Geometry and topology of nonholonomic structures

Texas A&M University, College Station TX

Investigators

Abstract

The nonholonomic structures of the title are modeled on physical systems with velocity constraints not arising from position constraints. The simplest example of such a system is a ball rolling without slipping on a hard surface. Nonholonomic structures and especially distributions that correspond to linear constraints for velocities, are common in robotics (car-like robots) and neuroscience (models of the motion of articulated bodies), as well as in areas of physics and mathematics. One of the thrusts of the project, devoted to inverse optimal control problems, will provide explicit algorithms for recovering a cost functional from the given set of optimal trajectories. Another thrust of the project, devoted to the Gromovs h-principle for distributions, will answer important questions of global existence of several new classes of distributions with prescribed differential properties and the possibility to avoid singularities of such distributions by continuous deformations. Other thrusts of the project are devoted to equivalence problems for new classes of distributions, CR structures, and other filtered structures. The principal investigator, in collaboration with his students, will continue to develop a special Maple based software package for computation of state-feedback and gauge invariants of various control and mechanical systems of practical interest, and for new variants of Tanaka prolongation needed in the project, which are beyond the scope of the existing standard packages. Broader impacts will include training and advising of several PhD students, and undergraduate research projects on topics of this project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.