Flying-Wing Aircraft Control Allocation

Chao Ma1 and Lixin Wang2

Beijing University of Aeronautics and Astronautics, Beijing, China, 100083

Flying-wing configuration aircraft has multi-control-effectors redundantly. Especially theinnovative drag rudders are implemented on it. So the control allocation is strongly nonlinear andmulti-axes coupling. The features and applicability in different flight tasks of several typical controlallocation methods were summarized. The multi-objectives optimization method based on nonlinearprogramming was utilized because of the new characteristics of the control allocation for the flying-wing configuration aircraft. The differences of control allocation results under different objectiveswere analyzed. And the attitude-tracking flight control system based on nonlinear dynamic inversetheory was designed, and the digital simulation using a six degree-of-freedom nonlinear model ofcertain flying-wing configuration aircraft was conducted, which demonstrate the applicability forflying-wing configuration aircraft. The results also indicated that the impacts of different optimizingobjectives on flight states were remarkable and different.

NomenclatureAMT = all-moving tipB = control effectiveness matrix ( n m× )

b = wing spanc = mean aerodynamic chordELEV = elevonH = flight heightL = total rolling momentM = total pitching momentMa = Mach number

N = total yawing momentPF = pitching flapq = dynamic pressure

S = wing areaSSD = slot-spoiler-deflectorV = flight velocityX = flight state parametersα = attack angle

β = sideslip angle

φ = rotation angle

θ = pitch angleψ = yawing angle

xω = rotation angle rate

yω = pitch angle rate

zω = yawing angle rate

1 PhD Student, School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics,Beijing, China, E-mail: machao@ase.buaa.edu.cn.2 Professor, School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics,Beijing, China, E-mail: bhu_wlx@tom.com

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-55

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

mC∆ = increment of pitching moment coefficient

lC∆ = increment of rolling moment coefficient

nC∆ = increment of yawing moment coefficient

DC∆ = increment of drag coefficient

LC∆ = increment of lift coefficient

dM = desired moments around three axes

aδ = aileron deflection angle

eδ = elevator deflection angle

rδ = rudder deflection angle

δ = control-surfaces deflection angle vector (1 n× )

desδ = distribution of control surfaces (1 n× )

maxiδ = upper limit of position of the i th control surface

miniδ = lower limit of position of the i th control surfaces

maxiρ = upper limit of deflection rate of the i th control surfaces

miniρ = lower limit of deflection rate of the i th control surfaces

I. Introductionodern high performance combat aircraft usually has multi-control-effectors. So the control allocation becomes

one of the most important parts of flight control system. Flying-wing is the ideal configuration for modern combat

aircraft because of the satisfied aerodynamics and stealth characteristics. The flying-wing configuration aircraft

cancels vertical tail and rudders which are implemented on the conventional aircraft. In stead of them, several

innovative drag rudders are designed for yawing control. So the control allocation of flying-wing configuration

aircraft is different from the conventional aircraft.

Most research focused on the accuracy and rapidity of the control allocation method, and which assumed the

control surface’s aerodynamic model globally or locally linear [1~2] . Aiming at the special problems of flying-wing

configuration aircraft, the classical nonlinear optimization method was used for control allocation in this paper, not

only to solve the two basic problems of strongly nonlinear and multi-axes coupling characteristics, but also to

indicate the principles how all the control surfaces were managed during control allocation. Based on the six-

freedom nonlinear aerodynamics and motion models of a typical flying-wing configuration aircraft, the accurate

digital simulation was conducted. The applicability of the control allocation method and the impact to flight states

were discussed in this paper.

M

II. Control allocation problems

A. Concepts and features

Conventional aircraft has three types of control surfaces, including elevator, aileron and rudder, each of which

produces control moment around the corresponding axis. So the required deflection angles are only defined if the

control moments are attainable. The control problem of conventional aircraft obeys the following relations:

0( )

( )

( )

q e

p r a r

p r a r

M qSc q eM M M M ML qSb p r a rL L L L LN qSb p r a rN N N N N

α α δ

β δ δ

β δ δ

α α δβ δ δβ δ δ

∆

= + + + += + + + += + + + +

& &

(1)

Where Mα , pL , Nβ et al are aerodynamic derivatives, ande

Mδ ,a

Lδ ,r

Nδ et al are control derivatives. 0M

is the body pitching moment at basic flight condition. α∆ is the increment of attack angle.

