trajectory mathematical distance applied to airspace major

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Trajectory Mathematical Distance Applied to AirspaceMajor Flows Extraction

Daniel Delahaye, Stéphane Puechmorel, Sameer Alam, Eric Féron

To cite this version:Daniel Delahaye, Stéphane Puechmorel, Sameer Alam, Eric Féron. Trajectory Mathematical Dis-tance Applied to Airspace Major Flows Extraction. EIWAC 2017, 5th ENRI International Workshopon ATM/CNS, ENRI, Nov 2017, Tokyo, Japan. pp. 51-66., �10.1007/978-981-13-7086-1_4�. �hal-01598864�

https://hal-enac.archives-ouvertes.fr/hal-01598864

https://hal.archives-ouvertes.fr

ENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2017)

[EN-A-018] Trajectory Mathematical Distance Applied toAirspace Major Flows Extraction

+ +D. Delahaye ∗ S. Puechmorel ∗ S. Alam ∗∗ E. Feron ∗∗∗∗ OPTIM, ENAC-LAB

French Civil Aviation University7 Avenue Edouard Belin, 31055, Toulouse, France

∗ [daniel.delahaye—stephane.puechmorel]@enac.fr

∗∗ Nanyang Technological UniversitySingapore

∗∗ [email protected]

∗∗∗ Georgia Technology InstituteAtlanta, USA

∗∗∗ [email protected]

Abstract: In this paper, the problem of aircraft trajectories representation and analysis is addressed. In manyoperational situations, there is a need to have a value expressing how trajectories are close to each other. Somemeasures have been previously defined, mainly for trajectory prediction applications, all of them being basedon distance computations at given positions in space and time. The approach presented here is to consider thetrajectory as a whole object belonging to a functional space and to perform all computations in this space. Anefficient algorithm for computing mathematical distance between trajectories is then presented and applied tothe major flows extraction in the French airspace.

Keywords: Distance, trajectory, homotopy,energy, hierarchical clustering, airspace, major flows.

1 IntroductionFuture Air Traffic Management relies, in part, on

the use of decision support tools (DST) to provide im-proved service to the user community under increas-ing traffic demand. Furthermore, this improvementhas to be validated by the mean of system perfor-mance metrics such as complexity, robustness, capac-ity. As aircraft fly 4D trajectories, there is a strongneed to quantify the associated trajectory accuracy inorder to validate aircraft models and trackers. Suchvalidation is usually based on a comparison betweenthe actual trajectory and a reference by the mean of atrajectory distance. This last point is the key elementof the whole process. Such trajectory distance is stillneeded for ATM applications and the goal of this pa-per is to present a new trajectory distance based onrigorous mathematical concepts. Although trajecto-ries are well understood and studied, relatively littleinvestigation on the precise comparison of trajecto-ries is presented in the literature. A key issue in per-formance evaluation of ATM decision support tools(DST) is the distance metric that determines the sim-ilarity of trajectories. Most existing measures [7, 22]compute a mean distance of the corresponding posi-

tions of two equal duration trajectories. Supplemen-tary statistics such as variance, median, minimum,and maximum distances are also suggested to extendthe description of similarity. In [17], Needman pro-posed an alignment based distance metric that revealsthe spatial transition and temporal shift between thegiven trajectories, and introduced an area based met-ric that calculates the total enclosed area between tra-jectories using trajectory intersection.

One main disadvantage of the existing approachesis that they are all limited to the equal duration (life-time) trajectories. By duration we refer to the num-ber of coordinate points that constitute the trajectory.These coordinates are sampled at different instances.Since the existing measures depend on the mutual co-ordinate correspondences, they cannot be applied totrajectories that have different durations. Conventionaldistance measures assume that the temporal samplingrates of trajectories are equal. They do not cope withthe uneven sampling instances, i.e. varying temporaldistance between the coordinates. Therefore, thereis a need to develop other alternatives that can ef-fectively measure the difference between unrestrictedtrajectories.There are a lot of ATM applications where

1

+D.Delahaye, S.Puechmorel, S.Alam, E. Feron

ΣSOLVER+

−Control

Reference

Error

Feedback Path

Figure 1 General simulation model. The solver gen-erate trajectories as close as posible to the referencetrajectory thanks to the feedback path.

such distance between trajectories is needed.Aircraft model InferenceAll aircraft models are based on ODEs (Ordinary

Differential Equation), including tabular ones (see Fig. 1).The aircraft model inference consists in answeringthe following question :Given a parametrized modeland a goal trajectory, can we infer the best param-eter values? A model can be viewed as a mappingfrom the control space into the trajectory space. Theway to answer the previous question is then given bythe closest model to the goal trajectory (see Fig. 2).In order to find the closest model in this trajectory

γ

Figure 2 Finding the best model from a given class.The green “grid” represents represents such class pro-duced by the model and γ is the goal aircraft trajec-tory.

space, a reliable trajectory distance is needed. Themodel inference problem has to solve the accuracy-smoothness dilemma :Over-fitted models are gen-erally poor predictors. The previous constructiongives the shortest path (and thus the distance) betweenthe goal trajectory and the trajectory set which can besynthesized by the model.

