Adaptive position tracking of VTOL UAVs

Transcription

1 Joint 48th IEEE Conference on Decision and Control and 8th Chinese Control Conference Shanghai, P.R. China, December 16-18, 009 Adaptive position tracking of VTOL UAVs Andrew Roberts and Abdelhamid Tayebi Abstract An adaptive position tracking control scheme is proposed for vertical thrust propelled unmanned airborne vehicles UAVs in the presence of external disturbances. As an intermediary step, the system attitude is used to direct the thrust towards the position target. Instrumental in our control design, an extraction method allowing to obtain the desired attitude and thrust from the required force driving the system towards the desired position, is proposed. Finally, the control torque is designed for the overall system to achieve the tracking objective. The proposed controller ensures global asymptotic stability of the overall closed loop system. I. INTRODUCTION This paper deals with the position tracking problem of a vertical take-off and landing VTOL unmanned airborne vehicle in the presence of external disturbances. For this class of underactuated mechanical systems, a common practice is to use the system attitude as a means to direct the thrust in order to control the system position and velocity. This is an intuitive choice, which offers promising results when used with the backstepping approach. However, despite the tremendous efforts of the research community, there are still some open-problems in terms of handling the external disturbances, coupling between the translational and rotational dynamics, singularities as well as achieving global stability results. Examples of models including disturbance forces include [1], [] and [3], where it is normally assumed that the disturbance force is constant, as is the case in this paper. However, future work may consider the fact that the disturbance is time-varying, especially if it is due to aerodynamic drag. A second common problem is related to which system inputs are used to define the control. In most cases, it is desired to obtain the control in terms of the torque applied to the rotational dynamics of the system. However, this becomes difficult, especially when the effect of external disturbances is included. The only case that we know of that expresses the control in terms of the torque, and deals with external disturbances, is in [3]. However, this result is local and deals with the regulation problem rather than tracking. Thirdly, a well known problem is due to the coupling between the rotational and translational dynamics. This coupling is usually in the form of the control torque or angular velocity acting on the rotational dynamics. This problem is well known and explored in more detail in [4]. This work was supported by the Natural Sciences and Engineering Research Council of Canada NSERC. The authors are with the Department of Electrical and Computer Engineering, University of Western Ontario, London, Ontario, Canada. The second author is also with the Department of Electrical Engineering, Lakehead University, Thunder Bay, Ontario, Canada. arober88@uwo.ca, tayebi@ieee.org Note that this coupling is system dependant, and is not always present in certain systems, for example the quadrotor aircraft. A linear position tracking controller for such an aircraft is discussed in [5]. As is the case in our paper, it is often assumed that the coupling is negligible and is thus omitted in the control design. However, as discussed in [3] and [4], depending on the strength of the coupling, this can lead to unexpected oscillations