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A Study of Variable Thrust, Variable Specific Impulse Trajectories for Solar System Exploration A Thesis Presented to The Academic Faculty by Tadashi Sakai In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of Aerospace Engineering Georgia Institute of Technology December 2004 A Study of Variable Thrust, Variable Specific Impulse Trajectories for Solar System Exploration Approved by: John R. Olds, Advisor Amy R. Pritchett School of Aerospace Engineering School of Industrial and Systems Engi- neering Robert D. Braun David W. Way School of Aerospace Engineering National Aeronautics and Space Adminis- tration Eric N. Johnson School of Aerospace Engineering Date Approved: November 2004 To Naomi ACKNOWLEDGEMENTS There are many people whom I would like to thank. First, I would like to express my sincere gratitude and thanks to my graduate advisor Dr. John Olds. It has been a privilege to work with you. Without your guidance, wisdom, encouragement, and patience, this research would have not been possible. I would also like to thank the other members of my committee, Dr. Robert Braun, Dr. Eric Johnson, Dr. Amy Pritchett, and Dr. David Way. Your advice and insight has proved invaluable. I would also like to thank Dr. Panagiotis Tsiotras for his advice. Thank you also to my friends and coworkers in the Space Systems Design Lab. I really had a great time working with you. Especially I would like to thank Tim Kokan, David Young, Jimmy Young, and Kristina Alemany for their proofreading of this thesis. Reading and correcting my English must have been tough work. Other than the members of SSDL, Yoko Watanabe's knowledge and comments were really helpful. I hope their research will be fruitful and successful. I would like to thank my family in Japan. My father Toshio, mother Takae, elder brother Hiroshi, and younger sister Maki; they have all supported me so much. I can not thank you enough. Finally, I must thank my wife, Naomi, who has provided endless encouragement and support during the most difficult times of my work. Having you in my life to share my dreams and experiences makes them all the more special. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS iv LIST OF TABLES viii LIST OF FIGURES x LIST OF SYMBOLS AND ABBREVIATIONS xv SUMMARY xvii I INTRODUCTION 1 1.1 Problems of Variable Thrust, Variable Isp Trajectory Optimization . 1 1.2 Motivation for Research . 3 1.3 Research Goals and Objectives . 5 1.4 Approach . 6 1.5 Organization of the Thesis . 7 II EXHAUST-MODULATED PLASMA PROPULSION 9 2.1 Overview . 9 2.2 Mechanism . 10 2.3 Choosing the Power Level . 13 2.4 Literature Review . 16 III PRELIMINARY STUDY: SIMPLE TRAJECTORIES 20 3.1 Problem Formulation . 20 3.2 Results . 23 IV INTRODUCTION TO OPTIMIZATION PROBLEMS 29 4.1 Solution Methods for Optimization Problems . 29 4.2 Indirect Methods { Calculus of Variations . 30 4.2.1 Problems without Terminal Constraints, Fixed Terminal Time . 30 4.2.2 Some State Variables Specified at a Fixed Terminal Time . 32 4.2.3 Inequality Constraints on the Control Variables . 34 4.2.4 Bang-off-bang Control . 35 v V OPTIMIZATION OF INTERPLANETARY TRAJECTORY 37 5.1 Assumptions . 37 5.2 Equations of Motion for Low Thrust Trajectories . 42 5.2.1 VSI { No Constraints on Isp . 42 5.2.2 VSI { Inequality Constraints on Isp . 43 5.2.3 CSI { Continuous Thrust . 44 5.2.4 CSI { Bang-Off-Bang Control . 45 5.3 Solving the High Thrust Trajectory . 46 5.4 Problems with Swing-by . 49 5.4.1 Mechanism . 49 5.4.2 Equations of Motion . 50 5.4.3 Powered Swing-by . 54 VI DEVELOPMENT OF THE APPLICATION \SAMURAI" 56 6.1 Overview . 56 6.1.1 Capabilities . 56 6.1.2 Performance Index for Each Engine . 59 6.2 C++ Classes . 60 6.3 Flow and Schemes . 64 6.3.1 SAMURAI Flowchart . 64 6.3.2 VSI Constrained Isp . 66 6.3.3 CSI Continuous Thrust . 67 6.3.4 CSI Bang-Off-Bang . 67 6.4 Examples of Input and Output . 69 6.5 Validation and Verification . 74 6.5.1 Validation of High Thrust with IPREP . 