Each dM has just single one distribution [ , , ]a e rδ δ δ . And each conventional control surface is of definite

function of control, and of little coupling effect.

Flying-wing configuration aircraft has more than three innovative control effectors. First, the control allocation

solves the problem that the control effectors is more than the desired control axes, and the problem that the

innovative control effectors are multi-axes coupling. The desired control moments required by control law should be

optimally allocated to all the control effectors.

Secondly, the innovative control effectors are implemented on the flying-wing configuration aircraft, especially

including drag rudders for yawing control. For example, the flying-wing configuration combat aircraft uses the slot-

spoiler-deflector (SSD). The wind tunnel test results of SSD were shown in Fig.1. The deflections of SSD at

different attack angles all would additively yield the large amount of lift and drag effect, which could not be ignored

during control allocation.

Fig. 1 Lift and Drag Coefficients Increment of SSD deflections.

Thirdly, flying-wing configuration aircraft does not all use conventional control surfaces which are of linear

control characteristic. The innovative control surfaces used are strongly nonlinear. So the conventional methods for

control allocation no longer apply. Fig.2 showed that the all-moving tip (AMT)’s control moments around three axes

at Ma=0.6 and α =5°. Not only the yawing moment was little nonlinear, but also the coupled rolling and pitching

moments were strongly nonlinear.

Fig. 2 Control Moment Coefficients of AMT around Three Axes.

Finally, drag rudders have the longer range of deflection. For example, AMT and SSD deflect from 0° to 60°. So

the dynamic characteristics of control effectors can not be ignored during control allocation.

B. Mathematical descriptions

The control allocation problem of flying-wing configuration aircraft is to solve the nonlinear equations with

nonlinear constrains, which is given by:

{ }maxmin( ) ( ( )) ( ), ( ) ( ) ( )d i iit g t t t t tMδ δ δ δ δ= ≤ ≤ (2)

0 10 20 30 40 50 60-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

Ma=0.6,α=5o

Cont

rol

mome

nts

coef

fici

ents

δ / ( o)

Yawi ngRol l i ngPi t chi ng

-100

1020

0

20

40

60-0.02

0

0.02

0.04

0.06

0.08

-100

1020

020

4060

-0.1

-0.05

0

0.05

0.1

LC∆ DC∆

/ ( )SSDδ o / ( )SSDδ o/ ( )α o / ( )α o

where ( )g δ is the aerodynamic model of the control surfaces. δ is the control vector, as shown in formula (3):

1 2( ) [ , ,..., ,..., ]i ntδ δ δ δ δ= � n >3� (3)

Position and rate constrains can be approximately calculated together by upper position limitmax

( )i tδ and lower

position limitmin

( )i tδ in single sampling period, as shown in formulas (4):

max maxmax

minmin max

( ) , ( ) )max(

( ) min( , ( ) )

ii ii

i ii i

t t T T

t t T T

δ ρδ δ

δ ρδ δ

= − +

= − − (4)

Where T is sampling period of the simulation, and only rate limit was concerned, as shown in formula (5):

min max( 1 ... )i ii i nρ ρδ< < =& (5)

Currently, dynamic nonlinear control allocation became a challenging research aspect [3] .

C. Attainable moments set

The set of all attainable control moment vectors yielded by control surfaces deflection at a certain flight state is

the attainable moment set( AMS ), which decided by every control surface’s aerodynamic characteristics and

position limit. When AMS is larger, the aircraft has stronger control capability, and the maneuver characteristics

are better.

In the vector space 3� of moment coefficients around three axes, because conventional aircraft always uses

only three control surfaces, the shape of AMS is hexahedron. But the flying-wing configuration aircraft’s AMS is

the polyhedron whose planes are more than 6, or the convex body considering the nonlinear control characteristic.