in the system states. There are some examples of control which do address the coupling problem. For instance, in [6], a nice change of coordinates is presented, however, only for a planar system. In [3], a change of coordinates is also presented that removes the coupling due to the control torque. A consequence of this change of coordinates is that a new coupling is introduced in terms of the system angular velocity, which can only be neglected if the system yaw rate is assumed to be zero, which is the case in [3]. However, in practice this would likely not be the case. Therefore, there still seems to be some potential room for improvement regarding this coupling term in future work. Lastly, to the best of our knowledge there are no results in the available literature achieving global asymptotic stability for the position tracking problem of UAVs in the presence of disturbance forces. Rather we have controllers achieving practical stability, such as in [7] and [8], or local stability, such as in [1] and []. In this paper, a new method is presented for extracting the magnitude and direction of the thrust, in terms of unitquaternion, from a given desired translational force. Usually, attitude extraction is a form of Wahba s problem, which is discussed in [9], where an expression for the system attitude is sought based on a set of vector measurements. In [10] an attitude extraction method is used that uses two pairs of vector observations from an accelerometer and magnetometer, that is used to recover a measurement of the actual system attitude. For the case where only one set of vector measurements is given, as is the case with our method, there are an infinite number of solutions for the attitude extraction. However, we present a method to solve this problem with almost no restrictions on the two vector measurements, except for a mild singularity that can be avoided. This method is beneficial since generally, solutions of Wahba s problem require singular value decomposition, least squares or some numerical method. Relying on the new quaternion extraction method, we present an adaptive tracking controller using the torque as a control input. The proposed controller, which uses an adaptive estimation method using a projection mechanism [11], [1], achieves global asymptotic stability in the presence of /09/$ IEEE 533

2 a bounded constant disturbance force. II. DYNAMIC MODEL AND PROBLEM FORMULATION In this paper we use the well known 6-DOF equations for a rigid body to develop the control laws. To represent the system attitude, we make use of two forms of attitude representation, namely the rotation matrix and unitquaternion. The rotation matrix is the map from the inertial frame to the body frame where R SO3 := {R R 3 3 ; detr = 1; RR T = R T R = I 3 3 }. The dynamics of the rotation matrix are Ṙ = SΩR, where S is the skew-symmetric matrix such that Suv = u v, and Ω R 3 is the body-referenced angular velocity of the rigidbody. The system attitude can also be represented by the unit quaternion Q = q 0, q Q = {R R 3 Q = 1} where R = I Sq q 0 Sq, which has the dynamics Q = 1 0 Q 1 Ω where the operator represents quaternion multiplication, or equivalently given Q, P Q where P = p 0, p we have p Q P = 0 q 0 q T p q 0 p + p 0 q + Sqp Using this representation, the well known rigid body model can be obtained from the principle of conservation of linear and angular momentum, which is given by ṗ = v 3 v = gẑ T m RT ẑ 1 ml RT Sẑu + 1 m F d 4 Q = 1 q T Ω 5 q 0 I Sq I b Ω = SΩIb Ω + ǫ M SẑRF d + u 6 where p, v R 3 denote the inertial referenced system