Automatica. Global trajectory tracking control of VTOL-UAVs without linear velocity measurements

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1 Automatica 46 (200) Contents lists available at ScienceDirect Automatica journal homepage: Brief paper Global trajectory tracking control of VTOL-UAVs without linear velocity measurements Abelkaer Abessameu a, Abelhami Tayebi a,b, a Department of Electrical an Computer Engineering, University of Western Ontario, Lonon, Ontario, Canaa, N6A 3K7 b Department of Electrical Engineering, Lakehea University, Thuner Bay, Ontario, Canaa P7B 5E a r t i c l e i n f o a b s t r a c t Article history: Receive 7 April 2009 Receive in revise form 8 February 200 Accepte 24 February 200 Available online 8 April 200 Keywors: Trajectory tracking Unmanne Aerial Vehicles Linear-velocity observer This paper eals with the position control of Vertical Take-Off an Laning (VTOL) Unmanne Aerial Vehicles (UAVs) without linear velocity measurements. We propose a multistage constructive proceure, exploiting the cascae property of the translational an rotational ynamics. More precisely, we consier the force as a virtual control input for the translational ynamics, from which we extract the require (esire) system attitue an thrust achieving the tracking objective. Thereafter, the control torque is esigne to rive the actual attitue to the esire one. A nonlinear observer, as well as some instrumental auxiliary variables are use to obviate the nee for the linear velocity. Global asymptotic stability of the overall close loop system is achieve. Simulation results are provie to show the effectiveness of the propose control scheme. 200 Elsevier Lt. All rights reserve.. Introuction The control of Unmanne Aerial Vehicles (UAV) has recently receive an increasing interest in the control community. This interest is motivate by their potential applications in areas such as surveillance, search an rescue missions, monuments inspections, etc. VTOL-UAVs, which are suitable for a broa range of applications requiring stationary flights, constitute an important class of thrust propelle UAVs. These vehicles are generally uneractuate, i.e., equippe with fewer actuators than egrees-offreeom. It is clear that one of most important components for reliable autonomous flights is an efficient attitue control an stabilization scheme. In fact, this problem has been the focus of extene research over the past years, resulting in a myria of successful attitue controllers, see for instance, Tayebi (2008) an Wen an Kreutz-Delgao (99). However, the position control of uner-actuate VTOL vehicles in SE(3) is more challenging than the attitue control problem since global asymptotic stability is ifficult to achieve for this class of mechanical systems. Several solutions have been reporte in the literature, such as the feeback The material in this paper was not presente at any conference. This paper was recommene for publication in revise form by Associate Eitor Yong-Yan Cao uner the irection of Eitor Toshiharu Sugie. Corresponing author at: Department of Electrical an Computer Engineering, University of Western Ontario, Lonon, Ontario, Canaa, N6A 3K7. Tel.: ; fax: aresses: aabessa@uwo.ca (A. Abessameu), tayebi@ieee.org (A. Tayebi). linearization metho in Koo an Sastry (998), the backstepping approach in Frazzoli, Dahleh, an Feron (2000) an Pflimlin, Soueres, an Hamel (2007), the sliing moe technique in Maani an Benallegue (2007), an other control strategies base on gain scheuling (Kaminer, Pascoal, Hallberg, & Silvestre, 998) or on a neste saturation technique (Kenoul, Lara, Fantoni, & Lozano, 2006). The authors in Hamel, Mahony, Lozano, an Ostrowski (2002) an Pflimlin et al. (2007) propose a hierarchical esign proceure for the position control of VTOL-UAVs. The iea consists in using the vehicle s orientation an the thrust as control variables to stabilize the vehicle s position, an then apply a classical backstepping proceure to etermine the torque-input capable of stabilizing the require orientation. In Hua, Hamel, Morin, an Samson (2009), a similar control architecture is applie to solve the trajectory tracking problem, where the angular velocity is use as an intermeiate variable instea of the orientation. The authors in Aguiar an Hespanha (2007) propose a backstepping esign for the trajectory tracking problem of a class of uner-actuate systems, incluing VTOL