Robust Anti?swing Control for Unmanned Helicopter Slung?Load System with Prescribed Performance

，，，*

1.College of Automation Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，P.R.China；2.College of Aerospace Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract: A robust anti-swing control method based on the error transformation function is proposed，and the problem is handled for the unmanned helicopter slung-load system（HSLS）deviating from the equilibrium state due to the disturbances in the lifting process. First，the nonlinear model of unmanned HSLS is established. Second，the errors of swing angles are constructed by using the two ideal swing angle values and the actual swing angle values for the unmanned HSLS under flat flight，and the error transformation functions are investigated to guarantee that the errors of swing angles satisfy the prescribed performance. Third，the nonlinear disturbance observers are introduced to estimate the bounded disturbances，and the robust controllers of the unmanned HSLS，the velocity and the attitude subsystems are designed based on the prescribed performance method，the output of disturbance observer and the sliding mode backstepping strategy，respectively. Fourth，the Lyapunov function is developed to prove the stability of the closed-loop system. Finally，the simulation studies are shown to demonstrate the effectiveness of the control strategy.

Key words：unmanned helicopter slung-load system；prescribed performance；anti-swing control；disturbance observer；sliding mode controller

0 Introduction

The unmanned helicopter slung system is wide?ly used in military transportation，disaster relief，etc.，because of its capabilities of low altitude flight，fast loading and unloading of cargo，crossing obstacles and fast arriving at destination［1］. In recent years，with the development and application of large and medium-sized unmanned helicopters in various countries，the volume and weight of lifting goods by unmanned helicopters are increasing，which makes the unmanned helicopter slung-load system（HSLS）becomes an important application use of unmanned helicopters［2］.

Compared with the unmanned HSLS，the ex?ternal hanging load of the unmanned helicopter is in?creased. The whole system becomes a highly cou?pled，underactuated，nonlinear system. The slungload may oscillate due to the factors，like the ma?neuvering of the unmanned helicopter or the exter?nal disturbances. How to effectively control the un?manned helicopter to suppress load oscillation has become a research hotspot in the control field［3］. Un?til now，many scholars have conducted many stud?ies on the HSLS. For instance，in view of the influ?ence of slung-load on the unmanned helicopter sys?tem，the load influence on various modes of the heli?copter by direct balancing method in Refs.［4-5］. In Ref.［6］， three dynamic equations of helicopter hoisting system were established，and the instruc?tion smoothing technology was used to suppress load oscillation. Based on the flight dynamics model of the helicopter/crane coupling system，the balanc?ing and linearization of the coupling system were re?alized in Ref.［7］，and the flight control law was de?signed to improve the operation stability characteris?tics of the helicopter/crane coupling system. To re?alize load oscillation suppression，a design strategy of the active disturbance rejection control was pro?posed based on the improved genetic algorithm in Ref.［8］，which realized the on-line estimation of the parameter uncertainties and the external distur?bances to reduce the control errors and achieve the load swing reduction. For the control of the helicop?ter trajectory tracking and the swing reduction，a nonlinear trajectory tracking controller（TTC）was designed by using the backstepping control strategy in Ref.［9］，and the load oscillation suppression was realized through disturbance rejection delay feed?back control. In Ref.［10］，the unmanned HSLS was divided into the helicopter subsystem and the hanging subsystem，and the sliding mode controller and the swing reduction controller were designed to achieve the purpose of the load oscillation suppres?sion. In order to achieve the load swing reduction，a nonlinear TTC was designed for the unmanned HSLS in Ref.［11］. In addition，many scholars have studied the four-rotor SLS and the bridge crane SLS，and a lot of significant research results were achieved in Refs.［12-13］，which has reference sig?nificance for the research on the unmanned HSLSs.

