# Rotation-Angle Solution and Singularity Handling of Five-Axis Machine Tools for Dual NURBS Interpolation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rotation-Angle Solution of Five-Axis Machine Tools for Dual NURBS Interpolation

#### 2.1. Dual NURBS Interpolation

#### 2.2. Generic Method of Rotation-Angle Solution of Five-Axis Machine Tools

#### 2.2.1. Typical Layouts of Rotary Axes of Five-Axis Machine Tools and Analysis

#### 2.2.2. Generic Method for Solving the Rotation Angles of Five-Axis Machine Tools Based on the Vector Inner Product

#### 2.3. Solution Space of the Generic Method

- (a)
- No solution. The tool orientation vector $O\left(u\right)$ cannot be realized by rotation axes as the tool orientation vector is beyond the reach of the machine tool when ${i}^{2}+{j}^{2}<{\eta}^{2}$.
- (b)
- Finite solutions. The C-axis angle has two sets of solutions when ${i}^{2}+{j}^{2}\ge {\eta}^{2}$ and ${i}^{2}+{j}^{2}>0$, as shown in Equation (13). The rotation angles are high-order continuous with respect to the parameter u, when the tool orientation vector is high-order continuous.
- (c)
- Infinite solutions. The C-axis angle has infinite sets of solutions when ${i}^{2}+{j}^{2}={\eta}^{2}=0$. The tool orientation vector is at the singularity point, which cannot be changed no matter the C-axis angle. The tool orientation and the layout of rotation axes must meet the following conditions.

#### 2.4. Singularity Handling

## 3. Experiments and Discussions

#### 3.1. Experiment on an Open-Pocket Tool Path

#### 3.2. Experiment on a Cardioid Curve

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameters | Values |
---|---|

NURBS Degree | 3 |

Knot Vector | (0, 0, 0, 0, 0.2, 0.4, 0.6, 0.8, 1, 1, 1, 1) |

Weight Vector | (1, 1, 1, 1, 1, 1, 1, 1) |

Control Points of the Tool Tip Point Curve | (5,0,0), (−10,20,0), (10,20,0), (20,30,0), (30,30,0), (40,30,0), (50,20,0), (55,0,0) |

Control Points of the Tool Axis Point Curve | (0,0,15), (−15,20,15), (5,25,15), (15,35,15), (30,35,15), (45,35,15), (55,25,15), (60,0,15) |

## Appendix B

Parameters | Values |
---|---|

NURBS Degree | 3 |

Knot Vector | (0, 0, 0, 0, 1/9, 2/9, 3/9, 4/9, 5/9, 6/9, 7/9, 8/9, 1, 1, 1, 1) |

Weight Vector | (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) |

Control Points of the Tool Tip Point Curve | (0,0,0), (−51,13.5,0), (−51,87,0), (−30,75,0), (−22.5,69,0), (−1.5,57,0),(1.5,57,0), (22.5,69,0), (30,75,0), (51,87,0), (51,13.5,0), (0,0,0) |

Control Points of the Tool Axis Point Curve | (0,2.25,9), (−45,15,9), (−48,81,9), (−30,75,9), (−22.5,69,9), (−3,60,9),(3,60,9), (22.5,69,9), (30,75,9), (48,81,9), (45,15,9), (0,2.25,9) |

