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 *un*controllable (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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