Since the unmanned HSLS will be affected by external disturbances in the lifting process，it is nec?essary to deal with the disturbances in order to en?sure the effectiveness of the designed controller.Among the existing disturbance rejection methods，the disturbance observer is one of the effective strat?egies to deal with external disturbances［14-18］. For in?stance，for the attitude system of the variable sweptwing aircraft with external disturbances，the attitude control method was proposed based on the finite time disturbance observer and the fixed time distur?bance observer in Ref.［14］. For the quadrotor sys?tem with external disturbances and uncertainties，an integral backstepping sliding mode control strategy based on the disturbance observer was proposed in Ref.［15］.In Ref.［16］，a sliding mode control meth?od was proposed based on the disturbance observer for the nonlinear systems with disturbances. For the unmanned HSLS，the excessive load swing angle will affect the flight quality of the unmanned helicop?ter and even endanger the flight safety.Thus，it is necessary to consider the control of limited swing an?gle. There are generally three methods to deal with the output constraint control problems，including the obstacle Lyapunov function［19］，the funnel con?trol method［20］and the error transfer function meth?od［21-23］. Among them，the error transfer function method was widely used in dealing with the output constraint control problems. For example，for the TTC design problem of the small unmanned heli?copter，a nonlinear TTC was designed by using the error transfer function method in Ref.［21］，so that the tracking error can meet the prescribed perfor?mance. In Ref.［22］，a distributed attitude coopera?tive control strategy was studied based on the pre?scribed performance method for a three degrees of freedom unmanned helicopter attitude constraint sys?tem with input saturation. A prescribed performance tracking controller was designed for the nonlinear systems with unknown control direction and input saturation.

Inspired by the literature mentioned above，the main contributions of this paper are as follows：

（1）A disturbance observer is adopted to deal with the anti-disturbance problem.

（2）A robust control method is designed based on the error transfer function for the unmanned HSLS with swing angle constraint and disturbance.

（3）The control scheme can guarantee that all signals in the closed-loop system are bounded and the load swing angle error can meet the prescribed performance.

The sections of this paper are as follows：Sec?tion 1 describes the coupling nonlinear dynamic model of the unmanned HSLS and the physical meaning of each parameter，and then gives the sys?tem control target and control flow chart；in section 2，the controller of lifting subsystem，velocity sub?system and attitude subsystem are designed；section 3 proves the stability of the closed-loop system；sec?tion 4 presents the numerical simulation results and analysis；and section 5 draws conclusions of the study.

1 Unmanned Helicopter Slung?Load System Model

In this section，a full-state coupling nonlinear mathematical model of the unmanned HSLS is es?tablished on the basis of the medium unmanned heli?copter flying flat at a low speed. The following as?sumptions are given［10，21］：

（1）The ground coordinate system is adopted as the inertial coordinate system.

（2）The mass distribution of the unmanned he?licopter is uniform，and the center of mass does not change.

（3）The unmanned helicopter is strictly sym?metrical，and the speed of the main rotor is un?changed.

（4）The rope is a rigid body，weightless and in a tight state at all times.

（5）The hanging point is directly below the center of the mass of the unmanned helicopter.

（6）The swing angle of the hanging load oscil?lates at a small angle.

Under the assumptions above，the dynamic equation of the medium-size unmanned helicopter is given，and it can be described as［21］

wherer= [x，y，z]TandΓ=[u，v，w]Tare the po?sition and the velocity vectors of the unmanned heli?copter under the inertial frame，respectively；Θ=[φ，θ，ψ]TandΩ= [p，q，r]Tthe three Euler angles of the unmanned helicopter in the inertial frame and the angular velocity in the body coordinate system，respectively；T=[0，0，Tmr]TandΣ=[L，M，N]Tthe input force and the torque vectors for the un?manned helicopter，respectively；Tmrrepresents the pull of the main rotor；Hthe transformation matrix from the attitude angular velocity to the Euler angu?lar velocity，andthe transformation matrix from the body coordinate system to the ground coordinate system；G= [0，0，g]Tthe gravitational accelera?tion vector，andgis the gravitational acceleration；J= [Jxx，Jyy，Jzz]Tthe vector of the rotational iner?tia，andmhthe mass of the unmanned helicopter.

