Air Traffic Complexity Map based on Linear Dynamical Systems

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Air Traffic Complexity Map based on Linear Dynamical Systems

Adrían García, Daniel Delahaye, Manuel Soler

To cite this version:

Adrían García, Daniel Delahaye, Manuel Soler. Air Traffic Complexity Map based on Linear Dynam- ical Systems. 2020. �hal-02512103�

Air Traffic Complexity Map based on Linear Dynamical Systems

Adr´ıan Garc´ıa

Lab. of Applied Maths., Computer Science and Automatics for Aeronautics (MAIAA) Ecole Nationale de l'Aviation Civile (ENAC), Toulouse, France

adrian.garcia-sanchez@alumni.enac.fr

Daniel Delahaye

Lab. of Applied Maths., Computer Science and Automatics for Aeronautics (MAIAA) Ecole Nationale de l'Aviation Civile (ENAC), Toulouse, France

delahaye@recherche.enac.fr

Manuel Soler

Department of Bioengineering and Aerospace Engineering Universidad Carlos III (UC3M), Madrid, Spain

masolera@ing.uc3m.es

Abstract

This paper presents a new air traffic complexity metric based on linear dy- namical systems, of which goal is to quantify the air traffic control difficulty.

Previous works have shown that the structure and organization of air traffic are important factors in the perception of the complexity of an air traffic situation, but they usually were not able to explicitly address any pattern organization. The new metric, by identifying the organization properties of trajectories in a traffic pattern, captures some of the key factors involved in ATC complexity. The key idea of this work is to find a linear dynamical sys- tem which fits a vector field as closely as possible to the observations given by the aircraft positions and speeds. This approach produces an aggregate complexity metric that enables to identify high (low) complexity regions of airspace and compare their relative complexity. The metric is very adapted to compare different traffic situations for any scale (sector or country).

Keywords: Complexity, Dynamical System, Air Traffic Disorder

Preprint submitted to Journal Name July 14, 2019

1. Introduction

Regarding air traffic management research, there have been 3 main ob- jectives of interest: reduce air flight delay, solve efficiently air traffic conflicts and mitigate airspace congestion. This work proposes a method to assess the later.

In the context of sectorization problems or traffic assignment problems, the measurements of operational congestion used to construct the objective function are too crude to correctly reflect the difficulty of managing traf- fic situations. The operational capacity of a control sector is measured by the maximum number of flights which may cross the sector in a given pe- riod. This measurement does not take into account the direction of the traffic, treating geometrically structured and disordered traffic in the same way. Thus, in certain situations, a controller may continue to accept traffic even if operational capacity has been reached (structured traffic situation);

in other cases, a controller may need to refuse airplanes even though the op- erational capacity has not been reached (disordered traffic situation). Thus, modeling airspace congestion using the number of airplanes per unit of time is insufficient to reflect the levels of difficulty involved in a traffic situation.

Air traffic control organizes air flows in order to ensure flight safety and to increase the capacity of the route network. Currently, about 7500 flights are registered everyday over France, which is a crossroad for the whole Euro- pean airspace. This traffic generates a huge amount of control workload and the airspace is then divided into elementary sectors which are managed by air navigation controllers. For several years, a constant increase of air traffic has induced more and more congestion in the control sectors. Two strategies can then be applied to reduce such a congestion. The first one consists in adapting the demand to the existing capacity (slot-route allocation, collab- orative decision making etc...). The second one adapts the capacity to the demand (modification of the air network, new design of the sectorisation, new airports, etc...). For the two preceding approaches, the capacity of a sector is measured by the number of aircraft flying across the sector during a given period of time.

The observation of controlled sectors shows that sometimes, controllers accept planes beyond the capacity threshold, while in other situations, they refuse traffic before the maximum capacity is reached. This phenomenon clearly shows that the operational metric alone cannot account for controllers workload. The goal of this study is to synthesize a traffic complexity indicator

in order to better quantify the congestion in air sector, which will be more relevant than a simple number of aircraft which is independent of the traffic configuration. More precisely, our objective is to build some metrics of the intrinsic complexity of the traffic distribution in the airspace, which relates to controller workload. Those metrics must capture the level of disorder (or organization) of any traffic distribution. Usually, metrics are focused on the speed vector distribution and the associated disorder metric captures only some features of the traffic complexity. The real objective of our work is to build a metric which measures the disorder or the organization of a set of trajectories in a 4D space (3D for the space and 1D for the time).

Those metrics are relevant for many applications in the air traffic manage- ment area. For instance, when a sectoring is designed [1], the sectors have to be balanced from the congestion point of view and nowadays, only the number of aircraft is used to reach this objective. Another example where a congestion metric is needed is the traffic assignment [2, 3] for which an opti- mal time of departure and a route are searched for each aircraft in order to reduce the congestion in the airspace. Complexity metric may also be used to design new air networks, for dynamic sectoring concept, to define future ATM concepts (Free Flight) etc... Complexity metrics enable to qualify and quantify the performance of the Air Traffic service providers and enable a more objective consultation between airlines and providers.

The work presented in this paper is based on dynamical systems modeling of the air traffic. A dynamical system describes the evolution of a given state vector. If such a vector is given by the position of aircraft X~ = [x, y, z]^T, a dynamical system associates a speed vectorX~˙ = [v_x, v_y, v_z]^T to each point in the airspace. The key idea is to find a dynamical system which modelizes the observed aircraft trajectories. Based on this dynamical system modeling, a trajectory disorder metric can be computed.

In a first part, this paper will summarize the previous related works. The second part will present a linear dynamical system modeling for which the complexity metric can be represented into a complex coordinate system. In this system, it is very easy to identify any speed vector organization pattern.

2. Previous Related Works

The airspace complexity is related to both the structure of the traffic and the geometry of the airspace. Different efforts are underway to measure the whole complexity of the airspace.

