From 26de9aa53dd346ee8ba5fb1c3078180275d321fe Mon Sep 17 00:00:00 2001 From: Connor Date: Sun, 2 Jan 2022 15:54:25 -0700 Subject: [PATCH] Began section 2 --- LaTeX/thesis.tex | 144 +++++++++++++++++++++++++++++++++++------------ 1 file changed, 108 insertions(+), 36 deletions(-) diff --git a/LaTeX/thesis.tex b/LaTeX/thesis.tex index 661f0d9..a9893fe 100644 --- a/LaTeX/thesis.tex +++ b/LaTeX/thesis.tex @@ -7,10 +7,14 @@ \usepackage{changepage} \usepackage{fontspec} \usepackage{titlesec} +\usepackage{unicode-math} % \setmainfont{Adamina} % \setmainfont{Alegreya} -\setmainfont[Scale=1.2]{Average} +\setmainfont[Scale=1.1]{Average} +\setmathfont[Scale=1.1]{Fira Math} + +\newcommand{\sectionbreak}{\clearpage} \titleformat{\section} {\bfseries\fontspec{Roboto}\LARGE} @@ -32,7 +36,7 @@ Monotonic Basin Hopping} \geometry{left=1in, right=1in, top=1in, bottom=1in} -\doublespacing +\setstretch{2.5} \begin{document} @@ -131,27 +135,95 @@ Monotonic Basin Hopping} \section{Introduction} - Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Halls thrusters, and - others can be a powerful tool in the design of space missions. They tend to be particularly - suited to missions which require very high total $\Delta V$ values and take place over a - particularly long duration. As such, they are well-suited to interplanetary missions. For - instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep Space 1 - missions. + Continuous low-thrust arcs utilizing technologies such as Ion propulsion, Hall thrusters, and + others can be a powerful tool in the design of interplanetary space missions. They tend to be + particularly suited to missions which require very high total change in velocity or $\Delta V$ + values and take place over a particularly long duration. Traditional impulsive thrusting + techniques can achieve these changes in velocity, but they typically have a far lower specific + impulse and, as such, are much less efficient and use more fuel, 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 fuels are well-suited to interplanetary + missions. - Provide some historical background, motivations, and discussion of the basic problems being - investigated. Also a brief overview how the thesis will be laid out. + For instance, low thrust ion propulsion was used on the Bepi-Colombo, Dawn, and Deep + Space 1 missions. In general, anytime an interplanetary trajectory is posed, it is advisable to + first explore the possibility of low-thrust technologies. In an interplanetary mission, the + primary downside to low-thrust orbits (that they require significant time to achieve large + $\Delta V$ changes) is made irrelevant by the fact that interplanetary trajectories take such a + long time as a matter of course. - \section{Trajectory Optimization} - This section will outline the foundational problem of trajectory optimization. + Another technique often leveraged by interplanetary trajectory designers is the gravity assist. + Gravity assists cleverly 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. + This technique lends itself very well to impulsive trajectories. 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, + optimization with impulsive trajectories and gravity assists are common. + + However, there is no physical reason why low-thrust trajectories can't also incorporate gravity + assists. The optimization problem becomes much more complicated. The separate problems of + optimizing flyby parameters (planet, flyby date, etc.) and optimizing the low-thrust control + arcs don't combine very easily. In this paper, a technique is explored by setting the + dual-problem up as a Hybrid Optimal Control Problem (HOCP). + + This thesis will explore these concepts in a number of different sections. Section + \ref{traj_opt} will explore the basic principles of trajectory optimization in a manner agnostic + to the differences between continuous low-thrust and impulsive high-thrust techniques. Section + \ref{low_thrust} will then delve into the different aspects to consider when optimizing a low + thrust mission profile over an impulsive one. Section \ref{interplanetary} provides more detail + on the interplanetary considerations, including force models and gravity assists. Section + \ref{algorithm} will cover the implementation details of the HOCP optimization algorithm + developed for this paper. Finally, section \ref{results} will explore the results of some + hypothetical missions to Saturn. + + \section{Trajectory Optimization} \label{traj_opt} + + Trajectory optimization is concerned with a narrow problem (namely, optimizing a spaceflight + trajectory to an end state) with a wide range of possible techniques, approaches, and even + solutions. In this section, the foundations for direct optimization of these sorts of problems + will be explored by first introducing the Two-Body Problem, then an algorithm for directly + solving for states in that system, then exploring approaches to Non-Linear Problem (NLP) solving + in general and how they apply to spaceflight trajectories. \subsection{The Two-Body Problem} - Propose the two-body problem as a differential equation. + The motion of a spacecraft in space is governed by a large number of forces. When planning and + designing a spacecraft trajectory, we often want to use the most complete (and often complex) + model of these forces that is available. However, in the process of designing these + trajectories, we often have to compute the path of the spacecraft many hundreds, thousands, or + even millions of times. Utilizing very high-fidelity force models that account for aerodynamic + pressures, solar radiation pressures, multi-body effects, and many others may be infeasible + for the method being used if the computations take too long. + + Therefore, a common approach (and the one utilized in this implementation) is