A coordinate-free description of robot dynamics

Abstract

We provide a coordinate-free description of the dynamics for general open and closed chain mechanisms. The mechanism is assumed to consist of rigid links, and subject only to holonomic constraints. For an m degree of freedom mechanism consisting of k links, we define an appropriate mapping f from the m-dimensional configuration space M to N=SE(3)x...x SE(3) (k copies). By endowing N with the Riemannian metric defined by the link kinetic energies, the generalized inertia matrix can then be regarded as the pullback metric with respect to f, and the Coriolis terms given by the Christoffel symbols (of the first kind) with respect to the pullback metric. The coordinate-free description elucidates the basic differences in the various coordinate-based dynamics formulations that have been proposed in the literature, and provides a high-level view of robot dynamics that provides insight on how best to choose coordinates for computations.