Communications in Information and Systems
Volume 12 (2012)
Underactuated control in parallel transported directions: the example of the three dimensional Heisenberg group
Pages: 301 – 323
The authors consider a natural reachability problem on a Riemannian manifold. Given an initial point on a manifold together with an initial subspace of the tangent space at that point, consider piecewise smooth curves such that the velocity at each point along the curve is tangent to the parallel translation of the given initial subspace. The problem is to determine or characterize the set of points reachable by such curves. The authors show that the problem can be formulated in terms of the standard control theory machinery of singular distributions and vector fields by lifting to the frame bundle. It is shown that if the initial velocity subspace is tangent to a totally geodesic submanifold, then the reachable set is contained in that submanifold. Thus our problem makes contact with the existence and uniqueness problem for totally geodesic submanifolds.
In the absence of a general result along these lines, it is natural to consider special cases. The authors consider the case where $M$ is the three dimensional Heisenberg group. We show that in this example, all points are reachable and further, the final configuration of the subspace carried along can be specified as well. This stronger result will be expressed in terms of the orthonormal frame bundle.