Trajectory Tracking of Vertical Take-off and Landing Unmanned Aerial Vehicles Based on Disturbance Rejection Control

Lu Wang and Jianbo Su

Trajectory Tracking of Vertical Take-off and Landing Unmanned Aerial Vehicles Based on Disturbance Rejection Control

Lu Wang and Jianbo Su

—We investigate the trajectory tracking problem of vertical take-off and landing(VTOL)unmanned aerial vehicles (UAV),and propose a practical disturbance rejection control strategy.Firstly,the nonlinear error model is established completely by the modified Rodrigues parameters,while considering dynamics of the servo actuators.Then,a hierarchical control scheme is applied to design the translational and rotational controllers based on the time-scale property of each subsystem, respectively.And the linear extended state observer and auxiliary observer are used to deal with the uncertainties and saturation. Atlast,globalstability ofthe closed-loop system is analyzed based on the singular perturbation theory.Simulation results show the effectiveness of the proposed control strategy.

Index Terms—Unmanned aerial vehicles(UAV),trajectory tracking control,extended state observer,singular perturbation theory.

I.INTRODUCTION

R ECENTLY,vertical take-off and landing(VTOL)unmanned aerial vehicles(UAV)have attracted increasing interest in researches and applications in both military and civilsociety,such as rescue in disasters,unmanned inspection, and road traffic supervision.The motivation also comes from academic research institutes,since it can be used as low cost testbeds forrobotics studies.However,the VTOL UAV model, which has been widely investigated in mostworks,is known as a class of underactuated system with nonholonomic constraints of second order.According to the necessity of Brockett,there is no gloss or time invariant controller that can stabilize the underactuated system to the equilibrium point[1].Hence,new methodology should be investigated for this kind of systems.

Trajectory tracking control of VTOL UAV is a challenging work due to its coupling property,external disturbances, system uncertainties,etc.Several inspired approaches have been investigated,such as backstepping control[2?4],sliding mode control[2,5?7],feedback linearization[5],model predictive control[8],neural networks[9],fuzzy control[10],observer based control[11?12],etc.However,there are stillsome prominent problems to be considered and resolved.

1)The attitude representation and desired attitude extraction.In most previous works[2-10,13],Euler angles are usedto representthe attitude of rigid body.However,the simplified kinematics is often used as

whereφ,θ,ψdenote the roll,pitch,and yaw angles.ωx,ωy,ωzare the angular velocity of the rigid body,respectively.It is pointed out that the original kinematics of Euler angles is described as[14]

The existence of transcendental functions in(2)makes it difficult to design a control strategy.Noticing that(1)is a simplified form of(2)with the assumption that the rigid body rotates only in one direction at a time,and the roll/pitch angle changes when the pitch/roll angle equals to 0°.However,the assumption above is an ideal instance,which is infeasible. Meanwhile,the simplification will decrease the control accuracy.Furthermore,we find that the system model based on Euler angles is not available when pitch angleθ= ±π/2. Especially,considering the error of sensors and calculation, both attitude estimation and control algorithm based on Euler angles cannot work near the state ofθ=±π/2.Moreover, mostworks provided the desired attitude,angularvelocity,and angular acceleration directly by the position controller.Only [12,15]present the analytical solution of the desired attitude information based on quaternion.

2)Stability of hierarchical control structure.Concerning with the hierarchial control strategy,the position and attitude controllers can be designed separately for translationaland rotationalsubsystems,respectively.Although the above strategy can be introduced for controller design,the stability should be analyzed based on the overall closed-loop system,since the attitude’s tracking is an asymptotical procedure,which makes the attitude error between the actual and desired one a necessary concern in the analysis.However,in[2?5,7?10,13], stability is only analyzed for each subsystem.

3)Dynamics of actuators and its influence on the closedloop stability.In[12,15?16]and[11,17],cascade theory and singular perturbation theory are used to acquire the stability of the closed-loop system.In[18],the relationship is given between actuators and controlinputofa VTOL UAV.However, dynamics of actuators is never considered.

