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Precision allocation method of large-scale CNC hobbing machine based on precision-cost comprehensive optimization

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Abstract

In modern machine tool design, precision is an important index to characterize machine tool performance and precision allocation has become a key task. Since the middle of the twentieth century, precision allocation methods using optimization technology to balance manufacturing cost and precision level have gradually developed, but most methods mainly take the cost minimization as the goal to optimize the precision allocation. As the precision and manufacturing costs are a pair of factors to be comprehensively considered, balance between them is needed to meet different design requirements. This paper proposes a comprehensive optimization method to trade-off between precision and cost. A multi-object precision allocation optimization model aiming at minimizing fuzzy manufacturing cost and comprehensive precision of machine tool is constructed. A multi-object optimization algorithm to solve the model is designed, combining the multi-objective gray wolf optimization algorithm with multi-objective decision analysis method Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). A case study based on a large-scale hobbing machine shows that the comprehensive optimization of manufacturing cost and machining precision is realized by using the proposed multi-object precision allocation optimization method.

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Abbreviations

\({S}_{x},{S}_{y},{S}_{z}\) :

Displacements of linear motion axes X, Y, and Z

\({\theta }_{A},{\theta }_{C},{\theta }_{M}\) :

Rotation angles of rotating motion axes A, C, and M

\({}_{X}\delta {}_{x}\left({S}_{x}\right),{}_{X}\delta {}_{y}\left({S}_{x}\right),{}_{X}\delta {}_{z}\left({S}_{x}\right)\) :

Position errors of axis X at \({S}_{x}\)

\({}_{X}\varepsilon {}_{x}\left({S}_{x}\right),{}_{X}\varepsilon {}_{y}\left({S}_{x}\right),{}_{X}\varepsilon {}_{z}\left({S}_{x}\right)\) :

Angle errors of axis X at \({S}_{x}\)

\({}_{Y}\delta {}_{x}\left({S}_{y}\right),{}_{Y}\delta {}_{y}\left({S}_{y}\right),{}_{Y}\delta {}_{z}\left({S}_{y}\right)\) :

Position errors of axis Y at \({S}_{y}\)

\({}_{Y}\varepsilon {}_{x}\left({S}_{y}\right),{}_{Y}\varepsilon {}_{y}\left({S}_{y}\right),{}_{Y}\varepsilon {}_{z}\left({S}_{y}\right)\) :

Angle errors of axis Y at \({S}_{y}\)

\({}_{Z}\delta {}_{x}\left({S}_{z}\right),{}_{Z}\delta {}_{y}\left({S}_{z}\right),{}_{Z}\delta {}_{z}\left({S}_{z}\right)\) :

Position errors of axis Z at \({S}_{z}\)

\({}_{Z}\varepsilon {}_{x}\left({S}_{z}\right),{}_{Z}\varepsilon {}_{y}\left({S}_{z}\right),{}_{Z}\varepsilon {}_{z}\left({S}_{z}\right)\) :

Angle errors of axis Z at \({S}_{z}\)

\({}_{A}\delta {}_{x}\left({\theta }_{A}\right),{}_{A}\delta {}_{y}\left({\theta }_{A}\right),{}_{A}\delta {}_{z}\left({\theta }_{A}\right)\) :

Position errors of axis A at \({\theta }_{A}\)

\({}_{A}\varepsilon {}_{x}\left({\theta }_{A}\right),{}_{A}\varepsilon {}_{y}\left({\theta }_{A}\right),{}_{A}\varepsilon {}_{z}\left({\theta }_{A}\right)\) :

Angle errors of axis A at \({\theta }_{A}\)

\({}_{C}\delta {}_{x}\left({\theta }_{C}\right),{}_{C}\delta {}_{y}\left({\theta }_{C}\right),{}_{C}\delta {}_{z}\left({\theta }_{C}\right)\) :

Position errors of axis C at \({\theta }_{C}\)

\({}_{C}\varepsilon {}_{x}\left({\theta }_{C}\right),{}_{C}\varepsilon {}_{y}\left({\theta }_{C}\right),{}_{C}\varepsilon {}_{z}\left({\theta }_{C}\right)\) :

Angle errors of axis C at \({\theta }_{C}\)

\({}_{M}\delta {}_{x}\left({\theta }_{M}\right),{}_{M}\delta {}_{y}\left({\theta }_{M}\right),{}_{M}\delta {}_{z}\left({\theta }_{M}\right)\) :

Position errors of axis M at \({\theta }_{M}\)

\({}_{M}\varepsilon {}_{x}\left({\theta }_{M}\right),{}_{M}\varepsilon {}_{y}\left({\theta }_{M}\right),{}_{M}\varepsilon {}_{z}\left({\theta }_{M}\right)\) :

Angle errors of axis M at \({\theta }_{M}\)

\({}_{X}{}^{C}\delta {}_{x},{}_{X}{}^{C}\delta {}_{y},{}_{X}{}^{C}\delta {}_{z}\) :

