# Calibration of a Rotary Encoder and a Polygon Using a Two-Autocollimator Method
## 3 Experiments
### Uncertainty Evaluation
The uncertainty distribution assumes a rectangular probability by a square root of twelve, called type B.  
\[u=0.27^{\prime\prime}/\sqrt{12}=0.0780^{\prime\prime}\]  
(5) Resolution of the SelfA rotary encoder. The resolution of the SelfA rotary encoder is 0.04\({}^{\prime\prime}\). The uncertainty distribution assumes a rectangular probability by a square root of twelve, called type B.  
\[u=0.04^{\prime\prime}/\sqrt{12}=0.0116^{\prime\prime}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Standard Uncertainty** & **Type** & \(\mathbf{U(X_{i})}\) & \(\frac{\partial\mathbf{u}}{\partial\mathbf{X}_{i}}\) & \(\left|\frac{\partial\mathbf{u}}{\partial\mathbf{X}_{i}}\right|\mathbf{u}\left(\mathbf{X_{i}}\right)\) & **Degree of Freedom** \\ \hline Measured value, \(\varepsilon\) & & 0.0910\({}^{\prime\prime}\) & 1 & 0.0910\({}^{\prime\prime}\) & 43 \\ Repeatability of angle error measurement & A & 0.0450\({}^{\prime\prime}\) & 1 & 0.0450\({}^{\prime\prime}\) & 5 \\ Resolution of the autocollimator & B & 0.0003\({}^{\prime\prime}\) & 1 & 0.0003\({}^{\prime\prime}\) & 50 \\ Traceability of the autocollimator & B & 0.0050\({}^{\prime\prime}\) & 1 & 0.0050\({}^{\prime\prime}\) & 60 \\ Angle error of autocollimator & B & 0.0780\({}^{\prime\prime}\) & 1 & 0.0780\({}^{\prime\prime}\) & 50 \\ Resolution of the rotary encoder & B & 0.0116\({}^{\prime\prime}\) & 1 & 0.0116\({}^{\prime\prime}\) & 50 \\ pitch angle deviation, \(\beta\) & A & 0.0450\({}^{\prime\prime}\) & 1 & 0.0450\({}^{\prime\prime}\) & 5 \\ setup error, \(U\) & B & 0.0205\({}^{\prime\prime}\) & 1 & 0.0205\({}^{\prime\prime}\) & 50 \\ \hline \multicolumn{4}{c}{Combined standard uncertainty (\(u\)): 0.23; Effective degrees of freedom (\(v_{\text{eff}}\)): 74; Expanded uncertainty (95% confidence level): 0.46\({}^{\prime\prime}\) (k = 2.00).} \\ \end{tabular}
\end{table} 
Table 1: Uncertainty budget for the two-autocollimator method. The four uncertainty sources, (2), (4), and (5), are type B. All relative standard uncertainty is estimated as 10% with 50 degrees of freedom. The angle error standard uncertainty \(u\) (\(\alpha\)) is 0.0910\", the root sum square of standard uncertainty sources, from (1) to (5). Using the Welch-Satterthwaite equation [25], the degree of freedom \(v_{\text{eff}}\) of the measured value uncertainty is 43. The pitch angle deviation uncertainty \(u\) (\(\beta\)) is evaluated by the six repeated measurements and standard deviations. The maximum standard deviation for all pitch angle deviations is 0.11\". The standard uncertainty is 0.11\"/\(\sqrt{6}\)= 0.0450\". The degree of freedom is 5, called type A. The misalignment between the rotary encoder and the polygon causes the setup error uncertainty \(u\) (\(U\)). According to Equations (9) and (10), the setup errors, \(U_{1}\) and \(U_{2}\), are calculated as \(-\)0.01\" and 0.09\", respectively. The maximum setup error is 0.09\". The uncertainty distribution assumes a rectangular probability by a square root of twelve, called type B. The standard uncertainty is 0.71\"/\(\sqrt{12}\)= 0.2050\" with 50 degrees of freedom, using the 10% relative standard uncertainty. In Table 1, the combined standard uncertainty of the two-autocollimator method is 0.23\", the root sum square of all uncertainty contributions, including the measured value uncertainty, the pitch angle deviation standard uncertainty, and the setup error standard uncertainty. The degree of freedom \(v_{\text{eff}}\) of the combined standard uncertainty is 74. According to the t distribution [25], the coverage factor is 2.00. The expanded uncertainty for a confidence level of 95% in \(U_{\text{two-autocollimator}}\) is given as follows:  
\[U_{\text{two-autocollimator}}=0.46\"\ (\text{k}=2.00) \tag{13}\]  
In order to investigate the consistency of calibration results, the En-value is computed. The En-value is less than 1, which indicates an acceptance criterion of comparison.