position and velocity, respectively, m is the system mass, g is the gravitational acceleration, F d is the inertial referenced constant disturbance force, I b is the constant body-referenced inertia tensor, u is the control torque input, l is the torque lever arm, T is the system thrust, ẑ = 0, 0, 1 and ǫ M is the lever arm that creates a disturbance torque due to F d. The model for the disturbance force and torque is similar to [3], since we assume that the disturbance force is applied to a point on the body-referenced z axis at a distance ǫ M away from the system center of gravity. Note that the coupling between the translational and rotational dynamics appears in the equation of v in the form of the control torque u. For the purposes of the control design, this term is neglected where we assume that ml >> 1. For convenience we define the unknown parameters θ a = m 1 F d, θ b = ǫ M F d and thrust control input u t = m 1 T. Using the model given by 3 through 5 our objective is to force the position p to track some time-varying reference rt, given it meets the following requirements: Assumption 1: The second, third and fourth derivatives wrt t of the reference trajectory rt are uniformly continuous. Furthermore, the second derivative of the reference trajectory is bounded such that r δ r and ẑ T r < δ rz < g. Having defined the reference trajectory rt this motivates the choice of the error signals p = p rt 7 ṽ = v ṙt 8 To use the system velocity v as a virtual control we must use the system attitude R and thrust u t as virtual inputs to the translational dynamics 3-4. Therefore, we define µ = gẑ u t R T ẑ 9 µ = µ µ d 10 where µ d will be defined later in the control design. The use of µ as a virtual control requires the extraction of the thrust T and the attitude R or Q from µ d. An attitude and thrust extraction method to this effect is presented in the next section. In order to achieve asymptotic stability, we must ensure that the control law we specify for µ d is bounded. As a consequence, we make the following assumption on the disturbances θ a and θ b : Assumption : The disturbances θ a and θ b are constant and bounded such that 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 }. To control the system rotational dynamics, in addition to the system thrust u we also specify a desired angular velocity Ω d and the error signal Ω = Ω Ω d 11 where the expression for Ω d is defined later in the control design. III. ATTITUDE EXTRACTION Let µ d be the virtual control input that achieves the position tracking objective for the translational dynamics. In order to realize µ d, the control input τ for the rotational dynamics should be designed to take care of driving the system attitude to the desired attitude. Therefore, we need to extract the thrust u t and the desired attitude R d or Q d for a given µ d satisfying µ d = colµ d1, µ d, µ d3 = gẑ u t R T d ẑ 1 Although, in general, there are an infinite number of solutions to 1, we propose one solution for this problem using the unit quaternion: Lemma 1: Given µ d and assuming that µ d / L, L = {µ d R 3 ; µ d = 0, 0, µ d3 ; µ d3 [g, } 13 then a solution for the system thrust and attitude, in terms of the unit quaternion Q d = q d0, q d, that satisfies 1 is given by u t = gẑ µ d 14 1 q d0 = 1 + g µ 1/ d3 15 gẑ µ d 1 q d = µd µ d1 0 T 16 gẑ µ d q d0 Proof: The proof is omitted. 534

