9.3 Exponential Map

The exponential map from $\mathrm{\mathfrak{se}}(3)$ to $\mathrm{SE}(3)$ has a closed form:

$\displaystyle \mathbf{u},\boldsymbol{\omega}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(119)
$\displaystyle \theta$	$\displaystyle \equiv$	$\displaystyle \sqrt{\boldsymbol{\omega}^{T}\boldsymbol{\omega}}$	(120)
$\displaystyle \exp\left(\mathrm{alg}\left(\begin{array}{c} \mathbf{u}\\ \boldsymbol{\omega} \end{array}\right)\right)$	$\displaystyle =$	$\displaystyle \exp\left(\begin{array}{c\vert c} \boldsymbol{\omega}_{\times} & \mathbf{u}\\ \hline 0 & 0 \end{array}\right)$	(121)
	$\displaystyle =$	$\displaystyle \mathbf{I}+\left(\begin{array}{c\vert c} \boldsymbol{\omega}_{\ti... ...oldsymbol{\omega}_{\times}^{2}\mathbf{u}\\ \hline 0 & 0 \end{array}\right)+...$	(122)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \exp\left(\boldsymbol{\omega}_{\times}\right) & \mathbf{V}\cdot\mathbf{u}\\ \hline 0 & 0 \end{array}\right)$	(123)
$\displaystyle \mathbf{V}$	$\displaystyle \equiv$	$\displaystyle \mathbf{I}+\left(\frac{1-\cos\theta}{\theta^{2}}\right)\boldsymbo... ...eft(\frac{\theta-\sin\theta}{\theta^{3}}\right)\boldsymbol{\omega}_{\times}^{2}$	(124)

Note that the rotation block is computed according to Eq. 103. The coefficients of $\mathbf{V}$ should be calculated with Taylor series when $\theta$ is small.

The logarithm of $\left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)\in\mathrm{SE}(3)$ can be determined by first computing $\boldsymbol{\omega}=\mathrm{alg}^{-1}\left(\log\left(\mathbf{R}\right)\right)$ , then building $\mathbf{V}$ and finding $\mathbf{u}=\mathbf{V}^{-1}\cdot\mathbf{t}$ .