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**Air Traffic Complexity : A new concept**

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**Why is a complexity metric needed ?**

Correlated with the controller workload. Airspaces comparison (US/Europe). ATM optimization

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**Previous related works**

Dynamic Density (Mitre,Nasa) Geometric metrics (Enac,Cena,Mit) Spanning trees and neural networks (Nasa) Structural metrics (Enac,Cena,Mit)

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**Our approach Mathematical feature associated with a dynamical system.**

It measures the ability to mix the trajectories. Example : Laminar flow : weak complexity (the relative distance between particules does not change with time). Turbulent flow : strong complexity (close particules follow very different trajectories).

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Dynamical System Principle : find a dynamical system which trajectories fit the aircraft observations (Least Squares Minimization). Problem : model is not unique. Solution : select the model with most regular trajectories.

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**Linear Dynamical System**

The eigenvalues of A, control the dynamics of the system

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**Eigenvalues properties**

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Example

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Full Convergence 8 aircraft converging at the same place

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**Kolmogorov Entropy (full cv)**

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Random convergence 41 aircraft converging at the same area

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**Kolmogorov Entropy (random cv)**

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**Full organized rotation**

8 aircraft moving in rotation

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**Kolmogorov entropy (circle)**

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Spiral moving 8 aircraft on a spiral moving

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**Kolmogorov entropy (spiral)**

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**General Dynamical Systems**

Linear models have limited behaviour. In real trafic, they are restricted to general trend (drift, rotation, expansion or contraction). Extension to non linear models of the form

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Vector Splines Vector fields obtained by adding a linear term and a sum of so-called spline functions. Optimal interpolation of observations with respect to a variational criterion. Easy computation by elementary linear algebra (Singular Value Decomposition).

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**Non Linear Dynamical System**

Based on vector spline function Solution of the equation : where are the aircraft positions and are adjusted parameters.

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**Eight Aircraft Circle Spline Model**

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**Two Four-Aircraft Circles Spline Model**

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Linear Model

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Spline Model

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**Spatial Distribution of Complexity**

Previous trafic situation. Complexity value for each point of the sector integrated over time. The peaks represent the two crossing points.

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**ATC Database ATC data come from many sources : Different formats.**

Real traffic. Arithmetic simulation. Real time simulation. Different formats. Different space and/or time referentials.

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**A Unified Descriptor : The UEL**

UEL means « Uniform Event Locator », analogy with the URLs. Unambiguous data location. Independance request format / database format.

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Example ATC:spheroid/fr/bord/TA?map=cautra4&timeZone=localtime&first=’12/01/ :00:00’ &last=’12/01/ :00:00’ Context. ATC is for real traffic. Earth model. Here spheroid. Space zone (named in this ex) fr/bord/TA means for TA sector from the Bordeaux ACC. Argument after ‘?’ give information about the referentials and the time interval. Here the space reference is ‘cautra4’, the time reference is the local time.

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Example : comments A UEL request insures that the data will be given in the right time and space referentials and will not be sensitive to the database format. If spheroid earth model is replaced by the WGS84 the format conversion will be automatic. In the same way the argument: mapType=stereographic&position= will produce stereographic projection data with a map centered on lat=20, long=0.

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**Database structure DB Interface DBManager DB Interface UELResolver**

CORBA XML UELResolver

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