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Emailed committee and made progress on Section 5

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\documentclass{article}
\documentclass[defaultstyle,11pt]{LaTeX/thesis}
\usepackage{graphicx}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{setspace}
\usepackage{changepage}
\usepackage{fontspec}
\usepackage{titlesec}
\usepackage{unicode-math}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{amsmath}
% \setmainfont{Adamina}
% \setmainfont{Alegreya}
\setmainfont[Scale=1.1]{Average}
\setmathfont[Scale=1.1]{Fira Math}
\title{Designing Optimal Low-Thrust Interplanetary Trajectories Utilizing Monotonic Basin Hopping}
\author{Richard C.}{Johnstone}
\otherdegrees{B.S., Unviersity of Kentucky, Mechanical Engineering, 2016 \\
B.S., University of Kentucky, Physics, 2016}
\degree{Master of Science}{M.S., Aerospace Engineering}
\dept{Department of}{Aerospace Engineering}
\advisor{Prof.}{Natasha Bosanac}
\reader{TBD: Kathryn Davis}
\readerThree{TBD: Daniel Scheeres}
\newcommand{\sectionbreak}{\clearpage}
\abstract{ \OnePageChapter
There are a variety of approaches to finding and optimizing low-thrust trajectories in
interplanetary space. This thesis analyzes one such approach, Sims-Flanagan transcriptions, and
its applications in a multiple-shooting non-linear solver for the purpose of finding valid
low-thrust trajectory arcs between planets given poor initial conditions. These valid arcs are
then fed into a Monotonic Basin Hopping (MBH) algorithm, which combines these arcs in order to
find and optimize interplanetary trajectories, given a set of flyby planets. This allows for a
fairly rapid searching of a very large solution space of low-thrust profiles via a medium
fidelity inner-loop solver and a well-suited optimization routine. The trajectories found by
this method can then be optimized further by feeding the solutions back, once again, into the
non-linear solver, this time allowing the solver to perform optimization.
}
\titleformat{\section}
{\bfseries\fontspec{Roboto}\LARGE}
{\thesection}
{0.5em}
{}
\titleformat{\subsection}
{\fontspec{Roboto}\Large}
{\thesubsection}
{0.5em}
{}
\titleformat{\subsubsection}
{\fontspec{Roboto}\large}
{\thesubsubsection}
{0.5em}
{}
\dedication[Dedication]{
Dedicated to some people.
}
\newcommand{\thesisTitle}{Designing Optimal Low-Thrust Interplanetary Trajectories Utilizing
Monotonic Basin Hopping}
\acknowledgements{ \OnePageChapter
This will be an acknowledgement.
}
\geometry{left=1in, right=1in, top=1in, bottom=1in}
\setstretch{2.5}
\LoFisShort
\emptyLoT
\begin{document}
\title{\thesisTitle}
\author{Richard Connor Johnstone \\
B.S., University of Kentucky, 2016}
\input LaTeX/macros.tex
\maketitle
\vspace{3.5in}
\begin{adjustwidth}{100pt}{100pt}
\begin{center}
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 Master of Science Department of
Aerospace Engineering Sciences \\ 2022
\end{center}
\end{adjustwidth}
\newpage
This will be the copyright page.
\newpage
\begin{center}
This thesis entitled:
\thesisTitle
has been approved for the Department of Aerospace Engineering Sciences
\end{center}
\centerline{\begin{minipage}{4in}
\begin{center}
\vspace{1.5in}
\hrule
Dr. Natasha Bosanac
\vspace{1.5in}
\hrule
Dr. Daniel J. Scheeres
\vspace{1.5in}
\hrule
Dr. Kathryn Davis
\vspace{1.5in}
\end{center}
\end{minipage}}
\hspace{4.25in} Date: \hrulefill
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.
\newpage
%TODO: This should be better
Dedicated to some people.
\newpage
Richard Connor Johnstone
\thesisTitle
%TODO: Don't directly copy Bryce's formatting
Thesis directed by Dr. Natasha Bosanac
%TODO: This is just a quick abstract for now. Rewrite more towards the end.
\begin{abstract}
There are a variety of approaches to finding and optimizing low-thrust trajectories in
interplanetary space. This thesis analyzes one such approach, Sims-Flanagan transcriptions, and
its applications in a multiple-shooting non-linear solver for the purpose of finding valid
low-thrust trajectory arcs between planets given poor initial conditions. These valid arcs are
then fed into a Monotonic Basin Hopping (MBH) algorithm, which combines these arcs in order to
find and optimize interplanetary trajectories, given a set of flyby planets. This allows for a
fairly rapid searching of a very large solution space of low-thrust profiles via a medium
fidelity inner-loop solver and a well-suited optimization routine. The trajectories found by
this method can then be optimized further by feeding the solutions back, once again, into the
non-linear solver, this time allowing the solver to perform optimization.
\end{abstract}
\newpage
\tableofcontents
\listoffigures
\newpage
\section{Introduction}
\chapter{Introduction}
Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Hall thrusters, and
others can be a powerful tool in the design of interplanetary space missions. They tend to be
@@ -179,7 +87,7 @@ Monotonic Basin Hopping}
developed for this paper. Finally, section \ref{results} will explore the results of some
hypothetical missions to Saturn.
\section{Trajectory Optimization} \label{traj_opt}
\chapter{Trajectory Optimization} \label{traj_opt}
Trajectory optimization is concerned with a narrow problem (namely, optimizing a spaceflight
trajectory to an end state) with a wide range of possible techniques, approaches, and even
@@ -188,7 +96,7 @@ Monotonic Basin Hopping}
solving for states in that system, then exploring approaches to Non-Linear Problem (NLP) solving
in general and how they apply to spaceflight trajectories.
\subsection{The Two-Body Problem}
\section{The Two-Body Problem}
The motion of a spacecraft in space is governed by a large number of forces. When planning and
designing a spacecraft trajectory, we often want to use the most complete (and often complex)
model of these forces that is available. However, in the process of designing these
@@ -233,7 +141,7 @@ Monotonic Basin Hopping}
Where $\mu = G m_1$ is the specific gravitational parameter for our primary body of interest.
\subsubsection{Kepler's Laws and Equations}
\subsection{Kepler's Laws and Equations}
% TODO: Can I segue better from 2BP to Keplerian geometry?
@@ -260,7 +168,7 @@ Monotonic Basin Hopping}
\pi \sqrt{\frac{a^3}{\mu}}$ where $T$ is the period and $a$ is the semi-major axis.
\end{enumerate}
\subsection{Analytical Solutions to Kepler's Equations}
\section{Analytical Solutions to Kepler's Equations}
Kepler was able to produce an equation to represent the angular displacement of an orbiting
body around a primary body as a function of time, which we'll derive now for the elliptical
@@ -333,7 +241,7 @@ Monotonic Basin Hopping}
($E$) which can be related to spacecraft position, and time, but we still need a useful
algorithm for solving this equation.
\subsubsection{LaGuerre-Conway Algorithm}\label{laguerre}
\subsection{LaGuerre-Conway Algorithm}\label{laguerre}
For this application, I used an algorithm known as the LaGuerre-Conway algorithm, which was
presented in 1986 as a faster algorithm for directly solving Kepler's equation and has been
in use in many applications since. This algorithm is known for its convergence robustness
@@ -342,7 +250,7 @@ Monotonic Basin Hopping}
This thesis will omit a step-through of the algorithm itself, but the code will be present
in the Appendix.
\subsection{Non-Linear Problem Optimization}
\section{Non-Linear Problem Optimization}
Now we can consider the formulation of the problem in a more useful way. For instance, given a
desired final state in position and velocity we can relatively easily determine the initial
@@ -375,7 +283,7 @@ Monotonic Basin Hopping}
system dynamics adds too much complexity to quickly optimize indirectly and the individual
optimization routines needed to proceed as quickly as possible.
