IN the recent past, the use of vertical take-off and landing

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1 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY Aaptive Position Tracking of VTOL UAVs Anrew Roberts, Stuent Member, IEEE, an Abelhami Tayebi, Senior Member, IEEE Abstract An aaptive position-tracking control scheme is propose for vertical take-off an laning (VTOL) unmanne airborne vehicles (UAVs) for a set of boune external isturbances. The control esign is achieve in three main steps. The first step is evote to the esign of an aprioriboune linear acceleration riving the translational ynamics towar the esire trajectory. In the secon step, we extract the require aprioriboune thrust an the esire attitue, in terms of unit quaternion, from the esire acceleration erive in the first step. In the last step, we esign the require torque for the rotational ynamics, allowing the system s attitue to be riven towar the esire attitue obtaine at the secon step. Two control laws for the system control torque are rigorously esigne. The first control law ensures that the position-tracking objective is satisfie for any initial conitions, whereas the secon ensures that the tracking objective is satisfie for a set of initial conitions, which is epenant on the control gains. The latter case is inclue, since it is less complicate than the former control law an may be avantageous from a practical point of view. Finally, simulation results are provie to illustrate the effectiveness of the propose control strategy. Inex Terms Aaptive control, unmanne airborne vehicle (UAV), vertical take-off an laning (VTOL). I. INTRODUCTION IN the recent past, the use of vertical take-off an laning (VTOL) unmanne airborne vehicles (UAVs) has seen a significant increase in popularity. These types of vehicles are esire for a variety of applications, incluing visual inspection of structures (builings, briges, etc.), search an recovery, efence, etc. Most commonly these systems are pilote by a remote operator, however, a number of researchers have been working to evelop systems that offer a higher egree of autonomy, thus reucing the complexity of the task presente to the operator. One example of a VTOL UAV is the ucte-fan aircraft, which is illustrate in Fig. 1. The system uses one or two main rotors/fans/propellers to generate vertical thrust, which provies the VTOL ability. This type of aircraft uses vectore thrust, which controls the movement of the system by irecting Manuscript receive January 27, 2010; revise June 17, 2010 an October 24, 2010; accepte November 10, Date of publication December 23, 2010; ate of current version February 9, This paper was recommene for publication by Associate Eitor K. Iagnemma an Eitor W. K. Chung upon evaluation of reviewer s comments. This work was supporte by the Natural Sciences an Engineering Research Council of Canaa. This work was presente in part at Proceeing of 48th IEEE Conference on Decision an Control an the 28th Chinese Control Conference, Shanghai, China, A. Roberts is with the Department of Electrical an Computer Engineering, University of Western Ontario, Lonon, ON N6A 5B9, Canaa ( arober88@uwo.ca). A. Tayebi is with the Department of Electrical an Computer Engineering, University of Western Ontario, Lonon, ON N6A 5B9, Canaa, an also with the Department of Electrical Engineering, Lakehea University, Thuner Bay, ON P7B 5E1, Canaa ( tayebi@ieee.org). Digital Object Ientifier /TRO Fig. 1. Exogenous forces. the thrust using a set of ailerons/wings/control surfaces that are locate at the unersie of the system. In previous work, for example, [2], the goal was to achieve a tracking control law for the system attitue. This paper extens further to eal with the position-tracking problem of a ucte-fan VTOL UAV in the presence of external isturbances. Due to the nature of this uneractuate system, a common practice is to use the system attitue as a means to irect the thrust in orer to control the system position an velocity. This choice is intuitive an offers promising results when use with traitional backstepping approaches. Position control of VTOL UAVs has also been the focus of a number of researchers (for example, [3] [7]), yet espite their tremenous efforts, there still exist some open problems in terms of hanling external isturbances, coupling between system ynamics, singularities, as well as achieving global stability results. Previous work that aress unknown isturbances inclue [3] [5]. In most cases, the isturbance force is require to satisfy some assumptions in orer to evelop the control laws (for example, the isturbance is constant in the inertial frame of reference), which is also the case in this paper. A secon common problem is relate to which system inputs are use to specify the control law. Usually, it is esire to obtain the control torque that is applie to the rotational ynamics of the system (that is generate by the vectore thrust). This goal can be challenging, especially in the presence of external isturbances, an as a result, the control law is often specifie in terms of the esire system angular velocity (one integrator away from the control torque). The esire angular velocity must then be implemente using high-gain feeback. There are few examples of controllers in the literature that use the control torque as a system input while consiering external isturbances. In [5], a result is achieve for the position regulation problem; however, this result only guarantees local stability an oes not consier position tracking. In [8], a neural network aaptive controller has been propose; however, it is not accompanie with proofs for stability, either locally or globally /$ IEEE

2 130 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 For this type of aircraft, the vectore thrust action prouces a translational force in aition to a moment. Due to this system characteristic, another well-known problem is ue to this coupling between the rotational an translational ynamics. This coupling is usually in the form of a perturbing term (given as a function of the control torque or angular velocity) that affects the translational acceleration of the system. This problem is iscusse in more etail in [9] an [10]. This coupling term is system epenant, an is not always present in certain VTOL systems, for example, the quarotor aircraft. In [11], a positiontracking controller is propose for the quarotor aircraft achieving practical stability. In aition, an attitue controller for the quarotor aircraft is propose in [12]. As is the case with this paper, it is assume that the coupling term is negligible an is thus omitte in the control esign. However, as iscusse in [9] an [10], epening on the strength of the coupling an the choice of control gains, this can lea to unexpecte oscillations in the system states. There are some examples of controllers in the literature which aress the coupling problem. For instance, in [10], a nice change of coorinates is presente; however, only for a planar system. In [5], a change of coorinates is also presente that removes the coupling ue to the control torque. A consequence of this change of coorinates is that a new coupling is introuce in terms of the system angular velocity, which can only be neglecte if the system yaw rate is assume to be zero, which is the case in [5]. However, in practice, this woul likely not be the case. Therefore, there still seems to be some potential for improvement regaring this coupling term in future work. Last, to the best of our knowlege, there are no results in the available literature to achieve almost global asymptotic stability for the position-tracking problem of UAVs in the presence of isturbance forces that use the control torque as the system input. In [6] an [11], controllers are propose that achieve practical stability. Position-tracking control laws, that o not aress isturbance forces, can be foun in [7], [11], an [13]. In [4], a nice result is obtaine to achieve almost global stability using a simple control law, which is given in terms of the system angular velocity. A necessary