帶機(jī)械爪的無人機(jī)設(shè)計(jì)與控制【全套含CAD圖紙、說明書】,全套含CAD圖紙、說明書,機(jī)械,無人機(jī),設(shè)計(jì),控制,全套,CAD,圖紙,說明書 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; ).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 http:/ieeexplore.ieee.org.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 http:/www.ieee.org/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 ou