A polar decomposition based displacement metric for a finite region of SE(n)

Abstract

An open research question is how to define a useful metric on SE(n) with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. A technique is presented for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group SO(n+1). This technique is based on the polar decomposition (denoted as 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 displacement. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two spatial displacements. The result is a PD based metric on SE(n) that is left invariant. Such metrics have applications in motion synthesis, robot calibration, motion interpolation, and hybrid robot control.