The slung-load is added on the basis of the model（1），and the output of the lifting subsystem is defined as the back swing angleθlof the load and the side swing angleφl. Then，the displacement of the load relative to the hanging point is defined asPl，and the specific form is as follows［10］

wherelis the length of the rope. Displacement of load relative to the helicopter center of massPLis given by wherePh= [0，0，lh]Tis the displacement of the hanging point to the center of mass of the helicop?ter，andlhthe distance between the hanging point and the center of mass of the helicopter.

Then the velocity of the slung-load under the inertial frame can be written as［9，10］

Furthermore，the forceFlof the slung-load act?ing on a helicopter，and the momentMlcan be rep?resented as wheremlis the load mass；he air resistance of the load；ρthe damping coefficient；andslthe equivalent windward area of the load.

In addition，the equilibrium equation by suspen?sion point moment is given as follows［9，10］

The dynamic model of the load can be obtained by solving Eq.（6），and one can obtain

wherek=0.5ρsis the equivalent damping coeffi?cient of the load；Sφl，Cφl，SθlandCθlare the short forms for sinφl，cosφl，sinθland cosθl，respectively.The swing angle of the lifting load is in[-π/2，π/2]， thusG(Λ) is reversible. From Eq.（7），the reaction forceFland momentMlof the load on the helicopter can be inversely solved.

The coupling nonlinear model of the unmanned HSLS under the effect of disturbances is shown as follows

whereFl= [Flx，Fly，Flz]T，Flx，FlyandFlzare the components ofFlalong the coordinate axisx，yandz，respectively；Ml= [Mlx，Mly，0]T，Mlx，Mlyare the components ofMlalong the coordinate axisxandy，respectively；di∈R3，i= 1，2，3 the external disturbances.

The control objectives of this paper are as fol?lows：According to the load stability swing angle of the unmanned HSLS under the states of balanceθldandφld，a robust anti-swing controller is designed for the slung-load subsystem to make the swing an?gles converge to a small neighborhood near the de?sired swing angles. Then a robust controller is de?signed to control the speed and attitude angles of the helicopter. Finally，the unmanned helicopter is in the state of flat flight，while the load swing angles are in the small neighborhood of the target swing an?gles. On the basis of the control objectives，the con?trol structure diagram of this paper is shown in Fig.1［10］.

Fig.1 Unmanned HSLS controller design

2 Unmanned Helicopter Slung?Load System Controller Design

For the convenience of the subsequent control?ler design，the following assumptions and lemmas are given.

Assumption 1［21］The helicopter pitch angleφand the roll angleθsatisfy the following inequali?ties that

Assumption 2［21］All states of the system are measurable.

Assumption 3［21］For the external distur?bancedi∈R3，i=1，2，3，assume that there are positive constantsδiensurewhererepresents the 2-norm of a vector.

Assumption 4［15，21］The desired signalΓd(t)of the speed subsystem and the desired signalΘdof the attitude subsystem are sufficiently smooth func?tions and satisfy thatΓd∈MΓ，Θd∈MΘ，and the

Lemma 1［17］For a bounded system with ini?tial conditions at time zero，if there exists a continu?ous positive definite Lyapunov function，one has

and

then the system solutionx(t) is ultimately uniformly bounded，whereπ1andπ2are theK∞class func?tion，C1andC2the positive constants.

2.1 Slung?load subsystem controller design

The load swing angle error vector is defined aseY=Y-Yd= [eY1，eY2]T，where the signals can be described asY=[θl，φl]TandYd=[θld，φld]T，eYiis theith element ofeY. To realize the restricted control of load swing angles，it is necessary to con?strain the swing angle errors. The specific form is given as follows

whereλ1i，λ2i∈(0，1] are the design parameters；χi(t) is the prescribed performance function，and the expression can be written as［21］

By designing the parametersδi＞0 andχi0＞χi∞＞0，the steady-state and the transient perfor?mance of the swing angle error can be constrained.According to the above definition，we haveχi(0) =χi0，，whereχi(t) is a smooth de?creasing function.

