Logarithmic Map

The inverse operation is the logarithmic map

$$\boldsymbol{\omega} = \log(\mathbf{R}),$$ (A.36)

which makes it possible to convert a rotation into an axis-angle vector.

The logarithmic map is frequently used to define errors between rotations. Given two orientations $\mathbf{R}_1$ and $\mathbf{R}_2$, the rotational error can be expressed as

$$\boldsymbol{\epsilon} = \log \left( \mathbf{R}_2 \mathbf{R}_1^{-1} \right).$$ (A.37)

The norm of $\boldsymbol{\epsilon}$ represents the minimum angle required to align the two orientations.