On the derivative constraints of input shaping control

Conventionally, derivative constraints have been added to the input shaper to increase robustness to modeling error in natural frequency and damping ratio, and the robustness of input shaping has been evaluated from the ratio of residual vibration amplitude with input shaping to that without input shaping. However, the derivative constraints used for the ZVD shaper and the derivative of the ratio of residual vibration amplitude are mathematically confused in the previous literatures, even if the conceptual explanation for both derivatives therein is generally acceptable. In this paper, the relationship of the derivative constraints used for ZVD shaper and the zero derivative of the ratio of residual vibration amplitude are derived and clarified mathematically, and the relationship between them is demonstrated using an example.