# Fuzzy Echo State Network-Based Fault Diagnosis of Remote-Controlled Robotic Arms
## 4 Experiment
### Experimental Results  
In this experiment, the covariance matrix adaptation evolution strategy (CMA-ES) is used to optimize the hyperparameters of the deep fuzzy echo state network (DFESN). The four hyperparameters evaluated are reservoir size, spectral radius, sparsity parameter, and the number of fuzzy clusters, with respective ranges of [100-1000], [0-1], [0-0.5], and [5-50]. In the CMA-ES algorithm, as four parameters have been chosen as decision variables, which means \(d=4\), and according to \(\lambda=(4+3\ln d)\times 10\), \(\lambda\) is found to be 80, i.e., the population size is 80, the search step length is set at 0.1, and the maximum number of iterations is 200. The values for step length and the maximum number of iterations are determined based on initial experiments to balance computational efficiency and optimization effectiveness. The step length of 0.1 was chosen to ensure a moderate exploration rate in the search space. A higher step length may lead to larger jumps and potentially miss local optima, while a lower step length might result in slow convergence. The maximum number of iterations was set to 200 to provide sufficient exploration without incurring excessive computational time, given the complexity of the model. Figure 5 illustrates the variation in Mean Squared Error (MSE) of the DFESN throughout the optimization process. Despite some fluctuations in the curve, an overall decreasing trend in the MSE can be observed. The MSE reached its minimum value at iteration 124, at which point the best hyperparameters are found to be a reservoir size of 500, a spectral radius of 0.1, a sparsity parameter of 0.01, and a number of fuzzy clusters of 30. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Representative Meaning** \\ \hline \(\theta_{1}\) & The angle of the base \\ \(\theta_{2}\) & The angle of the shoulder \\ \(\theta_{3}\) & The angle of the elbow \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter and meaning. \begin{table}
\begin{tabular}{c c} \hline
**Fault Type** & **Category Label** \\ \hline leader and follower work normally & 1 \\ communication fault & 2 \\ actuator fault & 3 \\ follower sensor1 fault & 4 \\ follower sensor2 fault & 5 \\ follower sensor3 fault & 6 \\ leader sensor1 fault & 7 \\ leader sensor2 fault & 8 \\ leader sensor3 fault & 9 \\ \hline \end{tabular}
\end{table} 
Table 2: The type of faults and category label. In addition, to assist users intending to apply this network to other datasets, we recommend starting with an optimization strategy similar to CMA-ES. Initial hyperparameter values can be set using the same ranges as described in this study--reservoir size [100-1000], spectral radius [0-1], sparsity [0-0.5], and number of fuzzy clusters [5-50]. These ranges have proven effective across a variety of fault types in this study and provide a reasonable starting point. The test sets for different fault scenarios were input into the trained DFESN model, and the results are shown in Table 3. DFESN model achieved high accuracy rates exceeding 90% across the 9 fault types. Notably, fault types 3, 4, 6, 8, and 9 achieved a classification accuracy of 100%. The confusion matrix presented in Figure 6 provides a detailed evaluation of the DFESN model's performance in fault classification across different fault types for the leader-follower robotic arm system. Each coordinate from 1 to 9 represents a distinct fault type as described in Table 3, including normal operation, communication faults, actuator faults, and different sensor faults. The DFESN model exhibits a high classification accuracy for most of the fault types, achieving a perfect accuracy of 1.00 for fault types 1, 3, 4, 5, 6, 8, and 9. This indicates that these faults were correctly identified without any misclassification, highlighting the model's robustness in diagnosing these particular types of faults. Notably, the perfect classification for sensor faults (types 4, 5, 6, 8, and 9) shows that the model can effectively handle fault detection that involves complex sensor data.