## 4 Case Study
### Application of the Models  
Once the two models are made available through the previous steps, they are applied to diagnose the state of health of a new robot in operation. However, the data acquisition for a new robot is not an easy task, but requires additional effort and time in order to create an artificial fault at another severity level. Therefore, in this study, more emphasis is given to explaining how the developed methodology can be applied by introducing a virtual measurement, developed by interpolating the features between the existing training data. Although the ANN and GPR models are nonlinear by nature with respect to the fault severities of the HD and TB, the target (true) HIs are also assumed by the interpolation of the training data ignoring the nonlinearity. In order to examine how much error will be introduced by this approach, virtual data are generated only for the 11 existing fault conditions by interpolation. For example, a new condition (HD at \(Lv_{1}\), TB at \(Lv_{1}\)), which is also present in the existing conditions, can be generated by interpolating the two conditions HD at \(Lv_{0}\), TB at \(Lv_{0}\) and HD at \(Lv_{2}\), TB at \(Lv_{2}\). These are considered in the evaluation. However, conditions such as the interpolation of HD at \(Lv_{0}\), TB at \(Lv_{0}\) and HD at \(Lv_{1}\), TB at \(Lv_{1}\) are avoided since they are not in the existing conditions. After investigation, the 11 new conditions are obtained, as shown in Table 3. Recall that there are 30 data points (30% of the whole dataset) for each fault condition in the test dataset. Then, one can construct \(30\times 30=900\) new virtual features by interpolation, which amounts to a total of \(11\times 900=9900\) data points. The trained ANN model is then applied to these data to predict the state of health of each component and calculate the error between the output and target HI. The resulting histograms for the errors of the 9900 data points are given in Figure 15. Comparing with Figure 13, it is found that the RMSEs are 0.091 and 0.195, which are greater by 3.3 and 2.2 times, for the HD and TB, respectively. However, the authors believe that these are acceptable for estimating the HI in the dimension of [0, 1], which is because of the somewhat lower degree of nonlinearity. With this understanding, let us now apply the model to a new case, in which the robot is operated in a condition halfway between the two existing conditions denoted by \(a\) (HD at 0.5, TB at 1) and \(b\) (HD at 1, TB at 2/3) in the HI. Then we will expect that the HIs of the HD and TB will be somewhere around the interpolated ones, that is, 0.75 and 0.83 respectively. Assume that the motor control signal is acquired under this condition, from which the five feature values are obtained by the signal processing as mentioned in Section 4.1, although these are in fact obtained by interpolating the values of conditions \(a\) and \(b\). Applying these to the ANN model, the HIs of each component are obtained as 0.62 and 0.87 for the HD and TB, respectively. Then, apply these to the GPR model to obtain the HI of system performance, which is 0.65 with the confidence bounds [0.51, 0.78]. This is summarized in Figure 16a in the form of a bar chart, in which the error bar indicates the confidence interval. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Virtual Data Fault Condition** & **Existing Fault Condition 1** & **Existing Fault Condition 2** \\
**(HD \(Lv\), TB \(Lv\))** & **(HD \(Lv\), TB \(Lv\))** & **(HD \(Lv\), TB \(Lv\))** \\ \hline (0, 1) & (0, 0) & (0, 2) \\ (1, 0) & (0, 0) & (2, 0) \\ (1, 1) & (0, 0) & (2, 2) \\ (0, 2) & (0, 1) & (0, 3) \\ (1, 1) & (0, 1) & (2, 1) \\ (1, 1) & (0, 2) & (2, 0) \\ (1, 2) & (0, 2) & (2, 2) \\ (1, 2)Once we have the GPR model, the advantage is that we can perform what-if analysis: how well the system performance will be restored when a component is replaced by a new one. This is performed by changing the HI of the component to 0 in the model.