Formal Verification of Conflict Detection Algorithms for Arbitrary Trajectories

This paper presents an approach for developing formally verifiable conflict detection algorithms for aircraft flying arbitrary, nonlinear trajectories. The approach uses a multivariate polynomial global optimization algorithm based on Bernstein polynomials. Since any continuous function on a closed interval, such as an aircraft trajectory within a closed interval of time, can be uniformly approximated by a Bernstein polynomial, this global optimization algorithm can be used to define conflict detection algorithms for arbitrarily complicated trajectories. Conflict detection algorithms developed using this approach can be formally verified in a mechanical theorem prover. This represents an improvement over standard approaches to conflict detection for complex trajectories that essentially search for conflicts by testing many future states and are therefore not guaranteed to detect a given conflict. The proposed approach is illustrated with a formally verified conflict detection algorithm.