Unless Bosanac has a last minute change, the paper is done!

LaTeX/approach.tex

@@ -264,7 +264,7 @@
 						\item The $v_{\infty,in}$ vector representing excess velocity at the
 							planetary flyby (or completion of mission) at the end of the phase
 						\item The time of flight for the phase
-						\item The unit-thrust profile in a sun-fixed frame represented by a
+						\item The unit-thrust profile in a sun-centered frame represented by a
 							series of vectors with each element ranging from 0 to 1.
 					\end{itemize}
 			\end{itemize}
@@ -397,18 +397,12 @@
 			non-powered flyby.
 
 			From these two velocity vectors the turning angle, and thus the periapsis of the flyby,
-			can then be calculated by Equation~\ref{turning_angle_eq} and the following equation:
-
-			\begin{equation}
-				r_p = \frac{\mu}{\vec{v}_{\infty,in} \cdot \vec{v}_{\infty,out}} \cdot \left(
-				\frac{1}{\sin(\delta/2)} - 1 \right)
-			\end{equation}
-
-			If this radius of periapse is then found to be less than the minimum safe radius
-			(currently set to the radius of the planet plus 100 kilometers), then the process is
-			repeated with new random flyby velocities until a valid seed flyby is found. These
-			checks are also performed each time a mission is perturbed or generated by the NLP
-			solver.
+			can then be calculated by Equation~\ref{turning_angle_eq} and
+			Equation~\ref{periapsis_eq}. If this radius of periapse is then found to be less than
+			the minimum safe radius (currently set to the radius of the planet plus 100 kilometers),
+			then the process is repeated with new random flyby velocities until a valid seed flyby
+			is found. These checks are also performed each time a mission is perturbed or generated
+			by the NLP solver.
 
 			The final requirement then, is the thrust controls, which are actually quite simple.
 			Since the thrust is defined as a 3-vector of values between -1 and 1 representing some

LaTeX/results.tex

@@ -121,7 +121,7 @@
 
 		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
+		launch energy required to effectively use 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

LaTeX/trajectory_design.tex

@@ -366,7 +366,7 @@
 
 			As an example, we may wish to determine the velocity relative to the planet that the
 			spacecraft has at the periapsis of its hyperbolic trajectory during the flyby. This
-			could be useful, perhaps, for sizing the $\Delta V<$ required during the insertion stage
+			could be useful, perhaps, for sizing the $\Delta V$ required during the insertion stage
 			of the mission if the spacecraft is intended to be captured into an elliptical orbit
 			around its target planet. For a given incoming hyperbolic $\vec{v}_\infty$, we can first
 			determine the specific mechanical energy of the hyperbola at infinite distance by using
@@ -398,20 +398,19 @@
 
 			This algorithm will assume that the initial trajectory at the beginning of the mission
 			will be some hyperbolic orbit with velocity enough to leave the Earth. That initial
-			$v_\infty$ will be used as a tunable parameter in the NLP solver. This allows the
-			mission designer to include the launch $C_3$ in the cost function and, hopefully,
+			$v_\infty$ will be used as a tunable parameter in the optimization routine. This allows
+			the mission designer to include the launch $C_3$ in the cost function and, hopefully,
 			determine the mission trajectory that includes the least initial launch energy. This can
 			then be fed back into a mass-$C_3$ curve for prospective launch providers to determine
 			what the maximum mass any launch provider is capable of imparting that specific $C_3$
 			to.
 
 			A similar approach is taken at the end of the mission. This algorithm doesn't attempt to
-			exactly match the velocity of the planet at the end of the mission. Instead, the excess
-			hyperbolic velocity is also treated as a parameter that can be minimized by the cost
-			function. If a mission is to then end in insertion, a portion of the mass budget can
-			then be used for an impulsive thrust engine, which can provide a final insertion burn at
-			the end of the mission. This approach also allows flexibility for missions that might
-			end in a flyby rather than insertion.
+			exactly match the velocity of the planet. Instead, the excess hyperbolic velocity is
+			also treated as a parameter that can be minimized by the cost function. If a mission is
+			to then end in insertion, a portion of the mass budget can then be used for an impulsive
+			thrust engine, which can provide a final insertion burn. This approach also allows
+			flexibility for missions that might end in a flyby rather than insertion.
 
 		\subsection{Gravity Assist Maneuvers}
 
@@ -472,7 +471,7 @@
 			flyby, however, can provide a useful check on what turning angles are possible for a
 			given flyby, since the periapsis:
 
-			\begin{equation}
+			\begin{equation}\label{periapsis_eq}
 				r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right]
 			\end{equation}
 
@@ -504,9 +503,9 @@
 				here for its robustness given any initial guess \cite{battin1984elegant}.
 