Comparison of AMS between a certain low-aspect-ratio flying-wing configuration aircraft and F-4 aircraft was

shown in Fig. 3. Every l mC C∆ −∆ section of the former is the regular rectangle, but the latter’s is anomalous

polygon. It is demonstrated that the longitudinal and lateral control of flying-wing configuration aircraft coupled.

The largest pitching and rolling control moments can not be realized simultaneously, because both of them depend

on the elevons. Fig. 3(a) also showed the distinct drag coefficient increment of flying-wing configuration aircraft,

which was yielded by deflections of drag rudders. However, the conventional control surfaces would not cause

additive drag basically.

(a)Certain Flying-Wing Configuration Aircraft (b)Aircraft F-4 H=4572m�Ma=0.6 H=10668m�Ma=0.6

Fig. 3 Comparison of AMS between two types of aircraft.

D. Solutions Analysis

Because the number of control surfaces is bigger than desired control directions, so the nonlinear equations

which described the control allocation problem are underdetermined. Fig.4 showed the different situations about the

solutions of control allocation.

Fig. 4 Solutions of control allocation problem.

When Md lays outside AMS , the magnitude and direction should be modified depending on the approximate

objective. Then the problem can switch to the situation that Md is inside AMS .

ApproximateObjectives

……

……Position and RateLimits

AMS

Md ∈ AMS

Num. of controlsurfaces > 3

Infinite AccurateSolutions

Min. Energy

Min. Drag

DirectionPreservation

Pitch Prioritization

Best Set ofApproximate

Solutions

OptimizationSolution

Md ∉ AMS Infinite ApproximateSolutions

nC∆

lC∆mC∆

nC∆

lC∆mC∆

DC∆

III. Methods of control allocation

A. Generalized inverses

Generalized inverse (GI) is the simplest method of control allocation. But it is based on the linear situation. The

minimum two-norm of control vector can be obtained from the M-P inverse of the control matrix, which is given by:

21

2min[ ]T

dTB MBBδ δ−= ⇒ (6)

Considering the privileges of the different control axes, the weighted GI method was advanced [4] . This method

can make the higher efficiency control surfaces avoid getting into saturation early. It is given by:

21 12

( ) [ ( ) ] minT Tu N BN BN BN N u− −= ⇒ (7)

Where N is the weight matrix.

GI method was first developed, because it could be realized easily. But the solutions based on these algorithms

can’t achieve all of the AMS , and the truncation error may happen during calculation because of the position and

rate limits of control surfaces. The redistribute pseudo-inverse method was advanced to reduce the unexpected risk.

This improved method proceeds by allocating the residual Md to the unsaturated control surfaces until all them

becoming saturated [5] . But these algorithms based on GI method still never enumerate all of the AMS .

B. Daisy-chaining

Daisy-chaining (DC) method [6] divides all the control effectors into several groups depending on the basic

function of each control surfaces, including a main and several minor control surface combinations (or conventional

and innovative). The next combination starts working in turn when the previous group gets saturated. So the

conventional control surfaces are furthest utilized, and the aircraft avoid deflecting unconventional control effectors

all the time. This method is represented as:

11 2

21 2( ) ( ) dM Mg gδδ δδ

= → (8)

Where iB and iδ are control effectiveness matrix and control vector of i th divided combination respectively.

Multistage GI is also one of the DC methods, which is given by:

1

1

[ ] ( )i

i T T T ji i i d

j

B B B Mδ δ−

=

= −∑ (9)

In addition, other algorithm can be used for the sub-allocation of the control effectors.

C. Direct allocation

The direct allocation [7] (DA) presented by Durham is a geometric space method based on AMS , which could

find the intersection of the radial along Md direction with the boundary of AMS by searching it. This method is

given by:

ˆ /d dM M M= (10)

( ){ }ˆdM M Mδ δ λ= ∂= I (11)

Where M∂ is the boundary of the AMS , and λ is the scaling coefficient of the desired control moment

vector. If 1λ ≥ , Md is attainable. Otherwise Md is unable to achieve, then the AMS should reduced

proportionately until just intersecting on Md . So the DA method can ensure the desired direction of control

moment vector.

