Notation and Standard Nomenclature

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\author{courtney.savytska}
\date{September 2024}

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{\LARGE \textbf{Notation and Standard Nomenclature}}
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\section*{Notation}

\begin{itemize}
\item \textit{Scalars} are denoted by lowercase symbols, $a$. 
\item \textit{Vectors} are denoted by a symbol with an overhead arrow, $\vec{a}$. 
\item \textit{Matrices} are denoted by bold symbols, $\mathbf{a}$ or $\mathbf{A}$.
\item \textit{Reference frames} are denoted by a cursive F (generated using \texttt{\textbackslash mathcal{}} command) with a subscript to differentiate between reference frames, $\mathcal{F}_a$. 
\item A reference frame $\mathcal{F}_a$ is defined by its \textit{unit vectors} $\lbrace \vec{a}_x, \vec{a}_y, \vec{a}_z \rbrace$. 
\item A \textit{vectrix} defines a matrix of the unit vectors of a reference frame and is denoted by the reference frame symbol with an overhead arrow, \textit{i.e.}, $\vec{\mathcal{F}}_a^\top = \begin{bmatrix} \vec{a}_x  & \vec{a}_y & \vec{a}_z \end{bmatrix}$.
\end{itemize}


\section*{Standard Nomenclature}

A standard nomenclature to be used across all lab publications is found below.\\

\noindent\textit{Inertial and relative states of each platform}:
\begin{itemize}
\item Inertial position (general): $\mathbf{x} = \begin{bmatrix} x  & y & \theta \end{bmatrix}^\top $
\begin{itemize}
    \item Chaser inertial position: $\mathbf{x}_c = \begin{bmatrix} x_c  & y_c & \theta_c \end{bmatrix}^\top $
    \item Target inertial position:  $\mathbf{x}_t = \begin{bmatrix} x_t  & y_t & \theta_t \end{bmatrix}^\top $
    \item Obstacle inertial position:  $\mathbf{x}_o = \begin{bmatrix} x_o  & y_o & \theta_o \end{bmatrix}^\top $
\end{itemize}
\item Target pose relative to chaser: $\mathbf{r}_{tc} = \mathbf{x}_t - \mathbf{x}_c $
\item Obstacle pose relative to chaser: $\mathbf{r}_{oc} = \mathbf{x}_o - \mathbf{x}_c $\\
\end{itemize}


\noindent\textit{Reference Frames} (using \texttt{\textbackslash mathcal{}} command):
\begin{itemize}
    \item Inertial reference frame: $\mathcal{F}_I$
    \item Body-fixed reference frame (general): $\mathcal{F}_B$
    \begin{itemize}
        % \item For each specific platform, the subscript specifies the platform. 
        \item The chaser platform has body-fixed reference frame $\mathcal{F}_{B_c}$
        \item The target platform has body-fixed reference frame $\mathcal{F}_{B_t}$
        \item The obstacle platform has body-fixed reference frame $\mathcal{F}_{B_o}$\\
    \end{itemize}
\end{itemize}


\noindent\textit{Reference Frame Unit Vectors}:
\begin{itemize}
    \item Inertial reference frame unit vectors: $\vec{I}_x$, $\vec{I}_y$, $\vec{I}_z$
    \item Body-fixed reference frame unit vectors (general): $\vec{B}_x$, $\vec{B}_y$, $\vec{B}_z$
    \begin{itemize}
       % \item For each specific platform, the subscript specifies the platform. 
       \item The chaser platform has body-fixed reference frame unit vectors $\vec{B}_{cx}$, $\vec{B}_{cy}$, $\vec{B}_{cz}$
       \item The target platform has body-fixed reference frame unit vectors $\vec{B}_{tx}$, $\vec{B}_{ty}$, $\vec{B}_{tz}$
       \item The obstacle platform has body-fixed reference frame unit vectors $\vec{B}_{ox}$, $\vec{B}_{oy}$, $\vec{B}_{oz}$
    \end{itemize}
\end{itemize}

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