Impulse Vectors for Input-Shaping Control: A Mathematical Tool to Design and Analyze Input Shapers

Input shaping is an open-loop control technique for reducing residual vibrations in computer-controlled machines that is implemented by convolving a sequence of impulses (an input shaper) with any desired command [1]. The timing instants and amplitudes of the impulses are determined from the natural frequency and damping ratio of a vibratory system by solving a set of constraint equations [2]. Because the input shaper is generally located outside the feedback loop (Figure 1), input-shaping control does not affect the stability of the overall control system, which is a benefit compared to conventional feedback control.