Citation: Ye, H.; Wei, X.; Zhuang, X.;
Miao, E. An Improved Robust
Thermal Error Prediction Approach
for CNC Machine Tools. Machines
2022, 10, 624. https://doi.org/
10.3390/machines10080624
Academic Editors: Fang Cheng,
Qian Wang, Tegoeh Tjahjowidodo
and Ziran Chen
Received: 4 July 2022
Accepted: 27 July 2022
Published: 29 July 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
An Improved Robust Thermal Error Prediction Approach for
CNC Machine Tools
Honghan Ye
1
, Xinyuan Wei
2,
* , Xindong Zhuang
3
and Enming Miao
4
1
Department of Statistics, School of Computer, Data & Information Sciences, College of Letters & Science,
University of Wisconsin-Madison, Madison, WI 53705, USA; hye42@wisc.edu
2
School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 230009, China
3
Hangzhou Hikauto Technology Co., Ltd., Hangzhou 310000, China; zhuangxindong@hikauto.com
4
School of Mechanical Engineering, Chongqing University of Technology, Chongqing 400054, China;
miaoem@163.com
* Correspondence: weixy@ahut.edu.cn; Tel.: +86-152-5295-7376
Abstract:
Thermal errors significantly affect the accurate performance of computer numerical control
(CNC) machine tools. In this paper, an improved robust thermal error prediction approach is proposed
for CNC machine tools based on the adaptive Least Absolute Shrinkage and Selection Operator
(LASSO) and eXtreme Gradient Boosting (XGBoost) algorithms. Specifically, the adaptive LASSO
method enjoys the oracle property of selecting temperature-sensitive variables. After the temperature-
sensitive variable selection, the XGBoost algorithm is further adopted to model and predict thermal
errors. Since the XGBoost algorithm is decision tree based, it has natural advantages to address the
multicollinearity and provide interpretable results. Furthermore, based on the experimental data
from the Vcenter-55 type 3-axis vertical machining center, the proposed algorithm is compared with
benchmark methods to demonstrate its superior performance on prediction accuracy with 7.05
µm
(over 14.5% improvement), robustness with 5.61
µm
(over 12.9% improvement), worst-case scenario
predictions with 16.49
µm
(over 25.0% improvement), and percentage errors with 13.33% (over 10.7%
improvement). Finally, the real-world applicability of the proposed model is verified through thermal
error compensation experiments.
Keywords:
adaptive LASSO; CNC machine tools; thermal errors; robustness; variable selection;
XGBoost
1. Introduction
Due to changes in heat sources internally and externally in a machining process,
thermal deformation of machine tools occurs and thus changes the relative position between
the tool and workpiece, which is known as thermal errors or thermally induced errors [
1
,
2
].
Thermal errors have become one of the most important factors affecting the accuracy of
computer numerical control (CNC) machine tools, which account for up to 75% of the
overall geometrical errors of machined workpieces [
3
]. Therefore, it is important and
imperative to reduce thermal errors to improve the accuracy of CNC machine tools.
To reduce thermal errors, in general, there are two main research directions. The
first direction is the numerical analysis, which establishes the analytical model and then
simulates and analyzes thermal error law. For example, Creighton et al. [
4
] proposed a
thermal error compensation model using the finite element analysis in a high-speed micro-
milling spindle. Xu et al. [
5
] established thermal behavior models using the finite element
method (FEM) for an air-cooling ball screw system to predict and compensate for thermal
errors. Li et al. [
6
] proposed an explicit analytical thermal error model for compensation
considering ambient temperature fluctuations and the model was verified by both FEM and
an experiment on the machine tool. Xaver et al. [
7
] proposed a structural model using FEM
for the ball screw axes of the machine such that up to 87% of the maximal thermo-elastic
Machines 2022, 10, 624. https://doi.org/10.3390/machines10080624 https://www.mdpi.com/journal/machines
