This paper examines geometric modelling of elastically coupled, spatial rigid body pairs. For a pair of elastically coupled rigid bodies there exist coincident, unique points on the bodies or their rigid extensions known as centers of stiffness at which translation and rotation are minimally coupled. The rigid body displacement of frames at the centers of stiffness can be associated with a twist displacement, a continuous rigid body motion. A potential function is defined that is a simple quadratic function of the twist displacement. Constitutive equations relating rigid body displacement and wrench are derived. While the problem of modelling and computer simulation is emphasized, the work is also relevant for active compliance control of robotic devices.