ATTITUDE AND ORBIT CONTROL USING THE SPACECRAFT CONTROL TOOLBOX V4.6 by the Staff of Princeton Satellite Systems, Inc. The software described in this document is furnished under a license agreement. The software may be used, copied or translated into other languages only under the terms of the license agreement. Attitude and Orbit Control Using the Spacecraft Control Toolbox v4.6 (December 2000) © Copyright 1996-2000 by Princeton Satellite Systems, Inc. All rights reserved. Any provision of Princeton Satellite System Software to the U.S. Government is with “Restricted Rights” as follows: Use, duplication, or disclosure by the Government is subject to restrictions set forth in subparagraphs (a) through (d) of the Commercial Computer Restricted Rights clause at FAR 52.227-19 when applicable, or in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software clause at DFARS 252.227-7013, and in similar clause in the NASA FAR Supplement. Any provision of Princeton Satellite Systems documentation to the U.S. Government is with Limited Rights. The contractor/manufacturer is Princeton Satellite Systems, Inc., 33 Witherspoon Street, Princeton, New Jersey 08542. MATLAB is a registered trademark of The MathWorks, Inc. Macintosh is a registered trademark of Apple Computer, Inc. Silicon Graphics is a registered trademark os Silicon Graphics Adobe, Acrobat and FrameMaker are trademarks of Adobe Systems Incorporated. All other trademarks are the property of their respective holders. All other brand or product names are trademarks or registered trademarks of their respective companies or organizations. Printing History December 2000 First Printing Princeton Satellite Systems, Inc. 33 Witherspoon Street Princeton, New Jersey 08542 Technical Support/Sales/Info: http://www.psatellite.com For Marilyn and Eric Table of Contents CHAPTER 1 Introduction 21 Spacecraft Control 21 Introduction 21 Mnemonics 22 Control System Design 22 Dynamics and Modeling of Systems 22 Software Design 23 User Interface Design 24 Spacecraft Operations 24 The Spacecraft Control Engineer’s Job 24 This Book 24 ACS Terms 24 Pointing Accuracy 25 Pointing Knowledge 25 Pointing Stability 25 Jitter 25 What A Spacecraft Control Engineer Needs To Know 26 CHAPTER 2 Design Process 29 Introduction 29 Design Issues 29 Introduction 29 Requirements Evaluation and Interpretation 29 Databases 30 Configuration Management of Software and Documentation 31 Test Plans 31 Design Process 31 Steps in the Process 31 General Comments 33 CHAPTER 3 Preliminary Designs 35 Introduction 35 Requirements Analysis 36 Direct Requirements 36 Attitude and Orbit Control Using the Spacecraft Control Toolbox 5 Indirect (or Derived) Requirements 37 Control System Requirements 37 Satellite Design 38 Selecting a Satellite Configuration 38 Selecting Actuators 39 Selecting Sensors 41 Cost 42 CHAPTER 4 Mathematics 45 Introduction 45 Vectors and Matrices 45 Notation 45 Vector and Matrix Representations of Operations 47 Matrix Operations 47 Special Matrices 48 The Matrix Inverse 48 Useful Matrix-Vector Identities 49 Floating Point Operations Required for Matrix Arithmetic 50 Spherical Geometry 50 Differential Equations 51 Introduction 51 Numerical Integration 52 Discontinuities 53 Stiff Equations 54 Probability 55 CHAPTER 5 Time 57 Introduction 57 Time Scales 57 CHAPTER 6 Coordinate Transformations 59 Coordinate Frames 59 Transformation Matrices 62 Quaternions 64 Introduction 64 Fundamental Properties of the Quaternion 65 Quaternion Nomenclature 66 Quaternion Operations 67 Quaternion Transformations 67 6 Attitude and Orbit Control Using the Spacecraft Control Toolbox Quaternion Derivative 68 Linearization of the Quaternion 69 Small Angles 73 Physical Interpretation of the Quaternion 73 CHAPTER 7 Control Design 75 Introduction 75 The General Control System 75 Fundamental Relationships 76 Tracking Errors 79 State Space Closed Loop Equations 81 Approaches to Robust Control 82 Introduction 82 Modeling Uncertainty 83 Control Structure Design 86 Nyquist Like Techniques 86 Quantitative Feedback Theory 87 LQG Methods 88 H∞ and m synthesis 89 Single-Input Single-Output Control Design 91 Introduction 91 Elementary Loop Compensation 91 Digital Control 98 Introduction 98 Modified Continuous Design 99 Flexible Spacecraft Control 114 Introduction 114 Two Coupled Inertias 114 Lead Compensation of the Minimum Phase System 118 Non-Collocated Sensor and Actuator 120 CHAPTER 8 Command Distribution 125 Introduction 125 The Optimal Torque Distribution Problem 125 Examples 127 Reaction Wheels 127 Gimbaled Thrusters 128 Simplex 129 Attitude and Orbit Control Using the Spacecraft Control Toolbox 7 CHAPTER 9 Attitude Dynamics 131 Introduction 131 Rigid Body 131 Multibody 133 Pivoted Momentum Wheel 135 Modeling Flexible Structures 139 CHAPTER 10 