Dynamics, Geometry, and Control of Discrete Mechanical Systems

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The objective of this project is to study discretizations of mechanical systems with constraints. This study is motivated by the facts that such discretizations may be used for numerical simulations of mechanical systems and that problems of studying periodic motions in mechanics and control are intrinsically discrete. The proposed research is aimed at two areas:(i) Structurally-stable and structure-preserving discretizations of systems with velocity constraints. The dynamics of systems with velocity constraints is remarkably diverse. The goal is the development of discrete dynamical systems that adequately represent the continuous-time constrained dynamics.(ii) The use of such discretizations in stabilization of periodic orbits of mechanical systems. It is anticipated that such control techniques will be especially useful in cases when the period is long and/or intrinsic structure of controlled systems should be preserved during numerical simulations.The dynamics of systems with constraints is important in various industrial and scientific applications. Such constraints include rolling and sliding constraints, chained rigid links, and various constraints found in thermodynamics and electromechanical systems. 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 periodic motions may be desirable--for example in achieving a desired robotic or autonomous vehicle motion. Our methods should be helpful in analyzing and prescribing motions of constrained systems, such as robotic and autonomous vehicles.