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Abstract

In view of the mission of the cooperative target capture with the space manipulator, this article proposes to establish pose measurement model of the space cooperative target capture based on thick lens model for the first time. First, a ground simulation system is established for the cooperative target capture with the space manipulator based on the zoom vision system, coordinate system is defined, and pose calculation process is given. Then, space coordinate measurement model and pose calculation model are established for the system. Finally, the simulation experiments are conducted and the results validate the algorithm model and the high accuracy of measurement.

Keywords

Zoom vision system space manipulator pose measurement space cooperative target

Introduction

In recent years, target pose measurement has become a research hotspot^1–3 and is potentially useful in more fields.^4–9 As the space mission becomes more complicated and diversified, the computer vision–based space manipulator system is more applied to space missions.^10–14 However, it is difficult for traditional fixed-focus camera to maintain a high image resolution in the whole motion process of the space manipulator and cooperative target,^15–17 and the measurement accuracy will be affected accordingly. That is one of the major challenges for accomplishing the above space mission with the computer vision–based system.

On the other hand, the application of multiple fixed-focus cameras to capture overall targets in space mission will result in increased complexity of the mechanism, decreased robustness, and more space energy consumption, bringing great loads to the spacecraft body. Therefore, it is evident that extensive application of the zoom vision–based space manipulator will show important values.

As early as in 1990, J Ma and S Olsen¹⁸ proved that the zoom lens can be used to obtain depth information of the measured target. On this basis, JM Lavest et al.¹⁹ and their vision team carried out further researches and proved that the thick lens model is an ideal model for the zoom camera. In the subsequent studies, the team systematically explicated the difficulties in the calibration of vision measurement system.^20–23 In 2000, Y-S Chen et al.²⁴ from National Taiwan University proposed a simple and effective method of calibrating electric zoom cameras’ internal parameters. In 2001, a laboratory, CVIP at the University of Louisville, calibrated the zoom camera parameters using neural network method.²⁵ In 2009, M Sarkis et al.²⁶ from Technical University of Munich (Germany) calibrated automatic zoom camera using the least square method. In 2010, the Health Science College of Milwaukee University (USA) and the Broadcasting Institute of Technology of Yonsei University (Korea) jointly corrected the radial distortion of the automatic zoom camera.²⁷ In 2012, L Alvarez et al.²⁸ of ULPGC University (Spain) proposed a new mathematical model to study the lens distortion when the focal length of the zoom lens changes. In the same year, H Gao et al.²⁹ of Shenyang Ligong University (China) calibrated zoom center of image plane. In 2013, J Kang et al.³⁰ from Institute of KBS (Korea) put forward a kind of optical axis compensation method based on the binocular zoom camera.

The major contributions of this article are as follows. In view of the mission of the cooperative target capture with the space manipulator on the space station, the hardware-in-the-loop (HWIL) simulation experimental system is set up in this article. Considering the imaging principle, thick lens model is introduced to present the imaging relation of the zoom camera. Pose measurement model is established for the space cooperative target capture based on the zoom vision system, and the simulation is conducted to validate the model.

Space monocular zoom visual system

System composition

According to the real mission of the space manipulator in on-orbit service, a simulation experiment system is established on the ground, as shown in Figure 1.

Figure 1.

System composition.

In Figure 1, the system consists of six parts: ground microgravity system, space station, space manipulator, zoom vision system, cooperative target, and ground telemetry system. The camera of the zoom vision system is located above the base of the space station, the space manipulator marker is located at the end effector, and the cooperative target marker is located at the cooperative target. When the space manipulator executes the cooperative target capture mission, the ground telemetry system will send commands to each system, while the zoom vision system will calculate the pose information of the end effector of the manipulator and the captured mechanism of cooperative target, and the space station will program the trace of the space manipulator according to the commands to capture cooperative target. On the ground, the system can achieve free floating state of the space manipulator and cooperative target, complete target grabbing, and releasing tests for the manipulator.

Visual measurement coordinate system

The coordinate systems are defined for the measurement model of the system based on the structure of the simulation experiment system, as shown in Figure 2.

