Getting pretty close on the final review
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Connor
@@ -1,47 +1,50 @@
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\chapter{Algorithm Overview} \label{algorithm}
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In this section, we will review the actual execution of the algorithm developed. As an
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overview, the routine was developed to enable the determination of an optimized spacecraft
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trajectory from the selection of some very basic mission parameters. Those parameters
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This thesis will attempt to develop an algorithm for the preliminary analysis of feasibility in
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designing a low-thrust interplanetary mission to an outer planet by leveraging a monotonic basin
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hopping algorithm. In this section, we will review the actual execution of the algorithm
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developed. As an overview, the routine was designed to enable the determination of an optimized
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spacecraft trajectory from the selection of some very basic mission parameters. Those parameters
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include:
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\begin{itemize}
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\item Spacecraft dry mass
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\item Thruster Specific Impulse
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\item Thruster Maximum Thrusting Force
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\setlength\itemsep{-0.5em}
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\item Spacecraft dry mass in kilograms
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\item Thruster Specific Impulse in seconds
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\item Thruster Maximum Thrusting Force in Newtons
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\item Thruster Duty Cycle Percentage
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\item Number of Thruster on Spacecraft
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\item Total Starting Weight of the Spacecraft
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\item A Maximum Acceptable $V_\infty$ at arrival and $C_3$ at launch
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\item The Launch Window Timing and the Latest Arrival
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\item Number of Thrusters on Spacecraft
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\item Total starting mass of the Spacecraft in kilograms
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\item A Maximum Acceptable $V_\infty$ at arrival in kilometers per second
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\item A Maximum Acceptable $C_3$ at launch in kilometers per second squared
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\item The Launch Window Boundaries
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\item The Latest Arrival Date
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\item A cost function relating the mass usage, $v_\infty$ at arrival, and $C_3$ at
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launch to a cost
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\item A list of flyby planets starting with Earth and ending with the destination
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\end{itemize}
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Which allows for extremely automated optimization of the trajectory, while still providing
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the mission designer with the flexibility to choose the particular flyby planets to
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investigate.
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Which allows for an automated approach to optimization of the trajectory, while still providing
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the mission designer with the flexibility to choose the particular flyby planets to investigate.
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This is achieved via an optimal control problem in which the ``inner loop'' is a
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non-linear programming problem to determine the optimal low-thrust control law and flyby
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parameters given a suitable initial guess. Then an ``outer loop'' monotonic basin hopping
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algorithm is used to traverse the search space and more carefully optimize the solutions
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found by the inner loop.
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This is achieved via an optimal control problem in which the ``inner loop'' involves solving a
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TPBVP to find the optimal solution given a suitable initial guess. Then an ``outer loop''
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monotonic basin hopping algorithm is used to traverse the search space and determine the global
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optima by repeated use of control perturbation and the inner loop.
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\section{Trajectory Composition}
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In this thesis, a specific nomenclature will be adopted to define the stages of an
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interplanetary mission in order to standardize the discussion about which aspects of the
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software affect which phases of the mission.
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interplanetary mission in order to standardize the discussion about which aspects affect
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which phases of the mission.
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Overall, a mission is considered to be the entire overall trajectory. In the context of
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this software procedure, a mission is taken to always begin at the Earth, with some
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initial launch C3 intended to be provided by an external launch vehicle. This C3 is not
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fully specified by the mission designer, but instead is optimized as a part of the
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overall cost function (and normalized by a designer-specified maximum allowable value).
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Overall, an end-to-end trajectory is considered to be the entire overall trajectory. In this
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context a trajectory begins at the Earth, with some initial launch C3 intended to be
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provided by an external launch vehicle. This C3 is not fully specified by the trajectory
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designer, but instead can be considered a part of the overall cost function for optimization
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of the Two-Point Boundary Value Problem.
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This overall mission can then be broken down into a variable number of ``phases''
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This overall trajectory can then be broken down into a variable number of ``phases''
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defined as beginning at one planetary body with some excess hyperbolic velocity and
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ending at another. The first phase of the mission is from the Earth to the first flyby
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planet. The final phase is from the last flyby planet to the planet of interest.
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@@ -59,11 +62,11 @@
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minimum altitude above the surface or atmosphere of the flyby planet.
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\end{enumerate}
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These conditions then effectively stitch the separate mission phases into a single
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coherent mission, allowing for the optimization of both individual phases and the entire
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mission as a whole. This nomenclature is similar to the nomenclature adopted by Jacob
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Englander in his Hybrid Optimal Control Problem paper, but does not allow for missions
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with multiple targets, simplifying the optimization.
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These conditions then effectively stitch the separate phases into a single coherent mission,
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allowing for the optimization of both individual phases and the entire trajectory as a whole.
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This nomenclature is similar to the nomenclature adopted by Jacob Englander in his Hybrid
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Optimal Control Problem paper, but does not allow for trajectories with multiple targets,
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simplifying the optimization.
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\section{Inner Loop Implementation}\label{inner_loop_section}
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@@ -77,15 +80,36 @@
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into an NLP, but there are essentially three primary routines involved in the inner
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loop. A given state is propagated forward using the LaGuerre-Conway Kepler solution
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algorithm, which itself is used to generate powered trajectory arcs via the
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Sims-Flanagan transcribed propagator. Finally, these powered arcs are connected via a
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multiple-shooting non-linear optimization problem. The trajectories describing each
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phase complete one ``Mission Guess'' which is fed to the non-linear solver to generate
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one valid trajectory within the vicinity of the original Mission Guess.
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Sims-Flanagan transcribed propagator. Finally, these powered arcs are connected using a
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multiple-shooting approach driven optimization. The trajectories describing each
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phase complete one ``Guess'' which is fed to the non-linear solver to generate
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one valid trajectory within the vicinity of the original Guess.
