Robust Aircraft Trajectory Planning under Wind Uncertainty using Optimal Control

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This is a postprint version of the following published document:

González-Arribas, D., Soler, M., Sanjurjo-Rivo, M.

(2018). Robust Aircraft Trajectory Planning Under

Wind Uncertainty Using Optimal Control.

Journal of

Guidance, Control, and Dynamics

, 41(3), pp. 673-688

DOI:

10.2514/1.G002928

© 2018 by Daniel González-Arribas, Manuel Soler, and Manuel

Sanjurjo-Rivo.

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Robust

Aircraft

Trajectory

Planning

under

Wind

Uncertainty

using

Optimal

Control

Daniel González-Arribasa_, _Manuel _Solerb _and _Manuel _{Sanjurjo-Rivo}c

Universidad Carlos III de Madrid, Leganés, Spain, 28911 Madrid

Uncertainty in aircraft trajectory planning and prediction generates major chal-lenges for the future Air Traffic Management system. Therefore, understanding and managing uncertaintywillbe necessaryto realizeimprovements in airtraffic capacity, safety, efficiency and environmental impact. Meteorology (and, in particular, winds) represents one of the most relevant sources of uncertainty. In the present work, a framework based on optimal control is introduced to address the problem of robust and efficient trajectory planning under wind forecast uncertainty, which is modeled with probabilisticforecastsgenerated byEnsemblePrediction Systems. The proposed methodology is applied to a flightpl anningsc enarioun dera fr ee-routingoperational paradigm and employed to compute trajectories for different sets of user preferences, exploringthetrade-offbetweenaverageflightc osta ndpr edictability.R esultss howhow the impact of wind forecast uncertainty in trajectory predictability at a pre-tactical planninghorizon canbenotonly quantified,bu tal sore ducedth roughth eapplication of the proposedapproach.

a _PhD_Candidate,_Department_of_{Bioengineering}_and_Aerospace_Engineering,_{dangonza@ing.uc3m.es} b_Assistant_Professor,_Department_of_{Bioengineering}_and_Aerospace_Engineering,_{masolera@ing.uc3m.es} c_Assistant_Professor,_Department_of_{Bioengineering}_and_Aerospace_Engineering,_{msanjurj@ing.uc3m.es}

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I. Introduction

The Air Traffic Management (ATM) system is nowadays too rigid and inefficient. An ATM paradigm shiftis being fostered by different national/regional programs, such as SESAR1 in Eu-rope,NextGen2intheUnitedStates,orCARATSin JAPAN,aimedatdeliveringimprovementsin theKeyPerformanceAreasofcapacity,efficiency,safety,andenvironmentalimpact. Acommon fea-tureisthereplacementofthehighlystructuredcurrentATMsystemwithamoretrajectory-centric frameworkwhereairspaceuserswillenjoygreaterflexibilityi nt hed esigno ftr ajectories.Therefore, thetrajectorywillbecomethekeyelementofa newset ofoperatingprocedures collectivelyknown as Trajectory-BasedOperations(TBO).ThecornerstoneoftheTBO conceptisthe“business tra-jectory”(inSESARterminology),whichevolvesoutofacollaborativeandlayeredplanningprocess where thebusiness trajectoryis planned,shared,agreed,andmodifiedwh enne cessaryin or derto meetairlinebusiness interestswhilerespectingcapacityconstraints.

Optimal control has been shown to be a powerful framework for the generation of optimized flightprofileswithinthementionedprocess.3_Considering_wind_forecasts_is_essential_in_efficient_flight planning dueto thehighimpactwindhasin trajectoryperformanceand cost. Severalapproaches for finding optimal trajectories using wind forecast data have been introduced in the literature,including analytical optimal control, e.g., [1–4], dynamic programming, e.g., [5], and numerical optimal control with direct methods, e.g., [6, 7]. However, these works rely on deterministic windforecastsanddonotconsideruncertainty.

Understandingandmanaging uncertaintyisnecessaryin ordertoincreaseATM predictability, which isa key driverof ATM performance[8,Chapter 4]. Uncertaintygeneratesmajor challenges for TBO,and meteorologicaluncertainty hasbeen identifiedas on eofth emo stim pactfulsources of uncertainty influencingth et r ajectory.In ou rvi ew,no ten oughat tentionha sbe enpa idto uncertainty inen-routemeteorologyattheplanning stage.

Inthecontextofmeteorologicalforecasting,uncertaintyisafundamentalfeature,anditssources range from uncertainty in initial and boundary conditionsto uncertaintiesin themodelization of

1_{http://www.sesarju.eu/vision}

2_{https://www.faa.gov/nextgen/}

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atmosphericprocessesandcomputationallimitations. Thesefactorsareamplifiedb yn onlinearand sometimeschaoticdynamics,andtheresultinguncertaintylimitstheusefulnessofsingle determin-isticforecasts. Inresponsetothischallenge, EnsemblePredictionSystem(EPS)weredevelopedby theNumericalWeatherPrediction(NWP)communitywiththeaimofprovidingprobabilistic fore-castsandestimatethemagnitudeandshapeofthisuncertaintyataregionalandglobalscaleanda shortandmedium-rangetimehorizon. AnEPSiscomposedbyasetofaround10to100“ensemble members”,thatis,simulationsproducedwithdifferentperturbationsoftheinitialconditionsorthe parametersofthephysicalprocesses. EPShavebeenoneofthepillarsoftheimprovementinNWP accuracyforthelastdecades[9]. WhileEPSforecastshavestartedtobeemployedinATMresearch [10], littleusagehasbeenmadeof themin a trajectoryplanning andoptimizationcontext. Tothe bestofourknowledge,theonlyexceptionistheworkof[11,12],whereEPSforecastsareemployed in ordertobuilda “ProbabilisticTrajectoryPrediction”system.

Thereis,therefore,a gapbetweendeterministicwind-optimaltrajectoryplanningbasedon op-timalcontrol,ononehand,andmeteorologicaluncertaintymodelingthroughtheuseofEPSonthe other. Thismotivatesthegoalofthepresentwork: todevelopEPS-basedoptimization methodolo-giestoquantifyandreducetheeffectsofmeteorologicaluncertainty,improvingthepredictabilityof aircrafttrajectorieswhilemaintainingacceptablelevelsofefficiency.

Methodsforintegratingnonlinearuncertaintyin optimalcontrolcanbe groupedinthreemain approaches.Werefertothefirstasthe“UncertaintyQuantification+OptimalControl”(UQ+OC) method.Itemploysanon-intrusivegeneralizedPolynomialChaosrule[13]todetermineasetofvalues oftheuncertainparametersandtheirassociatedweights.AstandarddeterministicOCPissolvedfor eachofthesevalues;then,thesolutionisstatisticallycharacterizedusingthePolynomialChaosrule. Oncetheuncertaintyintheparametershasbeenresolved,the(nowdeterministic)optimalsolution can then beinterpolated from these precomputedsolutions in a computationally inexpensiveform (seea sketchofthis methodologyin Figure1.a).Itis, therefore,aneffectiveapproachiftheactual controlruleisneededaftertheuncertaintyhasbeenrealized;otherwise,itdoesnotgenerateaclearly defined control rule. There are some successful examples of the application of this methodology in aerospace research: in [14], an aerospace vehicle is optimally

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Execution interval

OCP

Calculation implementationDecision

Uncertainty duration

(a) Uncertainty Quantification OCP

Execution interval

Uncertainty duration

OCP

Calculation implementationDecision

(b) Robust OCP Execution interval Uncertainty duration OCP Calculation Decision implementation (c) StochasticOCP

Fig.1Robustand stochasticoptimalcontrol methodologies.

routedinanenvironmentwithuncertainthreats;in[15],itisemployedforconflict-free4Dtrajectory optimization.