Figure 2.

Visual measurement coordinate system.

In Figure 2, the system contains world coordinate system $O_{w} X_{w} Y_{w} Z_{w}$ (at the mass center of the root joint of the space manipulator marker), space manipulator marker coordinate system $O_{t 1} X_{t 1} Y_{t 1} Z_{t 1}$ , end effector coordinate system $O_{m 1} X_{m 1} Y_{m 1} Z_{m 1}$ (its pose correlation with space manipulator marker coordinate system to be ensured by mechanical geometry), cooperative target marker coordinate system $O_{t 2} X_{t 2} Y_{t 2} Z_{t 2}$ , and captured mechanism coordinate system $O_{m 2} X_{m 2} Y_{m 2} Z_{m 2}$ . At the same time, the camera base coordinate system is defined as $O_{c 0} X_{c 0} Y_{c 0} Z_{c 0}$ , when the focal length of the zoom camera is in the minimum position $f_{0}$ . The camera moving coordinate system is defined as $O_{cn} X_{cn} Y_{cn} Z_{cn}$ , when the focal length of the zoom camera is in the position $f_{n}$ .

Calculation process of pose measurement

When the space manipulator executes the cooperative capture mission, it is proposed to define the pose of the end effector of the space manipulator in the world coordination system, define the pose of the captured mechanism of the cooperative target in the world coordinate system, and accordingly define the pose correlation between the end effector and the captured mechanism. Figure 3 demonstrates the calculation process.

Figure 3.

Process of pose measurement.

First, to place the zoom camera above the space station to collect five feature points on the space manipulator marker and six feature points on the cooperative target marker and acquire the corresponding image coordinates. Then, to solve the coordinates of the feature points in moving coordinate system $O_{cn} X_{cn} Y_{cn} Z_{cn}$ at current focus $f_{n}$ . At this point, also to calculate the coordinates of the feature points in the base coordinate system by use of the rotation and transform correlation between the zoom camera moving coordinate system and base coordinate system. The world coordinate system lies in the mass center of the root joint of the space manipulator, and the rotation and transform matrix between it and camera base coordinate system is given by the mechanism; thus, the world coordinates of each feature point can be obtained. Since the coordinates of the feature points in the corresponding marker coordinate system, the correlation between the space manipulator marker coordinate system and end effector coordinate system, and the correlation between cooperative target marker coordinate system and captured mechanism coordinate system are known, the world coordinate system can be obtained, respectively, for the end effector and the captured mechanism by use of transform correlation. Accordingly, the pose correlation between the captured mechanism and end effector can be defined.

Pose measurement model

Thick lens model

In space measurement missions, the zoom camera is hardly applied, because its internal and external parameters change with focal length, and such change can greatly increase the fluctuation in pose measurement. Studies show that thick lens model is considered the ideal for zoom lens. By establishing zoom camera model based on thick lens model and applying the model to space measurement missions, the disadvantage of pinhole camera can be overcome.

As shown in Figure 4, $H_{oxy}$ and $H_{ixy}$ are called principal plane, perpendicular to the optical axis. The distance between $H_{oxy}$ and $H_{ixy}$ changes with the focal length, that is, the two principal planes can move along the optical axis. If the camera coordinate system is set on plane $H_{oxy}$ , the internal and external parameters of pinhole model can be applicable. Based on thick lens model, the zoom vision system for pose measurement is modeled as follows.

Figure 4.

Thick lens model.

Zoom coordinate measuring model

According to Figure 3, in zoom camera measurement system, it is required to establish the imaging correlation between the zoom camera and manipulator marker feature points and the imaging correlation between the camera and cooperative marker feature points. See Figure 5 for image correlation of two feature points.

Figure 5.

The relationship between image coordinate system and camera moving coordinate system.

