A novel robust thermal error modeling and compensation for slant bed CNC machines tool using small dataset based feature selection

Abstract

Thermal errors have a significant impact on the precision of computer numerical control (CNC) machine tools, particularly in high-speed spindle systems such as the BL20-HSY. This study presents a novel and data-efficient framework for thermal error modeling and compensation, designed to enhance machining accuracy under thermal load conditions. An experimental platform was developed to collect comprehensive thermal data from critical locations on the spindle structure. To improve the selection for temperature-sensitive points, fuzzy C-means (FCM) clustering, optimized through ant colony optimization (ACO), was employed. Additionally, a multivariable grey correlation analysis (MGCA) was conducted to further refine sensor selection by identifying critical points that exhibited a high correlation with thermal deviations. To overcome the challenges associated with small datasets, a robust model-agnostic meta-learning (MAML) approach was integrated with the grey wolf optimizer (GWO), resulting in an adaptive model capable of generalizing across various operating conditions. Experimental validation conducted on the T65 CNC lathe demonstrated that the proposed MAML-GWO model achieved a reduction of over 90% in thermal deviation along the X and Y axes, with a predictive accuracy of R² values reaching 0.97. These findings indicate that the model is suitable for real-time thermal error compensation in structurally similar CNC machine tools.

Keywords

Thermal errors / Small dataset / Temperature sensitivity points / Slant bed computer numerical control (CNC) / Model-agnostic meta-learning (MAML) / Grey wolf optimizer (GWO)

References

[1] Ramesh R, Mannan MA, Poo AN. Error compensation in machine tools—a review: part I: geometric, cutting-force induced and fixture-dependent errors. Int J Mach Tools Manuf, 2000, 40(9): 1235-1256.

[2] Ramesh R, Mannan MA, Poo AN. Error compensation in machine tools—a review: part II: thermal errors. Int J Mach Tools Manuf, 2000, 40(9): 1257-1284.

[3] Fan C, Xue CY, Zhao J, et al.. Process planning and contour-based error compensation for precision grinding of miniature scalpels. Adv Manuf, 2024, 12(1): 108-123.

[4] Denkena B, Abele E, Brecher C, et al.. Energy efficient machine tools. CIRP Ann, 2020, 69(2): 646-667.

[5] Weck M, McKeown P, Bonse R, et al.. Reduction and compensation of thermal errors in machine tools. CIRP Ann, 1995, 44(2): 589-598.

[6] Bryan J. International status of thermal error research (1990). CIRP Ann, 1990, 39(2): 645-656.

[7] ISO 230-3 (2007) Test code for machine tools part 3: determination of thermal effects. International Organisation for Standardization

[8] Abele E, Altintas Y, Brecher C, et al.. Machine tool spindle units. CIRP Ann, 2010, 59(2): 781-802.

[9] Zverv I, Pyoun YS, Lee KB, et al.. An elastic deformation model of high speed spindles built into ball bearings. J Mater Process Technol, 2005, 170(3): 570-578.

[10] Liu K, Liu Y, Sun MJ, et al.. Spindle axial thermal growth modeling and compensation on CNC turning machines. Int J Adv Manuf Technol, 2016, 87(5): 2285-2292.

[11] Jiang S, Zhao ZG, Sun ML, et al.. Analysis on thermal dynamic characteristics of CNC machine tool spindle. J Tianjin Univ, 2013, 46(9): 846-850

[12] Xiang ST, Zhu XL, Yang JG. Modeling for spindle thermal error in machine tools based on mechanism analysis and thermal basic characteristics tests. Proc Inst Mech Eng C J Mech Eng Sci, 2014, 228(18): 3381-3394.

[13] Vyroubal J. Compensation of machine tool thermal deformation in spindle axis direction based on decomposition method. Precis Eng, 2012, 36(1): 121-127.

[14] Zhou H, Fan KG, Gao R. Fast heat conduction-based thermal error control technique for spindle system of machine tools. Int J Adv Manuf Technol, 2020, 107(1): 653-666.

[15] Ko TJ, Gim TW, Ha JY. Particular behavior of spindle thermal deformation by thermal bending. Int J Mach Tools Manuf, 2003, 43(1): 17-23.

