A wrapper approach-based key temperature point selection and thermal error modeling method

  • Feng TanEmail author
  • Congying Deng
  • Hong Xiao
  • Jiufei Luo
  • Shuang Zhao


A wrapper approach-based key temperature point selection and thermal error modeling method is proposed to concurrently screen the optimal key temperature points and construct the thermal error model. This wrapper approach can strengthen the intrinsic relation between the key temperature points and the thermal error model to ensure the strong prediction performance. On the whole, the least squares support vector machine (SVM) is used as the basic thermal error modeling method and the binary bat algorithm (BBA) is used as the optimization algorithm. The selection status of temperature points and the values of hyperparameters γ and σ2 of SVM are coded in separate binary parts of the artificial bat’s position vector of BBA. The cost function is designed by balancing the prediction error and the number of key temperature points. For verification, the thermal error experiment was conducted on a horizontal machining center. Feeding the collected experimental temperature data and thermal error data to the proposed method, three optimal key temperature points were screened out and the corresponding optimal hyperparameters were simultaneously searched. To verify the superiority of the proposed method, the prediction performance comparison analysis was conducted with the conventional filter-based method. Specifically, in the conventional method, the key temperature points were screened by combining fuzzy c means (FCM) clustering and correlation analysis, and the multiple linear regression (MLR), the backpropagation neural network (BPNN), and the SVM were used to build the thermal error model, respectively. Comparison results showed that the prediction accuracy of the proposed method increased by up to 44.0% compared to the conventional method, which suggests the superior prediction performance of the proposed method.


Key temperature points Thermal error Support vector machine Binary bat algorithm Prediction performance 



This research was financially supported by the Chongqing Research Program of Basic Research and Frontier Technology (grant no. cstc2019jcyj-msxmX0540), the National Natural Science Foundation of China (grant no. 51605064), the National Natural Science Foundation of China (grant no. 51705058), and the National Natural Science Foundation of China (grant no. 51705059).