To obtain the control performance above，an error transfer function is introduced to convert the constrained inequality into an unconstrained form［21］

whereβiis the variable of transformed error，andQ(?) satisfies the following properties［21］

It can be known from Eq.（14）thatβican be utilize to ensure the predetermined transient and steady-state performance. Thus，the control objec?tives of the system can be converted to that the error transfer variableβiis bounded under a designed con?troller［21-23］，then the swing angle error can meet the prescribed performance in Eq.（11）.

Considering the effect of external disturbanced1，the lifting subsystem model can be written as

To ensure that the swing angle of the slungload subsystem can be converged quickly and stabi?lized in the small neighborhood near the desired val?ue，the following design of the swing angle control?ler is given based on the prescribed performance function.

According to the result in Ref.［21］，the trans?fer errorβiis written as

Then，we define thatαi=α(eYi(0)/χi(0) )Thus，the transfererror can be written as

The derivative ofβican be obtained as follows

It can define thatβ= [β1，β2]T，Π=diag{Π1，Π2}andχ=diag{χ1，χ2}，then one has

Considering the error signaleY=Y-Yd，the slung-load subsystem model （15） can be trans?formed into the following form

whereχ，ΠandeYare all known and can be used in the controller design.

To estimate the unknown disturbance in the lifting subsystem，a nonlinear disturbance observer of the following form is designed by［14］

whereZ1∈R3is the internal variable of the nonlin?ear disturbance observer；the estimated val?ue of disturbanced1；P1(Λ) ∈R3the function to be designed with respect to the variableΛ；L1∈R3×3the gain function andL1=?P1(Λ)/?Λ.

The disturbance observer error is defined as

From Eq.（22），one has

The Lyapunov function is designed as

According to Assumption 3，and Eqs.（23，24）can be written as

whereI2is the second-order identity matrix. Then，the following backstepping method is used to design the slung-load subsystem controller.

Define the constrained variables?1and?2as

whereis the virtual control law to be designed.

On the basis of Eqs.（20，26），it can be de?duced as

Furthermore，the virtual control law is de?signed as

wherek1=＞0 is the design parameter matrix.Substituting Eq.（28）into Eq.（27），one has

Combining Eq.（20）and Eq.（26），one has

Then，the slung-load subsystem control law is designed as

wherek2=＞0 is the matrix to be designed；the helicopter acceleration measure re?quired for the oscillation reduction. The target veloc?ity is obtained by integratingP，[ud，vd]T，will be used in the controller design of the speed subsystem.

Substituting Eq.（31）into Eq.（30），one can obtain

For the slung-load subsystem，the Lyapunov function can be designed as

Considering Eqs.（29，32，33），we have

2.2 Speed subsystem controller design

On the basis of the model（8） for the un?manned HSLS，the affine nonlinear model of the ve?locity subsystem can be obtained as

where

To estimate the unknown disturbances in the velocity subsystem，a nonlinear disturbance observ?er consistent with Eq.（21）is designed by

whereZ2∈R3is the internal variable of the nonlin?ear disturbance observer；the estimated val?ue of disturbanceD2；P2(Γ) ∈R3the function to be designed with respect to the variableΓ， andL2∈R3×3the gain function andL2=?P2(Γ)/?Γ.

The disturbance observer error is defined as

From Eq.（37），the error of the disturbance ob?server can be obtained as

The form of Lyapunov function is designed by

Substituting Eq.（38）into Eq.（39），one can obtain

whereI3is the third-order identity matrix，andδˉ2=δ2/mh.

Then，the tracking error vector of the helicop?ter velocity is defined as

The Lyapunov function is given as

The first derivative of Eq.（42） can be de?scribed as

The control law of the speed subsystem is de?signed as

wherek3=is a positive definite symmetric ma?trix to be designed. Substituting Eq.（45） into Eq.（44），one can obtain

The attitude angles and the pull force of the he?licopter can be solved by the control quantityas fol?lows［21］

where the yaw angleψdis known.

