# Pose Estimation of Non-Cooperative Space Targets Based on Cross-Source Point Cloud Fusion

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## Abstract

**:**

## 1. Introduction

- (1)
- We propose a cross-source point cloud fusion algorithm, which uses the unified and simplified expression of geometric elements in conformal geometry algebra, breaks the traditional point-to-point correspondence, and constructs a matching relationship between points and spheres, to obtain the attitude transformation relationship of the two point clouds. We used the geometric meaning of the inner product in CGA to construct the similarity measurement function between the point and the sphere as the objective function.
- (2)
- To estimate the pose of a non-cooperative target, the contour model of the target needs to be reconstructed first. To improve its reconstruction accuracy, we propose a plane clustering method based CGA to eliminate point cloud diffusion after fusion. This method combines the shape factor concept and the plane growth algorithm.
- (3)
- We introduced a twist parameter for rigid body pose analysis in CGA and combined it with the Clohessy–Wiltshire equation to obtain the posture and other motion parameters of the non-cooperative target through the unscented Kalman filter.
- (4)
- We designed numerical simulation experiments and semi-physical experiments to verify our non-cooperative target measurement system. The results show that the proposed cross-source point cloud fusion algorithm effectively solves the problem of low point cloud overlap and large density distribution differences, and the fusion accuracy is higher than that of other algorithms. The attitude estimation of non-cooperative targets meets the requirement of measurement accuracy, and the estimation error of the angle of the rotating spindle is 30% lower than that of other methods.

## 2. Method

#### 2.1. Measurement System and Algorithm Framework

#### 2.2. Cross-Source Point Cloud Fusion

#### 2.2.1. Rotation Invariant Features Based on CGA

#### 2.2.2. Find the Corresponding Point and Sphere

- Calculate the centroids of the two point clouds and move the centroids to the origins of their respective coordinate systems: ${P}_{C}=\frac{1}{{n}_{p}}{\displaystyle \sum}_{i=1}^{{n}_{p}}{p}_{i}$, ${Q}_{C}=\frac{1}{{n}_{Q}}{\displaystyle \sum}_{j=1}^{{n}_{Q}}{Q}_{j}$.
- Taking the centroid as the starting point and using the farthest point sampling method, uniformly sample n points in the reference point cloud.
- Given the distance from the centroid of the reference point cloud to each sampling point, {${d}_{l}^{1},{d}_{l}^{2},{d}_{l}^{3},\dots \dots {d}_{l}^{n}\}$, look for the distance to the centroid in the moving point cloud to satisfy the relationship $\Vert {d}_{k}^{j}-{d}_{l}^{i}\Vert <T$. In this paper, T is the average distance between points of the moving point cloud.
- Use the ${d}^{4}$ descriptor from (13) to judge the n points sampled and retain the points that satisfy (13). Otherwise, continue sampling until the final number of retained sampling points reaches m pairs. (The value of m is discussed in Section 4 of the paper).
- According to Equations (10)–(12), construct the corresponding feature sphere in the moving point cloud and calculate the center and radius of the sphere.

#### 2.2.3. Identifying Optimal Registration

#### 2.3. Point Cloud Model Reconstruction

#### 2.4. Estimation of Attitude and Motion Parameters

Algorithm 1. UKF Algorithm
| |

Input: | State variable at time k ${X}_{k}={\left[{x}_{k},{W}_{k},{V}_{k}\right]}^{T}$, covariance matrix${P}_{k}=\left[\begin{array}{ccc}{P}_{k}& & \\ & {Q}_{k}& \\ & & {R}_{k}\end{array}\right]$ |

Output: | At time k+1 ${X}_{k+1}$, ${P}_{k+1}$ |

Initial value: | ${X}_{0}={\left[{X}_{0},0,0\right]}^{T}$,${P}_{0}=\left[\begin{array}{ccc}{P}_{0}& & \\ & {Q}_{0}& \\ & & {R}_{0}\end{array}\right]$ |

Iteration process | |

Step 1: | Convert the pose quaternion into the error twistor ${\widehat{B}}_{\u2206}$, and perform sigma sampling on it |

Step 2: | Convert the sigma sample point of the error twistor into a pose dual quaternion, and obtain a new prediction point ${x}_{k+1,k}^{i}$ through the state equation |

Step 3: | Determine the mean and covariance of the predicted points ${\widehat{X}}_{k+1,k}={\sum}_{i=0}^{2L}{W}_{i}{x}_{k+1,k}^{i}$, ${\delta}_{{X}_{k+1}}^{i}={x}_{k+1,k}^{i}-{\widehat{X}}_{k+1,k}$, and ${P}_{k+1,k}={\sum}_{i=0}^{2L}{W}_{i}{\delta}_{{X}_{k+1}}{\delta}_{{X}_{k+1}}{}^{T}$ |

