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48
LaTeX/approach.tex
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\chapter{Algorithm Overview} \label{algorithm}
This thesis will attempt to develop an algorithm for the preliminary analysis of feasibility in
designing a low-thrust interplanetary mission to an outer planet by leveraging a monotonic basin
hopping algorithm. In this section, we will review the actual execution of the algorithm
developed. As an overview, the routine was designed to enable the determination of an optimized
spacecraft trajectory from the selection of some very basic mission parameters. Those parameters
include:
This thesis focuses on designing a low-thrust interplanetary mission to an outer planet by
leveraging a monotonic basin hopping algorithm. This section will review the actual execution of
the algorithm developed. As an overview, the routine is designed to enable the determination of
an optimal spacecraft trajectory that minimizes propellant usage and $C_3$ from the selection of
some very basic parameters. Those parameters include:
\begin{itemize}
\setlength\itemsep{-0.5em}
@@ -25,7 +24,7 @@
\end{itemize}
Which allows for an automated approach to optimization of the trajectory, while still providing
the mission designer with the flexibility to choose the particular flyby planets to investigate.
the designer with the flexibility to choose the particular flyby planets to investigate.
This is achieved via an optimal control problem in which the ``inner loop'' involves solving a
TPBVP to find the optimal solution given a suitable initial guess. Then an ``outer loop''
@@ -133,12 +132,11 @@
The following pseudo-code outlines the approach taken for the elliptical case. The
approach is quite similar when $a<0$:
% TODO: Some symbols here aren't recognized by the font
\begin{singlespacing}
\begin{verbatim}
i = 0
# First declare some useful variables from the state
sig0 = (position ⋅ velocity) / √(mu)
sig0 = dot(position, velocity) / √(mu)
a = 1 / ( 2/norm(position) - norm(velocity)^2/mu )
coeff = 1 - norm(position)/a
@@ -184,7 +182,7 @@
\label{laguerre_plot}
\end{figure}
\subsection{Sims-Flanagan Propagator}
\subsection{Propagating with Sims-Flanagan Transcription}
Until this point, we've not yet discussed how best to model the low-thrust
trajectory arcs themselves. The Laguerre-Conway algorithm efficiently determines
@@ -228,11 +226,9 @@
and the mass flow rate (a function of the duty cycle percentage ($d$), thrust ($f$),
and the specific impulse of the thruster ($I_{sp}$), commonly used to measure
efficiency)\cite{sutton2016rocket}:
\begin{equation}
\Delta m = \Delta t \frac{f d}{I_{sp} g_0}
\end{equation}
Where $\Delta m$ is the fuel used in the sub-trajectory, $\Delta t$ is the time of
flight of the sub-trajectory, and $g_0$ is the standard gravity at the surface of
Earth. From knowledge of the mass flow rate, we can then decrement the mass
@@ -272,10 +268,9 @@
From this information, as can be seen in Figure~\ref{nlp}, we can formulate the mission
in terms of a non-linear programming problem. Specifically, the variables describing the
trajectory contained within the Guess object can be represented as an input vector,
$\vec{x}$, the cost function produced by an entire trajectory propagation as $F$, and
the constraints that the trajectory must satisfy as another function $\vec{G}$ such that
$\vec{G}(\vec{x}) = \vec{0}$.
trajectory from the free variable, $\vec{x}$, the cost function produced by an entire
trajectory propagation, $F$, and the constraints that the trajectory must satisfy as
another function $\vec{G}$ such that $\vec{G}(\vec{x}) = \vec{0}$.
This is a format that we can apply directly to the IPOPT solver, which Julia (the
programming language used) can utilize via bindings supplied by the SNOW.jl
@@ -333,12 +328,12 @@
\subsection{Random Trajectory Generation}\label{random_gen_section}
At a basic level, the algorithm needs to produce a guess (represented by all of the
values described in Section~\ref{inner_loop_section}) that contains random values within
reasonable bounds in the space. However, that still leaves the determination of 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.