3

Significant research interest in the concept of ATC complexity was gener- ated by the Free Flight operational concept. Integral to Free Flight was the notion of dynamic density. Conceptually, dynamic density is a measure of ATC complexity that would be used to define situations that were so complex that centralized control was required [4].

Wyndemere Inc. [5] proposed a measure of the perceived complexity of an air traffic situation. This measure is related with the cognitive workload of the controller with or without knowledge of the intents of the aircraft. The metric is human oriented and is then very subjective.

Laudeman et al. from NASA [6] have developed a metric called Dynamic Density which is more quantitative than the previous ones and is based on the flow characteristics of the airspace. The Dynamic Density is a weighted sum of the traffic density (number of aircraft), the number of heading changes (>15 degrees), the number of speed changes (>0.02 Mach), the number of al- titude changes (>750 ft), the number of aircraft with 3D Euclidean distance between 0−25 nautical miles, the number of conflicts predicted in 25−40 nautical miles. The parameters of the sums have been adjusted by showing different situations of traffic to several controllers. B. Sridhar from NASA [7], has developed a model to predict the evolution of such a metric in the near future. Efforts to define Dynamic Density have identified the impor- tance of a wide range of potential complexity factors, including structural considerations.

Airspace complexity depends on both structural and flow characteristics of the airspace [7]. The structural characteristics are fixed for a sector and depend on the spatial and physical attributes of the Sector such as terrain, number of airways, airway crossings and navigation aids. The flow charac- teristics vary as a function of time and depend on the number of aircraft, mix of aircraft, weather, separation between aircraft, etc... A combination of these structural and flow parameters influences the controller workload.

The traffic itself is not enough to describe the complexity associated with an airspace. A few previous studies have attempted to include structural consideration in complexity metrics, but have done so only to a restricted degree. For example, the Wyndemere Corporation proposed a metric that included a term based on the relationship between aircraft headings and dom- inant geometric axis in a sector [5]. The importance of including structural consideration has been explicitly identified in recent work at Eurocontrol. In a study to identify complexity factors using judgment analysis, Airspace De- sign was identified as the second most important factor behind traffic volume

[8]. The impact of the structure on the controller workload can be found on the paper [9, 10]. Those papers show how strong the structure of the traffic (airways, sectors, etc...) is related with the control workload through several traffic factors dependent on the instantaneous distribution of traffic (clus- tering, number of aircraft, distance between aircraft, relative speed between aircraft, etc...)

The previous models do not take into account the intrinsic traffic disorder which is related to the complexity. The first efforts related with disorder can be found in [11]. This paper introduces two classes of metrics which measure the disorder of a traffic pattern. The first class is based on geometrical properties and proposed new metrics which are able to extract features on the traffic complexity such as proximity (measures the level of aggregation of aircraft in the airspace), convergence (for close aircraft, this metric measures how strongly aircraft are closer to each other) and sensitivity (this metric measure how the relative distance between aircraft is sensible to the control maneuver). The second class is based on a dynamic system modeling of the air traffic and uses the topological entropy as a measure of disorder of the traffic pattern.

G. Aigoin has extended and refined the geometrical class by using a cluster based analysis [12]. Two aircraft are said to be in the same cluster if the product of their relative speed and their proximity (a function of the inverse of the relative distance) is above a threshold. For each cluster, a metric of relative dependence between aircraft is computed and the whole complexity of the cluster is then given by a weighted sum of the matrix norm. Those norms give an aggregated measure of the level of proximity of aircraft in clusters and the associated convergence. From the cluster matrix, it is also possible to compute the difficulty of a cluster (it measures how hard it is to solve a cluster). Multiple clusters can exist within a sector, and their interactions must also be taken into account. A measure of this interaction has been proposed by G. Aigoin [12]. This technique allows multiple metrics of complexity to be developed such as average complexity, maximum and minimum cluster complexities, and complexity speeds.

Another approach based on fractal dimension has been proposed by S.

Mondoloni and D, Liang in [13]. Fractal dimension is a metric comparing traffic configurations resulting from various operational concepts. It allows in particular to separate the complexity due to sectorization from the com- plexity due to traffic flow features. The dimension of geometrical figures is well-known: a line is of dimension 1, a rectangle of dimension 2, and so on.

5

Fractal dimension is simply the extension of this concept to more complicated figures, whose dimension may not be an integer. The block count approach is a practical way of computing fractal dimensions: it consists in describing a given geometrical entity in a volume divided into blocks of linear dimension d and counting the number of blocks contained in the entity N. The fractal dimension D0 of the entity is thus:

D₀ = lim

d→0

logN

logd (1)

The application of this concept to air route analysis consists in comput- ing the fractal dimension of the geometrical figure composed of existing air routes. An analogy of air traffic with gas dynamics then shows a relation between fractal dimension and conflict rate (number of conflicts per hour for a given aircraft). Fractal dimension also provides information on the num- ber of degrees of freedom used in the airspace: a higher fractal dimension indicates more degrees of freedom. This information is independent of sector- ization and does not scale with traffic volume. Therefore, fractal dimension is a measure of the geometrical complexity of a traffic pattern.

Some new geometrical metrics have been developed in [14] which are able to capture the level of disorder or the level of organization for some traffic patterns. For instance, in an artificial round about moving, the speed vector are very different even if the global moving is full organized without any changes in the relatives distance between aircraft. To capture those features, the covariance and the Koenig metrics have been developed. The first one is able to identify disorder or organization of translation movings. The second one identify organized curl moving.

3. Metric Description

3.1. Linear Dynamical System Modeling

All the previous metrics capture only one feature of the complexity and are not able to produce an aggregate metric which can capture all the pos- sible situations (high-low density, how-low convergence, translation organi- zation, round about organization, etc...). The investigated indicator will be the topological entropy (Kolmogorov entropy) of the air traffic represented as a dynamical system, yielding an intrinsic measure of complexity of the geometry of the traffic. This is the only metric which is able to capture most features of the complexity. The non-linear form is even able to identify

any trajectories organizations (aircraft following the same path at the same speed). This paper describes this metric in detail for the linear form.

