Why Controllability Of Rolling Motions May Fail

Abstract

We are interested in knowing whether or not the motion of two smooth surfaces rolling on each other, without slip or twist, can be controlled. We present a few cases of surfaces rolling on a tangent plane where we show that controllability fails and why. The control system associated to a rolling motion defines a distribution in the configuration space. If this rolling distribution is bracket generating, local controllability is guaranteed. After deriving the kinematic equations for rolling Euclidean submanifolds of co-dimension one, we derive a condition for local controllability.