# Vertical Machining Center Feed Axis Thermal Error Compensation Strategy Research
## 2 Research Object and Methodology
### Based on Data-Driven Model of Thermal Errors in the Feed Axis of CNC Machine Tools
#### 2.3.2 Data-Driven Modeling of Thermal Errors in Feed Axes of CNC Machine Tools
To establish the relationship between slope and temperature change to characterize the thermal error, the process is as follows: locating the primary error curve \(\rightarrow\) extracting the slope \(\rightarrow\) linear regression with temperature-sensitive points to establish the relationship model between slope and temperature rise. The required slope versus temperature statistics are presented in Table 4.  
\[\mathit{Regress}_{X}(\Delta T_{i},\Delta T_{B},\ldots)=0.0570-0.0215*\Delta TEMP 4+0.03784*\Delta TEMP7 \tag{8}\]  
\[\mathit{Regress}_{f}(\Delta T_{i},\Delta T_{B},\ldots)=-5.0333+5.9215*\Delta TEMP 4+22.5197*\Delta TEMP7 \tag{9}\]  
\[\Delta TEMP=\mathit{TEMP}-\mathit{TEMP}_{(temperature)} \tag{10}\]  
Import Equations (8)-(10) to \(e_{j}^{Li}(T_{i},\Delta T_{B},\ldots)\) in Equation (6). The final thermal error model is shown in Equation (11) below. \[ \begin{array}{c}e_{j}(P_{i},T_{i},\Delta T_{B},\ldots)=5.2357*10^{-8}P_{i}^{ 3}-8.8847*10^{-5}P_{i}^{2}+0.0374P_{i}^{2}-3.0184\\ -5.0333+5.9215*\Delta TEMP4+22.5197*\Delta TEMP7\end{array}  \tag{11}\]  
Through the analysis of the collection of X-axis temperature and displacement sequences of the VMC655H vertical machining center, the positioning error is used as the model reference for thermal deformation. The ambient temperature and thermal error sensitive point at the pre-moderate period is selected as the modeling parameters to construct \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Temperature Rise** & **Fitting Slope** & **Fitting Intercept** & \multicolumn{6}{c}{**Temperature T/\({}^{\circ}\)C**} \\ \hline
**t/min** & \(\beta\) & **b** &
\begin{tabular}{c} **Ambient** \\ **Temperature** \\ \end{tabular} & **TEMP4** & **TEMP7** & **ATEMP4** & **\(\Delta\)TEMP7** \\ \hline
0 & 0.0672 & 2.0286 & 14.5000 & 14.5625 & 14.5000 & 0.0625 & 0.3125 \\
30 & 0.0587 & 3.9114 & 14.5947 & 14.9638 & 14.5947 & 0.3691 & 0.2956 \\
60 & 0.0543 & 4.6029 & 14.6938 & 15.2628 & 14.6936 & 0.5690 & 0.2767 \\
90 & 0.0493 & 4.8486 & 14.8120 & 15.5033 & 14.8120 & 0.6913 & 0.2395 \\
120 & 0.0470 & 4.6657 & 14.9167 & 15.7171 & 14.9167 & 0.8004 & 0.2063 \\
150 & 0.0454 & 4.5600 & 14.9938 & 15.8677 & 14.9938 & 0.8739 & 0.1913 \\
180 & 0.0428 & 4.4171 & 15.0619 & 15.9986 & 15.0619 & 0.9367 & 0.1766 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Slope and temperature statistics table. Figure 12: Comparison chart of thermal error time positions with different motion parameters. its geometric error surface, thermal error slope fitting surface, and thermal error intercept fitting surface to predict the temperature values collected in all-time states, and its multiple linear regression overall thermal error model output flow of the way to separate geometric error and thermal error is shown in the data fitting surface in Figure 13.  
As seen in Figure 13 above, the thermal error model based on the slope fitting produced a significant displacement change, which was corrected by fitting the geometric error surface to the thermal error intercept fitting surface. The residual interval of the single-case linear regression thermal error model was \(-2.48\) to \(2.38\) mm, and the mean variance of the slope of each predicted curve is \(0.8080\). The maximum value of error over time is \(42.70\) mm, and the accuracy after prediction was \(4.83\) mm, which has excellent fitting accuracy and  
Figure 13: Multiple linear regression flow and residual comparison chart. (**a**) geometric error fitting; (**b**) geometric error fitting surface; (**c**) thermal error slope fitting surface; (**d**) surface of thermal error intercept fitting; (**e**) thermal error model fitting surface; (**f**) fitting residual plot. can improve the thermal deformation to 88.69%. Suppose that there is a relatively wide range of fluctuations in the motion parameters.