New trends in Air Traffic Complexity

New trends in Air Traffic Complexity D. Delahaye and S. Puechmorel Applied Math Laboratory, ENAC

March 8, 2010

D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trendsENAC) in Air Traffic Complexity

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Agenda

Why complexity metrics are needed ? Dynamical system modeling of aircraft trajectories How a complexity map is built ? Results


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Agenda



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Why complexity metrics are needed ?

Airspace design Airspace comparison (US-Europe) Evaluation of Airspace Organisation Schemes Implementation of flexible use of airspace policies


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Why complexity metrics are needed ?

Airspace design Airspace comparison (US-Europe) Evaluation of Airspace Organisation Schemes Implementation of flexible use of airspace policies 4D contract framework 4D Trajectory design. Forecasting of potentially hazardous traffic situations. Automated Conflict Solver enhancement (robustness of the solution).


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Complexity vs Workload

Workload Related to cognitive processes for human controllers. Easy/Hard forecasting of conflict occurence. Monitoring is a non negligible part of the workload.


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Complexity vs Workload

Workload Related to cognitive processes for human controllers. Easy/Hard forecasting of conflict occurence. Monitoring is a non negligible part of the workload. Complexity Related to traffic structure. Measure of intrinsic disorder of a set of trajectories. Increases with : Sensitivity to uncertainties. Interdependance of conflicts.


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Intrinsic part of complexity Sensitivity Uncertainties on positions, wind speed, intents induce inaccuracy on trajectory prediction. Depending on the situation, prediction error can grow exponentially fast ! In such a case, the situation is complex because nearly impossible to forecast.


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Intrinsic part of complexity Sensitivity Uncertainties on positions, wind speed, intents induce inaccuracy on trajectory prediction. Depending on the situation, prediction error can grow exponentially fast ! In such a case, the situation is complex because nearly impossible to forecast. Interdependance Solving a conflict may induce other ones. Even in conflict-less situations, interactions between trajectories can rise the perceived level of complexity. Complexity is related to mixing behaviour. D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trendsENAC) in Air Traffic Complexity

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Sensitivity-interdependance

No Sensitivity

Hight sensitivity

Hight sensitivity

No conflict

Potential conflicts without interaction between solutions

Potential conflicts with interactions between solutions

Easy situation

Hard situation


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Agenda



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Linear Dynamical System Modeling

The key idea is to model the set of aircraft trajectories by a linear dynamical system which is defined by the following equation : ~˙ = A · X ~ +B ~ X ~ is the state vector of the system : where X   x ~  X = y  z ~ are the parameters of the model. Matrix A and vector B


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Regression of a Linear Dynamical System

V1 V2 X1 X2

X3 V3


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Regression of a Linear Dynamical System Based on a set of observations (positions and speeds), one has to find a dynamical system which fits those observations.Suppose that N obervations are given : Positions :   xi ~ i =  yi  X zi and speeds :  vxi ~ i =  vyi  V vzi 


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Regression of a Linear Dynamical System Based on a set of observations (positions and speeds), one has to find a dynamical system which fits those observations.Suppose that N obervations are given : Positions :   xi ~ i =  yi  X zi and speeds :  vxi ~ i =  vyi  V vzi 

A LMS precedure is applied in order to extract the matrix A and the ~ vector B.


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Properties of the matrix A

When real part of the eigenvalues of matrix A is positive, the system is in expansion mode and when they are negative, the system is in contraction mode. Furthermore, the imaginary part of such eigenvalues are related with curl intensity of the field.


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Properties of the matrix A

full curl

Imaginary axis

Organized Traffic

Real Axis convergence

full curl

divergence


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Linear Dynamical System Modeling : An example

2 d 1

v

v 45° 45°

v 3


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Linear Dynamical System Modeling : An example

Situation 1

Situation 2

Situation 3

Situation 4

Parallel trajectories (relative speed=0)

Convergent trajectories

Divergent trajectories

Round about trajectories (relative speed=0)

y

y

2 v d 90°

1 v

x

y

y

v

2

1 v

2 x

d 3 v

v

v 1

x

v 3

2 v

v 1

x

3 v

v 3

Position of the eigenvalues of matrix A in the complex coordinate system eigenvalues Imaginary

Imaginary

1

−1

Imaginary

1

1 Real −1

−1

1 Real −1

Imaginary

1

−1

1

1 Real −1

−1

1 Real −1

unit v/d


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Linear Model limitations

Give a global tendency of the traffic situation. Do not fit exactly with all traffic situations. ⇒ Non Linear Extension


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Non Linear Extension in Space ~˙ = ~f (X ~) X

Optimization problem ~f ? such that : minE =

i=N X

~i − ~f (X ~ i )k2 kV

i=1

and  Z min

 k∆~f (~x )k2 d~x with ∆~f =   3

R


∂ 2 fx ∂x 2 ∂ 2 fy ∂x 2 ∂ 2 fz ∂x 2

+ + +

∂ 2 fx ∂y 2 ∂ 2 fy ∂y 2 ∂ 2 fz ∂y 2

+ + +

∂ 2 fx ∂z 2 ∂ 2 fy ∂z 2 ∂ 2 fz ∂z 2

   

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Non Linear Extension in Space

