# A Toolpath Planning Method for Optical Freeform Surface Ultra-Precision Turning Based on NURBS Surface Curvature

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## Abstract

**:**

## 1. Introduction

## 2. Reconstruction of the NURBS Surfaces

## 3. Toolpath Planning Method for Freeform Surfaces Based on NURBS Surface Curvature

#### 3.1. The Archimedes Spiral Method

#### 3.2. Calculation of Toolpath Parameters Base on Analysis of NURBS Surfaces

#### 3.2.1. Analysis of NURBS Surfaces

#### 3.2.2. Calculation of Toolpath Parameters

_{n}of the contact point, and machining row distance L concerning various surface contact areas.

_{n}of the contact point in the radial direction may be regarded as infinite. The mathematical model linking the tool radius R, the residual height h, and the machining row distance L is shown in Equation (17).

#### 3.3. Toolpath Based on NURBS Surface Curvature

_{i,j}is the contact point between the diamond tool and the surface, and the point P

_{i,j}is the center point of the radius of the diamond tool tip circle. However, Q

_{i,j}cannot be directly applied to machining programs on ultra-precision machines. Therefore, tool contacts need to be converted to tool arc center points by means of tool compensation.

_{i,j}$({x}_{ij},{y}_{ij},{z}_{ij})$. The direction of compensation is the normal vector of the tool contact point along the front tool surface projection direction, and the compensation amount is the arc radius R of the diamond tool, as shown in Figure 9. The components of the compensation quantity in the x and z directions are calculated in Equations (21) and (22).

_{i,j}$({{x}_{ij}}^{\prime},{{y}_{ij}}^{\prime},{{z}_{ij}}^{\prime})$ is the coordinate of the center of the arc radius of the tool. It is the toolpath point. The toolpath for freeform ultra-precision slow tool servo diamond turning is shown in Figure 10.

## 4. Simulation and Analysis

_{1}≠ L

_{2}≠ L

_{3}.

_{4}= L

_{5}= L

_{6}.

## 5. Experiment and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Results of data parameterization; (

**b**) the distribution direction of the NURBS surface.

**Figure 7.**The area of contact between the tool and the machined surface: (

**a**) the contact area between the tool and the machined surface; (

**b**) the contact area between the tool and the machined surface is flat; (

**c**) the contact area between the tool and the machined surface is convex; (

**d**) the contact area between the tool and the machined surface is concave.

**Figure 10.**Toolpath for ultra-precision slow tool servo diamond turning after tool nose radius compensation.

**Figure 11.**Comparison of simulation results: (

**a**) the toolpaths generated by Archimedes spiral projection; (

**b**) the toolpaths generated by the toolpath planning method based on NURBS surface curvature.

**Figure 13.**Comparison of the simulation results of the residual height: (

**a**) the residual height of the toolpath generated by Archimedes spiral projection; (

**b**) the residual height of the toolpath generated by the toolpath planning method based on NURBS surface.

**Figure 16.**Measurements of the residual height in different areas of the freeform surfaces: (

**a**) measurement results corresponding to area I in Figure 15; (

**b**) measurement results corresponding to area II in Figure 15; (

**c**) measurement results corresponding to area III in Figure 15; (

**d**) measurement results corresponding to area IV in Figure 15.

**Figure 17.**Analysis of the residual height of different areas of the freeform surfaces: (

**a**) the residual height corresponding to area I in Figure 15; (

**b**) the residual height corresponding to area II in Figure 15; (

**c**) the residual height corresponding to area III in Figure 15; (

**d**) the residual height corresponding to area IV in Figure 15.

Processing Parameters | Numeric Requirements |
---|---|

Spindle speed | 60 rpm |

Tool arc radius | 1.06 mm |

Tool material | Diamond |

Tool rake angle | 0° |

Tool back angle | 15° |

Surface radius | 12.5 mm |

Machining row spacing | 0.012 mm |

Depth of cut | 0.01 mm |

Materials | PMMA |

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**MDPI and ACS Style**

Wang, X.; Bai, Q.; Gao, S.; Zhao, L.; Cheng, K.
A Toolpath Planning Method for Optical Freeform Surface Ultra-Precision Turning Based on NURBS Surface Curvature. *Machines* **2023**, *11*, 1017.
https://doi.org/10.3390/machines11111017

**AMA Style**

Wang X, Bai Q, Gao S, Zhao L, Cheng K.
A Toolpath Planning Method for Optical Freeform Surface Ultra-Precision Turning Based on NURBS Surface Curvature. *Machines*. 2023; 11(11):1017.
https://doi.org/10.3390/machines11111017

**Chicago/Turabian Style**

Wang, Xuchu, Qingshun Bai, Siyu Gao, Liang Zhao, and Kai Cheng.
2023. "A Toolpath Planning Method for Optical Freeform Surface Ultra-Precision Turning Based on NURBS Surface Curvature" *Machines* 11, no. 11: 1017.
https://doi.org/10.3390/machines11111017