# Flatness of heavy chain systems

Auteurs: NICOLAS PETIT AND PIERRE ROUCHON.

SIAM J. of Control and Optimization. Vol 40, No. 2, pp. 475-495. 2001. DOI: 10.1137/S0363012900368636

Abstract. In this 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. 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 tra jectory 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.

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SIAM J. of Control and Optimization. Vol 40, No. 2, pp. 475-495. 2001. DOI: 10.1137/S0363012900368636

Abstract. In this 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. 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 tra jectory 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.

Download PDF