Trajectory predictionAir traffic management research and development

has provided a substantial collection of decision sup-port tools that provide automated conflict detectionand resolution [4, 1, 28],trial planning [13], controlleradvisories for metering and sequencing [26, 2], traf-

fic load forecasting [14, 12], weather impact assess-ment [9, 25, 5]. The ability to properly forecast futureaircraft trajectories is central in many of those deci-sion support tools. As a result, trajectory prediction(TP) and the treatment of trajectory prediction uncer-tainty is an active areas of research and development(eg [23, 27, 16, 21, 24]).

Accuracy of TP is generally defined as point spa-tial accuracy (goal attainment) or as trajectory follow-ing accuracy. The last one can be rigorously definedby the mean of trajectory space. The first one is a limitcase of the second by adding a weight function in theenergy functional.

When we refer to trajectory prediction errors fora specific DST, we are typically comparing the pre-dicted trajectory for a specific DST to the actual tra-jectory to be experienced by an aircraft. Discrepan-cies between these two types of trajectories typicallyaffect the performance of the DST.

Radar tracker evaluationThe goal of a radar tracker is to eliminate the resid-

ual noise coming from the radars. It is a key elementof the ATM system and its accuracy is one of the fac-tors which determines the separation norm. In orderto validate such trackers, an exact reference trajectoryis generated and perturbed by a white Gaussian noise.This perturbed trajectory is then used as input of thetested tracker. The tracker generates an estimated tra-jectory which is compared to the reference trajectory.In order to do such comparison, a reliable trajectorydistance is also needed.

Alternative route synthesisAirspace congestion is related to aircraft located

in the same area during the same period of time. Then,when congestion has to be minimized, algorithms haveto separate aircraft in time (slot allocation), in space(route allocation) or both (bi-allocation). When routeallocation is investigated, associated algorithms needalternative routes set in order to spread the traffic onthem. A route is said to be alternative to another if itis different enough based on a trajectory distance.

Major flows definitionWhen radar tracks are observed on a radar screen

over a long period of time in a dense area, it is veryeasy to see major flows connecting major airports.The expression ”major flows” is often used but neverrigorously defined. Based on an exact trajectory dis-tance and a learning classifier, it is possible to answerthe following questions :Given a set of observed tra-jectories, can we spit it into ”similar” trajectoryclasses? If yes, classes with highest number of ele-ments will rigorously define the major flows. Giventhose classes and a new trajectory, can we tell if itbelongs to a major flow and which one? The princi-ple of the major flows definition is to use shape spaceto represent trajectory shapes as points and to use ashape distance. (the shape of a trajectory is the pathfollowed by an aircraft, that is the projection in the 3Dspace of its 4D trajectory. The speed on the path hasno impact).

2


γ1 γ

1

γ1 γ

1

Highest Density

Figure 3 Four trajectories γ1,γ2,γ3 and γ4, are sharinga common central straight line. One can identify anaverage of two aircraft at each point of this line andonly one in the other segments. If we compute thehighest density associated with those trajectories, wewill extract the central segment which is flown by noaircraft.

Major flows have not to be confused with highestdensity in the airspace. As a matter of fact, some ap-proaches for major flows extraction consider the accu-mulated traffic in the airspace and build a kind of den-sity map for which the highest areas are considered asmajor flows. This approach may be completely falseas shown on figure 1. On this figure, four artificialtrajectories share a common highest density area but,as it can be seen on the figure, no aircraft is flying this“high density trajectory”.

Another approach consists in extracting major flowson a set of trajectories thank’s to an efficient HMI anda bundling algorithm [11]). The results produced bythis kind of algorithm are quite similar to the onespresented in this paper but it is done manually and wepropose to do it automatically.

As it has been shown in this section, mathemat-ical distance between trajectories is a real need formany ATM applications. The next section of this pa-per presents some current trajectory distance metricsand shows their limitations. The third part gives adetailed mathematical description of our new trajec-tory distance. The fourth part introduces the asso-ciated algorithms implementation. Finally, the fifthpart, presents the application of such algorithms to themajor flows extraction of the French airspace with '8000 trajectories.

