A Low-Thrust Transfer Strategy to Earth-Moon Collinear Libration Point Orbits
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A LOW-THRUST TRANSFER STRATEGY TO EARTH-MOON COLLINEAR LIBRATION POINT ORBITS A Thesis Submitted to the Faculty of Purdue University by Martin T. Ozimek In Partial Fulfillment of the Requirements for the Degree of Master of Science December 2006 Purdue University West Lafayette, IN ii For Mom, Dad, Sarah, and Bops iii ACKNOWLEDGMENTS I would like to thank my parents for their seemingly never ending support and confidence that I would persevere in my personal quest for an advanced degree. The decision to commit to a higher degree has been an adventure that I hope to continue along, and I can’t begin to explain the importance of that priceless feeling of simply knowing that someone is there when needed. Professor Kathleen Howell, my advisor, is also owed a great deal of gratitude, not only for posing the fateful words “low-thrust” one day in her office during a discussion about NASA’s potential Jupiter Icy Moons missions, but also for her personal standard for excellence that she instills in each of her many successful graduate students. I’ve always felt that Purdue University reached out to me and offered me that “extra” indefinable something from the moment I began seriously considering a graduate institution. In no other person is this ambiguous something “extra” exemplified than in Professor Howell, who has sought to ensure that my research efforts are guided and ultimately shared with others in the best possible way. This notion has also been exemplified by Professor James Longuski, whose door has always been open to me from day one, and whom I must also credit in heavily influencing my decision to attend Purdue University. On more than one occasion, Professor William Crossley has also had his door open to help along my path of solving what turned out to be a difficult optimization problem. I also owe many thanks to Daniel Grebow. Dan has been a close friend, colleague, and even roommate throughout my stay at Purdue, and this work is the direct continuation of a mission analysis that we worked on together. Often, many of the new ideas I have for current and future research are a result of simple dialogs that we frequently engage in. iv The idea to study mission applications toward lunar south pole coverage would never have originated had I not fortuitously been privileged to work at the NASA Goddard Spaceflight Center during the summers of 2005 (as a member of the NASA Academy by way of the Indiana Space Grant Consortium) and 2006. There, I benefited from the knowledge of some of the greatest libration point mission experts in the world, and owe particular thanks to my mentor David Folta. Support from NASA under contract numbers NNG05GM76G and NNX06AC22G is greatly appreciated. Finally, I would like to thank Purdue University for financial support, including the Andrews Fellowship, for the entirety of my M.S. program. v TABLE OF CONTENTS Page LIST OF TABLES............................................................................................................ vii LIST OF FIGURES ......................................................................................................... viii ABSTRACT........................................................................................................................ x 1 INTRODUCTION ............................................................................................................1 1.1 Historical Overview of the Three-Body Problem.......................................................3 1.2 Developments in Low-Thrust Transfer Trajectories ..................................................5 1.2.1 Optimal Control .................................................................................................. 5 1.2.2 Application to Orbit Problems ............................................................................ 6 1.3 Focus of this Work .....................................................................................................7 2 BACKGROUND ............................................................................................................10 2.1 The Circular Restricted Three-Body Problem..........................................................10 2.1.1 Assumptions...................................................................................................... 11 2.1.2 Geometry........................................................................................................... 11 2.1.3 Equations of Motion.......................................................................................... 12 2.1.4 Libration Points................................................................................................. 15 2.1.5 Formulation Relative to P2 in the CR3BP......................................................... 17 2.2 Natural Periodic Orbits in the CR3BP......................................................................18 2.2.1 First-Order Variational Equations Relative to the Collinear Points.................. 19 2.2.2 The State Transition Matrix .............................................................................. 23 2.2.3 The Fundamental Targeting Relationships ....................................................... 25 2.2.4 Periodic Orbits .................................................................................................. 28 2.3 Invariant Manifolds ..................................................................................................31 2.3.1 Stable and Unstable Manifolds Associated with the Collinear Points.............. 32 vi Page 2.3.2 Invariant Manifolds Relative to a Fixed Point .................................................. 35 2.3.3 Computation of Manifolds Corresponding to Fixed Points Along an Orbit..... 37 2.4 Optimal Control Theory ...........................................................................................39 2.4.1 Summary of the First Necessary Conditions for Optimal Control.................... 40 2.4.2 Tests for a Local Minimum Value of the Performance Index .......................... 43 3 LOW-THRUST TRANSFER ALGORITHM ................................................................45 3.1 Engine Model ...........................................................................................................46 3.2 Control Law Derivation............................................................................................48 3.3 Adjoint Control Transformation...............................................................................55 3.4 Numerical Solution via Direct Shooting: A Local Approach...................................59 3.5 Shotgun Method for Initial Conditions: A Global Approach...................................63 4 MISSION APPLICATIONS...........................................................................................66 4.1 Orbits for Line-of-Sight Lunar South Pole Coverage (CR3BP) ..............................67 4.1.1 Three-Dimensional Periodic Orbits in the CR3BP ........................................... 67 4.1.2 Families of Orbits for Lunar South Pole Coverage ........................................ 68 4.1.3 Mission Orbit Selection Criteria ....................................................................... 73 4.2 Optimal Transfers to the Earth-Moon Stable Manifold............................................75 4.2.1 Transfers to a 12-Day L1 Halo Orbit ................................................................. 78 4.2.2 Transfer to a 14-Day L1 Vertical Orbit ............................................................. 87 4.2.3 Transfer to a 14-Day L2 Butterfly Orbit............................................................ 91 5 SUMMARY AND RECOMMENDATIONS.................................................................96 5.1 Summary...................................................................................................................96 5.2 Recommendations for Future Work .........................................................................98 LIST OF REFERENCES...................................................................................................99 vii LIST OF TABLES Table Page Table 4.1 Dynamical and Propulsion Constants................................................................77 Table 4.2 12-Day L1 Halo Orbit Transfer Data Summary .................................................82 Table 4.3 12-Day L1 Halo Orbit Long Transfer Data Summary........................................86 Table 4.4 14-Day L1 Vertical Orbit Transfer Data Summary............................................91 Table 4.5 14-Day L2 Butterfly Orbit Transfer Data Summary ..........................................95 viii LIST OF FIGURES Figure Page Figure 2.1 Geometry in the Restricted Three-Body Problem............................................12 Figure 2.2 Equilibrium Point Locations for the CR3BP....................................................16 Figure 2.3 Geometry of P2-Centered Rotating Frame .......................................................18 Figure 2.4 Linearized L1 Periodic Orbit.............................................................................23 Figure 2.5 Basic Diagram for a Free Final Time Targeting Scheme.................................27 Figure 2.6 Targeting a Perpendicular X-axis Crossing in the CR3BP ..............................30 Figure 2.7 Several L1 Lyapunov Orbits Obtained Via Continuation.................................31 Figure 2.8 Stable and Unstable Manifold at X eq ...............................................................34 Figure 2.9 Global