Because the DA method could enumerate all the elements of the AMS , and has the distinct physical

implications, many algorithms based on it were developed. Durham also presented facet-search algorithm [8] and

bisecting edge searching algorithm [9] . Li presented the adjacent facets searching algorithm [10] . And the DA method

can be realized by linear programming algorithm [11 12]− .

D. Optimization based method

Because the innovative configuration and control surface appeared, the nonlinear problem became more and

more notable. Another problem caused by Md laying outside the AMS also should be solved. So the Optimization

based method developed rapidly.

Optimization based method has the most distinct characteristic that the accurate realization of the control

moments around three axes is regarded as the constraint, and can bring the requirements corresponding to the flight

task into the optimization objectives. The generalized method could be described by:

( )min , ( )f X tδ (12)

maxmin. . ( ( )) ( ), ( ) ( ) ( )d i iiS t g t t t t tMδ δ δ δ= ≤ ≤ (13)

Formulas (13) showed the feasible region of the optimization problem. If the feasible solution doesn’t exist, like

the situation Md lays outside the AMS , the equation constraint should be released. Meanwhile the unattainable

control moments should be controlled by the optimization objectives.

Optimization based method can realize multiple and variable objectives, such as the minimum drag during the

cruise phase [13] , the maximum lift during take-off phase, and the minimum radar reflecting cross section during the

maneuver penetration [14] . Nelson also presented the method of prioritizing the pitching-moment requirement to

ensure that the maximum amount of pitch could be attained during any maneuver [15] , which solved the problem that

the control moments lied outside the AMS , and the pilot preferred the method of pitch prioritization because of

avoiding the unstable feeling.

E. Dynamic allocation

In the actual flight control system, the desired control moments required by the control law are time-varying. So

Härkegård presented a generalized method for control allocation which adapted to dynamic situation [16] . And Liu

applied it to the control allocation of a certain tailless aircraft [17] . This method is described by:

( ) ( )22

2 2

1 2min ( ) ( ) ( ) ( )pJ W t t W t t Tδ δ δ δ= − + − − (14)

Where ( )t Tδ − is the distribution at the last sampling period, and ( )p tδ is the desired stationary distribution of

control effectors among the actuators and determines the actuator positions at trimmed flight [16] . Where 1W and

2W are the weighting coefficients, which satisfy the equation 2 2

1 2W W W= + . For the linear aerodynamic model

of control surfaces, the solution of this method can be easily obtained by:

( ) ( ) ( ) ( )p dt E t F t T GM tδ δ δ= + − + (15)

Where

2 21

2 22

1 1

( )

( )

( )

E I GB W W

F I GB W W

G W BW

−

−

− −

= −

= −

=(16)

Dynamic control allocation method takes the constraint of contribution result at the last sampling period in

account. So the jump of the control surfaces can be prevented. However, it equals the generalized inverse method

when 2W is the zero vector.

The dynamic characteristics of control surfaces can be regarded as the second-order dynamical systems, as

shown in equation (17):

2

2 22j

desj j js s

ωδ δ

ξ ω ω=

+ + (17)

Where jξ is the damping ratio, and jω is the frequency.

The dynamic characteristic of every type of control effector differs from each other. Venkataraman presented the

method which formulated the objectives of control allocation in the frequency domain by considering the Fourier

transform of the moment-error [18] . The results demonstrated that the slower effectors were allocated the lower

frequencies in the desired signal while the faster effectors were allocated the higher frequencies, and the accuracy of

moment-tracking was improved.

Several typical methods of control allocation were summarized in Table 1.

Table 1 Compare of several typical methods of control allocation

Methods GI DC DA Optimization based

Combination All used √ √ √Linear √ √ √ √Aerodynamic

models Nonlinear √ √AMS Achievable √ √

Objectives Others √ √Considering dynamic allocation √

When dM AMS∉ Uncertain riskDepending onalgorithm used

Preservingdirection of

dM

Depending onapproximate objectives

Flight tasks applied Hard to apply Cuise phase Maneuver AllEach control allocation method only adapts to the corresponding aircraft type or flight task. And the methods

should be integrated for the practical use. However, the optimization based method is of the best applicability, and

aims at the particular feature of the flying-wing configuration aircraft. Developing with the computer technique and

the more complicated flight task, the control allocation became the optimization based problem.

