A Dynamic Linearization Modeling of Thermally Induced Error Based on Data-Driven Control for CNC Machine Tools


This paper proposes a novel dynamic linearization modeling method for machine tool thermal errors based on data-driven control theory, with improved accuracy and robustness under various practical working conditions of machine tool. The nonlinear, quasi-static and pseudo-hysteric characteristics of the machine tool temperature field are identified as the main causes for poor robustness in conventional thermal error mathematical models. The theoretical and practical difficulties in applying conventional modeling approaches based on the model-based control theory are demonstrated using two types of CNC machine tools as examples. The data-driven control theory is applied to dynamic linearization modeling and the developed data model has shown significant improvement over the general dynamic model in terms of model accuracy and robustness. The feasibility and effectiveness of the proposed dynamic linearization modeling method has been verified using two experiments, demonstrating excellent robustness and ability to adapt to various machining conditions and to improve machining accuracy.

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  1. 1.

    Ramesh, R., Mannan, M. A., & Poo, A. N. (2000). Error compensation in machine tools—A review: Part II: Thermal errors. International Journal of Machine Tool & Manufacture, 40(9), 1257–1284.

    Article  Google Scholar 

  2. 2.

    Yang, H., & Ni, J. (2003). Dynamic modeling for machine tool thermal error compensation. Journal of Manufacturing Science and Engineering, 125(2), 245–254.

    Article  Google Scholar 

  3. 3.

    Fujishima, M., Narimatsu, K., Irino, N., & Mori, M. (2019). Adaptive thermal displacement compensation method based on deep learning. CIRP Journal of Manufacturing Science and Technology, 25, 22–25.

    Article  Google Scholar 

  4. 4.

    Li, Y., Zhao, W., Lan, S., Ni, J., Wu, W., & Lu, B. (2015). A review on spindle thermal error compensation in machine tools. International Journal of Machine Tools and Manufacture, 95, 20–38.

    Article  Google Scholar 

  5. 5.

    Mayr, J., Jedrzejewski, J., Uhlmann, E., et al. (2012). Thermal issues in machine tools. CIRP Annals - Manufacturing Technology, 61(2), 771–791.

    Article  Google Scholar 

  6. 6.

    Moriwaki, T. (1988). Thermal deformation and its on-line compensation of hydrostatically supported precision spindle. CIRP Annals - Manufacturing Technology, 37(1), 393–396.

    Article  Google Scholar 

  7. 7.

    Li, B., Zhu, D., Pang, J., & Yang, J. (2011). Quadratic curve heat flux distribution model in the grinding zone. The International Journal of Advanced Manufacturing Technology, 54(9–12), 931–940.

    Article  Google Scholar 

  8. 8.

    Wang, H., Wang, L., Li, T., & Han, J. (2013). Thermal sensor selection for the thermal error modeling of machine tool based on the fuzzy clustering method. The International Journal of Advanced Manufacturing Technology, 69(1–4), 121–126.

    Article  Google Scholar 

  9. 9.

    Brecher, C., Hirsch, P., & Weck, M. (2004). Compensation of thermo-elastic machine tool deformation based on control internal data. Annals of the CIRP, 53(1), 299–304.

    Article  Google Scholar 

  10. 10.

    Horejs, O., Mares, M., Kohut, P., Barta, P. & Hornych, J (2010) Compensation of machine tool thermal errors based on transfer functions. Modern Machinery (MM) Science Journal. 3; 162–165.

  11. 11.

    Zverev, I. A., Eun, I. U., Chung, W. J., & Lee, C. M. (2003). Thermal model of high-speed spindle units. KSME International Journal, 17(5), 668–678.

    Article  Google Scholar 

  12. 12.

    Lee, J., Kim, D. H., & Lee, C. M. (2015). A study on the thermal characteristics and experiments of high-speed spindle for machine tools. International Journal of Precision Engineering and Manufacturing, 16(2), 293–299.

    Article  Google Scholar 

  13. 13.

    Bossmanns, B., & Tu, J. F. (1999). A thermal model for high speed motorized spindles. International Journal of Machine Tools and Manufacture, 39(9), 1345–1366.

    Article  Google Scholar 

  14. 14.

    Wang, X., Yu, T., Sun, X., Shi, Y., & Wang, W. (2016). Study of 3D grinding temperature field based on finite difference method: considering machining parameters and energy partition. The International Journal of Advance Manufacturing and Technology, 84(5–8), 915–927.

    Google Scholar 

  15. 15.

