Hybrid optimization algorithm for thermal displacement compensation of computer numerical control machine tool using regression analysis and fuzzy inference

Abstract

During the machining process, the computer numerical control machine is susceptible to variations in ambient temperature, cutting heat, and friction within the transmission parts, which generate different heat sources. These heat sources affect the machine structure in different ways, causing deformation of the machine and displacement of the tooltip and workpiece position, ultimately resulting in deviations in machining accuracy. The amount of thermal drift depends on several factors, including the material of the machine components, the cutting conditions, the duration of the machining process, and the environment. This study proposes a hybrid optimization algorithm to optimize the thermal variables of computer numerical control machine tool spindles. The proposed approach combines regression analysis and fuzzy inference to model the thermal behavior of the spindle. Spindle speed and 16 temperature measurement points distributed on the machine are input factors, while the spindle's axial thermal error is considered an output factor. This study develops a regression equation for each speed to account for the different temperature rise slopes and spindle thermal variations at different speeds. The experimental results show that the hybrid thermal displacement compensation framework proposed in this study effectively reduces the thermal displacement error caused by spindle temperature variation. Furthermore, the study finds that the model can be adapted to significant variations in environmental conditions by limiting the machining speed range, which significantly reduces the amount of data needed for model adaptation and shortens the adaptation time of the thermal displacement compensation model. As a result, this framework can indirectly improve product yield. The effects observed in this study are remarkable.

Introduction

In recent years, the issue of carbon neutrality has received much attention, and the manufacturing industry is facing carbon tax and emission control, so how to help the industry to reduce carbon emission from the perspective of manufacturing production is an urgent issue to be met and solved. Therefore, by improving the stability of the computer numerical control (CNC) machine tool, processors can be helped to reduce the error rate and prevent the machining error of finished products from exceeding the accepted standard, increasing the machine running time.

Thermal deformation displacement significantly affects the machining accuracy of the machine tool due to the uneven temperature distribution on the machine. Thermal deformation for CNC machine tools typically includes spindle and mechanical structure deformation. Spindle deformation refers to the thermal expansion and contraction of the spindle, which can cause changes in the relative position between the spindle and the workpiece and affect machining accuracy. On the other hand, mechanical structure deformation refers to the deformation of the machine tool structure due to changes in the ambient temperature or due to the movement of the feed axis causing temperature increases, which can cause changes in the relative position between the tool and the workpiece and also affect machining accuracy.

To compensate for spindle deformation, the most common approach is to use a thermal sensor to measure the spindle temperature and then use a compensation model to calculate the expected spindle deformation, which can adjust the machining process accordingly. Other techniques, such as spindle cooling, can also be used. The cooling unit, such as the cooling runner or cooling chip, is used to remove the heat source and suppress the heat dispersion, thus suppressing the deformation of the machine tool and reducing the accuracy difference caused by temperature fluctuation.

Thermal deformation of the mechanical structure refers to the deformation of the structural components of the machine tool due to temperature changes. When a machine tool is subjected to thermal loads, such as during machining operations, the various parts of the machine tool will expand or contract due to thermal expansion. This can cause the machine tool structure to deform, resulting in changes in the relative position between the tool and the workpiece, which can lead to machining errors or inaccuracies. It is, therefore, essential to compensate for the thermal deformation of the mechanical structure to ensure machining accuracy. Finite element analysis is often used to model and analyze the thermal deformation of the machine tool structure. Making a structural thermal symmetry design can effectively disperse the heat evenly to maintain the temperature balance and symmetry of the machine.

Because of the temperature difference, the structure deformation causes the tooltip displacement. Therefore, the equipment is placed in a constant temperature room, the ambient temperature is controlled within a specific range to reduce the ambient temperature fluctuation, the machine is warmed up and operated, and the cutting fluid is turned on and used for some time, the purpose is to make the internal structure temperature of the machine tool uniform so that the internal and external temperature of the machine tool can be equalized to reduce the occurrence of thermal displacement during machining. Overall, combining these techniques can eliminate or minimize the errors caused by the spindle and mechanical structure deformation and improve the machining accuracy of CNC machine tools.

Huang et al. ¹ proposed to measure temperature changes and analyze thermal shifts in heavy-duty tooling machines using Bragg gratings. Because the fiber Bragg grating is immune to electromagnetic wave interference and has high sensitivity and multi-point measurement capability, this real-time measurement system can measure the temperature of large fields and analyze thermal shifts to improve the accuracy of heavy-duty tooling machines. Because the choice of temperature points has a significant impact on the model inference, Liu et al. ² proposed a principal component regression (PCR) approach to modeling to eliminate the effect of co-linearity between temperature points, and the PCR model reduces the impact of temperature sensitive point changes on the model accuracy and has good prediction accuracy and robustness.

