 Surface accommodation coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27322.2 Density models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27522.3 Drag using a Jacchia model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27522.4 Gravity gradient with z-axis rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27622.5 Difference between a dipole and Meade-Fairfield magnetic field model . . . . . . . . . . . . . . . . 27722.6 Torque due to a residual dipole in geosynchronous orbit . . . . . . . . . . . . . . . . . . . . . . . . 27822.7 Comparison between pure specular and pure absorptions . . . . . . . . . . . . . . . . . . . . . . . 279

Simulation23.1 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28823.2 Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28923.3 Smooth friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28923.4 Stiff Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29023.5 Simulation right-hand-side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29223.6 Simulation demo results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Control Design24.1 Bode plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29724.2 Nichols plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29724.3 Root locus plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

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24.4 Double integrator plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30024.5 Double integrator model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30924.6 Torque transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31024.7 Loop compensation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31324.8 Delay example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31624.9 Bode Magnitude Plot of the Rate Control Loop with a Delays of 0.5 and 1 second . . . . . . . . . . 31724.10 Comparison of Pulsewidth Modulator and Zero Order Hold . . . . . . . . . . . . . . . . . . . . . . 31824.11 Comparison of the Four Rate Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32124.12 Comparison of the Matched Pole Zero with the Estimator . . . . . . . . . . . . . . . . . . . . . . . 32324.13 Control of a double integrator with a lead network . . . . . . . . . . . . . . . . . . . . . . . . . . . 32424.14 Bode plot for the collocated sensor actuator transfer function. . . . . . . . . . . . . . . . . . . . . . 32524.15 Bode plot for the non-collocated sensor and actuator transfer function . . . . . . . . . . . . . . . . 32624.16 Lead Compensator with Flex Mode Added . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32724.17 Lead compensator providing flex damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32824.18 Bode plot for the non-collocated sensor and actuator transfer function . . . . . . . . . . . . . . . . 32824.19 Bode plot for the non-collocated sensor and actuator transfer function with a phase-lead controller

and a crossover at 0.05 rad/sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32924.20 Root Locus plot for the non-collocated sensor and actuator transfer function with a phase lead con-

troller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32924.21 Bode plot with a high frequency phase lead controller . . . . . . . . . . . . . . . . . . . . . . . . . 33024.22 Bode plot with a flex compensator zero at 0.6 rad/sec. . . . . . . . . . . . . . . . . . . . . . . . . . 33124.23 Model Following Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33224.24 Model Following Control Response with Increasing Gain . . . . . . . . . . . . . . . . . . . . . . . 33324.25 Second order response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33524.26 Effect of the integrator on a second order step response . . . . . . . . . . . . . . . . . . . . . . . . 33624.27 Phase plane controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33724.28 PID damping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33824.29 Windup compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Spacecraft Attitude Control25.1 Nutation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34725.2 Nutation dynamics with a rate damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34825.3 Step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Command Distribution26.1 Reaction Wheel Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35126.2 Gimbaled thrusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35326.3 North face simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35326.4 Single Gimbal CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Attitude Estimation27.1 Quaternion from unit vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37627.2 Errors due to misidentified starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37627.3 Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Orbit Estimation28.1 Orbit estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38628.2 Disturbance estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40728.3 Continuous discrete orbit estimation with process noise . . . . . . . . . . . . . . . . . . . . . . . . 40828.4 Continuous discrete orbit estimation without process noise . . . . . . . . . . . . . . . . . . . . . . 40928.5 Downhill Simplex Batch Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41028.6 Batch Least Squares Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41128.7 Doppler shift beat frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41228.8 Illumination variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

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Geosynchronous Satellite Control System Design29.1 Dual Spin Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Sun Nadir Pointing Control30.1 Sun-nadir yaw trajectory for a GPS orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Solar Sails31.1 Solar pressure force model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44831.2 Heliocentric locally optimal steering for semi-major axis control . . . . . . . . . . . . . . . . . . . 46031.3 Planet-centric locally optimal steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

Fault Detection34.1 Spinner detection filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50134.2 Orbit thruster detection filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50234.3 DC Motor Failure Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50834.4 Hardening spring failure simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

Mission PLanning36.1 Sun-Synchronous Inclination vs. Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52036.2 Repeat Ground-Track Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52136.3 Coverage Region Example with Constant Elevation Contours . . . . . . . . . . . . . . . . . . . . . 52636.4 Observation Time Windows Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52736.5 Coverage Region with Cone Projected onto the Earth . . . . . . . . . . . . . . . . . . . . . . . . . 530

Thermal37.1 MLI effective emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54037.2 Isothermal satellite in geosynchronous orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54137.3 Thermal example. The referenced functions are included as listings. . . . . . . . . . . . . . . . . . 54337.4 Solar panel example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54637.5 Solar panel example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

Power38.1 Power vs time of year for a geosynchronous satellite . . . . . . . . . . . . . . . . . . . . . . . . . . 54838.2 Power vs heliocentric radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54838.3 Solar cell IV curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54938.4 Solar cell efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54938.5 Eclipse in geosynchronous orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

Communication Links39.1 Shannon Information Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55539.2 Power from nose temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55539.3 Noise due to the sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55639.4 Noise due to the sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55639.5 Noise due to Mars and Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55739.6 Noise due to an attenuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55739.7 Noise due to a receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55839.8 Gain for a circular aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55939.9 Loss due to atmospheric gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55939.10 Loss due to pointing error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56039.11 Loss due to distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56039.12 Loss due to polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56139.13 Loss due to rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56139.14 Link analysis for earth to Mars link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56239.15 Link analysis for L1 to earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

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ProbabilityA.1 Dog birth problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581A.2 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581A.3 Gaussian PDF and CPDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

Standard AtmosphereC.1 Standard atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

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CHAPTER 1

INTRODUCTION

1.1 How to Use This Book

This book covers the subject of spacecraft attitude and orbit control. It includes discussions of the physical systemsthemselves and how to control those systems. This chapter provides an introduction to the topics in this book. Inthe next chapter a complete tutorial is provided on spacecraft attitude and orbit control. It is recommended that thereader who is new to the subject go through this tutorial in detail. Many MATLAB R© scripts are provided. All ofthese are available from Princeton Satellite Systems’ website including all supporting functions which are includedas compiled MATLAB code. You can explore each of the topics in the tutorial by customizing these scripts. In theremaining chapters examples are given in MATLAB using Princeton Satellite Systems’ Spacecraft Control Toolboxfunctions. If you have the Spacecraft Control Toolbox and MATLAB, you can cut and paste the examples into theMATLAB Command window and execute them. If you save them in a script you can try different parameters andcustomize the scripts to your own needs. There are very few scripts that will work without the Spacecraft ControlToolbox. However, the given results serve to illustrate the text, which is completely self contained. No additionalproducts from The MathWorksTM are required for any examples in this book.

A code example is shown in Example 1.1. Plot2D is a function from Princeton Satellite Systems’ Spacecraft ControlToolbox to generate 2D plots. The line PrintFig(1,1,1,’SimpleExample’) prints the figure into an encap-sulated Postscript file. These last lines are in each example just for output purposes and so that you can cut and pastethe code into the MATLAB command window and execute it as is.

Example 1.1 Simple code example

1 t = linspace(0,100);2

3 Plot2D( t, sin(t) );4

5 PrintFig(1,1,1,’SimpleExample’)

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y

x

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In this book we have not given many examples of operational hardware. The best source for up-to-date informationon current space systems is the internet. Virtually all aerospace companies have websites with elaborate productdescriptions. In addition there are many online industry newsletters and information websites that provide informationon existing systems.

