# Relative Pose Determination of Uncooperative Spacecraft Based on Circle Feature

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Architecture of the Proposed Method

#### 2.1. Coordinate Systems

#### 2.2. Relative Pose Calculation Model

#### 2.3. ToF Camera Measurement Model and Data Transformation

## 3. Initial Point Cloud Segmentation Assisted by the Gray Image

## 4. Feature Parameters Calculation of the Non-Ideal Circle

#### 4.1. Circle Feature Parameters Calculation of Each Group of Point Cloud

#### 4.2. Parameters Fusion Based on Point Cloud

## 5. Experiments and Numerical Simulations

#### 5.1. Circle Feature Point Cloud Segmentation Experiment

#### 5.2. Relative Pose Accuracy and Robustness of Non-Ideal Circle Features of Point Clouds

- (1)
- Standard model settings: The standard model is a point cloud of the frustum of a cone with a resolution of 2 mm (as shown in Figure 15). The upper bottom’s radius is 12 cm, which is fixed on the mounting surface. On the standard plane, the lower bottom’s outer radius is 24 cm and the inner radius is 20 cm, which is the end face. The height of the model is 10 cm. Furthermore, the standard position of the circle feature model is fixed to the origin in CRF, i.e., the center of the end face coincides with the origin $(0,0,0)$. The Euler angles are all 0 and the transformation matrix is the identity matrix I.
- (2)
- Noise addition: Different levels of noises (0.01–1 magnitude of the resolution of point cloud standard model) are added to the standard model to verify the algorithm’s strong anti-interference ability.
- (3)
- Model transformation: 100 groups of positions and Euler angles are generated randomly as nominal values. The final circle feature observation point cloud is generated by the noise-added standard model transformation according to the nominal values.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Data of Time-of-Flight (ToF) camera. (

**a**) The captured gray image; (

**b**) The captured point cloud in the field of view (76,800 points). Obviously, the circle feature only accounts for a small part of the point cloud.

**Figure 11.**Direct segmentation result of point cloud. (

**a**) the result after the planar background segmentation; (

**b**) the result after the clustering segmentation.

**Figure 13.**(

**a**) The result of point cloud segmentation assisted by gray images; (

**b**) the result after the clustering segmentation.

**Figure 14.**(

**a**) The final results of the direct segmentation; (

**b**) final results of the segmentation assisted by gray image.

Direct Method | Gray Image Auxiliary Method | Improvement (%) | |
---|---|---|---|

Ellipse Detection Time (s) | — | 0.047 | — |

Data Conversion Time (s) | 26.625281 | 1.764461 | — |

Circle Segmentation Time (s) | 7.687266 | — | |

Total Time (s) | 34.312547 | 1.811461 | 94.72 |

Divided Point Number | 76,800 | 4970 | 93.53 |

Noise(mm) | Error phi (${}^{\circ}$) | Error Theta (${}^{\circ}$) | Error Angle (${}^{\circ}$) | |||
---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | |

0.02 | 1.5986 × ${10}^{-5}$ | 2.1604 × ${10}^{-5}$ | 2.2504 × ${10}^{-5}$ | 7.1102 × ${10}^{-5}$ | 0.0014 | 0.0031 |

0.05 | 3.0365 × ${10}^{-5}$ | 6.8873 × ${10}^{-5}$ | 3.8961 × ${10}^{-5}$ | 6.6372 × ${10}^{-5}$ | 0.0025 | 0.0042 |

0.08 | 6.6854 × ${10}^{-5}$ | 9.6205 × ${10}^{-5}$ | 6.6996 × ${10}^{-5}$ | 0.0001 | 0.0052 | 0.0067 |

0.2 | 0.0001 | 0.0001 | 0.0002 | 0.0003 | 0.0099 | 0.0114 |

0.5 | 0.0002 | 0.0003 | 0.0004 | 0.0004 | 0.0225 | 0.0189 |

0.8 | 0.0003 | 0.0004 | 0.0010 | 0.0012 | 0.0424 | 0.0233 |

2 | 0.0009 | 0.0007 | 0.0017 | 0.0015 | 0.0921 | 0.0555 |

Noise(mm) | Position(mm) | |
---|---|---|

Mean | Variance | |

0.02 | 0.0250 | 0.0152 |

0.05 | 0.0620 | 0.0278 |

0.08 | 0.0977 | 0.0721 |

0.2 | 0.2909 | 0.1844 |

0.5 | 0.6009 | 0.3109 |

0.8 | 0.8829 | 0.4063 |

2 | 2.3774 | 1.0606 |

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Liu, Y.; Zhang, S.; Zhao, X.
Relative Pose Determination of Uncooperative Spacecraft Based on Circle Feature. *Sensors* **2021**, *21*, 8495.
https://doi.org/10.3390/s21248495

**AMA Style**

Liu Y, Zhang S, Zhao X.
Relative Pose Determination of Uncooperative Spacecraft Based on Circle Feature. *Sensors*. 2021; 21(24):8495.
https://doi.org/10.3390/s21248495

**Chicago/Turabian Style**

Liu, Yue, Shijie Zhang, and Xiangtian Zhao.
2021. "Relative Pose Determination of Uncooperative Spacecraft Based on Circle Feature" *Sensors* 21, no. 24: 8495.
https://doi.org/10.3390/s21248495