In a “robust” or “tychastic” methodology, a discrete set of values for the uncertain variables is again determined by a Polynomial Chaos rule (though other possibilities exist, such as the cu-bature approach of [16]). An augmented dynamical system is built by stacking one copy of the state variables for each computed value of the uncertain parameters. Then, an optimal control problem is formulated in order to search for the control sequence that minimizes a probabilistic functionalthataggregatesthecostamongallofthetrajectories(seeFigure1.b). Someexamplesof

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aerospace applications employingthis techniqueare robust trajectory optimization for anaircraft basedondynamic soaring[17,Chapters2,3],trajectoryoptimizationforasupersonicaircraftwith uncertaintiesintheaerodynamicdata[18]andoptimizationofa spacecraftreorientationmaneuver [16].

A finalfr amework,wh ichwe re ferto as“s tochasticop timalco ntrol”,mo delsth esy stemas a StochasticDifferential Equation (SDE),where theuncertainty is nowa dynamic variable andthe statevectorevolvesaccordingtotheSDE[19,Chapter11].Astochasticoptimalcontrolproblemcan be definedfo ra co ntrolledSD E;ho wever,th eso lutionto th ispr oblemisno lo ngeradeterministic controlsequenceu(t)asinthepreviousapproach,butaclosedfeedbacklawu=u(t,Xt)depending on the state. While the simplifiedli near-quadraticve rsionof th ispr oblemas be enextensively exploredintheliterature(asitsdeterministicversion),practicalnumericalmethodsforthesolution to a general stochastic optimal control problem are at an early development stage (see [20] and references therein). Inthe aviationfield, only discretizedversions (Markov Decision Processes)of this problemshave beenconsidered (see[21]),but notin a planning context. SeeFigure1.c) fora diagramofthemethod.

Inthescopeofthiswork,themostsuitablemethodologyisthe“robust”or“tychastic”approach. Therearethreemainreasonsforthischoice: first,onecannotassumethatmeteorologicaluncertainty hasbeenfullyrealizedatthepre-tacticalstage,sowehavenotconsideredtheUQ+OCPapproach; in second place, theuncertain model of the windprovidedby anEPS fitsth ero bustformulation betterthananSDE-basedmodelization;and,finally,t hefi xedop erationlawth atispr oducedas the solutionto a robustOCPfitscu rrentan din comingop erationalpr oceduresinA TMbe tterth ana feedbackpolicyproducedbysolving astochasticOCP,whichwoulddemandcontinuousflightplan modifications.

Thus, the contributions of the paper are threefold: in firstpl ace,we ge neralizeth e“robust optimal control”methodology to a widerclass of problems, such as theones generated in aircraft trajectory planning (directapplication is notpossibleforthe reasonsdiscussedin Section IIB). In secondplace,wedevelopaformulationoftheproblemofgeneratingflightp lanst hata reoptimized for efficiency and predictability that integrates this methodology with EPSforecasts. Finally, we

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show thatthis method can beused to increasepredictabilityat theplanning stage in free-routing scenarios as well as potential future concepts. Early results from this research were presented in [22].

The paper is structured as follows: we describe the employed methodology in Section II and the application to the trajectory optimization problem with EPS-derived uncertainty and current operational concepts in Section III. Weextend the methodology under a potential future concept inSection IV.Wepresentresultsfora casestudyinSectionVandconclude withabriefdiscussion oftheworkanditsimplicationsinSectionVI.

II. Methodology

Deterministic

Probabilistic

_(continuous)

Probabilistic

_{(discretized)}

Trajectories

Optimal

Control

Problems

Sto

chastic Quadrature Rule

(Section I

I.D)

Unique value Continuous random variable Discrete approximation Probabilistic Trajectory Unique

Trajectory TrajectoryEnsemble

Section II.F

Section II.A, II.B

Section II.E

Section II.C

Di

ﬀ

erential

Equations

Robust OCP

Deterministic OCP _{Robust OCP}Discretized

Random

Parameters

Fig.2Methodology

Inthissection,wedescribetherobustoptimalcontrolframeworkthatwewillemploy. Weintroduce thedynamicalsystemformulationwithuncertaintiesinSectionsIIAandIIB,anddefinetherobust

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optimalcontrolprobleminSectionIIC.Then,inordertodiscretizetheuncertainty,weintroducethe “stochasticquadraturerules”inSectionIIDandthecorresponding“trajectoryensemble”inSection IIE.Wefinallyusethetrajectoryensembletodiscretizetherobustoptimalcontrolproblemintoa largerdeterministicoptimalcontrolprobleminSectionIIF,whichcanthenbesolvedwithstandard techniques for deterministic optimal control. Figure 2 illustrates the relationships between the differentcomponentsofthemethodology,aswellastherelevantsectionsforeachpart.

A. Dynamical system formulation

Weconsideraclassofdynamicalsystemsdefinedbyar andomlyp arametrizedO DE(Ordinary DifferentialEquation)withtheadditionofalgebraicvariablesandconstraints.Thisclassofsystems arecalledtychasticdynamicalsystemsin[16]andrelatedwork.Uncertaintyisdescribedwiththeaid ofastandardKolmogorovprobabilityspace(,F,P);itiscomposedbyasamplespaceofpossible outcomes , a σ-algebra ofevents F containing sets ofoutcomes, and theprobability functionP that assignsa probability toeachof theseevents. Theuncertain parameters ofthesystemwillbe modeled asa continuousrandomvariableξ: →Rnξ _that_we_assume_to_be _constant_in _time_. _For eachpossibleoutcomeω∈,therandomvariablestakea differentvalueξ(ω).

Wedenotethestatevectorbyx∈_Rnx_,_the_control_vector_by_u_∈

Rnu,thealgebraicvariablesby

z∈Rnz _and_t_∈

Ristheindependentvariable(usuallytime). Foreachoutcomeω0∈,thereexist

a uniquetrajectory patht →(x(ω0,t),z(ω0,t),u(ω0,t))that corresponds to therealizationof the

randomvariablesξ(ω0). Thedynamicsofthesystemaregivenbythefunctionsf :Rnx_×

Rnz_× Rnu_× Rnξ_× R→Rnx_,_h_: Rnx_× Rnz_× Rnu_× Rnξ_× R→Rnh_,_and_g_: Rnx_× Rnz_× Rnu_× Rnξ_× R→Rng_, suchthatvalidtrajectories fulfillt hec onditionsa lmosts urely( i.e.w ithp robability1 )4:

d

dtx(ω, t) =f(x(ω, t),z(ω, t),u(ω, t), ξ(ω), t), (1) h(x(ω, t),z(ω, t),u(ω, t), ξ(ω), t) = 0, (2)

gL≤g(x(ω, t),z(ω, t),u(ω, t), ξ(ω), t)≤gU, (3)

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wheregU−gL>0elementwise andnz≤ nh.Thus, foreachpossiblerealizationoftherandom variableξ(ω),thetrajectorywillfollowthedeterministicdifferential-algebraicequations(1)and(2) insidetheregionsofstatespaceallowedbytheconstraints(3)5_._We_have_employed_the_notation_x₍_ω,

t)toemphasizethedependenceofthetrajectoriesontherandomvariables;henceforth,wemayuse theabbreviatednotationxforimprovedclarity.

Inordertofullydeterminethetrajectory,weneedboththerealizationoftheuncertain param-eters ξ and a control or guidance law that completely determines the controlsat each t, for each possibleoutcomeω.WeaddressthistopicinSectionIIB.

B. The state-tracking formulation

Inpreviousliteratureemployingthisapproach(see[16],[18]or[23]),thecontrollawisconsidered as only dependant on time u= uF(t), thus leading to an “open-loop” control scheme where the controlsareequal ineachscenario(i.e. u(ω1,t)=u(ω2,t)∀ω1,ω2∈).