IV. Aerodynamic modeling of control-surfaces

Different from the conventional aircraft, a tipical flying-wing configuration aircraft has multi-control-surfaces,

as shown in Figure 5. Two pairs of innovative yawing control surfaces are implemented, which are AMT and SSD.

Fig. 5 Typical allocation of control surfaces of flying-wing configuration combat aircraft.

The drag rudders used on the flying-wing configuration aircraft all have the strongly nonlinear and multi-axes

coupling aerodynamic characteristics. Based on the wind tunnel tests or computational results, the best square

approach algorithm from the complicated function space to the simpler function space was used for the more

accurate aerodynamic modeling of each control-surface. The polynomial approach algorithm applied, the six-

degrees-of-freedom aerodynamic forces and moments of each control-surface could be given by:

0

( , , )n

ji j

j

C k Ma α β δ=

∆ =∑ (18)

Where iC∆ is the increment of control force or moment coefficient around a certain axis, and jk is the fitting

coefficient at certain flight condition. n is the dimension of the polynomial space, decided by the number of the

control-surfaces deflection angles tested or computered..

Taking the all-moving tips for example, the dimension of function space was assumed 3, so the bases of the

polynomial function space were given by [ ]21 δ δ . At the flight condition of Ma =0.6 and α =4.2º, the

Leading Edge Flap(Only for take-off)

ELEV

PF

AMTSSD

polynomial fitting results of the tests data of AMT were showed in Fig. 6. The lift, rolling moment and pitching

moment coefficients, trended to increase at first and decrease then as the deflection angle increased, because the

local stall of AMT happened as the local attack angle increased.

0 10 20 30 40 50 60

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Pi t chi ng Moment

Yawi ng Moment

Rol l i ng MomentSi de For ce

Li f t

Dr agFo

rce

Coef

fici

ents

Incr

emen

ts

δ / ( o)

Fig. 6 Six degrees-of-freedom aerodynamic models of AMT.

The aerodynamic curves of control-surfaces are generally continuous and smooth, and may be monotonic or

single peak. No matter the density of the data grid, the polynomial fitting method can give the remarkably accurate

aerodynamic models of the control-surfaces. Because the gradient matrix can be obtained from the polynomial

function directly and accurately, and it is unnecessary to be solved approximately by the numerical method, the

optimization calculation becomes more robust and reliable.

V. Multi-objectives based nonlinear programming method

According to the special characteristics of flying-wing aircraft’s control allocation problem which was described

in Sec. І (A), the multi-objectives based nonlinear programming method was utilized in this paper. More objectives

could be realized corresponding to the different flight phases or tasks. This method could be given by:

' .min

1 1 .max .min

minm m

i ii i i

i i i i

f fJ f

f fω ω

= =

−= =−∑ ∑ (19)

S.tmaxmin

( ) ( )dg M tδδ δ δ

=≤ ≤

(20)

Where iω is the weighting coefficient which stands for the importance portion of the corresponding objective,

and should satisfy1

1m

i

i

ω=

=∑ . 'if is the positive equal-dimension value of the i th objective. .maxif and .minif are

respectively the maximum and minimum of the target function among the feasible region.

Solving the sub-problem of quadratic programming in every iterative step, the successive quadratic

programming (SQP) algorithm was utilized in this paper by Matlab toolbox.

It is assumed that the desired control moments coefficients [ ]l n mC C C∆ ∆ ∆ at certain time during a

complicated flight task were [-0.02, -0.02, -0.02]. Then two different optimization objectives were chose for

validating the multi-objectives based nonlinear programming method of control allocation, as the following

described.

