## 4 Milling Cutter Wear Experiment and Analysis of Recognition Results
Different selections of \(l\) and \(\xi\) affect the topological structure of the SDP image. However, optimizing the SDP transformation by adopting the DDP transformation can better avoid overlapping between the spiral arms of different signals, concealing features. Taking the vibration signals collected from the X, Y, and Z axes during the 101 st tool pass of working condition 1 as an example, after data processing, both SDP and DDP transformations are performed under the same parameters, as shown in Figure 13.  
Figure 11: Different wear states of milling cutter. Figure 12: Milling cutter wear curve. In Figure 13a-c, the amplitude range of vibration signals in three axes is not uniform. By merging three-axis signals into one visual image, misjudgment due to path variation when characterizing tool wear state with a single directional vibration signal in milling operations can be avoided. Meanwhile, Figure 13d,e underwent SDP and DDP transformations, respectively. Compared to the SDP transformation, the DDP transformation adopts non-symmetrical spiral arms to represent time-domain signals, increasing the range of spiral arm opening angles, which can reduce the overlapping phenomenon between adjacent arms. As the DDP image spiral arms are primarily influenced by parameters \(l\) and \(\xi\), selecting a reasonable parameter combination [\(l\), \(\xi\)] for DDP transformation makes the feature differences of DDP images under different wear states more pronounced, facilitating improved recognition accuracy of milling cutter wear states. Utilizing cluster analysis to extract template images for each wear state, the Euclidean distance \(d\) measures the difference in template images between different wear states, serving as an indicator for parameter optimization. Continuing with working condition 1 as an example, after performing DDP transformations on the processed vibration signals under different parameters, the template images for each wear state are described using cluster centers. The variation of the template images difference \(d\) is shown in Figure 14.  
From Figure 14, it is evident that different parameter combinations [\(l\), \(\xi\)] correspond to multi-peaked functions for template image difference \(d\), and there is a coupling effect between parameters \(l\) and \(\xi\) in DDP transformation. Therefore, optimizing based on a single influencing parameter and neglecting the interaction between the two results is a relatively optimal solution. Utilizing the global optimization capability of DE, iterative optimization of the parameter combination [\(l\), \(\xi\)] is conducted with template image difference \(d\) maximization as the indicator. According to the milling process parameters of condition 1, the range for the lag coefficient \(l\) is [0, 13], and the range for the aperture gain \(\xi\) is [0, 120]. The variation in the objective function value \(d\) of DE-DDP with the number of iterations \(g\) is shown in Figure 15. From the figure, DE-DDP converges from the 14 th iteration, with the optimal solution being \(l=13,\xi=84\). Figure 13: Comparison between SDP transformation and DDP transformation. In the process of parameter optimization, using cluster analysis to extract template images of different wear states strengthens common features and weakens unique features within the same wear state, aiding in the quantitative indicator during iterative optimization. The dataset of condition 1, under the optimal parameter combination, yields template images of different wear states as shown in Figure 16. From the figure, it is clear that template images under different wear states have distinct features, and the corresponding spiral arm differences vary in size. Therefore, using DDP transformation to merge three-axis directional vibration signals into one can circumvent the phenomenon where single-direction vibration signal features cannot fully reflect the differences in tool wear state information, limiting its applicability. Figure 16: Template images of different wear states.