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Closed-loop mode geometric error compensation of five-axis machine tools based on the correction of axes movements

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Abstract

The evaluation of the results of geometric error compensation is critical, and it can serve as the feedback of the compensation to help improve the precision of the compensation. In this paper, the closed-loop mode geometric error compensation of five-axis machine tools is presented by correcting the movements of all axes. At first, general geometric error modeling of the machine tool is proposed based on POE theory. The initial positions of linear axes and rotary axes relative to the machine tool are considered and the error twist of each axis containing position-independent errors is established. Second, the closed-loop mode is developed by analyzing the open-loop mode of geometric error compensation. The precision inverse feedback module is formed by introducing the designed tool poses. The output of the feedback is the integrated errors of the revised movements of axes relative to the ideal tool poses of nominal movements of axes. Third, the adaptive correction of movements of axes is proposed based on CSO (chicken swarm optimization). The initializing of the swarm, the Jacobian based moving of roosters, and the moving of chicks containing one mutation are developed. The fitness of the swarm is calculated using the integrated errors relative to the compensation goal in the precision inverse feedback. Finally, simulations by comparing with open-loop mode error compensation and real cutting experiments are proposed on one SmartCNC500_DRTD five-axis machine center to verify the effectiveness of the closed-loop mode geometric error compensation.

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Funding

This work was financially supported by the National Natural Science Foundation of China (No. 51805457), Sichuan Science and Technology Program (2019YJ0249), China Postdoctoral Science Foundation (2020M673211), Key Science and Technology Project of Sichuan Province (2020ZDZX0003), the National Natural Science Foundation of China (No. 51775452), and the Fundamental Research Funds for the Central Universities (2682019CX30).

Author information

Authors and Affiliations

  1. Engineering Research Center of Advanced Driving Energy-saving Technology, Ministry of Education, Southwest Jiaotong University, Chengdu, 610031, China

    Guoqiang Fu, Jinghao Shi, Yunpeng Xie & Hongli Gao

  2. School of Mechanical Engineering, Sichuan University, Chengdu, 610065, China

    Guoqiang Fu

  3. Department of Electromechanical Measuring and Controlling, School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Guoqiang Fu, Jinghao Shi, Yunpeng Xie & Hongli Gao

  4. Key Laboratory of Air-driven Equipment Technology of Zhejiang Province, Quzhou University, Quzhou, 324000, China

    Xiaolei Deng

Authors
  1. Guoqiang Fu
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  2. Jinghao Shi
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  3. Yunpeng Xie
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  4. Hongli Gao
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  5. Xiaolei Deng
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Correspondence to Guoqiang Fu.

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Appendices

POE theory

The twist and the twist coordinates are represented as:

$$ \hat{\upxi}=\left[\begin{array}{cc}\hat{\upomega}& v\\ {}0& 1\end{array}\right];\kern0.5em \upxi ={\left[{\upomega}^{\mathrm{T}},{v}^{\mathrm{T}}\right]}^{\mathrm{T}}={\left[{\upomega}_1,{\upomega}_2,{\upomega}_3,{v}_1,{v}_2,{v}_3\right]}^{\mathrm{T}} $$

where v = [v1, v2, v3]T represents translational velocity,\( \hat{\upomega} \) is a skew-symmetric matrix and vector ω = [ω1, ω2, ω3]T represents the rotational velocity. The exponential matrix of the twist is represented as [9]

$$ g={e}^{\uptheta \hat{\upxi}}=\left\{\begin{array}{c}\left[\begin{array}{cc}{\mathrm{I}}_{3\times 3}& \overrightarrow{v}\uptheta \\ {}0& 1\end{array}\right]\kern1.00em \begin{array}{cc}& \end{array}\kern0.5em if\left\Vert \upomega \right\Vert =0\\ {}\left[\begin{array}{cc}{e}^{\hat{\upomega}\uptheta}& \left(I-{e}^{\hat{\upomega}\uptheta}\right)\left(\overrightarrow{\upomega}\times \overrightarrow{v}\right)+{\overrightarrow{\upomega}\overrightarrow{\upomega}}^{\mathrm{T}}\overrightarrow{v}\\ {}0& 1\end{array}\right]\begin{array}{cc}\begin{array}{cc}& \end{array}& if\left\Vert \upomega \right\Vert \ne 0\end{array}\end{array}\right. $$

When ‖ω‖ ≠ 0,\( \uptheta =\sqrt{\upomega_1^2+{\upomega}_2^2+{\upomega}_3^2} \) represents the rotation angle. And when ‖ω‖ = 0,\( \uptheta =\sqrt{v_1^2+{v}_2^2+{v}_3^2} \), represents the translational distance. \( {e}^{\hat{\upomega}\uptheta} \) can be obtained by the means of triangular progression as

$$ {e}^{\hat{\upomega}\uptheta}=I+\frac{\hat{\upomega}}{\left\Vert \upomega \right\Vert}\sin \uptheta +\frac{{\hat{\upomega}}^2}{{\left\Vert \upomega \right\Vert}^2}\left(1-\cos \uptheta \right) $$