Step 4: | Obtain the new observation point mean and covariance through the observation equations ${\widehat{Z}}_{k+1,k}={\sum}_{i=0}^{2L}{W}_{i}{z}_{k+1,k}^{i}$, ${\delta}_{{Z}_{k+1}}^{i}={z}_{k+1,k}^{i}-{\widehat{Z}}_{k+1,k}$, and ${P}_{{z}_{k+1}}={\sum}_{i=0}^{2L}{W}_{i}{\delta}_{{Z}_{k+1}}{\delta}_{{Z}_{k+1}}{}^{T}$ |

Step 5: | Determine the Kalman gain for the UKF using ${P}_{XZ}^{k+1}={\sum}_{i=0}^{2L}{\delta}_{{X}_{k+1}}^{i}{\delta}_{{Z}_{k+1}}{}^{T}$ and ${K}_{k+1}={P}_{XZ}^{k+1}/{P}_{{z}_{k+1}}$ |

Step 6: | Update the status according to ${X}_{k+1}={X}_{k}+{K}_{k+1}\left({Z}_{k+1}-{\widehat{Z}}_{k+1}\right)$ and ${P}_{k+1}={P}_{k}-{K}_{k+1}{P}_{{z}_{k+1}}{K}_{k+1}{}^{T}$ |

## 3. Experimental Results and Analysis

#### 3.1. Numerical Simulation

- Step 1:
- Set the positions of the Kinect and LiDAR sensors in the Blendor software, and select the appropriate sensor parameters. Table 2 lists the sensor parameter settings for the numerical simulation in this study.
- Step 2:
- Rotate the satellite model once around the z axis (inertial coordinate system), and collect 16 frames of point cloud in 22.5° yaw angle increments.
- Step 3:
- Use the proposed fusion algorithm to register the point clouds collected by the two sensors.
- Step 4:
- Rotate the satellite model around the x axis or y axis, and repeat steps 2–4.
- Step 5:
- Adjust the distance between the two sensors and the satellite model, and repeat steps 2–5.
- Step 6:
- Use the root mean square error (RMSE) to verify the effectiveness of the algorithm.

#### 3.1.1. Registration Results

#### 3.1.2. Pose Estimation Results of Simulated Point Clouds

#### 3.2. Semi-Physical Experiment and Analysis

#### 3.2.1. Experimental Environment Setup

#### 3.2.2. Results of Semi-Physical Experiments

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Comparison of Kinect (upper) and LiDAR (lower) point clouds at different rotation angles. (

**b**) Comparison of the numbers of frame point clouds with LiDAR and Kinect at different distances.

**Figure 4.**Rotation-invariant features of the four geometries. (

**a**) * means the reference point cloud, and + means the target point cloud. (

**b**) Line segment composed of any two points ${p}_{1}\wedge {p}_{2}\wedge {e}_{\infty}$. (

**c**) Plane composed of any three points ${p}_{1}\wedge {p}_{2}\wedge {p}_{3}\wedge {e}_{\infty}$. (

**d**) Circumscribed sphere ${p}_{1}\wedge {p}_{2}\wedge {p}_{3}\wedge {p}_{4}$ made up of four points.

**Figure 6.**(

**a**–

**c**) When rotating around the x axis, y axis, and z axis, simulated point cloud fusion error for different rotation angles. (

**d**) RMSE change curve for registration at different distances.

**Figure 8.**(

**a**) Translational position, (

**b**) angular position, (

**c**) velocity, and (

**d**) angular velocity estimation results of the target spacecraft.

**Figure 9.**Experimental device for simulating the relative motion between the spacecraft and target spacecraft.

**Figure 10.**(

**a**–

**c**) When rotating around the x axis, y axis, and z axis, real point cloud fusion error for different rotation angles. (

**d**) RMSE change curve for registration at different distances.

**Figure 13.**Non-cooperative target pose estimation results of our method: (

**a**) translational position, (

**b**) rotational position, (

**c**) velocity, and (

**d**) angular velocity estimation results.

**Figure 14.**Comparison of the (

**a**) translational position, (

**b**) rotation angle, (

**c**) velocity, and (

**d**) angular velocity estimation errors of the proposed algorithm and a previously reported algorithm [57].

**Figure 15.**(

**a**–

**c**) The x axis, y axis, and z axis, respectively, as the main axis of rotation, time and RMSE variation of different rotation attitudes. (

**d**) Time and error changes when the service spacecraft approaches from far to near.

**Table 1.**Expression of the inner product of different geometric objects in CGA [42].

Geometric Objects | Geometric Shape | Inner Product Representation |
---|---|---|

Point and point | ${P}_{1}\xb7{P}_{2}=-\frac{1}{2}{\left({P}_{2}-{P}_{1}\right)}^{2}$ | |

