RECENTLY, vertical take-off and landing (VTOL) unmanned

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1 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. JANUARY 5 65 Trajectory Tracking of Vertical Take-off an Laning Unmanne Aerial Vehicles Base on Disturbance Rejection Control Lu Wang an Jianbo Su Abstract We investigate the trajectory tracking problem of vertical take-off an laning (VTOL) unmanne aerial vehicles (UAV) an propose a practical isturbance rejection control strategy. Firstly the nonlinear error moel is establishe completely by the moifie Rorigues parameters while consiering ynamics of the servo actuators. Then a hierarchical control scheme is applie to esign the translational an rotational controllers base on the time-scale property of each subsystem respectively. An the linear extene state observer an auxiliary observer are use to eal with the uncertainties an saturation. At last global stability of the close-loop system is analyze base on the singular perturbation theory. Simulation results show the effectiveness of the propose control strategy. Inex Terms Unmanne aerial vehicles (UAV) trajectory tracking control extene state observer singular perturbation theory. I. INTRODUCTION RECENTLY vertical take-off an laning (VTOL) unmanne aerial vehicles (UAV) have attracte increasing interest in researches an applications in both military an civil society such as rescue in isasters unmanne inspection an roa traffic supervision. The motivation also comes from acaemic research institutes since it can be use as low cost testbes for robotics stuies. However the VTOL UAV moel which has been wiely investigate in most works is known as a class of uneractuate system with nonholonomic constraints of secon orer. Accoring to the necessity of Brockett there is no gloss or time invariant controller that can stabilize the uneractuate system to the equilibrium point [. Hence new methoology shoul be investigate for this kin of systems. Trajectory tracking control of VTOL UAV is a challenging work ue to its coupling property external isturbances system uncertainties etc. Several inspire approaches have been investigate such as backstepping control [ 4 sliing moe control [ 5 7 feeback linearization [5 moel preictive control [8 neural networks [9 fuzzy control [ observer base control [ etc. However there are still some prominent problems to be consiere an resolve. ) The attitue representation an esire attitue extraction. In most previous works [- 3 Euler angles are use Manuscript receive October 3; accepte July 3 4. This work was supporte by National Natural Science Founation of China (63). Recommene by Associate Eitor Changyin Sun Citation: Lu Wang Jianbo Su. Trajectory tracking of vertical take off an laning unmanne aerial vehicles base on isturbance rejection control. IEEE/CAA Journal of Automatica Sinica 5 (): Lu Wang an Jiaobo Su are with the Department of Automation Key Laboratory of System Control an Information Processing Ministry of Eucation Shanghai Jiao Tong University Shanghai 4 China ( wanglu987xy@sjtu.eu.cn; jbsu@sjtu.eu.cn). to represent the attitue of rigi boy. However the simplifie kinematics is often use as φ = ω x θ = ω y ψ = ω z () where φ θ ψ enote the roll pitch an yaw angles. ω x ω y ω z are the angular velocity of the rigi boy respectively. It is pointe out that the original kinematics of Euler angles is escribe as [4 φ = θ ψ [ sin φ tan θ cos φ tan θ cos φ sin φ sin φ sec θ cos φ sec θ [ ωx ω y ω z. () The existence of transcenental functions in () makes it ifficult to esign a control strategy. Noticing that () is a simplifie form of () with the assumption that the rigi boy rotates only in one irection at a time an the roll/pitch angle changes when the pitch/roll angle equals to. However the assumption above is an ieal instance which is infeasible. Meanwhile the simplification will ecrease the control accuracy. Furthermore we fin that the system moel base on Euler angles is not available when pitch angle θ = ±π/. Especially consiering the