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In this formulation the cost function $F$ is a user provided function of the input Guess.
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The constraint function $G$ defines the following conditions that must be met:
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\begin{itemize}
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\item For every phase other than the final:
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\begin{itemize}
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\item The minimum periapsis of the hyperbolic flyby arc must be above some
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user-specified minimum safe altitude.
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\item The magnitude of the incoming hyperbolic velocity must match the magnitude
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of the outgoing hyperbolic velocity.
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\item The spacecraft position must match the planet's position (within bounds)
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at the end of the phase.
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\end{itemize}
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\item For the final phase:
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\begin{itemize}
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\item The spacecraft position must match the planet's position (within bounds)
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at the end of the phase.
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\item The final mass must be greater than the dry mass of the craft.
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\end{itemize}
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\end{itemize}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{flowcharts/nlp}
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\caption{A flowchart of the Non-Linear Problem Solving Formulation}
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\caption{A flowchart of the TPBVP Solution Approach}
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\label{nlp}
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\end{figure}
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@@ -141,7 +165,9 @@
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For example, from the orbital parameters of a certain state, the orbital period can
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be determined. If the system is then propagated for an integer multiple of the orbit
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period, the state should remain exactly the same as it began. In
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Figure~\ref{laguerre_plot} an example of such an orbit is provided.
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Figure~\ref{laguerre_plot} an example of such an orbit is provided in which the final
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state was tested against the initial state and found to be equal to the original to
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within $1 \times 10^{-12}$ in magnitude.
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\begin{figure}[H]
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\centering
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@@ -150,7 +176,6 @@
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approach to solving Kepler's Problem}
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\label{laguerre_plot}
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\end{figure}
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% TODO: Consider adding a paragraph about the improvements in processor time
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\subsection{Sims-Flanagan Propagator}
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@@ -179,10 +204,14 @@
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\label{sft_plot}
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\end{figure}
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Figure~\ref{sft_plot} shows that the Sims-Flanagan transcription model can be used
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to effectively model these types of orbit trajectories. In fact, the Sims-Flanagan
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model is capable of modeling nearly any low-thrust trajectory with a sufficiently
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high number of $n$ samples.
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Figure~\ref{sft_plot} shows that the Sims-Flanagan transcription model can be used to
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effectively model these types of orbit trajectories by plotting a very common ``spiral''
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trajectory in which the thrust is always on and the thrust direction is always in line
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with the direction of the velocity vector. As can be seen, this produces a spiraling
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trajectory in which the distance between spirals becomes increasingly larger as the
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trajectory achieves higher and higher distances from the Sun. In fact, the Sims-Flanagan
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model is capable of modeling nearly any low-thrust trajectory with a sufficiently high
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number of $n$ samples.
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Finally, it should be noted that, in any proper propagation scheme, mass should be
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decremented proportionally to the thrust used. The Sims-Flanagan Transcription
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@@ -199,7 +228,8 @@
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Where $\Delta m$ is the fuel used in the sub-trajectory, $\Delta t$ is the time of
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flight of the sub-trajectory, and $g_0$ is the standard gravity at the surface of
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Earth.
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Earth. From knowledge of the mass flow rate, we can then decrement the mass
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appropriately based on the magnitude of the thrust vector at each point.
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\subsection{Non-Linear Problem Solver}
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@@ -208,11 +238,11 @@
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a (proposed) trajectory. This trajectory need not be valid.
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For the purposes of discussion in this Section, we will assume that the inner-loop
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algorithm starts with just such a ''Mission Guess``, which represents the proposed
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algorithm starts with just such a ''Guess``, which represents the proposed
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trajectory. However, we'll briefly mention what quantities are needed for this
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input:
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A Mission Guess object contains:
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A Guess object contains:
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\begin{singlespacing}
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\begin{itemize}
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\item The spacecraft and thruster parameters for the mission
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@@ -233,24 +263,25 @@
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\end{itemize}
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\end{singlespacing}
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From this information, as can be seen in Figure~\ref{nlp}, we can formulate the
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mission in terms of a non-linear problem. Specifically, the Mission Guess object can
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be represented as a vector, $x$, the propagation function as a function $F$, and the
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constraints as another function $G$ such that $G(x) = \vec{0}$.
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From this information, as can be seen in Figure~\ref{nlp}, we can formulate the mission
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in terms of a non-linear programming problem. Specifically, the variables describing the
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trajectory contained within the Guess object can be represented as an input vector,
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$\vec{x}$, the cost function produced by an entire trajectory propagation as $F$, and
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the constraints that the trajectory must satisfy as another function $\vec{G}$ such that
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$\vec{G}(\vec{x}) = \vec{0}$.
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This is a format that we can apply directly to the IPOPT solver, which Julia (the
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programming language used) can utilize via bindings supplied by the SNOW.jl
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package\cite{snow}.
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IPOPT also requires the derivatives of both the $F$ and $G$ functions in the
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formulation above. Generally speaking, a project designer has two options for
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determining derivatives. The first option is to analytically determine the
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derivatives, guaranteeing accuracy, but requiring processor time if determined
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algorithmically and sometimes simply impossible or mathematically very rigorous to
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determine manually. The second option is to numerically derive the derivatives,
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using a technique such as finite differencing. This limits the accuracy, but can be
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faster than algorithmic symbolic manipulation and doesn't require rigorous manual
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derivations.
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IPOPT also requires the derivatives of both the $F$ and $G$ functions in the formulation
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above with respect to the input $\vec{x}$ vector. There are two options for determining
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derivatives. The first option is to analytically determine the derivatives, guaranteeing
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accuracy, but requiring processor time if determined algorithmically and sometimes
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simply impossible or mathematically very rigorous to determine manually. The second
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option is to numerically approximate the derivatives, using a technique such as finite
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differencing. This limits the accuracy, but can be faster than algorithmic symbolic
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manipulation and doesn't require rigorous manual derivations.
