# Calibration of a Rotary Encoder and a Polygon Using a Two-Autocollimator Method
## 1 Introduction  
Polygons and rotary encoders are considered to be full-circle angle standards. A polygon can be treated as a circumference divided by an equal angle, such as 15\({}^{\circ}\), 30\({}^{\circ}\), and 90\({}^{\circ}\). These are named 24-faced, 12-faced, or 4-faced polygons, respectively. A rotary encoder is made up of a glass disc with a suitable reader. The reader can determine the angular positions of the glass disc. Calibration of polygons and rotary encoders is essential to determine their accuracy at most national standards laboratories worldwide. The calibration values are standard angle values to transfer other instruments. Polygons and rotary encoders are implemented for various tasks such as accuracy testing and calibration of angle-measuring instruments, rotary tables, and so on. The calibration is based on the circle closure principle: the sum of all angle errors of a full circle must be zero. Therefore, the circle closure principle is widely used to calibrate polygons and rotary encoders. For the calibration of rotary encoders, many previous papers proposed using self-calibration. For the self-calibration method, it is unnecessary to use an external standard. Sensor heads and a rotary encoder are installed inside a rotary table. The rotary encoder is rotated, and the sensor heads measure angle values at different angle positions corresponding to the rotary encoder. According to the circle closure principle, the rotary encoder can self-calibrate. Many self-calibration methods have been proposed [1; 2; 3]. [PERSON] et al. [4; 5] developed an angle comparator that was constructed using a rotary encoder and 16 read heads to calibrate all angle errors of the rotary encoder. [PERSON] et al. [6; 7; 8; 9; 10; 11] developedthe SelfA rotary encoder for self-calibrating the rotary encoder. The SelfA rotary encoder consists of a rotary encoder and several groups of sensor heads. The SelfA rotary encoder can calibrate all angle errors of the rotary encoder. In contrast to self-calibration, using an external standard is another method to calibrate a rotary encoder. Many studies have demonstrated the use of a polygon as an external standard [12; 13]. [PERSON] et al. [14] and [PERSON] et al. [15] used a 36-faced polygon, [PERSON] et al. [16] used a 24-faced polygon, and [PERSON] et al. [17] used a 12-sided polygon and a 6-sided polygon. This method has to use an autocollimator to measure the polygon. The problem with this method is that the polygon must be calibrated before calibrating a rotary encoder. For the calibration of polygons, many papers proposed using a rotary encoder as an external standard [18; 19]. The rotary encoder has to calibrate by using self-calibration to transfer standard angle values. These standard angle values can calibrate a polygon. This method also has the disadvantage of using two experiments to calibrate the rotary encoder and the polygon. Using two autocollimators is another method to only calibrate a polygon [20; 21; 22]. The polygon is mounted on a rotary table. The two autocollimators measure the polygon. The angle errors of the rotary table do not influence the calibration of the polygon. Cross-calibration is used simultaneously, calibrating a rotary encoder and a polygon [23]. With this method, a rotary encoder is the external angle standard, and a polygon is the calibrated standard. All angle errors of the rotary encoder and all pitch angle deviations of the polygon are unknown. Based on the circle closure principle, the rotary encoder and the polygon can be calibrated simultaneously. However, the measurement cycle depends on how many faces the polygon has. [PERSON] et al. [24] proposed a 24-sided polygon as the calibrated angle standard. The SelfA rotary encoder is the external angle standard. The polygon and SelfA rotary encoder can be calibrated simultaneously. This method is time-consuming depending on how many faces the polygon has. The 24-sided polygon is used so the measurement cycle is 24. Therefore, we propose the two-autocollimator method.