305 lines
11 KiB
TeX
305 lines
11 KiB
TeX
\documentclass{article}
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\usepackage{mathtools}
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\usepackage{geometry}
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\usepackage{titlesec}
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{}
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\newcommand{\thesisTitle}{Designing Optimal Low-Thrust Interplanetary Trajectories Utilizing
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Monotonic Basin Hopping}
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\geometry{left=1in, right=1in, top=1in, bottom=1in}
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\doublespacing
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\begin{document}
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\title{\thesisTitle}
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\author{Richard Connor Johnstone \\
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B.S., University of Kentucky, 2016}
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\maketitle
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\vspace{3.5in}
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\begin{adjustwidth}{100pt}{100pt}
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\begin{center}
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A thesis submitted to the Faculty of the Graduate School of the University of Colorado in
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partial fulfillment of the requirements for the degree of Master of Science Department of
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Aerospace Engineering Sciences \\ 2022
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\end{center}
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\end{adjustwidth}
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\newpage
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This will be the copyright page.
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\newpage
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\begin{center}
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This thesis entitled:
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\thesisTitle
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has been approved for the Department of Aerospace Engineering Sciences
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\end{center}
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\centerline{\begin{minipage}{4in}
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\begin{center}
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\vspace{1.5in}
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\hrule
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Dr. Natasha Bosanac
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\vspace{1.5in}
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\hrule
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Dr. Daniel J. Scheeres
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\vspace{1.5in}
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\hrule
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Dr. Kathryn Davis
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\vspace{1.5in}
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\end{center}
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\end{minipage}}
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\hspace{4.25in} Date: \hrulefill
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The final copy of this thesis has been examined by the signatories, and we find that both the
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content and the form meet acceptable presentation standards of scholarly work in the above
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mentioned discipline.
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\newpage
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%TODO: This should be better
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Dedicated to some people.
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\newpage
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Richard Connor Johnstone
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\thesisTitle
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%TODO: Don't directly copy Bryce's formatting
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Thesis directed by Dr. Natasha Bosanac
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%TODO: This is just a quick abstract for now. Rewrite more towards the end.
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\begin{abstract}
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There are a variety of approaches to finding and optimizing low-thrust trajectories in
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interplanetary space. This thesis analyzes one such approach, Sims-Flanagan transcriptions, and
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its applications in a multiple-shooting non-linear solver for the purpose of finding valid
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low-thrust trajectory arcs between planets given poor initial conditions. These valid arcs are
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then fed into a Monotonic Basin Hopping (MBH) algorithm, which combines these arcs in order to
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find and optimize interplanetary trajectories, given a set of flyby planets. This allows for a
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fairly rapid searching of a very large solution space of low-thrust profiles via a medium
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fidelity inner-loop solver and a well-suited optimization routine. The trajectories found by
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this method can then be optimized further by feeding the solutions back, once again, into the
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non-linear solver, this time allowing the solver to perform optimization.
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\end{abstract}
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\newpage
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\tableofcontents
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\newpage
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\section{Introduction}
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Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Halls thrusters, and
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others can be a powerful tool in the design of space missions. They tend to be particularly
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suited to missions which require very high total $\Delta V$ values and take place over a
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particularly long duration. As such, they are well-suited to interplanetary missions. For
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instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep Space 1
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missions.
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Provide some historical background, motivations, and discussion of the basic problems being
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investigated. Also a brief overview how the thesis will be laid out.
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\section{Trajectory Optimization}
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This section will outline the foundational problem of trajectory optimization.
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\subsection{The Two-Body Problem}
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Propose the two-body problem as a differential equation.
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\subsubsection{Kepler's Equations}
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Detail Kepler's equations for astrodynamics.
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\subsection{Analytical Solutions to Kepler's Equations}
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Discuss how, since the 2BP is analytically solveable, there exists algorithms for solving
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these equations.
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\subsubsection{LaGuerre-Conway Algorithm}
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Step through the LaGuerre-Conway algorithm, as it's used in this implementation.
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\subsection{Non-Linear Problem Optimization}
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Discuss the concept of a non-linear problem as it related to the study of optimization.
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\subsubsection{Non-Linear Solvers}
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Mention common non-linear solvers (Ipopt, SNOPT, etc.).
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\subsubsection{Newton Trust-Region Method}
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Discuss how the Newton Trust-Region method works.
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\subsubsection{Linesearch Method}
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I may take this section out, because I'm not currently using a linesearch. But I would cover
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the additions of linesearch methods.
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\section{Low-Thrust Considerations}
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Highlight the differences between high and low-thrust mission profiles.
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\subsection{Low Thrust Overview}
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Dive deeper into the concept of low thrust trajectories, highlighting the added trouble with
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propagation/integration.
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\subsection{Sims-Flanagan Transcription}
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Reveal the advantages of Sims-Flanagan transcription as an alternative to higher-fidelity
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propagation models. Be sure to mention its uses in many legitimate places.