vehicles, where the states are guarantee to converge to a ball near the origin. While the above control schemes rely on the availability of the full state for feeback, only few work has been one in the case where the linear velocity is not available for feeback. For flying vehicles, velocity estimations can be obtaine via approximate erivation of the successive measurements from GPS sensors. For fast moving vehicles, the stanar proceure consists of integrating the acceleration, an coupling this result with the erivative of the GPS measurements (Benzemrane, Santosuosso, & Damm, 2007). This estimation metho suffers from several /$ see front matter 200 Elsevier Lt. All rights reserve. oi:0.06/j.automatica

2 054 A. Abessameu, A. Tayebi / Automatica 46 (200) problems, namely the fact that the errors inuce by a GPS system may reach many meters, an in practice, numerical integration along with measurement noise inuces a very fast growing velocity measurement error. There are several technical solutions to overcome these problems such as using high-precision sensors like a D-GPS. However, the GPS signal is not available in inoor an urban applications (structure/brige inspection for example) ue to signal blockage an attenuation, which may eteriorate the positioning accuracy. To solve the linear-velocity estimation problem without the use of a GPS, several authors have consiere the combination of artificial vision an the inertial sensors, see for example Cheviron, Hamel, Mahony, an Balwin (2007), Ronon, Salazar, Escareno, an Lozano (200) an references therein. Another solution is to use observers to estimate the missing states, as one in Do, Jiang, an Pan (2003), where the trajectory tracking problem of a planar-vtol is treate. It is worth mentioning, that in the case where a GPS is not available (in inoor applications for instance), there are several techniques that allow to obtain the UAV position, such as the combination of an inertial measurement unit (IMU) with a vision system; or the use of a network of Ultra-wieban (UWB) receivers which track a large number of small (inexpensive) UWB transmitters. The main contribution of this work is to provie a solution to the position tracking problem of VTOL UAVs without linear-velocity measurements. We exploit the cascae nature of the system an first esign an intermeiary control input for the translational ynamics of the vehicle, without linear velocity measurements, from which we can extract the magnitue of the necessary thrust input an the esire orientation (in terms of unit-quaternion) of the aircraft. The thrust input will be use to rive the translational ynamics of the aircraft, an the time-varying esire attitue will be consiere as a reference input to be tracke by the rotational ynamics with an appropriate esign of the torque input. In our approach, we ensure that the intermeiary translational control input is a priori boune an at least twice ifferentiable. This, as it will become clear later, will be a key ingreient guaranteeing the existence of a solution to the attitue an thrust extraction algorithm, as well as the bouneness of the actual control inputs. As a result, the propose control scheme guarantees global asymptotic trajectory tracking with an a priori boune thrust. 2. System moel In this paper, we consier a VTOL aircraft moele as a rigiboy. Let F i {ê, ê 2, ê 3 } enotes the inertial frame, an F b {ê b, ê 2b, ê 3b } enotes the boy-fixe frame of the aircraft. Let the position an linear velocity of the aircraft, expresse in the inertial frame F i, be enote by p R 3 an v R 3 respectively, an let its angular velocity, expresse in the boy-fixe frame F b, be enote by ω R 3. To represent the attitue (orientation) of the aircraft, we make use of the unit-quaternion representation (Shuster, 993). The unit-quaternion Q = (q T, η) T is a four-element vector, compose of a vector component q R 3 an a scalar component η R, satisfying the unity constraint: q T q + η 2 =. The orthogonal rotation matrix R(Q) SO(3) that efines the rotation of the boy frame by an angle γ about the axis escribe by the unit vector ˆk R 3, can be escribe by a unit-quaternion Q = (q T, η) T such that: q = ˆk sin(γ /2), an η = cos(γ /2). The rotation matrix R(Q), relate to the unit-quaternion Q, that brings the inertial frame into the boy frame, can be obtaine through the Roriguez formula as: R(Q) = (η 2 q T q)i 3 + 2qq T 2ηS(q), where