  In this note we identify a class of underactuated mechanical systems whose
desired constant equilibrium position can be globally stabilised with the
ubiquitous PID controller. The class is characterised via some easily
verifiable conditions on the systems inertia matrix and potential energy
function, which are satisfied by many benchmark examples. The design proceeds
in two main steps, first, the definition of two new passive outputs whose
weighted sum defines the signal around which the PID is added. Second, the
observation that it is possible to construct a Lyapunov function for the
desired equilibrium via a suitable choice of the aforementioned weights and the
PID gains and initial conditions. The results reported here follow the same
research line as [7] and [20]---bridging the gap between the Hamiltonian and
the Lagrangian formulations used, correspondingly, in these papers. Two
additional improvements to our previous works are the removal of a non-robust
cancellation of a potential energy term and the establishment of equilibrium
attractivity under weaker assumptions.
