The precise control of piezoelectric actuators is limited by inherent hysteresis, creep, and nonlinear behavior, which necessitate high-resolution displacement sensing for effective closed-loop operation. Although optical interferometers can achieve nanometer and sub-nanometer resolution, their practical implementation is often constrained by complex optical alignment, sensitivity to environmental disturbances, and limited robustness in high-speed measurements. Optical triangulation sensors offer a more robust and straightforward alternative; however, their resolution is typically insufficient for nanometer-scale displacement measurements. In this study, a novel optical triangulation sensor based on a two-stage geometric optical amplification scheme is proposed for measuring the expansion of piezoelectric stacks. The method relies purely on geometric optical amplification and does not require interferometric techniques or complex signal processing. Using off-the-shelf optical components and an industrial imaging sensor, the proposed system achieves a displacement resolution of 109.6 nm, a repeatability of 74.62 nm, and an accuracy of 98.81% with a maximum error of 207.14 nm under hysteresis measurements. The achieved resolution is primarily limited by the spatial resolution of the camera sensor, indicating that further improvements are possible through optimization of the optical configuration or the use of higher-resolution imaging devices. Owing to its simplicity and robustness, the proposed sensor is well suited for real-time closed-loop control of piezoelectric actuators.
Köksu et al. (Thu,) studied this question.