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Use of a Virtual Polyhedron for Interim Checking of the Volumetric and Geometric Errors of Machine Tools

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Abstract

Volumetric and geometric errors should be periodically checked to ensure that the accuracy of machine tools remains within the tolerable range. However, existing methods require complex devices, and are thus unsuitable for cost-effective interim error checks. We present a simple, rapid and cost-effective method for interim error checks. The measurement paths are constructed using a virtual polyhedron; volumetric errors are checked by calculating the coordinates of the vertices using the measured side lengths. The tool is sequentially moved to each vertex, and the side lengths are measured using a double ball-bar. As the virtual polyhedron is composed of virtual regular tetrahedrons, the relationships between the coordinates of the vertices and side lengths are unique. Linear scale and squareness errors are measured using an error synthesis model with a least-squares approach. The method was applied to a real machine tool, and performance was verified by confirming that the maximum L2 norm of volumetric error is improved from 57.6 to 32.8 μm after compensating for the measured geometric errors. Thus, the validity of the proposed method was confirmed by an improvement of 43% in volumetric error. The measurement results were confirmed by the circular tests of ISO 230-4; the peak-to-valley radial deviation improved from 16.0 to 11.2 μm after compensation, and the proposed method contributed to a 30% improvement in the radial deviation.

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Abbreviations

m :

Number of vertices of the virtual polyhedron

c i :

Linear scale error of the linear axis i (i = X, Y, Z) (μm/mm)

s ij :

Squareness error of linear axis j around axis i (i, j = X, Y, Z) (rad)

L :

Nominal side length of the virtual polyhedron (mm)

δL i,j :

Measured deviations of the side lengths between the i-th and j-th vertices of the virtual polyhedron (i, j = 1, …, m) (mm)

P i,n (x i,n, y i,n, z i,n):

Nominal coordinates of the i-th vertex of the virtual polyhedron (i = 1, …, m) (mm)

P i,c (x i,c, y i,c, z i,c):

Coordinates of the i-th vertex of the virtual polyhedron in {P} (i = 1, …, m) (mm)

P i,a (x i,a, y i,a, z i,a):

Coordinates of the i-th vertex of the virtual polyhedron in {R} (i = 1, …, m) (mm)

δP i,c (δx i,c, δy i,c, δz i,c):

Positional errors between Pi,c and Pi,n (i = 1, …, m) (mm)

δP i,a (δx i,a, δy i,a, δz i,a):

Volumetric errors at the i-th vertex (i = 1, …, m) (mm)

{i}:

Coordinate system of axis i, (i = X, Y, Z)

{P}, {R} :

Virtual polyhedron and reference coordinate systems, respectively

\({{\varvec{\uptau}}}_{i}^{j}\) :

4 × 4 homogeneous transformation matrix from the j to i coordinate system

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Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. 2020R1C1C100330011 and 2019R1A2C2088683).

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. 2020R1C1C100330011 and 2019R1A2C2088683).

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Authors and Affiliations

  1. School of Mechanical and Automotive Engineering, Kyungil University, 50, Gamasil-gil, Hayang-eup, Gyeongsan-si, Gyeongbuk, 38428, Republic of Korea

    Kwang-Il Lee

  2. School of Mechanical Engineering, Kyungpook National University, 80, Daehak-ro, Buk-gu, Daegu, 41566, Republic of Korea

    Heung-Ki Jeon & Seung-Han Yang

  3. Division of Mechanical Engineering Technology, Yeungnam University College, 170, Hyeonchung-ro, Nam-gu, Daegu, 42415, Republic of Korea

    Jae-Chang Lee

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  1. Kwang-Il Lee
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Correspondence to Seung-Han Yang.

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Lee, KI., Jeon, HK., Lee, JC. et al. Use of a Virtual Polyhedron for Interim Checking of the Volumetric and Geometric Errors of Machine Tools. Int. J. Precis. Eng. Manuf. 23, 1133–1141 (2022). https://doi.org/10.1007/s12541-022-00666-7

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