4)Controller design with internaluncertainties and external disturbances.Several researches related to controller design againstuncertainties have been studied based on sliding mode control[2,5?7],neutralnetworks[9],fuzzy systems[10],adaptivebackstepping control[19],disturbance observer[16],etc.However,the chartering of SMC,convergence rate of weights in neutralnetworks and fuzzy systems willlimitthe applications of these methods in practical.The adaptive algorithm can only deal with the external disturbances[19].The disturbance observer is adopted to dealwith the uncertainties[20],however, the saturation of the actuators is not considered.

Based on the above review,trajectory tracking control of a VTOL UAV is explored is this paper,taking both uncertainties and actuators'dynamics into account.The error model of VTOL UAV based on trajectory tracking task is established based on modified Rodrigues parameters(MRPs),based on which analytical expression of desired attitude information is given.Then,considering the dynamics of actuators,the overall system is divided into three subsystems according to their time-scale properties,based on which a hierarchical control structure is presented.Thereafter,anti-windup controllers against system uncertainties are proposed based on translational and rotational subsystems,respectively.Stability of the overallclosed-loop system is analyzed based on singular perturbation theory.In summary,the main contributions of the proposed control strategy are presented as follows:

1)A hierarchical control structure is proposed due to the cascade property between translational and rotational subsystems.Meanwhile,the analytical solution of desired attitude information based on MRPs is given.

2)A modified disturbance rejection controller is proposed for disturbance rejection performance as well as the input saturation of actuators.

3)The singular perturbation theory is employed with consideration of the actuators'dynamics,based on which the strictly Lyapunov stability conclusion is achieved.

The rest of this paper is organized as follows.In Section II,the trajectory tracking error model is established based on MRPs,and the analytical solution of desired attitude information is given.In Section III,the overall system is divided into three subsystems,based on which a hierarchical strategy is introduced.In Section IV,anti-windup controllers are proposed based on translational and rotational subsystems, respectively.In Section V,singular perturbation theory is introduced to analyzed the stability of the overallclosed-loop system.Simulations are presented in Section V to verify the effectiveness of the proposed controlstrategy,followed by the conclusions in Section VI.

II.SYSTEM MODEL AND PROBLEM FORMULATION

A.System Model

There are totally three coordinates used in this paper,earth frame Fe,body-fixed frame Fb,and orientated frame Fd.We choose MRPs to represent the attitude.MRPs are described as a three-dimension vector without restrictions,which is defined asσ=rrr tan(α/4),where rrr andαrepresent the unit vector of rotational axis and rotation angle of the rigid body, respectively.Due to the definition of MRPs,its kinematics is given as:

whereω∈R3denotes the angular velocity of the VTOL UAV. The matrix G(σ)is given as

where[σ×]is the skew-symmetric matrix ofσ,and I3represents the identity matrix with the dimension of three by three.Concerning with the MPRs problem,please see[21]for further details.

We consider the VTOL UAV as a rigid body without deformation,and the system model is described as

whereξ,vvv∈R3are the position and velocity in the earth frame,eee3=[0 0 1]Tis the unit vector of z axis,T is the controlled thrust andτ=[τ1τ2τ3]T∈R3is the controlled torque.m and J∈ R3×3denote the mass and inertia matrix of the rigid body.ddd1and ddd2are bounded externaldisturbances.The orthogonalattitude transition matrix is denoted by R∈S O(3).And R in terms ofthe MRPs vector is shown as

Remark 1.T andτare the resulted aerodynamic force and moment described in the body-fixed coordinate,which lead to motion of the VTOL UAV.Different UAVs have differenttypes of actuators,whose aerodynamic characteristics are also different.Without loss of generality,we consider the aerodynamic force and moment as the input of the flight control system.In the next subsection,the dynamics of the thrust and torque caused by the actuators are also taken into account.