According to Eq.（47），the attitude tracking signal of the attitude subsystem isΘd=[φd，θd，ψd]T，and the signal will be used in the controller design of the attitude subsystem.

2.3 Attitude subsystem controller design

On the basis of the model（8）of the unmanned HSLS，the affine nonlinear mathematical model of the attitude subsystem for the unmanned helicopter can be written as

where

To estimate the unknown disturbance in the at?titude subsystem，a nonlinear disturbance observer consistent with the form of Eq.（21）is designed as

whereZ3∈R3is the internal variable of the nonlin?ear disturbance observer；the estimated val?ue of disturbanceD3；P3(Ω) ∈R3the function to be designed with respect to the variableΩ；L3∈R3×3the gain function andL3=?P3(Ω)/?Ω.

The disturbance observer error is defined as

By taking the derivative of Eq.（50），we can have the error dynamic as

The Lyapunov function is designed as

According to Assumption 2，and Eqs.（50，52），one has

The attitude error vectors of the helicopter are defined as

whereΘd= [φd，θd，ψd]Tis the tracking signal for the attitude system；andΩdthe virtual control law.

In order to avoid directly using the derivatives of the signalsΘdandΩdin the subsequent controller design，the first order filter is introduced，and the variablesζΘandζΩcan be expressed as

whereμθandμΩare the filter time constants. Define the filter errors as

The derivative of Eq.（56）is obtained as fol?lows

and

The virtual control law is designed as

wherek4＞0 is the parameter to be designed.

Substituting Eq.（60）into Eq.（59），it can be obtain

Define the sliding surfaceas［10］

Combining Eqs.（48，61，62），one has

The sliding mode surface approach law is writ?ten as［25］

whereε＞0 is designed parameter， andk5=kT5＞0 the designed symmetric positive defi?nite matrix.

The control law of attitude acquisition subsys?tem is designed as follows

Considering Eqs.（63—65），one has

In order to avoid system chattering caused by t he sign function vector，the smooth function vectorγ1(s) is used to approximate sgn(s)，and the expres?sion is as follows［21］

whereμ1＞0 is the design parameter. Defining the difference between the sign function vector and the smooth function vector， then we haveγ0=sgn(s) -γ1(s). Since the symbolic function vector and the smooth function vector are bounded，γ0sat?isfies thatis the unknown positive con?stant. Therefore，the modified control law of the at?titude subsystem can be obtained as

The Lyapunov function is designed as

Considering Eqs.（57，58，66，69），we have

3 Stability Analysis

Based on the system model and the control ob?jectives in section 1，and the controller design in sec?tion 2，the following theorem is given.

Theorem 1For the unmanned HSLS in Eq.（8）with the limited load swing angle and exter?nal disturbances，the nonlinear disturbance observer（21）of the slung-load subsystem，the constrained robust controller（31）of the slung-load subsystem based on the error transfer function，the nonlinear disturbance observer（36）of the velocity subsystem and the robust controller（45）of the slung-load sub?system are designed. As well as the nonlinear distur?bance observer（48）of the attitude subsystem and the sliding mode backstepping controller（68），the signals of the closed-loop system are ultimately uni?formly bounded，and the load swing angle meets the prescribed performance（11）.

ProofThe Lyapunov function of the closedloop system is constructed as

According to Eqs.（25，34，40，46，53，70），one has

where

It can be known from Eq.（72）that the three controllers（31，45，68）designed for the unmanned HSLS can ensure all signals of the closed-loop sys?tem bounded.Then，we have

Considering Eq.（73），the following conclusion can be drawn that the system signal?1，?2，eΓ，s，eζθandeξΩare ultimately uniformly bounded. It can be obtained

According to Eq.（74），when the timet→∞，the signalβwill meet. Then the tracking error of hoisting subsystem can meet the preset performance in Eq.（11）. Similarly，other sig?nals?2，eΓ，s，eζθandeξΩare also bounded，so the theorem is proved.

