Collocated Adaptive Control of Underactuated Mechanical Systems

Collocated adaptive control of underactuated mechanical systems is still a concern for the control community. The main difficulty comes from the nonlinearity of the collocated inverse dynamics with respect to the base parameters, which forbids the direct application of classical adaptive control schemes. This paper extends and encompasses the Slotine's adaptive control, which was developed for fully actuated mechanical systems, to stabilize the collocated state space of an underactuated mechanical system. The key point is to define the sliding variable as the difference between the system's velocity and an exogenous state whose dynamics is considered as control input. We first revisit the Slotine's result in view of this definition and then show how to extend it to the underactuated case. Stability and convergence of time-varying reference trajectories for the collocated dynamics are shown to be in the sense of Lyapunov. Global well-posedness of the control laws is achieved by means of a new algebraic property of the mass matrix. Simulations, comparisons to existing control strategies, and experimental results on a two-link manipulator verify the soundness of the proposed approach.