Perturbed Rotations via Rotation Vector

Definition

Perturbation via rotation vector provides a new way of computing the Jacobian of a rotation matrix with respect to the rotation vector. The previous derivation requires to compute the left Jacobian which could be complicated.

Consider a left perturbation of a rotation matrix \(\mathbf{R}\) with Lie algebra vector \(\boldsymbol{\rho}\) by \(\Delta \mathbf{R}\) with Lie algebra vector \(\Delta \boldsymbol{\rho}\). The change of result relative to this disturbance for a vector \(\mathbf{v} \in \mathbb{R}^3\) is:

\[ \begin{align} \frac{\partial \left(\mathbf{R} \mathbf{v}\right)}{\partial \Delta \boldsymbol{\rho}} &= \lim_{\Delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{\exp \left[\Delta \boldsymbol{\rho} \right]_\times \exp \left[ \boldsymbol{\rho}\right]_\times \mathbf{v} - \exp \left[ \boldsymbol{\rho} \right]_\times \mathbf{v}}{\Delta \boldsymbol{\rho}} \\ &= \lim_{\Delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{\left(\mathbf{I} + \left[\Delta \boldsymbol{\rho} \right]_\times \right) \exp \left[ \boldsymbol{\rho} \right]_\times \mathbf{v} - \exp \left[ \boldsymbol{\rho} \right]_\times \mathbf{v}}{\Delta \boldsymbol{\rho}} \\ &= \lim_{\Delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{\left[\Delta \boldsymbol{\rho} \right]_\times \mathbf{R} \mathbf{v}}{\Delta \boldsymbol{\rho}} \\ &= \lim_{\Delta \boldsymbol{\rho} \rightarrow \mathbf{0}} \frac{-\left[ \mathbf{R} \mathbf{v} \right]_\times \Delta \boldsymbol{\rho}}{\Delta \boldsymbol{\rho}} \\ &= -\left[ \mathbf{R} \mathbf{v} \right]_\times. \end{align} \]

Note that the calculation of a left Jacobian, \(\mathbf{J}_l\) is omitted compared to the direct Lie algebra's derivation. This makes perturbation model much more practical.