[16] Wei X, Gao F, Li Y, et al.. Study on optimal independent variables for the thermal error model of CNC machine tools. Int J Adv Manuf Technol, 2018, 98(1): 657-669.

[17] Li Y, Zhao J, Ji S. A reconstructed variable regression method for thermal error modeling of machine tools. Int J Adv Manuf Technol, 2017, 90(9): 3673-3684.

[18] Zhang CX, Gao F, Meng ZH, et al.. A novel linear virtual temperature constructing method for thermal error modeling of machine tools. Int J Adv Manuf Technol, 2015, 80(9): 1965-1973.

[19] Zhao JL, Wu LY, Huang LK, et al.. Research on optimization and modeling of temperature measurement points based on improved fuzzy C means clustering algorithm. Modular Mach Tool Autom Manuf Technol, 2019, 6: 63-66

[20] Liu H, Miao EM, Wang JF, et al.. Temperature-sensitive point selection and thermal error model adaptive update method of CNC machine tools. Machines, 2022, 10(6): 427.

[21] Erb I (2020) Partial correlations in compositional data analysis. Applied Computing and Geosciences 6:100026. https://doi.org/10.1016/j.acags.2020.100026‏

[22] Gao F, Liu J, Yan L, et al.. Study on optimization of thermal key points for machine tools based on Fisher optimal segmentation method. Chin J Sci Instrum, 2013, 34(5): 1070-1075

[23] Cai LG, Li GP, Cheng Q. Reduction of thermal error temperature measurement points of machine tools based on rough set and partial correlation analysis. J Beijing Univ Technol, 2016, 42(7): 969-974

[24] Alataiqeh M, Shi H, Qu Q et al (2025) Thermal error modeling of Slant bed CNC lathe spindle based on BiLSTM with data augmentation and grey wolf optimizer algorithm. Case Studies in Thermal Engineering 70:106090. https://doi.org/10.1016/j.csite.2025.106090

[25] Li Y, Yu ML, Bai YM, et al.. A review of thermal error modeling methods for machine tools. Appl Sci, 2021, 11(11): 5216.

[26] Weng LT, Gao WG, Zhang DW, et al.. Analytical modelling of transient thermal characteristics of precision machine tools and real-time active thermal control method. Int J Mach Tools Manuf, 2023, 186. ArticleID: 104003

[27] Zhang JF, Feng PF, Chen C, et al.. A method for thermal performance modeling and simulation of machine tools. Int J Adv Manuf Technol, 2013, 68(5): 1517-1527.

[28] Li Y, Shi HX, Ji SJ, et al.. Thermal positioning error modeling of servo axis based on empirical modeling method. Micromachines, 2021, 12(2): 201.

[29] Kang CM, Zhao CY, Fu LX. Thermal error modeling of machining center spindle based on physical modeling method. J Northeast Univ Nat Sci, 2020, 41(4): 528.

[30] Liu YC, Li KY, Tsai YC. Spindle thermal error modeling based on LSTM deep learning for a CNC machine tool. Appl Sci, 2021, 11(12): 5444.

[31] Liu PL, Du ZC, Li HM, et al.. Thermal error modeling based on BiLSTM deep learning for CNC machine tool. Adv Manuf, 2021, 9(2): 235-249.

[32] Bezdek JC, Ehrlich R, Full W. FCM: the fuzzy C-means clustering algorithm. Comput Geosci, 1984, 10(2/3): 191-203.

[33] Dorigo M, Gambardella LM. Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput, 2002, 1(1): 53-66.

[34] Finn C, Xu K, Levine S (2018) Probabilistic model-agnostic meta-learning. arXiv. https://doi.org/10.48550/arXiv.1806.02817

[35] Mirjalili S, Mirjalili SM, Lewis A. Grey wolf optimizer. Adv Eng Software, 2014, 69: 46-61.

Funding

National Natural Science Foundation of China(U22B2086)

Xi'an Science and Technolog Plan Program(23ZDCYJSGG0012-2023)

Shaanxi Outstanding Youth Science Foundation(2025JC-JCQN-032)

RIGHTS & PERMISSIONS

Shanghai University and Periodicals Agency of Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature