# Calibration of a Rotary Encoder and a Polygon Using a Two-Autocollimator Method
## 2 Configuration and Theory of the Two-Autocollimator Method  
Figure 1a shows a schematic of the calibration setup. A 24-faced polygon is mounted at the center of a rotary encoder. The 24-faced polygon and the rotary encoder are rotated at the same time. In this paper, the 24-faced polygon is used to illustrate the related theory and conduct experiments. A polygon with any number of faces can be used. This polygon has 24 measuring faces. Two autocollimators are placed and fixed in front of the rotary encoder. The no. 1 autocollimator is collimated to the first face of the polygon. The no. 2 autocollimator is collimated to the second face of the polygon. As shown in Figure 1b, the polygon has the measuring faces (R) and the nominal faces (N). In the ideal polygon, the measuring faces are the same as the nominal faces. Between the measuring face and the nominal face is a slight deviation, called the pitch angle deviation, \(\beta_{\rm i}\). Each measuring face has a different pitch angle deviation. For the 24-faced polygon, one full circle is assumed to be divided into 24 nominal faces and 24 measuring faces. The angle interval between two consecutive nominal faces is 15\({}^{\circ}\)called the nominal pitch angle. According to the circle closure principle, the summation of all pitch angle deviations must be zero:  
\[\sum_{i=1}^{24}\beta_{i}=0 \tag{1}\]  
The rotary encoder is calibrated to determine the angle error \(\alpha\) (\(\theta\)), which is an angle deviation between the measured and true angles. The angle error \(\alpha\) (\(\theta\)) is the angle error of the rotary encoder at \(\theta\). The 24-faced polygon can be measured at rotary encoder angles \(\theta\) of ((i \(-\) 1)*15\({}^{\circ}\)), where i is from 1 to 24. Figure 2 shows the calibration value determination. The measured values contain the angle error of the rotary encoder, the pitch angle deviation of the 24-faced polygon, and the setup error. The setup error is the misalignment between the rotary encoder and the polygon, which remains constant at different angles. Different autocollimators have different setup errors. The setup error \(U_{1}\) is for the no. 1 autocollimator, and the setup error \(U_{2}\) is for the no. 2 autocollimator. As shown in Figure 2a, the rotary encoder is rotated at the starting angle of 0\({}^{\circ}\). The first face of the 24-faced polygon is adjusted to a starting angle of 0\({}^{\circ}\). The no. 1 autocollimator measures the angle error \(\alpha\) (0\({}^{\circ}\)), the pitch angle deviation \(\beta_{1}\), and the setup error \(U_{1}\). The rotary encoder is rotated in one step to an angle of 15\({}^{\circ}\). The no. 1 autocollimator collimates the second face of the 24-faced polygon. The no. 1 autocollimator then measures the angle error \(\alpha\) (15\({}^{\circ}\)), the summation of the pitch angle deviations (\(\beta_{1}+\beta_{2}\)), and the setup error \(U_{1}\). The rotary encoder is rotated in two steps to an angle of 30\({}^{\circ}\). The no. 1 autocollimator the third face of the 24-faced polygon. The no. 1 autocollimator then measures the angle error \(\alpha\) (30\({}^{\circ}\)), the summation of the pitch angle deviations (\(\beta_{1}+\beta_{2}+\beta_{3}\)), and the setup error \(U_{1}\). Overall, the rotary encoder is rotated in 23 steps to an angle of 345\({}^{\circ}\). The no. 1 autocollimator collimates the 24 th face of the 24-faced polygon. The no. 1 autocollimator measures the angle error \(\alpha\) (345\({}^{\circ}\)), the summation of the pitch angle deviations of the first to the 24 th faces, and the setup error \(U_{1}\). As shown in Figure 2b, the rotary encoder is at the starting angle of 0\({}^{\circ}\). The no. 2 autocollimator collimates the second face of the 24-faced polygon. The no. 2 autocollimator measures the angle error \(\alpha\) (0\({}^{\circ}\)), the pitch angle deviation \(\beta_{2}\), and the setup error \(U_{2}\). The rotary encoder is rotated in one step at 15\({}^{\circ}\). The no. 2 autocollimator collimates the third face of the 24-faced polygon. The no.