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REPRESENTATIONS of INVARIANT MANIFOLDS for APPLICATIONS in SYSTEM-TO-SYSTEM TRANSFER DESIGN a Thesis Submitted to the Faculty Of REPRESENTATIONS OF INVARIANT MANIFOLDS FOR APPLICATIONS IN SYSTEM-TO-SYSTEM TRANSFER DESIGN A Thesis Submitted to the Faculty of Purdue University by Christopher E. Patterson In Partial Fulfillment of the Requirements for the Degree of Master of Science May 2005 ii For Betty and Ruth. iii ACKNOWLEDGMENTS First, I thank my parents. They have been with me through some very difficult times and supported me in more ways than I can mention. I could not have succeeded in school or in life without their encouragement. Because research does not occur in a vacuum, I thank all the past and current members of my research group for their interest in my work and for their help. Their friendship is greatly appreciated and I hope I succeed in returning the favor. I thank my adviser, Professor K. C. Howell, for her guidence throughout my graduate career. Her motivation has been invaluable. Finally, I thank the School of Aeronautics and Astronautics at Purdue University for support of my research and my education. Portions of this work were supported by NASA and the Goddard Mission Services Evolution Center under contract numbers NCC5-358 and NAG5-11839. iv TABLE OF CONTENTS Page LIST OF FIGURES ................................ vii ABSTRACT .................................... x 1 Introduction ................................... 1 1.1 Problem Definition ............................ 2 1.2 Previous Contributions .......................... 5 1.2.1 Three-Body Problem ....................... 5 1.2.2 Approximations and Transfers .................. 8 1.3 Current Work ............................... 10 2 BACKGROUND: MATHEMATICAL MODELS ............... 12 2.1 The Circular Restricted Three-Body Problem ............. 13 2.1.1 Assumptions ............................ 13 2.1.2 Reference Frames ......................... 14 2.1.3 Characteristic Quantities ..................... 15 2.1.4 Equations of Motion ....................... 15 2.1.5 Particular Solutions ........................ 18 2.1.6 The State Transition Matrix ................... 19 2.1.7 Differential Corrections and Halo Orbits ............ 21 2.2 Invariant Manifold Theory ........................ 25 2.2.1 Invariant Manifolds ........................ 25 2.2.2 Periodic Orbits and Maps .................... 29 2.2.3 Orbital Stability ......................... 30 2.2.4 Stable and Unstable Manifolds of Periodic Orbits ....... 33 2.2.5 Eigenstructure of the Monodromy Matrix ............ 34 v Page 2.2.6 Computation of the Stable and Unstable Manifolds of Periodic Halo Orbits ............................ 35 2.3 Transition to an Ephemeris Model .................... 39 3 Manifold Approximations Using Cells ..................... 44 3.1 Cell Approximations ........................... 44 3.2 Cell Creation ............................... 55 3.2.1 Division of Manifold Data .................... 55 3.2.2 Cell Expansion .......................... 58 3.3 Applications Using Cell Approximations ................ 60 3.3.1 Determination of a Destination Halo Orbit ........... 60 3.3.2 Saving Computation and Storage ................ 63 4 Manifold Approximations as the Basis for Transfers ............. 65 4.1 Determination of Transfers to Periodic Halo Orbits .......... 65 4.1.1 Approaching the Halo Orbit ................... 65 4.1.2 Computation of Halo Orbit Insertion .............. 70 4.1.3 An Alternative to Direct Halo Orbit Insertion ......... 72 4.2 Generating System-to-System Transfers in the CR3BP ........ 76 4.2.1 Transition of CR3BP Transfers to an Ephemeris Formulation . 79 4.3 Adaptation for Lunar Encounters .................... 83 4.3.1 Cost Reduction .......................... 90 5 Transfers to Orbits of the Sun-Earth L2 Point ................ 95 5.1 Varying Size of Initial Earth-Moon Lissajous Orbit .......... 96 5.1.1 Comparison With Southern Earth-Moon L2 Orbits ...... 100 5.2 Varying Initial Time of Computation .................. 100 5.3 Transferring From an Earth-Moon L1 Orbit .............. 104 6 Conclusion .................................... 109 6.1 Summation ................................ 109 6.2 Suggestions for Future Work ....................... 109 vi Page LIST OF REFERENCES ............................. 111 vii LIST OF FIGURES Figure Page 2.1 Reference Frames ............................. 14 2.2 Location of the Libration Points ..................... 19 2.3 A Periodic Halo Orbit in the Earth-Moon System ........... 24 2.4 Stable and Unstable Manifolds Tangent to Subspaces ......... 28 2.5 Manifold Tubes in Earth-Moon CR3BP Collapsing on Two Halo Orbits ..................... 38 2.6 A “Halo-Like” Lissajous Trajectory ................... 41 3.1 Each Tube is Contained within a Successively Larger Tube ...... 