Robust machine tool thermal error compensation modelling based on temperature-sensitive interval segmentation modelling technology

  • Yunsheng Liu
  • Enming MiaoEmail author
  • Hui Liu
  • Yangyang Chen


Establishing a mathematical model that can reflect the relationship between temperature increase and thermal error during machining is the core of thermal error compensation technology for CNC machine tools. The collinearity between temperature-sensitive points and the correlation between temperature-sensitive points and thermal errors are important factors affecting the prediction accuracy and robustness of the thermal error compensation model. Based on the thermal error measurement experiments of the Leaderway-V450 CNC machine tool in different periods of the year, the principal component regression (PCR) modelling algorithm, which can eliminate the collinearity effect, is proposed to establish the thermal error compensation model of the machine tool on the basis of selecting the temperature-sensitive points by using the correlation coefficient. It is compared with the newly proposed ridge regression thermal error compensation modelling algorithm. The results show that the thermal error compensation modelling method proposed in this paper can basically control the Z-direction thermal error of the CNC machine tool spindle within 10 μm with only two temperature sensors and has higher engineering practicability. It is found that the thermal error compensation model of machine tools has a jump interval affected by the ambient temperature. This interval is called the temperature-sensitive interval, and a temperature-sensitive interval subsection point selection algorithm is proposed to build a subsection model on both sides of the segment point. The results show that the Z-direction thermal error of the spindle of CNC machine tools can be basically controlled within 5 μm with only two temperature sensors and that the model is highly robust and has great engineering application value.


CNC machine tool Thermal error Collinearity Principal component regression algorithm Temperature-sensitive interval Segmentation modelling 



This work is supported by the Key Project of the National Natural Science Foundation of China (No. 51490660/51490661) and the Scientific Research Foundation of Chongqing University of Technology.