In Figure 5, $O_{cn} X_{cn} Y_{cn} Z_{cn}$ is the zoom camera coordinate system in principal plane $H_{o} xy$ at focal length $f_{n}$ . $O_{1} xy$ is the physical image coordinate system and $O_{0} uv$ is the image pixel coordinate system. Suppose $P_{i}$ and $P_{j}$ are the two feature points of the same marker, their three-dimensional (3D) coordinates in the camera moving coordinate system are denoted as $P_{cn, i} = (X_{cn, i}, Y_{cn, i}, Z_{cn, i})^{T}$ and $P_{cn, j} = (X_{cn, j}, Y_{cn, j}, Z_{cn, j})^{T}$ . The distance between the two feature points is $d_{ij}$ ; the two feature points form an angle $α_{cn, ij}$ at optical center $O_{cn}$ ; their image points on image plane are $p_{cn, i}$ and $p_{cn, j}$ , respectively; their physical image coordinates, respectively, at $p_{cn, i} = (x_{cn, i}, y_{cn, i})^{T}$ and $p_{cn, j} = (x_{cn, j}, y_{cn, j})^{T}$ and the distance between them and optical point are $S_{cn, i}$ and $S_{cn, j}$ .

From perspective imaging correlation, equation (1) is obtained as

{\begin{cases} x_{c n, i} = f_{n} \frac{X_{c n, i}}{Z_{c n, i}}, y_{c n, i} = f_{n} \frac{Y_{c n, i}}{Z_{c n, i}} \\ x_{c n, j} = f_{n} \frac{X_{c n, j}}{Z_{c n, j}}, y_{c n, j} = f_{n} \frac{Y_{c n, j}}{Z_{c n, j}} \end{cases}

(1)

The angle $α_{cn, ij}$ is expressed as

\cos α_{c n, i j} = {\overset{⇀}{v}}_{c n, i} \cdot {\overset{⇀}{v}}_{c n, j}

(2)

where ${\overset{⇀}{v}}_{c n, i}$ and ${\overset{⇀}{v}}_{c n, j}$ are the unit vectors and denoted as

{\begin{array}{l} {\overset{⇀}{v}}_{c n, i} = \frac{1}{\sqrt{x_{c n, i}^{2} + y_{c n, i}^{2} + f_{n}^{2}}} (\begin{matrix} x_{c n, i} \\ y_{c n, i} \\ f_{n} \end{matrix}) \\ {\overset{⇀}{v}}_{c n, j} = \frac{1}{\sqrt{x_{c n, j}^{2} + y_{c n, j}^{2} + f_{n}^{2}}} (\begin{matrix} x_{c n, j} \\ y_{c n, j} \\ f_{n} \end{matrix}) \end{array}

(3)

The 3D coordinates of the feature points in the camera coordinate system are expressed as

{\begin{array}{l} P_{c n, i} = S_{c n, i} {\overset{⇀}{v}}_{c n, i} \\ P_{c n, j} = S_{c n, j} {\overset{⇀}{v}}_{c n, j} \end{array}

(4)

In $Δ O_{cn} P_{cn, i} P_{cn, j}$ , according to the law of cosines, equation (5) is obtained as

d_{ij}^{2} = S_{cn, i}^{2} + S_{cn, j}^{2} - 2 S_{cn, i} S_{cn, j} \cos α_{cn, ij}

(5)

Considering the research requirements on the cooperative target capture with the manipulator in the space station, five feature points are established for the space manipulator marker and six feature points for the cooperative target marker. Therefore, $S_{cn, i}$ and $S_{cn, j}$ can be calculated by three quadratic equations composed of three random feature points as

{\begin{matrix} d_{ij}^{2} = S_{cn, i}^{2} + S_{cn, j}^{2} - 2 S_{cn, i} S_{cn, j} \cos α_{cn, ij} \\ d_{ik}^{2} = S_{cn, i}^{2} + S_{cn, k}^{2} - 2 S_{cn, i} S_{cn, k} \cos α_{cn, ik} \\ d_{jk}^{2} = S_{cn, j}^{2} + S_{cn, k}^{2} - 2 S_{cn, j} S_{cn, k} \cos α_{cn, jk} \end{matrix}

(6)

$S_{cn, i}$ , $S_{cn, j}$ , and $S_{cn, k}$ can be obtained from equation (6). The calculated results can be optimized by use of the redundant feature points on the marker and then put in equation (4) to calculate the corresponding 3D coordinates of the feature points in zoom camera moving coordinate system $O_{cn} X_{cn} Y_{cn} Z_{cn}$ .

