Mathematical equivalence of two derivative constraints for robust input shapers

Derivative constraints and/or nonzero vibration constraint at model frequency have been widely used for the design of robust input shapers suppressing residual vibration under the condition with some modeling errors in natural frequency and damping ratio. However these constraints are confused mathematically in some previous literatures even if conceptual explanations of these constraints are acceptable. In this paper, robustness constraints are categorized. Then two types of derivative constraints for robust input shapers are clarified mathematically, and the relationship of the two derivative constraints are derived. Furthermore a modified robustness performance index is introduced. The validity of the assertion is demonstrated by means of simulation and experimental studies.