error of sensors an calculation both attitue estimation an control algorithm base on Euler angles cannot work near the state of θ = ±π/. Moreover most works provie the esire attitue angular velocity an angular acceleration irectly by the position controller. Only [ 5 present the analytical solution of the esire attitue information base on quaternion. ) Stability of hierarchical control structure. Concerning with the hierarchial control strategy the position an attitue controllers can be esigne separately for translational an rotational subsystems respectively. Although the above strategy can be introuce for controller esign the stability shoul be analyze base on the overall close-loop system since the attitue s tracking is an asymptotical proceure which makes the attitue error between the actual an esire one a necessary concern in the analysis. However in [ stability is only analyze for each subsystem. 3) Dynamics of actuators an its influence on the closeloop stability. In [ 5 6 an [ 7 cascae theory an singular perturbation theory are use to acquire the stability of the close-loop system. In [8 the relationship is given between actuators an control input of a VTOL UAV. However ynamics of actuators is never consiere. 4) Controller esign with internal uncertainties an external isturbances. Several researches relate to controller esign against uncertainties have been stuie base on sliing moe control [ 5 7 neutral networks [9 fuzzy systems [ aaptive

2 66 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. JANUARY 5 backstepping control [9 isturbance observer [6 etc. However the chartering of SMC convergence rate of weights in neutral networks an fuzzy systems will limit the applications of these methos in practical. The aaptive algorithm can only eal with the external isturbances [9. The isturbance observer is aopte to eal with the uncertainties [ however the saturation of the actuators is not consiere. Base on the above review trajectory tracking control of a VTOL UAV is explore is this paper taking both uncertainties an actuators ynamics into account. The error moel of VTOL UAV base on trajectory tracking task is establishe base on moifie Rorigues parameters (MRPs) base on which analytical expression of esire attitue information is given. Then consiering the ynamics of actuators the overall system is ivie into three subsystems accoring to their time-scale properties base on which a hierarchical control structure is presente. Thereafter anti-winup controllers against system uncertainties are propose base on translational an rotational subsystems respectively. Stability of the overall close-loop system is analyze base on singular perturbation theory. In summary the main contributions of the propose control strategy are presente as follows: ) A hierarchical control structure is propose ue to the cascae property between translational an rotational subsystems. Meanwhile the analytical solution of esire attitue information base on MRPs is given. ) A moifie isturbance rejection controller is propose for isturbance rejection performance as well as the input saturation of actuators. 3) The singular perturbation theory is employe with consieration of the actuators ynamics base on which the strictly Lyapunov stability conclusion is achieve. The rest of this paper is organize as follows. In Section II the trajectory tracking error moel is establishe base on MRPs an the analytical solution of esire attitue information is given. In Section III the overall system is ivie into three subsystems base on which a hierarchical strategy is introuce. In Section IV anti-winup controllers are propose base on translational an rotational subsystems respectively. In Section V singular perturbation theory is introuce to analyze the stability of the overall close-loop system. Simulations are presente in Section V to verify the effectiveness of the propose control strategy followe by the conclusions in Section VI. