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Low-Thrust Many-Revolution Trajectory Optimization Low-Thrust Many-Revolution Trajectory Optimization by Jonathan David Aziz B.S., Aerospace Engineering, Syracuse University, 2013 M.S., Aerospace Engineering Sciences, University of Colorado, 2015 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Aerospace Engineering Sciences 2018 This thesis entitled: Low-Thrust Many-Revolution Trajectory Optimization written by Jonathan David Aziz has been approved for the Department of Aerospace Engineering Sciences Daniel J. Scheeres Jeffrey S. Parker Jacob A. Englander Jay W. McMahon Shalom D. Ruben Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Aziz, Jonathan David (Ph.D., Aerospace Engineering Sciences) Low-Thrust Many-Revolution Trajectory Optimization Thesis directed by Professor Daniel J. Scheeres This dissertation presents a method for optimizing the trajectories of spacecraft that use low- thrust propulsion to maneuver through high counts of orbital revolutions. The proposed method is to discretize the trajectory and control schedule with respect to an orbit anomaly and perform the optimization with differential dynamic programming (DDP). The change of variable from time to orbit anomaly is accomplished by a Sundman transformation to the spacecraft equations of motion. Sundman transformations to each of the true, mean and eccentric anomalies are leveraged for fuel-optimal geocentric transfers up to 2000 revolutions. The approach is shown to be amenable to the inclusion of perturbations in the dynamic model, specifically aspherical gravity and third- body perturbations, and is improved upon through the use of modified equinoctial elements. An assessment of computational performance shows the importance of parallelization but that a single, multi-core processor is effective. The computational efficiency facilitates the generation of fuel versus time of flight trade-offs within a matter of hours. Many-revolution trajectories are characteristic of orbit transfers accomplished by solar electric propulsion about planetary bodies. Methods for modeling the effect of solar eclipses on the power available to the spacecraft and constraining eclipse durations are also presented. The logistic sunlight fraction is introduced as a coefficient that scales the computed power available by the fraction of sunlight available. The logistic sunlight fraction and Sundman-transformed DDP are used to analyze transfers from low-Earth orbit to geostationary orbit. The analysis includes a systematic approach to estimating the Pareto front of fuel versus time of flight. In addition to addressing many-revolution trajectories, this dissertation advances the utility of DDP in the three-body problem. Fuel-optimal transfers are presented in the Earth-Moon circular restricted three-body problem between distant retrograde orbits, between Lyapunov orbits and iv between Halo orbits. Those include mechanisms for varying the time of flight and the insertion point onto a target orbit. A multi-phase DDP approach enables initial guesses to be constructed from discontinuous trajectory segments. DDP is shown to leverage the system dynamics to find a heteroclinic connection between Lyapunov orbits, which is facilitated by the multi-phase approach. Dedication To Mom and Dad and the rest of you. vi Acknowledgements Thank you to my committee members, Daniel Scheeres, Jeffrey Parker, Jacob Englander, Jay McMahon and Shalom Ruben for your instruction, advising and mentoring. An honorary seat and my sincerest thanks is given to the late George Born, without whom I would not have enrolled at CU Boulder. This project took form in the low-thrust trajectory optimization research group with guidance from Jeffrey Parker and support from Jonathan Herman, Jeannette Heiligers, Nathan Parrish and Stijn De Smet. Thank you, low-thrust optimizers, and to the many visiting students who joined in the fun for a shorter period. This project reached maturity in the Celestial Mechanics and Spaceflight Mechanics Laboratory under the advising of Daniel Scheeres. Thank you Dan for welcoming me to CSML and sharing your wisdom