# A geometric error budget method to improve machining accuracy reliability of multi-axis machine tools

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## Abstract

Machining accuracy reliability is considered to be one of the most important indexes in the process of performance evaluation and optimization design of the machine tools. Geometric errors, thermal errors and tool wear are the main factors to affect the machining accuracy and so affect the machining accuracy reliability of machine tools. This paper proposed a geometric error budget method that simultaneously considers geometric errors, thermal errors and tool wear to improve the machining accuracy reliability of machine tools. Homogeneous transformation matrices, neural fuzzy control theory and a tool wear predictive approach were employed to develop a comprehensive error model, which shows the influence of the geometric, thermal errors and tool wear to the machining accuracy of a machine tool. Based on Rackwite–Fiessler and Advanced First Order and Second Moment, a reliability model and a sensitivity model were put forward, which can deal with the errors of a machine tool drawn from any distribution. Then, a geometric error budget method of multi-axis NC machine tool was developed and formed into a mathematical model. In such method, the minimum cost of machine tool was the optimization objective, the reliability of the machining accuracy was the constraint, and the sensitivity was to identify the geometric errors to be optimized. An example conducted on a five-axis NC machine tool was used to explain and validate the proposed method.

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## Abbreviations

$$\Delta x_{x}$$ :

Positioning error

$$\Delta y_{x}$$ :

Y direction of straightness error

$$\Delta z_{x}$$ :

Z direction of straightness error

$$\Delta \alpha _{x}$$ :

Roll error

$$\Delta \beta _{x}$$ :

Pitch error

$$\Delta \gamma _{x}$$ :

Yaw error

$$\Delta x_{y}$$ :

X direction of straightness error

$$\Delta y_{y}$$ :

Positioning error

$$\Delta z_{y}$$ :

Z direction of straightness error

$$\Delta \alpha _{y}$$ :

Pitch error

$$\Delta \beta _{y}$$ :

Roll error

$$\Delta \gamma _{y}$$ :

Yaw error

$$\Delta x_{z}$$ :

X direction of straightness error

$$\Delta y_{z}$$ :

Y direction of straightness error

$$\Delta z_{z}$$ :

Positioning error

$$\Delta \alpha _{z}$$ :

Pitch error

$$\Delta \beta _{z}$$ :

Yaw error

$$\Delta \gamma _{z}$$ :

Roll error

$$\Delta x_{B}$$ :

X direction run-out error

$$\Delta y_{B}$$ :

Y direction run-out error

$$\Delta z_{B}$$ :

Z direction run-out error

$$\Delta \alpha _{B}$$ :

Around the X-axis turning error

$$\Delta \beta _{B}$$ :

Turning error

$$\Delta \gamma _{B}$$ :

Around the Z-axis turning error

$$\Delta x_{A}$$ :

X direction run-out error

$$\Delta y_{A}$$ :

Y direction run-out error

$$\Delta z_{A}$$ :

Z direction run-out error

$$\Delta \alpha _{A}$$ :

Turning error

$$\Delta \beta _{A}$$ :

Around the Y-axis turning error

$$\Delta \gamma _{A}$$ :

Around the Z-axis turning error

$$\Delta x_\varphi$$ :

X direction run-out error

$$\Delta y_\varphi$$ :

Y direction run-out error

$$\Delta z_\varphi$$ :

Z direction run-out error

$$\Delta \alpha _\varphi$$ :

Around the X-axis turning error

$$\Delta \beta _\varphi$$ :

Around the Y-axis turning error

$$\Delta \gamma _\varphi$$ :

Turning error

$$\Delta \gamma _{{xy}}$$ :

X, Y-axis perpendicularity error

$$\Delta \beta _{xz}$$ :

X, Z-axis perpendicularity error

$$\Delta \alpha _{yz}$$ :

Y, Z-axis perpendicularity error

$$\Delta \gamma _{xB}$$ :

B-axis parallelism error in YZ plane

$$\Delta \alpha _{zB}$$ :

B-axis parallelism error in XY plane

$$\Delta \gamma _{yA}$$ :

A-axis parallelism error in XZ plane

$$\Delta \beta _{zA}$$ :

A-axis parallelism error in XY plane

$$\Delta y_{AB}$$ :