2 MATHEMATICAL DISTANCE BETWEEN TRA-JECTORIES

2.1 IntroductionIn a vector space, distances are very well defined.If we consider two points ~P1 = (x1, y1)T and ~P2 =

(x2, y2)T in a plane (see Fig. 4), the distance betweenthem can be computed with the classical formula ofthe euclidean distance (see Fig. 4) :

d(~P1, ~P2) =

√(x2 − x1)2 + (y2 − y1)2 (1)

P1

P2

γ2

1γDistance

Distance= ?

Figure 4 On the left, two points ~P1 and ~P2 has beendrawn for which the classical Euclidean distance isshown in red. On the right, two trajectories are drawn(γ1, γ2) for which one want to determine a mathemat-ical distance.

What is the distance, if now the points ~P1 and~P2 are replaced by two trajectories γ1 and γ2 ? Tra-jectories are infinite dimension mathematical objectswhich are not easy to manipulate. We are looking fora mathematical distance between trajectories (γ1 andγ2) with the following properties :

• d(γ1(t), γ2(t)) = 0 ⇒ γ1(t) = γ2(t)

• d(γ1(t), γ2(t)) = d(γ2(t), γ1(t))

• d(γ1(t), γ2(t)) + d(γ2(t), γ3(t)) ≥ d(γ1(t), γ3(t))

One of the main results of this paper is the estab-lishment of such mathematical distance between air-craft trajectories.

2.2 Current Trajectory DistancesAn aircraft trajectory is a time sequence of coor-

dinates representing the aircraft path over a periodof time and may be represented by a N-uple :T ={(x1, y1, z1, t1), (x2, y2, z2, t2), ..., (xN , yN , zN , tN)}whereN is the duration.

The simplest metric used for computing the dis-tance between a pair of trajectories is the mean of co-ordinate distance, which is given as

m1(T a,T b) =1N

N∑n=1

dn (2)

where the displacement between the positions is cal-culated using the Cartesian distance

dn = [(xan − xb

n)2 + (yan − yb

n)2 + (zan − zb

n)2]12 (3)

Note that, the mean of distance metric makes threecritical assumptions :

3


1. the durations of both trajectories are the same :Na =

Nb = N

2. the coordinates are synchronized tan = tb

n

3. the time sampling rate is constant tan+1 − ta

n =tam+1 − ta

m

It is evident that the mean of distance is very sensi-tive to the partial mismatches and cannot deal withthe distortions in time.

To provide more descriptive information, the sec-ond order statistics such as median, variance, min-imum and maximum distance may be incorporated.For instance variance trajectory distance is defined as

m2(T a,T b) =1N

N∑n=1

(dn − m1(T a,T b))2 (4)

Although these statistics supply extra information,they inherit (even amplify) the shortcomings of theordinary mean of distance metric m1. Besides, noneof the above metrics is sufficient enough by itself tomake an accurate assessment of the similarity.

Another possible candidate for the distance be-tween two trajectories γ1 and γ2 will simply be to takethe supremum norm (see Fig 5), that is :

d∞(γ1, γ2) = sups∈R‖γ1(s) − γ2(s)‖ (5)

d 8

γ1(t)

γ (t)2

OD

Figure 5 Supremum norm distance

Since γ1 and γ2 are constant outside bounded in-tervals of R, the supremum is well defined. How-ever, this metric is not sensitive to global propertiesof curves. In the Fig. 6, the curves γ1 and γ2 are at thesame distance from γ3 but have very different shapes.From an operational point of view, γ1 is just a shifted

γ

γ

3

γ2

1

Figure 6 Different trajectories with same sup distance

copy of γ3 while γ2 will probably not be realistic.

For trajectories γ1, γ2 with the same origin-destinationpairs, γ1−γ2 can be defined as a compactly supportedmapping and an area distance between trajectories canbe defined :

d2(γ1, γ2) =

(∫R‖γ1(t) − γ2(t)‖2 dt

) 12

(6)

d2 ~ area γ1(t)

γ (t)2

OD

Figure 7 Area distance between trajectories with thesame origin-destination pairs

An extension of such area based distance metricis proposed in [17]. The crossing points of two paths(where T a(pi) = T b(p j)) are used to define regionsQ j, j = 1, .., J between trajectories (see Fig. 8). For

��

��

��

��

1γ

2γ

1γ

2γ

Q 1

Q 1Q 2

Q 3

Figure 8 Area distance between trajectories with orwithout crossings.

each region, a polygon model is generated and the en-closed area is found by the parameterized shape. Theresulting distance is given by :

m3(T a,T b) =

N∑n=1

area(Q j) (7)

This metric can handle more complex trajectories, how-ever it is sensitive to entanglements of the trajectory,it discards the time continuity, and fails to distinguishtwo trajectories in opposite directions. Furthermore,it is not adapted to 3D trajectories.

In order to introduce our new mathematical dis-tance between trajectories, one must first give somerepresentation definitions.

