connor/thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Unless Bosanac has a last minute change, the paper is done!

Browse Source
This commit is contained in:
Connor
2022-03-15 20:34:31 -06:00
parent fca9f32ea7
commit d00b977581
6 changed files with 26 additions and 46 deletions
Download Patch File Download Diff File Expand all files Collapse all files

38
LaTeX/trajectory_design.tex
View File

@@ -366,7 +366,7 @@
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
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
@@ -398,20 +398,19 @@
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,
$v_\infty$ will be used as a tunable parameter in the optimization routine. 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.
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.
\subsection{Gravity Assist Maneuvers}
@@ -472,7 +471,7 @@
flyby, however, can provide a useful check on what turning angles are possible for a
given flyby, since the periapsis:
\begin{equation}
\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}
@@ -504,9 +503,9 @@
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
position, $\vec{r}_1$, and final position, $\vec{r}_2$, 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
with paths that traverse less than one full orbit. Therefore, 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
@@ -559,7 +558,7 @@
\begin{equation}
c_3 = \begin{cases}
\frac{\sqrt{\psi} - \sin sqrt{\psi}}{\psi^{3/2}} \quad &\text{if} \, \psi > 10^{-6} \\
\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}
@@ -578,7 +577,7 @@
flight:
\begin{equation}
\chi = sqrt{\frac{y}{c_2}}
\chi = \sqrt{\frac{y}{c_2}}
\end{equation}
\begin{equation}
@@ -691,7 +690,7 @@
$v_{eq}$, such that the thrust equation becomes:
\begin{align}
v_{eq} &= v_e - \frac{\Delta p A_e}{\dot{m}} \\
v_{eq} &= v_e + \frac{\Delta p A_e}{\dot{m}} \\
F &= \dot{m} v_{eq} \label{isp_1}
\end{align}
@@ -717,7 +716,7 @@
\subsection{Sims-Flanagan Transcription}
this thesis chose to use a model well suited for modeling low-thrust paths: the
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.
@@ -829,10 +828,3 @@
\caption{Graphic of an orbit-raising with a high efficiency cutoff}
\label{high_efficiency_fig}
\end{figure}
All of this is, of course, also true for impulsive trajectories. However, since the
thrust presence for those trajectories are generally taken to be impulse functions, the
control laws can afford to be much less complicated for a given mission goal, by simply
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.