B.Problem Formulation

The trajectory tracking problem of a VTOL UAV is investigated,and the objective in this work is to design a control thrust Tdand controltorqueτd,which enable the VTOL UAV to track a desired trajectory quickly and accurately.Define the system errors as:position error?ξ=ξ?ξd,and velocity error ?vvv=vvv?˙ξd.?σ,?ωin(7)are errors of MRPs and angularvelocity given as:

where‖·‖denotes the Euclidean norm of a vector.

The analytical solution ofσd,ωd,˙ωdis shown in Section III.And Lemma 1 holds for MRPs error.

Lemma 1[22].If the attitude variable pairs(σ,ω)and (σd,ωd)both satisfy MRPs kinematics in(3),their relative attitude variable pair also satisfies(3).

Denoting the nominal values of mass and rotational inertia as m0and J0,then their errors are given as‰

The thrust and torque inputs of VTOL UAV are obtained by the servo systems,such as motors,flapping angles,controlsurfaces,etc.,which may affect the stability of the overall closed-loop system.Assuming the controller of the actuators make the error dynamics of thrust and torque satisfy:

where?T=Td?T,?τ=τd?τ.KTand Kτare the control gains,and tTand tτare the time constants.

Based on descriptions above,the system error model is represented as

The compound disturbances on the system dynamics are given as

where J?denotes the vector form of the diagonalelements of J and,forω=[ω1ω2ω3]T,we have:

III.TIME-SCALE SEPARATION AND HIERARCHICAL STRATEGY

A traditional method to design guidance and control strategies in aeronautics is assuming that the controller will enable the rotationaldynamics to converge fasterthan the translational dynamics by using an attitude controller with higher gains.

From the VTOL UAV model shown above,we know that the attitude error will converge asymptotically after the convergence of actuators.The position error will converge asymptotically after the convergence of both attitude error and actuators.According to the convergent speed of the differentparts ofthe overallsystem,we regard the translational subsystem as slow subsystem,rotational subsystem as fast subsystem,and actuators'dynamics as ultra-fast subsystem. The time-scale property of each subsystem is shown in Fig.1.

Fig.1.Time-scale property of each system.

The singular perturbation theory can be used in the controller design and stability analysis based on the multi-timescale properties of VTOL UAV system.Two time-scale factors ε1andε2are introduced to formalize the time-scale separation. Introducing the following notations:

we finally get the error model of VTOL UAV as

where the coupling terms are defined as fff1=(Rdeee3?T)+ (I3? ?R)(Rdeee3Td)?(I3? ?R)(Rdeee3?T),fff2=??τ.

The purpose of these two time-scale factors is to adjustthe gain of the controller for each subsystem,whose convergence speed will be changed correspondingly.From the system model in(5),we find that the transition matrix R and control thrust T will affect the translational motion of VTOL UAV. Notice from(6)that the transition matrix R in terms of MRPs can also be regarded as the output of the rotational subsystem.This can be considered as the cascade property of VTOL UAV.A practical hierarchical strategy is introduced to implement the control system.Consequently,translational and rotational controllers can be designed separately.The translationalcontrolleris firstly designed to extractthe desired thrust Tdand attitude matrix Rd.The desired attitude information can enable a VTOL UAV to track the desired trajectory. Thereafter,the desired torque vectorτdis determined by the rotational controller with the desired attitude matrix.At last, Tdandτdcan be treated as the input for the actuators to implementthe whole controlsystem.The following Condition is assumed in the controller design.In the procedure of controllerdesign,the subsystems/subsystem whose convergent speeds/speed are/is higher than the corresponding subsystem to be controlled are/is already converged.However,the stability should also be analyzed based on the original error model, and this assumption is only used in the procedure of controller design.The diagram of hierarchicalcontrol strategy is shown in Fig.2.