This“open-loop”formulationis,however,notapracticalschemeforallgeneraloptimalcontrol problems. In some problems, the dynamic system could be unstable and the trajectories would divergetowards undesirableregions ofthe statespace; in others(suchas the one weface in com-mercial aircraft trajectory optimization), it is necessary to apply finalco nditionsan d/orha vea unique trajectory for some of thestates. In anaircraft trajectory planning context, one needsto finda un iquepa thin la titudean dlo ngitudeth aten dsat a sp ecifiedpoin tin spac e,as well as a fixeda irspeedschedule.

Insteadoflookingforanoptimalcontrol,then,wewilllookforanoptimalguidance: wedesignate some of the statesas “tracked” states and we replace the uniquecontrols uF(t) that are applied identically in all scenarios by scenario-specificco ntrolsu(ω,t) th aten sureth atth etr ackedstates follow a uniquetrajectory for all scenario values of therandom variables, as long as it is feasible withinthedynamics and constraintsof theproblem and therandomvariables arebounded. This is only appropriate for practical problems where the actual system has low-level controllers that

5_Note_that_this_system_is_a_special_case_of_a_stochastic_differential_equation_in_the_sense_of_[19,_Chapter_2]_where

therandom parameters arenot random processes, i.e., areconstant. Thismotivated [16] to employthe term “tychastic”,proposedinearlierwork,tofacilitatethedistinction.

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can track the desired trajectory in real time at a shorter timescale than that of the optimal control problem. In our practical context (long-range trajectory planning), the autopilot can compute the controls that are needed for the aircraft to follow a route at the calculated airspeeds and altitudes, and the omission of the short-term control dynamics (by assuming that the states are tracked exactly) does not introduce a significant error in performance.

We define the number of “control degrees of freedom” of the dynamical system asd=nz+nu−

nh. Letqx≤min{nx, d}be the number of tracked states; without loss of generality, we can assume that the tracked states are the firstqxstates (rearrange the state vector otherwise), i.e.

x= x1 . . . xqx xqx+1 . . . xnx T =     xq xr     ,

where xq is the tracked part of the state vector and xr is the untracked part. Let In be the identity matrix of shape n×n and 0n1,n2 be the zero matrix (i.e. a matrix with zeroes in all its

entries) of shapen1×n2. We define the matrixEx∈Rqx×nx as

Ex=

Iqx 0qx,nx−qx

.

This matrix transforms the state vector into the “tracked states” vectorxq =Exxthat contains only the states whose evolution is equal in all scenarios. In general, other combinations of states (represented by a general full-rank tracking matrix Ex that is not built as we have described, or even an analogous nonlinear transformationEx(·)) could also be tracked.

As emphasized earlier, controls are no longer unique for all scenarios. However, in order to completely determine the value of the controls at each moment, we need to close the remaining d−qxdegrees of freedom. We can do that by selectingqu controls andqz algebraic variables to be equal in all scenarios (as in the “open-loop” problem), as long as qx+qu+qz =d. In analogous fashion asEx, we can define the tracking matricesEzandEuin order to select the tracked algebraic variables and controls. The tracking scheme is then completely defined by the tuple{qx, qz, qu}and a rearranging of the dynamic system in the manner described above (i.e. tracked variables before untracked variables).

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With the aid of the tracking matrices, we can now define the tracking conditions (which, again, apply almost surely):

Ex(x(ω1, t)−x(ω2, t)) = 0, ∀t,∀ ω1, ω2∈Ω,

Ez(z(ω1, t)−z(ω2, t)) = 0,∀t, ∀ω1, ω2∈Ω,

Eu(u(ω1, t)−u(ω2, t)) = 0,∀t,∀ω1, ω2∈Ω.

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Thetrackingconditionsenforceequalityinthetrackedvariablesbetweenrealizations: notethat Ex(x(ω1,t)−x(ω2,t))isthevectorofdifferencesbetweenthetrackedstatesinoutcomeω1andthe

tracked statesin outcomeω2. Theother twoconditions areanalogous trackingconditions forthe

dependentvariablesandthecontrols.

Foranarbitrarydynamicsystemwithuncertainparameters,noteverytracking schemeofthis form is necessarily feasible; indeed, it is trivial to build examples where there are no values of the untracked controls and algebraic variables that fulfillt hed ynamice quationsa longa tracked trajectory for certain values of the uncertain parameters, or examples where there are several or infinitevaluesthatdo(thereforeleavingthecontrolsundeterminedduringoperationofthesystem). Wediscussthisconditionforlinearsystems intheAppendix.

Note that the “open-loop” robustcontrol formulation employedin [16, 18, 23][17, Chapter 2] constitutesaspecialcaseofthisformulation,withqx=qz=0.

C. Definition of the Robust Optimal Control Problem with Tracking

Let _E[·] be the expectation operator associated to the probability space (Ω,F, P). We define

the terminal cost or ‘Mayer term” Φ :R×R×Rnx_×

Rnx_→

R, the running cost or “Lagrange term”

L:Rnx_×

Rnz_×

Rnu_×

Rnξ_×

R→R. We define the cost functional to minimize as:

J =E Φ(t0, tf,x(t0),x(tf)) + Z tf t0 L(x,z,u, ξ, t)dt . (5)

We define the non-anticipative initial conditions function Ψ0:Rnξ×R→Rnx−qx. This equation

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allowing for uncertainty in the initial conditions:

xr(t0) = Ψ0(ξ, t0) (6)

Iftheinitialconditionsareknownwithcertainty(aswewillassumeinSectionIII),thenΩ0(ξ,t0)=

xr,0.

We also define the function Ψ :R×R×Rnx_×

Rnx _→

Rthat contains the remaining boundary

conditions:

E[Ψ(t0, tf,x(t0),x(tf))] = 0. (7)

While these conditions are imposed in average, the ones that depend only on tracked states collapse to boundary conditions that are imposed exactly (as the value of the tracked states at the endpoints is unique); otherwise, they remain probabilistic constraints.

The objective J and the boundary conditions Ψ are written in terms of mean value, but they can be easily generalized to other statistics under the expected value formulation. For example, the variance of a function G(ξ) can be written as E[(G−E[G])2] =E[G2]−E[G]2 using expected

values.

Wealsogroupthedifferential-algebraicequationsandconstraints(1)(2)(3):

d

dtx=f(x,z,u, ξ, t), h(x,z,u, ξ, t) = 0, gL≤g(x,z,u, ξ, t)≤gU.

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The robust optimal control problem with tracking (ROCT) can now be defined as:

minimize J(5)

subjecttodifferential-algebraicequations(8) boundaryconditions(7) trackingconditions(4)                          (ROCT)

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D. Stochastic quadrature rules

Inorder to solveProblem (ROCT),we will approximatetheuncertain parameters with a dis-creteprobabilitydistribution. Wedefinea “ stochasticq uadraturer ule”( SQR)a sa p rocedurethat generatesa finites eto fq uadraturep oints{ξk},k∈{1,...,N}a ndw eights{wk},k∈{1,...,N}, such that we can build anapproximation6 to a stochastic integral I =E[G(ξ)] =R G(ξ)dP with

thesum: QG= N X k=1 wkG(ξk). (9)

where G(ξ)is a well-behaved function. Basicstatistical quantities,such as averagesand vari-ances,canbeobtainedwiththisintegralbythecorrespondingfunctionchoices7_. _There_are_a num-berofapproacheswithdifferentapproximationtechniquesthatcanprovideastochasticquadrature rule; dependingontheproblemand thedesiredapproximationprecision,differentSQRsmightbe best. TheidealSQRcan reachthedesiredapproximationaccuracywitha lowN,because thesize of the discretized robust optimal control problem grows linearly on N (as it can be observed in SectionIIF).Welisthere someexamplesof SQRclasses:

1) In Monte Carlo methods, the pointsξkare randomly sampled from the probability distribution and weighted equally (wk=N−1). Under mild assumptions onG, the approximation error of the integral converges asymptotically at anO(1/√N) rate. This implies that precise solutions

can easily become computationally expensive; common practice in Monte Carlo simulations is to use a N of 104 _{to 10}5_{. Its main advantages are its simplicity and the fact that the}

asymptotic rate of convergence is independent of the dimension ofξ.

