An SVD-based projection method for interpolation on SE(3)

This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in ( ), a subgroup of the affine group in IR and 2) the projection of the trajectories onto (3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on ( ). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on (3). Third, from a computational point of view, the method we propose is less expensive than traditional methods.