Input Shaping for Multibody Oscillatory Systems Described by DAEs

Abstract

Input Shaping is a feedforward technology that do not induces unwanted dynamics for systems described by ordinary differential equations. This work examines the issue of Input Shaping for multibody oscillatory systems described by differential algebraic equations (DAEs). A Double Pendulum has been choosed as show case, because this multibody system without flexible bodies exhibits two oscillatory modes that Input shaping might cancel. Firstly, the constraint equations corresponding to the prismatic and the two revolute joints have been formulated. Thus the Jacobian corresponding to those equations has been carried out. Customary techniques for the numerical integration of the Differential Algebraic Equations formulated with basis in the Jacobian plus Lagrange multipliers to model the contact forces were developed. The response to a white noise feed into the trolley driver variable was obtained by undergoing such numerical integration. The time response to the white noise corresponding to the orientation of the local axis of each link was collected. Thus the FFT of both time signals allows the experimental estimation of the natural frequencies associated to both modes of vibration. Finally a two mode Input Shaper has been designed for the multimode system. Several comparisons were presented for the shaped and unshaped responses. Input Shaping a type of Finite Impulse Response (FIR) Filtering performs well for mitigating the oscillations. The difficulties to prove this statement, facing with DAEs, have been overcome by this novel work.