1.2 Spacecraft Control

Spacecraft control is usually synonymous with “Attitude Control,” the engineering discipline of keeping a satellite orspacecraft pointed in the right direction. However, in the last 20 years the umbrella of spacecraft control has expandedto include close orbit control including automatic rendezvous and docking, formation flying and close maneuvering.In the past orbit control was synonymous with “mission planning”, which can be thought of as low sampling ratefeedback control. In modern spacecraft it may be necessary to control relative position as tightly, if not more tightlythan, spacecraft attitude.

As an engineering discipline, spacecraft control embodies five distinct areas:

1. Control system design

2. Dynamics and modeling of systems

3. Software design

4. User interface design

5. Spacecraft operations

The last is rarely considered, but is of critical importance in satellite control system design. The five areas togethermake up the Attitude and Orbit Control System (AOCS). Usually the system aspects of a spacecraft control system aremore important than the control laws themselves and often much more difficult to implement.

This book discusses all aspects of spacecraft control. This book assumes introductory courses in physics and calculusthrough elementary differential equations. Some background in control theory would be helpful but is not essential.This section will continue to expand upon each of the five areas of the spacecraft control discipline.

1.2.1 Mnemonics

Table 1-1 gives some mnemonics used to denote spacecraft control systems.

Table 1-1. Attitude control system mnemonics

Mnemonic Meaning Where usedACS Attitude Control System GeneralAOCS Attitude and Orbit control System Inmarsat 3ADS Attitude Determination System GPS IIRADACS Attitude Determination and Control SystemADCS Attitude Determination and Control System

1.2.2 Control System Design

Attitude and orbit control system design can be further decomposed into:

1. Attitude determination

2. Orbit determination

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3. Attitude Control

4. Orbit Control

5. Control distribution

The first two involve sensor measurements, often using a combination of different sensors, to get an estimate of thespacecraft pointing and position. The third and fourth involve the design and implementation of the control loops. Thelast, distribution, involves taking control demands and converting them into demands on the actuators. For example, athruster may be pulsewidth modulated so torque or angular acceleration demand must be converted into pulsewidthsfor the thrusters. In many cases, control distribution and control are combined into one step. Early control systemswere analog and the sensors were designed, in some cases, to produce outputs that could be used directly. For example,a scanning earth sensor produces roll and pitch measurements and often autonomously makes adjustments for sun andmoon interference so that the sensor always gives valid measurements without the need for complex decision makinglogic.

1.2.3 Dynamics and Modeling of Systems

The second area of spacecraft control can be decomposed into:

1. Modeling

2. Simulation

Modeling is the creation of computer software to replicate the physical behavior of the system. Only those physicalcharacteristics that are relevant to the problem being studied should be modeled. For example, for the purposes ofattitude control design, the mechanical and thermal dynamics of an earth sensor are rarely of interest and are usuallyneglected. Simpler models are easier to test and debug and run faster. The numerical models are embedded in asimulation. Since it is rarely practical to test control systems on real satellites, the designer must rely on simulationto validate his or her designs. Simulations can range from all software running on the same platform, to softwaremodels with the control system running on a flight or equivalent board or one in which actuator and sensor hardwareare integrated into the simulation. Types of simulations are shown in Figure 1-1 on the following page.

In the first block diagram all of the models and the control software are part of the simulation. Everything is software.This is the easiest to set up and maintain. However simulations may involve hundreds of thousands of lines of codeand large databases of parameters. Configuration management to allow reproduction of simulation runs is not a trivialtask and not something that should be taken lightly. The second diagram uses the actual flight software running onits own processor. This allows testing of the flight software with the simulation and requires knowledge of hardwareinterfaces and networking. The final block diagram uses hardware actuators and sensors as well as the the flightsoftware. Figure 1-2 on page 5 shows the hierarchy of simulations and where in the lifecycle each would be used.

1.2.4 Software Design

The third area is the implementation of control systems in software. This is of critical importance. Ultimately, space-craft control engineers must also be software engineers, whether they write in C++ or use a block diagram language.Most of a spacecraft control system has little to do with control theory, but rather is related to how the satellite willoperate and interact with spacecraft operations. Aside from controlling the satellite, the software must:

1. Implement the user interface (command and telemetry)

2. Switch operational modes

3. Provide fault detection

4. Provide redundancy management

5. Plan maneuvers, operations, etc.

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Figure 1-1. Types of simulations

DynamicsActuators Sensors

Control Software

Simulation Computer

DynamicsActuators Sensors

Control Software

Simulation Computer

Flight Computer

DynamicsActuators Sensors

Control Software

Simulation Computer

Flight Computer

1.2.5 User Interface Design

The fourth area of spacecraft control is the most important. Most satellites that are lost are the result of operatorerror; operator error is often due to user interface problems. A user interface problem can be a wrong command, or acommand whose effect is not completely understood.

In the early days of satellite design, the user interface was never designed; rather, it was implemented in an ad hocfashion. Today some satellite manufacturers (by no means the majority) have recognized that this is an importantcomponent and must be considered early in the design phase.

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Figure 1-2. Applications of different types of simulations

All SoftwareDesign and

AnalysisOps

Support

Software + Flight

Computer

FSW Verfication and TestAnomaly

Investigations

Software + Flight Computer + Hardware

Fixed BasedMoving Base

Integration and Test

Anomaly Investigations

1.2.6 Spacecraft Operations

The last area in the discipline of spacecraft control is flying the spacecraft, which many designers do once their de-signs are launched. This involves planning maneuvers, monitoring the spacecraft during maneuvers and investigatinganomalies. Anomaly investigation ranges from post-mortems on lost spacecraft to fine-tuning the control systemperformance by uploading new gains, for example.

1.2.7 The Spacecraft Control Engineer’s Job

How much time does a satellite control system designer spend on these tasks? Roughly 5% of a spacecraft controlengineer’s time is in actual control loop design. About 10% of his or her time will be spent modeling the hardware.The other 85% will be spent writing software. This latter number includes writing analysis tools and simulations.

1.3 Control System Terms

Some attitude control terminology is not defined consistently throughout the aerospace industry. Textbooks and com-panies use different terms for the same concept. Orbit control, as opposed to mission planning, is a relatively newconcept and the terminology is still evolving. This section explains a few confusing terms, and gives the definitionsthat are used by Princeton Satellite Systems.

5

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1.4. WHAT A SPACECRAFT CONTROL ENGINEER NEEDS TO KNOW CHAPTER 1. INTRODUCTION

1.3.1 Pointing Accuracy

Also referred to as the attitude control accuracy. this generally refers to how well the attitude of the vehicle in questioncan be controlled with respect to a commanded direction. When given as a requirement, it is an absolute bound on theerror allowed in the spacecraft’s orientation with respect to the commanded orientation.

1.3.2 Pointing Knowledge

Also referred to as the attitude determination accuracy, this refers to how well the orientation of the spacecraft isknown with respect to an absolute reference. This term can be used for either real-time or after the fact orientationknowledge. When given as a requirement, it is an absolute bound on the error allowed in the knowledge of thespacecraft’s orientation with respect to an absolute reference.

Sensor accuracy is the sum of the ultimate accuracy of the measurement, determined by the object used by the sensors,and all the errors in the measurement such as mounting tolerances and thermal effects. Pointing knowledge is theoverall accuracy of attitude determination sensor suite. Stars are the most accurate source for a measurement followedby the Sun and Earth.

Pointing knowledge can sometimes be improved after the fact, such as by monitoring the earth’s magnetic field andlater post-processing the output of a magnetometer for better knowledge of a telescope’s pointing direction for aparticular photograph.

The pointing knowledge of a spacecraft is usually better than its pointing accuracy, especially if it is not required to beavailable in real-time. For example, control systems which use thrusters as the sole actuators generally have a pointingaccuracy limited to half the thrusters’ pulsewidth plus the attitude determination accuracy. Systems which incorporatewheels into the attitude control system can usually operate close to the attitude determination accuracy so that thepointing accuracy and the pointing knowledge are very close to each other.

