Autonomous on-orbit servicing is the future space activity which can be utilized to extend the satellite life. Relative pose estimation for a malfunctioned satellite is one of the key technologies to achieve robotic on-orbit servicing. In this paper, a relative pose determination method by using point cloud is presented for the final phase of the rendezvous and docking of malfunctioned satellites. The method consists of three parts: (1) planes are extracted from point cloud by utilizing the random sample consensus algorithm. (2) The eigenvector matrix and the diagonal eigenvalue matrix are calculated by decomposing the point cloud distribution matrix of the extracted plane. The eigenvalues are utilized to recognize rectangular planes, and the eigenvector matrix is the attitude rotation matrix from the sensor to the plane. The solution of multisolution problem is also presented. (3) An extended Kalman filter is designed to estimate the translational states, the rotational states, the location of mass center, and the moment-of-inertia ratios. Because the method only utilizes the local features without observing the whole satellite, it is suitable for the final phase of rendezvous and docking. The algorithm is validated by a series of mathematical simulations.

The on-orbit servicing technology, which is a very challenging research topic, is commonly regarded to be utilized to extend the life of malfunctioned satellites. Although several on-orbit servicing demonstration cases were performed [

Because the relative pose measurement for a malfunctioned satellite is the precondition of on-orbit servicing [

Therefore, other viable image processing method or sensors were utilized by some researchers. Terui et al. used binocular stereo vision to calculate the dense 3D points of target satellite [

However, Lichter and Dubowsky proposed a method to estimate the state, shape, and inertia parameters of noncooperative targets based on a series of distance images [

Inspired by literatures [

Because satellites are man-made objects, there are some regular components on satellites, such as the main body, solar panels, and docking ring. Although these components are not designed specifically to measure the relative pose of malfunctioned satellites, they have obvious geometric features for recognition and can be utilized to realize relative navigation. The rectangular feature is one of the typical geometric shapes on malfunctioned satellites. For example, the main body of most satellites and solar panels is composed of a variety of rectangles. Therefore, this paper proposes to select the rectangular planes as the identification object.

The random sample consensus (RANSAC) algorithm is utilized to extract planes from the point cloud dataset. The algorithm can be described as follows: firstly, select a mathematical model; secondly, find the so-called minimum set from the original point cloud sample, and then estimate the parameters of the model by using the minimum set of points; thirdly, check the consistency of the remaining points with the estimated model, and add these qualified points into the point set; and finally, the parameters of the model can be reestimated using all the points that conform to the model.

According to the aforementioned RANSAC algorithm, the specific steps for extracting the plane from the point cloud are proposed as follows:

Select three points

Calculate the parameters of the plane

Suppose the plane thickness as

Find out how many points are satisfying

Repeat step 1~step 4

Some components of a malfunctioned satellite are composed of rectangular planes. The dimensions of these rectangular planes are known for a specific malfunctioned satellite. After the plane is extracted from the point cloud, the point cloud distribution matrix is utilized to identify if it is a rectangle plane and determine its dimensions.

Before calculating the point cloud distribution matrix, the location of the geometric centroid of the extracted plane can be determined as

In literature [

The eigenvector matrix and the diagonal eigenvalue matrix can be obtained by utilizing a matrix decomposition operation as

The eigenvalue matrix reflects the point cloud distribution in the extracted plane body coordinates, which is described in Figure

The body coordinate system.

Similarly, in the body coordinate system, it is obvious that the eigenvalues matrix can be defined as

It is supposed that the feature points obey a uniform distribution in the feature plane coordinate system and the measurement noise of point cloud is the white Gaussian noise. Therefore,

Therefore, equation (

Suppose

Substitute equation (

If it is supposed that the extracted plane is a rectangular plane and the number of point

Equations (

Similarly,

Suppose

It is regarded that

Similarly,

It can be seen from the above derivation that the eigenvalue matrix

The matrix decomposition operation presented in equation (

Thus, the quaternion of the body coordinate system with respect to the chaser body coordinate system, which is denoted as

If the eigenvalues of

Description of the multisolution problem.

It is supposed that case (1) is the coordinate system of the present time (see Figure

Because the target attitude quaternion

The predicted

If the scalar part of

Cases (2), (3), and (4) need to be changed to case (1); the coordinate system should be rotated 180 degrees around the

If there are 2 or more extracted rectangular planes in a single measurement, the similar problem exists. It is supposed that there are 2 extracted rectangular planes. The 2 attitude quaternions are denoted as

When two satellites are nearly in the same circular orbit with a close distance, the Hill equations can perfectly describe the translational motion dynamics as

As the target satellite is a malfunctioned satellite, the angular velocity cannot be obtained directly. The attitude kinematics and dynamics can be written as

The geometrical relationship between the two satellites is described in Figure

Geometrical relationship between two satellites.

According to the geometrical relationship presented in Figure

Because the two satellites are nearly in same circular orbit with a close distance, the rotation matrix

Therefore, equation (

It is obvious that

As the target is a malfunctioned satellite which is noncooperative, no information including the attitude parameters can be provided to the service spacecraft. An available way to measure the target attitude is combining the relative attitude and the chaser attitude. According to equation (

A set of feature rectangles is tracked by the observer. As the target tumbling in orbit, new rectangular features are observed and must be added to the state vector, and some features disappear and can be only propagated in the filter. This strategy has been proven effective in literatures [

Define state variables as

The dynamic equations are made up of equations (

Define the state transfer matrix

Similarly, in equations (

Therefore,

The detailed information of the EKF can be concluded as

Initialization:

Time update:

Measurement update:

The estimated states

In order to evaluate the effectiveness of extracting a point cloud plane, a point cloud rectangular plane is simulated by the simulation software. The point cloud is randomly simulated obeying a uniform distribution in a rectangular plane, which ranges from -5 cm to 5 cm in the

The original point cloud.

