Generation of Desired Lissajous Curve-Like Vibrational Orbits for a Single Motor-Based Flexible Actuator

This study presents a vibrational orbit generation method for a flexible actuator. We propose a method for generating a desired Lissajous curve-like vibrational orbit by inputting a synthetic wave of two sine waves into the vibration source. First, we propose a dynamic model of the actuator, which consists of a flexible beam and a rotating motor. The beam has a rectangular cross-section with principal axes that are non-parallel to the rotation shaft, introducing structural anisotropy. Next, we analyze the relationship between input wave and vibrational orbit of the beam's tip. We demonstrate that single sine wave input generates an ellipse-like orbit, and the orientation of its major axis changes with the input frequency. By selecting two frequencies whose major axes are orthogonal and overlapping the two waves, we generate a Lissajous curve-like orbit with a desired frequency ratio. Finally, the theory is validated through experiments, where complex and diverse orbits are produced based on the frequency ratio and phase difference of the synthetic waves. This method contributes to the development of new underactuated robotic manipulation and locomotion systems, such as foot, wing, and fin mechanisms vibrating at high speeds.