3.3 Adjoint

The adjoint in $\mathrm{SE(3)}$ is computed from the generators, just as in $\mathrm{SO}(3)$ :

$\displaystyle \boldsymbol{\delta}=\left(\begin{array}{cc} \mathbf{u} & \boldsymbol{\omega}\end{array}\right)^{T}$	$\displaystyle \in$	$\displaystyle \mathrm{se(3)},\; C=\left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}\\ \hline \mathbf{0} & 1 \end{array}\right)\mathbf{\in}\mathrm{SE(3)}$	(78)
$\displaystyle C\cdot\exp\left(\boldsymbol{\delta}\right)$	$\displaystyle =$	$\displaystyle \exp\left(\mathrm{Adj}_{C}\cdot\boldsymbol{\delta}\right)\cdot C$
$\displaystyle \exp\left(\mathrm{Adj}_{C}\cdot\boldsymbol{\delta}\right)$	$\displaystyle =$	$\displaystyle C\cdot\exp\left(\boldsymbol{\delta}\right)\cdot C^{-1}$	(79)
$\displaystyle \mathrm{Adj}_{C}\cdot\boldsymbol{\delta}$	$\displaystyle =$	$\displaystyle C\cdot\left(\sum_{i=1}^{6}\boldsymbol{\delta}_{i}G_{i}\right)\cdot C^{-1}$	(80)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} \mathbf{R}\mathbf{u}+\mathbf{t}\times\mathbf{R}\boldsymbol{\omega}\\ \mathbf{R}\boldsymbol{\omega} \end{array}\right)$	(81)
$\displaystyle \implies\mathrm{Adj}_{C}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \mathbf{R} & \mathbf{t}_{\times}\ma... ...R}\\ \hline \mathbf{0} & \mathbf{R} \end{array}\right)\in\mathbb{R}^{6\times6}$	(82)

Note that moving a tangent vector via the adjoint mixes the rotation component into the translation component.