Professor Melvin Leok
Department of
Mathematics
University of
Michigan
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.
Curiously, while the geometric
structure of mechanical systems plays a critical role in the construction of
geometric control algorithms, these algorithms have typically been implemented
using numerical schemes that ignore the underlying geometry.
Geometric integration is the field of
numerical analysis that focuses on developing geometric structure-preserving
integrators, and computational geometric mechanics focuses on developing
geometric integrators for dynamical systems arising from mechanics.
We will discuss the application of
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 addition, a discrete analogue of
the method of controlled Lagrangians will be introduced, and the method is
applied to the construction of a digital, real-time, feedback controller that
stabilizes the inverted relative equilibrium of the cart-pendulum system.
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.
Friday, October 7,
2005
3:30 – 4:30
p.m.