Sphere Rolling on Sphere: Alternative Approach to Kinematics and Constructive Proof of Controllability

The kinematic equations for rolling a sphere on another sphere, subject to non-holonomic constraints of non-slip and non-twist, are known and can be found in [7]. Here we present an alternative approach to derive these kinematic equations which is also suitable for describing the rolling of more general manifolds embedded in Euclidean space. This approach consists on rolling each of the manifolds separately on a common affine tangent space and then using the transitive and symmetric properties of rolling maps to derive the kinematic equations of rolling one manifold on the other. We use this approach to derive the kinematic equations for rolling an n-dimensional sphere on another one with the same dimension. It is also well known that the sphere rolling on sphere system is controllable, except when the two spheres have equal radii. This is a theoretical result that guarantees the possibility to roll one of the spheres on the other from any initial configuration to any final configuration without violating the non-holonomic constraints. However, from a practical viewpoint it is important to know how this is done. To answer this more applied question, we present a constructive proof of the controllability property, by showing how the forbidden motions can be performed by rolling without slip and twist. This is also illustrated for 2-dimensional spheres.