# Calibration of a Rotary Encoder and a Polygon Using a Two-Autocollimator Method
## 5 Conclusions  
The proposed two-autocollimator method uses two autocollimators to calibrate the 24-sided polygon and the SelfA rotary encoder. The two autocollimators are set up outside the SelfA rotary encoder. The 24-faced polygon is on the SelfA rotary table. As discussed in the Theory of The Two-Autocollimator Method section, all pitch angle deviations of the 24-faced polygon and angle errors of SelfA rotary encoder can be calibrated simultaneously. To verify the proposed two-autocollimator method, the shift-angle method, based on cross-calibration, is used to measure the same 24-faced polygon, and self-calibration is used to measure the same SelfA rotary encoder. According to the shift-angle method, based on cross-calibration, an autocollimator calibrates all pitch angle deviations of the 24-faced polygon. The difference in pitch angle deviations is smaller than \(\pm\)0.28\". The SelfA rotary encoder comprises 12 read heads and calibrates using self-calibration. Angle errors of SelfA rotary encoder can be calibrated using the proposed two-autocollimator method and self-calibration. The difference in angle errors is smaller than \(\pm\)0.27\".  
Following the ISO Guide to the Expression of Uncertainty in Measurement [25], the evaluated expanded uncertainty of the proposed two-autocollimator method is 0.46\". For a 95% confidence level, the coverage factor is 2.00. The difference in pitch angle deviations and expanded uncertainty are used in the _En_-value for the proposed two-autocollimator and shift-angle methods. The maximum _En_-value is 0.58. The difference in angle errors and expanded uncertainty are used in the _En_-value to compare the proposed two-autocollimator method and self-calibration. The maximum _En_-value is 0.59. Both the _En_-values are less than 1, meaning that the proposed two-autocollimator method is practical.  
[PERSON] and [PERSON] planned the experimental architecture. [PERSON] analyzed the measurement results. [PERSON], [PERSON] and [PERSON] wrote the manuscript. [PERSON] performed the final article confirmation. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Institute of Advanced Industrial Science in Japan, the Bureau of Standards Metrology and Inspection in the Republic of China, and the Industrial Technology Research Institute in the Republic of China under the grant N407 EA1130.  
Not applicable.  
The data presented in this article are available on request from the corresponding author.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Calibration** & **Measurement Cycle** \\ \hline The proposed two-autocollimator method & A rotary encoder and a polygon & One \\ \hline The previous two-autocollimator studies & A polygon & One \\ \hline Self-calibration & A rotary encoder & One \\ \hline The shift-angle method (cross-calibration) & A rotary encoder and a polygon & It depends on how many faces of a polygon (ex.: a 24-faced polygon is 24 measurement cycles) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparisons between the proposed two-autocollimator method and related previous studies.