step for this type of problem is to obtain a metho to extract the magnitue an irection of the thrust from a given esire translational force. In the case where only one vector is use for attitue extraction, there exists an infinite number of solutions to this problem. However, similar to the work of [14], we utilize one solution to the attitue extraction problem in terms of the unit quaternion that has almost no restrictions on the emane acceleration, except for a mil singularity that can be avoie. Relying on this quaternion extraction metho, we present two aaptive tracking controllers using the torque as a control input. Both controllers epen on an aaptive estimation metho, which use a projection mechanism [15], [16]. The projection mechanism is require to avoi the singularity associate with the attitue extraction metho. The first propose controller achieves the position-tracking objective for any initial conition of the state, whereas the secon controller achieves the position-tracking objective for a set of initial conitions, which are epenant on the control gains. The latter controller is inclue, since it is less complicate than the prior case an may be more suitable to use in practice. During the process of eveloping the two control laws, the isturbance forces are assume to be constant in the inertial frame. In this case, both control laws are proven to achieve the position-tracking objective provie that an upper boun of the isturbance force is known apriori(although the actual magnitue of the isturbance force may be less than this limit). To evaluate the robustness of the propose controller when the isturbance force is not constant, simulation results are provie, which consiers a moel of the aeroynamic forces that are exerte on the system in the presence of a uniform external win, which is assume to have a constant velocity. II. BACKGROUND A. Attitue Representation In this paper, we make use of two well-known forms of attitue representation, the rotation matrix (irect cosine matrix) an the unit quaternion [17] [19]. The rotation matrix is the map from the inertial frame to the boy frame, where R SO(3) := {R R 3 3 ; et(r) =1;RR T = R T R = I 3 3 }. The ynamics of the rotation matrix are Ṙ = S(Ω)R, where S( ) is the skew-symmetric matrix such that S(u)v = u v, where enotes the vector cross prouct an u, v R 3. The vector Ω R 3 is the boy-reference angular velocity of the rigi boy. The system attitue can also be represente by the unit quaternion Q =(q 0,q) Q = {Q S 3, Q =1}. The rotation matrix corresponing to a unit quaternion Q =(q 0,q) is given by R(q 0,q)=I S(q) 2 2q 0 S(q). (1) Other attitue representations an their transformations, incluing transformations from unit quaternion to rotation matrices can be foun in [17]. Given two quaternion Q, P Q, where P =(p 0,p) we use the well-known noncommutative quaternion multiplication Q P = ( p 0 q 0 q T p, q 0 p + p 0 q + S(q)p ). (2) The unit quaternion has the inverse Q 1 =(q 0, q), which has the property Q Q 1 = Q 1 Q =(1, 0), where (1, 0) is known as the ientity quaternion, an Q (1, 0) =(1, 0) Q = Q. In general, the unit-quaternion ynamics is expresse using the boy-reference angular velocity Ω by ] = 1 2 A (q 0,q)Ω (3) Q = 1 [ 0 2 Q Ω [ q T A (q 0,q)= q 0 I S(q) ]. (4) B. System Moel Using the representation given in Section II-A, the wellknown rigi boy moel can be written as follows: ṗ = v (5) v = gẑ T m RT ẑ 1 ml RT S(ẑ)u + 1 m F (6)

3 ROBERTS AND TAYEBI: ADAPTIVE POSITION TRACKING OF VTOL UAVS 131 Q = 1 [ q T ] Ω (7) 2 q 0 I S(q) I b Ω = S(Ω)Ib Ω+ɛ M S(ẑ)RF + u (8) where p, v, R 3 enote the inertial reference system position an velocity, respectively, F enotes the inertial reference isturbance force, I b is the constant boy-reference inertia tensor, u is the control torque input, T is the system thrust, ẑ = col [0, 0, 1], m is the system mass, l is the torque lever arm, g is the gravitational acceleration, an ɛ M is the lever arm that creates a isturbance torque ue to F. The moel for the isturbance force an torque is similar to [5], since we assume that the isturbance force is applie to a point on the boyreference z-axis at a istance ɛ M away from the system center of gravity. The coupling between the translational an rotational ynamics appears in the equation of v in the form of the control torque u. To simplify notation, we efine two unknown parameters θ a an θ b, in aition to a scalar thrust control input u t, which are given by θ a = 1 m F, θ b = ɛ M F, u t = T m. (9) C. Mathematical Preliminary Throughout the paper, u enotes the Eucliian norm of the vector u, an M F enotes the Frobenius norm of the matrix M. We also consier a boune function h : R 3 R 3 such that 0 h(u) < 1 u, whose first an secon partial erivatives are enote as follows: u h(u) :=φ h(u), u φ h(u)v := f φ h (u, v). (10) We choose the following boune function: h(u) = ( 1+u T u ) 1/2 u (11) from which we evaluate the partial erivatives as follows: f φ h φ h (u) = ( 1+u T u ) 3/2 ( I3 3 S(u) 2) (12) (u, v) = ( 1+u T u ) 5/2 ( 3 ( S(u) 2 I 3 3 ) vu T + ( 1+u T u ) (2S(u)S(v) S(v)S(u)) ). (13) The eigenvalues of φ h (u) are given by ( ) 1+ u 2 3/2 λ (φ h (u)) = ( ) 1+ u 2 1/2 ( ) (14) 1+ u 2 1/2 an therefore, 0 < φ h (u) 1 u. III. MAIN PROBLEM FORMULATION Using the moel given by (5) (8), our objective is to force the position p to track some continuous time-varying reference r(t), given that it meets the following requirements. Assumption 1: The secon, thir, an fourth erivatives (w.r.t. t) of the reference trajectory r(t) are uniformly continuous. Furthermore, there exists positive constants δ r an δ rz such that the secon erivative of the reference trajectory is boune by r δ r an ẑ T r <δ rz <g. Although, in general, the isturbance force exerte on the aircraft can be time-varying, for the purposes of eveloping the control laws we consier (for the time being) that the isturbance force is constant in the inertial frame of reference (for example, this may be vali if the system is moving with a constant velocity in the presence of a constant an uniform win). Assumption 2: The isturbances θ a an θ b are constant an there exists positive constants δ a <gan δ b such that the isturbances are containe in the set θ a B a := {θ a R 3 ; θ a < δ a <g}, θ b B b := {θ b R 3 ; θ b <δ b }. Assumption 3: The control torque lever arm l is sufficiently large such that ml 1, an therefore, the coupling term (ml) 1 R T S(ẑ)u 0. At this point, we efine the error signals for the system states that will be use in the control esign. Some of the error signals involve virtual control laws that will not be efine until later in the paper. Given the reference trajectory r(t), we efine the error signals as follows: p = p r(t), position error (15) ṽ = v ṙ(t), velocity error. (16) To use the system velocity v as a virtual control, we must use the system attitue R an thrust u t as virtual inputs to the translational ynamics (5) an (6). Therefore, we efine μ = gẑ u t R T ẑ (17) which is the acceleration of the system ue to gravity an the system thrust expresse in the inertial frame. We efine μ as the esire virtual acceleration, which will be efine later as an intermeiate step in the control esign. The acceleration error can then efine as follows: μ = μ μ, acceleration error. (18) The use of μ as a virtual control requires the extraction of the thrust u t an the esire attitue (or esire orientation) Q =(q 0,q ) from μ. That is, we require a transformation χ u t : R 3 R an χ Q : R 3 S 3, such that the signals u t = χ u t (μ ) an (q 0,q )=χ Q (μ )=(χ q0 (μ ),χ q (μ )) satisfy μ = col [μ 1,μ 2,μ 3 ]=gẑ u t R T ẑ (19) where R = R (q 0,q ) as efine by (1). The choice of χ u t an χ Q is the focus of Section IV. Provie that the transformation χ Q is ifferentiable, it is also possible to efine the ynamics of the esire attitue Q. The erivative of the esire attitue can then be use to obtain the esire angular velocity Ω, base on the following well-known relationship: Q = 1 2 A (q 0,q )Ω. (20)

