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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 20151563Robust Control of Four-Rotor Unmanned AerialVehicle With Disturbance UncertaintyShafiqulIslam,Member,IEEE,PeterX.Liu,SeniorMember,IEEE,andAbdulmotalebElSaddik,Fellow,IEEEAbstractThis paper addresses the stability and track-ing control problem of a quadrotor unmanned flying robotvehicle in the presence of modeling error and disturbanceuncertainty. The input algorithms are designed for au-tonomous flight control with the help of an energy function.Adaptation laws are designed to learn and compensatethe modeling error and external disturbance uncertainties.Lyapunov theorem shows that the proposed algorithms canguarantee asymptotic stability and tracking of the linearand angular motion of a quadrotor vehicle. Compared withthe existing results, the proposed adaptive algorithm doesnot require an a priori known bound of the modeling errorsand disturbance uncertainty. To illustrate the theoreticalargument, experimental results on a commercial quadrotorvehicle are presented.Index TermsFour-rotor (quadrotor) unmanned aerialvehicle (UAV), Lyapunov method, robust adaptive control.I. INTRODUCTIONDURING the last decade, research on small-scale un-manned aerial vehicles (UAVs) has been carried out bymany researchers and industrials all over the world. The interestfor such small-scale vehicles is growing in military and civilianapplications, such as surveillance, inspection, and search-and-rescue missions in dangerous and awkward environments thatare inaccessible for human intervention. Most recent resultsin this area can be found in 20, 22, 23, and 27. Incontrast with a single-rotor helicopter, a quadrotor UAV hasmany advantages, such as low cost, hovering capability, verticaltakeoff and landing ability, small size, noiseless operation, andeasy maintenance. Autonomous flight control system designfor a small/micro-scale quadrotor UAV for both indoor andoutdoor environments is challenging because of its underac-tuated property, coupling between translational and rotationaldynamics, inherent nonlinearity associated with the dynamicalmodel, and external disturbances associated with uncertainflying environment as well as the effect of large payloadmass variation, nonlinear aerodynamic damping forces, andManuscript received August 23, 2013; revised February 20, 2014,June 6, 2014, and July 22, 2014; accepted August 30, 2014. Dateof publication October 29, 2014; date of current version February 6,2015. This work is supported by the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for first author Dr. Shafiqul Islam.S. Islam is with the University of Ottawa, Ottawa, ON K1N 6N5,Canada, and also with Carleton University, Ottawa, ON K1S 5B6,Canada (e-mail: sislamsce.carleton.ca; shafiqul11975).P. X. Liu is with Carleton University, Ottawa, ON K1S 5B6, Canada.A. El Saddik is with the University of Ottawa, Ottawa, ON K1N 6N5,Canada, and also with New York University Abu Dhabi, Abu Dhabi,United Arab Emirates.Color versions of one or more of the figures in this paper are availableonline at .Digital Object Identifier 10.1109/TIE.2014.2365441gyroscopic moments. To achieve control stability and desiredtracking of quadrotor aerial vehicle systems, various automaticflight control systems have been introduced in the literature26, 818, 21. These results can be classified intotwo categories as classical and adaptive control systems. Amodel-based proportionalintegralderivative (PID) and linearquadratic regulator control mechanism for a quadrotor systemcan be found in 1, 4, 5, 8, 10, 11, 18. Recently,Efe in 21 has proposed a neural network approach to thecontrol of a quadrotor UAV. The author showed that the neuralnetwork technique can be trained to provide the coefficients ofa finite-impulse-response-type approximator. More specifically,the idea of using a neural network was to approximate theresponse of an analog PIDcontroller with time-varyingaction coefficients and differintegration orders. The classicalalgorithms are based on a linear approximation model of thevehicle dynamics. These algorithms were developed to achievean autonomous hovering flight control for a quadrotor system.However, linear designs may not be robust, exhibiting poortracking performance in