Then, the metrics obtained following this approach are used to quantify the local level of disorder and interaction of a set of trajectories in a given zone of airspace. These metrics are based on the modeling of this set of trajectories by a linear dynamical system. This enables to identify different structures of organization of the aircraft speed vectors such as translation, curve organizations, divergence, convergence or a mix of them.

3.1.1. Model

A full vector field can be summarized by the equation of a dynamical system which describes and controls the evolution of a given state vector.

The key idea of this metric is to model the set of aircraft trajectories by a linear dynamical system which is defined by the following equation:

~˙

X =A·X~ +B~ (2)

X~ is the state vector of the system:

X~ =



 x y z



 (3)

Equation 2 associates a speed vectorX~˙ with each point in the state space X.~ This synthesis a particular vector field. The static behavior of this vector field is given by the vector B~ and the linear mapping between the speed vector X~˙ and the position vector X~ is given by matrix A. Therefore, the coefficients of matrix A determine the mode of evolution of the system in relation to its dynamics. More precisely, the eigenvalues of this matrix will determine the behavior of the system and will be used to obtain the complexity metric. The properties and relevance of these eigenvalues will be explained in section 3.1.4. For the time being, we focus on defining and modeling the linear dynamical system that will allow us to obtain matrix A.

Our problem therefore consists on determinig the dynamical model which is closest to the observations we have available at a given instant. For each of these observations, we have a position measurement (see Figure 1):

7

X_i =



 x_i yi

z_i





and a speed measurement:

V_i =



 v_xi v_yi vzi





Figure 1: Radar captures associated with three aircraft

We thus wish to find the vector field described by a linear equation X~˙ =A·X~ +B~

which is best fitted to our observations. To illustrate this aspect, we construct a grid over the airspace (see Figure 2) on which we carry out regression of a vector field in such a way as to minimize the error between the model and the observation.

In order to use matrix forms, we rewrite equation 2 as V =C·X, intro- ducing the following matrices:

X =





x1 x2 x3 · · · xn

y₁ y₂ y₃ · · · y_n 1 1 1 · · · 1



 (4)

V =

v_x1 v_x2 v_x3 · · · v_xn v_y1 v_y2 v_y3 · · · v_yn

(5)

C =

a₁₁ a₁₂ b₁

| {z }

A

a21 a22

|{z}

B~

b2

(6)

where X ∈ R^3×n, V ∈ R^2×n, C ∈ R^2×3, A ∈ R^2×2, ~B ∈ R^2×1 and n represents the number of observations at a given instant (number of aircraft present in a sector at a given instant). To facilitate visualisation and problem understanding, we are considering x and y components of the observations for the moment since air traffic controllers see 2 dimensions speed vectors on their screens.

Figure 2: Vector field produced by the linear dynamic system

3.1.2. Regression

Based on the observations of the aircraft (positions and speed vectors), the dynamical system has to be adjusted with the minimum error. This fitting has been done with a Least Mean Square minimization (LMS) method.

For each considered aircraft i, it is supposed that position X~_i = [x_i, y_i, z_i]^T and speed vector V~_i = [v_xi, v_yi, v_zi]^T are given. We then construct an error criterionE, between the dynamical system model and the observation, based on a norm (Euclidian, in our case) which should be minimized in relation to matrix A and vector B, and therefore in relation to matrix~ C, which represents the parameters of the model:

9

E = v u u t

i=n

X

i=0

V~_i−

A·X~_i+B~

2

= v u u t

i=n

X

i=0

V~_i−

C·X~_i

2

(7) We then consider the definition of the Frobenius norm of a matrixA:

kAk_F = sX

i

X

j

A²ij (8)

Since the Frobenius norm of a matrix is equal to the square root of the sum of the squares of its entries, theLM Scriterion ofEmay then be written in the following form:

E =kV −C·Xk_F (9)

The problem is to find the matrix C_opt which minimizes such a criterion.

Minimizing E is equivalent to minimizing E² =kV −C·Xk²_F. To do this, we will calculate the gradient of E² in relation to matrix C using matrix derivation properties:

∇_CE² =∇_CkV −C·Xk²_F (10)

We then work out the definition of the Frobenius norm of a matrix Ato obtain an equivalent expression:

kAk_F = s

X

i

X

j

A²ij =p

trace(A^T·A) (11) Therefore, taking into account the definition of the Frobenius norm showed in equation 11 and working out the right hand side of equation 10, we get:

kV −C·Xk²_F = r

traceh

(V −C·X)^T·(V −C·X)i²

=traceh

(V −C·X)^T·(V −C·X)i

(12)

By applying (A−B·C)^T = A^T −C^T·B^T

to equation 12, we get:

trace

V^T −X^T·C^T

·(V −C·X)

=trace

V^T·V −V^T·C·X

−X^T·C^T·V +X^T·C^T·C·X (13) Now, departing from equation 13 and considering that trace(M +N) = trace(M) +trace(N), we finally get:

kV −C·Xk²_F =trace

V^T·V

+trace

−V^T·C·X

+trace

−X^T·C^T·V +trace

X^T·C^T·C·X

(14) Recalling equation 10, we get:

∇_CE² =∇_Ctrace

V^T·V

+∇_Ctrace

−V^T·C·X

+∇_Ctrace

−X^T·C^T·V +∇_Ctrace

X^T·C^T·C·X

(15) We first consider the following properties concerning the derivatives of traces:

∂

∂Xtrace[A·X·B] =A^T·B^T (16)

∂

∂Xtrace

A·X^T·B

=B·A (17)

