Wang Lixin,Xu Zijian,Yue Ting

School of Aeronautics Science and Engineering,Beihang University,Beijing 100083,China

Dynamic characteristics analysis and flight control design for oblique wing aircraft

Wang Lixin,Xu Zijian,Yue Ting*

School of Aeronautics Science and Engineering,Beihang University,Beijing 100083,China

The movement characteristics and control response of oblique wing aircraft(OWA)are highly coupled between the longitudinal and lateral-directional axes and present obvious nonlinearity.Only with the implementation offlight control systems can flying qualities be satisfied.This article investigates the dynamic modeling of an OWA and analyzes its dynamic characteristics.Furthermore,a flight control law based on model-reference dynamic inversion is designed and verified.Calculations and simulations show that OWA can be trimmed by rolling a bank angle and deflecting the triaxial control surfaces in a coordinated way.The oblique wing greatly affects longitudinal motion.The short-period mode is highly coupled between longitudinal and lateral motion,and the bank angle also occurs in phugoid mode.However,the effects of an oblique wing on lateral mode shape are relatively small.For inherent control characteristics,symmetric deflection of the horizontal tail will generate not only longitudinal motion but also a large rolling rate.Rolling moment and pitching moment caused by aileron deflection will reinforce motion coupling,but rudder deflection has relatively little effect on longitudinal motion.Closed-loop simulations demonstrate that the flight control law can achieve decoupling control for OWA and guarantee a satisfactory dynamic performance.

1.Introduction

Oblique wing aircraft(OWA)can vary wing sweep for optimal configuration at various flight speeds and extension of their lf ight envelope.1,2Compared to conventional fixed-wing aircraft,an OWA maintains excellent low-speed,takeoff and landing performance at no sweep while being capable of large lift to drag ratio via high skew angle in supersonic flight.Cruise and maneuvering performance can also be enhanced when the wings are at a moderate oblique angle.As a result,OWA have the ability to adapt to multi-mission flight and possess higher operational efficiency than traditional fixed-wing designs.3

Compared to variable-sweep wing aircraft,the neutral point of OWA shifts in a small range with configuration change,reducing trim drag and load on the fuselage and empennage.The wave drag of OWA is also smaller in transonic and supersonic flight.The lift dependent wave drag and the volume dependent wave drag of OWA are 1/4 and 1/16 less than those of variable sweep aircraft,respectively.4Furthermore,oblique wing designs are simple and reliable in structure,thus reducing the complexity offuselage structure and related aerodynamic drag.5However,the asymmetry of OWA introduces special flight dynamics problems and challenges for flight control.6–8The asymmetric con figuration will produce side force,rolling moment and yawing moment in level flight,which do not exist in conventional aircraft,leading to severe coupling and nonlinearity of the aircraft.Moreover,aileron de flection induces not only rolling moment but also pitching moment,and rolling effectiveness is insufficient at high skew angle.Only with flight control systems can the longitudinal and lateral motions of OWA be decoupled to meet operational requirements of pilots and achieve good flight quality.

Currently,the flight controllers for OWA are mostly designed by linear control methods based on linear equations.9–15For example,Alag et al.13and Pahle14used linear quadratic optimal control theory based on model-following technology to design control laws.Clark and Letron15proposed a command and stability augmentation system where eigenstructure assignment techniques are combined with an optimization procedure to determine the feedback matrix for approximating the desired eigenstructure.Evaluation of these controllers on nonlinear equations of motion seems to be less desirable with certain time delays and great oscillations.16With regard to the research on applying modern control technologies,such as intelligent control theory,to OWA with high cross-coupling and nonlinearity,Pang16designed an attitude controller with a sliding mode control method for near-space vehicles with an oblique wing.However,the overshoot of this closed-loop system was relatively high,and no reports are available about OWA in the conventional flight envelope.

This paper investigates the nonlinear dynamic model of OWA,and dynamic response is numerically simulated and analyzed.According to the highly cross-coupled and nonlinear properties of OWA,model-reference dynamic inversion is used for flight control law design.Differential horizontal tail and ailerons are allocated for roll control.This approach can successfully maintain multivariate decoupling control for OWA.

2.Layout and aerodynamic characteristics

The OWA investigated in this paper is presented in Fig.1.The oblique wing is designed to pivot from 0°to 60°with the right wing forward.

