Citation: Wei, X.; Ye, H.; Feng, X.
Year-Round Thermal Error Modeling
and Compensation for the Spindle of
Machine Tools Based on Ambient
Temperature Intervals. Sensors 2022,
22, 5085. https://doi.org/10.3390/
s22145085
Academic Editors: Fang Cheng,
Qian Wang, Tegoeh Tjahjowidodo
and Ziran Chen
Received: 11 June 2022
Accepted: 5 July 2022
Published: 6 July 2022
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Article
Year-Round Thermal Error Modeling and Compensation for the
Spindle of Machine Tools Based on Ambient Temperature Intervals
Xinyuan Wei
1,†
, Honghan Ye
2,†
and Xugang Feng
1,
*
1
School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243032, China;
weixy@ahut.edu.cn
2
Department of Industrial and Systems Engineering, University of Wisconsin—Madison,
Madison, WI 53705, USA; hye42@wisc.edu
* Correspondence: fxg@ahut.edu.cn; Tel.: +86-13965390996
† These authors contributed equally to this work.
Abstract:
The modeling and compensation method is a common method for reducing the influence of
thermal error on the accuracy of machine tools. The prediction accuracy and robustness of the thermal
error model are two key performance measures for evaluating the compensation effect. However, it
is difficult to maintain the prediction accuracy and robustness at the desired level when the ambient
temperature exhibits strong seasonal variations. Therefore, a year-round thermal error modeling and
compensation method for the spindle of machine tools based on ambient temperature intervals (ATIs)
is proposed in this paper. First, the ATIs applicable to the thermal error prediction models (TEPMs)
under different ambient temperatures are investigated, where the C-Means clustering algorithm
is utilized to determine ATIs. Furthermore, the prediction effect of different numbers of ATIs is
analyzed to obtain the optimal number of ATIs. Then, the TEPMs corresponding to different ATIs in
the annual ambient temperature range are established. Finally, the established TEPMs of ATIs are
used to predict the experimental data of the entire year, and the prediction accuracy and robustness
of the proposed ATI model are analyzed and compared with those of the low and high ambient
temperature models. The prediction accuracies of the ATI model are 20.6% and 41.7% higher than
those of the low and high ambient temperature models, respectively, and the robustness is improved
by 48.8% and 62.0%, respectively. This indicates that the proposed ATI method can achieve high
prediction accuracy and robustness regardless of the seasonal temperature variations throughout
the year.
Keywords: CNC machine tool; thermal error; ambient temperature interval; model robustness
1. Introduction
In the machining process, changes in the internal and external heat sources such
as motor operation, friction, cutting heating, and environmental temperature result in
thermal errors, which represent 40–70% of the total errors of machine tools [1,2]. With the
development of high-precision CNC machine tools, the influence of thermal errors on the
tool performance is gradually becoming dominant [
3
]. To reduce such influence on machine
tools, there are two main approaches. The first approach is to establish the analytical model
and then to simulate and analyze thermal error laws. Although numerical analysis is
promising to compensate for thermal errors, it is extremely difficult for the numerical
method to build an exact structural model in practice and simulate the thermal deformation
of machine tools because of complicated deformation processes. Alternatively, software
compensation methods are commonly used to reduce the influence of thermal errors on
the accuracy of machine tools [
4
]. In this method, temperature sensors are first installed
on various locations of a machine, and a thermal error prediction model is established
based on the temperature information collected from those sensors. Then the established
model is embedded into the CNC system to realize thermal error compensation in real-time.
Sensors 2022, 22, 5085. https://doi.org/10.3390/s22145085 https://www.mdpi.com/journal/sensors