∂

∂Xtrace

A^T·X^T·B·X·A

=B^T·X·A·A^T +B·X·A·A^T (18) We then evaluate each term of the right hand side of equation 15:

∂

∂Ctrace

V^T·V

= 0 (19)

Applying the property showed in equation 16:

∂

∂Ctrace

−V^T·C·X

=−V ·X^T (20)

Applying the property showed in equation 17:

11

∂

∂Ctrace

−X^T·C^T·V

=−V ·X^T (21)

Applying the property showed in equation 18 and adding the identity matrix I to make the left hand side coherent with equation 18:

∂

∂Ctrace



X^T·C^T· I

|{z}

B in eq 18

·C·X



=I^T·C·X·X^T+I·C·X·X^T = 2·C·X·X^T (22) Adding up the results obtained in equations 19, 20, 21, 22, we get:

∇_CE² = 0−V·X^T −V·X^T + 2·C·X·X^T

=−2·V·X^T + 2·C·X·X^T (23) The optimum is given by cancelling the above: ∇_CE² = 0 ⇔ V·X^T = C_opt·X·X^T; which then allows us to calculateC_opt:

C_opt =V·X^T· X·X^T−1

| {z }

X⁺

(24) IfX·X^T is invertible, i.e., X columns are linearly independent, theLM S problem has an unique solution provided by Copt =V·X⁺, where X⁺ repre- sents the pseudo-inverse of matrix X. Knowing that X ∈ R^3×n ⇒ ∃X⁺ ∈ R^n×3. It is important to remark that having the third row of X full of 1s (see equation 4) allows us not only to rewrite A·X+B~ asC·X but also to always make X columns linearly independent. The only aircraft geometry that would not satisfy the linear independence of X columns is the situation in which two or more aircraft are located on the same 2 dimensions space grid point, which is imposible in real traffic unless they are in different flight levels; however, as it will be detailed in section 3.1.7, we are separating the flow of aircraft in different altitudes to compute the complexity metric and aircraft position uncertainty will also be applied. Therefore, matrix X⁺ is unique and does always exist.

The term X·X^T−1

might imply numerical problems when inversing the resulting matrix: some values can be marked as N aN. As it will be seen in section 3.1.5, matrixXcan be large (n >20), so to avoid numerical problems

concerning X·X^T−1

, we will not work out that term but we will use the properties of a pseudo-inverse matrix to calculate C_opt:

C_opt =V·X⁺ (25)

To do so, we apply the singular values decompositionSV Din no compact form to matrix X and X⁺ to reach a shortcut to compute X⁺ without numerical problems:

X =R·S·T^T (26)

X⁺ =T·S⁺·R^T (27)

where R ∈ R^3×3, S ∈ R^3×n, T^T ∈ R^n×n. Therefore, T ∈ R^n×n, S⁺ ∈ R^n×3, R^T ∈R^3×3.

It is important to notice that equation 27 comes from the fact that ma- trices R and T^T are square and ortogonal (i.e., R⁻¹ =R^T). Matrix S is the diagonal matrix of the singular values ofX. This decomposition allows us to control conditioning and avoid condition troubles by only inversing singular values that are not null. Then, computing the right hand side of equation 26 enables us to compute X⁺ from equation 27 and obtain C_opt from equation 25.

Firstly, we have to calculate matrices R, S and T^T. Matrix X·X^T is a 3-by-3 symmetric square matrix, and the square roots of its eigenvalues

λ^X_i ^·XT∈R, λ^X·X_i ^T ≥0

are the singular values of X σ_i^X : σ^X_i = +

q

λ^X·X_i ^T (28)

Once computed, the singular values of X are sorted in descending order (σ1 ≥σ2 ≥ · · ·σr), whereris equal to the rank ofX(r≤min(3, n)). It must be noticed that X·X^T and X^T·X possess the same r non null eigenvalues.

Therefore, there are r non null singular values of X.

Then, matrixS is defined as:

S =















σ₁ 0 · · · 0 0 0 σ2 · · · 0 0 ... ... . .. ... ... 0 0 · · · σ_r 0 0 0 · · · 0

|{z}n−r

0















3−r

(29)

13

where the additional n−r columns and 3−r rows elements are 0.

Matrices R and T^T are not unique. Matrix T^T = (τ₁, . . . , τ_n) with rows τ₁, . . . , τ_n being the normalised eigenvectors of X^T·X and sorted in the or- der given by σ^X_i . Matrix R = (ρ1, . . . , ρ3) with columns ρ1, . . . , ρr being calculated as:

ρ_i = 1

σ^X_i ·X·τ_i (30)

and columnsρ_r+1, . . . , ρ₃ (if any) being chosen so thatRis ortogonal (ap- plying Gram-Schmidt method to make ρ_r+1, . . . , ρ₃ ortonormal with respect to ρ₁, . . . , ρ_r).

Once computed,T andR^T are simply obtained transposingT^T andR an S⁺ is obtained as follows:

S⁺ =















1

σ1 0 · · · 0 0 0 _σ¹

2 · · · 0 0 ... ... . .. ... ... 0 0 · · · _σ¹

r 0

0 0 · · · 0

|{z}3−r

0















n−r

(31)

where the additional n−r rows and 3−r columns elements are 0.

For our case,R, S and T^T can be easily obtained from:

= 0 0

r 0

Figure 3: Singular values decomposition of a matrix X_3×n

Finally, matrix C_opt is thus given by:

Copt =V·T·S⁺·R^T (32) and then, matrixA is extracted from matrixC_opt as showed in equation 6. We then compute the eigenvalues of matrix A (2 eigenvalues are obtained since A is a 2-by-2 matrix).