The asymmetry of an OWA significantly affects its aerodynamic characteristics.The pressure distribution along the chord direction changes due to the spanwise flow of air.The aerodynamic load of the right forward-swept wing is concentrated on the wing root;thus,the leading-edge suction of the right wing tip and the lift coefficient decrease.The left backswept wing is the opposite:aerodynamic load is concentrated on the wing tip,and the leading-edge suction and lift coeff icient increase,which can be seen in Fig.2(a).Since leadingedge suction makes a greater contribution to lift,the left wing has a lift increment.As seen in Fig.2(b),the lift increment of left wing ΔLLis greater than the lift increment of right wing ΔLR,which produces nose-down pitching moment ΔM and rolling moment ΔL that make the right wing move downward.

Fig.1 Oblique wing aircraft.

Fig.2 Influence of oblique wing on aerodynamic force and moment.

The asymmetric wings of an OWA generate side force that symmetric wings do not have.In Fig.2(c),the velocity of airflow V0can be divided into velocity normal to the leading edge Vnand velocity parallel to the leading edge Vt.Regardless of viscosity,Vthas no effect on the pressure distribution of the wing surface.The drag Dncorresponding to Vncan be divided into Dxand Dyin the wind coordinate system.In symmetric swept back wings,the two side-components DyLand DyRoffset each other.But,for the oblique wing,the lack of symmetry introduces a side component that increases as the skew angle becomes larger.The side force is large enough that it cannot be ignored when contrasted with the magnitude of drag.

The leading edge suction of the left backswept wing is larger than that of the right forward-swept wing at subsonic speed,and the lift vector of the left wing inclines forward(see Fig.2(b)),which results in smaller drag of the left wing.Atsupersonic speed,the wave drag of the right wing is larger.Consequently,the drag of the left wing is larger than that of the right wing at subsonic and supersonic speed,i.e.DnL＜DnR.Fig.2(c)shows thattheasymmetricdragproducesayawingmoment ΔN ＞ 0 N·m that tends to reduce the skew angle.

The right aileron is ahead of the left on the oblique wing,and the difference of lift caused by aileron deflection introduces pitching moment.So,apart from the rolling moment caused by aileron deflection,the additional pitching moment should also be considered in flight trimming and maneuvering.Moreover,the aileron moment arm ladecreases when the wings are oblique(see Fig.3);thus,the rolling effectiveness of ailerons will decrease with the increase of skew angle Λ.This will lead to insufficient rolling effectiveness of the ailerons.It is therefore necessary to use horizontal tail to help roll control when the wings are highly skewed.

3.Modeling and dynamic characteristics analyses

3.1.Flight dynamic modeling

The aerodynamic forces and moments of OWA become highly nonlinear and cross-coupled as the skew angle becomes larger.In addition,the inertia is also cross-coupled between an aircraft’s longitudinal and lateral-directional axes.So the flight motionofOWA shouldbemodeledbysix-degree-offreedom nonlinear equations and cannot be divided into longitudinal and lateral equations.Compared with conventional fixed-wing aircraft,OWA’s asymmetry leads to great changes in moments of inertia and cross products of inertia;the cross products of inertia Ixyand Iyzare no longer zero.Therefore,the moment equations cannot be simpli fied like with conventional fixed-wing aircraft,and they are expressed in Eq.(1).5

Fig.3 Decrease in aileron moment arm caused by oblique wing.

where L,M and N are total moment components in body axes;p,q and r are roll,pitch and yaw angular rate in body axes respectively;Ix,Iy,Iz,Ixyand Izxare moments of inertia and cross products of inertia in body axes.

Due to its asymmetry,OWA wing rotation,lift,drag and pitching moment vary greatly,and asymmetrical side force,rolling moment and yawing moment are generated.The nonlinear aerodynamic forces and moments can be expressed as follows:

where X,Y and Z are aerodynamic force components in body axes;X0,Y0,Z0,L0,M0and N0are aerostatic forces and moments;u,α and β are the airplane speed,angle of attack and sideslip angle,respectively; δaL,δaR,δr,δeLand δeRare the control inputs of left aileron,right aileron,rudder,left horizontal tailand right horizontaltail,respectively;the partial derivatives Xi,Yi,Zi,Li,Miand Ni（i=p，q，r，δeL，δeR，δr，δaL，δaR，˙α）are aerodynamic derivatives corresponding to variables.The aerostatic forces and moments and dynamic derivatives are related to u,α,β and Λ can be obtained by interpolation during simulation.

Eq.(2)shows that OWA are highly coupled in aerodynamics,and triaxial state variables all have influences on longitudinal and lateral aerodynamic forces/moments.The cross static derivatives Xβ,Yα,Zβ,Lα,Mβ,Nαand cross dynamic derivatives Yq,Zp,Zr,Lq,Mp,Mr,Nqexist in OWA,but they are zero for conventional airplanes.The change of moment arm and aerodynamic efficiency of ailerons has obvious effects on rolling effectiveness.The left and right ailerons have distinctly different control effectiveness due to asymmetry of the flow field;thus,the left and right ailerons should be considered separately in the design of flight control laws.While the ailerons produce a pitching moment,i.e.Mδa,the asymmetric wing has little effect on the actions of the rudder and horizontal tail.

The rolling effectiveness of OWA is insufficient at high skew angles,so a differential horizontal tail is needed for roll control.For optimal control,ailerons and the differential horizontal tail should be allocated properly,which is an important problem for OWA in trimming and the design offlight control laws.This paper employs the concept of pseudo-controls17to facilitate the efficient use and combination of control power.Both ailerons and the differential horizontal tail are regarded as independent control variables.The allocation of control moments is the only factor to be considered in flight control surfaces.Therefore,moment equations can be written as

whereˉx=[p，q，r，α，β，μ，u，γ，H]Tare state variables related to control allocation,with γ and H are flight path angle and flight altitude;u=[δaL,δaR,δr,δeL,δeR]T;ff（ˉx）is a nonlinear threeelement vector function varied with state variables,and gf（ˉx）is a nonlinear 3×5 matrix function representing the control matrix.

To avoid a situation where deflection of the differential horizontal tail approaches full but deflection of ailerons is still small,Δ =diag(δaLmax,δaRmax,δrmax,δeLmax,δeRmax)is introduced for weighting.Therefore,Eq.(3)can be rewritten as follows:

where ^u= [δ/δ ，δ/δ ，δ/δ ，δ/δ ，δ/δ]T.

aLaLmaxaRaRmaxrrmaxeLeLmaxeReRmaxAccording to Eq.(4),the optimal control u can be gotten by pseudo-inverse.Thecontroldeflection hasthesmallest two-norm,corresponding to the smallest control power,theoretically.The expression is

where the superscript ‘+” symbolizes pseudo-inverse.

3.2.Dynamic characteristics

(1)Trimming in different skew angles

OWA can vary skew angle at different speeds for a multimission flight.The following three typical conditions are chosen for trimming and the trimming parameters are presented in Table 1,where φ is the bank angle.

The results in Table 1 show that:

(a)To trim the asymmetric aerodynamic moments caused by an oblique wing,the triaxial control surfaces should deflect in a coordinated way.At the same time,OWA make use gravity to balance the side force by rolling an angle.

(b)As the angle of attack for trimming decreases with the increase in flight speed,the resulting nose-down pitching moment decreases.Thus the horizontal tail takes up less to trim the pitching moment.

(c)The bank angle and the deflection of lateral control surfaces become larger,since the asymmetric aerodynamic forces and moments increase with the increase in flight speed and skew angle.

(d)Because of the allocation by pseudo-controls,the left and right ailerons hold different deflection areas based on different control effectiveness.The differential horizontal tail takes part in roll control;thus,it decreases the deflection of ailerons and facilitates their efficient use.

(2)Natural modes

The linear equations of motion can be derived from the sixdegree-of-freedom nonlinear equations by utilizing the smalldisturbance theory.Based on these linear equations,five modes corresponding to the modes of conventional airplanes can be obtained.Taking Condition 2 as an example,the eigenvalues of modes are presented in Table 2.

Five modes of OWA are similar but not identical to those of conventional airplanes.The short-period mode of straight wings is seen to be a motion in which longitudinal parameters α and q are present with signi ficant magnitude.But the shortperiod mode of OWA is highly longitudinal/lateral coupled,and all rotation parameters change signi ficantly,as shown in Fig.4(a).This phenomenon is mostly caused by the cross static derivatives Lαand Nα.In Fig.4(a),Δα,Δβ,Δθ and Δφ are variations of angle of attack,sideslip angle,angle of pitch and bank angle.In the phugoid mode,bank angle also changes,as seen in Fig.4(b).This is because the side force varies with flight speed;thus,the bank angle for balancing side force also changes.In Fig.4(b),ΔV is variation of airplane speed.Nevertheless,the changes in lateral mode shape are generally small.The motion responses are similar to those of the straight wing,and longitudinal parameters have no signi ficant variations in lateral modes.

Table 2 Eigenvalues of modes.

(3)Control characteristics

The open-loop control characteristics of OWA are analyzed for Condition 2.The responses to square wave input of asymmetric aileron Δδaasym=2°,rudder Δδr=2°and a symmetric horizontal tail Δδesym=2°are presented in Fig.5(pulse width Δt=2 s).

Fig.5 shows that an oblique wing makes the longitudinal and lateral motions highly coupled:

(a)Although the symmetric deflection of the horizontal tail only generates pitching moment,the response of OWA is a fairly large roll angular rate p.This is because OWA do not have the bilateral symmetry.The lift of the left wing is no longer equal to that of the right wing when the angle of attack changes;thus,the rolling moment changes greatly with variation in angle of attack.Moreover,the moment of inertia Ixxdecreases obviously with rotation of the wing.Consequently,symmetric deflection of the horizontal tail introduces a large rolling rate.

(b)As the right aileron is ahead of the left aileron on the oblique wing,ailerons generate rolling moment and pitching moment.The pitching moment changes angle of attack;thus,the rolling moment will be greatly influenced.This results in a more obvious coupling between longitudinal and lateral axes.

(c)Rudder deflection produces sideslip and rolling angles,but,since these have little effect on longitudinal aerodynamics,these result in little change to longitudinal motion parameters.

4.Flight control design based on model-reference dynamic inversion

Nonlinear dynamic inversion(NDI)is a kind of modern control method aimed directly at nonlinear motion models.Because unsteady aerodynamic forces of OWA are obvious and triaxial motion is highly cross-coupled and nonlinear,it is difficult to build an accurate motion model for OWA.Without an accurate model,the control effectiveness of pure NDI is unsatisfying.Hence,ideal models established according to the requirements for handling qualities are introduced based on NDI.OWA can track these models by NDI,thereby ensuring the controlled aircraft enjoys satisfactory flight qualities in quite a large flight envelope.18–20

4.1.Flight control structure

As shown in Fig.6,the flight control system based on modelreference dynamic inversion includes four parts:ideal reference models,compensators,NDI inner loop and NDI outer loop.Each part is introduced as follows:

(1)Ideal models form the ideal control response˙μRM,qRMand βRMunder the control command[˙μc，qc，βc]T,where ˙μ is the velocity bank angle.According to the low-order equivalent system method for assessing flying qualities,the low-order equivalent models are chosen as ideal models,and their parameters are determined by requirements for handling qualities.The ideal models are expressed in Eq.(6).

where the reciprocal of rolling time constant ωpis 4 rad/s; ωspand ξspare the frequency and damp of shortperiod mode,respectively,taken asωsp=5 rad/s,ξsp=1.2;time constant Tθ2can be solved by control anticipation parameters(CAP),and the value of CAP is 1;the desired response for sideslip angle is taken as the first order inertia link,and ωβ=3 rad/s.

(2)PI compensators are used to generate the control commands for the NDI loop and timely track the response of ideal models.It compensates for errors of loop and the external disturbance.βRM,˙μRMand the corresponding feedback signals pass through the compensator to generate˙βcmdand˙μcmdas the control command of outer loop while qRMis directly taken as the pitching control command of inner loop qc.

(3)The NDI outer loop is used to generate control commands of the inner loop.[˙βcmd，˙μcmd]Tis resolved to[pc,rc]T,which combines with the pitching control command of inner loop qcto generate the commands of inner loop.According to Ref.21,the control forces resulting from control surfaces are much smaller than aerodynamic forces;thus,the effect of these small perturbations on dynamics is negligible and can be canceled in steady state by incorporating integrators into the control law.So the control forces can be neglected when designing the outer loop,and the differential equation of[α,β,μ]Tcan be expressed as

Fig.5 Responses to square wave input of asymmetric aileron,rudder and a symmetric horizontal tail for Condition(2).

where fm（ˉx）is a nonlinear three-element vector function varied with state variables;and gm（ˉx）is a nonlinear 3×3 matrix function varied with α and β;fμ,fαand fβare continuous functions related to state variables.

According to the time-scale separation method,10,20,21the dynamic responses offast variables are considered to have arrived at steady state when designing the outer loop.Ref.21demonstrates that this approximation justifies slow-state control law.So the control commands of the inner loop can be solved by pseudo-inverse:

In Eq.(7),the equations of˙β and˙μ do not contain q,which means that the command of˙β and˙μ can be realized just by controlling p and r,and the pitching control command of inner loop qcis free from the influence of βcand˙μc.By making pitching control command αc=0°,qRMcan replace the pitching command generated by the outer loop,which is intended to be the pitching control command of inner loop.

(4)The NDI inner loop calculates the required control surfaces to track the command[pc,qc,rc]Tand achieve the desired control.Based on the principles of NDI,a control law of the inner loop can be obtained that allocates ailerons and differential horizontal tail by pseudocontrols.The expression is presented in Eq.(4).