 				Firstly, some geometric considerations must be accounted for. For any initial
-				position, $\vec{r}_0$, and final position, $\vec{r}_f$, and time of flight $\Delta
+				position, $\vec{r}_1$, and final position, $\vec{r}_2$, and time of flight $\Delta
 				t$, there are actually two separate transfer orbits that can connect the two points
-				with paths that traverse less than one full orbit. For each of these, there are
+				with paths that traverse less than one full orbit. Therefore, there are
 				actually then two trajectories that can connect the points
 				\cite{vallado2001fundamentals}. The first of the two will have a $\Delta \theta$ of
 				less than 180 degrees, which we classify as a Type I trajectory, and the second will
@@ -559,7 +558,7 @@
 
 				\begin{equation}
 					c_3 = \begin{cases}
-						\frac{\sqrt{\psi} - \sin sqrt{\psi}}{\psi^{3/2}} \quad &\text{if} \, \psi > 10^{-6} \\
+						\frac{\sqrt{\psi} - \sin \sqrt{\psi}}{\psi^{3/2}} \quad &\text{if} \, \psi > 10^{-6} \\
 						\frac{\sinh\sqrt{-\psi} - \sqrt{-\psi}}{(-\psi)^{3/2}} \quad &\text{if} \, \psi < -10^{-6} \\
 						1/6 \quad &\text{if} \, 10^{-6} > \psi > -10^{-6}
 					\end{cases}
@@ -578,7 +577,7 @@
 				flight:
 
 				\begin{equation}
-					\chi = sqrt{\frac{y}{c_2}}
+					\chi = \sqrt{\frac{y}{c_2}}
 				\end{equation}
 
 				\begin{equation}
@@ -691,7 +690,7 @@
 			$v_{eq}$, such that the thrust equation becomes:
 
 			\begin{align}
-				v_{eq} &= v_e - \frac{\Delta p A_e}{\dot{m}} \\
+				v_{eq} &= v_e + \frac{\Delta p A_e}{\dot{m}} \\
 				F &= \dot{m} v_{eq} \label{isp_1}
 			\end{align}
 
@@ -717,7 +716,7 @@
 
 		\subsection{Sims-Flanagan Transcription}
 
-			this thesis chose to use a model well suited for modeling low-thrust paths: the
+			This thesis chose to use a model well suited for modeling low-thrust paths: the
 			Sims-Flanagan transcription (SFT)\cite{sims1999preliminary}. The SFT allows for
 			flexibility in the trade-off between fidelity and performance, which makes it very
 			useful for this sort of preliminary analysis.
@@ -829,10 +828,3 @@
 				\caption{Graphic of an orbit-raising with a high efficiency cutoff}
 				\label{high_efficiency_fig}
 			\end{figure}
-
-			All of this is, of course, also true for impulsive trajectories. However, since the
-			thrust presence for those trajectories are generally taken to be impulse functions, the
-			control laws can afford to be much less complicated for a given mission goal, by simply
-			thrusting only at the moment on the orbit when the transition will be most efficient.
-			For a low-thrust mission, however, the control law must be continuous rather than
-			discrete and therefore the control law inherently gains a lot of complexity.

LaTeX/trajectory_optimization.tex

@@ -20,7 +20,7 @@
 
 			The other category is the direct methods. In a direct optimization problem, the cost
 			function itself provides a value that an iterative numerical optimizer can measure
-			itself against. The optimal solution is then found by varying the inputs $x$ until the
+			itself against. The optimal solution is then found by varying the inputs $\vec{x}$ until the
 			cost function is reduced to a minimum value, often determined by its derivative
 			jacobian. A number of tools have been developed to optimize NLPs via this direct method
 			in the general case.
@@ -48,7 +48,7 @@
 				University. It uses a sparse sequential quadratic programming algorithm as its
 				back-end optimization scheme.
 
-				Another common NLP optimization packages (and the one used in this implementation)
+				Another common NLP optimization package (and the one used in this implementation)
 				is the Interior Point Optimizer or IPOPT\cite{wachter2006implementation}. It  uses
 				an Interior Point Linesearch Filter Method and was developed as an open-source
 				project by the organization COIN-OR under the Eclipse Public License.
@@ -74,11 +74,6 @@
 				step the initial guess, now labeled $x_{k+1}$ after the addition of the ``step''
 				vector and iterates this process until predefined termination conditions are met.
 
-				In this case, the IPOPT algorithm was used, not as an optimizer, but as a solver. For
-				reasons that will be explained in the algorithm description in Section~\ref{algorithm} it
-				was sufficient merely that the non-linear constraints were met, therefore optimization (in
-				the particular step in which IPOPT was used) was unnecessary.
-
 			\subsubsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem}
 
 				One straightforward approach to trajectory corrections is a single shooting
@@ -154,8 +149,7 @@
 				very well to low-thrust arcs and, in fact, Sims-Flanagan Transcribed low-thrust arcs
 				in particular, because there actually are control thrusts to be optimized at a
 				variety of different points along the orbit. This is, however, not an exhaustive
-				description of ways that multiple shooting can be used to optimize a trajectory,
-				simply the most convenient for low-thrust arcs.
+				description of ways that multiple shooting can be used to optimize a trajectory.
 
 	\section{Monotonic Basin Hopping Algorithms}