Liu et al. ³ proposed a new integrated thermal error model using the workpiece inspection data from the production line and C_pk. C_pk is used in the modeling, and C_pk impact analysis is performed to select the thermal key point (TKP). The proposed model's thermal error is then compensated in real-time during the machining process. Due to the thermal error compensation, manual tool adjustment is no longer required, which helps to improve the quality of the workpiece and saves the factory cost.

Abdulshahed et al. ⁴ proposed to build a tool machine thermal displacement model by Adaptive Neuro-Fuzzy Inference System (ANFIS) method, and ranking and clustering the temperature points by gray system theory and fuzzy c-means (FCM), and the ANFIS-FCM model single speed can be controlled within ±4 μm. Jin et al. ⁵ proposed a method to evaluate the thermal characteristics of the feeding system under different operating conditions. The feed system's temperature rises, and positioning errors vary for different feed speeds, cutting loads, and preloads of the ball screw. It was found that the preloads were the most significant factor affecting the temperature rise of the bearing, while the load and feed rate were the most significant factors affecting the temperature rise of the nut of the ball screw. The wavelet neural network based on the Nonlinear Autoregressive Moving Average-L2 model (WNN-NARMAL2) model was used to establish the relationship between temperature rise at sensitive points and operating conditions.

Liu et al. ⁶ proposed a thermal error model using the ridge regression algorithm to suppress the adverse effect of temperature point co-linearity on the robustness of thermal error prediction. Fujishima et al. ⁷ proposed to evaluate the reliability of thermal error prediction by deep learning algorithm and to change the compensation weights adaptively to avoid extreme estimates due to sensor failures. Fu et al. ⁸ proposed a radial basis function (CSO-RBF) neural network based on the chicken swarm optimization algorithm applied to thermal error modeling. K-means clustering and radial basis function neural network methods are used to filter the optimal temperature point combinations. Second, the CSO-RBF neural network deals with the nonlinear relationship between temperature variables and thermal errors. Hou et al. ⁹ proposed a multi-objective genetic algorithm, which will combine model-based and data-based, based on a physical model of thermal expansion in the vertical direction of the lathe spindle box with a model based on thermal run-in experimental data, high accuracy, and robustness, and the residuals of the model estimation are within 15 μm. Shi et al. ¹⁰ proposed a thermal error modeling method based on a Bayesian neural network. The relationship between the temperature rise of the feed drive system and positioning error was investigated by measuring the machine's thermal characteristics, such as temperature field and positioning error. The maximum thermal error can be reduced from 18.2 to 5.14 μm by using a Bayesian neural network. Bao et al. ¹¹ proposed to use of fuzzy clustering theory and gray theory for temperature point selection and then thermal shift modeling through multivariate regression. Because the temperature change is time-dependent, Zhang et al. ¹² proposed to use long short-term memory (LSTM) neural network to predict the temperature change of generator brushes, and the prediction error of the coming day's brush temperature is stable within 0.4 °C. Li and Zhao ¹³ proposed using wired Spearman's rank correlation analysis to identify the location and number of critical temperature points, and the least square method was used to model the thermal deformation.

The rest of this article organizes as follows. The problem in this study descript in the “Problem description” section. Next illustrates the materials and method used in this study. Experimental planning and methodology depicted in the “Experimental planning and methodology” section. This study concludes in the “Conclusion” section.

Problem description

The CNC machine tool is widely used in the mold and automotive industry, from cell phone housings to automotive molds. For the automotive industry, engines used to be a crucial scene for CNC machine tool applications. The development of electric vehicles has also added new components such as motors and batteries, single-speed transmissions, and charging pads. The parts and components of electric vehicles require a higher level of precision. The slightest noise can be easily highlighted during the operation of electric vehicles, so excellent surface finish and higher precision machining quality are required. CNC machine tools require high speed, accuracy, and smoothness. In the part of machining accuracy, many factors affect the machining accuracy, shown as follows:

Geometrical errors: These are errors caused by the imperfect geometry of the machine, such as misalignment or wear of the axes, spindle, or tool holder.

Thermal errors: These are caused by changes in temperature, which can cause expansion or contraction of machine components, resulting in positional inaccuracies.