the presence of modeling errors.To improve the hovering performance, popular backsteppingcontrol techniques have been employed to address the problemof coupling in the pitchyawroll and the problem of couplingin kinematic and dynamic of the system 12, 13, 1517.In view of the control structure and their evaluation results, onecan notice that the design and stabilization of the hovering flightcontrollers were difficult and complicated. To simplify thisdesign, one may include an integral action with the backstep-ping technique so called the integral backstepping algorithm13. The idea of including the PID term with classical back-stepping design was to reduce the steady tracking errors and,at the same time, maintain asymptotic stability of the wholeclosed-loop system. However, the presence of uncertain envi-ronment and strong dynamic coupling between translationaland orientational dynamics and nonlinearities associated withpayload mass, aerodynamic, and gyroscopic effects necessi-tate an advanced controller to increase the performance andmaneuverability of the system. Specifically, most quadrotorrobots are very small and lightweight, making the systemsmore sensitive to the variation of the payloads and externaldisturbance uncertainty. As a result, additional payload mass,uncertain aerodynamic, and gyroscopic forces may changevehicle dynamics, effecting stability and tracking response ofthe system significantly.To deal with this problem, nonlinear control design has beenstudied by some researchers (see, for example, 2, 3, 6, 9,and 19). Das et al. in 2 proposed a dynamic inversion mech-anism based on the linearization technique for the hovering0278-0046 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.1564IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 2015flight control system for the quadrotor. Using the backsteppingprocedure, a direct adaptive tracking system is also presentedforquadrotoraerialvehicles19.In6,Rbinedarobust Htracking controller with the backstepping techniqueto control an uncertain quadrotor system. The backsteppingcontroller technique was employed to track the desired trajec-tory. A nonlinear robust Hcontroller was used to stabilizethe rotational dynamics of the quadrotor. Nonlinear adaptivecontrol using the backstepping technique was proposed byHuang et al. in 9 for an underactuated quadrotor system inthe presence of model parameter uncertainty. Their method canbe used to ensure the boundedness of the tracking errors ofthe position and yaw rotation via using the Lyapunov function.However, algorithm design, implementation procedure, andclosed-loop stability analysis is very complicated as the methodused nine steps and various augmented and auxiliary signals.Most recently, Cosmin and Macnab in 3 applied a fuzzyapproach to relax the model dynamics of the quadrotor system.They have employed an adaptive fuzzy technique to learn andcompensate uncertainty associated with the quadrotor modeland disturbances. However, the stability analysis relies on thefact that the fuzzy approximation errors, external disturbances,and the modeling error uncertainties are bounded by a smallpositive constant. In view of the universal fuzzy approximationtheorem, it is possible to find a fuzzy system with a largenumber of fuzzy membership functions to estimate any givenreal continuous function with a small fuzzy approximationerrors. In real-time application, the designer can only developa fuzzy system that uses a finite number of fuzzy rules andfuzzy membership functions as memory space for computationis limited in most practical applications. As a result, largefuzzy approximation errors may cause an unstable closed-loopsystem. In view of the existing designs and their stabilityanalysis, we can see that reported results demand an a prioriknown upper bound of the modeling error and uncertainty toensure stability of altitude and attitude dynamics in uncertainflying environment. In practice, it is not possible to know theexact values of the uncertainty associated with the environment(for example, wind gust), payload mass, moment of inertia,aerodynamic friction, and gyroscopic effects as they vary withdifferent flight missions for different flying environments. Asa matter of fact, unpredictable changes in outdoor flying envi-ronment increase the