10.3 Exponential Map

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

$\displaystyle v=\left(\begin{array}{c} \mathbf{u}\\ \boldsymbol{\omega}\\ \lambda \end{array}\right)$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{7}$	(135)
$\displaystyle \theta$	$\displaystyle \equiv$	$\displaystyle \sqrt{\boldsymbol{\omega}^{T}\boldsymbol{\omega}}$	(136)
$\displaystyle A$	$\displaystyle \equiv$	$\displaystyle \dfrac{\sin\theta}{\theta}$	(137)
$\displaystyle B$	$\displaystyle \equiv$	$\displaystyle \dfrac{1-\cos\theta}{\theta^{2}}$	(138)
$\displaystyle C$	$\displaystyle \equiv$	$\displaystyle \dfrac{1-A}{\theta^{2}}$	(139)
$\displaystyle D$	$\displaystyle \equiv$	$\displaystyle \dfrac{\frac{1}{2}-B}{\theta^{2}}$	(140)
$\displaystyle s^{-1}$	$\displaystyle \equiv$	$\displaystyle e^{-\lambda}$	(141)
$\displaystyle \alpha$	$\displaystyle \equiv$	$\displaystyle \dfrac{\lambda^{2}}{\lambda^{2}+\theta^{2}}$	(142)
$\displaystyle \beta$	$\displaystyle \equiv$	$\displaystyle \dfrac{s^{-1}-1+\lambda}{\lambda^{2}}$	(143)
$\displaystyle \gamma$	$\displaystyle \equiv$	$\displaystyle \frac{\frac{1}{2}-\beta}{\lambda}$	(144)
$\displaystyle X$	$\displaystyle \equiv$	$\displaystyle \dfrac{1-s^{-1}}{\lambda}$	(145)
$\displaystyle Y$	$\displaystyle \equiv$	$\displaystyle \alpha\cdot\beta+\left(1-\alpha\right)\cdot\left(B-\lambda C\right)$	(146)
$\displaystyle Z$	$\displaystyle \equiv$	$\displaystyle \alpha\cdot\gamma+\left(1-\alpha\right)\cdot\left(C-\lambda D\right)$	(147)
$\displaystyle \mathbf{R}$	$\displaystyle \equiv$	$\displaystyle \mathbf{I}+A\boldsymbol{\omega}_{\times}+B\boldsymbol{\omega}_{\times}^{2}$	(148)
$\displaystyle \mathbf{V}$	$\displaystyle \equiv$	$\displaystyle X\mathbf{I}+Y\boldsymbol{\omega}_{\times}+Z\boldsymbol{\omega}_{\times}^{2}$	(149)
$\displaystyle \exp\left(\mathrm{alg}\left(v\right)\right)=\exp\left(\begin{arra... ...dsymbol{\omega}_{\times} & \mathbf{u}\\ \hline 0 & -\lambda \end{array}\right)$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{V}\cdot\mathbf{u}\\ \hline \mathbf{0} & s^{-1} \end{array}\right)$	(150)

The coefficients of $\mathbf{R}$ and $\mathbf{V}$ should be calculated with Taylor series when $\theta$ or $\lambda$ is small.