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Think I'm gonna call it for now. May reassess, but almost completely done!

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\chapter{Results Analysis} \label{results}
\chapter{Sample Saturn Trajectory Analysis} \label{results}
The algorithm described in this thesis is quite flexible in its design and could be used as
a tool for a mission designer on a variety of different mission types. However, to consider
a relatively simple but representative mission design objective, a sample mission to Saturn
was investigated.
Ultimately, two optimized trajectories were selected. The results of those trajectories can
be found in Table~\ref{results_table} below:
\begin{table}[h!]
\begin{small}
\centering
\begin{tabular}{ | c c c c c c | }
\hline
\bfseries Flyby Selection &
\bfseries Launch Date &
\bfseries Mission Length &
\bfseries Launch $C_3$ &
\bfseries Arrival $V_\infty$ &
\bfseries Fuel Usage \\
& & (years) & $\left( \frac{km}{s} \right)^2$ & ($\frac{km}{s}$) & (kg) \\
\hline
EMS & 2024-06-27 & 7.9844 & 60.41025 & 5.816058 & 446.9227 \\
EMJS & 2023-11-08 & 14.1072 & 40.43862 & 3.477395 & 530.6683 \\
\hline
\end{tabular}
\end{small}
\caption{Comparison of the two most optimal trajectories}
\label{results_table}
\end{table}
\section{Mission Constraints}
The sample mission was defined to represent a general case for a near-future low-thrust
@@ -91,7 +66,17 @@
used by the mass at launch and the $C_3$ number is determined by dividing the $C_3$
at launch by the maximum allowed. These two numbers are then weighted, with the fuel
usage value getting a weight of three and the launch energy value getting a weight
of one. The values are summed and returned as the cost value.
of one. The values are summed and returned as the cost value, represented as the value
$J$ below:
\begin{equation}
J(\vec{x}, m_{dry}, C_{3,max}) = 3 \left| \frac{h(\vec{x})}{m_{dry}} \right| +
\left| \frac{k(\vec{x})}{C_{3,max}} \right|
\end{equation}
\noindent
Where $h(\vec{x})$ represents the total fuel mass used during the trajectory and
$k(\vec{x})$ represents the launch $C_3$ of the initial phase.
\subsection{Flybys Analyzed}
@@ -102,9 +87,12 @@
the mission.
For this particular mission scenario, the following flyby profiles were
investigated:
investigated (E: Earth, M: Mars, V: Venus, J: Jupiter, S: Saturn). These flyby choices
were initially sampled randomly, but as patterns were noticed during the previous runs,
certain trajectories were chosen to investigate phases that seemed promising.
\begin{itemize}
\setlength\itemsep{-0.5em}
\item EJS
\item EMJS
\item EMMJS
@@ -128,17 +116,17 @@
contrast to the usual dichotomy of low-thrust travel. The cost function used for this
analysis did not include the time of flight as a component of the overall cost, and yet
this trajectory still managed to be the lowest cost trajectory of all trajectories found
by the algorithm.
by the algorithm, meaning that it has merit for both a flyby mission as well as a capture
mission.
The mission begins in late June of 2024 and proceeds first to an initial gravity assist
with Mars after three and one half years to rendezvous in mid-December 2027.
Unfortunately, the launch energy required to effectively used the gravity assist with
Mars at this time is quite high. The $C_3$ value was found to be $60.4102$ kilometers
per second squared. However, for this phase, the thrust magnitudes are quite low,
raising slowly only as the spacecraft approaches Mars, allowing for a nearly-natural
trajectory to Mars rendezvous. Note also that the in-plane thrust direction was neither
zero nor $\pi$, implying that these thrusts were steering thrusts rather than
momentum-increasing thrusts.
