\documentclass[]{AVSSimReportMemo}
\usepackage{AVS}

\newcommand{\ModuleName}{inertial3DSpin}
\newcommand{\subject}{Guidance Module to Perform an Inertially Constant Spin}
\newcommand{\status}{Initial Version}
\newcommand{\preparer}{H. Schaub}
\newcommand{\summary}{Generate the reference attitude trajectory for a general 3D inertial spin with a constant inertial angular velocity vector.   }


\begin{document}


\makeCover


%
%	enter the revision documentation here
%	to add more lines, copy the table entry and the \hline, and paste after the current entry.
%
\pagestyle{empty}
{\renewcommand{\arraystretch}{1.1}
\noindent
\begin{longtable}{|p{0.5in}|p{4.5in}|p{1.14in}|}
\hline
{\bfseries Rev}: & {\bfseries Change Description} & {\bfseries By} \\
\hline
Draft & initial copy & H. Schaub \\
v1.1 & fixed typo above Eq.~(4) & H. Schaub \\
v1.2 & updated the documentation now that the module outputs only the reference frame states & H. Schaub \\
\hline

\end{longtable}
}

\newpage
\setcounter{page}{1}
\pagestyle{fancy}

\tableofcontents
~\\ \hrule ~\\


\section{Introduction}
This technical note discusses the guidance mathematics to perform a constant spin about an inertially fixed axis.  

\section{Setup}
This module requires that the initial inertial attitude $\bm\sigma_{R/N}$ of the reference frame $\mathcal{R}$ relative to the inertial frame $\mathcal{N}$ be set.  Further, the inertially constant $\mathcal{R}$ frame angular velocity vector $\leftexp{N}{\bm\omega}_{R/N}$ must be defined in inertial frame components.  




\section{Reference Frame Propagation}
The MRP differential kinematic equations are
\begin{equation}
	\dot{\bm\sigma}_{R/N} = \frac{1}{4} [B(\bm\sigma_{R/N})] \leftexp{R}{\bm\omega}_{R/N}
\end{equation}
where
\begin{equation}
	[B(\bm\sigma_{R/N})] = (1-\sigma_{R/N}^{2}) [I_{3\times 3}] + 2 [\tilde{\bm\sigma}_{R/N}] + 2 \bm\sigma_{R/N} \bm\sigma_{R/N}^{T}
\end{equation}
and $\sigma_{R/N} = |\bm\sigma_{R/N}|$.

If there inertially fixed reference angular velocity vector is given in inertial frame components, it must be rotated using the appropriate DCM:
\begin{equation}
	\leftexp{R}{\bm\omega_{R/N}} = [RN] \leftexp{N}{\bm\omega_{R/N}}
\end{equation}

\bibliographystyle{unsrt}
\bibliography{references}

\end{document}
