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+\chapter{Introduction}
+
+	Continuous low-thrust engines utilizing technologies such as Ion propulsion, Hall thrusters, and
+	others can be a powerful system in the enabling of long-range interplanetary missions with fuel
+	efficiencies unrivaled by those that employ only impulsive thrust systems. The challenge in
+	utilizing these systems, then, is the design of trajectories that effectively utilize this
+	technology. Continuous thrust propulsive systems tend to be particularly suited to missions
+	which require very high total change in velocity ($\Delta V$) values and take place over a
+	particularly long duration. Traditional impulsive thrusting techniques can achieve these changes
+	in velocity, but typically have a far lower specific impulse and, as such, are much less fuel
+	efficient, costing the mission valuable financial resources that could instead be used for
+	science. Because of their inherently high specific impulse (and thus efficiency), low-thrust
+	propagation systems are well-suited to interplanetary missions.
+
+	The first attempt by NASA to use an electric ion-thruster for an interplanetary mission was the
+	Deep Space 1 mission\cite{brophy2002}. This mission was designed to test the ``new'' technology,
+	first appearing as a concept in science fiction stories of the early 1900's and first tested
+	successfully during NASA's Space Electric Rocket Test (SERT) mission of
+	1964\cite{cybulski1965results}, on an interplanetary mission for the first time. The Ion
+	thruster used on Deep Space 1 allowed the mission to rendezvous with both an asteroid (9969
+	Braille) and later with a comet (Borrelly), when the technologies being tested, such as the ion
+	thruster, proved robust enough and efficient enough to allow for two mission extensions.
+
+	After this initial successful test, ion thrusters and other forms of low-thrust electric
+	propulsion have been used in a variety of missions. The NASA Dawn \cite{rayman2006dawn}
+	spacecraft in 2015 became the first spacecraft to successfully orbit two planetary bodies,
+	thanks in large part to the efficiency of its ion propulsion system. Also notable is the
+	joint ESA and JAXA spacecraft Bepi-Colombo\cite{benkhoff2010bepicolombo}, which was launched
+	in October 2018 and is projected to perform a flyby of Earth, two of Venus, and six of
+	Mercury before inserting into an orbit around that planet.
+
+	A common theme in mission design is that there always exists a trade-off between efficiency
+	(particularly in terms of fuel use) and the time required to achieve the mission objective. Low
+	thrust systems in particular tend to produce mission profiles that sacrifice the rate of
+	convergence on the target state in order to achieve large increases in fuel efficiency. Often a
+	low-thrust mission profile in Earth orbit will require multiple orbital periods to achieve the
+	desired change in spacecraft state. Interplanetary missions, though, provide a particularly
+	useful case for continuous thrust technology. The trajectory arcs in interplanetary space are
+	generally much, much longer than orbital missions around the Earth. Because of this increase,
+	even a small continuous thrust is capable of producing large $\Delta V$ values over the course
+	of a single trajectory arc.
+
+	Another technique often leveraged by interplanetary trajectory designers is the gravity assist.
+	Gravity assists utilize the inertia of a large planetary body to ``slingshot'' a spacecraft,
+	modifying the direction of its velocity with respect to the central body, the Sun. The gravity
+	assist maneuver itself can be modeled very effectively by an impulsive maneuver with certain
+	constraints, placed right at the moment of closest approach to the (flyby) target body. Because
+	of this, missions that combine largely natural trajectories, with impulsive maneuvers and
+	planetary flybys at strategic locations to optimize fuel use in achieving orbital velocity
+	changes are quite common.
+
+	However, the complexity of optimizing for fuel usage, time of flight, and other useful mission
+	parameters increases greatly when low-thrust propulsion and gravity assists are combined. The
+	separate problems of optimizing flyby parameters (planet, flyby date, etc.) and optimizing the
+	low-thrust control arcs don't combine very easily. This concept has been explored heavily by Dr.
+	Jacob Englander \cite{englander2014tuning}, \cite{englander2017automated},
+	\cite{englander2012automated} recently in an effort to develop a generalized and automated
+	routine for producing unconstrained, globally optimal trajectories for realistic interplanetary
+	mission development that utilizes both planetary flybys and efficient low-thrust electric
+	propulsion techniques. Similar studies have also been performed by a number of researchers
+	including a team from JPL\cite{sims2006} as well as a Spanish team\cite{morante}, among several
+	others.
+
+	This thesis will attempt to develop an algorithm for the optimization of low-thrust enabled
+	trajectories for initial feasibility analysis in mission design. The algorithm will utilize
+	a non-linear programming solver to directly optimize a set of control thrusts for the
+	user-provided flyby planets, for any provided cost function. A monotonic basin hopping algorithm
+	(MBH) will then be employed to traverse the search space in an effort to find additional local
+	optima. This approach differs from the work produced earlier by Englander and the other teams,
+	but is largely meant to explore the feasibility of such techniques and propose a few
+	enhancements. The approach defined in this thesis will then be used to investigate an example
+	mission to Saturn.
+
+	This thesis will explore these concepts in a number of different sections. Section
+	\ref{traj_dyn} will explore the basic dynamical principles of trajectory design, beginning the
+	with fundamental system dynamics, then exploring interplanetary system dynamics and gravity
+	flybys, and finally the dynamics that are specific to low-thrust enabled trajectories. Section
+	\ref{traj_optimization} will then discuss process of optimizing spacecraft trajectories in
+	general and the tool available for that. Section \ref{algorithm} will cover the implementation
+	details of the optimization algorithm developed for this paper. Finally, section \ref{results}
+	will explore the results of some hypothetical missions to Saturn.