THE DESIGN OF SYSTEM–TO–SYSTEM TRANSFER ARCS USING INVARIANT MANIFOLDS IN THE MULTI – BODY PROBLEM A Thesis Submitted to the Faculty of Purdue University by Germ´an Porras Alonso In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2006 Purdue University West Lafayette, Indiana ii A.M.D.G. Tambi´en dedico esta tesis a mi familia, sin dudad el mejor regalo que Dios me ha dado. Muy en especial se la dedico a mis padres, Germ´an y Ana, a mis hermanos, Miguel e Ignacio, y a mis padrinos, mi t´ıa Babi y mi t´ıo David (D.E.P.). iii ACKNOWLEDGMENTS I would first like to thank my family, in particular my parents, Germ´an and Ana andbrothers, MiguelandIgnacio, fortheirconstantsupportandencouragementwhile this work was being completed. I would also like to thank those friends, past and present, who, in addition to their support, also provided many opportunities to enjoy life outside of work. There are to many to mention, but they know who they are and they have my eternal gratitude for their friendship. TheothermembersinProfessorHowell’sresearchgrouphavealwaysbeenasource of inspiration and friendship. In particular, I would like to thank Masaki Kakoi, Chris Patterson and Raoul Rausch. Through their friendship, technical discussions and their extensive help with GENERATOR, they have made this work possible. The members of my committee, Professor Martin Corless, Professor Sptephen Heister and Professor James Longuski have been an invaluable source of advice in matters related and unrelated to this work. In addition, their classes have been among the high points of my academic years. Ofcourse,thereisonepersonwithwhomthisworkwouldneverhavebeenpossible, Professor Kathleen Howell. She was the first to introduce this novel, to me at least, andintriguingideaofusinganextracelestialbody, throughthethree–bodyproblem, to generate unconventional and exciting spacecraft trajectories. Over the years she has proven to be a constant source of expertise and encouragement, specially when problems arose. Finally I would like to thank the Purdue University School of Aeronautics and Astronautics and the recently formed Department of Engineering Education for their financial support throughout the duration of this work. iv TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Historical Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 The n – Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Three – Body Problem . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Circular Restricted Three – Body Problem . . . . . . . . . . . . 9 2.3.1 Non-Dimensional Quantities in the CR3BP . . . . . . . . . . . 10 2.3.2 Equations of motion of the CR3BP . . . . . . . . . . . . . . . 12 2.4 Equilibrium Solutions to The Circular Restricted Three – Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 The Jacobi Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5.1 The Zero – Velocity Curves . . . . . . . . . . . . . . . . . . . 15 2.6 Halo Orbits Associated with the Collinear Libration Points . . . . . . 18 2.7 Reference Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.8 The State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . 22 2.9 The Monodromy Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.10 Stability Issues in the Circular Restricted Three – Body Problem . . 25 2.11 Manifolds Associated with a Halo Orbit . . . . . . . . . . . . . . . . 29 3 General Characteristics of System-to-System Trajectory Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 v Page 3.1 The Differential Corrections Process . . . . . . . . . . . . . . . . . . . 33 3.2 General Characteristics of Trajectory Arcs . . . . . . . . . . . . . . . 38 3.3 Restricting the Differential Corrections Process to One Subset of the Solution Space . . . . . . . . . . . . . . . . . . . . 43 4 Preliminary Trajectory Design . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 The Apparent Jacobi Constant . . . . . . . . . . . . . . . . . . . . . 51 4.3 The Apparent Jacobi Constant and Preliminary Trajectory Design . . 57 4.4 Implementation of the Search Procedure . . . . . . . . . . . . . . . . 60 5 A Design Strategy for the Local Optimization of Spacecraft Trajectories . . 69 5.1 The Two–Level Differential Corrections Process . . . . . . . . . . . . 69 5.2 Modification of the Two–Level Differential Corrections Process to pro- duce System–to–System Transfer Arcs . . . . . . . . . . . . . . . . . 73 5.2.1 Converting the State Transition Matrix from a Rotating Frame to an Inertial Frame . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.2 Converting the State Transition Matrix from an Inertial Frame to a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.3 Numerical Verification of the State Transition Matrix Frame Conversion Theory . . . . . . . . . . . . . . . . . . . . 