Towards Automatic Flight Control forCommercial Airliners in Formation Flight ?

D. Büchner ∗ J.A.A. Engelbrecht ∗ J. Adams ∗∗

C. Redelinghuys ∗∗

∗ Stellenbosch University, Stellenbosch, South Africa∗∗ University of Cape Town, Cape Town, South Africa

Abstract: This paper presents research done towards the goal of achieving automatic flightcontrol for commercial airliners in formation flight. The motivation for this research is toultimately reduce fuel-consumption through a reduction in the drag of the follower aircraft,which is a result of the formation flight. Traditional aerodynamic equations for conventionalflight of fixed-wing aircraft are expanded to include formation flight interactions. A trim analysisuncovers risks, challenges and feasible trim regions for the formation follower to maintain.These regions include a potentially risky region which is sandwiched between two untrimmableregions, with respect to a maximum aileron setting, and an outside region which has only oneuntrimmable bound, making it less risky but with lower fuel-consumption benefit. Next, a statespace representation is constructed, allowing for a linear dynamics analysis. The poles and theirmovement as a function of the lateral and vertical separation of the follower aircraft relative tothe leader aircraft are shown, and indicate greater changes in flight dynamics due to verticalseparation than to lateral separation. The results of the trim analysis and linear dynamicsanalysis form the basis for the design of a formation flight control system.

Keywords: Aircraft control, autonomous control, flight control, formation flight, linearanalysis, non-linear models, stability analysis, state space models, trim analysis

NOMENCLATURE

a1 Tailplane lift coefficientAR Aspect ratiob, c̄ Wingspan, wing chordbf Double the tailfin heightbh Tailplane spanclα 2-D wing lift coefficient gradientCD Drag coefficientCL Lift coefficientCl Rolling moment coefficientCm Pitching moment coefficientCn Yawing moment coefficientCS Sideforce coefficient in stability frameCY Sideforce coefficient in body frameCX Longitudinal force in body frameCZ Vertical force in body frameg Gravitational accelerationh Mass centre positionh0 Wing aerodynamic centreIxx, Iyy, Izz Moments of inertia in body framem Aircraft mass (unloaded aircraft)M Mach numberq̄ Dynamic pressure ( 12ρV̄

2)rc Core radiusS, Sf Wing area, tailfin areaT Thrustu, v, w Linear velocity components

? The authors would like to thank Airbus and the NationalAerospace Centre of the University of the Witwatersrand, Johan-nesburg, for providing bursaries for Mr. Büchner and Mr. Adams.

p, q, r Angular velocity componentsV Freestream velocityV s Speed of sound in airVf ,VT Tail volume ratio, fin volume ratioα Angle of attackβ Sideslip angleδa, δe, δr Aileron, elevator, rudder deflection angles� Downwash angleη Lateral separation normalised to wingspanζ Vertical separation normalised to wingspan

ζfbfb

ζvzvb

ηhbhb

θ, φ, ψ Pitch, roll and yaw angleµ rcbρ Air densityσ Downwash influence factorτ Moment influence factor

Subscripts

c Conventional isolated flightf Tailfinf’ Formation flight conditionsj Lead aircraftk Follower aircraftt Trim flight conditions

1. INTRODUCTION

In the passenger air-travel industry, there is a growingdemand for the reduction of fuel-consumption with en-

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Copyright © 2014 IFAC 12188

vironmental and cost benefits in mind. Formation flightallows for a decrease in induced drag and a reductionin fuel consumption. Wind-tunnel tests have shown that,depending on the formation flight configuration, drag re-ductions of as high as 25% may be achieved (Blake andGingras 2004). An analysis by Bower et al. (2009) showedthat, when formation geometries and route optimisationare considered for commercial airliners, a 13% reductionin fuel consumption may be realised.

A previous study performed by Bizinos and Redelinghuys(2012) investigated the aerodynamic interaction of aircraftflying in formation. An aerodynamic model was derived forthe induced forces and moments experienced by a trailingaircraft due to the trailing vortices of the lead aircraft.It was found that a very non-linear relationship existsbetween these induced forces and moments and the sepa-ration between the two aircraft, and that the changes areparticularly steep near the optimum separation distance.

