# A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve

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## Abstract

**:**

## 1. Introduction

## 2. Trajectory Fitting Based on the NURBS Curve

## 3. Fairing of the NURBS

## 4. NURBS Interpolation

#### 4.1. Feed-Rate Planning

**Step 1:**considering that now there is no analytical solution to calculate the length of NURBS, the Simpson formula is used to obtain the estimation of the length through numerical iteration:

**Step 2**: the length between the two points corresponding to the parameters $up$ and $low$ is calculated:

**Step 3:**the error between the aforementioned two lengths is calculated:

**Step 4**: the parameter interval $\left(0,1\right)$ was sectioned by the equivalent distance $up-low$ to generate the knot vector $\left({u}_{0},{u}_{1},\dots ,{u}_{n}\right)$. Thus, the length of the NURBS is:

#### 4.2. NURBS Interpolation

## 5. Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The initial fitted polishing trajectory for fairing optimisation and interpolation (

**a**) the fitted polishing trajectory (

**b**) the partial enlarged view.

**Figure 4.**The end-point trajectory before, and after, fairing optimization: (

**a**) the end-point trajectory; and (

**b**) the partial enlarged view

**Figure 5.**The end-point trajectory interpolated by different methods: (

**a**) the end-point trajectory; and (

**b**) the partial enlarged view.

**Figure 6.**Runtime of the trajectory interpolated by the proposed method: (

**a**) runtime of the trajectory; and (

**b**) the partial enlarged view.

**Figure 7.**Runtime of the trajectory interpolated by the linear method: (

**a**) runtime of the trajectory; and (

**b**) the partial enlarged view.

**Figure 9.**Changes in surface errors (

**a**) before and (

**b**) after conducting the linear interpolation-based polishing.

**Figure 10.**Changes in surface errors (

**a**) before and (

**b**) after conducting the proposed interpolation-based polishing.

Trajectory | Before | After | ||
---|---|---|---|---|

Maximum Curvature | Maximum Shear Jerk | Maximum Curvature | Maximum Shear Jerk | |

End-point | 1.249 | 0.285 | 0.253 | 0.025 |

Reference-point | 1.380 | 0.325 | 0.506 | 0.032 |

Method | Interpolation Error (mm) | Runtime Error (s) | ||
---|---|---|---|---|

Maximum | Mean | Maximum | Mean | |

Proposed | 0.005 | 0.002 | 0.010 | 0.005 |

Linear | 0.091 | 0.076 | 0.130 | 0.076 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, D.; Guo, H.
A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve. *Appl. Sci.* **2018**, *8*, 1355.
https://doi.org/10.3390/app8081355

**AMA Style**

Zhao D, Guo H.
A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve. *Applied Sciences*. 2018; 8(8):1355.
https://doi.org/10.3390/app8081355

**Chicago/Turabian Style**

Zhao, Dong, and Hao Guo.
2018. "A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve" *Applied Sciences* 8, no. 8: 1355.
https://doi.org/10.3390/app8081355