[EN-023] Mathematical Models for Aircraft Trajectory ...dcsl.gatech.edu/papers/eiwac13.pdf · D. Delahaye, S. Puechmorel, E. Feron, P. Tsiotras question is then given by the closest

ENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2013)

[EN-023] Mathematical Models for Aircraft TrajectoryDesign : A Survey.

+ D. Delahaye S. Puechmorel ∗ E. Feron P. Tsiotras ∗∗∗ Applied Math Lab (MAIAA)

Ecole Nationale Aviation Civile(ENAC)Toulouse, France

[email protected]

∗∗ School of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, [email protected]

Abstract Air traffic management ensure the safety of flight by optimizing flows and maintaining separation betweenaircraft. After giving some definitions, some typical feature of aircraft trajectories are presented. Trajectories are ob-jects belonging to spaces with infinite dimensions. The naive way to address such problem is to sample trajectories atsome regular points and to create a big vector of positions (and or speeds). In order to manipulate such objects withalgorithms, one must reduce the dimension of the search space by using more efficient representations. Some dimen-sion reduction tricks are then presented for which advantages and drawbacks are presented. Then, front propagationapproaches are introduced with a focus on Fast Marching Algorithms and Ordered upwind algorithms. An exampleof application of such algorithm to a real instance of air traffic control problem is also given. When aircraft dynamicshave to be included in the model, optimal control approaches are really efficient. We present also some applicationto aircraft trajectory design. Finally, we introduce some path planning techniques via natural language processing andmathematical programming.

Keywords Air traffic conflict resolution, genetic algorithm, B-Spline approximation1 Introduction

Aircraft trajectory is one of the most fundamen-tal objects within the frame of ATM. However, partlydue to the fact that aircraft positions are most of thetime represented as radar plots, the time dependenceis generally overlooked so that many trajectory statis-tics conducted in ATM are spatial only. Even in themost favorable setting, with time explicitly taken intoaccount, trajectory data is expressed as an ordered listof plots labeled with a time stamp, forgetting the un-derlying aircraft dynamics. Furthermore, the collec-tion of radar plots describing the same trajectory canhave tenths more samples, nearly all of them redun-dant. From the trajectory design point of view, thisredundancy is real handicap for the optimization pro-cess. In this survey, alternative trajectory represen-tations are presented with a description of their ad-vantages and limits. Such new approaches may beapplied in many areas :

Aircraft trajectories data compression. As ithas been previously mentioned, ATM system manageaircraft trajectories and control them in order guarantysafety and airspace capacity. Currently those trajecto-ries are represented by the mean of plot lists whichare manipulated by ATM software. Every day, allaircraft trajectories are registered into large database

for which huge capacity is needed. Based on thisnew trajectory representation for which redundancyhas been removed, the trajectories database may bestrongly improved from the capacity point of view.This compressed trajectory format may also be usedfor improving the trajectories transmission betweenATM entities.

Aircraft trajectories Distance Computation. Al-though trajectories are well understood and studied,relatively little investigation on the precise compar-ison of trajectories is presented in the literature. Akey issue in performance evaluation of ATM decisionsupport tools (DST) is the distance metric that deter-mines the similarity of trajectories. Some proposedrepresentation may be used to enhance trajectory dis-tance computation.

Aircraft model Inference. All aircraft modelsare based on ODEs(Ordinary Differential Equation),including tabular ones. Control input includes con-dition and model parameters. The model refinement(and computational complexity) ranges from tabularto many degrees of freedom. The aircraft model in-ference consists in answering the following question:Given a parametrized model and a goal trajectory, canwe infer the best parameter values? A model canbe viewed as a mapping from the control space intothe trajectory space. The way to answer the previous

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D. Delahaye, S. Puechmorel, E. Feron, P. Tsiotras

question is then given by the closest model to the goaltrajectory.

Trajectory prediction. Air traffic managementresearch and development has provided a substantialcollection of decision support tools that provide auto-mated conflict detection and resolution [35, 22, 2],trialplanning [7], controller advisories for metering andsequencing [28, 55],traffic load forecasting [19, 26],weather impact assessment [25, 27, 24]. The ability toproperly forecast future aircraft trajectories is centralin many of those decision support tools. As a result,trajectory prediction (TP) and the treatment of trajec-tory prediction uncertainty continue as active areas ofresearch and development (eg [56, 63, 49, 61, 62]).Accuracy of TP is generally defined as point spatialaccuracy (goal attainment) or as trajectory followingaccuracy. The last one can be rigorously defined bythe mean of trajectory space. The first one is a limitcase of the second by adding a weight function in theenergy functional. Since we may prescribe smooth-ness accuracy of a simplified model relative to a finerone, may be computed.

Major flows definition. When radar tracks areobserved over a long period of time in a dense area,it is very easy to identify major flows connecting ma-jor airports. The expression ”major flows” is oftenused but never rigorously defined. Based on an exacttrajectory distance and a learning classifier, it is pos-sible to answer the following questions: Given a setof observed trajectories, can we split it into ”similar”trajectory classes? If yes, classes with highest num-ber of elements will rigorously define the major flows.Given those classes and a new trajectory, can we tell ifit belongs 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 the3D space of its 4D trajectory. The speed on the pathhas no impact).

Trajectory planning. To improve Air Traffic Man-agement, projects have been initialized in order to com-pel the aircraft in position and in time (4D trajectory)so as to avoid potential conflict and allow for someoptimality with respect to a given user cost index, en-vironmental criteria (noise abatement, pollutant emis-sion . . . ). Depending on the time horizon, severalkind of plannings can be designed:• At a strategical level, only macroscopic indi-

cators like congestion, mean traffic complexity,delays can be taken into account, consideringthe high level of uncertainty;

• at a pre-tactical level, the accuracy of previousindicators, specially congestion and complexityincreases while at the same time early conflictdetection can be performed;

• finally, at the tactical level, conflict resolutionis the major concern and optimality of the tra-jectories is only marginally interesting.

As we can see, there many areas of ATM wheretrajectories are the main objects that have to be ma-nipulate.

The first part of this survey presents some rele-vant features of aircraft trajectories. The second part,presents dimension reduction tricks for optimizationapproaches. The third part describes approaches basedon wave front propagation in isotropic and anisotropicenvironments. The fourth part presents automatic con-trol approaches with some application to air trafficcontrol. Finally, the fifth part introduces some pathplanning techniques via natural language processingand mathematical programming.

2 Some trajectories featuresIn the following all aircraft trajectories will be de-

scribed as mappings from a time interval [a, b] to astate space E with E either R3 or R6 depending on thefact that speed is assumed to be part of aircraft stateor not. Extension to trajectories on a sphere (typicallylong haul flights) will be sketched only.

2.1 Notations and terminologyThe reference for this section is [47]. Let γ[a, b]→

E be a trajectory. The origin of the trajectory is γ(a)and the destination is γ(b). Those two points are calledthe endpoints of the trajectory. All trajectories are as-sumed to be at least continuously differentiable (classC1) so that the length of a trajectory γ[a, b] → E iswell defined as:

l(γ) =

∫ b

a‖γ′

(t)‖dt

If ‖γ′

(t)‖ = 0 for some t ∈ (a, b) the point t issaid to be singular. A parametrized curve of class Cp

(or more concisely a Cp curve) will be a Cp map-ping from an open time interval (a, b) to the statespace E with no singular points. Any C1 curve can beparametrized by arclength. Let γ(a, b)→ E be such acurve. Defining the mapping s(a, b)→ (0, l(γ)) by:

s(t) =

∫ t

a‖γ′

(t)‖dt

we see that by the non singularity assumption onγ, s

′

(t) = ‖γ′

(t)‖ > 0 for any t ∈ (a, b), so that s is aninvertible mapping. Now, γ ◦ s−1 is a mapping fromthe open interval (0, l(γ)) to E satisfying:

‖(γ ◦ s−1)′

‖ = ‖(γ′

◦ s−1) ◦ (s−1)′

‖ = 1

In the following, we will simply write γ(s), s ∈(0, l(γ)) for a curve parametrized by arclength, drop-ping the variable t.

