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Getting pretty close on the final review

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@@ -81,6 +81,59 @@
\ddot{\vec{r}} = - \frac{\mu}{r^2} \hat{r}
\end{equation}
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as:
We may also wish to utilize the total orbital energy for a spacecraft within this model.
Since the spacecraft is acting only under the gravitational influence of the planet and
no other forces, we can define the total specific mechanical energy as
\cite{vallado2001fundamentals}:
\begin{equation} \label{energy}
\xi = \frac{v^2}{2} - \frac{\mu}{r}
\end{equation}
\noindent
Where the first term represents the kinetic energy of the spacecraft and the second term
represents the gravitational potential energy.
\subsection{Kepler's Laws}
Now that we've fully qualified the forces acting within the Two Body Problem, we can concern
@@ -282,13 +335,10 @@
\section{Interplanetary Considerations}\label{interplanetary}
The question of interplanetary travel opens up a host of additional new complexities. While
optimizations for simple single-body trajectories are far from simple, it can at least be
said that the assumptions of the Two Body Problem remain fairly valid. In interplanetary
travel, the primary body most responsible for gravitational forces might be a number of
different bodies, dependent on the phase of the mission. In fact, at some points along the
trajectory, there may not be a ``primary'' body, but instead a number of different forces of
roughly equal magnitude vying for ``primary'' status.
In interplanetary travel, the primary body most responsible for gravitational forces might
be a number of different bodies, dependent on the phase of the mission. In fact, at some
points along the trajectory, there may not be a ``primary'' body, but instead a number of
different forces of roughly equal magnitude vying for ``primary'' status.
In the ideal case, every relevant body would be considered as an ``n-body'' perturbation
during the entire trajectory. For some approaches, this method is sufficient and preferred.
@@ -296,7 +346,7 @@
can be applied in this case to simplify the model.
Interplanetary travel does not merely complicate trajectory optimization. The increased
complexity of the search space also opens up new opportunities for orbit strategies. The
complexity of the search space also opens up new opportunities for mission designers. The
primary strategy investigated by this thesis will be the gravity assist, a technique for
utilizing the gravitational energy of a planet to modify the direction of solar velocity.
@@ -306,45 +356,13 @@
search space, but some of these tools can also be leveraged by the automated optimization
algorithm.
\subsection{Launch Considerations}
Before considering the dynamics and techniques that interplanetary travel imposes upon
the trajectory optimization problem we must first concern ourself with getting to
interplanetary space. Generally speaking, interplanetary trajectories require a lot of
orbital energy and the simplest and quickest way to impart orbital energy to a satellite
is by using the entirety of the launch energy that a launch vehicle can provide.
In practice, this value, for a particular mission, is actually determined as a parameter
of the mission trajectory to be optimized. The excess velocity at infinity of the
hyperbolic orbit of the spacecraft that leaves the Earth can be used to derive the
launch energy. This is usually qualified as the quantity $C_3$, which is actually double
the kinetic orbital energy with respect to the Sun, or simply the square of the excess
hyperbolic velocity at infinity\cite{wie1998space}.
This algorithm and many others will take, essentially for granted, that the initial
orbit at the beginning of the mission will be some hyperbolic orbit with velocity enough
to leave the Earth. That initial $v_\infty$ will be used as a tunable parameter in the
NLP solver. This allows the mission designer to include the launch $C_3$ in the cost
function and, hopefully, determine the mission trajectory that includes the least
initial launch energy. This can then be fed back into a mass-$C_3$ curve for prospective
launch providers to determine what the maximum mass any launch provider is capable of
imparting that specific $C_3$ to.
A similar approach is taken at the end of the mission. This algorithm, and many others,
doesn't attempt to exactly match the velocity of the planet at the end of the mission.
Instead, the excess hyperbolic velocity is also treated as a parameter that can be
minimized by the cost function. If a mission is to then end in insertion, a portion of
the mass budget can then be used for an impulsive thrust engine, which can provide a
final insertion burn at the end of the mission. This approach also allows flexibility
for missions that might end in a flyby rather than insertion.
