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On the Optimal Conflict Resolution for Air Traffic Control Dept. Of Electrical Systems and Automation Pisa University Laboratory of Information and Decision Systems MIT Lucia Pallottino Eric Feron Antonio Bicchi JUP Meeting January 18-19, 2001

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Introduction ä Formulation as Optimal Control Problem (OCP): ä Extremal solution (PMP): â Unconstrained path; â Constrained path of zero length; â Constrained path of non-zero length; ä Conflict Resolution Algorithm; ä Formulation as a Mixed Integer Programming (MIP)

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Model of OCP Motion of aircraft subject to some constraint: u linear velocity parallel to a fixed axis on the vehicle u constant non negative linear velocities u bounded steering radius u 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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The OCP collision constraint :

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Unconstrained Optimal Solution for OCP Optimal solution will consist of concatenations of free and constrained arcs, e.g. Unconstrained Constrained

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u Extremal unconstrained path are concatenation of F s egment F Arc of a circle Unconstrained paths for OCP Type S Type C

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Extremal free paths of type CSC

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Extremal free paths of type CCC

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Velocities are parallel and the line joining the two aircraft can only translate Velocities are symmetric respect to the line joining the two aircraft Contact Configuration

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Conflict Resolution Algorithm

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Model of MIP Motion of aircraft subject to some constraint: u linear velocity parallel to a fixed axis on the vehicle u constant non negative linear velocities u Minimum distance between aircraft u 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.

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Configuration after maneuver: Maneuvre scenario for MIP A C B Initial Configuration:

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Nonintersecting direction of motion Constraints that are function of and are linear in

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Intersecting direction of motion A B B Constraints that are function of and are linear in

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are linear function in Linear constraint for MIP “big” positive number Boolean variables

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Minimum deviation problem: Aircraft variables constraints u The MIP problem

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CPLEX Solutions

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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

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