Dynamic Programming Applied to Electromagnetic Satellite Actuation

138 Pages·2013·4.4 MB·English

Dynamic Programming Applied to
Electromagnetic Satellite Actuation
Gregory Eslinger
ssl
May 2013 # 2-13
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Dynamic Programming Applied to
Electromagnetic Satellite Actuation
Gregory Eslinger
ssl
May 2013 # 2-13
This material is declared a work of the U.S. Government and is not subject to
copyright protection in the United States.
This work is based on the unaltered text of the thesis by Gregory Eslinger
submitted to the Department of Aeronautics and Astronautics in partial fulfillment
of the requirements for the degree of Master of Science in Aeronautics and
Astronautics at the Massachusetts Institute of Technology.
Acknowledgments
This work was performed primarily under contract Z660401 with the Defense Ad-
vanced Research Project Agency (DARPA) as part of the Resonant Inductive Near-
field Generation System (RINGS) program. The author gratefully thanks the spon-
sors for their generous support that enabled this research.
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Contents
1 Introduction 13
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.1 Spacecraft Formation Flight . . . . . . . . . . . . . . . . . . . 15
1.1.2 Electromagnetic Formation Flight . . . . . . . . . . . . . . . . 17
1.1.3 Resonant Inductive Near-Field Generation System . . . . . . . 22
1.1.4 RINGS Control . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.1.5 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . 25
1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.1 EMFF Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.2 EMFF Testbeds . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2.3 EMFF Control . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Dynamic Programming 33
2.1 Fundamentals of Dynamic Programming . . . . . . . . . . . . . . . . 35
2.2 Dynamic Programming Formulation Types . . . . . . . . . . . . . . . 37
2.2.1 Discounted Cost. . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.2 Stochastic Shortest Path . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 Average Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Approximate Dynamic Programming . . . . . . . . . . . . . . . . . . 40
2.3.1 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Cost Approximation . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Applying Dynamic Programming to a Physical System . . . . . . . . 46
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3 Formulating RINGS as a Dynamic Programming Problem 49
3.1 RINGS Dynamic Programming Formulations . . . . . . . . . . . . . . 50
3.1.1 Position State Reduction . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Attitude State Reduction . . . . . . . . . . . . . . . . . . . . . 53
3.2 Specific Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Static Axial Case . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 Rotating Axial Case . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.3 Planer Motion with Commanded Attitude . . . . . . . . . . . 57
3.2.4 Full Planer with Commanded Torque . . . . . . . . . . . . . . 58
4 Cost-to-Go For EMFF Systems 59
4.1 Cost-to-Go Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Aggregation Results . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 Cost Approximation Results . . . . . . . . . . . . . . . . . . . 60
4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.1 Aggregation Controller Performance . . . . . . . . . . . . . . . 63
4.2.2 Aggregation Balance . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.3 Cost Approximation Performance . . . . . . . . . . . . . . . . 65
4.2.4 Aggregation vs. Cost Approximation . . . . . . . . . . . . . . 65
5 Implementing A Dynamic Programming Controller 69
5.1 Control Design Considerations . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 Controller Development . . . . . . . . . . . . . . . . . . . . . 71
5.1.3 Controller Storage . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.4 Controller Operation . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.5 Use of Dynamic Programming . . . . . . . . . . . . . . . . . . 72
5.2 Dynamic Programming Implementation . . . . . . . . . . . . . . . . . 73
5.2.1 General Architecture . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.2 Direct Input Controller . . . . . . . . . . . . . . . . . . . . . . 77
5.2.3 Rollout Controller . . . . . . . . . . . . . . . . . . . . . . . . 77
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6 Nonlinear Programming Mass Property Identification for Space-
craft 81
6.1 Known Methods for Mass Identification . . . . . . . . . . . . . . . . . 82
6.1.1 Least Squares Methods . . . . . . . . . . . . . . . . . . . . . . 82
6.1.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3 Solving the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3.1 Computing the Gradient . . . . . . . . . . . . . . . . . . . . . 86
6.3.2 Gradient-Only Solvers . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Convergence Guarantees . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.1 Convexity of h(x) . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4.2 Convergence on Actual Mass Parameters . . . . . . . . . . . . 92
