\subsection{Positioning}
\shortdefinition[Pos Vec.]
$_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{ForestGreen}\fbox{W}}\,_{\color{red}\fbox{B}}$,
{\color{blue} Relative to Frame}, {\color{red} P. in other Frame}, {\color{ForestGreen} Start P. of vec in first F.},
\hl{$\sin = s$, $\cos = c$}

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

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

\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 Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
$\begin{smallmatrix}
        \psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
        \theta = \arcsin(-R_{31})\\
        \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
    \end{smallmatrix}$
{\scriptsize
$[\vec{n}]^\times = \begin{bmatrix}
        0    & -a_3 & a_2  \\
        a_3  & 0    & -a_1 \\
        -a_2 & a_1  & 0
    \end{bmatrix}$
}

\shortdefinition[Rot. Vec]
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$


\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$).

$\vec{q} = \begin{bmatrix}
        \vec{v}(\vec{q}), a(\vec{q})
    \end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')

\bi{Mult} {\scriptsize
    $\vec{q} \otimes \vec{p} = \begin{bmatrix}
            a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
            a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
        \end{bmatrix}$
}

\bi{To Rot Mat} {\scriptsize
    $R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
}


\shortdefinition[Transf. M]
{\scriptsize
    $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
    $\mat{T}_{AB} = \begin{bmatrix}
            \mat{R}_{AB}        & {_A}\vec{t}_B \\
            \mat{0}_{1\times 3} & 1
        \end{bmatrix};
        \mat{T}_{BA} = \mat{T}_{AB}^{-1} =
        \begin{bmatrix}
            \mat{R}_{AB}^\top    & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
            \mat{0}_{1 \times 3} & 1
        \end{bmatrix}$
}