while allowing me a great deal of academic freedom. Thank you to all CSML'ers. There are too many past and present fellow graduate students to acknowledge, so on behalf of all of you, I focus that gratitude to Sarah Melssen, Steve Hart, Annie Brookover and Carrie Simon for administrative support. I am grateful for the support I have received from a number of organizations. This work was supported by a NASA Space Technology Research Fellowship. I am indebted to Jacob Englander for helping to make that possible. Thank you for your unwaivering enthusiasm that began with the initial proposal and continues to the assembly of this document. Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Thank you Gregory Lantoine for playing host and stimulating our research collaboration. vii Contents Chapter 1 Introduction 1 1.1 Orbit Transfers . 2 1.2 Historical Low-Thrust Many-Revolution Solutions . 7 1.2.1 Indirect Methods . 7 1.2.2 Control Laws . 9 1.2.3 Direct Optimization . 10 1.3 Differential Dynamic Programming . 11 1.4 Dissertation Overview . 13 2 Low-Thrust Spaceflight Dynamics 16 2.1 Perturbed Keplerian Motion . 16 2.1.1 Power Modeling . 18 2.1.2 Control Variables . 18 2.2 Modified Equinoctial Elements . 20 2.3 The Circular Restricted Three-Body Problem . 22 2.3.1 Spacecraft State and Dynamics . 22 2.3.2 The Jacobi Constant . 25 2.3.3 Lagrange Points . 26 viii 3 Differential Dynamic Programming 28 3.1 Notation . 28 3.1.1 Tensor Notation . 29 3.2 Introduction . 30 3.3 Forward Pass . 30 3.4 Augmented Lagrangian Method . 31 3.5 Backward Sweep . 32 3.5.1 Stage Subproblems . 32 3.5.2 Second-Order Dynamics . 34 3.5.3 Stage Cost-To-Go Derivatives . 36 3.5.4 Stage Update Equations . 36 3.5.5 Inter-phase Subproblems . 37 3.5.6 Multiplier Update . 39 3.5.7 Parameter Update . 40 3.6 Trust-Region Quadratic Subproblem . 41 3.7 Iteration . 42 3.8 DDP and Indirect Optimization . 43 3.9 Example Earth-Mars Rendezvous . 44 3.9.1 Variable Departure and Arrival Times . 46 3.9.2 Monotonic Basin Hopping . 48 4 Sundman-Transformed Differential Dynamic Programming 53 4.1 The Sundman Transformation . 54 4.1.1 Sundman-Transformed Dynamics . 54 4.1.2 The Sundman-Transformed State Transition Matrix and Tensor . 55 4.2 GTO to GEO Transfer in Cartesian Coordinates . 56 4.2.1 Spacecraft State and Dynamics . 56 ix 4.2.2 Augmented Lagrangian Cost Function . 58 4.2.3 Trajectory Computation . 59 4.2.4 STM Computation . 59 4.2.5 Case B Transfer Results . 60 4.2.6 Case B with Perturbations . 64 4.3 Sundman-Transformed DDP in Modified Equinoctial Elements . 68 4.3.1 Sundman Transformations in Modified Equinoctial Elements . 71 4.3.2 The Augmented Modified Equinoctial Element State Vector . 71 4.4 GTO to GEO Transfer in Modified Equinoctial Elements . 73 4.4.1 Boundary Conditions . 73 4.4.2 Augmented Lagrangian Cost Function . 74 4.4.3 Numerical Setup . 75 4.4.4 GTO to GEO Results . 76 4.5 Conclusion . 80 5 A Smoothed Eclipse Model 85 5.1 Conical Eclipse Geometry . 85 5.2 Smoothed Sunlight Fraction . 87 5.3 Trajectory Optimization Example . 90 5.3.1 Constant power with smoothed eclipsing . 90 5.3.2 Boundary conditions and dynamics . 91 5.3.3 Initial results . 92 5.4 Penumbra Terminator Detection . 98 5.4.1 Calculation method for penumbra entry and exit . 99 5.4.2 Mesh adjustment for penumbra entry and exit . 101 5.5 Results with Penumbra Detection . 102 5.6 Conclusion . 104 x 6 A Penalty Method for Path Inequality Constraints 107 6.1 Case E Transfer . 109 6.1.1 Apsis Constraints . 110 6.1.2 Augmented Lagrangian Cost Function . 111 6.1.3 Case E Transfer Results . 111 6.2 Maximum Eclipse Duration . 115 6.2.1 Cylindrical Shadow Geometry . 119 6.2.2 Spacecraft Beta Angle . 121 6.2.3 Eclipse Maps . 122 6.2.4 The Shadow Quartic . 124 6.2.5 Evaluating the Eclipse Duration and its Derivatives . 126 6.2.6 Eclipse-Constrained Mars Orbit Transfer . 127 6.3 Conclusion . 132 7 Differential Dynamic Programming in the Three-Body Problem 134 7.1 Minimum-Radius Constraint . 135 7.2 Time Variables . 135 7.3 Transfers in the CRTBP . 136 7.3.1 Single-Revolution DRO Transfer . 138 7.3.2 Multi-Revolution DRO Transfers . ..
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