The lift and drag are usually inconsistent with each other, and the requirements of which depend on the flight

task. Considering the first situation: if the aircraft was required to ensure both the larger lift and the smaller drag

during the flight task, the objectives and weighting coefficients could be given by:

1 Df C=∆ � 1 1aω = (21)

2 Lf C=−∆ � 2 11 aω = − (22)

Where 1a is the weighting coefficient of the minimum drag objective. The results demonstrated that the lift and

drag both decreased with the weighting coefficient 1a increasing from 0 to 1, as shown in Fig. 7. When 1a equaled

0, both the PFs deflected the lowest position, as shown in Fig.8, and meanwhile the LC∆ was the largest. When 1a

equaled 1, the DC∆ was the smallest. The results also proved that only single side of the drag rudders deflected

automatically at this situation, as shown in Fig. 9.

0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.100.120.140.16

Forc

eCo

effi

cien

tsIn

crem

ents

a1

Dr agLi f t

Figure 7. Optimizing results of the lift and drag with weighting coefficient increasing in the first situation

0.0 0.2 0.4 0.6 0.8 1.0

-30

-20

-10

0

10

20

30

δ/

(o )

a1

Lef t el evonRi ght el evonLef t Pi t chi ng f l apRi ght Pi t chi ng f l ap

Figure 8. Distributions of longitudinal -lateral control surfaces in the first situation

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

δ/

(o )

a1

Lef t el evonRi ght el evonLef t SSDRi ght SSD

Figure 9. Distributions of directional control surfaces in the first situation

Considering the second situation: if the aircraft was required to ensure both the larger lift and drag during control

allocating, the objectives and weighting coefficients could be given by:

1 Df C=−∆ � 1 2aω = (23)

2 Lf C=−∆ � 2 21 aω = − (24)

Where 2a is the weighting coefficient of the maximum drag objective. The results demonstrated that the lift

trended to decrease and the drag trended to increase with the weighting coefficient 2a increasing from 0 to 1, as

shown in Fig. 10. When 2a equaled 0, both the PFs deflected the lowest position, as shown in Fig.11, and

meanwhile the LC∆ was the largest. When 2a equaled 1, the DC∆ was the largest. Meanwhile the results also

proved that both side of the drag rudders (AMTs and SSDs) began deflecting automatically as 2a increased, as

shown in Fig. 12.

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.04

0.08

0.12

0.16

Forc

eCo

effi

cien

tsIn

crem

ents

a2

Dr agLi f t

Figure 10. Optimizing results of the lift and drag with weighting coefficient increasing in the second situation

0.0 0.2 0.4 0.6 0.8 1.0

-30

-20

-10

0

10

20

30

δ/

(o )

a2

Lef t el evonRi ght el evonLef t pi t chi ng f l apRi ght pi t chi ng f l ap

Figure11. Distributions of longitudinal -lateral control surfaces in the second situation

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

δ/

(o )a

2

Lef t AMTRi ght AMTLef t SSDRi ght SSD

Figure12. Distributions of directional control surfaces in the second situation

Above results of control allocation demonstrated that the additive lift and drag caused by control allocation were

distinct and varied with the objectives for the flying-wing configuration aircraft. So the proper objectives of control

allocation should be defined according to the requirements of the flight task.

VI. Simulation

The control allocation method of nonlinear programming algorithm used for a certain flying-wing configuration

aircraft was simulated in the altitude-tracking flight control system based on the nonlinear dynamic inverse theory.

This flight control system is suitable for many complicated flight tasks: intercepting, air-to-ground strike, landing on

carrier, and so on, and could be directly integrated with other advanced flight control system [19] . The structure of

the flight control system was showed in Fig. 13.

