4.3 Exponential Map

The exponential map from $\mathrm{\mathfrak{se}}(2)$ to $\mathrm{SE}(2)$ has a closed form:

$\displaystyle v=\left(\begin{array}{c} x\\ y\\ \theta \end{array}\right)$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(50)
$\displaystyle \mathbf{R}$	$\displaystyle \equiv$	$\displaystyle \left(\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right)$	(51)
$\displaystyle \mathbf{V}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{cc} \frac{\sin\theta}{\theta} & -\frac{1-\cos... ...}\\ \frac{1-\cos\theta}{\theta} & \frac{\sin\theta}{\theta} \end{array}\right)$	(52)
$\displaystyle \exp\left(\mathrm{alg}\left(v\right)\right)=\exp\left(\begin{arra... ...ert c} 0 & -\theta & x\\ \theta & 0 & y\\ \hline 0 & 0 & 0 \end{array}\right)$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{V}\cdot\left(\... ...array}{c} x\\ y \end{array}\right)\\ \hline \mathbf{0} & 1 \end{array}\right)$	(53)

The elements of $\mathbf{V}$ should be calculated with Taylor series when $\theta$ is small.