modeling error uncertainty, significantlymaking the flight control system design even more complicated.In this paper, we focus on the stability and tracking con-trol problem of a quadrotor flying vehicle in the presence ofmodeling error and disturbance uncertainties associated withaerodynamic and gyroscopic effects, payload mass, and otherexternal forces/torques induced from uncertain flying environ-ment. The algorithm for position tracking design combinesgravity compensation, desired linear acceleration, and propor-tional derivative (PD)-like term with an adaptive control term.The attitude controller comprises PD-like terms and desiredangular acceleration terms with an adaptation control term.An adaptation law is used to learn and compensate uncertainchanges in dynamics as a result of modeling error and distur-bance uncertainties. Lyapunov stability analysis is employedto show the tracking convergence of the closed-loop system.Compared with the existing design, the proposed method doesnotrelyonanaprioriknownupperboundofthemodelingerrorand disturbance uncertainties. The bound can be obtained bydesigning an adaptive law. As a result, the design can be appliedto a quadrotor UAV with large uncertainty appeared from thevariation of the payload mass, flying environment, moment ofinertia, aerodynamic friction, and gyroscopic effects. Variousexperimental studies on a commercial quadrotor vehicle aregiven to demonstrate the effectiveness of the proposed designfor the real-world application.This paper is organized as follows. We begin the paper byintroducing the kinematic and dynamic model of the vehiclesin Section II. In Section III, we introduce an adaptive flightcontrol strategy. A detailed stability analysis is also given inSection III. Experimental results are presented in Section IV.Finally, conclusion is given in Section V.II. MODELDYNAMICS OFQUADROTORVEHICLEWe first derive the nominal dynamical behavior by devel-oping the mathematical model of the quadrotor flying vehicle12, 13. To derive the motion dynamics for the quadrotorrobot vehicle, let us consider two main reference frames asEarth-fixed inertial reference frame eand body-fixed framefattached to the vehicle. Then, the position of the vehicle isdefined as P(t) = x(t) y(t) z(t)Tand its attitude representedby three Euler angles as (t) = (t) (t) (t)T. The vehiclehas six degrees of freedom with three translational velocitiesas V (t) = V1(t) V2(t) V3(t)Tand three rotational velocitiesas (t) = 1(t) 2(t) 3(t)Twith respect to the body-fixedframe. Then, the relationship between velocitiesP,(t) and(V,) for the two frames can be written asP =Rt()V(1) =T()(2)where Rt ?33and T ?33are the transformation veloc-ity matrix and the rotation velocity matrix between eand f,given as follows:Rt=CCSSC CSCSC+ SSCSSSS+ CCCSC SSSSCCC(3)=10S0CCS0SCC(4)where S(.)and C(.)denote sin(. ) and cos(. ), respectively. Wenow take derivatives (1) and (2) to constitute the kinematicequations for the quadrotor vehicle, i.e.,P =RtV +RtV(5) =T +?T +T?.(6)UsingRt=RtS()withtheskew-symmetricmatrixS(),i.e.,S() =032301210(7)ISLAM et al.: ROBUST CONTROL OF FOUR-ROTOR UAV WITH DISTURBANCE UNCERTAINTY1565we can write (5) and (6) in the following form:P =Rt(V + V )(8) =T + C(,)(9)where C(,) is defined asC(,) =C S + CC SS C SC CS .(10)Applying Newtons laws in the body-fixed reference frame f,the dynamic equation of motion for the vehicle subjected toforces Ftand moments Ttapplied to the center of the masscan be derived asFt=mV + (mV )(11)Tt=I + (I)(12)where m ? and I = diagIx,Iy,Iz denotes the mass andsymmetric positive definite constant inertia matrix, respec-tively. The forces and torque moments developed in the centerof the mass of the vehicle along the direction of the frame fcan be expressed asFt=Ff Fd Fg(13)Tt=Tf Ta TG(14)where Ffis the force generated by the propellers as given bythe following equation:Ff=004i=1Fi(15)with Fi= 2iwith the lift constant 0, and Fdis theaerodynamic drag forces defined asFd= KdV(16)with Kd= diagKd1,Kd2,Kd3, where Kd1 0, Kd2 0,and Kd3 0. The force from the gravity effect can be derivedas Fg= mRtG with G = 0 0 gTand g = 9. 