## 3 Thermal Error Experiment  
To substantiate the robustness of the POA-CNN-LSTMNN model, experiments were conducted using a machine tool equipped with a high-velocity motorized spindle provided by the laboratory. Throughout the experiment, it was essential to simultaneously gather temperature data from multiple points around the spindle and spindle thermal error data. The measurements of temperature were obtained from various sources, including a high-precision PT100 temperature sensor with a patch design and a resolution of 0.15 \({}^{\circ}\)C, an industrial measuring device utilizing non-contact infrared probes for metering purposes, and a paperless recorder with 12 channels used for recording and data acquisition. The spindle s thermal error comprised axial errors in the Z direction and radial errors in the X/Y direction. Here, the eddy current displacement sensor was employed for collecting the most dominant axial thermal error data, and the change in radial thermal error was relatively small and could be ignored [42; 43; 44]. We arranged 11 temperature measurement locations in the spindle s front and rear bearings, motor, spindle box cavity, etc. Together with ambient temperature, 12 locations were designated for temperature measurement. The detailed arrangement of thermal measuring locations is illustrated in Figure 7.  
The process of acquiring temperature and thermal error data is depicted in Figures 8 and 9, respectively. The spindle s sensor setup was carried out in accordance with Table 2, enabling the collection of temperature data from the 12 designated measurement points. To guarantee the model s generalization and precision, the data were collected for seven different spindle speeds during this experiment. Each of the seven groups of experiments started with the machine in a cold state and allowed the spindle to run at a certain speed for four hours, with temperature and thermal error data collected at one-minute intervals. The rotational speeds of the seven groups of experimental spindles from low to high were V1 = 1000 r/min, V2 = 2500 r/min, V3 = 4000 r/min, V4 = 5500 r/min, V5 = 7000 r/min, V6 = 8500 r/min, and V7 = 10,000 r/min. The data obtained from the experiment are presented in Figure 10.  
Figure 7: Location map of the temperature measuring point. Figure 8: Paperless temperature acquisition recorder. \begin{table}
\begin{tabular}{l l} \hline \hline
**Temperature Measurement Point Number** & **Temperature Measurement Point Position** \\ \hline T1, T2, T3 & Outside of the front bearing \\ T4, T6 & Outside of the rear bearing \\ T5, T7, T8 & Built-in motor housing \\ T9 & Inside of the front bearing \\ T10 & Inside of the rear bearing \\ T11 & Spindle box cavity \\ T12 & Ambient temperature \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Distribution of temperature measuring points for the motorized spindle. Figure 9: Diagram of error collected: (**a**) axial thermal error acquisition of the spindle: (**b**) eddy current displacement sensor; (**c**) PC data acquisition interface diagram. The H-K-P method was employed for select points with temperature sensitivity from the 12 thermal measurement locations. When applying the K-medoids algorithm, it is essential to initially ascertain the quantity of cluster centers. In cases where we did not  
Figure 10: Temperature data curve and thermal error data curve of spindle at different rotations. know how many categories were good, we first used the hierarchical clustering algorithm to ascertain the quantity of groups. To cluster the temperature data obtained from the 12 measurement points, hierarchical clustering was employed. The resulting clustering outcomes are illustrated in Figure 11.  
From the graph of clustering results in Figure 11, it can be seen that the effective number of clusters for the 12 to-be-clustered measurement points needed to be clustered into four classes.