1.3.3 Pointing Stability

When given as a requirement, this is the maximum rate of change of angular orientation allowed.

1.3.4 Jitter

Jitter refers to the errors in attitude of a frequency too high to be controlled by the attitude control system. Whengiven as an attitude control performance requirement, it is a specified angle bound or angular rate limit on short-term,high-frequency motion of the spacecraft.

Each spacecraft has an inherent controller period required to sense an attitude error and implement a correction. Thisperiod determines the controller’s bandwidth by the relationship 1/t. Disturbances of a frequency above the bandwidthare not attenuated by the attitude control system. Sources of such high-frequency disturbances could include theinternal vibration of a sensor.

Example 1.2 on the facing page shows a spacecraft pointing history with jitter. The jitter can be seen separately byfiltering the data and subtracting the smoothed curve from the original.

1.4 What A Spacecraft Control Engineer Needs To Know

The following is an outline of topics with which a spacecraft control designer needs to be familiar.

1. DynamicsRigid body

6

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CHAPTER 1. INTRODUCTION 1.4. WHAT A SPACECRAFT CONTROL ENGINEER NEEDS TO KNOW

Example 1.2 Attitude jitter

1 tau = 3; % fundamental interval of controlsystem

2 w = 2*pi/tau;3 t = linspace(0,2.5*tau,1000);4

5 for k = 1:length(t)6 % Add limit cycling7 dist1(k) = 0.1*sin(w*t(k));8 % add low-frequency white noise and bias9 dist2(k) = 0.03+0.05*(sin(w/3*t(k))+cos(w

/1.5*t(k)));10 % Add jitter and high-frequency white noise11 dist3(k) = (0.03)*sin(30*w*t(k))+ randn

*0.015;12 end13

14 % Add desired angle15 %------------------16 theta = 30;17

18 % Plot19 %-----20 yL = {’\theta’ ’Standard Deviations’};21 Plot2D(t,[theta+dist1+dist2+dist3;dist3/sig],’

Time’,yL,’Attitude Jitter’);22 PrintFig(1,1,1,’JitterDemo’)

Attitude Jitter

0 1 2 3 4 5 6 7 829.8

29.9

30

30.1

30.2

30.3

θ

0 1 2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

Sta

ndar

d D

evia

tions

Time

AerodynamicsAeroelasticity (for launch vehicles)MultibodyElectromechanical systemsHydraulic systems (for launch vehicles and large vehicles like the Space Shuttle)

2. DisturbancesSolar pressureAerodynamic dragRadiation pressureAlbedo pressureMagnetic torquesOutgassing

3. KinematicsCoordinate framesTransformation matricesQuaternionsEuler angles

4. ControlSingle-Input-Single-Output (SISO)Multi-Input-Multi-Output (MIMO)Nonlinear

5. SensorsGyrosAccelerometersGPSEarth sensorsSun sensorsStar sensors

7

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1.5. PRODUCT INFORMATION CHAPTER 1. INTRODUCTION

MagnetometersPotentiometersTemperature sensorsCurrent sensors

6. ActuatorsRocket engines and other types of propulsionMotorsReaction wheelsControl moment gyrosMagnetic torquers

7. MathDifferential equationsLinearization of nonlinear modelsNumerical methodsProbability and statistics

This book covers most of these topics. In many cases, the topics are explored as part of extended examples.

1.5 Product Information

For MATLAB R© product information, please contact:

The MathWorks, Inc.3 Apple Hill DriveNatick, MA, 01760-2098 USATel: 508-647-7000Fax: 508-647-7001E-mail: [email protected]: www.mathworks.com

For additional information on the Spacecraft Control Toolbox, Formation Flying Module, Spin Axis Attitude Deter-mination Module, or Solar Sail Module, please contact:

Princeton Satellite Systems, Inc.33 Witherspoon St.Princeton, NJ 08542 USATel: 609-279-9606Fax: 609-279-9607E-mail: [email protected]: www.psatellite.com

8

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CHAPTER 2

DESIGN TUTORIAL

2.1 Introduction

This tutorial will walk you through elements of the design of a spacecraft. All of the scripts in this tutorial will workwith MATLAB R© without any additional Spacecraft Control Toolbox functions.

Our goal is to design a geostationary spacecraft. The spacecraft must meet the following requirements:

1. Deliver -126 dBW at the beam center in the Ku band

2. Maintain the orbital station to within 0.1 deg both in-plane and out-of-plane

3. Have a life of 7 years

The requirements flow among the communications, propulsion, power and attitude control subsystems is shown inFigure 2-1 on the next page. The payload requirements drive both power and attitude control. For example, morepower is required if the attitude pointing requirements are loosened. The life and box requirements drive the amountof fuel required and possibly the type of thrusters used. A long life requirement might drive the design to the use ofelectrically augmented or electric thrusters. Propulsion also drives the pointing requirements. The more precise thepointing the less fuel is required to perform maneuvers. Pointing affects both propulsion and communications in theform

f = a+ bθ (2-1)

where f is the power or the propellant required. The box size drives the communications subsystem since a large boxmight increase the range to the beam center or lead to the need for a steerable antenna. Power drives the disturbancessince a larger solar array produces larger disturbances. Larger disturbances affect pointing so there is a closed loopbetween power and attitude. Closed loops in the requirements may require design iterations. It is important to carefullythink out the requirements flow prior to starting the design lest unknown requirements result in design problems latein the design process.

We’ll take the following steps:

1. Produce a conceptual design based on past experience of what should work

2. State our simplifying assumptions in the design

3. Compute the power needed to meet the first two requirements. This will allow us to size our solar panels andbattery.

4. Perform a thermal and optical analysis. This will determine the actual size of our solar arrays and the distur-bances on the spacecraft.

5. Do the orbit control analysis to meet the lifetime requirement

9

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2.2. CONCEPTUAL DESIGN CHAPTER 2. DESIGN TUTORIAL

Figure 2-1. Requirements flow

Communications

Power on Ground

Attitude Control Power

Pointing Power

Propulsion

Life

Disturbances

Disturbances

Mission Requirements

Pointing

Box

6. Design the attitude control system

7. Simulate the entire system

2.2 Conceptual Design

A conceptual design for a satellite that would meet these requirements is shown in Figure 2-2 on the facing page. Themain aspects are:

1. A box with insulating foil on the outside

2. Two radiators on the north and south faces

3. Batteries

4. Transponder (a radio transmitter)

5. Antenna for the payload that points at the earth

6. Two solar panels that always face the sun

7. Twelve thrusters. Four are on the north face, four on the west face and four on the east faces

North, south, east and west refer to the directions you would see if you were on the zenith (away from the earth face)and looking at the earth.

2.3 Simplifying Assumptions

We will make some simplifying assumptions about the spacecraft.

1. The thrusters can be fired so that they can produce any torque and force vector.

10

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CHAPTER 2. DESIGN TUTORIAL 2.4. LINK ANALYSIS

Figure 2-2. Tutorial spacecraft

North

South

NadirZenith

Radiator

Battery

Transponder

Thruster

Fuel Tank

East West

Shunt

Solar Panels

+Y

+X

+Z

2. We have sensors that measure roll, pitch and yaw. Roll is the angle about the x-axis (the axis going from westto east). Pitch is the axis going from north to south (y) and yaw is the axis pointing at the earth (z).

3. The spacecraft can be treated as a rigid body with a constant inertia matrix and center-of-mass that doesn’tmove. This means that the rotation of the solar panels has no effect on center-of-mass or inertia.

4. There are no moving parts on the spacecraft. The rotation of the solar panels with respect to the core of thespacecraft will be ignored as discussed above.