81 m/s2. The totalmoments developed by the propellers can be defined asTf=d(F2 F4)d(F3 F1)d(F1 F2+ F3 F4)(17)where d is the distance from the center of the mass to therotor axes, and dis the drag factor. The aerodynamic frictiontorques Taare modeled asTa= Kf(18)with Kf= diagKf1,Kf2,Kf3, and Kf1, Kf2, and Kf3arethe positive constants of aerodynamic coefficients. In flyingvehicles, the gyroscopic effects appeared as a result of thevehicles body rotation in space and propeller rotation coupledwith the body rotation. It is assumed that the reaction torqueapplied to the airframe due to rigid body rotation is small. Then,the gyroscopic torques experienced by the rotors as they movealong the rotor mast with the body-fixed reference frame aredefined asTG= 4i=1 Iri(19)whereIristheinertiaoftherotorblade,andisaretheangularrotational velocities of the rotors. In view of (13)(19), we canderive the dynamical model of the quadrotor vehicle in thereference frame eas follows:Ff=mRTtP + KdRTtP + mRTtG(20)Tf=IT + IC(,) + KfT + T TI T 4i=1Iri.(21)The model can be simplified in the following form:P =xFf P Ga(22) =bu C(,) T TI T 4i=1Iri(23)with b=(IT)1, u=Tf, =T1, =I1Kf, x=(mRTt)1, = m1Kd, and Ga= EzG with Ez= 0,0,1T. From (22)and (23), one can see that the quadrotor helicopter containsunknown parameters of mass that may vary with different pay-loads in different flight missions. Notice also from (23) that theattitude dynamics associated with nonlinear centrifugal, corio-lis, nonlinear aerodynamic damping, and gyroscopic torques asa result of airframes and rotors. In addition to large modelingerror uncertainty, external disturbances from uncertain flyingenvironment bring more challenge to stabilize a small-scalequadrotor vehicle.III. ALGORITHMDESIGN ANDSTABILITYANALYSISHere, we introduce a robust adaptive flight control strategyfor a small-size quadrotor aerial vehicle. Our main objective isto develop a robust adaptive tracking algorithm that can forcethe aerial vehicle to track a desired task against modeling errorand disturbance uncertainties. In contrast with existing design,the proposed design does not require the upper bound of themodeling error and external disturbance uncertainty. To beginwith this development, we first consider that the translationaland rotational dynamics (22) and (23) are affected by externaldisturbance uncertainties as da(t) = dx(t),dy(t),dz(t)Tanddb(t) = d(t),d(t),d(t)T. Throughout our stability analy-sis, the following assumptions will be used.Assumption 1: The given desired task x1d, x3dand theirfirstandsecond derivatives arebounded andbelongs toaknowncompact set.Assumption 2: The position, orientation and their firstderivatives are available for measurement.Assumption 3: The translational transformation matrix Rtis bounded as ?Rt? rwith r 0.Assumption 4: The angular velocity transformation matrixT is also bounded as ?T? twith t 0.1566IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 2015Remark 1: Assumptions 3 and 4 exist as Eulers angles areconsidered as (/2) (/2), (/2)(/2), and 0, we then apply the following projectionadaptation algorithm to updateaandbas:a=Proj?a,asgnT(a)a?(38)b=Proj?b,bsgnT(b)b?.(39)This implies that if the parameter estimate starts in the seta(0) aand a(0) awill remain in a(t) aandISLAM et al.: ROBUST CONTROL OF FOUR-ROTOR UAV WITH DISTURBANCE UNCERTAINTY1567b(t) bt 0. For the system (36) and (37), we considerthe following composite Lyapunov-like functional:Vt= VL+ VA(40)with VL=(1/2)eTaPaea+(1/2)Ta1aTa, VA=(1/2)eTbPbeb+(1/2)Tb1bTb,Pa ?66andPb ?66arepositivedefinitematrices that satisfy the following Lyapunov equation:ATaPa+ PaAa= Qa,ATbPb+ PbAb= Qb(41)for the given Aa, Ab, Qa, and Qb, the matrix Paand Pbcanbe determined. Now, we differentiate (40) with respect to timealong with the tracking trajectory of the closed-loop systems(38) and (39). Then,Vtcan be written asVt=eTaPa ea+ eTbPb eb+Ta1aa+Tb1bb=eTaPa?Aaea+ Baasign(a)?+Ta1aa+ eTbPb?Abeb+ Bbbsign(b)?+Tb1bb. (42)Using (32) and (35) with (38) and (39),Vtcan be written asVt 12TQ(43)with = eTa,eTbTand Q = Qa0;0 Qb. Based on the aboveanalysis,wecanstateourmainresultsinthefollowingTheorem1.Theorem 1: Let assumptions 14 hold. Then, the closed-loop systems (36) and (37) along with the parameter projectionmechanism given in (38) and (39) are bounded, and the trackingerrors converge to zero as the time goes to infinity.Proof: Using (43) with (38) and (39), one can write thederivativeVtof (40) asVt 12TQ 0.