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An approach to enhancing machining accuracy of five-axis machine tools based on a new sensitivity analysis method

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Abstract

Identification of key geometric errors is an essential prerequisite for improving the machining accuracy of five-axis machine tools. This paper presents a new sensitivity analysis (SA) method to extract key geometric errors, and then to improve the machining performance of machine tools by compensating key geometric error components. Development of geometric error prediction model is involved to obtain geometric error values at arbitrary positions at first. Based on the multi-body system theory and flank milling theory, the machining error model is developed, which considers 37 geometric errors. Then, a new SA method is introduced by taking the machining error model as sensitivity analysis model and taking the geometric errors as analytical factors. Meanwhile, a sensitivity index, which has the characteristics of simple expression and clear physical meaning, is proposed, i.e., the peak value of the machining error caused by each geometric error. Moreover, the simulations analysis is carried out to obtain the sensitivity coefficient of each geometric error and the key error components. Finally, the validity and correctness of the proposed method are demonstrated by the experiments. Furthermore, the SA method can be extended to multi-axis machine tools.

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Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

Code availability

All code of this study is available from the corresponding author on reasonable request.

Abbreviations

SA:

Sensitivity analysis

GSA:

Global sensitivity analysis

PIGEs:

Position-independent geometric errors

PDGEs:

Position-dependent geometric errors

MBS:

Multi-body system

W:

Workpiece

T:

Tool

CMM:

Coordinate measuring machine

S i :

Sensitivity

SN i :

The sensitivity coefficient

G :

The geometric error vector

D :

Position vector of each moving axis

l :

Length of the cutting tool

E :

The volumetric error model

\({{\varvec{e}}}_{i}\) :

Machining error caused by ith geometric error

d :

The tool radius

\({-{\varvec{l}}}_{k}\) :

The tool length corresponding to the kth cutting point

\({}_{j}{}^{i}{T}_{p}\) :

The ideal static transformation matrices between the adjacent bodies of machine tool

\({}_{j}{}^{i}{T}_{pe}\) :

The actual static transformation matrices between the adjacent bodies of machine tool

\({}_{j}{}^{i}{T}_{s}\) :

The ideal motion transformation matrices between the adjacent bodies of machine tool

\({}_{j}{}^{i}{T}_{se}\) :

The actual motion transformation matrices between the adjacent bodies of machine tool

\({E}_{v}\left(v=x,y,z\right)\) :

The component of volumetric error in the v-direction

\({e}_{v}\left(v=x,y,z\right)\) :

The component of machining error in the v-direction

\({{\varvec{r}}}_{wjk}^{i}\) :

Ideal position vector of the tool center point corresponding to the kth cutting point in the workpiece coordinate system

\({{\varvec{r}}}_{tjk}\) :

Position vector of the tool center point corresponding to the kth cutting point in the tool coordinate system

\({{\varvec{n}}}_{sjk}\) :

Unit normal vector of the workpiece at the kth cutting point

\({{\varvec{r}}}_{wjk}\) :

Actual position vector of the tool center point corresponding to the kth cutting point in the workpiece coordinate system

\({{\varvec{u}}}_{jk}\) :

Actual cutting point of machined workpiece

\(H\) :

Position vector of the body reference coordinate system origin of each body

\({{\varvec{u}}}_{jk}^{i}\) :

Ideal cutting point of machined workpiece

\({{\varvec{r}}}_{w}\) :

Position vector of the tool center point in the workpiece coordinate system in the actual machining process

\({{\varvec{r}}}_{t}\) :

Position vector of tool center point in tool coordinate system

\({{\varvec{r}}}_{w}^{i}\) :

Position vector of the tool center point in the workpiece coordinate system in the ideal machining process

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Funding

This work is financially supported by the National Natural Science Foundation of China (No. 51775010) and Natural Science Foundation of Anhui Province of China (No. 2108085ME167).

Author information

Authors and Affiliations

  1. College of Mechanical Engineering, Anhui Science and Technology University, Chuzhou Anhui, 233100, People’s Republic of China

    Haohao Tao, Tongjie Li & Feng Chen

  2. Beijing Key Laboratory of Advanced Manufacturing Technology, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Jinwei Fan & Ri Pan

Authors
  1. Haohao Tao
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  2. Jinwei Fan
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Contributions

Haohao Tao: resources, supervision, methodology, validation, formal analysis, writing—original draft, writing—review and editing. Tongjie Li: project administration, funding acquisition. Feng Chen: conceptualization, investigation. Jinwei Fan: project administration, funding acquisition. Ri Pan: data curation, software.