3.1.3. Goodness of fit

The linear dynamical system generates a vector field that exactly fits the observations in case there are 3 or less aircraft in that sector: error E (see equation 7) becomes 0. This exact fitting is not always possible with a linear system: if there are more than 3 aircraft in the sector, an error is going to remain between the observations and the modeled vector field. Although the fitting of the linear system is not flawless, the LM S procedure determines the system which better fits the observations. The computed linear system suffices our needs since our purpose is to obtain a metric from that system in order to be able to estimate the traffic pattern, i.e., the local disorder and interaction of the set of trajectories.

A brief explanation of the flawless fit phenomenon is provided here. The vector field created by the linear dynamical system is intended to fit the aircraft speed vector (2-component vector) at a given time. Therefore, we have to solve a problem with 2·nunknowns, being 2 the number of unknowns each aircraft has (v_x and v_y) and being n the number of aircraft present at a given time. The linear dynamical system, in a 2D problem like the one developed in this paper, possesses 6 variables (see equation 6): matrix A contains 4 elements and vector B, 2 elements. So if there are 3 aircraft in~ the sector, the number of unknowns match the number of system variables, leading to an exact copy of the aircraft observations carried out by the linear dynamical system.

To assess the goodness of fit of the system, a coefficient of determination can be used. The R-square, which is the proportion of variance (%) in the dependent variables (the vector field evaluated at aircraft positions) that can be explained by the independent variables (aircraft speed observations). R- square measures how successful the fit is in explaining the variation of the speed vectors over the defined grid. So if R-quare is equal to 1, the fit is flawless. If n is higher than 3, R-square is going to be lower than 1 almost every time (exceotional situations may occur, such as 4 aircraft with same

15

speed over only 2 different trajectories).

For each speed componentv_x andv_y, a R-square assessment has been car- ried out in the following way: each speed set has n values marked y1, . . . , yn

(the observations) each associated with a predicted (or modeled) valuef₁, . . . , f_n. y= 1

n

n

X

i=1

y_i Mean of the observations SS_res = 1−X

i

(y_i−f_i)² Residual sum of squares SStot = 1−X

i

(yi−y)² Total sum of squares R² = 1− SS_res

SS_tot Coefficient of determination

Applying the above for v_x and v_y, the set of three aircraft is always generating both R-square equal to 1. Next, two examples of 4 and 5 aircraft are showed.

For the set displayed in Figure 4, R-square ofv_x was 0.9998 and R-square ofvy, 0.9988. As it can be seen, the fit is almost flawless. For the set displayed in Figure 5, R-square of v_x was 0.6705 and R-square of v_y, 0.7866; showing a worse fit but useful for our purpose as well.

Figure 4: Vector field produced for 4 aircraft

Figure 5: Vector field produced for 5 aircraft

3.1.4. Properties of Eigenvalues

The eigenvalues of matrix A describe and summarize the evolution of the system. These eigenvalues are complex numbers. Their real parts are related to the convergence or the divergence property of the system in the direction of the eigenvector. When such eigenvalues have positive real parts, the system is in expansion mode (produces divergence) and when they are negative, the system is in contraction mode (produces convergence). The absolute value of these real parts is proportional to the level of contraction or expansion of the system: the larger those real parts are in absolute value, the faster the evolutions. Furthermore, the imaginary part of the eigenvalues are related to the level of curve tendency of the system: tendency of the system to organize itself following a global rotation movement associated with each of the eigen axes. Depending of those eigenvalues, a dynamical system can evolve in contraction, expansion, rotation or a combination of those three modes.

We will refer to a full organized traffic pattern when the relative distances between aircraft do not change with time. For such a situation, the traffic is very predictable and very comfortable to address by a controller: trajectories do not present any difficulties. These patterns are translation, rotation or both.

Then, the evolution properties of the system related to the position of the eigenvalues can be summerized in the complex coordinate system (see Figure 6). In this coordinate system, it is then possible to identify the locus of the eigenvalues of matrix A associated with organized traffic situations:

17

the vertical strip around imaginary axis. Therefore, organized situations are located around the imaginary axis: when the relative distances between aircraft change slowly with time (this mean that the relative speeds between aircraft are close to zero and the traffic has no interaction), the eigenvalues of matrixA of the associated linear dynamical system have a small real part (in absolute value).

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦

𝑅𝑒𝑎𝑙 convergence

rotationrotation

divergence Organized Traffic

Figure 6: Impact of the eigenvalues of matrixAon the dynamics of the system

As an example (see Figure 7), the eigenvalues of matrix A have been calculated for a situation with three aircraft located on a circumference, for which only the orientation of the speed vectors is modified in order to create four traffic situations (pure translation, convergence, divergence and pure rotation). As we see in Figure 7, the pure translation and pure rotation are the two organized traffic situations since they have eigenvalues in the central band of the complex plane.

In the first case, the eigenvalues are null because the aircraft are flying in parallel, representing a translation: distances between aircraft remain un- changed with time. In the second case, the eigenvalues are real negative;

the system evolves in a contraction mode and the four aircraft are converg-

Translation Convergence Divergence Rotation

𝐗_𝟏 𝐕_𝟏

𝐗𝟐 𝐕𝟐 𝐗𝟑 𝐕𝟑 y

𝑥 120°

𝐝

𝐗_𝟏 𝐕_𝟏

𝐗𝟐 𝐕𝟐 𝐗𝟑 𝐕𝟑 y

𝑥

𝐗𝟏 𝐕𝟏

𝐗𝟐 𝐕𝟐 𝐗𝟑

𝐕_𝟑 y

𝑥 𝐗𝟏

𝐕𝟏

𝐗𝟑 𝐕𝟑 y

𝑥

𝐗𝟐 𝐕𝟐

Position of the eigenvalues of matrix A in the complex plane

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦

𝑅𝑒𝑎𝑙 𝜆1,2

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦

𝑅𝑒𝑎𝑙 𝜆1,2

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦

𝑅𝑒𝑎𝑙 𝜆1,2

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦

𝑅𝑒𝑎𝑙 𝜆1

𝜆2

Figure 7: Eigenvalues loci for 4 traffic situations

ing: the norms of the relative distances between aircraft diminish with time.

