Flatness of Heavy Chain Systems

In this paper the flatness [M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Internat. J. Control, 61 (1995), pp. 1327–1361, M. Fliess, J. Lévine, 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 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 [R. M. Murray, in Proceedings of the 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 by a one-dimensional (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. This parameterization results from symbolic computations. Replacing the time derivative by the Laplace variable s yields a second order differential equation in the spatial variable where s is a parameter. Its fundamental solution 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 Liouville transformation, this solution satisfies, modulo explicitly computable exponentials of s, the assumptions of the Paley– Wiener theorem. This solution is, in fact, the transfer function from the flat output (the position of the free end of the system) to the whole state of the system. Using an inverse Laplace transform, we end up with an explicit motion planning formula involving both distributed and punctual advances and delays operators.