Control of an Axially Moving Timoshenko Beam Attached to a Moving Base

This paper investigates the vibration control of a Cartesian robot consisting of a trolley and a flexible robotic arm, wherein the robotic arm is treated as an axially moving Timoshenko beam with time-varying length. A mathematical model describing the dynamics of the trolley and the beam is established based on the extended Hamilton principle. According to this dynamic model, a boundary control law is proposed to suppress the undesired transverse vibration of the beam. The stability of the closed-loop system is verified by using the Lyapunov method. Through simulation, the system's performance under the proposed control law is demonstrated, and the impressive efficiency of the control law is also shown.