The results of this study lead to questions about thestability and performance of the flight control system ofthe trailing aircraft in formation flight. The induced forcesand moments would require unconventional trim settingsfor the trailing aircraft’s control surfaces. For example,the ailerons would have a non-zero trim setting due tothe constant rolling moment experienced by the trailingaircraft. Since the changes in the forces and momentsare steep near the optimum separation distance, the trimsettings would also be very sensitive to small changes inthe position of the trailing aircraft relative to the leadingaircraft.

Formation flight also has implications for the feedbackcontrol gains of the flight control system. The presenceof the leading aircraft can be modelled as changes inthe aerodynamic coefficients of the trailing aircraft, whichmanifest as changes in the aerodynamic stability andcontrol derivatives on which the feedback gain calculationsare based. The changes in the aerodynamic stability andcontrol derivatives of the trailing aircraft due to thepresence of the leading aircraft would therefore lead toreduced flight control performance and even instability.The flight control gains would therefore have to be re-designed and gain scheduled as a function of the separationdistance.

The research presented in this paper is the initial stepstowards investigating the implications of formation flightfor the flight control systems of passenger airliners. Thetrim analysis and linearised dynamic model derived in thispaper will serve as the basis for evaluating the stabilityand performance of current flight control systems in for-mation flight. Once the baseline has been established, thespecialised requirements that formation flight place on theflight control system will be determined, and a new flightcontrol architecture will be proposed, implemented andevaluated.

Previous research on flight control in formation flight in-clude two papers by Brodecki et al.. Their research showsthe design of a control system that addresses the uniqueenvironment encountered by an aircraft flying in formationin the upwash of the formation leader. The control sys-tem uses an advanced extremum seeking algorithm whichutilises an EKF to estimate gradients, as the exact postion

of the sweet spot cannot practically be known (Brodeckiet al. 2013a). Furthermore, the emergent behaviour of thecontrol system is investigated. The desired echelon for-mation emerges consistently after formation is initializedat random points using a Monte Carlo scheme. This isachieved without inter-vehicle communication, using onlyminimal information about the other formation membersand the extremum seeking algorithm, which drives eachmember to the sweet spot for fuel consumption minimisa-tion (Brodecki et al. 2013b).

Studies, such as Zou et al. (2009), show a trend of interestin formation flight due to fuel consumption reductions,though complete formation flight interactions were not yetadequately modelled, or were not yet taken into account atthis stage. The assumption was made that an uncertaintyin the induced drag coefficient for formation followersexists, and an adaptation algorithm was developed toestimate the drag coefficient. A control algorithm wasdesigned to achieve formation flight within an arbitrarilysmall bounded tracking error.

2. AERODYNAMIC INTERACTIONS AND TRIMSETTINGS

2.1 Induced Forces and Moments

A model for the induced forces and moments experiencedby a trailing aircraft in formation flight was derived byBizinos and Redelinghuys (2012). In this paper, the modelis analysed to determine the required trim settings anddynamic response of the trailing aircraft as a functionof lateral and vertical separation relative to the leadingaircraft. The standard aerodynamic equations expressed inwind axes are expanded as shown in (1) with the formationflight effects included, denoted by subscript f ’.

CD = CDt,c + CDα (α− αt) + CDM V−V tV s + CDf ′ 〈η, ζ〉CL = CLt,c + CLα (α− αt) + CLM V−V tV s + . . .

CLα̇ α̇+c̄

2V tCLqq + CLδe (δe − δet) + CLf ′ 〈η, ζ〉

CY = CYββ +b

2V tCYpp+

b

2V tCYrr + CYδa δa + . . .

CYδr δr + CY f ′ 〈η, ζ〉Cl = Clββ +

b

2V tClpp+

b

2V tClrr + Clδa δa + . . .

