Closed-Form Pairwise Aircraft Conflict Resolution with First-Order Uncertainty Propagation

This paper presents a closed-form framework for detecting and resolving predicted losses of horizontal separation between two level aircraft. In the relative-velocity plane, the method derives the time and distance of closest approach, the first loss-of-separation time, the complete conflict-free track-angle intervals for an ownship-only maneuver, synchronized same-direction track changes, equal-and-opposite ground-speed changes, and a nominal separation-constrained return to the original trajectories. Existence conditions and degenerate geometries are stated for each maneuver family. A first-order Jacobian model propagates uncertainty from measured position, ground speed, track angle, and maneuver-execution time to the resolution variables. Direct trajectory tests, randomized heading sweeps, table recomputation, and nonlinear Monte Carlo propagation are used to evaluate the analytical solutions. Across the tested encounter grid, synchronized track changes reduce resolution-angle uncertainty relative to ownship-only maneuvers, with the largest reduction in reciprocal encounters and a smaller benefit at shallow intercept angles. The framework provides computationally inexpensive analytical candidates and explicit sensitivity information under its stated two-dimensional, instantaneous-maneuver, and input-error assumptions.