8.3 Exponential Map

The exponential map from $\mathrm{\mathfrak{so}}(3)$ to $\mathrm{SO}(3)$ has a closed form (also called the Rodrigues formula). The tangent vector $\boldsymbol{\omega}$ can be interpreted as an axis-angle representation of rotation: its exponential is the rotation around the axis $\boldsymbol{\omega}/\left\Vert \boldsymbol{\omega}\right\Vert$ by $\left\Vert \boldsymbol{\omega}\right\Vert$ radians:

$\displaystyle \boldsymbol{\omega}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(99)
$\displaystyle \theta$	$\displaystyle \equiv$	$\displaystyle \sqrt{\boldsymbol{\omega}^{T}\boldsymbol{\omega}}$	(100)
$\displaystyle \exp\left(\mathrm{alg}\boldsymbol{\omega}\right)$	$\displaystyle =$	$\displaystyle \exp\left(\boldsymbol{\omega}_{\times}\right)$	(101)
	$\displaystyle =$	$\displaystyle \mathbf{I}+\boldsymbol{\omega}_{\times}+\frac{1}{2!}\boldsymbol{\omega}_{\times}^{2}+\frac{1}{3!}\boldsymbol{\omega}_{\times}^{3}+...$	(102)
	$\displaystyle =$	$\displaystyle \mathbf{I}+\left(\frac{\sin\theta}{\theta}\right)\boldsymbol{\ome... ...s}+\left(\frac{1-\cos\theta}{\theta^{2}}\right)\boldsymbol{\omega}_{\times}^{2}$	(103)

The higher-order terms in Eq. 102 collapse because $\boldsymbol{\omega}_{\times}^{3}=-\theta^{2}\boldsymbol{\omega}_{\times}$ . The coefficients of $\mathbf{R}$ should be calculated with Taylor series when $\theta$ is small.

Given a rotation matrix $\mathbf{R}\in\mathrm{SO}(3)$ , the logarithm can be computed by first determining $\cos\theta=\frac{1}{2}\left(\mathrm{tr}\left(\mathbf{R}\right)-1\right)$ , and then computing $\boldsymbol{\omega}$ from symmetric differences (see the second term of Eq. 103).