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Machine Learning Regression for Estimating Characteristics of Low-Thrust Transfers

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Machine Learning Regression for Estimating Characteristics of Low-Thrust Transfers MACHINE LEARNING REGRESSION FOR ESTIMATING CHARACTERISTICS OF LOW-THRUST TRANSFERS A Thesis Presented to The Academic Faculty By Gene L. Chen In Partial Fulfillment of the Requirements for the Degree Master of Science in the School of Aerospace Engineering Georgia Institute of Technology May 2019 Copyright c Gene L. Chen 2019 MACHINE LEARNING REGRESSION FOR ESTIMATING CHARACTERISTICS OF LOW-THRUST TRANSFERS Approved by: Dr. Dimitri Mavris, Advisor Guggenheim School of Aerospace Engineering Georgia Institute of Technology Dr. Alicia Sudol Guggenheim School of Aerospace Engineering Georgia Institute of Technology Dr. Michael Steffens Guggenheim School of Aerospace Engineering Georgia Institute of Technology Date Approved: April 15, 2019 To my parents, thanks for the support. ACKNOWLEDGEMENTS Firstly, I would like to thank Dr. Dimitri Mavris for giving me the opportunity to pursue a Master’s degree at the Aerospace Systems Design Laboratory. It has been quite the experience. Also, many thanks to my committee members — Dr. Alicia Sudol and Dr. Michael Steffens — for taking the time to review my work and give suggestions on how to improve - it means a lot to me. Additionally, thanks to Dr. Patel and Dr. Antony for helping me understand their work in trajectory optimization. iv TABLE OF CONTENTS Acknowledgments . iv List of Tables . vii List of Figures . viii Chapter 1: Introduction and Motivation . 1 1.1 Thesis Overview . 4 Chapter 2: Background . 5 Chapter 3: Approach . 17 3.1 Scenario Definition . 17 3.2 Spacecraft Dynamics . 18 3.3 Choice Between Direct and Indirect Method . 19 3.3.1 Chebyshev polynomial method . 20 3.3.2 Sims-Flanagan method . 21 3.4 Monotonic Basin Hopping . 25 3.5 Indirect Optimization . 26 3.6 Design of Experiments . 31 Chapter 4: Experiments . 33 v 4.1 Computational Setup . 33 4.2 Experiment for Research Question 1 . 33 4.3 Experiment for Research Question 1a: Choice of Direct Solver . 35 4.4 Experimental Setup . 38 4.5 Indirect Solver . 39 4.6 Training Data . 40 4.7 Experiment for Research Question 2 . 42 Chapter 5: Conclusions . 54 5.1 Findings . 54 5.2 Future Work . 55 References . 60 vi LIST OF TABLES 4.1 Comparison between trial runs of 100 object pairs solved using SLSQP vs SNOPT7 . 38 4.2 Comparison between trial runs of 100 object pairs solved while varying basin hopping settings . 39 4.3 MAE and RMSE of the machine learning algorithms compared to that of the Lambert predictor for final spacecraft mass . 51 4.4 MAE and RMSE of the machine learning algorithms for time-of-flight . 51 vii LIST OF FIGURES 1.1 Cumulative number of Near-Earth Asteroids (NEAs) Discovered in Recent Years. (From [2]) . 2 1.2 Comparison between a notional impulsive and a continuous-thrust trajec- tory from Earth to Jupiter. 3 2.1 An example of a porkchop plot. Colors represent the ∆V for the transfer at the corresponding departure time and time-of-flight. (From [12]) . 6 2.2 Porkchop plot for Earth-Mars transfers in 2005. Blue contours indicate equal ∆V transfers, and red lines indicate transfers with equal time-of- flight (From [13]) . 7 2.3 Distribution of prediction errors for two machine learning regressors (blue) and a Lambert solver (green). (From [10]) . 8 2.4 Decision Tree. (From [29]) . 12 2.5 Random Forest. (From [31]) . 13 2.6 Gradient Boosting. (From [31]) . 14 2.7 Artificial neural network. (From [31]) . 15 2.8 Support vector regression process. (From [32]) . 16 3.1 Physical Meaning of Orbital Elements (from [34]) . 18 3.2 Basics of the Sims-Flanagan Method (from [39]) . 22 3.3 Sims-Flanagan Method. In this case, nforwards is 3 and nbackwards is 2. 24 3.4 Notional objective function topography, to minimize energy (from [40]) . 25 viii 3.5 The Latin Hypercube Design of Experiments (from [41]) . 32 4.1 An example run of the indirect solver. 35 4.2 Chebyshev polynomial-based trajectory depends heavily on the order of the polynomial. 36 4.3 Indirect solver’s solution to the Earth-Mars rendezvous problem. 40 4.4 A transfer between objects with a large difference in inclination. The space- craft starts at the magenta point and travels to the yellow point. Red arcs indicate thrusting segments and blue arcs indicate coasting segments. Axes are in units of AU and to scale with each other. 41 4.5 Time-of-flight vs. final spacecraft mass graphs . 43 4.6 Validation trajectory mass errors for artificial neural network and decision tree algorithms. 45 4.7 Validation trajectory mass errors for random forest regression and support vector regression. 46 4.8 Validation trajectory mass errors for kernel ridge regression and gradient boosting. 47 4.9 Validation trajectory mass errors for artificial neural network and decision tree algorithms. 48 4.10 Validation trajectory mass errors for random forest regression and support vector regression. 49 4.11 Validation trajectory mass errors for kernel ridge regression and gradient boosting. 