Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1993 Optimal low-thrust, Earth-Moon trajectories Craig Allan Kluever Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Aerospace Engineering Commons Recommended Citation Kluever, Craig Allan, "Optimal low-thrust, Earth-Moon trajectories " (1993). Retrospective Theses and Dissertations. 10466. https://lib.dr.iastate.edu/rtd/10466 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9321183 Optimal low-thrust, Earth-Moon trajectories Kluever, Craig Allan, Ph.D. Iowa State University, 1993 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 Optimal low-thrust, Earth-Moon trajectories by Craig Allan Kluever A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department: Aerospace Engineering and Engineering Mechanics Major: Aerospace Engineering Approved: Members Comnp^tee: Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1993 11 TABLE OF CONTENTS CHAPTER 1. INTRODUCTION 1 Historical Background 1 Thesis Topics 2 System Models 3 CHAPTER 2. OPTIMAL 2-D EARTH-MOON TRANSFER .... 5 Hierarchy of Sub-problems 5 Maximum-Energy Escape/Capture Trajectories 6 Sub-optimal Coasting Translunar Trajectories 17 Optimal LEO-to-LLO Transfer 26 Two-Point Boundary Value Problem Solution 36 Augmented State System 38 Verification of Direct/Indirect Method Solution 47 CHAPTER 3. OPTIMAL TRANSFER FOR LOWER T/W .... 49 Hierarchy of Sub-problems 49 Edelbaum Approximation for Quasi-Circular Transfer 53 Auxiliary Minimum-Time Transfers to Curve-fit Boundaries 54 Costate-Control Transformation 56 Numerical Results 59 Ill CHAPTER 4. OPTIMAL TRANSFERS USING A SWITCHING FUNCTION STRUCTURE 63 Two-Point Boundary Value Problem Formulation 63 Minimum-Fuel Transfer from Earth-Escape to the Lunar SOI 68 Numerical Results 72 CHAPTER 5. VEHICLE/TRAJECTORY OPTIMIZATION .... 81 Maximum Net Mass 81 Minimum Initial Mass in LEO 95 CHAPTER 6. OPTIMAL 3-D EARTH-MOON TRANSFERS ... 97 Necessary Conditions for Minimum-Fuel 3-D Trajectories 98 Minimum-Fuel 3-D Trajectories for Higher T/W 106 Final Meridian Constraints 114 Minimum-Fuel 3-D Trajectories for Lower T/W 116 CHAPTER 7. SUMMARY AND CONCLUSIONS 122 REFERENCES 126 ACKNOWLEDGMENTS 129 APPENDIX A. ADDITIONAL NECESSARY CONDITIONS ... 130 APPENDIX B. DIRECTION COSINES FOR OPTIMAL 3-D STEER­ ING 133 APPENDIX C. COSTATE-CONTROL TRANSFORMATION FOR 3-D STEERING 137 iv LIST OF TABLES Table 2.1: Terminal Boundary Conditions (wrt Moon) for Initial Guesses 25 Table 4.1: Switching Function Solutions 80 Table 4.2: Initial Costates for Switching Function Solutions 80 Table 4.3: Burn Arc Switch Times 80 Table 4.4: Coast Arc Switch Times 80 Table 6.1: Convergence History for Minimum-Fuel 3-D NEP Trajectory 119 LIST OF FIGURES Figure 2.1: Maximum-Energy Earth-Escape Spiral for tf =2.37 days ... 13 Figure 2.2: Optimal Thrust Direction and Flight Path Angles - Earth- Escape Phase 14 Figure 2.3: Final Velocity Components vs. Final Earth Radial Distance . 14 Figure 2.4: Final Escape Spiral Time vs. Final Earth Radial Distance . 15 Figure 2.5: Maximum-Energy Moon-Capture Spiral for tf =15.6 hours . 17 Figure 2.6: Optimal Thrust Direction and Flight Path Angles - Moon- Capture Phase 18 Figure 2.7: Final Radial Velocity vs Final Moon Radial Distance .... 18 Figure 2.8: Final Circumferential Velocity vs Final Moon Radial Distance 19 Figure 2.9: Capture Spiral Time vs Final Moon Radial Distance 19 Figure 2.10: Optimal Posigrade Trajectory - rotating coordinates 33 Figure 2.11: Optimal Posigrade Trajectory - inertia! coordinates 34 Figure 2.12: Optimal Thrust Direction and Flight Path Angles - Earth escape 35 Figure 2.13: Optimal Thrust Direction and Flight Path Angles - Moon capture 35 Figure 2.14: Optimal Retrograde Trajectory - rotating coordinates .... 36 Figure 2.15: Optimal Retrograde Trajectory - inertia! coordinates 37 vi Figure 3.1: Optimal Trajectory for NEP Vehicle - rotating coordinates . 60 Figure 3.2; Optimal Trajectory for NEP Vehicle - inertial coordinates . 