Controllability of rolling without twisting or slipping in higher dimensions

We describe how the dynamical system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\hat M$ without twisting or slipping, can be lifted to a nonholonomic system of elements in the product of the oriented orthonormal frame bundles belonging to the manifolds. By considering the lifted problem and using properties of the elements in the respective principal Ehresmann connections, we obtain sufficient conditions for the local controllability of the system in terms of the curvature tensors and the sectional curvatures of the manifolds involved. We also give some results for the particular cases when $M$ and $\hat M$ are locally symmetric or complete.