## References

- Li, D.D.; Zhang, W.M.; Zhou, W.; Shang, T.F.; Fleischer, J. Dual NURBS Path Smoothing for 5-Axis Linear Path of Flank Milling. Int. J. Precis. Eng. Man.
**2018**, 19, 1811–1820. [Google Scholar] [CrossRef] - Sun, Y.; Bao, Y.; Kang, K.; Guo, D. An adaptive feedrate scheduling method of dual NURBS curve interpolator for precision five-axis CNC machining. Int. J. Adv. Manuf. Technol.
**2013**, 68, 1977–1987. [Google Scholar] [CrossRef] - Xu, J.T.; Zhang, D.Y.; Sun, Y.W. Kinematics performance oriented smoothing method to plan tool orientations for 5-axis ball-end CNC machining. Int. J. Mech. Sci.
**2019**, 157, 293–303. [Google Scholar] [CrossRef] - Takeuchi, Y.; Watanabe, T. Generation of 5-axis control collision-free tool path and postprocessing for NC data. CIRP Ann.
**1992**, 41, 539–542. [Google Scholar] [CrossRef] - Lee, R.; She, C. Developing a postprocessor for three types of five-axis machine tools. Int. J. Adv. Manuf. Technol.
**1997**, 13, 658–665. [Google Scholar] [CrossRef] - Mahbubur, R.; Heikkala, J.; Lappalainen, K.; Karjalainen, J.A. Positioning accuracy improvement in five-axis milling by post processing. Int. J. Mach. Tools Manuf.
**1997**, 37, 223–236. [Google Scholar] [CrossRef] - She, C.; Chang, C. Development of a five-axis postprocessor system with a nutating head. J. Mater Process. Tech.
**2007**, 187, 60–64. [Google Scholar] [CrossRef] - She, C.; Huang, Z. Postprocessor development of a five-axis machine tool with nutating head and table configuration. Int. J. Adv. Manuf. Technol.
**2008**, 38, 728–740. [Google Scholar] [CrossRef] - She, C.; Chang, C. Design of a generic five-axis postprocessor based on generalized kinematics model of machine tool. Int. J. Mach. Tools Manuf.
**2007**, 47, 537–545. [Google Scholar] [CrossRef] - Peng, Y.; Ma, J.; Wang, L.; Yan, R.; Li, B. Post-processing Algorithm Based on Total Differential Method for Multi-axis Machine Tools with Arbitrary Configuration. J. Mech. Eng.
**2012**, 48, 121–126. [Google Scholar] [CrossRef] [Green Version] - Li, Z. Research and Application on Kinematic Generic Modeling Theory for Five-Axis; Southwest Jiaotong University: Chengdu, China, 2013. [Google Scholar] [CrossRef]
- He, Y.X.; Xu, Q.H.; Zhou, Y.H. Kinematics model and its solution for NC machines of arbitrary configuration. J. Mech. Eng.
**2002**, 38, 31–36. [Google Scholar] [CrossRef] - Tan, G.Z.; Chen, H.R.; Dai, Y.X. Study on Kinematics Analysis and Non-Linear Error of a TATC Five-Axis Machine Tool. Mach. Des. Manuf.
**2018**, 332, 229–232. [Google Scholar] [CrossRef] - My, C.A.; Bohez, E. A novel differential kinematics model to compare the kinematic performances of 5-axis CNC machines. Int. J. Mech. Sci.
**2019**, 163, 105–117. [Google Scholar] [CrossRef] - Xu, R.F.; Cheng, X.; Zheng, G.M.; Chen, Z.T. A tool orientation smoothing method based on machine rotary axes for five-axis machining with ball end cutters. Int. J. Adv. Manuf. Tech.
**2017**, 92, 3615–3625. [Google Scholar] [CrossRef] - Liu, Y.; Wan, M.; Xing, W.J.; Xiao, Q.B.; Zhang, W.H. Generalized actual inverse kinematic model for compensating geometric errors in five-axis machine tools. Int. J. Mech. Sci.
**2018**, 145, 299–317. [Google Scholar] [CrossRef] - Farouki, R.T.; Han, C.Y.; Li, S.Q. Inverse kinematics for optimal tool orientation control in 5-axis CNC machining. Comput. Aided. Geom. D
**2014**, 31, 13–26. [Google Scholar] [CrossRef] - Lin, T.K.; Lin, A.C. Conversion CL data to NC data using an instinctive method for non-orthogonal table-type 5 axis machines. In Mechatronics and Applied Mechanics II, Pts 1 and 2; 2nd International Conference on Mechatronics and Applied Mechanics (ICMAM2012); Wang, C.K., Guo, J., Eds.; Trans Tech Publications Ltd.: Bach, Switzerland, 2013; Volume 300–301, pp. 232–235. [Google Scholar]
- Yu, D.; Yan, G.R.; Fan, Q.X.; Ding, T.; Xu, X.Y. Research on Optimization of Rotation Angle and Singular Area Handling in Five-Axis Post-Processing. J. Graph.
**2016**, 37, 614–619. [Google Scholar] [CrossRef] - Hong, X.Y.; Hong, R.J.; Lin, X.C. Analysis and optimization of singular problem in five-axis machining. J. Nanjing Univ. Technol. (Nat. Sci. Ed.)
**2021**, 43, 58–64. [Google Scholar] [CrossRef] - Beudaert, X.; Lavernhe, S.; Tournier, C. Feedrate interpolation with axis jerk constraints on 5-axis NURBS and G1 tool path. Int. J. Mach. Tools Manuf.
**2012**, 57, 73–82. [Google Scholar] [CrossRef] [Green Version] - Lu, Y.A.; Wang, C.Y. Smoothing method of generating flank milling tool paths for five-axis flat-end machining considering constraints. Int. J. Adv. Manuf. Tech.
**2020**, 110, 3295–3309. [Google Scholar] [CrossRef] - Castagnetti, C.; Duc, E.; Ray, P. The Domain of Admissible Orientation concept: A new method for five-axis tool path optimisation. Comput. Aided. Design
**2008**, 40, 938–950. [Google Scholar] [CrossRef] - Wang, Q.R.; Feng, Y.X.; Zhang, Z.X.; Tan, J.R. Tool orientation sequence smoothing method based on the discrete domain of feasible orientations. Int. J. Adv. Manuf. Tech.
**2017**, 92, 4501–4510. [Google Scholar] [CrossRef] - Hu, P.C.; Tang, K. Improving the dynamics of five-axis machining through optimization of workpiece setup and tool orientations. Comput. Aided. Design
**2011**, 43, 1693–1706. [Google Scholar] [CrossRef] - Sorby, K. Inverse kinematics of five-axis machines near singular configurations. Int. J. Mach. Tools Manuf.
**2007**, 47, 299–306. [Google Scholar] [CrossRef] - Liu, H.; Liu, Q.; Sun, P.P.; Liu, Q.T.; Yuan, S.M. The optimal feedrate planning on five-axis parametric tool path with geometric and kinematic constraints for CNC machine tools. Int. J. Prod. Res.
**2017**, 55, 3715–3731. [Google Scholar] [CrossRef]

