Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem

Abstract

Given a submersion $\pi :Q\to M$ with an Ehresmann connection $\mathcal{H}$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a generalization of the magnetic force. This generalized force is described using the curvature of $\mathcal{H}$ along with a new form of parallel transport of covectors vanishing on $\mathcal{H}$. Using the Pontryagin maximum principle, we apply this theory to optimal control problems $M$ and $Q$ to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length.

31 pages

References in corpus (1)

• Rubber rolling over a sphere

Cited by in corpus (3)

• On the Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses
• Homogeneous geodesics in sub-Riemannian geometry
• Infinitesimally helicoidal motions with fixed pitch of oriented geodesics of a space form