Motion Control of Three Link Underactuated Manipulator by Bifurcation Control

Motion control of a three link underactuated manipulator, whose first joint has an actuator and a sensor and second and third joints do not have actuator or sensor, is theoretically proposed without feedback control with respect to the motion of the free links. By using high-frequency vertical excitation, so called Kapitza pendulum is stabilized at the upright position without state feedback control. The phenomenon can be regarded as a subcritical pitchfork bifurcation. On the other hand, it is known that the horizontal excitation causes supercritical pitchfork bifurcation in a pendulum. Also, the inclination of excitation from the horizontal and vertical directions produces the perturbation of the complete supercritical and subcritical pitchfork bifurcations, respectively. In this paper, we apply the method of multiple scales to obtain the averaged equations governing the motion of the free links. We perform the bifurcation analysis of the free links and clarify the equilibrium points in the free links and their stability. Then, we propose a strategy to swing up the free links and to stabilize them at the upright position by actuating the perturbation of the pitchfork bifurcations based on the change of the inclination of excitation.