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\contentsline {section}{\numberline {1}Introduction}{10}{}%
\contentsline {section}{\numberline {2}Trajectory Optimization}{12}{}%
\contentsline {subsection}{\numberline {2.1}The Two-Body Problem}{12}{}%
\contentsline {subsubsection}{\numberline {2.1.1}Kepler's Laws and Equations}{14}{}%
\contentsline {subsection}{\numberline {2.2}Analytical Solutions to Kepler's Equations}{15}{}%
\contentsline {subsubsection}{\numberline {2.2.1}LaGuerre-Conway Algorithm}{17}{}%
\contentsline {subsection}{\numberline {2.3}Non-Linear Problem Optimization}{18}{}%
\contentsline {subsubsection}{\numberline {2.3.1}Non-Linear Solvers}{18}{}%
\contentsline {subsubsection}{\numberline {2.3.2}Linesearch Method}{19}{}%
\contentsline {section}{\numberline {3}Low-Thrust Considerations}{21}{}%
\contentsline {subsection}{\numberline {3.1}Low Thrust Overview}{21}{}%
\contentsline {subsection}{\numberline {3.2}Sims-Flanagan Transcription}{21}{}%
\contentsline {section}{\numberline {4}Interplanetary Trajectory Considerations}{22}{}%
\contentsline {subsection}{\numberline {4.1}Patched Conics}{22}{}%
\contentsline {subsection}{\numberline {4.2}Gravity Assist Maneuvers}{22}{}%
\contentsline {subsection}{\numberline {4.3}Multiple Gravity Assist Techniques}{22}{}%
\contentsline {subsection}{\numberline {4.4}Ephemeris Considerations}{22}{}%
\contentsline {section}{\numberline {5}Algorithm Overview}{23}{}%
\contentsline {subsection}{\numberline {5.1}Trajectory Composition}{23}{}%
\contentsline {subsection}{\numberline {5.2}Inner Loop Implementation}{23}{}%
\contentsline {subsubsection}{\numberline {5.2.1}LaGuerre-Conway Kepler Solver}{23}{}%
\contentsline {subsubsection}{\numberline {5.2.2}Sims-Flanagan Propagator}{23}{}%
\contentsline {subsubsection}{\numberline {5.2.3}Non-Linear Problem Solver}{24}{}%
\contentsline {subsubsection}{\numberline {5.2.4}Monotonic Basin Hopping}{24}{}%
\contentsline {subsection}{\numberline {5.3}Outer Loop Implementation}{24}{}%
\contentsline {subsubsection}{\numberline {5.3.1}Inner Loop Calling Function}{24}{}%
\contentsline {subsubsection}{\numberline {5.3.2}Genetic Algorithm Description}{24}{}%
\contentsline {section}{\numberline {6}Results Analysis}{25}{}%
\contentsline {subsection}{\numberline {6.1}Sample Trajectory to Saturn}{25}{}%
\contentsline {subsubsection}{\numberline {6.1.1}Comparison to Less Optimal Solutions}{25}{}%
\contentsline {subsubsection}{\numberline {6.1.2}Cost Function Analysis}{25}{}%
\contentsline {subsubsection}{\numberline {6.1.3}Comparison to Impulsive Trajectories}{25}{}%
\contentsline {section}{\numberline {7}Conclusion}{26}{}%
\contentsline {subsection}{\numberline {7.1}Overview of Results}{26}{}%
\contentsline {subsection}{\numberline {7.2}Applications of Algorithm}{26}{}%
\contentsline {subsection}{\numberline {7.3}Recommendations for Future Work}{26}{}%