Figure 13. Flight control system based dynamic inverse theory

The total moments around three axes could be obtained by:

2 2

( ) ( )

( ) ( )

( ) ( )

x x z y z y xy z x y

y y x z x z xy y z x

z z y x y x xy x y

L I I I I

M I I I I

N I I I I

ω ω ω ω ω ωω ω ω ω ω ωω ω ω ω ω

= + − + −= + − − += + − − −

& &

& &

&

(25)

The angle rates around three axes could be obtained by:

desδDesired Control

Moments

ControlAllocation

AltitudeCommand

XControlSurfaces

AircraftDynamicInverse

δ

1 sin 00 cos cos sin0 sin cos cos

x

y

z

ω θ γω γ θ γ ψ

γ θ γω θ

=−

&

&

&

(26)

The desired dynamic models of altitude angles were all assumed the simple first-order delay systems in the

simulation, as given by:

11 1c sTφ

φφ =

+� 1

1 1c sTψ

ψψ

=+

� 11 1c s

Tθ

θθ =

+(26)

Where time constants Tφ , Tψ , and Tθ are assumed 0.33s�0.5s and 0.2s respectively.

Based on the results of wind tunnel tests, the six-degrees-of-freedom nonlinear motion and aerodynamics models

of a typical flying-wing configuration aircraft with low-aspect-ratio were utilized for simulation. The parameters of

control surfaces used were showed in Tab. 2.

Table 2. Characteristics of control surfaces

Control Surface ELEV PF AMT SSD

Deflection range -30º~30º 0º~90º

Rate limit ±100º/s

Time constant 0.05s

Rapid altitude-tracking flight control system could be practical for the combat use. When the intercepted target

aircraft located at the top right direction of the attacking aircraft, as shown in Fig. 14, and when the target aircraft

was ready to turn to the back of the attacking aircraft, the pilot should point the head of the aircraft to the target

aircraft rapidly, and preserve it locked and tracked for attacking at any moment. This rapid-pointing maneuver can

help attacking aircraft lock the target without changing flight path, so the response of the combat can be improved.

Figure 14. Rapid-pointing maneuver

Z X

Y

Above maneuver for combat was simulated by the altitude- tracking flight control system. The trapezium signals

of three altitude angles were carried out to evaluate the control allocation method, which maneuvered the aircraft to

the collimation altitude. Three objectives of control allocation were carried out: the minimum drag, the maximum

lift and the maximum drag. The results of the altitude tracking were shown from Fig. 15 to Fig. 17, which

demonstrated that the performance of the tracking system was satisfied, without overshooting and oscillation of both

pitching and yawing angles, and only with a small oscillation of rolling angle at the beginning and the ending of the

maneuver.

0 2 4 6 8 10 12 14

-12-10-8-6-4-202

φ/

(o )

T / ( s)

DesiredMin C

D

Max CL

Max CD

Figure 15. Tracking results of rolling angle

0 2 4 6 8 10 12 14

0

5

10

15

20

ψ/

(o )

T / ( s)

DesiredMin C

D

Max CL

Max CD

Figure 16. Tracking results of yawing angle

0 2 4 6 8 10 12 142468

10121416

θ/

(o )

T / ( s)

DesiredMin C

D

Max CL

Max CD

Figure 17. Tracking results of pitching angle

The time-domain responses of aerodynamic angles were showed in Fig. 18. The attack angle trimmed at pointing

altitude was the lowest under the objective of maximum lift, and the highest under the objective of maximum drag.

The sideslip angles responded uniformly with three different objectives, and all preserved the higher value in

maneuvering. The time-domain responses of velocity and height were showed in Fig. 19. It was demonstrated that

the flight height increased the fastest with the maximum lift, about 210m in 10s. And the flight velocity decreased

the fastest with the maximum drag, about 90m/s in 10s.

0 2 4 6 8 10 12 14-10123456789

10

α/

(o )

T / ( s)

Min CD

Max CL

Max CD

0 2 4 6 8 10 12 14-30

-25

-20

-15

-10

-5

0

5

β/

(o )

T / ( s)

Min CD

Max CL

Max CD

Figure 18. Responses of aerodynamic angles

0 2 4 6 8 10 12 14100

120

140

160

180

200

V/

(m/s

)

T / ( s)

Min CD

Max CL

Max CD

0 2 4 6 8 10 12 14

4000

4040

4080

4120

4160

4200

4240

H/

(m)

T / ( s)

Min CD

Max CL

Max CD

Figure 19. Responses of flight velocity and height

As a result, the simulation results proved the impacts of control allocation on the flight states depended on the

objectives used. For the general flight task, the objective of minimum drag should be selected, so the minimum

energy for control was required. If the flight task requires tracking with climbing concomitantly, the objective of

maximum lift should be selected for control allocation. If he flight task requires tracking with decelerating

concomitantly, the objective of maximum drag should be selected for control allocation.

