# A Method for Identifying the Wear State of Grinding Wheels Based on VMD Denoising and AO-CNN-LSTM
## 2 Grinding Wheel Wear State Recognition Based on VMD Denoising and AO-CNN-LSTM
### VMD Algorithm for Denoising AE Signals
#### 2.2.1 Aquila Optimizer
\[\dot{u_{k}}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum_{i=1,i<k}^{K}\hat{u}_{i}( \omega)+\frac{\lambda(\omega)}{2}}{1+2\alpha(\omega-\omegaConcerning \(\alpha\), a diminutive value may not adequately denoise the signal, while an exaggerated value might inaccurately eliminate essential components. After empirical observations and iterative experiments, it was determined that setting \(K\) to 6 yields a gradually stabilized central frequency. \(K\) values of 5 or lower do not yield stable central frequencies in comparison to \(K=6\). When \(K\) is set to 7, the central frequency mirrors that of \(K=6\). To prevent mode mixing, \(K=6\) was ultimately chosen. For enhanced noise suppression, \(\tau\) was set to 0.  
An excessively small \(\alpha\) may introduce oscillations in the decomposition results, whereas an excessively large \(\alpha\) might result in overly smooth decomposition outcomes. Therefore, \(\alpha\) was configured to be approximately twice the length of the sample to strike a balance between noise suppression and signal fidelity [13; 14].  
In the experimental phase, the VMD algorithm was applied to the AE signal dataset by substituting the optimal parameters \(K,\alpha\), and \(\tau\). The denoising process was executed, and Figure 1 illustrates the plot of intrinsic mode components obtained through the decomposition of a segment of experimentally collected AE signals using the VMD algorithm.  
To address the mode components containing noise after decomposition, a comparative experiment was conducted, exploring various threshold functions and different levels of wavelet packet decomposition. After careful evaluation, a hard threshold function, which effectively preserves signal features, was chosen to undergo three-level wavelet packet decomposition on the noisy components, yielding denoised mode components [15]. Subsequently, the denoised mode components were synthesized and reconstructed alongside the retained noise-free mode components to generate the denoised signal. Figure 2 illustrates a comparison with the original AE waveform before and after the denoising process.  
Figure 1: IMFs after VMD.  
Figure 2: Waveform comparison before and after VMD-wavelet denoising.  
As depicted in Figure 2, the VMD algorithm employed in this study showcases remarkable denoising effects. The post-denoising signal waveform exhibits increased stability, effectively eliminating noise while preserving the essential information of the original signal, aligning it seamlessly.  
Empirical Mode Decomposition (EMD) is a classic adaptive signal decomposition method that is simple and easy to use, but it may encounter mode mixing issues when dealing with certain complex signals. Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) is an improved version of the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method, also based on EMD. By introducing noise assistance, ICEEMDAN enhances the stability and reliability of EMD. However, compared to VMD, there may still be limitations in dealing with highly nonlinear and non-stationary signals during the grinding wheel wear process [16]. To quantitatively validate the superiority of the denoising effect of our method, this paper employs the Output Signal-to-Noise Ratio (\(SNR_{OUT}\)) and Mean Squared Error (MSE) of the denoised signal as metrics. The \(SNR_{OUT}\) and MSE of the denoised signals using EMD, ICEEMDAN, and the method employed in this paper are shown in Table 1. Formulas (7) and (8) illustrate the calculation, where \(n\) represents the sample quantity, \(x_{i}\) denotes the original signal, and \(\hat{x}_{i}\) represents the denoised signal.  
\[SNR_{OUT}(dB)=10\times\log_{10}\left(\frac{\sum_{i=1}^{n}{x_{i}}^{2}}{\sum_{i =1}^{n}(x_{i}-\hat{x}_{i})^{2}}\right) \tag{7}\]  
\[MSE=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2} \tag{8}\]  
The \(SNR_{OUT}\) obtained by our method was higher than that of EMD and ICEEMDAN denoising, while the MSE was lower than that of EMD and ICEEMDAN.