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\usepackage{changepage}
\usepackage{fontspec}
\usepackage{titlesec}
\usepackage{unicode-math}
% \setmainfont{Adamina}
% \setmainfont{Alegreya}
\setmainfont[Scale=1.2]{Average}
\setmainfont[Scale=1.1]{Average}
\setmathfont[Scale=1.1]{Fira Math}
\newcommand{\sectionbreak}{\clearpage}
\titleformat{\section}
{\bfseries\fontspec{Roboto}\LARGE}
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Monotonic Basin Hopping}
\geometry{left=1in, right=1in, top=1in, bottom=1in}
\doublespacing
\setstretch{2.5}
\begin{document}
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\section{Introduction}
Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Halls thrusters, and
others can be a powerful tool in the design of space missions. They tend to be particularly
suited to missions which require very high total $\Delta V$ values and take place over a
particularly long duration. As such, they are well-suited to interplanetary missions. For
instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep Space 1
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
particularly suited to missions which require very high total change in velocity or $\Delta V$
values and take place over a particularly long duration. Traditional impulsive thrusting
techniques can achieve these changes in velocity, but they typically have a far lower specific
impulse and, as such, are much less efficient and use more fuel, costing the mission valuable
financial resources that could instead be used for science. Because of their inherently high
specific impulse (and thus efficiency), low-thrust fuels are well-suited to interplanetary
missions.
Provide some historical background, motivations, and discussion of the basic problems being
investigated. Also a brief overview how the thesis will be laid out.
For instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep
Space 1 missions. In general, anytime an interplanetary trajectory is posed, it is advisable to
first explore the possibility of low-thrust technologies. In an interplanetary mission, the
primary downside to low-thrust orbits (that they require significant time to achieve large
$\Delta V$ changes) is made irrelevant by the fact that interplanetary trajectories take such a
long time as a matter of course.
\section{Trajectory Optimization}
This section will outline the foundational problem of trajectory optimization.
Another technique often leveraged by interplanetary trajectory designers is the gravity assist.
Gravity assists cleverly utilize the inertia of a large planetary body to ''slingshot`` a
spacecraft, modifying the direction of its velocity with respect to the central body, the Sun.
This technique lends itself very well to impulsive trajectories. The gravity assist maneuver
itself can be modeled very effectively by an impulsive maneuver with certain constraints, placed
right at the moment of closest approach to the (flyby) target body. Because of this,
optimization with impulsive trajectories and gravity assists are common.
However, there is no physical reason why low-thrust trajectories can't also incorporate gravity
assists. The optimization problem becomes much more complicated. The separate problems of
optimizing flyby parameters (planet, flyby date, etc.) and optimizing the low-thrust control
arcs don't combine very easily. In this paper, a technique is explored by setting the
dual-problem up as a Hybrid Optimal Control Problem (HOCP).
This thesis will explore these concepts in a number of different sections. Section
\ref{traj_opt} will explore the basic principles of trajectory optimization in a manner agnostic
to the differences between continuous low-thrust and impulsive high-thrust techniques. Section
\ref{low_thrust} will then delve into the different aspects to consider when optimizing a low
thrust mission profile over an impulsive one. Section \ref{interplanetary} provides more detail
on the interplanetary considerations, including force models and gravity assists. Section
\ref{algorithm} will cover the implementation details of the HOCP optimization algorithm
developed for this paper. Finally, section \ref{results} will explore the results of some
hypothetical missions to Saturn.
\section{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
solutions. In this section, the foundations for direct optimization of these sorts of problems
will be explored by first introducing the Two-Body Problem, then an algorithm for directly
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}
Propose the two-body problem as a differential equation.
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
trajectories, we often have to compute the path of the spacecraft many hundreds, thousands, or
even millions of times. Utilizing very high-fidelity force models that account for aerodynamic
pressures, solar radiation pressures, multi-body effects, and many others may be infeasible
for the method being used if the computations take too long.
