Stabilization of flexible one-link arm position: Stability domains in the space of feedback gains

A one-link flexible arm with a load at the end is considered. The arm can be rotated around its hub in a horizontal plane. This rotation is actuated by a DC motor. The dynamics of a controlled flexible arm are described by an infinite-dimensional mathematical model, which contains the Bernoulli-Euler beam equation, equations of the actuator motion and boundary con-ditions. The control voltage fed to the motor is a linear com-bination of arm angular position, integral, rotation speed, arm deformations, tip position, and velocity. The principal goal of the article is to find linear feedback gains such that the desired arm position is asymptotically sta-ble. This problem of stability arises when an elastic manipula-tor is designed. We have analytically described the boundaries of the domains of asymptotic stability in the space of feedback gains and constructed these domains in different cases. It is shown that the stability domains for a flexible arm belong to and are less than the corresponding domains for a rigid arm. We have shown that under feedback on tip position and veloc-ity, the one-link flexible manipulator is unstable if damping is small. 1