﻿ 具有同心圆特征的非合作目标超近距离姿态测量
 光电工程  2018, Vol. 45 Issue (8): 180126      DOI: 10.12086/oee.2018.180126

1. 北京理工大学光电学院，北京 100081;
2. 北京理工大学光电成像技术与系统教育部重点实验室，北京 100081;
3. 中国空间技术研究院钱学森实验室，北京 100094

Research on pose measurement between two non-cooperative spacecrafts in close range based on concentric circles
Wang Ke1,2, Chen Xiaomei1,2, Han Xu3
1. School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China;
2. Key Laboratory of Photoelectronic Imaging Technology and System, Beijing Institute of Technology, Ministry of Education of China, Beijing 100081, China;
3. China Academy of Space Technology, Qian Xuesen Laboratory, Beijing 100094, China
Abstract: Conventional measurement of relative poses between two non-cooperative spacecrafts in close range is derived from the iteration of monocular vision or three-dimensional reconstruction of binocular vision, which introduces errors in the process of feature matching, and the timeliness and accuracy are poor. Regarding the issues above, this article tries to do some researches on measurement of relative poses between two non-cooperative spacecrafts in close range based on concentric circles. Here, 'concentric circles' means the spatial parallel but not coplanar positional relationship between docking ring and engine nozzle. Through the binocular vision measurement model, the angle adaptability and the applicability are improved. Then, the algorithm of this model is simulated, and the simulated results show that the accuracy of the algorithm can reach higher than 0.5°.
Keywords: non-cooperative spacecrafts    relative poses measurement    binocular vision measurement    concentric circles

1 引言

2 双目姿态测量模型 2.1 坐标系定义

 图 1 世界坐标系、相机坐标系、图像坐标系、像素坐标系之间的关系 Fig. 1 The relationship among world coordinate system, camera coordinate system, image coordinate system and pixel coordinate system
2.2 测量原理 2.2.1 圆半径和圆心投影位置计算

 图 2 摄像机观察圆形平面的投影图 Fig. 2 The projection of the circular plane from the camera perspective

2.2.2 相对姿态计算

1) 目标卫星相对于追踪卫星姿态角(绕Y轴转动) $\gamma$求解，如图 4所示。

 图 4 XZ平面内的姿态测量模型 Fig. 4 Position measurement model on XZ plane

$\Delta A{C_1}{D_1}$中，$A{C_1}$可以求得：

 $A{C_1} = \frac{{{C_1}{D_1}}}{{\cos {\beta _1}}},$ (1)

$\Delta {C_1}{F_1}{G_1}$$\Delta {C_1}{F_1}{H_1}中，{\beta _1}$${\beta _2}$可以求得：

 ${\beta _1} = \arctan \left( {\frac{{{F_1}{G_1}}}{{{f_1}}}} \right),$ (2)
 ${\beta _2} = \arctan \left( {\frac{{{F_1}{H_1}}}{{{f_1}}}} \right),$ (3)

${C_1}{D_1}$${C_1}{E_1}可以通过小孔成像关系求得：  {C_1}{D_1} = {f_1}\frac{R}{{{R_1}}}, (4)  {C_1}{E_1} = {f_1}\frac{r}{{{r_1}}}, (5) 式中：R$$r$分别为对接环和喷嘴的半径，${R_1}$${r_1}分别为其对应成像大小。 \Delta {C_1}{E_1}B$$\Delta {C_1}{D_1}A$中，$B{E_1}$$A{D_1}可以求得：  B{E_1} = {C_1}{E_1} \cdot \tan {\beta _2}, (6)  A{D_1} = {C_1}{D_1} \cdot \tan {\beta _1}, (7) \Delta {O_1}{E_1}B中，{O_1}{D_1}$$d$(对接环到喷嘴的距离)可以求得：

 ${O_1}{D_1} = \frac{{A{D_1} \cdot ({C_1}{D_1} - {C_1}{E_1})}}{{B{E_1} - A{D_1}}},$ (8)
 $d = \sqrt {({O_1}{E_1}^2 + B{E_1}^2)} - \sqrt {({O_1}{D_1}^2 + A{D_1}^2)} 。$ (9)

Sabatini提出的模型仅能求解主点F1G1点斜下方时的姿态角。如果物距为3000 mm，基线为1800 mm，光轴与Z轴夹角为20°，此时主点F1G1点和H1点之间，用原模型求解姿态角存在较大误差，且只能在一个平面内(如XZ平面)改变相对姿态，改进的模型将分以下三种不同的位置关系求解姿态角，并给出了YZ平面内的姿态测量模型：