\subsection{Patched Conics}
The first hurdle to deal with in interplanetary space is the problem of reconciling
Two-Body dynamics with the presence of multiple and varying planetary bodies. The most
common method for approaching this is the method of patched
conics\cite{bate2020fundamentals}. In this model, we break the interplanetary trajectory
up into a series of smaller sub-trajectories. During each of these sub-trajectories, a
common method for approaching this is the method of patched conics
\cite{bate2020fundamentals}. In this model, we break the interplanetary trajectory up
into a series of smaller sub-trajectories. During each of these sub-trajectories, a
single primary is considered to be responsible for the trajectory of the orbit, via the
Two-Body problem.
@@ -356,8 +374,7 @@
Solar System, the spacecraft is either within the Sphere of Influence of a planetary
body or the Sun. However, there are points in the Solar System where the gravitational
influence of two planetary bodies are roughly equivalent to each other and to the
influence of the Sun. These are considered LaGrange points\cite{euler1767motu}, but are
beyond the scope of this initial analysis of interplanetary mission feasibility.
influence of the Sun.
\begin{figure}[H]
\centering
@@ -366,12 +383,74 @@
\label{patched_conics_fig}
\end{figure}
This effectively breaks the trajectory into a series of orbits defined by the Two-Body
problem (conics), patched together by distinct transition points. These transition
points occur along the spheres of influence of the planets nearest to the spacecraft.
Generally speaking, for the orbits handled by this algorithm, the speeds involved are
enough that the orbits are always elliptical around the Sun and hyperbolic around the
planets.
This effectively breaks the trajectory into a series of arcs each governed by a distinct
Two-Body problem patched together by distinct transition points. These transition points
occur along the spheres of influence of the planets nearest to the spacecraft.
Therefore, we must understand how to convert our spacecraft's state from the Sun frame
to the planetary frame as it crosses this boundary. An elliptical orbit about the sun
will have enough orbital energy to represent a hyperbolic orbit around the planet. So we
first need to determine the velocity of the spacecraft relative to the planet as it
crosses the SOI, which we can determine by subtraction \cite{vallado2001fundamentals}:
\begin{equation}
\vec{v}_{sc/p} = \vec{v}_{sc/sun} - \vec{v}_{planet/sun}
\end{equation}
Since the orbit around the planet is hyperbolic, in order to characterize the hyperbola
we must determine the velocity of the spacecraft when it has infinite distance relative
to the planet. Since this never occurs, a further approximation is made that the
velocity that the spacecraft has (relative to the planet) as it crosses the SOI can be
modeled as the $\vec{v}_\infty$ of that hyperbolic arc.
As an example, we may wish to determine the velocity relative to the planet that the
spacecraft has at the periapsis of its hyperbolic trajectory during the flyby. This
could be useful, perhaps, for sizing the $\Delta V<$ required during the insertion stage
of the mission if the spacecraft is intended to be captured into an elliptical orbit
around its target planet. For a given incoming hyperbolic $\vec{v}_\infty$, we can first
determine the specific mechanical energy of the hyperbola at infinite distance by using
Equation~\ref{energy}:
\begin{equation}
\xi = \frac{v^2}{2} - \frac{\mu}{r} = \frac{v_\infty^2}{2}
\end{equation}
We can then leverage the conservation of energy to determine the velocity at a
particular point, $r_{ins}$:
\begin{align}
\xi_{ins} &= \frac{v_{ins}^2}{2} - \frac{\mu}{r_{ins}} \\
\xi_{ins} &= \xi_\infty = \frac{v_\infty^2}{2} \\
v_{ins} &= \sqrt{\frac{2\mu}{r_{ins}} + v_\infty^2}
\end{align}
\subsection{Launch Considerations}
Generally speaking, an interplanetary mission begins with launch. For a satellite of
given size, a certain amount of orbital energy can be imparted to the satellite by the
launch vehicle. In practice, this value, for a particular mission, is actually
determined as a parameter of the mission trajectory to be optimized. The excess velocity
at infinity of the hyperbolic orbit of the spacecraft that leaves the Earth can be used
to derive the launch energy. This is usually qualified as the quantity $C_3$, which is
actually double the kinetic orbital energy with respect to the Sun, or simply the square
of the excess hyperbolic velocity at infinity\cite{wie1998space}.
