# Vertical Machining Center Feed Axis Thermal Error Compensation Strategy Research
## 1 Introduction  
In modern manufacturing, the requirements for CNC machine tools are high precision, efficiency, and stability, which is the industry's unwavering goal. Precision and ultraprecision machining are essential development directions of CNC machine tools. CNC machine tools are called \"industrial mother machines\", which are essential to industrial product quality. CNC machine tools are widely used in aerospace, military, automotive, and naval vessels, mainly for processing precision and complex parts. However, due to the increase in operating time, under the combined effect of vibration, thermal, and shock loads, the accuracy and stability of the machine tool decrease, seriously affecting the precision and stability of the machined parts. Errors that affect the precision of CNC machine tool machining can be divided into geometric, thermal, and cutting force errors. Research indicates that thermal errors are one of the most critical reasons that affect the accuracy of machine tool machining, and thermal errors account for 40-70% of total manufacturing deformations [1]. Since the thermal error was introduced at the end of the last century [2], the means have gradually changed from the error prevention method to the error compensation method. The error compensation method has received wide attention from machine tool researchers for its low cost. In the last decade, data-driven thermal error modeling hasbeen used to construct an empirical model between thermal error deformation and temperature rise variation by ensuring the statistical significance of the thermal error model in terms of considerable probability validity, so that its results are used to adjust the relative position between the workpiece and the tool for error compensation implementation. A unified thermal error modeling protocol was gradually developed: optimization of temperature-sensitive points, thermal error modeling, compensation implementation, and verification [3]. In the early research process, scholars agreed that uneven temperature field distribution was the leading cause of thermal errors [4]. Specific temperature measurement points have a certain correlation with the thermal errors of machine tools. These temperature measurement points are gradually collectively referred to as thermal critical points or temperature-sensitive points [5]. The selection of temperature-sensitive points is crucial to compensate for thermal errors in CNC machines. To improve the effectiveness of modeling temperature measurement points and thermal errors, cluster analysis, correlation analysis, and correlation coefficient analysis are often used for the detection of temperature-sensitive points. [PERSON] et al. [6] used a fuzzy clustering algorithm to optimize the grouping of temperature measurement points and select appropriate points sensitive to thermal error, reducing the number of independent variables in the modeling. In the study by [PERSON] et al. [7], based on the gray system theory of the gray correlation model, the temperature variables in the thermal error model were reduced and optimized from 16 to 4. [PERSON] et al. [8] used fuzzy C-mean clustering (FCM) and ISODATA to group temperature sensors. They established a thermal model of the artificial neural network backpropagation genetic algorithm to demonstrate its precision. [PERSON] et al. [9] used models based on the GM gray model and fuzzy C-class mean clustering methods to identify different groups of critical temperature points from IR thermographs, resulting in residual values less than \(\pm\)2 \(\upmu\)m for the FCM-ANFIS model. With further research, [PERSON] et al. [10] used principal component regression (PCR) modeling to eliminate the effect of multicollinearity among temperature variables and achieved good prediction precision and robustness. The prediction accuracy of the thermal error model ultimately determines the machine's machining precision. Therefore, data-driven models based on machine learning have started to be used for thermal error compensation.