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Correspondence to Tongjie Li.

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Appendices

Appendix 1

$$\left\{\begin{array}{l}{\delta }_{x}\left(x\right)=-2.64\times {10}^{-4}+3.02\times {10}^{-3}cos\left(5.78\times {10}^{-4}x\right)-9.76\times {10}^{-3}sin\left(5.78\times {10}^{-4}x\right)\\ \begin{array}{cc}\begin{array}{cc}& \end{array}& -2.95\times {10}^{-3}\mathrm{cos}\left(2x\times 5.78\times {10}^{-4}\right)+5.1\times {10}^{-3}\mathrm{sin}\left(2x\times 5.78\times {10}^{-4}\right)\end{array}\\ {\delta }_{y}\left(x\right)=2.17-1.041cos\left(5.28\times {10}^{-4}x\right)-3.561sin\left(5.28\times {10}^{-4}x\right)\\ \begin{array}{ccc}& & -1.912\mathrm{cos}\left(2x\times 5.28\times {10}^{-4}\right)+1.231\mathrm{sin}\left(2x\times 5.28\times {10}^{-4}\right)+0.7221\mathrm{cos}\left(3x\times 5.28\times {10}^{-4}\right)\end{array}\\ \begin{array}{ccc}& & +0.5917\mathrm{sin}\left(3x\times 5.28\times {10}^{-4}\right)+0.07389\mathrm{cos}\left(4x\times 5.28\times {10}^{-4}\right)-0.1703\mathrm{sin}\left(4x\times 5.28\times {10}^{-4}\right)\end{array}\\ {\delta }_{z}\left(x\right)=-2.14\times {10}^{-4}-1.16\times {10}^{-2}cos\left(1.27\times {10}^{-3}x\right)-7.83\times {10}^{-4}sin\left(1.27\times {10}^{-3}x\right)\\ \begin{array}{ccc}& & +2.97\times {10}^{-3}\mathrm{cos}\left(2x\times 1.27\times {10}^{-3}\right)-1.96\times {10}^{-3}\mathrm{sin}\left(2x\times 1.27\times {10}^{-3}\right)\end{array}\\ \begin{array}{ccc}& & +5.74\times {10}^{-3}\mathrm{cos}\left(3x\times 1.27\times {10}^{-3}\right)-4.6\times {10}^{-3}\mathrm{sin}\left(3x\times 1.27\times {10}^{-3}\right)\end{array}\\ {\varepsilon }_{x}\left(x\right)=-10.23+4.89cos\left(1.03\times {10}^{-3}x\right)+4.92sin\left(1.03\times {10}^{-3}x\right)+0.99cos\left(2x\times 1.03\times {10}^{-3}\right)\\ \begin{array}{ccc}& & +0.28\mathrm{sin}\left(2x\times 1.03\times {10}^{-3}\right)+2.2\mathrm{cos}\left(3x\times 1.03\times {10}^{-3}\right)-0.81\mathrm{sin}\left(3x\times 1.03\times {10}^{-3}\right)\end{array}\\ {\varepsilon }_{y}\left(x\right)=-1.29-1.1cos\left(1.05\times {10}^{-3}x\right)-1.07sin\left(1.05\times {10}^{-3}x\right)\\ \begin{array}{ccc}& & +0.05\mathrm{cos}\left(2x\times 1.05\times {10}^{-3}\right)+2.65\mathrm{sin}\left(2x\times 1.05\times {10}^{-3}\right)+1.97\mathrm{cos}\left(3x\times 1.05\times {10}^{-3}\right)\end{array}\\ \begin{array}{ccc}& & +3.68\mathrm{sin}\left(3x\times 1.05\times {10}^{-3}\right)-2.87\mathrm{cos}\left(4x\times 1.05\times {10}^{-3}\right)+0.52\mathrm{sin}\left(4x\times 1.05\times {10}^{-3}\right)\end{array}\\ {\varepsilon }_{z}\left(x\right)=-10.69+4.45cos\left(1.08\times {10}^{-3}x\right)+13.98sin\left(1.08\times {10}^{-3}x\right)\\ \begin{array}{ccc}& & +0.17\mathrm{cos}\left(2x\times 1.08\times {10}^{-3}\right)+4.54\mathrm{sin}\left(2x\times 1.08\times {10}^{-3}\right)+7.36\mathrm{cos}\left(3x\times 1.08\times {10}^{-3}\right)\end{array}\\ \begin{array}{ccc}& & +2.86\mathrm{sin}\left(3x\times 1.08\times {10}^{-3}\right)-3.01\mathrm{cos}\left(4x\times 1.08\times {10}^{-3}\right)+4.67\mathrm{sin}\left(4x\times 1.08\times {10}^{-3}\right)\end{array}\end{array}\right.\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& & \end{array}$$
(15)