2


Remark 1. One must be careful with the respec-tive definitions of trajectories and curves: a curve isdefined on an open interval and thus has no endpoints.Nevertheless, any trajectory γ[a, b] → E has an as-sociated curve, namely γ(a, b) → E. It is generallymore convenient to deal with curves to avoid specialtreatment of the endpoints.

Remark 2. The non singularity assumption on theunderlying curve is very natural when dealing withaircraft trajectories in R3 since it is not possible for anaircraft to stop except at the endpoints of the trajec-tory.

Remark 3. While the case E = R3 is very naturaland intuitive, care must be taken when E = R6 sinceall the preceding definitions apply in a completely dif-ferent setting: for example, the non singularity as-sumption does not implies nowhere zero speed, butonly that speed and acceleration cannot both vanishat the same time. The arclength parametrization al-lows to define very important geometrical quantitieswhen E = R3 .

Definition 1. Let:

γ(0, l)→ R3

be a C1 curve parametrized by arclength. The unittangent vector to γ at s ∈ (0, l) is :

τ(s) = γ′

(s)

It is clear from the definition of parametrizationby arclength that τ(s) is a unit vector.

Definition 2. Let:

γ(0, l)→ R3

be a C2 curve parametrized by arclength. The cur-vature of γ at s ∈ (0, l) is :

K(s) = ‖γ′′

(s)‖

The curvature can be explicitly computed even ifthe curve γ is not parametrized by arclength. The gen-eral formula is:

K(t) =‖γ′

(t) ∧ γ′′

(t)‖‖γ′ (t)‖3

with ∧ the vector cross product. Curvature is ofprimary importance for ATM related studies since asmentioned before aircraft trajectories are mainly madeof straight lines and arcs of circle and so have piece-wise constant curvature. If at point t the curvature isnot zero, the curve is said to be biregular at t. Fora curve γ parametrized by arclength, the unit normalvector ν(s) is defined at all biregular points by :

ν(s) =γ′′

(s)K(s)

Remark 4. A straight line has everywhere zerocurvature. However, it is clearly possible to define

a unit normal vector. At a biregular point, τ(s) andν(s) are well defined. Taking their cross product givesa new vector β(s) = τ(s) ∧ ν(s). If the curve γ isassumed to be C3, it can be shown that β(s) and ν(s)are collinear :

β(s) = T (s).ν(s)

The real number T (s) is called the torsion of thecurve at s and represents an obstruction for the curveto be planar. As for the curvature, it is possible tocompute the torsion even if the curve is not parametrizedby arclength :

T (t) = −det(~γ

′

(t), ~γ′′

(t), ~γ′′′

(t))‖~γ′ (t) ∧ ~γ′′ (t)‖2

Torsion is not so useful as curvature for en-routedata analysis since only a few number of trajectorieshave non zero torsion. However, it is very relevant interminal areas.

Remark 5. The E = R6 case is again very dif-ferent, since the geometric meaning of curvature andtorsion is not obvious in this setting. Furthermore,the extra degrees of freedom will impose using higherorder derivatives in order to build up an equivalentdescription. A complete treatment goes beyond thescope of the present paper and has little interest forour purpose (in practical applications, the speed infor-mation, when available, is used to improve estimatesof curvature and torsion and not to study a trajectoryin R6).

3 Trajectory Models for OptimizationThis section presents some dimension reduction

tricks in order to reduce the dimension of the statespace for which an optimization process is searchingfor an optimal vector of parameter.

This approach is summarized by the figure 1.The optimization process controls the parameter

vector which is then used to build the trajectory γ forevaluation.

Each coordinate can be considered separatly in or-der to build a given trajectory:~γ(t) = [x(t), y(t), z(t)]T .

In this section, several trajectory models are pre-sented and compared. Simple models are first pre-sented.

3.1 Straight line segmentsOne of the easiest way to design trajectory is to

use waypoints connected by straight lines (see fig-ure 2. This easy principle ensure continuity for thetrajectory but not for its derivatives. If one want to ap-proximate trajectory with many shape turns, one haveto increase the number of waypoints in order to reducethe error between the model of the real trajectory.

In order to improve concept Lagrange interpola-tion process adjust a polynomial function to a givenset of waypoints.

3


Figure 3 Ln(x) is represented by the black curve. Theothers curves are the polynomials li(x).

Reconstruction

Trajectory

Trajectory

Evaluation

Optimization

γ

X (parameters)

y=f(X)

Figure 1 .The oprimization process control the X vec-tor in order to build a trajectory γ for evaluation.

WP1

WP2

WP3

WP4

Figure 2 Trajectory defined by four waypoints con-nected by straight lines.

3.2 Lagrange interpolationGiven n + 1 real numbers yi,0 ≤ i ≤ n, and n + 1

distinct real numbers x0 < x1 < ... < xn, Lagrangepolynomial [43] of degree n (Ln(x)) associated with{xi} and {yi} is a polynomial of degree n solving theinterpolation problem :

pn(xi) = yi, 0 ≤ i ≤ n

Ln(x) =

n∑i=0

f (xi)li(x)

where

li(x) =∏j,i

(x − x j)(xi − x j)

When derivatives have also to be interpolated, Her-mite interpolation has to be used.

3.3 Hermite interpolationHermite interpolation [6] generalizes Lagrange in-

terpolation by fitting a polynomial to a function f thatnot only interpolates f at each knot but also interpo-lates a given number of consecutive derivatives of fat each knot.[

∂ jH(x)∂x j

]x=xi

=

[∂ j f (x)∂x j

]x=xi

for all j = 0, 1, ...,m and i = 1, 2, ..., kThis means that n(m + 1) values

(x0, y0), (x1, y1), . . . , (xn−1, yn−1),(x0, y′0), (x1, y′1), . . . , (xn−1, y′n−1),...

......

(x0, y(m)0 ), (x1, y

(m)1 ), . . . , (xn−1, y

(m)n−1)

must be known, rather than just the first n values re-quired for Lagrange interpolation. The resulting poly-nomial may have degree at most n(m + 1)1, whereasthe Lagrange polynomial has maximum degree n1.

These interpolation polynomials seem attractivebut they both induce oscillations between interpola-tion points (Runge ’s phenomenon. Runge’s phenomenonis a problem of oscillation at the edges of an inter-val that occurs when using polynomial interpolationwith polynomials of high degree (which is the casefor Lagrange and Hermite interpolation). An exam-ple of such Runge’s phenomenon is given on figure 4for which Lagrange interpolation has been used.

Figure 4 Lagrange interpolation result for a set ofaligned points.

We can conclude that interpolation with high de-gree polynomial is risky. In order to avoid this draw-back of high degree polynomial interpolation one mustuse piecewise interpolation.

4


x0 xi xi+1xi−1 xn

Initial slope

Figure 6 Piecewise quadratic interpolation.

3.4 Piecewise Linear InterpolationThis is the simplest piecewise interpolation method.Given n + 1 real numbers yi,0 ≤ i ≤ n, and n + 1

distinct real numbers x0 < x1 < ... < xn, we considerthe n linear curves li(x) = aix + bi on the intervals[xi, xi+1] for i = 0, ...n − 1.