Clδr δr + Clf ′ 〈η, ζ〉Cm = Cm0 + Cmαα+ Cmα̇ α̇+ CmM

V−V tV s

+ . . .

c̄

2V tCmqq + Cmδe δe + Cmf ′ 〈η, ζ〉

Cn = Cnββ +b

2V tCnpp+

b

2V tCnrr + Cnδa δa + . . .

Cnδr δr + Cnf ′ 〈η, ζ〉(1)

Where, CAB ≡ ∂CA∂BThe aerodynamic and physical parameters for the Boeing-747 were extracted from Heffley and Jewell (1972) for thecruise flight condition (M = 0.8, 40000 ft altitude). Theinteraction coefficients were determined for the Boeing-747using an approximate model which assumed a symmetricallift distribution (leading aircraft) with no sweep or dihed-eral. The approximations are assumed to be acceptable in

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developing the control system. Longitudinal separation isfixed at a distance of 10 wingspans. Longitudinal separa-tion variation around this point will have little effect asthe vortices stay nearly constant in strength and diameterfor small longitudinal changes.

The induced forces and moments are highly non-linear,which creates an interesting control and trim problem.The functions that describe these forces and moments arerepeated in (2) and (3) for the convenience of the reader.

The incremental drag and rolling moment coefficients arereproduced in Fig. 1. These plots clearly indicate that theoptimum position for drag reduction corresponds to thelargest rolling moments. The plots show results which aresignificant for the trim calculations shown later.

0 0.5 1 1.5 2

−0.03

−0.0075

0.015

0.0375

0.06

Lateral Separation η

CDf′

(a) CDf ′

0 0.5 1 1.5 2

−0.05

−0.0275

−0.005

0.0175

0.04

Lateral Separation η

Clf

′

(b) Clf ′

Fig. 1. Induced drag and rolling moment coefficient asfunctions of lateral separation η and vertical separa-tion ζ = 0

CDf ′ =2CL,kCL,jπ3AR σjk

CLf ′ =−clαCL,j

2π2AR σjk

CY f ′ =SfS

2CL,jπARζf

σjkf

Clf ′ =clαCL,j2π2AR τjk

Cmf ′ = CLf ′ (h− h0)− VTCLωhf ′(1− d�dα

)CLωhf ′ =

−2a1CL,jπ3ARηh

σjkωh

Cnf ′ =2CL,kCL,jπ3AR τjk − Vf

2CL,jπARζf

σjkf

(2)

σjk = ln

∣∣∣∣ ((η−(π/4))2+ζ2+µ2)((η+(π/4))2+ζ2+µ2)(η2+ζ2+µ2)2 ∣∣∣∣σjkf = ln

∣∣∣ (η−π/8)2+(ζ+ζv)2+µ2(η−π/8)2+(ζ+ζv−ζfπ/8)2+µ2

∣∣∣ . . .− ln

∣∣∣ (η+π/8)2+(ζ+ζv)2+µ2(η+π/8)2+(ζ+ζv−ζfπ/8)2+µ2

∣∣∣· · ·

(3)

τjk = −2√ζ2 + µ2

[tan−1

(η−π/4√ζ2+µ2

). . .

+ tan−1(

η+π/4√ζ2+µ2

)− 2 tan−1

(η√ζ2+µ2

)]. . .

−η ln∣∣∣∣ ((η−π/4)2+ζ2+µ2)((η+π/4)2+ζ2+µ2)(η2+ζ2+µ2)2 ∣∣∣∣ . . .

−π8 ln∣∣∣ (η+π/4)2+ζ2+µ2

(η−π/4)2+ζ2+µ2

∣∣∣σjkωh = ln

∣∣∣∣ (ζ2+(η−π8 −π8 ηh)2+µ2)(ζ2+(η+π8 +π8 ηh)2+µ2)(ζ2+(η−π8 +π8 ηh)2+µ2)(ζ2+(η+π8 −π8 ηh)2+µ2)∣∣∣∣

The reduction in induced drag through formation flightis achieved by taking advantage of the pair of trailingvortices generated as the lead aircraft produces lift. Whenpositioned outboard of this pair of trailing vortices, avarying upwash is induced along the lifting surfaces of thefollowing aircraft. This causes an increased effective angleof attack which both increases and rotates the resultingaerodynamic forces on the wing and empennage. Theresulting increase in lift and reduction of induced dragallow the aircraft to be re-trimmed for improved rangeperformance. At the optimal relative positioning however,the lateral moments and side force experienced require sig-nificant control surface deflections for trim which reducesobtainable formation flight benefit (Kless et al. 2012).

The strength of the induced flow is a function of thecirculation strength of the trailing vortices and the relativeseparation of these to position of the trailing aircraft.Depending on the methodology used, the region of op-timum lateral separation is predicted anywhere betweenb < η ≤ πb/4, with ζ = 0, and with all methods predictinga very small region of peak drag benefit on the order of10% of the span (Bower et al. 2009, Blake and Gingras2004). In a two aircraft formation, this optimum positioncorresponds to the position of highest induced rolling mo-ments which can be high enough to saturate the ailerons ofthe trailing aircraft. In order to take full advantage of theformation flight effects, the trailing aircraft must thereforebe controlled with a high degree of position accuracy whilecoping with large rolling moments.

2.2 Trim Actuator Settings

The required trim actuator settings can be calculatedover a range of vertical and lateral separations using theaerodynamic equations in (1), along with basic thrust andgravity models.

First, the trim settings for the conventional airliner inisolated flight is calculated. This is done under the assump-tions that trim angle of attack αt and thus the trim pitchangle θt are small, and that the lift is much larger thanthe drag. Equation (4) then shows how the trim angle ofattack and elevator settings are solved. Aileron and ruddersettings will be kept at 0◦ deflection.

[αtδet

]=

[CLα CLδeCmα Cmδe

]−1 [mgq̄tS− CL0−Cm0

](4)

The trim thrust requirement can then be calculated using(5).

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0 0.5 1 1.5 2−8

−5.5

−3

−0.5

2

Lateral Separation η

ElevatorDeflection(deg)

(a) Elevator Deflection

0 0.5 1 1.5 2−1

0.125

1.25

2.375

3.5

Lateral Separation η

Rudder

Deflection(deg)

(b) Rudder Deflection

0 0.5 1 1.5 2−0.5

0.25

1

1.75

2.5

Lateral Separation η

Sideslip(deg)

(c) Sideslip Angle

0 0.5 1 1.5 22

4

6

8

10

Lateral Separation η

Angle

ofAttack

(deg)

(d) Angle of Attack

Fig. 2. Trim settings and states for ζ = 0 (Full requiredsettings for formation flight)

Tt = q̄tSCDt cosαt − q̄tSCLt sinαt +mg sin θt (5)

Next, (1) is solved with all forces and moments in trim(i.e. lift force cancels gravity force, side-force and momentsall equal 0). The equations for formation flight effect’scontribution to the trim settings and states is solvedsimultaneously. Note that the formation and conventionalsettings can be superimposed as (1) is a linearised model.

Equation (5) is then used to find the total throttle settingusing full trim settings and states. The full, non-linearequation can be used here as it is given in (5).

Fig. 2 shows the resulting required trim settings and statesfor a vertical separation of zero. The required ailerontrim setting is shown with the required trim thrust inFig. 3. Note that the required aileron trim settings varybetween large negative and positive deflections. Such largeaileron deflections do not physically make sense. However,it should be noted that a linear model is used, and this istherefore simply an indication to which extent the aileronsare insufficient in certain regions of formation flight. Thephysical implications are that the rolling moment willoverpower the ailerons and the aircraft will be forced intoa roll and out of formation flight.

2.3 Trim Ranges

Next, it is necessary to analyse the trim throttle settingfunction as shown on Fig. 3. By comparing this to therequired setting for conventional flight, a range of fuel-consumption benefit is identified with the assumption thata lower throttle setting equates to lower fuel-consumption.It is interesting to note that if the follower aircraft isdirectly behind the leader aircraft, the formation flightinteractions adversely affect overall drag reduction. Thisis explained by the vortices pushing down on both wingsof the follower aircraft, effectively reducing lift withoutinducing moments.