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However, the Julia language has an excellent interface to a new technique, known as
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automatic differentiation\cite{RevelsLubinPapamarkou2016}. Automatic differentiation
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@@ -271,10 +302,10 @@
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\section{Outer Loop Implementation}
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Now we have the tools in place for, given a potential ''mission guess`` in the
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Now we have the tools in place for, given a potential ''guess`` in the
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vicinity of a valid guess, attempting to find a valid and optimal solution in that
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vicinity. Now what remains is to develop a routine for efficiently generating these
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random mission guesses in such a way that thoroughly searches the entirety of the
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random guesses in such a way that thoroughly searches the entirety of the
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solution space with enough granularity that all spaces are considered by the inner loop
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solver.
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@@ -293,14 +324,14 @@
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\label{mbh_flow}
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\end{figure}
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\subsection{Random Mission Generation}\label{random_gen_section}
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\subsection{Random Trajectory Generation}\label{random_gen_section}
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At a basic level, the algorithm needs to produce a mission guess (represented by all
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of the values described in Section~\ref{inner_loop_section}) that contains random
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values within reasonable bounds in the space. This leaves a number of variables open
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to for implementation. For instance, it remains to be determined which distribution
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function to use for the random values over each of those variables, which bounds to
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use, as well as the possibilities for any improvements to a purely random search.
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At a basic level, the algorithm needs to produce a guess (represented by all of the
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values described in Section~\ref{inner_loop_section}) that contains random values within
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reasonable bounds in the space. However, that still leaves the determination of which
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distribution function to use for the random values over each of those variables, which
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bounds to use, as well as the possibilities for any improvements to a purely random
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search.
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Currently, the first value set for the mission guess is that of $n$, which is the
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number of sub-trajectories that each arc will be broken into for the Sims-Flanagan
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@@ -351,22 +382,20 @@
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velocity was calculated. However, instead of multiplying the randomly generate unit
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direction vector by a random number between 0 and the square root of the maximum
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launch $C_3$, bounds of 0 and 10 kilometers per second are used instead, to provide
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realistic flyby values.
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realistic flyby values\cite{englander2014tuning}.
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The outgoing excess hyperbolic velocity at infinity is then calculated by again
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choosing a uniform random unit direction vector, then by multiplying this value by
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the magnitude of the incoming $v_{\infty}$ since this is a constraint of a
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non-powered flyby.
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From these two velocity vectors the turning angle, and thus the periapsis of the
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flyby, can then be calculated by the following equations:
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From these two velocity vectors the turning angle, and thus the periapsis of the flyby,
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can then be calculated by Equation~\ref{turning_angle_eq} and the following equation:
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\begin{align}
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\delta &= \arccos \left( \frac{\vec{v}_{\infty,in} \cdot
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\vec{v}_{\infty,out}}{|v_{\infty,in}| \cdot {|v_{\infty,out}}|} \right) \\
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r_p &= \frac{\mu}{\vec{v}_{\infty,in} \cdot \vec{v}_{\infty,out}} \cdot \left(
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\begin{equation}
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r_p = \frac{\mu}{\vec{v}_{\infty,in} \cdot \vec{v}_{\infty,out}} \cdot \left(
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\frac{1}{\sin(\delta/2)} - 1 \right)
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\end{align}
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\end{equation}
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If this radius of periapse is then found to be less than the minimum safe radius
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(currently set to the radius of the planet plus 100 kilometers), then the process is
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@@ -375,17 +404,96 @@
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solver.
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The final requirement then, is the thrust controls, which are actually quite simple.
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Since the thrust is defined as a 3-vector of values between -1 and 1 representing
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some percentage of the full thrust producible by the spacecraft thrusters in that
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direction, the initial thrust controls can then be generated as a $20 \times 3$
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matrix of uniform random numbers within that bound.
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Since the thrust is defined as a 3-vector of values between -1 and 1 representing some
|
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percentage of the full thrust producible by the spacecraft thrusters in that direction,
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the initial thrust controls can then be generated as a $20 \times 3$ matrix of uniform
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random numbers within that bound. The number 20 was chosen as the number of
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subtrajectories per phase to provide reasonable fidelity for allowing phases to run
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longer (on the order of 2 or 3 orbits) without sacrificing speed per Englander
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\cite{englander2012automated}. One possible improvement would be to choose the number
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more intelligently based on the expected number of revolutions.
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\subsection{Monotonic Basin Hopping}\label{mbh_subsection}
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Now that a generator has been developed for mission guesses, we can build the
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Now that a generator has been developed for guesses, we can build the
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monotonic basin hopping algorithm. Since the implementation of the MBH algorithm
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used in this paper differs from the standard implementation, the standard version
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won't be described here. Instead, the variation used in this paper, with some
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performance improvements, will be considered.
|
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|
||||
The aim of a monotonic basin hopping algorithm is to provide an efficient method for
|
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completely traversing a large search space and providing many seed values within the
|
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space for an ``inner loop'' solver or optimizer. These solutions are then perturbed
|
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slightly, in order to provide higher fidelity searching in the space near valid
|
||||
solutions in order to fully explore the vicinity of discovered local minima. This
|
||||
makes it an excellent algorithm for problems with a large search space, including
|
||||
several clusters of local minima, such as this application.
|
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|
||||
The algorithm contains two loops, the size of each of which can be independently
|
||||
modified (generally by specifying a ``patience value'', or number of loops to
|
||||
perform, for each) to account for trade-offs between accuracy and performance depending on
|
||||
mission needs and the unique qualities of a certain search space.
