Partially analytical extra-insensitive shaper for a lightly damped flexible arm

For a lightly damped flexible arm, an extra-insensitive (EI) shaper is an efficient logic to suppress the residual vibration of a flexible arm. Presently, the EI shaper is obtained approximately by a numerical solution on the assumption that damping ratio ζ and vibration tolerance V_tol are within some intervals. In this paper, we introduce a partially analytical EI shaper which is a closed form using one parameter for a specific damping ratio ζ and vibration tolerance V_tol. The validity of the partially analytical EI shaper is demonstrated by simulation and experimental studies using a flexible up-and-down motion control system.