Exact Solution (Amodei) ~f (X ~)=

N X

~ −X ~i k).~ai + A.X ~ +B ~ Φ(kX

i=1

with ~ −X ~i k) = Q(kX ~ −X ~i k3 ) Φ(kX and

2 6 Q =4

2 2 2 ∂xx + ∂yy + ∂zz 0 0

0 2 2 2 ∂xx + ∂yy + ∂zz 0

3 0 7 0 5 2 2 2 ∂xx + ∂yy + ∂zz


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Non Linear Extension in Space and time

~˙ = ~f (X ~ , t) X

Optimization problem We are looking for ~f such that :

minE =

i=N X k=K X

~ i (tk ) − ~f (X ~ i , tk )k2 kV

i=1 k=1

and Z min R3

∂~f k∆~f (~x )k2 + k k2 d~x dt ∂t t

Z


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Exact Solution (Puechmorel and Delahaye) ~f (X ~ , t) =

N X K X

~ (t) − X ~i (tk )k, |t − tk |).~ai,k + A.X ~ +B ~ Φ(kX

i=1 k=1

with Φ(r , t) = diag

σ √ .erf r. π

"

r 1 .p σ 2 + θ.|t|


#!

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Gradient close form ∂Φ(r , t) = (α − β).x ∂x with

2 2.σ 1 − 2 r σ .(2+θ.|t|) p α= 2 . .e r .π 2 + θ.|t|

Φ(r , t) β= r2 p r = x2 + y2 + z2


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Agenda



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Characterization of sensitivity How fast two neighboring dynnamical system trajectories diverge ? Let γ(t, s), t : [a, b], s ∈ V be a family of trajectories of the dynamical system in the neighborood V of a given point s0 . We assume that the nominal trajectory is t 7→ γ(t, s0 ). A perturbed trajectory is t 7→ γ(t, s) with s ∈ V . Divergence to nominal trajectory with respect to time is thus kγ(t, s0 ) − γ(t, s)k = D(t, s).


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Characterization of sensitivity How fast two neighboring dynnamical system trajectories diverge ? Let γ(t, s), t : [a, b], s ∈ V be a family of trajectories of the dynamical system in the neighborood V of a given point s0 . We assume that the nominal trajectory is t 7→ γ(t, s0 ). A perturbed trajectory is t 7→ γ(t, s) with s ∈ V . Divergence to nominal trajectory with respect to time is thus kγ(t, s0 ) − γ(t, s)k = D(t, s). Computing D(t, s) Main idea : when t 7→ γ(t, s) is the solution of a differential equation with initial condition γ(0, s) = s, it is possible to show that D itself satisfies a differential equation.


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local behaviour of trajectories

V s0 s

γ (t,s 0) D(t,s)

γ (t,s)


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Local behaviour of trajectories The variational equation I The value of D(t, s) is the result of a cumulative process. Assume that γ(t, s) is defined to be a flow : ∂γ(t, s) = F (t, γ(t, s)) γ(0, s) = s ∂t with F a smooth vector field. Given a nonimal trajectory γ(t, s0 ), then divergence of nearby trajectories can be found up to order one in ks − s0 k by solving : ∂D(t, s) = DF (t, γ(t, s0 )).D(t, s) ∂t

D(0, s) = ks − s0 k

with DF the jacobian matrix of F (with respect to s). D. Delahaye and S. Puechmorel (Applied Math Laboratory, New trendsENAC) in Air Traffic Complexity

March 8, 2010

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Lyapunov exponents

The variational equation II Since the previous equation is linear, it can be described by a matrix M(t) that obeys : dM(t) = DF (t, γ(t, s0 )).M(t) dt

M(0) = Id

This equation is called the variational equation of the flow. The variational equation describes in some sense a linear dynamical system “tangent” to the original one.


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Lyapunov exponents

Lyapunov exponents Let U t (t)Σ(t)V (t) = M(t) be the SVD decomposition of M(t). The Lyapunov exponents are mean values of the logarithms of the diagonal elements of Σ(t): 1 κ(s) = − T

Z

T

log(Σii (t))dt ∀Σii (t) ≤ 1 0


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Summary

Based on trajectories observations, a 4D non linear dynamical system model is buit. The associated 4D vector field is then computed on a cube of airspace. Lyapunov exponents are computed on each point of the cube in order to built a complexity map.


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Interpretation of Lyapunov exponents

Given an initial point, the Lyapunov exponents and the associated SVD decomposition provide us with a decomposition of space in principal directions and corresponding convergence/divergence rate. It is a localized version of the complexity based on linear systems.


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Agenda



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Results


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Results


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Results


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Conclusion

An intrinsic trajectories complexity metric has been developped. This metric can be computed as a map. Give the areas of airspace where the traffic is organized and the ones where there is desorder


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Current work

Parallel computation by the mean of Local linear Model ~ ~ ~ ~ ~i − X ~ )+ ~ i (X ~ i , ti ) = ~f (X ~ , t) + ∂ f (X , t) (ti − t) + ~f (X ~ , t) + ∂ f (X , t) (X V ~ ∂t ∂X ~i − X ~ k) O(|ti − t| + kX ~i − X ~ ) + O(|ti − t| + kX ~i − X ~ k) = ~a + ~b(ti − t) + C .(X min

N X

~a,~b,C i=1

~ i (X ~ i , ti ) − ~a + ~b(ti − t) + C .(X ~i − X ~ )k2 .ψ(ti − t, X ~i − X ~) kV

Robustness improvement by taking into account uncertainties on time position of aircraft on their trajectories


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