At a basic level, the algorithm needs to produce a guess for the free variable vector
(represented by all of the values described in Section~\ref{inner_loop_section}) that
contains random values within reasonable bounds in the space. However, that still leaves
the determination of 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
@@ -372,18 +367,18 @@
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
the trajectory 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
natural trajectory arc. For this Lambert's case, the trajectory 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
However, for the truly random trajectory 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
@@ -471,14 +466,11 @@
Because of this, the perturbation used in this implementation follows a
bi-directional, long-tailed Pareto distribution generated by the following
probability density function\cite{englander2014tuning}:
\begin{equation}
1 +
\left[ \frac{s}{\epsilon} \right] \cdot
\left[ \frac{\alpha - 1}{\frac{\epsilon}{\epsilon + r}^{-\alpha}} \right]
\end{equation}
\noindent
Where $s$ is a random array of signs (either plus one or minus one) with dimension
equal to the perturbed variable and bounds of -1 and 1, $r$ is a uniformly
distributed random array with dimension equal to the perturbed variable and bounds

7
LaTeX/conclusion.tex
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@@ -10,11 +10,8 @@
In performing this examination, two results were selected for further analysis. These
results are outlined in Table~\ref{results_table}. As can be seen in the table, both
resulting trajectories have trade-offs in mission length, launch energy, fuel usage, and
more. However, both results show very interesting trajectories that could indicate some
favorable possibilities for such a mission profile. Each of these trajectories should be
within the capabilities of existing launch vehicles in terms of $C_3$.
\section{Recommendations for Future Work}\label{improvement_section}
more. Each of these trajectories appear to be within the capabilities of existing launch
vehicles in terms of $C_3$.
In the course of producing this algorithm, a large number of improvement possibilities were
noted. This work was based, in large part, on the work of Jacob Englander in a number of

75
LaTeX/introduction.tex
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@@ -1,16 +1,16 @@
\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.
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,
@@ -29,16 +29,15 @@
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.
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,
@@ -58,24 +57,22 @@
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.
including a team from JPL\cite{sims2006}, among several others\cite{morante}.
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 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 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.
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.

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LaTeX/results.tex
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@@ -1,31 +1,29 @@
\chapter{Sample Saturn Trajectory Analysis} \label{results}
\chapter{Application: Designing a Trajectory To Saturn} \label{results}
The algorithm described in this thesis is quite flexible in its design and could be used as
a tool for a mission designer on a variety of different mission types. However, to consider
a relatively simple but representative mission design objective, a sample mission to Saturn
was investigated.
To consider a relatively simple but representative mission design objective, a sample mission to
Saturn was investigated.
\section{Mission Constraints}
\section{Mission Scenario}
The sample mission was defined to represent a general case for a near-future low-thrust
trajectory to Saturn. No constraints were placed on the flyby planets, but a number of
The sample mission is defined to represent a general case for a near-future low-thrust
trajectory to Saturn. No constraints are placed on the flyby planets, but a number of
constraints were placed on the algorithm to represent a realistic mission scenario.
The first choice required by the application is one not necessarily designable to the
initial mission designer (though not necessarily fixed in the design either) and is that
of the spacecraft parameters. The application accepts as input a spacecraft object
containing: the dry mass of the craft, the fuel mass at launch, the number of onboard
thrusters, and the specific impulse, maximum thrust and duty cycle of each thruster.
initial mission designer (though not necessarily fixed in the design either) and is that of
the spacecraft parameters. The application accepts as input a spacecraft object containing:
the dry mass of the spacecraft, the fuel mass at launch, the number of onboard thrusters,
and the specific impulse, maximum thrust and duty cycle of each thruster.
For this mission, the spacecraft was chosen to have a dry mass of only 200 kilograms for
a fuel mass of 3300 kilograms. This was chosen in order to have an overall mass roughly
in the same zone as that of the Cassini spacecraft, which launched with 5712 kilograms
of total mass, with the fuel accounting for 2978 of those kilograms\cite{cassini}. The
dry mass of the craft was chosen to be extremely low in order to allow for a variety of
''successful`` missions in which the craft didn't run out of fuel. That way, the
delivered dry mass to Saturn could be thought of as a metric of success, without
discounting mission that may have delivered just under whatever more realistic dry mass
one might set, in case those missions are in the vicinity of actually valid missions.
For this mission, the spacecraft was chosen to have a dry mass of only 200 kilograms for a
fuel mass of 3300 kilograms. This was chosen in order to have an overall mass roughly in the
same zone as that of the Cassini spacecraft, which launched with 5712 kilograms of total
mass, with the fuel accounting for 2978 of those kilograms\cite{cassini}. The dry mass of
the spacecraft was chosen to be extremely low in order to allow for a variety of
''successful`` missions in which the spacecraft didn't run out of fuel. That way, the
delivered dry mass to Saturn could be thought of as a metric of success, without discounting
mission that may have delivered just under whatever more realistic dry mass one might set,
in case those missions are in the vicinity of actually valid missions.
