Low Thrust Trajectory Optimization in Cislunar and Translunar Space by Nathan Luis Olin Parrish B.S., Aerospace Engineering, California Polytechnic State University, 2012 M.S., Aerospace Engineering Sciences, University of Colorado at Boulder, 2014 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 ii This thesis entitled: Low Thrust Trajectory Optimization in Cislunar and Translunar Space written by Nathan Luis Olin Parrish has been approved for the Department of Aerospace Engineering Sciences ____________________________________ Daniel J. Scheeres ____________________________________ Jeffrey S. Parker ____________________________________ Jay W. McMahon ____________________________________ Christoffer Heckman ____________________________________ Daniel Kubitschek 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 Parrish, Nathan L. O. (Ph.D., Aerospace Engineering Sciences) Low Thrust Trajectory Optimization in Cislunar and Translunar Space Thesis directed by Daniel J. Scheeres Low-thrust propulsion technologies such as electric propulsion and solar sails are key to enabling many space missions which would be impractical with chemical propulsion. With exhaust velocities 10x higher than chemical rockets, electric propulsion systems can deliver a spacecraft to its target state for a fraction of the fuel. Due to the low thrust, the control must remain active for weeks or even years. When three-body dynamics are considered, the change in dynamics over the course of a trajectory can be extreme. This greatly complicates low-thrust mission design and navigation in cislunar and translunar space, making it an area of active research. Deterministic strategies for trajectory design and optimization rely on linearizing the problem and solving a series of linearized problems. In regimes with simple or slowly-varying dynamics, the linearization holds “true enough”, and we can easily arrive at a solution. However, three-body environments readily provide real cases where the linearization for all but the most carefully-chosen problem descriptions break down. This thesis presents a few modifications to existing algorithms to improve convergence. This thesis then uses this fast, robust method for trajectory optimization to generate training samples for a machine learning approach to optimal trajectory correction. We begin with one optimal low-thrust transfer. Then, we optimize thousands of transfers in the neighborhood of the nominal transfer. These transfers are described in the language of indirect optimal control, with the optimal control given as a function of Lawden’s primer vector. We see that for a slightly iv different initial condition, the states and the costates both follow a slightly different trajectory to the target. A feedforward artificial neural network is trained to map the difference in states to the difference in costates, with a high degree of accuracy. Finally, we explore a potential application of this neural network: spacecraft that can navigate themselves autonomously in the presence of errors. We propose this as a method for future spacecraft that can optimally correct their trajectories without ground contacts. We demonstrate neural network navigation in two simplified dynamical environments: two-body heliocentric gravity, and the Earth-Moon circular restricted three body problem. v Acknowledgements This thesis is dedicated to two great men who each, in turn, believed in me at a critical juncture: Dr. Douglas A. O’Handley (1937-2016) and Dr. George H. Born (1939-2016). Both men were pioneers in the field of space exploration, students of the cosmos, and teachers dedicated entirely to raising up the next generation of scientists and engineers. Committed to the future and undeterred by illness or age, each worked tirelessly until the day he died during the course of this PhD. Ad astra per aspera. I also wish to thank the following: • Professors George Born, Web Cash, Jeff Parker, and Dan Scheeres for each, successively, serving as my advisor • Sarah Melssen, for her heroic administrative support • The Graduate Assistance in Areas of National Need (GAANN) Fellowship, for funding my first 2 years of research • The NASA Space Technology Research Fellowship (NSTRF), for funding years 3-5 and providing opportunities to collaborate with NASA personnel at Goddard and JPL • Steven Hughes, for his collaboration from Goddard Space Flight Center • Jon Sims and Aline Zimmer, for their collaboration from the Jet Propulsion Lab • Christopher Cartland, for sparking ideas about machine learning • The Bahá’í communities in Colorado and the entire Four Corners region, without whom I could never have reached this point with joy and purpose • Jessa Karlberg, for making the final months happier • My parents, who sacrificed a great deal to give me a quality education vi Contents Chapter 1. INTRODUCTION .................................................................................................................. 1 1.1. THE CHALLENGE & BENEFIT OF ELECTRIC PROPULSION ................................................. 1 1.2. THE CHALLENGE & BENEFIT OF N-BODY GRAVITY FIELDS ............................................ 3 2. BACKGROUND .................................................................................................................... 8 2.1. DYNAMICS ....................................................................................................................... 8 2.1.1. Circular Restricted Three Body Problem ................................................................ 9 2.1.2. Electric Propulsion ................................................................................................ 10 2.1.3. Solar Sails ............................................................................................................. 12 2.2. OPTIMAL CONTROL........................................................................................................ 13 2.2.1. Direct vs. Indirect Optimization ........................................................................... 14 2.2.2. Indirect Transcription............................................................................................ 16 2.2.3. Direct Transcription .............................................................................................. 20 2.2.4. Sequential Quadratic Programming ...................................................................... 21 2.2.5. Numerical Approaches.......................................................................................... 24 2.3. METHODS OF DIFFERENTIATION .................................................................................... 31 2.4. NEURAL NETWORKS ...................................................................................................... 34 3. EFFICIENT FORMULATION OF MULTIPLE SHOOTING ............................................ 40 3.1. DIRECT MULTIPLE SHOOTING ........................................................................................ 42 3.1.1. Two-Stage Differential Corrector ......................................................................... 43 3.1.2. Simplified SQP ..................................................................................................... 46 3.1.3. Mesh Refinement .................................................................................................. 48 vii 3.1.4. Quadratic Endpoint Constraints ............................................................................ 50 3.1.5. Line Search ........................................................................................................... 55 3.1.6. Initial Guesses for Direct Method ......................................................................... 58 3.2. INDIRECT SHOOTING ...................................................................................................... 68 3.2.1. Homotopy ............................................................................................................. 72 3.2.2. Initial Guesses for Indirect Method ...................................................................... 76 3.2.3. Example Transfers with Indirect Method ............................................................. 79 4. FAMILIES OF TRAJECTORIES ........................................................................................ 83 4.1. DRO TO DRO ................................................................................................................ 84 4.2. L2 HALO TO L2 HALO ..................................................................................................... 88 4.3. DRO TO L2 HALO .......................................................................................................... 94 5. NEURAL NETWORKS APPLIED TO OPTIMAL CONTROL ......................................... 97 5.1. UNCERTAINTY PROPAGATION ........................................................................................ 98 5.1.1. Polynomial Chaos ................................................................................................. 98 5.1.2. Comparison of Polynomial Chaos and Neural Networks ..................................... 99 5.2. OPTIMAL LOW THRUST TRAJECTORY CORRECTION