The airliner is not trimmable at the optimal fuel con-sumption location due to the fact that the induced rollingmoment requires aileron trim deflections which are outsidethe physical deflection limits.

0 0.25 0.5 0.75 1 1.25 1.50

150

300

450

600

Lateral Separation, η

Throttle

Setting(kN)

Full Formation ThrottleThrottle Conventional

0 0.25 0.5 0.75 1 1.25 1.5−200

−100

0

100

200

Lateral Separation, η

AileronDeflection(degrees)

Fig. 3. Comparison of trim regions for ζ = 0, showing trimaileron and throttle settings.

Inspecting Fig. 3, two valid trim regions are found however.The first is a “sandwich” region, which is a narrow regionsandwiched between two untrimmable regions. The secondis an “outer” region, which only has an untrimmable regionon its inside. The rectangle for sandwich region showsa range between maximum aileron settings of −25◦ to25◦. The rectangle for the outer region indicates an areabetween a maximum required aileron deflection of −25◦,and a chosen trim of −10◦ aileron deflection.Each region comes with its own advantages and disadvan-tages. The sandwich region has a better fuel-consumptionreduction due to a lower induced drag at trim. Further-more, it is possible to have a 0◦ aileron deflection, whichwill avoid unmodelled drag effects on the ailerons. Thesmall width of the sandwich region may be impractical forthe trailing aircraft to track in real atmospheric conditions.In the sandwich region, the core of the trailing vortex willbe impinging on the wing of the trailing aircraft, whichwill induce large angles of attack. This may invalidate theassumptions of the aerodynamic model. Furthermore, theinduced rolling moments at each extreme of this region isin the direction that would naturally roll the aircraft fur-ther into the untrimmable region, worsening the problem.

The outer region is a safer option however, as the aircraftdoes not have to stay in such a narrow following regionas for the sandwich region. Furthermore, the inducedrolling moment near the outer region is in a direction thatwill tend to naturally push the aircraft away from theuntrimmable region, giving it the potential for inherentfault recovery. However, the outer region will have a non-zero rolling moment, which results in the need for a non-zero aileron deflection which will introduce unmodelleddrag. Lastly, it may be a simpler task of initiating forma-tion flight for the outer trim region. It is thus clear thatthis is a risk versus benefit consideration.

Fig. 4 compares the required aileron and thrust settingsover a range of vertical separations for both trim regions.These plots indicate that the best fuel consumption gainfor both regions are at zero vertical separation. Further-more, it is evident that the trim is more sensitive to

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−1 −0.5 0 0.5 10

50

100

150

200

Vertical Separation ζ

Throttle

Setting(kN)

Formation − SandwichFormation − OuterConventional

(a) Throttle setting comparison

−1 −0.5 0 0.5 1−20

−10

0

10

20

30

40

Vertical Separation ζ

AileronDeflection(deg)

Sandwich RegionOuter Region

(b) Aileron setting comparison

Fig. 4. Comparison of sandwich (η = 0.713) and outer (η =1.33) region trim settings over vertical displacement

lateral separation changes than it is to vertical separationchanges.

3. LINEAR DYNAMICS ANALYSIS

Following the trim analysis, the next step is to derivethe linearised dynamic model of the aircraft about eachcalculated trim as a function of lateral and vertical dis-placement. An eigenvalue analysis of the linearised modelis then performed to observe how the dynamic response ofthe aircraft changes over the range of lateral and verticaldisplacements.

3.1 State Space Representation

The conventional dynamic model of an aircraft in isolatedflight is traditionally separated into sets of longitudinallyand laterally decoupled states. Any coupling present be-tween the states is insignificant enough to be neglectedand can be treated as a disturbance during the design ofthe flight control systems. Equations (6) shows the formatof the conventional linearised system.