Each li(x) has to connect two points ((xi, yi),(xi+1, yi+1)

yi = aixi + bixi yi+1 = aixi+1 + bixi+1

In order to associate a piecewise formulation ofthis interpolation method, the following “tent” func-tions are defined :

ψi(x) =

x−xi−1xi−xi−1

i f x ∈ [xi−1, xi]xi+1−xxi+1−xi

i f x ∈ [xi, xi+1]0 elsewhere

Then,

f (x) =

i=n∑i=0

yi.ψi(x)

An example of such a linear piecewise interpola-tion is given on figure 5

x0 xi xi+1xi−1 xn

Figure 5 Piecewise linear interpolation.

The derivative of the resulting curve is not con-tinuous. In order to fix this drawback, one can usepiecewise quadratic interpolation

3.5 Piecewise Quadratic InterpolationWe consider the n quadratic curves ψi(x) = qi(x) =

aix2 + bix + ci on the intervals [xi, xi+1] for i = 0, ...n−

1.Each qi(x) has to connect two points ((xi, yi),(xi+1, yi+1);⇒ yi = aix2

i + bixi + ci and yi+1 = aix2i+1 + bixi+1 + ci.

Furthermore, on each point, the derivative of the pre-vious quadratic has to be equal to the derivative of thenext one; ⇒ 2ai + bi = 2ai−1 + bi−1. For the firstsegment the term 2ai−1 + bi−1 is arbitrarily chosen(this will affects the rest of the curve). An exampleof piecewise quadratic interpolation is given on fig-ure 6. The main drawback of piecewise quadratic in-terpolation is linked to the effect induced on the curveby moving on point. As a matter of fact moving onepoint may totally change the shape of the interpolat-ing curve. The piecewise cubic interpolation avoidthis drawback.

3.6 Piecewise cubic interpolationThis interpolation is also called Hermite cubic in-

terpolation [37]. For this interpolation :

ψi(x) = Ci(x) = aix3 + bix2 + cix + di

and we have the following constraints :

Ci(xi) = yi Ci(xi+1) = yi+1C′i (xi) = y′i =

yi+1−yi−1xi+1−xi−1

C′i (xi+1) = y′i+1 =yi+2−yixi+2−xi

An example of piecewise cubic interpolation is givenon figure 7.

xi xi+1xi−1

iyyi−1

yi+1

yi+2

xi+2

h

slope in islope in i+1

Figure 7 Piecewise cubic interpolation.

Moving a point do not affect all the curve which isthe main advantage of this interpolation.The resultingcurve is C1 but not C2 (the second derivative is notcontinuous). The curvature radius of a curve may beexpressed by the following expression :

R =1 +

(d f (x)

dx

) 32

|(

d2 f (x)dx2

)|

The piecewise cubic interpolation do not insure thattrajectory curvature is continuous which is not adaptedfor aircraft trajectory mainly if TMA areas and cubicspline interpolation has to be used.

5


3.7 Cubic Spline InterpolationThis method has been developed by General Mo-

tor in 1964 [15]. For this piecewise interpolationpsii(x) = S i(x) with the following constraints :

S i(xi) = yi S i(xi+1) = yi+1

S′

i(xi) = S′

i−1(xi+1) S′

i(xi+1) = S′

i+1(xi+1)S′′

i (xi) = S′′

i−1(xi+1) S′′

i (xi+1) = S′′

i+1(xi+1)

One can show that S i(x) for x ∈ [xi, xi+1] is givenby :

S i(x) = σi6 .

(xi+1−x)3

xi+1−xi+ σi+1

6 . (x−xi)3

xi+1−xi

+ yi.xi+1−xxi+1−xi

−σi6 .(xi+1 − xi)(xi+1 − x)

+ yi+1.x−xi

xi+1−xi−

σi+16 .(xi+1 − xi)(x − xi)

where

σi =d2S i(x)

dx2

An example of such interpolation is given on fig-ure 8.

xi−1

yi−1 yi+2

xi+1

iy

xi

yi+1

S i (t)

xi+2

Figure 8 Cubic Spline Interpolation.

Such spline is also called natural spline because itrepresents the curve of a metal spline constrained tointerpolate some given points.

When interpolation is not a hard constraint, onecan use some control points which change the shapeof a given trajectory without forcing this trajectory togo through such control point; such approach is calledapproximation for which one of the famous methodsis the Bezier curve.

3.8 Bezier Approximation CurveBezier curves[30] were widely publicized in 1962

by the French engineer Pierre Bezier, who used themto design automobile bodies. But the study of thesecurves was first developed in 1959 by mathematicianPaul de Casteljau using de Casteljau’s algorithm [31],a numerically stable method to evaluate Bezier curves.A Bezier curve is defined by a set of control points ~P0

through ~Pn, where n is called its order (n = 1 for lin-ear, 2 for quadratic, etc.). The first and last controlpoints are always the end points of the curve; how-ever, the intermediate control points (if any) generally

do not lie on the curve. Given points ~P0 and ~P1, a lin-ear Bezier curve ~B(t) is simply a straight line betweenthose two points (see figure 9). The curve is given by :

~B(t) = ~P0 + t(~P1 − ~P0) = (1 − t)~P0 + t~P1 , t ∈ [0, 1]

P0

P1

Figure 9 Bezier Curve with 2 points.

With four points (~P0, ~P1, ~P2, ~P3), a Bezier curveof degree three can be built. The curve starts at ~P0

going towards ~P1 and arrives at ~P3 coming from thedirection of ~P2. Usually, it will not pass through ~P1 or~P2; these points are only there to provide directionalinformation (see figure ??).

BÉZIER CURVE

P3

P2

P1

P0

P0P1

P2

Figure 10 Cubic Bezier curve.

Properties

• The polygon formed by connecting the Bezierpoints with lines, starting with ~P0 and finishingwith ~Pn , is called the Bezier polygon (or con-trol polygon).

• The convex hull of the Bezier polygon containsthe Bezier curve.

• The start (end) of the curve is tangent to the first(last) section of the Bezier polygon.

The explicit form of the curve is given by :

~B(t) = (1−t)3~P0+3(1−t)2t~P1+3(1−t)t2~P2+t3~P3 , t ∈ [0, 1].

~B(t) =

n∑i=0

bi,n(t)~Pi, t ∈ [0, 1]

6


it =4

t i+1t i

B (t)i,0

X (t)0

0 1 2 3 5 6 7 8

Pi

9

1

Figure 11 Uniform B-Splines of Degree Zero

where the polynomials

bi,n(t) =

(ni

)ti(1 − t)n−i, i = 0, . . . n

are known as Bernstein basis polynomials of degreen. So, if there are many points, one has to manipulatepolynoms with high degree. In order to circumventthis weak point one must use Basis-Splines.

3.9 Basis SplinesA B-spline [21] is a spline function that has min-

imal support with respect to a given degree, smooth-ness, and domain partition. B-splines were investi-gated as early as the nineteenth century by NikolaiLobachevsky. A fundamental theorem states that ev-ery spline function of a given degree, smoothness,and domain partition, can be uniquely represented asa linear combination of B-splines of that same de-gree and smoothness, and over that same partition. Itis a powerful tool for generating curves with manycontrol points, B stands for basis.A single B-splinecan specify a long complicated curve and B-splinescan be designed with sharp bends and even “corners”.B-Spline interpolation is preferred over polynomialinterpolation because the interpolation error can bemade small even when using low degree polynomi-als for the spline. Furthermore, spline interpolationavoids the problem of Runge’s phenomenon whichoccurs when interpolating between equidistant pointswith high degree polynomials.

3.9.1 Uniform B-Splines of Degree Zero

We consider a node vector ~T = {t0, t1, ..., tn} witht0 ≤ t1 ≤, ...,≤ tn and n points ~Pi. One want to build a

curve ~X0(t) such that :

~X0(ti) = ~Pi

⇒ ~X0(t) = ~Pi ∀t ∈ [ti, ti+1].

