Maybe nearing the final commit

LaTeX/approach.tex

@@ -234,6 +234,15 @@
 			Earth. From knowledge of the mass flow rate, we can then decrement the mass
 			appropriately based on the magnitude of the thrust vector at each point.
 
+			In the implementation provided in this thesis, a Sims-Flanagan $n$ value of 20 was
+			chosen for each of the phases. This value, as tested by
+			Englander\cite{englander2012automated}, is sufficient for trajectories that maintain a
+			reasonably small number of solar revolutions. In the future, a more intelligent solution
+			could be to determine, roughly, the number of revolutions around the sun that the orbit
+			will likely take (perhaps by using the period of the orbit at the beginning of the
+			propagation and the time of flight), and scaling $n$ according to the number of
+			revolutions.
+
 		\subsection{Non-Linear Problem Solver}
 
 			Now that we have the basic building blocks of a continuous-thrust trajectory, we can
@@ -434,15 +443,24 @@
 			generator produces two random missions as described in
 			Section~\ref{random_gen_section} that differ only in that one contains random flyby
 			velocities and control thrusts and the other contains Lambert's-solved flyby
-			velocities and zero control thrusts. For each of these guesses, the NLP solver is
-			called. If either of these mission guesses have converged onto a valid solution, the
-			lower loop, the ``drill loop'' is entered for the valid solution. After the
-			convergence checks and potentially drill loops are performed, if a valid solution
-			has been found, this solution is stored in an archive. If the solution found is
-			better than the current best solution in the archive (as determined by a
-			user-provided cost function of fuel usage, $C_3$ at launch, and $v-\infty$ at
-			arrival) then the new solution replaces the current best solution and the loop is
-			repeated. Taken by itself, the search loop should quickly generate enough random
+			velocities and zero control thrusts.
+
+			The choice of adding a ``Lambert's solution-seeded'' random mission is an attempt to
+			improve upon previous low-thrust MBH iterations by more intelligently choosing some of
+			the initial guesses. Because the majority of the initial guesses that are truly random
+			don't converge, this provides a greater number of valid initial guesses for the MBH
+			algorithm to use to find better trajectories. However, each MBH loop does then take
+			twice the time, which increases the amount of time required to truly traverse the entire
+			space.
+
+			For each of these guesses, the NLP solver is called. If either of these mission guesses
+			have converged onto a valid solution, the lower loop, the ``drill loop'' is entered for
+			the valid solution. After the convergence checks and potentially drill loops are
+			performed, if a valid solution has been found, this solution is stored in an archive. If
+			the solution found is better than the current best solution in the archive (as
+			determined by a user-provided cost function of fuel usage, $C_3$ at launch, and
+			$v-\infty$ at arrival) then the new solution replaces the current best solution and the
+			loop is repeated. Taken by itself, the search loop should quickly generate enough random
 			mission guesses to find all ``basins'' or areas in the solution space with valid
 			trajectories, but never attempts to more thoroughly explore the space around valid
 			solutions within these basins.
@@ -489,4 +507,25 @@
 			iterations to perform without improvement in a row before ending the drill loop.
 			This process can be repeated essentially ''search patience`` number of times in
 			order to fully traverse all basins.
+			
+			It is worth validating whether the described Monotonic Basin Hopping algorithm
+			actually provides significant improvements in global cost-function minimization over
+			a single NLP-optimization run for the particular problem of low-thrust
+			interplanetary trajectories. Therefore, a simple sample Earth to Saturn trajectory
+			was selected and every single valid trajectory discovered by the MBH algorithm was
+			stored. In Figure~\ref{mbh_analysis}, the cost function values for each of these
+			valid trajectories is plotted in the order in which they were found. The line
+			indicates the minimum cost function discovered until that point in time. This
+			validates that the MBH, over sufficient number of iterations, will find optimal
+			values that the NLP optimization scheme could not discover on its own. It is also
+			worth noting that the ``basin drilling'' feature of the MBH scheme can be clearly
+			seen in the plot, where the algorithm continues to improve its own ``basin-best''
+			until it no longer can do so over a number of attempts.
+
+			\begin{figure}[H]
+				\centering
+				\includegraphics[width=\textwidth]{LaTeX/fig/mbh_analysis}
+				\caption{MBH cost function reduction over time}
+				\label{mbh_analysis}
+			\end{figure}