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

March 5, 2009

D. Delahaye and S. Puechmorel (Applied Math Laboratory
New trends
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March 5, 2009

1 / 32

AgendaWhy complexity metrics are needed ?

D. Delahaye and S. Puechmorel (Applied Math Laboratory
New trends
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March 5, 20092 / 32

AgendaWhy complexity metrics are needed ?
Dynamical system modeling of aircraft trajectories

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

March 5, 20092 / 32

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

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

March 5, 20092 / 32

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

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

March 5, 20092 / 32

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

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

March 5, 20093 / 32

Why complexity metrics are needed ?

Airspace design

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

Why complexity metrics are needed ?

Airspace design
Airspace comparison (US-Europe)

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

Why complexity metrics are needed ?

Airspace design
Airspace comparison (US-Europe)
Evaluation of Airspace Organisation Schemes

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

Why complexity metrics are needed ?

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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32

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).

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20094 / 32Complexity vs Workload

Workload

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March 5, 20095 / 32Complexity vs Workload

Workload
Related to cognitive processes for human controllers.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32Complexity vs Workload

Workload
Related to cognitive processes for human controllers.
Easy/Hard forecasting of conflict occurence.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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 :D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20095 / 32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.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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Intrinsic part of complexity
Sensitivity

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Intrinsic part of complexity
Sensitivity
Uncertainties on positions, wind speed, intents induce inaccuracy on
trajectory prediction.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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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 exponentiallyfast !

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20096 / 32

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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20096 / 32

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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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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 exponentiallyfast !
In such a case, the situation is complex because nearly impossible toforecast.
Interdependance
Solving a conflict may induce other ones.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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ENAC)
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March 5, 20096 / 32

Intrinsic part of complexity
Sensitivity
Uncertainties on positions, wind speed, intents induce inaccuracy on
trajectory prediction.
Depending on the situation, prediction error can grow exponentiallyfast !
In such a case, the situation is complex because nearly impossible toforecast.
Interdependance
Solving a conflict may induce other ones.
Even in conflict-less situations, interactions between trajectories canrise the perceived level of complexity.

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

March 5, 20096 / 32

Intrinsic part of complexity
Sensitivity
Uncertainties on positions, wind speed, intents induce inaccuracy on
trajectory prediction.
Depending on the situation, prediction error can grow exponentiallyfast !
In such a case, the situation is complex because nearly impossible toforecast.
Interdependance
Solving a conflict may induce other ones.
Even in conflict-less situations, interactions between trajectories canrise the perceived level of complexity.
Complexity is related to mixing behaviour.
D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20096 / 32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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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AgendaWhy complexity metrics are needed ?
Dynamical system modeling of aircraft trajectories
How a complexity map is built ?
Results

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 20098 / 32

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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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Regression of a Linear Dynamical System

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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Regression of a Linear Dynamical System
Based on a set of observations (positions and speeds), one has to finda dynamical system which fits those observations. Suppose that N
obervations are given :Positions :xi~ i =  yi Xzi
and speeds :vxi~ i =  vyi VvziD. Delahaye and S. Puechmorel (Applied Math Laboratory
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Regression of a Linear Dynamical System
Based on a set of observations (positions and speeds), one has to finda dynamical system which fits those observations. Suppose that N
obervations are given :Positions :xi~ i =  yi Xzi
and speeds :vxi~ i =  vyi VvziA LMS precedure is applied in order to extract the matrix A and the~vector B.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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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 incontraction mode.

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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 incontraction mode.
Furthermore, the imaginary part of such eigenvalues are related withcurl intensity of the field.

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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Linear Dynamical System Modeling : An example

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 systemeigenvalues
ImaginaryImaginaryImaginaryImaginary

1

1

1

1

−1

1 Real−1−1

1 Real−1−1

1 Real−1−1

1 Real−1unit v/d

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Linear Model limitations

Give a global tendency of the traffic situation.