The mission begins in late June of 2024 and proceeds first to an initial gravity assist with
Mars after three and one half years to rendezvous in mid-December 2027. Unfortunately, the
launch energy required to effectively used the gravity assist with Mars at this time is
quite high. The $C_3$ value was found to be $60.4102 \frac{\text{km}^2}{\text{s}^2}$. However,
for this phase, the thrust magnitudes are quite low, raising slowly only as the spacecraft
approaches Mars, allowing for a nearly-natural trajectory to Mars rendezvous. Note also that
the in-plane thrust angle was neither zero nor $\pi$, implying that these thrusts were
steering thrusts rather than momentum-increasing thrusts.
\begin{figure}[H]
\centering
@@ -158,15 +146,15 @@
\end{figure}
The second and final leg of this trip exits the Mars flyby and, initially burns quite
heavily along the velocity vector in order to increase it's semi-major axis. After an
initial period of thrusting, though, the spacecraft effectively coasts with minor
adjustments until its rendezvous with Saturn just four and a half years later in June of
2032. The arrival $v_\infty$ is not particularly small, at $5.816058$ kilometers per
second, but this is to be expected as the arrival excess velocity was not considered as
a part of the cost function. If capture was not the final intention of the mission, this
may be of little concern. Otherwise, the low fuel usage of $446.92$ kilograms for a
$3500$ kilogram launch mass leaves much margin for a large impulsive thrust to enter
into a capture orbit at Saturn.
heavily along the velocity vector in order to increase its semi-major axis. After an initial
period of thrusting, though, the spacecraft effectively coasts with minor adjustments until
its rendezvous with Saturn just four and a half years later in June of 2032. The arrival
$v_\infty$ is not particularly small, at $5.816058 \frac{\text{km}}{\text{s}}$, but this is
to be expected as the arrival excess velocity was not considered as a part of the cost
function. If capture was not the final intention of the mission, this may be of little
concern. Otherwise, the low fuel usage of $446.92$ kilograms for a $3500$ kilogram launch
mass leaves much margin for a large impulsive thrust to enter into a capture orbit at
Saturn.
\begin{figure}[H]
\centering
@@ -183,7 +171,7 @@
\label{ems_components}
\end{figure}
In this case the algorithm effectively realized that a higher-powered launch from
In this case the algorithm effectively discovered that a higher-powered launch from
the Earth, then a natural coasting arc to Mars flyby would provide the spacecraft with
enough velocity that a short but efficient powered-arc to Saturn was possible with
effective thrusting. It also determined that the most effective way to achieve this
@@ -202,12 +190,11 @@
\section{Slower, More Efficient Trajectory}
Next we'll analyze the nominally second-best trajectory. While the cost function
provided to the algorithm can be a useful tool for narrowing down the field of search
results, it can also be very useful to explore options that may or may not be of similar
"efficiency" in terms of the cost function, but beneficial for other reasons. By
outputting many different optimal trajectories, the MBH algorithm can allow for this
type of mission design flexibility.
Next we'll analyze the nominally second-best trajectory. While the cost function provided to
the algorithm can be a useful tool for narrowing down the field of search results, it can
also be very useful to explore options that may or may not have quite as small of a cost
function value, but beneficial for other reasons. By outputting many different optimal
trajectories, the MBH algorithm can allow for this type of mission design flexibility.
To highlight the flexibility, a second trajectory has been selected, which has nearly
equal value by the cost function, coming in slightly lower. However, this trajectory
@@ -276,81 +263,51 @@
While the fuel use is also slightly higher at $530.668$ kilograms, plenty of payload
mass is still capable of delivery into the vicinity of Saturn. Also, it should be noted
that the incoming excess hyperbolic velocity at arrival to Saturn is significantly
lower, at only $3.4774$ kilometers per second, meaning that less of the delivered
lower, at only $3.4774\frac{\text{km}}{\text{s}}$, meaning that less of the delivered
payload mass would need to be taken up by impulsive thrusters and fuel for Saturn orbit
capture, should the mission designer desire this.
Also, as mentioned before, the launch energy requirements are quite a bit lower. Having
a second mission trajectory capable of launching on a smaller vehicle could be valuable
to a mission designer presenting possibilities. According to an analysis of the Delta IV
and Atlas V launch configurations\cite{c3capabilities} in Figure~\ref{c3}, this
\section{Final Trajectory Analysis}
Ultimately, two optimized trajectories were selected to be excellent candidates for further
consideration. The resultant flyby selection, launch and arrival dates, and relevant cost
function input of those trajectories can be found in Table~\ref{results_table} below:
\begin{table}[h!]
