In our work we study mechanical systems from the Lagrangian point of view. In this talk we shall consider those systems whose Lagrangian is kinetic energy with respect to some Riemannian metric (i.e., systems with arbitrary kinetic energy) and which have a single input vector field which depends upon the system's configuration (more general treatments allow the input to depend on velocity and time as well). Earlier work by Lewis and Murray in the multi-input context has provided sharp conditions for ``local configuration accessibility'' for these systems and some weak sufficient conditions for ``local configuration controllability.'' As one of our goals is to strengthen the conditions for controllability, it seems natural to first investigate the single-input case. We show, using techniques of Sussmann, that in the single-input case all systems we consider are locally configuration uncontrollable (except in the trivial case where the configuration manifold has dimension one). Thus we are able to fully characterise the single-input case, something which has not been done for general control systems. This suggests that the systems under consideration may have a structured enough control Lie algebra that a complete characterisation of local configuration controllability may be possible in the multi-input case. In the talk we present two simple single-input examples (a robotic leg and a planar rigid body) and, using information provided by the proof of our general result, describe the locally reachable configurations.

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