Relations Between Collision Probability, Mahalanobis Distance, and Confidence Intervals for Conjunction Assessment

Some of the most common metrics for collision risk assessment are the probability of collision, miss distance in Mahalanobis space, and confidence intervals. Sometimes they are used in combination; for example, the miss distance is used as a prescreening method to identify potentially hazardous conjunctions; other times they are used as alternative means; that is, covariance ellipse overlapping checks are employed instead of computing the probability of collision. In this work, we show that the three risk indexes are intimately connected once a suitable distance is defined. We argue that Mahalanobis miss distance is a proper metric only when the sigma-normalized hard-body size is negligible; we thus investigate the minimum Mahalanobis distance between the hard-body circle and the combined position covariance as an alternative. Its computation is fully analytic, as the most complex operation is finding the roots of a quartic polynomial. When multiplied by the sigma-normalized hard-body radius, such distance provides an upper bound to the collision probability. When used to scale the covariance matrix, it provides the largest confidence interval supporting a noncollision event. Finally, when adopted as an actionable threshold, analytical bounds on the probability of miss detection and of false alarms can be computed.