CHOICE OF RIEMANNIAN METRICS FOR RIGID BODY KINEMATICS

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 possible choices of Riemannian metrics and affine connections on this manifold. In the first part of the paper we derive the semi-Riemannian metrics whose geodesics are screw motions and show that the only metrics are indefinite but non degenerate, and they are unique up to a choice of two scaling constants. 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 which achieves this. Further, we show that there is a family of metrics that are compatible with such connection. A metric in this family must be a product of the bi-invariant metric on the group of rotations and a positive definite constant metric on the group of translations.