(44)If ? = 0, then we conclude thatVtis negative in space. Thisimplies that (Vt,a,b) L. Since all the variables on theright-hand side of (36) and (37) are bounded, then we can alsoconclude that L. Hence, is uniformly continuous andbounded. We now take the integral (44) from 0 to T, we haveVt(T) Vt(0) T?0min. (Q)2?2.(45)Using (40), we can write the tracking error bound asT?0min. (Q)2?212(0)TQ(0) +12Ta(0)1aa(0)+12Tb(0)1bb(0)(46)with Vt(T) 0. This implies that L2. Since is uniformlycontinuous over the interval 0,) with T = , then usingBarbalats lemma 14, we can conclude that limtVt=0 and limt = 0 provided that the parameter estimationerrors are bounded by the projection scheme.Notice that the sign(. ) function may cause discontinuity in(31), (32), (34), (35), (38), and (39). To smooth out the controlinputs, we can estimate sgn(. ) by using the bounded inputatanh(a/?o1) andbtanh(b/?o2) with the small value of?o1 0, ?o2 0, and tanh(. ) is a smooth bounded satura-tion function, where tanh(k/?op) = tanh(k/?op)1),.,tanh(k/?op)n)Twith k = a,b, and p = 1, 2. Then, weintroduce the following Theorem 2.Theorem 2: Let assumptions 14 hold. Then, the closed-loopsystemsformulatedby(28),(30),(31),(33),(34),(38),and(39) are bounded, and the tracking errors converge to a small setthat is close to zero.Proof: The proof of Theorem 2 can be shown along theline of the proof of Theorem 1. We first replace the sgn(. ) func-tion in controller and adaptation laws given in (30), (31), (33),(34),(38),and(39).WethenfollowthestepsusedforTheorem1.After some manipulations, we can writeVtas follows:Vt 12min(Q)?2+ ?To?PoBo?c(47)where o= eTaeTbT, Bo= Ba0;0 Bb, c= (1o+ 2o),Po= Pa,0;0 Pb, ?atanh(a/?01) atanh(a/?01)? 1oand ?btanh(b/?02) btanh(b/?02)? 2o(ea,eb,Qda,Qdb,a,b) (c a b a b), ea c1,eb c2,c1= ea| eTaPaea c1,c2= eb| eTbPbeb c2withc1 0and c2 0,Qda da ?3n= x1d, x1d, x1dT,Qdb db ?3n= x3d, x3d, x3dT, a(t)a, b(t)bfor all t 0. We can further simplifyVtasVt 14min(Q)?o?2+2max(P0)min(Q)?Bo?22c.(48)This implies thatVtis negative outside the compact set as oo|?o? (2max(Po)/min(Q)?B0?c. This means thatthe tracking error signals are uniformly ultimately bounded asthe boundedness property of the parameter estimates a, b,a,andbare guaranteed by using the projection feature of theirlearning estimates.Remark 2: The proposed design does not depend on ana priori know upper bound of the modeling error and distur-bance uncertainties. The bound can be obtained by designingan adaptive law. Therefore, the design can be applied fora quadrotor UAV with large modeling error and disturbanceuncertainty associated with outdoor flying environment, such aswindgust,thepayloadmass,uncertainenvironment,momentofinertia, nonlinear aerodynamic friction, and gyroscopic effects.Remark 3: The design can be directly applied on a com-mercially available quadrotor UAV, such as 1 and 4, byjust adding an adaptive term with exiting PD controller. It isworth noting from the vehicle model that the lifting forces maysimply be designed by just compensating the gravity forces.However, mass parameters along with aerodynamic drag forcesmay vary with different flight missions in uncertain outdoorenvironment. As a result, an adaptation mechanism must beincluded in the design to deal with the modeling error anddisturbance uncertainties.1568IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 2015IV. EXPERIMENTALRESULTSTo validate the control algorithm developed in the previoussection, we conduct various experiments on a commercialquadrotor flying robot vehicle 1. For our control evaluation,we decompose the control design into two separate loops asinner and outer loops. The inner control loop is designed byusing the attitude control input to control the roll, pitch, andyawangles.Theouterloopcontainsaltitudeandvirtualpositioncontroller to calculate desired thruster and desired rolling aswell as the pitching motion. The inner loop runs on an onboardmicroprocessor at 1 kHz with access to IMU data at 300 Hz.The height of the vehicle is measured by an onboard barometricpressure sensor. For outdoor environment, the position of thevehicle is obtained by an onboard GPS, whereas the visualtracker provided by Phonenix Technologies Inc. is employedto measure the position of the vehicle in indoor environment.The angular velocity of the vehicle is measured by onboardgyros. Before