\subsubsection{Non-Linear Solvers}
\subsection{Non-Linear Solvers}
For these types of non-linear, constrained problems, a number of tools have been developed
that act as frameworks for applying a large number of different algorithms. This allows for
simple testing of many different algorithms to find what works best for the nuances of the
@@ -400,7 +308,7 @@ Monotonic Basin Hopping}
libraries that port these are quite modular in the sense that multiple algorithms can be
tested without changing much source code.
\subsubsection{Linesearch Method}
\subsection{Linesearch Method}
As mentioned above, this project utilized IPOPT which leveraged an Interior Point Linesearch
method. A linesearch algorithm is one which attempts to find the optimum of a non-linear
problem by first taking an initial guess $x_k$. The algorithm then determines a step
@@ -414,7 +322,7 @@ Monotonic Basin Hopping}
was sufficient merely that the non-linear constraints were met, therefore optimization (in
the particular step in which IPOPT was used) was unnecessary.
\section{Low-Thrust Considerations} \label{low_thrust}
\chapter{Low-Thrust Considerations} \label{low_thrust}
Thus far, the techniques that have been discussed can be equally useful for both impulsive and
continuous thrust mission profiles. In this section, we'll discuss the intricacies of continuous
@@ -423,7 +331,7 @@ Monotonic Basin Hopping}
trajectory as well as introduce the concept of a control law and the notation used in this
thesis for modelling low-thrust trajectories more simply.
\subsection{Low-Thrust Control Laws}
\section{Low-Thrust Control Laws}
In determining a low-thrust arc, a number of variables must be accounted for and, ideally,
optimized.
@@ -448,7 +356,7 @@ Monotonic Basin Hopping}
however, the control law must be continuous rather than discrete and therefore the control law
inherently gains a lot of complexity.
\subsection{Sims-Flanagan Transcription}
\section{Sims-Flanagan Transcription}
The major problem with optimizing low thrust paths is that the control law must necessarily be
continuous. Also, since indirect optimization approaches are quite difficult, the problem must
@@ -470,7 +378,7 @@ Monotonic Basin Hopping}
number of sub-arcs, one can rapidly approach a fidelity equal to a continuous low-thrust
trajectory within the Two-Body Problem, with only linearly-increasing computation time.
\section{Interplanetary Trajectory Considerations} \label{interplanetary}
\chapter{Interplanetary Trajectory Considerations} \label{interplanetary}
The question of interplanetary travel opens up a host of additional new complexities. While
optimizations for simple single-body trajectories are far from simple, it can at least be
@@ -488,7 +396,7 @@ Monotonic Basin Hopping}
technique for utilizing the gravitational energy of a planet to modify the direction of
solar velocity.
\subsection{Patched Conics}
\section{Patched Conics}
The first hurdle to deal with is the problem of reconciling the Two-Body problem with
the presence of multiple and varying planetary bodies. The most common method for
@@ -504,7 +412,7 @@ Monotonic Basin Hopping}
a series of orbits defined by the Two-Body problem (conics), patched together by
distinct transition points.
\subsection{Gravity Assist Maneuvers}
\section{Gravity Assist Maneuvers}
As previously mentioned, there are methods for utilizing the orbital energy of the other
planets in the Solar System. This is achieved via a technique known as a Gravity Assist,
@@ -529,7 +437,7 @@ Monotonic Basin Hopping}
turning angle of this bend. In doing so, one can effectively achieve a (restricted) free
impulsive thrust event.
\subsection{Multiple Gravity Assist Techniques}
\section{Multiple Gravity Assist Techniques}
Naturally, therefore, one would want to utilize these gravity flybys to reduce the fuel
cost to arrive at their destination target state. However, these flyby maneuvers are
@@ -574,31 +482,31 @@ Monotonic Basin Hopping}
of optimizing both flyby selection and thrust profiles, porkchop plots are less helpful,
and an algorithmic approach is preferred.
% \section{Genetic Algorithms}
% \chapter{Genetic Algorithms}
% I will probably give only a brief overview of genetic algorithms here. I don't personally know
% that much about them. Then in the following subsections I can discuss the parts that are
% relevant to the specific algorithm that I'm using.
