# 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
The experimental results indicate that using the VMD method for denoising the AE signals in the grinding wheel grinding process is superior to the denoising effects of EMD and ICEEMDAN, especially in handling highly nonlinear and non-stationary signals. In conclusion, the VMD algorithm utilized in this study was deemed effective in mitigating the noise level of the acoustic emission signal, laying a reliable foundation for subsequent processing by neural network models. ### VMD Algorithm for Denoising AE Signals  
#### 2.2.1 Aquila Optimizer  
The Aquila Optimizer (AO) algorithm is renowned for its high reliability and consistency, characterized by a robust capability to swiftly converge with strong stability during optimization. The optimization procedure of the Aquila algorithm unfolds through the following steps [17; 18]. Step (1): Vertical Dive--Preliminary Determination of Exploration Range. In this initial step, akin to an eagle soaring high in the sky, the algorithm performs a vertical dive to preliminarily identify the optimal hunting area globally. This step aims to \begin{table}
\begin{tabular}{c c c} \hline \hline
**Denoising Methods** & \(SNR_{OUT}\) & **MSE** \\ \hline EMD & 7.69 & 0.0723 \\ ICEEMDAN & 9.15 & 0.0092 \\ Our method & 10.34 & 0.0081 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The signal denoising indicators of different denoising methods. determine the search space wherein the optimal solution is positioned. The corresponding equation is expressed as  
\[\left\{ \begin{matrix}X_{1}(t+1)=X_{best}(t)\big{(}1-\frac{t}{t}\big{)}+\\ (X_{M}(t)-X_{best}(t)\varepsilon)\\ X_{M}(t)=\frac{1}{N}\sum\limits_{i=1}^{N}X_{i}(t),\forall j=1,2,\ldots,D_{im} \end{matrix} \right. \tag{9}\]  
In Equation (9), \(X_{1}(t+1)\) represents the solution for the \(t\) iteration of this method; \(X_{best}(t)\) is the best solution, indicating the nearest position of the target prey; \(t\) and \(T\) represent the current iteration and the maximum iteration times, respectively; \(X_{M}(t)\) represents the mean position of the current solution in the \(t\) iteration; \(\varepsilon\) is a random value between 0 and 1; and \(D_{im}\) is the dimensionality of the problem. \(N\) represents the total number of solutions used when computing the mean position in the formula. Step (2): Short Glide Attack--Narrowing the Exploration Range. Following the initial identification of the prey area, akin to an eagle hovering above the target prey after soaring from a high altitude, this step involves narrowing down the hunting area. In other words, it aims to reduce the search space of the optimal solution. The corresponding equation is expressed as  
\[X_{2}(t+1)=X_{best}(t)L(D)+X_{R}(t)+(y-x)\varepsilon \tag{10}\]  
\[y=r\times cos\theta,x=r\times sin\theta,r=r_{1}+U\times D_{1},\theta=\frac{3 \pi}{2}-\omega\times D_{1} \tag{11}\]  
\[L(D)=s\frac{\mu\sigma}{|v|^{\frac{1}{\beta}}} \tag{12}\]  
\[\sigma=\frac{\Gamma(1+\beta)\times sin\Big{(}\frac{\pi\beta}{2}\Big{)}}{\Gamma \Big{(}\frac{1+\beta}{2}\Big{)}\times\beta\times 2^{(\frac{\beta-1}{2})}} \tag{13}\]  
In Equations (10)-(13), \(X_{2}(t+1)\) represents the solution for the \(t+1\) iteration of this method; \(X_{R}(t)\) is a random solution within [1, N] for the population; \(x\), \(y\) are the spiral shapes during the search process; \(r\) is the radius during hovering, with \(r_{1}\) taking values from [1, 20]; \(\theta\) is the hovering angle; \(U\), \(\omega\) take fixed values 0.00565, 0.005, respectively; \(D_{1}\) takes values from [1, D], where \(D\) is the dimension space; \(L(D)\) is the hunting flight distribution function; \(s\), \(\beta\) take fixed values 0.01, 1.5, respectively; \(\mu\), \(\upsilon\) take random values within [0, 1]; and \(\Gamma\) is the gamma function. The values and their ranges mentioned above are based on the research method proposed by [PERSON] et al. in \"Aquila Optimizer: A novel meta-heuristic optimization Algorithm\" [17]. Step (3): Low-Altitude Flight--Expanding the Exploration Range.