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.