The third situation represents an expansion evolution for which the eigen- values are real positive and the aircraft are diverging: the relative distances increase with time. It must be noticed that in the two previous situations, the distance between aircraft changes with time, not being organized traffic patterns. The last situation is associated with full imaginary eigenvalues for which the aircraft stay at the same distance from each other in a curl mov- ing. The computation of matricesAand vectorsB~ is given for these previous examples:

Translation Convergence Divergence Rotation

A=

0 0 0 0

A= ^υ_d·

−1 0 0 −1

A= ^υ_d· 1 0

0 1

A= ^υ_d·

0 1

−1 0

B~ = υ

0

B~ = 0

0

B~ = 0

0

B~ = 0

0

~λ= ^υ_d· 0

0

~λ= ^υ_d· −1

−1

~λ = ^υ_d· +1

+1

~λ= ^υ_d· +j

−j

(33) 19

Looking at Figure 7, the small squares are the initial positions of aircraft at a given time (this represents the observation given by a radar for instance, with the associated speed vector). As it can be seen, the aircraft are initially located on a circumference with diameter 2·d. The positions of aircraft 2 and 3 are symmetric to the x axis. All the speed norms are the same and have been fixed at υ. The vector field associated to each one of these 4 situations is showed in Figure 8.

(a) Translation (b) Convergence

(c) Divergence (d) Rotation

Figure 8: Vector field produced for the 4 toy examples

A dynamical system can then also be considered as a mapT from the state space X to itself. Based on this mapping T, one can define the topological entropy (or Kolmogorov entropy) of the dynamical system which measures the level of mixing of X by T. This entropy is computed with the help of the eigenvalues of matrix A and is associated with the changes of the relative distances between points from X by T. The topological entropy of a dynamical system is then a disorder indicator of the distribution of the aircraft in the considered airspace: traffic with crossing trajectories has higher entropy. This entropy is not a statistical metric and is very adapted to the air traffic control system for which there are few aircraft in the sectors (statistical entropy needs a lot of samples in order to be reliable).

3.1.5. Extension with uncertainties

The vector field created by the linear dynamical model is based on obser- vations. There is always a level of uncertainty in these observations. There- fore, since we should account for these uncertainties, it has been considered that at every sample time of the analysis each aircraf that is within the current modelled airspace area can be ahead or behind its actual position.

Then, a time shift is applied to each aircraft (i.e., the reference one and the neighbours) to account for 10 forward and 10 aft positions: there will be 21 positions per aircraft at each given time, representing the uncertainty in the position of each aircraft. Each shift is equal to 12 seconds, so the most for- ward and aft positions are separated 2 minutes with respect to the reference position.

For the sake of simplicity, we consider 2 shifts in each direction (5 posi- tions in total per aircraft) instead of 10 only for the visualization of the un- certainties problem in this current section. These positions will be addressed as different aircraft that are flying in cluster, very close to each other. In Figure 9, the 3 aircraft from Figure 1 are expanded with their correspond- ing forward and aft positions after applying a time shifting, resulting in a 15-aircraft problem.

21

Figure 9: Uncertainties associated with three aircraft

Figure 10 shows the vector field produced by the linear dynamical system corresponding to the above situation.

Figure 10: Vector field of the uncertainties extension

Figure 11 compares the vector field of the situation with and without the uncertainties extension.

(a) Real traffic situation (b) Extension with uncertainties Figure 11: Traffic situation with and without uncertainties

Apparently, they look like the same. In the real traffic situation, both vx and vy R-square are equal to 1 since there are 3 aircraft and the linear dynamical system is capable of flawlessly fitting the observations. With un- certainties, R-square is equal to 0.992 and 0.985 respectively, which is close to 1 although a very small error remains in the fit. In the first case, ma- trix A eigenvalues are -0.673 ± 17.009j; and in the second case, -0.735 ± 16.764j. This shows that uncertainties make this situation a bit more con- vergent (higher real part, in absolute value) and less curly (lower imaginary part).

Having 5 aircraft flying very close to each other instead of one, allows us not only to account for the uncertainties in aircraft positioning but also to make the model more robust since these 5 aircraft would be following the same trajectory, and then, they would be providing the linear system with further information to understand the tendency of the actual aircraft and trajectory (realizing the direction of that trajectory).

This extension will make matrix X larger since from here on, n would be equal to 5 · number of aircraf t. As previously mentioned, SV D was applied to avoid numerical problems due to matrix X size.

3.1.6. Extension with 3D motion

We will apply a 3D model for the metric computation of real air traffic since measuring flow contraction in the vertical direction is also desirable for

23

a set of T M Aand en-route trajectories. We then will only apply this model to the case of real air traffic flying over real airspace, which is analysed in section 3.2.2.

Regarding the mathematical model, minor changes must be applied to account for the third dimension. New matricesX,V andCare the following:

X =









x₁ x₂ x₃ · · · x_n y₁ y₂ y₃ · · · y_n z1 z2 z3 · · · zn

1 1 1 · · · 1









(34)

V =





v_x1 v_x2 v_x3 · · · v_xn v_y1 v_y2 v_y3 · · · v_yn v_z1 v_z2 v_z3 · · · v_zn



 (35)

C =





a11 a12 a13 b1

a₂₁ a₂₂ a₂₃ b₂

| {z }

A

a₃₁ a₃₂ a₃₃

|{z}

B~

b₃



 (36)

Now matrix A is a 3-by-3 matrix, therefore, it possesses 3 eigenvalues instead of 2. With this model, if no uncertainties are considered, the linear system is able to perfectly fit the speed vector field of 4 aircraft (in 2D the linear system was able to model 3 aircraft vector field with no error). Then, v_x, v_y and v_z R-square are all equal to 1. As it can be seen in equation 36, matrix C possesses now 12 elements, matching the number of aircraft speed components for a 4-aircraft problem (3 speed components each).

