航天器碰撞概率简化计算方法的比较与应用
doi: 10.11728/cjss2025.01.2024-0031 cstr: 32142.14.cjss.2024-0031
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王宁 男, 2000年3月出生于山西省运城市. 现为北京控制工程研究所硕士研究生, 主要研究方向为航天器空间规避策略. E-mail: wnqwert123@163.com -
郭建新 男, 1978年出生于江西省. 现为北京控制工程研究所研究员, 硕士生导师, 主要研究方向为卫星自主导航与轨道控制. E-mail: gjxwanquan@163.com
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通讯作者:
Comparison and Application of Simplified Calculation Methods for Spacecraft Collision Probability
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摘要: 针对空间飞行器碰撞概率问题, 基于碰撞概率的假设条件与计算原理, 对现有常见方法进行分析, 归纳每种方法在线性与非线性、空间目标是否规则等方面的适用性及特点, 通过数值计算验证了上述方法, 并从计算速度与计算精度两个维度对每种方法进行评估. 评估结果表明, 中心概率密度法的计算速度最快,曲线积分法的相对误差最小, 基于辛普森公式的二维积分方法具有较均衡的效果, 三维体积积分适用于非线性场景,蒙特卡洛法最为精准但是运算速度最慢. 综合考虑计算精度、计算速度、相对速度、相对位置关系等因素, 根据每种方法的特点提出了多种场景下较为适用的碰撞概率简化计算方法, 研究结果可以为在轨碰撞概率的计算与分析提供参考.Abstract: With the rapid increase in the number of debris and spacecraft in space, spacecraft need to assess collision risks in real time. However, due to the limited on-board software and hardware resources, there is an urgent need for a collision probability calculation method with appropriate computational load and high assessment accuracy. Aiming at the problem of spacecraft collision probability, this paper firstly explains the assumptions and calculation principles of collision probability, then summarizes and analyzes the existing common methods, summarizes the applicability and characteristics of each method in terms of linearity and nonlinearity, whether the space target is regular, etc., and finally verifies the adaptability of the above methods through numerical calculation, and evaluates each method from the two dimensions of calculation speed and calculation accuracy. The evaluation results show that the central probability density method has the fastest calculation speed. The relative error of the curve integration method is the smallest, the two-dimensional integration method based on Simpson’s formula has a relatively balanced effect. 3D volume integration is suitable for nonlinear scenes, the Monte Carlo method is the most accurate, but the slowest. Considering the factors such as calculation accuracy, calculation speed, relative velocity, and relative position relationship, this paper proposes a simplified calculation method for collision probability in various scenarios according to the characteristics of each method, and the research results can provide a reference for the calculation and analysis of on-orbit collision probability.
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Key words:
- Collision probability /
- Covariance matrix /
- Space debris /
- Nonlinear relative motion /
- Space vehicles
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表 1 两个航天器在J2000惯性系下的初始位置速度
Table 1. Initial position velocity of two spacecraft in an inertial frame of J2000
Object x/km y/km z/km A –150.798437 1182.676987 –7143.77556 B –150.619960 1182.018386 –7143.78799 Object vx / (km·s–1) vy / (km·s–1) vz / (km·s–1) A 7.290071 –1.126282 –0.354014 B 5.915107 4.347936 0.643388 
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表 2 二维碰撞概率计算结果
Table 2. Calculation results of two-dimensional collision probability
方法 碰撞概率 相对误差率 计算时间/s 二维面积积分法 $ 7.066\times {10}^{-4} $ － 0.1097 中心概率密度面积法 $ 7.051\times {10}^{-4} $ 0.0021 0.0064 一重积分法 $ 7.066\times {10}^{-4} $ $ 8.9\times {10}^{-14} $ 0.1259 级数法 $ 7.051\times {10}^{-4} $ 0.0021 0.0126 矩形域近似法 $ 7.067\times {10}^{-4} $ $ 1.00\times {10}^{-4} $ 0.0392 曲线积分法 $ 7.066\times {10}^{-4} $ $ 5.3\times {10}^{-15} $ 0.1097 辛普森公式积分法 $ 7.066\times {10}^{-4} $ $ 5.64\times {10}^{-6} $ 0.0074
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表 3 蒙特卡罗方法计算结果
Table 3. Calculation results by Monte Carlo method
方法 碰撞概率 运算次数/次 计算时间/s 二维蒙特卡洛法 $ 7.068\times {10}^{-4} $ $ 1\times {10}^{10} $ 1224.6 三维蒙特卡洛法 $ 7.076\times {10}^{-4} $ $ 5\times {10}^{6} $ 67180.4
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表 4 不同应用情形下碰撞概率工程适用方法
Table 4. Calculation method applicable to the collision probability under different circumstances
相遇情形 方法 同轨 体积拼接积分法 异轨 辛普森公式积分法 多目标碰撞概率计算 中心概率密度面积法 编队绕飞 体积拼接积分法、曲线积分法 合作目标 曲线积分法 非合作目标 辛普森公式积分法 高精度计算 蒙特卡洛法、体积拼接积分法
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