Constructive feedback linearization of underactuated mechanical systems with 2-DOF

In the last years, the control community has developed several and powerful methods to control nonlinear systems, especially for underactuated mechanical systems. Thus, methods based on passivity, like Interconnection and damping assignment passivity-based control (IDA-PBC) and Controlled Lagrangians have solved many interesting control problems for particular full classes of systems. Usually, the solutions of these methods relies on solving a set of partial differential equations (PDEs), which is not always possible. This paper presents a constructive methodology to control underactuated mechanical systems with 2-DOF, by means of classical feedback linearization and Lyapunov design. The steps of the design are presented following a simple pseudo-code(1), that allows us to redesign a proposed fictitious output in a constructive way. The methodology has been tested with three very well-known underactuated mechanical systems: the inertia wheel pendulum, the pendulum on a cart and the rotary pendulum. The obtained solution for the inertia wheel pendulum takes into account the friction, since recent works have shown that it cannot be neglected. In the case of the planar pendulum on a cart, the solution is quite similar to the one obtained by Controlled Lagrangians but with better performance, and, furthermore, our planar pendulum solution paves the way to obtain a new solution for the rotary pendulum, or the so-called Furuta pendulum, that, to the best of our knowledge, has the largest attraction basin presented and experimentally tested so far. The attraction basin tends to the whole upper half plane by increasing a control gain. The only thing that hampers us to do this, is the actual saturation limit of the control input. In spite of the latter, successful experimental results for this rotary pendulum solution are given.