~X0(t) =∑

i

Bi,0(t).~Pi

where

Bi,0(t) =

{1 i f t ∈ [ti, ti+1]0 elsewhere

The shape of the ~X0(t) function in one dimension isgiven on figure 11.

3.9.2 Uniform B-Splines of Degree One

We are searching for a piecewise linear approxi-mation ~X1(t) for which :

~X1(t) =

(1 −

t − titi+1 − ti

)~Pi−1+

(1 −

t − titi+1 − ti

)~Pi ∀t ∈ [ti, ti+1]

One can write ~X1(t) :

~X1(t) =∑

i

Bi,1(t).~Pi

where

Bi,1(t) =

t−ti−1ti−ti−1

i f t ∈ [ti−1, ti]ti+1−tti+1−ti

i f t ∈ [ti, ti+1]0 elsewhere

The shape of the ~X1(t) function in one dimensionis given on figure 12.

it =4

t i+1t i

0 1 2 3 5 6 7 8

Pi

9

X (t)1

t i−1

i−1,1B (t)1

Figure 12 Uniform B-Splines of Degree One

7


3.9.3 Uniform B-Splines of Degree Three

Those B-Splines have been developed at Boeingin the 70s and represent one of the simplest and mostuseful cases of B-splines. Degree 3 B-Spline with n +1 control points is given by :

~X3(t) =

n∑i=0

Bi,3(t).~Pi 3 ≤ t ≤ n + 1

where Bi,3(t) = 0 if t ≤ ti or t ≥ ti+4.

~X3(t) =

j∑i= j−3

Pi.Bi,3(t) t ∈ [ j, j + 1], 3 ≤ j ≤ n

When a single control point ~Pi is moved, only theportion of the curve ~X3(t) with ti < t < ti+4 is changed⇒ local control. The basis functions have the follow-ing properties :

• They are translates of each other i.e Bi,3(t) =B0,3(t − i)

• They are piecewise degree three polynomial

• Partition of unity∑

i Bi(t) = 1 for 3 ≤ t ≤ n + 1

• The function ~Xi(t) are of degree 3 for any set ofcontrol points

Bi−2,3(t) =1h

(t − ti−2)3 if t ∈ [ti−2, ti−1]h3 + 3h2(t − ti−1) + 3h(t − ti−1)2 − 3(t − ti−1)3

if t ∈ [ti−1, ti]h3 + 3h2(ti+1 − t) + 3h(ti+1 − t)2 − 3(ti+1 − t)3

if t ∈ [ti, ti+1](ti+2 − t)3 if t ∈ [ti+1, ti+2]0 otherwise

Those basis functions are shown on figure 13.

B (t)2,3 3,3

B (t)B (t)1,3

1 2 3 54 6 7 8

B (t)4,3

2/3

Figure 13 Order 3 basis function

3.9.4 Principal Component Analysis

When trajectories samples are available (from radarfor instance), one can build a dedicated bases whichwill minimize the number of coefficient for trajectoryreconstruction. Principal component analysis (PCA)is a mathematical procedure that uses an orthogonaltransformation to convert a set of observations of pos-sibly correlated variables into a set of values of lin-early uncorrelated variables called principal compo-nents. The number of principal components is lessthan or equal to the number of original variables.

γ

ψ

Figure 14 The black trajectories represent registeredsamples for which 4 principal components are ex-tracted (in this artificial example) for minimum errorreconstruction process.

In the example presented in figure 14 a set of tra-jectories γi(t), i = 1..n are used to build K = 4 prin-cipal components (ψk(t)) which can be used to recon-struct the initial trajectories.

γi(t) =

k=K∑k=1

aikψk(t)

When probability density functions of the coeffi-cient aik can be identified, one can use this trick toplug a stochastic optimization process which gener-ates random coefficients in order to produce relevantrandom trajectory. More information about PCA canbe found in the reference [3].

3.9.5 Homotopy trajectory design

An easy way to build trajectory is to used refer-ence trajectory (regular trajectories used by aircraft)and to compute a weighted sum of such reference tra-jectories to build a new one. If we consider two (ormore) references trajectories joining the same origindestination pair (see figure 15) (past flown trajectoriesmay be considered) :

γ1, γ2

One can create a new trajectory γα by using an homo-topy :

8


γ2

γ1

γα

B

A

Figure 15 One new trajectory is built by using aweighted sum of two reference trajectories.

γα = (1 − α)γ1 + αγ2

In this example only one coefficient α has beenused but one can extend this principle to several pa-rameters.

After having review algorithms based on optimiza-tion, the next section presents wave front propagationapproaches.

4 Wavefront algorithms4.1 Generalities

It is a well known fact in physics that waves ofvery high frequencies tend to propagate along linesthat are geodetic with respect to the metric that yieldsas the length of a curve the propagation time. Theknowledge of the wave velocity at each point of spaceallows for the computation of wave fronts that are theset of all points reached by the wave at a given time,assuming it has started at a point source. The prin-ciple of the wavefront propagation algorithms is totranspose the previous physical model by assuminga velocity directly related to the criterion to be opti-mized. As an example, congestion will be taken intoaccount by velocity reduction, so that paths crossingcongested areas will be penalized. Dependind on thefact that the metric is or not isotropic (the later casebeing the one to be investigated when the wind is usedinto the criterion and has a non negligible velocity),two classes of algorithms are used.

4.2 Fast Marching AlgorithmsThe Fast Marching method, presented by Sethian [68]

is a part of the more general methods called LevelSet [52]. These techniques are designed to track theevolution of interfaces. The evolution of the wave-front can be compared to deform a curve or a surfacefrom a partial differential equation. The Fast March-ing method is used in the particular case where thewavefront speed is isotropic. It can still be applied ifthe anisotropy is low enough. For air trafic applica-tions, this last assumption is valid in some areas of

the airspace (of course it is not the case in the vicinityof jet streams).

In the particular case where the Fast Marchingmethod is applicable, the calculus of the minimumtime T to reach any points of the environment fromthe initial point is equivalent to solve the Eikonal equa-tion of the form :

|∇T (x)| =1

F(x), F(x) > 0, et T (xinitial) = 0

(1)where x ∈ R2 represents the position in space, T ∈ Rthe minimum time and F ∈ R the speed of propaga-tion.

In free space, the wave speed F is equivalent tothe aircraft speed. When we have forbidden areas,we force the propagation speed at zero in order toget a barrier value since the time to reach this pointwill be equal to infinity. Thereby, we have guaran-teed avoidance property for those areas. For the con-gestion, the method is different, we want to penalizesome areas where the congestion is high but we do notwant to ban aircrafts from driving through these areas.We just need to reduce the propagation speed. Thus,the time is increased proportionnaly to the congestionvalue, penalizing the crossing.

To design the optimal path between the arrival pointand the departure point, we can then perform a gradi-ent descent using the calculated values of T on thespace, from the arrival point to the initial point. Thereis no risk to get stuck on a local minimum since thefunction T has only one optimum which is global.

The numerical resolution is like the graph searchalgorithms. However, in opposition to these graphsearch algorithms, the Fast Marching method is con-sistent since when the grid is refined, the obtained so-lution converges on the exact solution of the Eikonalequation [68] that is a geodetic line.

4.3 Ordered upwind algorithm

When wind is to be taken into account and hasa non negligible speed with respect to the one of theaircraft, the propagation is no longer isotropic. Thespeed of the wavefront depends on the position andthe directions of wind. A specific algorithm, calledOrdered Upwind, has been developed to overcomethis problem in [69], at the expense of a higher algo-rithmic complexity. Basically, an extra parameter, theanisotropy ratio, is considered: it is the ratio of thefastest to slowest propagation speed for each points.Given a point in space, the algorithm first considersthe points on the current wavefront that are closest toit: it gives a time to travel when taking as propagation

9


speed the slowest. Now, the other points to be consid-ered at located no farther than the anistropy ratio timethe minimal distance. By maintaining a list of poten-tial points contributing to the information at the pointof interest, the ordered upwind algorithm can still beimplemented in a single pass.