4


2.3 RepresentationSince objects of interest are aircraft trajectories,

we need to find an adapted framework in which com-putations may be made on trajectories as a whole.There are basically two ways of understanding what atrajectory is :• The time/position approach. In this case, a tra-

jectory can be represented as a mapping froma bounded interval of R (the life time of thetrajectory) to R3 or R6 depending on whetherspeed is part of the data or not. Since there is anexplicit dependence on time, there is a need tocalibrate trajectories with time shifts for all ap-plications involving trajectory comparison. Wewill see in the following that there is neverthe-less a mean of reducing the problem so that ori-gin of time is automatically calibrated.

• The shape approach. Here, trajectories are un-derstood as paths and time is not directly rel-evant (from a more formal point of view, wetake the quotient of the trajectories understoodas mappings by the group of diffeomorphismsacting on time), so that we may assume that theunderlying life time of trajectories is always theinterval [0, 1]. This is the right framework fordealing with major flows estimation.

2.4 Trajectories as mappingsWe will assume in the following that trajectories

are given as mappings from a compact interval of R toR3. The case of mappings from R to R6 (that is withexplicit speed, for example as given by radar trackingfilter) can be derived with minor changes and thus willnot be addressed here. Since physical trajectories aresmooth unless there is a perturbing noise, we madethe choice to take all trajectories as smooth mappingsfrom a compact interval of R to R3.

The first point to deal with is the necessary cali-bration of the origin of time for trajectories compar-ison. Remembering that there is an explicit depen-dence on time, one cannot just time shift one trajec-tory in time in order to make it coincident with an-other in order to compare them : this will result inforgetting distortions in time, that is trajectories withthe same range (as mappings) but different positionsat different times may become equal.

Since we choose to compare trajectories as map-pings, a good candidate for computing the distancewill be to integrate over time (like for the area dis-tance) and to evaluate a mean error instead of the rawsum of squares :

dT (γ1, γ2)2 =1

2T

∫ T

−T‖γ1(t) − γ2(t)‖2dt (8)

with T > 0. Or, if we allow the mean to be weighted :

dT (γ1, γ2)2 =1T

∫R

h(t/T )‖γ1(t) − γ2(t)‖2dt (9)

and h a positive summable function such that :∫R

h(u)du = 1 (10)

This formula defines a semi-distance between trajec-tories γ1 and γ2 (see appendix A).

The previous family of semi-distances has nicefeatures because of the scaling ability, but since it isnot a single metric, it is difficult to use standard algo-rithms based on distances (for example, classificationalgorithms). There is thus a need for another defini-tion of proximity between trajectories that will yielda single value while capturing interesting global char-acteristics.

Before introducing our homotopic distance betweentrajectories one must introduce how do we cope withtime difference between trajectories.

2.5 Parametrization invarianceA very important constraint to take into account

is the parametrization invariance: the shape of an ob-ject is independent on the way its contour is followed.In its seminal paper, Kendall introduced the notion ofshape manifold [8]: the originality of its work was theuse of a differential geometry setting to implicitly en-force the invariance with respect to shape-preservingtransformations. Curves were represented as finitesequences of distinguished points, called landmarks.Some related algorithms were eventually designed forair traffic analysis applications. In a study conductedby the Mitre corporation on behalf of the Federal Avi-ation Authority (FAA) [3], a spectral clustering algo-rithm was applied to sampled trajectories. Only thedistance between landmarks was used, no invarianceunder euclidean transformations were imposed. Dueto the high computational complexity, a random pro-jection was first applied to the data in order to re-duce the dimension of the samples. The most impor-tant limitation of this approach is that the shape ofthe trajectories is not taken into account when apply-ing the clustering procedure unless a re-sampling pro-cedure based on arc-length is applied: changing thetime parametrization of the flight paths will induce achange in the classification. Methods based on timesseries as surveyed in [10, 20] are appealing, but turnout to be inadequate for the present application. Fi-nally, functional data statistics [6, 18] provides a pow-erful framework, still lacking the re-parametrizationinvariance. In this section, flight paths will be mod-eled as points in an infinite dimensional riemanianmanifold. An intrinsic notion of distance exists in thissetting and is defined as the infimum of the length ofthe paths connecting two points. Having this at handallows the use of standard, distance based algorithmslike k-means, k-mediods or hierarchical clustering.