4 132 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 Since A T A = I 3 3, one can obtain the esire angular velocity from Ω =2A T [ ] χq0 (μ ). (21) t χ q (μ ) Using the esire orientation Q, we also efine the attitue error Q =( q 0, q) as the rotation from the actual attitue Q to the esire attitue Q, which is given by Q = Q 1 Q, attitue error. (22) During the control esign, it is necessary to obtain the ynamics of the attitue error Q. Differentiating (22) along the trajectories (7) an (20), we obtain Q = 1 [ q T ] (Ω Ω). (23) 2 q 0 (Ω Ω )+S( q)(ω +Ω) Due to the use of the quaternion multiplication operator, the attitue error Q preserves the properties of the unit quaternion an can also be consiere as a rotation/orientation. Using this representation, one of the goals of the control esign is to force the attitue error Q to the ientity quaternion (1, 0), ortothe negate ientity quaternion ( 1, 0), since this represents a rotation of 2π from the esire attitue an is, therefore, the same physical orientation. A. Error Dynamics In this section, we efine the ynamics of the error signals given in the previous section. In light of Assumption 3, for the purposes of the control esign, we assume that the coupling term in the translational ynamics ue to the control torque is negligible. As a result of the earlier formulation, the system error ynamics are given by p =ṽ (24) ṽ = μ + μ + θ a r (25) Q = 1 [ q T (Ω Ω ] β) 2 q 0 (β + Ω Ω )+S( q)(β + Ω+Ω ) (26) I b Ω = S(Ω)Ib Ω+S(ẑ)Rθ b I b β + u (27) with Ω =Ω β, where β is a virtual control law for the angular velocity that is efine later in the control esign. More specifically, we fin two virtual control laws β = β 1 for the first control law, an β = β 2 for the secon control law. Base on this formulation, our control strategy can be separate into three tasks. 1) Specify the virtual control law for the esire virtual acceleration μ that satisfies the position- an velocity-tracking objectives. 2) Fin continuous an ifferentiable transformations χ u t an χ Q, which extract the esire thrust u t an esire attitue Q, respectively, that forces the system to the esire virtual acceleration specifie in step 1. 3) Specify the virtual control law for the angular velocity β that forces the system attitue Q to track the esire system attitue Q specifie in step 2, which is use to specify the control law for the system control torque u to force the system angular velocity Ω to track the esire angular velocity β. Section IV escribes the propose attitue extraction algorithm an efines the transformations χ u t an χ Q. Section V-A briefly restates the projection mechanism from [16]. Section V- B an C escribes the two propose control laws. A step-by-step proceure escribing the implementation of the two controllers is given in Section V-D. Finally, Section VI gives simulation results from the two propose controllers. IV. ATTITUDE EXTRACTION In this section, we efine the transformations χ u t an χ Q, which extract the system thrust u t, an esire attitue Q or R, from the virtual control law μ given by (19). In general, the attitue extraction problem for a single pair of vectors can be state as follows: given two vectors u an v, where u = v, we wish to fin an orientation R SO(3) that satisfies R u = v. (28) Note that (28) can also be expresse using the unit quaternion as follows: [ ] [ ] 0 = Q 1 0 v Q u (29) where instea we seek an expression for the quaternion Q. In [14], an intuitive solution to this problem is escribe. A similar solution which yiels a unit quaternion is state in the following lemma. Lemma 1: Given two vectors u an v, where u = v 0, an where u v; then, a solution for the unit quaternion Q =(q 0,q ) that satisfies (29) exists an is given by q 0 = 1 u 2 + u T v (30) u 2 q = 1 u 1 2( u 2 + u T S(v)u. (31) v) Proof: The proof is given in Appenix A. Remark 1: As iscusse by [14], this extraction uses the vector which is orthogonal to both u an v as the axis of rotation. Due to this choice, this metho extracts the solution to the problem that minimizes the angle of rotation. Given the solution of the unit quaternion by (30) an (31), a rotation matrix which satisfies (28) can also be foun using (1). We now apply the results of the quaternion extraction to our particular case. Recall (17) (19), an let μ be the virtual control input that achieves the position-tracking objective for the translational ynamics. We nee to extract the thrust u t an the esire orientation R or Q for a given μ satisfying (19). A solution for the system thrust u t = χ u t an esire attitue Q =(χ q0,χ q ) that satisfies (19), where the rotation matrix R is foun from (1) is given by the following lemma. Lemma 2: Given μ an assuming that μ / L L = {μ R 3 ; μ = col [0, 0,μ 3 ];μ 3 [g, )} (32)

5 ROBERTS AND TAYEBI: ADAPTIVE POSITION TRACKING OF VTOL UAVS 133 a solution for the transformations χ u t an χ Q, which extracts the system thrust u t = χ u t an esire attitue Q =(q 0,q )= (χ q0,χ q ), which satisfies (19), is given by χ u t = μ gẑ (33) ( ( 1 χ q0 = 1+ g μ )) 1/2 3 2 μ gẑ (34) μ 2 1 χ q = μ 1. 2 μ gẑ χ q0 0 (35) Proof: Note that (19) can also be written as R Tẑ = u 1 t (gẑ μ ). Choosing the system thrust input (33), then it is clear that this is similar to the problem escribe by Lemma 1 with v =ẑ an u =(gẑ μ ) / gẑ μ an u = v = 1. Applying these values of u an v to (30) an (31) results in the expressions given by (34) an (35). Remark 2: The singularity (32) correspons to the esire attitue being perfectly inverte such that the vertical acceleration of the vehicle is greater than or equal to the acceleration ue to gravity. Note that the singularity only applies to the esire attitue Q, an the actual attitue of the system can take any value without encountering a singularity. Since the singularity correspons to an unesirable operating moe of the aircraft, avoiing this singularity oes not significantly limit the normal operating moe of the system. Avoiing this singularity can be achieve by using a boune law for the esire virtual acceleration μ. This is the main purpose of using the function h escribe by (11) an the projection mechanism escribe by Section V-A. During the process of the control esign, it is necessary to obtain an expression for the angular velocity Ω, which can be foun using (21) an by ifferentiating the expression for the esire attitue Q.Thevalueof Q is obtaine by straightforwar ifferentiation of the extracte attitue given by (34) an (35). This proceure ultimately yiels the following expression: Ω = M(μ ) μ (36) 1 M(μ )= μ gẑ 2 c 1 μ 1 μ 2 μ μ gẑ c 1 μ 2 c 1 μ 2 1 μ gẑ c 1 μ 1 μ 2 μ 1 c 1 μ 2 μ gẑ μ 1 μ gẑ 0 where c 1 = μ gẑ + g μ 3. (37) V. POSITION-TRACKING CONTROL In this section, we propose two control laws in terms of the system torque control input u. The first control law guarantees the position-tracking objective is satisfie for any initial conitions. The secon control law guarantees the position-tracking objective for a set of initial conitions that is epenant on the control gains. The latter controller is inclue, since it is somewhat less complicate an may be beneficial from a practical perspective. The first step of the control esign is to choose the esire virtual acceleration μ. Base on the earlier formulation, there are some requirements for this control law that must be consiere. 1) To ensure a solution always exists for the esire orientation, Q, the esire virtual acceleration μ must be boune such that it oes not encounter the singularity (32). 2) In orer to satisfy the tracking objective, the expression for the esire virtual acceleration μ must contain an (aaptive) estimate of the isturbance force. 