5. The thermal and solar disturbance model will only include the solar arrays.

6. Only the RF subsystem consumes power.

7. The power model will only include the RF emitted by the antenna, the shunts which dump excess power and thebattery.

2.4 Link Analysis

The link analysis is needed to determine how much power we need for the payload. The payload drives all otherrequirements because it is the reason we are building the satellite. We are given a specification for beam power at thecenter the beam. We will simplify our model to assume that the two losses are the range loss and the pointing loss.The link equation is

PR = PTGTGRLFLD

(2-2)

PR is the received power, PT is the transmitted power, GR is the gain of the receive antenna, GT is the gain of thetransmit antenna, LF is the free-space loss due to the expansion of the wavefront with the square of the distance fromthe transmit antenna and LD is the depointing loss due to attitude errors. The link is shown in Figure 2-3 on the nextpage.

The equation for the maximum boresight gain of an antenna is

G =4πAλ2

(2-3)

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2.4. LINK ANALYSIS CHAPTER 2. DESIGN TUTORIAL

Figure 2-3. Link

Data

RF

Transmit antenna Receive antenna

Solar panel

Modulator

where λ is the wavelength and A is the antenna aperture area. The range or free-space loss is

LF =(

4πRλ

)2

(2-4)

where R is the range from the antenna to the beam center on the ground and λ is the wavelength. To compute thepointing loss we must first compute the 3 dB angle, that is the angle from the boresight for which the gain is down by3 dB. dB are defined as

dB = 10 log10(x) (2-5)

The 3 dB angle, θ3dB , is

θ3dB = 70λ

DT(2-6)

where the angle is in degrees and DT is the diameter of the transmit antenna. λ and DT must be in the same units.

The pointing loss is

LP = 10

„1.2“

θθ3dB

”2«

(2-7)

where θ is the beam pointing error in deg. Ku band for downlink has a frequency f = 12.7 GHz so the wavelength is

λ =c

f(2-8)

where c is the speed of light. For Ku band λ is 2.4 cm.

The power required drives the solar array design. The beam pointing error connects to the attitude error through theequations

φAZ = θroll +δAZ

180/πθyaw (2-9)

φEL = θpitch +δEL

180/πθyaw (2-10)

where δEL and δAZ are the beam center offsets.

The Link script generates the plot in Example 2.1 on the facing page. This shows the power, in dBW on the outputof the receive antenna given 1000 W at the input to the transmit antenna. The receive antenna radius is 0.5 m and thetransmit antenna radius is 1 m.

12

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CHAPTER 2. DESIGN TUTORIAL 2.5. THERMAL AND OPTICAL MODEL

Example 2.1 Link as a function of pointing error

Link

0 1 2 3 4 5 6 7 8 9

x 104

−129.5

−129

−128.5

−128

−127.5

−127

−126.5

−126

−125.5

−125

time (sec)

Pow

er (

dBW

)

Power on ground

2.5 Thermal and Optical Model

The interaction of the solar flux with the solar arrays is used in the orbit disturbance model, the attitude disturbancemodel and the power and thermal model. Figure 2-4 shows the thermal model.

Figure 2-4. Thermal model

solar fluxreflecte

d solar flu

x

Reemitted solar flux

+zα

Power

The thermal balance isρaASs

Tn− Pa − 2AσεT 4 = 0 (2-11)

where A is the panel area, ρa is the absorptivity of the panel and ranges from 0 to 1, ε is the emissivity of the panel(assumed the same for both sides) and ranges from 0 to 1, T is the panel temperature, Pa is the power extractedby the solar cells, S is the solar flux at the earth’s orbit (about 1367 W/m2) and σ is Boltzmann’s constant equal to5.67× 10−8 W/m2K4, s is the sun vector and n is the outward unit normal to the surface. The power produced by thepanels is

Pa = ηρaASsTn (2-12)

where η is the conversion efficiency and sTn is the dot product, equivalent to the cosine of the angle between s and n.

13

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2.5. THERMAL AND OPTICAL MODEL CHAPTER 2. DESIGN TUTORIAL

Consequently, the thermal balance is

(1− η)ρaASsTn− 2AσεT 4 = 0 (2-13)

Any energy that is not absorbed is reflected specularly. The sum of the specular and absorption coefficient must equalone

1 = ρa + ρs (2-14)

The force on the solar panel is related to the reflection coefficients. The solar pressure force is

F = −ScAsTn((2ρssTn)n+ ρas) (2-15)

where c is the speed of light in a vacuum equal to 2.99792458× 108 m/s. The reemitted flux is the same on both sidesso does not contribute to the net force. Let the solar array outward (cell) normal be

n =

0sin γcos γ

(2-16)

where γ is the average tilt of the array due to thermal distortion.

The sun vector is always in the yz-plane of the solar panel. At equinox it is along z and at solstice it is elevated. Letthe angle be β. Then the sun vector is

s =

0sinβcosβ

(2-17)

so thatsTn = cosβ cos γ + sinβ sin γ = δ (2-18)

The thermal balance is now(1− η)ρaASδ − 2AσεT 4 = 0 (2-19)

and the force isFs = −S

cAδ(2ρsδn+ ρas) (2-20)

The temperature of the solar array is

T =(

(1− η)ρaSδ2σε

)1/4

(2-21)

and the power produced by the array isPa = ηρaASδ (2-22)

We would like the absorptivity ρa to be has large as possible to reduce the mass of solar arrays we need to take toorbit. Modern multi-junction cells have efficiencies, η of 21.5%. Consequently our thermal design consists entirely ofpicking the emissivity of the panels. The higher the emissivity, the lower the panel temperature. Generally, the colderthe solar cell the better its efficiency. However, there may be a minimum desirable temperature (which will happenwhen β is a maximum). Therefore, this constraint allows us to determine our ε.

The script ThermalOptical plots the temperature, power and force as a function of sun angle as shown in Figure 2-5 on the facing page.

14

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CHAPTER 2. DESIGN TUTORIAL 2.6. SPACECRAFT ORBIT CONTROL

Figure 2-5. Solar effects

−50 0 50305

306

307

308

309

310

311

312

313

β (deg)

Tem

pera

ture

(de

g−K

)

Panel Temperature

−50 0 50525

530

535

540

545

550

555

560

565

570

575

β (deg)

Pow

er (

W)

Panel Power

−50 0 50−10

−8

−6

−4

−2

0

2

4

β (deg)

For

ce (µ

N)

Panel Force

xyz

2.6 Spacecraft Orbit Control

2.6.1 Dynamics

The disturbances on the spacecraft are due to the gravitational acceleration of the sun and moon, solar pressure and theeffect of the earth not being spherical. For this tutorial we will only look at the earth asymmetry and sun gravitationalacceleration. Actually the effect of the moon is larger than the sun but the orbit of the moon is too complex for thistutorial. The 3-body gravitational equations for the sun, earth and spacecraft are

ri = −3∑

j=1,j 6=i

µjr3ij

(ri − rj) (2-23)

where µj is the gravitational parameter of the jth body and where rij = ri − rj . ri is the acceleration of the ith bodywith respect to the reference center. Figure 2-6 on the next page shows the geometry.

It is useful to write the equations out explicitly.

r1 = −µ2r1 − r2

|r12|3− µ3

r1 − r3

|r13|3(2-24)

r2 = −µ1r2 − r1

|r21|3− µ3

r2 − r3

|r23|3(2-25)

r3 = −µ1r3 − r1

|r31|3− µ2

r3 − r2

|r32|3(2-26)

The denominator terms are always positive. Let 1 be the spacecraft, 2 the earth and 3 the sun. Note that rji = −rij .

15

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2.6. SPACECRAFT ORBIT CONTROL CHAPTER 2. DESIGN TUTORIAL

Figure 2-6. Three body gravity

r2

r3

r1

r21

Sun

Earth

We want the dynamical equations for r.

r1 − r2 = −(µ1 + µ2)r1 − r2

|r12|3− µ3

(r1 − r3

|r13|3+r3 − r2

|r23|3

)(2-27)

Let

r = r1 − r2 (2-28)d = r3 − r2 (2-29)

r1 − r3 = r − d (2-30)

r = −(µ1 + µ2)r

|r|3− µ3

(r − d|r − d|3

+d

|d|3

)(2-31)

since µ1 is much smaller than µ2, we can neglect µ1 and

r = −µ2r

|r|3− µ3

(r − d|r − d|3

+d

|d|3

)(2-32)

The influence of the sun is due to the offset of the spacecraft from the earth’s center and pushes in the direction of thevector from the sun to the spacecraft. As r → 0 the perturbation goes to zero.