50 4.12 Workflow of methodology to create the machine learning estimators. Red arrows indicate comparisons made between models during the experiment for Research Question 2. ........................... 53 ix SUMMARY Compared to chemical propulsion, electric propulsion is much more efficient in its use of propellant in space. An electric propulsion engine can easily provide an order of mag- nitude more velocity imparted, while using the same propellant mass. This advantage is applied by space mission designers: enabling greater payload mass delivered or mission flexibility. However, there are significant drawbacks for electric propulsion. Firstly, the achieved thrust is minuscule, which rules out its application in launch vehicles, since the force of gravity cannot be counteracted in nearly any situation. Also, the computational difficulty of finding a suitable low-thrust trajectory is much more than that of finding a trajectory where the thrust can be assumed to be applied instantaneously, which can be a reasonable assumption in missions involving chemical propulsion. This is due to the long thrusting durations typical of electric propulsion missions, requiring a computer program to generate and store spacecraft states and controls at many time points along the trajectory. The high computation time demanded by a low-thrust trajectory solver has the greatest impact on problems involving large search spaces, where there exists a large number of po- tential targets to visit, and the additional decision of timing for starting the maneuver burn. The infeasibility of calculating high-fidelity trajectory solutions to all transfer combinations would typically impact planning for missions to space debris or small Solar System bodies, where potential destinations number in the thousands. The goal of this work was to use machine learning techniques to create an accurate and computationally efficient manner of estimating the fuel and time cost of a transfer between a wide variety of orbits. A particular focus was placed on flyby-type problems, where the velocity at the target is unconstrained. This is of interest because little literature has investigated flyby transfers to targets that are of completely disparate orbit types. Existing literature tends to focus on transfers between objects that at least have a common orbital inclination, even if the longitude of ascend- ing nodes may differ (e.g. 9th Global Trajectory Optimization Competition (GTOC)). In x this thesis, sequential design choices were made for the solver used to create the training data. It was found that using the Sims-Flanagan method, a direct method, was the most practical way of finding valid trajectories for creating the training data. Several machine learning methods were used to predict fuel and time expenditures of the trajectories - sup- port vector regression and gradient boosting were among the highest-scoring techniques, but no one machine learning technique greatly outperformed any other. Thus, this thesis demonstrates that machine learning methods can produce quick and accurate-enough (for sequence generation problems, potentially) predictions of fuel and time expenditures of general low-thrust transfers. xi CHAPTER 1 INTRODUCTION AND MOTIVATION Trajectory planning is a time-consuming and difficult part of spacecraft mission design. There are often requirements for a spacecraft to visit specific points in space, with tight fuel margins. Therein lies the difficulty, as it is trivial to create an infeasible trajectory, but difficult to create an optimal one. Additionally, the ability for a spacecraft to visit multiple destinations is often desired by mission planners; however, this comes at a cost of making the trajectory planning problem much harder. This is because there is additional complexity in choosing not only the destinations the spacecraft would visit, but also the order in which the spacecraft visits them. At the same time, trajectories must be found between candidate destinations. Since the number of destinations can easily reach the tens of thousands and even climb, as with the number of known asteroids near the Earth, seen in Figure 1.1, a large dimensionality can be anticipated from the problem. Likewise, the European Space Agency (ESA) predicts that there are around 29,000 pieces of space debris larger than 10 cm in size. Further compounding the problem is the fact that the mass, power, and other properties of the spacecraft could change as it travels along its trajectory[1]. The extent of the difficulty of sequential trajectory planning is best exemplified in the Global Trajec- tory Optimization Competition (GTOC), an international competition, occurring every 1-2 years, to create the most optimal trajectory for the given problem. As put forth by Izzo [3], the trajectory optimization problem can be interpreted as a global optimization task. In general, the global optimization problems in GTOC are designed to be complex, and the GTOC 1 problem achieved this complexity by allowing for sequences of planetary flybys and having a long launch window[3].
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