# Calibration of a Rotary Encoder and a Polygon Using a Two-Autocollimator Method
## 3 Experiments  
### The/Two-Autocollimator Method  
Figure 3a shows a photograph of the experimental setup. The 24-faced polygon is used in the experiment, as expressed in Figure 3b. The polygon is mounted at the center of the SelfA rotary encoder (e-motionsystem, Inc., SCMS-127). A servo motor with an air bearing controls the SelfA rotary encoder. The SelfA rotary encoder and the 24-faced polygon are rotated at different angles from \(0^{\circ}\) to \(345^{\circ}\) in steps of \(15^{\circ}\). The no. 1 autocollimator (MOLLER-WEDEL OPTICAL GmbH, ELCOMAT 2000) and the no. 2 autocollimator (MOLLER-WEDEL OPTICAL GmbH, ELCOMAT 3000) are fixed outside the SelfA rotary encoder. Both the two autocollimators have the same specifications. The measuring range is from +1000\({}^{\circ}\) to \(-1000^{\circ}\), and the resolution is 0.001\({}^{\circ}\). The SelfA rotary encoder is rotated at the starting angle of \(0^{\circ}\). The no. 1 autocollimator is collimated to the first face of the 24-faced polygon. The no. 2 autocollimator is collimated to the second face of the 24-faced polygon. Calibration procedures are as follows. The SelfA rotary encoder is located at \(0^{\circ}\). The no. 1 autocollimator is aligned to the first face of the 24-faced polygon. The angle of the no. 1 autocollimator is adjusted to approximately 0. The no. 2 autocollimator is aligned to the second face. The angle of the no. 2 autocollimator is adjusted to approximately 0. Then, the SelfA rotary encoder is rotated by \(15^{\circ}\) in one step. The no. 1 autocollimator collimates the second face of the polygon. The no. 2 autocollimator collimates the third face of the polygon. The two autocollimators collimate successive faces of the polygon. Finally, the SelfA rotary encoder is rotated by \(345^{\circ}\) in 23 steps. The no. 1 autocollimator collimates the 24 th face of the polygon. The no. 2 autocollimator collimates the first face of the polygon. Both the no. 1 autocollimator and the no. 2 autocollimator measure 24 values corresponding to \(0^{\circ}\), \(15^{\circ}\),..., \(345^{\circ}\) in steps of \(15^{\circ}\). The measured values of the no. 1 autocollimator are as per Equation (4). The measured values of the no. 2 autocollimator are as per Equation (5). These 48 measured values are used to calculate the difference measured values in Equations (6) and (7). All pitch angle deviations of the polygon can be calculated using Equations (7) and (8), as depicted in  
Figure 3: (**a**) Photograph of the experimental setup. (**b**) Photograph of the 24-sided polygon. Figure 4. According to Equations (9) and (10), the setup errors, \(U_{1}\) and \(U_{2}\), are calculated as \(-0.02\)\" and \(0.71\)\", respectively. A shift-angle method, based on cross-calibration, for calibrating the 24-sided polygon is used to verify the calibration results. The detailed calculation is presented in the reference [24]. All pitch angle deviations of the polygon can be calculated using the shift-angle method, as depicted in Figure 4. Figure 5 depicts the difference in calibration results using the two-autocollimator method and shift-angle methods. The difference in calibration results is less than \(\pm\) 0.28\". Figure 6 shows the angle errors, from \(\alpha\) (0\({}^{\circ}\)) to \(\alpha\) (345\({}^{\circ}\)) in steps of 15\({}^{\circ}\), using Equations (4) and (5). The SelfA rotary encoder is used to verify the proposed two-autocollimator method. The theory of the SelfA rotary encoder is described in the literature [6; 7; 8; 9; 10; 11]. A previous study proposed the SelfA rotary encoder configuration [24] so we do not describe the details in this paper. The SelfA rotary encoder consists of twelve optical sensors and a rotary encoder. All optical sensors are divided into three groups which consist of three optical sensors, four optical sensors, and seven optical sensors, respectively. Optical sensors of each set are installed for the same rotary encoder at an equal interval, at angles of 120\({}^{\circ}\), 90\({}^{\circ}\), and 51.42\({}^{\circ}\), respectively.