The longitudinal drift must be computed from the gravitational potential. The gravitation potential can be written as aseries of spherical harmonic coefficients written in spherical coordinates

V = −µ2

r

∞∑n=2

[(ar

)n n∑m=0

(sn,m sinmλ+ cn,m cosmλ)Pn,m(sinφ)

](2-33)

where r is radius λ is longitude and φ is latitude. To find the acceleration we must take the gradient

a = −ir∂V

∂r− iλ

1r

∂V

∂λ− iφ

1r sinλ

∂V

∂φ(2-34)

Longitudinal drift is only caused by the second acceleration term and we are only interested in the terms up to n = 2.The potential becomes

V = −µ2

r

(ar

)2 2∑m=0

(sn,m sinmλ+ cn,m cosmλ)Pn,m(sinφ) (2-35)

16

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CHAPTER 2. DESIGN TUTORIAL 2.6. SPACECRAFT ORBIT CONTROL

The Legendre functions of interest are

P2,0 =12

(3 sin2 φ− 1) (2-36)

P2,1 = 3 sinφ cosφ (2-37)P2,2 = 3 cos2 φ (2-38)

When we are on the equator φ = 0 and therefore

P2,0 = −12

(2-39)

P2,1 = 0 (2-40)P2,2 = 3 (2-41)

We can then expand the sum

V = −µ2a2

r3

[3s2,2 sin 2λ+ 3c2,2 cos 2λ− 1

2c2,0

](2-42)

The longitudinal acceleration is

aλ = −1r

∂V

∂λ=

6µ2a2

r4[s2,2 cos 2λ− c2,2 sin 2λ] (2-43)

A comparison of the model with a 36 × 36 spherical harmonic model is shown in Figure 2-7. As can be seen theyagree closely.

Figure 2-7. Comparison of longitudinal accelerations for 36 × 36 harmonic model and single harmonic model

0 50 100 150 200 250 300 350 400

−4

−2

0

2

4

6

8x 10

−11

Longitude (deg)

Acc

eler

atio

n (k

m/s

2 )

Longitudinal acceleration

Simplified model36x36 Spherical harmonic model

2.6.2 Geo-synchronous Spacecraft Simulation

The M-file SimulationOrbit.m simulates a geo-synchronous orbit over two and a half days. The spacecraft isinitialized with the following position and velocity states in the earth-centered inertial reference frame:

~r =

42, 16700

, ~v =

03.074557

0

(2-44)

17

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2.6. SPACECRAFT ORBIT CONTROL CHAPTER 2. DESIGN TUTORIAL

where a = 42, 167 km is the semi-major axis for GEO. This is found by satisfying the orbital period equation,T = 2π

√|~r|3/µ, for a period of one day, or T = 86, 400 seconds. The inclination is zero, so the velocity is purely in

the ECI y direction. The magnitude of the velocity is found by the velocity equation for a circular orbit:

|~v| =√

µ

|~r|

where µ = 3.98600436e5 km3/s2 is the gravitational constant for earth.

The acceleration is computed in the RHSOrbit.m function. It includes gravitational acceleration from the Earthand the sun, including the longitudinal acceleration term from Eq. 2-43 on the previous page. The results from thesimulation are plotted in Figure 2-8. The lower left plot shows the motion of the satellite in the inertial frame. Theother 3 plots show the relative position of the satellite with respect to a moving coordinate system that rotates withthe earth. In the rotating coordinate system, which is shown in Figure 2-9 on the next page, y points along the orbitvelocity, x points zenith away from earth, and z points south.

Figure 2-8. Simulation of Spacecraft in Geosynchronous Orbit with sun and earth Gravity Disturbances

−1.5 −1 −0.5 0 0.5−2

0

2

4

6

∆ x (km)

∆ y

(km

)

Rotating frame

0 20 40 60−2

−1

0

1

2

3

Time (hr)

∆ z

(km

)

Rotating frame

0 20 40 60−5

0

5x 10

4

Time (hr)

Pos

ition

(km

)

ECI frame

0 20 40 60−2

0

2

4

6

Time (hr)

Pos

ition

(km

)

Rotating frame

∆ x∆ y

The relative coordinate system is convenient in that it enables us to view how the actual orbit of the spacecraft ismoving with respect to its desired station, directly above a fixed point on the earth. The motion in the y directionrepresents the East-West drift, and the motion in the z direction represents the North-South drift.

After the inertial state is integrated, the relative position is computed as follows:

∆~r(t) =

cos(ωt) sin(ωt) 0− sin(ωt) cos(ωt) 0

0 0 1

~r(t)− 42, 167

00

(2-45)

We first rotate from the ECI to the local frame, through the angle ωt, where ω is the angular rate of the earth and t isthe elapsed time. We then subtract the earth-fixed station position.

18

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CHAPTER 2. DESIGN TUTORIAL 2.7. ATTITUDE CONTROL SYSTEM DESIGN AND SIMULATION

Figure 2-9. Inertial and Rotating Coordinate Systems for Station-Keeping

x

y

YECI

ZECI

ω

z

XECI

2.6.3 North-South Stationkeeping

The North-South motion of the satellite is due to primarily to the gravitational acceleration of the sun on the spacecraft.The moon also contributes to this motion, but it is ignored in this tutorial. As the plot in Figure 2-8 on the facingpage indicates, this motion follows a growing oscillation, with a period of about 1 day. North South stationkeepingis required in order to keep the bounds of this oscillation within the acceptable limits, as dictated by the missionrequirements. The requirements for this example are to keep the oscillation between ±0.1 degrees. In terms ofdistance, this converts to:

d = 42167× 0.1× π/180 = 73.6 km

Our simulation shows that the magnitude of North-South oscillation grows at a rate of about 1.5 km/day. Therefore,we would have to apply an impulsive delta-v in the orbit normal direction about once every 50 days. The simulationresults also show that the North-South relative velocity grows at a constant rate of about 0.08 m/s per day. Thus, on the50th day, the delta-v required would be 4 m/s. This translates into an annual delta-v budget of 30 m/s. When the effectsof the moon are included, the North-South oscillation grows somewhat larger to about 2 km/day, which is 1 degree peryear. It then requires about 50 m/s of annual delta-v to perform North-South stationkeeping.

2.6.4 East-West Stationkeeping

The East-West motion is caused by accelerations acting in the xy plane of the rotating frame. This includes both thethe gravity from the sun and moon, as well as the gravitational harmonics of the asymmetric earth. The effect of earthoblateness depends on the longitude of the station, as seen in Eq. 2-43 on page 17. The general result is a drift in thelongitudinal or East-West direction. The plot in Figure 2-8 on the facing page shows the longitudinal displacement (ydirection) growing at a rate of about 2.6 km per day. It would therefore take 28 days in order to reach our 73.6 kmboundary. Periodic burns in the y direction are applied to bound this drift. The associated delta-v depends upon thelongitude and the stationkeeping requirement (allowable drift), but it is typically only about 2 m/s per year.

2.7 Attitude Control System Design and Simulation

The steps done in designing an attitude control system are

1. Create a pointing budget

2. Create a disturbance budget

3. Select actuators and sensors

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2.7. ATTITUDE CONTROL SYSTEM DESIGN AND SIMULATION CHAPTER 2. DESIGN TUTORIAL

4. Design the control algorithms

5. Test in simulation

We are assuming ideal actuators and sensors so we will not do steps 2-3. The pointing budget is already factored intothe link analysis and we require that the spacecraft point in roll and pitch to within 0.1 deg.