implementing the proposed design on an actualquadrotor vehicle, we conduct extensive simulation studies ona simulated model of the vehicle. The purpose of the simulationstudies is to determine the control gains and threshold values ofthe design parameters for experimental evaluation. The mass ofthe vehicle varies from 0.5 to 1 kg. The external forces/torquesfrom aerodynamic dynamic friction are generated by consider-ing the influence of the translational and rotational velocity ofthe air. The translational and rotational air velocities for threedirections are assumed to be 5,5,5Tm/s and 5,5,5Trad/s.The inertia components Ixx, Iyy, and Izzin x, y, and z arevaried from 3.9 to 4.5, 4.4 to 5.4, and 4.9 to 5.9 in gm2,respectively. The values of the parameters aand bin transla-tional and rotational dynamics are varied from 10 to 10 N andfrom 10 to 10 N m. With this design setup, we simulate theproposed method and obtain control design parameters that areused for experimental evaluation. For experimental evaluation,we first implement the PD terms of the proposed design on thevehicle and achieve hovering performance. Then, we integrateadaptive terms with the PD control terms and examine the sta-bility and tracking performance by varying the values of aandbfrom 5 to 5 N and from 5 to 5 N m. By using this setup,we perform an experimental test on the given quadrotor vehiclefor both indoor and outdoor flying environments. The controlgains are chosen by extensive simulation example as Kpa=diag20 20 20, Kda= diag20 20 20, K= diag20 20 20,Kpb= diag50 50 50, and Kdb= diag50 50 50. The valuesof Qaand Qbare chosen as Qa= Qb= I66. The learninggains are chosen as a= b= 1. The implementation blockdiagram of the proposed design is shown in Fig. 1.We now evaluate the proposed attitude control and stabilityon the given quadrotor vehicle. The performance of attitudecontrol is very important for quadrotor stability as it is directlyrelated to the performance of the actuators. Let us now applythe proposed attitude control algorithm to stabilize the attitudedynamics of the quadrotor in outdoor environment. Our aim inthisexperimentistostabilizeattitudedynamicsofthequadrotorin a free-flight mission. In other words, the attitude controlalgorithm has to ensure zero attitude angles in free flight againstexternal disturbance. The experimental results are depicted inFigs. 24. In view of these results, we can notice that roll, pitch,Fig. 1.Implementation block diagram of the proposed design on acommercial quadrotor flying vehicle.Fig. 2.Time history of roll output angle in degrees.Fig. 3.Time history of pitch output angle in degrees.and yaw angles are stabilized close to zero despite the externaldisturbance injecting along the , , and direction. All thecontrollers run on an onboard computer and use an onboardIMU and barometric pressure sensor to control height, position,roll, pitch, and yaw.We now apply the position control algorithm for autonomoustakeoff, landing, and tracking of the given desired trajectorieszd, xd, and yd. The evaluation results are depicted in Figs. 5and 6. Fig. 5 presents the time history of the desired andactual altitude tracking of the vehicle. Fig. 6 depicts the timehistory of the actual and desired trajectory of the position of thevehicle. In view of Fig. 5, we can observe the slight deviationbetween actual and desired takeoff (014 s) and landing phases(3840 s). This appears from actuator dynamics as the model isISLAM et al.: ROBUST CONTROL OF FOUR-ROTOR UAV WITH DISTURBANCE UNCERTAINTY1569Fig. 4.Time history of yaw output angle in degrees.Fig. 5.Desired altitude tracking zd(blue solid) and actual output z (reddash) in meters in outdoor flying environment.Fig. 6.Actual and desired path of quadrotor UAV in (y,x) in (m,m).Blue (y,x) is the actual path, and red (y,x) is the desired path.relatively slower in the takeoff and faster in the landing. On theother hand, oscillation during the transient phase can be seendue to the measurement error associated with the barometricpressure sensor. It should also be noted that the altitude flightin the z-direction is influenced by thrust and gravity forces.Therefore, the oscillation in the altitude tracking also appearedfrom battery voltage as the thrust forces are generated by themotors. In Fig. 6, we also observe the