6.5 Implementation Considerations . . . . . . . . . . . . . . . . . . . . . 93
7 System Characterization for Thruster-Based Spacecraft 95
7.1 SPHERES With Expansion Port . . . . . . . . . . . . . . . . . . . . 96
7.1.1 Predicted Changes . . . . . . . . . . . . . . . . . . . . . . . . 96
7.1.2 Mass Characterization Test . . . . . . . . . . . . . . . . . . . 98
7.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2 VERTIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2.1 Thruster Impingement . . . . . . . . . . . . . . . . . . . . . . 103
7.2.2 Mass Property Identification . . . . . . . . . . . . . . . . . . . 106
7.3 RINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3.1 Expected Results . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.2 Thruster Impingement . . . . . . . . . . . . . . . . . . . . . . 111
7.3.3 Mass Property Identification . . . . . . . . . . . . . . . . . . . 111
8 Model-Free State Estimation Using Line-of-Sight Transmitters 115
8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3 Position Determination . . . . . . . . . . . . . . . . . . . . . . . . . . 120
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8.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.3.2 Solving for Position . . . . . . . . . . . . . . . . . . . . . . . . 124
8.3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.4 Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9 Conclusion 131
9.1 Novel Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
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List of Figures
1-1 Scope of Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1-2 Example of a Mission Architecture That Uses Wireless Power Transfer 21
1-3 Example of Non-Keplerian Orbits Using EMFF . . . . . . . . . . . . 22
1-4 RINGS and SPHERES During an RGA Flight . . . . . . . . . . . . . 23
1-5 Linear Track EMFF Testbed . . . . . . . . . . . . . . . . . . . . . . . 27
1-6 3 DoF EMFF Testbed . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1-7 µEMFF Testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2-1 Simple Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2-2 Dynamic Programming Analysis of a Simple Markov Process . . . . . 34
2-3 Stochastic Shortest Path Markov Process . . . . . . . . . . . . . . . . 39
2-4 Illustration of State Aggregation . . . . . . . . . . . . . . . . . . . . . 41
2-5 Aggregation Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 42
2-6 Illustration of Aggregation Techniques . . . . . . . . . . . . . . . . . 42
2-7 Controller Development Using Dynamic Programming . . . . . . . . . 47
3-1 Definition of Two Coils in Proximity . . . . . . . . . . . . . . . . . . 50
3-2 General RINGS State Illustration . . . . . . . . . . . . . . . . . . . . 52
3-3 Definition of Coil Normal Vector . . . . . . . . . . . . . . . . . . . . . 54
3-4 Static Axial RINGS Setup . . . . . . . . . . . . . . . . . . . . . . . . 55
3-5 Rotating Axial RINGS Setup . . . . . . . . . . . . . . . . . . . . . . 56
3-6 Full Planer Motion RINGS Setup . . . . . . . . . . . . . . . . . . . . 58
4-1 Cost To Go Using Aggregation. . . . . . . . . . . . . . . . . . . . . . 60
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4-2 Cost To Go Using Cost Approximation With Sample Trajectories . . 62
4-3 Aggregation Performance Over Differing Number of Divisions . . . . 64
4-4 Aggregation Performance Over Differing Number of Divisions . . . . 66
5-1 Controller Development Flow . . . . . . . . . . . . . . . . . . . . . . 70
5-2 General Control Architecture . . . . . . . . . . . . . . . . . . . . . . 74
5-3 State Breakout Architecture . . . . . . . . . . . . . . . . . . . . . . . 75
5-4 RINGS Static Axial Breakout Architecture . . . . . . . . . . . . . . . 76
5-5 Direct Input Controller Architecture . . . . . . . . . . . . . . . . . . 78
5-6 Rollout Controller Architecture . . . . . . . . . . . . . . . . . . . . . 80
6-1 Hidden Markov Model . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6-2 Mass Property Identification Simulation . . . . . . . . . . . . . . . . 94
7-1 SPHERES With Expansion Port . . . . . . . . . . . . . . . . . . . . 96
7-2 Results of Thruster Characterization Test . . . . . . . . . . . . . . . 99
7-3 Results of Thruster Characterization Test . . . . . . . . . . . . . . . 101
7-4 The VERTIGO System with NASA Astronaut Tom Marshburn [75] . 103
7-5 VERTIGO Thruster Characterization Results . . . . . . . . . . . . . 105
7-6 Results of VERTIGO Mass Identification Maneuvers . . . . . . . . . 107
7-7 RINGS Center of Gravity Ground Test . . . . . . . . . . . . . . . . . 109
7-8 RINGS Thruster Characterization Results . . . . . . . . . . . . . . . 111
8-1 Example Transmitter Setup . . . . . . . . . . . . . . . . . . . . . . . 118
8-2 SPHERES Receiver Locations . . . . . . . . . . . . . . . . . . . . . . 118
8-3 Transmitter Distance Illustration . . . . . . . . . . . . . . . . . . . . 120
8-4 BearingAngleandRangeAdjustmentProbabilityDistributionFunctions122
8-5 Position Estimate Error Analysis . . . . . . . . . . . . . . . . . . . . 127
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