Computing the Logarithm of Homogenous Matrices in SE(3)

Abstract

We compute the logarithm of the elements of the Lie group SE(3), that leads to elements of the Lie algebra se(3). The approach is purely tensorial, including the case of the in SE(3) for which the rotation component is a symmetric matrix. This is an extension of the logarithm of orthogonal matrices from SO(3), that leads to skew-symmetric matrices in so(3). We give a Rodrigues-like formula for the map exp:se(3) SE(3), that is surjective, and give a method for computing its multivalued inverse map. The case of matrices with symmetric rotation component has an elegant solution that does not appear in the studies about this problem. The results have applications in direct and inverse kinematics in robotics.