Low-harmonic rational Bézier curves for trajectory generation of high-speed machinery

For a given high-speed machinery, a significant source of the internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the system is forced to follow. In this paper a special class of rational Bézier curves is presented that correspond to low-harmonic trajectory patterns. Harmonic Bernstein polynomials and harmonic deCasteljau algorithm are also introduced as two major tools for generating the harmonic rational Bézier curves. These curves can be used to synthesize trajectories within the dynamic response limitations of the actuators while avoiding the excitation of the natural modes of vibration of the system.