position errors due to thepresence of the measurement error associated with the GPS.For comparison purposes, we now evaluate the followingPID-like control algorithm:u(t) = Kpe(t) + Kd e(t) + Kit?0e(t)dt(49)whereKp,Kd,andKiarethepositivecontrolgains.Wechooseto use the above industrial control algorithm for comparisonpurposes as it has been widely used for controlling a quadrotorvehicle. The PID controller is individually designed for eachaxis by considering the motion dynamics of the quadrotorvehicle. To do that, the linear and angular acceleration dynamicof the vehicle is first simplified as follows: x =u1m(cossincos + sinsin)(50) y =u1m(cossinsin sincos)(51) z = g +u1m(coscos)(52) =u2Ixx, =u3Iyy, =u4Izz(53)where u2, u3, u4and u1are the inputs, Ixx, Iyy, and Izzrepresents the moment of inertia with respect to the x-, y-,and z-axes, respectively. The products of inertia are assumedto be zero as they are very small and Ixx Iyydue to sym-metry. The controller is designed by linearizing the quadro-tor model (50)(53) at the hover state x = x0, y = y0, z =z0, = 0, = 0, = 0, x = 0, y = 0, z = 0, = 0, =0, = 0. We first design the attitude controller by as-suming that the vehicle is close to the hovering flight statewith small roll and pitch angles. Hence, by assuming smallroll and pitch angles, we can write cos 1, cos 1,sin , and sin . Now, by assuming a hovering state,the PID-based inputs are used to calculate the control mo-ment for attitude dynamics as u2= Kpe(t) + Kd e(t) +Ki?t0e(t)dt, u3= Kpe(t) + Kd e(t) + Ki?t0e(t)dt,u4= Kpe(t) + Kd e(t) + Ki?t0e(t)dt with Kp 0,Kp 0, Kp 0, Kd 0, Kd 0, Kd 0, Ki0, Ki 0, Ki 0, e= (d ), e= (d ), e=(d ), e= (d), e= (d) and e= ( d ).The desired lifting/thruster force is then calculated by using thePID input algorithm as u1= Kpzez+ Kdz ez+ Kiz?t0ezdt,where ez= (zd z), ez= ( zd z), Kpz 0, Kdz 0, andKiz 0. Then, we evaluate the position and attitude inputon a given quadrotor vehicle in simulated and experimentalenvironment. The simulation is used to obtain the bound on thecontrol design parameters. The experimental evaluation processremains similar to a robust adaptive control method that wasillustrated in the previous section. We conduct an experimentin indoor flying environment as the vehicle drifts away in thepresence of external disturbances in outdoor environment. Theposition of the vehicle in indoor is measured from four precisevisual trackers provided by Phonenix Technologies Inc. Thevehicle is controlled to hover from rest, then from hoveringflight to takeoff, then free-flight control and from free flight tohovering flight. For the given task, the bound on the controldesign parameters is calculated by using simulation examplesthrough a trial-and-error search method. The control designparameters are selected as Kpx= 0. 4, Kpy= 0. 4, Kpz= 0. 4,Kdx= 0. 4, Kdy= 0. 4, Kdz= 0. 4, Kix= 0. 02, Kiy= 0. 02,and Kiz= 0. 02. The attitude dynamics are assumed to be fasterthan outer-loop position dynamics. Hence, the control gains forattitude dynamics are chosen as Kp= Kp= Kp= Kd=1570IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 2015Fig. 7.Desired altitude tracking zd(blue) and actual output z (red) inmeters with disturbance in indoor flying environment.Fig. 8.Actual and desired path of quadrotor UAV in (y,x) in (m,m)during from free flight to hovering flight. Blue (y,x) is the actual path,and red (y,x) is the desired path with disturbance in indoor flyingenvironment.Kd= Kd= Ki= Ki= Ki= 1. The external input dis-turbancesfortranslationalandrotationaldynamicsarealsocho-sen through a trial and search method. The maximum boundson the external input disturbances for the position dynamicsare selected as 0.5 N. The rotational input dynamics are keptfree from external disturbances. The experimental results aredepicted in Figs. 7 and 8. Figs. 7 and 8 show the time historyof the actual and desired trajectory of the altitude and positionof the vehicle under external disturbances. From these results,we can notice the large deviation between actual and desiredposition trajectories during takeoff, free flight, and hoveringflight states. We can also notice from Fig. 8 that the vehicledrifts away from desired free-flight state to hovering, resultingin very large tracking errors in the x-, y-, and z-directions.V. CONCLUSIONIn this