DISTANCE METRICS ON THE RIGID-BODY MOTIONS WITH APPLICATIONS TO MECHANISM DESIGN

In this article we examine the problem of designing a mechanism whose tool frame comes closet to reaching a set of desired goal frames. The basic mathematical question we address is characterizing the set of di stance metrics in SE(3), the Euclidean group of rigid-body motions. Us ing Lie theory, we show that no bi-invariant distance metric (i.e., on e that is invariant under both left and right translations) exists in SE(3), and that because physical space does not have a natural length scale, any distance metric in SE(3) will ultimately depend on a choice of length scale. We show how to construct left- and right-invariant d istance metrics in SE(3), and suggest a particular left-in-variant dis tance metric parametrized by length scale that is useful for kinematic applications. Ways of including engineering considerations into the c hoice of length scale are suggested, and applications of this distance metric to the design and positioning of certain planar and spherical mechanisms are given.