3) Due to the use of backstepping, the expression for μ must be twice ifferentiable. In orer to satisfy the first two requirements, the aaptive estimate of the isturbance force must be guarantee to be boune apriori. To meet this criteria, we use a projection-base estimation algorithm. Furthermore, in orer to satisfy the thir requirement, the solution for the isturbance estimates obtaine using the projection-base aaptation law must be twice ifferentiable. This motivates us to use the sufficiently smooth projection algorithm escribe by [16]. The use of other existing projection-base algorithms (for example, see [15] an [20]) is not irectly applicable since they are not always ifferentiable. As shown in [16], when utilizing projection, overparameterization is require when the system is of a sufficiently high orer. For this reason, a secon aaptive estimate ˆθ 2 is use ue to the unknown isturbance θ a, an a thir aaptive estimate ˆθ 3 is use ue to the unknown isturbance θ b. We begin by briefly escribing the sufficiently smooth projection algorithm of [16] in Section V-A. This is followe by the two control laws in Section V-B an C respectively. A. Aaptive Estimation Using Projection In this section, we restate the projection algorithm escribe by [16], which yiels aaptive estimates whose trajectories are sufficiently smooth. The use of the projection algorithm allows us to use an aaptive estimate of the isturbance force in the expression of the esire virtual acceleration μ, while ensuring that the esire attitue Q never encounters the singularity (32). Consier a constant unknown parameter θ p, which belongs to the set B p := { } θ p R 3 ; θ p <δ p, where the parameter δp is known apriori.letˆθ p be the corresponing aaptive estimate of θ p, an efine the error θ p = θ p ˆθ p. In general, the ieal aaptive estimation law is given by ˆθp = τ, which oes not necessarily guarantee that ˆθ p B p. Base on the ieal aaptive estimation law, a projection-base aaptation law efine in [16], which guarantees the boun of ˆθ p is given for our particular case by ˆθ p = τ + α(ˆθ p,δ p,τ) (38) α(ˆθ p,δ p,τ)= k α η 1 η 2 ˆθp (39) k α = (2(ɛ 2 α +2ɛ α δ p ) 2 δp 2 ) 1 (40)

6 134 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 η 1 = { (ˆθT p ˆθp δp 2 ) 2, if ˆθ p T ˆθ p >δp 2 0, otherwise } (41) η 2 = ˆθ p T τ +((ˆθ p T τ) 2 + δα) 2 1/2 (42) where ɛ α > 0, δ α > 0 an has the properties ˆθ p <δ p + ɛ α, θt p α 0, ˆθ p C 1. (43) B. Controller 1 Let ˆθ 1 enote the first aaptive estimate of the unknown parameter θ a. Using the boune function h(u) efine by (11), we propose the following law for the esire virtual acceleration: μ = r ˆθ 1 k p Γ 1 v h( p) (k v + k θ ) h(ṽ) (44) where k p,k θ,k v > 0, Γ v =Γ T v > 0, an ẑ = col [0, 0, 1].Using the parameters δ rz an δ a efine in Assumptions 1 an 2, we place the restriction k p ẑ T Γ 1 v +2k θ + k v + ɛ α <g δ rz δ a (45) where ɛ α > 0 is a control gain use in the projection algorithm. Applying the value of the esire virtual acceleration (44) to the thrust an attitue extraction algorithm specifie by (33) (35), we extract the system thrust an esire attitue u t = χ u t (μ ), q 0 = χ q0 (μ ), q = χ q (μ ) (46) from which we obtain the attitue error Q =( q 0, q) =Q 1 Q =(q 0, q ) Q. We propose the following estimation law for the first aaptive estimate: ˆθ 1 = γ (τ 2 + α(ˆθ 1,δ a + k θ,τ 2 )) (47) τ 2 =Γ v ṽ + k θ k p φ h (ṽ)γ 1 v γ ( + h( p)+ k θk v γ φ h (ṽ)h(ṽ) γ q (k θ + k v )φ h (ṽ)m(μ ) T 2u tk θ φ h (ṽ)r T S( q) γ ) q (48) q = q 0 ẑ + S(ẑ) q (49) where γ > 0, γ q > 0, φ h (u) is the partial erivative of h(u) as efine by (12), an α is the projection function efine by (39). We propose the following virtual control law for the angular velocity: β 1 = M(μ )(r (3) + w β ) 1 γ q Φṽ K q q (50) w β = k p k v φ h (ṽ)γ 1 v h( p) γ α(ˆθ 1,δ a + k θ,τ 2 ) (51) Φ=(γ γ q M(μ ) 2u t S( q)r)γ v + γ q k p M(μ )Γ 1 v φ h ( p) (52) where K q = Kq T > 0, an the matrix M(μ ) is given by (37). In general, the erivative of (50) is given by β 1 = f β1 + f β1 θ a (53) where the actual expressions for f β1 an f β1 are given in Appenix B. Let ˆθ2 enote the secon estimate of θ a, ˆθ3 enote the estimate of θ b, an let Ω =Ω β 1 enote the angular velocity error. We propose the following control law for the system torque control input: u = γ q q + S(Ω)I b Ω S(ẑ)Rˆθ 3 + I b f β1 + I b fβ1 ˆθ2 K ω Ω (54) where K ω = Kω T > 0, in aition to the following aaptation laws: ˆθ 2 = γ θ2 ( f β T 1 I b Ω+α(ˆθ2,δ a, f β T 1 I b Ω)) (55) ˆθ 3 = γ θ3 ( R T S(ẑ) Ω+α(ˆθ 3,δ b, R T S(ẑ) Ω)) (56) where γ θ2, 3 > 0, an δ b > 0 is given base on Assumption 2. Theorem 1: Consier the system escribe (24) (27), where we apply the control an estimation laws (47) an (54) (56). Provie that the following inequalities are satisfie: λ min (K q ) > 2 2k v ū t δ μ +2γ γ q (k v + k θ ) δ 2 μ + k2 v 2ɛ 1 λ min (Γ v ) > γ q k v ɛ 1 δμ 2 (57) where ɛ 1 > 0, an ū t = g + δ r + δ a + k p Γ 1 v + k v +2k θ + ɛ α (58) δ μ = g δ rz δ a k p ẑ T Γ 1 v k v 2k θ ɛ α (59) then the system thrust input u t is boune an nonvanishing such that 0 <δ μ <u t < ū t, the system states (p, v, Ω) are boune, an lim [p(t) r(t),v(t) ṙ(t), q(t), Ω(t)] = 0 (60) t for any initial conition. 1 Furthermore, the aaptive estimates ˆθ 1, ˆθ 2, ˆθ 3 are boune, an in particular, the estimate ˆθ 1 converges asymptotically to the constant unknown isturbance θ a. Proof: The proof is given in Appenix B. C. Controller 2 In this section, we propose a similar albeit simpler version of the control law escribe in Section V-B. The motivation to simplify the previous result is largely ue to the complexity of the virtual control law for the angular velocity β 1 from (50), an it is erivative of (53). It is possible to simplify this virtual control law, if an aitional constraint is satisfie that is base on the initial conitions of the state an the control gains. This simplifie version, which may be more suitable from a practical perspective, is escribe in the following. Using the previous law for the esire virtual acceleration from (44) uner the restriction (45), an the extraction of the 1 There are two equilibrium solutions (physically ientical), which satisfy the position-tracking objective, which are efine by ( p, ṽ, q, Ω) = 0, q 0 = ±1. The equilibrium solution characterize by q 0 =1is stable, while the one characterize by q 0 = 1 is unstable (repeller equilibrium). This is ue to the well-known topological obstruction on SO(3) uner strictly continuous feeback. For more etails regaring this topological obstruction (see [21]). One solution to this problem consists of using iscontinuous or hybri feeback control laws (see, for instance, [22]).

7 ROBERTS AND TAYEBI: ADAPTIVE POSITION TRACKING OF VTOL UAVS 135 thrust control input u t an esire attitue Q from (46), we obtain the attitue error Q =( q 0, q) =Q 1 Q.Letˆθ 1 enote the first estimate of θ a, where we use the estimation law (47). We propose the following virtual control law for the angular velocity: β 2 = M(μ )(r (3) + w β ) K q q (61) where K q = Kq T > 0, an w β is given by (51). The erivative of β 2 can be written as follows: β 2 = f β2 + f β2 θ a (62) where the actual expressions for f β2 an f β2 are given in Appenix D. Using the angular velocity error Ω =Ω β 2,we propose the following control law for the system control torque input: u = γ q q + S(Ω)I b Ω S(ẑ)Rˆθ 3 + I b f β2 + I b fβ2 ˆθ2 K ω Ω (63) where K ω = Kω T > 0. Using the new expression for angular velocity error Ω, we apply the aaptive estimation laws (55) an (56). Theorem 2: Consier the system escribe (24) (27), where we apply the control an estimation laws (47) an (63). Using the angular velocity error Ω =Ω β 2, where β 2 is obtaine using (61), we apply the estimation laws (55) an (56). Provie that the following inequalities are satisfie: λ min (K q ) > 2 2k v ū t δ μ +2γ γ q (k v + k θ ) δ 2 μ + k2 v 2ɛ 1 + δ2 1 2γ q ɛ 2 (64) λ min (Γ v ) > γ q k v ɛ 1 δμ 2 (65) ( ) 2γq γ 2γq k p δ 1 = 2ū t + Γ v + Γ 1 v (66) δ μ where ɛ 2 > 0, then the system thrust input u t is boune an nonvanishing such that 0 <δ μ <u t < ū t, the system states (p, v, Ω) are boune, an lim [p(t) r(t),v(t) ṙ(t), q(t), Ω(t)] = 0 (67) t for all system initial conitions that satisfy ) k p (1+ p(0) T p(0) X(0)T CX(0) ( 2 Δv < λ min (Γ v ) 1 ) ɛ 2 2 ( Δ v = k v Γ v γ ) q k v ɛ 1 δμ 2 I 3 3 δ μ X = col[ṽ, 1 q 0, q, Ω,θ a k θ h(ṽ) ˆθ 1, (68) (69) θ a ˆθ 2,θ b ˆθ 3 ] (70) C = iag[ Γ v I 3 3, 4γ q I 4 4, I b I 3 3,γ 1 θ 1 I 3 3, γ 1 θ 2 I 3 3,γ 1 θ 3 I 3 3 ]. (71) Furthermore, the aaptive estimates ˆθ 1, ˆθ 2, an ˆθ 3 are boune, an in particular, the estimate ˆθ 1 converges asymptotically to the constant unknown isturbance θ a. Proof: The proof is given in Appenix C. D. Implementation To implement the controllers given in Section V-B an C, consier the following iterative proceure. 