This algorithm will assume that the initial trajectory at the beginning of the mission
will be some hyperbolic orbit with velocity enough to leave the Earth. That initial
$v_\infty$ will be used as a tunable parameter in the NLP solver. This allows the
mission designer to include the launch $C_3$ in the cost function and, hopefully,
determine the mission trajectory that includes the least initial launch energy. This can
then be fed back into a mass-$C_3$ curve for prospective launch providers to determine
what the maximum mass any launch provider is capable of imparting that specific $C_3$
to.
A similar approach is taken at the end of the mission. This algorithm doesn't attempt to
exactly match the velocity of the planet at the end of the mission. Instead, the excess
hyperbolic velocity is also treated as a parameter that can be minimized by the cost
function. If a mission is to then end in insertion, a portion of the mass budget can
then be used for an impulsive thrust engine, which can provide a final insertion burn at
the end of the mission. This approach also allows flexibility for missions that might
end in a flyby rather than insertion.
\subsection{Gravity Assist Maneuvers}
@@ -392,6 +471,20 @@
the spacecraft arrives at the planet from one direction and, because of the influence of
the planet, leaves in a different direction\cite{negri2020historical}.
This can be visualized in Figure~\ref{grav_assist_fig}, which shows the bend in the
spacecraft's velocity due to the hyperbolic arc as it passes the planet. This turns the
direction of the spacecraft's velocity relative to the planet, which has an overall
effect on kinetic energy that can be seen by adding the two vectors to the velocity of
the planet relative to the sun. By passing in front of the planet or behind it (relative
to its velocity), energy can be removed or added to the spacecraft by the maneuver.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{fig/flyby}
\caption{Visualization of velocity changes during a gravity assist}
\label{grav_assist_fig}
\end{figure}
This effect can be used strategically. The ``bend'' due to the flyby is actually
tunable via the exact placement of the fly-by in the b-frame, or the frame centered at
the planet, from the perspective of the spacecraft at $v_\infty$. By modifying the
@@ -401,25 +494,22 @@
\subsection{Flyby Periapsis}
Now that we understand gravity assists, the natural question is then how to leverage
them for achieving certain velocity changes. This can be achieved via a technique called
``B-Plane Targeting''\cite{cho2017b}. But first, we must consider mathematically the
effect that a gravity flyby can have on the velocity of a spacecraft as it orbits the
Sun. Specifically, we can determine the turning angle of the bend mentioned in the
previous section, given an excess hyperbolic velocity entering the planet's sphere of
influence ($v_{\infty, in}$) and a target excess hyperbolic velocity as the spacecraft
leaves the sphere of influence ($v_{\infty, out}$):
them for achieving certain velocity changes\cite{cho2017b}. But first, we must consider
mathematically the effect that a gravity flyby can have on the velocity of a spacecraft
as it orbits the Sun. Specifically, we can determine the turning angle of the bend
mentioned in the previous section, given an excess hyperbolic velocity entering the
planet's sphere of influence ($\vec{v}_{\infty, in}$) and a target excess hyperbolic
velocity as the spacecraft leaves the sphere of influence ($\vec{v}_{\infty, out}$):
\begin{equation}
\delta = \arccos \left( \frac{v_{\infty,in} \cdot v_{\infty,out}}{|v_{\infty,in}|
|v_{\infty,out}|} \right)
\begin{equation}\label{turning_angle_eq}
\delta = \arccos \left( \frac{\vec{v}_{\infty,in} \cdot
\vec{v}_{\infty,out}}{|\vec{v}_{\infty,in}| |\vec{v}_{\infty,out}|} \right)
\end{equation}
From this turning angle, we can also determine, importantly, the periapsis of the flyby
that we must target in order to achieve the required turning angle. The actual location
of the flyby point can also be determined by B-Plane Targeting, but this technique was
not necessary in this implementation as a preliminary feasibility tool, and so is beyond
the scope of this thesis. The periapsis of the flyby, however, can provide a useful
check on what turning angles are possible for a given flyby, since the periapsis:
that we must target in order to achieve the required turning angle. The periapsis of the
flyby, however, can provide a useful check on what turning angles are possible for a
given flyby, since the periapsis:
\begin{equation}
r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right]
@@ -443,9 +533,121 @@
a time of flight between the two positions, what velocity was necessary to connect
the two states.
