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Use correct glossary entry
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@skulumani
skulumani committed Jul 6, 2018
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28 changes: 14 additions & 14 deletions tex/appendix/relative_eoms_derivation.tex
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Expand Up @@ -26,29 +26,29 @@ \section{Relative variations}\label{sec:relative_variations}
\subsection{Relative Position Variation}\label{sec:relative_position_variation}
The variation of the relative position given in~\cref{eq:relative_position} is defined as
\begin{align}
\delta \gls{sym:rpos} &= \left. \diff{}{\epsilon} \right|_{\epsilon=0} \gls{sym:Ra}[^T] \gls{sym:ipos}^\epsilon ,\nonumber \\
&= \gls{sym:Ra}[^T] \delta \gls{sym:ipos} .\label{eq:relative_position_variation}
\delta \gls{sym:rpos} &= \left. \diff{}{\epsilon} \right|_{\epsilon=0} \gls{sym:aatt}[^T] \gls{sym:ipos}^\epsilon ,\nonumber \\
&= \gls{sym:aatt}[^T] \delta \gls{sym:ipos} .\label{eq:relative_position_variation}
\end{align}

\subsection{Relative Attitude Variation}\label{sec:relative_attitude_variation}
The variation of the relative attitude given in~\cref{eq:relative_attitude} is expressed as
\begin{align}
\delta \gls{sym:ratt} &= \left. \diff{}{\epsilon} \right|_{\epsilon=0} \gls{sym:Ra}[^T] \gls{sym:iatt}^\epsilon = \gls{sym:Ra}[^T] \delta \gls{sym:iatt}, \nonumber\\
&= \gls{sym:Ra}[^T] \gls{sym:iatt} \gls{sym:iattvar} = \gls{sym:ratt} \gls{sym:iattvar}, \label{eq:relative_attitude_variation_1}
\delta \gls{sym:ratt} &= \left. \diff{}{\epsilon} \right|_{\epsilon=0} \gls{sym:aatt}[^T] \gls{sym:iatt}^\epsilon = \gls{sym:aatt}[^T] \delta \gls{sym:iatt}, \nonumber\\
&= \gls{sym:aatt}[^T] \gls{sym:iatt} \gls{sym:iattvar} = \gls{sym:ratt} \gls{sym:iattvar}, \label{eq:relative_attitude_variation_1}
\end{align}
where we used the fact that the asteroid attitude does not vary, \( \delta \gls{sym:Ra} = 0\), and the variation of the spacecraft attitude is given in~\cref{eq:inertial_att_variation}.
where we used the fact that the asteroid attitude does not vary, \( \delta \gls{sym:aatt} = 0\), and the variation of the spacecraft attitude is given in~\cref{eq:inertial_att_variation}.
The variation of the spacecraft attitude \( \gls{sym:iattvar} \in \so \) is related to the relative variation \( \gls{sym:rattvar} \in \so \) by
\begin{align}
\gls{sym:rattvar} &= \gls{sym:ratt} \gls{sym:iattvar} \gls{sym:ratt}[^T],\nonumber\\
&= \gls{sym:Ra}[^T] \gls{sym:iatt} \gls{sym:iattvar} \gls{sym:iatt}[^T] \gls{sym:Ra}.\label{eq:relative_attitude_variation}
&= \gls{sym:aatt}[^T] \gls{sym:iatt} \gls{sym:iattvar} \gls{sym:iatt}[^T] \gls{sym:aatt}.\label{eq:relative_attitude_variation}
\end{align}
We can invert~\cref{eq:relative_attitude_variation} to find
\begin{align}\label{eq:relative_attitude_variation_inverse}
\gls{sym:iattvar} &= \gls{sym:iatt}[^T] \gls{sym:Ra} \gls{sym:rattvar} \gls{sym:Ra}[^T] \gls{sym:iatt}.
\gls{sym:iattvar} &= \gls{sym:iatt}[^T] \gls{sym:aatt} \gls{sym:rattvar} \gls{sym:aatt}[^T] \gls{sym:iatt}.
\end{align}
Using this, the varation of the relative attitude is derived by substituting~\cref{eq:relative_attitude_variation_inverse} into~\cref{eq:relative_attitude_variation_1} as
\begin{align}\label{eq:relative_attitude_matrix_variation}
\delta \gls{sym:ratt} = \gls{sym:Ra}[^T] \gls{sym:iatt} \gls{sym:iattvar} = \gls{sym:rattvar} \gls{sym:Ra}[^T] \gls{sym:iatt} = \gls{sym:rattvar} \gls{sym:ratt}
\delta \gls{sym:ratt} = \gls{sym:aatt}[^T] \gls{sym:iatt} \gls{sym:iattvar} = \gls{sym:rattvar} \gls{sym:aatt}[^T] \gls{sym:iatt} = \gls{sym:rattvar} \gls{sym:ratt}
\end{align}