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Moel There are totally three coorinates use in this paper earth frame F e boy-fixe frame F b an orientate frame F. We choose MRPs to represent the attitue. MRPs are escribe as a three-imension vector without restrictions which is efine as σ = r tan(α/4) where r an α represent the unit vector of rotational axis an rotation angle of the rigi boy respectively. Due to the efinition of MRPs its kinematics is given as: σ = G(σ)ω (3) where ω R 3 enotes the angular velocity of the VTOL UAV. The matrix G(σ) is given as G(σ) = ( σ T ) σ I 3 + [σ + σσ T (4) where [σ is the skew-symmetric matrix of σ an I 3 represents the ientity matrix with the imension of three by three. Concerning with the MPRs problem please see [ for further etails. We consier the VTOL UAV as a rigi boy without eformation an the system moel is escribe as ξ = v m v = e 3 mg Re 3 T + σ = G(σ)ω J ω = ω Jω + τ + where ξ v R 3 are the position an velocity in the earth frame e 3 = [ T is the unit vector of z axis T is the controlle thrust an τ = [ τ τ τ 3 T R 3 is the controlle torque. m an J R 3 3 enote the mass an inertia matrix of the rigi boy. an are boune external isturbances. The orthogonal attitue transition matrix is enote by R SO(3). An R in terms of the MRPs vector is shown as R = I 3 4( σt σ) ( + σ T σ) [σ + 8 ( + σ T σ) [σ. (6) Remark. T an τ are the resulte aeroynamic force an moment escribe in the boy-fixe coorinate which lea to motion of the VTOL UAV. Different UAVs have ifferent types of actuators whose aeroynamic characteristics are also ifferent. Without loss of generality we consier the aeroynamic force an moment as the input of the flight control system. In the next subsection the ynamics of the thrust an torque cause by the actuators are also taken into account. B. Problem Formulation The trajectory tracking problem of a VTOL UAV is investigate an the objective in this work is to esign a control thrust T an control torque τ which enable the VTOL UAV to track a esire trajectory quickly an accurately. Define the system errors as: position error ξ = ξ ξ an velocity error ṽv = v ξ. σ ω in (7) are errors of MRPs an angular velocity given as: where σ an R = RR T (5) σ = σ σ ω = ω Rω (7) is the inverse of σ known as σ = σ is known as the error of attitue transition matrix. Operator enotes the prouction of MRPs which is escribe in (8) with two MRPs variables of σ an σ σ σ = ( σ )σ + ( σ )σ σ σ + σ σ σ T σ where enotes the Eucliean norm of a vector. The analytical solution of σ ω ω is shown in Section III. An Lemma hols for MRPs error. Lemma [. If the attitue variable pairs (σ ω) an (σ ω ) both satisfy MRPs kinematics in (3) their relative attitue variable pair also satisfies (3). Denoting the nominal values of mass an rotational inertia as m an J then their errors are given as { m = m m (9) J = J J. The thrust an torque inputs of VTOL UAV are obtaine by the servo systems such as motors flapping angles control (8)

3 WANG AND SU: TRAJECTORY TRACKING OF VERTICAL TAKE-OFF AND LANDING UNMANNED AERIAL VEHICLES 67 surfaces etc. which may affect the stability of the overall close-loop system. Assuming the controller of the actuators make the error ynamics of thrust an torque satisfy: T = K T T τ = K τ τ () t T t τ where T = T T τ = τ τ. K T an K τ are the control gains an t T an t τ are the time constants. Base on escriptions above the system error moel is represente as ξ = ṽv ṽv = ge 3 Re 3 T + m ξ σ = G( σ) ω ω = J ( L(ω)J + τ) + ( R ω [ ω Rω ) T = K T T τ = K τ τ. t T t τ () The compoun isturbances on the system ynamics are given as = m ( ṽv ge 3 + m m ξ ) =J [ J ω L(ω) J J( R ω [ ω Rω ) () where J enotes the vector form of the iagonal elements of J an for ω = [ ω ω ω 3 T we have: [ ω ω 3 ω ω 3 L(ω) = ω 3 ω ω 3 ω. (3) ω ω ω ω III. TIME-SCALE SEPARATION AND HIERARCHICAL STRATEGY A traitional metho to esign guiance an control strategies in aeronautics is assuming that the controller will enable the rotational ynamics to converge faster than the translational ynamics by using an attitue controller with higher gains. From the VTOL UAV moel