I. Introduction
With the development of science and technology, the requirements of displacement feedback technology in numerical control systems have become more strict. The field demands higher resolution, higher accuracy, and higher reliability. At present, the mainstream measurement mechanisms for displacement measurement include optical grating, magnetic grating, capacitive grating, and time grating. In the studies on capacitive grating, Anandan and George [1] proposed a wide-range capacitive sensor for linear and angular displacement measurement. Yu et al. [2] proposed a high-precision absolute angular-displacement capacitive sensor using three-stage time-grating in conjunction with a remodulation scheme. In the studies on time grating, Chen et al. [3] proposed a long-range time-grating sensor for displacement measurement and achieved nanometer accuracy. In the studies on magnetic grating, Park et al. [4] and Nguyen et al. [5] tackled high-precision magnetic grating in their studies. In addition, the optical grating displacement measurement technology is widely used due to its strong anti-interference feature, simple structure, and ability to easily perform large-range measurements. In optical grating of studies, effectively correcting the error is a key content, mainly contain: Ye et al. [6] proposed a precise phase demodulation algorithm, and achieved high measurement resolution and accuracy. Tan and Tang [7] proposed a method of online correction of measurement signals using a radial basis function. Lu and Trumper [8] proposed an automatic correction time measurement dynamic reverse (TDR) method for angular displacement measurement. Watanabe et al. [9] proposed an error calibration method with five reading heads evenly distributed in the circumference. Probst [10] proposed an error compensation method based on eight reading heads. However, in the measurement of optical grating, the error compensation is the most accurate method. It is of great significance to study the error compensation method.