# Calibration of Rotary Encoders Using a Shift-Angle Method
## 3 Experiments
### Uncertainty Evaluation  
According to the ISO Guide to the Expression of Uncertainty in Measurement (GUM) [21], measurement uncertainties originate from several sources. The first step is to build a function that relates the angle error \(\alpha\), the measured value \(\epsilon\), and the pitch angle deviation \(\beta\), as expressed in Equation (3). The combined standard uncertainty is expressed as follows. \[u_{c}^{2}(\alpha)=\biggr{(}\frac{\partial\alpha}{\partial\epsilon}\biggr{)}^{ 2}u(\epsilon)^{2}+\biggr{(}\frac{\partial\alpha}{\partial\beta}\biggr{)}^{2} u(\beta)^{2}+\biggr{(}\frac{\partial\alpha}{\partial U}\biggr{)}^{2}u(U)^{2} \tag{10}\]  
The measured value uncertainty \(u(\epsilon)\) is evaluated using the repeatability of angle error measurement, resolution of the autocollimator, traceability of the autocollimator, angle error of the autocollimator, resolution of the SelfA rotary encoder, and residual setup error calculation. (1) Repeatability of angle error measurement. Repeatability is determined using multiple measurements. The measured results of the shift-angle method, ranging from \(1^{\circ}\) to \(360^{\circ}\), are computed as the average of six measurement results. The maximum standard deviation of all angle errors is \(0.13^{\prime\prime}\). \[u=0.13^{\prime\prime}/\sqrt{6}=0.0531^{\prime\prime}\]  
The degree of freedom is 5. (2) Resolution of the autocollimator. The resolution of the autocollimator is 0.001\". The distribution assumes a rectangular probability. \[u=0.001^{\prime\prime}/2\sqrt{3}=0.0003^{\prime\prime}\]  
The relative uncertainty is estimated as 10% with 50 degrees of freedom. Figure 11: Calibration results for the rotary encoder obtained using self-calibration and shift-angle method. Self-calibration, red dash line; and shift-angle method, blue solid line. Figure 12: Difference in calibration results between self-calibration and shift-angle method. \[u=0.0001^{\prime\prime}/2\sqrt{3}=0.0001^{\prime\prime}\] (3) Traceability of the autocollimator. The calibration certificate declares that the uncertainty is 0.01\" with a coverage factor \(k\) of 2 (95% confidence level). \[u=0.01^{\prime\prime}/2=0.0050^{\prime\prime}\] The relative uncertainty is estimated as 0% with \(\infty\) degrees of freedom. (4) Angle error of the autocollimator. In the calibration certificate, the maximum error is 0.05\" in the measurement range of \(\pm\)1000\". The distribution assumes a rectangular probability. \[u=0.07^{\prime\prime}/\sqrt{3}=0.0404^{\prime\prime}\] The relative uncertainty is estimated as 10% with 50 degrees of freedom. (5) Resolution of SelfA rotary encoder. The resolution of the SelfA rotary encoder is 0.04\". The distribution assumes a rectangular probability. \[u=0.04^{\prime\prime}/2\sqrt{3}=0.0101^{\prime\prime}\] The relative uncertainty is estimated as 10% with 50 degrees of freedom. (6) Residual setup error calculation. The difference between the setup error calculation, in Equation (7), and curve fitting is less \(10^{-4\prime}\). The distribution assumes a rectangular probability. \[u=0.0001^{\prime\prime}/2\sqrt{3}=0.0001^{\prime\prime}\] The relative uncertainty is estimated as 10% with 50 degrees of freedom. The measured value uncertainty \(u(\varepsilon)\), which is the root sum square of all uncertainty contributions, is 0.0676\". According to the Welch-Satterthwaite equation [21], the effective degree of freedom \(\
u_{\text{eff}}\) of the measured value uncertainty is 12. The measured value of uncertainty \(u(\beta)\) is evaluated based on the repeatability of the pitch angle deviation measurement. (7) Repeatability of the pitch angle deviation measurement, \(\beta\). Repeatability is determined using multiple measurements. The pitch angle deviations, from the first to the 24 th face, are the average of the six measurement results. The maximum standard deviation for all pitch angle deviations is 0.12\". \[u=0.13^{\prime\prime}/\sqrt{6}=0.0531^{\prime\prime}\]0 The degree of freedom is 5.