According to the characteristic response of the first order inertia link to the step signal,the output[p,q,r]is designed to track the command[pc,qc,rc]Tasymptomatically by making[˙p，˙q，˙r]Tc=ωf[pc-p，qc-q，rc-r]T.ωf=diag(ωp,ωq,ωr)=diag(15,10,10)rad/s is taken to achieve good tracking results,where ωqand ωrare reciprocals of time constant offirst order inertia link.The structure of inner loop is shown in Fig.7.

Fig.6 Flight control system scheme based on model-reference dynamic inversion.

Fig.7 Structure of inner loop considering control saturation.

Fig.8 Closed-loop response to triaxial input.

4.2.Closed-loop flight simulation and analysis

To validate the control law based on model-reference dynamic inversion,the pitching,side slipping and rolling maneuvers are simulated.Condition 2 is taken as an example,and the dynamic response curves are presented in Fig.8.

In Fig.8(a),the velocity bank angle˙μ does not appear to oscillate;it tracks the commands quickly and almost matches with the response of the ideal model.Meanwhile,β does not exceed 0.5°in maneuvering flight.During the pitching maneuver,the curve of pitch angular rate q also coincided with that of the ideal model.β and p change little and recover to zero quickly,as seen in Fig.8(b).This demonstrates that the flight control system successfully maintains decoupling control between the longitudinal and lateral axes.In side slipping maneuvers,the response of sideslip angle has a little delay compared to the ideal model,but they are still pretty close,as seen in Fig.8(c).Therefore,the flight control system enables the OWA to track commands quickly and maintain decoupling control.

The actual responses are close to those of the ideal model,so good handling qualities can be achieved by these designs.According to the response of the control surfaces in Fig.8,it can be found that the differential horizontal tail takes part in roll control for the allocation by pseudo-controls.Thus it decreases the deflection of ailerons and facilitates their efficient use.

5.Conclusions

(1)Oblique wings change the lift,drag and pitching moment greatly,and generate side force,rolling moment and yawing moment that symmetric wings do not have.To trim the asymmetric aerodynamic moments,triaxial control surfaces should deflect in a coordinated way that balances the asymmetric moments.Meanwhile,OWA should roll on an angle to balance the side force.

(2)OWA have significant aerodynamic and inertial crosscoupling between the longitudinal and lateraldirectional axes.The short-period mode of an OWA is highly coupled,and all rotation parameters take on a substantial amount of variation.In the phugoid mode,the bank angle also changes.Nevertheless,the changes in lateral mode shape are generally small.The motion responses are similar to those of straight wings,and longitudinal parameters have no significant variations in lateral modes.

(3)The control characteristics of OWA are quite distinct from those of conventional aircraft.Although symmetric deflection of the horizontal tail only generates pitching moment,the response has a fairly large rolling rate p.Ailerons generate both rolling moment and pitching moment;thus,coupling between longitudinal and lateral axis becomes more obvious.The deflection of rudder introduces both sideslip and rolling angles,which have little effect on longitudinal motion parameters.

(4)The aileron moment arm shortens in OWA,which leads to insufficient rolling effectiveness.So they need a differential horizontal tail to assist roll control at high skew angles.Ailerons and the differential horizontal tail are allocated by pseudo-controls to facilitate efficient use and combine control power.This decreases the deflection of ailerons and has satisfactory control effect.

(5)The control law based on model-reference dynamic inversion maintains decoupling control for OWA.The actual response is close to that of the ideal model,and good handling qualities can be realized from ideal model designs.

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Wang Lixin is a professor and Ph.D.supervisor at Beihang University.His main research interests lie in aircraft design, flight dynamics,and flight control.

Xu Zijian received his B.S.from Beihang University and is now Ph.D.student.His main research interests are flight dynamics, flight simulation and flight control.

Yue Ting is an assistant professor at Beihang University.He received his B.S.,M.S.and Ph.D.in aircraft design from Beihang University in 2004,2006 and 2010 respectively.His main research interests are aircraft fight dynamics and control.

15 December 2015;revised 15 June 2016;accepted 17 August 2016

Available online 21 October 2016

Control allocation;

Decoupling;

Dynamic characteristics;

Model-reference dynamic inversion;

Oblique wing aircraft

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 10 82338821.

E-mail addresses:wlx_c818@163.com(L.Wang),xuzijian@buaa.edu.cn(Z.Xu),yueting_buaa@sina.com(T.Yue).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.10.010

1000-9361?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).