paper, we have presented a robust adaptive trackingsystem for a quadrotor flying robot vehicle in the presenceof the modeling error and disturbance uncertainties. The algo-rithms have been developed by using a Lyapunov-like energyfunction by assuming that all the states are available for mea-surement. The design can be used to reject the effects of themodeling errors and disturbances acting on the vehicle and, atthe same time, maintain the bounded stability and tracking con-trol property of the whole closed-loop system. Projection-basedadaptive schemes have been employed to ensure the bound-edness property of the parameters. Unlike existing design, theproposed algorithm does not require an a priori known boundof the modeling error and disturbance uncertainty. The boundcan be obtained by designing an adaptive law. The method canbe applied to a quadrotor system with large modeling error anddisturbance uncertainty associated with payload mass, strongwind gust, moment of inertia, nonlinear aerodynamic friction,and gyroscopic effects. Experimentation with comparison withother methods has been carried out on a commercial quadrotorvehicle to demonstrate the theoretical development of thispaper. It is noticed from our results that the IMU measure-ment is very noisy and oscillatory. We observe that a smallIMU measurement error can generate large displacement forhigh-speed motion of the quadrotor vehicle. To deal with thisproblem, an observer can be used to estimate the position andorientation. Among various existing observer designs, high-gain observers are a well-known and popular technique foraccurate and fast reconstruction of the actual state (see 24and references therein). However, as a result, the demand ofthe high gains to achieve satisfactory performance in high-gain-observer-based design may amplify the noise associated withthe state estimates. A potential solution to this problem is toadd an integral term of the output estimation error with theproportional error term 25.REFERENCES1 AscendingTechnologies(AscTec),Krailling,Germany.Online.Available: http:/www.asctec.de2 A. Das, F. Lewis, and S. Subbarao, “Dynamic neural network based robustbackstepping control approach for quadrotors,” presented at the AIAAGuidance, Navigation Control Conf. 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Control, vol. 41,no. 2, pp. 177188, Feb. 1996.8 L.A.J.Toledo,M.Sigut,andJ.Felipe,“Stabilizationandaltitudetrackingof a four-rotor microhelicopter using the lifting Operators,” IET ControlTheory Appl., vol. 3, no. 4, pp. 452464, Apr. 2009.9 M. Huang, B. Xian, C. Diao, K. Yang, and Y. Feng, “Adaptive track-ing control of underactuated quadrotor unmanned aerial vehicles viabackstepping,” in Proc. Amer. Control Conf., Baltimore, MD, USA,Jun. 30Jul. 2, 2010, pp. 20762081.10 P. E. I. Pounds, P. I. Corke, and R. E. Mahony, “Modelling and control ofa large quadrotor robot,” Control Eng. Pract., vol. 18, no. 7, pp. 691699,Jul. 2010.11 P. Castillo, A. Dzul, and R. Lozano, “Real-time stabilization and trackingof four-rotor mini rotorcraft,” IEEE Trans. Control Syst. Technol., vol. 12,no. 4, pp. 510516, Jul. 2004.12 S. Bouabdallah and R. Siegwart, “Backstepping and sliding-mode tech-niques applied to an indoor micro quadrotor,” in Proc. IEEE Int. Conf.Robot. Autom., Barcelona, Spain, Apr. 1822, 2005, pp. 22592264.13 S. Bouabdallah and R. Siegwart, “Full control of a quadrotor,” inProc. IEEE/RSJ Int. Conf. Intell. Robots Syst., San Diego, CA, USA,Oct. 29Nov. 2 2007, pp. 153158.ISLAM et al.: ROBUST CONTROL OF FOUR-ROTOR UAV WITH DISTURBANCE UNCERTAINTY157114 S. Sastry and M. Bosdon, Adaptive control: Stability, Convergence andRobustness.Upper Saddle River, NJ, USA: Prentice-Hall, 1989.15 T. Madani and A. Benallegue, “Control of a quadrotor via full statebackstepping technique,” in Proc. 45th IEEE Conf. Decision Control,San Diego, CA, USA, Dec. 1315, 2006, pp. 1315.16 T. Madani and A. Benallegue, “Backstepping control for a quadrotorHelicopter,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Beijing,China, 2006, pp. 32553260.17 T.MadaniandA.Benallegue,“Backsteppingslidingmodecontrolappliedto a miniature quadrotor flying robot,” in Proc. 32nd IEEE IECON, Paris,France, 2006, pp. 700705.18 T. Hamel, R. Mahony, and A. 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