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A method of sensitivity analysis and precision prediction for geometric errors of five-axis machine tools based on multi-body system theory

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Abstract

In order to ensure the mechanical performance and machining accuracy of the machine tool, the problem of mutual deviations of the machine tool moving parts in the machining process, which causes machining errors and subsequent accuracy prediction difficulties, is solved. Firstly, the structure and motion mechanism of the machine tool are analysed; a static accuracy model of the machine tool machining posture relationship is established using multi-body system theory and coordinate transformation; the measured deviation values are fitted and solved according to the formula; and the geometric error law affecting machining accuracy and the distribution of machining point error values in the machine tool motion space are explored. Then, the response surface method is used to simplify the solution. Finally, the blade is selected as the machined part, and the machining trajectory is extracted for experiments to obtain the error distribution range of the machining surface of the blade. The final experimental results surface, along the Z-directional component, and the integrated average compensation rate reached 42.6% and 89.6%, respectively, verifying the effectiveness of the method in this paper.

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Data availability

The datasets used or analysed during the current study are available from the corresponding author on reasonable request.

Code availability

The codes used or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

This study was supported by High-tech Key Research and Development Projects from Science and Technology Department of Sichuan Province, Grant No. 2021YFG0056 and funded by Intelligent Policing Key Laboratory of Sichuan Province, No. ZNJW2022KFQN004 and ZNJW2022ZZMS003.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Sichuan University of Science and Engineering, Sichuan, 643002, Yibin, China

    Zhou Tan, Yinghua Liao & Junchao Zhou

  2. Yibin Intelligent Terminal Digital Manufacturing Engineering Technology Research Center, Yibin, 644000, Sichuan, China

    Zhou Tan, Yinghua Liao & Jie Jiang

  3. Sichuan Yibin Zhiwei Technology Co., Ltd., Yibin, 644000, Sichuan, China

    Jie Jiang

  4. Intelligent Policing Key Laboratory of Sichuan Province, Luzhou, China

    Junchao Zhou

Authors
  1. Zhou Tan
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  2. Yinghua Liao
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  3. Jie Jiang
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  4. Junchao Zhou
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Corresponding author

Correspondence to Yinghua Liao.

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Ethics approval

This project is about China’s machine tool precision system. This project is taken from the five-axis vertical machining centre as the main research point. In this project, in order to ensure that these regulations are effectively implemented, the school’s academic ethics committee has to play a strict supervisory role and guarantee its behaviour.

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Appendix

Appendix

The additional equation is as follows:

