Approximately Bi-Invariant Metrics

Distance Metrics on SE(3) and SE(2) (Venkataramanujam, Larochelle)

There are various useful metrics for finding the distance between two points in Euclidean space. However, metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. We have developed a metric for a finite set of rigid body displacements which uses a mapping of the special Euclidean group SE(N-1). This technique is based on embedding SE(N- 1) into SO(N) via the polar decomposition of the homogeneous transform representation of SE(N-1). To yield a useful metric for a finite set of displacements appropriate for design applications, the principal frame and a characteristic length are used. A bi-invariant metric on SO(N) is then used to measure the distance between any two displacements in SE(N-1). A detailed algorithm for the application of this method was presented and illustrated by three examples. This technique has potential applications in mechanism synthesis and robot motion planning.