Additional display functions and section 6 draft done

LaTeX/thesis.bib

@@ -32,3 +32,33 @@
   number={GSFC-E-DAA-TN14154},
   year={2014}
 }
+
+@misc{cassini,
+  title={Cassini Mission Fact Sheet},
+  author={NASA Jet Propulsion Laboratory},
+  url={https://solarsystem.nasa.gov/missions/cassini/mission/quick-facts/},
+  journal={NASA Solar System Exploration},
+  year={2018}
+}
+
+@inproceedings{polk2001performance,
+  title={Performance of the NSTAR ion propulsion system on the Deep Space One mission},
+  author={Polk, J and Kakuda, R and Anderson, J and Brophy, J and Rawlin, V and Patterson, M and Sovey, J and Hamley, J},
+  booktitle={39th Aerospace Sciences Meeting and Exhibit},
+  pages={965},
+  year={2001}
+}
+
+@article{stough2021nasa,
+  title={NASA’s Space Launch System: Capabilities for Ultra-High C3 Missions},
+  author={Stough, Robert W and Robinson, Kimberly F and Holt, James B and Smith, David A and Hitt, W David and Perry, Beverly A},
+  year={2021}
+}
+
+@article{c3capabilities,
+author = {Vardaxis, George and Pitz, Alan and Wie, Bong},
+year = {2022},
+month = {02},
+pages = {},
+title = {AAS 12-128 CONCEPTUAL DESIGN AND ANALYSIS OF PLANETARY DEFENSE TECHNOLOGY (PDT) DEMONSTRATION MISSIONS}
+}

LaTeX/thesis.tex

@@ -955,32 +955,257 @@
 				in order to fully traverse all basins.
 
 	\chapter{Results Analysis} \label{results}
-		Simply highlight that the algorithm was tested on a sample trajectory to Saturn.
 
-		\section{Sample Trajectory to Saturn}
-			Give an overview of the trajectory that was ultimately chosen.
+		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.
 
-			\subsection{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.
+		\section{Mission Constraints}
 
-			\subsection{Cost Function Analysis}
-				Give some real-world context for the mass-use, time-of-flight, etc.
+			The sample mission was defined to represent a general case for a near-future low-thrust
+			trajectory to Saturn. No constraints were placed on the flyby planets, but a number of
+			constraints were placed on the algorithm to represent a realistic mission scenario. 
 