In order to keep the efficiency of the Fast March-ing algorithm, Petres proposed an extension of the al-gorithm Fast Marching in [54], he assumed the fieldis smooth. He applied this extension to plan a pathfor autonomous underwater vehicles taking underwa-ter currents into account. We propose here a simi-lar extension to Petres’s method of the Fast Marchingmethod, our extension is specific to aircraft trajecto-ries.

In the next section, we present an example of ap-plication of such wave propagation algorithm for airtraffic management problem.

4.4 Light Propagation Algorithm for Air TrafficManagement

In geometric optics, light behavior is modeled us-ing rays. Light emitted from a point is assumed totravel along such a ray through space. In an effortto explain the motion through space taken by rays asthey pass through various media, Fermat (1601-1665)developed his principle of least action [32]:

The path of a light ray connecting two points is theone for which the time of transit, not the length, is a

minimum.

We can make several observations as a result ofFermat’s principle :

• In a homogeneous medium, light rays are rec-tilinear. That is, within any medium where theindex of refraction is constant, light travels in astraight line.

• In an inhomogeneous medium, light rays fol-low smooth geodesic curves with minimum tran-sit time.

Light therefore tends to avoid high index areas whererays are slowed down. Light reaches lowest speed forthe highest encountered index.

Based on this principle of least action, we intro-duce an optimal path planning algorithm which com-putes smooth geodesic trajectories in environmentswith static or dynamic obstacles. This algorithm mim-ics light propagation between a starting point towardsa destination point, with obstacles modeled by high-index areas. By controlling the index landscape, it

is possible to ensure that the computed trajectoriesmeet the speed constraints and remain at a specifiedminimum distance from obstacles. Congestion andthe protection zone (volume surrounding the aircraftwhere no other aircraft may enter) of other aircraftwill be modeled as high-index areas. Our light propa-gation algorithm (LPA) is designed from a particularaircraft point of view. It is assumed that the aircraftknows the surrounding aircraft trajectories (the set oftrajectories of the other aircraft is a given input of thealgorithm).

We have applied successfully this algorithm to theaircraft conflict resolution problem. To address thisconflict resolution problem, aircraft are sequentiallyresolved using LPA. We assign a trajectory to the firstaircraft disregarding the other aircraft (without con-sidering any constraints). Then, LPA looks for a tra-jectory for the subsequent aircraft by considering thetrajectory of the first aircraft as a constraint, and so on,up to the mth aircraft which considers the m− 1 previ-ous aircraft trajectories as constraints. The trajectoryassigned to an aircraft in each resolution step mustavoid other aircraft trajectories that are considered asfixed constraints (known data). In our case, the air-craft ordering is chosen at random. In practice, someoperational criteria may also be used in order to se-lect a specific sequence (for instance: first-come first-served rule, some aircraft may have higher priority,trajectory length, etc.). LPA has been applied on a dayof traffic (August 12, 2008) with about 8000 flights.The initial trajectories (before conflict resolution) in-duce a total number of clusters, with some aircraftin real conflict, equal to 3344. The algorithm nearlysolves all conflicts, with only 28 situations for whichconflict-free trajectories have not been found. How-ever, these situations correspond to some aircraft be-ing already in conflict at the beginning of the simula-tion, for instance at their starting point.Only 1501 tra-jectories have been modified to reach such a conflict-free planning. In many cases, the new computed tra-jectories are shorter than the initial ones (those thatfollow waypoints), due to the fact that LPA is search-ing for the shortest path trajectories and proposes di-rect routes when possible.

5 Optimal Control for TrajectoryGeneration

5.1 Optimal Trajectory Generation

In the physical space, a trajectory is occasionallyrepresented as a four-dimensional flight path, follow-ing the tradition of air traffic control [20], with time

10


as the fourth dimension, in addition to the normallyused three-dimensional representation of a path. Gen-erating time-parameterized paths necessitates the in-corporation of the aircraft dynamics and/or kinemat-ics, which makes the problem much more difficultthan simply finding a path that avoids obstacles in thephysical three-dimensional space. Path-planning is aterm commonly used in the robotics and artificial in-telligence communities to refer to the problem of gen-erating an obstacle-free path to be followed by a ve-hicle (robot, aircraft, vehicle, etc) in a two or threedimensional space containing obstacles [44].

Because the vehicle dynamics are not taken intoaccount in these path-planning methods (the solutionof which only considers the geometric constraints ofthe problem) it is often the case that the resulting pathis infeasible, that is, it cannot be followed exactly oreven closely by the vehicle. One way to ensure thatthe resulting paths correspond to feasible trajectoriessatisfying the vehicle dynamics, is to use optimal con-trol theory. The objective of optimal control theory isto determine the control input(s) that will cause a pro-cess (i.e., the response of a dynamical system) to sat-isfy the physical constraints, while, at the same time,minimize (or maximize) some performance criterion.Feasibility of the trajectories is automatically ensuredusing this approach. The typical optimal control prob-lem (OCP) can be stated as follows:

Given initial conditions x0, final conditions x f ∈

X, and an initial time t0 ≥ 0, determine the final timet f > t0, the control input u(t) ∈ U and the correspond-ing state history x(t) for t ∈ [t0, t f ] which minimizethe cost function

J(x, u) =∫ t f

t0L(x(t), u(t)) dt, (2)

where x(t) and u(t) satisfy, for all t ∈ [t0, t f ] the dif-ferential and algebraic constraints

x(t) − f (x(t), u(t)) = 0,C(x(t), u(t)) ≤ 0. (3)

Optimal control has its roots in the theory of cal-culus of variations, which originated in the 17th cen-tury by Fermat, Newton, Liebniz, and the Bernoullis,and was subsequently further developed by Lagrange,Weirstrass, Legendre, Clebsch and Jacobi and othersin the 18th and 19th centuries [29]. Calculus of vari-ations deals with the problem of minimizing (2) sub-ject to the simple differential equality constraint of theform x(t) − u(t) = 0, and is not able to handle morecomplicated differential equality constraints such as(3) or algebraic constraints such as (3). It was not

until the middle of the 20th century when the Sovietmathematician L. S. Pontryagin developed a completetheory that could handle constraints such as (3) and(3). Simply put, Pontryagin’s celebrated MaximumPrinciple [53] states that the optimal control for thesolution of the problem (2)-(3) is given as the point-wise minimum of the so-called Hamiltonian function,that is,

uopt = argminu∈U H(t, x, λ, u), (4)

where H(t, x, λ, u) = L(x, u) + λT f (x, u) is the Hamil-tonian, and λ are the co-states, computed from

λ(t) = −∂H∂x

(x(t), λ(t), u(t)). (5)

subject to certain boundary (transversality) conditionson λ(t f ). Unfortunately, an analytic solution to theprevious problem is difficult. The optimal control for-mulation of a trajectory optimization problem usingPontryagin’s Maximum Principle (PMP) leads to aTwo-point Boundary Value Problem (TBVP), or a Multi-point Boundary Value Problem (MBVP) when the op-timal trajectory is composed of multiple phases. Nu-merical techniques such as shooting and multiple shoot-ing methods can be applied to solve accurately theseTBVP and MBVP, but their convergence is very sen-sitive to the choice of an initial guess for the solution.A software that solves the optimal control problem us-ing this approach is BNDSCO [50]. BNDSCO is anexample of a class of numerical optimization methodswhich are often referred to as indirect methods. Theterm indirect reflects the fact that in these methods asolution is sought not by maximizing (or minimizing)the cost (2) but, rather, by computing potential opti-mizers by solving the corresponding necessary opti-mality conditions (4)-(5).

