OPTIMIZING B-SPLINES USING GENETIC ALGORITHMS

APPLIED TO AIR-TRAFFIC CONFLICT RESOLUTION

Clement Peyronne, Daniel Delahaye

Applied Mathematic Laboratory (LMA), ENAC, 7 rue Edouard Belin, Toulouse, France

Marcel Mongeau

Universite de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathematiques de Toulouse, F-31062 Toulouse, France

CNRS, Institut de Mathematiques de Toulouse UMR 5219, F-31062 Toulouse, France

Laurent Lapasset

Air Trafﬁc Management, Capgemini, Toulouse, France

Keywords:

Air-trafﬁc conﬂict resolution, B-splines, Genetic algorithms.

Abstract:

Conﬂict resolution has always been a sensitive matter in air-trafﬁc management. Current European projects

aim partial or total automation of air trafﬁc control to deal with the constant growth of air trafﬁc. Techno-

logical advances on ﬂight management system allows us to consider an automatic conﬂict resolution using

continuous trajectories. In this paper, we present a new methodology that, ﬁrst, relies on B-splines to model

trajectories, secondly models air-trafﬁc conﬂict resolution as an optimization problem whose decision vari-

ables are the spline control points. Finally, we use genetic algorithms to tackle this optimization problem in

order to generate optimal conﬂict-free situations.

1 INTRODUCTION

From the beginning, the most critical point of air traf-

ﬁc control was to ensure safety separation between

airplanes. To achieve this goal a safety standard sep-

aration distance has been deﬁned: 5 Nm (Nautical

miles) horizontally and 1000 feet vertically. Air traf-

ﬁc controllers are responsible for ensuring the respect

of these separation rules.

1.1 Context

Air trafﬁc being constantly increasing, controllers in

charge of an aviation sector must handle more and

more ﬂights. Up to now, decreasing the size of avi-

ation sectors could compensate the growth of trafﬁc

but we are reaching the point where a decrease of the

size of sectors is no longer efﬁcient.

Nowadays, air trafﬁc management has already

used every available ressource to increase airspace ca-

pacity. However, from now to 2020, air trafﬁc is ex-

pected to be multiplied by two or three. Consequently,

air trafﬁc management (ATM) will have to deal with

this overload while ensuring at least equivalent stan-

dards of safety (Alliot and de Verdi

ere, 2003). The

SESAR European project aims at ﬁnding solutions

to this problem by providing a decision support to

air trafﬁc controllers in order to decrease their work-

load. Considering the technological advances on the

aircraft ﬂight management system (FMS), we will ex-

plore in this paper the possibility of a full automation

generating continuous trajectories that new FMS can

follow.

1.2 Previous Related Work

By the past, two kinds of method have become pre-

dominant in automatic air trafﬁc conﬂict resolution

for their good results : genetic algorithms (GA) (Du-

rand, 2004) and navigation functions (Zeghal, 1994).

Genetic algorithms are part of what is known as

natural computation or evolutionary methods. This

optimization method is based on the evolution theory

213

Peyronne C., Delahaye D., Mongeau M. and Lapasset L..

OPTIMIZING B-SPLINES USING GENETIC ALGORITHMS APPLIED TO AIR-TRAFFIC CONFLICT RESOLUTION.

DOI: 10.5220/0003082802130218

In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 213-218

ISBN: 978-989-8425-31-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

and uses concepts such as mutation, crossover and,

of course, selection. Each possible solution of our

optimization problem is encoded as a chromosome

via a speciﬁc encoding (for example, number and du-

ration of simple maneuvers). The algorithm creates

randomly a ﬁrst chromosome population. Each chro-

mosome ability to solve the problem (ﬁtness) is then

evaluated. The best individuals (according to their ﬁt-

ness) are selected and mutation, crossover are applied

to obtain a new population of chromosome (next gen-

eration). The user chooses the necessary generation

number for the algorithm to converge (this choice is

usually done empirically).

The obtained trajectories’ velocity stays within ac-

ceptable bounded range with respect to ATM oper-

ational constraints. However, these techniques have

not been tested yet with curved trajectories.

Navigation functions use a different representa-

tion. Indeed, the airspace is considered as a poten-

tial ﬁeld, and airplanes like particles navigating in it.

Naturally, negative charges represent obstacles to the

airplane (other aircrafts, congested areas) and positive

charges, its destination. As a result, each airplane is

attracted to its destination while being repulsed by ob-

stacles which enables to the automatic generation of

conﬂict-free trajectories.

Figure 1: Principle of navigation functions between two air-

crafts (left), between an aircraft and an obstacle (right).

It has been demonstrated that navigation function

methods ensure collision avoidance and connection

between departure and destination. The major draw-

back of this method is that the obtained solution does

not respect the ATM constraints such as limited speed,

or trajectory smoothness. Evenmore, they can lead to

major delays and overcosts as they tolerate any devi-

ation from the business trajectories (BT : the straight

line between the departure and arrival points).