Vibration: Vibrations can be caused by the machine structure, cutting forces, or spindle speed and can cause dimensional inaccuracies, surface roughness, and reduced tool life.

Tool wear and breakage: Wear or breakage of the cutting tool can result in inaccurate machining dimensions, surface roughness, and poor tool life.

Workpiece distortion: Workpiece distortion due to clamping or cutting forces can cause dimensional inaccuracies and poor surface finish.

Environmental factors: Environmental factors such as dust, moisture, or even operator behavior can affect the accuracy of the machining process.

Among them is the machine tool in the machining process, including motor, chip, tool, or mechanical friction, energy conversion, and heat generation. The internal and external heat sources by conduction, convection, and radiation cause uneven changes in the temperature of each component. The uneven distribution of heat sources and the complexity of the structure affect the relative position between the spindle tooltip and the table changes, reducing the geometry and positioning accuracy of the machine while causing deviations in the size of the workpiece, increasing the defective rate of the product and will also affect the effectiveness of carbon reduction.

The change in temperature of the machine tool seriously affects the machining accuracy. Therefore, thermal compensation is a crucial method for achieving high accuracy. This experiment uses a CNC tapping center as the test platform. A temperature sensor is attached to the machine's vital heat source, and the machine's temperature change and the displacement of the spindle tip point relative to the table are collected at different spindle speeds. The temperature points are sorted and filtered by Pearson's correlation coefficient and temperature–displacement curve to find the critical temperature points, and then multiple regression equations are established according to different spindle speeds. Fuzzy inference is used to assign weights to obtain the thermal compensation values at the rotational speeds without modeling. Finally, the model is validated in two ways: single speed and different operating conditions. Figure 1 shows the difference in the thermal compensation, and it can be observed that there is a significant difference in the machining tool marks and surface finish.

Figure 1.

Thermal compensation difference. (a) Thermal compensation OFF. (b) Thermal compensation ON.

Materials and methods

Regression analysis

This section briefly describes the method required for the modeling process, including the application of Pearson correlation coefficients, regression analysis, and fuzzy inference.

Pearson correlation coefficient is used to measure the correlation between two variables, X and Y. It is the ratio of the product of the covariance of the two variables and the standard deviation (as in equation (1)), and its coefficient will be between −1 and 1. The coefficient of 1 means that X and Y have a positive correlation, and the Y variable will increase with the X variable; the coefficient of −1 means a negative correlation, and the Y variable will decrease with the increase of the X variable. An absolute value of 0.5–1 indicates a high correlation between the two variables, 0.3–0.5 indicates a moderate correlation and 0.1–0.3 indicates a weak correlation

$$\rho x,y={\displaystyle \frac{cov(x,y)}{\sigma x\sigma y}}$$

(1)

where cov is the covariance, σx is the standard deviation of x, and σy is the standard deviation of y.

Regression analysis (RA)^14–16 is a statistical method to analyze the relationship between variables and variables, mainly to understand whether the independent and dependent variables are related and to build a mathematical model to predict our dependent variables. It can be divided into simple and complex regression based on complexity. Simple regression predicts the relationship between a single independent variable and a dependent variable, while complex regression investigates the relationship between a dependent variable and multiple independent variables.

The univariate linear regression equation is the relationship between a dependent variable and an independent variable, and its regression equation is expressed as follows:

$$Y={\beta }_0+{\beta }_1{x}_1+\epsilon$$

(2)

where ${\beta }_0$ is the constant, ${\beta }_1$ is the regression coefficient, and $\epsilon$ is the error.

The multivariate linear regression equation is the relationship between one dependent variable and multiple independent variables, and its regression equation is expressed as follows:

$$Y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+\epsilon$$

(3)

where ${\beta }_0$ is the constant, ${\beta }_1,{\beta }_2,{\beta }_3,\dots ,{\beta }_m$ is the regression coefficient, and $\epsilon$ is the error.

RA is often used in two significant areas: explanation and prediction. In terms of explanation, the regression equation is calculated from the obtained experimental data, and then the contribution of each independent variable to the dependent variable is known through the regression equation. In terms of prediction, since the regression equation is a linear relationship, the dependent variable can be brought by the change in the independent variable, so RA can be used to predict the future change in the dependent variable and further compensate or respond to it by the regression model.

