# Uncertainty Analysis of Spherical Joint Three-Dimensional Rotation Angle Measurement
## 4 Evaluation of Measurement Uncertainty at Specific Angles  
In the early data processing, the data group is divided into four regions. In the subsequent uncertainty analysis, points in four different regions were selected for evaluation. The sample data of angles at (\(-5^{\circ}\), \(-5^{\circ}\), \(0^{\circ}\)), (\(-5^{\circ}\), \(5^{\circ}\), \(5^{\circ}\)), (\(5^{\circ}\), \(-5^{\circ}\), \(0^{\circ}\)), and (\(5^{\circ}\), \(5^{\circ}\), \(5^{\circ}\)) were selected, and the measurement uncertainty was evaluated by MCM, including the three-dimensional rotary calibration table, ball head eccentricity error, sensor repeatability error, and network model introduction. Taking (\(-5^{\circ}\), \(-5^{\circ}\), \(0^{\circ}\)), the third quadrant midpoint, as an example, the evaluation process was as follows. 1. After optimizing the parameters of the measurement model based on GRNN, the \(\omega\) value was determined, which was the optimal network measurement model. 2. The probability density function of each input value of the sensor array is determined as S1\(\sim\)N (\(-1.9587\), \(0.00102\)), the input S2\(\sim\)N (\(-1.6779\), \(0.00122\)), the input S3\(\sim\)N (\(2.4394\), \(0.00102\)), and the input S4\(\sim\)N (\(-3.5227\), \(0.00112\)). 3. When setting random sampling times, M = \(10^{6}\), S1, S2, S3, and S4 are randomly selected through random M number generation in MATLAB and put into a well-trained neural network measurement model to obtain M (\(\alpha\), \(\beta\), and \(\gamma\)) sample values which contain both the sensor and the neural network's own uncertainty. 4. The above two sections show that the rotation errors caused by the errors of the three-dimensional rotary calibration table are respectively \(u_{1}(\alpha)\) subjected to U(\(-1^{\prime\prime}\), \(1^{\prime\prime}\)), \(u_{1}(\beta)\) subjected to U(\(-3.6^{\prime\prime}\), \(3.6^{\prime\prime}\)), and \(u_{1}(\gamma)\) subjected to U(\(-2^{\prime\prime}\), \(2^{\prime\prime}\)). The rotation errors caused by the ball head eccentricity error are as follows: \(u_{2}(\alpha)\) obeys the arcsine distribution (\(-2^{\prime}41^{\prime\prime}\), \(2^{\prime}41^{\prime\prime}\)), \(u_{2}(\beta)\) obeys the arcsine distribution (\(-2^{\prime}37^{\prime\prime}\), \(2^{\prime}37^{\prime\prime}\)), and \(u_{2}(\gamma)\) obeys the arcsine distribution (\(-1^{\prime}35^{\prime\prime}\), \(1^{\prime}35^{\prime\prime}\)). 5. M random simulation samples are performed separately, assuming that the items are independent of each other, and are calculated according to Equation (5), with the sample values (\(\alpha^{\prime\prime}\), \(\beta^{\prime\prime}\), \(\gamma^{\prime\prime}\)) of (\(\alpha\), \(\beta\), and \(\gamma\)). 6. According to the algorithm steps of the Monte Carlo method, the final result of calculating the average of 106 simulation calculations yields \(\alpha\), \(\beta\), \(\gamma\) at (\(-5^{\circ}\), \(-5^{\circ}\), \(0^{\circ}\)), the optimal estimate of the measured value at (\(-5^{\circ}\), \(-58^{\prime}51^{\prime\prime}\), \(-4^{\circ}59^{\prime}59^{\prime\prime}\), \(0^{\circ}11^{\prime}12^{\prime\prime}\)), at which point the standard uncertainty for each rotation angle is \(U_{c}(\alpha)\) = \(1^{\prime}33^{\prime\prime}\), \(U_{c}(\beta)\) = \(1^{\prime}31^{\prime\prime}\), and \(U_{c}(\gamma)\) = \(39^{\prime\prime}\). 7. The confidence probability \(p=95\%\) is set, and the sum and quantiles after numerical increments (\(\alpha^{\prime\prime}\), \(\beta^{\prime\prime}\), \(\gamma^{\prime\prime}\)) are obtained. Thus, the inclusion intervals of \(\alpha\), \(\beta\), and \(\gamma\) are \(\alpha\)\(\epsilon\) [\(-5^{\circ}1^{\prime}26^{\prime\prime}\), \(-4^{\circ}56^{\prime}19^{\prime\prime}\)], \(\beta\)\(\epsilon\) [\(-5^{\circ}2^{\prime}29^{\prime\prime}\), \(-4^{\circ}57^{\prime}30^{\prime\prime}\)], and \(\gamma\)\(\epsilon\) [\(10^{\prime}8^{\prime\prime}\), \(12^{\prime}17^{\prime\prime}\)], depending on (12): \[\text{U}(\text{y})=\frac{Y_{(1+p)\text{M}/2}-Y_{(1-p)\text{M}/2}}{2}\] (16)