The deflections of ELEVs and PFs were showed from Fig. 20 to Fig. 23. The ELEVs dissymmetrical deflect to

make the aircraft roll left at 3th second when the maneuver started. Meanwhile upward-deflection angle of the left

ELEV was larger than the downward-deflection angle of the right ELEV for aircraft tracking the nose up signal, and

the PFs deflected upwards symmetrically. Then the ELEVs became deflecting inversely, which produced the right

rolling moment to prevent aircraft rolling more left. When the aircraft’s altitude began reverting at 11th second, the

deflecting law of the elevons was opposite with the situation when maneuver started.

0 2 4 6 8 10 12 14-30-25-20-15-10-505

Left

ElEV

/(o )

T / ( s)

Min CD

Max CL

Max CD

0 2 4 6 8 10 12 14-30

-20

-10

0

10

20

Righ

tEl

EV/(

o )

T / ( s)

Min CD

Max CL

Max CD

Figure 20. Deflection of left elevon Figure 21. Deflection of right elevon

0 2 4 6 8 10 12 14-20

-10

0

10

20

30

Left

PF/(

o )

T / ( s)

Min CD

Max CL

Max CD

0 2 4 6 8 10 12 14-20

-10

0

10

20

30

Righ

tPF

/(o )

T / ( s)

Min CD

Max CL

Max CD

Figure 22. Deflection of left pitching flap Figure 23. Deflection of right pitching flap

The deflections of the drag rudders with the three objectives of control allocation were showed from Fig. 24 to

Fig.27. When the maneuver started at 3th second, the left AMT and SSD deflected small angles to make aircraft yaw

left. When the aircraft stayed at 20º yawing angle, the large negative sideslip angle appeared, because the flying-

wing aircraft studied was directional unstable, the left drag rudders deflected for balancing the right yawing moment

caused by the sideslip. When the aircraft’s altitude began reverting at 11th second, the deflecting law of the drag

rudders was opposite with the situation when maneuver started.

0 2 4 6 8 10 12 1405

10152025303540

Left

AMT

/(o )

T / ( s)

Min CD

Max CL

Max CD

0 2 4 6 8 10 12 140

5

10

15

20

25

Righ

tAM

T/(

o )

T / ( s)

Min CD

Max CL

Max CD

Figrure 24. Deflection of left all-moving tip Figure 25. Deflection of right all-moving tip

0 2 4 6 8 10 12 140

102030405060708090

Left

SSD

/(o )

T / ( s)

Min CD

Max CL

Max CD

0 2 4 6 8 10 12 140

102030405060708090

Righ

tSS

D/(

o )

T / ( s)

Min CD

Max CL

Max CD

Figure 26. Deflection of left slot-spoiler-deflector Figure 27. Deflection of right slot-spoiler-deflector

In summary, the pitching flaps deflected to the lowest position when the objective of control allocation was the

maximum lift. Both sides of drag rudders (AMTs and SSDs) automatically deflected to produce the large additive

drag when the objective of the maximum drag was selected.

VII. Conclusion

The control allocation methods should be utilized depending on the aircraft type and the flight task carried out.

The multi-objectives based nonlinear programming method was satisfied for the flying-wing configuration aircraft,

solving the special problems of strongly nonlinear and multi-axes coupling characteristics. According to the

requirements of flight tasks, the flight states could be adjusted by the properly selected objective for control

allocation.

However, in the altitude-tracking flight control system based on the dynamic inverse theory, the distributions of

control surfaces will cause the larger change of the flight states as the maneuver time gets longer, especially when

the objective of control allocation is the maximum drag. So the parameters of referenced dynamic models of altitude

angles should be adjusted in real time during the maneuver. Otherwise, the performance of the flight control system

may deteriorate.

Acknowledgments

This work was supported by the National Natural Science Foundation of China �No. 10477004).

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