Since the desired attitude informationσd,ωd,˙ωdis determined by the virtual controller Rd,we present the analytical solution of these information.

Theorem 1.By introducing the notation δ = [δ1δ2δ3]T? Rdeee3Td,it is always possible to extract the desired attitude as:

Fig.2. Control structure of the system.

We assume the virtual controller Rdeee3Tdis differentiable, the desired angular velocity is given as

whereγ=Td+δ3.

Then,the desired angular acceleration is described as

whereΓ'is a matrix with the size of 3×3.

Proof.We notateσd=[σd1σd2σd3]T.The vectorδ is described from the defi nition of MRPs as

Notice that there is a constraint‖δ‖=1,hence,only two degrees-of-freedom of rotation can be determined by this vector.We assumeσd3=0 to calculateσd1andσd2.Since‖σd‖≤1,we have:

Consequently,we can easily prove that(17)holds.From(4) and the following equations

we can obtain(18)～(21). □

IV.CONTROLLER DESIGN

Based on the above error model,a hierarchical control scheme is presentto exploitthe cascade property.The control design steps can be summarized as follows:

Step 1.The translational controller is designed based on subsystemΣ1under the assumption that fff1=0.A linear extended state observer(ESO)[23]is introduced to estimate and compensate the compound disturbances.The desired attitude information is extracted by Theorem 1 as the input of subsystemΣ2.

Step 2.The rotational controller is designed based on subsystemΣ2under the assumption that fff2=0.A linear ESO is also used to suppress the attitude error caused by the compound disturbances.

Step 3.Stability of the overall system is analyzed based on Lyapunov analysis,taking the dynamics of actuators into account.

A.Translational Controller Design

We notate the derivative of ddd'1as hhh1(t).Then,the secondorder linear ESO for translational subsystemΣ1is

where g1and g2are positive constants to be selected.

Transforming(26)to the frequency-domain using the Laplace transform,and substitute(14)into(26),we get:

where s is the Laplace operator.

From(27),we finally get

g1and g2should satisfy that the polynomial s2+g1s+g2is Hurwitz.Here,we simply choose g1=2ω0,g2= ω20. The ESO views both internaluncertainties and externaldisturbances as the extended state to be estimated and compensated in the controller.Hence,the ESO can reject the influence caused by both internaluncertainties and externaldisturbances.

The backstepping technique and an auxiliary observer are introduced to design the trajectory tracking controller.We firstly introduce the following variables:

where kα1,kα2are strictly positive matrices,and an auxiliary observer similar to[12]is given as

where k1,k2are positive matrices to be selected.

It is easy to verify that the thrust input is bounded as

and for a candidate Lyapunov function V1=12(eeeT1eee1+eeeT2eee2), its derivative is given as

where?ddd1?ddd'1??zzz2.

The derivative of the control input Rdeee3Tdis described as follows:

B.Rotational Controller Design

By introducing the notation B=G?1(?σ),we have?ω= B˙?σ.Then we get

In order to extract the bounded controller,we rewrite the compound disturbances as

Then,the system dynamics can be rewritten as

where hhh2(t)denotes the derivative of the compound disturbances ddd'2.

Then,the second-order linear ESO for rotational subsystem Σ2is proposed as

where g3and g4are positive constants to be selected.

Transforming(39)to the frequency-domain using the Laplace transform,and substitute(15)into(39),we have:

where s is the Laplace operator.

From(40),we finally get

g3and g4should satisfy that the polynomial s2+g3s+g4is Hurwitz.Here,we simply choose g3=2ω1,g4=ω21.

To design the attitude tracking controller,we introduce the following variables:

Then,the control inputτ'dis given as:

where kβ1,kβ2are strictly positive matrices,and the observer is described as

where k3,k4are positive matrices to be selected.

The controltorque of the attitude tracking problem is finally described as

For a candidate Lyapunov function V2=12(eeeT3eee3+eeeT4eee4), its derivative is as follows:

Remark 2.From (32),the translational controller is bounded.That is,the output of thrust is saturated.From(45) and the definition of tanh function,it is clear that the output of torque is saturated.