|
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|
||||
The first loop, the ``search loop'', first calls the random mission generator. This
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generator produces two random missions as described in
|
||||
Section~\ref{random_gen_section} that differ only in that one contains random flyby
|
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velocities and control thrusts and the other contains Lambert's-solved flyby
|
||||
velocities and zero control thrusts. For each of these guesses, the NLP solver is
|
||||
called. If either of these mission guesses have converged onto a valid solution, the
|
||||
lower loop, the ``drill loop'' is entered for the valid solution. After the
|
||||
convergence checks and potentially drill loops are performed, if a valid solution
|
||||
has been found, this solution is stored in an archive. If the solution found is
|
||||
better than the current best solution in the archive (as determined by a
|
||||
user-provided cost function of fuel usage, $C_3$ at launch, and $v-\infty$ at
|
||||
arrival) then the new solution replaces the current best solution and the loop is
|
||||
repeated. Taken by itself, the search loop should quickly generate enough random
|
||||
mission guesses to find all ``basins'' or areas in the solution space with valid
|
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trajectories, but never attempts to more thoroughly explore the space around valid
|
||||
solutions within these basins.
|
||||
|
||||
The drill loop, then, is used for this purpose. For the first step of the drill
|
||||
loop, the current solution is saved as the ``basin solution''. If it's better than
|
||||
the current best, it also replaces the current best solution. Then, until the
|
||||
stopping condition has been met (generally when the ``drill counter'' has reached
|
||||
the ``drill patience'' value) the current solution is perturbed slightly by adding
|
||||
or subtracting a small random value to the components of the mission.
|
||||
|
||||
The performance of this perturbation in terms of more quickly converging upon the
|
||||
true minimum of that particular basin, as described in detail by
|
||||
Englander\cite{englander2014tuning}, is highly dependent on the distribution
|
||||
function used for producing these random perturbations. While the intuitive choice
|
||||
of a simple Gaussian distribution would make sense to use, it has been found that a
|
||||
long-tailed distribution, such as a Cauchy distribution or a Pareto distribution is
|
||||
more robust in terms of well chose boundary conditions and initial seed solutions as
|
||||
well as more performant in time required to converge upon the minimum for that basin.
|
||||
|
||||
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
|
||||
of 0 and 1, $\epsilon$ is a small value (nominally set to $1e-10$), and $\alpha$ is
|
||||
a tuning parameter to determine the size of the tails and width of the distribution
|
||||
set to $1.01$, but easily tunable.
|
||||
|
||||
The perturbation function then steps through each parameter of the mission,
|
||||
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Otherwise, the drill counter increments and the process is repeated. Therefore, the
|
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id="tspan295443">False</tspan></text>
|
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id="tspan290574">False</tspan></text>
|
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|
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xml:space="preserve"
|
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transform="matrix(0.26458333,0,0,0.26458333,-34.454704,17.217553)"
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@@ -1189,6 +1061,42 @@
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id="tspan295445">True</tspan></text>
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style="fill:none;stroke:#000000;stroke-width:0.264583px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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|
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|
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Before Width: | Height: | Size: 46 KiB After Width: | Height: | Size: 42 KiB |
@@ -277,4 +277,23 @@ URL = {https://arc.aiaa.org/doi/abs/10.2514/6.2006-6746},
|
||||
eprint = {https://arc.aiaa.org/doi/pdf/10.2514/6.2006-6746}
|
||||
}
|
||||
|
||||
@article{battin1984elegant,
|
||||
title={An elegant Lambert algorithm},
|
||||
author={Battin, Richard H and Vaughan, Robin M},
|
||||
journal={Journal of Guidance, Control, and Dynamics},
|
||||
volume={7},
|
||||
number={6},
|
||||
pages={662--670},
|
||||
year={1984}
|
||||
}
|
||||
|
||||
@article{wales1997global,
|
||||
title={Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms},
|
||||
author={Wales, David J and Doye, Jonathan PK},
|
||||
journal={The Journal of Physical Chemistry A},
|
||||
volume={101},
|
||||
number={28},
|
||||
pages={5111--5116},
|
||||
year={1997},
|
||||
publisher={ACS Publications}
|
||||
}
|
||||
|
||||
@@ -4,6 +4,7 @@
|
||||
\usepackage{amssymb}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{float}
|
||||
\usepackage{xfrac}
|
||||
|
||||
|
||||
@@ -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.
|
||||
|
||||
|
||||
|
||||
@@ -1,61 +1,57 @@
|
||||
\chapter{Trajectory Optimization} \label{traj_optimization}
|
||||
|
||||
\section{Solving Boundary Value Problems}
|
||||
\section{Optimization of Boundary Value Problems}
|
||||
|
||||
This section probably needs more work.
|
||||
An approach is necessary, in trajectory optimization and many other fields, to optimize
|
||||
highly non-linear, unpredictable systems such as this. The field that developed to
|
||||
approach this problem is known as Non-Linear Programming (NLP) Optimization.
|
||||
|
||||
\section{Optimization}
|
||||
|
||||
\subsection{Non-Linear Problem Optimization}
|
||||
|
||||
Now we can consider the formulation of the problem in a more useful way. For instance, given a
|
||||
desired final state in position and velocity we can relatively easily determine the initial
|
||||
state necessary to end up at that desired state over a pre-defined period of time by solving
|
||||
Kepler's equation. In fact, this is often how impulsive trajectories are calculated since,
|
||||
other than the impulsive thrusting event itself, the trajectory is entirely natural.
|
||||
|
||||
However, often in trajectory design we want to consider a number of other inputs. For
|
||||
instance, a low thrust profile, a planetary flyby, the effects of rotating a solar panel on
|
||||
solar radiation pressure, etc. Once these inputs have been accepted as part of the model, the
|
||||
system is generally no longer analytically solvable, or, if it is, is too complex to calculate
|
||||
directly.
