# Calibration of Rotary Encoders Using a Shift-Angle Method
## 1 Introduction  
The angle standard is based on the circle closure principle: the sum of the angles of a full circle must be 2\(\upmu\)t. Therefore, the circle closure principle is widely used to calibrate many reference angle standards and working angle standards, such as angle gauges, polygons, and rotary encoders. Rotary encoders are widely used in laboratories and factories in the field of measurement and manufacturing. The angle errors of the rotary encoders affect the measurements and manufacturing quality. For the precise calibration of the rotary encoders, two methods are widely used: self-calibration and cross-calibration.  
In self-calibration, it is unnecessary to use an external standard to calibrate rotary encoders [1]. The sensor heads and rotary encoders are installed inside a rotary table. The sensor heads read out the measured angles distributed at different angles corresponding to the rotary encoder. According to the circle closure principle, a rotary encoder can self-calibrate. Many self-calibration methods have been proposed. [PERSON] et al. [2] proposed a calibration rotary encoder system in which six read-heads were unevenly spaced around a rotary encoder to self-calibrate its accuracy. [PERSON] et al. [3; 4; 5] developed an angle comparator that was constructed using one rotary encoder and 16 read-heads in which eight read-heads were uniformly distributed and eight read-heads were arranged pairs. In this system, a new method for self-calibration of divided circles calibrated the rotary encoders. In addition, [PERSON] et al. [6; 7] indicated an optimization-based arrangement of reading heads for self-calibration. [PERSON] et al. [8; 9; 10; 11] developed the SelfA rotary encoder for calibrating rotary encoders. The SelfA encoder consists of a rotary encoder and several groups of sensor heads.  
Cross-calibration is widely used to calibrate angle standards. With this method, an auxiliary reference angle standard is used in conjunction with an equation based on the circle closure principle. The angle errors of the reference angle standard and the calibratedangle standard can be measured simultaneously. Many studies have demonstrated that such measurements involve the use of a polygon as the reference angle standard. [PERSON] et al. [12; 13] proposed the four-sided polygon as the reference angle standard to calibrate the laser dynamic goniometer. [PERSON] et al. [14] indicated the 36-sided optical polygon as the reference angle standard to calibrate the index table. Many previous studies have employed different-faced polygons, such as 12 faces, 24 faces, and 36 faces, as the reference angle standard to calibrate the rotary encoder [15; 16; 17; 18; 19; 20]. However, this method has a measurement limitation in terms of the pitch angle of the polygon.  
In this paper, we present a new shift-angle method for calibrating rotary encoders at every angle. The introduction section presents the related research papers and techniques. Next, the theory and configuration section describes the proposed shift-angle method, which can be used to calibrate angles smaller than the pitch angle of the polygon. In the experiment, an autocollimator and a 24-faced polygon are used to calibrate the SelfA rotary encoder that is equipped with a rotary encoder and multi-arranged sensor heads. The SelfA rotary encoder can be calibrated by using self-calibration. The experiment shows the difference between the results of self-calibration and the proposed shift-angle method for the same SelfA rotary encoder. The results indicate that the proposed shift-angle method is feasible and can be validated. Then, the uncertainty evaluation of the shift-angle method is analyzed. Finally, a discussion and conclusions are presented.