The proliferation of space debris in LEO has become a major concern for the space industry. With the growing interest in space exploration, the prediction of potential collisions between objects in orbit has become a crucial issue. It is estimated that, in orbit, there are millions of fragments a few millimeters in size and thousands of inoperative satellites and discarded rocket stages. Given the high speeds that these fragments can reach, even fragments a few millimeters in size can cause fractures in a satellite's hull or put a serious crack in the window of a space shuttle. The conventional method proposed by Akella and Alfriend in 2000 remains widely used to estimate the probability of collision in short-term encounters. Given the small period of time, it is assumed that, during the encounter: (1) trajectories are represented by straight lines with constant velocity; (2) there is no velocity uncertainty and the position exhibits a stationary distribution throughout the encounter; and (3) position uncertainties are independent and represented by Gaussian distributions. This study introduces a novel derivation based on first principles that naturally allows for tight and fast upper and lower bounds for the probability of collision. We tested implementations of both probability and bound computations with the original and our formulation on a real CDM dataset used in ESA's Collision Avoidance Challenge. Our approach reduces the calculation of the probability to two one-dimensional integrals and has the potential to significantly reduce the processing time compared to the traditional method, from 80% to nearly real-time.

翻译：低地球轨道空间碎片的激增已成为航天工业的主要关切。随着对太空探索兴趣的日益增长，预测轨道物体间的潜在碰撞已成为关键问题。据估计，轨道上存在数百万个毫米级尺寸的碎片，以及数千个失效卫星和废弃火箭级。考虑到这些碎片所能达到的高速，即使毫米级碎片也可能导致卫星外壳破裂或在航天飞机舷窗上造成严重裂痕。Akella与Alfriend于2000年提出的传统方法至今仍被广泛用于估计短期遭遇的碰撞概率。鉴于遭遇时间极短，该方法假设：(1) 轨迹以恒定速度的直线表示；(2) 不存在速度不确定性，且位置分布在遭遇期间呈现平稳分布；(3) 位置不确定性相互独立并服从高斯分布。本研究提出一种基于第一性原理的新推导方法，可自然导出紧致且可快速计算的碰撞概率上下界。我们在欧空局碰撞规避挑战赛使用的真实CDM数据集上，分别测试了原始方法与本研究的概率计算及边界计算实现。我们的方法将概率计算简化为两个一维积分，与传统方法相比，有望将处理时间从80%显著降低至近实时水平。