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Optimal Orbit-Raising and Attitude Control of All-Electric Satellites

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Optimal Orbit-Raising and Attitude Control of All-Electric Satellites OPTIMAL ORBIT-RAISING AND ATTITUDE CONTROL OF ALL-ELECTRIC SATELLITES A Dissertation by Suwat Sreesawet Master of Science, Wichita State University, 2014 Bachelor of Engineering, Kasetsart University, 2009 Submitted to the Department of Aerospace Engineering and the faculty of the Graduate School of Wichita State University in the partial fulfillment of the requirements for the degree of Doctor of Philosophy December 2018 c Copyright 2018 by Suwat Sreesawet All Rights Reserved OPTIMAL ORBIT-RAISING AND ATTITUDE CONTROL OF ALL-ELECTRIC SATELLITES The following faculty members have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy, with a major in Aerospace Engineering. Atri Dutta, Committee Chair James E. Steck, Committee Member Roy Myose, Committee Member Animesh Chakravarthy, Committee Member John Watkins, Committee Member Accepted for the College of Engineering Steven Skinner, Interim Dean Accepted for the Graduate School Dennis Livesay, Dean iii DEDICATION To my family, my mother, father, and sister who have always supported me and provided me the warmth to my heart for entire life, even though I am on the other side of world. To my future wife who always cares for me, and to my friends with whom I have spent many great times together. iv ACKNOWLEDGMENTS First of all, I have many thanks for my family, my dad, mon and sister. They always keep supporting and encouraging me since the day I started breathing with my own nose. I always perceive their love and support even I have been in the opposite side of the world for many years. I would like to thank my adviser, Dr. Atri Dutta, for his guidance and support all along this dissertation. He has kindly and sincerely helped since the first day i met him. I have learned a variety of knowledge from him. I would also like to thank the committee members Dr. James E. Steck, Dr. John Watkins, Dr. Animesh Chakravarthy and Dr. Roy Myose for their advice and suggestions for this research. I would like to tan I would also like to thank Ministry of Science and Technology, Thailand for funding my graduate program. I would Thanks also to my colleagues and friends who helped with their suggestions during the course of this research work. v ABSTRACT Electric propulsion is gaining popularity among satellite operators due to its fuel efficiency. However, electric propulsion has the limitation of producing a small magnitude of thrust, meaning that the transfer time to geostationary orbit is of the order of several months. The long transfer time adds more complexity to the mission design process due to the long exposure to hazardous radiation belts. Obviously, thrusters require electric power that is generated from sunlight by satellite solar panels. Therefore, the earth's shadow significantly impacts the orbit-raising maneuver. This study proposes a novel, robust, and fast numerical methodology for generating low-thrust trajectories to the geosynchronous orbit. This methodology utilizes a new set of state variables that has a physical interpretation and exhibits slow variation under a small magnitude of thrust. The absence of mathematical singularities in the equatorial plane adds to the benefits. The new set of state variables, along with a closed-loop guidance scheme and direct optimization methodology, is used to optimize the satellite trajectory. An unconstrained optimization scheme is able to robustly and rapidly generate low-thrust orbit-raising trajectories for a variety of mission scenarios, various initial orbits, application of electric battery to allow thrusting during eclipses, and orbital perturbations due to the earth's oblateness or a third body. The proposed methodology can be seamlessly integrated into receding horizon control scheme, which recomputes the minimum time trajectory at regular intervals referred to as planning horizon. The attitude of spacecraft must also be maintained in a desired direction which can be time-varying. Regular satellites perform orbit-raising using stowed solar array. In contrast, all-electric satellites perform orbit-raising using deployed solar arrays. Therefore, simple inverse controllers for attitude control with a neural network-based observer has been studied and evaluated. We demonstrate that the performance of the inverse controller is drastically improved with the proposed observer. vi TABLE OF CONTENTS Chapter Page 1. INTRODUCTION ............................... 1 1.1 Motivation . 1 1.2 Research Objective . 3 1.3 Literature Review . 4 1.3.1 Low-Thrust Orbit-Raising . 4 1.3.2 Attitude Control . 6 1.4 Contributions . 8 1.5 Organization of Dissertation . 