Difference, Interpolation, and Perturbation

Definition

	Special Orthogonal Group	Special Euclidean Group
Difference	Given \(\mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_{12} \in SO(3)\), where \(\mathbf{R}_{12}\) is the attitude difference between \(\mathbf{R}_1\) and \(\mathbf{R}_2\), we have: \(\begin{align*} &\mathbf{R}_1 \mathbf{R}_{12} = \mathbf{R}_2 \\ &\boldsymbol{\rho}_{12} = \left[ \ln \mathbf{R}_{12} \right]_{-\times} = \left[ \ln \mathbf{R}^T_1 \mathbf{R}_2 \right]_{-\times} \end{align*}\)
Perturbation on Lie Group	Let \(\mathbf{R}, \Delta \mathbf{R} \in SO(3)\) and \(\boldsymbol{\rho}, \Delta \boldsymbol{\rho} \in \mathfrak{so}(3)\) the corresponding Lie algebras. Then: \(\begin{align*} \Delta \mathbf{R} \mathbf{R} &= \exp\left(\left[\Delta\boldsymbol{\rho} \right]_\times \right) \exp\left(\left[\boldsymbol{\rho} \right]_\times \right) \\ &= \exp \bigl( \left[ \boldsymbol{\rho} + \mathbf{J}^{-1}_l (\boldsymbol{\rho}) \Delta \boldsymbol{\rho} \right]_\times \bigr) \end{align*}\)
Perturbation on Lie Algebra	If we do an addition on Lie algebra vector \(\boldsymbol{\rho}\) by perturbing it with \(\Delta \boldsymbol{\rho}\), we can approximate the multiplication on the Lie group as: \(\begin{align*} \exp \left( \left[\boldsymbol{\rho} + \Delta \boldsymbol{\rho} \right]_\times \right) &= \exp \left( \left[ \mathbf{J}_l \Delta \boldsymbol{\rho} \right]_\times \right) \exp \left( \left[ \boldsymbol{\rho} \right]_\times \right) \\ &= \exp \left( \left[\boldsymbol{\rho} \right]_\times \right) \exp \left(\left[ \mathbf{J}_r \Delta \boldsymbol{\rho} \right]_\times \right). \end{align*}\)