# A geometric error budget method to improve machining accuracy reliability of multi-axis machine tools

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## Abstract

Machining accuracy reliability is considered to be one of the most important indexes in the process of performance evaluation and optimization design of the machine tools. Geometric errors, thermal errors and tool wear are the main factors to affect the machining accuracy and so affect the machining accuracy reliability of machine tools. This paper proposed a geometric error budget method that simultaneously considers geometric errors, thermal errors and tool wear to improve the machining accuracy reliability of machine tools. Homogeneous transformation matrices, neural fuzzy control theory and a tool wear predictive approach were employed to develop a comprehensive error model, which shows the influence of the geometric, thermal errors and tool wear to the machining accuracy of a machine tool. Based on Rackwite–Fiessler and Advanced First Order and Second Moment, a reliability model and a sensitivity model were put forward, which can deal with the errors of a machine tool drawn from any distribution. Then, a geometric error budget method of multi-axis NC machine tool was developed and formed into a mathematical model. In such method, the minimum cost of machine tool was the optimization objective, the reliability of the machining accuracy was the constraint, and the sensitivity was to identify the geometric errors to be optimized. An example conducted on a five-axis NC machine tool was used to explain and validate the proposed method.

## Keywords

Thermal errors Tool wear A comprehensive error model Machining accuracy reliability Geometric error budget## List of symbols

- \(\Delta x_{x}\)
Positioning error

- \(\Delta y_{x}\)
Y direction of straightness error

- \(\Delta z_{x}\)
Z direction of straightness error

- \(\Delta \alpha _{x}\)
Roll error

- \(\Delta \beta _{x}\)
Pitch error

- \(\Delta \gamma _{x}\)
Yaw error

- \(\Delta x_{y}\)
X direction of straightness error

- \(\Delta y_{y}\)
Positioning error

- \(\Delta z_{y}\)
Z direction of straightness error

- \(\Delta \alpha _{y}\)
Pitch error

- \(\Delta \beta _{y}\)
Roll error

- \(\Delta \gamma _{y}\)
Yaw error

- \(\Delta x_{z}\)
X direction of straightness error

- \(\Delta y_{z}\)
Y direction of straightness error

- \(\Delta z_{z}\)
Positioning error

- \(\Delta \alpha _{z}\)
Pitch error

- \(\Delta \beta _{z}\)
Yaw error

- \(\Delta \gamma _{z}\)
Roll error

- \(\Delta x_{B}\)
X direction run-out error

- \(\Delta y_{B}\)
Y direction run-out error

- \(\Delta z_{B}\)
Z direction run-out error

- \(\Delta \alpha _{B}\)
Around the X-axis turning error

- \(\Delta \beta _{B}\)
Turning error

- \(\Delta \gamma _{B}\)
Around the Z-axis turning error

- \(\Delta x_{A}\)
X direction run-out error

- \(\Delta y_{A}\)
Y direction run-out error

- \(\Delta z_{A}\)
Z direction run-out error

- \(\Delta \alpha _{A}\)
Turning error

- \(\Delta \beta _{A}\)
Around the Y-axis turning error

- \(\Delta \gamma _{A}\)
Around the Z-axis turning error

- \(\Delta x_\varphi \)
X direction run-out error

- \(\Delta y_\varphi \)
Y direction run-out error

- \(\Delta z_\varphi \)
Z direction run-out error

- \(\Delta \alpha _\varphi \)
Around the X-axis turning error

- \(\Delta \beta _\varphi \)
Around the Y-axis turning error

- \(\Delta \gamma _\varphi \)
Turning error

- \(\Delta \gamma _{{xy}}\)
X, Y-axis perpendicularity error

- \(\Delta \beta _{xz}\)
X, Z-axis perpendicularity error

- \(\Delta \alpha _{yz}\)
Y, Z-axis perpendicularity error

- \(\Delta \gamma _{xB}\)
B-axis parallelism error in YZ plane

- \(\Delta \alpha _{zB}\)
B-axis parallelism error in XY plane

- \(\Delta \gamma _{yA}\)
A-axis parallelism error in XZ plane

- \(\Delta \beta _{zA}\)
A-axis parallelism error in XY plane

- \(\Delta y_{AB}\)
An offset errors between A, B-axis along Y-axis

- \(\Delta z_{AB}\)
An offset errors between A, B-axis along Z-axis

- \(\Delta x_t\)
X direction of straightness error

- \(\Delta y_t\)
Y direction of straightness error

- \(\Delta z_t\)
Positioning error

- \(\Delta \alpha _t\)
Pitch error

- \(\Delta \beta _t\)
Yaw error

- \(\Delta \gamma _t\)
Roll error

## Notes

### Acknowledgments

The authors are most grateful to the National Natural Science Foundation of China (Nos. 51575010 and 51575009), Beijing Nova Program (Z1511000003150138), the Leading Talent Project of Guangdong Province, Open Project of State Key Lab of Digital Manufacturing Equipment & Technology (Huazhong University of Science and Technology), Shantou Light Industry Equipment Research Institute of science and technology Correspondent Station (2013B090900008), the National Science and Technology Major Project (2013ZX04013-011), the Jing-Hua Talents Project of Beijing University of Technology and the National Science and Technology Major Project (2013ZX04013) for supporting this research presented in this paper.

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