5


2.6 Trajectories registrationA flight path may be modeled as a smooth curve

γ : [a, b] → R3 that maps a time to a position. Twodistinct trajectories γ1, γ2 are most of the time definedon different time intervals, say [a1, b1] (resp. [a2, b2])for γ1 (resp. γ2), making the comparison betweenthem quite awkward. This issue is well known inthe field of functional data statistics as the registra-tion problem. In a formal sense, it amounts to finda pair (φ1, φ2) of strictly increasing diffeomorphismsφ1 : [0, 1] → [a1, b1], φ2 : [0, 1] → [a2, b2] such thatthe transformed curves γ1 ◦φ1, γ2 ◦φ2, defined on thecommon interval [0, 1], are as similar as possible. Thespecial problem instance:

minφ1,φ2

∫ 1

0‖γ1 ◦ φ1(t) − γ2 ◦ φ2(t)‖2 dt

gives the Frechet distance between γ1, γ2. Comput-ing the optimal φ1, φ2 is a difficult task, unless thecurves are assumed to be polygonal. Furthermore, asmentioned in [18], the registration procedure may re-move some important features from the data: the ex-tra degree of freedom provided by the so-called warp-ing functions φ1, φ2 may have the detrimental effectof registering curves that does not need it [19]. Adiscrete relative to the Frechet distance is known asdynamic time warping and may be used to comparesampled sequences. Nevertheless, it suffers from thesame drawback.

On the end of the other scale, a much simple pro-cedure is to select only affine transformations for thewarping functions. Given a trajectory γ : [a, b]→ R3,the affine registration is γ ◦ φ with:

φ : t ∈ [0, 1] 7→ a + (b − a)t

It amounts to shift the time origin so as to make itcoincident with 0, then to scale by the length b − a ofthe time interval.

In between, registration procedures based on timelandmarks or monotonic polynomial approximationmay be used [19]. Most of the time, a penalty crite-rion must be added to the similarity measure in orderto avoid the over-registration phenomenon. It worthmentioning a special procedure, that will be used inthe sequel, that is more in line with geometry. Givena smooth curve γ : [a, b] → R3, its arclength is thesmooth mapping:

s : t ∈ [a, b] 7→∫ t

a‖γ′(u)‖du

The length lγ of the curve is just s(b). Assuming thatγ′ never vanishes, s is strictly increasing, thus invert-ible. It induces a warping function:

ξ : t ∈ [0, 1] 7→ s−1(tlγ) ∈ [a, b]

that is characterized by the property:

∀t ∈]0, 1[, ‖Dtγ ◦ ξ(t)‖ = lγ

where Dt stands for the derivative with respect to t.This warping function is intimately related to the land-marks approach of [8], as sampling evenly in the in-terval [0, 1] will result in a geometric even samplingon the curve itself (with respect to arclength). It willbe denoted as the arclength warping in the sequel.

2.7 The manifold of pathsThe idea of representing curves as point on an infi-

nite dimensional manifold arises in the field of patternrecognition as an answer to the problem of assessing adegree of similarity between two shapes [15]. Withinthis frame, only closed curves were considered as theyrepresent objects contours. In the context of air traffic,flight paths are never closed, unless the aircraft takeoff and land at the same airport, which is a quite un-common for airliners. The initial mathematical modelmust be adapted to cope this specificity. For the sakeof simplicity, all trajectories are assumed to be definedon the time interval [0, 1].

Definition 1 The space of immersions Imm([0, 1],R3)is the set of smooth curves γ : [0, 1]→ R3 with nowherevanishing derivative in the interval ]0, 1[.

Generally speaking, an immersion will be a curve withnowhere vanishing derivative in the interior of its do-main. It is clear that for such a curve the arclength iswell defined and strictly increasing thus the geometricwarping function exists. It may be used to perform aregistration step to ensure that all curves are definedon [0, 1].

Given γ in Imm([0, 1],R3), its derivative norm‖Dtγ‖ is a continuous mapping on the compact in-terval [0, 1] and thus has a non-zero minimum valuem. If ε : [0, 1] → R3 is a smooth mapping such thatsup[0,1] ‖Dtε‖ < m, then γ+ε will have a nowhere van-ishing derivative and thus still belongs to Imm([0, 1],R3).This indicates that this space has locally the structureof a vector space (in fact a Banach space) and glob-ally the one of a differentiable manifold. To get ridof the influence of parametrization, the shape spaceis defined as a quotient with respect to all increasingdiffeomorphisms of the interval [0, 1]:

E = Imm([0, 1],R3)/Diff+([0, 1])

E inherits the manifold structure from Imm([0, 1],R3).A point in E will be denoted by [γ] and is an equiva-lence class of mappings γ ◦ φ with φ ∈ Diff+([0, 1]).A tangent vector at [γ] is a couple ([γ], v) where v isa smooth mapping from [0, 1] to R3. This mappingmust be understood as an infinitesimal displacementfield on the base curve γ. As usual, the set of tangentvector is called the tangent bundle of E, denoted by

6


TE. An riemanian metric can be introduced on E, inthe spirit of [15]:

g[γ](u, v) =

∫ 1

0〈u(t), v(t)〉(1 + Aκ2(t))‖Dtγ(t)‖dt

(11)

+ µ〈u(1), v(1)〉 − µ〈u(0), v(0)〉 (12)

where γ is a representative curve of [γ] and κ is thecurvature of γ at t. The parameters A, µ are strictlypositive real numbers that tune the respective impor-tance of the curvature and the endpoints. The riema-nian metric is invariant under a change of parametriza-tion and thus does not depend on the particular choiceof γ in the equivalence class [γ].

