Metrics and Connections for Rigid-Body Kinematics

Abstract

The set of rigid-body motions forms a Lie group called SE(3), the special Euclidean group in three dimensions. In this paper, we investigate possilble choices of Riemannian metrics and affine connections on SE(3) for applications to kinematic analysis and robot-trajectory planning. In the first part of the paper, we study metrics whose geodesics are screw motions. We prove that no Riemannian metrics call have such geodesics, and we show that the metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannan metrics. In the second part of the paper, we investigate affine connections which through the covariant derivative give the correct expression for the acceleration of a rigid body. We prove that there is a unique symmetric connection with this property. Furthermore, we show that there is a family of Riemannian metrics that are compatible with such a connection. These metrics are products of the bi-invariant metric on the group of rotations and a positive-definite constant metric on the group of translations.