Pose solution model

The transform correlation of the space feature points and the zoom camera moving coordinate system can be known from section “Zoom coordinate measuring model.” Set the coordinates of the feature points on the manipulator marker as $P_{t 1, ai}$ in the marker coordinate system and $P_{cn, ai}$ in the zoom camera moving coordinate system and set the coordinates of the feature points on the cooperative target marker as $P_{t 2, bi}$ in the marker coordinate system and $P_{cn, bi}$ in the zoom camera moving coordinate system, then

{\begin{matrix} P_{cn, ai} = R_{tcn, a} P_{t 1, ai} + T_{tcn, a} \\ P_{cn, bi} = R_{tcn, b} P_{t 2, bi} + T_{tcn, b} \end{matrix}

(7)

where $R_{tcn, a}$ is the orthogonal rotation matrix between the zoom camera moving coordinate system and end effector marker coordinate system and $R_{tcn, b}$ is the orthogonal rotation matrix between the zoom camera moving coordinate system and cooperative target marker coordinate system. $T_{tcn, a}$ and $T_{tcn, b}$ are the translation matrices, and the measurement model introduced in section “Zoom coordinate measuring model” can be used to solve equation (7).

Suppose the coordinates of the space feature points on the end effector marker are $P_{c 0, ai}$ and $P_{cn, ai}$ , respectively, in the zoom camera base coordinate system and zoom camera moving coordinate system, while the coordinates of the space feature points on cooperative target marker are $P_{c 0, bi}$ and $P_{cn, bi}$ , respectively, in the zoom camera base coordinate system and zoom camera moving coordinate system, then

{\begin{matrix} P_{c 0, ai} = R_{ccn} P_{cn, ai} + T_{ccn} \\ P_{c 0, bi} = R_{ccn} P_{cn, bi} + T_{ccn} \end{matrix}

(8)

where $R_{ccn}$ is the orthogonal rotation matrix between the zoom camera moving coordinate system and zoom camera base coordinate system; $T_{ccn}$ is the translation matrix and can be defined by means of camera calibration.

Then, the coordinates of the space feature points on the manipulator marker and cooperative target marker can be transformed from $P_{c 0, ai}$ and $P_{c 0, bi}$ (in the zoom camera base coordinate system) to $P_{w, ai}$ and $P_{w, bi}$ (in the world coordinate system) as follows

{\begin{matrix} P_{w, ai} = R_{cw} P_{c 0, ai} + T_{cw} \\ P_{w, bi} = R_{cw} P_{c 0, bi} + T_{cw} \end{matrix}

(9)

where $R_{cw}$ is the orthogonal rotation matrix between the zoom camera base coordinate system and world coordinate system; $T_{cw}$ is the translation matrix and can be defined according to the geometric features of aircraft body.

To establish correlation between the space manipulator end effector in the world coordinate system, the coordinate transformation between the end effector and space manipulator marker shall be known. Set the coordinates to be $P_{m 1, ai}$ in the end effector coordinate system and $P_{m 2, bi}$ in the captured mechanism coordinate system, then

{\begin{matrix} P_{t 1, ai} = R_{mt, a} P_{m 1, ai} + T_{mt, a} \\ P_{t 2, bi} = R_{mt, b} P_{m 2, bi} + T_{mt, b} \end{matrix}

(10)

where $R_{mt, a}$ is the orthogonal rotation matrix between the end effector coordinate system and manipulator marker coordinate system and $T_{mt, a}$ is the translation matrix, while $R_{mt, b}$ is the orthogonal rotation matrix between the captured mechanism coordinate system and cooperative target marker coordinate system and $T_{mt, b}$ is the translation matrix and can be defined according to the geometric features of aircraft body.

