The investigators plan to analyze the dynamics and control of constrained mechanical systems. Proposed research includes the study of the dynamics and stability of LR systems (systems with left-invariant Lagrangian and right-invariant constraints), the properties of nonholonomic integrators, navigation and stabilization problems for systems with controls applied to internal degrees of freedom, applications of variational integrators to the method of controlled Lagrangians, dynamics of infinite-dimensional nonholonomic systems, integrable nonholonomic systems, and the use of overdetermined coordinates in mechanics and control. The investigators also plan to use geometric methods to develop a systematic approach to the construction of coordinates in the phase space that take into account intrinsic properties of a system under investigation and thus allows the researcher to write down equations of motion in a simple and compact way. This approach should be helpful both in understanding the dynamics of a system and in doing numerical simulations which respect mechanical properties.
The dynamics of systems with constraints is important in various industrial and scientific applications. Examples of such constraints include rolling and sliding, chained rigid links, rotor dynamics on rigid bodies, and coupling between elastic rods and rigid bodies. There are numerous instances in industry, engineering and science where such constraints arise: robotics, the dynamics of wheeled vehicles, and the motion of satellites in space are examples. In applications, stabilization of steady-state motions (such as the straightforward or circular motion at a constant rate) is often desired -- for example in achieving a desired robotic or autonomous vehicle motion. Our methods should be helpful in achieving such motions and in general in analyzing and prescribing the motions or robotic and autonomous vehicles. We will also provide a framework which will be helpful in developing computer simulations of such systems.