**Figure 1.**Common structure layouts of machine tools with orthogonal rotary axes. (

**a**) Dual-head style, (

**b**) single-head and single-turntable style, (

**c**) dual-turntable style.

**Figure 2.**Common structure layouts of machine tools with pendulous rotary axes. (

**a**) Dual-head style, (

**b**) single-head and single-turntable style, (

**c**) dual-turntable style.

**Figure 6.**The rotation-angle results of the open-pocket curve. (

**a**) The rotation angles of the machine tool with dual orthogonal turntables. (

**b**) The rotation angles of the machine tool with dual pendulous turntables.

**Figure 8.**Solution results of rotation angles in the cardioid curve interpolation. (

**a**) Results of the A−axis angle. (

**b**) Results of the C−axis angle.

**Figure 9.**The optimization results of the cardioid curve feedrate. (

**a**) The feedrate curves in the parameter field. (

**b**) The C−axis angles with interpolation time.

**Figure 11.**The machining results of the cardioid curve interpolation. (

**a**) The rotation angles solved by the proposed method. (

**b**) The rotation angles solved by the inverse trigonometric method.

**Table 1.**Rotation-angle handling at the singularity point based on the derivatives of the tool orientation vector.

Conditions of the Tool Orientation | Equation of the Rotation Angle | Solutions |
---|---|---|

