ACKNOWLEDGEMENTS There are many people whom I would like to thank. First, I would like to express my

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Text of ACKNOWLEDGEMENTS There are many people whom I would like to thank. First, I would like to express my

  • A Study of Variable Thrust, Variable Isp

    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

    October 2004

  • A Study of Variable Thrust, Variable Isp

    Trajectories for Solar System Exploration

    Approved by:

    John R. Olds, Advisor

    Robert D. Braun

    Eric N. Johnson

    Amy R. Pritchett

    David W. Way

    Date Approved

  • To Naomi

    iii

  • 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

    DEDICATION iii

    ACKNOWLEDGEMENTS iv

    LIST OF TABLES ix

    LIST OF FIGURES xi

    LIST OF SYMBOLS OR ABBREVIATIONS xvi

    SUMMARY xviii

    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 A BRIEF DESCRIPTION OF LOW THRUST TRAJECTORY OPTI- MIZATION 9

    2.1 Trajectory Optimization in General . . . . . . . . . . . . . . . . . . . . . . 9

    2.2 Solution Methods for Optimization Problems . . . . . . . . . . . . . . . . . 9

    2.3 Indirect Methods – Calculus of Variations . . . . . . . . . . . . . . . . . . 11

    2.3.1 Problems without Terminal Constraints, Fixed Terminal Time . . . 11

    2.3.2 Some State Variables Specified at a Fixed Terminal Time . . . . . 12

    2.3.3 Inequality Constraints on the Control Variables . . . . . . . . . . . 15

    2.3.4 Bang-off-bang Control . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    III EXHAUST-MODULATED PLASMA PROPULSION SYSTEMS 21

    3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    3.2 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    3.3 Choosing the Power Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    v

  • IV PRELIMINARY STUDY: COMPARISON OF HIGH THRUST, CON- STANT ISP , AND VARIABLE ISP WITH SIMPLE TRAJECTORIES 29

    4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    V INTERPLANETARY TRAJECTORY OPTIMIZATION PROBLEMS 38

    5.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    5.2 Equations of Motion for Low Thrust Trajectories . . . . . . . . . . . . . . 43

    5.2.1 VSI – No Constraints on Isp . . . . . . . . . . . . . . . . . . . . . . 43

    5.2.2 VSI – Inequality Constraints on Isp . . . . . . . . . . . . . . . . . . 44

    5.2.3 CSI – Continuous Thrust . . . . . . . . . . . . . . . . . . . . . . . . 45

    5.2.4 CSI – Bang-Off-Bang Control . . . . . . . . . . . . . . . . . . . . . 46

    5.3 Solving the High Thrust Trajectory . . . . . . . . . . . . . . . . . . . . . . 47

    5.4 Problems with Swing-by . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    5.4.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    5.4.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    5.4.3 Powered Swing-by . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    VI DEVELOPMENT OF THE APPLICATION “SAMURAI” 57

    6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    6.1.1 Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    6.1.2 Performance Index for Each Engine . . . . . . . . . . . . . . . . . . 60

    6.2 C++ Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    6.3 Flow and Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    6.3.1 SAMURAI Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    6.3.2 VSI Constrained Isp . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    6.3.3 CSI Continuous Thrust . . . . . . . . . . . . . . . . . . . . . . . . . 68

    6.3.4 CSI Bang-Off-Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    6.4 Examples of Input and Output . . . . . . . . . . . . . . . . . . . . . . . . 70

    6.5 Validation and Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    6.5.1 Validation of High Thrust with IPREP . . . . . . . . . . . . . . . . 75

    6.5.2 Validation of CSI with ChebyTOP . . . . . . . . . . . . . . . . . . 75

    vi

  • VII PRELIMINARY RESULTS: PROOF OF CONCEPT 80

    7.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    7.2 Numerical Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    7.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    7.4 Relationship among Fuel Consumption, Jet Power, and Travel Distance . . 93

    VIIINUMERICAL EXAMPLES: “REAL WORLD” PROBLEMS 100

    8.1 Scientific Mission to Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    8.1.1 Venus Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    8.1.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    8.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    8.2 Human Mission to Mars: Round Trip . . . . . . . . . . . . . . . . . . . . . 109

    8.2.1 Mars Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

    8.2.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    8.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    8.3 JIMO: Jupiter Icy Moons Orbiter . . . . . . . . . . . . . . . . . . . . . . . 121

    8.3.1 JIMO Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    8.3.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 122

    8.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

    8.4 Uranus and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    8.5 Swing-by Trajectories with Mars . . . . . . . . . . . . . . . . . . . . . . . . 129

    IX CONCLUSIONS AND RECOMMENDED FUTURE WORK 138

    9.1 Conclusions and Observations . . . . . . . . . . . . . . . . . . . . . . . . . 139

    9.2 Research Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

    9.3 Recommended Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 142

    APPENDIX A — ADDITIONAL RESULTS FROM PRELIMINARY STUDY144

    APPENDIX B — EQUATIONS USED IN THE CODE 155

    APPENDIX C — NUMERICAL TECHNIQUES 159

    APPENDIX D — “SAMURAI” CODE MANUAL 177

    REFERENCES 180

    vii

  • VITA 186

    viii

  • LIST OF TABLES

    1 Examples of Spacecraft Propulsion Systems[4][42]. . . . . . . . . . . . . . . 1

    2 Trip Time Savings(Comparison by Distance): (VSI - CSI)/CSI × 100 (%) . 37

    3 Trip Time Savings(Comparison by Initial Mass): (VSI - CSI)/CSI × 100 (%) 37

    4 Orbital Data[37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    5 Transfer Orbit to Venus with 10MW Jet Power. . . . . . . . . . . . . . . . . 85

    6 Transfer Orbit to Saturn with 30MW Jet Power. . . . . . . . . . . . . . . . 86

    7 Time Until Spacecraft Mass (m0 = 100MT) Becomes Zero (in TU Sun). . . 86

    8 Fuel Consumption for VSI type II