ẋlong = Alongxlong +Blongulongẋlat = Alatxlat +Blatulat

(6)

For formation flight, the state vectors are expanded in (7)to include formation flight interaction states η, ζ and ∆ψ,which are the lateral separation, vertical separation, andthe difference in the heading angle between the formationleader and follower respectively. ∆ψ is required for thedescription of the formation flight differential equations in(8).

xlong =[V α q θ ζ

]Txlat = [β p r φ η ∆ψ]

T (7)

ζ̇ = V tb sin (θ − α) ≈V tb (θ − α)

η̇ = V tb sin (∆ψ) ≈V tb (∆ψ)

ψ̇ = q sinφ sec θ + r cosφ sec θ

(8)

Following initial derivations of the state space represen-tation, it was concluded that significant coupling existsbetween the lateral and longitudinal subsystems, and afull model was derived. Equation (9) shows the format ofthis.

ẋfull =[

Along Along-latAlat-long Alat

]xfull + Bfullufull (9)

Equations (10) - (13) show the sub-matrices. Large termsare indicated as partial derivatives and expanded in (14)- (16). Note that Bfull was omitted as its derivation isconsidered trivial and irrelevant to this particular analysis.

Along =

∂v̇∂v

∂v̇∂α 0

∂v̇∂θ

∂v̇∂ζ

∂α̇∂v

∂α̇∂α

∂α̇∂q

∂α̇∂θ

∂α̇∂ζ

∂q̇∂v

∂q̇∂α

∂q̇∂q 0

∂q̇∂ζ

0 0 1 0 0

0 −V tb 0V tb 0

(10)

Alat =

∂β̇∂β 0 −1

∂β̇∂φ

∂β̇∂η 0

∂ṗ∂β

∂ṗ∂p

∂ṗ∂r 0

∂ṗ∂η 0

∂ṙ∂β

∂ṙ∂p

∂ṙ∂r 0

∂ṙ∂η 0

0 0 1 tan θt 0 0

0 0 0 0 0 V tb0 0 sec θt 0 0 0

(11)

Along-lat =

0 0 0 0 ∂v̇∂η 0

0 0 0 0 ∂α̇∂η 0

0 0 0 0 ∂q̇∂η 0

0 0 0 0 0 0

0 0 0 0 0 0

(12)

Alat-long =

0 0 0 0 ∂β̇∂ζ0 0 0 0 ∂ṗ∂ζ0 0 0 0 ∂ṙ∂ζ0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

(13)

Longitudinal Elements∂v̇∂v = −

ρV tS(CDt)m −

q̄tSm

CDMVs

∂v̇∂α = −

Tm sin (αt)−

q̄tSm CDα + g

∂v̇∂θ = −g∂v̇∂ζ =

q̄tSCL,jmπ2AR

(2CL,kπ +

clααt2

)∂σj,k∂ζ

∣∣∣t

· · ·

(14)

