# Real-Time Compensation for Thermal Errors of the Milling Machine
## 1 Introduction  
Position accuracy is one of the critical performance factors affecting the machining precision of machine tools in operation. According to the studies [1, 2], thermal deformations may be responsible for 40%-70% of geometrical inaccuracies of machine tools. To reduce the influence of the thermal induced errors, some techniques associated with the compensation schemes have been developed and implemented in the machine tools, including implementation of thermally symmetrical machine design [3, 4, 5], the introduction of additional cooling systems [6, 7] and thermal deformation compensation [8, 9, 10]. Although optimization designs of machine tool structure are an effective way to reduce thermal errors, thermal displacement cannot be completely avoided since the external and internal heat sources under varying operating conditions cannot be accurately predicted at the design stage. Compensation techniques are considered essential methods for reducing the positioning errors and hence increasing the working accuracy [11, 12]. Thermal problems are more complicated than geometric problems because the temperature field of a machine tool changes constantly with the working cycle and the environmental conditions. The development of an effective thermal error compensation system greatly depends on the accurate prediction of the time-variant thermal errors. Most thermal error compensation studies have focused on the measurements of the thermal characteristics and then the establishment of thermal-induced error models with accurate prediction based on empirical approach [13, 14, 15]. Generally, the most common model for prediction of thermally induced displacements is obtained by multiple linear regressions [16, 17]. Recently, different kinds ofartificially intelligent approaches have also been applied to thermal error modeling in order to improve the accuracy and robustness of the error models. These artificial intelligence modeling techniques include artificial neural networks (ANNs) [18; 19; 20], fuzzy logic [19], adaptive neuro-fuzzy inference systems (ANFIS) [21] and integrations of the different modeling methods [22; 23]. For example, [PERSON] _et al._[18] applied the artificial neural network model to approximate complex multivariable non-linear relationships and hence effectively compensated for the thermal deformation of multiple axes machine tools. Their results showed the maximum deformation was reduced from 150 \(\upmu\)m to 15 \(\upmu\)m, and the machining error was reduced from 75 \(\upmu\)m to 16 \(\upmu\)m through the application of the neural model. [PERSON] _et al._[24] employed thermal transfer functions to develop an advanced thermal compensation in horizontal four-axis machine tools. Only four thermal sensors, instead of 21 original sensors, were used for calibration of measurements, which were found to decrease the error from 100 \(\upmu\)m to 10 \(\upmu\)m, and lessen the displacement error of the tool center point. In more recent years, [PERSON] _et al._[22] proposed a novel method combining the ANFIS and grey system theory to predict thermal errors in machining. Basing on this method, they further developed accumulation generation operation (AGO) to simplify the modeling procedures, which were demonstrated to have a stronger prediction power than the standard ANFIS model. In another work [23], they integrated the adaptive neuro fuzzy inference system (ANFIS) with fuzzy c-means clustering method (FCM) to forecast thermal error compensation on computer numerical control (CNC) machine tools. Results of their studies show that the ANFIS-FCM model produced better performance, achieving up to 94% improvement in errors with a maximum residual error of \(\pm\)4 \(\upmu\)m.  
Another important factor affecting the accuracy in thermal error modeling is the temperature variables, which are determined based on selection of appropriate locations for the temperature sensors.