1) Obtain the signals p, v, Q, Ω, an the esire reference trajectory r(t) an i /t i r(t), i =2, 3, 4, an calculate the error signals p, ṽ. 2) Calculate the virtual control law μ using (44), which is use to obtain the system thrust input u t from (33), esire attitue Q using (34) an (35), an the matrix M(μ ) from (37). 3) Calculate the error signals Q an μ from (22) an (18), respectively, which is use to obtain τ 2 from (48) an ˆ from (47). 4) Using the virtual control law for the esire angular velocity (50) (or (61) for controller 2), calculate the angular velocity error Ω =Ω β 1 ( Ω =Ω β 2 for controller 2). 5) Using the expression for the erivative β 1 ( β 2 for the secon control law) given by (103) an (104) [(101) an (102) for controller 2], apply the control law u from (54) (or (63) for controller 2), an the estimation laws ˆθ2 an ˆθ 3 from (55) an (56), respectively. VI. SIMULATIONS To ai in the evelopment of the control laws, some assumptions were require regaring the isturbance force F an the control torque lever arm l. Assumption 2 states that the isturbance forces were assume to be constant in the inertial frame, an Assumption 3 requires that the control torque lever arm is sufficiently large enough that an unesire coupling term can be omitte. However, these may be unlikely an/or unrealistic assumptions, since the isturbance forces are epenant on system aeroynamic forces cause by win an the motion of the system, an the control torque lever arm may not be sufficiently large. To aress these shortcomings, we emonstrate the robustness of the propose controller by incluing in the simulations a time-varying isturbance force (which is base on an aeroynamic moel) an the coupling term that was previously omitte (using reasonable values for the system mass m an the control torque lever arm l). The isturbance moel use is efine by F = F rag + F ram F rag = v w v R T C R (v w v) TρA F ram = 2 RT I xy R (v w v) where F rag are frictional rag forces that are proportional to the square of the external airflow, F ram is the ram-rag

8 136 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 Fig. 2. Torque-generating aeroynamic forces as a result of airflow in boyfixe frame. force, 2 I xy = iag (1, 1, 0), v w is an inertially reference unknown win velocity (which is assume to be uniform an constant in the inertial frame), ρ is the air ensity, A is the uct cross-sectional area, an C R 3 is a matrix that consists of system-epenant aeroynamic constants expresse in the boy-fixe frame. For the purposes of simulation, the aeroynamic-rag matrix an win velocity was chosen to be C = iag [0.1, 0.1, 0.05] kg/m an v w =[ 1, 1, 0] m/s. Assuming that the external airflow is uniform, ue to the cylinrical symmetry of the system, the net aeroynamic force is assume to be applie at a point on the boy-reference z-axis, locate at a constant istance of ɛ M =0.1 m from the system center of gravity, which is often referre to as the aeroynamic centerof-pressure [see Fig. 2]. In aition, to simulate uncertainty in the system inertia tensor, the controllers were implemente using an expecte value of I b = iag [0.5, 0.5, 0.25] kg m 2, where the actual value was specifie as I b = iag [0.6, 0.6, 0.3] kg m 2. For both simulations, the esire reference trajectory was specifie as r(t) =[10t, 30 sin(0.1t +3.48), 20 sin(0.1t +4.71)] T. 1) Other system parameters use in simulations: system mass m =5kg, gravitational acceleration g =9.81 m/s 2, control torque lever arm l =0.5 m, upper boun for θ a, δ a =5m s 2, upper boun for θ b, δ b =3N m, air ensity ρ =1.2kg/m 3, an uct cross-sectional area A =0.114 m 2 (correspons to 15 in uct inner iameter). 2) Initial conitions use in simulations: p(0) = col[50, 10, 0] m, v(0) = col [5, 0, 0.5] m/s, Q(0) = (q 0 (0), q (0)) = (0, 1, 0, 0), an Ω(0) = col [0, 0, 0]. 3) Control/aaptation gains for controllers 1 an 2: k p = 1, k v =0.1, Γ v = iag [0.2, 0.2, 0.8], k θ =1, γ q =10, K q =20I 3 3, K ω =20I 3 3, γ =0.2, an γ θ2 = γ θ3 = 1. The simulation results are given by Figs Although the secon controller is easier to implement, in situations where the system initial conitions are sufficiently far from the esire trajectory, some control gains are require to have extremely large values [as specifie by requirements (64) (68)]. These Fig. 3. Three-imensional plot of esire versus actual system trajectory for controller 1. Fig. 4. Three-imensional plot of esire versus actual system trajectory for controller 2. Fig. 5. Norm of position error p. 2 In aition to generating the thrust T along the boy-reference vertical axis ẑ, the change in momentum of the airflow (ue to the system rotors/propellers) can cause an aitional force when the external uct airflow velocity has a component, which is orthogonal to the thrust vector T ẑ. This force (which is also orthogonal to the thrust vector) is cause ue to the eceleration of the horizontal component of the airflow, an is known as the ram rag. For further information, see [23]. Fig. 6. Norm of velocity error ṽ. requirements are likely conservative, an simulations suggest that the omain of attraction is actually larger than the region specifie by (68).

9 ROBERTS AND TAYEBI: ADAPTIVE POSITION TRACKING OF VTOL UAVS 137 Fig. 7. Norm of attitue error q. Fig. 11. System thrust T. spons to the system being completely inverte. This is one to emonstrate the effectiveness of the control laws for extreme eviations in the system attitue. The control laws were effective, espite the time-varying isturbance ue to aeroynamic rag, the coupling term, which was omitte uring the control esign, an uncertainty in the inertia tensor. Fig. 8. Fig. 9. Fig. 10. Norm of angular velocity error Ω. Norm of estimation error θ 1. Norm of control torque u. The simulations show that both controllers are successful in forcing the system to the esire trajectory. For each case, the system attitue was initialize at Q =(0, 1, 0, 0), which corre- VII. CONCLUSION Two aaptive position-tracking controllers have been propose for a VTOL UAV in the presence of external isturbances. The secon control law is inclue since it is somewhat less complicate to implement than the first controller, which may be esirable from a practical point of view. The two control laws are epenant on a quaternion extraction algorithm, which allows the extraction of the system attitue an thrust from the esire virtual acceleration that is require to achieve the tracking objective. This quaternion extraction metho provies almost global results, with the exception of a mil singularity, which is easily avoie by using a boune control law for the esire virtual acceleration require to meet the tracking objective. To compensate for the external isturbances while ensuring the boun of the require acceleration, projection is use in the aaptive estimation algorithms. Although this improves the robustness of the propose controllers, this metho requires the use of an overparameterization of the parameters estimates. Simulations were performe for the two propose controllers. In both cases, the controllers successfully achieve the tracking objective even in the presence of the coupling term that is omitte in the control esign, time-varying isturbance ue to aeroynamic rag an uncertainty in the inertia tensor of the system. Due to the time-varying nature of the isturbances, a slight error was observe in the simulations. The implementation of the secon controller was less complicate; however, in general, high gains were require to guarantee stability, although simulations suggest that the stability requirements are quite conservative. Still, even when using reasonable gains, the secon controller was shown to be effective. There is also potential for improvement ue to the open problem relate to the coupling term that is neglecte in the control esign. Another improvement woul be to remove the overparameterization that is require ue to the projection algorithm.