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12192

∂α̇∂v = −

ρSCLtm −

qtSCLMV tmVs

∂α̇∂α = −

q̄tSCLαmV t

∂α̇∂q = 1−

q̄tSc̄CLQ

2V2

tm

∂α̇∂θ = −

g

V tsin θt

∂α̇∂ζ =

q̄tSCL,j

mV tπ2AR

(clα2 −

2CL,kαtπ

)∂σj,k∂ζ

∣∣∣t

∂q̇∂v =

q̄tSc̄CmMIyyVs

∂q̇∂α =

q̄tSc̄Iyy

Cmα∂q̇∂q =

q̄tSc̄2

2IyyV tCmQ

∂q̇∂ζ =

q̄tSc̄CL,jIyyπ2AR

[−clα

2

(∂σj,k∂ζ

∣∣∣t

)(h− h0) + · · ·

V t(1− d�dα

) (2a1πηh

∂σjkwh∂ζ

∣∣∣t

)]Lateral Elements∂β̇∂β =

q̄tS

mV t

(CYβ + CDt − CLtαt

)∂β̇∂p =

q̄tSb

2mV2

t

CYP∂β̇∂r =

q̄tS

mV tCYR − 1

∂β̇∂φ =

g

V tcos θt

∂β̇∂η =

2Sf q̄tCL,j

V tmπARζf

(∂σjkf∂η

∣∣∣t

)∂ṗ∂β =

q̄tSbIxx

Clβ∂ṗ∂p =

q̄tSb2

2IxxV tClP

∂ṗ∂r =

q̄tSb2

2IxxV tClR

∂ṗ∂η =

q̄tSbclαCL,j2Ixxπ2AR

(δτjkδη

∣∣∣t

)∂ṙ∂β =

q̄tSbIzz

Cnβ∂ṙ∂p =

q̄tSb2

2V tIzzCnP

∂ṙ∂r =

q̄tSb2

2V tIzzCnR

∂ṙ∂η =

2q̄tSbCL,jIzzπAR

(CL,kπ2

∂τjk∂η

∣∣∣t− V̄fζf

∂σjkf∂η

∣∣∣t

)

(15)

Coupling Elements∂v̇∂η =

q̄tSCL,jmπ2AR

(2CL,kπ +

clααt2

)∂σj,k∂η

∣∣∣t

∂α̇∂η =

q̄tSCL,j

mV tπ2AR

(clα2 −

2CL,kαtπ

)∂σj,k∂η

∣∣∣t

∂q̇∂η =

q̄tSc̄CL,jIyyπ2AR

[−clα

2

(∂σj,k∂η

∣∣∣t

)(h− h0) + · · ·

V t(1− d�dα

) (2a1πηh

∂σjkwh∂η

∣∣∣t

)]∂β̇∂ζ =

2Sf q̄tCL,j

V tmπARζf

(∂σjkf∂ζ

∣∣∣t

)∂ṗ∂ζ =

q̄tSbclαCL,j2Ixxπ2AR

(δτjkδζ

∣∣∣t

)∂ṙ∂ζ =

2q̄tSbCL,jIzzπAR

(CL,kπ2

∂τjk∂ζ

∣∣∣t− V̄fζf

∂σjkf∂ζ

∣∣∣t

)(16)

The partial derivatives with respect to η and ζ in (14)- (16) are written in terms of influence factors τjk, σjk,σjkf and σjkωh . These influence factors are described inBizinos and Redelinghuys. Their η and ζ derivatives arenot explicitly provided here for the sake of saving space,but can easily be determined by using a package such asMatlab’s symbolic toolbox.

This state space representation is partially verified bysetting the vertical and horizontal separation to verylarge values, with the hypothesis that this will simulate

conventional, isolated flight. The resulting poles are shownin Table 1. An eigenvector analysis proved that the polesare correct according to the mode that they describe.

Table 1: Conventional flight polesPhugoid mode −0.0019± 0.0706i

Short-period mode −0.3259± 0.9009iDutch roll mode −0.0197± 0.906i

Roll mode −0.6042Spiral mode −0.0109

The conventional flight poles in Table 1 were confirmed tobe in the correct order of magnitude through comparisonswith external sources including Caughy (2011) and Heffleyand Jewell (1972).

A controllability analysis revealed that the system iscontrollable over its defined operating regions using thederived state space model. Observability has not beenchecked yet, since the configuration of the sensor suiterequired for formation flight has not been investigated,and the set of available sensor measurements from whichthe formation flight states will be estimated has not beendefined.

3.2 Eigenvalue Analysis

The state space representation model was then used to findpoles for both the discussed trim regions. The result is rootloci with respect to lateral and vertical separations. Fig.5 shows the resulting lateral and vertical root loci, a plotof the movement of the poles or eigenvalues, for both trimregions. An analysis of this, accompanied by an eigenvectoranalysis indicated that the conventional modes are lost,and instead, new modes overpowering in roll angle androll rate are found. It is also clear that multiple formationflight modes are unstable – where all the conventionalmodes were stable. These findings were confirmed by alinear simulation, which indicated strong rolling behaviourand complete barrel rolls.