|
||||
|
||||
Therefore an approach is needed, in trajectory optimization and many other fields, to optimize
|
||||
highly non-linear, unpredictable systems such as this. The field that developed to approach
|
||||
this problem is known as Non-Linear Problem (NLP) Optimization.
|
||||
A Non-Linear Programming Problem is defined by an attempt to optimize 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.
|
||||
|
||||
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 declaring
|
||||
the solution optimal. These conditions then allow the non-linear problem (generally) to be
|
||||
reformulated as a two point boundary value problem. Solving this boundary value problem can
|
||||
provide a control law for the optimal path. Indirect approaches for spacecraft trajectory
|
||||
indirect methods, involve declaring a set of necessary and/or sufficient conditions for
|
||||
optimality. These conditions then allow the problem to be reformulated as a two point
|
||||
boundary value problem. Solving this boundary value problem involves determining a
|
||||
control law for the optimal path. Indirect approaches for spacecraft trajectory
|
||||
optimization have given us the Primer Vector Theory\cite{jezewski1975primer}.
|
||||
|
||||
The other category is the direct methods. In a direct optimization problem, the cost
|
||||
function itself is calculated to provide the optimal solution. The problem is usually
|
||||
thought of as a collection of dynamics and controls. Then these controls can be modified
|
||||
to minimize the cost function. A number of tools have been developed to optimize NLPs
|
||||
via this direct method in the general case. For this particular problem, direct
|
||||
approaches were used as the low-thrust interplanetary system dynamics adds too much
|
||||
complexity to quickly optimize indirectly and the individual optimization routines
|
||||
needed to proceed as quickly as possible.
|
||||
function itself provides a value that an iterative numerical optimizer can measure
|
||||
itself against. The optimal solution is then found by varying the inputs $x$ 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.
|
||||
|
||||
Both of these methods have been applied to the problem of low-thrust interplanetary
|
||||
trajectory optimization \cite{Casalino2007IndirectOM} to find local optima over
|
||||
low-thrust control laws. It has often been found that 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, discussed later, is leveraged to attempt to find all minima basins in
|
||||
the solution space by ``hopping'' around with different initial guesses. It is,
|
||||
therefore, 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.
|
||||
|
||||
\subsubsection{Non-Linear Solvers}
|
||||
For these types of non-linear, constrained problems, a number of tools have been developed
|
||||
that act as frameworks for applying a large number of different algorithms. This allows for
|
||||
simple testing of many different algorithms to find what works best for the nuances of the
|
||||
problem in question.
|
||||
|
||||
One of the most common of these NLP optimizers is SNOPT\cite{gill2005snopt}, which
|
||||
is a proprietary package written primarily using a number of Fortran libraries by
|
||||
the Systems Optimization Laboratory at Stanford University. It uses a sparse
|
||||
sequential quadratic programming approach.
|
||||
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
|
||||
number of Fortran libraries by the Systems Optimization Laboratory at Stanford
|
||||
University. It uses a sparse sequential quadratic programming algorithm as its
|
||||
back-end optimization scheme.
|
||||
|
||||
Another common NLP optimization packages (and the one used in this implementation)
|
||||
is the Interior Point Optimizer or IPOPT\cite{wachter2006implementation}. It can be
|
||||
used in much the same way as SNOPT and uses an Interior Point Linesearch Filter
|
||||
Method and was developed as an open-source project by the organization COIN-OR under
|
||||
the Eclipse Public License.
|
||||
is the Interior Point Optimizer or IPOPT\cite{wachter2006implementation}. It uses
|
||||
an Interior Point Linesearch Filter Method and was developed as an open-source
|
||||
project by the organization COIN-OR under the Eclipse Public License.
|
||||
|
||||
Both of these methods utilize similar approaches to solve general constrained non-linear
|
||||
problems iteratively. Both of them can make heavy use of derivative Jacobians and Hessians
|
||||
@@ -67,14 +63,14 @@
|
||||
libraries that port these are quite modular in the sense that multiple algorithms can be
|
||||
tested without changing much source code.
|
||||
|
||||
\subsubsection{Linesearch Method}
|
||||
\subsubsection{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
|
||||
of a non-linear problem by first taking an initial guess $x_k$. The algorithm then
|
||||
determines a step direction (in this case through the use of either automatic
|
||||
differentiation or finite differencing to calculate the derivatives of the
|
||||
non-linear problem) and a step length. The linesearch algorithm then continues to
|
||||
cost function) and a step length. The linesearch algorithm then continues to
|
||||
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.
|
||||
|
||||
@@ -83,15 +79,42 @@
|
||||
was sufficient merely that the non-linear constraints were met, therefore optimization (in
|
||||
the particular step in which IPOPT was used) was unnecessary.
|
||||
|
||||
\subsubsection{Multiple-Shooting Algorithms}
|
||||
\subsubsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem}
|
||||
|
||||
Now that we have software defined to optimize non-linear problems, what remains is
|
||||
determining the most effective way to define the problem itself. The most simple
|
||||
form of a trajectory optimization might employ a single shooting algorithm, which
|
||||
propagates a state, given some control variables forward in time to the epoch of
|
||||
interest. The controls over this time period are then modified in an iterative
|
||||
process, using the NLP optimizer, until the target state and the propagated state
|
||||
matches. This technique can be visualized in Figure~\ref{single_shoot_fig}.
|
||||
One straightforward approach to trajectory corrections is a single shooting
|
||||
algorithm, which propagates a state, given some control variables forward in time to
|
||||
the epoch of interest. The controls over this time period are then modified in an
|
||||
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:
|
||||
|
||||
\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
|
||||
\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}.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
@@ -108,15 +131,15 @@
|
||||
as this one.