80 5.2.4 Converting the Time Sensitivity Vector Between Inertial and Rotating Frames . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Type I Example: Transfer from an Earth-Moon Halo Orbit to a Sun-Earth Halo Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.1 Behavior of the Solution as a result of the Modified Two–Level Differential Corrections Process . . . . . . . . . . . . . . . . . 85 5.3.2 Ephemeris Model Verification of Results . . . . . . . . . . . . 89 5.4 Type II Example: Transfer from an Sun-Earth Halo Orbit to a Sun-Mars Halo Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4.1 Behavior of the Solution as a result of the Modified Two–Level Differential Corrections Process . . . . . . . . . . . . . . . . . 94 6 Summary and Recommendations . . . . . . . . . . . . . . . . . . . . . . . 101 vi Page LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 A General Characteristics of Trajectory Arcs Surfaces . . . . . . . . . . . . . 107 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 vii LIST OF TABLES Table Page 3.1 Number of iterations for the modified and unmodified differential cor- rector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1 Cost for the preliminary trajectory that serves as the basis for the example in Figure 4.7. The maneuver points for ∆V and ∆V appear 1 2 in Figure 4.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Reduced cost for the preliminary trajectory by a reduction in TOF compared to the trajectory in Figure 4.7. . . . . . . . . . . . . . . . . 66 4.3 Costs associated with the initial Sun-Earth to Sun-Mars halo orbits transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.1 Costs associated with the initial Earth-Moon to Sun-Earth halo orbits transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 LowercostasaresultofthecorrectionsprocedureappliedtotheEarth- Moon to Sun-Earth transfer. . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Lower cost as a result of the corrections procedure applied to the Sun- Earth to Sun-Mars transfer. . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Comparison of Sun-Earth to Sun-Mars system transfers. . . . . . . . 98 5.5 Lower cost as a result of shifting the start and end points of the link arc in the Sun-Earth to Sun-Mars transfer example. . . . . . . . . . . 98 viii LIST OF FIGURES Figure Page 2.1 Nomenclature in the n – Body Problem. . . . . . . . . . . . . . . . . 8 2.2 Definitions in the CR3BP. . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Lagrange points — The five equilibrium solutions in the CR3BP (not to scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Earth-Moon Zero – Velocity half-surface for a C = 3.15. . . . . . . . . 16 2.5 Halo around L in the Earth-Moon rotating frame. . . . . . . . . . . 17 1 2.6 Zero – Velocity curves on the xˆ−yˆ plane as C decreases. . . . . . . . 18 2.7 Halo around L in the Earth-Moon rotating frame. . . . . . . . . . . 19 1 2.8 Halo family in the vicinity of L in the Earth-Moon rotating frame. . 20 1 2.9 Halo family in the vicinity of L in the Earth-Moon rotating frame. . 20 1 2.10 Lissajous orbit in the vicinity of L in the Earth-Moon system [29]. . 21 1 2.11 Reference trajectory with associated STM, evaluated at various points along the path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.12 Monodromy matrix eigenvalue structure for a halo orbit. . . . . . . . 28 2.13 Stable manifold associated with an Earth-Moon L halo orbit; A ≈ 1 z 39,000 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.14 Earth-Moon manifolds corresponding to an L halo orbit. . . . . . . . 30 1 2.15 Trajectories belonging to a Sun-Earth stable manifold and an Earth- Moon unstable manifold. . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Visual representation of the variational relationships. . . . . . . . . . 36 3.2 Differential corrections process to reach the desired final position. . . 37 3.3 Alternate solution that possesses the same TOF. . . . . . . . . . . . . 37 3.4 Variable TOF trajectories in the Sun-Earth synodic frame. . . . . . . 39 3.5 Alternate views (a), (b), (c) of the surface formed by the low ∆V solutions in Figure 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Surface formed by the high ∆V solutions in Figure 3.4. . . . . . . . . 41 ix Figure Page 3.7 Relationship between ∆V and TOF for the two solution sets that ap- pear in Figures 3.5 and 3.6 . . . . . . . . . . . . . . . . . . . . . . . . 42 3.8 Modified differential corrector concept. . . . . . . . . . . . . . . . . . 