2) Quasi-Monte Carlo methods are similar to Monte Carlo methods, but the random samples are replaced by a deterministic low-discrepancy sequences that sample the outcome space in

6 _{Naturally, we expect the discrete approximation to converge to the true value as}_N_{→ ∞} 7 _{e.g. the variance of}_γ₌_G₍_ξ_{) can be computed as}

E[γ2]−(E[γ])2= N X k=1 wkG(ξk)2− N X k=1 wkG(ξk) !2 .

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amoreevenmanner[24]. Forcertainproblems,a rateofconvergenceofO(1/N)(fasterthan MonteCarlo)isobserved;severalexplanationshavebeenadvancedintheliterature[25][26].

3) GeneralizedPolynomialChaos(gPC)methodsrelyontheexpansionoftherandominputsand outputsonanorthogonalpolynomialbasis,thusallowingforrecoveryofsomestatistical quantitiesdirectlyfromtheexpansioncoefficients[13][27].Thenonintrusivestochastic collo-cationisthegPCvariantthatisusefulforrobustoptimalcontrol:itcharacterizestheoutputas aninterpolantatthesetofnodes{ξk}.Itisefficientforproblemswithlow-dimensionalrandom variables[28],butthenumberofrequirednodesgrowsquicklywiththedimensionofthe uncertainparametersevenwhenusinghigher-efficiencysparsegrids[29].Itcanalsobe combinedwithkrigingtechniques[15].

4) Cubature techniques are high-dimensionalanalogues of regular one-dimensional quadrature rulesthatlookforexactapproximationofcertainclassesoffunctions. Acompilationof cuba-turerulescanbefoundin[30]and[31]. Theirscalingandaccuracycharacteristicsaresimilar to stochastic collocation gPC. The “Hyper-Pseudospectral” cubature scheme introduced in [16]wasspecificallydeveloped forrobustoptimalcontrolproblems: itlooksto findthemost

efficient (in a certain sense)N points and weights for a given probability distribution.

5) Empirical distributions and opportunity discretizations: in certain practical settings, it might be the case that the uncertain parameters are already characterized as a discrete distribution: for example, if they are empirically estimated by a discrete set of measurements or simulated from a relatively small number of scenarios. EPS forecasts can be classified in this latter category, and we therefore use them “as given” with no other preprocessing (with equal weights for each member).

E. The trajectory ensemble

Letxq(t) :R→Rqx _{define a trajectory for the tracked states, and}_z

q(t) anduq(t) analogously. Suppose a SQR has been chosen, with a number of points N. For each one of these points ξk, the tracking trajectory (xq,zq,uq)(t) defines a unique trajectory given a full set of initial conditions; we will now collect each one of theseN trajectories in a trajectory ensemble. We define the trajectory

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ensemble associated to a tracking trajectory (xq,zq,uq)(t) as the set of trajectories{(xk,zk,uk)(t)} with k ∈ {1, . . . , N} such that the trajectory k is generated by the initial conditions xk(t0) =x0

and the tracking trajectory with ξ=ξk, i.e.

d dtxk =f(xk,zk,uk, ξk, t), h(xk,zk,uk, ξk, t) = 0, gL≤g(xk,zk,uk, ξk, t)≤gU, Exxk(t) =xq(t), Ezzk(t) =zq(t), Euuk(t) =uq(t). (10)

Note that the last three equations equate the tracked variables to their value in the tracked trajectory and thus we will not need to incorporate the tracking conditions yet.

We can now build an augmented dynamical system that comprises all of the trajectories in the trajectory ensemble, whose state (xE ∈ RnxN_{), control (}_u

E ∈ RnuN_{) and algebraic variables} (zE∈RnzN_{) contain the state, control and algebraic vectors of all the trajectories in the trajectory} ensemble: xE=         x1 .. . xN         ; zE=         z1 .. . zN         ; uE=         u1 .. . uN         . (11)

We define the differential equation, algebraic equations and inequality constraints of this aug-mented dynamical system as:

fE(xE,zE,uE, t) =         f(x1,z1,u1, ξ1, t) .. . f(xN,zN,uN, ξN, t)         , (12)

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hE(xE,zE,uE, t) =         h(x1,z1,u1, ξ1, t) .. . h(xN,zN,uN, ξN, t)         , (13) gE(xE,zE,uE, t) =         g(x1,z1,u1, ξ1, t) .. .         . (14) g(xN,zN,uN,ξN,t)

Note that, whilethis augmented dynamical systemrepresents (approximately) a system with uncertainty,itisadeterministicsystem. Thismeansthatwecanuseittoformulateadeterministic

optimalcontrolproblemthatapproximatesProblem(ROCT).

F. ROCT discretization with the trajectory ensemble

WenowproceedtobuildsuchadeterministicapproximanttoProblem(ROCT),whichwewill callDROCT(DiscretizedROCT).UsingthetrajectoryensembleandtheSQR,wedefinetheMayer andLagrangetermsoftheProblem(DROCT)asfollows:

JE= ΦE(xE(t0),xE(tf)) + Z tf t0 LE(xE,zE,uE, t)dt, (15) ΦE(xE(t0),xE(tf)) = N X k=1 wkφ(xk(t0),xk(tf)), (16) LE(xE,zE,uE, t) = N X k=1 wkL(xk,zk,uk, t); (17)

and discretize the boundary conditions as

ΨE(t0, tf,xE(ω, t0),xE(ω, tf)) = N

X

k=1

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ExN

Forconcisewritingofthediscretizationofthetrackingconditions(4),wewilldefinethematrix∈

Rqx(N−1)×nxN_._as: E_xN =         Ex . ._. Ex                     Inx −Inx Inx −Inx . ._. . ._. Inx −Inx             . (19) EN

z ∈ Rqz(N−1)×nzN and EuN ∈ Rqu(N−1)×nuN can be defined in analogous fashion. These matricesmaptheensemblestatevectortothedifferencesinthetrackedstatesbetweentrajectories.

MakinguseofEquations(10)-(14)aswellasEquations (15)-(19),wecan completenowthe formulationofthedeterministicapproximant:

minimize JE subject to x˙E=fE(xE,zE,uE, t) hE(xE,zE,uE, t) = 0 IGgL≤gE(xE,zE,uE, t)≤IGgU EN xxE= 0 EN z zE = 0 EN uuE = 0 E(t0, tf,xE(ω, t0),xE(ω, tf)) = 0                                                              (DROCT) whereIG= [Ing. . .Ing] T _∈ RngN×ng_. xxE Ez zE EuuE

This optimal control problem can now be solved with standard deterministic direct methods (see, for example, of [32, Chapter 4] or [33, 34]). From the point of view of implementation, definingandmodelingthetrackedvariablesineveryscenarioisnotnecessaryastheyareequal;asa consequence,implementationofthetrackingconstraintsEN ₌_0, N ₌₀_and N ₌₀_can alsobeomitted. Theresultingformulationconstitutesthe“compactform”ofProblem (DROCT). NotethatthesizeofthisproblemisproportionaltobothN andthedimensionofthetychastic

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dynamicalsystemminusthenumber oftrackedvariables(ifN 1),withthenumber ofvariables being O(N(nx+nh)) (in the compact form) and the number of function evaluations (of f, g, h and L) per node being O(N(nx+nh+ng+1)). That is, a DROCT is approximately N times larger thana similar deterministicproblem, but it preservesthelinearscaling of theproblemsize onthedimensionofthestatespace thatrepresents acoreadvantageofnumericaloptimalcontrol. Therefore, itavoids the“curse ofdimensionality”, i.e., theexponential scalingof theproblemsize andcostonthedimensionofthestatespacethat appearsin dynamicprogramming.

