## 3 High-Fidelity Digital Twin Construction Methods
### VMD-GRU Intelligent Monitoring Approach  
The digital twin system provides accurate and reliable intelligent services for manufacturing, operation, and maintenance stages to optimise the machining and manufacturing process to achieve smart manufacturing. Current research on digital twins focuses on the level of model frameworks and less on the development of actual intelligent services, which is an important reason why digital twins are difficult to promote. With the machining process, the critical components of the equipment change as the wear of the tool has an essential impact on the quality of the workpiece, so it is important to monitor the status of the device to ensure the quality of machining and improve productivity. Tool condition monitoring methods have direct and indirect ways; the implementation of the direct plan has high costs and affects the continuity of machining, so the application has been limited, while the indirect method is through the sensor acquisition of displacement, force, acoustic emission, spindle power, and other signals to monitor the amount of tool wear indirectly. In this paper, based on the digital twin system mentioned above, an intelligent monitoring service for the state of the device, a vital component of the machining system, is developed for the centring machining process. Firstly, the time-frequency domain edge components of the sensed data are extracted using the variational modal decomposition (VMD). Then the input eigenvectors are constructed by normalising them, and finally, the wear amount is predicted using the GRU's sequential model. Figure 7: The workflow of the OPC UA Client and Server. Sensor data have a low data value density due to a large amount of random background noise and repetitive, redundant data and also due to the complex coupling relationship between various signals. Therefore, the sensed data must be processed to accurately extract the practical components to construct suitable features as model inputs, which is crucial for the final effect of monitoring. Firstly, suitable signal processing methods are chosen to convert the signals. In the time series signal analysis, there are three main extraction methods commonly used: the method of time domain analysis, the technique of frequency domain analysis, and the method of time-frequency domain analysis. Among them, the process of the time domain analysis and frequency domain analysis is to decompose the signal in the time domain and frequency domain, respectively, which is suitable for dealing with smooth movements. In this paper, the monitoring of the components, due to the damage processed by the members, will change the frequency domain characteristics, which belong to the typical non-smooth signal. Therefore, this paper adopts the time-frequency domain analysis method for non-smooth signs. The vibration signal's inherent modal component, the IMF component, is extracted using the variational modal decomposition (VMD) method [27]. The basic idea is to assume that all IMFs are bandwidth-limited components near their centre frequencies so that the sum of the estimated bandwidths is minimised, which translates into the following constrained variational problem:  
It is realised as follows:  
\[\min\left\{\sum_{k=1}^{K}\left|\left|\partial_{t}\right|\left(\hat{g}(t)+ \frac{j}{\pi t}\cdot u_{k}(t)\right)e^{-j\omega kt}\right|\left|2\right.\right\} \tag{2}\]  
\[\text{s.t.}\sum_{k=1}^{K}u_{k}=s \tag{3}\]  
initialise the parameters. \[\left\{\hat{u}_{k}^{1}\right\},\left\{\omega_{k}^{1}\right\},\hat{\lambda}^{1 },n\gets 0\]  
\[\omega_{k}^{n+1}\leftarrow\frac{\int_{0}^{\infty}\omega\left|\hat{u}_{k}^{n+1 }(\omega)\right|^{2}d\omega}{\int_{0}^{\infty}\left|\hat{u}_{k}^{n+1}(\omega) \right|^{2}d\omega} \tag{4}\]  
2. update the parameters cyclically,  
\[\hat{\lambda}^{n+1}(\omega)\leftarrow\hat{\lambda}^{n}(\omega)+\tau(\hat{f}( \omega)-\sum_{k}\hat{u}_{k}^{n+1}(\omega)) \tag{5}\]  
3. judge whether the convergence condition is satisfied, then end and output the result; otherwise, repeat step 2.