The thruster was chosen to have a specific impulse of 3200 seconds, a maximum thrust of
250 millinewtons, and a 100\% duty cycle. This puts the thruster roughly in line with
@@ -308,6 +306,6 @@
\centering
\includegraphics[width=\textwidth]{fig/c3}
\caption{Plot of Delta IV and Atlas V launch vehicle capabilities as they relate to
payload mass \cite{c3capabilities} from a source from 2007}
payload mass \cite{c3capabilities} from Vardaxis, et al, 2007 }
\label{c3}
\end{figure}

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LaTeX/thesis.tex
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@@ -22,14 +22,15 @@
Much work has been performed recently to utilize the increasingly viable technology of
low-thrust electric propulsion systems on missions of interplanetary scope. This thesis analyzes
a technique for the initial analysis of feasibility of utilizing a combination of low-thrust
propulsion systems and natural gravity flybys for missions to the outer planets. First, a method
for finding local optima by utilizing an interior-point linesearch algorithm to directly
optimize the entire trajectory as a Non-Linear Programming problem is presented. Then, a
Monotonic Basin Hopping algorithm is utilized to traverse the search space, improve the local
optima determined by the internal optimizer, and determine the global optima. This allows for a
medium-fidelity, fully automated global optimization of the low thrust controls and flyby
parameters for a given mission objective.
a technique for designing trajectories for spacecraft with a low-thrust propulsion system that
also use natural gravity flybys for missions to the outer planets. Often, the goal is to find
feasible solutions that also minimize propellant mass requirements. First, locally optimal
solutions are constructed by using an interior-point linesearch algorithm, along with multiple
shooting techniques for optimization. Then, Monotonic Basin Hopping is utilized to traverse the
search space, improve the local optima determined by the internal optimizer, and determine the
global optima. This approach allows for a medium-fidelity, fully automated global optimization
of the low thrust controls and flyby parameters for a given target destination. As an
application of this method, two sample trajectories to Saturn are analyzed.
}

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LaTeX/trajectory_design.tex
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@@ -16,24 +16,24 @@
very high-fidelity force models that account for aerodynamic pressure, solar radiation
pressure, multi-body effects, and other forces may be too time intensive for a
particular application. Initial surveys of the solution space often don't require such
complex models in order to gain valuable insight.
complex models in order to gain valuable preliminary insight.
Therefore, a common approach (and the one utilized in this implementation) is to first
use a lower-fidelity dynamical model that captures only the gravitational force due to
the primary body around which the spacecraft is orbiting. This approach can provide an
A common approach (and the one utilized in this implementation) is to first use a
lower-fidelity dynamical model that captures only the gravitational force due to the
primary body around which the spacecraft is orbiting. This approach can provide an
excellent low-to-medium fidelity model that is useful as an underlying model in an
algorithm for quickly categorizing a search space for initial mission feasibility
explorations.
In order to explore the Two Body Problem, we must first examine the full set of
assumptions associated with the force model\cite{vallado2001fundamentals}. 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 point masses
with constant mass. This removes the need to account for non-uniform densities and
asymmetry. Finally, for convenience in notation at the end, we'll also assume 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.
are only concerned with the gravitational influence between the nominative two bodies:
the spacecraft and the planetary body around which it is orbiting. Secondly, both of
these bodies are modeled as point masses with constant mass. This removes the need to
account for non-uniform densities and asymmetry. Finally, for convenience in notation at
the end, we'll also assume 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.