3.1.7. Metric computation

To gain representativeness of the air traffic flow organization with respect to the set of trajectories, a local disorder metric has been assessed (how organised the traffic is locally). By assessing this problem in discrete time along one aircraft trajectory and considering an area of influence surrounding this aircraft to account for other aircraft, this metric represent the local disorder that is present in the vecinity of an aircraft along its trajectory at each time.

Having obtained matrixAand its eigenvalues, in order to produce a scalar metric, the local disorder metric is assigned according to the following:

ˆ The metric will be the addition, in absolute value, of the negative real parts of the 3 eigenvalues.

ˆ If any eigenvalue possesses a negative real part, then the metric will be 0.

With this definition we measure the intensity of the convergence tendency of a set of trajectories. If the metric is 0, the traffic will be organised and complexity will be 0. It must be noticed that if the resulting behaviour is a divergence motion, it is considered to be an organised pattern (metric equal to 0) because it will not affect the workload of the air traffic controller since divergent aircraft will not give rise to air traffic conflicts.

Therefore, the procedure that we apply consists in solving the LM S method while following an aircraft along all the position observations of its trajectory. This aircraft will be referred as the reference aircraft. As for the aircraft that are flying in its vecinity, they will be referred as neighbour aircraft. Every aircraft within the searching area of the reference aircraft at each time will be considered as a neighbour aircraft and therefore, its obser- vations will be used in the LM S computation (at each time, the neighbour aircraft may be different). This searching area is defined as a box window (24.8 x 24.8N M) centered at each reference aircraft position and will always possess the same dimension throughout the analysis period. This dimension is based on the longitudinal separation minima applicable to en-route aircraft multiplied by a scalar factor.

The uncertainty extension will be applied to both reference and neighbour aircraft. So matrix A is computed using the observations of the reference and neighbour aircraft and their uncertainties as well at each time. Figure 12 shows the geometry of the problem that is solved for a reference aircraft at a given time.

25

reference neighbour

neighbour

Figure 12: Example of local trajectory interaction at a given time

It has to be noticed that this problem is a 2D problem, so aircraft flying in different flight levels are not going to influence each other in the local disorder computation. Then we must desaggregate the traffic according to flown flight level. After applying the searching area criteria, a filtering has been done to account for aircraft that are flying in the same flight level;

however, an extension of this filter was necessary to account for the aircraft that were changing their altitude and then were susceptible of influencing the metric. The criteria applied to both the reference and neighbour aircraft to filter the possible aircraft affecting the local disorder can be seen in Figure 13.

(a) (b) (c) (d)

(e) (f) (g)

Figure 13: Aircraft altitude evolution that has an interaction with a certain FL

When the neighbour aircraft altitude (after applying the searching area principle) is close to the reference aircraft altitude, that neighbour aircraft is considered as an actual neighbour aircraft. In RV SM (Reduced Vertical Separation Minima) airspace, en-route aircraft vertical separation is 1000 f t between FL290 and FL410. Then, the filter is set to look for neighbour aircraft within a vertical separation, with respect to the reference one, equal to 3000 f t (30 FL). Therefore, aircraft flying within 30 flight levels above and below the reference aircraft will be considered for the metric.

The filter allows the following situations for both the reference and neigh- bour aircraft: (a) aircraft is climbing to the reference aircraft altitude, (b) aircraft is descending to reference aircraft altitude, (c) aircraft is about to climb from reference aircraft altitude, (d) aircraft is about to descend from reference aircraft altitude, (e) aircraft is climbing and will fly from a lower to a higher altitude than reference aircraft altitude, (f) aircraft is at reference aircraft altitude, (g) aircraft is descending and will fly from a higher to a lower altitude than reference aircraft altitude.

In case two aircraft are separated by 1000 f t (10 flight levels) and their directions are opposite to each other, opposite heading (see Figure 14), the interdependancy of those aircraft trajectories would be analized as well in our problem since those aircraft would be considered as neighbours and therefore their trajectories may interact with each other. Although they are separated one flight level and they do not represent an air traffic conflict for an air traf- fic controller, this situation may contribute to increase the local convergence tendency of a set of trajectories. When both aircraft maintain their corre- sponding altitudes, the linear system does not capture a convergent flow but a curl motion that becomes stronger when the aircraft meet at the same point in the airspace (both aircraft being separated 1000 f t in vertical direction).

Thus, if aircraft do not change their altitudes, flows at different flight levels will not interact with each other. This lack of interdependency will result in a non complex situation for an air traffic controller.

27

Figure 14: Lack of interdependency between two consecutive flight level trajectories

After solving this problem for one reference aircraft along its trajectory within the analysed airspace region, we will assign a new reference aircraft and repeat the exact same procedure until every aircraft within the analysed airspace has been analysed and has been assigned its local disorder metric at each time. It is important to remark that a former reference aircraft is susceptible to be a neighbour aircraft with respect to any other aircraft.

After solving this problem, each aircraft will possess an assigned local complexity value. Plotting this scalar metric on a multi-colour map through- out the analysis period, the evolution of the local disorder values will show whether certain regions of the analysed airspace are globally organized or not and how the organization of these regions affect adjacent ones (one conver- gent region may be surrounded by divergent ones).

3.2. Results

3.2.1. Toy Examples

As previously mentioned, for the following toy examples no uncertainties are considered and every aircraft is flying in the same flight level (2D prob- lem). To fully understand the properties of matrix A eigenvalues, a slight different metric has been assessed. This metric will represent the global be- haviour of the system, so it is computed once per time interval involving in its definition every aircraft present in the problem.

Having obtained matrixAand its eigenvalues, in order to produce a scalar metric, the disorder metric is assigned according to the following:

ˆ Both matrix A eigenvalues are negative: we record the most negative one (the most convergent).

