diff --git a/LaTeX/fig/porkchop.png b/LaTeX/fig/porkchop.png new file mode 100644 index 0000000..347b776 Binary files /dev/null and b/LaTeX/fig/porkchop.png differ diff --git a/LaTeX/thesis.tex b/LaTeX/thesis.tex index 88369fc..536b896 100644 --- a/LaTeX/thesis.tex +++ b/LaTeX/thesis.tex @@ -162,6 +162,7 @@ Monotonic Basin Hopping} right at the moment of closest approach to the (flyby) target body. Because of this, optimization with impulsive trajectories and gravity assists are common. + % TODO: Might need to remove the HOCP stuff However, there is no physical reason why low-thrust trajectories can't also incorporate gravity assists. The optimization problem becomes much more complicated. The separate problems of optimizing flyby parameters (planet, flyby date, etc.) and optimizing the low-thrust control @@ -470,23 +471,108 @@ Monotonic Basin Hopping} trajectory within the Two-Body Problem, with only linearly-increasing computation time. \section{Interplanetary Trajectory Considerations} \label{interplanetary} - Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an - interplanetary journey. + + 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 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. However, for other uses, a more + efficient model is necessary. The method of patched conics can be applied in this case to + simplify the model. + + Interplanetary travel does not simply negatively impact trajectory optimization. The + increased complexity of the search space also opens up new opportunities for orbit + strategies. 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. \subsection{Patched Conics} - Describe the method of patched conics. + + The first hurdle to deal with is the problem of reconciling the Two-Body problem with + the presence of multiple and varying planetary bodies. The most common method for + approaching this is the method of patched conics. 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. + + The transition point can be calculated a variety of ways. The most typical method is to + calculate the gravitational force due to the two bodies separately, via the Two-Body + models. Whichever primary is a larger influence on the motion of the spacecraft is + considered to be the primary at that moment. This effectively breaks the trajectory into + a series of orbits defined by the Two-Body problem (conics), patched together by + distinct transition points. \subsection{Gravity Assist Maneuvers} - Describe how a gravity assist maneuver would work in the framework of patched conics. Also - discuss the advantages of such a maneuver. + + As previously mentioned, there are methods for utilizing the orbital energy of the other + planets in the Solar System. This is achieved via a technique known as a Gravity Assist, + or a Gravity Flyby. During a gravity assist, the spacecraft enters into the + gravitational sphere of influence of the planet and, because of its excess velocity, + proceeds to exit the sphere of influence. Relative to the planet, the speed of the + spacecraft increases as it approaches, then decreases as it departs. From the + perspective of the planet, the velocity of the spacecraft is unchanged. However, the + planet is also orbiting the Sun. + + From the perspective of a Sun-centered frame, though, this is effectively an elastic + collision. The overall momentum remains the same, with the spacecraft either gaining or + losing some in the process (dependent on the directions of travel). The planet also + loses or gains momentum enough to maintain the overall system momentum, but this amount + is negligible compared to the total momentum of the planet. The overall effect is that + the spacecraft arrives at the planet from one direction and, because of the influence of + the planet, leaves in a different direction. + + 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 + turning angle of this bend. In doing so, one can effectively achieve a (restricted) free + impulsive thrust event. \subsection{Multiple Gravity Assist Techniques} - Discuss the advantages of chaining together multiple gravity assists and highlight the - difficulties in choosing these assists. Here I can mention porkchop plots, Lambert's problem, - etc. Here I can also talk about Hybrid Optimal Control Problems. - \subsection{Ephemeris Considerations} - I can quickly mention SPICE here and talk a bit about validation. + Naturally, therefore, one would want to utilize these gravity flybys to reduce the fuel + cost to arrive at their destination target state. However, these flyby maneuvers are + quite restricted. The incoming hyperbolic velocity must be equal in magnitude to the + outgoing hyperbolic velocity. Also, the turning angle $\delta$, in the following + equation, correlates with the radius of periapsis of the hyperbolic trajectory crossing + the planet: + + \begin{equation} + r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right] + \end{equation} + + Where $v_\infty$ is the magnitude of hyperbolic velocity. Naturally, the radius of + periapsis must not fall below some safe value, in order to avoid the risk of the + spacecraft crashing into the planet or its atmosphere. + + In order to visualize which trajectories are possible within these constraints, porkchop + plots are often employed, such as the plot in Figure~\ref{porkchop}. These plots outline + various incoming and outgoing qualities of the trajectory arc between two planetary + bodies. For instance, during an arc from launch at Earth to a flyby one might plot the + launch C3 against the Mars arrival $v_\infty$ for a variety of launch and arrival dates. + + \begin{figure} + \centering + \includegraphics[width=\textwidth]{LaTeX/fig/porkchop} + \caption{A sample porkchop plot of an Earth-Mars transfer} + \label{porkchop} + \end{figure} + + This is made possible by solving Lambert's problem for the planetary ephemeris at the + epochs plotted. Lambert's problem is concerned with determining the orbit between two + positions at two different times in space. There are a number of different Lambert's + problem algorithms that allow a mission designer to determine the velocity needed (and + thus the $\Delta V$) required to achieve a position at a later time. From this, the + designer can algorithmically determine trajectory properties in the porkchop plot for + easy visualization. + + 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. % \section{Genetic Algorithms} % I will probably give only a brief overview of genetic algorithms here. 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