A Multi-Mode Analytical Frequency Modulation Input-Shaping Control

In rest-to-rest maneuvers, input shapers like the double step (DS), zero vibration (ZV), and zero vibration derivative (ZVD) are widely utilized to eliminate residual vibrations in single-mode systems. These shapers can be used to eliminate residual oscillations in multimode systems, given that the higher frequencies are odd multiples of the system's fundamental frequency. However, the natural frequencies depend on the physical properties of the system, and such ratios cannot be guaranteed. In this study, an analytical frequency modulation technique is proposed to eliminate the residual oscillations of a double pendulum using a modified single-mode shaper. The proposed technique is based on altering the natural frequencies of the system, forcing the odd multiple ratio. This involves modifying a single-mode double-step (SMDS) input shaper by adding scaled state variables, first and second angles, to the original shaper. This addition allows the user to choose the first natural frequency and force the second natural frequency to be an odd multiple of the chosen frequency. To apply the proposed technique, the double pendulum nonlinear equations of motion are derived, linearized, and then solved analytically using modal analysis. The scaling parameters used to modify the natural frequencies are then solved analytically. To prove the concept, several numerical simulations with randomly selected parameters are presented and then experimentally tested on a scaled overhead crane. The numerical and experimental results demonstrate the effectiveness of the proposed technique.