3 We now desire to find the dynamics of the solutions 15 and 16 with respect to the change in µ d as stated in the following lemma: Lemma : Given that µ d is differentiable, and using the well known expression for quaternion dynamics given by Q d = 1 0 Q d 17 β a solution for the angular velocity β of the desired quaternion Q d is given by β = Φµ d µ d 18 Φµ d = 1 u t c µ d1µ d µ d + u tc 1 µ d c 1 µ d1 u tc 1 µ d1 µ d µ d1 c 1 1 µ d u t µ d1 u t 0 19 where c 1 = u t + g µ d3. Proof: The proof is omitted, yet this result is found from direct albeit tedius differentiation of Having obtained the desired quaternion Q d, we now define the attitude error Q = q 0, q, where Q = Q 1 d Q 0 where we recall that Q represents the actual system attitude. The dynamics of the error quaternion are obtained in light of 1 and 17 Q = 1 0 Q Ω 0 β Q IV. ADAPTIVE ESTIMATION USING PROJECTION 1 To avoid the singularity 13 the virtual control µ d that will be defined must be bounded such that ẑ T µ d = µ d3 < g. To achieve asymptotic stability, the control µ d must include some estimate of the disturbance force. Therefore, to ensure that µ d remains bounded, we utilize projection based adaptive estimation based on the second assumption. Given the error θ p = θ p ˆθ p and adaptive estimation law ˆθp = τ, one differentiable projection algorithm defined in [1] is given by ˆθ p = τ + αˆθ p, δ p, τ αˆθ p, δ p, τ = k α η 1 η ˆθp 3 k α = ǫ n+1 1 α + ǫ α δ p δ p 4 { ˆθT η 1 = ˆθ n+1 } p p δp if ˆθT p ˆθp > δp 5 0 otherwise η = ˆθ 1/ p T τ + ˆθT p τ + δ α 6 where ǫ α > 0, δ α > 0, which has the properties ˆθ p < δ p + ǫ α, θ T p α 0 and ˆθp C n V. GLOBAL POSITION TRACKING CONTROL Focusing on the translational dynamics first, we recall the position error p and velocity error ṽ from 7 and 8. Recall θ a = m 1 F d, then from 4, 9 and 10 where we neglect the coupling term, the velocity dynamics become v = µ + µ d + θ a 7 where we see that for equilibrium solution v, µ = 0, the control must have some non-zero steady state value of µ d = θ a. This motivates the use of a adaptive estimate of θ a which will be used in the control law µ d. We denote this adaptive estimate as ˆθ 1 and define the error where k θ > 0 and θ 1 = θ a ˆθ 1 k θ zṽ 8 zu := 1 + u T u 1/ u 9 φu := u zu = 1 + u T u 3/ I3 3 Su 30 where we note that 0 < λφu λφu 1, u, where λ and λ denote the smallest and largest eigenvalue of, respectively. The term k θ zṽ in 8 is used to force the error θ 1 to zero when t in order to achieve asymptotic stability for the position error p. Differentiating 7, 8 and 8 along 3-4 we obtain the translational error dynamics p = ṽ 31 ṽ = µ + µ d + θ a r = µ + µ d + θ 1 + ˆθ 1 + k θ zṽ r 3 θ 1 = k θ φṽ θ 1 + τ 1 ˆθ1 33 = k θ φṽ µ + ˆθ 1 + k θ zṽ r 34 τ 1 From 3 we define the virtual control µ d = r ˆθ 1 K p z p K v zṽ 35 where K p = k 1 Γ 1 v, K v = k v + k θ I, k 1 > 0, k v > 0, Γ v = Γ T v > 0, from which we obtain ṽ = fṽ1 + θ 1 = fṽ + θ a 36 fṽ1 = K p z p k v zṽ + µ 37 fṽ = K p z p K v zṽ + µ ˆθ 1 38 where the two separate functions fṽ1 and fṽ are defined for convenience. It is useful to note that the error signal µ = µ µ d can be expressed in terms of the vector part of the error quaternion q by µ = u t R T S q q 39 q = Sẑ q + q 0 ẑ 40 Using the expression for µ d given by 35 we find the system thrust input u t and the desired system attitude in terms of the unit quaternion Q d using 14, 15 and 16, respectfully. 535