V.STABILITY ANALYSIS

Theorem 2.Given the error model of a VTOL UAV for trajectory tracking problem in(11),with the compound disturbances shown in(12).Letthe thrustinput Tdand desired attitude information given by(30)and Theorem 1,respectively, with a linear ESO and an auxiliary observer proposed in(26) and(31).Then,let the torque inputτdin(45),with a linear ESO and an auxiliary observer designed in(39)and(4).There exist the time-scale factors such that the proposed control strategy can stabilize the system asymptotically.

Proof Consider the candidate Lyapunov functions V1and V2defined in Section IV,we define a new Lyapunov function

As shown later,‖β‖can converge to zero asymptotically. Then,the derivative of V is shown as

where

Since a1is a positive constant,the firsttwo minors of matrix Γis positive.Then,we should find the scopes ofε1,ε2such that the third to sixth minors are positive.For simplicity,let Γibe the matrix's minor of i.Then,the third to sixth minors are given as

From det(A3)＞0 and det(A4)＞0,we have:

Since det(A5)＞0,it follows that:

If det(A6)＞0,the following should be satisfied:

If the time-scale factorsε1andε2are selected based on the above requirements,matrixΓis positive.The unforced system is exponentially stable,that is,the system is input-tostate stable with the input

From(28)and(41),we know thatcan converge to ddd'1and ddd'2asymptotically,that is,can converge to zero asymptotically.Then,from Lemma 4.7 of[24],we know thatthe cascade overallsystem is asymptotically stable.Hence, limt→∞eee1=limt→∞eee2=limt→∞eee3=limt→∞eee4= limt→∞?T=limt→∞?τ=0.

From the descriptions above and the results in[25],the auxiliary observers in(31)and(44)are asymptotically stable.Consequently,limt→∞α=limt→∞˙α=limt→∞β=limt→∞˙β=0.Noticing that eee1to eee4are linear diffeomorphism of?ξ,?vvv,?σ,?ω,α,˙α,β and ˙β,hence,we have the following conclusion:

VI.SIMULATION RESULTS

Simulations are shown to illustrate the effectiveness of the proposed control strategy.We consider a VTOL UAV model with the parameters being set as:m = 4 kg, Jx= Jy= 0.08 kg·m2,and Jz= 0.14 kg·m2. The initial condition is given as:ξ(t0)=[2 3 5]Tm, vvv(t0)=[0 0 0]Tm/s,σ(t0)=[0 0 0]T,ω(t0)= [0 0 0]Trad/s.The parameters of the controller are given as follows:k2=3.5,k3=2,k4=0.2,kα1=kα2=1.5, kβ1=kβ2=2.5,ε1=0.1,ε2=0.05.

Tracking of a spiral rising trajectory with the existence of perturbation of parameters and unknown disturbance is accomplished in Matlab/Simulink.The desired trajectory is as follows:

The external disturbances acting on the translational and rotational dynamics are given as:

Simulation results are illustrated in Figs.3～7.The trajectory tracking effect of the VTOL UAV is illustrated in Fig.3. The tracking errors of position,velocity,MRPs and angular velocity are shown in Figs.4 and 5,while Fig.6 shows the estimation effects of linear ESO for both translational and rotational subsystems.The changing tendency of roll,pitch and yaw angles during the trajectory tracking are indicated in Fig.7.

Fig.3.Trajectory tracking effect.

Fig.4. Tracking error of position and velocity.

