Incremental Updates

In iterative algorithms, it is not convenient to update a rotation matrix directly using

$$\mathbf{R}_{k+1} = \mathbf{R}_k + \Delta\mathbf{R},$$ (A.38)

because the result might no longer belong to $SO(3)$.

Instead, an infinitesimal perturbation is used:

$$\delta\boldsymbol{\omega} \in \mathfrak{so}(3)$$ (A.39)

and the solution is updated according to

$$\mathbf{R}_{k+1} = \exp\!\left( [\delta\boldsymbol{\omega}]_\times \right) \mathbf{R}_k.$$ (A.40)

This approach automatically preserves the orthonormality of the matrix.

This formulation is widely used in extended Kalman filters, pose estimation, visual odometry, and bundle adjustment.