## 4 Case Study
### GPR Model Construction  
As addressed above, the robot system performance is defined by the root mean square (RMS) of the vibration in the z-direction at the end-effector axis. This is measured during robot operation under conditions of various faults of different severities for the HD and TB. Before construction of the GPR model, it is desirable to investigate the relative importance of each component in the system performance, including their interaction effect. For this purpose, analysis of variance (ANOVA) is performed, in which input factors with a \(p\)-value smaller than 0.05 are regarded as statistically significant. The results are given in Table 2, which states that all three factors, namely, the HD, TB, and their interaction, affect the system performance significantly. While the most significant is the HD, as shown in the F-value, it should be noted that the result should be interpreted as a result of the wider severity range that was imposed by the authors. If the severity is reduced, the value will decrease accordingly. Another note is that the influence of the TB and its interaction with the HD are similar in magnitude, which indicates that their interaction clearly exists and should be incorporated into the system model.  
Based on this understanding, a GPR model for the system performance is constructed, which is able to capture the interactive relation, with the input as the HI of the HD and TB, and the output as the RMS of the vibration. In the model, as for the components, the predicted RMS values are normalized into the HI with a range [0, 1] as the system  
Figure 13: Error histogram for HI by ANN regression (**a**) for HD with RMSE 0.0275 and (**b**) for TB with RMSE 0.0902.  
performance. The hyper-parameters \(\left(h,\ \sigma_{f}^{2},\ \sigma_{n}^{2}\right)\) are optimized at 0.9525, 0.0467, and 0.0130 with the training datasets by maximizing the marginal log-likelihood of the GPR. The result is given by the response surface plot along with the confidence bounds, as shown in Figure 14a, in which the five black dots at each fault condition indicate the HI values from the RMS of measured vibration, and the confidence bounds are given by the shaded surface. The trend of the RMS in terms of the respective HD and TB axes reveals that the HD affects the system performance much more than the TB, but this is as a result of a wider severity range for the HD. The nonplanar shape of the response surface indicates that the HD and TB affect the system performance interactively. Figure 14b,c are the mean responses as a function of the HD and TB severities while the other is fixed, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum Sq.** & **d.f.** & **Mean Sq.** & **F** & \(\mathbf{p}\)**-Value** \\ \hline TB & 0.0007 & 2 & 0.0003 & 4.09 & 0.0251 \\ HD & 0.0391 & 2 & 0.0196 & 233.6 & 0 \\ TB \(\times\) HD & 0.0018 & 4 & 0.0005 & 5.38 & 0.0017 \\ Error & 0.0030 & 36 & 0.0001 & & \\ Total & 0.0446 & 44 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: ANOVA table.  
Figure 14: (**a**) Response surface plot of GPR with confidence interval; (**b**) responses in terms of HD fault with TB fault fixed; and (**c**) responses in terms of TB fault with HD fault fixed.