## 2 Methodology  
The proposed methodology is illustrated in Figure 1, which addresses the system-level diagnosis framework. It consists of two steps: (1) training and (2) application. In the model training, two models are constructed to estimate the health of each component (\(H\)) by using the features (\(F\)) from the motor control signal and to estimate the resulting degradation of the system performance (\(S\)).  
In order to construct the models, artificial faults of different severities are seeded into the components, and this can simulate the health degradation of each component, which is given by \(H_{1}\), \(H_{2}\),... in the figure. The robots are operated under each fault condition, during which the informative features are extracted and selected from the  
Figure 1: System-level diagnosis framework: (**a**) diagnosis of component health; (**b**) estimation of system performance; (**c**) current state of health of components and system; (**d**) state of health after replacement of component A; and (**e**) state of health after replacement of component B.  
motor signals that control the motion. These are denoted by \(F_{1}\), \(F_{2}\),... in the figure. A vibration signal is also acquired from the end-effector by the accelerometer, by which the root mean square (RMS) values are computed as a measure of system performance. These are denoted by \(S_{1}\), \(S_{2}\),... in the figure. In the feature extraction, wavelet packet decomposition (WPD), which is appropriate for nonstationary signals, is employed, in which the signals are decomposed into the time-frequency domain. The statistical features and wavelet energy (WE) are extracted for each decomposed coefficient. A hybrid score based on the Spearman correlation and Fisher discriminant ratio (FDR) is then applied to select the most informative features among the features pool. Once the data are prepared by this method, the artificial neural network (ANN) model is constructed to relate the features set (\(F\)) with the health (\(H\)) of each component as shown in Figure 1a. A Gaussian process regression (GPR) model is also constructed to relate the health (\(H\)) of each component with the system performance (\(S\)) as shown in Figure 1b.  
After the construction of the models, they are applied to diagnose the health of a new robot by using the motor control signal. Introducing the health indicator (HI) as 0 and 1 for the normal and failure, respectively, the results may be represented by a bar chart as in Figure 1c, which shows the current state of health of each component as well as the system performance. In this figure, they are given by 0.4, 0.6, and 0.8 for the components A, B, and system as an illustration. Note that the degradation of system performance stems from these components. Once this is available, one can further develop the replacement strategy by conducting a what-if study: if only the component A or B is replaced, i.e., the HI of A or B is restored to 0, then the HI of the system will be restored to 0.2 or 0.4 as shown in Figure 1d,e, respectively. In this illustration, it is better to replace A since the system restores to the better condition 0.2 than the alternative, 0.4. Note that simply replacing the worse component B in HI, which is illustrated in Figure 1e in this case, does not always result in the better restoration of system performance. This is because of the nonlinear and interactive nature of components and system in terms of health degradation, which is why the system-level diagnosis framework is necessary.