% \subsection{Decision Vectors}
% \section{Decision Vectors}
% Discuss what a decision vector is in the context of an optimization problem.
% \subsection{Selection and Fitness Evaluation}
% \section{Selection and Fitness Evaluation}
% Discuss the costing being used as well as the different types of fitness evaluation that are
% common. Also discuss the concept of generations and ``survival''.
% \subsubsection{Tournament Selection}
% \subsection{Tournament Selection}
% Dive deeper into the specific selection algorithm being used here.
% \subsection{Crossover}
% \section{Crossover}
% Discuss the concept of crossover and procreation in a genetic algorithm.
% \subsubsection{Binary Crossover}
% \subsection{Binary Crossover}
% Discuss specific crossover algorithm used here.
% \subsubsection{Mutation}
% \subsection{Mutation}
% Discuss both the necessity for mutation and the mutation algorithm being used.
\section{Algorithm Overview} \label{algorithm}
\chapter{Algorithm Overview} \label{algorithm}
In this section, we will review the actual execution of the algorithm developed. As an
overview, the routine was developed to enable the determination of an optimized spacecraft
@@ -629,77 +537,127 @@ Monotonic Basin Hopping}
algorithm is used to traverse the search space and more carefully optimize the solutions
found by the inner loop.
\subsection{Trajectory Composition}
Discuss briefly the nomenclature used in defining these trajectories. Currently this isn't
``baked in'' to the code, so I have some freedom to adopt Englander's notation or use my own
(since my intended use case is a little simpler).
\section{Trajectory Composition}
\subsection{Inner Loop Implementation}
Give a better overview of the inner loop specifically. Probably this section will have a more
in-depth flowchart.
In this thesis, a specific nomenclature will be adopted to define the stages of an
interplanetary mission in order to standardize the discussion about which aspects of the
software affect which phases of the mission.
\subsubsection{LaGuerre-Conway Kepler Solver}
Overall, a mission is considered to be the entire overall trajectory. In the context of
this software procedure, a mission is taken to always begin at the Earth, with some
initial launch C3 intended to be provided by an external launch vehicle. This C3 is not
fully specified by the mission designer, but instead is optimized as a part of the
overall cost function (and normalized by a designer-specified maximum allowable value).
This overall mission can then be broken down into a variable number of ``phases''
defined as beginning at one planetary body with some excess hyperbolic velocity and
ending at another. The first phase of the mission is from the Earth to the first flyby
planet. The final phase is from the last flyby planet to the planet of interest.
Each of these phases are then connected by a flyby event at the boundary. Each flyby
event must satisfy the following conditions:
\begin{enumerate}
\item The planet at the end of one phase must match the planet at the beginning of
the next phase.
\item The magnitude of the excess hyperbolic velocity coming into the planet (at the
end of the previous phase) must equal the magnitude of the excess hyperbolic
velocity leaving the planet (at the beginning of the next phase).
\item The flyby ``turning angle'' must be such that the craft maintains a safe
minimum altitude above the surface or atmosphere of the flyby planet.
\end{enumerate}
These conditions then effectively stitch the separate mission phases into a single
coherent mission, allowing for the optimization of both individual phases and the entire
mission as a whole. This nomenclature is similar to the nomenclature adopted by Jacob
Englander in his Hybrid Optimal Control Problem paper, but does not allow for missions
with multiple targets, simplifying the optimization.
\section{Inner Loop Implementation}
The optimization routine can be reasonable separated into two separate ``loops'' wherein
the first loop is used, given an initial guess, to find valid trajectories within the
region of the initial guess and submit the best. The outer loop is then used to traverse
the search space and supply the initial loop with a number of well chosen initial
guesses.
