Smooth trajectory generation for industrial machines and robots based on high-order S-curve profiles

The generation of optimal reference trajectories poses a challenging problem for industrial machines due to the trade-off between execution speed and vibration amplitude. This paper presents a novel method for systematically planning smooth and robust S-curve trajectories of arbitrary order adapted to vibratory behaviour for high-speed operations. In particular, the minimum time trajectory featuring good continuity under kinematic bounds is designed utilizing an improved procedure based on the existence prediction of trajectory segments. A complete analytical solution for trajectories of orders up to five is then derived. Moreover, through an analysis of the dynamic response characteristics, a parameter tuning strategy based on modal properties is established, enabling further shaping of the acceleration profiles to remove the energy content around the resonances with lower sensitivity to modelling errors. Due to the embedded filtering effect, the optimized trajectories combine specific vibration cancellation with high-frequency attenuation while providing flexibility to balance competing objectives. Comparative experiments conducted on a robot transferring flexible beams demonstrate the effectiveness and robustness of the proposed methodology in reducing residual vibration and settling time under system uncertainties.