# A Geometric Accuracy Error Analysis Method for Turn-Milling Combined NC Machine Tool

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## Abstract

**:**

## 1. Introduction

## 2. Geometric Error Terms for Turn-Milling Combined NC Machine Tool

#### 2.1. The Topological Structure of Turn-Milling Combined NC Machine Tool

#### 2.2. Determination of Geometric Error Terms

## 3. Geometric Error Modeling Methods

#### 3.1. Analysis of Ideal Kinematics Model of Turn-Milling Combined NC Machine Tool

#### 3.2. Analysis of Actual Kinematics Model of Turn-Milling Combined NC Machine Tool

#### 3.2.1. The Actual Kinematics Model in Milling Mode

#### 3.2.2. The Actual Kinematics Model in Turning Mode

#### 3.3. Geometry Error Modeling of Turn-Milling Combined NC Machine Tool

#### 3.3.1. Geometry Error Modeling in Milling Mode

#### 3.3.2. Translation Axis Geometric Error Model

## 4. Geometric Precision Error Analysis

#### 4.1. Analysis of the Interval Sensitivity of Each Axis to Geometric Errors

#### 4.1.1. Interval Sensitivity Analysis of X-Axis Geometric Error to Spatial Error

#### 4.1.2. Interval Sensitivity Analysis of Y-Axis Geometric Error to Spatial Error

#### 4.1.3. Interval Sensitivity Analysis of Z-axis Geometric Error to Spatial Error

#### 4.1.4. Interval Sensitivity Analysis of Perpendicularity Errors to Spatial Errors

#### 4.2. Global Maximum Interval Sensitivity Analysis to Spatial Error of Geometric Error Sources

## 5. AN Experimental Study

^{−5}, 0.04744, and 9.09 × 10

^{−7}, respectively, which are at least 2 orders of magnitude smaller than ${\Delta}_{xY,}{\Delta}_{yY}\mathrm{and}{\Delta}_{zY}$. Among them, the value of ${\Delta}_{yY}$ is larger due to the error caused by the weight of Y-axis and B axis. And the reason for a larger value of ${\epsilon}_{yY}$ is synchronization error of Y-axis. The experimental results is close to the sensitivity analysis results of the geometric error mentioned above. That is, the positioning error and linearity error of Y-axis are highly sensitive, while the sensitivity of yaw error, pitch error, and roll error is low. The sensitivity analysis results of the turning-milling compound geometric error is consistent with the experimental error decomposition results, demonstrating the effectiveness of the sensitivity analysis method.

## 6. Conclusions and Discussion

#### 6.1. Discussion

#### 6.2. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\mathbf{Classical}\mathbf{Body}\mathit{i}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{L}}^{\mathbf{0}}\left(\mathit{i}\right)$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

${\mathit{L}}^{\mathbf{1}}\left(\mathit{i}\right)$ | 0 | 0 | 1 | 2 | 3 | 4 | 0 | 6/9 | 0 | 8 |

${\mathit{L}}^{\mathbf{2}}\left(\mathit{i}\right)$ | 0 | 0 | 0 | 1 | 2 | 3 | 0 | 0/8 | 0 | 0 |

${\mathit{L}}^{\mathbf{3}}\left(\mathit{i}\right)$ | 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |

${\mathit{L}}^{\mathbf{4}}\left(\mathit{i}\right)$ | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

${\mathit{L}}^{\mathbf{5}}\left(\mathit{i}\right)$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

$\mathbf{Perpendicularity}\mathbf{Error}/\left({}^{\xb0}\right)$ | $\mathbf{Position}\mathbf{Error}/\left(\mathit{m}\mathit{m}\right)$ | $\mathbf{Angle}\mathbf{Error}/\left({}^{\xb0}\right)$ | |
---|---|---|---|

X | ${\mathit{\beta}}_{\mathit{X}}$ | ${\mathit{\Delta}}_{\mathit{x}\mathit{X}};{\mathit{\Delta}}_{\mathit{y}\mathit{X}};{\mathit{\Delta}}_{\mathit{z}\mathit{X}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{X}};{\mathit{\epsilon}}_{\mathit{y}\mathit{X}};{\mathit{\epsilon}}_{\mathit{z}\mathit{X}}$ |