The actual numerical solution to this boundary value problem is not important to
include here, but there have been a large number of algorithms written to solve
Lambert's problem quickly and robustly for given inputs\cite{jordan1964application}.
There are many algorithms developed to solve Lambert's problem, but the universal
variable method is used here for its robustness in finding trajectories regardless
of geometry. This method is concerned with the determination of the variables $y$
and $A$ by a method of iterating $\psi$, which represent the square root of the
distance traveled between the two points. These variables can then be used to build
$f$ and $g$ functions, which can completely constrain the initial and final states.
This problem can be solved by any root-finding method, with bisection being used
here for its robustness given any initial guess \cite{battin1984elegant}.
Firstly, some geometric considerations must be accounted for. For any initial
position, $\vec{r}_0$, and final position, $\vec{r}_f$, and time of flight $\Delta
t$, there are actually two separate transfer orbits that can connect the two points
with paths that traverse less than one full orbit. For each of these, there are
actually then two trajectories that can connect the points
\cite{vallado2001fundamentals}. The first of the two will have a $\Delta \theta$ of
less than 180 degrees, which we classify as a Type I trajectory, and the second will
have a $\Delta \theta$ of greater than 180 degrees, which we call a Type II
trajectory. They will also differ in their direction of motion (clockwise or
counter-clockwise about the focus). This can be seen in Figure~\ref{type1type2}.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{fig/lamberts}
\caption{Visualization of the possible solutions to Lambert's Problem}
\label{type1type2}
\end{figure}
The iteration used in this thesis will start by first calculating the change in true
anomaly, $\Delta \theta$, as well as the cosine of this value, which can be found
by:
\begin{align}
\cos (\Delta \theta) &= \frac{\vec{r}_1 \cdot \vec{r}_2}{|\vec{r}_1| |\vec{r}_2|} \\
\Delta \theta &= \arctan(y_2/x_2) - \arctan(y_1/x_1)
\end{align}
The direction of motion is then chosen such that counter-clockwise orbits are
considered, as travelling in the same direction as the planets is generally more
efficient. Next, the variable $A$ is defined:
\begin{equation}
A = DM \sqrt{|r_1| |r_2| (1 - \cos(\Delta \theta))}
\end{equation}
A is independent of $\psi$, and therefore won't need updating as the iteration
proceeds. Then $\psi$ is initialized to any number within its bounds
($[-4\pi,4\pi^2]$), arbitrarily set to 0, representing a parabolic arc as a starting
point.
From here, the iteration loop can begin. Specifically, time of flight is calculated
at each step and compared to the expected value. The iteration proceeds until the
time of flight matches the expected value to within a provided tolerance. In order
to calculate the time of flight at each step, we must first calculate some useful
coefficients:
\begin{equation}\label{loop_start}
c_2 = \begin{cases}
\frac{1-\cos(\sqrt{\psi})}{\psi} \quad &\text{if} \, \psi > 10^{-6} \\
\frac{1-\cosh(\sqrt{-\psi})}{\psi} \quad &\text{if} \, \psi < -10^{-6} \\
1/2 \quad &\text{if} \, 10^{-6} > \psi > -10^{-6}
\end{cases}
\end{equation}
\begin{equation}
c_3 = \begin{cases}
\frac{\sqrt{\psi} - \sin sqrt{\psi}}{\psi^{3/2}} \quad &\text{if} \, \psi > 10^{-6} \\
\frac{\sinh\sqrt{-\psi} - \sqrt{-\psi}}{(-\psi)^{3/2}} \quad &\text{if} \, \psi < -10^{-6} \\
1/6 \quad &\text{if} \, 10^{-6} > \psi > -10^{-6}
\end{cases}
\end{equation}
\noindent
Where the conditions of this piecewise function represent the elliptical,
hyperbolic, and parabolic cases, respectively. Once we have these, we can calculate
another variable, $y$:
\begin{equation}
y = |r_1| + |r_2| + \frac{A (c_3 \psi - 1)}{\sqrt{c_2}}
\end{equation}
We can then finally calculate the variable $\chi$, and from that, the time of
flight:
\begin{equation}
\chi = sqrt{\frac{y}{c_2}}
\end{equation}
\begin{equation}
\Delta t = \frac{c_3 \chi^3 + A \sqrt{y}}{\sqrt{c_2}}
\end{equation}
Based on the value of this time of flight and how it compares to the expected value,
the bounds on $\psi$ are adjusted, a new $\psi$ is calculated at the midpoint
between the bounds, and the iteration begins again at Equation~\ref{loop_start}. If
the time of flight is sufficiently close to the expected value, the algorithm is
allowed to complete.
