A simple STLC test for mechanical systems underactuated by one control

We consider the controllability problem, i.e., the existence of a suitable control input that achieves a desired reconfiguration, for underactuated mechanical systems. Since there is no general analytic tool for investigating this natural controllability property in nonlinear systems, one possibility is to study small-time local controllability (STLC), a property which is sufficient for stating controllability. The available STLC conditions require the computation of Lie brackets on the classical state-space form of the dynamic model equations. In this paper, we provide a simple sufficient condition for testing STLC in underactuated mechanical systems with n degrees of freedom and n - 1 control inputs, directly based on the terms of the system inertia matrix. As an application, we analyze the STLC of planar robots with n rotational joints, one of which is passive.