A conservative analytical framework for UAV collision risk assessment under position uncertainty using noncentral chi-square distributions

In low-altitude air traffic management, trajectory-execution uncertainties introduce non-negligible mid-air collision risks, requiring computationally efficient and reliability-oriented risk assessment for short-term UAV encounters. This paper develops a conservative analytical framework for collision-risk evaluation under locally Gaussian position uncertainty. The instantaneous collision probability over sphere and ellipsoid collision zones is first formulated in terms of noncentral chi-square random variables. A minimum-eigenvalue-based approximation is then developed to provide a conservative upper bound, leading to the Nchi-S and Nchi-E methods for sphere and ellipsoid collision zones, respectively. From a reliability perspective, pairwise collision is interpreted as an encounter-level failure event, and survival-analysis concepts are used to relate instantaneous collision risk to finite-horizon collision-free performance. Under locally constant covariance and locally linear relative motion, the Maximum Instantaneous Collision Probability (MICP) is further used to characterize the critical risk level over a short encounter horizon. A Mahalanobis-distance-based separation criterion is subsequently established to link prescribed collision-risk requirements with uncertainty-aware separation design. Numerical experiments quantify the tightness and computational cost of the proposed conservative bound and show that it provides computationally efficient risk estimates without observed underestimation in the tested cases. The results further demonstrate that the Mahalanobis-distance-based criterion provides conservative separation boundaries with improved airspace efficiency over conventional CPA-based criteria under anisotropic uncertainty. The resulting framework provides a computationally efficient pairwise risk-evaluation module for tactical conflict screening and reliability-oriented low-altitude traffic management.