While we will notdo so, it is possible to partially parallelize this problem in order to speed it up. The computational cost of an optimal control problem solved through direct collocation canbe dividedinfunctionevaluations(whichincludes thecomputationofderivativematrices)and NLPalgorithm time. Theformercan be naturallyparallelized ina scenario-wisefashion, butit is harderto replaceNLPsolversbyparallelversions;dependingonwhichofthetwoparts ofthecost dominates,parallelizationwouldbemoreor lesseffective.

III. RobustOptimalFlightPlanning

In this section, we apply the methodology described in Section II to the problem of flight planningunderwindforecastuncertainty. WegiveanintroductiontoEnsemblePredictionSystems inSectionIIIA.WedescribethedynamicalmodelinSectionIIIB,performanecessaryreformulation inSectionIIIC,applythetrajectoryensembleformulationinSectionIIIDanddefinetheassociated Problem(DROCT) inSectionIIIE.

A. Ensemble Prediction Systems

UncertaintyinNumericalWeatherPredictionarisesmainlyfromincompleteorimprecise knowl-edgeofthestateoftheatmosphereatthetimeoftheforecast andmodeluncertainties[35].Because theseuncertaintiesarepropagatedthroughnonlinearandchaoticatmosphericdynamics,simple sta-tisticalcharacterizationisinadequatefortherepresentationoftheforecastuncertainty[9].Instead, themainapproachinNWPforthecharacterizationofuncertaintyisbasedonensembleforecasting: amethodologyforgeneratingprobabilisticforecastsbyrunningacertainnumber(generallybetween 10 and 100) of different simulations or “members”, as well as the techniques employed to set up

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each member.

Initial conditions Final state

a) Deterministic forecast b) Ensemble forecast with initial conditions perturbations

c) Ensemble forecast with parameter perturbations

d) Ensemble forecast with multiple perturbation methods

Fig.3 Deterministicforecasts and EPSforecasts

Severalmethodsareusedforproducingthedifferentmembers ofanEPS(Figure3). Themost importantincludeensembledataassimilation,wheretheinitialconditionsareperturbedaccording to theiruncertainty; singularandbreedingvectors, wherethe perturbationischosento excitethe fastest-growingdynamicalinstabilities;stochasticparametrizations,wheretheparametersofmodels of the atmospheric processesat subgrid spatiotemporal scalesare perturbed; andmultiphysics or multimodel schemes, where themembers are drawn from theoutput from differentmodels. Each NWP center maintains an Ensemble Prediction System that implements a different combination of these techniques. It should be noted that these methods aim to be “economic” by producing a goodrepresentation of the uncertainty from a limited number of simulations, since eachone is computationallyexpensive.

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TherearethreemainclassesofEPS.Indecreasingscopeandmaturityandincreasingresolution, theyareglobal scaleEPS,regionalscale(LimitedAreaModelor LAM)EPSand convectivescale EPS. For the purposes of considering wind uncertainty in medium-haul flights,bo thgl obaland LAM EPS can be used. However, since LAMs are usually produced in order to forecast surface weather ofa specificre gion(s uchas Eu ropeorNo rthAm erica),th eiroc eanicco verageislimited. Therefore, in this workwewill relyon globalEPS, whichcan be usedfor intercontinentalflights. Historical global EPS forecasts from major NWP centres around the world can be found at the TIGGEdataset8, hostedattheECMWF(EuropeanCentreforMedium-RangeWeatherForecasts) website. Outputfrom EuropeanLAMmodelsissimilarlycompiled intheTIGGELAMdataset.

Several probabilistic metrics can be obtained from EPS output[36]: some examples are the

ensemble mean (the averaged value of a variable across the members), the ensemble spread (the standard deviation) orthe probabilityof anevent (theproportion of themembers thatforecast a certain event ocurring). Each ensemble member is oftentaken as equally likely when computing these metrics; therefore, an EPS-based SQR has its N members as the quadrature points and wk=1/N asitsassociatedweights.

B. Dynamical Model

Weconsider a 3-DoF point-massmodel ofaircraft used widelyin ATM studies, BADA 3 [37]. Wewillrestrictourselvestotheanalysisofthecruisephaseforthesakeofsimplicity(notethatthe impactofwindforecastuncertainty iscumulative,and thusmoreimportantforlongerflights,and thecruise phasecomprises mostofa medium-haulor long-haulflight).I na ddition,w ec onsidera constant flightl evelb utw en otet hato urm ethodologyc anb ee xtendedt of ull4 Dp roblems.9_We consider anellipsoidalEarthasin theWGS84model,withradiiof curvatureofellipsoidmeridian and primevertical denoted byRM and RN respectively. Wetake windfromanEPS forecastand set remainingatmospheric parameters according to theInternationalStandard Atmosphere(ISA) model.

8 _{http://apps.ecmwf.int/datasets/data/tigge/}

9 _As_we _mentioned_in _the_{introduction,} _direct_methods _are _flexible_{e nough}_{t hat}_{t hey}_{c an}_{h andle}_{m ore}_complex

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We make a further simplificationb yt akingt heh eadinga sa c ontrolv ariablei nsteado fthe bank angle, thus allowing it to change instaneously (as it is done in mostrouting algorithms). In previouswork[38],wefoundthatomittingheadingandbankangledynamicsforafree-routingcruise trajectorywithoutsharpturnshasminimalimpactonsolutionaccuracy(wellbelowothersourcesof modelling error). Thedynamics of thesystem are, therefore,described by thefollowing systemof differentialequations(finEquation(8)):

˙ x= d dt             φ λ v m             =             (RN +h)−1(vcos(χ) +wx(φ, λ, t)) (RM+h)−1cos−1(φ)(vsin(χ) +wy(φ, λ, t)) (T−D(m, v))/m −η(v)T             , (20)

whereφisthelatitude,λisthelongitude,visthetrueairspeed,misthemass,histhegeodetic altitude,χistheheading,wxandwyarethezonalandmeridionalcomponentsofthewind10,Tisthe thrust force, D is thedrag force and η is thethrust-specific fuel consumption; both η and D are modelled according to the BADA 3 specification, assuming that L = mg. The control vector is composedbythethrustTandtheheadingχ.