shown above we know that the attitue error will converge asymptotically after the convergence of actuators. The position error will converge asymptotically after the convergence of both attitue error an actuators. Accoring to the convergent spee of the ifferent parts of the overall system we regar the translational subsystem as slow subsystem rotational subsystem as fast subsystem an actuators ynamics as ultra-fast subsystem. The time-scale property of each subsystem is shown in Fig.. Fig.. Time-scale property of each system. The singular perturbation theory can be use in the controller esign an stability analysis base on the multi-timescale properties of VTOL UAV system. Two time-scale factors an ε are introuce to formalize the time-scale separation. Introucing the following notations: ω = ω τ = τ K T = ε K T K τ = ε K τ we finally get the error moel of VTOL UAV as ξ = ṽv Σ Σ ṽv = ge 3 (R e 3 T f m ) + ξ (4) σ = G( σ) ω ω = J ( L(ω) J + f ) + + J τ ( R ω R[ ω ω ) (5) Σ 3 K T ε T = T t T ε τ = K τ t τ τ (6) where the coupling terms are efine as f = (R e 3 T ) + (I 3 R)(R e 3 T ) (I 3 R)(R e 3 T ) f = τ. The purpose of these two time-scale factors is to ajust the gain of the controller for each subsystem whose convergence spee will be change corresponingly. From the system moel in (5) we fin that the transition matrix R an 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 regare as the output of the rotational subsystem. This can be consiere as the cascae property of VTOL UAV. A practical hierarchical strategy is introuce to implement the control system. Consequently translational an rotational controllers can be esigne separately. The translational controller is firstly esigne to extract the esire thrust T an attitue matrix R. The esire attitue information can enable a VTOL UAV to track the esire trajectory. Thereafter the esire torque vector τ is etermine by the rotational controller with the esire attitue matrix. At last T an τ can be treate as the input for the actuators to implement the whole control system. The following Conition is assume in the controller esign. In the proceure of controller esign the subsystems/subsystem whose convergent spees/spee are/is higher than the corresponing subsystem to be controlle are/is alreay converge. However the stability shoul also be analyze base on the original error moel an this assumption is only use in the proceure of controller esign. The iagram of hierarchical control strategy is shown in Fig.. Since the esire attitue information σ ω ω is etermine by the virtual controller R we present the analytical solution of these information. Theorem. By introucing the notation δ = [ δ δ δ 3 T R e 3 T it is always possible to extract the esire attitue as:

4 68 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. JANUARY 5 Fig.. Control structure of the system. η = δ3 T + σ = T η ( + η ) [ δ δ. (7) We assume the virtual controller R e 3 T is ifferentiable the esire angular velocity is given as ω =Γ(δ) δ (8) Γ(δ)= T γ δ δ δ + T γ δ γ δ T γ δ δ δ γ (9) δ T δ T where γ = T + δ 3. Then the esire angular acceleration is escribe as ω = Γ(δ δ) δ + Γ(δ) δ. () Γ(δ δ) is the time-erivative of Γ(δ) which is Γ(δ δ) = (δt δγ + T δ T δ + T δ 3 ) T γ Γ(δ) + T γ Γ () where Γ is a matrix with the size of 3 3. Proof. We notate σ = [ σ σ σ 3 T. The vector δ is escribe from the efinition of MRPs as δ = T ( + σ T σ ) 8σ σ 3 4σ ( σ T σ ) 8σ σ 3 + 4σ ( σ T σ ) 4( σ σ + σ 3 ) + ( σt σ ). () Notice that there is a constraint δ = hence only two egrees-of-freeom of rotation can be etermine by this vector. We assume σ 3 = to calculate σ an σ. Since σ we have: δ3 η = + T = σ + σ = σt σ + σ T σ (3) an δ = 4T σ ( σ T σ ) ( + σ T σ ) δ = 4T σ ( σ T σ ) ( + σ T σ ). (4) Consequently we can easily prove that (7) hols. From (4) an the following equations T = δ T δ T (5) γ + δ + δ = T γ we can obtain (8) (). IV. CONTROLLER DESIGN Base on the above error moel a hierarchical control scheme is present to exploit the cascae property. The control esign steps can be