Revisiting energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint (shoulder). Different from previous energy-based control solutions, we obtain the following results: 1) we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameters such that the up-down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. 2) We prove that the up-down equilibrium point is saddle (hyperbolic and unstable) via an elementary proof by using the Routh-Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. This paper prove that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. The simulation results are provided to validate these results.