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Linear Model limitations

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

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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

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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

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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
R3


k∆~f (~x )k2 d~x with ∆~f = 

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22∂ 2 fx+ ∂∂yf2x + ∂∂zf2x∂x 2∂ 2 fy∂ 2 fy∂ 2 fy
+
+
2
2
∂x
∂y
∂z 2
∂ 2 fz∂ 2 fz∂ 2 fz+ ∂y 2 + ∂z 2
∂x 2






March 5, 200919 / 32

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

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Non Linear Extension in Space and Time~˙ = ~f (X~ , t)
X

Optimization problem
We are looking for ~f such that :D. Delahaye and S. Puechmorel (Applied Math Laboratory
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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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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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∂tt

Z

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Non Linear Extension in Space and Time

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

"

r
1
.p
σ
2 + θ.|t|D. Delahaye and S. Puechmorel (Applied Math Laboratory
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AgendaWhy complexity metrics are needed ?
Dynamical system modeling of aircraft trajectories
How a complexity map is built ?
Results

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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March 5, 200923 / 32

Characterization of sensitivity
Dynamical system trajectoriesVγ (t , s0)γ (t , s )D(t , s )Let t 7→ γ(t, s0 ) be a nominal dynamical system trajectory (s0 initialpoint)A perturbed trajectory is t 7→ γ(t, s) with s ∈ V (V is an open
neighborood of a given s0 ).
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Characterization of sensitivity

How fast two neighboring dynamical system trajectories diverge ?

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Characterization of sensitivity

How fast two neighboring dynamical system trajectories diverge ?
One has to compute the evolution of the distance to nominal
trajectory with respect to time : D(t, s) = kγ(t, s0 ) − γ(t, s)k.

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Characterization of sensitivity

How fast two neighboring dynamical system trajectories diverge ?
One has to compute the evolution of the distance to nominal
trajectory with respect to time : D(t, s) = kγ(t, s0 ) − γ(t, s)k.
Computing D(t, s)
Main idea : when t 7→ γ(t, s) is the solution of a differential equation withinitial 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

D0 D1 D2

D3

D4

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Local behaviour of trajectories

The variational equation
γ(t, s) being a flow :∂γ(t, s)= F (t, γ(t, s)) γ(0, s) = s∂twith F a smooth vector field.

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Local behaviour of trajectories

The variational equation
γ(t, s) being a flow :∂γ(t, s)= F (t, γ(t, s)) γ(0, s) = s∂twith F a smooth vector field.
then divergence of nearby trajectories can be found by solving :∂D(t, s)= DF (t, γ(t, s0 )).D(t, s)∂t

D(0, s) = ks − s0 kwith DF the jacobian matrix of F (with respect to s).

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Lyapunov exponents
The variational equation II
If the three axis are considered simultaneously, the previous equation
has the following matrix structure :dM(t)= DF (t, γ(t, s0 )).M(t)dt

M(0) = Id

This equation is called the variational equation of the flow.

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Lyapunov exponents
The variational equation II
If the three axis are considered simultaneously, the previous equation
has the following matrix structure :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 the local linear
approximation of the original non linear dynamical system.

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Lyapunov exponents
The variational equation II
If the three axis are considered simultaneously, the previous equation
has the following matrix structure :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 the local linear
approximation of the original non linear dynamical system.
Lyapunov exponents

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Lyapunov exponents
The variational equation II
If the three axis are considered simultaneously, the previous equation
has the following matrix structure :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 the local linear
approximation of the original non linear dynamical system.
Lyapunov exponents
Let U t (t)Σ(t)V (t) = M(t) be the SVD decomposition of M(t).

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Lyapunov exponents
The variational equation II
If the three axis are considered simultaneously, the previous equation
has the following matrix structure :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 the local linear
approximation of the original non linear dynamical system.
Lyapunov exponents
Let U t (t)Σ(t)V (t) = M(t) be the SVD decomposition of M(t).
The Lyapunov exponents are limit values of the logarithms of the
diagonal elements of Σ(t).
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Interpretation of Lyapunov exponents

Given an initial point, the Lyapunov exponents and the associatedSVD decomposition provide us with a decomposition of space inprincipal directions and corresponding convergence/divergence rate.

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Interpretation of Lyapunov exponents

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

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

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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Results

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Results

D. Delahaye and S. Puechmorel (Applied Math Laboratory
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ENAC)
in Air Traffic Complexity

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