74 6.5.2 Validation of CSI with ChebyTOP . 74 VII PRELIMINARY RESULTS: PROOF OF CONCEPT 79 7.1 Problem Description . 79 7.2 Numerical Accuracy . 82 7.3 Results and Discussion . 84 vi 7.4 Relationship among Fuel Consumption, Jet Power, and Travel Distance . 93 VIII NUMERICAL EXAMPLES: \REAL WORLD" PROBLEMS 99 8.1 Scientific Mission to Venus . 100 8.1.1 Venus Exploration . 100 8.1.2 Problem Description . 100 8.1.3 Results . 100 8.2 Human Mission to Mars: Round Trip . 108 8.2.1 Mars Exploration . 108 8.2.2 Problem Description . 109 8.2.3 Results . 109 8.3 JIMO: Jupiter Icy Moons Orbiter . 120 8.3.1 JIMO Overview . 120 8.3.2 Problem Description . 121 8.3.3 Results . 121 8.4 Uranus and beyond . 125 8.5 Swing-by Trajectories with Mars . 128 IX CONCLUSIONS AND FUTURE WORK 137 9.1 Conclusions and Observations . 138 9.2 Research Accomplishments . 141 9.3 Recommended Future Work . 142 APPENDIX A | RESULTS FROM PRELIMINARY STUDY 144 APPENDIX B | EQUATIONS USED IN THE PROGRAM SAMURAI 155 APPENDIX C | NUMERICAL TECHNIQUES 159 APPENDIX D | \SAMURAI" CODE MANUAL 177 REFERENCES 180 VITA 186 vii LIST OF TABLES 1-1 Examples of Spacecraft Propulsion Systems[4][42]. 1 3-1 Trip Time Savings(Comparison by Distance): (VSI - CSI)/CSI 100 (%) . 28 × 3-2 Trip Time Savings(Comparison by Initial Mass): (VSI - CSI)/CSI 100 (%) 28 × 7-1 Orbital Data[37]. 80 7-2 Transfer Orbit to Venus with 10MW Jet Power. 85 7-3 Transfer Orbit to Saturn with 30MW Jet Power. 86 7-4 Time Until Spacecraft Mass (m0 = 100MT) Becomes Zero (in TU Sun). 86 7-5 Fuel Consumption for VSI type II and CSI type I (kg). 89 8-1 Earth { Mars Round Trip Fuel Consumption for VSI type I. 111 8-2 Earth { Mars Round Trip Fuel Consumption. 113 8-3 Comparison of High Thrust ∆V for Earth { Jupiter: With and Without Mars Swing-by . 129 8-4 Comparison of High Thrust ∆V for Earth { Saturn: With and Without Mars Swing-by . 130 8-5 Comparison of Fuel Consumption for VSI type I: from Earth to Jupiter With and Without Mars Swing-by. 132 8-6 Comparison of Fuel Consumption for VSI type I: from Earth to Saturn With and Without Mars Swing-by. 132 8-7 Comparison of Fuel Consumption for VSI type II: from Earth to Jupiter With and Without Mars Swing-by. 134 8-8 Comparison of Fuel Consumption for VSI type II: from Earth to Saturn With and Without Mars Swing-by. 134 8-9 Comparison of Fuel Consumption for CSI type II: from Earth to Jupiter With and Without Mars Swing-by. 135 8-10 Comparison of Fuel Consumption for CSI type II: from Earth to Saturn With and Without Mars Swing-by. 135 A-1 Transfer Orbit to Venus with 20MW Jet Power. 144 A-2 Transfer Orbit to Venus with 30MW Jet Power. 145 A-3 Transfer Orbit to Mars with 10MW Jet Power. 146 A-4 Transfer Orbit to Mars with 20MW Jet Power. 147 A-5 Transfer Orbit to Mars with 30MW Jet Power. 148 A-6 Transfer Orbit to Asteroids with 10MW Jet Power. 149 viii A-7 Transfer Orbit to Asteroids with 20MW Jet Power. 150 A-8 Transfer Orbit to Asteroids with 30MW Jet Power. 151 A-9 Transfer Orbit to Jupiter with 10MW Jet Power. 152 A-10 Transfer Orbit to Jupiter with 20MW Jet Power. 152 A-11 Transfer Orbit to Jupiter with 30MW Jet Power. 153 A-12 Transfer Orbit to Saturn with 10MW Jet Power. 153 A-13 Transfer Orbit to Saturn with 20MW Jet Power. 154 ix LIST OF FIGURES 1-1 Future Interplanetary Flight with VASIMR[5]. 3 1-2 Transfer Trajectory from Earth to Mars with Thrust Direction. 4 2-1 VX-10 Experiment at Johnson Space Center[45]. 10 2-2 Synoptic View of the VASIMR Engine[22]. 12 2-3 Power Partitioning and Relationship between Thrust and Isp. 13 2-4 Thrust vs. Mass Flow Rate for a CSI Engine. 15 2-5 Thrust vs. Mass Flow Rate for a VSI Engine. 15 2-6 Thrust vs. Mass Flow Rate for a VSI Engine with Limitations. 16 3-1 Trajectories for Preliminary Study. 20 3-2 Preliminary Study: Isp and Fuel Consumptions for VSI and CSI. 21 3-3 Results: R = 100 DU, m0 = 20 MT, PJ = 500 kW. 23 3-4 Isp histories for VSI and CSI when CSI Isp is 3,000 sec: R = 100 DU, m0 = 20 MT, PJ = 500 kW. ..
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