In this paper, we present a new methodology that,

ﬁrst, relies on B-splines to model trajectories, sec-

ondly models air-trafﬁc conﬂict resolution as an op-

timization problem whose decision variables are the

spline control points. Finally, we use genetic algo-

rithms to tackle this optimization problem in order to

generate optimal conﬂict-free situations.

We present our trajectory model in Section 2. Sec-

tion 3 presents the optimization method we chose to

solve our optimization problem: genetic algorithms

(GA). We detail in Section 4 the objective function

(conﬂict detection) of our optimization problem. Nu-

merical results are presented in Section 5. We draw

conclusions in Section 6.

2 TRAJECTORY MODEL

B-spline was primarily introduced to ﬁnd a curve in-

terpolating a set of points of R

called control points.

It was later extended to approximation, thereby avoid-

ing the undesirable oscillation inherent to interpola-

tion. In our study, we shall focus on this use of splines

to approximate a set of control points. The control

polygon, the piecewise linear curve linking the control

points, completely deﬁnes the curve (Rabut, 2008).

Basically, B-splines are parameterized curves gen-

eralizing the Bezier curve concept. It is an efﬁcient

approximation tool which is constructed from poly-

nomial pieces joined at certain parameter’s values

called knots, stored in a knot vector. In a very sim-

pliﬁed way, if we consider a set of control points

) = P

∈ R

(i = 0, 1, ..., n), and a parameter u,

we can deﬁne the B-spline as follows :

C(u) = (σ

(u), σ

(u)), u ∈ [a, b]

where σ

(u) and σ

(u) are the B-splines approxi-

mations of the couples (i, X

)

i=0:n

and (i,Y

)

i=0:n

for

u ∈ [a, b] .

Theoritically, the curve is calculated by multiply-

ing the B-splines basis functions by the control points

coordinates. For σ

(u) =

n+1

∑

i=−1

(u) (1)

However, in practice, B

construction is made by

recurrence. Consequently, we used a more numeri-

cally convenient way to calculate the B-spline values

only using control points values. The B-spline and its

derivative values at knot u

are deﬁned by :

i+1

+4X

i−1

i+1

−X

i−1

i+1

−2X

i−1

000

i+2

−3X

i+1

+3X

−X

i−1

(2)

ICEC 2010 - International Conference on Evolutionary Computation

214

Therefore, by using Taylor series, we obtain the B-

spline value for all u ∈ [u

i+1

(u) = σ

+ (u − u

)(σ

+ (u − u

)(

+ (u − u

)

000

))

(3)

We choose to rely on B-spline modeling trajectories

because it is a very efﬁcient tool for curve approx-

imation in terms of both approximation quality and

computational time. Moreover, B-splines feature in-

teresting properties such as C

-continuity (crucial for

modeling smooth aircraft trajectories), robustness and

ﬂexibility (if one control point is displaced, only a

small part of the curve will be affected). The last

good point in favor of B-splines is its compatibility

with GA : the curve is completely determined by few

control points only, which will be the optimization pa-

rameters of the GA. In our study, we use a ﬁxed max-

imal number of control points between the departure

and the arrival points.

3 OPTIMIZATION METHOD

When several aircraft are involved in a conﬂict, the

conﬂict resolution problem has been shown to be NP-

hard (Durand, 2004). Moreover, the optimization

variables being the B-splines control-point location,

we shall see that our objective function (4) is not

differentiable with respect to these variables. Con-

sequently, we must rely on black box (direct) opti-

mization methods to address our problem. In this pa-

per, we choose to use a stochastic global optimization

method, genetic algorithms, to guide the control-point

location. As we mentioned in the previous part, GA

seem to be appropriate with B-splines. In this section,

we explain brieﬂy GA theory and then we shall detail

how we adapt the different operators (selection, muta-

tion and crossover) to our air trafﬁc conﬂict resolution

problem.

3.1 Basis of Genetic Algorithms

Evolutionary algorithms use techniques inspired by

evolutionary biology such as inheritance, mutation,

natural selection, and recombination (or crossover) to

ﬁnd approximate solutions to optimization problems

(Goldberg, 1989), (Koza, 1992). An individual, or so-

lution to the problem to be solved, is represented by

a list of parameters, called chromosome or genome.

Initially several such individuals are randomly gener-

ated to form the ﬁrst initial population (POP(k) with

k = 0 on Figure 2). Then each individual is evaluated,

and a value of ﬁtness is returned by a ﬁtness function

Figure 2: Genetic algorithm with tournament selection. The

ﬁrst step consists in the selection of the best individuals

from population POP(k). Afterwards, recombination op-

erators are applied in order to produce the POP(k + 1) pop-

ulation.

