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  \title{This Will Be The Title Page}
  \author{Richard Connor Johnstone}

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  This will be the copyright page

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  This will be the Acknowledgments

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  \begin{abstract}

    This will be the overview/summary/abstract page

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  \tableofcontents

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  \section{Introduction}
    Provide some historical background, motivations, and discussion of the basic problems being investigated. Also a brief overview how the thesis will be laid out.

  \section{Trajectory Optimization}
    This section will outline the foundational problem of trajectory optimization.

    \subsection{The Two-Body Problem}
      Propose the two-body problem as a differential 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 these equations.

      \subsubsection{LaGuerre-Conway Algorithm}
        Step through the LaGuerre-Conway algorithm, as it's used in this implementation.

    \subsection{Non-Linear Problem Optimization}
      Discuss the concept of a non-linear problem as it related to the study of optimization.

      \subsubsection{Non-Linear Solvers}
        Mention common non-linear solvers (Ipopt, SNOPT, etc.).

      \subsubsection{Newton Trust-Region Method}
        Discuss how the Newton Trust-Region method works.

      \subsubsection{Linesearch Method}
        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}
    Highlight the differences between high and low-thrust mission profiles.

    \subsection{Low Thrust Overview}
      Dive deeper into the concept of low thrust trajectories, highlighting the added trouble with propagation/integration.

    \subsection{Sims-Flanagan Transcription}
      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}
    Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an interplanetary journey.

    \subsection{Patched Conics}
      Describe the method of patched conics.

    \subsection{Gravity Assist Maneuvers}
      Describe how a gravity assist manuever would work in the framework of patched conics. Also discuss the advantages of such a maneuver.

    \subsection{Multiple Gravity Assist Techniques}
      Discuss the advantages of chaining together multiple gravity assists and highlight the difficulties in choosing these assists. Here I can mention porkchop plots, lambert's problem, etc. Here I can also talk about Hybrid Optimal Control Problems.

    \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.

    \subsection{Decision Vectors}
      Discuss what a decision vector is in the context of an optimization problem.

    \subsection{Selection and Fitness Evaluation}
      Dicuss 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.

    \subsection{Crossover}
      Discuss the concept of crossover and procreation in a genetic algorithm.

      \subsubsection{Binary Crossover}
        Discuss specific crossover algorithm used here.

      \subsubsection{Mutation}
        Discuss both the neccessity for mutation and the mutation algorithm being used.

  \section{Algorithm Overview}
    Highlight the algorithm at a high-level. This is likely where flowcharts and diagrams will go to give a high-level overview.

    \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).

    \subsection{Inner Loop Implementation}
      Give a better overview of the inner loop specifically. Probably this section will have a more in-depth flowchart.

      \subsubsection{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}
        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}
        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}
        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 dicuss the error handling.

      \subsubsection{Genetic Algorithm Decription}
        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}
    Simply highlight that the algorithm was tested on a sample trajectory to Saturn.

    \subsection{Sample Trajectory to Saturn}
      Give an overview of the trajectory that was ultimately chosen.

      \subsubsection{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}
        Give some real-world context for the mass-use, time-of-flight, etc.

      \subsubsection{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}
    \subsection{Overview of Results}
      Quick re-wording of the previous section in a paragraph or two for reader's convenience.

    \subsection{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}
      Recommend future work, obviously. There are a \emph{ton} of opportunities for improvement including parallellization, cluster computing, etc.

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