machines Article The Thermal Error Estimation of the Machine Tool Spindle Based on Machine Learning Yu-Cheng Chiu , Po-Hsun Wang and Yuh-Chung Hu *   Citation: Chiu,Y.-C.; Wang, P.-H.; Hu, Y.-C. The Thermal Error Estimation of the Machine Tool Spindle Based on Machine Learning. Machines 2021, 9, 184. https:// doi.org/10.3390/machines9090184 Academic Editor: Francisco J. G. Silva Received: 18 July 2021 Accepted: 27 August 2021 Published: 30 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 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/). Department of Mechanical and Electromechanical Engineering, National ILan University, Yilan 26041, Taiwan; b0722061@ms.niu.edu.tw (Y.-C.C.);b0722097@ms.niu.edu.tw (P.-H.W.) * Correspondence: ychu@niu.edu.tw; Tel.: +886-3-9317-469 Abstract: Thermal error is one of the main sources of machining error of machine tools. Being a key component of the machine tool, the spindle will generate a lot of heat in the machining process and thereby result in a thermal error of itself. Real-time measurement of thermal error will interrupt the machining process. Therefore, this paper presents a machine learning model to estimate the thermal error of the spindle from its feature temperature points. The authors adopt random forests and Gaussian process regression to model the thermal error of the spindle and Pearson correlation coefficients to select the feature temperature points. The result shows that random forests collocating with Pearson correlation coefficients is an efficient and accurate method for the thermal error modeling of the spindle. Its accuracy reaches to 90.49% based on only four feature temperature points—two points at the bearings and two points at the inner housing—and the spindle speed. If the accuracy requirement is not very onerous, one can select just the temperature points of the bearings, because the installation of temperature sensors at these positions is acceptable for the spindle or machine tool manufacture, while the other positions may interfere with the cooling pipeline of the spindle. Keywords: Gaussian process regression; machine learning; machine tool spindle; Pearson correlation coefficient; random forest; thermal error 1. Introduction The machining errors of machine tools mainly include geometrical errors, thermal errors, errors caused by cutting-force, fixture-dependent errors, etc. [1–4]. Bryan [1], Ramesh [2], and Li et al. [3] mentioned that thermal errors account for 40% to 70% of the total machining errors. The heat sources in the machining process of machine tools include two categories: internal heat sources, and external heat sources [2,4]. Internal heat sources include the heat produced by cutting, the heat induced by the friction of bearings, spindle, gearbox, and motion guides; the heat generated in the motor; and the heating or cooling effects produced by the cooling system. The external heat sources include ambient temperature changes, solar radiant heat, and human body radiant heat. Being a key component of the machine tool, the spindle will generate a lot of heat during the machining process and thereby result in a thermal deformation/error of itself. The main heat sources of the spindle are the friction of bearings and the heat generated by the motor. Takabi [5] mentioned that the heat-generation of the bearings is affected by the bearing type, the torque and preload applied on the bearing, the lubrication, etc. The heat transfer of the spindle involves complex physical phenomena such as conduction, convection, and radiation because the spindle is composed of many components of different materials and it rotates at high speeds during machining process. The design and manufacture of machine tools cannot completely solve the afore- mentioned diverse machining errors appearing in machining process [1,2,4]. A feasible solution is error compensation in the machining process. For compensation of thermal errors during the machining process, the machine tool should be capable of knowing Machines 2021, 9, 184. https://doi.org/10.3390/machines9090184 https://www.mdpi.com/journal/machines