\subsection{Relative Angular Velocity Variation}\label{sec:relative_angular_velocity_variation}
Expand Down Expand Up @@ -94,8 +94,8 @@ \subsection{Relative Angular Velocity Variation}\label{sec:relative_angular_velo
where we used~\cref{eq:relative_angular_velocity}.
Applying~\cref{eq:attitude_kinematics_derivative,eq:hatRxR} into~\cref{eq:relative_attitude_derivative_1} gives
\begin{align}
\dot{R}_r &= \hat{\Omega}_A^T R_A^T \gls{sym:iatt} + \gls{sym:Ra}[^T] \gls{sym:iatt} \parenth{\gls{sym:ratt}[^T] \gls{sym:rangvel}}^\wedge, \nonumber \\
&= \hat{\Omega}_A^T R_A^T \gls{sym:iatt} + \gls{sym:Ra}[^T] \gls{sym:iatt} \gls{sym:ratt}[^T] \hat{\Omega}_r \gls{sym:ratt}, \nonumber \\
\dot{R}_r &= \hat{\Omega}_A^T R_A^T \gls{sym:iatt} + \gls{sym:aatt}[^T] \gls{sym:iatt} \parenth{\gls{sym:ratt}[^T] \gls{sym:rangvel}}^\wedge, \nonumber \\
&= \hat{\Omega}_A^T R_A^T \gls{sym:iatt} + \gls{sym:aatt}[^T] \gls{sym:iatt} \gls{sym:ratt}[^T] \hat{\Omega}_r \gls{sym:ratt}, \nonumber \\
&= - \hat{\Omega}_A \gls{sym:ratt} + \hat{\Omega}_r \gls{sym:ratt}. \label{eq:relative_attitude_derivative}
\end{align}
From~\cref{eq:relative_attitude_variation} and the product rule, the time derivative of \( \gls{sym:rattvar} \) is given by
Expand All @@ -119,14 +119,14 @@ \subsection{Relative Angular Velocity Variation}\label{sec:relative_angular_velo
\subsection{Relative velocity variation}\label{sec:relative_velocity_variation}
The variation of the relative linear velocity is computed from~\cref{eq:relative_velocity} as
\begin{align}\label{eq:relative_velocity_variation_1}
\delta \gls{sym:rvel} &= \left. \diff{}{\epsilon} \right|_{\epsilon=0} \gls{sym:Ra}[^T] \dot{x}^\epsilon , \nonumber \\
\delta \gls{sym:rvel} &= \left. \diff{}{\epsilon} \right|_{\epsilon=0} \gls{sym:aatt}[^T] \dot{x}^\epsilon , \nonumber \\
&= R_A^T \delta \dot{\gls{sym:ipos}},
\end{align}
since we assume that the asteroid is in a constant rate of rotation.
We can redefine~\cref{eq:relative_velocity_variation_1} in terms of the relative variables by finding the derivative of~\cref{eq:relative_position_variation} as
\begin{align}\label{eq:relative_position_variation_derivative_1}
\delta \dot{x}_r &= \dot{R}_A^T \delta \gls{sym:ipos} + \gls{sym:Ra}[^T] \parenth{\delta \dot{\gls{sym:ipos}}}, \nonumber \\
&= \parenth{\gls{sym:Ra} \hat{\Omega}_A}^T \delta\gls{sym:ipos} + \gls{sym:Ra}[^T] \parenth{\delta \dot{\gls{sym:ipos}}}.
\delta \dot{x}_r &= \dot{R}_A^T \delta \gls{sym:ipos} + \gls{sym:aatt}[^T] \parenth{\delta \dot{\gls{sym:ipos}}}, \nonumber \\
&= \parenth{\gls{sym:aatt} \hat{\Omega}_A}^T \delta\gls{sym:ipos} + \gls{sym:aatt}[^T] \parenth{\delta \dot{\gls{sym:ipos}}}.
\end{align}
Rearranging~\cref{eq:relative_position_variation_derivative_1} for \(\delta \dot{\gls{sym:ipos}} \) and substituting into~\cref{eq:relative_velocity_variation_1} gives
\begin{align}\label{eq:relative_velocity_variation}
Expand All @@ -135,7 +135,7 @@ \subsection{Relative velocity variation}\label{sec:relative_velocity_variation}

In summary, the variation of the relative variables are defined as
\begin{align*}
\delta \gls{sym:rpos} &= \gls{sym:Ra}[^T] \delta \gls{sym:ipos}, \\
\delta \gls{sym:rpos} &= \gls{sym:aatt}[^T] \delta \gls{sym:ipos}, \\
\delta \gls{sym:ratt} &= \gls{sym:rattvar} \gls{sym:ratt} ,\\
\parenth{\delta \gls{sym:rangvel}}^\wedge&= \dot{\eta}_r - \hat{\Omega}_r \gls{sym:rattvar} + \hat{\Omega}_A \gls{sym:rattvar} + \gls{sym:rattvar} \hat{\Omega}_R - \gls{sym:rattvar} \hat{\Omega}_A, \\
\delta \gls{sym:rvel} &= \dot{x}_r + \hat{\Omega}_A \delta \gls{sym:rpos} .
Expand Down

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