4 If we recall 17 and 18 it is now possible to define the desired attitude dynamics Q d in terms of the angular velocity β by obtaining µ d. Differentiating 35 in light of 31 and 3 we find µ d = f µd + g µd θ a 41 f µd = r 3 ˆθ1 K p φ pṽ K v φṽfṽ 4 g µd = K v φṽ 43 which is used with 18 and 41 to obtain β = Φµ d h β + Φµ d g µd θ1 44 h β = f µd + g µd ˆθ1 + k θ zṽ 45 Applying the quaternion multiplication to 1 we obtain the attitude error dynamics Q = q0, q q 0 = 1 qt β Ω 46 q = 1 q 0 Ω β + S qβ + Ω 47 Using the above results, we now consider the first Lyapunov function V 1 = k pt p 1/ + 1 ṽt Γ v ṽ + 1 q θt 1 θ1 48 where > 0, > 0, which we differentiate along and 46 to find V 1 τ = k θ θt 1 φṽ θ 1 k v ṽ T Γ v zṽ + q T u t S qrγ v ṽ + Ω Φµ d h β + θ 1 T τ γ 1 θ ˆθ = γ 1 θ 1 τ 1 + Γ v ṽ g T µ d Φµ d T q = k θ φṽfṽ1 + Γ v ṽ + φṽk v Φµ d T q 50 If we recall from 11, Ω = Ω Ωd, the next step in the design procedure requires defining the virtual control law Ω d from 49. From 4 we see that f µd depends on the value of ˆθ1. Since we intend to use projection for the adaptive estimate, we must define the adaptation law ˆθ 1 before proceeding further. Therefore, we define the first adaptation law as ˆθ 1 = τ + α ˆθ1, δ a + k θ, τ 51 Evaluating 45 from 37, 4 and 43 we group terms into two parts h β = h β1 + h β with h β1 = r 3 Γ v ṽ α ˆθ1, δ a + k θ, τ ṽ K p φ p + k v φṽk p z p 5 h β = W 1 zṽ + W q 53 W 1 = k vφṽ 54 W = k v u t φṽr T S q φṽk v Φµ d T 55 where the function h β is chosen since it is bounded functions of ṽ and q, and do not necessarily require cancelation using the virtual control. We now define the virtual control input Ω d = Φµ d h β1 + u t S qrγ v ṽ K q q 56 where K q = K T q > 0 which results in V 1 = k θ θt 1 φṽ θ 1 k v ṽ T Γ v zṽ + q T Ω q T Φµ d W 1 zṽ q T K q + Φµ d W q θ 1 T α ˆθ1, δ a + k θ, τ 57 Note that the term in 57 involving α is non-positive due to the properties of the smooth projection. To study the bound of Φµ d, W 1 and W we first recall from 35 the control law for µ d. Due to the use of projection, using we can guarantee that ˆθ 1 < δ a +k θ. Therefore, due to assumption 1, µ d is bounded such that µ d µ d where µ d δ r + δ a + K p + k v + k θ + ǫ α 58 where ǫ α > 0 is a gain used in the smooth projection function. Note the third component of µ d is bounded by µ d3 µ d3 where µ d3 = k 1 ẑ T Γ 1 v + k v + k θ + δ a + δ rz + ǫ α 59 Due to the constraint 13 we require that µ d3 < g, therefore we define δ µd = g µ d3 > 0. From 14, 35, 58 and 59 we find the thrust is bounded by δ µd u t ū t where ū t = g + µ d. Calculating the Frobenius norm of the matrix given by 19 we find Φµ d F = u t u t + µ d3 g + 1 u 60 t which we use to find the bound Φµ d F δ 1 µ d. Using 60 and Young s Inequality, we now study the bound of the term involving W 1 in 57, or q T Φµ d W 1 zṽ kv q T q + kvǫ q ṽ T zṽ61 ǫ q δ µ d where ǫ q > 0. From 57 and 61, we define v q = k v Γ v kv ǫ q δµ I 6 d k v ū t = K q + γ θ 1 δ µd δµ K v + k v I d ǫ q 63 where we choose the control gains k v, Γ v and K q such that v and q are positive definite, which guarantees that the indefinite terms are always dominated. Therefore, 57 becomes V 1 k θ θt 1 φṽ θ 1 ṽ T v zṽ q T q q + q T Ω

5 In order to proceed with determining the control torque u, we differentiate 56 obtaining Ω d = f Ωd + g Ωd θ a 65 where the exact expressions for f Ωd and g Ωd are given in the appendix. As shown by 65 we must still deal with the unknown parameter θ a. Since we have already defined our first adaptation law, we must use over-parameterization of the unknown parameter, and now define a new estimate ˆθ and the error θ = θ a ˆθ. Also, we now denote ˆθ 3 as the adaptive estimate of the unknown parameter θ b and define the error θ 3 = θ b ˆθ 3. We now consider the second Lyapunov candidate V = V Ω T I b Ω + 1 γ θ θt θ + 1 γ θ3 θt 3 θ3 66 where γ θ > 0, γ θ3 > 0, which we differentiate to obtain V k θ θt 1 φṽ θ 1 v T v zṽ q T q q + Ω T q SΩI b Ω + SẑRˆθ 3 + u I b f Ωd I b