Appendix 2

Please see Tables

Table 6 Ideal and actual static homogeneous transformation matrices for the adjacent bodies
Full size table

6,

Table 7 Ideal and actual motion homogeneous transformation matrices for the adjacent bodies
Full size table

7 and

Table 8 Position vector of the body reference coordinate system origin of each body
Full size table

8.

Appendix 3

$$\left\{\begin{array}{l}\begin{array}{l}e_x=\left(\delta_x\left(x\right)+\right.\delta_x\left(y\right)-l_kcos\left(b\right)\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xc}+\varepsilon_{xz}\right)+h_{4z}\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)\right)\\\;\;\;\;-\left(\varepsilon_y\left(b\right)+\delta_y\left(c\right)\right)sin\left(c\right)-\varepsilon_z\left(x\right)h_{3y}-h_{4y}\left(\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_{xy}\right)+\varepsilon_{xz}z+\delta_x\left(c\right)cos\left(c\right)\\\;\;\;\;+h_{5z}\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xz}\right)+h_{6z}\left(\begin{array}{l}\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xc}+\varepsilon_{xz}\\+\varepsilon_y\left(c\right)cos\left(c\right)+\varepsilon_x\left(c\right)sin\left(c\right)\end{array}\right)+\varepsilon_y\left(x\right)z\\\;\;\;\;-h_{5y}(\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_z\left(z\right)+\varepsilon_{xy})-l_kcos\left(b\right)cos\left(c\right)\left(\varepsilon_y\left(b\right)+\varepsilon_y\left(c\right)+\varepsilon_x\left(c\right)\right)-\varepsilon_{xy}y\\\;\;\;\;+\delta_x\left(b\right)cos\left(b\right)cos\left(c\right)+\left(\delta_z\left(b\right)-l_k\right)cos\left(c\right)sin\left(b\right)-\varepsilon_z\left(x\right)y-\varepsilon_{bz}l_kcos\left(b\right)sin\left(c\right)+\varepsilon_y\left(y\right)z\\\;\;\;\;-\left(\varepsilon_z\left(c\right)+\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_z\left(z\right)+\varepsilon_{xy}\right)\left(h_{6y}cos\left(c\right)+h_{6x}sin\left(c\right)\right)+\varepsilon_y\left(x\right)h_{3z}+\delta_x\left(z\right)\\\;\;\;\;-\varepsilon_x\left(b\right)l_ksin\left(c\right)+\left(\varepsilon_z\left(c\right)+\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_z\left(z\right)+\varepsilon_{xb}+\varepsilon_{xy}\right)l_ksin\left(b\right)\left.sin\left(c\right)\right)dn_{sjkx}\end{array}\\\begin{array}{l}e_y=\left(\delta_y\left(x\right)\right.+\delta_y\left(y\right)+\delta_y\left(z\right)+h_{5x}\left(\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_{xy}+\varepsilon_z\left(z\right)\right)+h_{4x}\left(\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_{xy}\right)\\\;\;\;\;\;+\delta_z\left(b\right)sin\left(b\right)sin\left(c\right)+\varepsilon_z\left(x\right)h_{3x}-\varepsilon_{yz}z+h_{6x}cos\left(c\right)\left(\varepsilon_z\left(c\right)+\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_z\left(z\right)+\varepsilon_{xy}\right)\\\;\;\;\;\;-h_{5z}\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)+\varepsilon_x\left(z\right)+\varepsilon_{yz}\right)-\varepsilon_x\left(x\right)z-\varepsilon_x\left(y\right)z+\left(\varepsilon_{bz}-\varepsilon_x\left(c\right)\right)l_kcos\left(b\right)cos\left(c\right)\\\;\;\;\;\;-h_{6z}\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)+\varepsilon_x\left