Residual vibration reduction for a flexible positioning system using input shaping technique

Elastic elements, such as timing belts, have become more attractive where long travel length is required. However, the compliance of the belt causes mechanical resonance which reduces stability margins, forces servo gains down and deteriorates machine performance. In order to suppress the vibration of a linear belt-driven system and improve the transient response performance, the mathematical model of the belt-driven system is deduced and a double closed-loop control system is constructed in this paper. Then the mechanical resonance of the belt-driven system is analyzed and classified into two categories: low frequency resonance and high frequency resonance, which is crucial for guiding the design of controller for vibration suppression. A notch filter is adopted to attenuate the high-frequency vibration and increase servo gains. A simple PID controller accompanied with input shapers is proposed to suppress the low-frequency vibration. The input shaper is designed on the z-plane and its analytic solution is deduced, which can plan an appropriate reference profile with robustness to the variations of both natural frequency and damping ratio so that the fast point-to-point motions with minimum residual vibration can be obtained. At last, the robustness of the input shaping method is discussed. Simulation results demonstrate the effectiveness of the methods proposed in this paper.