Therefore, a common approach (and the one utilized in this implementation) is to first look
simply at the single largest force governing the spacecraft in motion, the gravitational force
due to the primary body around which it is orbiting. This can provide an excellent
low-to-medium fidelity model that can be extremely useful in categorizing the optimization
space as quickly as possible. In many cases, including the algorithm used in this paper, it is
unlikely that local cost-function minima would be missed due to the lack of fidelity of the
Two Body Problem.
In order to explore the Two Body Problem, we must first examine the full set of assumptions
associated with the force model. Firstly, we are only concerned with the nominative two
bodies: the spacecraft and the planetary body around which it is orbiting. Secondly, both of
these bodies are modeled as simple point masses. This removes the need to account for
non-uniform densities and asymmetry. The third assumption is that the mass of the spacecraft
($m_2$) is much much smaller than the mass of the planetary body ($m_1$) and enough so as to be
considered negligible. The only force acting on this system is then the force of gravity that
the primary body enacts upon the secondary. Lastly, we'll assume a fixed inertial frame. This
isn't necessary for the formulation of a solution, but will simplify the derivation.
Reducing the system to two point masses with a single gravitational force acting between them
(and only in one direction) we can model the force on the secondary body as:
\begin{equation}
\ddot{\vec{r}} = - \frac{G \left( m_1 + m_2 \right)}{r^2} \frac{\vec{r}}{\left| r \right|}
\end{equation}
\subsubsection{Kepler's Equations}
Detail Kepler's equations for astrodynamics.
\subsection{Analytical Solutions to Kepler's Equations}
Discuss how, since the 2BP is analytically solveable, there exists algorithms for solving
Discuss how, since the 2BP is analytically solvable, there exists algorithms for solving
these equations.
\subsubsection{LaGuerre-Conway Algorithm}
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I may take this section out, because I'm not currently using a linesearch. But I would cover
the additions of linesearch methods.
\section{Low-Thrust Considerations}
\section{Low-Thrust Considerations} \label{low_thrust}
Highlight the differences between high and low-thrust mission profiles.
\subsection{Low Thrust Overview}
@@ -181,7 +253,7 @@ Monotonic Basin Hopping}
Reveal the advantages of Sims-Flanagan transcription as an alternative to higher-fidelity
propagation models. Be sure to mention its uses in many legitimate places.
\section{Interplanetary Trajectory Considerations}
\section{Interplanetary Trajectory Considerations} \label{interplanetary}
Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an
interplanetary journey.
@@ -200,31 +272,31 @@ Monotonic Basin Hopping}
\subsection{Ephemeris Considerations}
I can quickly mention SPICE here and talk a bit about validation.
\section{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.
% \section{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}
Discuss what a decision vector is in the context of an optimization problem.
% \subsection{Decision Vectors}
% Discuss what a decision vector is in the context of an optimization problem.
\subsection{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''.
% \subsection{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}
Dive deeper into the specific selection algorithm being used here.
% \subsubsection{Tournament Selection}
% Dive deeper into the specific selection algorithm being used here.
\subsection{Crossover}
Discuss the concept of crossover and procreation in a genetic algorithm.
% \subsection{Crossover}
% Discuss the concept of crossover and procreation in a genetic algorithm.
\subsubsection{Binary Crossover}
Discuss specific crossover algorithm used here.
% \subsubsection{Binary Crossover}
% Discuss specific crossover algorithm used here.
\subsubsection{Mutation}
Discuss both the necessity for mutation and the mutation algorithm being used.
% \subsubsection{Mutation}
% Discuss both the necessity for mutation and the mutation algorithm being used.
\section{Algorithm Overview}
\section{Algorithm Overview} \label{algorithm}
Highlight the algorithm at a high-level. This is likely where flowcharts and diagrams will go to
give a high-level overview.
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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}
\section{Results Analysis} \label{results}
Simply highlight that the algorithm was tested on a sample trajectory to Saturn.
\subsection{Sample Trajectory to Saturn}
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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}
\section{Conclusion} \label{conclusion}
\subsection{Overview of Results}
Quick re-wording of the previous section in a paragraph or two for reader's convenience.