The resulting $f$ and $g$ functions (and the derivative of $g$, $\dot{g}$) can then
be calculated:
\begin{align}
f &= 1 - \frac{y}{|r_1|} \\
g &= A \sqrt{\frac{y}{\mu}} \\
\dot{g} &= 1 - \frac{y}{|r_2|}
\end{align}
And from these, we can calculate the velocities of the transfer points as:
\begin{align}
\vec{v}_1 &= \frac{\vec{r}_1 - f \vec{r}_2}{g} \\
\vec{v}_2 &= \frac{\dot{g} \vec{r}_2 - \vec{r}_1}{g}
\end{align}
\noindent
Fully constraining the connecting orbit.
\subsubsection{Planetary Ephemeris}
@@ -454,13 +656,12 @@
many packages have been developed for this purpose. The most commonly used for this
is the SPICE package, developed by NASA in the 1980's. This software package, which
has ports that are widely available in a number of languages, including Julia,
contains many useful functions for astrodynamics.
contains many useful functions for astrodynamics.
The primary use of SPICE in this thesis, however, was to determine the planetary
ephemeris at a known epoch. Using the NAIF0012 and DE430 kernels, ephemeris in the
ecliptic plane J2000 frame could be easily determined. A method for quickly
determining the ephemeris using a polynomial fit was also employed as an option for
faster ephemeris-finding, but ultimately not used.
ecliptic plane J2000 frame could be easily determined for a given epoch, provided as
a decimal Julian Day since the J2000 epoch.
\subsubsection{Porkchop Plots}
@@ -492,42 +693,123 @@
\end{figure}
However, this is an impulsive thrust-centered approach. The solution to Lambert's
problem assumes a natural trajectory. However, to the low-thrust designer, this is
needlessly limiting. A natural trajectory is unnecessary when the trajectory can be
modified by a continuous thrust profile along the arc. Therefore, for the hybrid problem
of optimizing both flyby selection and thrust profiles, porkchop plots are less helpful,
and an algorithmic approach is preferred.
problem assumes a natural trajectory. A natural trajectory is unnecessary when the
trajectory can be modified by a continuous thrust profile along the arc. Therefore,
for the hybrid problem of optimizing both flyby selection and thrust profiles,
porkchop plots are less helpful, and an algorithmic approach is preferred.
\section{Low Thrust Considerations} \label{low_thrust}
Thus far, the techniques that have been discussed can be equally useful for both impulsive and
continuous thrust mission profiles. In this section, we'll discuss the intricacies of continuous
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.
In this section, we'll discuss the intricacies of continuous 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{Specific Impulse}
The primary advantage of continuous thrust methods over their impulsive counterparts is
in their fuel-efficiency in generating changes in velocity. Put specifically, all
thrusters are capable of translating a mass flow (the rate of mass ejection from the
thruster during operation) to a thrust imparted to the craft. Low thrust techniques
suffer from limitations in the amount of thrust they can produce, but benefit from high
efficiency by means of achieving that thrust by means of very low mass ejection rates.
This efficiency is often captured in a single variable called specific impulse, often
denoted as $I_{sp}$. We can derive the specific impulse by starting with the rocket
thrust equation\cite{sutton2016rocket}:
\begin{equation}
F = \dot{m} v_e + \Delta p A_e
\end{equation}
\noindent
Where $F$ is the thrust imparted, $\dot{m}$ is the fuel mass rate, $v_e$ is the exhaust
velocity of the fuel, $\Delta p$ is the change in pressure across the exhaust opening,
and $A_e$ is the area of the exhaust opening. We can then define a new variable
$v_{eq}$, such that the thrust equation becomes:
\begin{align}
v_{eq} &= v_e - \frac{\Delta p A_e}{\dot{m}} \\
F &= \dot{m} v_{eq} \label{isp_1}
\end{align}
\noindent
And we can then take the integral of this value with respect to time to find the total
impulse, dividing by the weight of the fuel to derive the specific impulse:
\begin{align}
I &= \int F dt = \int \dot{m} v_{eq} dt = m_e v_{eq} \\
I_{sp} &= \frac{I}{m_e g_0} = \frac{m_e v_{eq}}{m_e g_0} = \frac{v_{eq}}{g_0}
\end{align}
Plugging Equation~\ref{isp_1} into the previous equation we can derive the following
formula for $I_{sp}$:
\begin{equation} \label{isp_real}
I_{sp} = \frac{F}{\dot{m} g_0}
\end{equation}
\noindent
Which is generally taken to be a value with units of seconds and effectively represents
the efficiency with which a thruster converts mass to thrust.