Fuzzy inference^17–19

Zadeh introduced the theory of fuzzy sets by introducing levels in the conditions so that the conditions are not just 0 or 1. Fuzzy systems consist of techniques and methods for processing imprecise, undefined, or non-specific information from the system environment. Fuzzy systems allow describing of intrinsically ambiguous phenomena by passing knowledge to the system processed in symbolic form and recorded in the rule base using if-then conditions. Figure 2 shows the fuzzy controller.

Figure 2.

Fuzzy controller.

Experimental planning and methodology

Thermal compensation is a crucial method for achieving high accuracy in machining processes. In the past, various data acquisition methods and algorithms have been used for modeling. Temperature sensors, tool length measurements, displacement meters, and thermographs have been used to collect relevant data for compensation. In the algorithm part, regression analysis, K-MEANS, support vector regression (SVR), and LSTM algorithms have been used to select key temperature points and build the compensation model. The aim is to improve processing accuracy and reduce costs. Recent developments in placing sensors inside the spindle have enabled real-time monitoring and compensation for changes in spindle displacement. The compensation structure can be either directly through the control or through the IPC. Over time, the compensation model may require correction. Therefore, there is a development of remote compensation model correction through the cloud.

Most of the previous research has been focused on compensating for temperature changes, but the heat energy generated by the spindle is different at different spindle speeds. Therefore, this study has incorporated the spindle speed factor and collected data on temperature changes and spindle displacement at different speeds. Thermal compensation values were obtained using multiple regression equations and fuzzy inference.

In this study, 16 temperature sensors were placed on the machine to collect temperature and spindle axial thermal deformation data for 4 h of spindle operation at different rotational speeds, and data were collected every 5 s. Pearson's correlation coefficient selected the critical temperature points, and six multivariate regression equations were established based on the relationship between the temperature at the critical temperature points and the spindle axial thermal deformation at 3000, 6000, 9000, 12,000, 15,000, and 18,000 r/min. The estimated spindle thermal deformation was obtained by calculating the attribution values with the fuzzy inference of the membership function at speed without the model. The experimental plan is shown in Figure 3.

Figure 3.

Experimental flowchart.

The experimental platform is shown in Figure 4, which includes a tapping center from Precision Machinery Research & Development Center (PMC) with a built-in spindle and a SIEMENS 840Dsl controller. Figure 5 shows that the spindle tool holder type is BT30, and the spindle operation condition is planned as shown in Figure 6. In the measurement equipment, the displacement sensor spec is Pulsotronic KJ2-M8MB40-ANU, and the temperature sensor is AD592, which is fixed to the machine by solid magnet and clay. The sampling time is every 5 s to record the spindle axis's thermal deformation and the machine's temperature. Figure 7 shows the relative position of the spindle tool and fixture.

Figure 4.

Tapping center and Siemens 840Dsl controller.

Figure 5.

BT30 tool holder test bar.

Figure 6.

Spindle rotation cycle.

Figure 7.

Measurement of spindle thermal deformation.

The thermal deformation estimated by the model is written to the Siemens 840Dsl controller mechanical parameter address 43,900 through PLC or OPC-UA communication to compensate for the thermal deformation in real-time, ²⁰ as shown in Figure 8.

Figure 8.

Controller thermal compensation architecture.

Experimental results

Because the temperature change of each temperature point varies with the heat generation at different rotational speeds, the critical temperature points are selected using a temperature–displacement curve with a Pearson correlation coefficient, and the coefficient threshold is set above 0.97. The critical temperature points were selected as T6, T8, and T9. The number of crucial temperature points is controlled to less than 4, which helps improve the machine tool's accuracy and stability through the low cost and high precision technology. Figure 9 shows the relationship between temperature and thermal deformation for 4 h of operation at each speed. The amount of heat generated by the operation at different speeds is different, so the time for the thermal deformation to reach saturation is also different. The thermal energy generated by the spindle varies at different speeds, resulting in a varying time required to reach thermal equilibrium. To provide information on this, Table 1 presents the time required to reach thermal equilibrium for spindle speeds ranging from 3000 to 18,000 r/min.

Figure 9.

Temperature and displacement relationships for different rotational speeds.

Table 1.

Thermal equilibrium time at different speeds.

Spindle speed (r/min)	Time to thermal equilibrium (s)
3000	10,160
6000	9675
9000	6895
12,000	6615
15,000	5385
18,000	5560

To account for common sources of heat generation in CNC machine tools, 16 temperature sensors are positioned at various points around the tool. These points include the spindle's motor bases, the spindle's front and rear bearings of the spindle, the fixed and support side bearings, and the feed screw nut for the feed axis.