$\{\begin{array}{l}{i}^{2}+{j}^{2}=0\\ {i}_{u}^{2}+{j}_{u}^{2}>0\end{array}$ | $\mathrm{sin}\omega \left({i}_{u}\mathrm{cos}\theta +{j}_{u}\mathrm{sin}\theta \right)=0$ | $\{\begin{array}{l}{\theta}_{1}=\mathrm{arctan}2\left(-{i}_{u},{j}_{u}\right)\\ {\theta}_{2}=\mathrm{arctan}2\left({i}_{u},-{j}_{u}\right)\end{array}$ |

$\{\begin{array}{c}{i}^{2}+{j}^{2}=0\\ {i}_{u}^{2}+{j}_{u}^{2}=0\\ {i}_{uu}^{2}+{j}_{uu}^{2}>0\end{array}$ | $\mathrm{sin}\omega \left({i}_{uu}\mathrm{cos}\theta +{j}_{uu}\mathrm{sin}\theta \right)=0$ | $\{\begin{array}{l}{\theta}_{1}=\mathrm{arctan}2\left(-{i}_{uu},{j}_{uu}\right)\\ {\theta}_{2}=\mathrm{arctan}2\left({i}_{uu},-{j}_{uu}\right)\end{array}$ |

$\{\begin{array}{c}{i}^{2}+{j}^{2}=0\\ {i}_{u}^{2}+{j}_{u}^{2}=0\\ {i}_{uu}^{2}+{j}_{uu}^{2}=0\\ {i}_{uuu}^{2}+{j}_{uuu}^{2}>0\end{array}$ | $\mathrm{sin}\omega \left({i}_{uu}\mathrm{cos}\theta +{j}_{uu}\mathrm{sin}\theta \right)=0$ | $\{\begin{array}{l}{\theta}_{1}=\mathrm{arctan}2\left(-{i}_{uuu},{j}_{uuu}\right)\\ {\theta}_{2}=\mathrm{arctan}2\left({i}_{uuu},-{j}_{uuu}\right)\end{array}$ |

… | … | … |

**Table 2.**The kinematic and dynamic constraints of the five-axis machine tool and the geometric constraint.

Axis | Maximum Velocity (Unit/s) | Maximum Acceleration (Unit/s^{2}) | Maximum Jerk (Unit/s^{3}) |
---|---|---|---|

X/Y/Z (mm) | 100 | 500 | 3000 |

A (degree) | 22.9 | 28.6 | 85.9 |

C (degree) | 45.8 | 28.6 | 85.9 |

Maximum Chord Error (mm) | 0.125 | ||

Interpolation Cycle Time (s) | 0.002 |

Feedrate (mm/s) | Spindle Speed (rpm) | Cutting Depth (mm) | Cutting Width (mm) |
---|---|---|---|

20 | 6000 | 9 | 0.2 |

Method | Ra (μm) | Rq (μm) | Rz (μm) |
---|---|---|---|

The proposed method | 1.26 | 1.55 | 6.89 |

The simple inverse trigonometric method | 5.67 | 7.48 | 33.78 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sun, P.; Liu, Q.; Wang, J.; Yin, Z.; Wang, L.
Rotation-Angle Solution and Singularity Handling of Five-Axis Machine Tools for Dual NURBS Interpolation. *Machines* **2023**, *11*, 281.
https://doi.org/10.3390/machines11020281

**AMA Style**

Sun P, Liu Q, Wang J, Yin Z, Wang L.
Rotation-Angle Solution and Singularity Handling of Five-Axis Machine Tools for Dual NURBS Interpolation. *Machines*. 2023; 11(2):281.
https://doi.org/10.3390/machines11020281

**Chicago/Turabian Style**

Sun, Pengpeng, Qiang Liu, Jian Wang, Zhenshuo Yin, and Liuquan Wang.
2023. "Rotation-Angle Solution and Singularity Handling of Five-Axis Machine Tools for Dual NURBS Interpolation" *Machines* 11, no. 2: 281.
https://doi.org/10.3390/machines11020281