\chapter{Introduction}

	Continuous low-thrust engines utilizing technologies such as Ion propulsion, Hall thrusters, and
	others enable 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 propulsion 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 is 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 transfer
	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}, among several others\cite{morante}.

	This thesis focuses on optimization of low-thrust enabled trajectories that use gravity assists.
	The approach uses 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) is then 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 is then used to design low thrust trajectories
	with gravity assits from the Earth to Saturn.

	This thesis is organized as follows: 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.