(generally, the objective function of our optimization

problem). This initial population undergoes a selec-

tion process which identiﬁes the most adapted indi-

viduals. The selection process which has been used in

our experiments is a deterministic (λ, µ)-tournament

selection. This selection begins by randomly select-

ing λ individuals from the current population POP(k)

and keeps the µ bests (λ > µ). These two steps are re-

peated until a new intermediate population (POP

) is

completed. Following selection, one of the three fol-

lowing operators is applied : nothing, crossover, and

mutation. The associated probability of application

are respectively (1− p

− p

)), p

and p

. Crossover

results in two new child chromosomes, which are

added to the next generation population. The chro-

mosomes of the parents are mixed during crossover.

These processes ultimately result in the next gener-

ation population of chromosomes (POP(k+1) on Fig-

ure 2) that is different from the initial generation. This

generational process is repeated until a termination

condition has been reached. The next section presents

the application of this algorithm to our speciﬁc prob-

lem.

3.2 Chromosome Encoding and Fitness

Calculus

In this section we detail the link between our model-

ing method and genetic algorithms. Indeed, our op-

timization method is using B-splines to calculate the

aircraft trajectories in order to determine each individ-

uals ﬁtness. To deﬁne this interaction, we ﬁrst deﬁne

an encoding.

A chromosome will represent the trajectories of

N aircrafts using a matrix of N × N

where N

is the

maximal number of control points we wish to con-

OPTIMIZING B-SPLINES USING GENETIC ALGORITHMS APPLIED TO AIR-TRAFFIC CONFLICT RESOLUTION

215

sider. For example, the case where N = 20 airplanes

and N

= 2 we encode the trajectories as follows :

Airplane1 P

Airplane2 P

... ... ...

Airplane19 P

Airplane20 P

Obviously, departure and arrival points are not en-

coded in the chromosome as they are not meant to

move, they are not optimization variable.

Let us now deﬁne precisely how we specify the

control points P

. We want a compromise between

allowing the trajectory to deviate freely (in any direc-

tions) from the business trajectory (BT) and staying

as close as possible from it. For that purpose, we de-

ﬁne a ﬁxed width band (D

max

), depending on the BT

length (D

tot

), around the BT trajectory where the con-

trol points are allowed to lie.

A matter for the control point location is that, if

there are all gathered in the same region, it can create

oscillations on the resultant trajectory. Consequently,

we distribute uniformly the control points along the

BT.

To summarize, a control point can be deﬁned by

a single parameter representing a bandwidth percent-

age (±

max

% ).

Moreover, it is desirable to be able to have

a different number of control points for a given

chromosome. In order to satisfy this goal, we

used a convention: percentage are generated within

[−200%;200%], if the percentage is between −100%

and 100%, then we create a control point, if it belongs

to [−200%;−100%[∪[100%;200%], we do not. Con-

sequently, our chromosome is a N × N

matrix of per-

centage. Several cases are presented in Figure 3 and

Figure 3: Chromosome encoding and corresponding B-

spline with one percentage exceeding 100%.

3.3 Crossover and Mutation

Let us now describe how we adapted the genetic op-

erators we use in our GA.

Figure 4: Another chromosome encoding and its

corresponding B-spline with both percentages within

[−100%, −100%].

3.3.1 Slicing Crossover

The crossover operator aims at ﬁnding better solu-

tions by mixing two good individuals of the previ-

ous generation. Therefore, we decide to keep trajec-

tory consistencies by inverting only complete trajec-

tories using a slicing crossover (we do not separate

control points belonging to the same trajectory). Ac-

cordingly, the crossover consists in picking a plane

number randomly and inverting the parent trajecto-

ries corresponding to the next aircrafts. An example

is given in Figure 5 :

Figure 5: Parent and children chromosomes.

3.3.2 Mutation

The mutation operator aims at diversifying the genes

in the population in order to explore as much as

possible the problem space. Thus, mutation con-

sists in choosing randomly one control point in the

chromosome and to assign to it a new random num-

ber (using a uniform distribution) in the interval

[−200%;200%]. (see Figure 6).

ICEC 2010 - International Conference on Evolutionary Computation

216

Figure 6: Initial and mutated chromosome.

4 OBJECTIVE FUNCTION :

CONFLICT DETECTION

METHOD

In order to evaluate each chromosome ﬁtness, we de-

code it into N trajectory curves (one per aircraft) and

evaluate two quantities. First, how many conﬂicts the

situation engenders and secondly what is the total ex-

tra distance induced with respect to the BT. To cal-

culate these quantities, we discretize the airspace into

square cells of size half the standard separation (noted

d). Our conﬂict detection is performed in two steps :

• First, for each airplane, we store the grid’s cells

through which the aircraft ﬂies, the aircraft num-

ber (its label), the entry and exit times in and out

each of the stored cell,

• Then, we select each stored cell and we check

whether any other airplane goes through any of

the eight neighboring cells for other airplanes. If

there are such airplanes, we check the time to see

whether there is a conﬂict between these two air-

planes. If so, we calculate the conﬂict duration.

