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LaTeX/thesis.tex
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\degree{Master of Science}{M.S., Aerospace Engineering}
\dept{Department of}{Aerospace Engineering}
\advisor{Prof.}{Natasha Bosanac}
\reader{TBD: Kathryn Davis}
\readerThree{TBD: Daniel Scheeres}
\reader{Kathryn Davis}
\readerThree{Daniel Scheeres}
\abstract{ \OnePageChapter
There are a variety of approaches to finding and optimizing low-thrust trajectories in
@@ -573,7 +573,7 @@
Englander in his Hybrid Optimal Control Problem paper, but does not allow for missions
with multiple targets, simplifying the optimization.
\section{Inner Loop Implementation}
\section{Inner Loop Implementation}\label{inner_loop_section}
The optimization routine can be reasonable separated into two separate ``loops'' wherein
the first loop is used, given an initial guess, to find valid trajectories within the
@@ -761,13 +761,117 @@
in order to achieve maximum precision.
\section{Outer Loop Implementation}
Overview the outer loop. This may require a final flowchart, but might potentially be too
simple to lend itself to one.
\subsection{Inner Loop Calling Function}
The primary reason for including this section is to discuss the error handling.
Now we have the tools in place for, given a potential ''mission guess`` in the
vicinity of a valid guess, attempting to find a valid and optimal solution in that
vicinity. Now what remains is to develop a routine for efficiently generating these
random mission guesses in such a way that thoroughly searches the entirety of the
solution space with enough granularity that all spaces are considered by the inner loop
solver.
\subsection{Monotonic Basin Hopping}
Once that has been accomplished, all that remains is an ''outer loop`` that can generate
new guesses or perturb existing valid missions as needed in order to drill down into a
specific local minimum. In this thesis, that is accomplished through the use of a
Monotonic Basin Hopping algorithm. This will be described more thoroughly in
Section~\ref{mbh_subsection}, but Figure~\ref{mbh_flow} outlines the process steps of
the algorithm.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{flowcharts/mbh}
\caption{A flowchart visualizing the steps in the monotonic basin hopping
algorithm}
\label{mbh_flow}
\end{figure}
\subsection{Random Mission Generation}
At a basic level, the algorithm needs to produce a mission guess (represented by all
of the values described in Section~\ref{inner_loop_section}) that contains random
values within reasonable bounds in the space. This leaves a number of variables open
to for implementation. For instance, it remains to be determined which distribution
function to use for the random values over each of those variables, which bounds to
use, as well as the possibilities for any improvements to a purely random search.
Currently, the first value set for the mission guess is that of $n$, which is the
number of sub-trajectories that each arc will be broken into for the Sims-Flanagan
based propagator. For this implementation, that was chosen to be 20, based upon a
number of tests in which the calculation time for the propagation was compared
against the accuracy of a much higher $n$ value for some known thrust controls, such
as a simple spiral orbit trajectory. This value of 20 tends to perform well and
provide reasonable accuracy, without producing too many variables for the NLP
optimizer to control for (since the impulsive thrust at the center of each of the
sub-trajectories is a control variable). This leaves some room for future
improvements, as will be discussed in Section~\ref{improvement_section}.
The bounds for the launch date are provided by the user in the form of a launch
window, so the initial launch date is just chosen as a standard random value from a
uniform distribution within those bounds.
A unit launch direction is then also chosen as a 3-length vector of uniform random
numbers, then normalized. This unit vector is then multiplied by a uniform random
number between 0 and the square root of the maximum launch $C_3$ specified by the
user to generate an initial $\vec{v_\infty}$ vector at launch.
Next, the times of flight of each phase of the mission is then decided. Since launch
date has already been selected, the maximum time of flight can be calculated by
subtracting the launch date from the latest arrival date provided by the mission
designer. Then, each leg is chosen from a uniform distribution with bounds currently
set to a minimum flight time of 30 days and a maximum of 70\% of the maximum time of
flight. These leg flight times are then iteratively re-generated until the total
time of flight (represented by the sum of the leg flight times) is less than the
maximum time of flight. This allows for a lot of flexibility in the leg flight
times, but does tend toward more often producing longer missions, particularly for
missions with more flybys.
Then, the internal components for each phase are generated. It is at this step, that
the mission guess generator splits the outputs into two separate outputs. The first
is meant to be truly random, as is generally used as input for a monotonic basin
hopping algorithm. The second utilizes a Lambert's solver to determine the
appropriate hyperbolic velocities (both in and out) at each flyby to generate a
natural trajectory arc. For this Lambert's case, the mission guess is simply seeded
with zero thrust controls and outputted to the monotonic basin hopper. The intention
here is that if the time of flights are randomly chosen so as to produce a
trajectory that is possible with a control in the vicinity of a natural trajectory,
we want to be sure to find that trajectory. More detail on how this is handled is
available in Section~\ref{mbh_subsection}.
However, for the truly random mission guess, there are still the $v_\infty$ values
and the initial thrust guesses to generate. For each of the phases, the incoming
excess hyperbolic velocity is calculated in much the same way that the launch
velocity was calculated. However, instead of multiplying the randomly generate unit
direction vector by a random number between 0 and the square root of the maximum
launch $C_3$, bounds of 0 and 10 kilometers per second are used instead, to provide
realistic flyby values.
The outgoing excess hyperbolic velocity at infinity is then calculated by again
choosing a uniform random unit direction vector, then by multiplying this value by
the magnitude of the incoming $v_{\infty}$ since this is a constraint of a
non-powered flyby.
From these two velocity vectors the turning angle, and thus the periapsis of the
flyby, can then be calculated by the following equations:
\begin{align}
\delta &= \arccos \left( \frac{\vec{v}_{\infty,in} \cdot
\vec{v}_{\infty,out}}{|v_{\infty,in}| \cdot {|v_{\infty,out}}|} \right) \\
r_p &= \frac{\mu}{\vec{v}_{\infty,in} \cdot \vec{v}_{\infty,out}} \cdot \left(
\frac{1}{\sin(\delta/2)} - 1 \right)
\end{align}
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 percentage of the full thrust producible by the spacecraft thrusters in that
direction, the initial thrust controls can then be generated as a $20 \times 3$
matrix of uniform random numbers within that bound.
\subsection{Monotonic Basin Hopping}\label{mbh_subsection}
Outline the MBH algorithm, going into detail at each step. Mention the long-tailed PDF being
used and go into quite a bit of detail. Englander's paper on the MBH algorithm specifically
should be a good guide. Mention validation.
@@ -798,7 +902,7 @@
Talk a bit about why this work is valuable. Missions that could have benefited, missions that
this enables, etc.
\section{Recommendations for Future Work}
\section{Recommendations for Future Work}\label{improvement_section}
Recommend future work, obviously. There are a \emph{ton} of opportunities for improvement
including parallelization, cluster computing, etc.