A Guaranteed Geometric Approach to Covariance-Ellipsoid Separation for Satellite Collision Avoidance

Abstract A novel geometric collision risk assessment method is introduced that determines the unique minimum separation distance between two covariance ellipsoids representing the positional uncertainty of space objects. By reformulating the problem to find the unique point at which the normal vectors of the uncertainty envelopes are parallel, we eliminate the need for complex sextic polynomial root-finding. The methodology employs a characteristic bisection algorithm generalizable to n dimensions, guaranteeing a real solution with O (1) complexity regardless of dimensionality. This provides a critical enabling technology for next-generation AI-enabled space traffic management (STM), facilitating both real-time autonomous decision-making and the integration of high-dimensional non-spatial risk factors.