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Time-Suboptimal Trajectories for Vibration-Free Positioning of Undamped Flexible Systems

Time-suboptimal reference trajectories are presented in this paper for vibration-free positioning of 1-DOF undamped linear flexible systems. The trajectories are obtained by applying shaping techniques; that is, firstly the profiles are parameterized in time domain, and then the parameter values are determined to satisfy conditions specified both in time and frequency domains. Closed-form solutions are available in the undamped case. Since time-optimal problems subject to acceleration and velocity constraints are solved, positioning time of the resulting trajectories are minimized; positioning time shorter than the well-known UM-ZV shaper can be achieved in some cases. Implementation issues which arise when a discrete-time controller is used to generate the proposed trajectory are also investigated and solved. The trajectories are implemented in a positioning controller and the vibration suppression performance is verified through experiments as well as numerical simulations.