\begin{figure}[H]
\centering
@@ -45,7 +45,6 @@
Under these assumptions, the force acting on the body due to the law of universal
gravitation is:
\begin{align}
F_2 &= - \frac{G m_1 m_2}{r^2} \frac{\vec{r}}{\left| r \right|} \\
F_1 &= \frac{G m_2 m_1}{r^2} \frac{\vec{r}}{\left| r \right|}
@@ -53,7 +52,6 @@
And by Newton's second law (force is the product of mass and acceleration), we can
derive the following differential equations for $r_1$ and $r_2$:
\begin{align}
m_2 \ddot{\vec{r}}_2 &= - \frac{G m_1 m_2}{r^2} \frac{\vec{r}}{\left| r \right|} \\
m_1 \ddot{\vec{r}}_1 &= \frac{G m_2 m_1}{r^2} \frac{\vec{r}}{\left| r \right|}
@@ -65,7 +63,6 @@
inertial frame. $G$ is the universal gravitational parameter, $m_1$ is the mass of the
planetary body, and $m_2$ is the mass of the spacecraft. From these equations, we can
then determine the acceleration of the spacecraft relative to the planet:
\begin{equation}
\ddot{\vec{r}} = \ddot{\vec{r}}_2 - \ddot{\vec{r}}_1 =
- \frac{G \left( m_1 + m_2 \right)}{r^2} \frac{\vec{r}}{\left| r \right|}
@@ -76,27 +73,19 @@
negligible $m_2$ term. We can also introduce, for convenience, a gravitational parameter
$\mu$ which represents the gravity constant for the system about the center of motion
($\mu = G (m_1 + m_2) \approx G m_1$). Doing so and simplifying produces:
\begin{equation}
\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:
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
ourselves with more practical applications of it as a force model. It should be noted,
firstly, that the spacecraft's position and velocity (given an initial position and velocity
@@ -105,6 +94,8 @@
one-dimensional equations (one for each component of the three-dimensional space) and
three unknowns (the three components of the second derivative of the position).
\subsection{Kepler's Laws}
In the early 1600s, Johannes Kepler produced just such a solution, by taking advantages of
what is also known as ``Kepler's Laws'' which are\cite{murray1999solar}:
@@ -113,68 +104,61 @@
expanded to any orbit by re-wording as ``all orbital paths follow a conic section
(circle, ellipse, parabola, or hyperbola) with a primary mass at one of the foci''.
Specifically the path of the orbit follows the trajectory equation:
The conic trajectory equation explains this observation and offers a description
of the path as:
\begin{equation}
r = \frac{\sfrac{h^2}{\mu}}{1 + e \cos(\theta)}
\end{equation}
Where $h$ is the angular momentum of the satellite, $e$ is the
where $h$ is the angular momentum of the satellite, $e$ is the
eccentricity of the orbit, and $\theta$ is the true anomaly, or simply
the angular distance the satellite has traversed along the orbit path.
the angular distance the satellite has traversed along the orbit path from
periapsis.
\item The area swept out by the imaginary line connecting the primary and secondary
bodies increases linearly with respect to time. This implies that the magnitude of the
orbital speed is not constant. For the moment, we'll just take this
value to be a constant:
\begin{equation}\label{swept}
\frac{\Delta t}{T} = \frac{k}{\pi a b}
\end{equation}
Where $k$ is the constant value, $a$ and $b$ are the semi-major and
where $k$ is the constant value, $a$ and $b$ are the semi-major and
semi-minor axis of the conic section, and $T$ is the period. In the
following section, we'll derive the value for $k$.
\item The square of the orbital period is proportional to the cube of the semi-major
axis of the orbit, regardless of eccentricity. Specifically, the relationship is:
axis of the orbit, regardless of eccentricity. For an elliptical orbit this
observation connects to the following known expression for the orbit period:
\begin{equation}
T = 2 \pi \sqrt{\frac{a^3}{\mu}}
\end{equation}
Where $T$ is the period and $a$ is the semi-major axis.
where $T$ is the period and $a$ is the semi-major axis.
\end{enumerate}
\subsection{Kepler's Equation}
Kepler was able to produce an equation to represent the angular displacement of an
orbiting body around a primary body as a function of time, which we'll derive now for
the elliptical case\cite{vallado2001fundamentals}. Since the total area of an ellipse is
the product of $\pi$, the semi-major axis, and the semi-minor axis ($\pi a b$), we can
relate (by Kepler's second law) the area swept out by an orbit as a function of time, as
we did in Equation~\ref{swept}. This leaves just one unknown variable $k$, which we can
determine through use of the geometric auxiliary circle, which is a circle with radius
equal to the ellipse's semi-major axis and center directly between the two foci, as in
Figure~\ref{aux_circ}.