In the simulation,we fi nd that with the existence of both external disturbances and internal uncertainties,the proposed controller can enable the VTOL UAV to track a time-varying trajectory quickly and accurately.The linear ESO can estimate the disturbances and compensate for them in the control scheme to improve the control performance,whereas the proposed controller can enable the VTOL UAV to track a desired trajectory effectively.We also carry out the adaptive backstepping method in[19]for comparison.In Table I,the root-mean-square(RMS)errorofthe proposed controlstrategy is compared with that of adaptive backstepping method.It is shown that with time-varying disturbances and internal uncertainties,the control performance of the adaptive backstepping is not as well as our proposed control strategy.It is shown in Fig.6 thatthe linear ESO estimates the disturbance accurately, and the estimation error converges quickly.Figs.4 and 5 also show that the proposed control strategy have good tracking performance.

VII.CONCLUSION

In this paper,the trajectory tracking control of a VTOL UAV is investigated.The MRPs based system error model is established and the hierarchical control strategy is introduced based on the time-scale property.Then,a practicaldisturbancerejection controller is proposed with linear ESO for both translationaland rotational subsystems,respectively.The auxiliary observer is implemented to guarantee the boundedness of the control output.At last,stability conclusion of the overall system is given based on singular perturbation theory. Simulation results verify that the proposed control strategy can successfully enable the VTOL UAV to track a desired trajectory.The designed linear ESO can also estimate the compound disturbances caused by both external and internal uncertainties for higher accuracy of tracking.

Fig.5.Tracking error of MRPs and angular velocity.

TABLE I COMPARISON OF CONTROL PERFORMANCE (RMS ERROR)

Fig.6. Disturbance estimation performance.

Fig.7. Equivalent control effect of Euler angles.

REFERENCES

[1]Brockett R W.Asymptotic stability and feedback stabilization.Differential Geometric Control Theory.Boston:Birkh¨auser,1983.181?191

[2]Bouabdallah S,Siegwart R.Backstepping and sliding-mode techniques applied to an indoor micro quadrotor.In:Proceedings of the 2005 IEEE International Conference on Robotics and Automation.Barcelona, Spain:IEEE,2005.2247?2252

[3]Mian A A,Wang D B.Modeling and backstepping-based nonlinear control strategy for a 6 DOF quadrotor helicopter.Chinese Journal ofAeronautics,2008,21(3):261?268

[4]Zuo Z.Trajectory tracking control design with command-filtered compensation for a quadrotor.IET Control Theory and Applications,2010, 4(11):2343?2355

[5]Lee D,Kim H J,Sastry S.Feedback linearization vs.adaptive sliding mode controlfor a quadrotor helicopter.InternationalJournalof Control, Automationand Systems,2009,7(3):419?428

[6]Xu R,¨Ozg¨uner¨U.Sliding mode control of a class of underactuated systems.Automatica,2008,44(1):233?241

[7]Efe M ¨O.Battery power loss compensated fractionalorder sliding mode control of a quadrotor UAV.Asian Journal of Control,2012,14(2): 413?425

[8]Raffo G V,Ortega M G,Rubio F R.An integral predictive/nonlinear control structure for a quadrotor helicopter.Automatica,2010,46(1): 29?39

[9]Efe M O.Neural network assisted computationally simple PlλDμcontrolofa quadrotor UAV.IEEETransactionsonIndustrialInformatics, 2011,7(2):354?361

[10]Zemalache K M,Maaref H.Controlling a drone:comparison between a based model method and a fuzzy inference system.Applied Soft Computing,2009,9(2):553?562

[11]Bertrand S,Gu′enard N,Hamel T,Piet-Lahanier H,Eck L.A hierarchical controller for miniature VTOL UAVs:design and stability analysis using singular perturbation theory.ControlEngineeringPractice,2011,19(10): 1099?1008

[12]Abdessameud A,Tayebi A.Globaltrajectory tracking controlof VTOLUAVs without linear velocity measurements.Automatica,2010,46(6): 1053?1059

[13]Nicol C,Macnab C J B,Ramirez-Serrano A.Robust adaptive control of a quadrotor helicopter.Mechatronics,2011,21(6):927?938

[14]Ginsberg J.Engineering Dynamics.Cambridge:Cambridge University Press,2008.