g Ωd ˆθ + θ T gω T 1 d I b Ω ˆθ γ θ + θ 3 T R T Sẑ Ω 1 ˆθ3 67 γ θ3 From 67 we define the control and estimation laws u = q + SΩI b Ω SẑRˆθ 3 + I b f Ωd +I b g Ωd ˆθ K w Ω 68 ˆθ = γ θ gω T d I b Ω + α ˆθ, δ a, gω T d I b Ω 69 ˆθ 3 = γ θ3 R T Sẑ Ω + α ˆθ3, δ b, R T Sẑ Ω 70 Note that the use of projection is not required for the estimation laws 69-70, yet it is used to improve robustness. Applying to 67 gives the negative semi-definite result V k θ θt 1 φṽ θ 1 ṽ T v zṽ q T q q Ω T K w Ω 71 from which we conclude that p, ṽ, Ω are bounded. The adaptive estimates were bounded a priori due to the use of projection in the adaptive estimation laws, and the signal q is bounded by convention. It follows from Barbalats lemma that provided the second, third and fourth derivatives of the reference trajectory rt are uniformly continuous, then the signals ṽ, θ 1, q and Ω converge asymptotically to zero. Also, since we have ṽ 0 then p 0, which satisfies the trajectory tracking objective. If we consider 41 and 51 given by µd ˆθ 1 = f µd + g µd θ a 7 τ + α ˆθ1, δ a + k θ, τ and note that τ depends on µ d, and f µd depends on ˆθ 1, it becomes apparent that the defined control is dynamic in nature due to the interaction of µ d and ˆθ 1. However, since we know that α is Lipschitz continuous, then the RHS of 7 is also Lipschitz continuous, and thus a solution for the control in terms of µ d and ˆθ 1 always exists. Theorem 1: Consider the system given by 3-6 and the reference trajectory rt where assumptions 1 and hold. Using 35, if we specify the desired attitude, Q d, using 15 and 16, and specify the desired angular velocity, Ω d, and system thrust, u t, using 56 and 14, and apply the control torque 68 in addition to the adaptation laws 51, 69 and 70, then the state variables pt, ṽt, qt, Ωt are globally bounded and converge asymptotically to zero. Proof: The proof is given above as a direct result of the constructive control design procedure. VI. CONCLUSION An adaptive position tracking controller achieving global asymptotic stability, has been proposed for a VTOL-UAV in the presence of external disturbances. The proposed control scheme relies on the use of a quaternion extraction method allowing the extraction of the desired system attitude and thrust from the required force achieving the tracking objective. The quaternion extraction method provides almost global results, except for a mild singularity that can be easily avoided. To facilitate the bounded control, projection is used in the adaptive estimation algorithms. Although, this improves the robustness of the proposed controller, this method requires the use of over-parameterization of the unknown disturbances. APPENDIX Recalling 30, we differentiate to obtain f φ u, v = / u φuv where f φ u, v = 1 + u T u 5/ [ 3 Su I vu T u T u SuSv SvSu ] 73 Note the function Φµ d can be expressed as Φµ d = γ Φ Ψµ d where γ Φ = u 1 tc 1 and Ψµd is the matrix component of Φµ d. We find u t = α T 1 µ d where α 1 = µ d gẑ. We define c 1 = u t + g µ d3 and note that ċ 1 = α µ d T µ d where α µ d = α 1 µ d ẑ, then u 1 t µ d Φµ d v = Z 1 µ d, v = Ψµ d vm γ + γ Φ Λ 1 µ d, v M γ = γ Φ gẑ µ d T 3c 1 I SẑSgẑ µ d Λ 1 = M 1 + T c 1 v c 1 v 1 µ d v 1 µ d1 v α1 µ d T + u t v + µ d v 3 u t v 1 µ d1 v 3 0 T α µ d T M 1 = M 1A M 1B M 1A = T µ d v 1 v 1 µ d1 + v µ d c 1 v 3 u t v M 1B = T µ d1 v 1 v µ d + c 1 v 3 µ d1 v u t v 1 537

