# A Review of Thermal Error Modeling Methods for Machine Tools
## 2 Thermal Error Modeling Method for Machine Tools
### Other Modeling Methods  
In the paper of [PERSON] et al. [90] published in 2011, a thermal error modeling method based on the optimal selection of the time series model was proposed. This model integrated the dynamic and historical information of the machine tool and had higher modeling  
Figure 4: Comparison of SVM, RBF NN, and hybrid model for modeling and predicting. accuracy than the general least square method. It also had stronger anti-interference ability against random factors, which indicated that the time series model can further improve the prediction accuracy of the thermal error model. In the research of [PERSON] et al. [91], a machine tool thermal error modeling method based on the state space (SS) model was mentioned. With the use of the SS model for compensation, the thermal error of the lathe was reduced from 19.40 \(\upmu\)m to 4.20 \(\upmu\)m. In 2015, [PERSON] et al. [92] proposed a natural exponential model method to symmetrically predict the thermal error of the spindle. Based on the model, the constant of thermal equilibrium time and steady-state thermal error at random speeds were calculated according to the finite element analysis results. Compensation tests have been carried out on the HDBS-63 horizontal machining center. It was found that the maximum axial error was reduced from 55 \(\upmu\)m to 16 \(\upmu\)m, and the radial error was reduced from 15 \(\upmu\)m to 6 \(\upmu\)m after compensation. In 2018, [PERSON] and his colleagues [93] proposed a new method for machine tool error modeling based on the Kalman filter. It regarded the regression coefficient of the statistical model as the state vector and the statistical model as the observation equation. The modeling accuracy of the Kalman filter method was 10.5% and 1.8% higher than that of the least square method and LSSVM method, respectively. Besides, the modeling time was 0.9% and 6.8% less than that of least square method and LSSVM method, respectively. In the paper of [PERSON] et al. [94] in 2020, an improved method based on transfer functions was presented. Its modelling and calculation speed were suitable for real-time applications. When the method was applied on the 5-axis machine tool, it was found out that the thermal error reduction in the \(X\), \(Y\) and \(Z\) directions were 62%, 56%, and 73%, respectively, after compensation. Early in 2021, a new dynamic linear modeling method of machine tool thermal error based on data driven control theory was proposed by [PERSON] et al. [95]. The prediction error of the new thermal error model of the CNC machine tool was less than 4%. Moreover, fuzzy clustering combined with principal component regression (PCR) was applied for thermal error modeling by [PERSON] et al. [96]. The prediction accuracy of the thermal error model was about 86%. In the paper of [PERSON] et al. [97], based on the bond graph method, a thermal characteristic model was developed. The maximum error of thermal deformation predicted by the model was 1.45 \(\upmu\)m, which indicated that this method has great advantages in real-time prediction of spindle thermal behavior, and can be used in industrial applications. In recent years, deep learning has been performing well in various fields. It takes the advantage of large datasets and efficient training algorithms to outperform other approaches in various machine learning [98]. The principle of machine learning is shown in Figure 5.  
Figure 5: The principle of machine learning. Particularly, deep learning has also become an important means to realize thermal error compensation of machine tools. For example, based on bidirectional short-term memory (BILSTM) deep learning, [PERSON] et al. [99] completed thermal error modeling and applied it on the machine tool, reducing the thermal error by 85%. In the paper of [PERSON] et al. [100], a thermal prediction model based on a self-organizing deep neural network (DNN) was developed to facilitate accurate-based training for thermal error modeling of heavy-duty machine tool-foundation systems.