I 3 is the 3-by-3 ientity matrix an the matrix S(x) is the skew-symmetric matrix such that S(x)u = x u for any vector u R 3, where enotes the vector cross prouct. The quaternion multiplication between two unit quaternion, Q = (q T, η ) T an Q 2 = (q T 2, η 2) T, is efine Fig.. VTOL aircraft. by the following non-commutative operation: Q Q 2 = (η q 2 + η 2 q + S(q )q 2 ) T, η η 2 q T q T 2. The inverse or conjugate of a unit quaternion is efine by, Q = ( q T, η) T, with the quaternion ientity given by (0, 0, 0, ) T, (Shuster, 993). Using this representation of the attitue, the equations of motion of the VTOL aircraft can be escribe by {ṗ = v, (Σ ): v = gê 3 T m R(Q)T ê 3, Q = ηi3 + S(q) (Σ 2 ): 2 q T ω, I f ω = τ S(ω)I f ω, where m an g are the aircraft mass an the gravitational acceleration. I f R 3 3 is a symmetric positive efinite constant inertia matrix of the vehicle with respect to F b. The scalar T an the vector τ represent respectively the magnitue of the thrust applie to the vehicle in the irection of ê 3b, an the external torque applie to the system expresse in F b. An example of a VTOL aircraft consiere in this paper is illustrate in Fig.. 3. Problem formulation Our objective in this work is to esign global control laws for the thrust T (t) an the torque τ(t) allowing the VTOL aircraft to track a esire trajectory p (t). We assume that the linear-velocity vector is not available for feeback. In other wors, we woul like to esign a linear-velocity-free global control law guaranteeing the bouneness an the asymptotic convergence to zero of the following position an linear-velocity tracking errors e(t) = p(t) p (t), ṽ(t) ė(t) = v(t) ṗ (t). (2) Due to the uner-actuate nature of the system, the esign of the thrust an torque inputs for this class of systems is not straightforwar. In the following, we will present the control esign methoology that we aopt in this work to achieve our objectives. 3.. Thrust an esire attitue extraction Consier the translational ynamics (Σ ) in (), which can be rewritten as {ṗ = v, (Σ ): v = F T m f (Q, Q (3) ), with f (Q, Q ) ( R(Q) T R(Q ) T ) ê 3, (4) F gê 3 T m R(Q ) T ê 3, (5) ()

3 A. Abessameu, A. Tayebi / Automatica 46 (200) where the variable F is an intermeiary control input to the translational ynamics, an Q (t) = (q T, η ) T is the unit quaternion representing the esire attitue of the vehicle. In the following Lemma, (Roberts & Tayebi, 2009), we will present a thrust an esire attitue extraction algorithm (in terms of the unit-quaternion) from the expression of the intermeiary control input, given in (5). Lemma. Consier Eq. (5) an let the vector F (µ, µ 2, µ 3 ) T. It is always possible to extract the thrust magnitue an the esire system s attitue from (5) as T = m gê 3 F, (6) η = 2 + m(g µ 3), q = m µ2 µ, 2T 2T η 0 (7) uner the conition that F (0, 0, x), for x g, (8) where enotes the Eucliean norm of a vector. In aition, uner the conition that the intermeiary control F is ifferentiable, we can write the esire angular velocity of the aircraft as ω = (F)Ḟ, (9) (F) = γ 2 γ 2 µ µ 2 µ 2 + γ 2 γ 2 µ 2 γ 2 µ 2 γ γ 2 µ µ 2 µ γ 2, (0) µ 2 γ µ γ 0 with γ = (T /m) an γ 2 = γ + (g µ 3 ). Proof. A similar proof can be foun in Abessameu an Tayebi (2009) an Roberts an Tayebi (2009). The result in Lemma states that if one is able to esign an appropriate intermeiary control input F(t) for the translational ynamics, that satisfies conition (8) for all t > 0, then the positive thrust magnitue T (t), an the aircraft esire attitue Q (t), can be extracte as in (6) an (7) respectively. Note that the solution (6) an (7) is singularity-free Attitue error ynamics Provie that the aircraft esire attitue Q is etermine, we efine the attitue tracking error, escribing the iscrepancy between the vehicle s attitue an its esire attitue, namely Q = ( q T, η) T, by Q = Q Q, () governe by the unit-quaternion ynamics q = 2 ( ηi 3 + S( q)) ω, η = 2 qt ω, (2) ω = ω R( Q)ω, (3) where ω is the angular velocity error vector an ω is the esire angular velocity of the aircraft given in (9). Matrix R( Q) is the rotation matrix relate to Q, an is given by R( Q) = R(Q)R(Q ) T, (Shuster, 993). With the above efinition, we can see that attitue tracking is achieve when Q coincies with Q, or Q = (0 T, 3 ±)T, an ω = 0 3, with 0 3 = col[0, 0, 0]. Note that ue to the inherent reunancy of the quaternion representation, Q an Q represent the same physical orientation however, one is rotate 2π relative to the other about an arbitrary axis. Accoringly, Q = (0 T, 3 ±)T correspon to the same physical point. From () an the efinition of R(Q), it can be easily shown that the function f (Q, Q ) in (4) can be expresse in terms of the elements of Q as f (Q, Q ) = R(Q) T ( I 3 R( Q) ) ê 3 = 2R(Q) T S( q) q, (4) with q = ( q, q 2, q 3 ) T, q = ( q 2, q, η) T. In aition, it is easy to verify that R(Q) T S( q). 