the elliptical case\cite{vallado2001fundamentals}. Because the total area of an ellipse
is the product of $\pi$, the semi-major axis, and the semi-minor axis ($\pi a b$), we
can relate (by Kepler's second law) the area swept out by an orbit as a function of
time, as we did in Equation~\ref{swept}. This leaves just one unknown variable $k$,
which we can determine through use of the geometric auxiliary circle, which is a circle
with radius equal to the ellipse's semi-major axis and center directly between the two
foci, as in Figure~\ref{aux_circ}.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{fig/kepler}
\caption{Geometric Representation of Auxiliary Circle}\label{aux_circ}
\caption{Geometric representation of auxiliary circle}\label{aux_circ}
\end{figure}
In order to find the area swept by the spacecraft\cite{vallado2001fundamentals}, $k$, we
can take advantage of the fact that that area is the triangle $k_1$ subtracted from the
elliptical segment $PCB$:
\begin{equation}\label{areas_eq}
k = area(seg_{PCB}) - area(k_1)
\end{equation}
\noindent
Where the area of the triangle $k_1$ can be found easily using geometric formulae:
\begin{align}
area(k_1) &= \frac{1}{2} \left( ae - a \cos E \right) \left( \frac{b}{a} a \sin E \right) \\
&= \frac{ab}{2} \left(e \sin E - \cos E \sin E \right)
@@ -186,7 +170,6 @@
can find the area for the elliptical segment $PCB$ by first finding the circular segment
$POB'$, subtracting the triangle $COB'$, then applying the fact that an ellipse is
merely a vertical scaling of a circle by the amount $\frac{b}{a}$.
\begin{align}
area(PCB) &= \frac{b}{a} \left( area(POB') - area(COB') \right) \\
&= \frac{b}{a} \left( \frac{a^2 E}{2} - \frac{1}{2} \left( a \cos E \right)
@@ -197,26 +180,20 @@
By substituting the two areas back into Equation~\ref{areas_eq} we can get the $k$ area
swept out by the spacecraft:
\begin{equation}
k = \frac{ab}{2} \left( E - e \sin E \right)
\end{equation}
Which we can then substitute back into the equation for the swept area as a function of
time (Equation~\ref{swept}) for period of time since the spacecraft left periapsis:
\begin{equation}
\frac{\Delta t}{T} = \frac{t_2 - t_{peri}}{T} = \frac{E - e \sin E}{2 \pi}
\end{equation}
Which is, effectively, Kepler's equation. It is commonly known by a different form:
\begin{equation}
M = \sqrt{\frac{\mu}{a^3}} \Delta t = E - e \sin E
\end{equation}
Where we've defined the mean anomaly as $M$ and used the fact that $T =
\sqrt{\frac{a^3}{\mu}}$. This provides us a useful relationship between Eccentric Anomaly
where we've defined the mean anomaly as $M$ and used the fact that $T =
\sqrt{\frac{a^3}{\mu}}$. This provides us a useful relationship between eccentric anomaly
($E$) which can be related to spacecraft position, and time, but we still need a useful
algorithm for solving this equation in order to use this equation to propagate a
spacecraft.
@@ -224,34 +201,25 @@
\subsection{LaGuerre-Conway Algorithm}\label{laguerre}
For this thesis, the algorithm used to solve Kepler's equation was the general numeric
root-finding scheme first developed by LaGuerre in the 1800s and first applied to
Kepler's equation by Bruce Conway in 1985\cite{laguerre_conway}. In his paper, Conway
makes a compelling argument for utilizing the less common LaGuerre method over higher
order Newton or Newton-Raphson methods.
The Newton-Raphson methods, while found to generally have quite impressive convergence
rates (generally successfully solving Kepler's equation correctly within 5 iterations),
were prone to failures in convergence given certain specific initial conditions.
Therefore LaGuerre's algorithm is proposed as an alternative.
The algorithm can be relatively easily derived by examining the polynomial equation with
$m$ roots:
root-finding scheme first developed by LaGuerre in the 1800s and first applied to Kepler's
equation by Bruce Conway in 1985\cite{laguerre_conway}. In his paper, Conway makes a
compelling argument for utilizing the less common LaGuerre method over higher order Newton
or Newton-Raphson methods. The Newton-Raphson methods, while found to generally have quite
impressive convergence rates (generally successfully solving Kepler's equation correctly
within 5 iterations), were prone to failures in convergence given certain specific initial
conditions. Therefore LaGuerre's algorithm is proposed as an alternative.