-			\subsection{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.
+			The first choice required by the application is one not necessarily designable to the
+			initial mission designer (though not necessarily fixed in the design either) and is that
+			of the spacecraft parameters. The application accepts as input a spacecraft object
+			containing: the dry mass of the craft, the fuel mass at launch, the number of onboard
+			thrusters, and the specific impulse, maximum thrust and duty cycle of each thruster.
+
+			For this mission, the spacecraft was chosen to have a dry mass of only 200 kilograms for
+			a fuel mass of 3300 kilograms. This was chosen in order to have an overall mass roughly
+			in the same zone as that of the Cassini spacecraft, which launched with 5712 kilograms
+			of total mass, with the fuel accounting for 2978 of those kilograms\cite{cassini}. The
+			dry mass of the craft was chosen to be extremely low in order to allow for a variety of
+			''successful`` missions in which the craft didn't run out of fuel. That way, the
+			delivered dry mass to Saturn could be thought of as a metric of success, without
+			discounting mission that may have delivered just under whatever more realistic dry mass
+			one might set, in case those missions are in the vicinity of actually valid missions.
+
+			The thruster was chosen to have a specific impulse of 3200 seconds, a maximum thrust of
+			250 millinewtons, and a 100\% duty cycle. This puts the thruster roughly in line with
+			having an array of three NSTAR ion thrusters, which were used on the Dawn and Deep Space
+			1 missions\cite{polk2001performance}.
+
+			Also of relevance to the mission were the maximum $C_3$ at launch and $v_\infty$ at
+			arrival values. In order to not exclude the possibility of a non-capture flyby mission,
+			it was decided to not include the arrival $v_\infty$ term in the cost function and,
+			because of this, the maximum value was set to be extremely high at 500 kilometers per
+			second, in order to fully explore the space. In practice, though, the algorithm only
+			looks at flybys below 10 kilometers per second in magnitude. The maximum launch $C_3$
+			energy was set conservatively to 200 kilometers per second squared. This is upper limit
+			is only possible, for the given start mass, using a heavy launch system such as the
+			SLS\cite{stough2021nasa} or the comparable SpaceX Starship, though at values below about
+			half of this maximum, it begins to become possible to use existing launch solutions.
+
+			Finally, the mission is meant to represent a near future mission. Therefore the launch
+			window was set to allow for a launch in any day in 2023 or 2024 and a maximum total time
+			of flight of 20 years. This is longer than most typical Saturn missions, but allows for
+			some creative trajectories for higher efficiency.
+
+			It should be noted that each of these trajectories was found using an $n$ value of 20 as
+			mentioned previously, but in post-processing, the trajectory was refined to utilize a
+			slightly higher fidelity model that uses 60 sub-trajectories per orbit. This serves to
+			provide better plots for display, higher fidelity analyses, as well as to highlight the
+			efficacy of the lower fidelity method. Orbits can be found quickly in the lower fidelity
+			model and easily refined later by re-running the NLP solver at a higher $n$ value.
+
+			\subsection{Cost Function}
+
+				Each mission optimization also allows for the definition of a cost function. This
+				cost function accepts as inputs all parameters of the mission, the maximum $C_3$ at
+				launch and the maximum excess hyperbolic velocity at arrival.
+
+				The cost function used for this mission first generated normalized values for fuel
+				usage and launch energy. The fuel usage number is determined by dividing the fuel
+				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.
+
+			\subsection{Flybys Analyzed}
+
+				Since the algorithm itself makes no decisions on the actual choice of flybys, that
+				leaves the mission designer to determine which flyby planets would make good
+				potential candidates. A mission designer can then re-run the algorithm for each of
+				these flyby plans and determine which optimized trajectories best fit the needs of
+				the mission.
+
+				For this particular mission scenario, the following flyby profiles were
+				investigated:
+
+				\begin{itemize}
+					\item EJS
+					\item EMJS
+					\item EMMJS
+					\item EMS
+					\item ES
+					\item EVMJS
+					\item EVMS
+					\item EVVJS
+				\end{itemize}
+
+		\section{Faster, Less Efficient Trajectory}
+
+			In order to showcase the flexibility of the optimization algorithm (and the chosen cost
+			function), two different missions were chosen to highlight. One of these missions is a
+			slower, more efficient trajectory more typical of common low-thrust trajectories. The
+			other is a faster trajectory, quite close to a natural trajectory, but utilizing more
+			launch energy to arrive at the planet.
+
+			It is the faster trajectory that we'll analyze first. Most interesting about this
+			particular trajectory is that it's actually quite efficient despite its speed, in
+			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.
+
+			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 thrusters are almost entirely turned