4. Control esign proceure Base on the above extraction algorithm, the main iea in our esign metho is to exploit the cascae nature of the system () an consier a control esign proceure that can be summarize in the following points: () Consier the translational ynamics (Σ ) in (3), an esign the intermeiary translational control input, F, for the aircraft that satisfies conition (8) without linear-velocity measurements. Then, using the extraction algorithm in Lemma, we extract the necessary thrust T (t) an the aircraft esire attitue Q (t). The magnitue of the thrust will be the input to the subsystem (Σ ). (2) Consier the rotational ynamics (Σ 2 ) in (), an Q (t) as a time varying esire attitue, an esign a linear-velocityfree torque input such that the attitue tracking error q(t) converges asymptotically to zero. (3) Show the global asymptotic stability of the overall system. 4.. Step : intermeiary position control esign To esign an intermeiary control F for the translational ynamics (3) that achieves our control objectives without linearvelocity measurements, we have to take into consieration some important requirements. First, it is important to notice that for conition (8) to be satisfie, the thir element of the control input F must be boune a priori. Hence, to use the extraction algorithm escribe in Lemma, it is sufficient to esign an a priori boune intermeiary control F. On the other han, we can see that the term T m f (Q, Q ) in (3) constitutes a nonlinear perturbation to the translational ynamics, which is completely unknown at this stage of the control esign. Fortunately, we know that f (Q, Q ) is boune since it is function of orthogonal rotation matrices. In aition, we can see from (6) that the esign of an a priori boune intermeiary control input is necessary to guarantee a boune thrust input an hence a boune perturbation term. Secon, we can notice from the expression of ω in (9) that ω is function of F. It is clear that to implement a trajectory tracking attitue controller, that necessarily requires the knowlege of ω (t) an ω (t), we nee to ensure that these signals are boune. In aition, the intermeiary control F must be at least twice ifferentiable. Moreover, if a nonlinear observer or a partial state feeback is consiere in the esign of F using the position tracking errors, Ḟ an F will necessarily be function of v an v, respectively, which are not available for feeback. To achieve our objectives, an solve the above problems, we introuce the following new variables ξ = e θ, z = ξ = ṽ θ, (5) where θ R 3 is a esign variable to be etermine later. The translational error ynamics can then be written as ż = T m f (Q, Q ) + F θ p. (6)

4 056 A. Abessameu, A. Tayebi / Automatica 46 (200) Consier the following positive efinite function V t = ( z T z + k p ξ T ξ + k (ξ ψ) T (ξ ψ) ), (7) 2 where k p, k are strictly positive scalar gains, an the esign variable ψ is the output of an auxiliary system which plays the role of an estimator of the linear velocity at this stage of the control esign. The time erivative of V t is given as ( V t = z T T ) m f (Q, Q ) + F θ p Taking + z T ( k p ξ + k (ξ ψ) ) k ψ T (ξ ψ). (8) F θ = p k p ξ k (ξ ψ), (9) an using (4), we obtain V t = 2T m zt R(Q) T S( q) q k ψ T (ξ ψ). (20) It is clear that with a simple esign of ψ, the last term in the right han sie of (20) can be guarantee to be negative. In aition, to guarantee the stability result of the overall system, as we will see in the proof of our result, the first term in the right han sie of (20) shoul be consiere in the torque input esign. In view of (9), we propose the linear-velocity-free intermeiary control input for the VTOL vehicle, F, an the auxiliary variable θ as F = p k θ tanh(θ) k θ2 tanh( θ), (2) θ = k θ tanh(θ) k θ2 tanh( θ) + k p ξ + k (ξ ψ), (22) where k θ an k θ2 are strictly positive scalar gains, an θ(0) an θ(0) can be selecte arbitrarily. It is worth noticing that the iea behin the introuction of the new variable θ is