-
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-
Preprints of the 19th World CongressThe International Federation
of Automatic ControlCape Town, South Africa. August 24-29, 2014
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
19th IFAC World CongressCape Town, South Africa. August 24-29,
2014
12189
-
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).
19th IFAC World CongressCape Town, South Africa. August 24-29,
2014
12190
-
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
19th IFAC World CongressCape Town, South Africa. August 24-29,
2014
12191
-
−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)
19th IFAC World CongressCape Town, South Africa. August 24-29,
2014
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
19th IFAC World CongressCape Town, South Africa. August 24-29,
2014
12193
-
−2 −1 0 1 2
−2
−1
0
1
2
Real
Imag
inar
y
(a) Lateral sandwich region locus
−2 −1 0 1 2
−2
−1
0
1
2
Real
Imag
inar
y
(b) Lateral outer region locus
−2 −1 0 1 2
−2
−1
0
1
2
Real
Imag
inar
y
(c) Vertical sandwich region locus
−2 −1 0 1 2
−2
−1
0
1
2
Real
Imag
inar
y
(d) Vertical outer region locus
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
19th IFAC World CongressCape Town, South Africa. August 24-29,
2014
12194