An offset errors between A, B-axis along Y-axis

$$\Delta z_{AB}$$ :

An offset errors between A, B-axis along Z-axis

$$\Delta x_t$$ :

X direction of straightness error

$$\Delta y_t$$ :

Y direction of straightness error

$$\Delta z_t$$ :

Positioning error

$$\Delta \alpha _t$$ :

Pitch error

$$\Delta \beta _t$$ :

Yaw error

$$\Delta \gamma _t$$ :

Roll error

## References

1. Cai, L. G., Zhang, Z. L., Cheng, Q., Liu, Z. F., & Gu, P. H. (2015). A geometric accuracy design method of multi-axis NC machine tool for improving machining accuracy reliability. Eksploatacja i Niezawodnosc-Maintenance and Reliability, 17(1), 143–155.

2. Cai, L. G., Zhang, Z. L., Cheng, Q., Liu, Z. F., Gu, P. H., & Qi, Y. (2016). An approach to optimize the machining accuracy retainability of multi-axis NC machine tool based on robust design. Precision Engineering, 43, 370–386.

3. Carlos, C. A., & Hoffbauer, N. L. (2009). An approach for reliability-based robust design optimisation of angle-ply composites. Composite Structures, 90, 53–59.

4. Chen, J. S. (1995). Computer-aided accuracy enhancement for multi-axis CNC machine tool. International Journal of Machine Tools and Manufacture, 35(4), 593–605.

5. Cheng, Q., Qi, Z., Zhang, G. J., Zhao, Y. S., Sun, B. W., & Gu, P. H. (2016). Robust modelling and prediction of thermally induced positional error based on grey rough set theory and neural networks. International Journal of Machine Tools and Manufacture, 83(5), 753–764.

6. Cheng, Q., Zhao, H., Zhao, Y., et al. (2015a). Machining accuracy reliability analysis of multi-axis machine tool based on Monte Carlo simulation. Journal of Intelligent Manufacturing. doi:10.1007/s10845-015-1101-1.

7. Cheng, Q., Zhang, Z. L., Zhang, G. J., Gu, P. H., & Cai, L. G. (2015b). Geometric accuracy allocation for multi-axis CNC machine tools based on sensitivity analysis and reliability theory. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 229(6), 1134–1149.

8. Choudhury, S. K., & Rath, S. (2000). In-process tool wear estimation in milling using cutting force model. Journal of Materials Processing Technology, 99(1), 113–119.

9. Ding, W., Zhou, M., Huang, X., et al. (2007). Study on accuracy design of multi-axis machine tools oriented to remanufacturing. Journal of Basic Science and Engineering, 15(4), 559–568.

10. Dorndorf, U., Kiridena, V. S. B., & Ferreira, P. M. (1994). Optimal budgeting of quasistatic machine tool errors. Journal of Engineering for Industry, 116(1), 42–53.

11. Dufour, P., & Groppetti, R. (1981). Computer aided accuracy improvement in large NC machine-tools. In Proceedings of the 2006 international conference on the MTDR (Vol. 22, pp. 611–618).

12. El Ouafi, A., Guillot, M., & Bedrouni, A. (2000). Accuracy enhancement of multi-axis CNC machines through on-line neurocompensation. Journal of Intelligent Manufacturing, 11(6), 535–545.

13. Fan, K. C., Chen, H. M., & Kuo, T. H. (2012). Prediction of machining accuracy degradation of machine tools. Precision Engineering, 36(2), 288–298. doi:10.1016/j.precisioneng.2011.11.002.

14. Fu, G. Q., Fu, J. Z., Xu, Y. T., Chen, Z. C., & Lai, J. T. (2015). Accuracy enhancement of five-axis machine tool based on differential motion matrix: Geometric error modeling, identification and compensation. International Journal of Machine Tools and Manufacture, 89, 170–181.

15. Glass, K., & Colbaugh, R. (1996). Real-time tool wear estimation using cutting force measurements. Proceedings—IEEE International Conference on Robotics and Automation, 4, 3067–3072.

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

17. Guo, S. (2012). Research on modeling and compensation of machining error induced by tool wear. Nanjing: Nanjing University of Aeronautics and Astronautics.