61 Figure 3.3: Optimal Thrust Direction and Flight Path Angles - Earth escape 62 Figure 3.4: Optimal Thrust Direction and Flight Path Angles - Moon capture 62 Figure 4.1: Minimum-Fuel Trajectory for t/=.5.16 days - rotating coordi­ nates 73 Figure 4.2: Switching Function vs. Time - i/=5.16 days 74 Figure 4.3: Thrust Direction and Flight Path Angles - tf=5.1Q days ... 75 Figure 4.4: Minimum-Fuel Trajectory for tf=3.8 days - rotating coordinates 76 Figure 4.5: Switching Function vs. Time - i/=3.8 days 77 Figure 4.6: Minimum-Fuel Trajectory for tf=2.6 days - rotating coordinates 77 Figure 4.7: Switching Function vs. Time - tf=2.6 days 78 Figure 4.8: Optimal Final Mass vs. Trip Time 79 Figure 5.1: Maximum Net Mass Trajectory for tf = 72.5 days - rotating coordinates 89 Figure 5.2: Thrust Direction and Flight Path Angles 89 Figure 5.3: Optimal Payload Fraction vs. Trip Time 91 Figure 5.4: Optimal I,p and Power vs. Trip Time 92 Figure 5.5: Maximum Net Mass Trajectory for tf = 65 days - rotating coordinates 93 Figure 5.6: Maximum Net Mass Trajectory for = 200 days - rotating coordinates 93 vil Figure 5.7: Maximum Net Mass Trajectory ÎOT TF = 200 days - inertial coordinates 94 Figure 6.1: Optimal Mass in LLO vs. Lunar Inclination 108 Figure 6.2: Minimum-Fuel 3-D Trajectory (x-y Plane) - rotating frame . 109 Figure 6.3: Minimum-Fuel 3-D Trajectory (x-z Plane) - rotating frame . 109 Figure 6.4: Minimum-Fuel 3-D Trajectory - rotating frame 110 Figure 6.5: Optimal Thrust Steering Angles - Earth escape 112 Figure 6.6: Optimal Thrust Steering Angles - Moon capture 113 Figure 6.7: Optimal Mass in Polar LLO - Meridian Constraints 115 Figuré 6.8: Minimum-Fuel 3-D Trajectory (x-y Plane) - rotating frame . 120 Figure 6.9: Minimum-Fuel 3-D Trajectory (x-z Plane) - rotating frame . 120 1 CHAPTER 1. INTRODUCTION Historical Background Low-thrust propulsion systems have been identified as an efficient means for performing space missions. Spacecraft propelled by low-thrust engines are capable of delivering a greater payload fraction compared to spacecraft using conventional chem­ ical propulsion systems. The National Aeronautical Space .Administration (NASA) is currently investigating several applications of low-thrust propulsion including a manned Mars mission [1], scientific missions to Jupiter, Uranus, Neptune and Pluto [2], and lunar missions leading to a permanent lunar colony. Recently, Aston [3] demonstrated the merits and feasibility of using low-thrust propulsion to ferry cargo between low-Earth orbit and low-lunar orbit. The study of optimal trajectories for low-thrust spacecraft is an integral part of these research efforts. Much of the earlier work in low-thrust trajectories concerned orbital transfers in an inverse-square gravity field. Edelbaum [4-5] investigated optimal orbit-raising, rendezvous, and station keeping manueavers using low-thrust propulsion. Early stud­ ies of low-thrust interplanetary trajectories were investigated by Melbourne and Sauer [6] and Break well aiid Rauch [7]. Melbourne and Sauer studied optimal Earth-Mars low-thrust trjectories where only the gravitational field of the sun is considered. The Breakwell and Rauch effort involved patching together trajectories where the space­ 2 craft is in a central gravity field with respect to a planet or the sun. By comparison, the volume of published work on low-thrust trajectories for an Earth-Moon mission is somewhat limited. Early preliminary studies were performed by London [8] and Stuhlinger [9]. In each study, the low-thrust trajectory is divided and treated separately with respect to the Earth or Moon using central gravity motion and patched together at the Sphere of Influence. More recent work on optimal low- thrust Earth-Moon trajectories has been performed by Golan and Breakwell [10] and Enright [11]. Both studies utihzed trajectories influenced by the simultaneous gravity fields of the Earth and Moon.