A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition

An open research question is how to define a useful metric on the special Euclidean group SE(n) with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of SE(n) with elements of the special orthogonal group SO(n+ 1). This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of SE(n). The embedding of the elements of SE(n) into SO (n+ 1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO (n+ 1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n)