# 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  
1. Thermal error modeling based on multiple linear regression theory  
From the previous Figure 10, the shape of the error curve remains unchanged, while the slope of the error curve changes with increasing temperature. The current mainstream linear regression error modeling method separates the basic shape of the error curve to express geometric error and thermal deformation. It is roughly considered that the slope of the curve is caused by thermal deformation, so its pitch is considered. Referring to the Siemens, Huazhong CNC machine tool look-up table type of a thermal error compensation scheme [20], the separation of geometric and thermal errors for modeling is further considered [21]. Its multiple linear regression modeling process for thermal error is as follows:  
This subsection intends to assume that the positioning error of the machine tool is all considered as a mapping function of thermal error and the temperature-sensitive points of temperature change \(T_{i}\), position \(P_{i}\) and ambient temperature difference \(\Delta T_{B}\) as independent variables. \[\left\{ \begin{aligned} e_{j}(P_{i},T_{i},\Delta T_{B},\ldots)& \cong e_{j}^{Li}(T_{i},\Delta T_{B},\ldots)+e_{j}^{Lc}(P_{i},T_{i}, \Delta T_{B},\ldots)\\ e_{j}^{Li}(T_{i},\Delta T_{B},\ldots)&=Regress_{X}( \Delta T_{i},\Delta T_{B},\ldots)(P_{i}-P_{0})+Regress_{I}(\Delta T_{i}, \Delta T_{B},\ldots)\\ e_{j}^{Lc}(P_{i},\ldots)&=Fitting(P_{i}-P_{0}) \end{aligned} \right. \tag{6}\]  
In Equation (6), \(e_{j}(P_{i},T_{i},\Delta T_{B},\ldots)\) represents the integrated positioning error of the machine feed axis and the approximate thermal error of machine feed axis. \(e_{j}^{Li}(T_{i},\Delta T_{B},\ldots)\) indicates the thermal error, and \(Regress_{X}(T_{i},\Delta T_{B},\ldots)\) multiple regression of temperature rise values is used to obtain the mapping relationship between temperature rise variation and the slope of thermal deformation of machine feed axis. \(Regress_{j}(\Delta T_{i},\Delta T_{B},\ldots)\) indicates the relationship between the thermal deformation intercept of the feed axis of the machine tool established by the temperature increase variation. \(e_{j}^{Lc}(P_{i},T_{i},\Delta T_{B},\ldots)\) denotes the geometric error, \(Fitting(P_{i}-P_{0})\) one-element multiple linear fits are used to obtain the position-related deformation, and \(Fitting\) finds the evaluation deformation of each measurement point by the mean method. \(P_{i}\) denotes the motion coordinates of the machine feed axis, \(P_{0}\) denotes the machine origin coordinates, and \(\Delta T_{i}\) denotes the different temperature sensors selected. The geometric error deformations obtained at different periods in Figure 10 are converted into horizontal curves with a one-time fitted slope of 0. The reference curves are shown in Figure 12.  
\[e_{j}^{Lc}(P_{i},\ldots)=5.2357*10^{-8}P_{i}^{3}-8.8847*10^{-5}P_{i}^{2}+0.037 4P_{i}-3.0184 \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Time & TEMP1 & TEMP2 & TEMP3 & TEMP4 & TEMP5 \\ \hline Degree of & & & & & \\ association & 0.9344 & 0.9691 & 0.8663 & 0.9667 & 0.9596 \\ \hline Time & TEMP6 & TEMP7 & TEMP8 & TEMP9 & TEMP10 \\ \hline Degree of & & & & & \\ association & 0.9654 & 0.9684 & 0.6111 & 0.9533 & 0.9648 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Degree of gray correlation between the slope of the PE curve and the rate of temperature. The mean curve of the geometric error was regressed by using a cubic linear regression function for the curve regression, and the primary deformation trend of the curve could be regressed. It is verified that the standard deviation of the three fitted parts is 0.8561, which has a good predictive capacity.