summarize as follows: Step. The translational controller is esigne base on subsystem Σ uner the assumption that f =. A linear extene state observer (ESO) [3 is introuce to estimate an compensate the compoun isturbances. The esire attitue information is extracte by Theorem as the input of subsystem Σ. Step. The rotational controller is esigne base on subsystem Σ uner the assumption that f =. A linear ESO is also use to suppress the attitue error cause by the compoun isturbances. Step 3. Stability of the overall system is analyze base on Lyapunov analysis taking the ynamics of actuators into account. A. Translational Controller Design We notate the erivative of as h (t). Then the seconorer linear ESO for translational subsystem Σ is ẑz = ge 3 Re 3 T m + ẑz ξ + g (ṽv ẑz ) ẑz = g (ṽv ẑz ) (6) where g an g are positive constants to be selecte. Transforming (6) to the frequency-omain using the Laplace transform an substitute (4) into (6) we get: { sẑz = (ẑz ) + sṽv + g (ṽv ẑz ) (7) sẑz = g (ṽv ẑz ) where s is the Laplace operator. From (7) we finally get g ẑz = s + g s + g. (8)

5 WANG AND SU: TRAJECTORY TRACKING OF VERTICAL TAKE-OFF AND LANDING UNMANNED AERIAL VEHICLES 69 g an g shoul satisfy that the polynomial s + g s + g is Hurwitz. Here we simply choose g = ω g = ω. The ESO views both internal uncertainties an external isturbances as the extene state to be estimate an compensate in the controller. Hence the ESO can reject the influence cause by both internal uncertainties an external isturbances. The backstepping technique an an auxiliary observer are introuce to esign the trajectory tracking controller. We firstly introuce the following variables: { e = ξ α (9) e = ṽv α + k e. Then the control input is esigne as e3 T = m g ξ + k α tanh(α) + k α tanh( α) m [e 3 g ξ + k α tanh(α) + k α tanh( α) R e 3 = T (3) where k α k α are strictly positive matrices an an auxiliary observer similar to [ is given as α = k α tanh(α) k α tanh( α) + ẑz + (k + k )e + ( k )e where k k are positive matrices to be selecte. It is easy to verify that the thrust input is boune as (3) T m ( ξ + k α + k α + g) (3) an for a caniate Lyapunov function V = (et e + e T e ) its erivative is given as V λ min (k ) e λ min (k ) e + e ( + f m ) (33) where ẑz. The erivative of the control input R e 3 T is escribe as follows: δ = m [ ξ δ = m [ ξ (4) + k α tanh(α) (α) + k α tanh( α) α + k α α ( α) tanh(α) tanh(α) α + k α α (α) (α) + tanh( α)... tanh( α) k α α + k α α ( α) ( α) with... α being... tanh(α) α = k α (α) (34) tanh( α) α k α α + g (ṽv ẑz )+ ( α) (k + k )(e 3 g Re 3 T + ẑz ξ α+ m k e k e ) + ( k )(e k e ). (35) B. Rotational Controller Design By introucing the notation B = G ( σ) we have ω = B σ. Then we get σ = (J B) f + + (J B) τ. (36) In orer to extract the boune controller we rewrite the compoun isturbances as = (J B) [ J ω ω Jω J(Ḃ σ R[ ω ω + R ω ). (37) Then the system ynamics can be rewritten as σ = (J B) τ + ḋ = h (t) (38) where h (t) enotes the erivative of the compoun isturbances. Then the secon-orer linear ESO for rotational subsystem Σ is propose as ẑz 3 = τ + ẑz 4 + g 3 ( ω ẑz 3 ) ẑz 4 = g 4 ( ω ẑz 3 ) (39) where g 3 an g 4 are positive constants to be selecte. Transforming (39) to the frequency-omain using the Laplace transform an substitute (5) into (39) we have: { sẑz 3 = (ẑz 4 ) + s σ + g 3 ( σ ẑz 3 ) sẑz 4 = g 4 ( σ (4) ẑz 3 ) where s is the Laplace operator. From (4) we finally get g 4 ẑz 4 = s + g 3 s + g. (4) 4 g 3 an g 4 shoul satisfy that the polynomial s + g 3 s + g 4 is Hurwitz. Here we simply choose g 3 = ω g 4 = ω. To esign the attitue tracking controller we introuce the following variables: { e3 = σ β e 4 = σ (4) β + k 3 e 3. Then the control input τ is given as: [ τ = J B k β tanh(β) k β tanh( β) (43) where k β k β are strictly positive matrices an the observer is escribe as β = [ k β tanh(β) k β tanh( β)+ ẑz 4 + (k 3 + k 4 ) e 4 + ( k 3) e 3 (44) where k 3 k 4 are positive matrices to be selecte. The control torque of the attitue tracking problem is finally escribe as τ = J B [ k β tanh(β) k β tanh( β). (45)