ˆ One is negative (convergent) and the other is null (organized): we record the negative one.

ˆ One is positive (divergent) and the other is null (organized): we record the positive one.

ˆ Both eigenvalues are positive: we record the largest one (the most divergent).

Six traffic situations will be analysed to compare the level of difficulty (order of complexity) as a function of the level of predictability and of in- terdependency between trajectories. These full trajectories of aircraft are plotted on the following figures and the initial positions and speeds are sym- bolized with points and arrows.

Figure 15 shows a parallel flow of aircraft. It represents an easy situation since the flow produces a low sensitivity scenario and no conflicts. There will always be null eigenvalues; therefore, null entropy and consequently, null complexity (see equation 33). It must be noticed that this is valid only in this addresed case, when every aircraft possesses the same parallel speed;

otherwise there would be one null eigenvalue and one non null eigenvalue.

Figure 15: Parallel flow

Figure 16 shows a full symmetric convergence of eight aircraft flying at the same speed. It represents an average situation with high sensitivity and conflicts with no interaction between solutions.

29

Figure 16: Eight aircraft converging at the same point

It must be noticed that the metric begins to be negative, showing the situation is globally converging. If we continue evaluating that situation along time, after the crossing of the aircraft, the metric will become positive and the situation will be globally diverging.

-900 -720 -540 -360 -180 0 180 360 540 720 900

Figure 17: Scalar metric for eight aircraft converging at the same point

While moving from the negative real axis to the positive real axis, the eigenvalues will cross the imaginary axis, acquiring null real part. As it can

be seen in Figure 17, at that moment there is a discontinuity in the scalar metric that clearly identifies the conflict: every aircraft is in the exact same position, rapidly changing from infinite convergence to infinite divergence.

This phenomenon is explained next. From equation 33, one can see that when tha aircraft are converging and their distance to the centre dis infinitesimal, the eigenvalues become a negative infinite number. Then, immediately after the aircraft meet at the centre, being dinfinitesimal again, the aircraft begin to diverge and the eigenvalues become a positive infinite number. Since the eigenvalues are always real in this case, no curl moving is detected by the metric.

Figure 18 shows a random convergence of aircraft in the same area. Air- craft do not necessarily have the same speed. It represents a difficult situ- ation with high sensitivity and potential conflicts with interactions between solutions.

Figure 18: Random aircraft converging in the same area

In this case, the metric first identifies a soft converging pattern. Then, it reaches a minimum in the central area of convergence (matching the max- imum convergence tendency) and begins to be positive, meaning that the aircraft are diverging at that moment (see Figure 19) and reaching a maxi- mum (matching the maximum divergence tendency). The jump of the metric is not as sharp as the previous case and the overall entropy is also lower.

31

-50 -40 -30 -20 -10 0 10 20 30 40 50

Figure 19: Scalar metric for random aircraft converging in the same area

The locus of the complex eigenvalues associated to this case is shown in Figure 20. In this situation a rotation tendency is present since the eigen- values are complex numbers. It can be checked that the two eigenvalues are conjugate. This locus begins in the convergence region, negative real axis, and captures some rotation tendency of the flow (while the imaginary value increases) to finish in the divergence region, positive real axis.

-40 -30 -20 -10 0 10 20 30 40 Real

-65 -55 -45 -35 -25 -15 -5 5 15 25 35 45 55 65

Imaginary

Figure 20: Loci of the eigenvalues for random aircraft converging in the same area

Then, the interception between North-South and West-East flows is as- sessed. It can be seen that the highest entropy is found in the orthogonal crossing: a high scalar metric corresponds to a high air traffic complexity. It must be noticed that the lower the crossing angle, the lower the entropy and consequently, the lower the complexity.

-4 -3 -2 -1 0 1 2 3 4

Figure 21: N-S and W-E flows crossing at 30 deg

33

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Figure 22: N-S and W-E flows crossing at 60 deg

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Figure 23: N-S and W-E flows crossing at 90 deg

It is worth mentioning that in these 3 cases, although each aircraft pos- sesses a different speed, the eigenvalues are always real: no curl tendency is perceived since the set of aircraft is flying only along 2 different trajectory directions within the same horizontal plane.

3.2.2. Real Airspace

In this section, we assess real air traffic over France FIR airspace. The complexity problem is analysed within this airspace region so aircraft are filtered and only being considered as part of the model when they fly within FIR boundary.

4. Conclusion

The Kolmogorov entropy associated to air traffic distributions has been shown to be relevant to measure the intrinsic complexity of the speed vectors of aircraft following a set of trajectories. Extrapolated to real air traffic con- trol situations, the traffic complexity can be defined as an intrinsic measure- ment of the complexity associated with a traffic situation. This measurement is independent of the system in charge of the traffic and is solely dependent on the geometry of trajectories. It is linked to sensitivity to initial condi- tions and to the interdependency of conflicts. This interdependency between conflicts is linked to the level of mixing between trajectories.

This approach, based on linear dynamical systems, thus enables to pro- duce a local aggregate metric associated with any traffic situation because it is sensitive to relative distances between aircraft. Such a metric is able to recognize any global organization pattern, i.e., a speed vector pattern that represents the level of organization of a set of trajectories. Then, the analysis of such metric may be used in order to compare several traffic situations. As the number of degrees of freedom of the linear model is reduced, an error re- mains between the model and the observations when we increase the number of measurements.

Although relying on a dynamical system, it does not measure conflicts between aircrafts, but between flows. Topological entropy measures flux complexity but not individual aircrafts interactions. The number of aircraft in an air sector does not encompass the strong complexity of the mental workload involved in the control process, even if those two entities are very dependent.

If detail metric is needed in order to indentify high (low) complexity areas, a non-linear dynamical system has to be used.

35

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