Hello Learning Adjoint representation for Modern Robotics
Adjoint representation
Given \(T = (R, p) \in SE(3)\) \[ \left[\operatorname{Ad}_{T}\right]=\left[\begin{array}{cc}R & 0 \\ {[p] R} & R\end{array}\right] \in \mathbb{R}^{6 \times 6} \]
Adjoint map
The Adjoint map associated with \(T\) is: \[ \mathcal{V}^{\prime}=\left[\operatorname{Ad}_{T}\right] \mathcal{V} \] which is sometimes also written as \[ \mathcal{V}^{\prime}=\operatorname{Ad}_{T} (\mathcal{V}) \] In terms of the matrix form \([\mathcal{V}] \in se(3) \) of \(\mathcal{V} \in \mathbb{R}^{6}\) \[ \left[\mathcal{V}^{\prime}\right]=T[\mathcal{V}] T^{-1} \]
Properties
\( \operatorname{Ad}_{T_{1}}\left(\operatorname{Ad}_{T_{2}}(\mathcal{V})\right)=\operatorname{Ad}_{T_{1} T_{2}}(\mathcal{V}) \) or \( \left[\operatorname{Ad}_{T_{1}}\right]\left[\operatorname{Ad}_{T_{2}}\right] \mathcal{V}=\left[\operatorname{Ad}_{T_{1} T_{2}}\right] \mathcal{V} \) and \[ \left[\operatorname{Ad}_{T}\right]^{-1}=\left[\operatorname{Ad}_{T^{-1}}\right] \]