9 2. SPACECRAFT TRANSLATIONAL DYNAMICS ............ 11 2.1 Two-Body Problem . 11 2.2 Perturbation . 14 2.2.1 Engine Thrust . 14 2.2.2 Third Body . 15 2.2.3 Earth Oblateness . 16 2.3 Set of Orbital Parameters . 17 2.3.1 Classical Orbital Elements . 17 2.3.2 Modified Equinoctial Orbital Element . 19 2.3.3 Spherical Coordinates . 21 2.4 Proposed Set of Orbital State Variables . 22 2.4.1 Reference Frames . 23 2.4.2 State Variables and Transformation . 27 2.4.3 Variation of Proposed State Variables . 30 3. SPACECRAFT ROTATIONAL DYNAMICS ............... 38 3.1 Reference Frames . 38 3.2 Attitude Representation . 39 3.2.1 Euler's Angles & Direction Cosine Matrix . 39 3.2.2 Single Rotation and Axis of Rotation . 41 3.2.3 Quaternion or Euler's Parameters . 42 3.3 Attitude Kinematics . 44 3.4 Attitude Motion . 49 4. NUMERICAL TRAJECTORY OPTIMIZATION ............. 54 4.1 Unconstrained Optimization . 54 vii TABLE OF CONTENTS (continued) Chapter Page 4.1.1 Orbit Segmentation and Eclipse Consideration . 55 4.1.2 Optimization Sub-Problem . 57 4.1.3 Terminal Conditions . 62 4.1.4 Special Case of Planar Transfer . 62 4.2 Receding Horizon Control . 64 4.2.1 Constrained Non-Linear Optimization Problem . 65 4.2.2 Initial Guess Generation . 67 5. ATTITUDE CONTROLLER DESIGN ................... 69 5.1 Inverse Controller . 69 5.2 Uncertainty and Compensation . 70 5.2.1 The Design of Modified State Observer . 72 6. SIMULATION RESULTS ........................... 79 6.1 Results of Low-Thrust Orbit-Raising . 79 6.1.1 Planar Transfers . 80 6.1.2 Non-Planar Transfers . 86 6.1.3 Sub-Optimality of Computed Solutions . 91 6.2 Receding Horizon Control . 92 6.3 Attitude Controller . 95 7. ATTITUDE CONTROL TESTBED DESIGN ............... 105 7.1 Background . 105 7.1.1 State of Spherical Air Bearing Testbed . 106 7.2 Testbed Design . 107 7.2.1 Mechanical Design . 108 7.2.2 Electrical Design . 113 8. CONCLUSIONS ................................ 126 8.1 Thesis Summary . 126 8.2 Future Work . 128 8.2.1 Low-Thrust Trajectory Generation . 128 8.2.2 Attitude Controller . 129 8.2.3 Attitude Control Testbed . 130 REFERENCES .................................... 131 viii TABLE OF CONTENTS (continued) Chapter Page Appendices ....................................... 141 ix LIST OF TABLES Table Page 3.1 List of possible rotational sequences and singularities. 39 −1 3.2 List of function S (θ2; θ3) for all Euler's angle sequence. 46 6.1 Summary of orbit-raising results. 91 6.2 Optimality gap of solutions computed using proposed methodology. 92 x LIST OF FIGURES Figure Page 1.1 Van Allen radiation belts [1]. 2 2.1 Radius vectors in two-body problem. 12 2.2 Perturbation from third body. 16 2.3 3-1-3 sequential rotation of classical orbital parameter. 18 2.4 Spherical coordinates. 22 2.5 First rotation about the basis vector J^ of frame I. 24 2.6 Second rotation about the basis vector I^0 of frame I0. 25 2.7 Third rotation about the basis vector k^ of frame O. 26 3.1 Reference frames to describe actual and desired spacecraft orientations. 38 3.2 Rotation of a reference frame about an axis. 41 3.3 Structure of spacecraft with masses. 50 3.4 Multiple-spin body. 51 4.1 Spiral shape of low-thrust trajectory (not to scale). 55 4.2 Segmentation of the trajectory over a revolution. 56 4.3 Model of the Earth's shadow. 57 4.4 Algorithm flowchart. 64 4.5 Spacecraft path planning. 65 4.6 Flowchart of receding horizon scheme. 68 5.1 Block diagram of inverse feedback control and observer. 73 5.2 Structure of neural network. 74 6.1 Result of the circular LEO to GEO transfer. 81 xi LIST OF FIGURES (continued) Figure Page 6.2 Result of GTO to GEO planar transfer. 82 6.3 Result of sub-GTO to GEO planar transfer. 83 6.4 Result of planar super-GTO to GEO planar transfer. 84 6.5 Result of circular LEO to GEO Transfer with energy storage. 85 6.6 Result of non-planar circular LEO to GEO transfer. 87 6.7 Result of the non-planar GTO to GEO Transfer. 89 6.8 Result of the non-planar LEO to GEO case with the energy storage. 90 6.9 Result of the receding horizon control in earth orbit. 94 6.10 Result of the receding horizon control in Enceladus's orbit. 95 6.11 Comparison of performance between with and without MSO. 97 6.12 Control input torque with MSO. 97 6.13 Performance of MSO . 98 6.14 Comparison of performance between the controllers with and without MSO. 100 6.15 Control input and its transient period. 101 6.16 Performance of MSO prediction . 101 6.17 Problem configuration. 102 6.18 Comparison of performance between the controllers with and without MSO. 103 6.19 Control input. 104 6.20 Performance of MSO prediction . 104 7.1 Shapes of spherical air bearing. 107 7.2 Examples of spherical attitude control testbeds. 108 xii LIST OF FIGURES (continued) Figure Page 7.3 Air bearing structure..
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