A smooth path between two points [γ1], [γ2] in Eis represented by a smooth homotopy Φ ∈ Imm([0, 1],R3),that is a smooth mapping from [0, 1]2 to R3 such that:

• Φ(0, •) = γ1(•),Φ(1, •) = γ2(•)

• ∀(s, t) ∈ [0, 1]2, DtΦ(s, t) , 0

The derivative of Φ with respect to the homotopy pa-rameter s, denoted by DsΦn is a smooth curve on[0, 1], so that for a given s, the couple ([Φ(s, •)],DsΦ(s, •))is a tangent vector in TE[phi(s,•)]. An visual representa-tion of an smooth homotopy along with the associatedtangent vectors is given in Fig. 9.

(a, )1(t)γ ’

(t)γ ’2(b, )

1γ ’(u)

du1

γ ’(u)d( )

dudvu

vu initial point

Figure 9 Smooth path between two curves

Using the riemanian metric (11) on TE, the energyof a path Φ can be defined in the usual way:

E(Φ) =

∫ 1

0g[φ(s,•)] (DsΦ(s, •),DsΦ(s, •)) ds (13)

It is equivalent for a path to minimize the energy orthe length, the former is preferred as it saves a squareroot in the expression. The critical points of E arecalled geodesic paths. Since it is only a local condi-tion (vanishing derivative), it may not correspond toa minimum of E. If such a global minimum exists, apath realizing it, is called a minimizing geodesic. Inthe finite dimensional setting, the Hopf-Rinov theo-rem may be invoked to prove the existence of a mini-mizing geodesic between arbitrary points. Unfortu-nately, it doesn’t hold generally for infinite dimen-sional manifolds. It turns out that in the framework

defined above, a minimizing geodesic exits betweenany two curves, thus making possible the definitionof a distance on E:

For any couple ([γ1], [γ2]) in E2, the distance be-tween [γ1] and [γ2] is given by:

d =

∫ 1

0

√g[φ(s,•)] (DsΦ(s, •),DsΦ(s, •))ds (14)

where Φ is any homotopy between γ1, γ2 realiz-ing the minimum of E.

The distance d turns E into a metric space and canbe used in any distance-based clustering algorithm.

3 ALGORITHM

3.1 Distance AlgorithmIn order to compute the distance between two tra-

jectories (γ1,γ2), a time regularization is first appliedto both trajectories. Then, an homotopy Φ betweenγ1,γ2 is built for which a discrete grid is built in orderto minimize its associate energy.

Let a be the origin of the trajectory γ. We have :γ(t) = a +

∫ t0 γ′(s)ds ,so a couple (a, γ′) (∈ W) with

γ′ compactly supported defines a trajectory.An homotopy between (a, γ′1) and (b, γ′2) is a con-

tinuous mapping Φ : [0, 1] → W such that Φ(0) =(a, γ′1) , Φ(1) = (b, γ′2). Intuitively, an homotopy is acontinuous deformation between two trajectories.c

The deformation energy between γ1 and γ2 is linkedto the distance between those trajectories and can becomputed with the energy of the homotopy betweenγ1 and γ2 :

E(Φ) =

∫ 1

0

(∥∥∥∥∥∂vu

∂u

∥∥∥∥∥2

+

∫R

∥∥∥∥∥∂γ′u(s)∂u

∥∥∥∥∥2

ds)

du (15)

In the case of a linear homotopy (which is the sim-plest one), the associated energy is given by :

Φ0(u, s) =([(1 − u).a + u.b] ,

[(1 − u).γ′1(s) + u.γ′2(s)

])(16)

E(Φ0) = ‖b − a‖2 +

∫R

∥∥∥γ′1(s) − γ′2(s)∥∥∥2

ds (17)

There is an infinite number of homotopies shiftingfrom γ1 to γ2 and our problem is to find the one withthe minimum energy.

The deformation energy of a shape homotopy isobtained with a slight change in the expression fortrajectories.

7


Figure 11 On this metric space each trajectory is rep-resented by a point (blue point)..