\subsection{Sims-Flanagan Transcription}
this thesis chose to use a model well suited for modeling low-thrust paths: the
Sims-Flanagan transcription (SFT)\cite{sims1999preliminary}. The SFT allows for
flexibility in the trade-off between fidelity and performance, which makes it very
useful for this sort of preliminary analysis.
First the continuous arc is subdivided into a number ($N$) of individual consistent
timesteps of length $\frac{tof}{N}$ where the $tof$ represents the total length of time
for that particular mission phase. The control thrust is then applied as an impulsive
maneuver at the center of each of these time steps. This approach can be seen visualized
in Figure~\ref{sft_fig}.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{fig/sft}
\caption{Example of an orbit raising using the Sims-Flanagan Transcription with 7
Sub-Trajectories}
\label{sft_fig}
\end{figure}
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 numerical integration algorithms, by simply solving
Kepler's equation (using the LaGuerre-Conway algorithm introduced in
Section~\ref{laguerre}) $N+1$ times. First, the state is propagated to the middle of the
first arc. Then a discontinuity is allowed in the velocity at that point and the state
is propagated again to the middle of the next arc. That process is repeated $N-1$ times,
and then finally, the last half-arc is propagated after applying the final velocity
change.
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\cite{sims1999preliminary}.
\subsection{Low-Thrust Control Laws}
In determining a low-thrust arc, a number of variables must be accounted for and, ideally,
optimized. Generally speaking, this means that a control law must be determined for the
thruster. This control law functions in exactly the same way that an impulsive thrust
control law might function. However, instead of determining the proper moments at which
to thrust, a low-thrust control law must determine the appropriate direction, magnitude,
and presence of a thrust at each point along its continuous orbit.
In determining a low-thrust arc, a number of variables must be accounted for and,
ideally, optimized. Generally speaking, this means that a control law must be determined
for the thruster. This involves determining the appropriate direction, magnitude, and
presence of a thrust at each point along its continuous orbit.
\subsubsection{Angle of Thrust}
Firstly, we can examine the most important quality of the low-thrust control law, the
direction at which to point the thrusters while they are on. The methods for determining this
direction varies greatly depending on the particular control law chosen for that
mission. Often, this process involves first determining a useful frame to think about
the kinematics of the spacecraft. In this case, we'll use a frame often used in these
low-thrust control laws: the spacecraft $\hat{R} \hat{\theta} \hat{H}$ frame. In this
frame, the $\hat{R}$ direction is the radial direction from the center of the primary to
the center of the spacecraft. The $\hat{H}$ hat is perpendicular to this, in the
direction of orbital momentum (out-of-plane) and the $\hat{\theta}$ direction completes
the right-handed orthonormal frame.
Firstly, we can examine the direction at which to point the thrusters while they are on.
The methods for determining this direction varies greatly depending on the particular
control law chosen for that mission. Often, this process involves first determining a
useful frame to think about the kinematics of the spacecraft. In this case, we'll use a
frame often used in these low-thrust control laws: the spacecraft $\hat{R} \hat{\theta}
\hat{H}$ frame. In this frame, the $\hat{R}$ direction is the radial direction from the
center of the primary to the center of the spacecraft. The $\hat{H}$ hat is
perpendicular to this, in the direction of orbital momentum (out-of-plane) and the
$\hat{\theta}$ direction completes the right-handed orthonormal frame.
This frame is useful because, for a given orbit, especially a nearly circular one, the
$\hat{\theta}$ direction is nearly aligned with the velocity direction for that orbit at
@@ -538,34 +820,28 @@
direction for most effectively increasing (or decreasing if negative) the angular
momentum and orbital energy of the trajectory.
