# Calibration of Rotary Encoders Using a Shift-Angle Method
## 4 Discussion and Conclusions  
In this study, we propose a shift-angle method for calibrating rotary encoders. With this method, we use the autocollimator and the 24-faced polygon as an example. In practice, the polygon with any number of faces can be used. The traditional method uses an autocollimator and a 24-faced polygon, and the angles at which it can be calibrated are limited by the pitch angle of the polygon. However, by using the shift-angle method, the angles smaller than the pitch angle of the polygon can be calibrated.  
We set-up the autocollimator and the 24-faced polygon on the SelfA rotary encoder to verify the proposed shift-angle method. Twelve sensor heads and one rotary encoder are installed in the SelfA rotary encoder. The SelfA rotary encoder can also be calibrated using a self-calibration. Therefore, the SelfA rotary encoder is calibrated using the proposed shift-angle method and self-calibration. The difference in calibration results obtained using  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Standard Uncertainty** & \(\mu(X_{i})\) & \(\frac{\partial\alpha}{\partial X_{i}}\) & \(\left|\frac{\partial\alpha}{\partial X_{i}}\right|u(X_{i})\) & **Degree of Freedom** \\ \hline measured value uncertainty, \(\epsilon\) & 0.0676\({}^{\circ}\) & 1 & 0.0676\({}^{\circ}\) & 12 \\ Repeatability of angle error measurement & 0.0531\({}^{\circ}\) & 1 & 0.0531\({}^{\circ}\) & 5 \\ Resolution of the autocollimator & 0.0003\({}^{\circ}\) & 1 & 0.0003\({}^{\circ}\) & 50 \\ Traceability of the autocollimator & 0.0050\({}^{\circ}\) & 1 & 0.0050\({}^{\circ}\) & 50 \\ Angle error of autocollimator & 0.0404\({}^{\circ}\) & 1 & 0.0404\({}^{\circ}\) & 50 \\ Resolution of the rotary encoder & 0.0101\({}^{\circ}\) & 1 & 0.0101\({}^{\circ}\) & 50 \\ Residual setup error calculation & 0.0001\({}^{\circ}\) & 1 & 0.0001\({}^{\circ}\) & 50 \\ Repeatability of the pitch angle deviation & 0.0490\({}^{\circ}\) & 0.0401 & 0.0020\({}^{\circ}\) & 5 \\ measurement & & & 0.0144\({}^{\circ}\) & 50 \\ setup error uncertainty & 0.0144\({}^{\circ}\) & 1 & 0.0144\({}^{\circ}\) & 50 \\ \hline \hline \end{tabular}
\begin{tabular}{c} Combined standard uncertainty (\(\mu\)): 0.07”, effective degrees of freedom (\(\
u_{\rm eff}\)): 13, expanded uncertainty (95% confidence level): 0.15” (\(k=2.16\)). \\ \end{tabular}
\end{table}
Table 3: Combined uncertainty for shift-angle method.  
the two methods is less than \(\pm 0.1\)\". According to the ISO GUM, the expanded uncertainty of the proposed shift-angle method is 0.15\". According to the En number evaluation, the proposed shift-angle method is feasible for practical use.  
Conceptualization, T.-H.H. and T.W.; methodology, T.-H.H. and T.W.; validation, T.-H.H. and T.W.; data curation, T.-H.H., T.W. and P.-E.H.; writing--original draft preparation, T.-H.H.; writing--review and editing, T.-H.H. and T.W. All authors have read and agreed to the published version of the manuscript.  
This work was supported by Industrial Technology Research Institute and National Institute of Advanced Industrial Science and Technology.  
Not applicable.  
The authors declare no conflict of interest.