Fourier Analysis on Motion Using Screw Parameters

In this paper, the author presents the bi-invariant (Haar) integral for the group of rigid-body motions (Euclidean group) in three-dimensional space in terms of finite screw parameters, and proceeds to develop the matrix elements of irreducible unitary representations in this parameterization. This allows one to integrate and expand functions of motion described in terms of screw parameters. In robot kinematics, the screw-parameter description of a finite rigid-body motion is well known. This description of motion in three-dimensional space (which follows from Ball’s work on finite screws) provides an elegant way to view rigid-body kinematics. In contrast, the theoretical physics community usually uses Euler angles and spherical coordinates to parameterize rigid-body motions. It therefore comes as no surprise that in the field of Fourier analysis on groups, which has been developed in large part by theoretical physicists, that the Euler-angle/spherical coordinate description of rigid-body motions is most common. The contribution of this paper is to formulate Fourier analysis on the group of rigid-body motions in terms of screw parameters. The geometrically meaningful nature of the screw parameters combined with the group Fourier transform provides a tool for insight into problems that can be posed as convolutions on the Euclidean group. Such problems include workspace generation of serial linkages, kinematic error propagation, and the statistical mechanics of macromolecules.