Figure~\ref{nlp} provides an overview of the process of breaking a mission guess down
into an NLP, but there are essentially three primary routines involved in the inner
loop. A given state is propagated forward using the LaGuerre-Conway Kepler solution
algorithm, which itself is used to generate powered trajectory arcs via the
Sims-Flanagan transcribed propagator. Finally, these powered arcs are connected via a
multiple-shooting non-linear optimization problem. The trajectories describing each
phase complete one ``Mission Guess'' which is fed to the non-linear solver to generate
one valid trajectory within the vicinity of the original Mission Guess.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{LaTeX/flowcharts/nlp}
\caption{A flowchart of the Non-Linear Problem Solving Formulation}
\label{nlp}
\end{figure}
\subsection{LaGuerre-Conway Kepler Solver}
Discuss how the LaGuerre-Conway algorithm is used in the code to provide a fundamental
``natural trajectory'' between two quantized, but not necessarily close points. Mention
validation.
\subsubsection{Sims-Flanagan Propagator}
\subsection{Sims-Flanagan Propagator}
Discuss how this algorithm can then be expanded by using SFT to propagate any number of
low-thrust steps over a specific arc. Mention validation. Here I can also mention the ``Sc''
object and talk about how those parameters were chosen and effected the propagator.
\subsubsection{Non-Linear Problem Solver}
\subsection{Non-Linear Problem Solver}
Mention the package being used to solve NLPs and how it works, highlighting the trust region
method used and error-handling. Mention validation.
\subsubsection{Monotonic Basin Hopping}
\section{Outer Loop Implementation}
Overview the outer loop. This may require a final flowchart, but might potentially be too
simple to lend itself to one.
\subsection{Inner Loop Calling Function}
The primary reason for including this section is to discuss the error handling.
\subsection{Monotonic Basin Hopping}
Outline the MBH algorithm, going into detail at each step. Mention the long-tailed PDF being
used and go into quite a bit of detail. Englander's paper on the MBH algorithm specifically
should be a good guide. Mention validation.
\subsection{Outer Loop Implementation}
Overview the outer loop. This may require a final flowchart, but might potentially be too
simple to lend itself to one.
\subsubsection{Inner Loop Calling Function}
The primary reason for including this section is to discuss the error handling.
\subsubsection{Genetic Algorithm Description}
Similar to the MBH section, there are a lot of implementation details to go over here. Many
will have already been discussed in the background sections above. But I can step through
each of the decisions, similar to Englander's paper on this.
\section{Results Analysis} \label{results}
\chapter{Results Analysis} \label{results}
Simply highlight that the algorithm was tested on a sample trajectory to Saturn.
\subsection{Sample Trajectory to Saturn}
\section{Sample Trajectory to Saturn}
Give an overview of the trajectory that was ultimately chosen.
\subsubsection{Comparison to Less Optimal Solutions}
\subsection{Comparison to Less Optimal Solutions}
I should have a number of elite but less-optimal solutions. Honestly, I may write the
algorithm to keep all of the solutions to provide many points of comparison here.
\subsubsection{Cost Function Analysis}
\subsection{Cost Function Analysis}
Give some real-world context for the mass-use, time-of-flight, etc.
\subsubsection{Comparison to Impulsive Trajectories}
\subsection{Comparison to Impulsive Trajectories}
I may also remove this section. I could do a quick comparison (using porkchop plots) to
similar impulsive trajectories. Honestly, this is a lot of work for very little gain,
though, so probably the first place to chop if needed.
\section{Conclusion} \label{conclusion}
\subsection{Overview of Results}
\chapter{Conclusion} \label{conclusion}
\section{Overview of Results}
Quick re-wording of the previous section in a paragraph or two for reader's convenience.
\subsection{Applications of Algorithm}
\section{Applications of Algorithm}
Talk a bit about why this work is valuable. Missions that could have benefited, missions that
this enables, etc.
\subsection{Recommendations for Future Work}
\section{Recommendations for Future Work}
Recommend future work, obviously. There are a \emph{ton} of opportunities for improvement
including parallelization, cluster computing, etc.
% \bibliography{biblio}{}
% \bibliographystyle{plain}
\bibliographystyle{plain} % or "siam", or "alpha", etc.
\nocite{*} % list all refs in database, cited or not
\bibliography{LaTeX/refs} % Bib database in "refs.bib"
\appendix
% \input appendixA.tex
\end{document}