6 7 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. JANUARY 5 For a caniate Lyapunov function V = (et 3 e 3 + e T 4 e 4 ) its erivative is as follows: V λ min(k 3 ) e 3 λ min(k 4 ) e 4 + ε ( ) f e 4 + λ max (J B) (46) where ẑz 4. Remark. From (3) the translational controller is boune. That is the output of thrust is saturate. From (45) an the efinition of tanh function it is clear that the output of torque is saturate. V. STABILITY ANALYSIS Theorem. Given the error moel of a VTOL UAV for trajectory tracking problem in () with the compoun isturbances shown in (). Let the thrust input T an esire attitue information given by (3) an Theorem respectively with a linear ESO an an auxiliary observer propose in (6) an (3). Then let the torque input τ in (45) with a linear ESO an an auxiliary observer esigne in (39) an (4). There exist the time-scale factors such that the propose control strategy can stabilize the system asymptotically. Proof Consier the caniate Lyapunov functions V an V efine in Section IV we efine a new Lyapunov function as: V = V + V + T + τ T τ. Notice that I 3 R = sin α 4 σ 4( e 3 + β ) E ξ + ṽv R e 3 T = T. As shown later β can converge to zero asymptotically. Then the erivative of V is shown as V a ( e + e ) a ( e 3 + e 4 ) a 3 T ε a 4 τ + e (5 ε m T + 4T max σ )+ λ max (J B) e 4 τ + e + e 4 (47) where a = min{λ min (k ) λ min (k )} a = min{λ min (k 3 ) λ min (k 4 )} a 3 = K T a 4 = K (48) τ. t T t τ Define the general error vector of the VTOL UAV as: E T = [ e T e T e T 3 e T 4 T τ T an notice that: b = λ max(p )T max m b = 5λ max(p ) (49) m b 3 = λ max (J B). Then the erivative of V can be rewritten as V E T ΓE + e + e 4 (5) where a a b b a b ε Γ = a ε b 3. (5) a b 3 ε b 3 Since a is a positive constant the first two minors of matrix Γ is positive. Then we shoul fin the scopes of ε such that the thir to sixth minors are positive. For simplicity let Γ i be the matrix s minor of i. Then the thir to sixth minors are given as et(a 3 ) = a b + a a et(a 4 ) = a a ε et(a 5 ) = a a a 3 ε ε a a b a a b ε et(a 6 ) =a a b b 3 + a a a 3 a 4 a a a 4 b ε ε a 4 ε a a a 3 b ε ε ε a a a 3 b 3 ε + a a 3 b b 3ε a a a 3 a 4 b ε ε. (5) From et(a 3 ) > an et(a 4 ) > we have: Since et(a 5 ) > it follows that: < a a b. (53) ε a a a 3 a 3 b a b. (54) If et(a 6 ) > the following shoul be satisfie: Aε + Bε + C > (55) where A = a a b b 3ε 3 B = a a 3 b b 3ε 4 a a a 3 b 3ε 3 a a a 4 b C = a a a 3 a 4 a a a 3 a 4 b. Consiering that < aa we have A > C > B <. b B 4AC = (a a 3 b b 3ε 4 a a a 3 b 3ε 3 + a a a 4 b ) >. Define the variable ε = B B 4AC A an consier that B 4AC < B. Hence ε is positive. At this time ε < ε can make et(a 6 ) >. ε is finally etermine as { a a a 3 a 3 b } ε min a b ε. (56) If the time-scale factors an ε are selecte base on the above requirements matrix Γ is positive. The unforce system is exponentially stable that is the system is input-tostate stable with the input an. From (8) an (4) we know that ẑz an ẑz 4 can converge to an asymptotically that is an can converge to zero asymptotically. Then from Lemma 4.7 of [4 we know that the cascae overall system is asymptotically stable. Hence lim t e = lim t e = lim t e 3 = lim t e 4 = lim t T = limt τ =. From the escriptions above an the results in [5 the auxiliary observers in (3) an (44) are asymptotically stable. Consequently lim t α = lim t α = lim t β =

7 WANG AND SU: TRAJECTORY TRACKING OF VERTICAL TAKE-OFF AND LANDING UNMANNED AERIAL VEHICLES 7 lim t β =. Noticing that e to e 4 are linear iffeomorphism of ξ ṽv σ ω α α β an β hence we have the following conclusion: lim ξ = lim ṽv = lim σ = lim ω =. (57) t t t t VI. SIMULATION RESULTS Simulations are shown to illustrate the effectiveness of the propose control strategy. We consier a VTOL UAV moel with the parameters being set as: m = 4 kg J x = J y =.8 kg m an J z =.4 kg m. The initial conition is given as: ξ(t ) = [ 3 5 T m v(t ) = [ T m/s σ(t ) = [ T ω(t ) = [ T ra/s. The parameters of the controller are given as