|
||||
|
||||
However, some problems require the use of a more flexible algorithm. In these cases,
|
||||
sometimes a multiple-shooting algorithm can provide that flexibility and allow the
|
||||
NLP solver to find the optimal control faster. In a multiple shooting algorithm,
|
||||
rather than having a single target point at which the propagated state is compared,
|
||||
the target orbit is broken down into multiple arcs, then end of each of which can be
|
||||
seen as a separate target. At each of these points we can then define a separate
|
||||
control. The end state of each arc and the beginning state of the next must then be
|
||||
equal for a valid arc, as well as the final state matching the target final state.
|
||||
This changes the problem to have far more constraints, but also increased freedom
|
||||
due to having more control variables.
|
||||
sometimes a multiple-shooting algorithm can provide that flexibility and reduced
|
||||
sensitivity. In a multiple shooting algorithm, rather than having a single target
|
||||
point at which the propagated state is compared, the target orbit is broken down
|
||||
into multiple arcs, then end of each of which can be seen as a separate target. At
|
||||
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.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
@@ -136,81 +159,21 @@
|
||||
|
||||
\section{Monotonic Basin Hopping Algorithms}
|
||||
|
||||
% TODO: This needs to be rewritten to be general, then add the appropriate specific
|
||||
% implementation details to the approach chapter
|
||||
The techniques discussed thus far are useful for finding local optima. However, we would
|
||||
also like to traverse the search space in an attempt to determine the global optima over the
|
||||
entire search space. One approach to this would be to discretize the search space and test
|
||||
each point as an input to a local optimization scheme. In order to achieve sufficient
|
||||
coverage of the search space, however, this often requires long processing times in a
|
||||
high-dimensional environment.
|
||||
|
||||
The aim of a monotonic basin hopping algorithm is to provide an efficient method for
|
||||
completely traversing a large search space and providing many seed values within the
|
||||
space for an ''inner loop`` solver or optimizer. These solutions are then perturbed
|
||||
slightly, in order to provide higher fidelity searching in the space near valid
|
||||
solutions in order to fully explore the vicinity of discovered local minima. This
|
||||
makes it an excellent algorithm for problems with a large search space, including
|
||||
several clusters of local minima, such as this application.
|
||||
|
||||
The algorithm contains two loops, the size of each of which can be independently
|
||||
modified (generally by specifying a ''patience value``, or number of loops to
|
||||
perform, for each) to account for trade-offs between accuracy and performance depending on
|
||||
mission needs and the unique qualities of a certain search space.
|
||||
|
||||
The first loop, the ''search loop``, first calls the random mission generator. This
|
||||
generator produces two random missions as described in
|
||||
Section~\ref{random_gen_section} that differ only in that one contains random flyby
|
||||
velocities and control thrusts and the other contains Lambert's-solved flyby
|
||||
velocities and zero control thrusts. For each of these guesses, the NLP solver is
|
||||
called. If either of these mission guesses have converged onto a valid solution, the
|
||||
lower loop, the ''drill loop`` is entered for the valid solution. After the
|
||||
convergence checks and potentially drill loops are performed, if a valid solution
|
||||
has been found, this solution is stored in an archive. If the solution found is
|
||||
better than the current best solution in the archive (as determined by a
|
||||
user-provided cost function of fuel usage, $C_3$ at launch, and $v-\infty$ at
|
||||
arrival) then the new solution replaces the current best solution and the loop is
|
||||
repeated. Taken by itself, the search loop should quickly generate enough random
|
||||
mission guesses to find all ''basins`` or areas in the solution space with valid
|
||||
trajectories, but never attempts to more thoroughly explore the space around valid
|
||||
solutions within these basins.
|
||||
|
||||
The drill loop, then, is used for this purpose. For the first step of the drill
|
||||
loop, the current solution is saved as the ''basin solution``. If it's better than
|
||||
the current best, it also replaces the current best solution. Then, until the
|
||||
stopping condition has been met (generally when the ''drill counter`` has reached
|
||||
the ''drill patience`` value) the current solution is perturbed slightly by adding
|
||||
or subtracting a small random value to the components of the mission.
|
||||
|
||||
The performance of this perturbation in terms of more quickly converging upon the
|
||||
true minimum of that particular basin, as described in detail by
|
||||
Englander\cite{englander2014tuning}, is highly dependent on the distribution
|
||||
function used for producing these random perturbations. While the intuitive choice
|
||||
of a simple Gaussian distribution would make sense to use, it has been found that a
|
||||
long-tailed distribution, such as a Cauchy distribution or a Pareto distribution is
|
||||
more robust in terms of well chose boundary conditions and initial seed solutions as
|
||||
well as more performant in time required to converge upon the minimum for that basin.
|
||||
|
||||
Because of this, the perturbation used in this implementation follows a
|
||||
bi-directional, long-tailed Pareto distribution generated by the following
|
||||
probability density function:
|
||||
|
||||
\begin{equation}
|
||||
1 +
|
||||
\left[ \frac{s}{\epsilon} \right] \cdot
|
||||
\left[ \frac{\alpha - 1}{\frac{\epsilon}{\epsilon + r}^{-\alpha}} \right]
|
||||
\end{equation}
|
||||
|
||||
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
|
||||
of 0 and 1, $\epsilon$ is a small value (nominally set to $1e-10$), and $\alpha$ is
|
||||
a tuning parameter to determine the size of the tails and width of the distribution
|
||||
set to $1.01$, but easily tunable.
|
||||
|
||||
The perturbation function then steps through each parameter of the mission,
|
||||
generating a new guess with the parameters modified by the Pareto distribution.