E(Φ) =

∫ 1

0

(∥∥∥∥∥∂vu

∂u

∥∥∥∥∥2

+

∫R

∥∥∥∥∥∂γ′u(s)∂u

∥∥∥∥∥2

.∥∥∥γ′u(s)

∥∥∥ ds)

du

(18)In order to compute such energy, a grid on the

homotopy connecting γ1 to γ2 is built, as shown onFig. 10.

(a, )1(t)γ ’

(t)γ ’2

(b, )

x,y,z

Figure 10 Structure of the grid used for homotopy en-ergy minimization. Each red point has 2D coordinates(x,y) for which an optimization algorithm is used forsearching the z coordinates which minimize the en-ergy of the homotopy connecting γ1 to γ2.

This grid help us to compute an approximation ofsummation used in E(Φ). The optimization algorithmis searching for the z coordinate of each grid pointin order to minimize E(Φ). One can show that suchproblem is convex (from the optimization theory pointof view) and gradient like method can be used to findthe associated minimum (quadratic programming hasbeen used to solved this problem efficiently).

3.2 Clustering AlgorithmWe consider a set of trajectories extracted from

the radar track database of a given airspace. Hav-ing defined a distance between trajectories, one cangather together such trajectories in order to create clus-ters by using an adaptive clustering algorithm (hierar-chical clustering). Such a clustering algorithm aimsto partition the trajectory set into K clusters. To reachthis goal, trajectories are consider as points in the as-sociated metric space (see Fig. 11).

This algorithm uses two parameters, dmin and dmax,to respectively fuse clusters and create new clusters.Initially, each trajectory is considered as the centroidof a cluster. We then apply the three following princi-ples one after the other:

• if two centroids are at a distance lower thandmin, we fuse them into a single cluster, of whichthe resulting centroid is the barycenter of thetwo initial centroids. The barycenter is com-puted the following way :

µi =1N

i=N∑i=1

γi (19)

• a new individual is aggregated to a cluster if itsdistance from the closest centroid is lower thandmax and in this case we compute the new globalcentroid.

• Otherwise, we create a new cluster containingthe single trajectory.

The number of clusters is also a result of the al-gorithm. An example of clustering resust is given onFig. 12.

For each cluster c, one can compute also the fol-lowing features :

• Number of trajectories in the cluster Nc;

• Mean trajectory which is the cluster centroid(γc);

• Dispersion of the cluster:

Nc∑i=1

∥∥∥γ j − γc

∥∥∥2(20)

where ‖.‖ is the norm in the trajectory metricspace.

The overall processing can be summarized by theFig. 13

8


γ1 γ

Ν

Trajectory 1 Trajectory N

Registration

Clustering

Trajectories

Distance

Computation

Clusters

Registration

Figure 13 Overall structure of the algorithm

Figure 14 Radar tracks of the France traffic of June,27, 2015. This traffic correspond to the upper airspace

Figure 12 In this example the algorithm find elevenclusters with different features.

4 RESULTSThe algorithm has been applied to the French airspace

with an heavy traffic of 8764 flights corresponding toJune 27, 2015. This traffic has been extracted fromthe radar track database. Each trajectory being sam-pled every ten seconds, one has to manipulate about 5million points, each on them having four coordinates(x, y, z, t). The traffic is represented on Fig. 14.

The initial step consists in computing the trajec-tory registration in order to remove the absolute timedependency. Then, the dmin and dmax distance havebeen fixed in order to apply the hierarchical cluster-ing algorithm. Those distances have been establishduring experimentations (dmin = 2.30 dmax = 4.5).Based on those distances, the hierarchical clusteringalgorithm has extracted 47 major flows for this day asshown on Fig. 15.

The algorithm has been implemented into C++and executed on a IntelXeon3.2Ghz PC computer withan executing time of 30 seconds for extraction themajor flow associated to the 8764 flights of June 27,2015.

5 CONCLUSIONThis paper has shown that distance between tra-

jectories is a real need for ATM applications. Severalways of computing distances on the space of trajecto-ries have been presented with their limitations. Thisfamily of metrics, scale based, is mainly useful fordescriptive purpose and to quickly analyze a set oftrajectories (for example, as a tool complementary tostandard descriptive statistics).

Then, a concept originating from functional anal-ysis has been introduced in order to work directly ontrajectories as a whole. For more in depth analysisof trajectories, a new kind of distance has been in-troduced that is based on the energies of homotopiesjoining pairs of trajectories. This yield to a variationalproblem that cannot be solved directly, but may be re-duced to a quadratic optimization problem. This kindof distance allows computations to be done on trajec-tories understood as shapes (or embeddings).

Based on this new distance between trajectories,an efficient major flows extraction has been developedwith nice results on the French airspace for severalthousands of trajectories.

9


Figure 15 Major flows extracted from the hierarchicalalgorithm.