$$\begin{array}{c}\Delta {E}_{x}:{L}_{2}+{\delta }_{xc}-{\varepsilon }_{zc}+cos\left({\theta }_{a}\right)+sin\left({\theta }_{a}\right)-{\delta }_{yy}\left({\varepsilon }_{zc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{yc}\mathrm{sin}\left({\theta }_{c}\right)\right)+{\varepsilon }_{xy}\\ {\varepsilon }_{xy}\left({\varepsilon }_{zc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{yc}\mathrm{sin}\left({\theta }_{c}\right)\right)-{\varepsilon }_{zy}\left({\varepsilon }_{zc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{yc}\mathrm{sin}\left({\theta }_{c}\right)\right)+{\delta }_{xy}cos\left({\theta }_{a}\right)+\\ sin\left({\theta }_{a}\right)\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{c}\right)\right))+{\delta }_{yz}(sin\left({\theta }_{a}\right)+cos\left({\theta }_{a}\right)({\varepsilon }_{yc}cos\left({\theta }_{c}\right)-{\varepsilon }_{{\text{z}}c}\\ sin\left({\theta }_{c}\right)))+{\varepsilon }_{xy}\left(\mathrm{sin}\left({\theta }_{a}\right)+\mathrm{cos}\left({\theta }_{a}\right)\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{c}\right)\right)\right)+{\varepsilon }_{yy}(cos\left({\theta }_{a}\right)+\\ sin\left({\theta }_{a}\right)\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{c}\right)\right))-{\varepsilon }_{yy}(sin\left({\theta }_{a}\right)+cos\left({\theta }_{a}\right)({\varepsilon }_{yc}cos\left({\theta }_{c}\right)-{\varepsilon }_{zc}\\ sin\left({\theta }_{c}\right)))-{\varepsilon }_{zc}cos\left({\theta }_{c}\right)-{\varepsilon }_{zy}\left(\mathrm{cos}\left({\theta }_{a}\right)+\mathrm{sin}\left({\theta }_{a}\right)\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{c}\right)\right)\right)+\\ {\varepsilon }_{yc}sin\left({\theta }_{c}\right)+cos\left({\theta }_{a}\right)\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{c}\right)\right)+sin\left({\theta }_{a}\right)\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{c}\right)\right)\end{array}$$
(31)
$$\begin{array}{c}\Delta {E}_{y}:{\delta }_{yc}-cos\left({\theta }_{c}\right)+{L}_{2}{\varepsilon }_{zc}+{\varepsilon }_{xy}\left(\mathrm{cos}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{sin}\left({\theta }_{c}\right)\right)+{\varepsilon }_{zy}(cos\left({\theta }_{c}\right)-{\varepsilon }_{zy}\\ sin\left({\theta }_{c}\right))+{\delta }_{xy}\left({\varepsilon }_{zc}\mathrm{cos}\left({\theta }_{c}\right)+\mathrm{sin}\left({\theta }_{c}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right)\right)+{\delta }_{zy}({\varepsilon }_{zc}\\ sin\left({\theta }_{a}\right)+cos\left({\theta }_{a}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right))+{\varepsilon }_{xy}({\varepsilon }_{zc}sin\left({\theta }_{a}\right)+cos\left({\theta }_{a}\right)\\ \left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right))+{\varepsilon }_{zc}cos\left({\theta }_{a}\right)+{\varepsilon }_{yy}({\varepsilon }_{zc}cos\left({\theta }_{a}\right)+sin\left({\theta }_{a}\right)\\ \left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right))-{\varepsilon }_{yy}\left({\varepsilon }_{zc}\mathrm{sin}\left({\theta }_{a}\right)+\mathrm{cos}\left({\theta }_{a}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right)\right)\\ -{\varepsilon }_{zy}\left({\varepsilon }_{zc}\mathrm{cos}\left({\theta }_{a}\right)+\mathrm{sin}\left({\theta }_{a}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right)\right)-{\varepsilon }_{xc}sin\left({\theta }_{c}\right)+{\varepsilon }_{zc}sin\left({\theta }_{a}\right)\\ cos\left({\theta }_{a}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right)+sin\left({\theta }_{a}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right)+1\end{array}$$