10 138 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 APPENDIX A PROOF OF ATTITUDE EXTRACTION One efinition of a unit quaternion as it pertains to a rotational transformation is given by Q =(cos(δ/2), sin(δ/2)ˆk), where δ is an angle of rotation about the normalize axis of rotation ˆk. One possible solution for the angle an axis of rotation can be foun by using the scalar an vector proucts, i.e., u T v = u v cos(δ), an S(v)u = u v sin(δ)ˆk. From the efinition of the scalar prouct, we fin cos(δ) =u T v/ u 2,whereweassume u = v. The result sin(δ) =1/ u 2 ( u 2 + u T v)( u 2 u T v) follows from the fact that sin 2 (δ) =1 cos 2 (δ). Applying the ouble angle formula cos(δ) =1 2sin 2 (δ/2), we obtain sin(δ/2) = u 1/ u 2 u T v 2. Subsequently, the normalize axis of rotation is given by ˆk =(( u 2 + u T v)( u 2 u T v)) 1/2 S(v)u. Therefore, the solution for the vector an scalar parts of the quaternion are given by ˆk sin (δ/2) = 1 1 u 2( u 2 + u T v) S(v)u cos(δ/2) = sin(δ) 2sin(δ/2) = 1 u 2 + u T v. u 2 APPENDIX B PROOF OF THEOREM 1 In the following sections, we present the proof of the control law propose in Section V-B. The proof is complete in a number of stages. In subsection 1, we focus on the upper an lower bouns for the system thrust as a result of the propose control law. In subsection 2, we analyze the system translational ynamics, the ynamics of the estimation error, an the ynamics of the angular velocity associate with the quaternion Q. The parts of the proof containe in Appenix B-1 an 2 are the same for both propose control laws. Appenix B-3 finalizes the proof for the first propose control law, where the proof for the secon control law can be foun in Appenix C. Appenix D provies erivatives of a number of functions that are necessary to implement the controller. 1) Boune Control: The propose control laws are epenent on an attitue extraction algorithm that encounters a singularity efine by (32). Fortunately, the singularity can be avoie if the thir component of the virtual control law μ is boune such that μ 3 =ẑ T μ <g. Recall the expression for the signal μ given by (44). Due to Assumption 1, the acceleration of the reference trajectory is boune such that ẑ T r <δ rz <g an r <δ r, where the parameters δ r an δ rz are known apriori. Given the projection-base estimation law (47) an the property (43), the isturbance estimate ˆθ 1 is boune such that ˆθ 1 <δ a + k θ + ɛ α. In aition, the function h( ), efine by (11), is boune such that 0 h( ) < 1. Consequently, the signal μ is also boune by μ < μ = k p Γ 1 v + δ r + δ a +2k θ + k v + ɛ α (72) ẑ T μ < μ 3 = k p ẑ T Γ 1 v + δ rz + δ a +2k θ + k v + ɛ α (73) where ue to (45), the boun on the thir component of μ is limite to ẑ T μ = μ 3 < μ 3 <g. As a result, we efine δ μ = g μ 3 > 0. (74) As a result of the bouns (72) (74), the system thrust, given by (33), is also boune such that δ μ <u t < ū t, where ū t = μ + g. Therefore, the system thrust never vanishes an the singularity in the attitue an thrust extraction (32) is avoie. 2) Translational an Quaternion Dynamics: Recall the expression for the velocity error ynamics efine by (25), an let θ 1 enote the following estimation error function: θ 1 = θ a ˆθ 1 k θ h(ṽ). (75) Given the virtual control law μ efine by (44) an the estimation error θ 1, the time erivative of the velocity error can now be written as follows: ṽ = μ K 1 h( p) K 2 h(ṽ) ˆθ 1 + θ a = μ K 1 h( p) k v h(ṽ)+ θ (76) where K 1 = k p Γ 1 v, an K 2 =(k v + k θ ) I 3 3. Furthermore, in light of Assumption 2 an the velocity error (76), the time erivative of (75) is given by θ 1 = k θ φ h (ṽ) θ 1 + τ 1 ˆ (77) τ 1 = k θ φ h (ṽ)( μ K 1 h( p) k v h(ṽ)). (78) The expressions for ṽ an are epenant on the error function μ = μ μ. A more convenient notation is to express the error function μ in terms of the attitue error Q =( q 0, q) =Q 1 Q, where Q is the esire attitue efine by (34) an (35). This can be achieve if we consier the rotation matrix R = RR T (which correspons to the unit quaternion Q) an the fact that R = I S( q) 2 2 q 0 S( q) to obtain μ = W T 1 q W 1 = 2u t S( q)r q = S(ẑ) q + q 0 ẑ. (79) Consequently, the expressions for ṽ an can be written as functions of the attitue error q. At this point, we focus our attention on the ynamics of the attitue error in terms of the quaternion scalar q 0.From (26), we note that the time erivative of q 0 is given by q 0 = 1/2 q T (Ω Ω), where from (36), we recall Ω = M(μ ) μ. Differentiating (44) in light of (47) an (76), we fin the erivative μ to be given by μ = r (3) + w β + W 2 h(ṽ)+w 3 q + W 4 ṽ (k θ + k v ) φ h (ṽ) θ 1 (80) W 2 = kv 2 φ h (ṽ) W 3 = γ γ q (k θ + k v ) φ h (ṽ)m(μ ) T k v φ h (ṽ)w1 T W 4 = γ Γ v k p Γ 1 v φ h ( p) w β = k v k p φ h (ṽ)γ 1 v h( p) γ α(ˆθ 1,δ a + k θ,τ 2 ) (81)

11 ROBERTS AND TAYEBI: ADAPTIVE POSITION TRACKING OF VTOL UAVS 139 from which we obtain the esire attitue ynamics Ω = M(μ )(r (3) + w β + W 2 h(ṽ)+w 3 q + W 4 ṽ) (k θ + k v )M(μ )φ h (ṽ) θ 1. (82) Since Ω is not entirely known (ue to the presence of the signal θ 1 ), it is necessary to stuy the upper boun of the unesire terms in (82). For the most part, this analysis is straightforwar except for the matrix M(μ ). To etermine an upper boun for this matrix, we apply the Frobenius norm to the expression for M(μ ) given by (37). For convenience, we let ξ = col [ξ 1,ξ 2,ξ 3 ]=μ gẑ an fin the value of the norm to be given by M(μ ) F = 2/( ξ 2 + ξ ξ 3 )+1/ ξ 2. Since inf{ ξ } =inf{ ξ 3 } = δ μ, we obtain M(μ ) F 2 δ μ. (83) We now propose the following Lyapunov function caniate: V 1 = k p ( 1+ p T p 1) + 1 2ṽT Γ v ṽ +2γ q (1 q 0 )+ 1 2γ θt 1. (84) Given (24), (26), (76) (78), (82), an the aaptive estimation law (48), we ifferentiate V 1 to obtain V 1 = k v ṽ T Γ v h(ṽ) k θ θt γ 1 φ h (ṽ) θ 1 θ 1 T α(ˆθ 1,δ a + k θ,τ 2 ) + q T (Φṽ + γ q (Ω M(μ )r (3) M(μ )(w β + W 2 h(ṽ)+w 3 q))) (85) Φ=W 1 Γ v γ q M(μ )W 4. (86) 3) Angular Velocity Error Dynamics Controller 1: Recall the expression for the angular velocity error is given by Ω = Ω β 1. Applying the virtual control law β 1, which is given by (50), to V 1 efine by (85), we obtain V 1 = k v ṽ T Γ v h(ṽ) k θ θt γ 1 φ h (ṽ) θ 1 + γ q q T Ω θ 1 T α(ˆθ 1,δ a + k θ,τ 2 ) γ q q T M(μ )W 2 h(ṽ) γ q q T (K q + M(μ )W 3 ) q. (87) To further simplify this result, using (14) an (83), we apply Young s inequality to obtain the following upper boun: γ q q T M(μ )W 