Position errors between X and C axes

\({}_{X}{}^{C}\varepsilon {}_{x},{}_{X}{}^{C}\varepsilon {}_{y},{}_{X}{}^{C}\varepsilon {}_{z}\) :

Angle errors between X and C axes

\({}_{Z}{}^{X}\delta {}_{x},{}_{Z}{}^{X}\delta {}_{y},{}_{Z}{}^{X}\delta {}_{z}\) :

Position errors between Z and X axes

\({}_{Z}{}^{X}\varepsilon {}_{x},{}_{Z}{}^{X}\varepsilon {}_{y},{}_{Z}{}^{X}\varepsilon {}_{z}\) :

Angle errors between Z and X axes

\({}_{A}{}^{Z}\delta {}_{x},{}_{A}{}^{Z}\delta {}_{y},{}_{A}{}^{Z}\delta {}_{z}\) :

Position errors between A and Z axes

\({}_{A}{}^{Z}\varepsilon {}_{x},{}_{A}{}^{Z}\varepsilon {}_{y},{}_{A}{}^{Z}\varepsilon {}_{z}\) :

Angle errors between A and Z axes

\({}_{Y}{}^{A}\delta {}_{x},{}_{Y}{}^{A}\delta {}_{y},{}_{Y}{}^{A}\delta {}_{z}\) :

Position errors between Y and A axes

\({}_{Y}{}^{A}\varepsilon {}_{x},{}_{Y}{}^{A}\varepsilon {}_{y},{}_{Y}{}^{A}\varepsilon {}_{z}\) :

Angle errors between Y and A axes

\({}_{M}{}^{Y}\delta {}_{x},{}_{M}{}^{Y}\delta {}_{y},{}_{M}{}^{Y}\delta {}_{z}\) :

Position errors between M and Y axes

\({}_{M}{}^{Y}\varepsilon {}_{x},{}_{M}{}^{Y}\varepsilon {}_{y},{}_{M}{}^{Y}\varepsilon {}_{z}\) :

Angle errors between M and Y axes

\({M}_{\mathrm{1,2}},{M}_{\mathrm{2,3}},{M}_{\mathrm{3,4}},{M}_{\mathrm{4,5}},{M}_{\mathrm{5,6}},{M}_{\mathrm{6,7}}\) :

Motion transformation matrices

\({E}_{\mathrm{1,2}}^{m},{E}_{\mathrm{2,3}}^{m},{E}_{\mathrm{3,4}}^{m},{E}_{\mathrm{4,5}}^{m},{E}_{\mathrm{5,6}}^{m},{E}_{\mathrm{6,7}}^{m}\) :

Motion axis error transformation matrices

\({E}_{C,X}^{P},{E}_{X,Z}^{P},{E}_{Z,A}^{P},{E}_{A,Y}^{P},{E}_{Y,M}^{P}\) :

Inter axis error transformation matrices

\({M}_{\mathrm{1,7}}\) :

Ideal motion transformation matrix of hobbing

\({M}_{\mathrm{1,7}}^{e}\) :

Motion transformation matrix of hobbing considering errors

\(E\) :

Total error matrix of hobbing machine

\(\delta x,\delta y,\delta z,\varepsilon x,\varepsilon y,\varepsilon z\) :

Total errors of hobbing machine

\({\sigma }_{\delta x},{\sigma }_{\delta y},{\sigma }_{\delta z},{\sigma }_{\varepsilon x},{\sigma }_{\varepsilon y},{\sigma }_{\varepsilon z}\) :

Error distribution variances

\({I}_{\delta x},{I}_{\delta y},{I}_{\delta z},{I}_{\varepsilon x},{I}_{\varepsilon y},{I}_{\varepsilon z}\) :

Precisions at \(3\sigma\)

\({F}_{X},{F}_{Y},{F}_{Z},{F}_{A},{F}_{C},{F}_{M},F{A}_{ij}\) :

Fuzzy manufacturing costs of moving axes

\(a,b,c,d,{m}_{ij}\) :

Precision-cost function coefficients

\(F\left(Error\right)\) :

Fuzzy cost optimization objective

\(Ip\left(Error\right),Ia\left(Error\right)\) :

Precision optimization objectives

\({W}_{Ip},{W}_{Ia},{W}_{F}\) :

Optimize decision weights

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Funding

This work was supported by the Key Project of National Natural Science Foundation of China (Grant No. 51635003).

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  1. The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400030, China

    Zongyan Hu, Shilong Wang & Chi Ma

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  1. Zongyan Hu
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Contributions

Zongyan Hu contributed to the conception of the study and wrote the manuscript. Shilong Wang contributed to the conception of the study, funding acquisition, and supervision. Chi Ma contributed to analysis and manuscript review.

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Correspondence to Shilong Wang.

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Hu, Z., Wang, S. & Ma, C. Precision allocation method of large-scale CNC hobbing machine based on precision-cost comprehensive optimization. Int J Adv Manuf Technol 126, 3453–3474 (2023). https://doi.org/10.1007/s00170-023-11303-6

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