Metrics and connections for rigid-body kinematics

The set of rigid-body motions forms a Lie group called SE(3), the special E uclidean group in three dimensions. In this paper; we investigate possible choices of Riemannian metrics and affine connections on SE(3)far applicatio ns to kinematic analysis and robot-trajectory planning. In the first part o f the paper we study metrics whose geodesics are screw motions. We prove th at no Riemannian metric can have such geodesics, and we show that the metri cs whose geodesics are screw motions form a two-parameter family of semi-Ri emannian metrics. in the second part of the paper we investigate affine con nections which through the covariant derivative give the correct expression for the acceleration of a rigid body. We prove that there is a unique symm etric connection with this property. Furthermore, we show that there is a f amily of Riemannian metrics that are compatible with such a connection. The se metrics are products of the bi-invariant metric on the group of rotation s and a positive-definite constant metric on the group of translations.