An Eecient, Geometric Approach to Rigid Body Motion Interpolation

This paper develops a method for generating tra-jectories for a rigid body with speciied boundary conditions at the end points and intermediate locations. It is well known that SE(3), the set of all rigid body positions and orientations, is a non Euclidean space. In general, the problem of determining trajectories that are optimal with respect to some meaningful metric does not have an analytical solution. Thus it is diicult to develop eecient solutions for real time applications in computer graphics and robotics. In this paper, we consider SE(n?1) to be a sub-manifold (and a subgroup) of the GL(n; IR). Our method involves the generation of optimal trajec-tories in GL(n; IR), and then projecting the tra-jectories to SE(n ? 1). We prove several results that are relevant to this method. First, the natural Euclidean metric in GL(n ? 1; IR), when restricted to SO(n ? 1), is shown to yield the standard bi-invariant metric on SO(n ?1). Second, the natural Euclidean metric applied to the subgroup of GL(n; IR) that describes aane transformations of IR n?1 , when restricted to SE(n ? 1), yields the natural scale-dependent left invariant metric discussed in 10, 15]. Third, we show how the metric on GL(n; IR), allows projections on SO(n) and SE(n ? 1). We use standard algorithms for deriving the optimal curves in the ambient space. The projection operator allows us to project the optimal curves onto SO(n) and SE(n) to yield motions that are near optimal. Finally, we present results that illustrate the near optimality of the projected curves.