This repo provides exponential and log maps for special orthogonal groups (SO3) and special Euclidian groups (SE3).
Quick and intuitively, the rotation matrix is nonlinear. As R is in nonlinear space, instantaneous rotation is not dR. Rather, the instantaneous rotation is more like the information from the gyroscope in x, y, and z axes.
For more, refer to Chapter2.
The 3D vector needs to be mapped as a matrix. The mapping from 3D vector to the rotation matrix is exponential mapping.
The inverse mapping of exponential mapping. The log maps the rotation matrix to 3D vector.
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}