GodwitNavSOLUTIONS

Research

Closed-form conflict resolution, with its uncertainty stated.

Given two level aircraft on constant tracks, when will they come too close, which maneuvers fix it, and how much should we trust those answers given what we don’t know precisely about where the aircraft are and how fast they are going?

Animated illustration of conflict detection and resolution

  1. DetectPredict the closest approach
  2. ResolveBoth aircraft turn together
  3. ReturnRejoin the original tracks
Head-on encounter
  • Aircraft A
  • Aircraft B
  • 2.5 NM ring: rings touch at 5 NM
An illustration of the maneuver families in our paper, at about 60× speed. Both aircraft fly 250 kt. The common turn is found numerically for this page with a 25% margin; it is not our published closed-form solver.

The problem, in the relative-velocity plane

Assume both aircraft hold constant velocity and are currently separated by more than Sm. Fix the intruder and let the ownship move with the relative velocity. A loss of separation is predicted when that relative track passes inside the intruder’s protected circle. The two tangent lines to the circle bound the conflict, and every resolution maneuver can be read off this picture.

tCPA = −(r · vrel) / |vrel|²
dCPA = |r + vrel tCPA|
predicted conflict when tCPA > 0 and dCPA < Sm

Here r is the ownship’s position relative to the intruder and vrel the ownship’s velocity relative to the intruder, taken as nonzero. A negative tCPA means the closest approach is already past.

Relative-velocity geometry of a pairwise conflict The intruder sits at the center of a circle of radius S m. From the ownship, two tangent lines to the circle bound the conflict cone. The current relative track passes inside the circle, so the miss distance d CPA is smaller than S m. Ownship Intruder Sm dCPA relative track tangent (boundary) tangent (boundary)
Original schematic drawn for this site. The shaded wedge is the set of relative tracks that lose separation; turning the relative track outside either tangent line resolves the conflict.

What the paper derives

Every result is closed-form, with existence conditions and degenerate geometries stated for each maneuver family.

  • Detection

    Time and distance of closest approach, and the time separation is first lost.

  • Ownship-only track changes

    The complete conflict-free track-angle intervals when only one aircraft maneuvers.

  • Synchronized track changes

    Same-direction track changes by both aircraft, executed together.

  • Speed changes

    Equal-and-opposite ground-speed changes that restore separation without turning.

  • Return to track

    A nominal, separation-constrained return to the original trajectories after the conflict.

  • Uncertainty propagation

    A first-order Jacobian model that carries uncertainty in nine measured inputs (positions, ground speeds, track angle, and maneuver-execution time) into every resolution variable.

The main finding

When both aircraft maneuver together, the uncertainty in the required track change shrinks. The reduction is about 54% for reciprocal encounters and 44% at a 60° intercept, falling to about 10–21% at a shallow 20° intercept.

Synchronized maneuvering reduced resolution track-angle uncertainty relative to ownship-only maneuvering in every tabulated case. For head-on tracks the bearing uncertainty dominates; as the intercept angle narrows, ground-speed uncertainty becomes a large share of the total. These figures describe the tested encounter grid, not a universal bound.

How it was checked

Derivations are cheap to get wrong. The validation supplement publishes the methods and the data so anyone can repeat them.

Direct trajectory tests
A suite of 28 automated tests compares the analytic solutions with direct simulation. Every boundary resolution is tangent to the protected circle to within 10−8 NM, on both tangent branches.
Randomized heading sweeps
15,000 randomized encounter geometries, swept across the full heading circle, with no interval misclassification at the tested 0.25° heading resolution.
Nonlinear Monte Carlo
200,000 samples per geometry and three seeds. Across all 144 seed-level runs, the largest spread between seeds of the same case was 0.63%.
First-order vs Monte Carlo
Agreement within about 2% for synchronized track changes at every tabulated intercept angle. For ownship-only changes, about 2% at 90° and above, about 5% at 60°, and about 10% at shallow 40°–20° intercepts.
Table audit
A 372-comparison audit of a draft revision of the 15 numerical tables isolated a systematic sign defect in the draft uncertainty totals, traced to a negative prediction-time entry in the calculation worksheets.

Assumptions and limits

The results hold under these conditions, and the paper says so.

  • Two-dimensional, level flight. Vertical maneuvers and vertical separation are out of scope.
  • Track and speed changes are instantaneous; synchronized solutions assume both aircraft execute exactly and simultaneously.
  • The protected zone is a circle, and separation at the maneuver instant must exceed its radius.
  • The speed-change solution has no practical value when the required change exceeds about 30 kt, and is undefined for reciprocal and bisector-aligned geometries.
  • Uncertainty propagation is first-order and treats inputs as independent. It is invalid near the existence boundary of one resolution branch, where options should be gated or assessed by Monte Carlo.
  • The numerical study rests on two base geometries (reciprocal and 60° intercept) extended over six intercept angles.

Publications and filings

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

    Aydin Jalilov (GodwitNav Solutions LLC) and Firdovsi Jalilov (Independent Researcher)

    Preprint v2 with validation supplement, Zenodo, 2026. DOI 10.5281/zenodo.21956633. CC BY-NC-ND 4.0.

    Preprint public

  • Journal manuscript 2026-08-D0673

    Aydin Jalilov and Firdovsi Jalilov

    Submitted to the AIAA Journal of Air Transportation, August 2026.

    Submitted

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

    Aydin Jalilov and Firdovsi Jalilov

    Preprint v1, Zenodo, 2026. Superseded by v2 (v1 record; all versions).

    Superseded

  • U.S. Provisional Patent Applications 64/127,389 and 64/134,632

    Filed August 2026; assigned to GodwitNav Solutions LLC. Patent pending.

    Pending