# Calibration of a Rotary Encoder and a Polygon Using a Two-Autocollimator Method
## 3 Experiments
### Uncertainty Evaluation  
The uncertainty evaluation is based on the ISO Guide to the Expression of Uncertainty in Measurement [25]. The measured value \(\varepsilon\) is related to \(n\) independent and uncorrelated influence sources, \(X_{i}\), \(k=1,2,\,\ldots,n\), via the functional relation. According to Equation (3), the measurement uncertainty of the two-autocollimator method originates from several influence sources to contribute to the measured value \(\varepsilon\), including the angle error \(\alpha\), the pitch angle deviation \(\beta\), and the setup error \(U\). \[\varepsilon=f\left(\alpha,\ \beta,\ U\right) \tag{11}\]  
Table 1 shows the uncertainty budget for the proposed two-autocollimator method. The influence sources contribute to calibrating the polygon and the rotary encoder. Some influence quantities are directly measured and estimated, called Type A. Other influence quantities use specification knowledge to estimate using a set of probability distributions, called Type B. The combined standard uncertainty is expressed as follows:  
\[U_{\text{c}}^{2}\ (\varepsilon)=\left(\frac{\partial\varepsilon}{\partial \alpha}\right)^{2}u\ (\alpha)^{2}+\left(\frac{\partial\varepsilon}{\partial\beta}\right)^{2}u( \beta)^{2}+\left(\frac{\partial\varepsilon}{\partial U}\right)^{2}u(U)^{2} \tag{12}\]  
where \(u(\alpha)\), \(u(\beta)\), and \(u(U)\) are called standard uncertainties. The parameters \(\partial\varepsilon\ /\ \partial\alpha\), \(\partial\varepsilon\ /\ \partial\beta\) and \(\partial\varepsilon\ /\ \partial U\) are known as both sensitivity coefficients and weighting factors for the standard uncertainties. All sensitivity coefficients equal 1. Figure 6: Calibration results for the SelfA rotary encoder are obtained using the two-autocollimator method (circle label) and self-calibration (star label). Figure 7: Difference in calibration results for the SelfA rotary encoder between the two-autocollimator method and self-calibration. The angle error standard uncertainty \(u(\alpha)\) is evaluated using five uncertainty sources:  
(1) Repeatability of angle error measurement. Repeatability is an uncertainty source from six multiple measurements. The measured results of the two-autocollimator method, ranging from 0\({}^{\circ}\) to 345\({}^{\circ}\) in steps of 15\({}^{\circ}\), are computed as the average measurement results and the standard deviations. The standard uncertainty is a maximum standard deviation of 0.11\({}^{\prime\prime}\) by a square root of six, called type A.  
\[u=0.11^{\prime\prime}/\sqrt{6}=0.0450^{\prime\prime}\]  
The degree of freedom is 5. (2) Resolution of the autocollimator. According to the specification, the resolution of the no. 1 autocollimator and the no. 2 autocollimator is 0.001\({}^{\prime\prime}\). The standard uncertainty distribution assumes a rectangular probability by a square root of twelve, called type B.  
\[u=0.001^{\prime\prime}/\sqrt{12}=0.0003^{\prime\prime}\]  
(3) Traceability of the autocollimator. The calibration certificates of the no. 1 autocollimator and the no. 2 autocollimator declare the same expanded uncertainty of 0.01\({}^{\prime\prime}\) with a coverage factor \(k\) of 2 (95% confidence level). The standard uncertainty is obtained by dividing 0.01\({}^{\prime\prime}\) by 2, called type B.  
\[u=0.01^{\prime\prime}/2=0.0050^{\prime\prime}\]  
According to the t distribution [25], the coverage factor \(k\) of 2 equals the degrees of freedom 60. (4) Angle error of the autocollimator. The no. 1 autocollimator and no. 2 autocollimator are calibrated in the same measurement range of \(\pm 1000\)\({}^{\prime\prime}\). According to two calibration reports, the calibration error of the no. 1 autocollimator is 0.11 (peak-to-valley), and the calibration error of the no. 2 autocollimator is 0.27 (peak-to-valley). The maximum error is 0.27\({}^{\prime\prime}\).