In this experiment, the temperature sensors are located in three main areas: the spindle motor, the spindle front and rear bearings, and the spindle coolant inlet and outlet points. The specific locations of the sensors are shown in Table 2.

Table 2.

Temperature sensors location.

Sensor number	Description of position
1	Spindle coolant exit point on the right side
2	Spindle coolant exit point on the right side
3	Spindle coolant exit point on the right side
4	Right spindle coolant outlet
5	Spindle coolant outlet
6	The motor base for the spindle
7	Right spindle coolant outlet
8	Rear spindle bearing
9	Rear spindle bearing
10	Spindle front, behind tool magazine
11	Spindle coolant inlet
12	Spindle coolant inlet on the left side
13	Under the spindle head, near the z-axis rail
14	Spindle coolant inlet on the left side
15	Spindle coolant inlet on the left side
16	Spindle motor base

These sensor locations provide comprehensive coverage for monitoring the temperature of the relevant heat sources in the CNC machine tool.

Figure 10 shows the 16 temperature points distributed on the tooling machine. The critical temperature points are T6, T8, and T9 obtained from Pearson correlation coefficients, and it can be observed that the spindle coolant enters the cooling spindle at temperature point 13, leaves at temperature point 5, and returns to the spindle oil cooler. The temperature change is affected by the spindle coolant.

Figure 10.

16 temperature detecting position on a spindle.

Multivariate regression equations were developed using three critical temperature points, T6, T8, and T9, and the amount of spindle heat deflection in equations (4) to (9), where z is the z-axis compensation and x₁, x₂, and x₃ are the critical temperature points T6, T8, and T9, respectively. Table 3 shows the root mean square error indices after thermal deformation model training.

Regression equation of spindle speed 3000 r/min

$$\begin{array}{rl}z=& -1.1445x_1+16.379x_2+11.929x_3+6.7627x_1{x}_2+43.326x_1{x}_3-21.979x_2{x}_3\\ & -24.897x_1^2+8.383x_2^2-13.967x_3^2+0.22246x_1{x}_2{x}_3\end{array}$$

(4)

Regression equation of spindle speed 6000 r/min

$$\begin{array}{rl}z=& -2.32453742511533+11.3467687617435x_1-19.9307750037137x_2\\ & +10.3432573581011x_3\\ & -6.43659093842458x_1{x}_2+19.0558494586453x_1{x}_3-20.9172444943820x_2{x}_3\\ & -5.94064741858319x_1^2+13.5401343403343x_1^2+0.0262497073636531x_1{x}_2{x}_3\end{array}$$

(5)

Regression equation of spindle speed 9000 r/min

$$\begin{array}{rl}z& =0.528390325264109-2.15318687371120x_1+3.47384546867739x_2\\ & -2.62110090389524x_3-2.86615588722068x_1{x}_2+3.63799814150919x_1{x}_3\\ & +14.2040334908842x_2{x}_3-4.78362469814143x_2^2-10.2634411338967x_2^2\\ & -0.0240377854229422x_1{x}_2{x}_3\end{array}$$

(6)

Regression equation of spindle speed 12,000 r/min

$$\begin{array}{rl}z& =0.175605295473975+3.70764825274616x_1-4.67769067812112x_2\\ & +0.899301527059086x_3\\ & -4.70793902106517x_1{x}_2-10.2019475631673x_1{x}_3+2.65756619062512x_2{x}_3\\ & +6.46969917714535x_2^2+1.43524278179576x_2^2+4.22757205781204x_3^2\end{array}$$

(7)

Regression equation of spindle speed 15,000 r/min

$$\begin{array}{rl}z& =0.331059658703662+14.7151395681008x_1-6.28070903266030x_2\\ & -10.2329828193017x_3\\ & +15.9812880789332x_1{x}_2+2.66650475455280x_1{x}_3-17.4467493487219x_2{x}_3\\ & -10.0571783018933x_1^2+8.82039726050525x_3^2+0.0101106557616022x_1{x}_2{x}_3\end{array}$$

(8)

Regression equation of spindle speed 18,000 r/min

$$\begin{array}{rl}z=& -1.80856013608194+8.38839945991452x_1+0.0734396201625894x_2\\ & -13.2356998897624x_3-5.21566193154433x_1{x}_2-3.94323567100124x_1{x}_3\\ & +5.82630858333127x_2{x}_3+3.94766091189969x_1^2\\ & -0.0382398938885583x_1{x}_2{x}_3\end{array}$$

(9)

Table 3.

