6.3 Adjoint

$\displaystyle \boldsymbol{\delta}=\left(\begin{array}{ccc} \mathbf{u} & \boldsymbol{\omega} & \lambda\end{array}\right)^{T}$	$\displaystyle \in$	$\displaystyle \mathrm{sim(3)},\; T=\left(\begin{array}{c\vert c} \mathbf{R} & \... ...{t}\\ \hline \mathbf{0} & s^{-1} \end{array}\right)\mathbf{\in}\mathrm{Sim(3)}$	(201)
$\displaystyle T\cdot\exp\left(\boldsymbol{\delta}\right)$	$\displaystyle =$	$\displaystyle \exp\left(\mathrm{Adj}_{T}\cdot\boldsymbol{\delta}\right)\cdot T$
$\displaystyle \exp\left(\mathrm{Adj}_{T}\cdot\boldsymbol{\delta}\right)$	$\displaystyle =$	$\displaystyle T\cdot\exp\left(\boldsymbol{\delta}\right)\cdot T^{-1}$	(202)
$\displaystyle \mathrm{Adj}_{T}\cdot\boldsymbol{\delta}$	$\displaystyle =$	$\displaystyle T\cdot\left(\sum_{i=1}^{7}\boldsymbol{\delta}_{i}G_{i}\right)\cdot T^{-1}$	(203)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} s\left(\mathbf{R}\mathbf{u}+\mathbf{t}\tim... ...athbf{t}\right)\\ \mathbf{R}\boldsymbol{\omega}\\ -\lambda \end{array}\right)$	(204)
$\displaystyle \implies\mathrm{Adj}_{T}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c\vert c} s\mathbf{R} & s\mathbf{t}_{\... ... \hline \mathbf{0} & \mathbf{0} & 1 \end{array}\right)\in\mathbb{R}^{7\times7}$	(205)