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Analysis of intrersecting flows of agents

Eric Feron & David Dugail

Mini MURI 03/02/02

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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

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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?

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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

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-100-80-60-40-20020406080100

-100

-80

-60

-40

-20

0

20

40

60

80

100

x (nm )

y (n

m)

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

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Heading afterconflict

resolution

Originalheading

Position afterconflict resolution

Original position

W

w

Conflict area

Conflict area Conflict area

Heading Change vs. Offset Maneuver Models

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d

d tan

Offset maneuver (2/2) (2 flows, decentralized)

• 2 successive heading changes

• Proof by contradiction• Upper bound on lateral

displacement

dsep : min. separation dist.

N

sepdL 22

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Conflict geometry

Eastboundflow

Southboundflow

N

S

W E

z

z

T

T

(b, )

(a, )

b

a

O

R

+

Airw ays

Control volume

Entry points

Cone of decision

Decision bound

• 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

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Conflicting flows

-1 0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Deviation angle (deg)

Po

pu

latio

n

Population of deviation angles taken by aircraft

Distribution of deviations

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Conflict analysis

N

W

Protection-providingzone "U"

Protection-providingaircraft

New comer

An

Ai

PCZ of Ai

PCZ of An

Deviation angle is

bounded by T:

2

2

12

Ttan

where =Dsep/R

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Other results• Bounds on lateral displacements for arbitrary encounter angle,

speed, distance to conflict. Define: : Encounter angle, v2/v1 Then lateral displacement maneuver amplitude for stream 1is less than

with

• Bounds on displacement for longitudinal/lateral maneuvers(Independent utility functions for each aircraft)

)tan)tancos(sin,0max(cos 12

ddd

cos2

cos1cos 1

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-2 0 0 -1 5 0 -1 0 0 -5 0 0 5 0 1 0 0 1 5 0 2 0 0-2 0 0

-1 5 0

-1 0 0

-5 0

0

5 0

1 0 0

1 5 0

2 0 0

x (nm )

y (nm )

F low 3

F low 1 F low 2

Three flows (1/4)

• Decentralized resolution– diverges

N

• Centralized resolution (using mixed integer programming)

– stable– exhibits particular structure

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• 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

Three flows (2/4)

Flow 1

Flow 2Flow 3

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– a systematic way to deal with conflicts

– only 30% higher deviation compared with MIP

Three flows (3/4)

Structured solutionMixed Integer Prog. solution

Results

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• Control structure– flow independent– optimized

Flow 1

Flow 2Flow 3

Three flows (4/4)Flow 1

Flow 2Flow 3

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Flow management : a 3-D approach

How geometry and flow management merge ?

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Analysis of robustness

• Maneuver imprecision– leads to divergence in

some scenarios

• Aircraft position uncertainties

• 3-D geometrical tool may help

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Towards Free Flight...

• Limited information

• Situation is obtained by onboard device (radar)– control volume is attached to

each aircraft– events happen anytime

• Stability & performance ?