|
||||
After this perturbation, the NLP solver is then called again to find a valid
|
||||
solution in the vicinity of this new guess. If the solution is better than the
|
||||
current basin solution, it replaces that value and the drill counter is reset to
|
||||
zero. If it is better than the current total best, it replaces that value as well.
|
||||
Otherwise, the drill counter increments and the process is repeated. Therefore, the
|
||||
drill patience allows the mission designer to determine a maximum number of
|
||||
iterations to perform without improvement in a row before ending the drill loop.
|
||||
This process can be repeated essentially ''search patience`` number of times in
|
||||
order to fully traverse all basins.
|
||||
To solve this problem, a technique was described by Wales and Doye in 1997
|
||||
\cite{wales1997global} called a basin-hopping algorithm. This algorithm performs a random
|
||||
perturbation of the input states, optimizes using a local optimizer, then either accepts the
|
||||
new coordinates and performs a further perturbation, or rejects them and tests a new set of
|
||||
randomly-generated inputs.
|
||||
|
||||
This allows the algorithm to test many different regions of the search space in order to
|
||||
determine which ``basins'' contain local optima. If these local optima are found, the
|
||||
algorithm can attempt to improve the local optima based on parameters defined by the
|
||||
algorithm designer (by perturbation and re-applying the local optimization scheme) or search
|
||||
for a new basin in the search space.
|
||||
|
||||
17
julia/lamberts_fig.jl
Normal file
17
julia/lamberts_fig.jl
Normal file
@@ -0,0 +1,17 @@
|
||||
using PlotlyJS: savefig
|
||||
|
||||
r1 = oe_to_xyz([Earth.a, 0.6, 0.001, 0., 0., deg2rad(10.)], Sun.μ)[1:3]
|
||||
r2 = oe_to_xyz([Earth.a, 0.6, 0.001, 0., 0., deg2rad(140.)], Sun.μ)[1:3]
|
||||
tof = 0.5year
|
||||
v1 = Thesis.lamberts1(r1,r2,tof)[1]
|
||||
v2 = Thesis.lamberts2(r1,r2,tof)[1]
|
||||
state1 = [ r1; v1; 1000. ]
|
||||
state2 = [ r1; v2; 1000. ]
|
||||
path1 = prop(state1, tof)
|
||||
path2 = prop(state2, tof)
|
||||
plot([path2, path1],
|
||||
title="Lambert's Problem Solutions",
|
||||
colors=["#F00", "#0FF"],
|
||||
labels=["Type I", "Type II"],
|
||||
mode="light")
|
||||
|
||||
@@ -71,6 +71,120 @@ function lamberts(planet1::Body,planet2::Body,leave::DateTime,arrive::DateTime)
|
||||
|
||||
end
|
||||
|
||||
function lamberts1(r1::Vector{Float64},r2::Vector{Float64},tof_req::Float64)
|
||||
μ = Sun.μ
|
||||
r1mag = norm(r1)
|
||||
r2mag = norm(r2)
|
||||
|
||||
cos_dθ = dot(r1,r2)/(r1mag*r2mag)
|
||||
dθ = atan(r2[2],r2[1]) - atan(r1[2],r1[1])
|
||||
dθ = dθ > 2π ? dθ-2π : dθ
|
||||
dθ = dθ < 0.0 ? dθ+2π : dθ
|
||||
DM = -1
|
||||
A = DM * √(r1mag * r2mag * (1 + cos_dθ))
|
||||
dθ == 0 || A == 0 && error("Can't solve Lambert's Problem")
|
||||
|
||||
ψ, c2, c3 = 0, 1//2, 1//6
|
||||
ψ_down = -4π ; ψ_up = 4π^2
|
||||
y = r1mag + r2mag + (A*(ψ*c3 - 1)) / √(c2) ; χ = √(y/c2)
|
||||
tof = ( χ^3*c3 + A*√(y) ) / √(μ)
|
||||
|
||||
i = 0
|
||||
while abs(tof-tof_req) > 1e-2
|
||||
y = r1mag + r2mag + (A*(ψ*c3 - 1)) / √(c2)
|
||||
while y/c2 <= 0
|
||||
# println("You finally hit that weird issue... ")
|
||||
ψ += 0.1
|
||||
if ψ > 1e-6
|
||||
c2 = (1 - cos(√(ψ))) / ψ ; c3 = (√(ψ) - sin(√(ψ))) / √(ψ^3)
|
||||
elseif ψ < -1e-6
|
||||
c2 = (1 - cosh(√(-ψ))) / ψ ; c3 = (-√(-ψ) + sinh(√(-ψ))) / √((-ψ)^3)
|
||||
else
|
||||
c2 = 1//2 ; c3 = 1//6
|
||||
end
|
||||
y = r1mag + r2mag + (A*(ψ*c3 - 1)) / √(c2)
|
||||
end
|
||||
χ = √(y/c2)
|
||||
|
||||
tof = ( c3*χ^3 + A*√(y) ) / √(μ)
|
||||
tof < tof_req ? ψ_down = ψ : ψ_up = ψ
|
||||
ψ = (ψ_up + ψ_down) / 2
|
||||
|
||||
if ψ > 1e-6
|
||||
