Skipped 1, but otherwise finished dynamics

LaTeX/flowcharts/kepler.svg

@@ -7,7 +7,7 @@
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@@ -34,12 +34,12 @@
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@@ -537,5 +537,33 @@
          style="stroke-width:0.264583"
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          y="-221.37309">asin(E)</tspan></text>
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LaTeX/trajectory_design.tex

@@ -81,7 +81,7 @@
 				\ddot{\vec{r}} = - \frac{\mu}{r^2} \hat{r}
 			\end{equation}
 
-			\subsubsection{Kepler's Laws and Equations}
+		\subsection{Kepler's Laws}
 
 			Now that we've fully qualified the forces acting within the Two Body Problem, we can concern
 			ourselves with more practical applications of it as a force model. It should be noted,
@@ -132,18 +132,17 @@
 					Where $T$ is the period and $a$ is the semi-major axis.
 			\end{enumerate}
 
-		\subsection{Analytical Solutions to Kepler's Equations}
+		\subsection{Kepler's Equation}
 
 			Kepler was able to produce an equation to represent the angular displacement of an
-			orbiting body around a primary body as a function of time, which we'll derive now
+			orbiting body around a primary body as a function of time, which we'll derive now for
-			for the elliptical case\cite{vallado2001fundamentals}. Since the total area of an
+			the elliptical case\cite{vallado2001fundamentals}. Since the total area of an ellipse is
-			ellipse is the product of $\pi$, the semi-major axis, and the semi-minor axis ($\pi
+			the product of $\pi$, the semi-major axis, and the semi-minor axis ($\pi a b$), we can
-			a b$), we can relate (by Kepler's second law) the area swept out by an orbit as a
+			relate (by Kepler's second law) the area swept out by an orbit as a function of time, as
-			function of time, as we did in Equation~\ref{swept}.
+			we did in Equation~\ref{swept}. This leaves just one unknown variable $k$, which we can
-
+			determine through use of the geometric auxiliary circle, which is a circle with radius
-			This leaves just one unknown variable $k$, which we can determine through use of the
+			equal to the ellipse's semi-major axis and center directly between the two foci, as in
-			geometric auxiliary circle, which is a circle with radius equal to the ellipse's semi-major
+			Figure~\ref{aux_circ}.
-			axis and center directly between the two foci, as in Figure~\ref{aux_circ}.
 
 			\begin{figure}[H]
 				\centering
@@ -151,13 +150,15 @@
 				\caption{Geometric Representation of Auxiliary Circle}\label{aux_circ}
 			\end{figure}
 
-			In order to find the area swept by the spacecraft, $k$, we can take advantage of the fact
+			In order to find the area swept by the spacecraft\cite{vallado2001fundamentals}, $k$, we
-			that that area is the triangle $k_1$ subtracted from the elliptical segment $PCB$:
+			can take advantage of the fact that that area is the triangle $k_1$ subtracted from the
+			elliptical segment $PCB$:
 
 			\begin{equation}\label{areas_eq}
 				k = area(seg_{PCB}) - area(k_1)
 			\end{equation}
 
+			\noindent
 			Where the area of the triangle $k_1$ can be found easily using geometric formulae:
 
 			\begin{align}
@@ -165,8 +166,11 @@
 									&= \frac{ab}{2} \left(e \sin E - \cos E \sin E \right)
 			\end{align}
 
-			Now we can find the area for the elliptical segment $PCB$ by first finding the circular
+			\noindent
-			segment $POB'$, subtracting the triangle $COB'$, then applying the fact that an ellipse is
+			Where $E$, notably, is the eccentric anomaly, or the angular distance from the
+			periapsis to the vertical projection of the spacecraft on the auxiliary circle. Now we
+			can find the area for the elliptical segment $PCB$ by first finding the circular segment
+			$POB'$, subtracting the triangle $COB'$, then applying the fact that an ellipse is
 			merely a vertical scaling of a circle by the amount $\frac{b}{a}$.
 
 			\begin{align}
@@ -185,10 +189,10 @@
 			\end{equation}
 
 			Which we can then substitute back into the equation for the swept area as a function of
-			time (Equation~\ref{swept}):
+			time (Equation~\ref{swept}) for period of time since the spacecraft left periapsis:
 
 			\begin{equation}
-				\frac{\Delta t}{T} = \frac{E - e \sin E}{2 \pi}
+				\frac{\Delta t}{T} = \frac{t_2 - t_{peri}}{T} = \frac{E - e \sin E}{2 \pi}
 			\end{equation}
 
 			Which is, effectively, Kepler's equation. It is commonly known by a different form:
@@ -200,18 +204,81 @@
 			Where we've defined the mean anomaly as $M$ and used the fact that $T =
 			\sqrt{\frac{a^3}{\mu}}$. This provides us a useful relationship between Eccentric Anomaly
 			($E$) which can be related to spacecraft position, and time, but we still need a useful
-			algorithm for solving this equation.
+			algorithm for solving this equation in order to use this equation to propagate a
+			spacecraft.
 
