中文摘要：

空间目标碰撞概率的计算是航天器进行空间碎片预謦和规避机动的基础．为了简化计算，目前国内外在计算碰撞概率问题过程中大多基于线性相对运动的条件，将三维碰撞概率计算积分问题简化为位置误差概率密度函数在圆域内的二维积分问题．但是当空间目标相对运动速度较小时，这种线性运动条件不再成立，就需要在真实的非线性相对运动状态下重新考虑碰撞概率的计算问题．研究了非线性相对运动状态下碰撞概率计算的方法，并通过算例分析，验证了其正确性，适用于相对运动位置及其协方差随时间变化的情况．研究结果表明：任飞行器编队绕飞的椭圆相对运动过程中，相对速度在100m／s以下时，采用线性方法与非线性方法得到的结果之间的相对误差超过5％；同时在作圆轨道运动的两物体之间的接近分析过程中，相对速度在10m／s以下时，相对误差也超过1％．这些情况下需要考虑采用非线性方法进行碰撞概率的计算，为我国碰撞预警系统的完善提供一些有益的补充．

英文摘要：

The calculation of collision probability is the foundation of collision detection and avoidance maneuver for space objects. Now an assumption of linear relative motion is usually applied in the calculation of collision probability and then the complex 3-dimensional problem can be reduced to a 2-dimensional integral of probability density fimction over the area of circle. However, if the relative velocity value is very small, the term of linear relative motion is not valid. So it is necessary to consider the calculation of collision probability for nonlinear relative motion. The method used to calculate collision probability for nonlinear relative motion is studied, and test cases are designed to prove the validity of this method. It is applicable to collision probability problems involving relative velocity and error co- variance varying with time. The results indicate that it is necessary to calculate collision probability with this nonlinear method under certain circumstances. For example, for el- liptical relative motions in Satellite Formation Flying, when the relative velocity is below 100 m/s, the relative error between the linear method and tile nonlinear method exceeds 5%; for the problem of conjunction analysis of two satellites with circular orbit, when the relative velocity is below 10 m/s, the relative error is also larger than 1%. Some significant conclusions are obtained for collision detection system of our country.