The algorithm can be derived by examining the polynomial equation with $m$ roots:
\begin{equation}
g(x) = (x - x_1) (x - x_2) ... ( x - x_m)
\end{equation}
\noindent
We can then generate some useful convenience functions as:
\begin{align}
\ln|g(x)| &= \ln|(x - x_1)| + \ln|(x - x_2)| + ... + \ln|( x - x_m)| \\
\frac{d\ln|g(x)|}{dx} &= \frac{1}{x - x_1} + \frac{1}{x - x_2} + ... + \frac{1}{x -
x_m} = G_1(x)
\end{align}
and
\begin{align}
\frac{-d^2\ln|g(x)|}{dx^2} &= \frac{1}{(x - x_1)^2} + \frac{1}{(x - x_2)^2} + ... +
\frac{1}{(x - x_m)^2} = G_2(x)
@@ -259,42 +227,32 @@
Now we define the targeted root as $x_1$ and make the approximation that all of the
other roots are equidistant from the targeted root, which means:
\begin{equation}
x - x_i = b, i=2,3,...,m
\end{equation}
\noindent
We can then rewrite $G_1$ and $G_2$ as:
\begin{align}
G_1 &= \frac{1}{a} + \frac{n-1}{b} \\
G_2 &= \frac{1}{a^2} + \frac{n-1}{b^2}
\end{align}
\noindent
Which may be solved for $a$ in terms of $G_1$, $G_2$:
\begin{equation}
a = \frac{n}{G_1 \pm \sqrt{(n-1)(nG_2 - G_1^2)}}
\end{equation}
\noindent
With corresponding iteration function:
\begin{equation}
x_{i+1} = x_i - \frac{n g(x_i)}{g'(x_i) \pm \sqrt{(n-1)^2 f'(x_i)^2 - n (n-1) f(x_i)
f''(x_i)}}
\end{equation}
This iteration scheme can be shown to be globally convergent, regardless of the initial
guess. More relevantly, Conway also showed that the application of this method to
Kepler's equation was shown to converge with similar speed to many of the best common
higher order Newton-Raphson solvers. However, LaGuerre's method was also found to be
incredibly robust, converging to the correct value for every one of Conway's 500,000
tests. Because of this robustness, it is very useful for propagating spacecraft states.
guess. Conway also showed that the application of this method to Kepler's equation was shown
to converge with similar speed to many of the best common higher order Newton-Raphson
solvers. However, LaGuerre's method was also found to be incredibly robust, converging to
the correct value for every one of Conway's 500,000 tests. Because of this robustness, it is
useful for solving Kepler's equation.
\section{Interplanetary Considerations}\label{interplanetary}
\section{Interplanetary Trajectories}\label{interplanetary}
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
@@ -346,14 +304,15 @@
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.
occur along the spheres of influence of the planets nearest to the spacecraft. A
conceptual example of this process, labeled the method of patched conics, appears in
Figure~\ref{patched_conics_fig}.
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}
@@ -361,8 +320,8 @@
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.
velocity of the spacecraft (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
@@ -371,14 +330,12 @@
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} \\
@@ -387,14 +344,13 @@
\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}.
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
@@ -405,12 +361,12 @@
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. 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. This approach also allows
flexibility for missions that might end in a flyby rather than insertion.
A similar approach is taken at the end of the trajectory. This algorithm doesn't attempt
to exactly match the velocity of the planet. Instead, the excess hyperbolic velocity is
also treated as a parameter that can be minimized by the cost function. If a trajectory
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. This approach also
allows flexibility for missions that might end in a flyby rather than insertion.
\subsection{Gravity Assist Maneuvers}
@@ -441,7 +397,7 @@
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{fig/flyby}
\caption{Visualization of velocity changes during a gravity assist}
\caption{Velocity changes during a gravity assist}
\label{grav_assist_fig}
\end{figure}
@@ -451,7 +407,7 @@
turning angle of this bend. In doing so, one can effectively achieve a (restricted) free
impulsive thrust event.
\subsection{Flyby Periapsis}
\subsection{Flyby Periapsis Altitude}
Now that we understand gravity assists, the natural question is then how to leverage
them for achieving certain velocity changes\cite{cho2017b}. But first, we must consider
@@ -460,7 +416,6 @@
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}\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)
@@ -470,12 +425,10 @@
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}\label{periapsis_eq}
r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right]
\end{equation}
Cannot be lower than some safe value that accounts for the radius of the planet and
cannot be lower than some safe value that accounts for the radius of the planet and
perhaps its atmosphere if applicable.