-		\subsubsection{LaGuerre-Conway Algorithm}\label{laguerre}
+	\subsection{LaGuerre-Conway Algorithm}\label{laguerre}
 
-			For this application, I used an algorithm known as the LaGuerre-Conway
+		For this thesis, the algorithm used to solve Kepler's equation was the general numeric
-			algorithm\cite{laguerre_conway}, which was presented in 1986 as a faster and more
+		root-finding scheme first developed by LaGuerre in the 1800s and first applied to
-			robust algorithm for directly solving Kepler's equation and has been in use in many
+		Kepler's equation by Bruce Conway in 1985\cite{laguerre_conway}. In his paper, Conway
-			applications since. This algorithm is known for its convergence robustness and also
+		makes a compelling argument for utilizing the less common LaGuerre method over higher
-			its speed of convergence when compared to higher order Newton methods.
+		order Newton or Newton-Raphson methods.
 
-			This thesis will omit a step-through of the algorithm itself, but psuedo-code for
+		The Newton-Raphson methods, while found to generally have quite impressive convergence
-			the algorithm will be discussed briefly in Section~\ref{conway_pseudocode}.
+		rates (generally successfully solving Kepler's equation correctly within 5 iterations),
+		were prone to failures in convergence given certain specific initial conditions.
+		Therefore LaGuerre's algorithm is proposed as an alternative.
+
+		The algorithm can be relatively easily derived by examining the polynomial equation with
+		$m$ roots:
+
+		\begin{equation}
+			g(x) = (x - x_1) (x - x_2) ... ( x - x_m)
+		\end{equation}
+
+		\noindent
+		We can then generate some useful convenience functions as:
+
+		\begin{align}
+			\ln|g(x)| &= \ln|(x - x_1)| + \ln|(x - x_2)| + ... + \ln|( x - x_m)| \\
+			\frac{d\ln|g(x)|}{dx} &= \frac{1}{x - x_1} + \frac{1}{x - x_2} + ... + \frac{1}{x -
+			x_m} = G_1(x)
+		\end{align}
+
+		and
+
+		\begin{align}
+			\frac{-d^2\ln|g(x)|}{dx^2} &= \frac{1}{(x - x_1)^2} + \frac{1}{(x - x_2)^2} + ... +
+			\frac{1}{(x - x_m)^2} = G_2(x)
+		\end{align}
+
+		Now we define the targeted root as $x_1$ and make the approximation that all of the
+		other roots are equidistant from the targeted root, which means:
+
+		\begin{equation}
+			x - x_i = b, i=2,3,...,m
+		\end{equation}
+
+		\noindent
+		We can then rewrite $G_1$ and $G_2$ as:
+
+		\begin{align}
+			G_1 &= \frac{1}{a} + \frac{n-1}{b} \\
+			G_2 &= \frac{1}{a^2} + \frac{n-1}{b^2}
+		\end{align}
+
+		\noindent
+		Which may be solved for $a$ in terms of $G_1$, $G_2$:
+
+		\begin{equation}
+			a = \frac{n}{G_1 \pm \sqrt{(n-1)(nG_2 - G_1^2)}}
+		\end{equation}
+
+		\noindent
+		With corresponding iteration function:
+
+		\begin{equation}
+			x_{i+1} = x_i - \frac{n g(x_i)}{g'(x_i) \pm \sqrt{(n-1)^2 f'(x_i)^2 - n (n-1) f(x_i)
+			f''(x_i)}}
+		\end{equation}
+
+		This iteration scheme can be shown to be globally convergent, regardless of the initial
+		guess. More relevantly, Conway also showed that the application of this method to
+		Kepler's equation was shown to converge with similar speed to many of the best common
+		higher order Newton-Raphson solvers. However, LaGuerre's method was also found to be
+		incredibly robust, converging to the correct value for every one of Conway's 500,000
+		tests. Because of this robustness, it is very useful for propagating spacecraft states.
 
 	\section{Interplanetary Considerations}\label{interplanetary}