43 3.9 Intermediate step in a modified differential corrector example. . . . . 45 3.10 Final result of a modified differential corrector example. . . . . . . . . 46 4.1 System–to–System transfer concept. . . . . . . . . . . . . . . . . . . . 48 4.2 Potential-transfers between Sun-Earth and Earth-Moon manifold tubes. 49 4.3 Jacobi Constant computed in the Sun-Earth rotating frame for the manifold trajectories in Figure 2.15. . . . . . . . . . . . . . . . . . . . 52 4.4 Contributions to the apparent Jacobi Constant. . . . . . . . . . . . . 53 4.5 Earth-Moon Trajectory associated with Figure 4.4. . . . . . . . . . . 54 4.6 Contributions to the apparent Jacobi Constant of the Sun-Earth man- ifold trajectory in Figure 4.5. . . . . . . . . . . . . . . . . . . . . . . 55 4.7 Sun-Earth Trajectory associated with Figure 4.6. . . . . . . . . . . . 56 ¯ 4.8 Maneuver ∆V at a point, P, where two trajectories intersect. . . . . 57 4.9 Transfer arc between two manifolds showing the maneuver geometry at the departure and arrival points. . . . . . . . . . . . . . . . . . . . 59 4.10 Two Sun-Mars manifold tubes plotted in an inertial frame. . . . . . . 61 4.11 Expandedviewofthebeginningandendportionsofthemanifoldtubes in Figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.12 Transfer angle and manifold trajectories. . . . . . . . . . . . . . . . . 63 4.13 Jacobi Constant values to design an Earth-Mars transfer: (a) Sun- Earth rotating frame; (b) Sun-Mars rotating frame. . . . . . . . . . . 64 4.14 Jacobi Constant computed in the Sun-Earth rotating frame. . . . . . 65 5.1 Two–Level Differential Corrections process. . . . . . . . . . . . . . . . 70 5.2 Notation for the intermediate point. . . . . . . . . . . . . . . . . . . . 70 5.3 Two–Level Differential Corrections Process adapted for the Planet- Moon System-to-System transfer concept. . . . . . . . . . . . . . . . 74 5.4 Two–Level Differential Corrections Process adapted for the Planet- Planet (Moon-Moon) System-to-System transfer concept. . . . . . . . 75 5.5 Relative orientation of the reference frames. . . . . . . . . . . . . . . 76 x Figure Page 5.6 Comparison of numerical and analytical STM representative elements: rotating to inertial frame. . . . . . . . . . . . . . . . . . . . . . . . . 81 5.7 Earth-Moon unstable manifold trajectory, linking arc, and Sun-Earth stable manifold trajectory: Sun-Earth rotating frame. . . . . . . . . . 84 5.8 Earth-Moon halo orbit and the associated unstable manifold trajec- tory; the link arc is plotted in an Earth-Moon rotating frame. . . . . 85 5.9 Pre-corrected (green) and post-corrected trajectory and patch points in a Sun-Earth rotating frame. . . . . . . . . . . . . . . . . . . . . . . 86 5.10 Pre-corrected (green) and post-corrected Earth-Moon portion of the trajectory and patch points in an Earth-Moon rotating frame. . . . . 87 5.11 (a) Total ∆V cost per iteration; and, (b) ∆V cost per iteration at each patch point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.12 Modified Two–Level Differential Corrections process trajectory and fi- nal result in the four – body ephemeris model. . . . . . . . . . . . . . 90 5.13 Modified Two–Level Differential Corrections process trajectory and fi- nal trajectory in the four – body ephemeris model: Earth-Moon rotat- ing frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.14 Sun-Earth unstable manifold trajectory and Sun-Mars stable manifold trajectory: (a) inertial frame; (b) Sun-Earth rotating frame; (c) Sun- Mars rotating frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.15 Pre-corrected (green) and post-corrected (red) Sun-Earth to Sun Mars transfer: inertial frame. Note that (b) and (c) are not to scale. . . . . 96 5.16 (a) Total ∆V cost per iteration; and (b) ∆V cost per iteration at each patch point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.17 Lower cost Sun-Earth unstable manifold trajectory and Sun-Mars sta- ble manifold trajectory: inertial frame. . . . . . . . . . . . . . . . . . 99 A.1 Surface formed by the low ∆V solutions in Figure 3.4. Correctly scaled version of Figure 3.5 (a). . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.2 Alternate view of the surface formed by the low ∆V solutions in Figure 3.4. Correctly scaled version of Figure 3.5 (b). . . . . . . . . . . . . . 108 A.3 Alternate view of the surface formed by the low ∆V solutions in Figure 3.4. Correctly scaled version of Figure 3.5 (c). . . . . . . . . . . . . . 109 A.4 Surface formed by the high ∆V solutions in Figure 3.4. Correctly scaled version of Figure 3.6. . . . . . . . . . . . . . . . . . . . . . . . 110