18. Hocken, R., Simpson, J. A., Borchardt, B., et al. (1977). Three dimensional metrology. Annals of the CIRP, 26(2), 403–408.

19. Kang, Y., & Chang, C. W. (2007). Modification of a neural network utilizing hybrid filters for the compensation of thermal deformationin machine tools. International Journal of Machine Tools and Manufacture, 47, 376–387.

20. Khan, A. W., & Chen, W. Y. (2011). A methodology for systematic geometric error compensation in five-axis machine tools. International Journal of Advanced Manufacturing Technology, 53, 615–628.

21. Kiridena, V., & Ferreira, P. M. (1993). Mapping the effects of positioning errors on the volumetric accuracy of five-axis CNC machine tools. International Journal of Machine Tools and Manufacture, 33(3), 417–437.

22. Kong, D. D., Chen, Y. J., Li, N., & Tan, S. L. (2016). Tool wear monitoring based on kernel principal component analysis and v-support vector regression. International Journal of Machine Tools and Manufacture. doi:10.1007/s00170-016-9070-x.

23. Lee, D. M., & Yang, S. H. (2010). Mathematical approach and general formulation for error synthesis modeling of multi-axis system. International Journal of Modern Physics B, 4(15–16), 2737–2742.

24. Li, B., Hong, J., & Liu, Z. (2014). Stiffness design of machine tool structures by a biologically inspired topology optimization method. International Journal of Machine Tools and Manufacture, 84, 33–44.

25. Li, S. Y., Dai, Y. F., Yin, Z. Q., et al. (2007). Accuracy modeling technology of precision and ultra-precision machine tool. Changsha: National University of Defense Technology Press.

26. Liu, Y. W., Zhang, Q., Zhao, X. S., & Zhang, Z. F. (2002). Multi-body system based technique for compensating thermal errors in machining centers. Chinese Journal of Mechanical Engineering, 38(1), 127–130.

27. Lu, Y., Yao, Y. X., & Xu, H. Y. (1999). Intelligent identification of tool wear based on the state information state fusion of various parameters. Aviation Precision Manufacturing Technology, 35(6), 26–30.

28. Lu, Z. Y., Song, S. F., Li, H. S., et al. (2009). Structural reliability and reliability sensitivity analysis organization. Beijing: Sciences Press.

29. Portman, V. T. (1981). A universal method for calculating the accuracy of mechanical devices. Soviet Engineering Research, 1(7), 11–15.

30. Sarkar, S., & Dey, P. P. (2015). Tool path generation for algebraically parameterized surface. Journal of Intelligent Manufacturing, 26(2), 415–421.

31. Stryczek, R. (2014). A metaheuristic for fast machining error compensation. Journal of Intelligent Manufacturing. doi:10.1007/s10845-014-0945-0.

32. Tao, P. Y., Yang, G., Sun, Y. C., Tomizuka, M., & Lai, C. Y. (2012). Product-of-exponential (POE) model for kinematic calibration of robots with joint compliance. In IEEE/ASME international conference on advanced intelligent mechatronics (AIM) (Vol. 75, pp. 1–4).

33. Tortuma, A., Yayla, N., & Gökdag, M. (2009). The modeling of mode choices of intercity freight transportation with the artificial neural networks and adaptive neuro-fuzzy inference system. Expert Systems with Applications, 36, 6199–6217.

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

35. Xu, X. (2013). Research and application on accuracy distribution and optimization of high-grade CNC machine tools. Hangzhou: Zhejiang University.

36. Yan, J. Y., & Yang, J. G. (2009). Application of synthetic grey correlation theory on thermal, point optimization for machine tool thermal error compensation. International Journal of Advanced Manufacturing Technology, 43, 1124–1132.

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

38. Yang, J. (2012). Research on structure reliability calculation method and sensitivity analysis. Dalian: Dalian University of Technology.

39. Yang, J. G., Wang, X. S., & Zhao, H. T. (2005). Temperature measurement points optimization selection of machine tools. In Chinese Mechanical Engineering Society annual conference (pp. 627–632).

40. Yu, Z. M., Liu, Z. J., Ai, Y. D., & Xiong, M. (2013). Geometric error model and precision distribution based on reliability theory for large CNC gantry guideway grinder. Journal of Mechanical Engineering, 49(17), 142–151.