  1. 1.
    Ramesh R, Mannan MA, Poo AN (2000) Error compensation in machine tools — a review Part II: thermal errors. Int J of Mach Tools Manuf 40(9):1257–1284CrossRefGoogle Scholar
  2. 2.
    Mayr J, Jedrzejewski J, Uhlmann E, Alkan Donmez M, Knapp W, Hartig F, Wendt K, Moriwaki T, Shore P, Schmitt R (2012) Thermal issues in machine tools. CIRP Ann Manuf Technol 61(2):771–791CrossRefGoogle Scholar
  3. 3.
    Li Y, Zhao WH, Lan SH, Ni J, Wu WW, Lu BH (2015) A review on spindle thermal error compensation in machine tools. Int J of Mach Tools Manuf 95(8):20–38CrossRefGoogle Scholar
  4. 4.
    Cao HR, Zhang XW, Chen XF (2017) The concept and progress of intelligent spindles: A review. Int J Mach Tools Manuf 112(1):21–52CrossRefGoogle Scholar
  5. 5.
    Liu JL, Ma C, Wang SL, Wang SB, Yang B, Shi H (2019) Thermal-structure interaction characteristics of a high-speed spindle-bearing system. Int J Mach Tools Manuf 137(2):42–57CrossRefGoogle Scholar
  6. 6.
    Tan F, Wang L, Yin M, Yin GF (2019) Obtaining more accurate convective heat transfer coefficients in thermal analysis of spindle using surrogate assisted differential evolution method. Appl Therm Eng 149(2):1335–1344CrossRefGoogle Scholar
  7. 7.
    Sun LJ, Ren MJ, Hong HB, Yin YH (2017) Thermal error reduction based on thermodynamics structure optimization method for an ultra-precision machine tool. Int J Adv Manuf Technol 88(5-8):1267–1277CrossRefGoogle Scholar
  8. 8.
    Liu T, Gao WG, Zhang DW, Zhang YF, Chang WF, Liang CM, Tian YL (2017) Analytical modeling for thermal errors of motorized spindle unit. Int J of Mach Tools Manuf 112(1):53–70CrossRefGoogle Scholar
  9. 9.
    Creighton E, Honegger A, Tulsian A, Mukhopadhyay D (2010) Analysis of thermal errors in a high-speed micro-milling spindle. Int J of Mach Tools Manuf 50(4):386–393CrossRefGoogle Scholar
  10. 10.
    Wang LP, Wang HT, Li TM, Li FC (2015) A hybrid thermal error modeling method of heavy machine tools in z-axis. Int J Adv Manuf Technol 80(1-4):389–400CrossRefGoogle Scholar
  11. 11.
    Du ZC, Yao XD, Hou HF, Yang JG (2018) A fast way to determine temperature sensor locations in thermal error compensation. Int J Adv Manuf Technol 97(1-4):455–465CrossRefGoogle Scholar
  12. 12.
    Vyroubal J (2012) Compensation of machine tool thermal deformation in spindle axis direction based on decomposition method. Precis Eng 36(1):121–127CrossRefGoogle Scholar
  13. 13.
    Abdulshahed AM, Longstaff AP, Fletcher S (2015) The application of ANFIS prediction models for thermal error compensation on CNC machine tools. Appl Soft Comput 27(2):158–168CrossRefGoogle Scholar
  14. 14.
    Liu H, Miao EM, Zhuang XD, Wei XY (2018) Thermal error robust modeling method for CNC machine tools based on a split unbiased estimation algorithm. Precis Eng 51(1):169–175CrossRefGoogle Scholar
  15. 15.
    Guo QJ, Xu RF, Yang TY, He L, Cheng X, Li ZY, Yang JG (2016) Application of GRAM and AFSACA-BPN to thermal error optimization modeling of CNC machine tools. Int J of Adv Manuf Technol 83(5):995–1002CrossRefGoogle Scholar
  16. 16.
    Ma C, Zhao L, Mei XS, Shi H, Yang J (2017) Thermal error compensation of high-speed spindle system based on a modified BP neural network. Int J Adv Manuf Technol 89(9):3071–3085CrossRefGoogle Scholar
  17. 17.
    Liu Q, Yan JW, Pham DT, Zhou ZD, Xu WJ, Wei Q, Ji CQ (2016) Identification and optimal selection of temperature-sensitive measuring points of thermal error compensation on a heavy-duty machine tool. Int J Adv Manuf Technol 85(1-4):345–353CrossRefGoogle Scholar
  18. 18.
    Hey J, Sing TC, Liang TJ (2018) Sensor selection method to accurately model the thermal error in a spindle motor. IEEE T Ind Inform 14(7):2925–2931CrossRefGoogle Scholar
  19. 19.
    Miao EM, Liu Y, Liu H, Gao ZH, Li W (2015) Study on the effects of changes in temperature-sensitive points on thermal error compensation model for CNC machine tool. Int J Mach Tools Manuf 97(10):50–59CrossRefGoogle Scholar
  20. 20.
    Cheng Q, Qi Z, Zhang GJ, Zhao YS, Sun BW, Gu PH (2016) Robust modelling and prediction of thermally induced positional error based on grey rough set theory and neural networks. Int J Adv Manuf Technol 83(5):753–764CrossRefGoogle Scholar
  21. 21.
    Yang J, Shi H, Feng B, Zhao L, Ma C, Mei XS (2015) Thermal error modeling and compensation for a high-speed motorized spindle. Int J Adv Manuf Technol 77(5-8):1005–1017CrossRefGoogle Scholar
  22. 22.
    Li Y, Zhao J, Ji SJ, Liang FS (2019) The selection of temperature-sensitivity points based on K-harmonic means clustering and thermal positioning error modeling of machine tools. Int J of Adv Manuf Technol 100(9-12):2333–2348CrossRefGoogle Scholar
  23. 23.
    Yin Q, Tan F, Chen HX, Yin GF (2019) Spindle thermal error modeling based on selective ensemble BP neural networks. Int J Adv Manuf Technol 101(5-8):1699–1713CrossRefGoogle Scholar
  24. 24.
    Suykens JAK, Van Gestel T, De Brabanter J, De Moor B, Vandewalle J (2002) Least squares support vector machines. World ScientificGoogle Scholar
  25. 25.
    230-3 ISO (2007) Test code for machine tools part 3: determination of thermal effects. ISO copyright office, GenevaGoogle Scholar
  26. 26.
    Mirjalili S, Mirjalili SM, Yang X (2014) Binary bat algorithm. Neural Comput Appl 25(3-4):663–681CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Feng Tan
    • 1
    Email author
  • Congying Deng
  • Hong Xiao
  • Jiufei Luo
  • Shuang Zhao
  1. 1.School of Advanced Manufacturing EngineeringChongqing University of Posts and TelecommunicationsChongqingPeople’s Republic of China

Personalised recommendations