The POE formula of the forward kinematics of an open chain robot is as [9]:

figure a

The twist and the exponential matrix of position-dependent errors

The twist of position-dependent errors can be expressed as the unit twist and the motion angle as

$$ {\displaystyle \begin{array}{c}{\xi}_{ke}={\left[{\varepsilon}_{xk},{\varepsilon}_{yk},{\varepsilon}_{zk},{\delta}_{xk},{\delta}_{yk},{\delta}_{zk}\right]}^{\mathrm{T}}=\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}\cdotp {\xi}_{\mathrm{uke}}\\ {}=\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{\mathrm{zk}}^2}\cdotp {\left[\frac{\varepsilon_{xk}}{\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}},\frac{\varepsilon_{yk}}{\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}},\frac{\varepsilon_{zk}}{\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}},\frac{\delta_{xk}}{\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}},\frac{\delta_{yk}}{\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}},\frac{\delta_{zk}}{\sqrt{\varepsilon_{xk}^2+{\varepsilon}_{yk}^2+{\varepsilon}_{zk}^2}}\right]}^{\mathrm{T}}\ \end{array}} $$

Then, the corresponding exponential matrix can be calculated based on POE theory as

$$ {\displaystyle \begin{array}{c}{e}^{{\hat{\upxi}}_{\mathrm{ke}}}={e}^{\sqrt{\upvarepsilon_{\mathrm{x}\mathrm{k}}^2+{\upvarepsilon}_{\mathrm{y}\mathrm{k}}^2+{\upvarepsilon}_{\mathrm{z}\mathrm{k}}^2}\times {\hat{\upxi}}_{\mathrm{uke}}}\\ {}=\left[\begin{array}{cccc}1+\frac{\left(-1+\cos m\right)\left({\varepsilon}_{\mathrm{y}\mathrm{k}}^2+{\upvarepsilon}_{\mathrm{z}\mathrm{k}}^2\right)}{m^2}& -\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{x}\mathrm{k}}{\upvarepsilon}_{\mathrm{y}\mathrm{k}}}{m^2}-\frac{\varepsilon_{\mathrm{z}\mathrm{k}}\sin m}{m}& \frac{\varepsilon_{\mathrm{y}\mathrm{k}}\sin m}{m}-\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{x}\mathrm{k}}{\upvarepsilon}_{\mathrm{z}\mathrm{k}}}{m^2}& {l}_{\mathrm{x}}\\ {}-\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{x}\mathrm{k}}{\upvarepsilon}_{\mathrm{y}\mathrm{k}}}{m^2}+\frac{\varepsilon_{\mathrm{z}\mathrm{k}}\sin m}{m}& 1+\frac{\left(-1+\cos m\right)\left({\varepsilon}_{\mathrm{x}\mathrm{k}}^2+{\upvarepsilon}_{\mathrm{z}\mathrm{k}}^2\right)}{m^2}& -\frac{\varepsilon_{\mathrm{x}\mathrm{k}}\sin m}{m}-\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{y}\mathrm{k}}{\upvarepsilon}_{\mathrm{z}\mathrm{k}}}{m^2}& {l}_{\mathrm{y}}\\ {}-\frac{\varepsilon_{\mathrm{y}\mathrm{k}}\sin m}{m}-\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{x}\mathrm{k}}{\upvarepsilon}_{\mathrm{z}\mathrm{k}}}{m^2}& \frac{\varepsilon_{\mathrm{x}\mathrm{k}}\sin m}{m}-\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{y}\mathrm{k}}{\upvarepsilon}_{\mathrm{z}\mathrm{k}}}{m^2}& 1+\frac{\left(-1+\cos m\right)\left({\varepsilon}_{\mathrm{x}\mathrm{k}}^2+{\upvarepsilon}_{\mathrm{y}\mathrm{k}}^2\right)}{m^2}&\;{l}_{\mathrm{z}}\\ {}0& 0& 0& 1\end{array}\right]\end{array}} $$

