On the Optimal Conflict Resolution

for Air Traffic Control

Dept. Of Electrical Systems and Automation

Pisa UniversityLaboratory of Information and Decision Systems

MIT

Lucia Pallottino

Eric Feron

Antonio Bicchi

JUP Meeting January 18-19, 2001

Introduction

Formulation as Optimal Control Problem (OCP): Extremal solution (PMP):

Unconstrained path; Constrained path of zero length; Constrained path of non-zero length;

Conflict Resolution Algorithm;

Formulation as a Mixed Integer Programming (MIP)

Model of OCP

Motion of aircraft subject to some constraint: linear velocity parallel to a fixed axis on the

vehicle constant non negative linear velocities bounded steering radius Minimum distance between aircraft

GOAL: given an initial and a final configuration for each aircraft, find the collision free paths of minimum total time.

NiqTqqTq

NjitD

NiR

u

Nisinuy

uxJ

gi

gii

si

sii

ji

i

ii

ii

iii

iii

,...,1)(,)(

,...,1,,0

,...,1

,...,1

cosmin

,

The OCP

0))()(())()(()( ,22

, jijijiji dtytytxtxtD

collision constraint:

Unconstrained

Optimal Solution for OCPOptimal solution will consist of concatenations of free and

constrained arcs, e.g.

UnconstrainedUnconstrained

UnconstrainedUnconstrained

ConstrainedConstrainedConstrained

Extremal unconstrained path are concatenation of

segment

Arc of a circle

Unconstrained paths for OCP

Type S

Type C

Extremal free paths of type CSC

Extremal free paths of type CCC

Velocities are parallel and the line joining the two aircraft can only translate

Velocities are symmetricrespect to the line joining the two aircraft

Contact Configuration

Conflict Resolution Algorithm

Conflict Resolution Algorithm

Model of MIP

Motion of aircraft subject to some constraint: linear velocity parallel to a fixed axis on the vehicle constant non negative linear velocities Minimum distance between aircraft Maneuver: heading angle instantaneous change

GOAL: given an initial and a final configuration for each aircraft, find a single “minimum” maneuver to avoid all possible conflict.

),,( iiii pyx Configuration after maneuver:

Maneuvre scenario for MIP

A

C

B

Initial Configuration:

),,( iii yx

Nonintersecting direction of motion

Constraints that are function of

and are linear in

ip

),,( iii yx

Intersecting direction of motion

A

B

B

Constraints that are function of

and are linear in

ip

),,( iii yx

are linear function in

33

22

11

),,,(

),,,(

),,,(

bpyxg

or

bpyxg

or

bpyxg

ig jp

Linear constraint for MIP

2

),,,(

),,,(

),,,(

321

333

222

111

fff

bMfpyxg

bMfpyxg

bMfpyxg

“big” positive number

Boolean variables

if

M

Minimum deviation problem:

Aircraft

variables constraints

Booleanf

bfApA

pt

21

min

1t

n

iip

1

min

The MIP problem

t inip

,...,1maxmin

2

2

23

7

n

nn

CPLEX Solutions

CPLEX Solutions

CPLEX Solutions

CPLEX Solutions

CPLEX Solutions

CPLEX Solutions

n (Aircraft) Time (sec.) Length (nm) Delta (nm)

5 0.34 120 0.25

7 1.18 120 0.55

10 5.91 200 0.45

11 10.4 200 0.79

CPLEX Simulation