Y | ${\mathit{\alpha}}_{\mathit{Y}};{\mathit{\gamma}}_{\mathit{Y}}$ | ${\mathit{\Delta}}_{\mathit{x}\mathit{Y}};{\mathit{\Delta}}_{\mathit{y}\mathit{Y}};{\mathit{\Delta}}_{\mathit{z}\mathit{Y}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{Y}};{\mathit{\epsilon}}_{\mathit{y}\mathit{Y}};{\mathit{\epsilon}}_{\mathit{z}\mathit{Y}}$ |

Z | — | ${\mathit{\Delta}}_{\mathit{x}\mathit{Z}};{\mathit{\Delta}}_{\mathit{y}\mathit{Z}};{\mathit{\Delta}}_{\mathit{z}\mathit{Z}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{Z}};{\mathit{\epsilon}}_{\mathit{y}\mathit{Z}};{\mathit{\epsilon}}_{\mathit{z}\mathit{Z}}$ |

B | ${\mathit{\alpha}}_{\mathit{B}};{\mathit{\gamma}}_{\mathit{B}}$ | ${\mathit{\Delta}}_{\mathit{x}\mathit{B}};{\mathit{\Delta}}_{\mathit{y}\mathit{B}};{\mathit{\Delta}}_{\mathit{z}\mathit{B}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{B}};{\mathit{\epsilon}}_{\mathit{y}\mathit{B}};{\mathit{\epsilon}}_{\mathit{z}\mathit{B}}$ |

C | ${\mathit{\alpha}}_{\mathit{C}};{\mathit{\beta}}_{\mathit{C}}$ | ${\mathit{\Delta}}_{\mathit{x}\mathit{C}};{\mathit{\Delta}}_{\mathit{y}\mathit{C}};{\mathit{\Delta}}_{\mathit{z}\mathit{C}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{C}};{\mathit{\epsilon}}_{\mathit{y}\mathit{C}};{\mathit{\epsilon}}_{\mathit{z}\mathit{C}}$ |

W | ${\mathit{\alpha}}_{\mathit{W}};{\mathit{\beta}}_{\mathit{W}}$ | ${\mathit{\Delta}}_{\mathit{x}\mathit{W}};{\mathit{\Delta}}_{\mathit{y}\mathit{W}};{\mathit{\Delta}}_{\mathit{z}\mathit{W}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{W}};{\mathit{\epsilon}}_{\mathit{y}\mathit{W}};{\mathit{\epsilon}}_{\mathit{z}\mathit{W}}$ |

Adjacent Body | Position Error Transformation Matrices | Motion Error Transformation Matrices |
---|---|---|

O-C | $\Delta {\mathit{T}}_{\mathbf{06}\mathit{p}}=\left[\begin{array}{cccc}\mathbf{1}& \mathbf{0}& {\mathit{\beta}}_{\mathit{C}}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}& -{\mathit{\alpha}}_{\mathit{C}}& \mathbf{0}\\ -{\mathit{\beta}}_{\mathit{C}}& {\mathit{\alpha}}_{\mathit{C}}& \mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ | $\Delta {\mathit{T}}_{\mathbf{06}s}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\epsilon}}_{\mathit{z}\mathit{C}}& {\mathit{\epsilon}}_{\mathit{y}\mathit{C}}& {\mathit{\Delta}}_{\mathit{x}\mathit{C}}\\ {\mathit{\epsilon}}_{\mathit{z}\mathit{C}}& \mathbf{1}& -{\mathit{\epsilon}}_{\mathit{x}\mathit{C}}& {\mathit{\Delta}}_{\mathit{y}\mathit{C}}\\ -{\mathit{\epsilon}}_{\mathit{y}\mathit{C}}& {\mathit{\epsilon}}_{\mathit{x}\mathit{C}}& \mathbf{1}& {\mathit{\Delta}}_{\mathit{z}\mathit{C}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ |

O-W | $\Delta {\mathit{T}}_{\mathbf{08}\mathit{p}}=\left[\begin{array}{cccc}\mathbf{1}& \mathbf{0}& {\mathit{\beta}}_{\mathit{W}}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}& -{\mathit{\alpha}}_{\mathit{W}}& \mathbf{0}\\ -{\mathit{\beta}}_{\mathit{W}}& {\mathit{\alpha}}_{\mathit{W}}& \mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ | $\Delta {\mathit{T}}_{\mathbf{08}\mathit{s}}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\epsilon}}_{\mathit{z}\mathit{W}}& {\mathit{\epsilon}}_{\mathit{y}\mathit{W}}& {\mathit{\Delta}}_{\mathit{x}\mathit{W}}\\ {\mathit{\epsilon}}_{\mathit{z}\mathit{W}}& \mathbf{1}& -{\mathit{\epsilon}}_{\mathit{x}\mathit{W}}& {\mathit{\Delta}}_{\mathit{y}\mathit{W}}\\ -{\mathit{\epsilon}}_{\mathit{y}\mathit{W}}& {\mathit{\epsilon}}_{\mathit{x}\mathit{W}}& \mathbf{1}& {\mathit{\Delta}}_{\mathit{z}\mathit{W}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ |

O-Z | $\Delta {\mathit{T}}_{\mathbf{01}\mathit{p}}=\mathit{I}$ | $\Delta {\mathit{T}}_{\mathbf{01}s}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\epsilon}}_{\mathit{z}\mathit{Z}}& {\mathit{\epsilon}}_{\mathit{y}\mathit{Z}}& {\mathit{\Delta}}_{\mathit{x}\mathit{Z}}\\ {\mathit{\epsilon}}_{\mathit{z}\mathit{Z}}& \mathbf{1}& -{\mathit{\epsilon}}_{\mathit{x}\mathit{Z}}& {\mathit{\Delta}}_{\mathit{y}\mathit{Z}}\\ -{\mathit{\epsilon}}_{\mathit{y}\mathit{Z}}& {\mathit{\epsilon}}_{\mathit{x}\mathit{Z}}& \mathbf{1}& {\mathit{\Delta}}_{\mathit{z}\mathit{Z}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ |

Z-X | $\Delta {\mathit{T}}_{\mathbf{12}\mathit{p}}=\left[\begin{array}{cccc}\mathbf{1}& \mathbf{0}& {\mathit{\beta}}_{\mathit{X}}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{0}\\ -{\mathit{\beta}}_{\mathit{X}}& \mathbf{0}& \mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ | $\Delta {\mathit{T}}_{\mathbf{12}\mathit{s}}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\epsilon}}_{\mathit{z}\mathit{X}}& {\mathit{\epsilon}}_{\mathit{y}\mathit{X}}& {\mathit{\Delta}}_{\mathit{x}\mathit{X}}\\ {\mathit{\epsilon}}_{\mathit{z}\mathit{X}}& \mathbf{1}& -{\mathit{\epsilon}}_{\mathit{x}\mathit{X}}& {\mathit{\Delta}}_{\mathit{y}\mathit{X}}\\ -{\mathit{\epsilon}}_{\mathit{y}\mathit{X}}& {\mathit{\epsilon}}_{\mathit{x}\mathit{X}}& \mathbf{1}& {\mathit{\Delta}}_{\mathit{z}\mathit{X}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ |

X-Y | $\Delta {\mathit{T}}_{\mathbf{23}\mathit{p}}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\gamma}}_{\mathit{Y}}& \mathbf{0}& \mathbf{0}\\ {\mathit{\gamma}}_{\mathit{Y}}& \mathbf{1}& -{\alpha}_{\mathit{Y}}& \mathbf{0}\\ \mathbf{0}& {\alpha}_{\mathit{Y}}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ | $\Delta {\mathit{T}}_{\mathbf{23}\mathit{s}}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\epsilon}}_{\mathit{z}\mathit{Y}}& {\mathit{\epsilon}}_{\mathit{y}\mathit{Y}}& {\mathit{\Delta}}_{\mathit{x}\mathit{Y}}\\ {\mathit{\epsilon}}_{\mathit{z}\mathit{Y}}& \mathbf{1}& -{\mathit{\epsilon}}_{\mathit{x}\mathit{Y}}& {\mathit{\Delta}}_{\mathit{y}\mathit{Y}}\\ -{\mathit{\epsilon}}_{\mathit{y}\mathit{Y}}& {\mathit{\epsilon}}_{\mathit{x}\mathit{Y}}& \mathbf{1}& {\mathit{\Delta}}_{\mathit{z}\mathit{Y}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ |

