Global Stabilisation of Underactuated Mechanical Systems via PID Passivity-Based Control

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.