Existence and design of reduced-order stable controllers for two-link underactuated planar robots

This paper concerns reduced-order stable controllers in a unified way for two typical examples of underactuated robots called the Acrobot and the Pendubot which are two-link planar robots moving in a vertical plane with a single actuator at joints 1 and 2, respectively. The main contributions of this paper are threefold. First, using properties of the mechanical parameters of two-link planar robots, this paper proves via a constructive procedure that stable controllers exist for a local stabilization of the upright equilibrium point (two links are in upright position) of each robot by designing a specific output with an adjustable parameter and employing the pole-zero relation of the linearized model around the equilibrium point. Second, the paper presents a design method to obtain explicitly a formula of second-order (reduced-order) stable controllers by choosing the adjustable parameter. Third, in terms of strong stabilization (stabilization with a stable controller), some new features of the Acrobot and the Pendubot are revealed and are compared with those of the cart-pendulum system. Simulation results for the two robots are presented to validate the theoretical results.