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LaTeX/thesis.tex

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 Monotonic Basin Hopping}
 
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 \begin{document}
 
@@ -131,27 +135,95 @@ Monotonic Basin Hopping}
 
   \section{Introduction}
 
-    Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Halls thrusters, and
+    Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Hall thrusters, and
-    others can be a powerful tool in the design of space missions. They tend to be particularly
+    others can be a powerful tool in the design of interplanetary space missions. They tend to be
-    suited to missions which require very high total $\Delta V$ values and take place over a
+    particularly suited to missions which require very high total change in velocity or $\Delta V$
-    particularly long duration. As such, they are well-suited to interplanetary missions. For
+    values and take place over a particularly long duration. Traditional impulsive thrusting
-    instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep Space 1
+    techniques can achieve these changes in velocity, but they typically have a far lower specific
-    missions. 
+    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
+    For instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep
-    investigated. Also a brief overview how the thesis will be laid out.
+    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}
+    Another technique often leveraged by interplanetary trajectory designers is the gravity assist.
-    This section will outline the foundational problem of trajectory optimization.
+    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}
@@ -170,7 +242,7 @@ Monotonic Basin Hopping}
         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}
+  % \section{Genetic Algorithms}
-    I will probably give only a brief overview of genetic algorithms here. I don't personally know
+  %   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
+  %   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.
+  %   relevant to the specific algorithm that I'm using.
 
-    \subsection{Decision Vectors}
+  %   \subsection{Decision Vectors}
-      Discuss what a decision vector is in the context of an optimization problem.
+  %     Discuss what a decision vector is in the context of an optimization problem.
 
-    \subsection{Selection and Fitness Evaluation}
+  %   \subsection{Selection and Fitness Evaluation}
-      Discuss the costing being used as well as the different types of fitness evaluation that are
+  %     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''.
+  %     common. Also discuss the concept of generations and ``survival''.
 
-      \subsubsection{Tournament Selection}
+  %     \subsubsection{Tournament Selection}
-        Dive deeper into the specific selection algorithm being used here.
+  %       Dive deeper into the specific selection algorithm being used here.
 
-    \subsection{Crossover}
+  %   \subsection{Crossover}
-      Discuss the concept of crossover and procreation in a genetic algorithm.
+  %     Discuss the concept of crossover and procreation in a genetic algorithm.
 
-      \subsubsection{Binary Crossover}
+  %     \subsubsection{Binary Crossover}
-        Discuss specific crossover algorithm used here.
+  %       Discuss specific crossover algorithm used here.
 
-      \subsubsection{Mutation}
+  %     \subsubsection{Mutation}
-        Discuss both the necessity for mutation and the mutation algorithm being used.
+  %       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.
 
@@ -268,7 +340,7 @@ Monotonic Basin Hopping}
         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}
@@ -286,7 +358,7 @@ Monotonic Basin Hopping}
         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.