Interpolation schemes for rigid body motions

Computer-Aided Design, Elsevier, Vol 30, Num 3, pp.179-189, 1998

MESHING
RESEARCH
CORNER

California Institute of Technology. Pasadena. California. USA

Abstract
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 find a smooth motion that interpolates a given set of positions and orientations is therefore the same as finding 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 differential geometry. It is necessary to choose inertial and body- fixed reference frames to describe the position and orientation of the rigid body. We first 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 finding 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 efficient numerical method for finding 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.