[15]Abdessameud A,Tayebi A.Formation control of VTOL Unmanned Aerial Vehicles with communication delays.Automatica,2011,47(11): 2383?2394

[16]Wang L,Jia H M.The trajectory tracking problem of quadrotor UAV: globalstability analysis and controldesign based on the cascade theory. AsianJournal of Control,2014,16(2):574?588

[17]Esteban S,Gordillo F,Aracil J.Three-time scale singular perturbation controland stability analysis foran autonomous helicopteron a platform. International Journal of Robust and Nonlinear Control,2013,23(12): 1360?1392

[18]Michael N,Mellinger D,Lindsey Q,Kumar V.The GRASP multiple micro-UAV testbed.IEEE Robotics and Automation Magazine,2010, 17(3):56?65

[19]Cabecinhas D,Cunha R,Silvestre C.A nonlinear quadrotor trajectory tracking controller with disturbance rejection.Control EngineeringPractice,2014,16:1?10

[20]Wang L,Su J B.Global trajectory tracking of VTOL UAV based on disturbance rejection control.In:Proceedings of the 32nd Chinese Control Conference.Xi'an,China:IEEE,2013.4270?4275

[21]Tsiotras P.Further passivity results for the attitude control problem. IEEE Transactionson AutomaticControl,1998,43(11):1597?1600

[22]Cong B L,Liu X D,Chen Z.Distributed attitude synchronization of formation flying via consensus-based virtual structure.Acta Astronautica, 2011,68(11?12):1973?1986

[23]Zheng Q,Dong L L,Lee D H,Gao Z Q.Active disturbance rejection control for MEMS Gyroscopes.IEEE Transactions on Control Systems Technology,2009,17(6):1432?1438

[24]Khalil H K.Nonlinear Systems(Third edition).Upper Saddle River, New Jersey:Prentice Hall,2002.

[25]Olfati-Saber R.Globalconfiguration stabilization for the VTOL aircraft with strong input coupling.IEEE Transactions on Automatic Control, 2002,47(11):1949?1952

Lu Wang Ph.D.candidate in the Departmentof Automation,Shanghai Jiao Tong University,China.His research interests include disturbance rejection control,disturbance observer,nonlinear system control, unmanned aerial system and VTOL UAV control. Corresponding author of this paper.

received the B.S.degree in automatic control from Shanghai Jiao Tong University,China in 1989,the M.S.degree in pattern recognition and intelligent system from the Institute of Automation, Chinese Academy of Science,China in 1992,and the Ph.D.degree in control science and engineering from Southeast University,China in 1995.

Manuscript received October 10,2013;accepted July 23,2014.This work was supported by National Natural Science Foundation of China(61221003). Recommended by Associate Editor Changyin Sun

:Lu Wang,Jianbo Su.Trajectory tracking of vertical take off and landing unmanned aerial vehicles based on disturbance rejection control. IEEE/CAA Journal of AutomaticaSinica,2015,2(1):65?73

Lu Wang and Jiaobo Su are with the Department of Automation,Key Laboratory of System Control and Information Processing,Ministry of Education,Shanghai Jiao Tong University,Shanghai 200240,China(e-mail: wanglu1987xy@sjtu.edu.cn;jbsu@sjtu.edu.cn).

He joined the faculty of the Department of Automation,Shanghai Jiao Tong University in 1997, where he has been a full professor since 2000.His research interests include robotics,pattern recognition,and human-machine interaction.In these areas, he has published three books,more than 190 technical papers,and is the holder of 15 patents.

Dr.Su is a memberofthe TechnicalCommittee of Networked Robots,IEEE Robotics and Automation Society,a member of the Technical Committee on Human-Machine Interactions,IEEE System,Man,and Cybernetics Society, and a standing committee member of the Chinese Association of Automation. He has served as an associate editor for IEEE Transactions on Cybernetics since 2005.