Point cloud with 100 outliers.

Point cloud with 500 outliers.

Point cloud with 1250 outliers.

In this simulation, the repeat number is set to be 300 times, and the distance threshold is 0.005. The extraction results are shown in Table

Plane extraction from the point cloud.

Outliers | The number of extracted points | Time (ms) |
---|---|---|

0 | 2500 | 20 |

100 | 2400 | 20 |

500 | 2000 | 30 |

1250 | 1252 | 70 |

It can be seen in Table

The point cloud declines in number as the distance between the target and the sensor increases. In order to validate the effectiveness of a rectangular feature recognition formula as the number of points varies, the number of points in the rectangular plane is set to be 10000, 5000, 2500, 1250, and 625, respectively. The geometric centroid and the point cloud distribution matrix of the extracted plane are calculated, and the three eigenvalues are obtained by using the matrix decomposition operation. The theoretical value of the geometric centroid is (0, 0, 6.5), and the eigenvalues are 8.3333, 27, and 0, respectively. The geometric centroid and the 3 eigenvalues of the rectangular point cloud are calculated and presented in Table

The geometric centroid and eigenvalues.

Number (points) | The center of gravity | Eigenvalue distribution |
---|---|---|

10000 | (0.0154, -0.0390, 6.5000) | (8.4102, 26.7072, 0) |

5000 | (-0.0079, -0.0090, 6.5000) | (8.3106, 27.3320, 0) |

2500 | (-0.0801, -0.0033, 6.5000) | (8.5595, 26.7923, 0) |

1250 | (0.1542, -0.0438, 6.5000) | (8.4415, 26.4924, 0) |

625 | (0.0906, -0.3373, 6.5000) | (8.2372, 27.7920, 0) |

And then, the random noise of 0.01 cm, 0.1 cm, and 1 cm is added to the rectangular point cloud plane which contains 2500 points. It can be seen in Table

The location error of the geometric centroid.

Error (cm) | The geometric centroid (cm) | The location error (cm) |
---|---|---|

0 | (-0.0801, -0.0033, 6.5000) | (-0.0801, -0.0033, 0) |

0.01 | (0.0817, -0.0833, 6.4999) | (0.0817, -0.0833, 0.0001) |

0.1 | (0.0541, 0.0661, 6.4984) | (0.0541, 0.0661, 0.0016) |

1 | (-0.0261, 0.0210, 6.5254) | (-0.0261, 0.0210, 0.0254) |

The theoretical eigenvalues and the eigenvalues computed by using the matrix decomposition operation are presented in Table

The eigenvalues.

Error (cm) | Eigenvalues (simulation) | Eigenvalues (theory) |
---|---|---|

0 | (8.5595, 26.7923, 0) | (8.3333, 27, 0) |

0.01 | (8.5898, 26.9769, 0.0001) | (8,3334, 27.0001, 0.0001) |

0.1 | (8.3124, 26.9526, 0.0103) | (8.3433, 27.01, 0.01) |

1 | (9.4664, 27.7494, 0.9765) | (9.3333, 28, 1) |

In order to validate the effectiveness of the relative navigation method, digital simulation software has been written in VC++. In this simulation, it is assumed that the target and the chaser are in the same orbit with different orbital phases. The chaser locates at the position which is about 25 m behind the target satellite. The inertia matrix of the target satellite is set as

The PCL (Point Cloud Library) is employed to extract planes. And then, the proposed method is utilized to recognize the feature rectangular plane and determine the relative position and attitude between the feature rectangular plane and the chaser. The relative position and attitude errors are shown in Figures

Position errors.

Attitude errors with multisolution.

Attitude errors removing multisolution.

It can be seen from Figure

In the relative navigation filter,

It can be seen from Figures

Relative position errors.

Relative velocity errors.

Location errors of the mass center.

The estimation errors of the attitude, the angular velocity, and the moment-of-inertia ratios of the target satellite are presented in Figures

Attitude errors of the target.

Angular velocity errors of the target.

Moment-of-inertia ratio errors.

A new method is proposed in this paper to estimate target pose by utilizing local feature rectangular planes when the whole satellite cannot be observed. The plane extraction method, the rectangular feature recognition method, and the solution of multisolution problem are presented. Finally, the Kalman filter is designed to estimate the relative position and the attitude of the target satellite. Therefore, this paper provides a feasible algorithm for relative navigation during rendezvous and docking to malfunctioned satellites.

The data used to support the findings of this study are included within the article. The data is generated by using the simulation software which is described in Simulation and Evaluation. The data used to support the findings of this manuscript: “Pose Determination for Malfunctioned Satellites Based on Depth Information”, written by Feng Yu, Yi Zhao, and Yanhua Zhang, is generated by simulation software. The detailed simulation conditions are presented in detail in this manuscript. The simulation software is available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This study was supported by the “Fundamental Research Funds for the Central Universities,” No. NS2016084. The authors fully appreciate the financial support.