In recent years, direct methods have become in-creasingly popular for solving trajectory optimizationproblems, the major reason being that direct methodsdo not require an analytic expression for the necessaryconditions, which for complicated nonlinear dynam-ics can be intimidating. Moreover, direct methods donot need an initial guess for the co-states whose timehistories are difficult to predict a priori. As mentionedearlier, direct methods, do not try to satisfy the neces-sary conditions of optimality from PMP, instead, theyminimize directly (2) subject to (3)-(3).

The main idea behind direct methods is to dis-cretize the states and controls of the original continuous-time optimal control problem in order to obtain a finite-dimensional nonlinear programming problem (NLP).The solution of this NLP, which consists of discrete

11


variables, is used to approximate the continuous con-trol and state time histories. Typical direct methodsare collocation methods, which discretize the ordi-nary differential equations (ODEs) of the problem us-ing collocation or interpolation schemes [60, 77, 23,36]. They introduce the collocation conditions as NLPconstraints together with the initial and terminal con-ditions. The so-called “pseudospectral” methods useorthogonal polynomials to choose the collocation pointsand are very efficient, exhibiting superlinear conver-gence when the solution is smooth. Numerical opti-mal control software packages that implement directmethods for the solution of OCPs include SOCS [12],RIOTS [67], DIDO [59], PSOPT [8], GPOPS [57],MTOA [40] and DENMRA [81] among many oth-ers. A recent survey of numerical optimal controltechniques for trajectory optimization can be foundin [10].

In all these direct methods, the convergence rateand the quality of solution depends on the grid usedto discretize the equations, the cost and the problemconstraints. Uniform or fixed grid methods tend toperform poorly, especially when the problem has sev-eral discontinuities or irregularities. Not surprisingly,adaptive grid methods have been developed to accu-rately capture any discontinuities or switchings in thestate or control variables. The main idea behind allthese adaptive grid methods is to use a high resolution(dense) grid only in the vicinity of control switches,constraint boundaries etc, and a coarse grid elsewhere.Examples of such adaptive gridding techniques forthe solution of optimal control problems are [11, 14,67, 13, 41, 33].

A major issue with almost all current trajectoryoptimization solvers (direct or indirect) is the fact thattheir computational complexity is high and their con-vergence dependents strongly on the initial conditions,unless certain rather stringent convexity conditions hold.As a result, the solution of trajectory optimization prob-lem in real-time is still elusive. A common line of at-tack for solving trajectory optimization problems inreal time (or near real time) is to divide the prob-lem into two phases: an offline phase and an onlinephase [71, 46, 42, 79]. The offline phase consists ofsolving the optimal control problem for various refer-ence trajectories and storing these reference trajecto-ries onboard for later online use. These reference tra-jectories are used to compute the actual trajectory on-line via a neighboring optimal feedback control strat-egy typically based on the linearized dynamics. An-other strategy for computing near-optimal trajectoriesin real-time is to use a receding horizon (RH) ap-

proach [51, 9, 78]. In a receding horizon approacha trajectory that optimizes the cost function over aperiod of time, called the planning horizon, is de-signed first. The trajectory is implemented over theshorter execution time and the optimization is per-formed again starting from the state that is reached atthe end of the execution time. A third approach is touse a two-layer architecture, where first an acceptable(in terms of length, safety, etc) path is computed usingcommon path-planning techniques, and then an opti-mal time-parameterization is imposed on this path toyield a feasible trajectory. As mentioned earlier suchan approach needs to be carefully designed to ensurecompatibility of the resulting path with the vehicle dy-namics. However, when successful, such an approachis numerically very efficient and can be implementedin real-time with current computer hardware. Evenif the resulting trajectory is not exactly feasible, it isoften close to a feasible trajectory, or it can be madeas such using smoothing techniques [83]. As a re-sult, alternatively, the final trajectory can be used asa good initial guess for a follow-up optimal trajectorygenerator. The next section summarizes this approachfor applications related to aircraft maneuvering understrict time and fuel constraints. For more details theinterested reader is referred to [82, 5, 80].

6 Trajectory Optimization Methods inATM

Aircraft maneuvering was one of the first areaswhere optimal control theory was used to generate op-timal trajectories. Not surprisingly, traditionally, mostwork has been focused on military aircraft. Relevantreferences on this subject are too many to enumeratehere. We just mention the work on fuel and range op-timization studied in [34, 72, 70], and the minimum-time, three-dimensional aircraft trajectory optimiza-tion problem considered in [66]. In the latter, an ap-proximation of the aircraft dynamics using an energystate was used to reduce the dimension of the problemfor better convergence. This type of model reductiontechnique is commonly used for aircraft trajectory op-timization [1].

Some of these results have been extended to com-mercial airline operations. For example, the work of[18] deals with the problem of minimum-fuel trajecto-ries with fixed time-of-arrival (TOA) for several civilaviation aircraft including B737, B747 and B767. Tra-jectory planning problems have also been studied inthe context of air traffic management (ATM) and au-tomation. Reference [38] performed a sensitivity anal-

12


ysis of trajectory prediction for ATM. The aircraft tra-jectory synthesis problem is studied in [73] to providesome basic tools for air traffic automation. Somewhatrelated is the recent work of Sridhar [74], in whichhe considered the generation of wind-optimal trajec-tories for cruising aircraft while avoiding the regionsof airspace that facilitate persistent contrails forma-tion. A shooting method was employed to solve theassociated optimal control problem by minimizing aweighted sum of flight time, fuel consumption, and aterm penalizing the contrail formation. The airspaceavoidance problem has also been considered in Ref. [39].In that reference, the avoidance of restricted airspaceis formulated as a non-convex constrained trajectoryoptimization problem; it is claimed that with a fea-sible starting guess, the efficiency of the optimiza-tion algorithm is not degraded too much by the non-convex airspace constraints. Finally, some researchershave used ideas from stochastic optimal control todeal with issues related to the unpredictability of fu-ture trajectory, wind effects, presence of additionalaircraft in the ATM airspace etc [45, 76].

In this work we deal with the problem of efficientlygenerating minimum-time and minimum-fuel (with fixed TOA)landing trajectories for commercial aircraft. The for-mer problem is of relevance in case of an on-boardemergency where the pilot has to land the airplanequickly and safely to the closest airport or airfield; thelatter problem is of interest for typical terminal ATCphase applications. Prior work in emergency land-ing includes the abort landing problem in the pres-ence of windshear [16, 17, 48], and emergency land-ing during loss of thrust [75]. The latter referencesgenerate feasible trajectories using segments of tra-jectories corresponding to selected trim condition ma-neuvers. The search results are however limited tothose that can be generated by connecting trim statetrajectory segments with stable transitions. Becausethe unstable flight conditions are not considered inthe search, the algorithm cannot identify any feasi-ble trajectories containing unstable flight modes. Fur-thermore, the path length is used as the search cri-terion, which is less appropriate when compared toflight time for emergency landing, or fuel consump-tion for normal flight. Related work includes the in-vestigation of Atkins et al [4], where the problem ofemergency landing due to the loss-of-thrust was stud-ied using a hybrid approach. A two-step landing-siteselection/trajectory generation process was adoptedto generate safe emergency plans in real time undersituations that require landing at an alternate airport.In the trajectory generation routine, a heuristic path

planner was used to generate a three-dimensional tra-jectory connecting the current position of the aircraftto the runway, which consists of straight lines and cir-cular arcs. This method is fast and simple. However,it is limited to conservative aircraft maneuvers (typi-cally Dubins paths) in order to reduce the chance ofobtaining an infeasible trajectory. As a result, the op-timality of the generated trajectory could be unaccept-able for emergency landing, and further research isnecessary to reduce such a conservatism.