follows: k = 3.5 k 3 = k 4 =. k α = k α =.5 k β = k β =.5 =. ε =.5. Tracking of a spiral rising trajectory with the existence of perturbation of parameters an unknown isturbance is accomplishe in Matlab/Simulink. The esire trajectory is as follows: ξ = [.5t 5 sin( πt πt 5 ) 5 cos( 5 ) T m. (58) The external isturbances acting on the translational an rotational ynamics are given as:. sin(πt) +.5 cos( πt = ). sin(πt) + cos( πt ) N (59). sin(πt) +.5 cos( πt ). sin(πt) +. cos( πt = ). sin(πt) +. cos( πt ) N m (6). sin(πt) +. cos( πt ) Simulation results are illustrate in Figs The trajectory tracking effect of the VTOL UAV is illustrate in Fig. 3. The tracking errors of position velocity MRPs an angular velocity are shown in Figs. 4 an 5 while Fig. 6 shows the estimation effects of linear ESO for both translational an rotational subsystems. The changing tenency of roll pitch an yaw angles uring the trajectory tracking are inicate in Fig. 7. In the simulation we fin that with the existence of both external isturbances an internal uncertainties the propose controller can enable the VTOL UAV to track a time-varying trajectory quickly an accurately. The linear ESO can estimate the isturbances an compensate for them in the control scheme to improve the control performance whereas the propose controller can enable the VTOL UAV to track a Fig. 3. Fig. 4. Trajectory tracking effect. Tracking error of position an velocity. esire trajectory effectively. We also carry out the aaptive backstepping metho in [9 for comparison. In Table I the root-mean-square (RMS) error of the propose control strategy is compare with that of aaptive backstepping metho. It is shown that with time-varying isturbances an internal uncertainties the control performance of the aaptive backstepping is not as well as our propose control strategy. It is shown in Fig. 6 that the linear ESO estimates the isturbance accurately an the estimation error converges quickly. Figs. 4 an 5 also show that the propose control strategy have goo tracking performance. VII. CONCLUSION In this paper the trajectory tracking control of a VTOL UAV is investigate. The MRPs base system error moel is establishe an the hierarchical control strategy is introuce base on the time-scale property. Then a practical isturbance

8 7 IEEE/CAA JOURNAL OF AUTOMATICA SINICA VOL. NO. JANUARY 5 TABLE I COMPARISON OF CONTROL PERFORMANCE (RMS ERROR) rejection controller is propose with linear ESO for both translational an rotational subsystems respectively. The auxiliary observer is implemente to guarantee the bouneness of the control output. At last stability conclusion of the overall system is given base on singular perturbation theory. Simulation results verify that the propose control strategy can successfully enable the VTOL UAV to track a esire trajectory. The esigne linear ESO can also estimate the compoun isturbances cause by both external an internal uncertainties for higher accuracy of tracking. Fig. 5. Tracking error of MRPs an angular velocity. Fig. 7. Equivalent control effect of Euler angles. R EFERENCES [ Brockett R W. Asymptotic stability an feeback stabilization. Differential Geometric Control Theory. Boston: Birkha user [ Bouaballah S Siegwart R. Backstepping an sliing-moe techniques Propose strategy Aaptive backstepping Fig. 6. ξ ξ ξ Disturbance estimation performance. applie to an inoor micro quarotor. In: Proceeings of the 5 IEEE International Conference on Robotics an Automation. Barcelona Spain: IEEE [3 Mian A A Wang D B. Moeling an backstepping-base nonlinear control strategy for a 6 DOF quarotor helicopter. Chinese Journal of