45 3.2 Cells are Defined to Contain Small Regions of Manifold Trajectories . 46 3.3 Manifold Data is Transformed to a Cell Frame with Origin Q.....¯ 47 3.4 A Manifold Data Point Represented in Spherical Coordinates Relative to a Cell Frame ......................... 48 3.5 The N˜ and B˜ Vectors Placed at the Center of a Section of Data Points Directed Toward Two Potential Cell Origins .............. 50 3.6 (a) Using Q¯b as the Origin of a Cell Frame (b) Using Q¯n as the Origin of a Cell Frame .............. 51 3.7 Manifold Surfaces and Velocity Flow Approximated Within a Cell . 53 3.8 Sun-Earth L1 Manifold Position and Velocity Fit Errors ........ 54 3.9 Dividing a Tube into Cells ........................ 56 3.10 Dividing a Band Into Sectors ...................... 56 3.11 Five Tubes through a Set of Cells .................... 59 3.12 Locating the Intersection Point within a Cell .............. 61 4.1 Trust Region Around the Surface Approximations in a Cell ...... 67 4.2 Improvement of an Approach to a Halo Orbit Using Scale Factors ............................ 69 4.3 Transfer Determined using Cells ..................... 72 viii Figure Page 4.4 Initial Approach Arc to Halo ....................... 73 4.5 Attempt to Target a Candidate HOI Point ............... 74 4.6 Multi-Point Targeting To Stay Near the Halo Orbit .......... 76 4.7 Earth-Moon L1 Manifold Paths Intersect Sun-Earth L2 Manifold Cells ....................... 77 4.8 Transfer from the Earth-Moon L1 Halo Orbit to a Sun-Earth L2 Halo Orbit ...................... 80 4.9 Transition of Earth-Moon L1 Manifold Arc to Ephemeris Model . 81 4.10 Transition of the Earth-Moon to Sun-Earth Transfer to the Ephemeris Model ......................... 82 4.11 Earth-Moon L2 Tube Intersecting Sun-Earth L2 Tube Both Tubes Computed in CR3BP Models ............... 84 4.12 Earth-Moon L2 Manifold Surface Intersecting Sun-Earth L2 Tube (Earth-Moon Surface Computed in Ephemeris Formulation) ..... 85 4.13 Earth-Moon Manifold in Ephemeris Model; Transfer Arc and Halo Orbit in a Sun-Earth Circular Restricted Model ......... 87 4.14 The Entire Transfer is Converged in a Full Ephemeris Formulation . 88 4.15 A Transfer using Multi-Point Targeting Near the Halo Orbit is Converged in a Full Ephemeris Formulation .............. 89 4.16 Transfer Computed with No Cost from HOI Targeting Routine ....................... 91 4.17 Transfer Computed with No Cost from Multi-Point Halo Targeting Routine ................ 92 4.18 Earth-Moon Lissajous Orbit Changes Shape as Transfer Cost is Reduced ......................... 93 5.1 Case 3a. Ephemeris Earth-Moon Manifold and CR3BP Sun-Earth Transfer Arc to Halo Orbit ........................ 98 5.2 Case 3b. Sun-Earth View: Transfer Arc to LPO in Ephemeris Model 99 5.3 Case 3c. Free Ephemeris Transfer Between Systems .......... 99 5.4 Case 8b & c. Cost Reduction Facilitated by Large Changes in in Libration Point Orbits ......................... 102 ix Figure Page 5.5 Case 9b & c. Less Variation in Libration Point Orbits is Necessary to Reduce a Lower-Cost Preliminary Transfer ............... 103 5.6 Case 10b & c. Departing the Vicinity of the Earth-Moon L1 Point in a Direction Toward the Earth ....................... 105 5.7 Manifolds of LPO Near Earth-Moon L1 Point Directed Toward the Moon ........................ 106 5.8 Case 16b. Ephemeris Model Transfer Departing Earth-Moon System Via Lunar Flyby From an LPO Near the Earth-Moon L1 Point .... 108 x ABSTRACT Patterson, Christopher E. M.S., Purdue University, May, 2005. Representations of Invariant Manifolds for Applications in System-to-System Transfer Design. Major Professor: Kathleen C. Howell. The Lunar L1 and L2 libration points have been proposed as gateways granting inexpensive access to interplanetary space. To date, only individual solutions to the problem of traveling between libration point orbits in different three-body systems have been computed. The methodology to solve the problem for transfers between arbitrary three-body systems and even entire families of orbits is under investigation. For preliminary design, manifolds associated with periodic halo orbits in the circular restricted problem are well-suited as the foundation for such transfers. This work presents an initial approach to solving the general problem of system- to-system transfers. A method is detailed for approximating and storing invariant manifold data. The method is based on a unique cell structure whereby invariant man- ifolds associated with halo orbits in the same family and occupying large volumes of space are approximated within smaller, overlapping cells. The approximations within the cells are stored in a manner that allows the parameterization of halo orbits, a capability that is very useful in a transfer design procedure, freeing the designer to select any orbit that can be reached from any manifold
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