2 h(ṽ) γ q k 2 v 2ɛ 1 q T q + γ q k 2 v ɛ 1 δ 2 μ ṽ T h(ṽ) (88) where ɛ 1 > 0. Furthermore, ue to S( q) 1 an u t < ū t = g + μ, we also fin M(μ )W 3 2 2k v ū t δ μ + 2γ θ 1 γ q (k θ + k v ) δ 2 μ. (89) Due to the bouns (88) an (89), the Lyapunov function erivative (87) is boune by V 1 ṽ T Δ v h(ṽ) γ q q T Δ q q k θ γ θt 1 φ h (ṽ) θ 1 θ 1 T α(ˆθ 1,δ a + k θ,τ 2 )+γ q q T Ω, (90) ( Δ v = k v Γ v γ ) q k v ɛ 1 δμ 2 I 3 3 (91) ( 2 ) 2k v ū t δ μ Δ q = K q +2γ γ q (k θ + k v ) δμ 2 + k2 v I ɛ 1 (92) Provie that (57) is satisfie, then Δ v an Δ q are positive efinite matrices. Introucing the error signals θ 2 = θ a ˆθ 2 an θ 3 = θ b ˆθ 3, we introuce the secon Lyapunov function V 2 = V Ω T 1 I b Ω+ θt 2γ 2 θ2 + 1 θt θ2 2γ 3 θ3 θ3 = k p ( 1+ p T p 1) XT CX (93) X = col[ṽ, 1 q 0, q, Ω, θ 1, θ 2, θ 3 ] (94) C = iag[γ v, 4γ q I 4 4,I b,γ I 3 3,γ θ2 I 3 3,γ θ3 I 3 3 ]. (95) The time erivative of (93) is subsequently foun using (90), in aition to the control an estimation laws efine by (54) (56), to obtain the following result: V 2 ṽ T Δ v h(ṽ) γ q q T Δ q q k θ γ θt 1 φ h (ṽ) θ 1 Ω T K ω Ω θt 1 α(ˆθ 1,δ a + k θ,τ 2 ) θ T 2 α(ˆθ 2,δ a, f T β 1 I b Ω) θt 3 α(ˆθ 3,δ b, R T S(ẑ) Ω). Due to the property of the projection law given by (43), θ T i α> 0, an the Lyapunov function erivative can be simplifie as follows: V 2 ṽ T Δ v h(ṽ) γ q q T Δ q q k θ θt γ 1 φ h (ṽ) θ 1 Ω T K ω Ω. Therefore, V2 0 an the states ( p, ṽ, Ω) are boune. The attitue error Q is boune by efinition, an the aaptive estimation error,2,3 are boune ue to Assumption 2 an ue to the property of the projection mechanism (43). Applying Barbalat s Lemma, V2 is boune ue to Assumption 1, which shows that ( p, ṽ, q, Ω, θ 1 ) 0 as t. Since θ 1 0 an ṽ 0, then ˆθ 1 θ a. In aition, since ṽ 0, an ṽ = W1 T q K 1 h( p) (k v + k θ )h(ṽ) ˆθ 1 + θ a = K 1 h( p) =0, then p 0, which satisfies the tracking objective. APPENDIX C PROOF OF THEOREM 2 In this section, we present the proof of the control law propose in Section V-C. The two propose controllers iffer ue

12 140 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 to the choice of the virtual control law for the angular velocity. Consequently, the sections pertaining to the boune control an translational an quaternion ynamics are similar to both proofs. Therefore, before proceeing further see Appenix B-1 an 2. Base on the framework outline in Appenix B-1 an 2, the following section finalizes the proof of Theorem 2. Appenix D provies erivatives of a number of functions that are necessary to implement the controller. Using the secon control scheme, the angular velocity error is now efine as Ω =Ω β 2, where the virtual control law β 2 is given by (61). To stuy the stability of the system using the secon controller, we use the same Lyapunov function caniate V 1 given by (84). Using the expression for V 1 given by (85), in aition to the virtual control law β 2, the bouns efine by (88) an (89), an the matrices (91) an (92), the upper boun of the V 1 is now given by V 1 k θ θt γ 1 φ h (ṽ) θ 1 θ 1 T α(ˆθ 1,δ a + k θ,τ 2 )+ q T Φṽ ṽ T Δ v h(ṽ) γ q q T Δ q q + γ q q T Ω. Using (86) in aition to (79) an (81), the expression for Φ can also be written as follows: Φ=(γ γ q M(μ ) 2u t S( q)r)γ v + γ q k p M(μ )Γ 1 v φ h ( p). Due to boun of the matrix M(μ ) given by (83) an the fact u t < ū t = g + μ, we fin the upper boun of the matrix Φ given by ( ) 2γq γ 2γq k p Φ +2ū t Γ v + Γ 1 v. δ μ From the efinition of δ 1 given by (66), we see that Φ δ 1, an therefore, using Young s inequality, we fin q T Φṽ δ2 1 q T q + ɛ 2 2ɛ 2 2 ṽt ṽ for any ɛ 2 > 0. Therefore, the time erivative of the Lyapunov function V 1 is moifie as follows: δ μ V 1 k θ θt γ 1 φ h (ṽ) θ 1 θ 1 T α(ˆθ 1,δ a,τ 2 )+γ q q T Ω q T Δq q ṽ T Δv ṽ (96) where we efine the matrices Δ q = γ q Δ q δ2 1 2ɛ 2 I 3 3 Δ v = 1 (1 + ṽ T ṽ) 1/2 Δ v ɛ 2 2 I 3 3. (97) Using the two estimation error functions θ 2 = θ a ˆθ 2 an θ 3 = θ b ˆθ 3, we introuce the following Lyapunov function: V 2 = V Ω T 1 I b Ω+ θt 2γ 2 θ2 + 1 θt θ2 2γ 3 θ3 θ3 = k p ( 1+ p T p 1) XT CX (98) where X an C are given by (94) an (95), respectively. In light of (96), the estimation laws (55) an (56) an the control law (63), we fin the following upper boun for V 2 : V 2 ṽ T Δv h(ṽ) γ q q T Δq q k θ γ θt 1 φ h (ṽ) θ 1 Ω T K ω Ω θt 1 α(ˆθ 1,δ a + k θ,τ 2 ) θ T 2 α(ˆθ 2,δ a, f T β 2 I b Ω) θt 3 α(ˆθ 3,δ b, R T S(ẑ) Ω). Due to the property of the projection law (43), θ i T α>0, an V 2 ṽ T Δv h(ṽ) γ q q T Δq q k θ θt γ 1 φ h (ṽ) θ 1 Ω T K ω Ω. Given the requirements (64) an (65) are satisfie, then Δ q > 0 an Δ v > 0. However, in light of (97), to ensure that Δ v > 0, we must also satisfy the following inequality: ṽ 2 < 4 ɛ 2 2 Δ v 2 1. Due to the efinition of the Lyapunov function (98), the following inequality is always satisfie: 1 2 λ min(γ v ) ṽ 2 V 2 k p ( 1+ p T p 1) + X T CX where C is given by (71), which we can further simplify to obtain ṽ 2 2λ min (Γ v ) 1 (k p ( 1+ p T p 1) + X T CX). Therefore, to ensure V 2 0, it is sufficient to have 2λ min (Γ v ) 1 (k p ( 1+ p(0) T p(0) 1) + X(0) T CX(0)) < 4 ɛ 2 2 Δ v 2 1 which is satisfie ue to (68), an consequently, Δ v is positive efinite. Therefore, V 2 0 an the states ( p, ṽ, Ω) are boune. The attitue error Q is boune by efinition, an the aaptive estimation error θ 1,2,3 are boune ue to Assumption 2 an ue to the property of the projection mechanism (43). Applying Barbalat s lemma, V2 is boune ue to Assumption 1, which shows that ( p, ṽ, q, Ω, θ 1 ) 0 as t. Since θ 1 0 an ṽ 0, then ˆθ 1 θ a. In aition, since ṽ 0, an ṽ = W1 T q K 1 h( p) K 2 h(ṽ) ˆθ 1 + θ a = K 1 h( p) = 0, then p 0, which satisfies the tracking objective. APPENDIX D DERIVATIVES OF ANGULAR VELOCITY VIRTUAL CONTROL LAWS β 2 AND β 1 In this section, we obtain the erivatives of the two virtual control laws for the angular velocity β 1 an β 2, which are given by (50) an (61), respectively. Due to the complexity of the virtual control laws, we begin by evaluating the erivatives of several signals before continuing to the erivatives of the virtual control laws. We first focus on the partial erivative of the matrix M(μ ) an its transpose. Let μ = col [μ 1,μ 2,μ 3 ] an

