基于空间压缩和无穷级数的空间碎片碰撞概率快速算法

A Rapid Algorithm of Space Debris Collision Probability Based on Space Compression and Infinite Series

空间碎片碰撞概率的计算可以转化为2维正态分布概率密度函数(PDF)在圆域内的积分问题,针对以往方法的不足,首先通过空间压缩将不等方差PDF在圆域内的积分化为等方差PDF在椭圆区域内的积分,并将椭圆区域用与其面积相等的圆区域近似,化为等方差PDF在圆域内的积分,对引入的误差进行了讨论.该积分可以表示为首项与递推公式已知的无穷级数形式,可取级数前任意有限项的和作为概率积分的近似,并且给出了截断误差上限的表达式.根据近似结果分析了最大概率的表达式和此时的位置误差参数.以一次实际发生的碰撞为例对该方法进行了验证,并与另外两种进行了比较,结果表明该方法是一种精度高速度快的方法.

The calculation of collision probability between spacecraft and space debris can be transformed to the problem of 2-dimensional integral of anisotropic Gaussian distributed probability density function （PDF） within circle region. With the help of space compression, the integral of anisotropic PDF within circle region is transformed to the integral of isotropic PDF within ellipse region. The ellipse region can be approximated by a circle region which occupies the same area, and the upper-bound of error introduced is expressed analytically. Thus, the integral of isotropic PDF within circle region is obtained. This integral can be expressed as infinite series whose recursion is known. The partial sum of the infinite series could be taken as the approximation of the integral, and the expression of upper-bound of truncation error is presented. The approximate analytical expression is used to analyze the expression of maximum probability. In the end, a space debris collision case is analyzed with this algorithm; the results are compared with those from direct numerical integral algorithm and the algorithm which directly make anisotropic PDF isotropic. It is shown that the algorithm is accurate and rapid.