4 Simulation Results and Analysis

To verify the effectiveness of the designed con?troller，the medium-sized unmanned helicopter is se?lected as the simulation research object in this sec?tion. Specific simulation parameters are as follows：The unmanned helicopter model qualitymh=1 000 kg，the slung-load qualityml=100 kg，the length of the ropel=10 m，the gravitational accel?erationg=9.8 m/s2，the triaxial moment of inertiaJxx=180 kg ?m2，Jyy=200 kg ?m2andJzz=220 kg ?m2，the equivalent damping coefficient of the loadk=0.2，the external disturbances are assumed to be thatd1=[sin10t+cos10t，sin10(t+1) +cos 10(t+1) ]T，D2=[cos10t，cos10(t+1) ，cos10(t+2 )]T， andD3=[sin10t，sin 10(t+1)，sin 10(t+2)]T. The performance function of hoisting subsystem is selected asχ1(t) =0.4exp(-4t) +0.01，χ2(t) =0.3exp(-4t) +0.01. Assume that the system is in a low-speed flat flight state，and the flight speed of the unmanned helicopter is[ud，vd，wd]T=[10，5，2]T，=0. The load sta?bility swing angles can be obtained by balancing[θld，φld]T≈[0.15 rad，0.08 rad]T.

After full debugging，the controller parameters are set asλ1=λ2=0.5，k1=diag{25，20}，k2=diag{15，15}，k3=1 000I3，k4=50，ε=5，k5=10I3，μθ=μΩ=0.1，u1=5，L1=80I3，L2=100I3，L2=120I3.

Fig.2 Curves of slung-load angle

Fig.3 Error of eY1 with prescribed performance

Fig.4 Error of eY2 with prescribed performance

Fig.5 Tracking curves of unmanned helicopter speed

Fig.6 Curves of unmanned helicopter angular velocity

The simulation results are shown in Figs.2—11. Fig.2 shows that the two swing anglesθlandφlof the slung-load can be stabilized quickly to the ide?al swing angles of 0.15 rad and 0.08 rad under the action of the designed backstepping controller. The prescribed performance of the two tracking errorseY1andeY2of slung-load swing angles are fully re?flected in Fig.3 and Fig.4，respectively，and it can be seen that the transient and steady-state perfor?mance of tracking errors can be guaranteed by ad?justing the parameters of the prescribed performance function. From Figs.5—7，the designed robust con?troller（45）of the speed subsystem and the robust controller of the attitude subsystem controller（68）have good control performance，and the system state of the unmanned helicopter can quickly con?verge to the target values. The estimation perfor?mance of the disturbance observers（21，36，49）are shown in Figs.8—10. We can see that the de?signed disturbance observers can quickly estimate the actual disturbances，and the estimation error is stable within a small range. Furthermore，the trans?fer error variablesβ1andβ2are also convergent，as shown in Fig.11. In conclusion，the simulation re?sults verify the effectiveness of the designed control method.

Fig.7 Curves of unmanned helicopter Euler angle

Fig.8 Tracking curves of disturbance observer in slungload subsystem

Fig.9 Tracking curves of disturbance observer in speed sub?system

Fig.10 Tracking curves of disturbanceobserver in attitude subsystem

Fig.11 Curves of transfer error variables β1 and β2

5 Conclusions

The damping control of the coupling nonlinear unmanned HSLS in the lifting process has been studied. Two ideal swing angles of the slung-load under the flat flight for the unmanned helicopter have been obtained off-line. Considering the distur?bance of the slung-load subsystem，a prescribed ro?bust controller of load swing angles have been con?structed to ensure that the load can converge quickly and be stabilized near the ideal swing angles with the anti-disturbance ability. Aiming at the speed and attitude subsystem subject to external disturbance，the robust controllers have been designed for the speed and the attitude of the unmanned helicopter based on the disturbance observer. Then the stabili?

ty of the closed-loop system is proved by construct?ing the Lyapunov function. Finally，the simulation results show that the designed control scheme can achieve the expected control goal.In the future re?search，the robust anti-swing control based on the neural networks will be studied for the unmanned helicopter slung-load system with system uncertain?ties.