Thus, the coordinates of the manipulator end effector and captured mechanism of cooperative target can be obtained as

{\begin{matrix} P_{w, ai} = R_{mw, a} P_{m 1, ai} + T_{mw, a} \\ P_{w, bi} = R_{mw, b} P_{m 2, bi} + T_{mw, b} \end{matrix}

(11)

where $R_{mw, a} = R_{cw} R_{ccn} R_{tcn, a} R_{mt, a}$ , $T_{mw, a} = R_{cw} R_{ccn} R_{tcn, a} T_{mt, a} + R_{cw} R_{ccn} T_{tcn, a} + R_{cw} T_{ccn} + T_{cw}$ , $R_{mw, b} = R_{cw} R_{ccn} R_{tcn, b} R_{mt, b}$ , and $T_{mw, b} = R_{cw} R_{ccn} R_{tcn, b} T_{mt, b} + R_{cw} R_{ccn} T_{tcn, b} + R_{cw} T_{ccn} + T_{cw}$ .

Finally, the relation between the captured mechanism of the cooperative target and space manipulator end effector can be expressed as

P_{m 1, ai} = R_{mw, a}^{- 1} R_{mw, b} P_{m 2, bi} + R_{mw, a}^{- 1} (T_{mw, b} - T_{mw, a})

(12)

where $R_{mw, a}^{- 1} R_{mw, b}$ is the rotation matrix while $R_{mw, a}^{- 1} (T_{mw, b} - T_{mw, a})$ is the translation matrix. They are attitude matrix and position matrix of the captured mechanism relative to the manipulator end effector.

Simulation and analysis

Main simulation conditions

Condition 1. Camera pixel resolution: 1400 × 1024; principal point position: u₀ = 700, v₀ = 512; pixel size: 7.4 µm × 7.4 µm; zoom lens focus range: 16–64 mm; camera at focus f = 24.

Condition 2. $P_{t 1, a 1} = (100, 0, 0)^{T}$ , $P_{t 1, a 2} = (200, 0, 100)^{T}$ , $P_{t 1, a 3} = (300, 0, 200)^{T}$ , $P_{t 1, a 4} = (400, 0, 100)^{T}$ , $P_{t 1, a 5} = (500, 0, 0)^{T}$ , $P_{t 2, b 1} = (100, 0, 0)^{T}$ , $P_{t 2, b 2} = (200, 0, 100)^{T}$ , $P_{t 2, b 3} = (300, 0, 200)^{T}$ , $P_{t 2, b 4} = (400, 0, 200)^{T}$ , $P_{t 2, b 5} = (500, 0, 100)^{T}$ , $P_{t 2, b 6} = (600, 0, 0)^{T}$ , $R_{ccn} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ , $T_{ccn} = [\begin{matrix} 0 \\ 0 \\ 7.4343 \end{matrix}]$ , $R_{cw} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0.8660 & 0.5000 \\ 0 & - 0.5000 & 0.8660 \end{matrix}]$ , and $T_{cw} = [\begin{matrix} 0 \\ 1000 \\ 200 \end{matrix}]$ .

Condition 3. Set the transformation correlation between the end effector coordinate system and captured mechanism coordinate system as follows: the translation in three directions is $(100, 200, 500)^{T}$ , the rotation angle is 0 both in the x- and y-directions, and varies in the range $[- 20 \circ, 20 \circ]$ in the z-direction, with 1° step.

Simulation steps

Step 1. To generate image points of each feature point and apply noise of 0.03, 0.1, and 0.3 pixel standard deviation, respectively.

Step 2. To solve the coordinates $P_{cn, ai}$ and $P_{cn, bi}$ of the feature points in the camera moving coordinate system based on equations (1)–(6).

Step 3. To solve $R_{tcn, a}$ , $R_{tcn, b}$ , $T_{tcn, a}$ , and $T_{tcn, b}$ based on equation (7).

Step 4. To obtain the feature point coordinates $P_{c 0, ai}$ and $P_{cn, ai}$ in the camera moving coordinate system based on equation (8).