41. Yu, Z. M., Liu, Z. J., Ai, Y. D., et al. (2014). Thermal error modeling of NC machine tool using neural fuzzy control theory. Chinese Journal of Mechanical Engineering, 25(16), 2224–2231.

42. Zeroudi, N., & Fontaine, M. (2015). Prediction of tool deflection and tool path compensation in ball-end milling. Journal of Intelligent Manufacturing, 26(3), 425–445. doi:10.1007/s10845-013-0800-8.

43. Zhang, D., Wang, L., Gao, Z., & Su, X. (2013). On performance enhancement of parallel kinematic machine. Journal of Intelligent Manufacturing, 24(2), 267–276.

44. Zhang, H. T., Yang, J. G., & Jiang, H. (2009). Application of fuzzy neural network theory in thermal error compensation modeling of NC machine tool. Journal of Shanghai Jiangtong University, 43(12), 1950–1952, 1961.

45. Zhang, Y., Yang, J. G., & Jiang, H. (2012). Machine tool thermal error modeling and prediction by grey neural network. International Journal of Advanced Manufacturing Technology, 59, 1065–1072.

## Acknowledgments

The authors are most grateful to the National Natural Science Foundation of China (Nos. 51575010 and 51575009), Beijing Nova Program (Z1511000003150138), the Leading Talent Project of Guangdong Province, Open Project of State Key Lab of Digital Manufacturing Equipment & Technology (Huazhong University of Science and Technology), Shantou Light Industry Equipment Research Institute of science and technology Correspondent Station (2013B090900008), the National Science and Technology Major Project (2013ZX04013-011), the Jing-Hua Talents Project of Beijing University of Technology and the National Science and Technology Major Project (2013ZX04013) for supporting this research presented in this paper.

## Author information

Correspondence to Qiang Cheng.

