\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[Angle-Axis]
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
To quat: $\vec{q} = [\vec{n}, \alpha]$


\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\\
	$\mat{T}_{AB} = \begin{bmatrix}
			\mat{R}_{AB}        & {_A}\vec{t}_B \\
			\mat{0}_{1\times 3} & 1
		\end{bmatrix};
		\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
		\begin{bmatrix}
			\mat{R}_{AB}^\top    & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
			\mat{0}_{1 \times 3} & 1
		\end{bmatrix}$
}
