###### Abstract  
The thermal error modeling technology of computer numerical control (CNC) machine tools is the core of thermal error compensation, and the machining accuracy of CNC machine tools can be improved effectively by the high-precision prediction model of thermal errors. This paper analyzes several methods related to thermal error modeling in the latest research applications, summarizes their deficiencies, and proposes a thermal error modeling method of CNC machine tool based on the improved particle swarm optimization (PSO) algorithm and radial basis function (RBF) neural network, named as IPSO-RBFNN. By introducing a compression factor to make the PSO algorithm balance between global and local search, the structure parameters of RBF neural network are optimized. Furthermore, in order to pick up the temperature-sensitive variables, an improved model, which combines the K-means clustering algorithm and correlation analysis method based on back propagation (BP) neural network is proposed. After the temperature-sensitive variables are selected, the IPSO-RBFNN method is adopted to establish the thermal error model for CNC machine tool. Based on the experimental data of the CNC machine tool under the name of DMG-DMU65, the predictive accuracy of the IPSO-RBFNN model in Z direction reaches 2.05 \(\upmu\)m. Compared with other neural network method, it is improved by 10.48%, which indicates that it has better prediction ability. At last, the experiment verification for different thermal error terms at different velocities proves that this model has stronger robustness. CNC machine tools; thermal error; particle swarm optimization; radial basis function; neural network +
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predictive model of thermal errors and using for compensation [8]. In comparison with the way of error prevention, it is more economical and appropriate to decrease the thermal errors through compensation [9]. The aim of thermal error compensation is to establish thermal error model with high predictive performance for different working conditions. The modeling method for thermal errors of the CNC machine tool mainly includes two aspects: empirical modeling and theoretical modeling [10]. The theoretical modeling method requires analysis of the thermal mechanism, the distribution of temperature field of overall machine tool and the main parts, and calculation of the boundary conditions. Because of the complex deformation process, it is difficult to establish a strict model to simulate the thermal deformation of machine tool. In addition, when the terminal conditions are hard to determine, the predictive performance of the theoretical model may be poor. Therefore, empirical modeling method is more frequently used to establish the thermal errors of machine tool. To build a thermal error prediction model for machine tool through empirical modeling method, sensors are installed in multiple locations on machine tools to detect the temperature variations, taken as input variables of the model. Contrastively, thermal errors are considered as output variables. By analyzing the correlation between temperature measurements and thermal errors based on statistical models, thermal errors can be predicted by temperature measurements. In this way, thermal error compensation is realized and the precision of CNC machine tool is remarkably promoted. According to this exploratory direction, tremendous endeavors have been devoted to build predictive models statistically over the last two decades, such as least squares (LS), multivariable regression analysis (MRA), gray system theory (GS), support vector machine (SVM), neural network, and so on. LS is one of the most widely used methods for thermal error compensation of machine tool, which is applied to find the best function matching to data by finding the sum of the squares of errors minimal [11]. The LS method is theoretically mature and structurally simple. However, the predictability of this method under complex conditions is limited because of fewer independent variables.