## Appendix

### Appendix

Parameters Definitions
$$T_j$$ The temperatures set of the jth temperature point, which are the scalar variables that can be directly obtained
$$T_{jk}$$ The kth data of the jth temperature point, which is a scalar variable that can be directly obtained
$$W_k$$ The kth thermal error, which is an intermediate scalar variable
$$\overline{{\varvec{W}}}$$ The mean value of the thermal error, which is an intermediate scalar variable
$$\overline{T_j }$$ The mean value of the jth temperature point, which is an intermediate scalar variable
$$\chi _j$$ Principal factor, which is an intermediate scalar variable
$$v_i$$ A fuzzy variable, which is a scalar variable that can be directly obtained
$$T(v_i)$$ Fuzzy division set, which is an intermediate scalar variable
$$m_i$$ The number of division, which is an intermediate scalar variable
$$Q_i^j$$ The jth fuzzy subset of $$v_i$$ in the whole fuzzy set $$H_i$$, which is an intermediate scalar variable
$$\mu _{Q_i^j } (v_i )$$ Membership function of $$Q_i^j$$, which is an intermediate scalar variable
$$\varepsilon _i$$ The weight of the ith rule, which is an intermediate scalar variable
$${{\varvec{V}}}$$ The temperature input, which is a vector variable that can be directly obtained
$${{\varvec{W}}}$$ The thermal error output, which is an intermediate vector variable
$$N_{1}=n$$ The total number of the first layer, which is an intermediate scalar variable
$$N_{2}$$ The total number of the second layer, which is an intermediate scalar variable The fitness of each fuzzy rule, which is an intermediate scalar variable The whole fitness, which is an intermediate scalar variable
$$W_i$$ The output of the ith sub-networks, which is an intermediate scalar variable
$$\omega _{jl}^i$$ The connection weight of the latter network, which is an intermediate scalar variable
a,c The membership function coefficients of the premise network, which is an intermediate scalar variable
ES The square error of actual output and ideal output, which is an intermediate scalar variable
$$\vartheta$$ The learning rate, which is an intermediate scalar variable
$$\Delta t$$ The time interval between two measurements of the tool wear, which is a scalar variable that can be directly obtained
$$\Delta TW(\Delta t)$$ The change of the tool flank wear after the time interval $$\Delta t$$, which is an intermediate scalar variable
$$\mathrm{B}$$ A constant related to the tool and workpiece in a certain cutting condition, which is an intermediate scalar constant
 n The spindle speed, which is a scalar variable that can be directly obtained v The cutting feed, which is a scalar variable that can be directly obtained $$a_p$$ The cutting depth, which is a scalar variable that can be directly obtained D The tool diameter and the time interval of cutting process, which is a scalar constant that can be directly obtained $$\hbar (\eta )$$ The height of the cutting point before the tool wear, which is a scalar variable that can be directly obtained $$\xi _i$$ The indexes of these variables, which is an intermediate scalar variable $$\theta$$ The rake angles of the tool, which is a scalar constant that can be directly obtained $$\varsigma$$ The relief angles of the tool, which is a scalar constant that can be directly obtained $$\Delta R(\Delta t)$$ The change of the radial radius of the tool cutting edge, which is an intermediate scalar variable $$\eta$$ The angle between the axial direction and the line which connects the cutting point and center point O of the tool before the tool wear, which is a scalar variable that can be directly obtained $$\eta ^{\prime }$$ The angle between the axial direction and the line that connects the cutting point and center point O of the tool after the tool wear, which is an intermediate scalar variable $$R_0$$ The radius of the tool before tool wear, which is a scalar constant that can be directly obtained $$r_0$$ The radial radius of the tool before tool wear, which is a scalar constant that can be directly obtained $$r(\hbar )$$ The radial radius of the tool after the tool wear, which is a scalar variable that can be directly obtained P A point in the cutting edge before the tool wear, which is a vector variable that can be directly obtained $$P^{\prime }$$ The location of the point P after the tool wear, which is an intermediate vector variable $$\hbar (\eta )$$ The height of the selected cutting point, which is a scalar variable that can be directly obtained $$\wp$$ The error induced by the tool wear, which is an intermediate scalar variable $$\wp _x\quad \wp _y\quad \wp _z$$ The errors induced by tool wear in X, Y, Z directions, which are the intermediate scalar variables G A performance function, which is an intermediate scalar variable $$x_i$$ The uncorrelated non-normal variable, which is a scalar variable that can be directly obtained
 $$F_i(x_i)$$ The distribution function, which is a scalar variable that can be directly obtained $$f_i (x_i )$$ The density function, which is a scalar variable that can be directly obtained $$x_i^\prime$$ The equivalent normal random variable, which is an intermediate scalar variable $$\mu _{{x_i}}^\prime$$ The mean value of $$x_i^\prime$$, which is an intermediate scalar variable $$\sigma _{{x_i}}^\prime$$ The standard deviation of $$x_i^\prime$$, which is an intermediate scalar variable $$\Phi (\cdot )$$ The distribution function of the standard normal distribution, which is a scalar variable that can be directly obtained $$\phi (\cdot )$$ The density function of the standard normal distribution, which is a scalar variable that can be directly obtained $$\alpha _i$$ The sensitivity coefficient, which is an intermediate scalar variable $$\beta$$ The reliability index, which is an intermediate scalar variable $$P_f$$ The possibility of failure in single failure mode, which is an intermediate scalar variable $$\frac{\partial P_f }{\partial \sigma _{{x_i}}^\prime }$$ The reliability sensitivity, which is an intermediate scalar variable $$P_h$$ The possibility of failure under multiple failure modes, which is an intermediate scalar variable $$h_i$$ The ith performance function, which is an intermediate scalar variable $$\gamma _{12}$$ The correlation coefficient of these two failure modes, which is an intermediate scalar variable $$g_i$$ Geometric error parameter, which can be as a scalar variable that can be directly obtained, an intermediate scalar variable and a target scalar variable $$MC( {g_i } )$$ The cost of manufacturing the geometric error parameter $$g_i$$, which is an intermediate scalar variable MC The total manufacturing cost, which can be as an intermediate scalar variable and a target scalar variable s The number of the selected coordinate points, which is a scalar constant that can be directly obtained $$\frac{1}{s}\sum \limits _{t=1}^s {P_h (t)}$$ The mean value of the possibility of failure of the 16 coordinate points, which can be as an intermediate scalar variable and a target scalar variable $$\max P_h (t)$$ The maximum value of the possibility of failure of the 16 coordinate points, which can be as an intermediate scalar variable and a target scalar variable

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