6 Similarly, one can find µ d Φµ d T v = Z µ d, v 74 Z µ d, v = Ψµ d T vm γ + γ Φ Λ µ d, v Λ = M + µ d v 3 c 1 v c 1 v 1 µ d1 v 3 0 T α1 µ d T + T u t v u t v 1 µ d v 1 µ d1 v α µ d T M = M A M B M A = T v µ d1 µ d v 1 µ d v u t v 3 c 1 v M B = T u t v 3 µ d1 v 1 µ d1 v µ d v 1 c 1 v 1 Recall 47 and denote q = f q + g q θ a, where f q = 1 q 0I + S qω + 1 S q q 0IΦµ d f µd 75 g q = 1 S q q 0IΦµ d g µd 76 Recall 10 we find µ = f µ + g µ θ a where f µ = I + ut 1 R T ẑ µ d gẑ T f µd u t R T SΩẑ 77 g µ = I + u 1 t R T ẑ µ d gẑ T g µd 78 where θ 0 = δ p + k θ. We also find q = f q + g q θ a f q = Sẑf q + 1 ẑ qt Φµ d f µd 1 ẑ qt Ω 86 g q = Sẑg q + 1 ẑ qt Φµ d g µd 87 From 5 we differentiate to obtain ḣβ 1 = f hβ1 + g hβ1 θ a with f hβ1 = r 4 Γ v fṽ f α K p f φ p, ṽṽ K p φ pfṽ + k v f φ ṽ, K p z pfṽ +k v φṽk p φ pṽ 88 g hβ1 = Γ v g α K p φ p +k v f φ ṽ, K p z p 89 Finally, in light of the above results we can find f Ωd = Z 1 µ d, f β1 f µd + Φµ d f hβ1 K q f q + [ u 1 t S qrγ v ṽ µ d gẑ T f µd u t SRΓ v ṽf q u t S qsωrγ v ṽ +u t S qrγ v fṽ ] 90 g Ωd = Z 1 µ d, f β1 g µd + Φµ d g hβ1 K q g q + [ u 1 t S qrγ v ṽ µ d gẑ T g µd u t SRΓ v ṽg q + u t S qrγ v ] 91 Recall 3, from which we find fṽ1 = f fṽ1 +g fṽ1 θ a, where f fṽ1 = K p φ pṽ k v φṽfṽ + f µ 79 g fṽ1 = k v φṽ + g µ 80 From 50 we find τ = f τ + g τ θ a where f τ g τ = Γ v fṽ k θ f φ ṽ, fṽ1 fṽ φṽk p φ pṽ k v φṽ fṽ + φṽf µ + fφ ṽ, K v Φµ d T qfṽ + φṽk v Z µ d, qf µd +φṽk v Φµ d T f q 81 = Γ v k θ fφ ṽ, fṽ1 k v φṽ + φṽg µ + fφ ṽ, K v Φµ d T q + φṽk v Z µ d, qg µd +φṽk v Φµ d T g q 8 In light of the work presented in [1], we differentiate the smooth projection. We define θ 0 = δ a + k θ and αˆθ 1, θ 0, τ = f α + g α θ a where f α = k α η 1 η ˆθ1 k α η 1 η ˆθ1 η k α η 1 η ˆθ τ T ˆθ 1 + ˆθ 1 T 1 τ f τ ˆθ1 83 η g α = k α η 1 η ˆθ ˆθ 1ˆθT 1 Tτ 1 g τ 84 { 4 ˆθT η 1 = 1 ˆθ1 θ0 ˆθT 1 ˆθ1 if ˆθ 1 > θ otherwise 538 REFERENCES [1] M.-D. Hua, P. Morin, and C. Samson, Balanced-force-control of underactuated thrust-propelled vehicles, in Proc. 46th IEEE Conference on Decision and Control, 1 14 Dec. 007, pp [] M.-D. Hua, T. Hamel, P. Morin, and C. Samson, Control of thrustpropelled underactuated vehicles, INRIA Technical Report, 008. [3] J. Pflimlin, P. Soueres, and T. Hamel, Position control of a ducted fan vtol uav in crosswind, International Journal of Control, vol. 80, no. 5, pp , 007. [4] J. Hauser, S. Sastry, and G. Meyer, Nonlinear control design for slightly non-minimum phase systems: Application to v/stol aircraft, Automatica, vol. 8, no. 4, pp , 199. [5] S. Salazar-Cruz, A. Palomino, and R. Lozano, Trajectory tracking for a four rotor mini-aircraft, in Decision and Control, 005 and 005 European Control Conference. CDC-ECC th IEEE Conference on, Dec. 005, pp [6] R. Olfati-Saber, Global configuration stabilization for the vtol aircraft with strong input coupling, IEEE Trans. Autom. Control, vol. 47, no. 11, pp , Nov. 00. [7] A. P. Aguiar and J. P. Hespanha, Position tracking of underactuated vehicles, in Proc. American Control Conference the 003, vol. 3, 4 6 June 003, pp [8] E. Frazzoli, M. A. Dahleh, and E. Feron, Trajectory tracking control design for autonomous helicopters using a backstepping algorithm, in Proc. American Control Conference the 000, vol. 6, 8 30 June 000, pp [9] M. Shuster and S. Oh, Three-axis attitude determination from vector observations, Journal of Guidance and Control, vol. 4, pp , [10] N. Metni, J. M. Pflimlin, T. Hamel, and P. Soueres, Attitude and gyro bias estimation for a flying uav, in Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems IROS 005, 6 Aug. 005, pp [11] I. P. and J. Sun, Robust Adaptive Control. Prentice Hall, Inc., [1] Z. Cai, M. S. de Queiroz, and D. M. Dawson, A sufficiently smooth projection operator, IEEE Trans. Autom. Control, vol. 51, no. 1, pp , Jan. 006.