Reducing residual vibration in systems with time-varying resonances

Robots which perform rapid motions tend to excite system resonant frequencies. In order to perform a sequence of tasks more quickly, both move time and vibration settling time must be reduced. In addition, this must be accomplished in the presence of time-varying resonant frequencies. In this paper, functions are derived which accomplish open-loop moves rapidly with greatly reduced residual vibration amplitude, as resonant frequencies are allowed to vary by ±10%. First, a direct relationship is established between the frequency spectrum of the input functions and the resulting residual acceleration amplitude. This establishes a specification on spectral magnitude at resonance which must be satisfied in order to bring residual vibration within acceptable bounds. Then functions are derived as a series expansion of ramped sinusoid functions with coefficients chosen so as to minimize spectral magnitude in a frequency band surrounding resonance. Finally, some simulations are performed to indicate that these functions can reduce residual vibration considerably even when the resonant frequency changes by 10%.