An online angular self-calibration method for installation errors of rotary grating encoders

Installation-induced eccentricity and tilt are major sources of angular measurement error in rotary grating encoders, while existing self-calibration methods often require multiple reading heads or external reference instruments. This paper proposes an online angular self-calibration method for a polar-coordinate encoder that simultaneously measures rotation angle and radial displacement using a single polar-coordinate reading head. A coupled eccentricity–tilt error model is established, and the installation parameters are identified online from radial displacement measurements by solving an overdetermined linear system. The identified parameters are then incorporated into an angular compensation model for online error correction under varying mounting conditions. Conditioning and perturbation analyses demonstrate that the proposed identification method has good numerical stability and robustness. Experiments performed on an air-bearing calibration platform show that, under a fixed installation condition, the maximum angular error is reduced from 8.65 arcsec to 0.84 arcsec after compensation, corresponding to an approximately 90% reduction. In five repeated reinstallation tests, the residual angular errors after compensation range from 0.67 arcsec to 0.88 arcsec, with a mean value of 0.78 arcsec. A measurement uncertainty analysis based on the experimental results gives a combined standard uncertainty of 0.15 arcsec and an expanded uncertainty of 0.30 arcsec (k = 2). These results demonstrate that the proposed method provides an effective and practical solution for online suppression of installation-induced angular errors in high-precision rotary grating encoders.