\begin{small}
\centering
\begin{tabular}{ | c c c c c c | }
\hline
\bfseries Flyby Selection &
\bfseries Launch Date &
\bfseries Mission Length &
\bfseries Launch $C_3$ &
\bfseries Arrival $V_\infty$ &
\bfseries Fuel Usage \\
& & (years) & $\left( \frac{km}{s} \right)^2$ & ($\frac{km}{s}$) & (kg) \\
\hline
EMS & 2024-06-27 & 7.9844 & 60.41025 & 5.816058 & 446.9227 \\
EMJS & 2023-11-08 & 14.1072 & 40.43862 & 3.477395 & 530.6683 \\
\hline
\end{tabular}
\end{small}
\caption{Comparison of the two most optimal trajectories}
\label{results_table}
\end{table}
As mentioned before, the launch energy requirements of the second trajectory are quite a bit
lower. Having a second mission trajectory capable of launching on a smaller vehicle could be
valuable to a mission designer presenting possibilities. According to an analysis of the
Delta IV and Atlas V launch configurations\cite{c3capabilities} in Figure~\ref{c3}, this
reduction of $C_3$ from around 60 to around 40 brings the sample mission to just within
range of both the Delta IV Heavy and the Atlas V in its largest configuration, neither
of which are possible for the other result, meaning that either different launch
vehicles must be found or mission specifications must change.
range of both the Delta IV Heavy and the Atlas V in its largest configuration, neither of
which are possible for the other result, meaning that either different launch vehicles must
be found or mission specifications must change.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig/c3}
\caption{Plot of Delta IV and Atlas V launch vehicle capabilities as they relate to
payload mass}
payload mass \cite{c3capabilities} from a source from 2007}
\label{c3}
\end{figure}
\chapter{Conclusion} \label{conclusion}
\section{Overview of Results}
A mission designer's job is quite a difficult one and it can be very useful to have
tools to automate some of the more complex analysis. This paper attempted to explore one
such tool, meant for automating the initial analysis and discovery of useful
interplanetary, low-thrust trajectories including the difficult task of optimizing the
flyby parameters. This makes the mission designer's job significantly simpler in that
they can simply explore a number of different flyby selection options in order to get a
good understanding of the mission scope and search space for a given spacecraft, launch
window, and target.
In performing this examination, two results were selected for further analysis. These
results are outlined in Table~\ref{results_table}. As can be seen in the table, both
resulting trajectories have trade-offs in mission length, launch energy, fuel usage, and
more. However, both results should be considered very useful low-thrust trajectories in
comparison to other missions that have launched on similar interplanetary trajectories,
using both impulsive and low-thrust arcs with planetary flybys. Each of these missions
should be feasible or nearly feasible (feasible with some modifications) using existing
launch vehicle and certainly even larger missions should be reasonable with advances in
launch capabilities currently being explored.
\section{Recommendations for Future Work}\label{improvement_section}
In the course of producing this algorithm, a large number of improvement possibilities
were noted. This work was based, in large part, on the work of Jacob Englander in a
number of papers\cite{englander2014tuning}\cite{englander2017automated}
\cite{englander2012automated} in which he explored the hybrid optimal control problem of
multi-objective low-thrust interplanetary trajectories.
In light of this, there are a number of additional approaches that Englander took in
preparing his algorithm that were not implemented here in favor of reducing complexity
and time constraints. For instance, many of the Englander papers explore the concept of
an outer loop that utilizes a genetic algorithm to compare many different flyby planet
choice against each other. This would create a truly automated approach to low-thrust
interplanetary mission planning. However, a requirement of this approach is that the
monotonic basin hopping algorithm algorithm must converge on optimal solutions very
quickly. Englander typically runs his for 20 minutes each for evolutionary fitness
evaluation, which is over an order of magnitude faster than the implementation in this
paper to achieve satisfactory results.
Further improvements to performance stem from the field of computer science. An
evolutionary algorithm such as the one proposed by Englander would benefit from high
levels of parallelization. Therefore, it would be worth considering a GPU-accelerated or
even cluster-computing capable implementation of the monotonic basin hopping algorithm.
These cluster computing concepts scale very well with new cloud infrastructures such as
that provided by AWS or DigitalOcean.
Finally, the monotonic basin hopping algorithm as currently written provides no
guarantees of actual global optimization. Generally optimization is achieved by running
the algorithm until it fails to produce newer, better trajectories for a sufficiently
long time. But it would be worth investigating the robustness of the NLP solver as well
as the robustness of the MBH algorithm basin drilling procedures in order to quantify
the search granularity needed to completely traverse the search space. From this
information, a new MBH algorithm could be written that is guaranteed to explore the
entire space.