# Uncertainty Analysis of Spherical Joint Three-Dimensional Rotation Angle Measurement
## 3 Uncertainty Analysis of Spherical Joint Space Rotation Measurement System
### Uncertainty Caused by Network Model and Sensor  
The output measurement data of the eddy current sensor will be processed by the neural network measurement model. Its own uncertainty will be carried into the final measurement results through calculation. The sensor repeatability error and the uncertainty introduced by the network model can be evaluated by MCM. The neural network measurement model can be expressed as:  
\[\mathrm{y}^{\prime}(\alpha,\beta,\gamma)=\mathrm{f}(\mathrm{v}_{1},\mathrm{v }_{2},\mathrm{v}_{3},\mathrm{v}_{4},\mathrm{\omega}) \tag{15}\]  
Figure 6: Eccentricity error. \(\
u_{1}\),\(\
u_{2}\),\(\
u_{3}\),\(\
u_{4}\) are the output values of four eddy current sensors, and \(\omega\) represents the network parameters in the neural network measurement model. Inputs \(\
u_{1}\),\(\
u_{2}\),\(\
u_{3}\),\(\
u_{4}\)\(\omega\) are the main factors affecting the uncertainty of the model. \(\omega\) is the smoothing factor in the neural network, which is the overall factor affecting the measurement uncertainty in the transmission process. The smoothing factor \(\omega\) was used to determine the best measurement model. The \"\(\ell\)\" represents the GRNN model. The equation shows that the measured \(\alpha\), \(\beta\), and \(\gamma\) can be obtained through the mathematical model. The mathematical model means the measurement model trained by GRNN. The uncertainty of the measured 3D angle is also calculated indirectly by each input through a mathematical model. The steps for evaluating the measurement system using MCM are shown in Figure 7.  
The output of the sensor changes continuously with the range detected, and the error shows a random state. According to prior analysis, it can be preliminarily assumed that it is subject to an approximately normal distribution. The normal test of the input is carried out, so the output data of the sensor in a period is sampled. For (\(-5^{\circ}\), \(-5^{\circ}\), \(0^{\circ}\)), the third quadrant midpoint, 10 data samples were taken from the 4 sensor arrays, with a single sample size of 3448. Given that the sample size is less than 5000, the Shapiro-Wilk test [25], which is suitable for small sample sizes, was used. The S-W test was performed on the output samples of four sensors, S1, S2, S3, and S4, and the results are shown in Table 1. The data of S1, S2, S3, and S4 were the output of the sensor. According to Table 1, for samples S1, S2, S3, and S4, the statistical values of the normality test results were 0.994, 0.999, 0.998, and 0.997, indicating strong correlations. The data do not meet the normal distribution, but their absolute kurtosis values are all less than 10, and their absolute skewness values are all less than 3. Therefore, the normal distribution histogram and the P-P diagram can be combined for further analysis. The distribution of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Consor** & **The** & **The Standard** & **Skewness** & **Kurtosis** & **S–W Test** \\
**Model** & **Average** & **Deviation** & & & \\ \hline S1 & 6.627 & 0.003 & 0.12 & \(-0.523\) & 0.994 (0.000 ***) \\ S2 & 3.470 & 0.001 & 0.049 & 0.075 & 0.999 (0.010 **) \\ S3 & 2.190 & 0.001 & 0.146 & \(-0.085\) & 0.998 (0.000 ***) \\ S4 & 6.098 & 0.001 & \(-0.085\) & \(-0.207\) & 0.997 (0.000 ***) \\ \hline \hline \end{tabular} Note: ***, ** represent the significance level of 1%, 5% respectively.  
\end{table} 
Table 1: Sensor output distribution table. Figure 7: Uncertainty evaluation process based on MCM. the four sensors is similar. Therefore, taking sensor S1 as an example, the normality test histogram of S1 is shown in Figure 8.  
The normal graph presents a bell shape (high in the middle and low at both ends), indicating that although the data are not absolutely normal, they can be accepted as having a normal distribution. A P-P graph is a type of chart drawn according to the relationship among the cumulative proportion of variables and the cumulative proportion of the specified distribution.