Where m, lx, ly, and lz are shown as

$$ {\displaystyle \begin{array}{c}m=\sqrt{\varepsilon_{\mathrm{xk}}^2+{\upvarepsilon}_{\mathrm{yk}}^2+{\upvarepsilon}_{\mathrm{zk}}^2}\\ {}{l}_x=\frac{1}{m^2}{\varepsilon}_{\mathrm{xk}}\left({\updelta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{xk}}+{\updelta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{yk}}+{\updelta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{zk}}\right)+\frac{1}{m^2}\left(-{\delta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{xk}}+{\updelta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{zk}}\right)\left(\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{yk}}}{m^2}+\frac{\varepsilon_{\mathrm{zk}}\sin m}{m}\right)\\ {}\kern0.5em +\frac{1}{m^2}\left({\updelta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{xk}}-{\updelta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{yk}}\right)\left(-\frac{\varepsilon_{\mathrm{yk}}\sin m}{m}+\frac{\left(-1+\cos m\right){\upvarepsilon}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{zk}}}{m^2}\right)-\frac{\left(-1+\cos m\right)\left({\delta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{yk}}-{\updelta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{zk}}\right)\left({\upvarepsilon}_{\mathrm{yk}}^2+{\upvarepsilon}_{\mathrm{zk}}^2\right)}{m^4}\\ {}{l}_y=\frac{1}{m^2}{\varepsilon}_{\mathrm{yk}}\left({\updelta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{xk}}+{\updelta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{yk}}+{\updelta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{zk}}\right)+\frac{1}{m^2}\left({\delta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{yk}}-{\delta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{zk}}\right)\left(\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{yk}}}{m^2}-\frac{\varepsilon_{\mathrm{zk}}\mathrm{s} inm}{m}\right)\\ {}\begin{array}{cc}& \end{array}+\frac{1}{m^2}\left({\updelta}_{\mathrm{yx}}{\upvarepsilon}_{\mathrm{xx}}-{\updelta}_{\mathrm{xx}}{\upvarepsilon}_{\mathrm{yx}}\right)\left(\frac{\upvarepsilon_{\mathrm{xk}}\sin m}{m}+\frac{\left(-1+\cos m\right){\upvarepsilon}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{zk}}}{m^2}\right)+\frac{\left(-1+\cos m\right)\left({\delta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{xk}}-{\delta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{zk}}\right)\left({\upvarepsilon}_{\mathrm{xk}}^2+{\upvarepsilon}_{\mathrm{zk}}^2\right)}{m^4}\\ {}{l}_z=\frac{1}{m^2}{\varepsilon}_{\mathrm{zk}}\left({\updelta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{xk}}+{\updelta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{yk}}+{\updelta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{zk}}\right)+\frac{1}{m^2}\left({\delta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{yk}}-{\delta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{zk}}\right)\left(\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{zk}}}{m^2}+\frac{\varepsilon_{\mathrm{yk}}\sin m}{m}\right)\\ {}\begin{array}{cc}& \end{array}+\frac{1}{m^2}\left(-{\delta}_{\mathrm{zk}}{\upvarepsilon}_{\mathrm{xk}}+{\updelta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{zk}}\right)\left(-\frac{\varepsilon_{\mathrm{xk}}\sin m}{m}+\frac{\left(-1+\cos m\right){\varepsilon}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{zk}}}{m^2}\right)-\frac{\left(-1+\cos m\right)\left({\delta}_{\mathrm{yk}}{\upvarepsilon}_{\mathrm{xk}}-{\delta}_{\mathrm{xk}}{\upvarepsilon}_{\mathrm{yk}}\right)\left({\upvarepsilon}_{\mathrm{xk}}^2+{\upvarepsilon}_{\mathrm{yk}}^2\right)}{m^4}\end{array}} $$

Due to the small angle of angular errors, sin m = m and cos m = 1 according to the small-angle approximation. Then, the exponential matrix of the error twist can be expressed as

$$ {e}^{{\hat{\upxi}}_{\mathrm{ke}}}=\left[\begin{array}{cccc}1& -{\varepsilon}_{\mathrm{zk}}& {\varepsilon}_{\mathrm{yk}}& \frac{\delta_{\mathrm{xk}}\left({\upvarepsilon}_{\mathrm{xk}}^2+{\upvarepsilon}_{\mathrm{yk}}^2+{\upvarepsilon}_{\mathrm{zk}}^2\right)}{m^2}\\ {}{\varepsilon}_{\mathrm{zk}}& 1& -{\varepsilon}_{\mathrm{xk}}& \frac{\delta_{\mathrm{yk}}\left({\upvarepsilon}_{\mathrm{xk}}^2+{\upvarepsilon}_{\mathrm{yk}}^2+{\upvarepsilon}_{\mathrm{zk}}^2\right)}{m^2}\\ {}-{\varepsilon}_{\mathrm{yk}}& {\varepsilon}_{\mathrm{xk}}& 1&\;\frac{\delta_{\mathrm{zk}}\left({\upvarepsilon}_{\mathrm{xk}}^2+{\upvarepsilon}_{\mathrm{yk}}^2+{\upvarepsilon}_{\mathrm{zk}}^2\right)}{m^2}\\ {}0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cccc}1& -{\varepsilon}_{\mathrm{zk}}& {\varepsilon}_{\mathrm{yk}}& {\delta}_{\mathrm{xk}}\\ {}{\varepsilon}_{\mathrm{zk}}& 1& -{\varepsilon}_{\mathrm{xk}}& {\delta}_{\mathrm{yk}}\\ {}-{\varepsilon}_{\mathrm{yk}}& {\varepsilon}_{\mathrm{xk}}& 1&\;{\delta}_{\mathrm{zk}}\\ {}0& 0& 0& 1\end{array}\right] $$

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Fu, G., Shi, J., Xie, Y. et al. Closed-loop mode geometric error compensation of five-axis machine tools based on the correction of axes movements. Int J Adv Manuf Technol 110, 365–382 (2020). https://doi.org/10.1007/s00170-020-05793-x

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