Anti‐Swing control of the Pendubot using damper and spring with positive or negative stiffness

Abstract

In this article, we study the anti‐swing control of the Pendubot which is a 2‐link planar robot with a single actuator driving the first joint, whose control objective is to stabilize the Pendubot to the downward equilibrium point (DEP) with the two links in the downward position for all its initial states with the exception of a set of Lebesgue measure zero. To achieve this control objective, with respect to the angle of the first joint, we present a proportional‐derivative (PD) controller using a damper and a linear spring with positive stiffness, and a sinusoidal‐derivative (SD) controller using a damper and a nonlinear spring (force being sinusoidal function of the displacement) allowing even negative stiffness. We prove that the control objective is achieved if some conditions on the stiffness of linear or nonlinear spring are satisfied. We present analytical solutions to the optimal design of the derivative gain and the stiffness which minimizes the real parts of the dominant poles of the linearized model of the corresponding closed‐loop systems around the DEP for all Pendubots in terms of their five physical parameters without any constraint. Our discussion shows that the SD controller is superior to the PD controller. We provide simulation results for a Pendubot to show that the presented SD controller can stabilize the Pendubot to the DEP faster than the presented PD controller.

Specific Point in Time Excitation Control Method for Spatial Multi-Degree-of-Freedom Systems under Continuous Operation

“…A proportional derivative (PD) controller using a damper and a linear spring with positive stiffness and a sine derivative (SD) controller using a damper and a nonlinear spring with negative stiffness are proposed. Finally, it is proved that when the linear or nonlinear stiffness of a spring meets certain conditions, the control target can be achieved [12]. Some scholars have proposed two switching control methods to realize the control of self-assembled inverted pendulum, aiming at solving two important problems of self-assembled inverted pendulum: oscillating and stabilizing in its upright equilibrium position.…”