2.7.1 Design the control algorithms

The dynamical equations for the spacecraft are Euler’s equations

Iω + ω×Iω = T (2-46)

where I is the inertia matrix of the spacecraft, ω is the body rate measured about the body axes with respect to theinertial frame and T is the sum of all external torques on the spacecraft. ω× is the equivalent of the cross product ω×.Written out it is

ω× =

0 −ωz ωyωz 0 −ωx−ωy ωx 0

(2-47)

It allows us to write the vector differential equations as matrix differential equations. The vector form is not tied to aparticular coordinate frame but the matrix form is tied to a frame. In this case the equations are written in the bodyframe so the two forms are equivalent. The spacecraft is nominally earth pointing and our control system will correctsmall deviations from that pointing.

Figure 2-10. Spacecraft reference frame

ωo

θz

z

y

To earth center

Positive Orbit Normal

x, v

North

South

East

Westxs

ys

The attitude kinematics come from the small angle approximation

ω = θ + (1− θ×)

0−ωo

0

(2-48)

and

θ× =

0 −θz θyθz 0 −θx−θy θx 0

(2-49)

20

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CHAPTER 2. DESIGN TUTORIAL 2.8. POWER AND THERMAL SUBSYSTEM

or

ωx = θx − ωoθz (2-50)ωy = θy − ωo (2-51)

ωz = θz + ωoθx (2-52)

For small angular deviations these equations reduce to

θ = I−1T (2-53)

where θ is a 3-vector (a vector with 3 elements) of small angular changes. We can then replace I−1T with a giving usthe final form that we will use for attitude control system design.

θ = a (2-54)

We have converted 3 coupled nonlinear first-order dynamical equations and the, not shown, 3 nonlinear first-order dif-ferential equations that relate attitude to rate with 3 second order uncoupled linear differential equations. Surprisingly,these equations are used for the design of stationkeeping control systems on real satellites. We can now design ourcontrol system very simply. Let

a = −K(τ θ + θ) (2-55)

τ is the time constant and K is the forward gain. This is known as a proportional-derivative control system sinceone part is proportional to the measurement which is θ and one part is proportional to the derivative of the measure-ment. Note that the units are consistent. That is, if we replace the expression inside the parenthesis above with thecorresponding units of τ and θ we have:

sec1

sec+ 1 (2-56)

Clearly, the units of time cancel, so that both terms are dimensionless. The second order system can be written in theform

θ + 2ζσθ + σ2θ = 0 (2-57)

where ζ is the damping ratio and σ is the undamped natural frequency. That is if ζ were zero the system would oscillateat frequency σ. We can equate the gains with these parameters if

K = σ2 (2-58)

τ = 2ζ√K

(2-59)

Generally, we want ζ = 1 to get a smooth, non-oscillatory response so this becomes

K = σ2 (2-60)

τ = 21√K

(2-61)

We multiply the acceleration, computed above, by the inertia matrix to get the control torque

a = −IK(τ θ + θ) (2-62)

where I is the inertia matrix.

2.8 Power and Thermal Subsystem

The power and thermal subsystems for the spacecraft shown in Figure 2-11 on the next page.

21

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2.9. SIMULATION CHAPTER 2. DESIGN TUTORIAL

Figure 2-11. Power and thermal subsystems

RF

Radiated Heat

Battery

Transponder

Solar Panels

RadiatorShunt

Radiated Heat

Sun

We discuss these two subsystems together because they are both involved in the thermodynamic equilibrium for thespacecraft. The source of both power and heat is the sun. All incoming solar flux, whether converted to electricity ornot, must be radiated back into space. Electric power may be radiated as radio frequency emissions for the payload ortemporarily stored in a battery. The power balance equation is

Pa = Prf + Eb + Ps (2-63)

where Pa is the power coming in from the solar array, Prf is the RF power emitted by the antenna Eb is the energystored in the battery and Ps is the power transferred by the shunts to the radiator. To simplify the analysis we willignore the rest of the spacecraft and concern ourselves only with the solar panels and the power system describedabove. This is consistent with our solar pressure disturbance model which only models the solar panels. This equationhas two forms. When the battery is fully charged

Pa = Prf + Ps (2-64)

when it is partially chargedPa = Prf + Eb (2-65)

2.9 Simulation

2.9.1 Dynamical Model

We’ll write out the full dynamical model in state space form. We will be repeating some of the equations given above.All of these equations go into our right-hand-side. We are only going to simulate attitude control but we will includethe power system and orbital dynamics so we can look at the pointing performance over several orbits.

Attitude dynamics:ω = I−1

(r×n Fn + r×s Fs + Tc − ω×Iω

)(2-66)

where rn is the vector to the north array and rs is the vector to the south array.

22

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CHAPTER 2. DESIGN TUTORIAL 2.9. SIMULATION

External forces:

Fn = −ScAδn (2ρsδnnn + ρas) (2-67)

Fs = −ScAδs (2ρsδsns + ρas) (2-68)

where the subscripts s and n refer to north and south. F = Fn + Fs.

Attitude kinematics:

θx = ωx + ωoθz (2-69)θy = ωy + ωo (2-70)

θz = ωz − ωoθx (2-71)

Orbit dynamics:

r = −µ2r

|r|3− µ3

(r − d|r − d|3

+d

|d|3

)− alu+

Fn + Fsm

(2-72)

where where the longitudinal acceleration is

aλ = −1r

∂V

∂λ=

6µ2a2

r4[s2,2 cos 2λ− c2,2 sin 2λ] (2-73)

and m is the mass

Orbit kinematics:r = v (2-74)

Power system dynamics with the battery saturated:

Eb = 0 (2-75)

and when the battery is partially charged:Eb = Pa − Prf (2-76)

wherePa = ηρaAS (δn + δs) (2-77)

The temperature of the solar arrays is

Ts =(

(1− η)ρaSδs2σε

)1/4

(2-78)

Tn =(

(1− η)ρaSδn2σε

)1/4

(2-79)

The power delivered to the ground by the antenna is

θ3dB = 70λ

DT(2-80)

G =4πAλ2

(2-81)

LR =(

4πRλ

)2

(2-82)

LP = 101.2“

θθ3dB

”2

(2-83)

PT =PrfGTGRLRLP

(2-84)

The state vector is

x =[ωx ωy ωz θx θy θz vx vy vz rx ry rz Eb

]T(2-85)

23

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2.9. SIMULATION CHAPTER 2. DESIGN TUTORIAL

2.9.2 Control System

Our control system in discrete form is

Tc = −IK(θk + τ

θk − θk−1

∆t

)(2-86)

where K is the forward gain and τ is the rate gain time constant. ∆t is the time step between control updates. Thiscontrol system approximates rate by a first difference approximation.

fracθk − θk−1∆t (2-87)

In practice this is a bad idea since this leads to amplifying noise. For example, if the attitude has not changed betweensamples but the measurement is noisy this term will cause the control system to respond only to noise. In practice theentire measurement should pass first through a noise filter or a rate filter should be added to the rate term.

We’ll use the analog gains computed above because we will sample the control system fast enough (relative to the timeconstant of the model) that the discrete nature of the updates is not important. We are also assuming that our attitudesensor measures the angles relative to the local vertical frame. An earth sensor can measure pitch and roll easily butyaw is more difficult. Nonetheless we will assume that a yaw measurement is available.

2.9.3 Integrated Simulations

The Simulation script implements the integrated simulation. The two cases given in this chapter are run by chang-ing the following single line of code. You can change the variable simName to “equinox” to get an equinox simulation.

simName = ’Solstice’;

Winter Solstice Simulation

The first simulation will be 48 hours at solstice when the sun vector is at its largest angle to the orbit plane. Thiscauses increased disturbances if both solar panels curve in towards the sun vector. Figure 2-12 on the facing page andFigure 2-13 on the next page show the results. The orbit results show the oscillatory nature of the orbit but the plotsdo not show the effect of the disturbance force shown in the lower plot. In this simulation the sun and triaxial earthharmonics are perturbations along with solar pressure on the solar panels.

