Vibration suppression of multi-DOF walking piezo actuator stages using spatial iterative learning and step-length feedback control

Walking piezo actuator (WPA) stages enable centimeter-level travel ranges with nanometer-level positioning resolution, making them well suited for high-precision motion in semiconductor equipment. Despite these advantages, the achievable precision is constrained by two main factors. First, the walking motion of each WPA introduces periodic vibrations, leading to micrometer-level displacement errors. Second, these vibrations cause step-length inconsistencies among individual WPAs, which in turn induce axis-coupling errors. To address these issues, a novel control framework for WPA stages is developed by integrating spatial iterative learning control (SILC)–based pre-compensation with step-length feedback (SLFB) control. The SILC-based pre-compensation modifies the driving waveforms of the WPAs to smooth the foot-switching process, thereby reducing periodic vibration errors during walking motion. An inverse Hammerstein hysteresis model is incorporated into the SILC framework to compensate for rate-dependent hysteresis effects and to maintain effective vibration suppression over a wide range of actuation frequencies. In addition, the SLFB controller is designed using a particle swarm optimization algorithm and regulates the step length of each WPA in real time, reducing axis-coupling errors caused by step-length inconsistencies. Experimental results are presented, with the conventional driving-waveform-based actuation (DWA) method and the commutation-angle iterative learning control (CAILC) method reported in the literature used as reference methods for comparison. With the proposed SILC-SLFB method, vibrations in all degrees of freedom of the stage are effectively suppressed, achieving angular errors of 0.76 μrad and 0.59 μrad in the Rx and Ry directions, respectively, and a translational error of 0.038 μm in the Z direction. Compared with the DWA method, the proposed SILC-SLFB method reduces the displacement errors in the Rx, Ry, and Z directions by 99.31%, 99.72%, and 99.98%, respectively. Relative to the CAILC method, the corresponding error reductions are 98.17%, 98.83%, and 99.62%, respectively.