13 ROBERTS AND TAYEBI: ADAPTIVE POSITION TRACKING OF VTOL UAVS 141 v = col [v 1,v 2,v 3 ] enote two arbitrary vectors, an efine the functions Z 1,Z 2 : R 3 R 3 R 3 such that Z 1 (μ,v):= M(μ )v Z 2 (μ,v):= M(μ ) T v. μ μ From the efinition of M(μ ) given by (37), after some straightforwar albeit teious calculations, we evaluate these functions to be Z 1 (μ,v)=γm 1 M(μ )vf γ + γ M Λ 1 (μ,v) Z 2 (μ,v)=γm 1 M(μ ) T vf γ + γ M Λ 2 (μ,v) f γ = γm 2 (gẑ μ ) T (3c 1 I S(ẑ)S(gẑ μ )) where γ M = μ gẑ 2 c 1 1, c 1 = μ gẑ + g μ 3, an Λ 1 = c 1 v 2 c 1 v 1 α 1 (μ ) T μ 2 v 1 μ 1 v 2 μ 2 v 1 μ 1 v 1 2v 2 μ 2 + c 1 v v 1 μ 1 + v 2 μ 2 c 1 v 3 μ 1 v 2 0 u t v 2 u t v 1 0 +(u t v 2 + μ 2 v 3 u t v 1 μ 1 v 3 0) T α 2 (μ ) T, 2v 2 μ 1 μ 2 v 1 u t v 3 μ 1 v 1 0 Λ 2 = μ 2 v 2 u t v 3 μ 1 v 2 2μ 2 v 1 0 c 1 v 2 c 1 v 2 0 μ 2 v 3 c 1 v 2 + c 1 v 1 μ 1 v 3 α 1 (μ ) T 0 +( u t v 2 u t v 1 μ 2 v 1 μ 1 v 2 ) T α 2 (μ ) T with α 1 =(μ gẑ) / μ gẑ an α 2 = α 1 ẑ. In orer to obtain the erivative of the projection law α, we first focus on obtaining the erivative of the signal τ 2. Leaing up to this goal, we first ifferentiate several signals. Due to the unknown parameter θ a, in general, we group the erivative of an arbitrary signal x into known an unknown components as ẋ = f x + f x θ a. Recall the expression for the signal μ given by (80). This result can also be written as μ = f μ + f μ θ a, where the functions f μ an f μ are given by f μ = r (3) + w β +(W 2 + k θ K 2 φ h (ṽ)) h(ṽ)+w 3 q + W 4 ṽ + K 2 φ h (ṽ)ˆθ 1 f μ = K 2 φ h (ṽ) where K 2 =(k θ + k v ) I 3 3. Similarly, in light of (26), the erivative of q can be written as q = + f q f q θ a using the following expressions: f q = 1 2 ( q 0I + S( q)) Ω (S( q) q 0I)(M(μ )f μ ) f q = 1 2 (S( q) q 0I) ( M(μ ) f μ ). From the efinition of μ given by (18), the erivative μ = f μ + f μ θ a is obtaine, where f μ = (I + u 1 t R T ẑ(μ gẑ) T )f μ u t R T S(Ω)ẑ f μ = (I + u 1 t R T ẑ(μ gẑ) T ) f μ. The expression for ṽ, previously given by (76), can also be given by ṽ = fṽ1 + θ 1 = fṽ2 + θ a, where the functions fṽ1 an fṽ2 are given by fṽ1 = K 1 h( p) k v h(ṽ)+ μ fṽ2 = K 1 h( p) K 2 h(ṽ)+ μ ˆθ 1. Since we require the erivative of the signal fṽ1, we also fin fṽ1 = f fṽ + f fṽ, where f fṽ = K 1 φ h ( p)ṽ k v φ h (ṽ)fṽ2 + f μ f fṽ = k v φ h (ṽ)+ f μ. At this point we require the erivative of the signal τ 2,given by (48). Using the partial erivative of φ(u) given by (13), we obtain τ 2 = f τ 2 + f τ 2 θ a, where f τ 2 =Γ v fṽ2 k θ (f φ γ h (ṽ, fṽ1 )fṽ2 φ h (ṽ)k 1 φ h ( p)ṽ k v φ h (ṽ) 2 fṽ2 + φ h (ṽ)f μ ) + γ q (f φ h (ṽ, K 2 M(μ ) T q)fṽ2 + φ h (ṽ)k 2 Z 2 (μ, q)f μ f τ 2 + φ h (ṽ)k 2 M(μ ) T f q ) (99) =Γ v k θ (f φ γ h (ṽ, fṽ1 ) kv 2 φ h (ṽ) 2 + φ h (ṽ) f μ ) + γ q (f φ h (ṽ, K 2 M(μ ) T q)+φ h (ṽ)k 2 Z 2 (μ, q) f μ + φ h (ṽ)k 2 M(μ ) T f q ). (100) In light of the work presente in [16] an using the aforementione erivatives, we now ifferentiate the projection law α. Using the projection algorithm efine by (39) (42), in aition to the erivative of τ 2 as efine by (99) an (100), we obtain α(ˆθ 1,δ a + k θ,τ 2 )=f α + f α θ a, where the functions f α an f α are given by f α = k α η 1 η 2 ˆ k α η 1 η 2 ˆ η 2 k α η 1 η 2 ˆθ (τ2 T ˆθ 1 + ˆθ 1 T f τ 2 )ˆθ 1 1 τ 2 η 2 f α = k α η 1 η 2 ˆθ ˆθ 1 Tτ 1 ˆθT 1 fτ 2 2 { 4(ˆθ 1 η 1 = T ˆθ 1 θ0 2 )ˆθ 1 T ˆθ 1, if ˆθ 1 2 >θ0 2 0, otherwise. Having obtaine the erivative of α, we ifferentiate the signal w β given by (51) to obtain ẇ β = f w β + f w β θ a with (ṽ, f w β = k p k v f φ h Γ 1 v h( p) ) fṽ2 + k p k v φ h (ṽ)γ 1 v φ h ( p)ṽ γ f α f w β = k p k v f φ h (ṽ, Γ 1 v h( p) ) γ fα.

14 142 IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 1, FEBRUARY 2011 Using the expression for q given by (49), we also fin q = f q + f q θ a, where f q = + 1 S(ẑ)f q 2ẑ qt M(μ )f μ 1 2ẑ qt Ω f q = S(ẑ) f q + 1 2ẑ qt M(μ ) f μ. In light of the earlier results, we finally obtain the erivative of the virtual control law for the secon controller β 2 = f β2 + f β2 θ a, where f β2 = Z 1 (μ,r (3) + w β )f μ + M(μ )(r (4) + f w β ) K q f q (101) f β2 = Z 1 (μ,r (3) + w β ) f μ + M(μ ) f w β K q f q (102) which is use to specify the erivative of the virtual control law for the first controller β 1 = f β1 + f β1 θ a, where f β1 = f β2 γ Z 1 (μ, Γ v ṽ) f μ γ M(μ )Γ v fṽ2 k p Z 1 (μ, Γ 1 v φ h ( p)ṽ)f μ k p M(μ )Γ 1 v f φ h ( p, ṽ)ṽ k p M(μ )Γ 1 v φ h ( p)fṽ2 + 2 S( q)rγ v ṽ (μ gẑ) T f μ γ q u 2u t S(RΓ v ṽ)f q t γ q 2u t S( q)s(ω)rγ v ṽ + 2u t S( q)rγ v fṽ2 (103) γ q γ q f β1 = f β2 γ Z 1 (μ, Γ v ṽ) f μ γ M(μ )Γ v ( k p Z 1 μ, Γ 1 v φ h ( p)ṽ ) fμ k p M(μ )Γ 1 v φ h ( p) + 2 S( q)rγ v ṽ (μ gẑ) T fμ γ q u 2u t S(RΓ v ṽ) t γ f q q + 2u t γ q S( q)rγ v. (104) REFERENCES [1] A. Roberts an A. Tayebi, Aaptive position tracking of vtol uavs, in Proc. 48th IEEE Conf. Decision Control an 28th Chin. 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Teel, Robust global asymptotic attitue stabilization of a rigi boy by quaternion-base hybri feeback, in Proc. 48th IEEE Conf. Decision Control 28th Chin. Control Conf., 2009, pp [23] A. Ko, O. Ohanian, an P. Gelhausen, Ducte fan UAV moeling an simulation in preliminary esign, in Proc. AIAA Moel. Simulation Technol. Conf. Exhibit, 2007, AIAA Paper Anrew Roberts (S 08) receive the B.Sc. egree in electrical engineering an the M.Sc. egree in control engineering, in 2005 an 2007, respectively, from Lakehea University, Thuner Bay, ON, Canaa. He is currently working towar the Ph.D. egree with the Department of Electrical Engineering, University of Western Ontario, Lonon, ON. His current research interests inclue nonlinear control theory with applications involving the attitue estimation an control of unmanne aircraft. Abelhami Tayebi (SM 04) receive the B.Sc. egree in electrical engineering from Ecole Nationale Polytechnique Alger, Algeria, in 1992, the M.Sc. (DEA) egree in robotics from Université Pierre an Marie Curie, Paris, France, in 1993, an the Ph.D. egree in robotics an automatic control from Université e Picarie Jules Verne, Amiens, France, in December In December 1999, he joine the Department of Electrical Engineering, Lakehea University, Thuner Bay, ON, where he is currently a Professor. His research interests inclue linear an nonlinear control theory incluing aaptive control, robust control, an iterative learning control, with applications to robot manipulators an aerial vehicles. Prof. Tayebi serves as an Associate Eitor for IEEE TRANSACTIONS ON SYS- TEMS, MAN, AND CYBERNETICS PART B, Control Engineering Practice, an IEEE CSS Conference Eitorial Boar. He is the Founer an the Director of the Automatic Control Laboratory, Lakehea University.