connor/thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Section 4 is finished. Not great though

Browse Source
This commit is contained in:
Connor
2022-01-23 23:08:25 -07:00
parent 03825ea558
commit aa45f5f744
6 changed files with 177 additions and 84 deletions
Download Patch File Download Diff File Expand all files Collapse all files

BIN
LaTeX/fig/porkchop.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 201 KiB

106
LaTeX/thesis.tex
View File

@@ -162,6 +162,7 @@ Monotonic Basin Hopping}
right at the moment of closest approach to the (flyby) target body. Because of this,
optimization with impulsive trajectories and gravity assists are common.
% TODO: Might need to remove the HOCP stuff
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
@@ -470,23 +471,108 @@ Monotonic Basin Hopping}
trajectory within the Two-Body Problem, with only linearly-increasing computation time.
\section{Interplanetary Trajectory Considerations} \label{interplanetary}
Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an
interplanetary journey.
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
said that the assumptions of the Two Body Problem remain fairly valid. In interplanetary
travel, the primary body most responsible for gravitational forces might be a number of
different bodies, dependent on the phase of the mission. In the ideal case, every relevant
body would be considered as an ``n-body'' perturbation during the entire trajectory. For
some approaches, this method is sufficient and preferred. However, for other uses, a more
efficient model is necessary. The method of patched conics can be applied in this case to
simplify the model.
Interplanetary travel does not simply negatively impact trajectory optimization. The
increased complexity of the search space also opens up new opportunities for orbit
strategies. The primary strategy investigated by this thesis will be the gravity assist, a
technique for utilizing the gravitational energy of a planet to modify the direction of
solar velocity.
\subsection{Patched Conics}
Describe the method of 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
approaching this is the method of patched conics. In this model, we break the
interplanetary trajectory up into a series of smaller sub-trajectories. During each of
these sub-trajectories, a single primary is considered to be responsible for the
trajectory of the orbit, via the Two-Body problem.
The transition point can be calculated a variety of ways. The most typical method is to
calculate the gravitational force due to the two bodies separately, via the Two-Body
models. Whichever primary is a larger influence on the motion of the spacecraft is
considered to be the primary at that moment. This effectively breaks the trajectory into
a series of orbits defined by the Two-Body problem (conics), patched together by
distinct transition points.
\subsection{Gravity Assist Maneuvers}
Describe how a gravity assist maneuver would work in the framework of patched conics. Also
discuss the advantages of such a maneuver.
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,
or a Gravity Flyby. During a gravity assist, the spacecraft enters into the
gravitational sphere of influence of the planet and, because of its excess velocity,
proceeds to exit the sphere of influence. Relative to the planet, the speed of the
spacecraft increases as it approaches, then decreases as it departs. From the
perspective of the planet, the velocity of the spacecraft is unchanged. However, the
planet is also orbiting the Sun.
From the perspective of a Sun-centered frame, though, this is effectively an elastic
collision. The overall momentum remains the same, with the spacecraft either gaining or
losing some in the process (dependent on the directions of travel). The planet also
loses or gains momentum enough to maintain the overall system momentum, but this amount
is negligible compared to the total momentum of the planet. The overall effect is that
the spacecraft arrives at the planet from one direction and, because of the influence of
the planet, leaves in a different direction.
This effect can be used strategically. The ``bend'' due to the flyby is actually
tunable via the exact placement of the fly-by in the b-frame, or the frame centered at
the planet, from the perspective of the spacecraft at $v_\infty$. By modifying the
turning angle of this bend. In doing so, one can effectively achieve a (restricted) free
impulsive thrust event.
\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.
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
quite restricted. The incoming hyperbolic velocity must be equal in magnitude to the
outgoing hyperbolic velocity. Also, the turning angle $\delta$, in the following
equation, correlates with the radius of periapsis of the hyperbolic trajectory crossing
the planet:
\begin{equation}
r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right]
\end{equation}
Where $v_\infty$ is the magnitude of hyperbolic velocity. Naturally, the radius of
periapsis must not fall below some safe value, in order to avoid the risk of the
spacecraft crashing into the planet or its atmosphere.
In order to visualize which trajectories are possible within these constraints, porkchop
plots are often employed, such as the plot in Figure~\ref{porkchop}. These plots outline
various incoming and outgoing qualities of the trajectory arc between two planetary
bodies. For instance, during an arc from launch at Earth to a flyby one might plot the
launch C3 against the Mars arrival $v_\infty$ for a variety of launch and arrival dates.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{LaTeX/fig/porkchop}
\caption{A sample porkchop plot of an Earth-Mars transfer}
\label{porkchop}
\end{figure}
This is made possible by solving Lambert's problem for the planetary ephemeris at the
epochs plotted. Lambert's problem is concerned with determining the orbit between two
positions at two different times in space. There are a number of different Lambert's
problem algorithms that allow a mission designer to determine the velocity needed (and
thus the $\Delta V$) required to achieve a position at a later time. From this, the
designer can algorithmically determine trajectory properties in the porkchop plot for
easy visualization.
However, this is an impulsive thrust-centered approach. The solution to Lambert's
problem assumes a natural trajectory. However, to the low-thrust designer, this is
needlessly limiting. A natural trajectory is unnecessary when the trajectory can be
modified by a continuous thrust profile along the arc. Therefore, for the hybrid problem
of optimizing both flyby selection and thrust profiles, porkchop plots are less helpful,
and an algorithmic approach is preferred.
% \section{Genetic Algorithms}
% I will probably give only a brief overview of genetic algorithms here. I don't personally know