\subsection{Multiple Gravity Assist Techniques}
@@ -511,7 +464,9 @@
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}.
counter-clockwise about the focus). This can be seen in Figure~\ref{type1type2},
where both of the Lambert's solutions are presented for sample points in an orbit
around the Sun.
\begin{figure}[H]
\centering
@@ -523,7 +478,6 @@
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)
@@ -532,7 +486,6 @@
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}
@@ -547,7 +500,6 @@
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} \\
@@ -555,7 +507,6 @@
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} \\
@@ -563,23 +514,18 @@
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}
@@ -592,22 +538,17 @@
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.
Fully describing the connecting path with the specified flight time.
\subsubsection{Planetary Ephemeris}
@@ -620,8 +561,8 @@
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 (ICRF) could be easily determined for a given epoch, provided as
a decimal Julian Day since the J2000 epoch.
International Celestial Reference Frame could be easily determined for a given
epoch, provided as a decimal Julian Day since the J2000 epoch.
\subsubsection{Porkchop Plots}
@@ -641,10 +582,9 @@
Using porkchop plots such as the one in Figure~\ref{porkchop}, mission designers can
quickly visualize which natural trajectories are possible between planets. Using the
fact that incoming and outgoing $v_\infty$ magnitudes must be the same for a flyby,
a savvy mission designer can even begin to work out what combinations of flybys
might be possible for a given timeline, spacecraft state, and planet selection.
a mission designer can even begin to work out what combinations of flybys might be
possible for a given timeline, spacecraft state, and planet selection.
%TODO: Create my own porkchop plot
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig/porkchop}
@@ -652,13 +592,7 @@
\label{porkchop}
\end{figure}
However, this is an impulsive thrust-centered approach. The solution to Lambert's
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}
\section{Modeling Low Thrust Control} \label{low_thrust}
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
@@ -666,7 +600,7 @@
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}
\subsection{Engine Model}
The primary advantage of continuous thrust methods over their impulsive counterparts is
in their fuel-efficiency in generating changes in velocity. Put specifically, all
@@ -678,45 +612,34 @@
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
In this thesis the following approach is used 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.
@@ -752,7 +675,7 @@
continuous low-thrust trajectory within the Two-Body Problem, with only
linearly-increasing computation time\cite{sims1999preliminary}.
\subsection{Low-Thrust Control Laws}
\subsection{Low-Thrust Control Vector Description}
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
@@ -765,24 +688,23 @@
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.
frame often used in these low-thrust control laws: the spacecraft-centered $\hat{R}
\hat{\theta} \hat{H}$ 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 triad.
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
that moment. This allows us to define a set of two angles, which we'll call $\alpha$ and
$\beta$, to represent the in and out of plane pointing direction of the thrusters. This
convention is useful because a $(0,0)$ set represents a thrust force more or less
directly in line with the direction of the velocity, a commonly useful thrusting
direction for most effectively increasing (or decreasing if negative) the angular
momentum and orbital energy of the trajectory.
convention is useful because, in a near-circular path, a $(0,0)$ set represents a thrust
force more or less directly in line with the direction of the velocity, a commonly
useful thrusting direction for most effectively increasing (or decreasing if negative)
the angular momentum and orbital energy of the trajectory.
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) \\
@@ -791,12 +713,12 @@
\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, limited by the maximum thrust available
to the thruster. Not all control laws allow for this fine-tuned control of the thruster.
There is 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, 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 does vary the magnitude of the thrust control. In
The approach 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

42
LaTeX/trajectory_optimization.tex
View File

@@ -6,10 +6,11 @@
highly non-linear, unpredictable systems such as this. The field that developed to
approach this problem is known as Non-Linear Programming (NLP) Optimization.
A Non-Linear Programming Problem is defined by an attempt to optimize a function
A Non-Linear Programming Problem involves finding a solution that optimizes a function
$f(\vec{x})$, subject to constraints $\vec{g}(\vec{x}) \le 0$ and $\vec{h}(\vec{x}) = 0$
where $n$ is a positive integer, $x$ is any subset of $R^n$, $g$ and $h$ can be vector
valued functions of any size, and at least one of $f$, $g$, and $h$ must be non-linear.
valued functions of any size, and at least one of $f$, $\vec{g}$, and $\vec{h}$ must be
non-linear.
