Flatness of heavy chain systems

This paper gives an overview of the results of Petit and Rouchon (2001). Furthermore it contains some previously unpublished material concerning the homogeneous chain carrying a load. In the above paper the flatness of heavy chain systems, i.e. trolleys carrying a fixed length heavy chain that may carry a load, is addressed in the partial derivatives equations framework. We parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem, namely steering from one state to another state. When considered as a finite set of small pendulums these systems were shown to be flat in Murray (1996). Our study is an extension to the infinite dimensional case. Under small angle approximations, these heavy chain systems are described by a 1D partial differential wave equation. Dealing with this infinite dimensional description, we show how to get the explicit parameterization of the chain trajectory using (distributed and punctual) advances and delays of its free end.