69
thesis_pdf/thesis.aux
View File

@@ -10,39 +10,42 @@
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Geometric Representation of Auxiliary Circle}}{16}{}\protected@file@percent }
\newlabel{aux_circ}{{1}{16}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.2.1}LaGuerre-Conway Algorithm}{17}{}\protected@file@percent }
\newlabel{laguerre}{{2.2.1}{17}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Non-Linear Problem Optimization}{18}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.1}Non-Linear Solvers}{18}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.2}Linesearch Method}{19}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.1}Non-Linear Solvers}{19}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.2}Linesearch Method}{20}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3}Low-Thrust Considerations}{21}{}\protected@file@percent }
\newlabel{low_thrust}{{3}{21}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Low Thrust Overview}{21}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Sims-Flanagan Transcription}{21}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4}Interplanetary Trajectory Considerations}{22}{}\protected@file@percent }
\newlabel{interplanetary}{{4}{22}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Patched Conics}{22}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Gravity Assist Maneuvers}{22}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Multiple Gravity Assist Techniques}{22}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4}Ephemeris Considerations}{22}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5}Algorithm Overview}{23}{}\protected@file@percent }
\newlabel{algorithm}{{5}{23}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Trajectory Composition}{23}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Inner Loop Implementation}{23}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.1}LaGuerre-Conway Kepler Solver}{23}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.2}Sims-Flanagan Propagator}{23}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.3}Non-Linear Problem Solver}{24}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.4}Monotonic Basin Hopping}{24}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Outer Loop Implementation}{24}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.3.1}Inner Loop Calling Function}{24}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.3.2}Genetic Algorithm Description}{24}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6}Results Analysis}{25}{}\protected@file@percent }
\newlabel{results}{{6}{25}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Sample Trajectory to Saturn}{25}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.1}Comparison to Less Optimal Solutions}{25}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.2}Cost Function Analysis}{25}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.3}Comparison to Impulsive Trajectories}{25}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7}Conclusion}{26}{}\protected@file@percent }
\newlabel{conclusion}{{7}{26}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Overview of Results}{26}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Applications of Algorithm}{26}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.3}Recommendations for Future Work}{26}{}\protected@file@percent }
\gdef \@abspage@last{26}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Low-Thrust Control Laws}{21}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Sims-Flanagan Transcription}{22}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4}Interplanetary Trajectory Considerations}{23}{}\protected@file@percent }
\newlabel{interplanetary}{{4}{23}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Patched Conics}{23}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Gravity Assist Maneuvers}{24}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Multiple Gravity Assist Techniques}{25}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A sample porkchop plot of an Earth-Mars transfer}}{26}{}\protected@file@percent }
\newlabel{porkchop}{{2}{26}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4}Ephemeris Considerations}{26}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5}Algorithm Overview}{27}{}\protected@file@percent }
\newlabel{algorithm}{{5}{27}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Trajectory Composition}{27}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Inner Loop Implementation}{27}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.1}LaGuerre-Conway Kepler Solver}{27}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.2}Sims-Flanagan Propagator}{27}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.3}Non-Linear Problem Solver}{28}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.4}Monotonic Basin Hopping}{28}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Outer Loop Implementation}{28}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.3.1}Inner Loop Calling Function}{28}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.3.2}Genetic Algorithm Description}{28}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6}Results Analysis}{29}{}\protected@file@percent }
\newlabel{results}{{6}{29}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Sample Trajectory to Saturn}{29}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.1}Comparison to Less Optimal Solutions}{29}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.2}Cost Function Analysis}{29}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.3}Comparison to Impulsive Trajectories}{29}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7}Conclusion}{30}{}\protected@file@percent }
\newlabel{conclusion}{{7}{30}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Overview of Results}{30}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Applications of Algorithm}{30}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.3}Recommendations for Future Work}{30}{}\protected@file@percent }
\gdef \@abspage@last{30}

1
thesis_pdf/thesis.lof
View File

@@ -1 +1,2 @@
\contentsline {figure}{\numberline {1}{\ignorespaces Geometric Representation of Auxiliary Circle}}{16}{}%
\contentsline {figure}{\numberline {2}{\ignorespaces A sample porkchop plot of an Earth-Mars transfer}}{26}{}%

