Interpolation schemes for rigid body motions

This paper investigates methods for computing a smooth motion that interpolates a given set of positions and orientations. The position and orientation of a rigid body can be described with an element of the group of spatial rigid body displacements, SE(3). To nd a smooth motion that interpolates a given set of positions and orientations is therefore the same as nding an interpolating curve between the corresponding elements of SE(3). To make the interpolation on SE(3) independent of the representation of the group, we use the coordinate-free framework of diierential geometry. It is necessary to choose inertial and body-xed reference frames to describe the position and orientation of the rigid body. We rst show that trajectories that are independent of the choice of these frames can be obtained by using the exponential map on SE(3). However, these trajectories may exhibit rapid changes in the velocity or higher derivatives. The second contribution of the paper is a method for nding the maximally smooth interpolating curve. By adapting the techniques of the calculus of variations to SE(3), necessary conditions are derived for motions that are equivalent to cubic splines in the Euclidean space. These necessary conditions result in a boundary value problem with interior-point constraints. A simple and eecient numerical method for nding a solution is then described. Finally, we discuss the dependence of the computed trajectories on the metric on SE(3) and show that independence of the trajectories from the choice of the reference frames can be achieved by using a suitable metric.