  1. 1.
    Bryan J (1990) International status of thermal error research (1990). CIRP Ann 39(2):645–656. CrossRefGoogle Scholar
  2. 2.
    Aronson RB (1996) The war against thermal expansion. Manuf Eng 116(6):45Google Scholar
  3. 3.
    Miao E, Gong Y, Niu P, Ji C, Chen H (2013) Robustness of thermal error compensation modeling models of CNC machine tools. Int J Adv Manuf Technol 69(9):2593–2603CrossRefGoogle Scholar
  4. 4.
    Jianguo Y (1998) Error synthetic compensation technique and application for nc machine tools. Shanghai Jiao Tong University. (In Chinese)Google Scholar
  5. 5.
    En-ming M, Ya-yun G, Lian-chun D, Ji-chao M (2014) Temperature-sensitive point selection of thermal error model of CNC machining center. Int J Adv Manuf Technol 74(5):681–691CrossRefGoogle Scholar
  6. 6.
    Liu H, Miao EM, Wei XY, Zhuang XD (2017) Robust modeling method for thermal error of CNC machine tools based on ridge regression algorithm. Int J Mach Tools Manuf 113:35–48CrossRefGoogle Scholar
  7. 7.
    Farrar DE, Glauber RR (1967) Multicollinearity in regression analysis: the problem revisited. Rev Econ Stat 49(1):92–107CrossRefGoogle Scholar
  8. 8.
    Zhang D, Liu X, Shi H, Chen RY (1995) Identification of position of key thermal susceptible points for thermal error compensation of machine tool by neural network. Proceedings of SPIE - The International Society for Optical EngineeringGoogle Scholar
  9. 9.
    Krulewich DA (1998) Temperature integration model and measurement point selection for thermally induced machine tool errors. Mechatronics 4:395–412CrossRefGoogle Scholar
  10. 10.
    Attia MH, Fraser S (1999) A generalized modelling methodology for optimized real-time compensation of thermal deformation of machine tools and CMM structures. Int J Mach Tools Manuf 39(6):1001–1016CrossRefGoogle Scholar
  11. 11.
    Lo C, Yuan J, Ni J (1999) Optimal temperature variable selection by grouping approach for thermal error modeling and compensation. Int J Mach Tools Manuf 39(9):1383–1396CrossRefGoogle Scholar
  12. 12.
    Lee J, Yang S (2002) Statistical optimization and assessment of a thermal error model for CNC machine tools. Int J Mach Tools Manuf 42(1):147–155CrossRefGoogle Scholar
  13. 13.
    Jianguo Y, Deng W, Ren Y, Li Y, Xiaolong D (2004) Grouping optimization modeling by selection of temperature variables for the thermal error compensation on machine tools. China Mech Eng 15(6):478–481Google Scholar
  14. 14.
    Wang K (2006) Thermal error modeling of a machining center using Grey System Theory and HGA-trained neural network. IEEEGoogle Scholar
  15. 15.
    Abdulshahed AM, Longstaff AP, Fletcher S, Myers A (2015) Thermal error modelling of machine tools based on ANFIS with fuzzy c-means clustering using a thermal imaging camera. Appl Math Model 39(7):1837–1852CrossRefGoogle Scholar
  16. 16.
    Wei X, Gao F, Li Y, Zhang D (2018) Study on optimal independent variables for the thermal error model of CNC machine tools. Int J Adv Manuf Technol 98(1):657–669CrossRefGoogle Scholar
  17. 17.
    Chen JS, Yuan J, Ni J (1996) Thermal error modelling for real-time error compensation. Int J Adv Manuf Technol 12(4):266–275CrossRefGoogle Scholar
  18. 18.
    Yun WS, Kim SK, Cho DW (1999) Thermal error analysis for a CNC lathe feed drive system. Int J Mach Tools Manuf 39(7):1087–1101CrossRefGoogle Scholar
  19. 19.
    Mize CD, Ziegert JC (2000) Neural network thermal error compensation of a machining center. Precis Eng 24(4):338–346CrossRefGoogle Scholar
  20. 20.
    Lee J, Lee J, Yang S (2001) Thermal error modeling of a horizontal machining center using fuzzy logic strategy. J Manuf Process 3(2):120–127CrossRefGoogle Scholar
  21. 21.
    Pahk H, Lee SW (2002) Thermal error measurement and real time compensation system for the CNC machine tools incorporating the spindle thermal error and the feed axis thermal error. Int J Adv Manuf Technol 20(7):487–494CrossRefGoogle Scholar
  22. 22.
    Ramesh R, Mannan MA, Poo AN (2002) Support vector machines model for classification of thermal error in machine tools. Int J Adv Manuf Technol 20(2):114–120CrossRefGoogle Scholar
  23. 23.
    Zhang Y, Yang J, Jiang H (2012) Machine tool thermal error modeling and prediction by grey neural network. Int J Adv Manuf Technol 59(9):1065–1072CrossRefGoogle Scholar
  24. 24.
    Creighton E, Honegger A, Tulsian A, Mukhopadhyay D (2010) Analysis of thermal errors in a high-speed micro-milling spindle. Int J Mach Tools Manuf 50(4):386–393CrossRefGoogle Scholar
  25. 25.
    Mian NS, Fletcher S, Longstaff AP, Myers A (2011) Efficient thermal error prediction in a machine tool using finite element analysis. Meas Sci Technol 22(8):85107CrossRefGoogle Scholar
  26. 26.
    Li Y, Zhao W, Wu W, Lu B, Chen Y (2014) Thermal error modeling of the spindle based on multiple variables for the precision machine tool. Int J Adv Manuf Technol 72(9):1415–1427CrossRefGoogle Scholar
  27. 27.
    Yin Q, Tan F, Chen H, Yin G (2019) Spindle thermal error modeling based on selective ensemble BP neural networks. Int J Adv Manuf Technol 101(5):1699–1713CrossRefGoogle Scholar
  28. 28.
    ISO 230-3 (2007) Test code for machine tools part 3: determination of thermal effects. ISO Copyright Office, SwitzerlandGoogle Scholar
  29. 29.
    Tseng P (1997) A real-time thermal inaccuracy compensation method on a machining centre. Int J Adv Manuf Technol 13(3):182–190CrossRefGoogle Scholar
  30. 30.
    Hoerl AE, Kennard RW (2000) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 42(1):80–86CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Yunsheng Liu
    • 1
  • Enming Miao
    • 1
    Email author
  • Hui Liu
    • 2
  • Yangyang Chen
    • 1
  1. 1.College of Mechanical EngineeringChongqing University of TechnologyChongqingChina
  2. 2.School of AutomationXian University of Posts & TelecommunicationsXianChina

Personalised recommendations