Analysis of intrersecting flows of agents

1. Analysis of intrersecting flows of agents Eric Feron & David Dugail Mini MURI 03/02/02

2. Research motivation & goals • Analyses of conflict resolution usually involve pairs or a finite number of aircraft • Need to address the fear of domino effect • one conflict resolution triggers a new conflict elsewhere possibly leading to divergence in the system • Analysis of aircraft flows • worst-case standpoint • Stability & performance • simulations for insight on the system dynamics • analytical proofs

3. Background • A very rich literature on management of conflicts involving 2,3 or more but finite number of aircraft - Erzberger, Krozel, Kuchar, Niedringhaus, Sastry, Tomlin, Zeghal …. • An equally rich literature on conflict management with aircraft flows - eg gas models - Bakker, Blom, Simpson… Open-loop probabilistic models. • Very little available from current robotics literature (Recent research by Ruspini, Devasia, Meyer,…otherwise computational complexity results, eg Reif & Sharir) • How does one prove stability, and bound required aircraft deviations, for conflict resolution over a class of closed-loop aircraft interactions in a deterministic setting?

4. A "Control Volume" approach • Motivation: Infinite # of aircraft flow in and out • Analysis of completely random aircraft flows is difficult • Need to structure the flows & flow behaviors

5. A control volume approach 2-D Structured converging flows pre-determined points of entry regular or random entry Aircraft make 1 maneuver when entering maneuver is minimal offset, heading change maneuvers N

6. w Position after conflict resolution Original position Original heading a Heading after conflict resolution W Conflict area Conflict area Conflict area Heading Change vs. Offset Maneuver Models

7. 2 successive heading changes Proof by contradiction Upper bound on lateral displacement dsep : min. separation dist. N Offset maneuver (2/2) (2 flows, decentralized)

8. Conflict geometry • 2-D • Structured converging flows • pre-determined points of entry • regular or random entry • Aircraft make 1 maneuver when entering • maneuver is minimal • offset, heading change maneuvers

9. Conflicting flows Distribution of deviations

10. Conflict analysis Deviation angle is bounded by T: where d=Dsep/R

11. Other results • Bounds on lateral displacements for arbitrary encounter angle, speed, distance to conflict. Define: q : Encounter angle, m = v2/v1 Then lateral displacement maneuver amplitude for stream 1 is less than with • Bounds on displacement for longitudinal/lateral maneuvers (Independent utility functions for each aircraft)

12. N Three flows (1/4) • Centralized resolution (using mixed integer programming) • stable • exhibits particular structure • Decentralized resolution • diverges

13. Three flows (2/4) • Idea • create a control structure • independent of flow • optimize it to decrease lateral distance from original flight path • Concept • three-flow compatible • aircraft are assigned conflict free spots

14. Three flows (3/4) • a systematic way to deal with conflicts • only 30% higher deviation compared with MIP Mixed Integer Prog. solution Structured solution Results

15. Three flows (4/4) • Control structure • flow independent • optimized

16. Flow management : a 3-D approach How geometry and flow management merge ?

17. Analysis of robustness • Maneuver imprecision • leads to divergence in some scenarios • Aircraft position uncertainties • 3-D geometrical tool may help

18. Towards Free Flight... • Limited information • Situation is obtained by onboard device (radar) • control volume is attached to each aircraft • events happen anytime • Stability & performance ?