Space target collision risk analysis algorithm based on the square Mahalanobis distance
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摘要:
针对在空间目标碰撞风险预报中概率稀释现象导致碰撞概率的判断结果可靠性降低的问题，提出了基于平方马氏距离的碰撞风险评估方法，以提高碰撞风险预报结果的可靠性。基于 $ 3\sigma $法则建立平方马氏距离阈值模型，实现碰撞风险的快速判别；对联合球体半径、位置误差、误差椭圆长短轴比及旋转角等因素进行建模和分析，优化平方马氏距离的模型参数，提高平方马氏距离的计算效率；在此基础上，基于平方马氏距离发展碰撞风险预报评估模型，解决概率稀释问题。仿真结果表明：所提方法可有效降低高风险事件的漏检率。
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关键词:
- 空间目标 /
- 碰撞风险分析 /
- 平方马氏距离 /
- 碰撞概率 /
- $ 3\sigma $法则
Abstract:A collision risk assessment method based on the square Mahalanobis distance was proposed to increase the reliability of collision risk prediction results in order to address the issue that probability dilution in space target collision risk prediction reduces the reliability of collision probability. Firstly, the threshold of the square Mahalanobis distance was obtained according to the $ 3\sigma $ rule to determine whether an event has collision risk. Then, the factors such as combined hard-body radius, position error, the aspect ratio of the error ellipse, and the rotation angle were modeled and analyzed, and the model parameters of the square Mahalanobis distance were optimized to improve the computational efficiency. Finally, a collision risk prediction and evaluation model based on the square Mahalanobis distance was developed to solve the probability dilution problem. The outcome of the simulation demonstrates that the suggested approach can successfully lower the high-risk event missed detection rate.
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图 2 $ {P}_{{c}} $和 $ {M}^{2} $随误差椭圆的变化
Figure 2. $ {P}_{{c}} $ and $ {M}^{2} $ vary with the error ellipse
图 3 $ {P}_{{c}} $和 $ {M}^{2} $随位置误差变化( $ {A}_{\text{r}}=2.378\;9 $)
Figure 3. $ {P}_{{c}} $ and $ {M}^{2} $ vary with the position error ( $ {A}_{\text{r}}=2.378\;9 $)
图 4 $ {A}_{\text{r}} $不同，极值点位置随之变化
Figure 4. Position of the extreme point changes with different $ {A}_{\text{r}} $
图 5 $ \theta $变化导致极值点变化
Figure 5. Position of the extreme point changes with different $ \theta $
图 8 $ {\sigma }_{x} $对 $ {V}_{\text{r}} $的影响
Figure 8. Influence of $ {\sigma }_{x} $ on $ {V}_{\text{r}} $
图 9 固定 $ {\sigma }_{x} $，不同 $ \theta $下 $ {V}_{\text{r}} $随 $ {A}_{\text{r}} $和 $ \varDelta /R $的变化
Figure 9. Change of $ {V}_{\text{r}} $ with $ {A}_{\text{r}} $ and $ \varDelta /R $ in different $ \theta $ ( $ {\sigma }_{x} $= 120 m)
图 10 固定 $ {\sigma }_{x} $，不同 $ {A}_{\text{r}} $下 $ {V}_{\text{r}} $随 $ \theta $和 $ \varDelta /R $的变化
Figure 10. Change of $ {V}_{\text{r}} $ with $ \theta $ and $ \varDelta /R $ in different $ {A}_{\text{r}} $( $ {\sigma }_{x} $= 120 m)
表 1 $ {{M}}^{2} $和 $ {M}_{\min }^{2} $之间的偏差
Table 1. Deviation between $ {{M}}^{2} $ and $ {M}_{\min }^{2} $
R/m $ \varDelta $/m $ {\sigma }_{x} $/m $ \theta $/rad $ {A}_{\text{r}} $ 10 50 120 π/36 200 
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表 2 碰撞风险预报评估方法中的参数
Table 2. Parameters in collision risk prediction and evaluation method
参数 数值 $ {U}_{\text{DR}} $ 100 $ {U}_{\text{vr}} $ 1.2 $ {E}_{\text{ac}} $ 10.8 $ [{A}_{\text{r}\min } \;\;\;{A}_{\text{r}\max }] $ [ 0.0550 18.1900 ]$ [{\theta }_{\min }\;\;\; {\theta }_{\max }] $/ rad [ 0.1141 1.4567 ]+ kπ/2(k = 0, −1)
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表 3 FLOCK 4Y-25和SL-3 DEB的TLE
Table 3. TLE of FLOCK 4Y-25 and SL-3 DEB
卫星/碎片 两行轨道根数 FLOCK 4Y-25 1 55027U 23001U 24010.16624038 .00012377 00000+0 52739-3 0 9990 2 55027 97.4492 72.8718 0010226 14.6525 345.5006 15.23053222 56530 SL-3 DEB 1 14928U 83075F 24010.30073432 .00006854 00000+0 32096-3 0 9991 2 14928 97.7779 279.1809 0026055 294.6725 65.1796 15.19768913182859
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表 4 卫星FLOCK 4Y-25和碎片SL-3 DEB间的碰撞风险分析
Table 4. Collision risk analysis between Satellite FLOCK 4Y-25 and Debris SL-3 DEB
危险
区间区间起止时间 危险
持续
时长/s最接近
时刻最接近距
离/km$ {{M}}^{2} $ $ {M}_{\min }^{2} $ 是否
计算
最小值$ \theta $/rad $ {A}_{\text{r}} $ $ {P}_{{c}}(2\text{D}) $ $ {P}_{{c}}(\max ) $ $ {\sigma }_{d}/{\sigma }_{x} $ $ R/{\left| \boldsymbol{C}\right| }^{1/4} $ 评估
结果第①
段01-13T18:26:40.000—
01-13T18:26:41.0001 01-13T18:26:40.405 24.794 8 10 817 10 808 是 −0.023 5
(N)0.016 7
(N)−∞ −∞ 2.937 5 0.005 4 安全 第②
段01-13T19:13:56.000—
01-13T19:13:57.0001 01-13T19:13:56.765 22.828 8 103.231 6 103.141 1 否 −0.017 7
(N)0.157 4
(Y)−∞ −∞ 2.991 3 0.001 8 安全 第③
段01-13T20:01:21.000—
01-13T20:01:24.0003 01-13T20:01:22.701 0.025 5 4.4804 ×
10−71.6338 ×
10−7否 0.229 1
(Y)0.521 4
(Y)−7.730 4 −1.529 9 5.342 9×
10−41.928 9×
10−4危险 第④
段01-13T20:48:39.000—
01-13T20:48:40.0001 01-13T22:48:39.056 24.421 8 0.020 6 0.020 6 否 −0.077 5
(N)0.441 5
(Y)−9.119 3 −7.562 8 0.101 6 3.918 2×
10−5安全
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