Our conﬂict detection procedure send back the

chromosome ﬁtness to the GA. Here is the formula

we use to calculate the ﬁtness:

f (X ) = −(CN + (

∑

i=1

− N)) (4)

where CN is the number of conﬂict, NR

the length

of the new route calculated by the algorithm for the

aircraft i, while BT

the length of the business tra-

jectory (straight line from the departure to the arrival

point) and N the number of aircraft. High ﬁtness cor-

responds to good individuals. Indeed, the lower are

the number of conﬂict and the route lengthening, the

better is the chromosome.

5 RESULTS AND PERSPECTIVES

In this section, we present several results we obtained

using our methodology. First, we test our method on

the roundabout test problem which is a very common

test on automatic conﬂict-detection method. It con-

sists in making a certain number of planes ﬂy to the

diametrically opposed point at a common speed (each

point on the circle has an outgoing and an incoming

trajectory).

We choose to ﬂy N = 16 aircrafts for results’

readability) equidistributed on a circle of 100Nm(=

185200m) radius To address this conﬂict resolution,

we use the following parameters :

• Number of generation : 100.

• Size of the population : 500.

• Mutation probability : p

= 0.3.

• Crossover probability : p

= 0.6.

• Maximal number of control points : N

= 2.

To compute the conﬂict resolution, we used a

2.53GHz Intel(R) Core(TM)2 Duo on a Windows

Vista Operating System and we coded in Java. The

obtained conﬂict-free situation is presented in meter-

scale (1Nm = 1852m) in Figure 7. This whole resolu-

tion computation required approximately 4 minutes.

Figure 7: The roundabout conﬁguration after resolution.

The ﬁtness’ evolution with respect to generations

is shown in Figure 8. The ﬁtness is meant to increase

to 0 as formula (4) shows. Moreover, when the ﬁt-

ness is in [−1, 0] the situation is conﬂict-free. Conse-

quently, the algorithm stops only when the best indi-

vidual ﬁtness is in [−1, 0].

Although one can easily solve intuitively this aca-

demic problem due to its symmetry, our automatic

implementation does not exploit any symmetry here.

This result shows that our methodology is promising

as the obtained conﬂict resolution is consistent with

experts’ experience.

We also test our methodology on a more realistic

operational problem where N = 15 aircrafts are in-

volved in a fuzzy convergence. In this situation, for

OPTIMIZING B-SPLINES USING GENETIC ALGORITHMS APPLIED TO AIR-TRAFFIC CONFLICT RESOLUTION

217

Figure 8: The ﬁtness evolution with respect the generation.

each aircraft, we have its departure point, its head-

ing and its speed. To calculate the arrival points, we

make the 15 aircrafts ﬂy for one hour at a constant

speed, on a straight line following the initial direction

(heading). We present the conﬂict-free situations our

algorithm obtains is displayed in Figure 9.

Figure 9: The operational problem after resolution.

We can see on Figure 10 that on an operational

application, our method converges far more rapidly in

three generations (90 generations for the roundabout

test problem). This results shows that our method can

be very efﬁcient from an operational point of view.

Indeed, in operational problems, there are never more

than four aircrafts involved in the same conﬂict. Con-

sequently, our algorithm will be able to deal very efﬁ-

ciently with operational air trafﬁc. For this situation,

we used a different number of generations because of

its extremely quick convergence (number of genera-

tion: 20). Moreover, we can see that few aircrafts

are deviated, which is consistent with a controller be-

haviour, who would prefer to deviate largely one air-

craft instead of deviating lightly several aircrafts to

solve conﬂicts.

6 CONCLUSIONS

We have shown in this paper that the combination of

B-splines and genetic algorithms can be a promising

methodology for automatic conﬂict resolution in air

trafﬁc control. However, we have in mind several

Figure 10: The ﬁtness evolution with respect the generation.

developments to improve our approach such as us-

ing sharing in GA (deals with equirepartition of the

population on the different maximums) in our GA

or implementing a self-adaptative GA (every param-

eter such as bandwidth, control points’ number, etc

will be considered within the chromosome encoding,

as proper optimization variables). Furthermore, we

plan to exploit our B-spline model of trajectory to ad-

dress the conﬂict resolution problem with determinis-

tic derivative-free optimization methods (Conn et al.,

2009). Indeed, despite the local aspect of these meth-

ods, they can also be adapted to global optimization.

ACKNOWLEDGEMENTS

This work has been supported by French National Re-

search Agency (ANR) through COSINUS program

(project ID4CS n°ANR-09-COSI-005) and Capgem-

ini through the ﬁrst author PhD funding.

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