Y-B | $\Delta {\mathit{T}}_{\mathbf{34}\mathit{p}}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\gamma}}_{B}& \mathbf{0}& \mathbf{0}\\ {\mathit{\gamma}}_{B}& \mathbf{1}& -{\alpha}_{B}& \mathbf{0}\\ \mathbf{0}& {\alpha}_{B}& \mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ | $\Delta {\mathit{T}}_{\mathbf{34}\mathit{s}}=\left[\begin{array}{cccc}\mathbf{1}& -{\mathit{\epsilon}}_{\mathit{z}\mathit{B}}& {\mathit{\epsilon}}_{\mathit{y}\mathit{B}}& {\mathit{\Delta}}_{\mathit{x}\mathit{B}}\\ {\mathit{\epsilon}}_{\mathit{z}\mathit{B}}& \mathbf{1}& -{\mathit{\epsilon}}_{\mathit{x}\mathit{B}}& {\mathit{\Delta}}_{\mathit{y}\mathit{B}}\\ -{\mathit{\epsilon}}_{\mathit{y}\mathit{B}}& {\mathit{\epsilon}}_{\mathit{x}\mathit{B}}& \mathbf{1}& {\mathit{\Delta}}_{\mathit{z}\mathit{B}}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$ |

Error Measurement Result | Error Separation Result | Experimental Fitting Results | |||
---|---|---|---|---|---|

Error Term | Value | Error Term | Value | Error Term | Value |

${\mathit{\Delta}}_{\mathit{x}\mathit{Y}}$ | 0.98254 | ${\mathit{\Delta}}_{\mathit{x}\mathit{Y}}$ | −1.0092 | ${\mathit{\Delta}}_{\mathit{x}\mathit{Y}}$ | −1.0092 |

${\mathit{\Delta}}_{\mathit{y}\mathit{Y}}$ | 4.2648 | ${\mathit{\Delta}}_{\mathit{y}\mathit{Y}}$ | 4.2666 | ${\mathit{\Delta}}_{\mathit{y}\mathit{Y}}$ | 4.2666 |

${\mathit{\Delta}}_{\mathit{z}\mathit{Y}}$ | 17.693 | ${\mathit{\Delta}}_{\mathit{z}\mathit{Y}}$ | 36.712 | ${\mathit{\Delta}}_{\mathit{z}\mathit{Y}}$ | 36.714 |

/ | / | ${\mathit{\epsilon}}_{\mathit{x}\mathit{Y}}$ | 6.91 × 10^{−5} | ${\mathit{\epsilon}}_{\mathit{x}\mathit{Y}}$ | 6.91 × 10^{−5} |

/ | / | ${\mathit{\epsilon}}_{\mathit{y}\mathit{Y}}$ | 0.04744 | ${\mathit{\epsilon}}_{\mathit{y}\mathit{Y}}$ | 0.042089 |

/ | / | ${\mathit{\epsilon}}_{\mathit{z}\mathit{Y}}$ | 9.09 × 10^{−7} | ${\mathit{\epsilon}}_{\mathit{z}\mathit{Y}}$ | 9.09 × 10^{−7} |

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## Share and Cite

**MDPI and ACS Style**

Li, P.; Zhao, R.; Luo, L.
A Geometric Accuracy Error Analysis Method for Turn-Milling Combined NC Machine Tool. *Symmetry* **2020**, *12*, 1622.
https://doi.org/10.3390/sym12101622

**AMA Style**

Li P, Zhao R, Luo L.
A Geometric Accuracy Error Analysis Method for Turn-Milling Combined NC Machine Tool. *Symmetry*. 2020; 12(10):1622.
https://doi.org/10.3390/sym12101622

**Chicago/Turabian Style**

Li, Pengzhong, Ruihan Zhao, and Liang Luo.
2020. "A Geometric Accuracy Error Analysis Method for Turn-Milling Combined NC Machine Tool" *Symmetry* 12, no. 10: 1622.
https://doi.org/10.3390/sym12101622