In our approach, we start with the assumption thatthe path to be followed by the aircraft is given. Notethat this does not mean that the trajectory to be fol-lowed is given. A trajectory requires a time-parameterizedpath and it is, indeed, the main goal of this approachto provide such a time parameterization so as to meetfeasibility along with certain optimality specifications.The assumption that the path is given is not as un-usual or atypical as one may initially think. Com-mercial airliners during the terminal landing phase,are required to follow strict Air Traffic Control (ATC)rules, which guide the airplanes to follow “virtual”three-dimensional corridors all the way to the landingstrip. Furthermore, since our approach leads to veryfast computation of feasible trajectories, one can usethe approach over new, locally modified paths repeat-edly till a satisfactory path is found.

To this end, let a path in the three-dimensionalspace, parameterized by the path coordinate s, be givenas follows: x = x(s), y = y(s), z = z(s), wheres ∈ [s0, s f ]. The main objective is to find a time-parameterization along the path, i.e., a function s(t),where t ∈ [0, t f ] such that the corresponding time-parameterized trajectory (x(s(t)), y(s(t)), z(s(t))) min-imizes either the flight time t f (emergency landingcase) or fuel (terminal landing operation). As shownin [80] all control (thrust, angle of attack, load fac-tor, etc) constraints can be mapped into constraintsinvolving the specific kinetic energy of the aircraft,E = v2/2 where v is the aircraft velocity of the form

gw

(s) ≤ E(s) ≤ gw(s),

for some path-dependent functions gw

(s) and gw(s).The original problems therefore reduce to the follow-ing simplified problems:

For the Minimum-Time Problem we have

minT

∫ s f

s0

ds√

2E(s)(6a)

subject to E′(s) =T (s)

m− D(E(s), s) − g sin γ(s),(6b)

13


gw

(s) ≤ E(s) ≤ gw(s), (6c)

Tmin ≤ T (s) ≤ Tmax, (6d)

and for the Minimum-fuel Problem with fixed TOAwe have

minT

∫ s f

s0

T (s) ds, (7a)

subject to E′(s) =T (s)

m− D(E(s), s) − g sin γ(s),(7b)

t′(s) =1

√2E(s)

, (7c)

gw

(s) ≤ E(s) ≤ gw(s) (7d)

Tmin(s) ≤ T (s) ≤ Tmax(s), (7e)

where D(E(s), s) is the drag, T is the thrust, γ is theflight-path angle, and where prime denotes differenti-ation with respect to path length s. The main advan-tage of these problem formulations is the dimension-ality reduction of the problem that can be leveraged tosolve both of these problems very efficiently and re-liably. In fact, these OCPs are simple enough so thatthe optimal switching structure of the optimal solutioncan be unraveled using the necessary conditions fromPMP. For the minimum-fuel problem the switchingstructure varies depending on the given TOA. How-ever, for a given path and a fixed TOA, the structureis uniquely determined. This helps tremendously theconvergence properties of the algorithm.

The overall architecture for optimal on-line tra-jectory generation is shown in Fig. 16. As shown in

Landing

Task

Geometric

Path Planner

Feasible?

Back-up

Trajectory

Numerical

Optimal Control

Optimal

Trajectory

No

Yes

Time

Parameterization

Path

Smoothing

Figure 16 Schematic of landing trajectory optimiza-tion.

this figure, the method first generates a trajectory byassigning an optimal time parameterization along thepath given by the geometric path planner via the so-lution of one of the previous two optimization prob-lems. If the trajectory is feasible, then it is used as aninitial guess for the numerical optimal control solver.Meanwhile, such a feasible trajectory is also storedas a back-up plan in case of the failure of the NLPsolver. If the trajectory generated by the time-optimalpath tracking method is not feasible, then the path isrevised using the path smoothing method describedin [83], and the optimization is applied again to thesmoothed path. The process is repeated until eitherthe trajectory is feasible, or the maximum number ofiterations is reached. If no feasible trajectory can beobtained after reaching the iteration limit, the infeasi-ble trajectory is passed to the numerical optimal con-trol algorithm, which makes a last attempt to producea feasible trajectory. If this last attempt is not suc-cessful, then it does not exist a feasible trajectory thatsolves the problem.

7 Path planning techniques via naturallanguage processing and

mathematical programming: Aparadigm for future aircraft

trajectory managementAircraft trajectory planning has reached enough

maturity to shift the trajectory planning problem fromthe mathematical optimization of the aircraft trajec-tory to the automated parsing and understanding ofdesired trajectory goals, followed by their re-formulationin terms of a mathematical optimization program. Toa large extent, we propose a possible evolution of theFlight Management System currently used on all trans-port aircraft to become a full-fledged autonomous logicthat may also be used on unmanned aerial vehiclesas an alternative to the current -and bulky- Remotely-Piloted Vehicle (RPV) paradigm. What follows is adirect extension of work originally presented in [65].

The overall proposed architecture is shown in Fig. 17.It consists of several nested feedback loops. At theoperator level, the information is presented to the op-erator in the form of sentences expressed in naturallanguage (eg that used by air traffic control phrase-ology). At the level of trajectory planning automa-tion, the information is presented as a mix of continu-ous parameters (aircraft position and speed), and dis-crete parameters describing mission status (completedtasks, tasks remaining to be completed). The operator

14


Figure 17 Basic Autonomous Mission ManagementLoop for future Aircraft and UAVs

formulates the vehicles’s goals (eg flight plans) us-ing natural language. A natural language interpreterand task scheduler transforms the operator’s require-ments into tractable mathematical optimization pro-grams that may be executed by the the vehicle throughits flight control computer. The vehicle’s innermostdynamics (that consist of raw vehicle dynamics andstability augmentation system), although critical to ve-hicle stability, are not shown.

7.1 Natural language parsing and generation

The main goal of using a Natural Language Inter-face (NLI) for interacting with a computer-based sys-tem is to minimize the workload on the operator. Us-ing normal English sentence commands and reportsindeed allows an operator to communicate efficientlyand effectively with the an aircraft, as if it were ahuman pilot. The NLI module that we have devel-oped for demonstration purposes consists of two ma-jor components. The first one takes sentence com-mands from the operator (presumably an air trafficcontroller) and turns them into a formally coded com-mand that looks like the formulation of an optimiza-tion problem. The second component takes a codedcommand set from the aircraft and generates naturallanguage responses for the air traffic controller to in-terpret. A sample dialog between the human operatorand the machine could be as follows:Controller: Flight AA1234, this is Air Traffic Con-trol.Aircraft: Go ahead, Air Traffic Control.Controller: Add new waypoint. Proceed to waypointEcho-Charlie 5 in minimum time. and wait for furtherinstructions after the task is completedAircraft: Roger. Acknowledge task information - pro-ceeding to waypoint Echo-Charlie 5.Controller: AA1234, out.

The NLI module analyzes the natural sentences

produced by the air traffic controller using parsing,which is the process of converting an input sentence,e.g. “Proceed to waypoint Echo-Charlie 5 in mini-mum time,” into a formal representation. The lat-ter is typically a tree structure which can in turn betranslated into an explicit formal command. In oursystem, parsing consists of first applying entity ex-traction to all the individual concepts (e.g. “FlightAA1234” or “Echo-Charlie 5”) and then combiningthese concepts through cascades of finite-state trans-ducers using techniques derived from those describedin [58]. While natural language processing representsour progress so far, it is easy to imagine that it couldnow be completed by a voice recognition device tofurther ease the level of communication between con-troller (or operator) and aircraft.