Inaddition,thefollowingconstraints(asg(·)inEquation(8))apply:

vCAS,stall ≤vCAS(v)≤vCAS,max,

M(v)≤Mmax, (21)

Tidle(v)≤T≤Tmax,

where M is the Mach number, vCAS is calibrated airspeed (CAS), vCAS,stall and vCAS,max are the lower and upper bounds on calibrated airspeed, Mmax is the maximum operating Mach number,andTidleandTmaxarethethrustlimits.Itisadvantageousforcomputationalpurposesand forclarityofexposition,aswewillexplaininSectionIIIC,toreformulatethisdynamicalsystemasa differential-algebraicsystem(DAE)withtheadditionofthegroundspeedvGasanalgebraic

10_Contrary_to_the_usual_definition,_{w e}_{t ake}_w

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variable and the courseψas a control variable, linked to the remaining variables by two new equalityconstraints (see Figure 4 for a graphical explanation of Equation 23). The reformulated systemisgivenbythedynamicalfunction:

d dt             φ λ v m             =             (RN +h)−1vGcos (RM +h)−1cos−1(φ)vGsin (T−D(m, v))/m −η(v)T             , (22)

andtheequalityconstraints(hin Equation(8)):

vGcosψ=vcos(χ) +wx(φ, λ, t), vGsinψ=vsin(χ) +wy(φ, λ, t). (23) True Airspeed (TAS)

W

Heading Wind speed Ground speed

E

S

N

Course

Fig. 4 Relationship between airspeed, groundspeed, wind, heading and course

Theinequalityconstraints(21)applyinthesamefashion, withtheadditionof

vG≥0 (24)

∗

G

∗

G

toensureuniquenessofvGandψ(otherwise,(−v,ψ∗ +π/2)wouldproducethesameleft-handsideof Equation(23)as(v,ψ∗)).

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C. Coordinate transformation

Intherobustoptimalcontrolframeworkthatwehavepresented,theindependentvariable(time) isuniqueinallscenarios(itvariesinthesamerange). Therefore,thedirectapplicationofthe state-trackingformulationwoulddemandthepositionoftheaircrafttobeinafixedschedulewithrespect to time in all scenarios, therefore absorbingall of theuncertainty with constant airspeedchanges and associatedfuel burnvariation. Thisis inconvenient forseveral reasons: bynotallowingsome of the uncertainty to manifest in delay, the computed solutions could be very inefficient in some scenarios, or the feasiblerange ofaverage speedscould be reducedso that the mostunfavourable scenarios havesufficientmarginofvariationtotrackthetrajectorywith aschedule thatisfixedin time. Nevertheless,wewill notethatthis typeofsolutions canalsobeobtained withourapproach andtheextensionpresentedinSectionIV;theformulationthat wepresentgeneralizesandincludes thiskindoffixed-schedulesolutions.

Instead,wewilladoptaformulationthatismoreconsistentwithcurrentflightp roceduresand Flight ManagementSystem (FMS)technology,as theyare notexpected tochangein the short-to-medium termevenafter theintroductionoffree-routing. Wewill employ distanceflowna longthe route(denotedas s)as theindependentvariable,becauseitsinitialandfinalvaluea ret hes amein allscenariosthatfollowa uniqueroute. Asaconsequence,thetimetbecomes astatevariableand thenewdynamicalfunctioncanbe obtainedbydividingthetimederivatives byds/dt=vG:

d ds                 φ λ v m t                 =                 (RN+h)−1cos (RM+h)−1cos−1(φ) sin (mvG)−1(T−D(m, v)) −v−_G1η(v)T v−_G1                 . (25)

All equality and inequality constraints remain the same as in the untransformed system of differential-algebraic equations. Note that, by employing vG as an algebraic variable (instead of a function of wind, airspeed and heading), it will be computed only once at each node in the NLP iterations and produce sparser derivative matrices.

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D. Trajectory Ensemble

Wecannowdiscretizetheuncertaintyandcreatethetrajectoryensemble. Anensembleforecast contains a set of ensemble members, each one defininga d ifferentw indf orecast( and,therefore, different functionswx andwy). IftheensemblecontainsN members, we defineN s cenarios,each one having weight wk = 1/N and the windfunction that corresponds to the respective member; ourstochasticquadratureruleis,therefore,a simpleempiricalaverage. Wewillwritethecompact form ofthetrajectory ensembledirectly.

We choose to track the course ψand the true airspeed v, i.e., the functions ψ(s) and v(s)

are the same in every scenario (so we donot need to implement scenario-specific versions). As a consequence of (25),thisimplies thatthe evolutionof thelatitudeφis unique(asit onlydepends on theevolution of theunique variable ψ)and λ is alsounique (as itonly depends onφ and ψ). Therefore, the position variables act like tracked variables too, which is both relevant from the implementationpoint ofview (because wedonotneedto create copiesof themforeach scenario) andadesiredgoal(sincewewanttoobtainauniqueroute). Wealsodefinea:=dv/dsforpractical purposes, in order to combine the derivative of v and its tracking condition in a single set of constraints. Itsphysical interpretationistheslopeoftheairspeedprofile.

Takingadvantageofthesemanipulations,wecandefinethedynamicalsystemassociatedtothe trajectory ensemblewiththedynamicalfunction:

dxE ds = d ds                                 φ λ v t1 .. . tN m1 .. . mN                                 =                                 cos(ψ)/(RN+h) sin(ψ)/(RN +h)/cosφ a 1/vG,1 .. . 1/vG,N η(v)T1/vG,1 .. . η(v)TN/vG,N                                 , (26)

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with the control vector: uE= a T1 . . . TN χ1 . . . χN T ,

the equality constraints:

                                vG,1cos(ψ) .. . vG,Ncos(ψ) vG,1sin(ψ) .. . vG,Nsin(ψ) a·vG,1 .. . a·vG,N                                 =                                 vcos(χi) +wy,1(φ, λ, t) .. . vcos(χi) +wy,N(φ, λ, t) vsin(χi) +wx,1(φ, λ, t) .. . vsin(χi) +wx,N(φ, λ, t) (T1−D(v, m1))/m1 .. . (TN −D(v, mN))/mN                                 , (27)

and the inequality constraints:

vCAS,stall ≤vCAS(v)≤vCAS,max,

M(v)≤Mmax, Tidle(v)≤Tk ≤Tmax 0≤vG,k        ∀k∈ {1, . . . N} . (28)

E. DROCT of the Robust Flight Planning problem

Wewill nowfinisht hef ormulationo ft heR obustF lightP lanningp roblemb ya ddinga n opti-mizationcriterionandboundaryconstraints. Wedefinetwoadditionalscalaroptimizationvariables: the earliest arrival time t(sf)min and the latest arrival time as t(sf)max, which together define a

“window of arrival”. We also define two user-specified parameters: the Cost Index CI11, which

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represents the user’s preferences for reduced flight time versus reduced fuel burn, and the “disper-sion penalty” DP, which represents the user’s preferences for increased predictability versus average efficiency. We define the cost functional as:

JRF P = 1 N N X i=1 ti(sf) + CI· 1 N N X i=1 mi(sf) + DP·(t(sf)max−t(sf)min) (29)

and the boundary conditions:

(φ, λ, v)(0) = (φ0, λ0, v0), (φ, λ, v)(sf) = (φf, λf, vf), tk(0) = 0 mk(0) =m0                  ∀k∈ {1, . . . N}. (30) tf,min≤tk(sf)≤tf,max

TheProblem(DROCT)associatedtotheRobustFlightPlanningproblemcannowbe defined

as:

minimize JRFP (29)

subjectto dynamicalequation(26) equality constraints(27) inequalityconstraints(28) boundaryconditions(30)                                    (DRFP)

WesolveProblem(DRFP)inanexamplescenarioinSectionV.Notethat,withminimalchanges tothecostfunctionalandboundaryconditions,similarproblemscanbesolved(suchasfindingthe fuel-optimal trajectorythatarrives ina specifiedt imewindow).

IV. DynamicAirspeedAdjustment

In Section III, we have described a formulation whose solution generates a flight plan that is consistentwithcurrentflightproceduresandexistingFMSandTrajectoryPrediction(TP) technol-ogy: alateralpath(thatcan bediscretizedtoasequenceofsegmentsdelimitedbywaypoints)and

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anairspeedprofile( thatc ana lsob ed iscretizedt oa s equenceo fa irspeedsa ssociatedt oe achleg). Inthis section, weconsider a potentialfuture concept wheretheairspeed isdynamicallyadjusted inresponsetotimeleadsorlagsinordertoincreaseadherencetothepredictedflybytimesa teach pointinthetrajectory,andtheadjustmentisdoneaccordingtoafeedbacklawthatisincorporated totheflight plan;wewill namethis concept“dynamic airspeedadjustment”orDASA.