Orbits 143 Representations of Orbits 143 Propagating Orbits 144 CHAPTER 11 Budgets 147 Introduction 147 Pointing Budgets 147 Methodology 147 Beam Pointing Accuracy 148 Example Pointing Budget 148 Propellant Budgets 149 Introduction 149 Example Propellant Budget 150 CHAPTER 12 Sensors 153 Introduction 153 Types of Sensors 153 Optical Sensors 154 Star Trackers and Cameras 154 Horizon Sensors 155 Earth Sensors 156 Analog Sun Sensors 157 Digital Sun Sensors 158 Gyros 158 Other Sensors 159 Magnetometers 159 Accelerometers 159 CHAPTER 13 Actuators 161 Introduction 161 8 Attitude and Orbit Control Using the Spacecraft Control Toolbox Types of Actuators 161 Stepping Motor Model 162 Introduction 162 Electromechanical Dynamics 162 Reaction Wheel Model 164 Introduction 164 Motor Model 164 Reaction Wheel State Equations 165 Tachometer 166 Control Moment Gyro 166 Introduction 166 Modeling 167 Torque Distribution 167 Thrusters 168 Introduction 168 Physics of Thrusters 168 Pulsewidth Modulation 168 Minimum Impulse Bit 169 Time Constants 169 Fuel System 170 Magnetic Torquers 170 Solenoids 171 Introduction 171 Derivation of the equations of motion for a dual coil solenoid 172 Derivation of the equations of motion for a single coil solenoid 176 CHAPTER 14 Simulation 181 Introduction 181 Linear 182 Creating a State Space System 182 Zero Order Hold 183 First Order Hold 183 Nonlinear 184 CHAPTER 15 Disturbances 185 Introduction 185 External Disturbances 185 Introduction 185 Gravity Gradient 186 Residual Dipole 187 RF 187 Solar Pressure 187 Thruster Plumes 188 Attitude and Orbit Control Using the Spacecraft Control Toolbox 9 Internal Disturbances 189 Introduction 189 Fourier Series Representation 189 CHAPTER 16 Attitude Estimation 191 Introduction 191 Introduction to Estimation Theory 191 Gyro Model 193 Conversion of Continuous Time Data to Discrete 195 Simulation of Stochastic Processes 195 The Kalman Filter Algorithm 196 Batch Methods 197 Vector Measurements 199 CHAPTER 17 Geosynchronous Satellite Control 201 Introduction 201 Requirements 201 The Design Process 202 Transfer Orbit 202 Mission Orbit Design 203 The Geometry 205 Acquisition 205 Summary 207 CHAPTER 18 Geosynchronous Spacecraft Control Design 209 A Mission Architecture 209 Design Steps 211 Spacecraft Overview 211 Spinning Transfer Orbit 213 Dynamics 213 Actuators and Sensors 214 Changing the Spin Rate 216 Attitude Determination 217 Delta V Engine Firing 218 The Transfer Orbit Generator 218 Acquisition Using The Dual Spin Turn 219 Dynamics 219 Actuators and Sensors 219 10 Attitude and Orbit Control Using the Spacecraft Control Toolbox Initialization 219 Simulation 219 Pitch Acquisition 220 Disturbances 220 Dynamics 221 Introduction 221 Normal Operations 221 Dual Spin Stability 223 Stationkeeping Operations 223 Actuators and Sensors 225 Control System Organization 225 Modes 226 Earth Sensor 226 Gyros 227 Noise Filtering 227 Momentum Wheel Pitch and Tachometer Loops 227 Low Bandwidth Roll/Yaw Control 228 Thruster Control 231 High Bandwidth Roll/Yaw and Pitch Control 231 Magnetic Torquer Control 232 Thruster Control 232 Actuator Saturation 233 Thruster Resolution 233 Summary 234 CHAPTER 19 Sun Nadir Pointing Control 235 Introduction 235 Coordinate Frames 236 Sun Nadir Pointing 237 Components 239 Sensors 239 Actuators 241 Attitude Determination 242 Roll 242 Pitch 243 Sun Sensor Eye Pre-Processing 243 Solar Array Pitch 243 Yaw 244 Control 245 Reaction Wheel Loop 245 Attitude Loop 246 Solar Array Control 246 Momentum Control 246 Attitude and Orbit Control Using the Spacecraft Control Toolbox 11 CHAPTER 20 ACS Testing 249 Introduction 249 Industry Examples 249 Introduction 249 Telecom 1 250 MOS-1 251 IRAS 253 INSAT I 254 Intelsat V 257 Space Telescope 258 MBB 258 MSTI and DC-X 261 Summary 262 A Testing Methodology 263 Introduction 263 Requirements Flow and Testing 263 Testing Life-Cycle for the ACS Flight Software 264 Flight Vehicle Control System Testing 266 Test Levels (Preflight) 268 Test Levels (Flight) 269 CHAPTER 21 Fault Detection 271 Introduction 271 Failures 271 Failure Analysis 273 Fault Detection Isolation and Reconfiguration 274 Introduction 274 Approaches to Redundancy Management 276 The Fault Detection and Isolation System 278 An Example Using Detection Filters 281 CHAPTER 22 References 285 Introduction 285 Dynamics 285 Attitude Estimation 286 Control 286 Spacecraft Control 286 Astrodynamics 286 Propulsion 287 Real-Time Systems 287 12 Attitude and Orbit Control Using the Spacecraft Control Toolbox Devices 287 Software 287 Ephemeris 287 Heat Transfer 287 Testing 288 Coordinates 288 Failures 289 CHAPTER 23 Glossary 291 Index 295 Attitude and Orbit Control Using the Spacecraft Control Toolbox 13 14 Attitude and Orbit Control Using the 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