The attitude plots show the steady control effort to cancel the disturbance torque. There is an offset in attitude becausethe control is a proportional derivative controller. The offset is determined by the gain of the proportional term and theinertial torque. This torque is caused by the bending of the solar panels towards the sun. The battery starts unchargedbut since we are not modeling eclipses it quickly reaches the fully charged state. For the same reason the paneltemperature does not change. The RF power changes because the distance to the point of measurement changes.

Spring Equinox Simulation

At equinox the sun is in the orbit plane. With our model there are no disturbances on the spacecraft. Figure 2-14 onthe facing page and Figure 2-15 on page 26 show the results. Two differences are observed. The sun is in the orbitplane and does not produce an out-of-plane force. As a consequence there is no perturbation in the z direction. Inaddition, because the sun is in the orbit plane the disturbance torque is also zero and there is no attitude offset.

24

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CHAPTER 2. DESIGN TUTORIAL 2.9. SIMULATION

Figure 2-12. Solstice simulation orbit results

0 5 10 15 20 25 30 35 40 45 50−5

0

5x 10

4

Pos

ition

(km

)

Solstice Orbit

xyz

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

Vel

ocity

(km

/s)

xyz

0 5 10 15 20 25 30 35 40 45 50−20

−15

−10

−5

0

For

ce (µ

N)

Time (hr)

xyz

−1.5 −1 −0.5 0 0.5−1

0

1

2

3

4

5

6

∆ x (km)

∆ y

(km

)

Rotating frame

0 10 20 30 40 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time (hr)

∆ z

(km

)

Rotating frame

0 10 20 30 40 50−5

0

5x 10

4

Time (hr)

Pos

ition

(km

)

ECI frame

0 10 20 30 40 50−2

0

2

4

6

Time (hr)

Pos

ition

(km

)

Rotating frame

∆ x∆ y

Figure 2-13. Solstice simulation attitude, power, and thermal results

0 5 10 15 20 25 30 35 40 45 50−2

0

2

4x 10

−3

θ (d

eg)

Solstice Attitude

xyz

0 5 10 15 20 25 30 35 40 45 50−10

−5

0

5x 10

−5

ω (

rad/

s)

xyz

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

Tor

que

(µ N

m)

Time (hr)

xCyCzCxDyDzD

0 10 20 30 40 500

200

400

600

800

1000

1200B

atte

ry (

W−

s)Solstice Thermal and Power

0 10 20 30 40 50304.6

304.8

305

305.2

305.4

305.6

305.8

306

Pan

el T

empe

ratu

re (

deg−

K)

NS

0 10 20 30 40 501077

1077.5

1078

1078.5

1079

1079.5

Pow

er (

W)

Time (hr)0 10 20 30 40 50

−126.068

−126.0675

−126.067

−126.0665

−126.066

−126.0655

−126.065

−126.0645

RF

pow

er (

dBW

)

Time (hr)

Figure 2-14. Equinox simulation orbit results

0 5 10 15 20 25 30 35 40 45 50−5

0

5x 10

4

Pos

ition

(km

)

Equinox Orbit

xyz

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

Vel

ocity

(km

/s)

xyz

0 5 10 15 20 25 30 35 40 45 50−20

−15

−10

−5

0

For

ce (µ

N)

Time (hr)

xyz

−1.5 −1 −0.5 0 0.5−1

0

1

2

3

4

5

6

∆ x (km)

∆ y

(km

)

Rotating frame

0 10 20 30 40 50−1

−0.5

0

0.5

1

Time (hr)

∆ z

(km

)

Rotating frame

0 10 20 30 40 50−5

0

5x 10

4

Time (hr)

Pos

ition

(km

)

ECI frame

0 10 20 30 40 50−2

0

2

4

6

Time (hr)

Pos

ition

(km

)

Rotating frame

∆ x∆ y

25

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2.9. SIMULATION CHAPTER 2. DESIGN TUTORIAL

Figure 2-15. Equinox simulation attitude, power and thermal results

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

θ (d

eg)

Equinox Attitude

xyz

0 5 10 15 20 25 30 35 40 45 50−8

−6

−4

−2

0x 10

−5

ω (

rad/

s)

xyz

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

Tor

que

(µ N

m)

Time (hr)

xCyCzCxDyDzD

0 10 20 30 40 500

200

400

600

800

1000

1200B

atte

ry (

W−

s)Equinox Thermal and Power

0 10 20 30 40 50310.5

311

311.5

312

312.5

313

Pan

el T

empe

ratu

re (

deg−

K)

NS

0 10 20 30 40 501174

1174.5

1175

1175.5

1176

1176.5

Pow

er (

W)

Time (hr)0 10 20 30 40 50

−126.068

−126.0675

−126.067

−126.0665

−126.066

−126.0655

−126.065

−126.0645

RF

pow

er (

dBW

)

Time (hr)

26

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601

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INDEX

Actuatorcontrol moment gyro, 37, 245

single axis, 248magnetic torquer, 34, 250, 433, 439minimum impulse bit, 248momentum wheel, 90, 424, 489reaction wheel, 34, 89, 242, 350, 442single gimbal cmg, 353solenoid, 251stepping motor, 257Thruster, 433thruster, 34, 248, 351, 421

electrothermal hydrazine, 414pulsewidthmodulation, 248

Type, 241Attitude Dynamics

dual spin turn, 424flexible structures, 76, 453geosynchronous spacecraft, 425gyrostat, 75inertia, 69

positive definite, 69rigid body, 72

Attitude Estimation, 357batch methods, 364

Bayesian, 365differential corrector, 365maximum likelihood, 365

gyro model, 359iterated extended Kalman Filter, 360Kalman Filter, 359simple, 357spin axis, 368star identification, 382Unscented Kalman Filter, 361

Budgets, 187mass, 189pointing, 187power, 191propellant, 189

Command Distribution, 349gimbaled thrusters, 351

logic, 354optimal torque, 349reaction wheels, 350single gimbal CMG, 353

Control, 2Bode plot, 296control limiting, 336cross-axis coupling, 338digital, 312

continuous to discrete transformations, 318modified continuous design, 312

double integrator, 300, 304, 309, 311, 320, 324, 333,336

feedback, 295flexible structure, 322, 453

collocated, 325lead compensation, 326non-collocated, 326, 327

generalized integrator, 309geosynchronous, 413Linear Quadratic Gaussian methods, 306model following, 330momentum, 443Nichols plot, 296noise filtering, 430nutation, 432phase plane, 336PID, 333, 335, 336, 433rate control, 296robust, 303roll/yaw, 430root locus, 297, 327stationkeeping, 427Sun nadir, 435uncertainty, 303

Design Process, 27configuration management, 28databases, 28requirements, 27test plans, 29

Disturbancesaerodynamic, 272diffuse reflection, 278, 281, 446

602

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INDEX INDEX

external, 271Internal, 283Plume, 281

accommodation coefficients, 282method of characteristics, 281

radio frequency, 277residual dipole, 276solar pressure, 278, 445specular reflection, 445thermal, 280, 446

Fault Detection, 487detection filters, 494examples, 497failures, 488neural nets, 495online approximator, 495parameter estimation, 494parity relations, 493reasoning, 496reference models, 492satellite, 487

Flight Operations, 511example, 516mission control center, 515organizations, 511preparation, 513team organization, 514timeline, 511

