on the optimal conflict resolution for air traffic control dept. of electrical systems and...
TRANSCRIPT
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.
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NiR
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Nisinuy
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gi
gii
si
sii
ji
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ii
ii
iii
iii
,...,1)(,)(
,...,1,,0
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cosmin
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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