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An Optimal Method for Improving Volumetric Error Compensation in Machine Tools Based on Squareness Error Identification

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Abstract

This paper proposes a new method to identify the squareness errors of machine tools and gives an optimal method to improve the volumetric compensation accuracy based on the identified squareness errors. The volumetric error model based on screw theory is proposed and the effects of the squareness errors on the volumetric errors have been analyzed to clarify the possibility of improving volumetric error compensation by proper and accurate squareness errors. Then the identification method of the squareness errors is deduced in theory considering a 3-axis horizontal machine tool. With the identified squareness errors, an optimal method to improve volumetric error compensation in machine tools has been proposed. Experiments have verified that the 3 identified squareness errors are in accordance with the measured squareness errors within the accuracy range. Moreover, compared with the traditional measurement of squareness errors not considering the PDGEs, the proposed method is more effective and shows error compensation improvement in the volumetric error compensation process.

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Abbreviations

\( x,y,z \) :

Motion commands of X-, Y- and Z-axis

\( \hat{\xi }_{i} \) :

Motion twist for the i-axis \( (i = {\text{X,}}\,{\text{Y,}}\,{\text{Z)}} \)

\( \delta_{ij} \) :

Translational error of the j-axis in the i direction \( (i,j = {\text{X,}}\,{\text{Y,}}\,{\text{Z)}} \)

\( \varepsilon_{ij} \) :

Rotational error of the j-axis in the i direction \( (i,j = {\text{X,}}\,{\text{Y,}}\,{\text{Z)}} \)

\( S_{ij} \) :

Squareness error of the j-axis relative to the i-axis \( (i = X,\;j = {\text{Y,Z}}\;{\text{or}}\;i = Y,\;j = {\text{Z)}} \)

\( g_{bj}^{k} (0) \) :

HTM of tool \( (j = {\text{t)}} \) or worktable \( (j = {\text{w)}} \) coordinate frame relative to base coordinate frame in the reference configuration in actual \( (k = {\text{a)}} \) or ideal \( (k = {\text{i)}} \) condition

\( g_{ij}^{k} \) :

HTM of tool \( (j = {\text{t)}} \) or worktable \( (j = {\text{w)}} \) coordinate frame relative to base \( (i = {\text{b)}} \) or worktable \( (i = {\text{w)}} \) coordinate frame in actual \( (k = {\text{a)}} \) or ideal \( (k = {\text{i)}} \) condition

\( \Delta i \) :

Geometric errors of the i-aixs \( (i = {\text{X,}}\,{\text{Y,}}\,{\text{Z)}} \)

\( p_{0} \) :

The homogeneous coordinate of the point on the tool relative to the tool frame \( p_{0} = (0,\;0,\;0,\;1)^{T} \)

\( (x_{t0} ,\;y_{t0} ,\;z_{t0} ) \) :

The coordinate of the initial investigated point on the tool relative to the tool coordinate frame

\( (x_{i0} ,\;y_{i0} ,\;z_{i0} ) \) :

The coordinate of initial rotation center point of error screw motion of X-, Y- and Z-axis \( (i = 1 ,\; 2 ,\; 3 ) \)

\( S_{Vji} \) :

Global sensitivity coefficient of the error parameter in x, y and z direction in the global workspace, i is the number of geometric error parameter shown in Table 1\( (j = {\text{X,}}\;{\text{Y,}}\;{\text{Z,}}\;\;{\text{i}} = 1 , 2 ,\ldots , 2 1 ) \)

\( e_{i} \) :

Geometric error parameter with the symbol i representing the number of geometric error parameters as shown in Table 1

\( (e_{x1di} ,\;e_{y1di} ,\;e_{z1di} ) \) :

The volumetric error of the first body diagonal line at point i \( (i = 1,2, \ldots ,n ) \)

\( (x_{ul1} ,\;y_{ul1} ,\;z_{ul1} ) \) :

The unit vector of the moving direction of the first body diagonal line

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Acknowledgements

This research was supported by the National Natural Science Fund under Grant Nos. 51775210 and 51775212, and the National Science and Technology Major Project of China under Grant No. 2015ZX04000016.

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Authors and Affiliations

  1. National NC System Engineering Research Center, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Xuemin Zhong, Hongqi Liu, Xinyong Mao & Bin Li

  2. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Bin Li

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  1. Xuemin Zhong
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Correspondence to Hongqi Liu.

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Zhong, X., Liu, H., Mao, X. et al. An Optimal Method for Improving Volumetric Error Compensation in Machine Tools Based on Squareness Error Identification. Int. J. Precis. Eng. Manuf. 20, 1653–1665 (2019). https://doi.org/10.1007/s12541-019-00191-0

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