\subsection{Positioning}
\shortdefinition[Position Vector]
$_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{ForestGreen}\fbox{W}}\,_{\color{red}\fbox{B}}$,
{\color{blue} Original Frame}, {\color{red} End point}, {\color{ForestGreen} Target Frame},
\hl{$\sin = s$, $\cos = c$}

\shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors

\shortdefinition[Rot. Mat.] $\mat{R}_{z} = \begin{bmatrix}
        c(\psi) & -s(\psi) & 0 \\
        s(\psi) & c(\psi)  & 0 \\
        0       & 0        & 1
    \end{bmatrix}$\\
$\mat{R}_y(\theta) = \begin{bmatrix}
        c(\psi)  & 0 & s(\psi) \\
        0        & 1 & 0       \\
        -s(\psi) & 0 & c(\psi) \\
    \end{bmatrix}
    \mat{R}_x(\varphi)
    \begin{bmatrix}
        1 & 0       & 0        \\
        0 & c(\psi) & -s(\psi) \\
        0 & s(\psi) & c(\psi)
    \end{bmatrix}$

\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$

\shortlemma $\mat{R}_{BW} = \mat{R}_{WB}^{-1} = \mat{R}_{WB}^\top$, $\det(\mat{R}_{WB}) = 1$ (orth.)

\shortremark Cols of $\mat{R}_{WB}$ are basis vec. of Frame $\underset{\rightarrow}{\cF}{_B}$ in $\underset{\rightarrow}{\cF}{_W}$

\shortdefinition[Euler Angles] Yaw ($z$), Pitch ($y$), Roll ($x$), mult. rotation matrices, e.g.
$\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$, \hl{bound.}