+			off, allowing for a nearly-natural trajectory to Mars rendezvous.
+
+			\begin{figure}
+				\centering
+				\includegraphics[width=\textwidth]{fig/EMS_plot}
+				\caption{Depictions of the faster Earth-Mars-Saturn trajectory found by the
+				algorithm to be most efficient; planetary ephemeris arcs are shown during the phase
+				in which the spacecraft approached them}
+				\label{ems}
+			\end{figure}
+
+			\begin{figure}
+				\centering
+				\includegraphics[width=\textwidth]{fig/EMS_plot_noplanets}
+				\caption{Another depiction of the EMS trajectory, without the planetary ephemeris
+				arcs}
+				\label{ems_nop}
+			\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.
+
+			In this case the algorithm effectively realized 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
+			flyby was to increase orbital energy in the beginning of the arc, when increasing
+			the semi-major axis value is most efficient. All of these concepts are known to skilled
+			mission designers, but finding a trajectory that combined all of these concepts would
+			have required much time-consuming analysis of porkchop plots and combinations of
+			mission-design techniques. This approach is far more automatic than the traditional
+			approach.
+
+			The final quality to note with this trajectory is that it shows a tangible benefit of
+			the addition of the Lambert's solver in the monotonic basin hopping algorithm. Since the
+			initial arc is almost entirely natural, with very little thrust, it is extremely likely
+			that the trajectory was found in the Lambert's Solution half of the MBH algorithm
+			procedure.
+
+		\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.
+
+			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
+			appears to offer some benefits to the mission designer who would like to capture into
+			the gravitational field of Saturn or minimize launch energy requirements, perhaps for a
+			smaller mission, at the expense of increased speed.
+
+			The first leg of this three-leg trajectory is quite similar to the first leg of the
+			previous trajectory. However, this time the launch energy is considerably lower, with a
+			$C_3$ value of only $40.4386$ kilometer per second squared. Rather than employ an almost
+			entirely natural coasting arc to Mars, however, this trajectory performs some thrusting
+			at about the apoapsis point of its orbit in order to raise the periapsis enough to
+			rendezvous at roughly the same incidence angle in Mars' orbit, but one revolution ahead.
+			In this case, the launch was a bit earlier, occurring in November of 2023, with the Mars
+			flyby occurring in mid-April of 2026. This will prove to be helpful in comparison with
+			the other result, as this mission profile is much longer.
+
+			\begin{figure}
+				\centering
+				\includegraphics[width=\textwidth]{fig/EMJS_plot}
+				\caption{Depictions of the slower Earth-Mars-Jupiter-Saturn trajectory found by the
+				algorithm to be the second most efficient; planetary ephemeris arcs are shown during
+				the phase in which the spacecraft approached them}
+				\label{emjs}
+			\end{figure}
+
+			\begin{figure}
+				\centering
+				\includegraphics[width=\textwidth]{fig/EMJS_plot_noplanets}
+				\caption{Another depiction of the EMJS trajectory, without the planetary ephemeris
+				arcs}
+				\label{emjs_nop}
+			\end{figure}
+
+			The second phase of this trajectory also functions quite similarly to the second phase
+			of the previous trajectory. In this case, there is a little bit more thrusting necessary
+			simply for steering to the Jupiter flyby than was necessary for Saturn rendezvous in the
+			previous trajectory. However, most of this thrusting is for orbit raising in the
+			beginning of the phase, very similarly to the previous result. In this trajectory, the
+			Jupiter flyby occurs late July of 2029.
+
+			Finally, this mission also has a third phase. The Jupiter flyby provides quite a strong
+			$\Delta V$ for the spacecraft, allowing the following phase to largely be a coasting arc
+			to Saturn almost one revolution later. Because of this long coasting period, the mission
+			length increases considerably during this leg, arriving at Saturn in December of 2037,
+			over 8 years after the Jupiter flyby.
+
+			However, there are many advantages to this approach relative to the other trajectory.
+			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
+			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
+			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.
+
+			\begin{figure}
+				\centering
+				\includegraphics[width=\textwidth]{fig/c3}
+				\caption{Plot of Delta IV and Atlas V launch vehicle capabilities as they relate to
+				payload mass}
+				\label{c3}
+			\end{figure}
 
 	\chapter{Conclusion} \label{conclusion}
+
 		\section{Overview of Results}
+
 			Quick re-wording of the previous section in a paragraph or two for reader's convenience.
 
 		\section{Applications of Algorithm}
+
 			Talk a bit about why this work is valuable. Missions that could have benefited, missions that
 			this enables, etc.
 
 		\section{Recommendations for Future Work}\label{improvement_section}
+
 			Recommend future work, obviously. There are a \emph{ton} of opportunities for improvement
 			including parallelization, cluster computing, etc.