Flatness of heavy chain systems

In this paper the flatness [ M. Fliess, J. Levine, P. Martin, and P. Roucho n, Internat. J. Control, 61 ( 1995), pp. 1327-1361, M. Fliess, J. Levine, P . Martin, and P. Rouchon, IEEE Trans. Automat. Control, 44 ( 1999), pp. 922 -937] of heavy chain systems, i.e., trolleys carrying a fixed length heavy chain that may carry a load, is addressed in the partial derivatives equati ons framework. We parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem, namely, steering fro m one state to another state. When considered as a finite set of small pend ulums, these systems were shown to be at [R. M. Murray, in Proceedings of t he IFAC World Congress, San Francisco, CA, 1996, pp. 395-400]. Our study is an extension to the infinite dimensional case. Under small angle approximations, these heavy chain systems are described b y a one-dimensional (1D) partial differential wave equation. Dealing with t his infinite dimensional description, we show how to get the explicit param eterization of the chain trajectory using (distributed and punctual) advanc es and delays of its free end. This parameterization results from symbolic computations. Replacing the tim e derivative by the Laplace variable s yields a second order differential e quation in the spatial variable where s is a parameter. Its fundamental sol ution is, for each point considered along the chain, an entire function of s of exponential type. Moreover, for each, we show that, thanks to the Liou ville transformation, this solution satis es, modulo explicitly computable exponentials of s, the assumptions of the Paley-Wiener theorem. This soluti on is, in fact, the transfer function from the at output ( the position of the free end of the system) to the whole state of the system. Using an inve rse Laplace transform, we end up with an explicit motion planning formula i nvolving both distributed and punctual advances and delays operators.