Complete Analytical Solutions for Pairwise Aircraft Conflict Resolution and First-Order Error-Propagation Assessment of Solution Uncertainty

Abstract

Air traffic controller workload for separation assurance grows with traffic density, moti- vating automation of conflict detection and resolution (CD&R). This paper develops a com- plete, case-exhaustive closed-form treatment of the resolution of predicted losses of horizontal separation between a pair of level aircraft, unifying ownship-only, synchronized-track, and synchronized-speed maneuvers in a single relative-velocity formulation. Working in the rel- ative velocity plane, in which the intruder is stationary and the ownship moves with the relative ground-speed vector, and starting from the established encounter parameters (the time and distance of closest approach and the time of first loss of separation), the paper derives: (i) the complete family of conflict-free track angles for an ownship-only maneuver, as explicit angular intervals delimited by the tangent lines to the protected circle of radius Sm; (ii) closed-form solutions for a synchronized, same-direction track change by both air- craft; (iii) closed-form solutions for a synchronized, equal-and-opposite ground-speed change without altering either track; and (iv) an analytical geometry for a safe return to the original trajectory once the conflict is cleared. Because resolution maneuvers are computed in ad- vance of execution, the solution variables inherit the uncertainty of the measured positions, ground speeds, and track angles and of the maneuver execution time. A first-order sta- tistical error-propagation model traces these input uncertainties, taken as 95%-containment bounds, through the resolution formulas; numerical results for representative geometries with prediction (look-ahead) times of five and eight minutes quantify the growth of track-angle so- lution uncertainty as the intercept angle decreases, and show that synchronized maneuvering roughly halves the resolution track-angle uncertainty relative to an ownship-only maneuver.