Exact solution and infinite-dimensional stability analysis of a single flexible link in collision

There is an increasing interest in examining the use of flexible link manipulators in tasks where there is contact with the environment. Presently, there has been limited work examining the stability of force control strategies for such manipulators, especially in the case where there is a switching transition between the unconstrained and constrained environments. In this paper, the modeling and stability of a single flexible link under proportional derivative control contacting an environment is studied. Intuitively, since the system only has passive elements, one would expect the system to be stable. With a few very reasonable assumptions, the problems associated with finite-dimensional approximations are solved by using a novel infinite-dimensional approach. The resultant infinite-dimensional switching system is shown to be asymptotically stable using an energy-based method.