① 当主点F1在点G1的斜下方，此时${\beta _2} > {\beta _1}$，在$\Delta A{C_1}B$中可以求得：

 $\sin {\alpha _1} = \frac{{A{C_1}}}{d}\sin ({\beta _2} - {\beta _1}),$ (10)
 ${\alpha _1} = {\alpha _2} + {\beta _2} - {\beta _1},$ (11)
 ${\gamma _1} = {\alpha _1} - {\beta _2} = {\alpha _2} - {\beta _1}。$ (12)

② 当主点F1G1点和H1点之间，在$\Delta A{C_1}B$中可以求得：

 $\sin {\alpha _1} = \frac{{A{C_1}}}{d}\sin ({\beta _2} + {\beta _1}),$ (13)
 ${\alpha _1} = {\alpha _2} + {\beta _1} + {\beta _2},$ (14)
 ${\gamma _1} = {\alpha _2} + {\beta _1} = {\alpha _1} - {\beta _2}。$ (15)

③ 当主点F1H1点斜上方，此时${\beta _2} < {\beta _1}$，在$\Delta A{C_1}B$中可以求得：

 $\sin {\alpha _1} = \frac{{A{C_1}}}{d}\sin ({\beta _1} - {\beta _2}),$ (16)
 ${\alpha _1} = {\alpha _2} + {\beta _1} - {\beta _2},$ (17)
 ${\gamma _1} = {\alpha _1} + {\beta _2} = {\alpha _2} + {\beta _1}。$ (18)

 $\gamma = \frac{{\left| {{\gamma _1} - {\gamma _2}} \right|}}{2}。$ (19)

2) 目标卫星相对于追踪卫星姿态角(绕X轴转动)$\theta$求解，如图 5所示。

 图 5 YZ平面内的姿态测量模型 Fig. 5 Position measurement model on YZ plane

$\Delta {C_1}{O_1}{G_1}$$\Delta {C_1}{O_1}{H_1}$中：

 $2 \cdot {C_1}{O_1} \cdot {C_1}{G_1} \cdot \cos {\alpha _2} = {C_1}{O_1}^2 + {C_1}{G_1}^2 - {O_1}{G_1}^2,$ (20)
 $2 \cdot {C_1}{O_1} \cdot C{H_1} \cdot \cos {\alpha _1}\\~~~ = {C_1}{O_1}^2 + {C_1}{H_1}^2 - {({O_1}{G_1} + {G_1}{H_1})^2},$ (21)

 ${C_1}{G_1} = \frac{{{f_1}}}{{\cos {\beta _1}}},{C_1}{H_1} = \frac{{{f_1}}}{{\cos {\beta _2}}},$ (22)
 ${\theta _1} = \arccos \left( {\frac{{{f_1}}}{{{C_1}{O_1}}}} \right)。$ (23)

 $\theta = \frac{{\left| {{\theta _1} - {\theta _2}} \right|}}{2}。$ (24)
3 实验结果与分析

 图 6 仿真流程 Fig. 6 The process of simulation

 图 7 左、右像面成像示意图。(a)目标卫星相对于追踪卫星在XZ面内顺时针转动1°；(b)目标卫星相对于追踪器在YZ面内顺时针转动1° Fig. 7 The imaging diagrams of left and right image planes. (a) The target satellite rotates 1° clockwise in the XZ plane relative to the tracking satellite; (b) The target satellite rotates 1° clockwise in the YZ plane relative to the tracking satellite

 图 8 改变基线、光轴与Z轴夹角时姿态误差。(a)误差与基线的关系；(b)误差与光轴夹角的关系 Fig. 8 The error of the attitude angle when changing the baseline and the angle between optical axis and z axis. (a) The relationship between the error and the baseline; (b)The relationship between the error and the angle of optical axis

 图 9 单目、双目测量下改变绕X轴、Y轴转角对应的姿态误差。(a)误差与绕Y轴转角的关系；(b)误差与绕X轴转角的关系 Fig. 9 The error of the attitude angle by single and binocular measurements respectively when changing the angle of X axis or Y axis. (a) The relationship between the error and the angle around Y axis; (b) The relationship between the error and the angle around X axis

 图 10 同时改变绕X轴、Y轴转角时姿态误差 Fig. 10 The error of the attitude angle when changing the angle of X axis and Y axis
4 结论

1) 该模型不但可以求出对接环到喷嘴的距离，而且可以确定对接环和喷嘴圆心在像面上投影点的位置，提高了模型的适用性；

2) 该模型可以测量三维的姿态角信息，提高了测量的可靠性；

3) 该模型分三种不同的位置关系求解姿态角，完善了模型的角度适应性问题；

4) 该模型提高了姿态测量精度。

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