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A Semi Distance beetween trajectories

dT (γ1, γ2)2 =1

2T

∫ T

−T‖γ1(t) − γ2(t)‖2dt (21)

with T > 0. Or, if we allow the mean to be weighted :

dT (γ1, γ2)2 =1T

∫R

h(t/T )‖γ1(t) − γ2(t)‖2dt (22)

and h a positive summable function such that :∫R

h(u)du = 1 (23)

Now, since the derivative of ‖γ1(t) − γ2(t)‖2 is :

2(γ1(t) − γ2(t), γ′1(t) − γ′2(t)) (24)

((., .) denotes the usual scalar product) we have :

‖γ1(t) − γ2(t)‖2 = ∆−+∫ t

−∞

2(γ1(s) − γ2(s), γ′1(s) − γ′2(s))ds

with ∆− = limt→−∞ ‖γ1(t) − γ2(t)‖2 (this limit alwaysexists since we have assumed at the beginning that tra-jectories are mappings from compact intervals). Theexpression of the mean square error becomes :

dT (γ1, γ2)2 = ∆−+

1T

∫R

h(t/T )(∫ t

−∞

(γ1(s) − γ2(s), γ′1(s) − γ′2(s))ds)

dt

By fubini theorem, we have :

dT (γ1, γ2)2 = ∆−+

1T

∫ −T

−∞

(γ1(s) − γ2(s), γ′1(s) − γ′2(s))ds+∫ T

−T

(∫ T

sh(t)(γ1(s) − γ2(s), γ′1(s) − γ′2(s))dt

)ds

and finally :

d(γ1, γ2)2 = ∆−+∫ −T

−∞

(γ1(s) − γ2(s), γ′1(s) − γ′2(s))ds+∫ T

−T(γ1(s) − γ2(s), γ′1(s) − γ′2(s))(

∫ 1

s/Th(u)du)ds

Limits as T → 0,T > 0 and T → +∞ can be ob-tained. If T → 0,T > 0, since

∫ 1s/T h(u)du is bounded,

we have :

limT→0+

dT (γ1, γ2)2 = ∆−+∫ 0

−∞

(γ1(s) − γ2(s), γ′1(s) − γ′2(s))ds

This is in fact ‖γ1(0) − γ2(0)‖2.γ1, γ2 are constant mappings outside a compact

interval of R, so both γ′1 and γ′2 are compactly sup-ported. Furthermore the support of the mapping s →(γ1(s)−γ2(s), γ′1(s)−γ′2(s)) is included in the union ofthe supports of γ′1 and γ′2, thus A > 0 exists such that

11


this mapping vanishes outside [−A,+A]. This meansthat for T > A the first integral in the previous expres-sion vanishes. The second has value :∫ T

−T

((∫ 1

s/Th(u)du)(γ1(s) − γ2(s), γ′1(s) − γ′2(s))

)ds

(25)and here again, as soon as T > A this reduced to :∫ A

−A

((∫ 1

s/Th(u)du)(γ1(s) − γ2(s), γ′1(s) − γ′2(s))

)ds

(26)so :

limT→+∞

dT (γ1, γ2)2 = ∆−+

limT→+∞

∫ A

−A

((∫ 1

s/Th(u)du)(γ1(s) − γ2(s), γ′1(s) − γ′2(s))

)ds

by dominated convergence theorem, this gives :

limT→+∞

dT (γ1, γ2)2 = ∆−+K∫ A

−A(γ1(s)−γ2(s), γ′1(s)−γ′2(s))ds

(27)with K =

∫ 10 h(u)du. and finally, using again the fact

that γ′1, γ′2 are compactly supported, this is :

limT→+∞

dT (γ1, γ2)2 = ∆−+K∫R

(γ1(s)−γ2(s), γ′1(s)−γ′2(s))ds

(28)

Now, using the fact that :∫R

(γ1(s) − γ2(s), γ′1(s) − γ′2(s))ds = ∆+ − ∆− (29)

with ∆+ = lims→+∞ ‖γ1(s) − γ2(s)‖2, we have :

limT→+∞

dT (γ1, γ2)2 = (1 − K)∆− + K∆+ (30)

So the limit case T → +∞ is a convex combinationof the initial and final differences (this will be 0 if tra-jectories are on the same origin destination, thus thelimit of the dT is not a distance). Letting the weight-ing function h depend on time shift τ yields the finaldefinition of a family of metrics :

dT,τ(γ1, γ2) = supτ∈R

1T

∫R

h((t − τ)/T )‖γ1(t) − γ2(t)‖2dt

(31)with the property that the limit case T → 0,T > 0reduces to the supremum distance. This can be seenas a scale base distance, with T parameter being thescaling factor.

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trajectory mathematical distance applied to airspace major

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