# Fuzzy Echo State Network-Based Fault Diagnosis of Remote-Controlled Robotic Arms
## 1 Introduction
[24] presented a deep-learning-based fault diagnosis method for leader-follower systems using CNN with multisensor data fusion. [PERSON] et al. [25] proposed a fault diagnosis method for asynchronous motors using Long Short-Term Memory (LSTM) neural networks to directly process raw vibration data, achieving high accuracy without feature engineering. Additionally, some researchers tried to incorporate the fuzzy method into deep learning models to handle data with high uncertainty and noise. For example, [PERSON] et al. [26] proposed a hierarchical fused fuzzy deep neural network (FDNN) to enhance deep learning capability for data classification. One significant challenge faced by typical deep neural networks during the backpropagation process is the vanishing gradient problem. This issue arises when gradients become exceedingly small during training, particularly in deep architectures, which hinders the effective updating of weights and slows down the learning process. The echo state network (ESN) is a novel type of recurrent neural network that forms part of reservoir computing models [27]. In ESN, the reservoir has fixed, random weights. This unique structure eliminates the need for backpropagation through the recurrent layers, thereby mitigating the vanishing gradient problem. Covariance matrix adaptation evolution strategy (CMA-ES) is a continuous optimization algorithm designed to solve non-linear and non-convex optimization problems, particularly effective in exploring complex, high-dimensional search spaces without requiring gradient information [28]. The algorithm iteratively adapts a multivariate normal distribution, using the covariance matrix to guide the search towards promising regions, thereby enabling efficient exploration and exploitation of the hyperparameter space. Applying CMA-ES [28] to optimize deep fuzzy echo state networks (DFESNs) [29] helpsaddress the complexity involved in tuning multiple hyperparameters that significantly influence model performance. The covariance matrix captures the dependencies among hyperparameters, generating well-suited candidates for improving model accuracy and efficiency. In general, three challenging points remain in fault diagnosis for leader-follower robotic arm systems. First, model uncertainty due to the nonlinear and dynamic nature of the system poses significant difficulties. In addition, data variability and noise can obscure fault signals, complicating the diagnosis process. Finally, real-time diagnosis is critical as delays can result in significant performance degradation or system failures. To address the aforementioned challenges, this paper proposes a fault diagnosis method for remote-controlled robotic arm systems based on a deep fuzzy echo state network (DFESN) [29], whose hyperparameters are optimized using the covariance matrix adaptation evolution strategy (CMA-ES) [28]. To overcome the difficulty of modeling nonlinear and dynamic systems, the proposed method does not rely on an explicit mathematical model of the robot but instead leverages the reservoir computing ability of the ESN to capture system dynamics directly from data. To tackle the challenge of data variability and noise, a fuzzy logic mechanism is incorporated into the ESN to enhance robustness and generalization. This allows the model to effectively identify fault patterns in noisy and uncertain teleoperation environments. The method is specifically designed to support online real-time fault diagnosis by maintaining low computational complexity, making it suitable for high-frequency feedback control in robotic systems. Compared to [PERSON]'s original DFESN framework [29], which focused on general machinery fault diagnosis, our approach expands the application to a more complex and interactive leader-follower architecture. Moreover, we introduce CMA-ES for the simultaneous optimization of multiple key hyperparameters--including reservoir size, spectral radius, sparsity, and the number of fuzzy clusters--which significantly improves diagnostic accuracy and adaptability.