There are, however, two categories of approaches to solving an NLP. The first category,
indirect methods, involve declaring a set of necessary and/or sufficient conditions for
@@ -20,10 +21,10 @@
The other category is the direct methods. In a direct optimization problem, the cost
function itself provides a value that an iterative numerical optimizer can measure
itself against. The optimal solution is then found by varying the inputs $\vec{x}$ until the
cost function is reduced to a minimum value, often determined by its derivative
jacobian. A number of tools have been developed to optimize NLPs via this direct method
in the general case.
itself against. The optimal solution is then found by varying the inputs $\vec{x}$ until
the cost function is reduced to a minimum value, often determined by its derivative
jacobian. A number of tools have been developed to formulate NLPs for optimization via
this direct method in the general case.
Both of these methods have been applied to the problem of low-thrust interplanetary
trajectory optimization \cite{Casalino2007IndirectOM} to find local optima over
@@ -40,7 +41,7 @@
Therefore, a direct optimization method was leveraged by transcribing the problem into
an NLP and using IPOPT to find the local minima.
\subsubsection{Non-Linear Solvers}
\subsection{Non-Linear Solvers}
One of the most common packages for the optimization of NLP problems is
SNOPT\cite{gill2005snopt}, which is a proprietary package written primarily using a
@@ -63,7 +64,7 @@
libraries that port these are quite modular in the sense that multiple algorithms can be
tested without changing much source code.
\subsubsection{Interior Point Linesearch Method}
\subsection{Interior Point Linesearch Method}
As mentioned above, this project utilized IPOPT which leveraged an Interior Point
Linesearch method. A linesearch algorithm is one which attempts to find the optimum
@@ -74,7 +75,7 @@
step the initial guess, now labeled $x_{k+1}$ after the addition of the ``step''
vector and iterates this process until predefined termination conditions are met.
\subsubsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem}
\subsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem}
One straightforward approach to trajectory corrections is a single shooting
algorithm, which propagates a state, given some control variables forward in time to
@@ -82,31 +83,22 @@
iterative process, using the correction scheme, until the target state and the
propagated state matches.
As an example, we can consider the Two-Point Boundary Value Problem (TPBVP) defined
by:
As an example, we can consider the one-dimensional Two-Point Boundary Value Problem
(TPBVP) defined by:
\begin{equation}
y''(t) = f(t, y(t), y'(t)), y(t_0) = y_0, y(t_f) = y_f
\end{equation}
\noindent
We can then redefine the problem as an initial-value problem:
\begin{equation}
y''(t) = f(t, y(t), y'(t)), y(t_0) = y_0, y'(t_0) = x
y''(t) = f(t, y(t), y'(t)), y(t_0) = y_0, y'(t_0) = \dot{y}_0
\end{equation}
\noindent
With $y(t,x)$ as a solution to that problem. Furthermore, if $y(t_f, x) = y_f$, then
the solution to the initial-value problem is also the solution to the TPBVP as well.
Therefore, we can use a root-finding algorithm, such as the bisection method,
Newton's Method, or even Laguerre's method, to find the roots of:
\begin{equation}
F(x) = y(t_f, x) - y_f
\end{equation}
\noindent
To find the solution to the IVP at $x_0$, $y(t_f, x_0)$ which also provides a
solution to the TPBVP. This technique for solving a Two-Point Boundary Value
Problem can be visualized in Figure~\ref{single_shoot_fig}.
@@ -114,7 +106,7 @@
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig/single_shoot}
\caption{Visualization of a single shooting technique over a trajectory arc}
\caption{Single shooting over a trajectory arc}
\label{single_shoot_fig}
\end{figure}
@@ -133,8 +125,8 @@
each of these points we can then define a separate control, which may include the
states themselves. The end state of each arc and the beginning state of the next
must then be equal for a valid arc (with the exception of velocity discontinuities
if allowed for maneuvers at that point), as well as the final state matching the
target final state.
if allowed for maneuvers or gravity assists at that point), as well as the final
state matching the target final state.
\begin{figure}[H]
\centering
@@ -144,7 +136,7 @@
\end{figure}
In this example, it can be seen that there are now more constraints (places where
the states need to match up, creating an $x_{error}$ term) as well as control
the states need to match up, creating an $\vec{x}_{error}$ term) as well as control
variables (the $\Delta V$ terms in the figure). This technique actually lends itself
very well to low-thrust arcs and, in fact, Sims-Flanagan Transcribed low-thrust arcs
in particular, because there actually are control thrusts to be optimized at a