# Calibration of Rotary Encoders Using a Shift-Angle Method
## 3 Experiments  
### Shift-Angle Method  
Figure 4 shows a photograph of the experimental setup. The working environment is stabilized in which the temperature is 20 \(\pm\) 1 \({}^{\circ}\)C and relative humidity is 45 \(\pm\) 10%. The 24-faced polygon is mounted at the center of the SelfA rotary encoder (e-motionsystem, Inc., Shinagawa City, Japan, SCMS-127). The SelfA rotary encoder is equipped with 12 head sensors and a rotary encoder, which has 36,000 graduation lines at intervals of 36\({}^{\circ}\). The SelfA rotary encoder is controlled by a servo motor with an air bearing. The autocollimator (MOLLER-WEDEL OPTICAL GmbH, ELCOMAT 3000) is fixed outside the SelfA rotary encoder and aligned to the polygon. Figure 4: Photograph of the experimental setup. Figure 3: Relationship between the sensor heads and the grating lines of the rotary encoder [10]. The shift-angle \(S\) is 0\({}^{\circ}\). The first face of the 24-faced polygon is the starting face located at 0\({}^{\circ}\) of the SelfA rotary encoder, as illustrated in Figure 5a. The collimated beam of the autocollimator is aligned to the first face. The angle of the autocollimator is adjusted to approximately 0. Then, as the SelfA rotary encoder is rotated by 15\({}^{\circ}\) in one step, the autocollimator collimates the second face of the polygon. The autocollimator collimates successive faces of the polygon. The autocollimator measures 24 values corresponding to 0\({}^{\circ}\), 15\({}^{\circ}\),..., 345\({}^{\circ}\) in steps of 15\({}^{\circ}\). The measured values are listed in the second row of Table 1. Then, as illustrated in Figure 5b, the second face is aligned with the 0\({}^{\circ}\) of the SelfA rotary encoder, and the autocollimator should not be moved. The SelfA rotary encoder and the autocollimator repeat the previously described procedure. The autocollimator measures 24 values corresponding to 0\({}^{\circ}\), 15\({}^{\circ}\),..., 345\({}^{\circ}\) in steps of 15\({}^{\circ}\), which are listed in the third row of Table 1. Each face of the polygon is set as the starting face. The measured values (576 total) are listed in Table 1. These 576 measured values are used to calculate the different measured values listed in Table 2. All the  
As depicted in Figure 8a, the shift-angle is \(5^{\circ}\). As the starting face, the first face is located at \(5^{\circ}\) of the SelfA rotary encoder. The SelfA rotary encoder is rotated in 23 steps. The angle positions are \(5^{\circ}\), \(20^{\circ}\),..., \(350^{\circ}\) in steps of \(15^{\circ}\). The second row of Table 1 lists the measured values containing the angle errors, the pitch angle deviations, and the setup error. The pitch angle deviations are calculated in the previous measurement, where \(S\) is \(0^{\circ}\). The setup error \(U\) can be calculated using Equation (7). Therefore, the angle errors, corresponding to \(5^{\circ}\), \(20^{\circ}\),..., \(350^{\circ}\) in steps of \(15^{\circ}\), can be calculated, as illustrated in Figure 7. According to Figure 8b, the shift-angle \(S\) is \(10^{\circ}\). As the starting face, the first face is located at \(10^{\circ}\) of the SelfA rotary encoder. The SelfA rotary encoder rotates in 23 steps. The angle positions are \(10^{\circ}\), \(35^{\circ}\),..., \(355^{\circ}\) in steps of \(15^{\circ}\). SelfA angle errors, from \(10^{\circ}\), \(35^{\circ}\),..., \(355^{\circ}\) in steps of \(15^{\circ}\), are calculated by following the same calculation procedure, as illustrated in Figure 7.  
To check the setup error \(U\), another method uses the curve fitting least square method. The angle error can be expressed as a Fourier series because the angle error is a periodic function in one rotation. The first term on the right-hand side of Equation (9) averages the measured angular value by the two heads.