Therefore, at each point, the first controls of a control-law, whichever frame or
convention is used to define them, need to represent a direction in 3-dimensional space
that the force of the thrusters will be applied.
Using these conventions, we can then redefine our thrust vector in terms of the $\alpha$
and $\beta$ angles in the chosen frame:
\begin{align}
F_r &= F \cos(\beta) \sin (\alpha) \\
F_\theta &= F \cos(\beta) \cos (\alpha) \\
F_h &= F \sin(\beta)
\end{align}
\subsubsection{Thrust Magnitude}
However, there is actually another variable that can be varied by the majority of
electric thrusters. Either by controlling the input power of the thruster or the duty
cycle, the thrust magnitude can also be varied in the direction of thrust, limited by
the maximum thrust available to the thruster. Not all control laws allow for this
fine-tuned control of the thruster. Generally speaking, it's most efficient either to
thrust or not to thrust. Therefore, controlling the thrust magnitude may provide too
much complexity at too little benefit.
cycle, the thrust magnitude can also be varied, limited by the maximum thrust available
to the thruster. Not all control laws allow for this fine-tuned control of the thruster.
The algorithm used in this thesis, however, does allow the magnitude of the thrust
control to be varied. In certain cases it actually can be useful to have some fine-tuned
control over the magnitude of the thrust. Since the optimization in this algorithm is
automatic, it is relatively straightforward to consider the control thrust as a
3-dimensional vector in space limited in magnitude by the maximum thrust, which allows
for that increased flexibility.
\subsubsection{Thrust Presence}
The alternative to this approach of modifying the thrust magnitude, is simply to modify
the presence or absence of thrust. At certain points along an arc, the efficiency of
thrusting, even in the most advantageous direction, may be such that a thrust is
undesirable (in that it will lower the overall efficiency of the mission too much) or,
in fact, be actively harmful.
The algorithm used in this thesis does vary the magnitude of the thrust control. In
certain cases it actually can be useful to have some fine-tuned control over the
magnitude of the thrust. Since the optimization in this algorithm is automatic, it is
relatively straightforward to consider the control thrust as a 3-dimensional vector in
space limited in magnitude by the maximum thrust, which allows for that increased
flexibility.
For instance, we can consider the case of a simple orbit raising. Given an initial orbit
with some eccentricity and some semi-major axis, we can define a new orbit that we'd
@@ -599,61 +875,3 @@
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{Direct vs Indirect Optimization}
As previously mentioned, there are two different approaches to optimizing non-linear
problems such as trajectory optimizations in interplanetary space. These methods are the
direct method, in which a cost function is developed and used by numerical root-finding
schemes to drive cost to the nearest local minimum, and the indirect method, in which a
set of sufficient and necessary conditions are developed that constrain the optimal
solution and used to solve a boundary-value problem to find the optimal solution.
Both of these methods have been applied to the problem of low-thrust interplanetary
trajectory optimization \cite{Casalino2007IndirectOM}. The common opinion of the
difference between these two methods is that the indirect methods are more difficult to
converge and require a better initial guess than the direct methods. However, they also
require less parameters to describe the trajectory, since the solution of a boundary
value problem doesn't require discretization of the control states.
In this implementation, robustness is incredibly valuable, as the Monotonic Basin
Hopping algorithm is leveraged to attempt to find all minima basins in the solution
space by ``hopping'' around with different initial guesses. Since these initial guesses
are not guaranteed to be close to any particular valid trajectory, it is important that
the optimization routine be robust to poor initial guesses. Therefore, a direct
optimization method was leveraged by transcribing the problem into an NLP and using
IPOPT to find the local minima.
\subsection{Sims-Flanagan Transcription}
The major problem with optimizing low thrust paths is that the control law must necessarily be
continuous. Also, since indirect optimization approaches are, in the context of
interplanetary trajectories including flybys, 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)\cite{sims1999preliminary}.
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. This approach can be seen visualized in Figure~\ref{sft_fig}.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{fig/sft}
\caption{Example of an orbit raising using the Sims-Flanagan Transcription with 7
Sub-Trajectories}
\label{sft_fig}
\end{figure}
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.