Computation without Singularity for Logarithms of Homogeneous Matrices and Orthogonal Dual Tensors

This paper presents a systematic approach to the elements of the Lie algebra of rigid body displacements, denoted as se (3) by computing the logarithm of elements from the Lie group SE (3). The methodology is entirely based on tensor calculus and explicitly addresses cases where the rotation component within an SE (3) matrix is symmetric. This development generalizes the classical computation of logarithms for orthogonal matrices in SO (3), which correspond to skewsymmetric matrices in the Lie algebra so (3). A Rodrigues-type formulation is provided for the exponential map into se (3) of SE (3), which is shown to be surjective. Additionally, we propose a procedure for evaluating its multivalued inverse. These methodologies are extended to other parameterizations of rigid body displacements: orthogonal dual tensors. The calculations are presented in closed form and are free from singularities.