## 5 Measurement Uncertainty Analysis  
Benefiting from the simple measurement principle and the compact structure design of the inclination sensor, the measurement uncertainty of the proposed dual-axis precision level is mainly affected by the performance of the QPD and the electronic device. Thus, it is possible to evaluate the measurement uncertainty of the proposed precision dual-axis level based on the data obtained by the experiment proposed in Section 4. The combined standard measurement uncertainty of the inclination angle around the X axis \(u_{ex}\) could be obtained by:  
\[u_{ex}=\sqrt{u_{\alpha l_{x}}^{2}+u_{res\_x}^{2}+u_{drf\_x}^{2}+u_{rep\_x}^{2}} \tag{5}\]  
The first influential factor, \(u_{\alpha l_{ex}}\), is the standard uncertainty of the system calibration, which can be obtained from Figure 6. Taking the P-V data of the \(\pm\)100 arc-sec range and using a rectangular distribution, the standard uncertainty of this part was evaluated to be 0.652 arc-sec. The second factor, \(u_{res\_x}\), is the standard uncertainty of the resolution, which can be obtained from the steady-state signal variation in 60 min. From the experiment, the P-V data was 0.09 arc-sec in a rectangular distribution, so that its uncertainty was 0.057 arc-sec. The third factor, \(u_{drf\_ex}\), is the standard uncertainty of the drift due to environmental effect. According to Figure 11, the maximum drift in the X-direction was 0.22 arc-sec in one hour and was in a rectangular distribution. Its standard uncertainty was 0.127 arc-sec. The fourth factor, \(u_{rep\_ex}\), is the standard uncertainty of the measurement repeatability, and its probability distribution is Gaussian. From Figure 9a, it was evaluated to be 0.351 arc-sec. Table 1 summarizes the evaluated results of the combined uncertainty from these four sources. It should be noticed that the uncertainty of calibration \(u_{\alpha l_{ex}}\) has already taken the systematic error sources into consideration, such as the systematic error of the autocollimator measurement method (including the manufacturing error of the focusing lens and the defocus error of the QPD installation), the nonlinearity caused by the initial refraction angle of the reference laser, and the nonlinearity error of the QPD sensor in the large-range error measurement. Therefore, the used optical components must meet a certain manufacturing accuracy, and the standard operating procedure (SOP) should be followed to ensure that the systematic error caused by the assembly errors can be suppressed. The combined standard measurement uncertainty of the inclination angle around the Y axis \(u_{ey}\) could be obtained in the same way; the evaluated combined standard uncertainty \(u_{ey}\) and its sources are listed in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**Sources of Uncertainty** & **Symbol** & **Type** & **Probability Distribution** & **Standard Uncertainty** \\ \hline Calibration of the sensor (including the & & & & \\ contribution of systematic error and the & \(u_{\alpha l_{ex}}\) & B & Rectangular & 0.663 arc-sec \\ uncertainty of the reference autocollimator) & & & & \\ Resolution of sensor & \(u_{res\_x}\) & B & Rectangular & 0.057 arc-sec \\ Drift & \(u_{drf\_ex}\) & B & Rectangular & 0.127 arc-sec \\ Repeatability & \(u_{exp\_x}\) & A & Gaussian & 0.351 arc-sec \\ Standard uncertainty of \(\varepsilon_{x}\) & \(u_{ex}\) & & & 0.762 arc-sec \\ \hline \end{tabular}
\end{table} 
Table 1: Uncertainty of the inclination angle around the X-axis of the proposed precision level.