(32)
$$\begin{array}{c}\Delta {E}_{z}:{\varepsilon }_{xy}\left({\varepsilon }_{yc}\mathrm{sin}\left({\theta }_{a}\right)-{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{a}\right)\left(\mathrm{cos}\left({\theta }_{c}\right)+{\varepsilon }_{xc}\mathrm{sin}\left({\theta }_{c}\right)\right)\right)-{\varepsilon }_{yc}cos\left({\theta }_{a}\right)-\\ {\varepsilon }_{yy}\left({\varepsilon }_{yc}\mathrm{cos}\left({\theta }_{a}\right)-\mathrm{sin}\left({\theta }_{a}\right)\left(\mathrm{cos}\left({\theta }_{c}\right)+{\varepsilon }_{xc}\mathrm{sin}\left({\theta }_{c}\right)\right)\right)+{\varepsilon }_{yy}({\varepsilon }_{yc}sin\left({\theta }_{a}\right)-\\ cos\left({\theta }_{a}\right)\left(\mathrm{cos}\left({\theta }_{c}\right)+{\varepsilon }_{xc}\mathrm{sin}\left({\theta }_{c}\right)\right))+{\varepsilon }_{xc}cos\left({\theta }_{c}\right)+{\varepsilon }_{zy}({\varepsilon }_{yc}cos\left({\theta }_{a}\right)-sin\left({\theta }_{a}\right)\\ \left(\mathrm{cos}\left({\theta }_{c}\right)+{\varepsilon }_{xc}\mathrm{sin}\left({\theta }_{c}\right)\right))-{\varepsilon }_{yc}sin\left({\theta }_{a}\right)+cos\left({\theta }_{a}\right)\left(\mathrm{sin}\left({\theta }_{c}\right)+{\varepsilon }_{xc}\mathrm{cos}\left({\theta }_{c}\right)\right)+\\ sin\left({\theta }_{a}\right)\left(\mathrm{cos}\left({\theta }_{c}\right)+{\varepsilon }_{xc}\mathrm{sin}\left({\theta }_{c}\right)\right)\end{array}$$
(33)
$$\Delta x=\left\{\begin{array}{c}\Delta {x}_{\text{x}}=6.63{e}^{-8}{x}^{3}-3.31{e}^{-5}{x}^{2}+0.03x+1.11\\ \Delta {y}_{x}=-8.85{e}^{-10}{x}^{4}+9.53{e}^{-7}{x}^{3}-2.99{e}^{-4}{x}^{2}+0.02x-10.87\\ \Delta {z}_{x}=-2{e}^{-9}{x}^{4}+2.2{e}^{-6}{x}^{3}-5.9{e}^{-4}{x}^{2}-0.01{\text{x}}-9.76\\ \Delta {\alpha }_{\text{x}}=1.68{e}^{-10}{x}^{4}+1.67{e}^{-7}{x}^{3}+1.96{e}^{-5}{x}^{2}+0.01x-1.08\\ \Delta {\beta }_{x}=3.72{e}^{-10}{x}^{4}-4.31{e}^{-7}{x}^{3}+1.46{e}^{-4}{x}^{2}+0.01x+0.43\\ \Delta {\gamma }_{x}=5.24{e}^{-11}{x}^{4}-6.92{e}^{-9}{x}^{3}+7.46{e}^{-4}{x}^{2}-0.01x+2.4\end{array}\right\}$$
(34)
$$\Delta y=\left\{\begin{array}{c}\Delta {x}_{\text{y}}=-6.65{e}^{-10}{x}^{4}-1.01{e}^{-6}{x}^{3}-4.66{e}^{-4}{x}^{2}+0.05x-1.1\\ \Delta {y}_{y}=-4.88{e}^{-10}{x}^{4}+6.37{e}^{-7}{x}^{3}-2.72{e}^{-4}{x}^{2}+0.07x-0.45\\ \Delta {z}_{y}=3.72{e}^{-10}{x}^{4}-4{e}^{-7}{x}^{3}+8.85{e}^{-5}{x}^{2}+0.03{\text{x}}-21.6\\ \Delta {\alpha }_{y}=1.78{e}^{-10}{x}^{4}+1.22{e}^{-7}{x}^{3}+3.2{e}^{-5}{x}^{2}+0.1x-2.08\\ \Delta {\beta }_{y}=-7.97{e}^{-8}{x}^{3}+8.02{e}^{-5}{x}^{2}+0.02x+0.1\\ \Delta {\gamma }_{y}=9.25{e}^{-10}{x}^{4}-1.11{e}^{-6}{x}^{3}+4.09{e}^{-4}{x}^{2}-0.04x2.4\end{array}\right\}$$
(35)
$$\Delta z=\left\{\begin{array}{c}\Delta {x}_{z}=-3.17{e}^{-9}{x}^{4}-2.69{e}^{-6}{x}^{3}-7.04{e}^{-4}{x}^{2}+0.05x-0.32\\ \Delta {y}_{z}=-6.06{e}^{-10}{x}^{4}+3.29{e}^{-7}{x}^{3}-5.05{e}^{-7}{x}^{2}+0.003x-0.97\\ \Delta {z}_{z}=1.35{e}^{-9}{x}^{4}-1.06{e}^{-7}{x}^{3}+2.55{e}^{-5}{x}^{2}+0.06{\text{x}}-1.16\\ \Delta {\alpha }_{z}=7.45{e}^{-10}{x}^{4}-6.5{e}^{-7}{x}^{3}+1.61{e}^{-5}{x}^{2}+0.02x-0.02\\ \Delta {\beta }_{z}=-4.41{e}^{-8}{x}^{3}+1.21{e}^{-7}{x}^{3}-7.27{e}^{-5}{x}^{2}+0.01x-0.02\\ \Delta {\gamma }_{z}=-1.91{e}^{-9}{x}^{4}+1.93{e}^{-6}{x}^{3}-6.19{e}^{-4}{x}^{2}+0.05x+0.05\end{array}\right\}$$
(36)

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Zhou, T., Yinghua, L., Jie, J. et al. A method of sensitivity analysis and precision prediction for geometric errors of five-axis machine tools based on multi-body system theory. Int J Adv Manuf Technol 123, 3497–3512 (2022). https://doi.org/10.1007/s00170-022-10495-7

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