The proposed schemeis consistent with the conceptof making small tactical speed changes12 in order to increasepredictability. This ideahasbeen explored as a potentialfuture ATM system concept;forexample,in[39] andrelatedwork. Underthe“subliminalcontrol”paradigmexamined there, pilots or automated systems wouldimplement tactical speed changes in the interval [-6%, 3%] (under future technology) in order to reduce the risk of conflicts;t hism odificationwo uldbe small enough that it wouldfall within theuncertainty observed bythe air traffic controller (thus the “subliminal” label). In the present work, a speed change of a similar magnitude would be individually triggered by delays or leads following a pre-computed rule insteadof a periodic and centralized sector-wide calculation. Therefore, the impact on conflictsw ouldb ei ndirectthrough increasedadherencetothescheduledflybyt imes( whichwoulda llowf ore arlierdeconfliction).

A. The DASA law

Weconsidera simplecontrollaw13_:

v(s)−¯v(s) =K·(t(s)−¯t(s)), (31)

where ¯v(s) is a fixed airspeed schedule, ¯t(s) is the expected flyby time at position s, and K is a constant. This scheme constitutes a control law analogous to a proportional regulator, where the airspeed increments or decrements are proportional to the accumulated time lead or lag with regards to the expected trajectory. Under this law, the pilot or the FMS would change the planned airspeed according to the delay or lead compared to the scheduled times. We will optimize this law

12_{Note that, in this work, the rule implemening these speed changes is still computed at the pre-tactical planning} stage.

13_{Note that}_v_{is not a control variable, but the dynamics of airspeed tracking would again happen at a timescale of} much smaller characteristic time than that of the optimal control problem.

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jointly with the flight plan at the planning stage (i.e., the gainKwill be a result of the optimization process).

B. DASA formulation

InSection IIB,we brieflymentionedthe possibilityof substituting thetrackingconditionsby more general linearor nonlinear transformations of dimensiond; these transformations may have free parameters whose valuewill be implicitlyoptimized by theNLPsolver. Wewill employ this extensionofthetrackingformulationinordertoimplementtheplanningpartoftheDASAconcept: byreplacingthetrackingconditiononthetrueairspeed withtheDASAcontrollaw.

We start from the framework describedin Section III. Since theairspeed will nowbe specific to eachensemblemember, wedon’t collapseitinto a singlevariable; instead,weletv1,...,vN be themember-specifica irspeedsw itht hea ssociatedd ynamicfunction:

dvk

ds = (Tk−D(v, mk))/(mk·vG,k),k∈ {1, . . . N},

and we define a new state variable ¯v with dynamics:

dv¯ ds =a.

We also define the algebraic variable ¯t, the average flyby time, related to the state variables by the algebraic constraint: ¯ t= 1 N N X k=1 tk.

We define the scalar variable K, which will become a free optimization variable of the optimal control problem (therefore, its value will be computed by the NLP solver). We replace the tracking condition on the airspeed by the algebraic condition implementing the control law:

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The dynamical equation is now: dxE ds = d ds                                             φ λ ¯ v v1 .. . vN t1 .. . tN m1 .. . mN                                             =                                             cos(ψ)/(RN+h) sin(ψ)/(RM +h)/cosφ a (T1−D(v, m1))/(m1·vG,1) .. . (TN−D(v, mN))/(mN·vG,N) 1/vG,1 .. . 1/vG,N η(v)T1/vG,1 .. . η(v)TN/vG,N                                             . (33)

The equality constraints now include the control law:

                                    ¯ t vG,1cos(ψ) .. . vG,Ncos(ψ) vG,1sin(ψ) .. . vG,Nsin(ψ) v1−¯v .. . v1−¯v                                     =                                     N−1_ΣN k=1tk vcos(χi) +wy,1(φ, λ, t) .. . vcos(χi) +wy,N(φ, λ, t) vsin(χi) +wx,1(φ, λ, t) .. . vsin(χi) +wx,N(φ, λ, t) K·(t1−¯t) .. . K·(tN −¯t)                                     . (34)

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vCAS,stall≤vCAS(vk)≤vCAS,max M(vk)≤Mmax Tidle(vk)≤Tk≤Tmax 0≤vG,k                          ∀k∈ {1, . . . N}, (35)

and the boundary conditions now depend on ¯v instead ofv:

(φ, λ,v¯)(0) = (φ0, λ0, v0), (φ, λ,v¯)(sf) = (φf, λf, vf), tk(0) = 0 mk(0) =m0 tf,min≤tk(sf)≤tf,max                  ∀k∈ {1, . . . N}. (36)

Therefore,the(DROCT)associatedtotheDASAproblemcanbedefinedas: minimize JRFP (29)

subjectto dynamicalequation(33) equality constraints(34) inequalityconstraints(35) boundaryconditions(36)                                    (DASA)

Wepresentsome resultsunder thisformulationinSection VC.

V. CaseStudy

We consider a cruise flightf romt hev erticalo fN ewYorkt ot hev erticalo fL isbona tFL380. Theaircraftis anA330-301(BADA3 code A333)with aninitialmassof 200 tons,andweemploy the 6-hours-lead-time forecast from the Météo France PEARP ensemble for the 20th of January, 2016atthepressurelevel of200hPa. Aswementionin SectionIII,we consideruncertaintyinthe windfieldd erivedf romt heE PS.I no rdert of acilitatev isualizationa ndu nderstanding,weemploy a static weather picture; nevertheless, the methodology allows for the usage of dynamic weather forecasts.

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We employ the procedure presented in [38] for building a smooth approximation to the wind withgoodaccuracy. Theoptimalcontrolproblem isformulatedas inSection II,discretized witha trapezoidalscheme(see[32,Chapter4])intoanonlinearoptimizationproblem,whichisthensolved bytheIPOPTsolver[40]. WeemployPyomo[41]as theNLPinterface.

Inorderto analyzeandunderstand thedifferentforms inwhichthe impactofaleatory uncer-taintycanbereducedwiththeflightp lan,wew illfi rstst udyth epr oblemwi tha co nstantairspeed restriction(TAS=240m/s),whichwewilldenominateCaseA,andthenstudythefullproblemas describedinSectionIII,whichwewilldenominateCaseB.WefinallydiscussresultsfortheProblem (DASA),whichwewillsummarizeasCaseC.Wewillsolveeachoneoftheseproblemsmultipletimes, fordifferentvaluesofCI14andDP.

Forallthecasesthatwediscuss,theNLPgeneratedbyeveryindividualoptimization problem (for a given CI and DP) take around 2 - 20 seconds of CPU time to solve with IPOPT on our workstation, which is equipped with an Intel Xeon E3-1240 v5 CPU running at 3.5 GHz. This number does not include the cost of the preprocessing and initialization described in [38], which takesaroundaminuteandwehavenotoptimizedforspeedasitneedstoberunonlyonce. Table1 shows thefinals izeo ft her esultingn onlinearo ptimizationp roblemsa ftert het ranscriptiono fthe DROCT.

# Constant TAS Variable TAS DASA

NLP variables 11144 14104 16912

Equality constraints 11039 14016 16782

Inequality constraints 7565 11437 16877

Number of nonzeros in ...

Eq. constr. Jacobian 57271 88189 115945

Ineq. constr. Jacobian 7849 11673 17113

Lagrangian Hessian 18616 44496 60656

Table 1 Problem size.