Step 5. To obtain the feature point coordinates $P_{w, ai}$ and $P_{w, bi}$ in the world coordinate system based on equation (9).

Step 6. To obtain $P_{m 1, ai}$ and $P_{m 2, bi}$ based on equation (10).

Step 7. To calculate attitude matrix $R_{mw, a}^{- 1} R_{mw, b}$ and translation matrix $R_{mw, a}^{- 1} (T_{mw, b} - T_{mw, a})$ of the captured mechanism of the cooperative target relative to the manipulator end effector.

Simulation experiments

The sub-pixel location technique will be applied in the actual system, and 0.03, 0.1, and 0.3 pixel are the typical standard deviations for such location. Therefore, the corresponding noise level is assumed in the simulation.

Figures 6 and 7 show the 6 degree-of-freedom (DOF) errors at 0.03 pixel standard deviation. In Figure 6, the attitude error results of x-, y-, and z-axes are given, respectively, by three curves. Among them, the z-axis attitude error is less than 0.01°, which is better than x- and y-axes. In Figure 7, the translation error results of x-, y-, and z-axes are given, respectively, by three curves. It can be seen that the x-axis and y-axis translation error is less than 0.02 mm.

Figure 6.

The attitude error in 0.03 pixel standard deviation.

Figure 7.

The translation error in 0.03 pixel standard deviation.

Figures 8 and 9 show the 6-DOF errors at 0.1 pixel standard deviation. In Figure 8, the attitude error results of x-, y-, and z-axes are given, respectively, by three curves. Among them, the z-axis attitude error is less than 0.03°, which is better than x- and y-axes. In Figure 9, the translation error results of x-, y-, and z-axes are given, respectively, by three curves. It can be seen that the x-axis and y-axis translation error is less than 0.6 mm.

Figure 8.

The attitude error in 0.1 pixel standard deviation.

Figure 9.

The translation error in 0.1 pixel standard deviation.

Figures 10 and 11 show the 6-DOF errors at 0.3 pixel standard deviation. In Figure 10, the attitude error results of x-, y-, and z-axes are given, respectively, by three curves. Among them, the z-axis attitude error is less than 0.1°, which is better than x- and y-axes. In Figure 11, the translation error results of x-, y-, and z-axes are given, respectively, by three curves. It can be seen that the x-axis and y-axis translation error is less than 1.5 mm.

Figure 10.

The attitude error in 0.3 pixel standard deviation.

Figure 11.

The translation error in 0.3 pixel standard deviation.

Overall, the maximum angle error is 0.04° and the maximum displacement error is 0.6 mm at 0.03 pixel standard deviation. The maximum angle error is 0.15° and the maximum displacement error is 2.0 mm at 0.1 pixel standard deviation. The maximum angle error is 0.8° and the maximum displacement error is 5.0 mm at 0.1 pixel standard deviation. The angle error of 1.2° and the displacement error of 8 mm is allowed in space missions, so the results are acceptable.

Conclusion

This article proposes a pose measurement model for the cooperative target capture with the space manipulator based on the zoom vision system. A zoom coordinate measurement model is established based on thick lens model to complete the pose solution model. Finally, simulation experiments are conducted to validate the model, and the results show that the maximum angle error is 0.04° and the maximum displacement error is 0.6 mm at 0.03 pixel standard deviation, the maximum angle error is 0.15° and the maximum displacement error is 2.0 mm at 0.1 pixel standard deviation, and the maximum angle error is 0.8° and the maximum displacement error is 5.0 mm at 0.3 pixel standard deviation, all of which are within the tolerance. The model established herein will play an instructive and reference role in the space target measurement mission.

Footnotes

Academic Editor: Nasim Ullah

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China under grant no. 61171189.

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Pose measurement model of space cooperative target capture based on zoom vision system

Abstract

Keywords

Introduction

Space monocular zoom visual system

System composition

Visual measurement coordinate system

Calculation process of pose measurement

Pose measurement model

Thick lens model

Zoom coordinate measuring model

Pose solution model

Simulation and analysis

Main simulation conditions

Simulation steps

Simulation experiments

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References