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Geometric error identification method for machine tools based on the spatial body diagonal error model

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Abstract

For the identification of the geometric error in the CNC machine tools, this paper proposes a geometric error identification method for machine tools based on the spatial body diagonal error model. A body diagonal error measurement method is proposed to obtain the spatial error, which avoids repeated installation errors. In order to address the pathological problem in geometric error identification, a two-step regularization method based on the spatial body diagonal error model is proposed. Firstly, the mapping relationship between the geometric error of the machine tools and the spatial body diagonal error, which is measured by a multibeam laser interferometer, is established. Secondly, a two-step regularization method is adopted to improve the identification accuracy and the stability. Finally, the body diagonal error measurement method is applied to a horizontal three-axis machining center, and the measurement results show that the method is feasible. Simulations and experiments are performed to compare the two-step regularized identification method with the least square identification method, and the results prove the good reliability and validity of the proposed method. Comparing the identification results with the measurement results, the absolute error is less than 4.8584 μm, and the relative error is less than 25.6062%. It shows that the spatial body diagonal error model proposed in this paper is comprehensive, and the geometric error identification method is systematic and universal.

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Abbreviations

x, y, z :

The displacement in x, y and z axes

\(\begin{aligned}&{^{W}\boldsymbol{T}_{Z}},\;{^{Z}\boldsymbol{T}_{{Z}_{0}}},\;{^{{Z}_{0}}\boldsymbol{T}_{O}},\;{^{O}\boldsymbol{T}_{{X}_{0}}},\\&{^{{X}_{0}}\boldsymbol{T}_{X}},\;{^{X}\boldsymbol{T}_{{Y}_{0}}},\;{^{{Y}_{0}}\boldsymbol{T}_{Y}},\;{^{Y}\boldsymbol{T}_{T}}\end{aligned}\)  :

The macroscopic displacement matrix between the components of the machine tool

\(\begin{aligned}&{^{W}y_{Z}},\;{^{{Z_{0}}}x_{O}},\;{^{O}y_{{X_{0}}}},\\&{^{O}z_{{X_{0}}}},\;{^{X}y_{{Y_{0}}}},\;{^{X}z_{{Y_{0}}}},\;{^{Y}z_{T}}\end{aligned}\)  :

The distance between the origin of the coordinate system for each component

\(\begin{aligned}&{^{W}\Delta_{Z}},\;{^{Z}\Delta_{{Z_{0}}}},\;{^{{Z_{0}}}\Delta_{O}},{^{O}\Delta_{{X_{0}}}},\\&{^{{X_{0}}}\Delta_{X}},\;{^{X}\Delta_{{Y_{0}}}},\;{^{{Y_{0}}}\Delta_{Y}},\;{^{Y}\Delta_{T}}\end{aligned}\)  :

The microscopic displacement matrix between the components of the machine tool

\(\delta_{i} (j),\;\varepsilon_{i} (j)\) :

The position-dependent geometric error of the machine tool, i, j = x, y, z

\(\alpha, \;\beta,\; \gamma\) :

The squareness error of the machine tool

\(\theta_{Y0},\;\theta_{X1},\;\theta_{Z2}\) :

The rotation angle during the transformation of the body diagonal coordinate system and the bed coordinate system

\(l_{x},\;l_{y},\;l_{z}\) :

The working stroke of x, y, and z axes

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

The coordinate value of the space point under the measurement coordinate system

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

The coordinate value of the space point under the bed coordinate system

P w :

The workpiece machining point coordinate vector in the worktable coordinate system

P t :

The tool tip point coordinate vector in the tool coordinate system

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

The workpiece machining point coordinate value in the worktable coordinate system

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

The tool tip point coordinate value in the tool coordinate system

\({\boldsymbol{T}}_{1}\) :

The coordinate transformation matrix between different body diagonal coordinate systems and the bed coordinate system

\(^{M}{\boldsymbol{P}}_{w},\;^{M}{\boldsymbol{P}}_{t}\) :

The coordinate vector of the workpiece machining point and the tool tip point under the measurement coordinate system

\(^{W}{\boldsymbol{T}}_{T},\;^{W}{\boldsymbol{T}}_{T}^{\prime }\) :

The coordinate transformation matrix from the worktable coordinate system to the tool coordinate system under the ideal and actual conditions