9 WANG AND SU: TRAJECTORY TRACKING OF VERTICAL TAKE-OFF AND LANDING UNMANNED AERIAL VEHICLES 73 Aeronautics 8 (3): 6 68 [4 Zuo Z. Trajectory tracking control esign with comman-filtere compensation for a quarotor. IET Control Theory an Applications 4(): [5 Lee D Kim H J Sastry S. Feeback linearization vs. aaptive sliing moe control for a quarotor helicopter. International Journal of Control Automation an Systems 9 7(3): [6 Xu R Özgüner Ü. Sliing moe control of a class of uneractuate systems. Automatica 8 44(): 33 4 [7 Efe M Ö. Battery power loss compensate fractional orer sliing moe control of a quarotor UAV. Asian Journal of Control 4(): [8 Raffo G V Ortega M G Rubio F R. An integral preictive/nonlinear control structure for a quarotor helicopter. Automatica 46(): 9 39 [9 Efe M O. Neural network assiste computationally simple Pl λ D µ control of a quarotor UAV. IEEE Transactions on Inustrial Informatics 7(): [ Zemalache K M Maaref H. Controlling a rone: comparison between a base moel metho an a fuzzy inference system. Applie Soft Computing 9 9(): [ Bertran S Guénar N Hamel T Piet-Lahanier H Eck L. A hierarchical controller for miniature VTOL UAVs: esign an stability analysis using singular perturbation theory. Control Engineering Practice 9(): 99 8 [ Abessameu A Tayebi A. Global trajectory tracking control of VTOL- UAVs without linear velocity measurements. Automatica 46(6): [3 Nicol C Macnab C J B Ramirez-Serrano A. Robust aaptive control of a quarotor helicopter. Mechatronics (6): [4 Ginsberg J. Engineering Dynamics. Cambrige: Cambrige University Press 8. [5 Abessameu A Tayebi A. Formation control of VTOL Unmanne Aerial Vehicles with communication elays. Automatica 47(): [6 Wang L Jia H M. The trajectory tracking problem of quarotor UAV: global stability analysis an control esign base on the cascae theory. Asian Journal of Control 4 6(): [7 Esteban S Gorillo F Aracil J. Three-time scale singular perturbation control an stability analysis for an autonomous helicopter on a platform. International Journal of Robust an Nonlinear Control 3 3(): [8 Michael N Mellinger D Linsey Q Kumar V. The GRASP multiple micro-uav testbe. IEEE Robotics an Automation Magazine 7(3): [9 Cabecinhas D Cunha R Silvestre C. A nonlinear quarotor trajectory tracking controller with isturbance rejection. Control Engineering Practice 4 6: [ Wang L Su J B. Global trajectory tracking of VTOL UAV base on isturbance rejection control. In: Proceeings of the 3n Chinese Control Conference. Xi an China: IEEE [ Tsiotras P. Further passivity results for the attitue control problem. IEEE Transactions on Automatic Control (): [ Cong B L Liu X D Chen Z. Distribute attitue synchronization of formation flying via consensus-base virtual structure. Acta Astronautica 68( ): [3 Zheng Q Dong L L Lee D H Gao Z Q. Active isturbance rejection control for MEMS Gyroscopes. IEEE Transactions on Control Systems Technology 9 7(6): [4 Khalil H K. Nonlinear Systems (Thir eition). Upper Sale River New Jersey: Prentice Hall. [5 Olfati-Saber R. Global configuration stabilization for the VTOL aircraft with strong input coupling. IEEE Transactions on Automatic Control 47(): Lu Wang Ph. D. caniate in the Department of Automation Shanghai Jiao Tong University China. His research interests inclue isturbance rejection control isturbance observer nonlinear system control unmanne aerial system an VTOL UAV control. Corresponing author of this paper. Jianbo Su receive the B. S. egree in automatic control from Shanghai Jiao Tong University China in 989 the M. S. egree in pattern recognition an intelligent system from the Institute of Automation Chinese Acaemy of Science China in 99 an the Ph. D. egree in control science an engineering from Southeast University China in 995. He joine the faculty of the Department of Automation Shanghai Jiao Tong University in 997 where he has been a full professor since. His research interests inclue robotics pattern recognition an human-machine interaction. In these areas he has publishe three books more than 9 technical papers an is the holer of 5 patents. Dr. Su is a member of the Technical Committee of Networke Robots IEEE Robotics an Automation Society a member of the Technical Committee on Human-Machine Interactions IEEE System Man an Cybernetics Society an a staning committee member of the Chinese Association of Automation. He has serve as an associate eitor for IEEE Transactions on Cybernetics since 5.