In this note we identify a class of underactuated mechanical systems whosedesired constant equilibrium position can be globally stabilised with theubiquitous PID controller. The class is characterised via some easilyverifiable conditions on the systems inertia matrix and potential energyfunction, which are satisfied by many benchmark examples. The design proceedsin two main steps, first, the definition of two new passive outputs whoseweighted sum defines the signal around which the PID is added. Second, theobservation that it is possible to construct a Lyapunov function for thedesired equilibrium via a suitable choice of the aforementioned weights and thePID gains and initial conditions. The results reported here follow the sameresearch line as [7] and [20]---bridging the gap between the Hamiltonian andthe Lagrangian formulations used, correspondingly, in these papers. Twoadditional improvements to our previous works are the removal of a non-robustcancellation of a potential energy term and the establishment of equilibriumattractivity under weaker assumptions.
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