Computational Geometric Mechanics and its Applications to Geometric Optimal Control Theory

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the control of the 3D pendulum system, and more generally, the applications of discrete mechanics and geometry to the discretization of optimal control problems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space, without the use of local charts, reprojection, or constraints.

In addition, we will introduce a numerically robust shooting based optimization algorithm that relies on the conservation properties of geometric integrators to accurately compute sensitivity derivatives, thereby yielding an optimization algorithm for the control of mechanical systems that is exceptionally efficient.

This is joint work with Anthony Bloch (Math, UM), Mathieu Desbrun (CS, Caltech), Anil Hirani (CS, UIUC), Islam Hussein (Aero, UM), Taeyoung Lee (Aero, UM), Jerrold Marsden (CDS, Caltech), N. Harris McClamroch (Aero, UM), Amit Sanyal (MAE, ASU), Alan Weinstein (Math, Berkeley), and Dmitry Zenkov (Math, NCSU).

The research has been supported in part by NSF grant DMS-0504747, and a Rackham faculty fellowship and grant from the University of Michigan.