to first look + simply at the single largest force governing the spacecraft in motion, the gravitational force + due to the primary body around which it is orbiting. This can provide an excellent + low-to-medium fidelity model that can be extremely useful in categorizing the optimization + space as quickly as possible. In many cases, including the algorithm used in this paper, it is + unlikely that local cost-function minima would be missed due to the lack of fidelity of the + Two Body Problem. + + In order to explore the Two Body Problem, we must first examine the full set of assumptions + associated with the force model. Firstly, we are only concerned with the nominative two + bodies: the spacecraft and the planetary body around which it is orbiting. Secondly, both of + these bodies are modeled as simple point masses. This removes the need to account for + non-uniform densities and asymmetry. The third assumption is that the mass of the spacecraft + ($m_2$) is much much smaller than the mass of the planetary body ($m_1$) and enough so as to be + considered negligible. The only force acting on this system is then the force of gravity that + the primary body enacts upon the secondary. Lastly, we'll assume a fixed inertial frame. This + isn't necessary for the formulation of a solution, but will simplify the derivation. + + Reducing the system to two point masses with a single gravitational force acting between them + (and only in one direction) we can model the force on the secondary body as: + + \begin{equation} + \ddot{\vec{r}} = - \frac{G \left( m_1 + m_2 \right)}{r^2} \frac{\vec{r}}{\left| r \right|} + \end{equation} \subsubsection{Kepler's Equations} Detail Kepler's equations for astrodynamics. \subsection{Analytical Solutions to Kepler's Equations} - Discuss how, since the 2BP is analytically solveable, there exists algorithms for solving + Discuss how, since the 2BP is analytically solvable, there exists algorithms for solving these equations. \subsubsection{LaGuerre-Conway Algorithm} @@ -170,7 +242,7 @@ Monotonic Basin Hopping} I may take this section out, because I'm not currently using a linesearch. But I would cover the additions of linesearch methods. - \section{Low-Thrust Considerations} + \section{Low-Thrust Considerations} \label{low_thrust} Highlight the differences between high and low-thrust mission profiles. \subsection{Low Thrust Overview} @@ -181,7 +253,7 @@ Monotonic Basin Hopping} 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. - \section{Interplanetary Trajectory Considerations} + \section{Interplanetary Trajectory Considerations} \label{interplanetary} Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an interplanetary journey. @@ -200,31 +272,31 @@ Monotonic Basin Hopping} \subsection{Ephemeris Considerations} I can quickly mention SPICE here and talk a bit about validation. - \section{Genetic Algorithms} - I will probably give only a brief overview of genetic algorithms here. I don't personally know - that much about them. Then in the following subsections I can discuss the parts that are - relevant to the specific algorithm that I'm using. + % \section{Genetic Algorithms} + % I will probably give only a brief overview of genetic algorithms here. I don't personally know + % that much about them. Then in the following subsections I can discuss the parts that are + % relevant to the specific algorithm that I'm using. - \subsection{Decision Vectors} - Discuss what a decision vector is in the context of an optimization problem. + % \subsection{Decision Vectors} + % Discuss what a decision vector is in the context of an optimization problem. - \subsection{Selection and Fitness Evaluation} - Discuss the costing being used as well as the different types of fitness evaluation that are - common. Also discuss the concept of generations and ``survival''. + % \subsection{Selection and Fitness Evaluation} + % Discuss the costing being used as well as the different types of fitness evaluation that are + % common. Also discuss the concept of generations and ``survival''. - \subsubsection{Tournament Selection} - Dive deeper into the specific selection algorithm being used here. + % \subsubsection{Tournament Selection} + % Dive deeper into the specific selection algorithm being used here. - \subsection{Crossover} - Discuss the concept of crossover and procreation in a genetic algorithm. + % \subsection{Crossover} + % Discuss the concept of crossover and procreation in a genetic algorithm. - \subsubsection{Binary Crossover} - Discuss specific crossover algorithm used here. + % \subsubsection{Binary Crossover} + % Discuss specific crossover algorithm used here. - \subsubsection{Mutation} - Discuss both the necessity for mutation and the mutation algorithm being used. + % \subsubsection{Mutation} + % Discuss both the necessity for mutation and the mutation algorithm being used. - \section{Algorithm Overview} + \section{Algorithm Overview} \label{algorithm} Highlight the algorithm at a high-level. This is likely where flowcharts and diagrams will go to give a high-level overview. @@ -268,7 +340,7 @@ Monotonic Basin Hopping} will have already been discussed in the background sections above. But I can step through each of the decisions, similar to Englander's paper on this. - \section{Results Analysis} + \section{Results Analysis} \label{results} Simply highlight that the algorithm was tested on a sample trajectory to Saturn. \subsection{Sample Trajectory to Saturn} @@ -286,7 +358,7 @@ Monotonic Basin Hopping} similar impulsive trajectories. Honestly, this is a lot of work for very little gain, though, so probably the first place to chop if needed. - \section{Conclusion} + \section{Conclusion} \label{conclusion} \subsection{Overview of Results} Quick re-wording of the previous section in a paragraph or two for reader's convenience.