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A. Constant Airspeed (Case A)

Figure5displaysthegeographicalroutesfordifferentvaluesofthedispersionpenaltyDP.We also plot regions of higher uncertainty, which we have defined aspσ2

u+σv2, withσu and σv being the standard deviation of theuandvcomponents of wind across different members. It can be seen that routes computed with higher DP tend to avoid the high uncertainty zone in the Atlantic in order to increase predictability, at the cost of taking a more indirect route that is longer on average.

Fig.5(CaseA)OptimaltrajectoriesfromNewYorktoLisbon,forvaluesofDPfrom0Kg/s to 23Kg/s. Higherbrightnessinthe trajectorycolorindicates highervaluesofDP.

Figure 6.ashows theevolution ofthestate andcontrolvariablesalong theMaximum Average Efficiency (MAE)trajectory(corresponding to DP=0,theblack line in Figure5). Eachindividual line represents the trajectory that corresponds to an ensemble member. It can be seen that the spread in the ensembletimes, ensembleheadings, andensemble groundspeeds increasesmarkedly when the aircraftcrosses the area of high uncertainty near the destination airport (see Figure 5). Figure6.bshowstheevolutionofthestateandcontrolvariablesalongthehighpredictability(HP) trajectory(correspondingtoDP=23Kg/s,theyellowlineinFigure5). Itcanbeseenthatthespread in times and ground speedsare comparatively lower thanin theprevious case and thetrajectory ensembleistighter.

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−150 −100 −50 0 50 100 150 200 Time lead or lag (s) DP = 0 Kg/s DP = 23 Kg/s 240 250 260 270 280 290 300 310 Speed (m/s) Groundspeed Groundspeed 0 1000 2000 3000 4000 5000 Distance (km) 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Orientation (rad) Course Heading 0 1000 2000 3000 4000 5000 Distance (km) Course Heading

Fig. 6 (Case A) State-space evolution of the variables in the MAE and the HP trajectories. Time leads and lags are defined with respect to the average trajectory.

DP DP = 0

DP = 70

Fig. 7 (Case A) Pareto frontier of the problem.

Finally, Figure 7 shows the Pareto frontier of the problem, obtained by solving problems with different penalties DP (from DP = 0 to DP = 70). For the MAE case (DP = 0), the time dispersion at the final fix is above 4.5 minutes, whereas for the very high predictability (VHP) case (DP = 70), the time dispersion at the final fix is slightly above 1.5 minutes. In other words, around three minutes reduction in time uncertainty could be achieved by flying the VHP trajectory; however, there is a heavy cost, with the flight lasting for an additional 30 minutes and burning an additional 2500 Kg of fuel. A smaller reduction in arrival time dispersion, from 4.5 to 3.5 minutes, can be

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achieved at a cost of around 500 Kg.

B. Variable airspeed (Case B)

We now proceed to analyze the full problem, as described in Section III. Figure 8 shows the geographical path of theoptimized flightp lansf orC I=0 .A si nt hec onstanta irspeedc ase,the horizontalprofileso ft het rajectoriest hatp lacea h igherweighto nr educingu ncertaintya refarther from thehighuncertainty zone.

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Figure 9 shows the airspeed profiles for different values of the CI and DP parameters. As expected, trajectories optimized for a higher CI tend to feature higher speeds. However, there is another relevantdifference betweentrajectories: as the penaltyin uncertainty grows, the optimal airspeedwhen the aircraft crosses uncertainregions increases. Weattributethis resulttotheidea that,byflyingatahigherspeed,therelativeimportanceofthewindonthegroundspeedisreduced15_.

215 220 225 230 235 240 245 250 255 T AS (m/s) CI = 5 DP=0 DP=20 CI = 10 DP=0 DP=20 0 1000 2000 3000 4000 5000 Distance (km) 215 220 225 230 235 240 245 250 255 T AS (m/s) CI = 20 DP=0 DP=20 0 1000 2000 3000 4000 5000 Distance (km) CI = 40 DP=0 DP=20

Fig. 9 (Case B) Airspeed profiles.

These airspeed changes reduce uncertainty more efficiently than direct avoidance, as it can be observed in Figure10. For a given CI, there are two branches in the locus of the solutions for different DP values: in the left branch, uncertainty reduction is achieved mostly with airspeed increments in the uncertain zone; in the right branch, uncertainty is reduced mostly through horizontal path deviations. As discussed, reducing arrival time dispersion in the left branch is cheaper (100 to 150 Kg when reducing the arrival window size from 5 minutes to 4.5 minutes) than in the right branch (around 500 Kg when reducing from 3.5 minutes to 3 minutes).

15_{Consider the following illustrative example: the along-track wind can take one of two constant values,} _w 1 or w2, on a region where an aircraft flies a distance x at the airspeedv. The difference in arrival times is ∆t= x |w1−w2|

v2_{+ (}_w

1+w2)v+w1w2

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27000 27500 28000 28500 29000 29500 Fuel consumption (Kg) 310 315 320 325 330 335 340 345 Flight time (min) 5.0 minutes 4.5 minutes 4.0 minutes 3.5 minutes 3.0 minutes CI=5 CI=10 CI=15 CI=20 CI=30 CI=40

Fig.10 (CaseB) Paretofrontiersofthe variablespeedproblem. EachCI linerepresentsthe averageefficiencyforincreasingvaluesofDP,while thedashedlinesrepresentpointsofequal time dispersiononarrival.

Finally, we show thestate-spacetrajectories oftwo scenarios in Figure 11. The patternis similar totheconstantairspeedone (seeFigure6),withtheadditionoftheairspeedprofile.

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−150 −100 −50 0 50 100 150 200 Time lead or lag (s) MAE (DP=0 Kg/s) HP (DP=20 Kg/s) 210 220 230 240 250 260 270 280 290 300 Speed (m/s) True Airspeed Groundspeed True Airspeed Groundspeed 0 1000 2000 3000 4000 5000 Distance (km) 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Orientation (rad) Course Heading 0 1000 2000 3000 4000 5000 Distance (km) Course Heading

Fig. 11 (Case B) State-space evolution of the variables in the MAE and the HP trajectories for a CI=10 scenario. Time leads and lags are defined with respect to the average trajectory.

C. DASA Results (Case C)

Werun thesame scenariowiththeDASA formulation,values ofCI rangingfrom 5to 60,and values ofDPrangingfrom 104_Kg/s _to_0.2_Kg/s.16 _We_don’t_sweep _the_DP_parameter_in _the_same range as in Cases A andB because we can achieve lowtime dispersions for much lowervalues of DP. Theresultsoftheoptimizationsuggestthat,inCaseC,itischeapertoreduce uncertaintyby increasingthefeedbackstrengthK (and,therefore,absorbingsome uncertaintyinairspeed instead ofgroundspeed)thanbymodifyingthegeographicalpath: Figure12showsthatgeographicalpaths arealmostexactly equalin thisDPrange.

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Fig.12(Case C)Sameas Figure5, forCaseC.

Figure 13 shows the evolution of the state variables for different values of DP. The lowest predictability scenario hasa similar profile to the MAE trajectoriesin Figure 11; however, as DP increases,someofthewinduncertaintystartsbeingcompensatedbyairspeedvariationsinsteadof being fully passedonto the groundspeed. As a consequence of the TAS dispersion, the fuel burn showsmoredispersionasarrivaltimeuncertaintyisdecreased;weillustratethisresultinFigure14.

−150 −100 −50 0 50 100 150 200 Time lead or lag (s) DP = 0.0001 Kg/s DP = 0.004 Kg/s DP = 0.01 Kg/s 0 2500 5000 Distance (km) 200 220 240 260 280 300 Speed (m/s) Groundspeed True Airspeed 0 2500 5000 Distance (km) Groundspeed True Airspeed 0 2500 5000 Distance (km) Groundspeed True Airspeed