c2 = (1 - cos(√(ψ))) / ψ ; c3 = (√(ψ) - sin(√(ψ))) / √(ψ^3)
|
||||
elseif ψ < -1e-6
|
||||
c2 = (1 - cosh(√(-ψ))) / ψ ; c3 = (-√(-ψ) + sinh(√(-ψ))) / √((-ψ)^3)
|
||||
else
|
||||
c2 = 1//2 ; c3 = 1//6
|
||||
end
|
||||
|
||||
i += 1
|
||||
i > 500 && return [NaN,NaN,NaN],[NaN,NaN,NaN]
|
||||
end
|
||||
|
||||
f = 1 - y/r1mag ; g_dot = 1 - y/r2mag ; g = A * √(y/μ)
|
||||
v0t = (r2 - f*r1)/g ; vft = (g_dot*r2 - r1)/g
|
||||
return v0t, vft, tof_req
|
||||
|
||||
end
|
||||
|
||||
function lamberts2(r1::Vector{Float64},r2::Vector{Float64},tof_req::Float64)
|
||||
μ = Sun.μ
|
||||
r1mag = norm(r1)
|
||||
r2mag = norm(r2)
|
||||
|
||||
cos_dθ = dot(r1,r2)/(r1mag*r2mag)
|
||||
dθ = atan(r2[2],r2[1]) - atan(r1[2],r1[1])
|
||||
dθ = dθ > 2π ? dθ-2π : dθ
|
||||
dθ = dθ < 0.0 ? dθ+2π : dθ
|
||||
DM = 1
|
||||
A = DM * √(r1mag * r2mag * (1 + cos_dθ))
|
||||
dθ == 0 || A == 0 && error("Can't solve Lambert's Problem")
|
||||
|
||||
ψ, c2, c3 = 0, 1//2, 1//6
|
||||
ψ_down = -4π ; ψ_up = 4π^2
|
||||
y = r1mag + r2mag + (A*(ψ*c3 - 1)) / √(c2) ; χ = √(y/c2)
|
||||
tof = ( χ^3*c3 + A*√(y) ) / √(μ)
|
||||
|
||||
i = 0
|
||||
while abs(tof-tof_req) > 1e-2
|
||||
y = r1mag + r2mag + (A*(ψ*c3 - 1)) / √(c2)
|
||||
while y/c2 <= 0
|
||||
# println("You finally hit that weird issue... ")
|
||||
ψ += 0.1
|
||||
if ψ > 1e-6
|
||||
c2 = (1 - cos(√(ψ))) / ψ ; c3 = (√(ψ) - sin(√(ψ))) / √(ψ^3)
|
||||
elseif ψ < -1e-6
|
||||
c2 = (1 - cosh(√(-ψ))) / ψ ; c3 = (-√(-ψ) + sinh(√(-ψ))) / √((-ψ)^3)
|
||||
else
|
||||
c2 = 1//2 ; c3 = 1//6
|
||||
end
|
||||
y = r1mag + r2mag + (A*(ψ*c3 - 1)) / √(c2)
|
||||
end
|
||||
χ = √(y/c2)
|
||||
|
||||
tof = ( c3*χ^3 + A*√(y) ) / √(μ)
|
||||
tof < tof_req ? ψ_down = ψ : ψ_up = ψ
|
||||
ψ = (ψ_up + ψ_down) / 2
|
||||
|
||||
if ψ > 1e-6
|
||||
c2 = (1 - cos(√(ψ))) / ψ ; c3 = (√(ψ) - sin(√(ψ))) / √(ψ^3)
|
||||
elseif ψ < -1e-6
|
||||
c2 = (1 - cosh(√(-ψ))) / ψ ; c3 = (-√(-ψ) + sinh(√(-ψ))) / √((-ψ)^3)
|
||||
else
|
||||
c2 = 1//2 ; c3 = 1//6
|
||||
end
|
||||
|
||||
i += 1
|
||||
i > 500 && return [NaN,NaN,NaN],[NaN,NaN,NaN]
|
||||
end
|
||||
|
||||
f = 1 - y/r1mag ; g_dot = 1 - y/r2mag ; g = A * √(y/μ)
|
||||
v0t = (r2 - f*r1)/g ; vft = (g_dot*r2 - r1)/g
|
||||
return v0t, vft, tof_req
|
||||
|
||||
end
|
||||
|
||||
function porkchop(planet1::Body, planet2::Body, departures::Vector{DateTime}, arrivals::Vector{DateTime})
|
||||
v∞_in = [ norm(lamberts(planet1, planet2, depart, arrive)[2]) for depart in departures, arrive in arrivals ]
|
||||
v∞_out = [ norm(lamberts(planet1, planet2, depart, arrive)[1]) for depart in departures, arrive in arrivals ]
|
||||
|
||||
@@ -89,14 +89,22 @@ function standard_layout(limit::Float64, title::AbstractString; mode="dark")
|
||||
plot_bgcolor="rgba(255,255,255,0.0)",
|
||||
scene_aspectmode = "data",
|
||||
scene_xaxis = attr(
|
||||
title = "X (km)",
|
||||
exponentformat = "power",
|
||||
autorange = true,
|
||||
color="rgb(0,0,0)"
|
||||
),
|
||||
scene_yaxis = attr(
|
||||
title = "Y (km)",
|
||||
exponentformat = "power",
|
||||
autorange = true,
|
||||
color="rgb(0,0,0)"
|
||||
),
|
||||
scene_zaxis_visible = false
|
||||
scene_zaxis = attr(
|
||||
title = "Z (km)",
|
||||
exponentformat = "power",
|
||||
visible = false,
|
||||
),
|
||||
)
|
||||
end
|
||||
end
|
||||
@@ -287,7 +295,7 @@ function plot(paths::Vector{Matrix{Real}};
|
||||
traces = [ trace... ]
|
||||
for i = 2:length(paths)
|
||||
color = colors === nothing ? random_color() : colors[i]
|
||||
trace, new_limit = gen_plot(paths[i],label=labels[i],color=colors[i],markers=markers)
|
||||
trace, new_limit = gen_plot(paths[i],label=labels[i],color=color,markers=markers)
|
||||
push!(traces, trace...)
|
||||
limit = max(limit, new_limit)
|
||||
end
|
||||
|
||||
Reference in New Issue
Block a user