(z\right)+\varepsilon_{yc}+\varepsilon_{yz}+\varepsilon_x\left(c\right)cos\left(c\right)-\varepsilon_y\left(c\right)sin\left(c\right)\right)+\delta_y\left(b\right)cos\left(c\right)\\\;\;\;\;\;-l_kcos\left(b\right)\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)+\varepsilon_x\left(z\right)+\varepsilon_{yc}+\varepsilon_{yz}\right)+cos\left(b\right)sin\left(c\right)\left(\delta_x\left(b\right)+\varepsilon_y\left(c\right)-\varepsilon_y\left(b\right)\right)\\\;\;\;\;\;-h_{6y}sin\left(c\right)\left(\varepsilon_z\left(c\right)+\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_z\left(z\right)+\varepsilon_{xy}\right)-h_{4z}\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)\right)+l_k\varepsilon_x\left(b\right)cos\left(c\right)\\\;\;\;\;\;\;+\delta_y\left(c\right)cos\left(c\right)+\delta_x\left(c\right)sin\left(c\right)-l_kcos\left(c\right)sin\left(b\right)\left(\varepsilon_z\left(c\right)+\varepsilon_z\left(x\right)+\varepsilon_z\left(y\right)+\varepsilon_z\left(z\right)+\varepsilon_{xb}+\varepsilon_{xy}\right)\\\left.\;\;\;\;\;-\varepsilon_x\left(x\right)h_{3z}\right)dn_{sjky}\end{array}\\\begin{array}{l}e_z=\left(\delta_z\left(c\right)\right.+\varepsilon_x\left(x\right)y-\varepsilon_y\left(x\right)h_{3x}+l_kcos\left(c\right)sin\left(b\right)\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xc}+\varepsilon_{xz}\right)\\\;\;\;\;+sin\left(b\right)\begin{pmatrix}\left(\varepsilon_y\left(b\right)l_k+\varepsilon_y\left(c\right)l_k-\delta_x\left(b\right)\right)-l_ksin\left(c\right)\\\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)+\varepsilon_x\left(z\right)+\varepsilon_{yc}+\varepsilon_{yz}\right)\end{pmatrix}-h_{5x}\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xz}\right)\\\;\;\;\;+h_{6y}\left(\varepsilon_x\left(c\right)+\varepsilon_{yc}\cos\left(c\right)+\sin\left(c\right)\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xc}+\varepsilon_{xz}\right)\right)-h_{4x}\left(\varepsilon_y\left(x\right)+\varepsilon_y\left(y\right)\right)\\\;\;\;\;+h_{5y}\left(\varepsilon_{yz}+\varepsilon_x\left(y\right)+\varepsilon_x\left(z\right)+\varepsilon_x\left(x\right)\right)+\delta_z\left(x\right)+\delta_z\left(y\right)+\delta_z\left(z\right)+h_{4y}\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)\right)\\\;\;\;\;\;+h_{6x}\begin{pmatrix}sin\left(c\right)\left(\varepsilon_x\left(x\right)+\varepsilon_x\left(y\right)+\varepsilon_x\left(z\right)+\varepsilon_{yc}+\varepsilon_{yz}\right)-cos\left(c\right)\\\left(\varepsilon_y\left(y\right)+\varepsilon_y\left(z\right)+\varepsilon_{xc}+\varepsilon_{xz}+\varepsilon_y\left(x\right)\right)-\varepsilon_y\left(c\right)\end{pmatrix}+\varepsilon_x\left(b\right)\varepsilon_y\left(z\right)\varepsilon_{xb}l_kcos\left(c\right)\\\left.\;\;\;\;+\delta_z\left(b\right)cos\left(b\right)+\varepsilon_x\left(x\right)h_{3y}\right)dn_{sjkz}\end{array}\end{array}\right.$$
(16)

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Tao, H., Fan, J., Li, T. et al. An approach to enhancing machining accuracy of five-axis machine tools based on a new sensitivity analysis method. Int J Adv Manuf Technol 124, 2383–2400 (2023). https://doi.org/10.1007/s00170-022-10365-2

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