to moify (uring the transient) the esire trajectory, an esign the control scheme such that lim t ξ = 0 an lim t z = 0. Once this is achieve, the variable θ an its time erivative are force to converge to zero, ensuring hence the tracking of the original esire trajectories. As a result, this variable allows the esign of an a priori boune control input, F, such that F p + 3(k θ + k θ2 ). (23) Before we procee further in our control esign, we nee the following assumption on the esire trajectory an the control gains. Assumption. The secon, thir an fourth time-erivatives of the esire trajectory are boune. The elements of the esire acceleration vector p (t) := ( p, p 2, p 3 ) T, an the positive control gains k θ an k θ2 shoul satisfy one of the following conitions: (a) k θ + k θ2 < p (t), t 0, (b) k θ + k θ2 < p 2 (t), t 0, (c) p 3 (t) α, t 0, an k θ + k θ2 g α, () p (t) δ, t 0, an k θ + k θ2 (g 3 δ) with α > 0 an 0 δ < g. It is straightforwar to verify that if one of the above cases is met, conition (8) is satisfie, an the intermeiary control F can be use in the extraction metho in Lemma. In fact, cases (a) an (b) ensure that µ 0 an µ 2 0 for all t 0 respectively, case (c) is consiere such that µ 3 < g for all t 0, an finally, the more restrictive case () guarantees that F < g for all t 0. Since F is boune, an if conition (8) is satisfie for all time, the extracte value of the thrust T, in (6), is guarantee to be positive an a priori boune as ( T m g + δ + ) 3(k θ + k θ2 ) Λ, (24) with δ = p (t) an Λ a positive constant. Also, the extracte esire attitue of the vehicle, Q (t), is guarantee to be realizable Step 2: attitue control esign Now, we consier the orientation ynamics an esign a torque input for the vehicle that guarantees tracking of the esire attitue Q (t) given in (7). It is important to see that the intermeiary control F in (2) oes not epen explicitly on the position tracking error. As a result, the esire angular velocity, ω (t), erive in (9), oes not epen on the linear-velocity signal. However, the time erivative of the esire angular velocity, ω (t), will be given as ω = 2 z, (25) where the vector an matrix 2 are erive in the Appenix for the sake of clarity of presentation. It is clear that ω is function of the signal z which epens explicitly on the linear-velocity of the aircraft. Note that without the introuction of the new variable θ, the use of the position tracking error explicitly in the expression of F results in ω being function of the linear-velocity an ω being function of the linear-acceleration, which will make the control esign quite complicate. To esign the attitue tracking torque, we introuce the following variable = ω β, (26) with ω efine in (3) an β being a esign parameter to be etermine later. Exploiting the rotational ynamics (Σ 2 ) in () an expression (25), we can easily show that I f = τ H( ) + Ɣz If β, (27) with: H( ) = S(ω)I f ω I f S( ω)r( Q)ω +I f R( Q), Ɣ = I f R( Q) 2, an Q is given in (). Note that the angular velocity error ynamics (27) epen on the vector z, an hence on the aircraft linearvelocity, which is not available for feeback. To esign a torque input in (27) without linear-velocity measurements, we introuce the following nonlinear observer that generates estimates of the linear-velocity vector, ẑ, as { ẑ ˆξ = ν L p ξ, ν = + Ɣ T L 2 ξ, (28) v where L p an L v are strictly positive scalar gains, ξ (ˆξ ξ) an F θ p T m f (Q, Q ). It is important to mention that at this stage of the control esign, all the signals require for the observer are well etermine, among which are F, θ, Q an ω. Define the observation error vector as, z ξ = ẑ z. Using (6), the observation error ynamics can be written as z = L p z L 2 v ξ + Ɣ T. (29) To esign the necessary torque input, we consier the following positive efinite function V a = T z + L v ξ z + L v ξ L vl p ξ T ξ + 2k q ( η) + 2 T I f, (30) with k q > 0. The time erivative of V a, using (2), (26), (27) an (29), is given as V a = (L p L v ) z T z L 3 v ξ T ξ + ( z + L v ξ) T Ɣ T + k q q T ( + β) + T ( τ H( ) + Ɣz I f β ). (3) In view of this last equation, an using the observe states, we propose the following torque input for the rotational ynamics τ = H( ) + I f β k q q k Ω Ɣ ẑ + L v ξ, (32) with k Ω > 0, to obtain V a = (L p L v ) z T z L 3 v ξ T ξ + k q q T β k Ω T. (33) It is important to notice that β cannot be function of ẑ, since its time erivative will give rise to z.