29
thesis_pdf/thesis.log
View File

@@ -1,4 +1,4 @@
This is XeTeX, Version 3.141592653-2.6-0.999993 (TeX Live 2021/Arch Linux) (preloaded format=xelatex 2022.1.8) 8 JAN 2022 19:20
This is XeTeX, Version 3.141592653-2.6-0.999993 (TeX Live 2021/Arch Linux) (preloaded format=xelatex 2022.1.8) 23 JAN 2022 23:05
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -673,26 +673,29 @@ Package fontspec Info: Font family 'Roboto(0)' created for font 'Roboto' with
File: LaTeX/fig/kepler.png Graphic file (type bmp)
<LaTeX/fig/kepler.png>
LaTeX Font Info: Font shape `TU/Average(1)/m/n' will be
(Font) scaled to size 11.00006pt on input line 296.
(Font) scaled to size 11.00006pt on input line 297.
LaTeX Font Info: Font shape `TU/Average(1)/m/n' will be
(Font) scaled to size 7.92003pt on input line 296.
(Font) scaled to size 7.92003pt on input line 297.
LaTeX Font Info: Font shape `TU/Average(1)/m/n' will be
(Font) scaled to size 6.38004pt on input line 296.
(Font) scaled to size 6.38004pt on input line 297.
[15]
[16] [17] [18] [19] [20] [21
] [22
] [22] [23
] [23
] [24]
File: LaTeX/fig/porkchop.png Graphic file (type bmp)
<LaTeX/fig/porkchop.png>
[25] [26] [27
] [24] [25
] [28] [29
]
LaTeX Font Warning: Font shape `TU/Average(0)/m/it' undefined
(Font) using `TU/Average(0)/m/n' instead on input line 528.
(Font) using `TU/Average(0)/m/n' instead on input line 675.
[26
[30
] (./thesis.aux)
@@ -703,12 +706,12 @@ LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.
)
Here is how much of TeX's memory you used:
9616 strings out of 476257
197565 string characters out of 5816920
9636 strings out of 476257
198054 string characters out of 5816920
715810 words of memory out of 5000000
30172 multiletter control sequences out of 15000+600000
30191 multiletter control sequences out of 15000+600000
406997 words of font info for 84 fonts, out of 8000000 for 9000
1348 hyphenation exceptions out of 8191
73i,10n,114p,266b,391s stack positions out of 5000i,500n,10000p,200000b,80000s
Output written on thesis.pdf (26 pages).
Output written on thesis.pdf (30 pages).

56
thesis_pdf/thesis.toc
View File

@@ -5,32 +5,32 @@
\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 {subsubsection}{\numberline {2.3.1}Non-Linear Solvers}{19}{}%
\contentsline {subsubsection}{\numberline {2.3.2}Linesearch Method}{20}{}%
\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}{}%
\contentsline {subsection}{\numberline {3.1}Low-Thrust Control Laws}{21}{}%
\contentsline {subsection}{\numberline {3.2}Sims-Flanagan Transcription}{22}{}%
\contentsline {section}{\numberline {4}Interplanetary Trajectory Considerations}{23}{}%
\contentsline {subsection}{\numberline {4.1}Patched Conics}{23}{}%
\contentsline {subsection}{\numberline {4.2}Gravity Assist Maneuvers}{24}{}%
\contentsline {subsection}{\numberline {4.3}Multiple Gravity Assist Techniques}{25}{}%
\contentsline {subsection}{\numberline {4.4}Ephemeris Considerations}{26}{}%
\contentsline {section}{\numberline {5}Algorithm Overview}{27}{}%
\contentsline {subsection}{\numberline {5.1}Trajectory Composition}{27}{}%
\contentsline {subsection}{\numberline {5.2}Inner Loop Implementation}{27}{}%
\contentsline {subsubsection}{\numberline {5.2.1}LaGuerre-Conway Kepler Solver}{27}{}%
\contentsline {subsubsection}{\numberline {5.2.2}Sims-Flanagan Propagator}{27}{}%
\contentsline {subsubsection}{\numberline {5.2.3}Non-Linear Problem Solver}{28}{}%
\contentsline {subsubsection}{\numberline {5.2.4}Monotonic Basin Hopping}{28}{}%
\contentsline {subsection}{\numberline {5.3}Outer Loop Implementation}{28}{}%
\contentsline {subsubsection}{\numberline {5.3.1}Inner Loop Calling Function}{28}{}%
\contentsline {subsubsection}{\numberline {5.3.2}Genetic Algorithm Description}{28}{}%
\contentsline {section}{\numberline {6}Results Analysis}{29}{}%
\contentsline {subsection}{\numberline {6.1}Sample Trajectory to Saturn}{29}{}%
\contentsline {subsubsection}{\numberline {6.1.1}Comparison to Less Optimal Solutions}{29}{}%
\contentsline {subsubsection}{\numberline {6.1.2}Cost Function Analysis}{29}{}%
\contentsline {subsubsection}{\numberline {6.1.3}Comparison to Impulsive Trajectories}{29}{}%
\contentsline {section}{\numberline {7}Conclusion}{30}{}%
\contentsline {subsection}{\numberline {7.1}Overview of Results}{30}{}%
\contentsline {subsection}{\numberline {7.2}Applications of Algorithm}{30}{}%
\contentsline {subsection}{\numberline {7.3}Recommendations for Future Work}{30}{}%