Section 3

LaTeX/thesis.tex

@@ -332,7 +332,7 @@ Monotonic Basin Hopping}
 				($E$) which can be related to spacecraft position, and time, but we still need a useful
 				algorithm for solving this equation.
 
-      \subsubsection{LaGuerre-Conway Algorithm}
+			\subsubsection{LaGuerre-Conway Algorithm}\label{laguerre}
 				For this application, I used an algorithm known as the LaGuerre-Conway algorithm, which was
 				presented in 1986 as a faster algorithm for directly solving Kepler's equation and has been
 				in use in many applications since. This algorithm is known for its convergence robustness
@@ -357,9 +357,22 @@ Monotonic Basin Hopping}
 
 			Therefore an approach is needed, in trajectory optimization and many other fields, to optimize
 			highly non-linear, unpredictable systems such as this. The field that developed to approach
-      this problem is known as Non-Linear Problem (NLP) Optimization. In these cases, generally
+			this problem is known as Non-Linear Problem (NLP) Optimization.
-      speaking, direct methods are impossible, so a number of tools have been developed to optimize
+
-      NLPs in the general case, via indirect, iterative methods.
+			There are, however, two categories of approaches to solving an NLP. The first category,
+			indirect methods, involve declaring a set of necessary and/or sufficient conditions for declaring
+			the problem optimal. These conditions then allow the non-linear problem (generally) to be
+			reformulated as a two point boundary value problem. Solving this boundary value problem can
+			provide a control law for the optimal path. Indirect approaches for spacecraft trajectory
+			optimization have given us the Primer Vector Theory.
+
+			The other category is the direct methods. In a direct optimization problem, the cost function
+			itself is calculated to provide the optimal solution. The problem is usually thought of as a
+			collection of dynamics and controls. Then these controls can be modified to minimize the cost
+			function. A number of tools have been developed to optimize NLPs via this direct method in the
+			general case. For this particular problem, direct approaches were used as the low-thrust
+			system dynamics adds too much complexity to quickly optimize indirectly and the individual
+			optimization routines needed to proceed as quickly as possible.
 
 			\subsubsection{Non-Linear Solvers}
 				For these types of non-linear, constrained problems, a number of tools have been developed
@@ -401,15 +414,60 @@ Monotonic Basin Hopping}
 				the particular step in which IPOPT was used) was unnecessary.
 
 	\section{Low-Thrust Considerations} \label{low_thrust}
-    Highlight the differences between high and low-thrust mission profiles.
 
-    \subsection{Low Thrust Overview}
+		Thus far, the techniques that have been discussed can be equally useful for both impulsive and
-      Dive deeper into the concept of low thrust trajectories, highlighting the added trouble with
+		continuous thrust mission profiles. In this section, we'll discuss the intricacies of continuous
-      propagation/integration.
+		low-thrust trajectories in particular. There are many methods for optimizing such profiles and
+		we'll briefly discuss the difference between a direct and indirect optimization of a low-thrust
+		trajectory as well as introduce the concept of a control law and the notation used in this
+		thesis for modelling low-thrust trajectories more simply.
+
+		\subsection{Low-Thrust Control Laws}
+
+			In determining a low-thrust arc, a number of variables must be accounted for and, ideally,
+			optimized.
+
+			Firstly, we must determine the presence or absence of thrust. Often, this is a	question of
+			preference in the arsenal of the mission designer. Generally speaking, there are points along
+			an orbit at which thrusting in order to achieve the final orbit are more or less efficient.
+			For instance, in a classic orbit raising, if increasing the semi-major axis is the only goal,
+			then thrusting nearer to the periapsis is far more efficient than thrusting near the apoapsis.
+			For this reason, a mission designer may choose to reduce the thrust or turn it off altogether
+			during certain segments of the trajectory.
+
+			Secondly, the direction of thrust must also be determined. The methods for determining this
+			direction varies greatly depending on the particular control law chosen for that mission.
+			Generally speaking, a control law determines these two parameters: thrust presence and thrust
+			direction, at each point along the arc.
+
+			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.
 
 		\subsection{Sims-Flanagan Transcription}
-      Reveal the advantages of Sims-Flanagan transcription as an alternative to higher-fidelity
+
-      propagation models. Be sure to mention its uses in many legitimate places.
+			The major problem with optimizing low thrust paths is that the control law must necessarily be
+			continuous. Also, since indirect optimization approaches are quite difficult, the problem must
+			necessarily be reformulated as a discrete one in order to apply a direct approach. Therefore,
+			this thesis chose to use a model well suited for discretizing low-thrust paths: the
+			Sims-Flanagan transcription (SFT).
+
+			The SFT is actually quite a simple method for discretizing low-thrust arcs. First the
+			continuous arc is subdivided into a number ($N$) of individual consistent timesteps of length
+			$\frac{tof}{N}$. The control thrust is then applied at the center of each of these time steps.
+
+			Using the SFT, it is relatively straightforward to propagate a state (in the context of the
+			Two-Body Problem) that utilizes a continuous low-thrust control, without the need for
+			computationally expensive numeric integration algorithms, by simply solving Kepler's equation
+			(using the LaGuerre-Conway algorithm introduced in Section~\ref{laguerre}) $N$ times. This
+			greatly reduces the computation complexity, which is particularly useful for cases in which
+			low-thrust trajectories need to be calculated many millions of times, as is the case in this
+			thesis. The fidelity of the model can also be easily fine-tuned. By simply increasing the
+			number of sub-arcs, one can rapidly approach a fidelity equal to a continuous low-thrust
+			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