Point and plane | $P\xb7\pi =P\xb7n-d$ | |

Plane and sphere | $\pi \xb7S=n\xb7S-d$ | |

Sphere and sphere | ${S}_{1}\xb7{S}_{2}=\frac{1}{2}\left({r}_{1}^{2}+{r}_{2}^{2}\right)-\frac{1}{2}\left({s}_{2}^{2}-{s}_{1}^{2}\right)$ |

Sensor | Parameter | Value |
---|---|---|

Kinect 1.0 | Resolution | 640 px × 480 px |

Focal Length | 4.73 mm | |

FOV (V,H) | (43°, 57°) | |

Scan Distance | 1–6 m | |

Velodyne HDL-64 | Scan Resolution | (0.08°–0.35°) H |

0.4° V | ||

Scan Distance | 120 m | |

FOV (V) | 26.9° |

Parameter | Symbol | Value |
---|---|---|

Initial relative position | $\left({\mathrm{P}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}},{\mathrm{P}}_{\mathrm{z}}\right)$ | (0, 0, 0) m |

Initial relative velocity | $\left({\mathrm{V}}_{\mathrm{x}},{\mathrm{V}}_{\mathrm{y}},{\mathrm{V}}_{\mathrm{z}}\right)$ | (0, 0, 0) m/s |

Orbit altitude (circular orbit) | $\mathrm{r}$ | 0.2 m |

Initial Euler angle | $\left(\mathrm{Pitch},\mathrm{Roll},\mathrm{Yaw}\right)$ | (0, 0, 0) |

Initial angular velocity | $\left({\mathrm{W}}_{\mathrm{x}},{\mathrm{W}}_{\mathrm{y}},{\mathrm{W}}_{\mathrm{z}}\right)$ | (0, 0, 5) °/s |

Sensor acquisition frequency | $\mathrm{f}$ | 0.22 Hz |

Frame number | $\mathrm{N}$ | 16 |

Cloud | Depth | Range of Detection | Depth Uncertainty | Angle | |||
---|---|---|---|---|---|---|---|

Resolution | FPS | Resolution | FPS | Horizontal | Vertical | ||

640 × 480 | 30 | 320 × 240 (16 bit) | 30 | 0.8–6.0 m | 2–30 mm | 57° | 43° |

Channel Number | Rotation Speed | Scan Distance | FOV | Angular Resolution | ||
---|---|---|---|---|---|---|

Vertical | Horizontal | Vertical | Horizontal | |||

32 nonlinear | 300/600/1200 rpm | 0.4–200 m | −25°~+15° | 360° | 0.33° | 0.1°~0.4° |

Average Number of Point Clouds in a Frame (Kinect/LIDAR) | Our Method (s) | FMR [34] (s) | CBD [36] (s) | ||||||
---|---|---|---|---|---|---|---|---|---|

Rotate Around x axis | Rotate Around y axis | Rotate Around z axis | Rotate Around x axis | Rotate Around y axis | Rotate Around z axis | Rotate Around x axis | Rotate Around y axis | Rotate Around z axis | |

1239/126 | 3.06 | 3.46 | 3.60 | 2.06 | 2.01 | 2.56 | 5.79 | 5.60 | 6.17 |

1724/217 | 3.18 | 3.57 | 3.91 | 2.15 | 2.96 | 2.96 | 5.41 | 6.54 | 6.29 |

2653/358 | 3.43 | 3.31 | 4.18 | 2.40 | 3.29 | 3.06 | 6.32 | 6.38 | 6.33 |

4496/641 | 4.20 | 3.78 | 3.73 | 2.43 | 3.75 | 3.76 | 6.70 | 6.25 | 7.05 |

7921/951 | 4.10 | 4.35 | 4.27 | 3.18 | 3.09 | 3.92 | 6.19 | 6,06 | 7.10 |

8105/828 | 3.99 | 3.89 | 4.51 | 3.26 | 4.56 | 4.75 | 6.82 | 6.14 | 7.24 |

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## Share and Cite

**MDPI and ACS Style**

Li, J.; Zhuang, Y.; Peng, Q.; Zhao, L.
Pose Estimation of Non-Cooperative Space Targets Based on Cross-Source Point Cloud Fusion. *Remote Sens.* **2021**, *13*, 4239.
https://doi.org/10.3390/rs13214239

**AMA Style**

Li J, Zhuang Y, Peng Q, Zhao L.
Pose Estimation of Non-Cooperative Space Targets Based on Cross-Source Point Cloud Fusion. *Remote Sensing*. 2021; 13(21):4239.
https://doi.org/10.3390/rs13214239

**Chicago/Turabian Style**

Li, Jie, Yiqi Zhuang, Qi Peng, and Liang Zhao.
2021. "Pose Estimation of Non-Cooperative Space Targets Based on Cross-Source Point Cloud Fusion" *Remote Sensing* 13, no. 21: 4239.
https://doi.org/10.3390/rs13214239