    Mayr, J., Ess, M., Weikert, S. & Wegener, K. (2009). Compensation of thermal effects on machine tools using a FDEM simulation approach. Proceedings of the 9th Lamdamap. 38–47.

  16. 16.

    Wang, K. C., & Tseng, P. C. (2010). Thermal error modeling of a machine tool using data mining scheme. Journal of Advanced Mechanical Design Systems and Manufacturing, 4(2), 516–530.

    Article  Google Scholar 

  17. 17.

    Guo, Q., Yang, J., & Wu, H. (2010). Application of ACO-BPN to thermal error modeling of NC machine tool. The International Journal of Advanced Manufacturing Technology, 50(5–8), 667–675.

    Article  Google Scholar 

  18. 18.

    Zhang, H. T., Jiang, H., & Yang, J. G. (2010). An online modeling method for real-time thermal error compensation on high-speed machines based on RBF neural network theory. Key Engineering Materials, 455, 606–611.

    Article  Google Scholar 

  19. 19.

    Li, J. W., Zhang, W. J., Yang, G. S., Tu, S. D., & Chen, X. B. (2009). Thermal-error modeling for complex physical systems: the-state-of-arts review. The International Journal of Advanced Manufacturing Technology, 42(1–2), 168–179.

    Article  Google Scholar 

  20. 20.

    Lin, Z. C., & Chang, J. S. (2007). The building of spindle thermal displacement model of high speed machine center. The International Journal of Advanced Manufacturing Technology, 34(5–6), 556–566.

    Article  Google Scholar 

  21. 21.

    Yang, H., & Ni, J. (2005). Dynamic neural network modeling for nonlinear, nonstationary machine tool thermally induced error. International Journal of Machine Tools and Manufacture, 45(4–5), 455–465.

    Article  Google Scholar 

  22. 22.

    Yang, H., & Ni, J. (2005). Adaptive model estimation of machine-tool thermal errors based on recursive dynamic modeling strategy. International Journal of Machine Tools & Manufacture, 45, 1–11.

    Article  Google Scholar 

  23. 23.

    Blaser, P., Pavlicek, F., Mori, K., Mayr, J., Weikert, S., & Wegener, K. (2017). Adaptive Learning Control for Thermal Error Compensation of 5-Axis Machine Tools. Journal of Manufacturing Systems, 44, 302–309.

    Article  Google Scholar 

  24. 24.

    Hou, Z. S., & Wang, Z. (2013). From model-based control to data-driven control: survey, classification and perspective. Information Sciences, 235(235), 3–35.

    MathSciNet  Article  Google Scholar 

  25. 25.

    Chi, R., Hou, Z., Jin, S., Wang, D., & Hao, J. (2013). A data-driven iterative feedback tuning approach of ALINEA for freeway traffic ramp metering with PARAMICS simulations. IEEE Transactions on Industrial Informatics, 9(4), 2310–2317.

    Article  Google Scholar 

  26. 26.

    Lu, Y.C., & Yeh, S.S. (2018). Using the segmented iterative learning control method to generate volumetric error-compensated part programs for three-axis CNC milling machine tools. Journal of Manufacturing and Materials Processing. 2(3).

  27. 27.

    Haas, T., Lanz, N., Keller, R., Weikert, S., & Wegener, K. (2016). Iterative learning for machine tools using a convex optimisation approach. Procedia CIRP, 46, 391–395.

    Article  Google Scholar 

  28. 28.

    Hou, Z. & Huang, W. (1997). Model-free learning adaptive control of a class of SISO nonlinear systems. Proceedings of the 1997 IEEE American Control Conference, 343–344.

  29. 29.

    Hou, Z., & Jin, S. (2010). A novel data-driven control approach for a class of discrete-time nonlinear systems. IEEE Transactions on Control Systems Technology, 19(6), 1549–1558.

    MathSciNet  Article  Google Scholar 

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The research is supported by the National Natural Science Foundation of China (No. 51527806), the National Key R&D Program of China (No.2018YFB1701204) and the Shanghai Civil-Military Integration Project (No.2016-63).

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Correspondence to Zhengchun Du.

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Table 2 List of terms adopted in the modeling process

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Liu, P., Yao, X., Ge, G. et al. A Dynamic Linearization Modeling of Thermally Induced Error Based on Data-Driven Control for CNC Machine Tools. Int. J. Precis. Eng. Manuf. 22, 241–258 (2021). https://doi.org/10.1007/s12541-020-00463-0

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  • CNC machine tool
  • Thermal error
  • Dynamic modeling
  • Data-driven control
  • Data model