7.2 Task Scheduling and Communications Inter-facing

The task scheduling and communications process-ing components are designed to centralize all of theaircraft mission processing in one module. Togetherwith the Natural Language Interface, it provides flex-ibility for an operator to insert and change flight planduring the flight. The aircraft software keeps track ofthe flight tasks, waypoint locations and known obsta-cles to pass on to the guidance algorithm. The com-munications processing component provides the airtrafic controller or operator with the authority to sendcommands and receive status updates, threat or ob-stacle avoidance information and acknowledgementmessages. It also provides remote pilots monitoringthe flight with the ability to override the guidance sys-tem in the event of an emergency or error. The systemsends threat and override information to the air trafficcontroller before any status or update information inan effort to send the most important data relevant tothe demonstration before any auxiliary information.Input/Output data are processed every 1 Hz frame be-fore the task planner and guidance step to ensure thatthe most up-to-date information is used by the aircrafttrajectory planner. The task scheduling componentoperates like a Flight Management System and allowsthe aircraft operator or the air traffic manager to entera flight plan using a pre-defined list or as programmedduring a mission. Many additional features may beadded to such a task scheduler, such as orders to fol-low loiter patterns, ‘take me home’ or low-emissionapproaches functionalities. In addition, he or she hasthe option of providing (in real- time via the NLI) theoptimization metric used by the trajectory generation

15


algorithm (i.e. minimum time, minimum fuel, or theamount of time to finish the flight). Next, the opera-tor can either give the aircraft a new plan or changethe current plan it is performing. A “New Plan” com-mand is added to the end of the aircraft task list andis executed after all of the tasks currently scheduledhave been completed. A “Change Plan” command, onthe other hand, modifies the current task performed bythe aircraft. Once a task is completed, it is removedfrom the list. After each of these actions, an acknowl-edgement is sent to the air traffic controller and theupdated task information is included in the data sentto the Trajectory Generation Module.

7.3 Trajectory planning

After the Natural Language Interface and FlightPlanning and Scheduling components have convertedthe flight plan into a series of tasks for the aircraftto perform, the Trajectory Generation Module guidesthe vehicle from one task to the next, i.e. from an ini-tial state to a desired one, through an obstacle fieldwhile optimizing a certain objective. The latter canbe to minimize time, fuel or a combination of both.Much of the functionality described below becomesincreasingly available in today’s avionics systems, andalso include such real-world factors as weather andwind conditions. For our purposes, 2D scenarios wereconsidered in which special-use airspace and otherno-fly zones are viewed as obstacles and detected whilethe flight proceeds. The environment is always fullycharacterized inside a certain detection region D aroundthe aircraft. The resulting formulation can, however,be easily generalized to account for any detection shape,such as a radar cone, and for unknown areas withinthat shape. Since trajectories must be dynamicallyfeasible, the aircraft dynamics and kinematics shouldbe accounted for in the planning problem. For opti-mization purposes, the vehicle is characterized by adiscrete time, linear state space model (A, B) in an in-ertial 2D coordinate frame (east-north). As such, thestate vector x consists of the east-north position (x, y)and corresponding inertial velocity (x, y). Dependingon the particular model, the input vector u is an iner-tial acceleration or reference velocity vector. In bothcases, however, combined with additional linear in-equalities in x and u, the state space model must cap-ture the closed-loop dynamics that result from aug-menting the aircraft with a waypoint tracking con-troller. Since the environment is only partially-knownand further explored in real-time, a receding horizonplanning strategy is used to guide the vehicle towardsthe desired destination.

The latter is denoted by x f and is an ingress/egressstate of a waypoint with a corresponding inertial ve-locity vector. At each time step, a partial trajectoryfrom the current state towards the goal is computedby solving the trajectory optimization problem over alimited horizon of length T. Because of the computa-tion delay, the initial state x0 = (x0, y0, x0, y0) in theoptimization problem should be an estimate xestim ofthe position and inertial velocity of the aircraft whenthe plan is actually implemented. The solution to theoptimization problem provides a sequence of way-points (xi, yi) and corresponding inertial reference ve-locities (xi, yi) to the aircraft for the next T time steps.Typically, however, only the first waypoint and refer-ence velocity of this sequence are given to the way-point follower, and the process is repeated at the nexttime step. As such, new information about the stateof the vehicle and the environment can be taken intoaccount at each time step. By introducing a cost func-tion JT over the T time steps, the general trajectoryoptimization problem can be formulated as to

Minimizexi,ui JT =

T∑i=1

fi(xi, ui, x f ) + fT (xT , x f )

Subject to

xi+1 = A xi + Bui, i = 0, . . . ,T − 1x0 = xestimxi ∈ X0, i = 1, . . . ,Tui ∈ U0, i = 0, . . . ,T − 1(xi, yi) ∈ D0, i = 1, . . . ,T(xi, yi)\∈ O0, i = 1, . . . ,T

The objective function consists of stage costs fi(xi, ui, x f )corresponding to each time step i, and a terminal costterm fT (xT , x f ) that accounts for an estimate of thecost-to-go from the last state xT in the planning hori-zon to the goal state x f . The sets X0 andU0 representthe (possibly non-convex) constraints on the vehicledynamics and kinematics,such as bounds on velocity,acceleration and turn rate. Here, the 0-subscript de-notes the fact that these constraints can be dependenton the initial state. Lastly, the expressions (xi, yi) ∈D0 and (xi, yi)\∈ O0 capture the requirement that theplanned trajectory points should lie inside the knownregionD0, but outside the obstacles O0 as given at thecurrent time step i = 0. Note that they are assumedto hold for x0; if not, the trajectory optimization prob-lem would be infeasible from the start. As demon-strated in [64], however, despite the detection regionand avoidance constraints, the above receding hori-zon strategy has no safety guarantees regarding avoid-ance of obstacles in the future. Namely, the algorithmmay fail to provide a solution in future time steps

16


due to obstacles that are located beyond the surveil-lance and planning radius of the vehicle. For instance,when the planning horizon is too short and the maxi-mum turn rate relatively small, the aircraft might ap-proach a no-fly zone too closely before accounting forit in the trajectory planning problem. As a result, itmight not be able to turn away in time, which trans-lates into the optimization problem becoming infea-sible at a future receding horizon iteration. In [64],a safe receding horizon scheme was therefore pro-posed based on maintaining a known feasible trajec-tory from the final state xT in the current planninghorizon towards an obstacle-free holding pattern. Thelatter must lie in the region D0 of the environmentthat is fully characterized at the current time step, andis computed and updated online. Assuming that theplanned trajectories can be accurately tracked, at eachtime step, the remaining part of the previous plan to-gether with the holding pattern can then always serveas an a priori safe backup or “rescue” plan. In prac-tice, we have found that formulating the problem offinding the nominal and rescue trajectories optimiza-tion as mixed-integer programs (MIP) works very wellin practice, though such choices are not mandatoryand may be replaced by the other techniques discussedin this paper.

8 ConclusionThis survey has shown several approaches for tra-

jectory modeling. As it has been mentioned, trajec-tories are belonging to infinite dimension space forwhich dimension reduction has to be implemented.Using trajectory samples vectors is really redundantand inefficient. After having presented some defini-tions and some features of aircraft trajectories, somedimension reductions methods have been presented inorder to be included in an optimization process. In-terpolation and approximation technique have beenpresented and some information have also been givenabout Principal Component Analysis and Homotopy.The next section has presented wave front propaga-tion approaches and gives some details on Fast March-ing Algorithms and Ordered upwind algorithm. Anapplication to real problem coming from air trafficmanagement has also been given with the descrip-tion of the light propagation algorithm (LPA). Thefourth part has focused on methods coming automaticcontrol for which vehicle dynamics are explicitly in-cluded in the models. Some applications to air trafficmanagement have also been presented. Finally, thefifth part has presented an original path planning tech-

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[EN-023] Mathematical Models for Aircraft Trajectory ...dcsl.gatech.edu/papers/eiwac13.pdf · D. Delahaye, S. Puechmorel, E. Feron, P. Tsiotras question is then given by the closest