A direct Lyapunov approach for a class of underactuated mechanical systems

A Lyapunov direct method is proposed for a class of underactuated, mechanical systems. The direct method is derived in general for systems having n degrees of freedom of which only m < n are actuated. The applications consist of a class of systems where the elements of the mass/inertia matrix and the gravitational forces/torques are either constants or functions of a single generalized position variable and where n is two and m is one. The time derivative of the candidate Lyapunov function produces a relation that is solved via a matching method. Some of the matching equations consist of linear differential and partial differential equations. It is shown for this class of systems, that the solutions of these linear differential and partial differential equations necessary for assuring asymptotic stability can be evaluated numerically as part of the feedback process. Examples are presented involving an inverted pendulum cart and an inertia wheel pendulum.