5 A. Abessameu, A. Tayebi / Automatica 46 (200) Step 3: stability of the overall system Now, we can state our main result in the following theorem. Theorem. Consier the VTOL-UAV moel (), an let the esire trajectory p (t) an the controller gains k θ an k θ2 satisfy Assumption. Let the thrust input T (t) an the esire attitue Q (t) be given, respectively, by (6) an (7), with F (µ, µ 2, µ 3 ) T given by (2) an (22). Let the torque input be as in (32) with the observer (28). Let the parameters ψ an β, in (22) an (26) respectively, be given by ψ = λ(ξ ψ), (34) β = k β q + 2T k q m S( q)t R(Q)ν, (35) with λ > 0, k β > 0, ν = ẑ + L p ξ an q efine in (4). Pick the control an observer gains as follows L p L v > σ, L 3 v > σ 2, k q k β > Λ2 + L 2 p, (36) m 2 σ for some σ > 0, σ 2 > 0, an Λ given in (24). Then, starting from any initial conitions, all signals are boune an lim t ) ζ(t) = 0, with ζ(t) T = (e(t) T, ṽ(t) T, ξ(t) T, z(t) T, q(t) T, ω(t) T. Proof. First, it is easy to check that if the esire trajectory an the controller gains k θ an k θ2 satisfy Assumption, conition (8) is always satisfie, an hence it is always possible to extract the magnitue of the thrust an the esire attitue from (6) an (7) respectively for the VTOL vehicle. Consier the following Lyapunov function caniate V = V t + V a. (37) Using (34) an (35), the time erivative of V evaluate along the system trajectories, in view of (20) an (33), is obtaine as V = k λ(ξ ψ) T (ξ ψ) (L p L v ) z T z L 3 v ξ T ξ k Ω T k q k β q T q + 2T m qt S( q) T R(Q)( z + L p ξ). (38) Using the fact that R(Q) T S( q) an L p > L v from (36), an upper boun of V can be obtaine as V k λ ξ ψ 2 (L p L v σ ) z 2 k Ω 2 ( ) (L 3 v σ 2) ξ 2 k q k β Λ2 + L 2 p q 2, (39) m 2 σ where Λ is given in (24), an we have use the fact that for any real numbers a an b, we have 2ab a 2 /σ + σ b 2, for some σ > 0. Therefore, V is negative semi-efinite if conition (36) is satisfie. Hence, we can conclue that z, ξ, ψ, q,, z an ξ are boune. Consequently, θ, ψ, ż, z an ν are boune. Also, we can see that ( ξ ψ) is boune. Since q an ν are boune, we can see that β is boune, an hence ω is boune from (26). Hence, we can conclue that q is boune. In aition, we can easily verify that is boune using the expressions of τ in the close loop ynamics (27). As a result, V is boune. Hence, invoking Barbalat lemma, we can conclue that (ξ ψ) 0, ξ 0, z 0, 0 an q 0, an therefore we conclue that R( Q) I3 an η ±. Using the above results, we know that ξ ψ = ż λ( ξ ψ) is boune, an since we have shown that (ξ ψ) 0, we can conclue from Barbalat lemma that ( ξ ψ) 0. Consequently, we can conclue that ξ = z 0, since ψ = λ(ξ ψ) 0. Also, since σ 2 σ 2 Table Simulation parameters. p(0) = ( 2, 5, ), v(0) = (0, 0, 0), q(0) = (0, 0, 0, ), ω(0) = θ(0) = θ(0) = ˆξ(0) = ν(0) = (0, 0, 0), g = 9.8, k p = 0.3, k = 0.5, λ =, k θ =.5, k θ2 =.5, k β = 40, k q = 40, k Ω = 30, L p =.5, L v = 0.8. z an z converge to zero, it is clear that ẑ tens to zero, an with the limit of ξ, one can conclue that ν 0, an consequently, β 0 implying that ω 0. Exploiting the above bouneness results, an since q 0 an (ξ ψ) 0, we can conclue from the translational error ynamics (6), with (9) an (4), that ż 0 using the extene version of Barbalat lemma (See, for instance, Lemma 2 in Hua et al. (2009)). Hence, we conclue from (6) that ξ 0 an consequently ψ 0. To show the convergence of e an ṽ, we have to investigate the bouneness an asymptotic convergence to zero of the variables θ an θ. It can be seen that (22) can be rewritten as: θ = k θ tanh(θ) k θ2 tanh( θ) + χ, with χ = k p ξ + k (ξ ψ). Using the above results, we can easily verify that χ is boune an tens to zero as t goes to infinity. Therefore, we can show that θ an θ are boune an converge asymptotically to zero, an as a result, e 0 an ṽ 0 asymptotically. To complete the proof, we must show that the input torque in (32) is boune. Exploiting the above bouneness results, we can easily show that τ is boune if ω, ω an β are boune. By taking the time erivative of (35), we can show that t (S( q)t R(Q)) is boune if ω, ω are boune, an since ν an ν are boune, we know that β is boune if T an ω are boune. It is important to note from the time erivative of (35) that β use in (32) is completely known, since ω an ν are available from (3) an (28) respectively. As a result, we conclue that τ is boune if T, ω an ω are boune. Using the above bouneness results, it is clear that Ḟ an F are boune, an lim t F = p, lim t Ḟ = p (3) an lim t F = p (4), where p(3) an p (4) are, respectively, the thir an fourth erivatives of the esire trajectory, which are assume to be boune. Hence, using (6), (9) an (25), with (0), (A.2), (A.3) an Assumption, we can conclue that T, ω an ω are boune, an this ens the proof. 5. Simulation results Simulation results are presente to illustrate the effectiveness of the propose control scheme. Using SIMULINK, we consier a VTOL UAV moele as a rigi boy of mass m = 3 kg an with inertia matrix I f = iag(col(0.3, 0.3, 0.04)) kg m 2. The simulation parameters are illustrate in Table. The esire trajectory is given by p (t) T = (0 cos(0.t + 2), 0 sin(0.t + 2.4), t) m. Note that the controller gains are selecte to satisfy case (c) of Assumption an conition (36). The obtaine results are illustrate in Figs Figs. 2 an 3 illustrate the three components of the position an velocity tracking errors. Fig. 4 shows the attitue tracking error an Fig. 5 illustrates the esire an actual angular velocity of the aircraft. It is clear from these figures that asymptotic convergence to zero is guarantee after few secons. To illustrate the vehicle s position tracking, a 3-D plot of the vehicle s position with the esire trajectory is given in Fig. 6, as well as the projections of the curves in the ifferent planes.