\(^{W}{\boldsymbol{E}}_{T}\) :

The error motion matrix between the worktable coordinate system and the tool coordinate system

\(\Delta {P}_{X},\;\Delta {P}_{Y},\;\Delta {P}_{Z}\) :

The position error in X, Y and Z directions, respectively, under the measurement coordinate system

\(\Delta {V}_{X},\;\Delta {V}_{Y},\;\Delta {V}_{Z}\) :

The angular error in X, Y and Z directions, respectively, under the measurement coordinate system

\(\Delta {P}_{x},\;\Delta {P}_{y},\;\Delta {P}_{z}\) :

The position error in x, y and z directions, respectively, under the bed coordinate system

\(\Delta {V}_{x},\;\Delta {V}_{y},\;\Delta {V}_{x}\) :

The angular error in x, y and z directions, respectively, under the bed coordinate system

T :

The spatial coordinate transformation matrix

\(\Delta {\boldsymbol{r}}\) :

The spatial error matrix of the machine tool

A :

The mapping matrix between the spatial error and the twenty-one geometric errors of the machine tool

ε :

The twenty-one geometric errors matrix of the machine tool

\(\delta_{ij,k},\;\boldsymbol\varepsilon_{ij,k}\) :

The k-th polynomial coefficient in the position error and the angular error \(i,j = x,y,z\); \(k = 1,2, \cdots ,n\)

\({\boldsymbol{p}}_{{\delta_{i} (j)}},\;{\boldsymbol{p}}_{{\varepsilon_{i} (j)}}\) :

The polynomial coefficient vector of the position error and the angular error, \(i,j = x,y,z\)

\({\boldsymbol{h}}_{{\delta_{i} (j)}}, \;{\boldsymbol{h}}_{{\varepsilon_{i} (j)}}\) :

The position matrix corresponding to the position error and angular error, \(i,j = x,y,z\)

H :

The mapping matrix consisting of the machine tool position coordinates

p :

The geometric error polynomial coefficient vector

M :

The mapping matrix between the spatial error and the geometric error polynomial coefficient

m :

The number of measurement points involved in identification

n :

The polynomial order

\(\hat{\boldsymbol{p}}\) :

The identification result of p

R :

The regularization matrix

\(\alpha\) :

The regularized parameter

TIKHONOV:

The regularization theory

\(MESM(\hat{\boldsymbol{p}})\)  :

The mean square error matrix of \(\hat{\boldsymbol{p}}\)

\(\hat{\sigma }_{0}^{2}\) :

The unit weight variance of \(\hat{\boldsymbol{p}}\)

\(\xi\) :

The deviation between \(\hat{\boldsymbol{p}}\) and \(\boldsymbol p\)  

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Funding

This research work is supported by the National Key R&D Program of China (No. 2018YFB1701201) and the National Natural Science Foundation of China (No. 51675378).

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Authors and Affiliations

  1. Key Laboratory of Mechanism Theory and Equipment Design, Ministry of Education, Tianjin University, Tianjin, 300354, China

    Sitong Wang, Gaiyun He, Dawei Zhang, Yichen Yan & Ran Xie

  2. School of Marine Science and Technology, Tianjin University, Tianjin, 300354, China

    Wenjie Tian

  3. School of Engineering, University of Warwick, Coventry, CV4 7DL, UK

    Yumeng Song

Authors
  1. Sitong Wang
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  7. Ran Xie
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Contributions

Sitong Wang: Conceptualization, methodology, software, investigation, writing—original draft. Gaiyun He: Conceptualization, supervision, funding acquisition, resources. Wenjie Tian: Data curation, resources, visualization. Dawei Zhang: Validation, formal analysis, project administration. Yumeng Song: Writing, review and editing; visualization. Yichen Yan: Data curation, writing review. Ran Xie: Experiment, writing—review and editing.

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Correspondence to Gaiyun He.

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This article does not contain any studies with human participants or animals performed by any of the authors. In this experiment, we did not collect any samples of human and animals.

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Wang, S., He, G., Tian, W. et al. Geometric error identification method for machine tools based on the spatial body diagonal error model. Int J Adv Manuf Technol 121, 7997–8017 (2022). https://doi.org/10.1007/s00170-022-09633-y

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