The root mean square error (RMSE) after training model.

Spindle speed (r/min)	RMSE
3000	0.639065688265986
6000	0.417983815694071
9000	0.277440897959544
12,000	0.282955611082021
15,000	0.780215707688656
18,000	0.420818375577536

This system is designed to predict the thermal displacement output corresponding to spindle speeds that do not fall within the training data intervals of 3k, 6k, 9k, 12k, 15k, and 18k r/min. To achieve this, the fuzzy system is combined with a regression equation.

For instance, if the target speed is 4000 r/min, which falls within the range of 3000–6000 r/min, the difference between the target speed and the speed range is calculated for 1000 and 2000 r/min, respectively. The fuzzy inference of the membership function is then used to determine the corresponding membership values of 0.334 and 0.666, respectively. These membership values are used to estimate compensation of 4000 r/min. To calculate the membership value, Figure 11 demonstrates the use of the speed difference.

Figure 11.

Calculation of the membership function.

In the model verification part, there are two cases of single-speed and multiple working conditions for verification. The single-speed thermal compensation model verifies five speeds from low speed to high speed, including 4500, 8000, 10,500, 13,500, and 16,500 r/min, as shown in Figure 12, and the running time is from 2 to 4 h, and the residuals of each speed are shown in Table 4.

Figure 12.

Model verification single speed.

Table 4.

Residuals by speed.

Spindle speed (r/min)	Residual (μm)
4500	10.0287
8000	5.6419
10,500	5.3598
13,500	3.4792
16,500	9.2334

The multi-condition thermal displacement compensation model is verified, the spindle speed is shown in Figure 13, and the running time is 4.5 h. The relationship between spindle speed change and temperature and residuals was observed in Figure 14. The maximum elongation of the spindle before compensation was 23.121 μm and after compensation, the maximum extension of the spindle was controlled at 14.5899 μm under various working conditions, resulting in a 36.90% improvement in machining accuracy. The maximum elongation of the spindle before compensation was 16.4835 μm at 4500 r/min, and after compensation, the maximum extension of the spindle was controlled at 10.0287 μm, improving machining accuracy by 39.16%. At 13,500 r/min, the maximum elongation before compensation is 24.222 μm, and after compensation, the maximum spindle elongation is controlled at 3.4792 μm, improving machining accuracy by 85.64%. Significantly when the spindle speed is changed, there is a significant change in the axial displacement due to the centrifugal force. Since this article is currently using the BT30 handle, using the restraint tool holder can be tested to improve this phenomenon.

Figure 13.

Multi-operation spindle operation.

Figure 14.

Validation model in different conditions for spindle speed. (a) Temperature. (b) Residual.

Conclusion

This article proposes the RAFI hybrid optimization algorithm to develop a predictive model for the thermal displacement of the CNC tapping center. The validity of the proposed model is verified by single-speed and multiple-speed conditions. The proposed RAFI method can effectively reduce thermal error. The model needs to be adjusted when the environmental conditions vary too much or when mechanical wear caused by long processing time affects the thermal transfer behavior. The RAFI method can change the model according to the required speed range, reducing the amount of data collected and the time necessary and avoiding the impact on factory productivity.

To ensure that the data collected was comprehensive and accurate, temperature sensors were installed at various heat sources in the machine. These heat sources include the front and rear bearings of the main shaft, the base, the front and rear bearings of the feed axes, the motor seat of each axis, and the ambient temperature. Placing the temperature sensors at the primary heat sources enabled the collecting of relevant and accurate data. The choice of temperature points on a machine tool usually depends on the type of machine and the operating conditions. Still, measurements are generally taken at the front and rear spindle bearings, the base, the front, and rear feed axis bearings, the motor block of each axis, and the ambient temperature. These locations are the primary sources of the machine's thermal deformation. Measuring the temperature at these locations provides a relatively accurate understanding of the overall temperature variation of the machine so that adjustments can be made to the relevant parameters.

Because the experiment was only tested on the PMC tapping center with a built-in spindle, it is still being determined whether it will apply to different CNC machine tools and spindle types. When the spindle is heated and tilted, the difference in the length of the tool holder will affect the thermal displacement compensation result, which will significantly affect the precision machining. In the future, the radial thermal displacement measurement of the spindle will be added for analysis. In this paper, the fuzzy inference uses the spindle speed as an input factor to estimate the membership function. In future work, the input factors could be increased to improve the robustness of fuzzy inference.