6 058 A. Abessameu, A. Tayebi / Automatica 46 (200) Fig. 2. Position tracking error, with e = (e, e 2, e 3 ) T. Fig. 5. Elements of the aircraft angular velocity an the esire angular velocity vectors, with ω T = (ω, ω 2, ω 3 ) an ω T = (ω, ω2, ω3 ). Fig. 3. Velocity tracking error, with ṽ = (ṽ, ṽ 2, ṽ 3 ) T. Fig. 6. 3D plot of the VTOL vehicle trajectory (blue) with the esire trajectory (re). (For interpretation of the references to colour in this figure legen, the reaer is referre to the web version of this article.) Fig. 4. Attitue tracking error, with q = ( q, q 2, q 3 ) T. 6. Concluing remarks We aresse the trajectory tracking problem of a class of uner-actuate systems, VTOL-UAVs, without linear-velocity measurements. A separate translational an rotational control esign was presente, an global asymptotic stability of the overall close loop system was shown. At the first stage of the control esign, the requirement of the linear velocity has been obviate with the introuction of new control variables reshaping the esire trajectory uring the transient. In the secon stage of the control esign, a nonlinear observer has been erive an use to esign a linear-velocity-free control torque guaranteeing the tracking of the esire attitue erive at the first stage of the control esign. To the best of our knowlege, the propose linear-velocity-free control scheme is a new result proviing global asymptotic tracking for the class of uner-actuate systems uner consieration. Furthermore, the propose control strategy can be easily moifie to solve the trajectory tracking problem of VTOL-UAVs in the case where the linear-velocity is available for feeback, an global results can be shown. This result constitutes on its own right an interesting contribution since global results are ifficult to obtain for this class of uner-actuate systems. In the case where the linear velocity is available, a global result has been obtaine in Frazzoli et al. (2000) for the trajectory tracking problem, where the control is guarantee to be smooth provie that the rotation angle of the attitue error is ifferent from π/2. Also, the thrust input is efine as a solution to a secon orer ifferential equation. The authors in Frazzoli et al. (2000), first etermine an optimal esire thrust input an esire orientation, then using the backstepping proceure, a thrust an input torque are etermine. A conceptually similar approach is consiere in Hamel et al. (2002) to solve the stabilization problem. The main ifference between the propose approach an the work of Frazzoli et al. (2000) an Hamel et al. (2002), besies the non-availability of the linear velocity, is the aopte singularity-free attitue extraction metho (in terms of unit-quaternion) as well as the a priori bouneness of the virtual translational control input an the system s thrust. Acknowlegements This work was supporte by the Natural Sciences an Engineering Research Council of Canaa (NSERC).