Formation Flyingassignment methods, 155control, 145

analytic solution for circular orbits, 147linear programming for circular orbits, 148linear programming for eccentric orbits, 150

coordinate systems, 121definition, 119Gauss variational equations, 132Gauss’ variational equations, 133geometric parameters, 126, 136, 140geometric parameters for circular orbits, 136geometric parameters for eccentric orbits, 140, 142guidance, 152, 155halo orbit, 135Hills equations, 128, 129Hills frame, 124Lawdens equations, 130, 131Libration points, 133mean and osculating elements, 122missions, 119motivation, 119multiple team framework, 153orbit regimes, 120orbital element differences, 123

orbital elements, 121relative dynamics, 128relative frames, 124

Geosynchronous, 413acquisition, 416attitude determination, 423dual spin turn, 424mission orbit, 414requirements, 413spinning transfer orbit, 420transfer orbit, 414

ISS, 461proximity operations requirements, 464spacecraft, 461

ATV, 461Progress, 462Shuttle, 462Soyuz, 462

Kinematicscoordinate transformations, 57Euler angles, 58quaternion, 60

derivative, 62interpretation, 66linearization, 63

small angles, 66transformation matrices, 58

Launch and Reentry, 161azimuth angle, 170hypersonic lift and drag, 170Lambert’s method, 172launch guidance, 173orbital plane, 171three dimensional bank and flight path angle, 168three dimensional rotating Earth, 167two dimensional flat Earth, 164two dimensional optimal transfer, 161vehicles, 161

Linkantenna, 558budget, 559communication system, 553ISL, 563

constellations, 565implemented ISLs, 565lower layer protocols, 568networking, 566OSI reference model, 566

L1, 560line of sight, 563losses, 558

603

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INDEX INDEX

atmosphere, 559free space, 559pointing error, 559polarization, 559rain, 559

noise, 554attenuator, 556planet, 556receiver, 557sky, 556Sun, 555

Shannon Information Theorem, 554

Mathcalculus, 45differential equations, 286floating point operations, 44Laplace Transforms, 45, 587matrix, 41matrix identities, 44matrix operations, 43numerical integration, 287

stiff equations, 289probability, 579

axiomatic, 579binomial theorem, 580distributions, 580measurements, 583multivariate normal distributions, 584outliers, 585random signals, 584

spherical geometry, 44law of cosines, 45

vector, 41math

numerical integrationdiscontinuities, 289

Mission Planning, 517MissionPlanning

attitudeprofile, 529groundcoverage, 527launchopportunities, 521observation, 523orbitdesign, 517repeatgroundtrack, 519synsynchronous, 518

Multibody Dynamics, 83pivoted momentum wheel, 90topological tree, 83two body translational, 85

Optics, 221diffraction limit, 226errors, 228

geometry, 223imaging chip

APS, 234CCD, 234CID, 234

imaging chips, 239light gatherin, 227nomenclature, 221performance, 226pinhole camera, 230radiometry, 231radiosity, 233star trackers, 239telescopes, 222

Orbit, 95cartesian coordinates, 96cylindrical coordinates, 97equinoctial elements, 98gravity model, 101Kepler, 100Keplerian elements, 97linearized, 104numerical integration, 100propagation, 100

Orbit Estimation, 385autonomous, 393

GPS, 394MAGNAV, 394MANS, 395satellite to satellite tracking, 395SMART-1, 398

continuous discrete Kalman filter, 386disturbance estimator, 385

Orbit Maneuvers, 107A*, 114coplanar, 108Lambert law, 110low thrust, 111non planar, 110

Orbit Measurements, 398

Power, 547bus, 551generator, 547

auxiliary power unit, 551fuel cell, 550nuclear reactor, 550radioisotope thermal generator, 550solar, 547

load, 551Preliminary Design, 31

processor, 38propulsion, 39requirements, 31

604

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INDEX INDEX

Preliminary designcost, 39

Preliminary Designsatellite configuration, 33Propulsion, 259

electrothermal hydrazine, 414engines, 259fusion, 265Hall thruster, 264ion engine, 264low power electric, 264nozzle, 262nuclear thermal, 265rocket equation, 260

Rendezvous Sensors, 201antenna gain, 209CNR, 214Examples

ADLT, 205Apollo LM, 202AVGS, 202Cloud Profiling Radar, 205DART, 201ETS-VII, 203Kurs, 204NEAR, 205Orbital Maneuvering Vehicle, 204PRISMA, 201Relavis, 202Sandia, 205Space Shuttle, 203VGS, 202XSS-11, 201

ladar, 213beam scanning, 214optics, 217radiometry, 232

radar, 206analysis, 211bands, 207configuration, 208cross-section, 211losses, 212signal to noise ratio, 209

range equation, 215range resolution, 210vision, 216

Sensor, 193accelerometer, 200angle encoder, 38, 200Earth

scanning, 196earth, 38, 194, 421, 429

gyro, 38, 199, 359, 421, 429, 438horizon, 38, 194, 421magnetometer, 38, 200potentiometer, 38, 200stellar, 38, 438Sun, 38, 199, 368, 421, 438tachometer, 430

Sensor:horizon sensor, 369Sensors

navigation, 398landmark, 401telescope, 404

ranging, 399star trackers, 396

Active Pixel Sensors, 397CCD, 397CID, 397

Simulation, 3, 285control system verification, 290discontinuities, 289dual spin turn, 424linear, 285stochastic processes, 291

solar flux, 447Solar Sail, 445

cone and clock angle, 457flexibility, 453force model, 446guidance, 457semi-major axis control, 459sliding mass, 451vanes, 450

Spacecraft Control, 2Spacecraft Operations, 5Standard Atmosphere, 589Sun Nadir, 34, 435

attitude determination, 440momentum control, 443pointing, 436sensors, 438solar array, 441

TestingDC-X, 478INSAT 1, 476Intelsat V, 477IRAS, 472MBB, 478methodology, 481

flight vehicle, 482life-cycle, 482requirements flow, 481test levels, 485

MOS-1, 472

605

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INDEX INDEX

MSTI, 478SMART-1, 479Space Telescope, 477Telecom 1, 472

Thermalconductive, 537heat pipe, 538multilayer insulation, 539radiation, 536radiative, 537radiator, 538thermal balance, 535

TimeEarth rotation, 50Greenwich Mean Time, 51Julian date, 51scales, 49

Trajectory Optimization, 175constraints, 178continuation methods, 178downhill simplex, 177examples

Solar Polar Imager, 182Zermelo’s problem, 181

genetic algorithms, 177Hamilton’s, 176locally optimal, 458methods, 176problem classes, 180simulated annealing, 177

User Interface, 4

606

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Princeton Satellite Systems, Inc.

Spacecraft Attitude and Orbit Control is a state-of-the-art reference covering the latest advances in spacecraft attitude and orbit control. It is based on 30 years of experience in spacecraft control and includes cutting edge work that is unavailable in any other book. The book is intended to be fully self-contained and includes all required mathematics and control theory needed to delve into spacecraft control. The book approaches spacecraft control from a broader perspective by covering relative spacecraft position control as well as attitude control.

The book includes complete chapters on spacecraft examples including solar sails, formation flying, geosynchronous spacecraft and sun-nadir pointing spacecraft. Topics that influence attitude control, including orbit and attitude estimation, thermal control, power systems and communications links, are also included. Other chapters include spacecraft operations, fault detection and the spacecraft design process.

Numerous examples with code from Princeton Satellite Systems’ Spacecraft Control Toolbox for MATLAB® are included. A tutorial chapter provides a complete spacecraft design and includes all of the Matlab code needed to replicate the results. Owners of the toolbox can run every example and the book is included with each Professional and Classroom purchase. However, the textbook is fully self-contained and it is not necessary to own the toolboxes to learn the material.

The textbook is suitable for college juniors and seniors, graduate students and aerospace professionals.

SPACECRAFTATTITUDE and ORBIT

CONTROL

Michael Paluszek, Pradeep Bhatta, Paul Griesemer,

Joseph Mueller and Stephanie Thomas

S E C O N D E D I T I O N