Considering that the root loci for lateral and vertical sep-aration on Fig. 5 are plotted over comparable separationvariations for each trim region individually, it is evidentthat the dynamics change to a much larger extent for ver-tical separation variation compared to lateral separationvariation. In the outer region, the dynamics stay nearlyconstant for lateral separation variation, as can be seen onFig. 5b. Note that the root loci are plotted for lateral andvertical separation variations with ranges of 0.1 wingspans,centred across the outer and sandwich trim regions so thatthey are comparable.

4. CONCLUSION

It was found that there are challenges with trimmingthe follower aircraft at certain relative positions in theleader’s wake vortices. Specifically, it is not possible forthe representative airliner to counter the large inducedrolling moments at these following positions, including atthe optimum region.

Two trimmable regions were found however: the sand-wich region, which gives the greatest fuel consumptionreduction benefit, but with more risk and practical chal-lenges, and the outer region, which is less risky and more

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Fig. 5. Separation loci around trim regions. For vertical separation variation, moving from dark to light indicatesupward moving trim changes. For lateral separation variation, moving from dark to light indicates inward movingtrim changes. Conventional flight poles are marked with red crosses, and relate to formation flight at infiniteseparation distances.

practically viable, but with less benefit. Trim and lineardynamics analyses revealed interesting equilibrium anddynamics behaviours for the two regions.

In the sandwich region, the trim changes significantly forlateral separation changes, but less so for vertical separa-tion changes. In the outer region however, the trim is lesssensitive to lateral and vertical separation changes. Finally,the dynamics for both regions change significantly forvertical separation variation, but not for lateral separationvariation.

Furthermore, the extreme non-linearity of the inducedforces and moments present challenges with dynamicschanging as a function of spatial separation. These factorsindicate that the control system will need to be robustto large changes in the systems characteristics and shouldbe able to disengage the aircraft from formation withoutendangering the aircraft.

The necessary basis for the design of a formation flightcontrol system has been formed. The next step will be toevaluate a conventional flight controller’s performance ina formation flight scenario. Following this, a specialisedformation flight controller can be designed.

ACKNOWLEDGEMENTS

The authors would like to thank Andy Williams, for actingas a technical contact point with Airbus.

REFERENCES

Bizinos, N., Redelinghuys, C. (2012) Tentative studyof passenger comfort during formation flight within

atmospheric turbulence. AIAA, Journal of Aircraft. doi:10.2514/1.C032018

Blake, W., Gingras D.R. (2004) Comparison of predictedand measured formation flight interference effects. Jour-nal of Aircraft, volume 41, pp. 201–207.

Bower, G., Flanzer T., Kroo, I. (2009) Formation geome-tries and route optimization for commercial formationflight. AIAA, paper 2009-3615.

Brodecki, M., Kamesh, S., Qi-Ping, C. (2013a) Formationflight control system for in-flight sweet spot estimation.AIAA, Aerospace Sciences Meeting including the NewHorizons Forum and Aerospace Exposition, AIAA 2013-1037.

Brodecki, M., Kamesh, S., Qi-Ping, C. (2013b) Emergentbehavior of multi-vehicle formations using extremumseeking. AIAA, Aerospace Sciences Meeting includingthe New Horizons Forum and Aerospace Exposition,AIAA 2013-1033.

Caughy, D.A. (2011) Introduction to Aircraft Stabilityand Control Course Notes for M&AE 5070 SibleySchool of Mechanical & Aerospace Engineering, CornellUniversity.

Heffley, R.K., Jewell, W.F. (1972) Aircraft handlingqualities data. NASA.

Kless, J., Aftosmis, M.J., Ning, S.A., Nemec, M. (2012)Inviscid analysis of extended formation flight. SeventhInternational Conference on Computational Fluid Dy-namics

Zou, Y., Pagilla, P.R., Ratliff, R.T. (2009) Distributedformation flight control using constraint forces. Journalof Guidance, Control and Dynamics, volume 32, no. 1.doi:10.2514/1.36826

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