7 A. Abessameu, A. Tayebi / Automatica 46 (200) Appenix We erive in the following the expressions of the variables an 2 in Eq. (25). From (9), we can write ω = (F)Ḟ + (F) F, where (F) can be irectly obtaine from Ḟ with γ = γ ( µ µ 2 (g µ 3 ) ) Ḟ. In view of the esign (2), we have Ḟ = p (3) k θ h(θ) θ k θ2 h( θ) θ F = p (4) k θ h(θ) θ (k θ h(θ) + k θ2 h( θ)) θ k θ2 h( θ){(k + k p )z k ψ k θ h(θ) θ k θ2 h( θ) θ} (A.) where p (3) an p (4) are, respectively, the thir an fourth erivatives of the esire trajectory, an for u = (u, u 2, u 3 ) T R 3, we have efine h(u) = iag(v, v2, v3) an h(u) = iag(v, 2 v2, 2 v3 2 ), with v i = ( tanh2 (u i )) an v i = 2 ( 2 u i( tanh 2 (u i )) tanh(u i )), for i =, 2, 3, an iag is the iagonal matrix operator. Hence we can rewrite ω as in (25) with = (F)Ḟ + (F){p (4) k θ h(θ) θ k θ2 h( θ)( k ψ k θ h(θ) θ k θ2 h( θ) θ) (k θ h(θ) + k θ2 h( θ j )) θ}, 2 = k θ2 (k p + k ) (F)h( θ). References (A.2) (A.3) Abessameu, A., & Tayebi, A. (2009). Formation control of VTOL UAVs. In Proceeings of the 48th IEEE conference on ecision an control (pp ). Aguiar, A. P., & Hespanha, J. P. (2007). Trajectory-tracking an path-following of uneractuate autonomous vehicles with parametric moeling uncertainty. IEEE Transactions on Automatic Control, 52(8), Benzemrane, K., Santosuosso, G. L., & Damm, G. (2007). Unmanne aerial vehicle spee estimation via nonlinear aaptive observers. In Proceeings of the American control conference (pp ). Cheviron, T., Hamel, T., Mahony, R., & Balwin, G. (2007). Robust nonlinear fusion of inertial an visual ata for position, velocity an attitue estimation of UAV. In Proceeings of the IEEE international conference on robotics an automation (pp ). Do, K. D., Jiang, Z. P., & Pan, J. (2003). On global tracking control of a VTOL aircraft without velocity measurements. IEEE Transactions on Automatic Control, 48(2), Frazzoli, E., Dahleh, M. A., & Feron, E. (2000). 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Output tracking control esign of a helicopter moel base on approximate linearization. In Proceeings of the 37th IEEE conference of ecision an control (pp ). Maani, T., & Benallegue, A. (2007). Backstepping control with exact 2-sliing Moe estimation for a quarotor unmanne aerial vehicle. In Proceeings of the 2007 IEEE/RSJ international conference on intelligent robots an systems (pp. 4 46). Pflimlin, J. M., Soueres, P., & Hamel, T. (2007). Position control of a ucte fan VTOL UAV in crosswin. International Journal of Control, 80(5), Roberts, A., & Tayebi, A. (2009). Aaptive position tracking with external isturbance estimation for a VTOL-UAV. In Proceeings of the 48th IEEE conference on ecision an control (pp ). Ronon, E., Salazar, S., Escareno, J., & Lozano, R. (200). Vision-base position control of a two-rotor VTOL miniuav. Journal of Intelligent an Robotic Systems, 57, Shuster, M. D. (993). A survey of attitue representations. Journal of Astronautical Sciences, 4(4), Tayebi, A. (2008). Unit quaternion base output feeback for the attitue tracking problem. IEEE Transactions on Automatic Control, 53(6), Wen, J. T.-Y., & Kreutz-Delgao, K. (99). The attitue control problem. IEEE Transactions on Automatic Control, 36(0), Abelkaer Abessameu receive his Engineer egree in Electrical Engineering from Institut National Electricité et Electronique, INELEC, Algeria, in 995, an his M.Sc. (Magister) in robotics from Ecole Militaire Polytechnique e Borj-El-Bahri, Algeria, in 999. From 200 to 2007, he serve as a Lecturer (Maître assistant chargé e cours) an Research Assistant at épartement Automatisation es procéés inustriels at the University of Boumeres, Algeria. He is currently a Ph.D. stuent in the epartment of Electrical an Computer Engineering at the University of Western Ontario, Canaa. His research interest focuses on the control of autonomous aerial vehicles, attitue synchronization an cooperative control of multi-vehicle systems. Abelhami Tayebi receive his B.Sc. in Electrical Engineering from Ecole Nationale Polytechnique Alger, Algeria in 992, his M.Sc. (DEA) in robotics from Université Pierre & Marie Curie, Paris, France in 993, an his Ph.D. in Robotics an Automatic Control from Université e Picarie Jules Verne, France in December 997. He joine the epartment of Electrical Engineering at Lakehea University in December 999 where he is presently a Professor. He is a Senior Member of IEEE an serves as an Associate Eitor for IEEE Transactions on Systems, Man, an Cybernetics Part B; Control Engineering Practice an IEEE CSS Conference Eitorial Boar. He is the founer an Director of the Automatic Control Laboratory at Lakehea University. His research interests are mainly relate to linear an nonlinear control theory incluing aaptive control, robust control an iterative learning control, with applications to robot manipulators an aerial vehicles.