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\section{Interplanetary Trajectory Considerations}
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Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an
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interplanetary journey.
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\subsection{Patched Conics}
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Describe the method of patched conics.
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\subsection{Gravity Assist Maneuvers}
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Describe how a gravity assist maneuver would work in the framework of patched conics. Also
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discuss the advantages of such a maneuver.
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\subsection{Multiple Gravity Assist Techniques}
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Discuss the advantages of chaining together multiple gravity assists and highlight the
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difficulties in choosing these assists. Here I can mention porkchop plots, Lambert's problem,
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etc. Here I can also talk about Hybrid Optimal Control Problems.
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\subsection{Ephemeris Considerations}
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I can quickly mention SPICE here and talk a bit about validation.
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\section{Genetic Algorithms}
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I will probably give only a brief overview of genetic algorithms here. I don't personally know
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that much about them. Then in the following subsections I can discuss the parts that are
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relevant to the specific algorithm that I'm using.
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\subsection{Decision Vectors}
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Discuss what a decision vector is in the context of an optimization problem.
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\subsection{Selection and Fitness Evaluation}
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Discuss the costing being used as well as the different types of fitness evaluation that are
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common. Also discuss the concept of generations and ``survival''.
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\subsubsection{Tournament Selection}
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Dive deeper into the specific selection algorithm being used here.
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\subsection{Crossover}
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Discuss the concept of crossover and procreation in a genetic algorithm.
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\subsubsection{Binary Crossover}
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Discuss specific crossover algorithm used here.
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\subsubsection{Mutation}
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Discuss both the necessity for mutation and the mutation algorithm being used.
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\section{Algorithm Overview}
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Highlight the algorithm at a high-level. This is likely where flowcharts and diagrams will go to
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give a high-level overview.
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\subsection{Trajectory Composition}
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Discuss briefly the nomenclature used in defining these trajectories. Currently this isn't
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``baked in'' to the code, so I have some freedom to adopt Englander's notation or use my own
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(since my intended use case is a little simpler).
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\subsection{Inner Loop Implementation}
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Give a better overview of the inner loop specifically. Probably this section will have a more
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in-depth flowchart.
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\subsubsection{LaGuerre-Conway Kepler Solver}
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Discuss how the LaGuerre-Conway algorithm is used in the code to provide a fundamental
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``natural trajectory'' between two quantized, but not necessarily close points. Mention
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validation.
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\subsubsection{Sims-Flanagan Propagator}
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Discuss how this algorithm can then be expanded by using SFT to propagate any number of
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low-thrust steps over a specific arc. Mention validation. Here I can also mention the ``Sc''
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object and talk about how those parameters were chosen and effected the propagator.
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\subsubsection{Non-Linear Problem Solver}
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Mention the package being used to solve NLPs and how it works, highlighting the trust region
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method used and error-handling. Mention validation.
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\subsubsection{Monotonic Basin Hopping}
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Outline the MBH algorithm, going into detail at each step. Mention the long-tailed PDF being
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used and go into quite a bit of detail. Englander's paper on the MBH algorithm specifically
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should be a good guide. Mention validation.
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\subsection{Outer Loop Implementation}
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Overview the outer loop. This may require a final flowchart, but might potentially be too
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simple to lend itself to one.
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\subsubsection{Inner Loop Calling Function}
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The primary reason for including this section is to discuss the error handling.
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\subsubsection{Genetic Algorithm Description}
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Similar to the MBH section, there are a lot of implementation details to go over here. Many
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will have already been discussed in the background sections above. But I can step through
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each of the decisions, similar to Englander's paper on this.
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\section{Results Analysis}
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Simply highlight that the algorithm was tested on a sample trajectory to Saturn.
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\subsection{Sample Trajectory to Saturn}
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Give an overview of the trajectory that was ultimately chosen.
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\subsubsection{Comparison to Less Optimal Solutions}
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I should have a number of elite but less-optimal solutions. Honestly, I may write the
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algorithm to keep all of the solutions to provide many points of comparison here.
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\subsubsection{Cost Function Analysis}
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Give some real-world context for the mass-use, time-of-flight, etc.
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\subsubsection{Comparison to Impulsive Trajectories}
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I may also remove this section. I could do a quick comparison (using porkchop plots) to
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similar impulsive trajectories. Honestly, this is a lot of work for very little gain,
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though, so probably the first place to chop if needed.
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\section{Conclusion}
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\subsection{Overview of Results}
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Quick re-wording of the previous section in a paragraph or two for reader's convenience.
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\subsection{Applications of Algorithm}
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Talk a bit about why this work is valuable. Missions that could have benefited, missions that
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this enables, etc.
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\subsection{Recommendations for Future Work}
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Recommend future work, obviously. There are a \emph{ton} of opportunities for improvement
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including parallelization, cluster computing, etc.
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% \bibliography{biblio}{}
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% \bibliographystyle{plain}
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\end{document}
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